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UNITED STATES DEPARTMENT OF COMMERCE

C. R. Smith, Secretary

0 / y . NATIONAL BUREAU OF STANDARDS, A. V. Astin, Director „L W A t

MAY 9 1974

Electron Impact

Excitation of Atoms

B. L. Moiseiwitsch

Department of Applied Mathematics

The Queen’s University of Belfast

Belfast, Northern Ireland

AND

S. J. Smith

Laboratory Astrophysics Division National Bureau of Standards

Boulder, Colorado 80302

[

Reprinted from

Reviews of Modern Physics

Vol. 40, No. 2, April 1968 ]

NSRDS-NBS 25

National Standard Reference Data SeriesNational Bureau of Standards 25

Issued August 1968

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price $2
Foreword

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li

A. V. Astin, Director.
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hi
Abstract

The experimental and theoretical literature about the electron impact

excitation of atoms is reviewed. Theoretical methods ranging from the Bethe

and Born approximations to the close coupling approximations are discussed

and intercompared. Where possible, on theoretical grounds or through

intercomparison, the reliability of the various methods is discussed.

A general critique of the optical method of measuring excitation functions is given, with the objective of promoting higher quality future ex- perimental work. A critical study of existing experimental work leads to the conclusion that most workers have ignored important physical and

instrumental effects, and it may be presumed that the data in the literature

is subject to many unrecognized systematic errors. The literature on hydrogen and helium is discussed critically. The literature on the alkalis,

heavy rare gases, mercury, cadmium and zinc is surveyed but the quality

of the literature does not support critical review beyond some general comments about the physics of these atoms.

Key words: Atom, electron, excitation, experimental, helium, hydrogen,

impact, review, theoretical.

Contents

Page

Foreword hi

Preface v

Contents to paper 1 Introduction 1

IV
Electron Impact Excitation of Atoms

B. L. MOISEIWITSCH

Department of Applied Mathematics, The Queen's University of Belfast, Belfast, Northern Ireland

S. J. SMITH*

Joint Institute for Laboratory Astrophysics,] Boulder, Colorado CONTENTS

I. Theory 1

1. Introduction 1 2. Hydrogen Atoms 2

2.1. First Born Approximation 2

2.2. Bethe Approximation 7

2.3. Second Born Approximation 8

2.4. Exchange Scattering 9

2.5. Born-Oppenheimer Approximation 10

2.6. First-Order Exchange Approximation 10

2.7. Ochkur Approximation 11

2.8. Distorted Waves Approximation 12

2.9. Modified Bethe Approximation 14

2.10. Impact Parameter Method 14

2.11. Classical Approximation 15

2.12. Vainshtein, Presnyakov, Sobelman Approximation 16

2.13. Partial Wave Analysis 17

2.14. Unitarized Born Approximation 18

2.15. Close Coupling Approximations 18

3. Helium 21

3.1. First Born Approximation 21

3.2. Exchange Approximations 23

3.3. Distorted Waves Approximation 26

3.4. Close Coupling Approximation 26

3.5. Double Excitation of Helium 27

4. Neon and Argon. 28

5. Mercury 29

6. Alkali Metal Atoms 29

7. Oxygen and Nitrogen Atoms 31

8. Positive Ions 33

8.1. Hydrogenic Ions 33

8.2. Helium Ion 34

8.3. Helium-Like Ions 35

8.4. Lithium-Like Ions 35

8.5. Sodium-Like Ions 36

8.6. Potassium-Like Ions 36

8.7. Positive Ions Having p

2

, p

3

, or p

i Configurations 36

9. Polarization of Radiation 38

10. Theoretical Summary 39

II. Experiment 40

11. Introduction 40

12. Experimental Methods 41

12.1. Pressure Dependence of Excitation Functions 42

12.2. The Effects of Polarization 46

12.3. Electric Field Effects 49

12.4. Absolute Measurements of Optical Excitation

Functions 50

12.5. Electron Beam Measurements 51

13. Other Reviews of Electron Impact Excitation .... 51

14. Atomic Hydrogen 51

14.1. Excitation of H(2p) 52

14.2. Excitation of 77 (2s) 54

14.3. Excitation of Balmer Lines of Hydrogen. ... 56

* Staff member, Laboratory Astrophysics Division, National Bureau of Standards.

t Of the National Bureau of Standards and the University of Colorado.

15. Excitation of Helium Positive Ions 56

16. Helium 56

16.1. Helium w

1 /

>—»21 A Transitions 56

16.2. Helium w

1 5—>21 P Transitions 63

16.3. Helium n

1 D-^21 P Transitions 65

16.4. Helium n

3 S—*23 P Transitions 66

16.5. Helium n

3 P—*23 S Transitions 67

16.6. Helium ra

3 D—>23 P Transitions 67

16.7. Excitation of the Helium n = 2 Levels 67

16.8. Thresholds of Helium Excitation Functions. . 70

17. Heavy Rare Gases 71

18. Alkali Atoms 72

19. Heavy Two-Electron Systems 77

20. Experimental Summary 79

Acknowledgments 80

Bibliography 81

Tables 85

I. THEORY

1. Introduction

In the first part of this review we direct our attention

to the theoretical aspects of the excitation of atoms by

electron impact. We examine the atoms in order of

increasing complexity beginning with the simplest of

all atomic systems, atomic hydrogen, and then in turn

consider helium, the heavier inert gases, mercury, the

alkali metals, oxygen, and nitrogen, as well as certain

positive ions. Because a hydrogen atom has only a

single electron, it has been the subject of the most detailed investigations. We therefore begin by developing

the theory for atomic hydrogen leaving the generalization to more complex atoms until later. It is important

to realize however that even for the case of atomic

hydrogen, the work involved in obtaining accurate

excitation cross sections is formidable and that indeed no exact solution for the problem of the excitation of

the two lowest states of atomic hydrogen, the 2s and 2p

states, has yet been obtained. The only method by

which we can assess the accuracy of the various approximations which have been employed is by comparing

them with each other and with the available experimental data. At the present time the situation continues

to be very uncertain and a great deal of investigation,

both theoretical and experimental, remains to be done.

Our procedure will be to commence with the simplest

approximations and to introduce progressively the more involved and, in general, more accurate and re- vealing approximations. However, before proceeding

with this endeavor we draw attention to previous re- views of the theoretical work on the electron impact

excitation of atomic systems which include the im1
portant articles by Massey (1956) and by Peterkop and

Veldre (1966).

2. Hydrogen Atoms

We consider a system composed of two electrons

denoted by 1,2 moving in the Coulomb field of a proton which, to sufficient approximation, we may

regard as possessing infinite mass. Then the Schrodinger

equation describing this system takes the form —ftp e2 e2 e — 2 1 (Vx2 +V2

2

) + Et ¥(*, r2) =0, 2m r\ r2 r12 J CD

where ri, r2 are the position vectors of electrons 1, 2 referred to the proton as origin, rx2 is the interelectron

distance, ET is the total energy of the system, and

*(ri, r*) is the total wave function characterizing the

two electrons.

We now expand ^ in terms of the orthogonal and

normalized set of hydrogen atom wave functions pn (r)

satisfying the equation

[(— fi2/2m) V2 - (e2/r) - £„]0„( r) =0, (2)

where the En are the associated eigenenergies. Setting

*(ri, r2)=SFp(r2)^,(r1) l (3) p where the symbol S denotes a summation over the dis- crete states and an integration over the continuum

states, substituting into (1), multiplying across by

0n*(ri), and integrating with respect to ii, we obtain

2m C

[V2

2 +£„2]F„(r2) = —J

0„*(rx)F(rx , r2)*(rx, r2)<frx ,

(4)

where

YiiuUI-Wral-Wr,) (5)

is the interaction energy between electron 2 and a hydrogen atom composed of electron 1 and the proton,

and the wave number kn is given by

£„2 =(2m/h*){Er-EJ. (6)

Let us now suppose that electron 2 impinges upon a hydrogen atom in the ground Is state which we will

denote by 1, the direction of incidence being given by

the wave vector ki. Then the asymptotic behavior of the

function Fn ( r) for large r takes the form

F„(r)~exp (tki*r)5„i+

_1 exp ( iknr)fn (d, 0), (7)

where

/

n (0, 0) is the scattering amplitude corresponding

to the excitation of the wth state of the hydrogen atom

and 6, 0 are the polar angles of r referred to the direction of incidence as polar axis. Introducing the Green’s function G( r, r2) for a free

particle satisfying the equation

(VM-£n

2 )G(r, r2 )=5(r-r2 ) (8)

we may write

2m f f

F„(r) =exp (fkx *r)5nl-|- —JJ

G(r, r2 )^n *(r!)

XF(rx , r2)^(rx , r2 )*idr2. (9)

In order that Fn should have the correct asymptotic

form (7) we choose

G(r, r2) = — exp (ikn | r-r2 |)/(4tr

| r-r2 1). (10)

Now for large r

kn | r—

2 |~^nr-kn -r2, (11)

where k„ is a wave vector in the direction of r, and so we

see that

2m f f

fn (fit 0) = - ( 47T)-1 —JJ

exp ( — fkn -r2 )0n *(r1 ) XV(r1 ,r2)«r (r1,r2)Jr1rfr2. (12)

The differential cross section for the excitation of the. «th state of a hydrogen atom is given in terms of the

scattering amplitude fn {6, 0) , corresponding to incident

electrons scattered through angles 9, 0, by the formula

/n(M)=(W |/«(0,0) I 2 (13)

while the total cross section takes the form

Qn== ~j~ff I /»(0|0) | 2 sin 9d8d4>. (14)

2.1. First Born Approximation

We now make the simplest of all approximations, the

first Born approximation, which depends upon the

assumption that the incident electron interacts only

slightly with the target atom so that its wave function

may be closely approximated by the plane wave

exp (fki*r) which would be the correct function in the

absence of all interaction. We may anticipate that this

approximation should be valid when the speed of the

incident electron is great in comparison with that of the

electron in the target atom. This is essentially equivalent to the requirement that k\ay> 1, where a is the

range of the electron-atom interaction. Under these

circumstances we may substitute

*(rx , r2)=exp (ik1 -r2)01 (r1 ) (15)

into the right-hand side of (12). This yields the first Bom approximation to the scattering amplitude

2m r /n (0,0)=-(47r)-1 —J

exp{f(k1 -kn)-r2 }F„i(r2)c?r2 ,

(16)

2
where

VM(r2) = f^*(h)V(r,, rJMti)dri. (17)

Although the application of this approximation can

only be justified at high impact energies, it appears to yield quite useful results even for electron energies

which are fairly close to the threshold energy for excitation. Denoting the momentum change of the incident

electron by the vector UK, where K=ki—k„, and using

Bethe’s integral

/i^r != ^

exp( ri)’ ( }

state of hydrogen may be written in the form

2ir Q»=r* \fn{K)\>KdK, (25) h

2 ->Km m where Kmin = ki—kn and Km*x =ki+k„. Since

k?-k 2 = (2m/K) (En -E1) (26)

by the conservation of energy, we see that for high

electron impact energies

Kmia^(m/h2k1)(En-E1), Km^2kx

. (27)

As illustrative examples let us now consider the excitation of the 25 and 2p states of atomic hydrogen from

the ground 15 state. It can be readily shown that we may express the scattering amplitude in the form

fn (K) = -(2me2/h2K2)d{l-n) (ra^l), (19)

where

0(1 —n) = yV»*(r) exp (*K-r)0i(r)dr, (20)

the term arising from the electron-proton interaction

vanishing here because of the orthogonality of the

atomic wave functions.

We see from formulas (13), (19), and (20) that

(ki/kn)In (9, 0) depends only on K in the Born approximation, a result which has been verified experimentally

by Lassettre and his collaborators for the 2

1 S and 2l P

excitations of helium (Silverman and Lassettre, 1964;

Lassettre and Jones, 1964; Lassettre, Krasnow, and

Silverman, 1964).

It is sometimes useful to write the differential cross

section in terms of a generalized oscillator strength

fin (AT) = ( 2m/fi2

) [

(

Ei— En) /K2~]

| 0 ( 1 — w) | 2 (21)

which can be shown to satisfy the sum rule

£fc.(X)-l. (22)

n As the momentum change fiK tends to zero, UK)

approaches the optical oscillator strength given by

F

(El~£") J0„(r)z0i(r)<fr (23)

where the z coordinate of the atomic electron is measured in the direction of K.

Now

K2 = ki2+&n2 — 2&1&„ cos 9

and so KdK= k\kn sin Odd

from which it follows that

In (9, 0) sin 9d6=kr2 \fn (K) | 2 KdK

_

2we4 fm (K) dK

m?Ex-En K‘ 1 J

The total cross section for the excitation of the nth

I /2,(K) | 2 = 128flo2/ (ATW+f-) 6 (28)

and

\f2pm(K) \ 2 =2SSd0m/K2 (KW+$)t

, (29)

where m is the magnetic quantum number of the 2 p

state referred to the momentum change vector fiK

as polar axis. We see at once that according to the first Born approximation the cross sections vanish for the

excitation of 2p states with magnetic quantum numbers

m = ± 1 referred to the direction of the momentum

change vector fiK as polar axis. We also note that for

the I 5—>25 excitation the differential cross section is weakly dependent upon K for small angles of scattering

while for the I 5—>2^0 excitation it behaves like K~2

. For large angles of scattering the differential cross

sections for the I 5—>25 and I5—>2/>0 excitations decay

like K~12 and K~u

, respectively; these are much more

rapid decreases with angle than for elastic scattering

which falls off like K~4

. The differential cross sections

for the I5—>25 and I 5—>2/> excitations at an electron

impact energy of 100 eV calculated according to the

first Born approximation are illustrated in Fig. 1. Substituting expressions (28) and (29) into (25) and

performing the integrations over K yields at high impact

energies

@2^128(1) 107t/5&i2 (30)

and

Q^-[256(l)»F/ifei*] In (4flofti), (31)

where kx is the wave number of the incident electron,

from which it follows that the total cross sections for the

I5—>25 and I5—>2p excitations decay as Er1 and Erl In E,

respectively, for high energies E of impact. The total cross sections for the I5—>25 and I5—>2p excitations of

atomic hydrogen calculated according to the first Born

approximation are displayed in Fig. 2 . We see that the

cross section for the I5—>20 excitation, which corres- ponds to an optically allowed transition, is considerably

larger than the cross section for the I5—>25 excitation,

which is associated with an optically forbidden transition. In addition the peak of the I5—>20 excitation cross section occurs at a higher energy than that for the

I5—>25 excitation, and decays less rapidly with in3
creasing energy, as was already quite clear from the

asymptotic behavior of the cross sections given by

formulas (30) and (31).

It is also instructive to make a partial wave analysis

of the total excitation cross section:

Qn=tQ*'=t SC*"', (32)

(=0 (=0 (

,c=0 where the partial cross section Qnw is the contribution

to the total cross section arising from incident electrons

having azimuthal quantum number i and scattered elec- trons having quantum number l'

, that is incident and

scattered electrons with angular momenta ft[/(/+l)]1/2 and 1)]1/2

, respectively, while

e.'= io/ l

f =0

is the partial cross section corresponding to incident

electrons having angular momentum given by the

quantum number l and scattered electrons having all possible values of the angular momentum. For s-^s

transitions the only nonvanishing partial cross sections are those for which while for s— transitions

only the partial cross sections and Qn

t,(+1 are nonzero. Since

exp ik*r= j^(2/-f l)i(j( (kr)

P

( (cosd)

,

(33)

t=0

where jt(p) is a spherical Bessel function, it follows that

for s^s transitions the first Born approximation to the

^th-order partial cross section is given by

Qj=^{kJh){2(+\){qn0\ (34)

where

qJ= ^

f

jt(kir)ji(knr)V„i{r)r2dr. (35)

U J o A partial wave analysis has been carried out for the

Is

+2.S and Is—»3s excitations (Bates and Miskelly,

1957) and the Is—>2p excitation (Vainshtein, 1961) of

hydrogen atoms using the first Born approximation.

The partial cross sections QJ for the Is—»2s and Is—»2/>

Fig. 1. Differential cross sections for the excitation of atomic

hydrogen at 100-eV electron impact energy, calculated using the

first Bom approximation. Curve 1; Is—>2s, Curve 2: Is—>2/>.

Fig. 2. Total cross sections foF the excitation of atomic hydrogen, calculated using the first Bom approximation (Omidvar,

1964). Curve 1: Is—*2s, Curve 2: Is—*2p.

excitations are displayed in Figs. 3 and 4, respectively.

It can be readily seen that at a given impact energy

many more partial waves make a significant contribution to the total Is—>2p excitation cross section than to the Is—>2s cross section.

The contrasting behavior of the Is—»2s and Is—»2/> excitation cross sections arises from the fact that their

respective interaction potentials have differing asymptotic forms for large radial distances r. Thus F^.i*

decays exponentially with increasing r while F2P ,i» falls off much more slowly with an r~2 dependence for

large r. This provides the explanation for the different

angular distributions given by (28) and (29), the

different high-energy behaviors given by (30) and (31)

,

and the different convergences of the partial wave

expansions illustrated in Figs. 3 and 4. The first Born approximation has been employed by a number of investigators to calculate the total excitation cross sections for various transitions connecting

different states of hydrogen. Suppose that n', V, w!

and n, l, m are the principal, azimuthal, and magnetic

quantum numbers of the initial and final states of the

hydrogen atom, respectively, and let us denote the

excitation cross section for the transition by

Qn'i’m'.nim. We are mainly concerned with the average

of this cross section over m' and its sum over m, that

is with the excitation cross section for the transition

n'l'^nl given by

(2('+ l)-EEe.-. (36) m

Occasionally we require the excitation cross section for

the transition m between the levels with principal

quantum numbers n' and n\ this is obtained by aver- aging over l' and summing over l to yield

Qn',n = (1AO Xl^SCW.nl

V l = (

W

2

) (37)

V l In Tables I—III are presented the values of the excitation cross sections calculated by Vainshtein (1965) for

the n's-+ns, n's—mp, and n's—+nd transitions of atomic

hydrogen with n'<n and n', n= 1

? • • 9. For the sake

h
Fig. 3. Partial cross sections Qif for the Is—>2s excitation

of atomic hydrogen, calculated using the first Born approximation (Omidvar, 1964) . 24 denotes

1=0 Sy denotes

Iq»*.

t=

0

of convenience the energy of the incident electron is given in threshold units, that is, in terms of the quantity

e=E/(En —En ’)=kn

> 2/AEnn >, (38)

where AEnn> is the excitation energy in rydberg units

and kn

> is the wave number of the incident electron.

Veldre and Rabik (1965) have calculated cross sec- tions for the Is—*ns, ls—*np, and ns—>(w+l).s excitations of atomic hydrogen using the first Born approximation. Their excitation cross-section curves together

with those for the lower vaiues of n derived from the

calculations of Vainshtein (1965) are displayed in Figs. 5-7, respectively. Inspection of these figures re- veals that according to the first Born approximation

the value cm of e for which the cross section attains its maximum value is insensitive to n for both the Is—ms

and the 1s-^np excitations, although for the ns—*

(w+l)s excitations cm increases significantly with in- creasing n. Further we note that for each set of transitions, the smaller the value of cm, the more gradually

does the excitation cross-section decay with increasing €. Calculations of the excitation cross sections for a number of transitions in atomic hydrogen have been

also carried out by Milford and his collaborators

(McCoyd, Milford, and Wahl, 1960; Fisher, Milford,

and Pomilla, 1960; Milford, Morrissey, and Scanlon,

1960) using the first Born approximation. They in- vestigated all transitions between states with principal

quantum numbers n' = 3 and n= 4; all optically allowed

transitions between states with n' = 3 and n = 5, between states with n' = 4 and n — 5, and between states with n’ = 5 and n = 6; together with a few other transitions. Some of their excitation cross section curves are displayed in Figs. 8-10,

E(eV) Fig. 4. Partial cross sections Q2p* for the \s—+2p excitation

of atomic hydrogen, calculated using the first Born approximation (Omidvar, 1964) . S4 denotes

2 Qip1

, 1=

0

Sr denotes

00 S Qiv

l

. t=

0

Fig. 5. Total cross sections for the Is—>«s excitations of atomic

hydrogen for « = 2, •••, 11, calculated using the first Born ap- proximation [e is electron energy in threshold units defined by

Eq. (38)] (a) Curve 1: Is—>2s, Curve 2: Is—»3s, Curve 3: Is—*4s. (b) Curve 1: Is—>5s, Curve 2: Is—>6s, Curve 3: Is—>7s. (c) Curve

1; Is—>8s, Curve 2: Is—>9s, Curve 3; Is—>10s, Curve 4: Is—>11;.

5
Fig 6. Total cross sections for the ls-*np excitations of atomic

hydrogen for n = 2, • • •

, 9, calculated using the first Born approximation [e is electron energy in threshold units defined by Eq.

(38)3 (Vainshtein, 1965; Veldre and Rabik, 1965). (a) Curve 1: ls—>2p, Curve 2: Is—>3/>, Curve 3: Is—>4/>, Curve 4: Is—>5/>. (b) Curve 1: Is—>6/>, Curve 2: Is—*7p, Curve 3: Is—>8p, Curve 4: Is—>9/>.

Fig. 7. Total cross sections for the ns—»(n-}-l)s excitations of atomic hydrogen for « = 2, • • •, 9, calculated using the first Born

approximation [e is electron energy in threshold units defined by

Eq. (38)] (Vainshtein, 1965; Veldre and Rabik, 1965). (a) Curve 1: 2s—>3s, Curve 2: 3s—>4s, Curve 3: 4s—>5s, Curve 4: 5s—>6s. (b) Curve 1: 6s—>7s, Curve 2: 7s—>8s, Curve 3: 8s—>9s, Curve 4: 9s—»10s.

Fig. 8. Total cross sections for the excitation of atomic hydrogen, calculated using the first Born approximation and the Bethe

approximation (McCoyd, Milford, and Wahl, 1960). Curve 1: 3s—>Ap, Curve 2: 3£—>4d, Curve 3: 3d—> 4/. Solid curves denote

first Born approximation while dashed curves denote Bethe ap- proximation.

Fig. 9. Total cross sections for the excitation of atomic hydrogen, calculated using the first Born approximation and the Bethe

approximation (McCoyd, Milford, and Wahl, 1960). Curve 1: 3s—>5p, Curve 2: 3p-+5d, Curve 3: 3d—>5/. Solid curves denote

first Bom approximation while dashed curves denote Bethe ap- proximation.

Fig. 10. Total cross sections for the excitation of atomic hydrogen, calculated using the first Born approximation and the Bethe

approximation (Fisher, Milford, and Pomilla, 1960). Curve 1: 4s—>5p, Curve 2: 4£—>5d, Curve 3: 4d—>5/, Curve 4: 4/—»5g. Solid curves denote first Bom approximation while dashed curves de- note Bethe approximation.

6
Detailed calculations have also been performed on the

excitation of various levels of atomic hydrogen. (McCarroll, 1957; Boyd, 1959; McCrea and McKirgan,

1960; Omidvar, 1965; Kingston and Lauer, 1966a).

Cross sections for the transitions 1s—m ( n = 2, •••, 10),

2s—m, 2p0-^n, 2p±l—m ( n = 3 , •••, 10), and 3—>n (n = 4, • • *, 8) are given in Tables IV-VIII.

The eigenenergy En of the rath level of a hydrogen

atom is — 1/ra2 rydbergs so that for large values of the

principal quantum number n we have

dn = \rfidEn .

Hence the cross section for the excitation of atomic

hydrogen to all energy levels within the range En to En+dEn rydbergs is given by

\n*QndEn .

If Qi(E')dE' denotes the cross section for the ejection

of an electron into the continuum with energy between

E' and E'-\-dE' rydbergs, it follows that \nzQn re- garded as a function of En is continuous with Qi(Ef ). In Fig. 11 we illustrate this continuity for transitions

emanating from the ground state of atomic hydrogen

(McCarroll, 1957).

2.2. Bethe Approximation

A simplification of the first Born approximation

due to Bethe can be readily derived by expanding

exp (r'K-r) in the formula (20). Choosing the z axis

along the momentum change vector EK, which is therefore the direction of quantization of the atomic wave functions, yields

0(1 -nlm)=8omj

^„zo*(r)^i(r)

X(l+iKz-%KW+---)dr, (39)

where \pn i0 is the wave function of the state of the hydrogen atom with principal and azimuthal quantum numbers n, l and magnetic quantum number m = 0. Setting

(z*)i.m= Un iQ*{r)zspl (x)dx (40)

we obtain

0(l-nIO) =iK(z) linI -K2 (z2

) 1 . »i+ * * • (41)

and so

\fl.nlm{K) | 2 =(48om/KW) | (z)i,»,

-H(*K) (z2 )i.„H | 2 (42)

which converges provided Ka0<iCl since ao is the approximate range of the atomic wave functions. For

Kad£>l we note that |/i,n zo(K) | 2 provides a negligible

contribution to the excitation cross section Qn i. Let us first consider optically allowed transitions for

which the final state is a p state so that 1=1. Then

(z)i,nz is nonvanishing and we may put

| fl,nlm(K) \^(Uom/KW) | (*)l .nZ | 2 (1=1). (43)

Hence integrating from Km[ a = (m/h^ki) (

E

n — Ei) to Kmax = 2ki we find that

v

MifW W ln

2W

m(En —Ei) (Z=l). (44)

Thus we see that the ls—+np excitation cross section

decays as Erl ln E for large values of the impact energy

E=ELk^J2m in accordance with the Born approximation result obtained previously.

For optically forbidden transitions (z)i,n * = 0. If however (z2)i,„j does not vanish we can write

\fl,nlm(K) | 2^(Wao2

) ! (z2 )l.nZ | 2

- (45)

Neglecting the contribution from Kao>l and using ex- pression (45) for Koq^I we obtain

<2„z=(x/£iW)[| (z2 )i,„z | 2/oo2] (46)

so that the excitation cross section for a monopole

transition Is— or a quadrupole transition 1 s-+nd

decays like E~l for large values of E as found before.

The Bethe approximation may also be derived by

replacing the interaction energy F(rx , r2) by its asymptotic form for large radial distances r2 of the incident

electron. In the case of 1s—^nlm transitions this can be

achieved by substituting (ril/r2l+1

) Pi (cos 0i2 ) for l/ri2 in formula (5), where On is the angle between Tx and

r2, and then evaluating the scattering amplitude by

using formulas (16) and (17). This approach yields the same expression for/lin j m as the method described previously except when 1=0 for which case it gives a

vanishing result. The upper limit of the K integration in formula (25)

may be replaced by an adjustable parameter Kc which

is chosen so as to produce agreement with the first Born approximation at high electron impact energies.

Calculations along these lines have been performed by

Milford and his associates (McCoyd, Milford, and

Fig. 11. Illustration of the continuity of %n*Qn as a function of En with Qi as a function of E' (McCarroll, 1957). En is the

(negative) energy of an excited electron in level n\ E' is the

(positive) energy of an ejected electron in the continuum. The

crosses refer to levels with n = 2, 3, 4, 5. The numbers on the curves are the incident electron energies in rydbergs.

7
Wahl, 1960; Fisher, Milford, and Pomilla, 1960;

Milford, Morrissey, and Scanlon, 1960; McCoyd and

Milford, 1963; Milford, 1960; Scanlon and Milford,

1961), their Bethe approximation cross-section curves being compared with those obtained using the first Born

approximation in Figs. 8-10. Inspection of these figures

shows that the Bethe formula is in reasonably satis- factory accord with the first Bom approximation down

to electron impact energies which are not greatly above

the energy where the Born cross section attains its maximum value. Since the Born approximation itself ceases to be valid below this energy, the Bethe approximation provides a valuable alternative formula for

determining excitation cross sections as long as it is not employed at low impact energies.

McCoyd and Milford (1963) have used the best

available values of the cutoff parameter Kc to determine

the constants Cn'i',ni and D„'i',ni in the Bethe formula

for the excitation cross section

Qn'l’,nl = (Cn'V.nl/E) In {Dn'i> , n lE) (47)

for all transitions satisfying n—n'= 1 or 2 and l— l' —l

with 1 <»'< 10. The values of these constants when the

impact energy E is given in eV are displayed in Tables

IX and X. An extension of these calculations has been

made by Kingston and Lauer (1966a; 1966b) who investigated the case n—n'= 1 and 2 for 1<«'<6 and

all permissible values of V and l. They also evaluated

the excitation cross section Qn>,n for transitions between

the levels with principal quantum numbers n' and n. As long as the impact energy is not too low this may be

satisfactorily approximated by the formula

Qn’.n= (Cn»,*/£) In (Dn’,n E), (48)

the values of the constants Cn\n and Dn>,n calculated by

Kingston and Lauer (1966a; 1966b) being displayed in

Table XI.

2.3. Second Born Approximation

The first Born approximation neglects coupling to

all states other than the initial state and the final state

of the transition. One method of allowing for coupling

to other states is to employ the second Born approximation to the scattering amplitude. This approximation can be derived in the following manner. We consider

the case of the excitation of the «th state of atomic

hydrogen from the ground state and insert expression

(15) for SF into the right-hand side of Eq. (9) . Then we

find that to the first order in the interaction energy V

2m f C F»(r) = exp (ik!-r)5„i+ —JJ

G{ r, r2)^w *(r1 ) X V(

h r2 ) Pi ( r2) exp (tki • r2) diidr2. (49)

Now expanding 'F according to (3) , using formula (49)

for Fn and substituting into the right-hand side of expression (12), we see that to the second order in V:

Me, 0) =/.»(*, <t>)+M2) (e, 0), (50)

where /n

(1) is just the first Born approximation to the

scattering amplitude given by (16) and

/

n

(2) is given

by the formula

/„(2) (0,0) = (4tt) 1 sJJ exp {i(ki-r2'— k»*r2)}

X

exp (ikp | r2 —

2

' |) 4?r | r2 —

2

'

I Vnp (

2) VPi (

2

') dr2dt2. (51)

Expression (50) is known as the second Born approximation to the scattering amplitude. We see from for- mula (51) for/„(2) that it involves a summation over

all the states of hydrogen including the continuum

states. This makes it very difficult to evaluate exactly.

However it can be simplified by setting kp =k for all p, where k is independent of p, in which case the summation can be performed by using the closure relation

SPP*(T1)Pp(Ti')=d(T1 -T1'). (52)

It leads to the result

/»

(2) {e, 4>) = (47t) x J~^j JJ

exp fi(ki-r2'—k„*r2)}

exp (ik j r2 -r2

' |) X

4?r I r2 —

2

' Unl (r2, T2')dT2dT2 , (53)

where

u.i(n, U) = U.*(r.) J \|ri—

2 | rj

l &

X (|T^7 (54)

Expression (53) for /n

(2) has been used by Rothenstein

(1954) to investigate the \s-2p excitation of hydrogen

atoms by electron impact. Unfortunately this simplification of the second Bom approximation does not produce reliable scattering amplitudes.

Defining the quantities a = Re (/n(2)//»(1) ) and d =

Im ( /r.

(2)//n(1) ) we see that the differential cross section

given by the second Born approximation is

ln(0, 0) =(kn/ki)\M»(e, 0) I 2 {(l+a)*+/3*}. (55)

We note however that this neglects a term of the fourth

order in the interaction energy V contributed by the

real part of the third Bom approximation correction

term /„(3) while including other terms of the same order

and thus may well give rise to considerable error. Consequently it should be more satisfactory to neglect all fourth-order terms and use the formula

U6, 4>)={kjki) | /.«(*, 0) | 2 (l+2a), (56)

8
Fig. 12. Total cross sections for the lx—>2x excitation of atomic

hydrogen. Curve 1: first Born approximation (Omidvar, 1964). Curve 2: third order ls-2s coupling approximation (Kingston,

Moiseiwitsch, and Skinner, 1960). Curve 3: third order \s-2s-2p

coupling approximation (Kingston, Moiseiwitsch, and Skinner,

1960). Curve 4: ls-2s close coupling approximation neglecting

exchange (Smith, Miller, and Mumford, 1960). Curve 5: \s-2s-2p

close coupling approximation neglecting exchange (Omidvar,

1964).

which is correct to the third order in the interaction

energy.

The third-order approximation formula (56) has

been used to investigate the 1x-h>2x excitation (Kingston, Moiseiwitsch, and Skinner, 1960) and the lx—>2p

excitation (Moiseiwitsch and Perrin, 1965) employing

a simplification of the full expression (51) obtained by

truncating the summation at the states 2s, 2p0, 2p±l

with principal quantum number 2 and neglecting all states with higher principal quantum numbers. The

results of these calculations are given in Figs. 12 and 13. We see that for the lx—>2x excitation the over-all

effect of allowing for coupling to the initial and final states, often referred to as distortion, is to increase the

excitation cross section above that given by the first Born approximation, while allowance for coupling to the

2/>0 and 2/>±l states, which should be a very important

contributor to the effect known as polarization arising

from coupling to all states other than the initial and

final states, reduces the excitation cross section below

the first Born approximation curve. These conclusions

regarding the effect of distortion and polarization are

in general accordance with the results of close coupling

calculations with exchange neglected (Smith, Miller,

and Mumford, 1960; Omidvar, 1964b) which will be

discussed in Sec. 2.15. With regard to the lx—>2p excitation we see from Fig.

13 that the over-all effect of the coupling to the lx, 2x,

2p0, and 2p±\ states is to increase the excitation cross

section above the first Born approximation curve at the

lower electron impact energies but to reduce it below the

first Born approximation curve at the higher energies,

again in general accordance with the behavior revealed

by close coupling calculations with exchange neglected

(Omidvar, 1964b).

Allowance for coupling to higher levels has been made

recently by Holt and Moiseiwitsch (1968) who have

evaluated exactly the contribution to the second Born

approximation scattering amplitude from all intermediate levels up to p = 5 and have estimated the contribution from levels with p> 6 by setting kp = h and

using the closure relation (52). They find that the

effect of the couplings to the levels with p>3 is to bring the excitation cross sections closer to the first Born approximation at high energies and that the

lx—>2p excitation cross section given by the second Born

approximation now lies slightly below the first Born

approximation curve at all energies.

2.4. Exchange Scattering

So far we have not taken electron exchange into

account and because of its very great importance,

particularly at low electron impact energies, we now

turn our attention toward methods by which its in- clusion may be achieved. We express the total space wave function describing the system in the symmetrized

forms

^±(rx , r2)=1F(r1, r2)±^(r2, rx), (57)

where ^(r^ r2 ) is given by the expansion (3) and the

positively and negatively symmetrized functions Sfr+ and 'F~ are associated with singlet and triplet scattering, respectively. Then the entire wave function of

the system composed of a spin as well as a space part

is totally antisymmetric with respect to interchange

of all the coordinates, both space and spin, of elec- trons 1 and 2 thus satisfying the Pauli exclusion principle.

Substituting (57) into the Schrodinger equation

(1) , multiplying across by \pn *(Ti), and integrating with

respect to ri, yields

(

V

22 +£„2 ) Fb ±( r2) = (2m/h2) SRnp^ , (58)

P where ^np± (r2) =Fp±(r2

) Jp

n*(ri)F2 (r: ,

±&»(*2

) J

^n*(ri) { Fi(ri, r2) - (fi

2/2m) (Vr+ V) I (59)

Fig. 13. Total cross sections for the lx—>2p excitation of atomic

hydrogen. Curve 1: first Bom approximation (Omidvar, 1964).

Curve 2: third-order ls-2s-2p coupling approximation (Moiseiwitsch and Perrin, 1965). Curve 3: unitarized Born approximation

(Burke and Seaton, 1961). Curve 4: impact parameter method

(Seaton, 1962). Curve 5: ls-2s-2p close coupling approximation

neglecting exchange (Omidvar, 1964)

.

9
with

Vi ( rh r2 ) = ( e

2/rn) - (

e

1/

x ) ( 60)

being the interaction energy between electron 1 and a hydrogen atom composed of electron 2 and the proton,

and

V2 (ri, r2 ) = (

e

2/rn) - ( e

2/r2 ) (61)

being the interaction energy between electron 2 and a hydrogen atom composed of electron 1 and the proton.

Now using the Green’s function (10) for a free particle we obtain the following expression for the scat- tering amplitude:

2m r fnHG, 4>)=~ (47T )-1 — S

/ exp ( — ik„ • r2) Rnp± (r2)dr2 . n

2 pJ

(62)

The 1—m excitation cross sections Q„+ and Qn~

cor- responding to singlet and triplet scattering are given by

Qn±= J IJ ^ I 2 sin ed9d(t> ( 63)

and since they have spin statistical weighting factors

\ and f, respectively, the total 1— excitation cross section takes the form

e.=i<2»++ie»- (64)

2.5. Born-0ppenheimer Approximation

We now attempt to make allowance for exchange to the same order of accuracy as that used to derive the

first Born approximation. Again considering the case

of electrons incident upon a target hydrogen atom in its ground state, we set

jFV^r) = exp (fki*r)

FPHr)=0 (p* 1) (65)

in formula (59) for Rnp± and substitute the resulting

expression into (62). Then we obtain the so called

Born-Oppenheimer approximation to the scattering

amplitude which is given by

2m f C /n ±

(0, <t>) = — (47t) -1 —JJ

exp {f(k1 -kn)-r2 }^„ :i: (r1) X F2 (ri, r2)Mh)dTidT2 2m C f

T(47r)-1 —JJ

exp {^(kiTi-kn-ra)}

Xpn *(Ti) Fi(r: , r2)iMr2) *idr2 . (66)

The first term on the right-hand side of this formula for

the scattering amplitude is just the first Born approximation while the second term is the additional contribution arising from exchange according to the BornOppenheimer approximation.

Closed analytical expressions for the scattering amplitude corresponding to the Is—>2s and 1.?—

>2p0 excitations have been obtained by Corinaldesi and Trainor

(1952) using the Feynman parametrization technique,

the axis of quantization of the atomic wave functions

10

being chosen along the direction of incidence of the free electron. They found the exchange contribution to the

scattering amplitude for the Is—>2/>±l excitation to be

zero but this is not so, the correct expression being

given by Bell (1965) for this case. Calculations of the

Born-Oppenheimer cross sections for the Is—>2s, Is—

2p0, and Is—>2^± 1 excitations have been performed by

Wu (1960) but contain numerical errors, by Bell and

Moiseiwitsch (1963) who took the exchange amplitude

for the Is—>2/>±l excitation to vanish, and by Bell

(1965) who used the correct expression for the Is—

2/>±l exchange amplitude. The results of these calculations are presented in Tables XII and XIII. We see that for the Is—>2s excitation, the Born-Oppenheimer

approximation yields a cross section which is much

larger close to the threshold energy than that given by

the first Born approximation. This cannot be correct: thus for electrons with 13.6-eV energy corresponding to &i=l in units of Oo

_1

, the zero-order partial cross sec- tion given by the Born-Oppenheimer approximation

has magnitude 1.59 woo

2 whereas the maximum possible

cross section for the /th-order partial wave imposed by

the conservation theorem of Bohr, Peierls, and Placzek

(Mott and Massey, 1965a) is (2/+l)7rao2/&i2 so that the

Born-Oppenheimer approximation violates the theoretical maximum for P=0 at k\= 1. As we see later, this

gross overestimation of the excitation cross section produced by the Born-Oppenheimer approximation is a

typical occurrence for 5—*S transitions. On the other

hand, by referring to Table XIII, we see that the

Born-Oppenheimer approximation predicts a rather

smaller exchange effect for the Is—>2p excitations.

2.6. First-Order Exchange Approximation

The Born-Oppenheimer approximation (66) suffers

from the undesirable feature that the addition of a constant to the interaction potential F2 produces a nonvanishing alteration in the exchange scattering

amplitude owing to the lack of orthogonality between

the approximate wave function ^i(ri) exp (tki*r2) describing the initial state of the system and the approximate wave function \kn (r2) exp (fkn *ri) describing the

state of the system after an exchange event involving

the excitation of the atom has occurred. This arbitrari- ness of the Born-Oppenheimer approximation can be

seen to arise because (V2 +^i2)Fi± vanishes for F1± =

exp (fki*r) whereas it is clear from (58) that this term

is actually of the first order in the interaction potential.

Thus neglecting exchange and the coupling of the

initial state to all other states of the system we see that equation (58) gives us (V2

2 +&i2

) F1± (r2 ) = (2m/S! )F,±(r2

) J

|*,(r,) |* V,(r,, r (67)

[[Note that the argument can be carried through without these simplifying assumptions (Bell and Moiseiwitsch, 1963) .J
Replacing (V2 +&i2 )Fi± by the right-hand side of (67) and then making approximation (65), we now get in place of the Born-Oppenheimer approximation:

2

/yyi r r 2iyi r r /n±(0, 0) =-(4tt)-1 —JJ

exp {i(ki-kB )-r2 }^«*(r1 )F2 (ri, (4*)-1 —JJ

exp ^(k^n-k^ra) }

X^n *(r1 )^i(r2

) |Fi(ri, r2)— J

| hW) I 2 Fi(r/, r2 ) c/r/J dx^, (68)

which remains unaltered if a constant is added to the

interaction potential V\. This formula was first derived

by Feenberg (1932). It differs from the Born-Oppenheimer approximation by the replacement of the elec- tron-proton interaction e

2/r\ in V\ by

f l'S'i(ri') | J rn

which is the average of the electron-electron interaction

£

2Ai2 over the initial state of the hydrogen atom. The

Feenberg approximation can be derived alternatively

by using a variational principle for the scattering amplitude together with distorted wave trial functions, and

subsequently retaining only those terms of the scat- tering amplitude which are of the first order in the in- teraction energy. It seems therefore to be more securely

founded than the Born-Oppenheimer approximation

which follows from the use of a variational principle in

conjunction with plane wave trial functions. However

the attention of the reader is directed to the treatise by

Mott and Massey (1965b) where a different attitude is taken.

If instead of using equation (67) we employ the true equation satisfied by F^, the resulting scattering amplitude contains certain additional small terms which

have been excluded from formula (68). With these

terms retained the approximation to the scattering

amplitude is known as the first-order exchange approximation. It has been employed to calculate the cross

sections for the Is—>2s and Is—>2/> excitations by Bell

and Moiseiwitsch (1963) and by Bell (1965). The

results of these calculations are displayed in Tables

XII and XIII where they are compared with those obtained with the Born-Oppenheimer approximation. It

is seen that for the Is—»2s excitation the use of the

first-order exchange approximation produces a considerable reduction in the cross section below that given

by the Born-Oppenheimer approximation at low elec- tron impact energies. Moreover the zero-order partial cross section given by the first-order exchange approximation has magnitude 0.39 ttOq2 so that, unlike the

Born-Oppenheimer approximation, the upper bound

imposed by the conservation theorem is not violated.

Thus the first-order exchange approximation is the

better approximation for the case of the Is—>2s excitation of atomic hydrogen. We shall see later that this

appears to be generally so for £—>S transitions.

For the Is—>2p0 and 2/>±l excitations the difference

between the cross sections given by the Born-Oppenheimer and the first-order exchange approximations is negligible except at energies close to the threshold.

Inspection of Tables XII and XIII shows that ac- cording to the first-order exchange and the BornOppenheimer approximations, the effect of exchange is to reduce the Is—>2s and Is—>2/> excitation cross sec- tions for electron impact energies away from the

threshold.

2.7. Ochkur Approximation

Another approach to the problem of overcoming the

deficiencies of the Born-Oppenheimer approximation is due to Ochkur (1963). He expands the exchange scat- tering amplitude

2tn# f f

gjd, 0)=-(4tt)-1 -^- JJexp {i(ki*ij— k„*r2 )

}

X^»*(r0 (ji-T

1 —r{~1

) \pi (

r

2) diidi^ (69)

given by the Born-Oppenheimer approximation in

powers of krl and retains only the leading term which

behaves as ki~2

. The electron-proton interaction c

2/rx results in a contribution which decays more rapidly

than kr2 and so is neglected.

To obtain the leading term arising from the electronelectron interaction e

2/rn we note that

(47rr12 )~1 = ^-y-3 Jexp {iq- (rx-r2 )

} ^

. (70)

Since the main contribution to gn comes from q^ki,

we have to the lowest order in k-r1

(

r

12) -1=( Air/kF) 5 ( ri — r2) (71)

remembering that the Dirac 8 function can be expressed

in the form

$( f) = J^—3 J

exp (iq-r)</q. (72)

Hence

gn(6,<l>)=- ^-2 J

exp ( iK • r2) \pn *

(

r2) \pi ( r2) dr2, (73)

where K=ki—

k

n , giving for the entire scattering amplitude according to the Ochkur approximation:

/n±(0,0) = -(2/ao)[Al/^2 )±(lAi2)>(l-«). (74)

11
The calculations using this approximation have been

confined so far to helium excitation and are discussed

in Sec. 3.2. [Note added in proof. Recently Inokuti (1967)

has used the Ochkur approximation to calculate the

Is

>2s and Is—>2p excitation cross sections for atomic

hydrogen. As we see there, the Ochkur approximation

appears to be not only a very useful one but also rather

simple to apply since it is no more difficult to evaluate

than the first Born approximation.]

A modification of the Ochkur approximation based on a variational principle has been derived by Rudge

(1965). He has applied his formula to the Is—»2s

excitation of atomic hydrogen, obtaining a cross-section curve which is substantially smaller than that given by

the first Born approximation and also by the Is— 2s

close coupling calculation with exchange included

(Omidvar, 1964b).

2.8. Distorted Waves Approximation

We have already examined the use of the second Born

approximation in making partial allowance for the

effect of distortion. Full allowance for the distortion of

the incident and scattered electron waves can be

achieved in the following manner. We make a two-state

approximation involving just the initial state, which we

suppose to be the ground state 1 of the hydrogen atom,

and the final state n, and neglect coupling to all other

states. Then the infinite set of coupled equations (58)

reduce to the pair of equations

V2

2 +&i2 — ~ Fn(r2

) J F1± (r2) T —J

An(rx, r2) Fi± (ri)<iri— — j

Fx„(r2) Fn± (r2) ±JKin (Ti, r2)Fn± (ri)dri •

(75)

V2

2 +&n2 — — F„„(r2

) j

F„± (r2) zF —J

nn { ri, *2) Fn± (r1 )dii= — |VnX (r2) F1± (r2) ±j

r2) Fi± (ri)Jr1 j,

(76)

where the matrix elements FnX (r2) of the interaction

energy F(rx , r2 ) are given by the formula (17) and the

kernels iT„x (rx , r2) have the form

Kni(r1} r2) =0x (r2)0„*(rx) { (e2/r12)-F£x +En -E). (77)

If we know the solution of the reduced equation

2m 1 ~ V2

2 +&„2 — — Fnn (r2)

I 3

: „± (r2 )

“F — JK„n (Ti, r2) ffn± (ri)<irx=0 (78)

obtained by setting the right-hand side of (76) equal to zero and choosing the asymptotic behavior for large r to be

£F„±(r)~exp (— fk**r)-j-/'-1 exp {iknr)£n±{0, <p) (79)

we may express the scattering amplitude /„(0, 0) in

the form

s 1 2m

X

|

Vni (

2) Fi± (2) ±JK„i (

x , r2) F (

x ) drx |

dr2. ( 80)

We cannot obtain without solving the pair of

coupled equations (75) and (76). Since this requires

considerable computational effort, we introduce here

the so called distorted waves approximation which is based upon the assumption that the back coupling of

the final state n to the initial state 1 is small so that

Vu and Kin can be set equal to zero on the right-hand

side of Eq. (75) which now takes the reduced form

[vi'+W-^KuWl^r,)

T —j*Ku (Ti, T2)-Ji ± (ri)dri-0, (81)

where CFA is chosen to have the same asymptotic behavior as Fi* for large r: SvKO^exp (ikx -r) exp (ikxr) <f>). (82)

Replacing Tq* by 3

rx

=t in (80) yields the distorted waves

approximation to the scattering amplitude:

fn

±

(6, <t>) = — ( 4tt)

1 —J

^n±(Ti)

X |F„x (r2)3:x

:t (r2)±J

ni( rx , r2)3:x± (rx )drx|

<fr2. (83)

To solve equations (78) and (81) it is necessary to make

a partial waves analysis of and TA in which case the

total excitation cross section Q„ may be expressed in

terms of the partial cross sections Qna' according to the

formula (32). The earliest calculations employing the

distorted waves method were performed by Erskine

and Massey (1952) with, as well as without, the in- clusion of exchange. They dealt with the H—>25 excitation of atomic hydrogen and considered only the l =

12
Fic. 14. Total cross sections for the \s—*2p excitation of atomic

hydrogen. Curve 1: first Born approximation (Omidvar, 1964). Curve 2: \s-2p close coupling approximation including exchange

(Omidvar, 1964). symbols: distorted waves approximation

including exchange (Khashaba and Massey, 1958).

f =0 partial wave, the distorted wave functions being

determined by variational methods. The exchange dis- torted waves calculations were subsequently repeated

by Ochkur (1958) who however determined the dis- torted wave functions numerically using the integral

equations method developed by Drukarev (1953). The

zero-order partial cross sections calculated using

the method of distorted waves are compared with the

results obtained employing other forms of approximation in Table XIV. When exchange is neglected we see that the method of distorted waves produces zero-order

partial cross sections which are quite close to those

found using the first Born approximation, both being

in satisfactory accordance with the ls-25 close coupling

calculations of Bransden and McKee (1956). However

with exchange taken into account, not only are the

partial cross sections <22S0± resulting from the two applications of the distorted waves method very different

from those obtained using the Born-Oppenheimer

approximation but also they are considerably different

from each other. This is probably due to the sensitivity

of the partial cross sections to the distorted wave functions which were determined in different ways by

Erskine and Massey and by Ochkur. Moreover comparison with the ls-2s close coupling calculations with

exchange included carried out by Marriott (1958)

shows that the distorted waves approximation fails to produce accurate zero-order partial cross sections and

so we infer that the back coupling arising from ex- change is not sufficiently weak to be neglected here.

Calculations uave also been carried out by Khashaba

and Massey (1958) on the Is—>2p excitation of atomic

hydrogen treating the /=0 and the t = 0 and 1 partial waves by the distorted waves method and the higher

order partial waves by the Born-Oppenheimer approximation, variational methods being employed to determine the distorted wave functions. They found that the

Qip1 - 0 partial cross section was greatly reduced below

that given by the Born-Oppenheimer approximation

as a consequence of making allowance for distortion.

Ochkur has repeated the calculation of the 1, 0 partial cross section using an T = 0 distorted wave function for the scattered electron, determined with the aid of the

integral equation method due to Drukarev, and using an undistorted 1= 1 wave function for the incident

electron. The results of Ochkur’s calculations are not in satisfactory accordance with those obtained by Khashaba and Massey.

In Fig. 14 we make a comparison between the \s—>2p

total excitation cross section calculated by Khashaba

and Massey and the \s-2p close coupling approximation

cross section. The agreement between the two approximations is quite good, both of them leading to total excitation cross sections lying somewhat below the

first Born approximation curve. Vainshtein (1961) has examined the Is—>2s, Is—>2/>, Is—>3s, Is—>3/>, Is—>3rf, Is—>4s, Is—Ap, and 2s—^3p,

2/>—>3s excitations of hydrogen atoms treating the

partial cross sections with (, t'<6 by the distorted waves method, but neglecting exchange, and calculating

the higher order partial cross sections using the Born

approximation. The resulting total cross sections are displayed in Table XV. A noteworthy feature of these

values is that the distorted waves method without ex- change gives larger cross sections than the Born approximation at low energies. This occurs because allowance

for distortion involves the introduction of an attractive

potential so that the wave functions are pulled in and

the cross sections increased as a consequence.

We also show the partial cross sections obtained by

Vainshtein for the Is—>2s and \s-^2p excitations using

the distorted waves method in Figs. 15 and 16. It can

be readily seen that for the ls-+2p excitation the 1, 0

Fig. 15. Partial cross sections for the Is—>2s excitation of atomic

hydrogen calculated using the distorted waves method neglecting

exchange (Vainshtein, 1961). he is electron energy in threshold

units defined by Eq. (38).] Curve 1: QtP, Curve 2: Q^1

, Curve 3: Q-J, Curve 4:

c Curve 5: 2 Qi/.

r-o

13
and the 0, 1 partial cross sections make a negligible

contribution to the total excitation cross section, the

1, 0 curve being too small to be distinguished on the

figure. This is quite different from the result obtained

with the first Born approximation which gives the 1, 0

partial cross section as the primary contributor to the

total cross section. At low electron energies the dis- torted waves method gives the 2, 1 partial cross section

as the most important. Also, as one would expect, it is found that the (, i— 1 partial cross sections are much

larger than the l, t-\- 1 cross sections.

2.9. Modified Bethe Approximation

In all the previous sections we have assumed that the

coupling between the initial and final states is weak. In

certain cases however this coupling is obviously very

strong. An example of very strong coupling is the 2s—*2p

excitation of atomic hydrogen. It is found that when the

Born approximation is applied to it, the partial cross sections Q

(

, associated with incident electrons having

angular momentum 1) U

1/2

, exceed the theoretical maximum (2/-f l)7rOo2/&22 imposed by the conservation theorem (Mott and Massey, 1965a) when l

is small. A method of dealing with this difficulty has

been developed by Seaton (1955a). Let U be the value

of l for which

0Bethe^<l(2/+l)Tao2/^22

( l~> fi) >i(2/+l)7rao2A2

2 (/<4>), (84)

where Qsethe* is the partial cross section given by the

Bethe approximation. The modification of the Bethe

approximation suggested by Seaton sets

(2* = (?Bethe* (^>4) = i(2m)xao2/W<4) (85)

and is based on the plausible assumption that the partial cross sections for t< can be replaced by their

average value [§(2/+l)7rOo2 /£2

2] without undue error. Using this approximation Seaton has calculated the

rate coefficients for the 2s-^2pm and 2s—>2/>3/2 transitions in atomic hydrogen at temperatures of 10 000°K

and 20 000°K assuming a Maxwellian velocity dis- tribution of electrons and taking the energy differences

between the upper and lower states to be 0.0354 and

0.327 cm-1

, respectively. The rate coefficients are given

in Table XVI where they are compared with the values

obtained with the Born approximation, the usual

Bethe approximation and a semi-classical approximation (Purcell, 1952).

Allowance for strong coupling by means of the modified Bethe approximation decreases the rate coefficients

only slightly for the case under consideration because

most of the contribution to the cross sections comes from

distant encounters so that large values of l dominate

and the first Born approximation is fairly accurate. The effect of very strong coupling is most important

Fig. 16. Partial cross sections for the Is—>2p excitation of atomic

hydrogen calculated using the distorted waves method neglecting

exchange (Vainshtein, 1961). [e is electron energy in threshold

units defined by Eq. (38).] Curve 1: Qip2 ' 1

, Curve 2: Qop3 - 2, Curve 3: (?2p

4,3

, Curve 4: Q2pM, Curve 5: Qip

® ,5

} Curve 6: Q

0 ' 1

,

6 Curve 7: 2 Q^1

, Curve 8 :Q>P

. o

for n—»ra+l transitions of atomic hydrogen discussed

in the succeeding section and, for example, in the case

of the 3s—>3/> excitation of sodium examined in Sec. 6.

2.10. Impact Parameter Method

In this section we shall be concerned with the use of the semiclassical method introduced by Seaton ( 1962)

.

Known as the impact parameter method, it is based

upon the assumption that the path of the incident

electron can be taken to be a straight line whose

perpendicular distance from the atomic nucleus is the

impact parameter p. If Pn\{p) denotes the probability

that an incident electron having impact parameter p

produces a transition from the ground state 1 to an

excited state n of the target atom, the total \—m excitation cross section can be expressed in the form

Qn = 2ir f Pni(p)pdp. (86)

The transition probability Pn\ is evaluated using first- order time-dependent perturbation theory which gives

Pn1 = exp { i ( En — Ei) t/fi } Vni (t) dt

2

1 (87)

where Ex and En are the energies of the initial and final states of the atom, and V„i (t) is the matrix element of

the Coulomb interaction between the incident elec- tron and the target atom associated with the initial and final states. In the case of hydrogen

F„,(() = j

*.•(!,) Mr0*1, (88)

where \px and \(/„ are the atomic wave functions corres- ponding to the states 1 and n, respectively. The elec- tron-electron interaction e*/\ ri— r2 1 may be expressed

lU
as a multipole expansion using the formula

| n— r2 1"1 * £ Px(cos 0) , (89)

where r< and r> are the lesser and the greater of ri and

r2 , 0 is the angle between ri and r2, and the P\ (cos 0) are

the Legendre polynomials. For optically allowed transitions Is—>np, the only term of the multipole expansion

which provides a nonvanishing contribution to V„i is the dipole term. When p is large we may replace Vni by

its asymptotic form for large separations. Since the

dipole term corresponding to X = 1 takes the form

e

1*! *r2/r2

3 for large r2, we see that

p<pi, in which case (86) can be rewritten as

(?»=27rpi2 +27r/" Pnl (p)pdp. (95)

•'pi The procedure employed by Seaton (1962) was to adopt the smaller of the two cross sections obtained

using formulas (93) and (95) at each electron energy.

The excitation cross section calculated by him for the

Is—>2p transition of hydrogen is displayed in Fig. 13. Saraph (1964) has employed essentially the same method as that introduced by Seaton (1962) to evaluate

the excitation cross sections Qn , n+i for the n-*n-fl

optically allowed transitions of hydrogen. Her calculated values of the quantity 1 ,„ arising in the formula

ff.*(r,)rih(r,)*, (90)

J

in which case we obtain

4/(», 1)

Pn1 = AE

,

'Til rJ —

o

exp (iAEn\t) dt rf{t)

(91)

where f(n, 1) is the oscillator strength for the 1—

m

transition given by

<h

2f(n, 1)

AEnl

(92)

AEni is the excitation energy in rydberg units, and r2 is in units of the Bohr radius Oq. Remembering that the

path of the incident electron can be taken to be a

straight line, we have r2

2 (/) = p

2 +z>2

/2

, where v is the

speed of the electron. Then the integration occurring

in formula (91) can be performed analytically, the

resulting expression for Pn\ being given in terms of the

modified Bessel functions K0 and K\.

When the coupling between the initial and final states is strong, Pn\{p) becomes large for small values

of p and may even exceed the maximum possible value

unity when first-order perturbation theory is employed

to evaluate the transition probability. Two different

methods can be used to overcome this difficulty:

(i) A cutoff parameter p0 may be introduced which

is independent of electron energy and is chosen so as to give agreement with the Bom approximation at high

impact energies. Then

Qn = 2ir

f

a>Pni(p)pdp. (93)

^ PO

(ii) When the coupling is very strong, one can in- troduce a parameter pi chosen so that

Pnl(p)>h (P<Pl), (94)

where Pni is evaluated using first-order perturbation

theory. Since the actual transition probability when

the coupling is very strong is an oscillating function of p

for p<pi, v/e may replace Pni by its mean value \ for

Qn,n+l = Xr^.l,„2irrn

2

, (96)

where r„= j(5«2 +l)co is the mean radius, are pre- sented in Table XVII. Seaton’s impact parameter

method is particularly valuable for cases of very strong

coupling such as the n—m+1 optically allowed transitions for large n and the cross sections calculated by

Saraph ( 1964) are to be preferred to those given by the

Born approximation for w>3. It largely supercedes the

modified Bethe approximation discussed in the pre- ceding section.

2.11. Classical Approximation

The earliest application of classical mechanics to the

inelastic scattering of electrons by atoms was made by

Thomson (1912) who regarded the collision as a classi- cal binary impact between the incident electron and

one of the atomic electrons which was assumed to be

initially at rest. Clearly, at low impact energies, the

neglect of the motion of the atomic electron must be an

inadequate approximation. Consequently a more re- fined classical theory was introduced by Gryzinski

( 1959) making allowance for the velocity of the atomic

electron although still treating the collision as an impact

between two free electrons, the influence of the nucleus

being ignored.

Gryzinski (1959) evaluated the total cross section

for the collision of two free electrons, whose initial velocities Vi and v2 make an angle 6 with each other,

such that the energy of the atomic electron changes by

an amount AE. Then replacing the relative velocity V

of the two electrons by the mean value V= (vi2 -\-v2

2

) 1/2 he integrates over 9 assuming an isotropic velocity

distribution for the atomic electron. This yields the

total cross section cr(AE) for the transfer of an amount

of energy AE from the incident electron to the atom.

Classically we may suppose that excitation from an

initial state 1 to an excited state n of the target atom

takes place if AE lies in the range of energies

En— Ei< AE< En+i— E\. (97)

Integrating over this range of AE therefore gives the

classical approximation to the total 1—m excitation cross

15
section, all final angular momenta of the atom consistent with the specified range of energy transfer

being included. It is found that it decays as E*1 with

the energy E of the incident electron, which is a more

rapid fall off than that predicted by quantum theory:

Er1 In E. Stabler (1964) has found a method of integrating over 6 without having to make the approximation of replacing V by V. This actually leads to some- what simpler analytical expressions for the total 1—

excitation cross section than given by the original

Gryzinski evaluation. Another modification of the

classical approximation has been introduced by Kingston ( 1964) who, on the basis of the known slow variation n

zQn with principal quantum number n of the

upper level (see the end of Sec. 2.1) , expresses the 1—*n

excitation cross section in the form

Qn = n-za(En -E1). (98)

for the atomic electron given by (99) is quite different

from the actual quantum theoretical distribution.

Another attempt at obtaining the correct high-energy

behavior of the excitation cross section has been made

by Burgess (1964) . His approach to the problem was to use the semiclassical impact parameter method described in Sec. 2.10 for impact parameters p>po and

the classical approximation for p<po- Taking V=V

and assuming that the relative probabilities for the excitation of the 2s and 2p

states of atomic hydrogen are determined by / | pu | 2

| pis | 2 dr and / | pu \ 2 \pip | 2 dr,

Burgess (1964) has calculated the Is—>2s and Is—>2p

excitation cross sections. His cross section for the 1—>2 transition is displayed in Fig. 17. We note that it exhibits an interesting maximum just above threshold

as well as the main maximum at higher electron energies.

A valuable review of classical approximations has

been written recently by Burgess and Percival ( 1968)

.

In Fig. 17 we display the total cross sections for the 1—>2 excitation of atomic hydrogen calculated using the

various forms of the classical approximation as well as the Bom approximation. Due to the rapid fall off with

electron energy of the classical approximation cross

sections, they are necessarily smaller than the Born

approximation cross section at high energies. We see from Fig. 17 that this relationship persists down to low

energies for the 1—>2 transition. However for n—w+1

transitions with larger values of n, the classical approximation cross section becomes larger than the Born

approximation cross section at low energies.

In a more recent formulation of the classical scat- tering problem, Gryzinski (1965a; 1965b; 1965c) employs a continuous distribution of velocities for the

atomic electron having the form

/O) =[(*—2) !]~1 (^oA)* exp ( — Vo/v). (99)

2.12. Vainshtein, Presnyakov, Sobelman Approximation

A new method which makes explicit allowance for

the all important repulsion between the atomic and

incident electrons in the wave function characterizing

the collision has been introduced by Vainshtein, Pres- nyakov, and Sobelman ( 1963)

.

They express the total wave function in the form

*(rx , r2 ) =^i(r1)g(r1, r2), (100)

where pi is the wave function of the initial state of the

hydrogen atom, and then substitute into the Schrodinger equation (1). Changing to the new coordinates

R= Kr2+ ri)> j>= 2 ( r2 —

i) (101)

and introducing an approximation which is closely

equivalent to replacing the interaction e

2/^ by e2/R,

they arrive at the equation

The special case s=3 gives rise to a total excitation

cross section which decays according to the law Er1 In E

for large E in accordance with quantum theory. However it should be noted that the distribution of velocities

Fig. 17. Total cross sections for the 1—>2 excitation of atomic

hydrogen calculated using various forms of the classical approximation. Curve 1: classical impulse approximation (Stabler,

1964). Curve 2: Gryzinski approximation (Kingston, 1964).

Curve 3: semi-classical approximation (Burgess, 1964). Curve 4:

first Born approximation (Omidvar, 1965)

.

[

h

2 e2 e2 S (WV+2M+--- *(R,e)=o. (102)

We see that it describes the scattering of two free elec- trons by each other and the motion of their center of mass in the electrostatic field of the proton. Because

the variables R and p are separated in Eq. (102) its solution g(R, p) can be expressed exactly in terms of

confluent hypergeometric functions F{a,b,z). It can be

readily verified that

g(R, g)=N exp {ikx- (R+p)

}

XF(iv, 1; ik\R— iki*R)

XF( — iv, 1 ; ikip— iki • p) , (103)

where v = kc1 and iV = r(l— iV)r(l+iv).

Vainshtein, Presnyakov, and Sobelman (1963) now

note that the electron-proton interaction — e

2/^ gives no contribution to the scattering amplitude for excitation collisions when the Born approximation is used.

They therefore neglect this interaction in their own

16
approximation and are able, after the application of a peaking approximation, to express the scattering amplitude in the form

fn (K) = - (

2

me2/h2K2

) Ad{\-n), ( 104)

where A is a dimensionless quantity given by

NK2 f A= / F(iu, 1-ikir—iki‘T)

7T J XF( — iu, 1 ; ikir — zki • r) Xr~l exp ( 2iK • r) dr. (105)

We see that if A is set equal to unity we regain the

first Born approximation.

Unfortunately the terms which have been neglected

in deriving equation (102) give rise to divergences in the limit k\—>0. By introducing an effective charge f, Vainshtein et al. show that these divergences can be

eliminated if the parameter v is redefined according to the formula

v=(kl +E1 ''2 )-\ (106)

To evaluate A, Vainshtein el al. replaced exp (2zK*r)

by exp ( — 2?K*r) in (105). However Omidvar (1967)

has discovered that this further approximation is unnecessary and has given a method of evaluating A

exactly. Crothers and McCarroll (1965) have introduced a modification of the original approach of

Vainshtein et al. which also enabled them to evaluate A

without approximation but they found that the effec- tive charge and the corresponding value of v become

complex which seems undesirable.

The cross sections for the Is—>2s and Is—>2p excitation calculated by Omidvar using his expression for the

scattering amplitude with exchange neglected show

subsidiary maxima close to the threshold energy

similar to those found by Crothers and McCarroll. We

note that a subsidiary maximum close to the threshold was also obtained by Burgess (1964) for the Is—>2/> excitation cross section using a semiclassical approximation wdiich likewise regards the scattering problem

as a collision between two free electrons.

There are so many approximations of an uncertain

character which are made in the Vainshtein et al. approach, even as treated by Omidvar, that it seems impossible to assess properly the validity of the method.

[In a recent note by Crothers (1967) it is claimed that

there is an error in the work of Omidvar (1967) arising

from an incorrect analytic continuation. Correction of

this error leads to the removal of the subsidiary maxima

found previously.]

We suppose that the atomic hydrogen electron has

principal quantum number n and angular momentum

quantum numbers k, md, Wi* and that the free electron

has wave number kn and angular momentum quantum

numbers /2 , nt2

l

, m2\ The angular momenta of the two

electrons may then be coupled together to yield the

total orbital angular momentum with quantum numbers L, Ml and the total spin angular momentum with

quantum numbers S, Ms. Since the total orbital and

spin angular momenta are separately conserved

throughout the collision, their quantum numbers re- main unaltered.

Introducing the spacial function

,LMlS (T1 ,T2)= X) Cmi ‘m2 lMLhhL

'Pnl 1m l

l (ri) mil ,m2l X F/im ,*(02 , (tH)rrx Fnh i 2LS {r2), (107)

where Cmi lm 2

l ML llhL is a Clebsch-Gordan vector coupling coefficient, i/',u 1mi z ( ri) is an atomic hydrogen wave

function and Y

i

2m2

l is a spherical harmonic, we may ex- pand the complete spacial wave function in the symmetrized forms * ±L*LS( ri , r2 j =2-1/2 X(^>''

LM *"s

(i‘i, r2 )

v> ±4v^(r2 , fl)}, (108)

where v denotes the set of quantum numbers nl\l2. The positive sign is associated with the singlet .$ = 0

scattering case while the negative sign corresponds to the triplet S= 1 scattering case. Substituting ( 108)

into the Schrodinger equation (1) then yields the set

of coupled integrodifferential equations for the radial

functions FvLS (r)

:

d

2

, , 2 ^2 + 1 ) dr2

' n r

2 F„LS (r)

+ (2m/h2 )Z[V^L (r)-Wyv AS( r)jF^(r)=0, (109)

y' where the Vvv

> L (r) are potential interaction terms and

the Wvv

> LS (r) are exchange interaction terms whose detailed forms have been given by Percival and Seaton

(1957). These equations are independent of Ml and

Ms and exhibit no coupling between states of different

L, S quantum numbers.

Let us suppose now that the initial state of the whole

system of electron and hydrogen atom is v and that

the final state is v. Then the asymptotic behavior of the

radial function FvLS (r

)

may be chosen to have the form

at large r. FvLS (r)~kn

1/2 {sin (kn r—\l2Tr)bvv

'

2.13. Partial Wave Analysis

At the present stage it is convenient to examine the

partial wave analysis in more detail and from a different

point of view. The approach we shall follow is that due

to Percival and Seaton (1957).

-fcos (knr— |/27r) Ryv >\ (110)

or alternatively

FyLS (r) r^krT112

{ exp[— i{knr— §l2ir) ]<5 — exp [i (

k

nr— \l2tt) } , (111)

17
where Rvv

> and Svv> are the elements of the reactance matrix R and the scattering matrix S, respectively. It can

be readily verified that these matrices are related ac- cording to the formula

5=(l+iR)/(l-iR)- (112)

Since the reactance matrix is real and symmetric we

have

R* = R, R = R (113)

and so S = S, S*S = 1 (114)

which means that the scattering matrix is symmetric

and unitary. The unitary property of the scattering

matrix is equivalent to the conservation theorem of

Bohr, Peierls, and Placzek (Mott and Massey, 1965a)

.

The cross section can be conveniently expressed in

terms of the transmission matrix T defined by the

formula

T=l — S = — [2iR/(l— *R)]. (115)

Thus we have

Qn'w.m^^inh, nV)/lkn'*2(2h'+ 1)], (116)

where the collision strength £2 is given by

ft(«/i, n'li

)

= h Z (25+1) (2L+1) | T( nlik SL, n'l\U SL) | 2

. hh'SL

(117)

2.14. XJnitarized Born Approximation ( Sometimes

Known as the Born II Approximation)

The usual Bom approximation to the n'li'—>nli excitation cross section follows from the analysis summarized in the previous section by setting

T=-2iRB, (118)

where the matrix R# is the Born approximation to the

R matrix and has elements

Rb.,s = - ~ (knknl )^rjl2 (knr) Vvv A{r) h

2 J

0

Xji’2 (kn>r)r2dr. (119)

Expression (118) for the T matrix corresponds to an S

matrix which does not satisfy the unitary condition

S*S = 1 so that the conservation theorem (Mott and

Massey, 1965a) is not obeyed although this is not a

serious matter if T is small.

Seaton (1961) has pointed out that the unitary condition can be satisfied by setting R = R# in formula (115)

for T which yields the unitarized Born aporoximation

to the transmission matrix

T = — 2iRb/ ( 1 — iRs) • (120)

Calculation of the Born approximation

matrix for transitions involving the states

V nli h L

1 Is l l

2 2s l l

3 2P i- 1 l

4 2P M-l l

5 2 P (

have been performed by Lawson, Lawson, and Seaton

(1961). Using these matrix elements together with

formula (120) for the transmission matrix, V. M.

Burke and Seaton (1961) have evaluated the collision

strengths £2(ls, 2s), £2(1 s, 2 p), £2(1$, 2p0), £2(2s, 2pv<P), and £2 (2s, 2/>3/2 ). Whereas formula (118) for the trans- mission matrix makes no allowance for coupling between

the initial state or the final state and other states,

formula (120) does include some allowance for such

couplings. In addition, since the conservation theorem

is automatically satisfied by taking the transmission

matrix to have the form (120), one might expect the

resulting cross sections should be more accurate than

those obtained by employing the usual form of the

Born approximation. That this is not necessarily so can be verified by inspection of Figure 13 where the

cross sections for the Is—»2/> excitation of atomic hydrogen, calculated using the unitarized Born approximation and the \s-2s-2p close coupling approximation

with exchange neglected, are compared. The inadequacy of the unitarized Born approximation for the

Is

>2s and Is—>2p excitations occurs because the couplings involved are insufficiently strong.

Coupling to the 3s, 3p, and 3d states has been in- cluded in an investigation carried out by Somerville

(1962; 1963) using the unitarized Bom approximation

but the resulting changes in the cross sections were

small.

2.15. Close Coupling Approximation

A much more satisfactory approach to the problem of

determining excitation cross sections is to obtain the

transmission matrix elements by means of the exact

solution of the coupled integrodifferential equations

(109) using numerical techniques. Since one cannot

solve an infinite set of coupled equations, the atomic

states which enter into the expansion of the total wave

function (107) are confined to just a few of the lowestlying states giving rise to a finite set of coupled integrodifferential equations. This is called a close coupling

approximation, the order of the approximation depending upon the atomic states which are retained in

the expansion of the total wave function.

The first calculations along these lines were concerned

with the Is—>2s excitation and neglected coupling to

all states other than the initial and final states. We

refer to this as the Is— 2s close coupling approximation.

Bransden and McKee (1956) performed a numerical

integration of the pair of coupled equations associated

18
with the /=0 partial wave in which exchange was

neglected. Subsequently Marriott (1958) performed a

similar investigation but included full allowance for exchange. The resulting values of the Is—>2s excitation

cross sections for the zero-order partial wave, (V*

and 028o±

, where the absence of a sign indicates exchange

neglected and the positive and negative signs denote

singlet and triplet scattering, respectively, are given

in Table XIV. As we have noted earlier, when exchange

is neglected, the first Born and the distorted waves

approximations both provide partial cross sections

(V which are in fairly satisfactory agreement with the

results of the numerical integration of the coupled

differential equations. However, when exchange is included, the Born-Oppenheimer and the distorted waves approximations give rise to partial cross sections

Q2a0± which are considerably different from those calculated using the ls-2s close coupling approximation.

The calculations of Bransden and McKee (1956),

in which exchange was neglected, were then extended

by Smith, Miller, and Mumford (1960) to all partial waves making a significant contribution to the total

Is—>2s excitation cross section for electron impact

energies up to 54 eV their cross-section curve being

displayed in Fig. 18. They also included coupling to the

3s state, which was found to have the effect of slightly

reducing the total Is—>2s excitation cross sections; in

addition they calculated the Is—»3s excitation cross section using the ls-3s and the ls-2s-3s close coupling

approximations. Subsequently Smith (1960) extended

the calculations of Marriott (1958) on the ls-^-2s excitation, in which exchange was taken into account,

to the 1=1 and 2 partial waves.

Referring to the list of states (121) we note that the

states v=l, 2, 3, 4 are of the same parity and lead to four coupled integro-differential equations for L> 0

and to three coupled equations for L = 0 because the

v = 3 state ceases to exist for this special case. Since the

v=5 state has a different parity to the other states, it is uncoupled to them and gives rise to a separate equation. The solution of this equation is required only for

elastic scattering and for transitions between the 2p

states with j =

\

and j = §

.

These coupled integrodifferential equations have

been solved numerically by several investigators for a number of values of L (P. G. Burke, V. M. Burke,

Percival, and McCarroll, 1962; V. M. Burke and McCarroll, 1962; Damburg, 1962; Damburg and Peterkop,

1962 a, b, c and 1963; Burke, Schey, and Smith, 1963;

Gailitis and Damburg, 1963; Damburg and Gailitis,

1963, Burke, Ormonde, and Whitaker, 1966; Burke,

Ormonde, Taylor, and Whitaker, 1967; and Omidvar,

1964) . At the time of writing the most extensive calculations have been carried out by Burke, Schey, and

Smith (1963) and by Omidvar (1964) . In Tables XVIII

and XIX are displayed the partial cross sections Q

L

for the Is—>2s and ls—>2p excitations of atomic hydrogen calculated by Burke, Schey, and Smith (1963)

according to the ls-2s-2p close coupling approximation

with allowance made for exchange in which all four

integrodifferential equations connecting the v = 1, 2 , 3, 4

states are solved for each value of L. These values of the

singlet and triplet partial cross sections include the spin

weighting factors £ and f, respectively. All L values

making a significant contribution to the total cross sections were included. Presented in Tables XX and

XXI are the total Is—>2s and ls-+2p excitation cross sections calculated by Burke, Schey, and Smith (1963)

employing the ls-2s-2p close coupling approximation

with exchange taken into account. It is instructive to compare these total excitation cross sections with

those calculated by Omidvar (1964) according to the

ls-2s-2p close coupling approximation with exchange

neglected. This comparison is made in Figs. 18 and 19; we see that the effect of allowing for exchange is to reduce substantially both the Is—>2s and Is—>2p excitation cross sections close to the threshold energy al- though the behavior becomes more complicated a few

volts above threshold for the Is—*2s excitation. To

show the effect of coupling to states other than the

initial and final states of the transitions, we also dis- play in Figs. 18 and 19 the total ls-^2s excitation cross section calculated using the ls-2s close coupling approximation and the total Is—>2/> excitation cross sec- tion calculated according to the ls-2p close coupling

approximation, exchange being included in both cases. We see that coupling to the states different from the

initial and final states of the transition causes a reduction in the excitation cross sections, except for the

Is—>2s excitation close to the threshold energy where

in fact the coupling to the 2p state produces a small

increase.

Because of the probable importance of coupling to higher states, Burke (1963) investigated the con- vergence of the close coupling approximation by solving

the seven coupled integrodifferential equations involved

in the ls-2s-2p-3s-3p close coupling approximation

Fig. 18. Total cross sections for the Is—->2s excitation of atomic

hydrogen. Curve 1: first Bom approximation (Omidvar, 1964).

Curve 2: ls-2s close coupling approximation neglecting exchange

(Smith, Miller, and Mumford, 1960). Curve 3: ls-2s-2p close coupling approximation neglecting exchange (Omidvar, 1964).

Curve 4: ls-2s close coupling approximation including exchange

(Omidvar, 1964) . Curve 5: \s-2s-2p close coupling approximation

including exchange (Burke, Schey, and Smith, 1963)

.

19
at 16.5 eV. The results of his calculations for the Is—>2.s, Is—>2p, and Is—>3p excitation cross sections are dis- played in Table XXII. Clearly the effect of coupling

to the 3s and 3p states reduces the cross sections substantially, particularly for the Is—»2s excitation case. It is to be anticipated that the still higher states will

have a profound effect upon the total cross sections.

Burke, Schey, and Smith ( 1963) have also calculated

the Is—>3p excitation cross section using the ls-3/>

close coupling approximation. They find a large peak

in the neighborhood of 15 eV as can be seen from Fig. 20

where their cross-section curve is compared with the

Born approximation. However the \s-2s-2p-3s-3p close

coupling calculation performed by Burke (1963) shows

that coupling to the 2s, 2p, and 3s states is important,

reducing the Is—>3/> excitation cross section from 0.319

7rOo2 to 0.250 7rOo2 at 16.5 eV. Coupling to higher states may well be important also. Very interesting calculations in the neighborhood of

the threshold energy for the excitation of the 2s and 2p

states of atomic hydrogen have been performed by

Damburg and Gailitis (1963) and Gailitis and Damburg

(1963). They employed the \s-2s-2p close coupling

approximation with exchange included to determine the

Is—>2s and Is—>2p partial cross sections for L = 0.1, 2. These are displayed in Table XXIII. Using the \s-2s-2p

close coupling approximation with exchange neglected,

they also calculated the partial cross sections for L>3

so obtaining the total cross sections for the Is—»2s

and Is

>2p excitations given in Table XXIV. We see that the L= 0, 1, 2 partial cross sections as well as the

total cross sections for the Is—>2s and Is—>2p excitations

remain finite in the vicinity of the threshold energy.

This behavior can be understood by virtue of the

following considerations. The effect of the close coupling

between the degenerate 2s and 2p states is to introduce an additional potential —a/r2 acting upon the free electron, where a is a symmetrical matrix whose

diagonal elements are zero. The centrifugal term

^(AfT) A

2 occurring in the radial equation (109) then

Fig. 19. Total cross sections for the Is—>2p excitation of atomic

hydrogen. Curve 1: first Born approximation (Omidvar, 1964).

Curve 2: \s-2s-2p close coupling approximation neglecting ex- change (Omidvar, 1964). Curve 3: \s-2p close coupling approximation including exchange (Omidvar, 1964). Curve 4: \s-2s-2p

close coupling approximation including exchange (Burke, Schey,

and Smith, 1963).

Fig. 20. Total cross sections for the Is—>Zp excitation of atomic

hydrogen. Curve 1: first Born approximation (Vainshtein, 1965).

Curve 2: ls-3p close coupling approximation including exchange

(Burke, Schey, and Smith, 1963).

becomes replaced by X(X+1) A

2

, where a = X(X+l) is a

characteristic root obtained by diagonalizing the matrix

1) —a so that

X = — i+d+a) 2

. The threshold behavior of | T(v, v') | 2

, where T is the

transmission matrix, is given by | &2X+1/2

1 2 so that provided X-f-^ is real the partial cross section for the

v'—*v excitation vanishes in the limit as in the

usual way. However if there are negative values of a< — \ then = where /x is real and so

I £2*+i/2 |2 = | £2.V |2 = ! which means that the excitation cross section remains

finite in the neighborhood of the threshold.

An extension of the above argument shows also that

the cross sections oscillate in the vicinity of the threshold, this being confirmed by the \s-2s-2p close coupling calculations performed by Damburg and Gailitis

(1963) which reveal that the Is—»2/> excitation cross section exhibits a minimum just above the threshold

energy.

Very lengthy computations on the behavior of the

Is—»2s and \s—>2p excitation cross sections in the

vicinity of the n = 2 threshold employing the 6 state ls-2s-2p-3s-3p-3d close coupling approximation have

been carried out recently by Burke, Ormonde, and

Whitaker (1966, 1967). Their cross sections are compared with those obtained using the 3-state ls-2s-2p

close coupling approximation in Figs. 21 and 22. The

effect of adding a number of correlation terms to the 3- state close coupling expansion has been investigated

by Burke and Taylor (1966) and more extensively by

Taylor and Burke (1967). The results obtained with this

correlation approximation employing 20 correlation

terms, which allow explicitly for the interelectron separation /12 in a similar fashion to that done in bound state problems, are also displayed in Figs. 21 and 22.

Just above the n = 2 threshold at &i

2 = 0.75 the 3-state

approximation, with and without allowance for correlation, is dominated by an S—0, L— 1 resonance which

presumably would also be found using the 6-state ap20
proximation if extended to sufficiently low energies.

Above &i

2 = 0.85 the excitation cross sections given by

the 6-state approximation are dominated by resonances

arising from the n = 3 level. An interesting feature of the three calculations whose

results are presented in Figs. 21 and 22 is their satis- factory accordance over the energy range from just

above the n = 2 threshold to &i

2 = 0.85. This has led

Burke and his collaborators to make the hypothesis

that the close coupling approximation is valid for all energies which are insufficient to excite resonances and

channels associated with the lowest level excluded from

the expansion. In particular they suggest that the 6- state approximation should be quite accurate for all energies below that required to excite the resonances

arising from the n — 4 level, that is below about 12.5 eV.

3. Helium

Having examined the various approximations which

have been used to investigate the excitation of atomic

hydrogen, we now direct our attention to their application to the study of atomic systems with more than one

electron. The simplest of these is helium which is a

particularly interesting case to treat in detail since it has been the subject of the most thorough experimental

investigation.

state, the first Born approximation to the scattering

amplitude is given by

O o - fn\K) = J

exp (iKz)fn *(x)Mi)dT, (124)

where the z axis has been chosen along the momentum

change vector hK. If we now make use of the Schrodinger equations satisfied by the exact wave functions

ipi and 0„ of the initial and final states of the atom, we

find that this expression can be replaced by the alternative formula:

fn

Il (K) = —i K{E1 -En ) J

exp ( iKz)

X 0»*- h-0i-0»* Ur. (125)

dz dz

Calculations have been performed by Altshuler (1952;

1953) on the VS->2l P, 1

l S-*3l P, and VS-+2'

S

excitations of helium using both forms for the scattering

amplitude, suitably modified to hold for the case of two

electron atomic systems. To represent the ground state

of helium he took the simple Hylleraas function (122)

while for the n

x P states of helium he chose

0*i»( ri, r2 ) — 2

_1/2 {0i(2 I r1)0Bht (l | r2 )

3.1. First Born Approximation

As for the case of atomic hydrogen we begin by

looking at the Born approximation. The earliest calculations using this approximation were performed by

Massey and Mohr (1931; 1933b) who applied it to the

excitation of a number of singlet states of helium. In

the treatment of helium excitation we need to use approximate wave functions to describe the atomic elec- trons. For the ground state of the helium atom Massey

and Mohr chose the very simple Hylleraas (1929)

function

0i(fi, r2) =</>i (Z | ri)<f>i(Z | r2) , ( 122)

where

0i (X ] r) — (X3/7r) 1/2 exp ( —Xr) (123)

the parameter X being determined by the variational

method to be Z= 1.6875. (We express radial distances

in units of the Bohr radius a0 .) To represent the excited

states of helium they used symmetrized functions composed of products of hydrogenic orbitals.

Because of the approximate nature of the helium

atom wave functions there is an additional uncertainty

in these calculations beyond that arising from the application of the Born approximation. Bates, Fundaminsky,

Leech, and Massey (1950) have pointed out that information regarding the error introduced by the use of approximate wave functions can be obtained by employing

an alternative expression for the scattering amplitude

which weights the various regions of coordinate space

differently. We have already shown that for the excitation of the nth. state of a hydrogen atom from the ground

+01 (2 I rp) 1 I Ti)}, (126)

where 0„;m (X | r) is the hydrogenic wave function for

the state with quantum numbers n, l, m corresponding

to an effective charge X. For the 2

1 .5’ state of helium the

function (126) is inadequate because it is not orthogonal

to the ground-state wave function p\. However this

orthogonalization was achieved by taking

02oo+i, rt) =2-1/2 {01 (a | 1-00200 (/3 | r2) +0i (« | r2 )020o(/3 | fi) } +‘401+1, * 2 ) , ( 127)

where A was chosen so as to orthogonalize 02o 0 and 0i

while the screening parameters were determined by the

variational method to be a=1.98, |6=1.20. The total

excitation cross sections obtained by Altshuler are displayed in Figs. 23, 24, and 25. We see that although

the absolute magnitudes of the total excitation cross sections calculated with the aid of formulas I and II

differ somewhat, their dependence upon energy are rather similar.

The helium wave functions (122) and (126) give

rise to the very poor value 0.187 for the FS-h>2 X P

oscillator strength compared to the value 0.266 calculated by Wheeler (1933) using an accurate wave function for the ground state of helium obtained by Hylleraas (1930) as well as an improved wave function for

the 2

l P state. Lassettre and Jones (1964) have calculated the differential cross section for the FS—>2l P

excitation of helium employing the same atomic wave

functions as Wheeler. An interesting feature of their

calculations, first pointed out by Miller and Platzmann

21
(1957), is that although there is very poor accordance

between the differential cross section calculated by

Lassettre and Jones using the accurate wave functions

of Wheeler and that calculated using the crude hydrogen-like functions (122) and (126) together with for- mula I for the scattering amplitude, there is satisfactory

agreement between the refined calculations of Lassettre

and Jones and the differential cross section calculated

by Altshuler (1952) using formula II and the hydrogen-like functions. The calculations of Lassettre and

Jones were extended to larger values of K2 by Silverman and Lassettre (1964) who compared them with

their experimental data for 500-eV electrons. It can be

seen from Fig. 26 that the agreement between the shape

of the theoretical curve and the experimental data is remarkably good.

Fox (1965) has also used the Born approximation to calculate cross sections for the FS—*nl S excitations of

helium corresponding to n=2, •••, 10. For the groundstate wave function he chose the Hartree self-consistent

field function given by

M'h ri) =0i(ri)<£i(r2 ), (128)

fitting the one-electron orbital fa to a sum of two ex- ponential terms, while for the excited n

l S state of

helium Fox took wave functions having the form

\pn-\-A\pi, where A is a constant chosen so as to orthogonalize against fa, and

fa(rh r2) = 2“1/2

{ ( 2 | ri)<£B (r2)

E (ryd) Fig. 22. Total cross sections for the ls-+2p excitation of atomic

hydrogen (Burke, Ormonde, Taylor, and Whitaker 1967).

Curve 1: \s-2s-2p close coupling approximation. Curve 2: cor- relation approximation. Curve 3: ls-2s-2p-3s-3p-3d close coupling approximation.

Fig. 23. Total cross sections for the PS—»2lP excitation of helium calculated using formulas I and II for the first Bom approx- imation to the scattering amplitude (Altshuler, 1952)

.

+</>i(2 | r2 )fa(ri) }, (129)

where fa is a truncated Whittaker function. In Fig. 27 we compare the differential cross section for the

F.S—>2X S excitation calculated by Fox (1965) using the

Born approximation with the experimental data of

Lassettre, Krasnow, and Silverman (1964) for 500-eV

electrons. The accordance is very satisfactory. The total cross sections for the excitation of the n'S states of

Fig. 21. Total cross sections for the Is—»2s excitation of atomic

hydrogen (Burke, Ormonde, Taylor, and Whitaker, 1967).

Curve 1: \s-2s-2p close coupling approximation. Curve 2: correlation approximation. Curve 3: \s-2s-2p-3s-3p-3d close coupling approximation.

Fig. 24. Total cross sections for the FS—»3*P excitation of helium calculated using formulas I and II for the first Bom approximation to the scattering amplitude (Altshuler, 1952)

.

Fig. 25. Total cross sections for the FS—*2l S excitation of helium calculated using formulas I and II for the first Born approximation to the scattering amplitude (Altshuler, 1952)

.

22
Fig. 26. Differential cross section for the 1

X S—>2; 1 P excitation of helium at 500 eV. Solid curve: first Bom approximation (Lassettre

and Jones, 1964; Silverman and Lassettre, 1964). symbols: ex- perimental data for approximately 500 eV (Silverman and

Lassettre, 1964).

helium for n= 2, •••, 10 calculated by Fox (1965) are displayed in Fig. 28. Fox (1966; 1967) has also investigated the FS—>3l D

excitation of helium using the Born approximation,

taking the Eckhart (1930) function for the ground

state of helium given by

M'h r2) =exp { — (aTi+/3r2)

}

+exp { - (art+prx) } ( 130)

with a = 2.14, 0=1.19 and the function (126) for the

3

l D state of helium. He obtained the total excitation

cross section displayed in Fig. 29 employing the standard form I of the Born approximation and the alternative form II. In the same way as for the excitations of the 2

1 .S, 2

1 P, and 3l P states of helium treated by

Altshuler (1952; 1953), the alternative form of the

Born approximation gives rise to the larger cross section. By calculating the scattering in the forward direction, Fox (1966) shows further that the Eckhart

ground-state helium wave function produces a cross

section which is larger than that resulting from the use

of a Hartree self-consistant field function for the ground

state. The Born approximation has also been used by

Moiseiwitsch (1957) to calculate cross sections for the

2

3 S—>23P, 3

3 P, 3

3D, and 4

3D excitations of helium, the wave functions being chosen to have the general form

'I'nimi

T

h r2) = 2

_1/2 {0x(2 | ri)0n *m (r2) -0i(2|r2)0nim ( ri)}. (131)

For the 2

3 S—>23P, 3

3D, and 4

3 D excitations, the helium

2s orbital was chosen to be

<hs(r)=N{l-±(a+l3)r}e-i,r (132)

with a =2.00, 0 = 0.695 while 02pm, 0Mm, and 04(jm were assumed to be hydrogenic. The 2

3 S-+33P excitation

cross section was found to be very sensitive to the

parameters of both the initial and final orbitals. Consequently, for this particular case, use was made of the more accurate atomic orbitals derived by Morse, Young,

and Haurwitz (1935) and by Goldberg and Clogston

(1939);

02.(r) =N(re~flT

03pm (r) =M(re-Xr -£er6r)r •

-Ae~ar

) (133)

cos 6 (134)

(1/V2) sin The results of these Born approximation calculations

are displayed in Table XXV.

In the case of the 2

3 S-^23 P excitation, the coupling

between the initial and final states is rather strong. For

this reason the Seaton modification of the Bethe approximation (1961) was also used by Moiseiwitsch

(1957) to calculate the total cross section for the

2

3 S—>23 P transition, the results being compared with

those given by the Born approximation in Table

XXV. We see that the maximum value of the cross section is reduced from the Born approximation value of nearly 3007rOo2 to about 1157ra0

2

, the position of the

maximum being displaced to higher electron impact

energies and the shape of the peak being altered substantially.

3.2. Exchange Approximations

The approximations which we are dealing with under

the present heading are the Born-Oppenheimer approximation, the first-order exchange approximation, and

the Ochkur approximation.

One of the earliest investigations using the BornOppenheimer approximation was concerned with the

l

1 i'—=>23 S, 2

3 P, 33 P, 33D, and 43 P excitations and was

carried out by Massey and Mohr (1931; 1933b).

These transitions all involve a change in the total

spin of the helium atom and can only occur as the

result of electron exchange. The exchange scattering

amplitude was obtained by expanding in a series of

spherical harmonics. Their calculations were for 100-,

200-, and 400-eV incident electrons where the cross sections are all very small. Further calculations were performed by Bates et al. ( 1950) who, in particular, in- vestigated the difference in the cross sections arising

from using the post and prior interaction forms of the

Born-Oppenheimer approximation, often referred to as the post-prior discrepancy. The prior and post interactions are those between the incident electron and the

atom, and between the scattered electron and the atom,

respectively. If an exchange of electrons has taken

place, these two interactions will be different. However,

the scattering amplitudes given by the Born-Oppenheimer approximation associated with the post and

prior forms of interaction are still the same provided

exact atomic wave functions are employed which means that for the case of scattering by atomic hydrogen

there is no post-prior discrepancy. On the other hand,

since we have to use approximate atomic wave functions for scattering by helium, the two forms of BornOppenheimer approximation will lead to different re- sults in this case. This is illustrated in Fig. 30 where the

23
Fig. 27. Differential cross section for the 1

1 5-^21 5 excitation of helium at 500 eV. Solid curve: first Born approximation (Fox,

1965). symbols: experimental data for approximately 500 eV

(Lassettre, Krasnow, and Silverman, 1964)

.

Fig. 30. Total cross sections for the excitation of the 2

l S and 2l P

states of helium (a—

1 S, b

2

1 P) (Bates, Fundaminsky, Leech,

and Massey, 1950). Curve 1: Born-Oppenheimer approximation

using post interaction. Curve 2: Bom-Oppenheimer approximation using prior interaction. Curve 3: first Born approximation.

(b)

Fig. 28. Total cross sections for the FS—

>n}S excitations of helium for n = 2, 3, • •

•

, 10 calculated using the first Born approximation (Fox, 1965).

Fig. 29. Total cross sections for the FS—>31D excitation of helium calculated using formulas I and II for the first Born approx- imation to the scattering amplitude (Fox, 1966).

total cross sections for the TS—>21 5 and VS—->2l P

excitations of helium calculated using the post and

prior forms of the Born-Oppenheimer approximation

are displayed. We see that the discrepancy is very

great for the S^>S transition near the threshold but

relatively small for the S—>P transition. This is in

accordance with the observation made previously that

the Born-Oppenheimer approximation is very poor

for S—>S transitions, typically giving rise to greatly

overestimated cross sections.

More recent calculations using the Born-Oppenheimer approximation have been carried out by Massey

and Moiseiwitsch (1954; 1960) and by Bell, Eissa, and

Moiseiwitsch (1966) for the excitation of the 2

3 S and

2

3P states of helium. The calculations performed by

Massey and Moiseiwitsch used the simple Hylleraas

ground-state helium wave function (122) while those

carried out by Bell, Eissa, and Moiseiwitsch used the

variational function obtained by Green, Mulder, Lewis,

and Woll (1954) which has the form (128) with

<t>i(r)=N(e-Zr+ce~2Zr

) (135)

and Z= 1.4558, c—0.6. This is a rather good analytical

fit to the Hartree self-consistent field function for the

ground state of helium. Both groups of investigators

employed the wave functions for the 2

3 S and 23 P

states obtained by Morse, Young, and Haurwitz (1935)

possessing the form (131) with fa being given by

formula (133) and fapm being a hydrogen-like orbital

with a variationally determined screening parameter.

2U
Fig. 31. Total cross sections for the excitation of the 2

3 5 state

of helium. Curve 1: Born-Oppenheimer approximation (Bell, Eissa, and Moiseiwitsch, 1966). Curve 2: first-order exchange ap- proximation (Bell, Eissa, and Moiseiwitsch, 1966). Curve 3: Ochkur approximation (Ochkur and Bratsev, 1965b). Curve 4: 3-state close coupling approximation (Marriott, 1964)

Even though different ground-state wave functions were

employed, the Born-Oppenheimer approximation cross- sections calculated by the two groups of investigators were in good agreement. The cross section curves for the

VS—^23 S and ES—>23 P excitations calculated using

the Born-Oppenheimer approximation are displayed

in Figs. 31 and 32 where they are compared with the

Ochkur approximation calculations and with more

sophisticated calculations which are discussed in subsequent sections.

The values of the Born-Oppenheimer approximation

cross sections calculated by Bell, Eissa, and Moiseiwitsch are given in Tables XXVI and XXVII where a comparison is made with the first-order exchange approximation cross sections evaluated by the same authors. We see at once that for electron energies < 150 eV

the use of the first-order exchange approximation re- duces the VS-+23 S excitation cross section given by

the Born-Oppenheimer approximation by about a

factor of 10 although at higher energies the two approximations tend to each other. Inspection of Table

XXVII reveals that for the l

1 6”—>23 P (M = 0) excitation of helium, the cross section given by the BornOppenheimer approximation is somewhat larger than

that given by the first-order exchange approximation

for electron energies <50 eV but that above this energy

the two approximations yield similar results. For the

excitation of the 2

3 P (M = ± 1) states of helium the

two approximations are identical.

Calculations on the excitation of the 2

3 S state of

helium using a variety of forms of interaction between

the free electron and the atom have been carried out by

Joachain and Mittleman (1965) employing the BornOppenheimer approximation and modifications thereof.

They used the simple Hylleraas ground-state wave

function (122) together with the 2

3 S state wave function given by (131) and hydrogen-like orbitals having

variationally determined parameters. The cross sec- tions obtained by them using the post and prior forms

of the Born-Oppenheimer approximation are in reason- ably satisfactory accordance with each other and with

the cross section calculated by Massey and Moiseiwitsch

using the Born-Oppenheimer approximation. On the

other hand the cross sections calculated by Joachain

and Mittleman using post and prior forms of the first- order exchange approximation are in total disagreement

with each other. Moreover their prior interaction cross section is more than a factor of 2 greater than the first- order exchange approximation cross section calculated by Bell, Eissa, and Moiseiwitsch (1966) who also

employed the prior interaction but with a rather better

ground-state wave function.

We now come to the interesting calculations carried

out by Ochkur (1963) and by Ochkur and Bratsev

( 1965; 1966) based on the so called Ochkur approximation already discussed in Sec. 2.7. An important fea- ture of this approximation is that it cannot give rise to a post-prior discrepancy since it satisfies the principle

of detailed balancing. The cross sections calculated by

Ochkur (1963) for the ES—>23 S and l

l S—^23 P excitations, using atomic wave functions having the forms

(128) and (129) with the one-electron orbitals being

the simple analytical functions given by Veselov,

Antonova, Bratsev, and Kirillova (1961), are displayed

in Figs. 31 and 32. We see that the Ochkur approximation results in total excitation cross sections which are much smaller than the Born-Oppenheimer approximation cross sections and indeed considerably smaller

than the first-order exchange approximation cross sec- tions calculated by Bell, Eissa, and Moiseiwitsch.

Ochkur and Bratsev (1965) have extended the

Ochkur approximation calculations to the excitation of

many other states of helium, both triplet and singlet,

using Hartree-Fock atomic wave functions. Their re- sults are collected together in Tables XXVIII and

XXIX. It can be verified from Table XXVIII that

the cross sections for the excitation of the triplet states

all decay with electron impact energy E according to the law E~3 which is much faster than the laws for the

excitation of singlet states: E~l for optically forbidden

Fig. 32. Total cross sections for the excitation of the 2

3 P states

of helium. Curve 1: Born-Oppenheimer approximation (Bell, Eissa, and Moiseiwitsch, 1966). Curve 2: first-order exchange ap- proximation (Bell, Eissa, and Moiseiwitsch, 1966). Curve 3: Ochkur approximation (Ochkur and Bratsev, 1965b). Curve 4: distorted waves approximation including exchange (Massey and

Moiseiwitsch, 1960). Curve 5: distorted waves approximation

including exchange (Lashmore-Davies, 1965).

25
transitions and E~l In E for optically allowed transitions. Also the cross sections for the excitation of the

states of a given series are proportional to n~3

, where n

is the principal quantum number. Although the validity

of the n~3 law is justifiable only for large values of n it is found that it holds quite well even for low values of n. Because the cross sections calculated by Ochkur and

Bratsev for the excitation of the singlet states of helium

include allowance for exchange they should be slightly

smaller than the corresponding cross sections resulting

from the use of the first Born approximation.

Calculations of the excitation cross sections for the

2

3 S—>n3 L and 2

3 S^nx L transitions of helium have also

been performed by Ochkur and Bratsev (1966) using

the Ochkur approximation together with Hartree-Fock

atomic wave functions. The values of their cross sections are displayed in Tables XXX and XXXI. Satisfactory

accordance with the earlier calculation of Moiseiwitsch

(1957) on the excitation of the 3

3 P, 33D, and 4

3 D states

of helium using the Born approximation was found.

3.3. Distorted Waves Approximation

The earliest calculations using the distorted waves

approximation were performed by Massey and Mohr

(1933a) who investigated the US—>2 l P, US—>23 P excitations of helium.

More recently Massey and Moiseiwitsch ( 1954; 1960)

have calculated cross sections for the excitation of the

2

1 S, 2

3 S, and 2

3 P states of helium using the simple

Hylleraas ground-state wave function (122) while

Lashmore-Davies ( 1965) has recalculated the total cross section for the US—>23 P excitation using the

ground state wave function obtained by Green et al. (1954). For the excited states of helium both Massey

and Moiseiwitsch (1954; 1960) and Lashmore-Davies

(1965) used the helium wave functions obtained by

Morse, Young, and Haurwitz (1935). Unfortunately

their 2

1 S state wave function is not completely orthogonal ^o the ground-state wave function (122) and so

the accuracy of the cross section for the excitation of

the 2

: S state is correspondingly reduced.

The most interesting feature of the distorted waves

calculation for the excitation of the 2

3 S state of helium

is the appearance of the very sharp peak in the zero- order partial cross section just above the threshold

energy. This peak seems to arise as the consequence of the existence of a doubly excited state (D252

) 2 S of He~

with energy slightly less than that of the (ls2s) 3 S

metastable state of helium, producing a resonance in

the 5 wave scattering of electrons by the 2

3 S state of

helium.

Directing our attention now to the excitation of the

2

3 P states of helium, we see from Fig. 32 that the dis- torted waves calculations of Massey and Moiseiwitsch

(1960) and Lashmore-Davies (1965) are in good agreement with each other although the latter investigator

used a rather better ground-state wave function for the helium atom. We also note that allowance for distortion reduces the cross section somewhat below that

given by the Born-Oppenheimer approximation (but

by not nearly so much as that found in the case of the

US—»23 S excitation) to give cross sections which are not greatly dissimilar from that obtained with the

first-order exchange approximation. However we see from Fig. 32 that the cross section curves obtained

with the distorted waves approximation both lie well

above that given by the Ochkur approximation.

3.4. Close Coupling Approximation

Marriott (1964) has carried out an investigation of the scattering of electrons by helium atoms in which

he makes full allowance for the coupling between the

US, 2'S, and 2

3 S states. For the US ground state he

employed the simple Hylleraas wave function given by

(122), for the 2

3 S state he used the wave function

determined by Morse, Young, and Haurwitz (1935)

while for the 2'S state he used the wave function derived by Marriott and Seaton (1957) which is orthogonal to the Hylleraas ground-state function (122). The

US and 2

3 S wave functions are the same as those employed by Massey and Moiseiwitsch (1960) but the

2

X S state wave function should be superior since the

2

l S function used by Massey and Moiseiwitsch was not

completely orthogonal to the Hylleraas ground-state

wave function. The partial and total cross sections for the excitation of the 2

3 S and 2

! S states of helium obtained by Marriott using the US-2! S-23 S close coupling approximation with exchange included are given

in Tables XXXII and XXXIII, respectively. The total cross section for the US—>23 S excitation of helium obtained with the 3-state close-coupling approximation

is also displayed in Fig. 31 where it is compared with the

results obtained using less sophisticated methods.

We note that coupling to the 2

X P and 2

3 P states of helium has been entirely neglected in the above calculations. Because of the strong likelihood that these couplings could have an important effect upon the excitation cross sections, Burke, Cooper, and Ormonde

(1966) have carried out a calculation allowing for

close coupling to all the five states: VS, 2*S, 2

3 S, 2'P,

23 P. They employed Hartree-Fock atomic wave functions. Their cross sections for the excitation of the

2

1 S, 2

3 S, 2l P, and 2

3 P states of helium from the ground

state are displayed in Fig. 33.

Marriott (1957; 1966) has also investigated the conversion of the 2*S state of helium to the 2

3 S state by

electron impact:

He (2l S) +e—>He (23 S) +e+0.78 eV. ( 136)

The cross section for this superelastic collision has been

determined experimentally by Phelps (1955) who finds

3X1(7~14 cm2 (3407rao2

) for thermal electrons. However

using the 2

l S-23 S close coupling approximation to- gether with the atomic wave functions for the 2

X S and

2

3 S states of helium derived by Morse, Young, and

26
Fig. 33. Total cross sections for the excitation of helium calculated using the five-state close coupling approximation (Burke,

Cooper, and Ormonde, 1966). Curve 1: H-S—>23 5, Curve 2:

l

l S-*VS, Curve 3: VS-+VP, Curve 4: 1

1 5->21 P.

Haurwitz (1935), the cross section for the superelastic

collision (136) has been calculated by Marriott (1957)

to be 607tOo2 for 0.026-eV electrons which is much

smaller than the observed cross section. In a more

recent investigation using the l

1 S-21 S-23 S-33 S close

coupling approximation together with the Marriott

and Seaton (1957) wave function for the 2'S state of

helium and a reasonably accurate function for the 3

3 S

state, he calculated the cross section for the superelastic

collision ( 136) to be 1507tOo2 at 0.026 eV which is only

about a half of that found experimentally by Phelps.

This increase in the calculated total cross section at thermal energies is entirely owing to the use of the improved 2*S wave function, the effect of including the

helium ground state function being small. In Table

XXXIV we display the partial and total cross sections

for the superelastic conversion of the 2

1 5 state to the

2

3 S state of helium calculated using the l

1 .S-21 S-23 .5- 3

3 S close-coupling approximation. Marriott (1966)

has also calculated the cross section for the 2

3 S—»33 S

excitation by electron impact for the quartet total spin

state using the close coupling approximation.

An investigation of the excitations of the 2

l P, 31 P} and 43P states of helium from the ground state has been

carried out by Vainshtein and Dolgov (1959) using the

\

l S-nl P close-coupling approximation with exchange

neglected. The importance of the higher-order partial

waves, even for energies quite close to the threshold,

is easily verified by inspecting Table XXXV where the

partial and total cross sections for the 1

1 S-^31 P excitation calculated by Vainshtein and Dolgov (1959) are

given.

3.5. Double Excitation of Helium

The Born approximation has been used by Massey

and Mohr (1935) to calculate cross sections for the

excitation of the (2s2)

1 S, (2s2p) l P, (2s3p) 1 P, (2s4p) 1 P,

and (3s2p) l P doubly excited states of helium. For

the ground state of helium they took the Eckhart function (130) while for the doubly excited states they

chose the wave function

^»m,(ri, r2) =N{4>ni (<x

| ri)0B1 (jS| r2 )

+<£ni(a I r2 )$„ 2 (/3 1 ri) }

,

(137)

where <£n (X | r) is a hydrogenic function corresponding

to an effective charge X. For the (2s2

)

1

.S' and (2s2p) 1 P

states they took a = 2.00, /3=1.58; for the (2s3p) l P

and ( 2sApyP states they took a = 2.00, /3=1.00; while

for the ( 3s2p) l P state they chose a = 1.20, 6 = 2.00. The

excitation cross sections calculated by Massey and Mohr

(1935) are displayed in Table XXXVI. The smallest

cross section occurs for the excitation of the (2s2

)

1 5

state corresponding to an optically forbidden transition,

while the largest cross sections are found for the excitation of the ( 2s2p) l P and ( 3s2p) l P states corresponding

to optically allowed transitions.

A much more detailed investigation of the excitation

of doubly excited states of helium has been carried out

by Becker and Dahler ( 1964) . They began their study

by using the Born-Oppenheimer approximation to calculate the cross sections for the excitation of the

(2p

2

)

3 P, (2p3p) l P, (2p3d) 1 D, and (2p3d) z D states of helium, none of which are subject to autoionization.

For the ground state they used the simple Hylleraas

function (122) and the Eckhart function (130) while

for the doubly excited states they employed symmetrized wave functions composed of products of hydrogenic orbitals. Their cross section curves are given in

Fig, 34. By far the largest cross section arises from the

excitation of the (2p

2

)

3 P state, the lowest of the doubly

excited states which is stable towards autoionization.

Becker and Dahler also used the distorted waves method and the two state close coupling approximation,

Fig. 34. Cross sections for the excitation of doubly excited states

of helium calculated using the Born-Oppenheimer approximation

(Becker and Dahler, 1964). Curve 1: (2p

2

) 3P excitation using

Hylleraas ground-state function. Curve 2: (2p2

)

3 P excitation using Eckhart ground-state function. Curve 3: (2p'6p)l P excitation. Curve 4: {2p3d) l D excitation. Curve 5: (2p3d) 3D excitation.

27
both with exchange included, to investigate in further

detail the excitation of the (2p

2

)

3 P state of helium,

the ground state being represented by the simple Hylleraas wave function (122) . The resulting cross sections are displayed in Fig. 35. We see that the distorted waves method and the two state close coupling approximation give rise to practically indistinguishable cross section curves. The implication is that the coupling

between the ground state and the doubly excited state

is fairly weak, a conclusion which is supported by the

smallness (^lO-20 cm2

) of the excitation cross section.

4. Neon and Argon

Extensive calculations using the Born approximation

have been carried out by Veldre, Lyash, and Rabik

(1965a; 1965b) on the excitation of neon and argon by

electron impact.

We first consider the case of neon excitation. Because

of the lack of Hartree-Fock wave functions, hydrogenic

orbitals were used for the different electron shells in

the neon atom. Total excitatiqn cross sections were cal- culated for transitions between all the terms of the configurations (y)2p, (t)35, ( 7)3/», ( 7)%, ( 7) 3d, ( 7) 4p, and ( 7) 5$, where 7 signifies the core configuration

ls22s22p5

. Because the formulation was based upon the

assumption that LS coupling prevails for all configurations of the neon atom, and this is known to be invalid

for excited inert gas atoms, the total excitation cross sections calculated by Veldre, Lyash, and Rabik

(1965b) could be seriously in error. LS coupling holds for the ground ^0 state of an

inert gas atom; however for moderately excited states

of inert gas atoms jl coupling holds while for highly

excited statejj coupling occurs. Realizing this, Veldre,

Lyash, and Rabik (1965a) investigated the conse- quences of choosing different types of coupling ( LS,jl, jj) between the optical electron and the core of the

excited atom. The wave function of the optical electron was calculated by solving the radial equation for a

particle in the field produced by Gaspar’s potential

Fig. 35. Cross sections for the excitation of the doubly excited

(2i?yP state of helium (Becker and Dahler, 1964). Curve 1: Bom-Oppenheimer approximation. Curve 2: distorted waves ap- proximation. Curve 3: 2 state close coupling approximation (Curve

3 is identical to Curve 2)

.

Fig. 36. Total cross sections for the excitations of neon and

argon from their ground states calculated using the first Bom

approximation. is electron energy in threshold units defined by

Eq. (38) .J Curve 1: ls22s22ps3pl So excitation of neon using LS

coupling for the ground and excited states. When jl coupling is used for the excited state, cross section is reduced by factor of 3. Curve 2: \s2 2s2 2ph3plD2 excitation of neon using LS coupling for the ground and excited states. When jl coupling is used for the

excited state, cross section is reduced by factor of 15. Curve 3:

ls22s22p63s23pb4p1 So excitation of argon using LS coupling

for the ground and excited states. When jl coupling is used

for the excited state, cross section is reduced by factor of 3. Curve 4:

ls2 2s2 2p*3s23p64p1 D2 excitation of argon using LS coupling for the ground and excited states. When jl coupling is used for the

excited state, cross section is reduced by factor of 15.

(1952), which has an analytical form and yields good wave functions for atoms with fairly large nuclear

charge such as neon and argon. They derived wave

functions for the neon configurations ( 7) 2p, ( 7)3s}

( 7)3p, ( 7) 45, ( 7) 3^, ( 7) 4p, and ( 7) 5$, where 7 denotes

the core configuration ls2 2s22p5

, and for the argon configurations (y')3p, (7') 45, (7') 4P, (7')3d, (y')5s,

(y')5p, and ( 7') 6s, where 7

' signifies the core configuration \s22s2 2pe3s23p5

. For the excitation from the ground state to an excited state, LS coupling was used to characterize the

ground state while LS,jl, and jj couplings were chosen

for the excited state of the inert gas atom. Their re- sults show that the largest cross section is obtained

when LS coupling is chosen for the excited state while

the least cross section occurs with jl coupling. This can

be seen from Fig. 36 where the total cross sections for

the excitation of the states of the configuration ( 7)3p

from the ground 1 So state of neon and the excitation of

the states of the configuration ( 7') 4p from the ground

^0 state of argon, using LS and jl coupling for the

excited states, are compared.

In the case of transitions between excited states of the inert gas atom, the same type of coupling between

the optical electron and the core of the atom was used

for both states, either LS or jl coupling.

As a consequence of the fact that a change in the

type of coupling between the initial and final states of

the inert gas atom produces a considerable reduction in

the total excitation cross section, Veldre, Lyash, and

Rabik (1965a) found that the cross sections for the

excitation of ground-state inert gas atoms have much

28
smaller values than the cross sections for the excitation

of excited atoms.

Calculations of the excitation cross sections for (y) 2p—+(y)3p transitions of neon have also been carried

out by Boikova and Fradkin (1965) using the Ochkur

approximation together with LS, jl, and jj coupling

representations of the (y)3p excited states of the neon

atom.

5. Mercury

The excitation of mercury to the 6

3 P0 , 6

3 P\, 63 P2 , and 6'Pi states from the ground 6

x So state has been

investigated by Penney (1932a) using the Born-Oppenheimer approximation. He assumed that the wave function of the core remained unaltered during the transition and used the Coulomb approximation to determine

the radial wave functions of the two electrons outside

the core. The total wave functions of the 5" and P

states of mercury were then expressed according to the

formulae derived by Houston (1929). Penney found

that his wave functions yielded oscillator strengths for the 6' Pi—>6‘6’o and t)

3 P\—>6’ So transitions which were

about twice the experimentally determined values. He

therefore only gave relative cross sections for the excitation of the P states of mercury.

The calculations are not very accurate. Thus it should be noted that the thresholds for the excitations

of the 3 P0 , 3 Pi, and 3 P2 states are not in the order of

increasing energy as they should be. However, the cal- culations clearly indicate that the cross section for the

excitation of the 6

l Pi state is considerably larger than

those for the excitation of the 3 P states, particularly

for electron impact energies above about 15 eV, and

that it has a broad maximum located some distance

from the threshold energy whereas the cross sections

for the excitation of the 3 P states are more sharply

peaked and positioned close to the threshold.

The above calculations have been repeated by Yavorskii (1947) employing precisely the same method and wave functions. His excitation cross sections, which

were given in absolute units, are not in satisfactory

accordance with the relative cross sections calculated

by Penney. However, the threshold energies obtained

by Yavorskii are in the correct order. Regrettably the

values 3.71X10~16 cm2 and 1.05X10-16 cm2 for the

6

1 S0—*6l Pi and 6

1 So~^63 Pi excitation cross sections at

their maxima, quoted by Yavorskii in the text of his

paper, are somewhat greater than the values of the

maximum cross sections obtainable from his figure.

Recently more reliable calculations of the excitation

cross sections for the 6

3 P0 , 63 Pi, 6

3 P2 , and 6l Pi states

of mercury, employing the Ochkur approximation to- gether with Coulomb approximation functions yielding

oscillator strengths for the 6

1 Pr~>6' So and 6

3 Pi—>6* So

transitions in good agreement with the experimentally

determined values, have been performed by McConnell

and Moiseiwitsch (1968). Their cross sections are dis- played in Fig. 37.

6. Alkali Metal Atoms

We devote this section to the consideration of the

alkali metal atoms each of which is composed of a single

electron outside a core of completely filled electron

shells. In the ground state of an alkali metal atom the

outer electron is in an ns state with n— 2, 3, 4, 5, 6 for lithium, sodium, potassium, rubidium, and cesium,

respectively. We are mainly concerned with ns-+np

excitations of the outer electron of the alkali metal

atoms.

Because of the very strong coupling between the

initial and final states of the resonance transition ns—Hip, the Born approximation produces excitation

cross sections which are too large close to the threshold

energy. Born approximation calculations have been

carried out by Bates et al. (1950) and by Vainshtein

(1965) on the 3s—»3/> transition of sodium, and bv

Vainshtein, Opykhtin, and Presnyakov (1964b) on the resonance transitions of lithium, sodium, potassium,

rubidium, and cesium whose excitation cross sections are presented in Table XXXVII. We note the very

large values of these Born approximation cross sections

at their maxima. Vainshtein has also applied the dis- torted waves approximation with exchange neglected

to the 3s—+3p transition of sodium finding that allow- ance for distortion introduces an oscillation into the

excitation cross section with several maxima. Some of the partial cross sections obtained by Vainshtein exceed

the theoretical upper bound imposed by the unitary

property of the S matrix, clearly demonstrating the

strength of the coupling.

To overcome the difficulties arising from the very

strong coupling between the 3s and 3p states of sodium,

Seaton (1955a) has used his modification of the Bethe

approximation, previously discussed in section 2.9, to calculate the total excitation cross section for the

3s—>3

p

resonance transition of sodium. The cross sec- tion obtained by Seaton is displayed in Fig. 38. We see from this figure that the modified Bethe approximation

produces a considerable reduction of the excitation cross section near the threshold energy.

Calculations on the 3s—*3p resonance transition of

sodium using the semi-classical impact parameter

method have also been carried out by Seaton (1962).

He obtains an excitation cross section which is much

below that given by the Born approximation at low

impact energies, in accordance with the modified Bethe

approximation.

Salmona and Seaton (1961) have used the unitarized

Born approximation, discussed earlier in Sec. 2.14, to calculate the total excitation cross section for the 3s—>3

p

transition of sodium. In this method the Born approximation to the reactance matrix R is substituted into

the formula for the transmission matrix, thus ensuring

that the unitary property of the scattering matrix is satisfied. A further approximation made by Salmona

and Seaton was the neglect of all short range interac29
tions, just the long range interaction S/r2 being re- tained. The line strength 5 has been determined by /-

value measurements to be 19 atomic units which again

emphasizes the great strength of the coupling between

the 35 and 3p states of sodium. The neglect of the short

range interactions is valid for large values of the angular

momenta of the incident electron while the use of an

approximation which satisfies the conservation theorem

ensures that the small angular momenta are treated

without undue error. Inspection of Fig. 38 shows that

the method used by Salmona and Seaton (1961) provides a cross section curve which is in satisfactory ac- cordance with the modified Bethe approximation.

More sophisticated calculations which take account

of the very strong coupling between the 3s and 3p

states of sodium although neglecting exchange have

been carried out by Lane and Lin (1964) and by

Barnes, Lane, and Lin (1965). Designating the total

angular momentum quantum number by L and the

angular momentum quantum numbers of the atomic

and free electrons by l\ and k, respectively, the fol- lowing three states of the total system enter into the

excitation problem if we neglect coupling to all states

of the sodium atom other than the initial state 3s and

the final state 3p:

V k L

1 0 l l

2 1 l- 1 l

3 1 m L (138)

There is also a fourth state of the total system with

quantum numbers /x = 1, 1^=1, L = ( but this has a

different parity to the other three states and so need

not be considered here. For each value of l we have

therefore three coupled differential equations to solve.

Because the energies of the 3s and 3p states of sodium

are only 2.10 eV apart, Lane and Lin (1964) made the

reasonable approximation that the initial and final states are in exact resonance thus enabling them to decouple the differential equations. In addition, fol- lowing Salmona and Seaton (1961), they neglected

all the short-range interactions retaining only the longFig. 37. Total cross sections for the excitation of mercury cal- culated using the Ochkur approximation (McConnell and Moiseiwitsch, 1968). Curve 1: 6

1 S0—>63 P0 excitation. Curve 2: 6

1 S0—

6

3 Pi excitation. Curve 3: 6

l So—>63 Po excitation. Curve 4: 6

3 So—> 6

3 Pi excitation.

Fig. 38. Total cross sections for the 3s—>3p excitation of sodium.

Curve 1: first Born approximation (Karule and Peterkop, 1965).

Curve 2: modified Bethe approximation (Seaton, 1955). Curve 3: impact parameter method (Seaton, 1962). Curve 4: unitarized Born approximation (Salmona and Seaton, 1961). Curve 5: resonance distortion method (Lane and Lin, 1964). Curve 6: 3s-3p close coupling approximation neglecting exchange (Barnes,

Lane, and Lin, 1965).

range interaction S/r2

. Then employing the exact resonance solution corresponding to the incident channel y=l they performed a single iteration to obtain more accurate solutions of the other two differential

equations, the exact resonance approximation being

discarded. All the partial waves from 1=2 to 15 were

treated in this manner. However, because of the great

strength of the S/r2 interaction, they were unable to carry out the same procedure for the t=0 and 1 waves and so arbitrarily chose their partial cross sections to be

^7t(2/+1)Ai2 which is half the theoretical maximum

imposed by the conservation theorem. For the partial cross sections associated with i> 15 they used the

Bethe approximation. The resulting total cross section curve is displayed in Fig. 38. We see that it is in close

accordance with the cross section obtained by Seaton

using the modified Bethe approximation.

Barnes, Lane, and Lin (1965) have solved the three

coupled differential equations corresponding to the

states v = 1, 2, 3 of the total system by numerical integration. They included the short-range interactions in

this investigation using hydrogenic wave functions for

the optical electron with an effective charge of 2.92

and performed the calculation for all values of i up to

15. The partial cross sections obtained by them to- gether with their total cross sections, the contribution

from the partial waves with l> 15 being estimated with

the Bethe approximation, are displayed in Tables

XXXVIII and XXXIX. We also give their total cross section curve in Fig. 38. It is somewhat lower than the

cross section curves obtained with the less accurate

methods discussed above.

Close coupling calculations on the resonance transitions ns-^np of lithium, sodium, potassium and cesium

have been carried out by Karule and Peterkop (1964;

1965) for electron impact energies close to the threshold.

The first calculations performed by Karule and Peter- kop (1964) were concerned with the 2s—+2p excitation

of lithium and were carried out at energies of 2.0, 2.5

30
and 3.0 eV both including and neglecting exchange.

They used the Is and 2s analytical wave functions

derived by Clementi, Roothaan and Yoshimine (1962)

and a hydrogenic 2p wave function. Karule and Peter- kop (1965) subsequently repeated the lithium calculations using Hartree-Fock wave functions (without

allowance for the effect of the outer electron on the

core) (Ivanova and Ivanova, 1964) but found that

the changes in the partial cross sections were slight.

In addition they extended their investigation to sodium,

potassium, and cesium using semi-empirical atomic wave functions for the outer electron. Their calculations were carried out for the L=0, 1, •••, 8 partial waves neglecting exchange, for the L= 0, 1, 2, 3 partial waves including exchange for sodium and potassium,

and for the L = 0, 1, 2, 3, 4 partial waves including ex- change for cesium. The agreement with the calculations

of Barnes, Lane, and Lin (1965) for sodium with ex- change neglected is satisfactory at 4 eV, apart from the

L = 0 and 1 partial waves. The lack of accordance for these low order partial waves is presumably the conse- quence of the use of different atomic sodium wave

functions.

Karule and Peterkop (1965) also calculated the

partial sum

±<?

L=

by adding the partial cross sections for the low L values,

obtained including exchange, to the partial cross sec- tions for the higher L values, obtained neglecting

exchange. Since the partial cross sections corresponding

to L> 8 make a significant contribution to the total cross section, this means that their values of the partial

sum

iffL=

are considerably lower than the values of the total cross

section. This can be exemplified by considering the

6s—>6p transition of cesium for which Hansen (1964)

has carried out a modified Bethe approximation calculation. His total excitation cross sections are displayed

in Table XL. We see that at 5 eV his total 6s—>6p excitation cross section is 114 ttOq2 whereas the partial sum

iff

£=0

obtained by Karule and Peterkop is only 73 iraip. Finally we come to the cross sections for other excitations of the alkali metal atoms. Born approximation

calculations on the 3s—*3d and 3s—^Ap excitations of

sodium have been carried out by Bates et al. (1950)

using Hartree-Fock wave functions; on the 3.s—>4$,

3s—>4/>, 4s—>4p, 5s—>6p, and 5s-+lp excitations of

sodium by Vainshtein (1965) using semi-empirical wave functions for the optical electron; and on an immense number of transitions in lithium, sodium, potassium, rubidium, and cesium by Vainshtein, Opykhtin,

and Presnyakov (1964a) also using semi-empirical wave

functions for the outer electron. The Born approximation excitation cross sections obtained by Vainshtein

(1965) and by Vainshtein, Opykhtin, and Presnyakov

(1964a) are displayed in Tables XLI-XLV.

7. Oxygen and Nitrogen Atoms

In the present section we are mainly concerned with

the excitation of the 1 D and l S terms of the lowest configuration \s22s22p4 of atomic oxygen from the ground 3 P

term of this configuration.

The first calculations of the cross sections for these

excitations were carried out by Yamanouchi, Inui, and

Amemiya (1940) who employed a simplified version

of the distorted waves approximation. They found that

their 3 P-^D and 3 P-J>

1 S excitation cross sections were dominated entirely by the p wave contribution arising

from the incident electron. This being so Bates et al.

( 1950) were able to establish that at their maxima these

excitation cross sections were factors of 89 and 66 times

greater, respectively, than the upper bound imposed

by the conservation theorem.

As a consequence of the above situation, Seaton

(1953a) was led to make a detailed investigation of the

oxygen excitation problem for the incident p wave 2 P

scattering case. He showed that the assumption of weak

coupling, inherent in the distorted waves approximation, was quite invalid for the excitation problem under

consideration. Seaton therefore decided upon an entirely different approach based upon the fact that the

excitation energies of the l D and 1 S terms of the lowest

configuration of oxygen are rather small, as can be

readily verified from the following table:

Term n

Statistical

weight

U)n

Excitation energy

(rydbergs)

Observed HartreeFock

3p 1 9 0 0

1 D 2 5 0.145 0.152

1 A 3 1 0.308 0.378.

(139)

On expanding the Coulomb interactions e

2/r,;

- between the electrons of the oxygen atom using the multipole expansion formula (89) it is found that the

energies of the 3 P, 1 D, and X S states of the lowest configuration depend only upon the A = 0, 1, 2 terms of the

expansion, the main contribution arising from the

spherically symmetrical terms associated with A = 0. The neglect of the terms corresponding to A = 2 produces equal energies for the 3 P, 1D, and *S states and so

31
the approximation in which the X = 2 terms are omitted

is referred to as the exact-resonance approximation.

There are also exchange terms between the 2p electrons

and the Is and 2s electrons belonging to the core of the

oxygen atom which correspond to \ — 1. When the A = 2 terms are omitted, it is found that these X = 1 exchange

terms do not give rise to energy differences.

With the above consideration in mind, Seaton

(1953a) introduced the following approximations in

dealing with the oxygen excitation problem which to- gether compose his approximation I:

(i) the energy differences between the 3 P, l D, and X S

states of the lowest configuration can be neglected,

(ii) the 2p orbitals of the three states of the lowest

configuration can be taken to be the same and equal to the Hartree-Fock function for the X D state,

(iii) all the X = 2 terms in the integrodifferential

equations describing the scattering can be neglected,

(iv) all the X = 1 terms corresponding to exchange between the incident p wave electron and the Is and 2s core electrons can be neglected.

Approximation I enables the three integrodifferential

equations describing the excitation problem for the p wave 2 P scattering case to be uncoupled and an exact

solution to the equations derived. Using this solution as a first approximation, Seaton then discarded approximation (i) and obtained a better approximation to the

solution of the three original coupled integrodifferential

equations by carrying out a single iteration. This he

called approximation II. To test the accuracy of approximations I and II, Seaton proceeded to carry out more detailed calculations at a single electron impact energy corresponding

to the value of the wave number k\ of the incident elec- tron given by £i

2 = 0.3. This energy was deliberately

chosen to be insufficiently high to excite the X S term

of the ground configuration, so that only two coupled

equations are involved if the polarization resulting

from the presence of the X S term is neglected. He first carried out the same procedure as for approximation II but discarding approximations (iii) and (iv). This

Seaton refers to as approximation III. Finally he solved

the original pair of coupled equations exactly to give

him his approximation IV. The values of the collision

strengths obtained by Seaton for &i

2 = 0.3 are displayed

in Table XLVI, the numerals, 1, 2, 3 signifying the 3 P,

l D, l S terms, respectively. We see that exchange between the incident p electron and the core Is and 2s

electrons is fairly important and that approximation

III, which allows for this exchange, provides collision

strengths which are in rather good agreement with the

values given by the exact treatment of the coupled

equations. However we also note that ftp (l, 2) and

12p

( 2, 1) differ by about 16% in approximation IV

whereas detailed balancing requires them to be equal.

This is just the so called post-prior discrepancy and is an indication of the accuracy of the wave functions

Fig. 39. Collision strengths ^(1, 2), Q?{ 1, 3), and ^(2, 3) for transitions connecting the 3 P, l D, X S terms of the lowest configuration of atomic oxygen (Seaton, 1953a).

« which were used for the oxygen atom. Since it was not

possible to decide which of £2P

(1, 2) and f2p

(2, 1) provide the most reliable values for the collision_strength, 1 Seaton chose the geometric mean given by 12(1, 2) =

[£2(1,2)S2(2,

With regard to the p wave 2D scattering case, Seaton

used the distorted waves method to calculate the collision strength 0^(1, 2) at &i

2 = 0.3. He obtained the

value 0.007 which is less than 1% of the corresponding

value for the p wave 2 P scattering case. In addition

the d wave contribution to the collision strength f2d

( 1, 2) was found by Yamanouchi, Inui, and Amemiya (1940)

to increase slowly from zero at the threshold to the

value 0.077 at k^=\ and to a maximum value of 0.95

at &i

2 = 4; thus we see that 12d (l, 2) is unimportant

within the range of electron impact energies concerning

us here.

The final collision strengths I2J? (1, 2), 12p (l, 3), and

£2P (2, 3) obtained by Seaton in his first paper (1953a)

are displayed in Fig. 39. They were derived by multiplying the approximation II values ofS2p (l,2),fip (l,3),

and 12^(2, 3) by the factor 1.31 which yields agreement

with the approximation III value at &i

2 = 0.3. A satis- factory feature of these values is that they satisfy the

conservation theorem

S2*(w, ri) < (2£+l)co<, (140)

where o>< is the lesser of the statistical weights oj„, ov of

the states n, ri. We note that a discontinuity in the

slope of Op (l, 2) occurs at the threshold energy for the

excitation of the X S state owing to the neglect of this state below its excitation threshold.

In a subsequent paper Seaton (1955a) further investigated the accuracy of his calculations on the f2p (l, 3)

and f2p

(2, 3) collision strengths and introduced minor

alterations. He also calculated the O'* (2, 3) collision

strength finding that it was not unimportant. The final values of his collision strengths (Seaton, 1956) are displayed in Table XLVII, the corresponding cross sec- tions being given by the formulas

0(», ri) = (unkn

2/ir)Qn,n' ~ (<0n'kn'

2/ir) Qn’ ,n- (141)

The appropriate statistical weights are to be found in

( 139) . It has been estimated by Seaton that his calcu32
Fig. 40. Total cross sections for the Is—»2s excitation of hydrogenic ions of charge Z calculated using the Coulomb-Born approximation (first Born approximation for Z= 1) (Tully, 1960). [e is electron energy in threshold units defined by Eq. (38).]

lated collision strengths are accurate to within about

40% while their relative values should be still more

reliable. The extrapolated values given in italics in Table XLVII are probably accurate to within about a

factor of 2. A prescription for obtaining the collision strengths

for transitions between the 4 S, 2D, 2 P terms of the

lowest configuration ls2 2s22p3 of atomic nitrogen has

been given by Seaton (1956). Denoting these terms by

1, 2, 3 respectively, the collision strengths fi(l, 2),

1, 3) , and 0(2, 3) can be obtained from Table XLVII

by multiplying the corresponding oxygen collision

strengths by 25/24, 25/8, and 225/16, respectively.

More accurate calculations on the excitation of atoms

having a ground state 2p

q configuration involving the

solution of coupled equations are in progress and a paper

describing the formulation has been published by Smith,

Henry, and Burke (1966).

The probability per unit time for an atom in a state n

to make a n—*n' transition as the result of electron

impact can be expressed in the form ann >n{e), where

n(e) is the number density of electrons. Assuming a Maxwellian velocity distribution of electrons, Seaton

(1955; 1956) has calculated the deactivation coefficients

<*21 , <*3i, and a32 for atomic oxygen and atomic nitrogen.

These are displayed in Table XLVIII.

Finally we note that p wave collision strengths for the excitation of the 2p3

{ i S)3p 3 P and 6 P states of

oxygen from the ground 2pi 3 P state have been calculated by Percival (1957) using the distorted waves

method.

8. Positive Ions

The excitation of positive ions is dominated by the

Coulomb field which acts upon the incident electron

throughout its trajectory and indeed distorts it from a

linear path even at infinite separation. The effect of the

Coulomb field is so powerful that the threshold behavior

of the excitation cross section Q is entirely different to that exhibited by the neutral atom case. This is best

illustrated by the hydrogen-like ions towards which we

now direct our attention.

8.1. Hydrogenic Ions

We consider an ion having nuclear charge Z and

possessing a single electron only. The Is—>2s, Is—>3s, and Is—»2/> excitations of such ions have been examined

bv Tully (1960), Burgess (1961), and Gailitis (1963)

using the Coulomb-Born approximation. This is analogous to the customary Born approximation with,

however, Coulomb waves representing the incident

and scattered electrons instead of plane waves. Because

the Coulomb field exerts such a strong force upon the

incident electron, the Coulomb-Born approximation is more accurate for scattering by positive ions than the

Born approximation is for scattering by neutral atoms.

In Figs. 40 and 41 we display Z4Q plotted against the

energy of the incident electron in threshold units for the

Is—>2$ and Is—>2/> transitions calculated by Tully

(1960) and by Burgess (1961), respectively. We see immediately that for positive ions the total excitation

cross section monotonically increases to a nonzero value

at threshold whereas for neutral atoms the Born approximation yields a cross section which vanishes at threshold.A useful formula which is sometimes employed to estimate excitation cross sections for optically allowed

transitions, especially for the case of positive ions, is based upon the replacement of the potential Vn\ by

its asymptotic form (90) at large separations. Intro- ducing the oscillator strength f(n, 1) given by (92),

we find that the total excitation cross section can be

written

C» =

kn a0

2f(n, 1)

k\ AEnl

2do, (142)

where Fi and Fn are the wave functions describing the

incident and scattered electrons, AEnl is the excitation

energy in rydberg units, and do is an element of solid

angle in the direction of scattering.

If we now make use of the free-free Kramers Gaunt

factor g

defined according to the formula

2do =

87T2

yftkikn g' (143)

where the wave numbers k\ and kn of the incident and

Fig. 41. Total cross section for the Is—>2/> excitation of hydrogenic ions of charge Z calculated using the Coulomb-Born approximation (first Born approximation for Z = 1) (Burgess, 1961).

[e is electron energy in threshold units defined by Eq. (38).]

33
scattered electrons are given in units of a0

1

, we obtain

Q„= (87t/\^i2)[/(«, \)g/AEnl

~] (in ira0

2) = (1.28X10-*/E)lf(n, l)g/AEn{] (in cm2 ), (144)

where the unit of energy is the rydberg. This formula

yields unsatisfactory estimates of the excitation cross

section. It may be improved by introducing a cut-off

into the relevant integration in which case g becomes re- placed by an effective Gaunt factor g. Values of g for 1 s—mp excitations of hydrogen ions

are displayed in Table XLIX.

8.2. Helium Ion

The hydrogen-like ion which has been the subject of

the closest examination is the positive ion of helium

corresponding to Z = 2. The first detailed investigation of the excitation of

He+ was carried out by Bransden, Dalgarno, and King

(1953) who calculated the zero-order partial cross section for the Is—>2s excitation using the CoulombBorn approximation, the Coulomb-Born-Oppenheimer

approximation and the distorted Coulomb waves approximation both neglecting and including exchange.

The \s-2s-2p close coupling approximation has been

used recently by Burke, McVicar and Smith (1964a)

and by McCarroll (1964) to calculate the total excitation cross sections for the Is—>2s and Is—>2p transitions

of He+. Where comparison is possible the two investigations are in good agreement. The partial cross section

Q

L

for the Is—>2s and Is—>2p excitations calculated by

Burke, McVicar, and Smith (1964a), who carried out

the more extensive investigation, are displayed in

Tables L and LI, respectively. The values of the triplet

and singlet partial cross sections include the spin

weighting factors f and j, respectively. It is of interest

to note that the partial cross sections calculated neglecting exchange are good approximations to the sums of the

triplet and singlet partial cross sections for the L> 4

partial waves, indicating that exchange can be justifiably neglected for these higher-order partial waves.

Fig. 42. Total cross sections for the Is—>2s excitation of He+

ions. Curve 1: \s-2s-2p close coupling approximation (Burke,

McVicar, and Smith, 1964a) . Curve 2: Coulomb-Born approximation (Tully, 1960). Curve 3: unitarized Coulomb-Born approximation (Burgess, Hummer, and Tully, 1961)

.

Fig. 43. Total cross sections for the Is—>2p excitation of He+

ions. Curve 1: \s-2s-2p close coupling approximation (Burke,

McVicar, and Smith 1964a). Curve 2: Coulomb-Born approximation (Burgess, 1961). Curve 3: unitarized Coulomb-Bom approximation (Burgess, Hummer, and Tully, 1961).

To obtain the total excitation cross sections use was made of the higher L value partial cross sections calculated by Tully (1960) and by Burgess (1961) employing

the Coulomb-Born approximation. Very slow conver- gence with L value was found for the Is—>2p excitation,

it being necessary to include partial cross sections for

L values up to 27 for &i = 4. The ls—>2p partial cross sections for L> 15 were obtained by an extrapolation

procedure which made use of the fact that the graph of In Q

L against L is linear for the higher values of L.

The total cross sections for the Is—>2s and Is—»2/> excitations of He+ calculated by Burke, McVicar and

Smith (1964a) are displayed in Figs. 42 and 43 where

they are compared with the excitation cross section

calculated by Tully (1960) and by Burgess (1961)

using the Coulomb-Born approximation.

To assess the accuracy of the \s-2s-2p close coupling

approximation, Burke, McVicar, and Smith (1964a)

included the 3s and 3p states in the close coupling expansion for &i

2 = 5 corresponding to 68-eV incident electron

energy. The resulting zero-order partial cross sections are given in Table LII where they are compared with

the values obtained using the ls-2s-2p close coupling

approximation. We see that the effect of allowing for the 3s and 3p states is quite slight for the helium positive

ion in contrast to the case of atomic hydrogen for which

the analogous effect at 16.5-eV impact energy was found

to be considerable as can be verified by referring to Table XXII. Thus it seems that the close coupling approximation has nearly converged to the exact solution

with the inclusion of the Is, 2s, 2pstates for the helium

positive ion. In any event the \s-2s-2p close coupling

approximation is certainly better for the case of the

helium positive ion than for atomic hydrogen.

Burke, McVicar, and Smith (1964b) have also used

the \s-2s-2p close-coupling approximation to calculate

the positions and widths of the first few autoionizing

states of helium from the resonances which occur in the

5 and P wave phase shifts for elastic scattering of elec- trons by He+

, these resonances making their appearance

by a rapid variation in the appropriate phase shift by ir
radians. The calculated positions of these resonances

agree very well with the experimental data of Madden

and Codling (1963) and Simpson, Mielczarek, and

Cooper ( 1964) as can be seen from Table LIII where a comparison is made for some 1 S, i P, and 3 P autoionizing

levels. It should be remarked however that even when

exchange is neglected, the close coupling calculations

still provide fairly satisfactory values for the positions of

the resonances, which therefore do not seem very sensitive to the detailed assumptions made by the theory.

Calculations of the partial cross sections for the

1s—>3p excitation of the helium positive ion have also

been carried out by Burke, McVicar, and Smith

( 1964b) using the 1 s-3p close coupling approximation.

Partial and total 1 s—+3p excitation cross sections are displayed in Table LIV.

8.3. Helium-like Ions

The Coulomb-Born approximation has been employed by Sural and Sil (1966) to obtain a differential cross section formula for the P-S—>21 5 excitation of

positive ions belonging to the helium isoelectronic

sequence using the simple wave functions derived by

Morse, Young, and Haurwitz (1935). They carried out

detailed calculations for the special case of the l

l S—>2* A

excitation of Li+ at the threshold energy. An integration over all angles of scattering gave 0.304 X lQr2ira0

2 for the

total cross section at threshold.

8.4. Lithium-like Ions

Bely, Tully, and Van Regemorter (1963) have used

the Coulomb-Born approximation to calculate the

total excitation cross sections for the 2s—+2p resonance

transitions of the positive ions Be+

, C3+, 05+, and Mg9*

all of which belong to the lithium isoelectronic sequence.

They employed the simple analytical wave functions

derived by Veselov et al. (1961) for the 2s and 2

p

orbitals of the optical electron. The excitation cross sections obtained by them are presented in Figs.

44—47. Bely (1962) has calculated the effective free-free

Kramers Gaunt factor g for the resonance transition of

O5* at the threshold energy obtaining <7 = 0.79.

Bely, Tully, and van Regemorter (1963) also used

the 2s-2p unitarized Coulomb-Born approximation

(also known as the Coulomb-Born II approximation)

to investigate the resonance transitions of the lithium

sequence of positive ions. Only for the Be* ion does the

unitarized Coulomb-Born approximation produce significantly different results from those found with the

ordinary Coulomb-Born approximation.

A detailed investigation of the 2s—*2p excitation of

the N4+ positive ion member of the lithium isoelectronic

sequence has been carried out by Burke, Tait, and Lewis

(1966) using the Hartree-Fock functions obtained by

Weiss (1963). The following approximations were employed:

(i) the Coulomb-Born approximation,

(ii) the unitarized Coulomb-Born approximation,

Fig. 44. Total cross sections for the 2s—*2p excitation of Be+

(Bely, Tully, and Van Regemorter, 1963). is electron energy in threshold units defined by Eq. (38).] Curve 1: Coulomb-Born

approximation. Curve 2: unitarized Coulomb-Born approximation.

Fig. 45. Total cross sections for the 2s—>2/> excitation of C3+ (Bely, Tully, and Van Regemorter, 1963) . is electron energy in threshold units defined by Eq. (38).] Curve 1: Coulomb-Born

approximation. Curve 2: unitarized Coulomb-Born approximation.

Fig. 46. Total cross sections for the 2s—>2/> excitation of 05+ (Bely, Tully, and Van Regemorter, 1963) . [t is electron energy in threshold units defined by Eq. (38).] Curve 1: Coulomb-Born

approximation. Curve 2: unitarized Coulomb-Born approximation.

Fig. 47. Total cross sections for the 2s—>2p excitation of Mg94"

(Bely, Tully, and Van Regemorter, 1963). [e is electron energy in threshold units defined by Eq. (38).] Curve 1: Coulomb-Born

approximation. Curve 2: unitarized Coulomb-Born approximation.

35
(iii) the 2s-2p close-coupling approximation neglecting exchange,

(iv) the 2s-2p close-coupling approximation including exchange,

(v) the 2s-2p-3s-3p-3d close-coupling approximation neglecting exchange.

All of these approximations were found to yield the same 2s—>2p excitation cross section within a few percent. The cross section is displayed in Fig. 48. We conclude from this result that the resonance excitation

cross sections for the other positive ions of the lithium

isoelectronic sequence are also given accurately by the

Coulomb-Born approximation, except for Be+ whose

cross section may be overestimated somewhat.

The 2s—*3p excitation of N4+ was also investigated

with considerable thoroughness by Burke, Tait, and

Lewis (1966). Although they found that the excitation

cross sections given by the Coulomb-Born approximation and the 2s-3p close coupling approximation with

exchange neglected were in good accordance, the effect

of allowing for exchange and for the coupling to the 3s

and 3d states were found to be rather important, both

increasing the excitation cross section considerably at the threshold. The application of the unitarized Coulomb-Born approximation gave rise to misleading re- sults since it produced a slight reduction in the 2s-^3p

excitation cross section below that given by the ordinary

Coulomb-Born approximation. We conclude from this

that the predictions of the unitarized Coulomb-Born

approximation should be treated with caution and it seems likely that it is a useful approximation only for resonance transitions with large line strengths. The

cross sections for the 2s-^3p excitation of N4+ are dis- played in Fig. 49.

Burke, Tait, and Lewis (1966) have also employed

the Coulomb-Born approximation to calculate excitation cross sections for other transitions connecting the n = 2 and n = 3 levels of N4+. These are presented in

Table LV.

8.5.

Sodium-like Ions

The Coulomb-Born approximation has been used by

Bely, Tully, and Van Regemorter (1963) to calculate

the excitation cross sections for the 3s^3p transitions

Fig. 48. Total cross section for the 2s—>2p excitation of N4+ calculated using the 2s-2p-3s-3p-3d close coupling approximation

(Burke, Tait, and Lewis, 1966).

36

Fig. 49. Total cross sections for the 2s—>3

p

excitation of N4+ (Burke, Tait, and Lewis, 1966) . Curve 1: Coulomb-Bom approximation. Curve 2: 2s-3p close coupling approximation neglecting

exchange. Curve 3 : 2s-3p close coupling approximation including

exchange. Curve 4: 2s-2p-3s-3p-3d close coupling approximation

neglecting exchange. Curve 5: best estimate of 2s—>3p excitation

cross section.

of the positive ions Mg+, Si3+

, and Fe1H" which are all members of the sodium isoelectronic sequence. These

excitation cross sections are given in Figs. 50-52. We

see that the cross section for the resonance transition

of Mg+ is decreased considerably by employing the

unitarized Coulomb-Bom approximation; however for

the other ions the reduction is much less important.

Krueger and Czyzak (1965) have carried out calculations on the 3s—>3p and 3p—*3d excitation of Fe15+ using the Coulomb-Born approximation. Their results

for the 3s—>3p transition are in close accordance with

those obtained by Bely, Tully, and Van Regemorter

( 1963) . The excitation cross section for the 3p—>3d transition of Fe15+ calculated by Krueger and Czyzak

(1965) is displayed in Fig. 53.

8.6.

Potassium-like Ions

The only member of the potassium isoelectronic

sequence to be studied so far is Ca+. Van Regemorter

(1960a; 1960b; 1961) has calculated total excitation

cross sections for the 45—Ap, 3d-M:p, and 45—»3d transitions of this positive ion using the Coulomb-Born approximation as well as the unitarized Coulomb-Bom

approximation. His values are displayed in Table LVI.

The total excitation cross sections for the 452 Si/2—

4/>2 .Pi/2 and 452 6’i/2—>4/?

2 P3/ 2 transitions corresponding

to the H and K lines of Ca+

, respectively, are given by

Q(H) =IQ(4:S-Ap) = 19.397rao2

a-

'Vs II O = l7.357rao2 II p

Q{K) =lQ{As-^p) =38. 787rao2 'Vs II 0 = 34.7(W II d—4

where kf is the wave number of the scattered electron.

8.7.

Positive Ions Having p

2

, p

3

, or p

4 Configurations

Seaton (1953b; 1955b; 1956) has carried out a detailed investigation of the excitation of forbidden transitions between the terms of the lowest p

2

, p

3

, p

4 con-
Fig. 50. Total cross section for the 3s—>3/> excitation of Mg+

(Bely, Tully, and Van Regemorter, 1963). [e is electron energy in threshold units defined by Eq. (38).] Curve 1: Coulomb-Born

approximation. Curve 2: unitarized Coulomb-Born approximation.

Fig. 51. Total cross section for the 3s-^3p excitation of Si3+ (Bely, Tully, and Van Regemorter, 1963). [e is electron energy in threshold units defined by Eq. (38).] Curve 1: Coulomb-Born

approximation. Curve 2: unitarized Coulomb-Born approximation.

Fig. 52. Total cross section for the 3s—>3/> excitation of Fe15 'h (Bely, Tully, and Van Regemorter, 1963) . [e is electron energy in threshold units defined by Eq. (38).] Curve 1: Coulomb-Born

approximation. Curve 2: unitarized Coulomb-Born approximation.

Fig. 53. Total cross section for the 3p—>3d excitation of Fel5+ (Krueger and Czyzak, 1965). [e is electron energy in threshold

units defined by Eq. (38).] Curve 1: Coulomb-Born approximation. Curve 2: unitarized Coulomb-Born approximation.

figurations of certain positive ions. The various terms

which occur for these configurations are:

n p

2 and p

4 p

3

3

2 P

2

l D 2D

1

3 P 4 S, (146)

where we have denoted the terms of a given configuration by n— 1, 2, 3 in the order of increasing eigenenergy.

Because the variation with impact energy of the

excitation cross section for a positive ion is rather slow,

Seaton’s calculations were performed at the threshold

energy for excitation. The terms corresponding to n— 1, 2, 3 very nearly have the same energy for a positive ion and so the basic approximation used by Seaton was the exact resonance approximation which neglects

the energy differences between the terms. By far the

largest contribution to the collision strengths for the

1—>2 and 1—>3 transitions, which both involve spin

change and thus include exchange coupling only, is made

by the p wave, the other angular momenta providing

unimportant contributions. Detailed calculations were

performed by Seaton on the positive ions N+

, 0+, 02+

, and Ne2+ using Hartree-Fock wave functions. The

collision strengths 12(1, 2) and 12(1, 3) obtained by

Seaton (1953b) employing the exact resonance approximation together with certain small corrections are given in Table LVII. The values given in italics for the

other ions were found by interpolation and extrapolation and are estimates only.

The collision strengths for the 2—>3 transitions of

0+

, 02+, and S

+ ions have also been calculated by Seaton

( 1955b) . Because no change of spin occurs in the 2—>3 transitions, potential interaction terms arise in the

integrodifferential equations describing the excitation

collisions. Both potential and exchange interactions

have to be taken into account in the case of the pwave. The basic approximation used by Seaton (1955b)

for the p wave was the exact resonance approximation,

the X = 2 terms arising from the expansion of the Coulomb interaction e

2/rt_,- between the electrons being

allowed for by the use of first-order perturbation

theory. The presence of the potential interaction terms

results in partial waves other than the p wave being

important for 2—>3 transitions. The collision strengths

corresponding to these partial waves were calculated by

Seaton (1955b) using the distorted waves approximation but neglecting exchange interaction terms. Values

of the total collision strength 12(2, 3) determined by

Seaton ( 1955b) are displayed in Table LVII.

The exact resonance approximation has also been

applied by Czyzak and Krueger (1964) to calculate the

collision strengths 12(1, 2), 12(1, 3) and 12(2, 3) for the

3p2 ions S

2+

, Cl3+, and Ar4+ as well as the 3p3 ion S

+. Their values for S+ differ from those calculated by

Seaton (1953b; 1955b), the discrepancy being rather

large for 12(2, 3). Extensive new calculations on the

cross sections for the excitation of positive ions having

37
ground state 2

p

q and 3pq configurations have been re- ported by Saraph, Seaton, and Shemming (1966) and

Czyzak and Krueger (1967).

9. Polarization of Radiation

The dipole radiation emitted by an atom after excitation by an electron beam is polarized. Let us suppose

that the direction of the incident beam of electrons is parallel to the O2 axis of a rectangular frame of reference

0(x, y, z). Then the radiation may be regarded as being

due to an electric dipole in the Oz direction and two

equal electric dipoles in the Ox and Oy directions. Denoting the intensity of radiation per unit solid angle in a

direction perpendicular to the Oz axis due to the electric dipoles parallel and perpendicular to Oz by I\\ and

7a, respectively, we define the percentage polarization P

by the formula

P= 100(7,, —/x)/ (7||-f/±). (147)

Then it can be shown that the intensity of radiation

per unit solid angle in a direction subtending an angle 6 with the Oz axis is given by

1(6) =3(100-

cos2 6)1/ (300- P), (148)

where 47r/ is the total intensity resulting from integrating 1(6) over all solid angles. Since I = I\\-p 2/x we may

express the percentage polarization in the alternative

form

i’=100(3/ll -7)/(/ll +J). (149)'

The first theoretical treatment of the polarization of radiation was given by Oppenheimer (1927a; 1927b;

1928) and was further developed and applied by

Penney (1932b). Recently the theory has been re- examined and extended by Percival and Seaton (1958)

.

Supposing that the atom can be characterized according to the Russell-Saunders coupling scheme, we let L'S'J', LSJ, and L" S" J" be the orbital, spin, and

total angular momentum quantum numbers of the

initial, upper, and final levels of the atom, respectively.

We shall confine our considerations to those instances

for which the initial state of the atom has zero orbital

angular momentum so that L' — 0. We first examine the two electron case of helium

whose nucleus has zero spin and concern ourselves

with the polarization of the light emitted in the line

LSJ—>J

and in the multiplet LS—>L"S. The total

spin S of the helium atom can be 0 or 1. The polarization is zero if the upper level is an S state so that L = 0. If L= 1, corresponding to an upper P state, the per- centage polarization takes the form

P=mG(Q0 -Q1 )/(hoQ0 +h1Q1 ) (150)

while if L= 2, corresponding to an upper D state, we

have

P= lOOG^Qo+Qi- 2Qi) / (hoQ0

-\- hiQi-ph^Q-i)

,

( 151)

where Q\ml\ is the cross section for the excitation of the

state LSMlThe values of the constants G, ho, hi, hi obtained by Percival and Seaton using the Oppenheimer-Penney theory are given in Tables LVIII and

LIX.

We now examine the polarization of the Lyman a

line of hydrogen, produced by the electron impact

excitation of the 2 p states, and the resonance lines of the

alkali metal atoms. These lines all arise from a

2 5

initial state, a

2 P upper state and a2 5 final state. For

the 2 Pn‘r-PS transition the polarization of the radiation

is zero. For the 2 Pzn— transition and the 2 P—^S

multiplet, the percentage polarization according to the

Oppenheimer-Penney theory is given by formula ( 150)

,

the constants G0 , h0 , hi being presented in Tables LX

and LXI. The spin of the atomic nucleus is not neces- | sarily zero for these cases and so the constants involved | in the polarization formula were calculated by Percival ; and Seaton for the values 1 = 0, 1, § of the nuclear

spin quantum number.

Unfortunately, the Oppenheimer-Penney theory : yields results which depend upon the representation

used to characterize the state of the atom. This occurs because the probabilities of exciting the upper states

of the atom by electron impact and the optical transition

probabilities from these states are calculated independently in the Oppenheimer-Penney theory. The more

sophisticated treatment of Percival and Seaton con- , siders the probability of polarized photons being emitted

by the whole system of atom and incident electron. As

in the simpler theory, the polarization is zero for the

2/>i/2— transition of atomic hydrogen. For the case of the 2p3/r-+ls transition of atomic hydrogen Percival

and Seaton find

P(2pm) = 100(e0 -Qi)/(1.694Q0 +2.388Q1 ) (152)

which is quite close to the expression

P(2pm) = 100(G)— <2i)/(1.667<2o+2.333<2i) (153)

obtained using the Oppenheimer-Penney theory with

1=0, although rather different from that found taking

I = \. The polarization of the total radiation from the

2p—>1$ multiplet of atomic hydrogen is found by

Percival and Seaton to be

P(2p) = 100(G-<2i)/(2.375<2o+3.749G) (154)

which is not very different from the expression

P(2p) = 100(G-G)/(2.333G+3.667G) (155)

derived using the Oppenheimer-Penney theory with

7 = 0 but dissimilar to that obtained with I = \.

For the case of atomic hydrogen the exact expression

for the polarization is close to the Oppenheimer-Penney

formula with 7 = 0 because the hyperfine structure separations are small. However for atoms other than hydrogen, the hyperfine structure separations are significant

and then the theory including hyperfine structure must

be employed.

In a recent paper by Flower and Seaton (1967) the

effect of allowing for hyperfine structure in the calcula-
tion of the threshold polarizations of the resonance

lines of 6Li, 7Li, and 23Na has been investigated. The

percentage polarizations of the resonance lines calculated using the measured values of the hyperfine struc- ture are given in Table LXII where they are compared

with the values determined experimentally by Hafner,

Kleinpoppen, and Kruger (1965). The agreement between theory and observation is seen to be very good.

In general the polarization at the threshold energy for excitation can be evaluated without detailed calculations of the cross sections since the scattered electron

has zero velocity and therefore zero orbital angular

momentum at the excitation threshold so that only

states with Ml= 0 can be excited. If we set (A = 0 in the

Oppenheimer-Penney formula (155) for 7 — 0 we obtain

P{2p) =42.9 at the threshold for the polarization of the

2p—>ls multiplet while the more elaborate theory of

Percival and Seaton yields P( 2p) =42.1. However, as a consequence of the close coupling between the 2s and

2p states of hydrogen, this has been shown to be an

unjustifiable procedure by Damburg and Gailitis (1963)

who find that P(2p) oscillates near the threshold and

does not appear to tend towards a definite limit at the

threshold. Their values of P(2p) near the threshold

energy are given in Table LXIII.

At energies above the threshold the polarization of the

2p-^ls multiplet has been calculated by Burke, Schey,

and Smith (1963) using their cross sections calculated

with the \s-2s-2p close coupling approximation. These

values of P{2p) are also given in Table LXIII.

10. Theoretical Summary

The methods which have been employed to calculate

excitation cross sections can be conveniently classified according to the strengths of the couplings between the

initial and final states of the target atom and between

these states and other states of the atom.

For high-electron impact energies the weak coupling

approximations are appropriate. They are based upon

the assumption that the back coupling of the final state to the initial state, and the couplings of these

states to other states of the atom, are small and may be

neglected. The most widely used of the weak coupling

approximations is the first Born approximation which

represents the free electron by a plane wave and may be

employed without making undue error for electron

impact energies sufficiently far above that for which the

excitation cross section attains its maximum value.

A still simpler approximation which may be used at moderate as well as high energies is the Bethe approximation which, in the case of optically allowed transitions, replaces the matrix element F„i, coupling the

initial state 1 to the final state n, by its asymptotic

form for large radial distances. A more elaborate weak

coupling approximation is the distorted waves approximation which represents the free electron by functions

describing its motion in the static fields of the atom

before and after the excitation process. The use of this

approximation generally leads to an increase in the excitation cross section above that given by the first Born

approximation. Neither the first Born nor the distorted waves approximations make allowance for coupling to

states other than the initial and final states. Such couplings can be allowed for by using the second Born

approximation, provided they are fairly weak, which

should be a valid procedure at moderately high impact

energies.

The weak coupling approximations discussed so far do not take account of the Pauli exclusion principle

and thus neglect the role of electron exchange. Its importance is emphasized by the fact that the Born

approximation yields zero for the cross sections for

singlet—^triplet excitations of helium which arise entirely as the consequence of exchange. Now the BornOppenheimer approximation which, like the first Born

approximation, represents the free electron by plane

waves, is unsatisfactory since it gives rise to excitation

cross sections which are gross overestimates at low and

moderately high impact energies for the case of S—>S

transitions. The failure of this approximation comes

about by the neglect of certain terms of the first order

in the interaction energy between the free electron

and the atom. If these terms are retained one arrives at the Feenberg or first-order exchange approximation

which produces excitation cross sections that are substantially less than those given by the Born-Oppenheimer approximation for S-+S transitions. An alternative approach is to expand the exchange scattering

amplitude given by the Born-Oppenheimer approximation in powers of kf1 where ki is the wave number of

the incident electron, retaining only the leading term of

order k{~". This is the Ochkur approximation. It appears

to yield satisfactory results even for quite low electron

energies and has the attractive feature of being no more

difficult to apply than the first Born approximation.

The distorted waves approximation can also be generalized to take account of electron exchange. Although

this is the most accurate of the weak coupling approximations, its application requires an expansion in partial waves and the greatest computational effort. We now turn our attention to the case of strong coupling between the initial and final states of the target

atom but relatively weak coupling between these

states and other states of the atom. This case generally

arises at low impact energies for transitions between

states with a small energy difference, such as the reso- nance transitions of the alkali metal atoms and the

ns->(n-{-\)p transitions of atomic hydrogen for n

large. All the weak coupling approximations discussed

hitherto fail badly for such cases since they produce

excessively large cross sections near the threshold

energy. This behavior is a consequence of the fact that

the partial cross sections associated with low values of

the angular momentum of the incident electron are

grossly overestimated by weak coupling approximations

in a strong coupling situation, often exceeding the

39
theoretically determined maximum value imposed by

conservation considerations. Several strong coupling

approximations have been introduced by Seaton to overcome this difficulty. The modified Bethe approximation uses the Bethe approximation for the partial cross sections which are less than half the theoretical maximum value Qmax* and replaces all Bethe partial cross

sections which exceed this value by A somewhat

analogous procedure based upon a semi-classical approach and known as the impact parameter method has

also been used. Another method of treating strong

coupling cases is to employ the exact expression for the

excitation cross section given in terms of the reactance

matrix R which is then approximated by using the

Born approximation. This automatically ensures that

the conservation theorem, or unitarity, is obeyed and

has been termed the unitarized Born approximation.

Under strong coupling conditions the three approximations discussed above lead to a considerable reduction

in the excitation cross section below that given by the

Born approximation at low impact energies, and,

judging from the case of the 3s—>3p resonance transition

of sodium, the results are quite satisfactory. However,

if the coupling is insufficiently strong, it should be

noted that the very strong coupling approximations

may, in certain instances, give rise to misleading

results.

If the energy difference between the initial and final states of the transition is very small another approach

to the problem may be made. This is called the exact resonance approximation and is based upon the assumption that this energy difference may be taken to vanish

thus enabling the coupled equations describing the

scattering to be solved exactly. The neglected terms can

then be taken into account by an iterative procedure.

This method has been applied, for example, to the

3 P—^D and 3 P— excitations involving the terms of the ground configuration of atomic oxygen, and to the

3s-*3p excitation of sodium for which it yields a cross section which is in substantial agreement with that

obtained using the modified Bethe approximation and

the other strong coupling approximations.

In many instances the coupling between the initial and final states of the target atom may be fairly weak

while the coupling of the initial or final state to certain

other states of the atom may be strong. An example of

such a case is the Is—>2s excitation of atomic hydrogen

for which the coupling between the Is and 2s states is weak although the coupling of the 2s state to the 2p

states, with which it is degenerate, is very strong.

Moreover the coupling between the 2s state and the np

states with principal quantum numbers n>3 is also

strong and can be expected to have an important effect

upon the Is—»2s excitation collision. The most satis- factory approach to the problem of determining excitation cross sections for such cases is to perform an exact

solution of the coupled integrodifferential equations

connecting the initial and final states and the most

important other states of the target atomic system. This

is referred to as a close coupling approximation. However 0

it appears that many more states of the atom may make

a significant contribution to the scattering than was

originally anticipated so that even the \s-2s-2p-3s3p-3d close coupling calculations performed on the i 1 Is—»2j and ls-+2p excitations of atomic hydrogen may

be some distance from convergence to the actual solution, except for electron impact energies below that

required to excite the resonances arising from the n= 4

level where the 6 state approximation should be quite

accurate. Lastly we note that classical mechanics has also been i used to calculate excitation cross sections. The classical

approach produces the wrong decay of the excitation

cross section with increasing impact energy although

it seems to yield cross sections of about the correct magnitude at moderate energies. A synthesis of the

classical method and the semi-classical impact parameter method has been introduced which produces excitation cross sections having the correct energy fall off.

II. EXPERIMENT

11. Introduction

[Note added in proof. In this article we discuss se- lected measurements from the literature. An extensive

compilation of measured electron collision cross sections

is being prepared by the JILA Information Center.]

In the study of an atomic collision process the theoretician has the means, in principle, to obtain descriptions adequate for ordinary purposes through application of the Schrodinger equation to the complete

physical system. In low-energy atomic physics the

quantum-mechanical principles are usually assumed to be fully understood, but the technical difficulties involved in applying them in full to a complete atomic

system have been insurmountable. As we have seen in

preceding sections it is necessary to simplify the atomic

system conceptually and to develop mathematical

approximations. These approximations tend to fail in

the range below about one hundred electron volts,

which is the energy range of greatest interest in the

microscopic approach to the study of hot gases (stellar

atmospheres, plasmas, etc.)

.

This regime is then an area of opportunity for the

experimental physicist. He may be able to provide much

needed data which cannot be calculated accurately or he may be able to provide accurate measurements on a few systems which can be used as a check on the theoretical approximations. In the remainder of this paper

we consider the available experimental data on electron

impact excitation of atoms, commenting on quality

where possible.

The experimentalist, as well as the theoretician, has

a problem of simplifying the physical system which he

studies. First, it must be simple enough so that it can be
described accurately and concisely. Second, for purposes

of comparison with theoretical work, and to fill the current need for basic microscopic data, the observed

phenomena should be a good approximation to the

single process usually studied by the theoretician. In

this review of experimental work we emphasize those

experimental results which seem to be relevant to the

isolated electron-atom system. In considering an ex- perimental result, of necessity, the first concern will be

with the evidence that the atomic system studied is isolated: (1) that the results are independent of pres- sure or may be extrapolated reliably to zero pressure, and (2) that the atomic system is not irretrievably

influenced by the presence of macroscopic electric and

magnetic fields. The general criteria for isolation are dis- cussed further in Secs. 12.1, 12.2, and 12.3. In these

sections we have tried to identify and discuss those

physical properties of atoms, and of the electron impact

excitation process, which must be considered by the

experimentalist if systematic errors of serious proportion

are to be avoided. This discussion is the heart of the

experimental part of the review. It is our contention

that with the physical understanding of electron impact

excitation now represented in the literature and with

the experimental technology now available a very

significant improvement in the quality of results could

be realized in any future work. The general characteristics of excitation functions have been established. The

interest in the immediate future will be in providing more accurate measurements for detailed comparison

with calculated cross sections and for applications such

as in plasma diagnostics and astrophysical models.

The remainder of the discussion is a rather broad

survey of the apparently better experimental results,

but few of the descriptions in the literature are complete. The great majority of authors do not mention some of the more important physical and instrumental

effects discussed in Secs. 12.1-12.3. In many cases it is clear that the physics involved in the excitation meas- urements could well have been given more careful

attention and measurements could have been more

adequately described in publication.

12. Experimental Methods

Experimental methods which have been used to study inelastic collisions between electrons and atom fall into three main categories.

The first and most important of these includes those

experiments in which intensities of spectral lines excited

by an electron beam are used as a measure of the excitation process. The result of such a measurement is a photon-excitation cross section: the probability for producing a photon in a given spectral line, per atom

and per unit electron particle flux. We note at this point that the photon-excitation cross section produced by the experimentalist, even if he

succeeds in meeting strict criteria for isolated atoms,

in general cannot be directly compared with the levelexcitation cross section produced by the theoretician.

The differences are due to (1) cascading from higher

levels excited by electron impact, (2) the availability

of several channels for radiative decay of excited states,

and (3) the anisotropic property of the radiation pattern with respect to the electron beam axis [see formula

(148)]. The third of these effects is determined by the

relative probabilities for exciting the atomic dipoles

along the perpendicular to the electron beam axis. Therefore, the photon-excitation cross section as observed in a given direction should be compared with an

appropriate theoretical statement written in terms of cross sections for exciting the various magnetic substates of the atom and the transition probabilities which

determine the radiative decay scheme.

This distinction between the nature of the primary

results of experiment and theory has led to the extensive use of the term optical excitation function. We use this

term to describe the dependence on electron energy of

the cross section for producing photons in a given line. The term apparent excitation function may be used to describe the excitation function as observed in a particular direction without some correction (such as for

anisotropic properties of the radiation pattern).

Figure 54 is a schematic diagram of a typical apparatus used for the measurement of optical excitation

functions. Electrons emitted by the heated cathode are passed through the collision chamber in a parallel beam.

Usually, the geometry of the electron beam is controlled

visually by adjusting anode and grid potentials. In

most work the radiation excited by the electrons is sufficient to delineate the electron beam visually.

Focussing is usually accomplished purely electrostatically, although a few workers have used axial magnetic

fields.Through the energy range from first appearance of

visible light up to the ionization potential the beam is subject to space-charge spreading. However, in most

work, the optical system is arranged so the image of the

electron beam has its axis perpendicular to the entrance

slit of a monochromator and only partially covers the

Fig. 54. Electron beam apparatus used for the study of optical excitation functions of helium (St. John, Bronco, and Fowler,

1960).
slit length. Under these circumstances the signal is usually not noticeably sensitive to minor variations in

electron beam diameter.

Above the ionization potential, the beam often can be

maintained parallel with a diameter defined by the

aperture of the electron gun. This evidently occurs when ion densities are sufficient to neutralize the elec- tron beam space charge.

Absolute calibrations of the optical system are carried

out by placing a standard tungsten strip lamp in the

position of the electron beam, virtually or actually,

and calibrating the response of the optical system to the known flux. Absolute calibrations will be discussed

in more detail in Sec. 12.4. The optical method discussed in the preceding paragraphs is only suitable for short-lived states. For states with very long lifetimes a second method based on the

ejection of electrons from a metal target has been

applied by a number of workers. The metastable atom is deactivated at the target, the excitation energy going

into kinetic energy of the ejected electron and into over- coming the work function of the target material. The

yield of ejected electrons provides information about

the long-lived state similar to that which the optical

excitation function provides about the short-lived state. The optical calibration problem is replaced by the problem of determining the efficiency of collection and conversion of metastable atoms at the metal target.

A closely related method is based on deactivation in a gas of lower ionization potential, the Penning ionization

process. The third method for studying inelastic electronatom collisions is through the study of the energy-loss

spectrum of scattered electrons. An electron beam of definite energy is passed through a gas. An electron

energv-analyzer is used to measure the energy spectrum

of electrons scattered at a particular angle with the

beam axis. Pressures are reduced to the point that

multiple collisions can be neglected.

With the analyzer set to accept electrons of zero energy loss an electron current due to elastic scattering

is observed. As the energy loss setting is increased only

a background of stray electrons is observed until the

energy-loss component transmitted corresponds to the

first excitation potential. At this setting a peak in the

electron current is observed due to inelastic scattering

leaving the atom in the first excited state. Peaks cor- responding to excitation to higher excited states can be

observed as the energy resolution of the instrument may

permit.

The method appears to have some interesting advantages. Where energy resolution permits, an energy-loss

component represents the direct excitation to the cor- responding excited state, with no complications due to cascading, or due to the lifetime of the excited state. In

practice, however, there are difficulties in applying it to the measurement of excitation cross sections as a function of energy. The major problem is in the necessity

for taking data at many angles as well as over a range of energies. The angular distribution of electrons must be

investigated in detail and used as the basis of integration

in order to obtain a total cross section.

Much significant work has been accomplished with

this method, nevertheless. Lassettre and his co-workers

have used the method, at relatively high incident elec- tron energies, to study angular distributions of inelasti- cally scattered electrons, and some of these results have

been used earlier in this paper (3.1) in comparison with

theoretical results. Relative magnitudes of differential cross sections for different transitions may also be obtained at a particular angle and energy.

Several groups have exploited this method using

high-resolution electron spectrometers at relatively low

electron energies (Schulz and Philbrick, 1964; Chamberlain, Simpson, and Kuyatt, 1965) observing peaks

corresponding to many doubly excited states lying

beyond the ionization limit as well as peaks corresponding to singly excited states. Schulz and Philbrick

(1964) and Chamberlain (1965) have carried out highresolution studies of inelastic processes by observing a

particular loss component as a function of incident elec- tron energy. These studies do not yield reliable relative cross sections, but they have revealed the occurrence of sharp resonances in the excitation cross sections, which

are apparently associated with excitation through intermediate short-lived negative ion states.

12.1. Pressure Dependence of Excitation Functions

Our goal in this review is to evaluate the present

status of information about the excitation of free atoms

by incident electrons. Consequently, we are interested

primarily in experiments the results of which are independent of pressure variations or which can be corrected

to zero effective pressure. This requirement removes

from further consideration many of the older results,

including some from well-known works still frequently

used.

The major processes which have been recognized as giving rise to pressure dependence of excitation functions are (1) imprisonment of resonance radiation and

(2) collisional transfer of excitation. Since much of the

following discussion is based on studies of helium excitation, a helium energy level diagram is shown in Fig. 55. The importance of imprisonment of resonance radiation to measurements of excitation functions was fully

recognized by Phelps (1958), who applied the theories

of Holstein (1947; 1951), and of Biberman (1947) to the analysis of experimental observations of excitation

in helium. In the interpretation of observed excitation

functions it is convenient to be able to assume that the

photons produced in the de-excitation are subject to a

negligible amount of reabsorption in the gas. This is a

safe assumption for transitions between excited states, at experimentally practical temperatures, since the

populations of excited states are low. Reabsorption in

these transitions is negligible. However, for the reso-
Fig. 55. Term diagram for atomic helium.

nance lines, here defined as those connecting to the

ground state, great care is necessary to ensure that the

gas density is low enough so that reabsorption is negligible.

Reabsorption of photons by atoms in the ground state

effectively lengthens the life of the excited state, and

spreads the excited state population over a larger vol- ume. The longer effective lifetime of the upper state

results in an increased probability for intervention of

collisional processes, and for conversion through radiative transitions to lower levels other than the ground

state. In helium, for example, the 3

l P—>2 l S excitation

function is pressure dependent because of reabsorption

in the 3

1 P-^l1 S line, resulting in conversion of 3

1 P—>1 1 S

photons to 3

l P—>21 S photons.

The possible importance of collisional transfer of

excitation, as well as of imprisonment of resonance

radiation, was considered by Lees and Skinner in 1932.

In collisional transfer, an excited atom is de-excited in a

collision with a ground-state atom, with a transfer of

excitation energy and possibly changes in the values of

angular momentum and spin associated with the excitation energy. An example of such a reaction may be

written

He(w1 P)+He(l1 5)-^He(l1 5)+He(»3 F). (156)

In helium the 1 P levels have large cross sections for excitation by electron impact. Therefore, the most

noticeable effect of excitation transfer is the cascading

resulting from conversion of n

l P state atoms into states which are nearly degenerate with the n

x P state but

which are less heavily populated by direct excitation.

Thermal energies are available to overcome small energy

deficiencies. Lees (1932) observed a change with pres- sure in the shape of the helium 3 D excitation function,

with the addition of a component similar in energy

dependence to the 1 P excitation functions, this component increasing in size with pressure. He also observed an increase in the volume from which light from 3 D

levels was emitted. A comparable increase was observed

in the volume in which radiative decay from X P levels occurred.

Wolf and Maurer (1940) irradiated an isolated sample of helium with helium radiation excited by electron

impact. The sample should absorb only in lines con- nected to the ground state. However, they observed

emission from the sample corresponding to transitions

from states not optically connected to the ground state including 3D, 1 D, 3 P, 3 S, and 'S states. These effects were explained in terms of direct excitation transfer,

providing a convincing confirmation of the excitation

transfer theory put forward by Lees and Skinner.

A graphic display of the severe distortion of excitation functions by excitation transfer at moderate pres- sures was obtained by Lin and St. John (1962). Their

results are reproduced in Fig. 56. There have been a number of investigations of the

mechanics of excitation transfer. Gabriel and Heddle

(1960) studied the pressure dependence of helium

lines excited by electron impact. They noted that in view of the spin conservation rule collisional excitation

transfer between singlet and triplet systems at lower

levels should be improbable, and they concluded that

the increase of population of 1 D and 3 D states with

pressure was probably from excitation of higher n

x P

states by way of collisional transfer followed by cas- cading.

St. John and Fowler (1961) arrived at the same conclusion independently. They noted that the spin conservation rule based on LS coupling, which should pre- clude intersvstem collisional transfer at low values of

the principle quantum number, does not rigorously

prevent intersystem transfer for the highly excited

states for which LS coupling is weak. Furthermore

the 3

3D level was observed to be particularly sensitive

to collisional transfer, and by comparing pressure

dependences of different lines they concluded that the

dominant transfer mechanism is from l P to 3 F states at levels n = 4 through 15. Lin and Fowler (1961) suggested that a moderately strong selection rule AL = 2 applies to excitation transfer. Lin and St. John (1962)

showed that the pressure dependence of the 4

l D state population as well as of the 4

3D state population is consistent with this theory. Kay and Hughes (1967)

used time-resolved spectroscopy to study the apparent

lifetimes of the n'P, 3

l D, 4

1 D, and 3

3D levels. They
Fig. 56. The sensitivity of the helium X D and Z D excitation functions to excitation transfer effects

is illustrated in these curves obtained by Lin and St. John ( 1962)

.

The abscissas represent electron energy, 0-500 eV. Operating pres- sures are indicated in millimeters

of mercury.

concluded that partial breakdown of LS coupling occurs

for n = 4 and is essentially complete for n> 6. They

find that n

l P—+nF cross sections go approximately as n

4

, and that the reverse process is about ten percent of

the forward process.

It has been suggested that another mechanism for pressure dependence may be important near threshold.

Bogdanova and Geitsi (1964) have discovered that

structure in the shapes of some helium excitation functions near threshold is pressure-dependent and, furthermore, may depend on the presence of impurities. Figure

57 shows excitation functions for the helium lines at 4713 A (43 S—>23 P) and 5876 A (33£>-+23 P) with

mercury impurities present in various concentrations.

The helium pressures are about 10-2 Torr for these

curves. The threshold peaks are absent in pure helium,

but are evident with 10~4 Torr of the mercury impurity.

Krypton and hydrogen impurities produce similar

effects. Bogdanova, Bochkova, and Frish (1964) have suggested that the threshold pressure dependence is due

to the Hornbeck-Molnar process in the following

sequence:

He+e->He*(A)+e

He*(A)+He—*He2++e

He2++e->He*(B)+He. (157)

The last step in this process is rapid for slow electrons,

which are available from excitation processes due to electrons of near threshold energies. The relationship

of the impurity component to this proposed process is not clear, but it has been noted that the effective impurities have relatively high elastic scattering cross sections for slow electrons and may help confine the low

energy electrons to the vicinity of the electron beam.

Bogdanova and Marusin (1966) have shown that the

pressure-dependent component of the excitation function is delayed in time ( > 10~7 sec) with respect to the

pressure-independent component.

In response to this interpretation, Heddle (1967)

has discussed these threshold effects in terms of space

charge due to an intense electron beam. In general,

the space charge fields may cause a washing out of structure in the threshold region of an excitation function. The introduction of atoms of low ionization potential may lead to space charge neutralization and

permit the observation of structure. Heddle claims that

this interpretation is consistent with all available experimental data. He suggests that the time delay observed

by Bogdanova and Marusin corresponds to the time

necessary to accumulate a positive ion space charge to neutralize the electron space charge. Final resolution of

this question will require some additional laboratory

work since the evidence is sketchy, but Heddle’s ex- planation, presented with quantitative justification, is much less speculative than the suggestion of Bogdanova

and Geitsi.

Multiple scattering of electrons is another pressuredependent-type process which can affect excitation

measurements. Elastically and inelastically scattered

Fig. 57. Threshold behavior of the helium excitation function Q

in arbitrary units for the 4713- A line and the 5876- A line in pure

helium at 10-2 Torr (curves 1) and with partial pressures of mercury ranging from 1X1CF6 to 1X10-3 Torr (Curves 2-6).

The electron beam current density was 7X10~4 A/cm2 and the

effective width of the electron energy distribution was about 0.3 eV. To improve the display the curves are displaced successively

greater distances upward from the abscissas. [Bogdanova and

Geitsi, 1964].

bh
electrons could in principle destroy the spatial defini- tion of the beam and thus cause photometric complications. Such scattered electron current would not have

the same axial symmetry as the primary beam so that

the state of polarization of radiation excited by scattered

electrons would be different from that due to the primary beam. The inelastically scattered electron current, and ionization current if large, would produce

excitation characteristic of lower energies than that of the primary electron beam, so that a distortion of the

excitation functions would occur. These effects would be pressure-dependent, and therefore are not important in any pressure range where the

excitation function can be shown to be independent of pressure. Since the secondary processes are energy

dependent the safe pressure regimes may be expected

to vary with electron energy, however.

The number of atoms per cc at 10~3 Torr and 293°K

is ~3.8X1013

. Consideration of typical elastic and in- elastic collision cross sections indicates the extent to which gas scattering could be a significant problem at pressures commonly used. The peak of the cross section

for ionization of helium is about 3.5X10-17 cm2 at ~d00 eV (Kieffer and Dunn 1964). Other inelastic processes in helium total approximately 3X10-17 cm2 according to Gabriel and Heddle (1960). The probability for inelastic scattering of a 100-eV electron per

centimeter of helium at 10-3 Torr is then of the order of only two-tenths of one percent.

However, for some other atoms the peak ionization

cross section is an order of magnitude larger. Excitation cross sections may also be larger. Krypton, xenon, mercury, and the heavier alkalis are among the atoms

having large ionization cross sections, according to Kieffer and Dunn. The electron loss per centimeter

through inelastic processes might then amount to

several percent. One possible result is a significant atten- uation of the electron beam before it reaches the collector, resulting in an absolute error as well as a distortion. Another possible error is the distortion due to excitation by the slow inelastically scattered electrons.

Magnetic confinement of radially scattered electrons

could help make this effect important.

In the literature there are many examples of measure- ments demonstrating the sensitivity of excitation functions to pressure-dependent effects. Effects of imprisonment and excitation transfer more or less comparable to those just discussed occur for excitation functions of any atomic species, and a demonstration of the dependence of excitation functions on pressure must be a part

of any valid measurement. Much more useful to the

reader than the usual statement that signal is proportional to pressure, is a presentation of the apparent

excitation as a function of pressure, preferably at

several electron energies. In order to illustrate the nature of effects observed in different lines we show in Figs. 58 and 59 results obtained in helium by Heddle and Lucas (1963). Figure

58 shows the rapid increase with pressure of the magnitude of the apparent excitation function for several

helium lines. For 3'P excitation the increase is primarily

due to imprisonment of 3

l P—+VS radiation. Collisional

transfer loss of 3

] P atoms is believed to be negligible

and the 3'P population at increased pressure is controlled by the probability for the 3‘P—>2 X S transition.

At higher principal quantum numbers, however, the

excitation transfer collision may be more nearly competitive with the radiative decay loss. This would result

in a somewhat shorter lifetime with conversion of 'P

atoms into other angular momentum states. We note

that the primary effect observed would be a change of magnitude.

Of the other excitation functions considered in Fig.

58, the 4

3 D—>23 P function is most pressure sensitive.

This observation is consistent with the suggestion of

St. John and Fowler (1961) and the theory of Lin and

Fowler (1961) to the effect that the 3D states are heavily

fed by excitation transfer to the 3 F states with cascading

to the 3 D state. The 42D—J>2 l P function is subject to the same type of feeding through collisional transfer to the higher X F

states, but the pressure dependence is relatively less because the 4

l D direct excitation cross section is larger

than that for the 4

3 D state (Lin and St. John, 1962).

The 4

3 A—>23 P and 33 P^>23 S lines are relatively insensitive to pressure as indicated in Fig. 58. Presumably

this insensitivity occurs because the 4

3 S and 33 P states are not strongly populated by cascading from the 3 F

states. The 4'S— line is seen in Fig. 58 to be quite

pressure sensitive, as would be expected since the 'A

states are fed by cascading from the X P states and therefore the apparent cross section would show the indirect

effect of imprisonment of resonance radiation. Collisional transfer of excitation is not believed to play a

significant role in the pressure sensitivity of this excitation function because of the energy difference between

the 4' S and 4

l P states. The data on polarization of radiation show the same

general sensitivities to pressure as do the excitation

function data. The curves shown in Fig. 59 indicate

that the contributions from secondary processes are

effectively depolarized. The pressure dependences of

polarization of radiation from several states, notably the

3

3 P and 4l D states, in fact, seem to be greater than for the corresponding excitation functions. This probably

occurs because the data for excitation and polarization

refer to different energies of the exciting electron.

The Heddle and Lucas pressure dependence curves

represent the behavior of the excitation functions and

of the polarization of helium lines excited at specific

electron energies. Pressure dependences of the l P and

l S excitation functions due to the trapping of resonance

radiation arise through an independent de-excitation

mechanism and therefore are independent of energy of

the exciting electrons. In the case of the 3

l P—>2[ S

function of helium, for example, a pressure-independent
correction to the magnitude may be estimated, if quoted pressures are accurate, on the basis of the theory

of imprisonment of resonance radiation. In general, effects on the final state excitation function of collisional excitation transfer depend on the

electron energy. The shape of the excitation function

for the initial state is carried over into the final state excitation function. In helium, the 1 P states which have

the largest excitation cross sections, are the major

sources for excitation transfer, and the pressure-dependent components of the recipient excitation functions

reflect the broad maxima at about 100 eV of the l P

functions.

12.2. The Effects of Polarization

Most experimental measurements of excitation functions are carried out by observing 7(90°), the intensity

per unit solid angle of radiation coming off perpendicuPressure (lorr)

Fig. 58. Pressure dependence of excitation functions for several helium lines. Electron energies used are indicated on the figure

(Heddle and Lucas, 1963)

.

Fig. 59. Pressure dependence of measured polarizations for several helium lines (Heddle and Lucas, 1963).

lar to the electron beam axis. In general, this radiation

is polarized and the angular distribution is anisotropic.

The theory of polarization of electron impact radiation

is discussed in Sec. 9. The angular distribution 1(6)

for electric dipole radiation is related to the percentage

polarization as described by Eq. (148)

1(6) =3(100— P cos2 6)1/ (300— P)

,

where 6 is the angle of observation with respect to the

axis of symmetry provided by the electron beam re- ferred to as the Oz axis in Sec. 9. I is the intensity per

unit solid angle averaged over all solid angle. The

percentage polarization P is defined by Eq. (147),

P=100(/ll -7x)/(/„+/i),

where I\\ and 7x are the intensities, observed at 90° to the electron beam axis, of the components with electric vectors parallel and perpendicular to the electron

beam axis. The polarization of radiation emitted in a transition

from a state of definite orbital angular momentum can

be expressed in terms of the cross sections for exciting

the various magnetic substates Ml• For example, in

the case of excitation of a P state, the cross section for

excitation of the \Ml\=Q substate is different from

the cross section for the | Ml | = 1 substates. The optical

transition probabilities are independent of Ml, so that

the relative values of the cross sections, Q0 and Qi,

determine the angular distribution and the polarization

of the emitted radiation.

The primary experimental data on excitation functions are related, ignoring cascading, to some combination of excitation cross sections for the magnetic substates (not to any one of them and not to the total cross section). The specific functional dependence of
the polarization on the values of Q(Ml) depends on the

fine and hyperfine structure. This dependence was dis- cussed in Sec. 9, where expressions are given for the

polarization of several lines of helium and hydrogen in

terms of Q0 and Q\. Special cases occur for excitation to the 6” states, and at threshold for excitation of any

level. For 5 states, with only the substate | Ml |=0,

the radiation is isotropic and unpolarized. At threshold

the outgoing electron carries away no angular momentum. The incident electron only has angular momentum

perpendicular to the electron beam axis. Conservation

of orbital angular momentum requires that only

| Ml j=0 states may be excited at threshold from an

initial state of zero orbital angular momentum.

The data for 1(d) can be used to obtain I if the polarization P is known. In the case of most experimental

data the angle of observation is 6 = 90° and the expression for I (0) reduces to

/ = 7(90°) [l-( P/300) ]. (158)

P(E) of the incident radiation, which is in turn a function of electron energy E. Therefore, the measured

excitation function is distorted. The incident intensity

Ii(E) may be written in terms of components I\\(E)

and I x(E) with electric vectors parallel and perpendicular to the axis of the electron beam. Then

/.•(£) =/„(E)+/x(E) (160)

and the transmitted intensity

It (E) = Tu Ili (E) + TxIj.(E), (161)

where T\\ and Tx are the transmissions of the instrument for the two components measured separately.

It is useful to write, from Eq. (147) and Eq. (160)

/,,(£) =§[/*(£)]{ 1+[P(P)/100]} (162)

/x(P)=i[/l-(P)]{l-[P(P)/100]}. (163)

Then use of Eqs. (162) and (163) in (161) leads to

Equation (158) is the basis for the definition of an

apparent cross section Qx, often used in the discussion

of experimental results. 7 bears the same relationship to the total photon-excitation cross section Q which

7(90°) bears to Qx, and

e=e^ci- (-P/300)]. (i 59) However, use of the “correction” of Qx in presenting

experimental results tends to reduce the quality of the

data, introducing the additional uncertainties of the

polarization measurements or calculations. If the primary experimental results are published in addition to values corrected for such effects as polarization and

cascading, the maximum worth of the measurement is preserved, independent of changing information about

polarization and transition probabilities. Comparisons

of theoretical calculations with experimental results

may then be based on separate calculations for the cross

sections for the magnetic substates, with subsequent

comparison with the primary experimental data for Qx

and with the primary experimental data for the polarization, taking appropriate notice of cascading effects. The most serious omission found throughout the

literature is failure to make proper allowance for instrumental polarization. The observation of the excitation

function for a spectral line usually requires the use of a prism or grating monochromator. Usually, the instrument modifies the state of polarization of the light.

This may occur, for example, at diffraction gratings

where components with electric vectors normal and

parallel to the rulings may have different effective

transmissions, and at prism surfaces where partial

reflection of obliquely incident light may result in

modification of the state of polarization.

It can be shown, quite simply, that the ratio of the

transmitted intensity to the incident intensity is not

constant, but is a function of the state of polarization

It (E)=Ii (E)h(Tu+Tx)

Tw-Tx\P(EY

T\\ + TxJ 100 _

'

(164)

The second term in the square brackets represents an

energy dependence in the ratio of the transmitted to the

incident intensity. If Tu ?±Tx and P(E) 0, the measured intensity is not proportional to the excitation

function. The fractional error in the measured excitation function [in addition to the effect described by

Eq. (158)], is AIt /It= ( T\ , - Tx) /( r, |+ Tx) • P(E)/m. ( 165)

This error can be very large, in fact, since severe instrumental asymmetry is known to occur.

It is important to notice in this connection that the

instrumental transmission coefficients TM and Tx are functions of wavelength, and that the instrumental

error will be different, in general, from one spectral

line to another. Figure 60 shows the polarization of

natural light produced by a half-meter Ebert-type

grating spectrograph blazed at 4000 A. Other examples

are given in a recent review by Heddle (in press).

Frc. 60. Instrumental polarization

(Tm&x Fml„ )/(Tm& + rmin )

of a half-meter Ebert-type grating spectrograph, calculated from measurements by G. Dunn (private communication).

hi
Various devices for depolarizing monochromators

can be applied. One method uses a partially polarizing

transmission element, such as an oblique quartz plate,

which is adjusted at each wavelength to the angle at which it exactly compensates for the instrumental

polarization. Another method relies on rotation of the

axis of polarization of the incident light by an angle such

that Tu = Tx. This method also requires readjustment

at each wavelength, and may be erroneously applied

for some longer-lived states if rotation of the axis of

polarization by stray magnetic fields is neglected

[see Eq. (171)3-

A method avoiding the anisotropy associated with

polarization is to observe the radiation, not at 90° to the electron beam, but at the “magic angle,” that angle

at which the total intensity is equal to the average

intensity over the total solid angle. From Eq. (148) we

may write this condition

I (6m) =i

or 3 ( 100 -

cos2 6M) =300-

which leads to cos 0a/= ("v(3) -1 and 6m = 54°46'. However, the polarization of light emitted at angle 6 is related to the polarization of light emitted at 90°

through the expression

P(e) [/ii(90

o

) — 7±(90°)3 sin2 9

100 /n(90°) sin2 0+/±(9O°) (cos2 0+1)

1

and this is finite at 0a/ = 54°46'. Therefore, there remain

the problems of instrumental polarization.

Burrows and Dunn (private communication) have

suggested a more complete solution: to observe at 90°

to the electron beam axis but to permit only one component of polarization, at 35°14' to the electron beam

axis, to pass into the monochromator. A Polaroid sheet

is used to accomplish this. As a result, the light incident on the monochromator is composed of two polarizations

according to the relation

I

cc I\ | cos2 6m-\-I -i- sin2 6m = 3-7| i T§7x = \1. (167)

The response is proportional to the total cross section.

Furthermore, since the radiation is linearly polarized at a fixed angle, 9 = 35° 14', the output is independent of

instrumental polarization effects. The method by-passes

almost all the problems arising from polarization and

permits a direct measurement of the total cross section,

but information about polarization is lost. Again there

is a possible error for the long-lived states due to Larmor

precession about residual magnetic fields. Measurement of polarization requires measurement

of the individual components 7|| and 7x. This is accomplished by use of an analyzer in the light beam, ahead

of the monochromator. The observed polarization P0b3 is then expressed in terms of the transmitted signals

/liTn and I±Tx, pob8/ioo=(/„r1 i-/xrx)/(/„r11 +/xrx). (168)

Use of Eqs. (162) and (163) leads to the form

p*b. =

(P/ioo)+[(rl i-r-L)/(7~ii+7~j.)]

ioo i+(p/ioo)-urll -r.L)/(rll +ri);]'

where P is defined by Eq. (147). The possibility for errors due to instrumental effects is clearly indicated

by Eq. (169). Successful measurement of the polarization P requires depolarization of the instrument at each wavelength or an explicit correction for instrumental transmission asymmetry, based on Eq. (169).

Heddle (private communication) has suggested that

depolarization of the instrument may be accomplished

using a polaroid analyzer set parallel or perpendicular

to the electron beam axis, but preceded by a half-wave

plate which may be rotated so that the analyzer passes

either I\\ or 7x through the monochromator. Both components are subject to the same instrumental trans- mission in this method and the error is eliminated.

However, a different half-wave plate is required at the

wavelength of each excitation function studied.

As has been suggested in the preceding paragraphs

Larmor precession is potentially an important source

of systematic error. Precession, about residual magnetic

fields, of atoms with nonzero orbital angular momentum

may result in depolarization, and in rotation of the

polarization axis. The Larmor precession frequency is L = gn0H/2Trh= 1.4X 106 gH cps, (170)

where n0 is the magnetic moment of one Bohr magneton,

g

is the Lande gfactor, 2irfi is Planck’s constant, and

27 is the magnetic field in gauss. Since gis approximately

unity and the terrestrial magnetic field is about 0.5 G,

the angular precession frequency, 2tL, may well be of

the order of 107 rad/sec. The mean lifetimes of a number

of excited states are 10~7 sec or longer so that significant

disorientation will occur if there is a large unattenuated

component of the earth’s field or some other field perpendicular to the beam axis. For some experimental work magnetic fields are used

to confine the electron beam. If these fields are homogeneous and aligned with the electron beam and are

large compared to the earth’s field no disorientation

occurs. For inhomogeneous fields | dH/dr | = ^ | dH/dz | and disorienting radial fields may become significant

off the axis. On the other hand, most experimental work

does not involve deliberate use of magnetic fields. In

fact magnetic shielding is frequently used to minimize

deflection of the electron beam. This is especially com- mon for measurements involving elements with low

excitation thresholds. For the electron in a magnetic

field H the radius of curvature is 3.36 (eV) 1,2/H cm,

where eV is the electron energy in electron volts. For

situations in which electron energies of interest are not

too low it is common practice to accept a small residual

field: terrestrial and stray fields of as much as half a gauss could be tolerated in some work without undue

curvature of electron trajectories.
40

Fig. 61. Rotation of the axis of polarization about a magnetic

field perpendicular to the axis of the exciting electron beam, shown

for several states of helium as a function of magnetic field. Observation is along the magnetic field. The curves are calculated from

Eq. (171) of the text.

Methods of calculating the effect of Larmor precession on polarization are readily available in the litera- ture (Ruark and Urey, 1930, p. 359). For the special case of a residual magnetic field perpendicular to the

electron beam along the direction of observation, the

axis of polarization is rotated through angle <f>, where

tan 2$ = 4xL/r. (171)

Here T is the reciprocal lifetime of the excited state. The precession is accompanied by a decrease in polarization of the observed radiation given by

P/P0 = [1+ (47r£/rj2]“1/2

, (172)

where P0 is the percent polarization in the absence of a magnetic field. The effect is illustrated in Figs. 61 and

62 showing the rotation of axis and the depolarization

of the rO-D states of helium. For states which live as long as the 6

X D states, residual magnetic fields may

cause serious errors. On the other hand, short-lived

states connecting to the ground state are not significantly affected by precession.

12.3. Electric Field Effects

Stray electric fields exist in any practical electron

beam apparatus. These may include space-charge fields due to the electron beam, especially at high currents and in magnetically confined beams; fields due to contact potentials; fields due to charges accumulated on

insulating surfaces; and penetration of fields originating

in electrostatic electron optical elements. The question

of residual electrostatic fields is a difficult and controversial one. The magnitudes of fields to be expected from space

charge in an unconfined beam can be estimated from

a simple calculation based on an infinitely long beam

of uniform density. In this case

£radiai^0.3[/MA/(eF) 1 /2](Pmm)-1 V/cm (173)

at the outer edge of the beam and has a linear dependence on radius within the beam, vanishing at the origin.

Here eV is the electron energy in electron volts, Rmm

is the beam radius in millimeters, and 7mA is the beam

current in microamperes. The volume average field within the homogeneous beam is two-thirds of 2tradiai at the outer edge. From the experimental conditions

described in the literature it is clear that radial fields of one or two volts per centimeter are commonly tolerated,

and substantially higher fields may be achieved. These

electron space-charge fields may be effectively canceled

by positive ion space charge for energies above the

ionization threshold. The problem of space charge has

been discussed in greater detail by Heddle (1967).

Jongerius (1961) has noted that a significant distortion of the electron energy scale may occur at the ioni- zation potential. This is due to the process just cited

whereby the positive ion space charge may build up to a

level at which the electron space charge is neutralized.

Within a small range of applied accelerating potential

the actual electron energy may change by more than a

volt as the negative space-charge depression is over- come. Jongerius presented this in connection with a study of the dependence of mercury excitation functions on gas pressure and electron beam density.

Bethe and Salpeter (1957, p. 284ff) have discussed

the mixing of orbital angular momentum states of

atomic hydrogen by electric fields. Because of the

degeneracy of states with different orbital angular

momenta but equal total angular momenta, very small

electric fields suffice for mixing in atomic hydrogen.

For the j = m= f, n = 3 levels electric fields must be

Fig. 62. Depolarization of radiation excited by electron impact,

due to Larmor precession, shown for several states of helium as a function of the magnetic field component perpendicular to the

electron beam. Observation is along the perpendicular magnetic

field component. The curves are calculated from Eq. (172). The

mean natural lifetime r for each of the 1 D states is indicated on the

figures.

h9
much less than Fc = 1.9 V/cm to avoid mixing. Critical

values of the field strengths, Fr , fall off roughly as nr5

, so mixing in higher n values is certainly important in any experimental apparatus. The critical field strengths

for mixing of the/ = § states is much larger because of

the Lamb shift, but even here the critical field falls as n~h and is down to ~1.7 V/cm for n = 6. In other atoms the degeneracies are removed in the

lower levels, but the very high levels are likely to be

subject to broadening and mixing by small fields. However, for most lines amenable to study, the effect of

electric field mixing on lifetimes and polarization of

lines can probably be neglected except for atomic

hydrogen.

12.4. Absolute Measurements of Optical

Excitation Functions

The determination of absolute values for optical

excitation measurements may be accomplished by

calibrating the optical system using a tungsten strip

lamp. It is important that this standard lamp be unpolarized. A calibrated lamp of known spectral radiance

may be used, or the spectral radiance may be calculated from Planck’s law if the temperature of the strip can be accurately determined with an optical pyrometer. The use of pyrometers and strip lamps is discussed

in some detail by Kostkowski and Lee in NBS Monograph 41 (1962), entitled “Theory and Methods of

Optical Pyrometry.” That paper adequately points

out the difficulties of attaining precision and accuracy

in these calibrations, and emphasizes the considerable

care and experience which must accompany the use of

strip lamps and pyrometers. Optical pyrometry is an

involved subject and is not discussed in detail here.

However, disappearing filament-type pyrometers cur- rently calibrated in a national laboratory and used with

care may give results accurate within about ±7°C.

Since pyrometers are universally calibrated on the

International Practical Temperature Scale, a correction to the Thermodynamic Temperature Scale is necessary. Radiant flux in the visible spectrum from a tungsten strip lamp, calculated on the basis of such

temperature measurements has associated errors of

about ±5%, allowing for about 2% uncertainty in the

emissivity of tungsten. This is a statement of minimum

error based on a current, high quality calibration and

skilled operation. Pyrometers not currently calibrated

may well be several times worse because of drift in

pyrometer lamps. Pyrometers calibrated through additional transfers from national standards are subject

to additional errors. There remains the problem of illuminating the optical

system to be calibrated with the same effective pattern

of radiation as is provided by the electron beam source. The usual technique is to place the strip lamp at the

position of the electron beam. The same condenser

lens may be used to focus the radiation from the strip

lamp onto the monochromator entrance slit as is used

for the electron beam source. In this case the effective

solid angle presented to the source by this lens need not

be known, but it must be small enough so that the

optics within the monochromator are not exceeded in

either mode of illumination.

The calibration is achieved by comparing the signal

output produced by the radiation from that area of the

tungsten strip which is focused through the monochromator entrance slit, with the signal produced by

the volume of electron beam from which radiation is transmitted by the entrance slit. No general statement

of the geometrical calibration factor can be given since

the geometrical analysis varies with the exact arrangement of the optical system. The electron beam and

tungsten strip are usually perpendicular to the slit, and the images do not cover the slit length. The source

lengths contributing to transmitted radiation are determined by the slit width and the linear magnification.

The tungsten strip width and Wien’s law are used to calculate the radiant flux per unit source length per

unit solid angle per unit bandwidth. Multiplication by

the monochromator bandwidth, the product of exit

slit width and reciprocal dispersions, yields radiant

flux per unit solid angle per unit source length contributing to the detector signal. This number permits the

signal output from the electron beam excitation mode to be converted to total radiant flux per unit solid angle

per unit length of electron beam. If the gas density,

polarization of the radiation, and total electron current

are known, the radiant flux emitted per unit solid

angle can be converted to an absolute photon excitation

cross section. The electron beam diameter and current

distribution are irrelevant for the geometrical analysis, as are the dimensions of the optical system and an exact

knowledge of the effective source length.

If different slit widths are used with the two sources, then the source lengths differ and the ratio of source

lengths must be accurately known. Also if removable

reflectors are used so that the calibrated lamp can be

placed virtually at the position of the electron beam

then the effect of extra reflection losses must be considered.

Several points at which measurement limitations are

likely to affect the accuracy of the absolute measure- ments are:

(1) Determination of radiant flux. This always is of

the order of ±5% in the best of circumstances and is entirely likely to be much larger. Photoelectric pyrometers, capable of more accurate results, have not yet been

applied in this field. (2) Gas density measurement. This problem has been

discussed recently by Kieffer and Dunn (1966) in a

critical review of ionization measurements. They emphasize that density measurement techniques are in a very unsatisfactory state. Determination of density to

50
within a few percent is impossible and there are indications that errors of 5 and 10% are common.

(3) Determination of angular distribution of radiation from electron beam excitation.

(4) Determination or elimination of instrumental

polarization Pinst .. (5) Photoelectric detectors usually are not uniformly

sensitive over the receiver surface. Small systematic

changes of electron beam geometry may cause errors.

12.5. Electron Beam Measurements

Kieffer and Dunn (1966) have discussed many of the

problems associated with measurements based on the use of electron beams in static gases. Much of their

discussion is relevant for excitation function measurements, and will not be repeated in this review. Accurate

electron beam measurements depend on proper precautions with respect to such effects as gas scattering,

reflection, secondary emission, and ionization, which

can cause incomplete collection of the electron beam or

false electron currents. These effects are energy-dependent so they may result in errors in shapes as well as absolute values of excitation functions, and are as important here as they are for the ionization measure- ments discussed by Kieffer and Dunn.

13. Other Reviews of Electron Impact Excitation

Among the various reviews of electron-impact excitation work which have been written, two are of special

interest here:

“The Measurement of Optical Excitation

Functions” by D. W. 0. Heddle, in Methods

of Experimental Physics, L. Marton, Ed. (Academic Press Inc., New York, to be published),

Vol. 8. “Measurements of Electron Excitation Functions” by D. W. O. Heddle and R. G. W. Keesing, in Advances in Atomic and Molecular

Physics, D. R. Bates and I. Estermann, Eds.

(Academic Press Inc., New York, 1968),

Vol. 4. The spirit of these reviews is much the same as that of the preceding sections. The authors are intent upon

elucidating the physical and instrumental effects which

must be controlled in high-quality measurements of

excitation functions.

Some other experimental reviews of interest are:

“Excitation and Ionization of Atoms by Electron Impact,” by H. S. W. Massey, Handhuch

der Physik, S. Fliigge, Ed. ( Springer-Verlag,

Berlin, 1956), Vol. XXXVI, Atoms II. “The Measurement of Collisional Excitation

and Ionization Cross Sections,” by W. S. Fite,

in Atomic and Molecular Processes, D. R.

Bates, Ed. (Academic Press Inc., New York,

1962).

These latter will serve the reader as excellent supplements to the sections to follow, in which we attempt to summarize and discuss excitation functions to be found

in the literature for atomic hydrogen, atomic helium,

the noble gases, and the atoms for which the unfilled

shells are one and two electron structures.

14. Atomic Hydrogen

The excitation of atomic hydrogen has received much

more attention from the theoretical physicist than has

the excitation of any other atomic species, due to the

relative simplicity of the system and the availability of accurate free-atom wave functions. The experimental

situation is not quite so fortunate in this respect. It is helium, rather than hydrogen, which is convenient for experimental purposes. The technical problem of producing dissociated hydrogen in the laboratory in an

adequately pure state and in sufficient concentration

has provided a barrier to experimental work with atomic

hydrogen.

Some early work on the excitation of atomic hydrogen was carried out by Ornstein and Lindeman (1933),

who recorded the relative intensities of the first three

Balmer lines as a function of bombarding electron

energies. The atomic hydrogen was produced in a Woods

discharge in hydrogen gas. Atom rich hydrogen was pumped out through a side-tube across the path of an

electron beam. Light excited in the region of electronatom interaction was studied with a spectrograph using

photographic recording. The authors recognized the

major difficulty in this method: recombination of

atomic hydrogen on metallic surfaces is so effective that

the use of electrodes to define potentials in the interaction region was precluded.

No further progress was made for more than twenty

years. In 1958 Fite and his collaborators published a

series of papers describing work accomplished using

beams of electrons and hydrogen atoms crossed in high vacuum. The excitation work presented at that time

(Fite and Brackmann, 1958) was based on the observation of Lyman-a flux from the electron-atom interaction volume. This region was shielded from electric

fields in order to avoid quenching the long-lived 2s

hydrogen atoms, which then moved out of the interaction volume in a time short compared to the mean

life of the atom. The radiation observed coming from

the interaction volume was all due to radiative decay of

the short-lived 2p state. A few months later Lichten and Schultz ( 1959) published a measurement of the 2s excitation cross section,

based on detection of free electrons ejected from a platinum surface on which the metastable atoms were

incident. Other atomic hydrogen excitation measure- ments have appeared since that time. All are crossed

beam measurements, based on improvements in high

vacuum and electronic measurement technology.

The hydrogen atom crossed beam measurements do

not suffer from the pressure dependent effects discussed

51
in the previous section. The pressure is low enough in

any of these beam experiments so that resonance radiation trapping and collisional transfer can be ignored.

The problems of distortion arising from instrumental

polarization cannot be ignored but they are easier to handle where broad band optical filters are used rather

than prism or grating monochromators. On the other

hand, there is a whole new set of problems. Signals to be

detected are indeed minute. Interaction geometries are complicated. Electron optical problems may be more

severe at the extremely low pressures required, since

pump fluids and other contaminants migrate readily

in a hard vacuum. Progress in the crossed-beam work

has been difficult and time consuming. As a result only a few excitation measurements have appeared.

There remains a need for additional work to confirm

the measurements now in the literature, and to extend

the range of information available. There is also a need

for more detailed and quantitative treatment of errors, so that the limitations of a given measurement are well

defined. A measurement is complete only if the relia- bility and accuracy of the results are stated quantitatively. Systematic geometrical and physical errors should be identified and evaluated. The statistical errors of the measurement should be stated separately

in well defined terms. The statement of a statistically

determined probable error coupled with quantitative

estimates of the magnitudes of systematic errors is valuable. A presentation of error bars two or three times

the statistical probable error in the expectation that

it covers a host of vaguely defined systematic effects provides little information.

There have been no accurate absolute measurements

of hydrogen atom excitation cross sections. [The paper

by Lichten and Schultz (1959) would seem to stand as a

contradiction to this statement. The authors estimated

<r(2.y) =0.28±0.147tUo2 on the basis of a measurement of

detector yields and other factors. However, in a note

added in proof, they pointed to a possible error and

referred to the Born approximation at approximately

40 V as the basis for an absolute scale. Thus the esti- mated accuracy of the absolute measurement is unspecified.] Normalization to the Born approximation is frequently used as the basis for quoting absolute values,

and this sometimes obscures the fact that the measure- ments are relative. The major significance of tne experimental excitation work has been to reveal the characteristics of cross sections in the lower energy range

where the conventional approximations used in quantum theoretical calculations are generally found to be

unreliable.

14.1. Excitation of H(2p)

A schematic diagram of the apparatus developed by

Fite and Brackmann (1958) for high vacuum crossed

beam work is shown in Fig. 63. The technique utilizes a chopped atom beam crossed with a direct current beam

of electrons in a field free region. An ionization chamber

Furnace-^

Fig. 63. Schematic diagram of the major elements of the crossed-beam apparatus used by Fite and Brackmann (1958) to study the excitation of Lyman-a radiation.

is used to detect the Lyman-a photons and the arrangement is such that the signal of interest must alternate

synchronously with the beam chopping frequency.

Microphonic signals as well as signals due to pressure

modulation originating with the chopped beam may

appear in addition to the signal due to Lyman-a photons

originating from primary beam atoms. Care is required

to guarantee the absence of such effects. Another problem is the geometrical variation of the

electron beam with energy. An excitation function can

exhibit the correct energy dependence only if the integral of atom density over the electron trajectory is independent of electron energy. In a beam experiment

it is necessary to ensure that all electrons pass through

the full depth of the atom beam at all energies. It is also important to evaluate the effect on the integrated

atom density traversed, of variations in the angular

divergence of the electron beam. Further, consideration

is needed of possible changes in photon collection and

detection efficiency with such changes as may occur in

the distribution of electron trajectories. If control of this

geometry is not demonstrated in some way the value

of the measurement is degraded. Structural details of the

cross section might still be of interest but a comparison

with theoretical shapes over a wide energy range would

be of limited value. Other problems include atom beam

purity, and the degree of spectral isolation of the radiation under investigation from radiation in other atomic

and molecular lines which may be excited.

In Fig. 64 are shown excitation functions for the

Ip—>ls transition in atomic hydrogen, determined by

observation of Lyman-a flux produced in a region free

of electric fields, emitted in a direction perpendicular to the electron beam axis. The experimental results displayed include the results of the original measurement

by Fite and Brackmann (1958), but exclude points near threshold now believed to be in error because they

did not reproduce in subsequent careful measurements

by Fite, Stebbings, and Brackmann (1959). The results

of the latter set of authors have also been included, but

52
Fig. 64. Relative measurements of the cross section for excitation of the 2/>—>ls transition in atomic hydrogen [Fite and

Brackmann (1958), O; Fite, Stebbings, and Brackmann (1959),

A; and Long, Cox, and Smith (1967), compared with the Born approximation result for Qj. (see text and Table XIII)

without cascading (Curve 1) and with cascading (Curve 2), and

compared with the result of a ls-2s-2p close coupling calculation

for Qj. (Burke, Schey, and Smith, 1963) (Curve 3) without cas- cading. Error bars shown for Fite et al. (1959) are confidence

limits. Fite and Brackmann (1958) give ±12% confidence limits. Long et al. (1967) results are limited by systematic errors of about 3%.

we have renormalized them by a small factor to a better

fit with recent results of Long, Cox, and Smith (1967).

Long, Cox, and Smith attempted to improve on the

results of Fite et al. through a strenuous effort to reduce

systematic and statistical errors to the order of a few

percent at most. In the course of this work they found

that the transmission of the lithium fluoride windows

used on the Lyman-a detectors varied with electron

energy, becoming more opaque at higher electron energies and with longer continuous exposure. The induced

absorption decayed with a time constant of several

seconds. Heath and Sacker (1966) had shown that

lithium fluoride windows become opaque to Lyman-a

when bombarded with 2-MeV electrons. Heath and

Sacker also found that magnesium fluoride windows are much less sensitive. Long, Cox, and Smith found that

insertion of a magnesium fluoride window ahead of the

detector eliminated the time-history-dependent effect,

while insertion of another lithium fluoride window did

not. The conclusion to be drawn is that the lithium

fluoride window transmission is sensitive to x rays produced by electrons with energies of only about a hundred volts (no detailed study of the phenomena was

carried out). These latter workers expected to obtain

results different from the earlier results of Fite et al. when the x-ray effect was removed, but as shown in Fig. 64 the results are in complete agreement, within

experimental errors, except for the small renormalization already mentioned. It appears that the effect was

minimized but not identified in the work by Fite et al. We suggest that it may have been responsible for the

original failure near threshold in the Fite and Brackmann results, which has not otherwise been explained.

Moderately high-energy electrons may have been pres- ent in the accelerating stages of their electron gun.

This confirmation of the observed 2/>— excitation

function permits a more definite conclusion that the

theoretical methods described in Sec. 2.15 do not lead

to an accurate description of the 2p—>ls excitation

function in the energy range below about 50 to 60 eV

except in the immediate vicinity of threshold. In Fig.

64 we have included the cross sections Q± calculated

using the Born approximation and the \s-2s-2p close

coupling approximation [Burke, Schey, and Smith

(1963)3 together with the formula

Q± = 0.918<32p+0.246«32po (174)

derived by Burke and Seaton ( 1960) . The experimental

results are normalized to the Born approximation, in- cluding a small allowance for cascading, at 200 eV.

The measurements and the values calculated by the

Born approximation seem to fit very well above about

120 eV, and it appears that the close coupling results

may fit the measured values down to about 60 eV, a

distinct improvement. However, below 50 eV the departure of the close coupling results is quite striking.

This discrepancy leads to the speculation that the

inclusion of the higher states in the close coupling

scheme might reduce the cross section in this region.

The calculations become extremely complex and expensive, and there is as yet no conclusive theoretical demonstration of what the effect of including coupling to all states might be. Burke (1963) showed that inclusion of

close coupling to the is and ip states reduced the value

obtained for the total cross section at 16.5 eV to 0.907

from the value of 1.094 obtained from a ls-2s~2p close

coupling calculation. This is significantly higher than

the experimental results on the basis of normalization

to Born approximation at 200 eV.

Some intensive calculations have been carried out in

the immediate vicinity of threshold by Burke, Ormonde,

Taylor, and Whitaker (1967). These authors show that

inclusion of close coupling to the n = i level does not

produce much of a change in the cross section at energies

in the range from the n = 2 threshold up to an energy

somewhat below the n — i threshold. However, it may

be that at higher energies the cumulative effect of coupling to all the discrete levels with n>i and the con- tinuum states would account for the difference between

the \s-2s-2p close coupling approximation results and

the observed behavior of the cross section.

Attempts to determine the polarization of Lyman-a

radiation by comparing the excitation functions at two

different angles (for example, 90° and 45° to the elec- tron beam axis) have failed to yield reproducible and

physically plausible results. The calculation of polarization from intensities at two angles involves the small

differences in the intensities. Relative errors become

very large and small systematic geometrical errors,

53
such as result from the use of rather extended source volumes, have large effects. Long, Cox, and Smith (1967) have concluded that

the major difficulty derives from the sensitivity of

lithium fluoride windows to low-energy x rays. The

window transmissions at a fixed position are found to vary with electron energy and with time of exposure.

Furthermore, the behavior of the windows would vary

with position since the exposures are dependent on

position. Therefore, consistent measurements of polarization cannot be obtained using lithium fluoride-covered

detectors.

A new technique, direct analysis of polarization by

reflection at the Brewster angle from a sheet of lithium

fluoride, seems promising, and preliminary results have

been obtained by Ott, Kauppila, and Fite (1967).

Chamberlain, Smith, and Heddle (1964) studied the

behavior of the 2 p

excitation cross section in the immediate vicinity of threshold. Working with a carefully

controlled electron beam geometry they obtained the

experimental points for Qx shown in Fig. 65. The ac- curacy of these results depended on knowledge of the

energy distribution of the electron beam. The energy

distribution was obtained by analyzing the axial portion

of the beam with a spherical retarding analyzer. The

assumption was made that this energy distribution

applied over the whole of the electron beam. Modification of beam energies by space charge fields was small

with the 1.5-mA beam used. Also shown in Fig. 65 is'

a

curve based on recent threshold calculations for Q± by

Fig. 65. Relative measurements of the cross section for excita- tion near threshold of the 2p—>ls transition in atomic hydrogen

[Chamberlain, Smith, and Heddle (1964) , A; and Fite, Stebbings,

and Brackmann (1959), O] compared with a composite (Burke,

Taylor, and Ormonde, 1967) of a 3-state+20 correlation term

calculation, below 11.4 eV, and a 6-state close coupling calculation,

above 11.4 eV (Burke, Taylor, Ormonde, and Whitaker, 1967).

The dashed curve was obtained by folding the electron energy

distribution from the work of Chamberlain et al. into the composite

curve. Error bars for Fite et al. are confidence limits. Error bars

for Chamberlain et al. are probable errors.

Fig. 66. Intercomparison of relative measurements for excita- tion of hydrogen atoms into the metastable 2s state, including

cascade contributions [Hils, Kleinpoppen, and Koschmieder

(1966), ; Lichten and Schultz (1959), solid line; Stebbings,

Fite, Hummer, and Brackmann (1961) A]. Hils et al. show error bars equal to twice rms errors plus 15% systematic errors. Stebbings et al. show confidence limits. Lichten and Schultz quoted a 15% error for the total cross section, the rms sum of all systematic errors and three times the probable statistical error. The dashed curve is the exchange cross section obtained by

Lichten and Schultz, shown in the ratio to the total cross section determined by them to within 5%.

Burke, Taylor, Ormonde, and Whitaker (1967). The

dashed curve shows the result of folding the experimental energy distribution from Chamberlain et al. into the calculated resonant threshold. The experimental points have been scaled on the abscissa and

shifted on the ordinate, this being allowable within

reasonable limits since no accurate determination was made of the contact potentials involved. The resulting

fit is quite good, noticeably better than can be achieved

with a step function cross section or using the earlier

results of Dambrug and Gailitis (1963).

Also shown in the figure are some of the points obtained by Fite, Stebbings, and Brackmann (1959) from

their remeasurement of the threshold behavior. While

their results do not reveal the threshold structure, they

are consistent with the results of Chamberlain, Smith,

and Heddle when account is taken of a larger energy

distribution and larger error bars.

14.2. Excitation of H(2s)

Three measurements of the excitation of the 2s state

of atomic hydrogen are shown in Fig. 66. We have re- normalized the three available experimental results to

facilitate direct comparison of the shapes. These are seen to be in fairly good agreement, considering experimental errors. Three aspects of the experimental results

deserve particular notice: (1) the structure in the verylow-energy range best shown in the results by Lichten

and Schultz (1959) ; (2) the gross structure through the

medium-and high-energy range found by Hils, Kleinpoppen, and Koschmeider (1966) and which seems to be

reflected in the work by Stebbings, Fite, Hummer, and

Brackmann (1960, 1961); and (3) the over-all energy

54
dependence of the measurements of Stebbings et al. and of Hils et al., which departs to a remarkable extent

from that predicted by the Born approximation.

The Lichten and Schultz measurement, already

briefly described, differed in basic principle from the

other two measurements. The latter two relied on

electric field quenching of the metastable atoms in free space within the field-of-view of a Lyman-a detector.

In both cases the metastables were allowed to move out

of the excitation region down the beam into a quench

region.

The Stebbings, Fite, Hummer, and Brackmann meas- urement was a point-by-point comparison of the 2s and

2 p

excitation cross sections. The detector was mounted

on a trolley so that the field-free excitation region and a down-stream region, in which a quenching field was

applied, could be sequentially observed. The results were presented as a 2s excitation cross section. The

reliability of their result is dependent on the quality of the 2p

excitation cross section used, and this was not

explicitly described.

The method of measurement by Hils, Kleinpoppen,

and Koschmieder was similar to that used by Stebbings

et al. An iodine counter with lithium fluoride window

was used to detect the Lyman-a emitted from a region

in which a quenching field was applied to the hydrogen

beam. No chopper was used in this case, the application

of the quenching field serving to differentiate between

the signal and background.

The paper by Stebbings et al. describes in considerable

detail the care taken to minimize errors due to the

angular distribution of metastable atoms resulting from

recoil, and due to spatial extent of the quenching field. They studied the angular distribution with which

metastables came from the electron-beam-atom-beam

intersection and found that the fraction of atoms with

recoil angles much greater than 20° was very small at all electron energies. Their apparatus was designed to subject all atoms within a 45° angle to quenching within the

field of view of the Lyman-a detector. The only com- ment we can add is to express the reservation that the

field of view of the type of detector used is not homogeneous, and that a systematic error is to be expected

from the variation with electron energy of the spatial

distribution of metastables. At low energies the source volume would be more extended than at high electron

energies.

It may also be noted that the soft x-ray mechanism

for modifying lithium fluoride window transmission

mentioned in the preceding section could possibly have

been operative in the measurements of Stebbings et al., and of Hils et al. This effect is somewhat more likely in

the former case because the measurement of 2p

excitation used for normalization involved direct observation

of the region traversed by the electron beam. A depression of the 2p

signal due to induced opacity of the

lithium fluoride could have led to a spuriously high

relative value for 2s excitation at higher energies.

Stebbings et al. do not present their primary data for

the 2p and 2s excitation observed in this work.

The departure of the measured 2s excitation cross sections from the Born approximation is so striking that

normalization of experimental results to the theory at high energy is questionable. The usual procedure has

been to normalize to the Born approximation plus a cascading correction at the highest energies for which

measurements are available. The problem here is that

the observed cross sections are so structured that one is reluctant to trust such a normalization. Hils et al. find a good fit over the range from 200 to 500 eV, but there

is an abrupt departure from the Born result below 200

eV which does not occur for 2p excitation.

Stebbings et al. found a good fit over the range from

400 to 700 eV. We would conclude from consideration

of their error bars that a reasonable fit is possible from

150 to 700 eV, which results in a higher normalization,

but this normalization is inconsistent with Hils et al. at lower energies. Stebbings et al. chose to normalize via

the 2p

measurement to the Born approximation for 2 p

excitation. The measured 2p excitation fits the Born

approximation to a much lower energy than is the case

for 2s excitation, and there is less uncertainty about the

cascading correction for the 2p cross section. On this

basis, Stebbings et al. find their results for 2s excitation

in the 400- to 700-eV range lie about 50% higher than

the Born cross section plus cascading. Their error bars

fail to overlap the Born result at the three points in this range.

Fig. 67. The relative cross section near threshold for excitation

of hydrogen atoms to the metastable 2s state, measured by Lichten and Schultz (1959) , represented by the bars, is compared with

the results of close coupling calculations. The long dashed curve

is the result of folding an electron-energy distribution of Gaussian

shape and 0.2 eV width into the theoretical curve. The lower set of curves are the corresponding results for the exchange cross

section. The relative height of the two folded cross sections is consistent with the ratio determined by Lichten and Schultz. (Burke, Taylor, and Ormonde, 1967).

55
Either of the above methods of normalization leads

to a value for the peak near threshold between 0.1 and

0.157rao2

. In Fig. 67 we show the experimental results

in the threshold region, due to Lichten and Shultz,

in comparison with recent close coupling calculations by

Burke, Taylor, Ormonde, and Whitaker (1967). It is the suggestion of Burke et al. that normalization to the

peak at about 11.7 eV may be valid. This is based on the

observation that inclusion of the higher states has a

small effect on the magnitude of the cross section at the

peak, where the value is ~0.227rOo2. This normalization

of the measurements leads to values at 500 to 700 eV

which are about twice as large as predicted by the Born

approximation. This measured peak-to-high energy

ratio is inconsistent with that predicted by the close

coupling calculation, which presumably converges to the

Born approximation at high energies.

In the case of 2p excitation the measurements of the

magnitude at threshold relative to the higher energy

measurements, by S. J. Smith (1965), are consistent

with the results of the close coupling calculation (Burke,

Taylor, Ormonde, and Whitaker, 1967). The 2s excitation cross section is in a relatively unsatisfactory state,

and the theoretical and experimental results must both

be suspect until the situation is clarified by further work.

14.3. Excitation of Balmer Lines of Hydrogen

Recent preliminary measurements by Kleinpoppen

and Kraiss (1967) appear to have superseded the meas- urements of Ornstein and Lindeman (1933) for the

Balmer-a line. The details of this relative measure- ment have not been published in full, but the principle

difficulties lie in the necessity for differentiating the

electron induced signal in the atom beam from the

background of continuous radiation from the hydrogen

disssociator. The Balmer-a excitation function shown

in Fig. 68 includes contributions from populations in

all angular momentum states of the n = 3 level. The

peak at threshold may be attributed to excitation of the

3s and 3d states while the broad maximum is mainly

due to 3p excitation.

Figure 69 shows the measured optical polarization of

the Balmer-a lines excited by electron impact (Kleinpoppen and Kraiss, 1967) . The fact that these observations include contributions from all available angular

momentum states and that these states are sensitive to mixing by electric fields of magnitudes which may

easily be produced by space-charge effects and field penetration, means that an interpretation of these observations in terms of excitation of the states of different

orbital angular momenta cannot be quantitative.

15. Excitation of Helium Positive Ions

The excitation of He+ (l.s) differs from the excitation

of atomic hydrogen in that the nuclear charge and mass

are different. The principal physical difference is the

influence of the long-range Coulomb field on the motion

of incident electrons. The incident electron gains kinetic

energy as it approaches the ion with the result that the

25—Ts excitation cross section is finite at threshold in

the simplest Coulomb-Born approximation. For atomic

hydrogen the finite threshold results from coupling to other states. Dance, Harrison, and Smith (1966) have

measured the cross section for excitation of He+ from

the ground state He+ (l5) to the metastable state He+ (25) using crossed charged beams in high vacuum. The experimental technique required was quite complicated because of the necessity for discriminating between the true excitation signal and several sources of

background signal. Modulation techniques were used,

but these were not capable of discriminating against the

effects of space-charge modulation of one beam by the

other. The correction for this effect was complicated by

a dependence of the modulated ion beam geometry on

the quenching field used for metastable detection.

The experimental results are shown in Fig. 70. In- cluded in this figure is a trial cross section from the

paper by Dance et al., chosen to give a good fit to the

experimental points near threshold when folded with

the electron energy distribution. The threshold points

have been omitted from the figure for clarity. The ex- perimental result is compared with the result of a close

coupling calculation by Burke, McVicar, and Smith

(1964a). There appears to be a serious discrepancy between the threshold behavior predicted by the close

coupling calculations and that found experimentally.

16. Helium

Most of the existing helium excitation function data

have been for transitions of the type n(S, P, D)—

2(5, P) ranging in wavelength from 2945 to 10 829 A.

Transitions to the ground state are in the range from

500 to 600 A, and are outside the range of conventional

spectroscopy. Transitions terminating on levels of

principle quantum number n>3 are generally at wavelengths longer than 10 000 A, where photographic and

photoelectric detection methods are insensitive. A few

such lines are at wavelengths in the range 8000 to 10 000 A, but have not been studied, presumably be- cause of low intensities due to comparatively small

branching ratios. For the same reasons there are no observations of transitions corresponding to increasing

orbital angular momentum, except, of course, those re- sulting from excitation of 5 states. As a result there has

been no more than one observed excitation function

corresponding to each level of helium. It is understood

that excitation functions refer to lines resulting from

excitation of ground-state atoms.

16.1. Helium n

l P-^21 S Transitions

Figure 71 presents relative measured apparent excitation functions for the 3

l P—>2l S transitions at 5016 A,

taken directly from the published tables or figures. All

observations were made at 90° to the electron beam axis.

56
6

-

n : ) : : i ! I I I ! 1-1.1 -J !

1

10 20 50 100 200 300 EfeV)

Fig. 68. Excitation function of the hydrogen Balmer-a line corrected for cascading. The errors are three times rms errors. (Kleinpoppen and Kraiss, 1967).

Fig. 69. Measured polarization of the hydrogen Balmer-a line as a function of electron energy. The errors are three times the rms errors plus 2% systematic errors. (Kleinpoppen and Kraiss,

1967).

Fig. 70. Measured excitation function for He+ (2s) from the He+ (l.s) ground state (Dance, Harrison, and Smith, 1966) compared with the result of a close coupling calculation by Burke,

McVicar, and Smith (1964a) to which has been added an esti- mated cascade contribution (Curve 3). Curve 1 is a trial cross

section chosen to give a good fit to the experimental points when

folded with the electron energy distribution. Threshold points

have been omitted from this figure. Curve 2 is derived by sub- traction from Curve 1 of estimated cascading. The error bars repre- sent 90% confidence limits.

One of the major factors contributing to the systematic differences among the 3'P excitation functions

shown in Fig. 71 is the polarization of the directly ex- cited component of the 3}P—>21 S line. The sources of such systematic differences have been discussed: distortion due to instrumental polarization, and pressure

depolarization occurring to different degrees in the

different measurements.

As has been mentioned, this line is extremely pres- sure sensitive due to imprisonment of radiation in the resonance line, 3

1 P—>1 l S. This should not have a drastic

effect on the shapes of the excitation functions. The

major effect of imprisonment is an increase in the in- tensity of the 3

l P—*21 S line, due to effective modification of the lifetime of the 3

l P state and of the branching

ratios. Some distortion of shapes due to imprisonment

may occur through modification of cascading patterns,

since transitions to the ground state are suppressed.

Imprisonment of nlP—^V-S radiation, where w> 5, will

increase the cascade contributions to the 3

l P population from higher n

x P states through 1 S and l D states resulting in deformation of the 3

l P—>2l S excitation

function near threshold. Away from threshold the dis- tortion will be small because of the similarity of the n

l P functions. Although collisional excitation transfer at the n = 3 level may not be important, excitation

transfer at higher levels may have some effect on the

pattern of cascading into the 3

l P state. Figure 72 shows

an estimate of the percentage cascade contribution to the 3

l P—>2X S line in the limit of low pressures. In this

simplified calculation the n

l P functions were assumed

to have the same threshold for all n. Figure 72 also

shows an estimate of cascade contribution at higher

pressures where imprisonment is complete, but does not

take into account the possible effects of excitation

transfer. We can estimate the extent to which the anisotropies

in the radiation pattern may lead to the systematic

differences in the excitation functions shown in Fig. 71. The measurements, carried out at different pressures, were subject to different degrees of pressure depolarization. Polarization and anisotropy are related in Eq.

( 148) , which simplifies at 6 = 90° to

</(«, E) )av = I(90°, E) { 1 — [P(£)/300]) , (175)

where we have indicated dependence on energy E. Since

polarization is energy-dependent, pressure depolarization causes a distortion in the shape of the apparent

excitation function at 0 = 90° from that which would be

obtained at zero pressure. A comparison of the various 3

X P-+2X S excitation

functions which is more meaningful than that shown in Fig. 71 can be attempted by calculating (1(6, E) )Av for each of the measured excitation functions 7(90°, E).

This requires a knowledge of the polarization P(E).

For this purpose we have used the measurements of

Heddle and Lucas shown in Fig. 73, where polarization

measurements by several authors are compared. We

have again referred to the results of Heddle and Lucas

57
E (eV)

Fig. 71. Measured excitation functions for the 5016- A. helium line (31 P—>21 S), normalized to unity at 200 eV: St. John, Miller, and

Lin (1964), 5.9 n, Curve 1 and ; Lees (1932), 44 n, Curve 6 and + ; Thieme (1932), 5 /x, Curve 2 and A; Heddle and Lucas (1963)

,

0.25 ix, Curve 3 and O; Yakhontova (1959), Curve 5; Zapesochnyi and Feltsan (1965), <0.04 tx, Curve 7; and Smit, Heideman, and

Smit (1963), Curve 4 and X.

for the variation of polarization with pressure, and have

applied an appropriate polarization reduction factor

from Fig. 59 for each of the excitation functions. The

resulting estimates of 1(E) are shown on a relative

basis in Fig. 74.

Fig. 72. The cascade fractions for the 5016-A helium line

(3! P—>21 5) are estimated for the optically thin (1) and optically thick (2) cases. Excitation functions from Heddle and Lucas

(1963), absolute values from Gabriel and Heddle (1960), and

helium transition probabilities from Wiese, Smith, and Glennon

(1966) are used as the basis for the estimate. Excitation functions and cross sections are assumed to scale as rrz

, where n is the principal quantum number, with no adjustment of threshold values. For the optically thick case, transitions to the ground state are suppressed.

58

The polarization corrections to the curves by Lees;

Thieme; Yakhontova; St. John, Miller, and Lin; and

Smit, Heideman, and Smit were small (~5%) since

they were very largely depolarized at the pressures used.

These pressures ranged from about 5 to 44 ju. It is particularly useful to intercompare these five curves. Since the radiation is almost completely depolarized,

instrumental polarization errors should be small. Furthermore, as indicated earlier in this section, the effects

of pressure on the shape of the 3

1 P excitation function

are not expected to be serious. The intercomparison of

these five curves, then, should provide an indication of

the scale on which other types of errors may occur. The results of St. John et al. at low energies, and of

Thieme at higher energies provide the largest departures from what would otherwise amount to a reasonable

degree of consistency. For the St. John, Miller, and Lin

curves in this and other figures of this section on

helium, we have used tabular data published in Miller’s

thesis ( 1964) . These data were taken from the contin- uous scan chart records of the excitation functions

published at St. John et al. The quality of the chart

records is not adequate to support a very accurate

determination of threshold behavior for any of the

helium excitation functions. The emphasis was on
E (eV) E(eV)

Fig. 73. The results of measurements of polarization of four helium lines excited by electron impact are intercompared [Heddle

and Lucas (1963) , Curves 1 and ; McFarland and Soltysik (1962) , Curves 2 and A, except \

X D which is McFarland (1967) ; Moustafa

Moussa (1967), Curves 3 and O; and Dolgov (1959), Curves 4.]

behavior over a wider energy range (0-500 eV) . In fact, a detailed comparison of excitation functions for other

helium lines, singlets, and triplets, shows that Miller’s

data is characterized by smaller initial slopes, and for the more resonant excitation functions, lower peaks

shifted to higher energies than for the results of other

workers, some of which were particularly concerned

with threshold behavior. The shifts are so pronounced

that they cannot easily be explained as due to electron

energy distribution or space charge effects. Thieme’s excitation function differs from all the

others in that his results fall off much more rapidly at the higher energies. Again this is characteristic of all Thieme’s results for the singlet and triplet lines of helium. We have no explanation for this difference.

However, the Thieme high-energy dependence stands

alone against the fairly consistent results obtained by

Yakhontova, Lees, St. John, et al., and Heddle and

Lucas. Furthermore, a recent experimental result by

H. R. Moustafa Moussa (1967), primarily concerned

with high energy dependence, exhibits a much smaller

slope than Thieme’s results and is much more nearly

consistent with the results of the other authors just

cited.

Moustafa Moussa also showed that the high-energy

slopes of the l P excitation functions are determined by

the optical oscillator strengths, and that his own results

for the excitation functions are consistent with the

theoretical predictions.

The Heddle and Lucas and the Zapesochnyi and

Feltsan curves in Fig. 71 were obtained at such low

pressures that the full polarization correction is re- quired. While this correction brings the front end of the

Heddle and Lucas curve fairly well into line in Fig. 74,

the tendency of their curve to dip at medium energies can reasonably be assumed to result from inaccurate

polarization measurements at these values. Heddle and

Lucas summarized their polarization measurements by

drawing a straight line going to zero at 200 eV. The

scatter of the data is such that other choices are possible.

The comparison of Fig. 74 suggests that their polarization values below 65 eV are rather good, but that their curve should fall more rapidly at higher energies. In this

medium energy region the data of McFarland and

59
Fig. 74. The excitation functions shown in Fig. 71 are corrected for the polarization anisotropy using the polarization data and pres- sure depolarization data of Heddle and Lucas ( 1963) [St. John, Miller, and Lin, Curve 1 ; Thieme, Curve 2 ; Heddle and Lucas, Curve 3

Smit, Heideman, and Smit, Curve 4; Yakhontova, Curve 5; Lees, Curve 6; and Zapesochnyi and Feltsan, Curve 7].

Soltysik (1962) would have provided a more consistent

correction.

Zapesochnyi and Feltsan, who worked at the lowest

pressures, obtained results which are in very poor agreement with all the others. The polarization correction

makes the situation worse. This discrepancy cannot be

definitely reconciled, but it appears possible that their

monochromator, which according to their schematic

diagram has a large number of oblique reflections and

transmissions in the same plane, may have had a very

high transmission asymmetry. This interpretation is consistent with the results for other helium lines; the

results of Zapesochnyi and Feltsan are inconsistent with

the results of other workers for the polarized lines, but

are more nearly consistent for the unpolarized lines. It is also worth noting that according to Fig. 72 their

result and that by Heddle and Lucas may include a

smaller contribution from cascading than the other

results, because of the low pressure at which they

worked.

In Fig. 75 the excitation functions of Lees and Yakhontova, which we have corrected for cascading, have

been compared with the energy dependence of the 3

l P

excitation cross section calculated by Ochkur and

Bratsev (1965b). Also included are results by H. R.

Moustafa Moussa (1967), of measurements made at low pressure and published corrected for polarization

and cascading.

The accurate absolute measurement of the magnitude

of the 3

l P—*2*S excitation function is especially difficult because of the effect of imprisonment of resonance radiation. Figure 76 based on the work of Phelps (1958) , is a graph showing high and low estimates of the effect of

imprisonment of resonance radiation on the intensity

of the 3

l P—>2l S line in helium. At zero pressure the

fraction /3_2 of 3

l P atoms giving rise to 3

l P—»2! S radiation is determined solely by the Einstein coefficients; i.e., /3_2=d3_2/(yl3-2+^43-i) =0.023. In the very-low-pressure range the fraction

/

3_2 in Fig. 76 can be represented

approximately by a linear relation:

/3_ 2=0.023+0.04 p(px 103

) , (176)

where p is the radius in centimeters of the cylindrical

reaction chamber and p is the helium pressure in Torr.

Helium pressures in the vicinity of a few times 10-5 Torr evidently are required to obtain absolute values for

the 3

l P—>2l S excitation function which approach to within a few percent of the zero pressure value.

Equation (176) may be difficult to apply to obtain

corrections in the low-energy range because of the question of definition of the effective radius p of the collision chamber in an apparatus for which only a fraction

of the diameter of the reaction chamber is focussed onto

the monochromator entrance slit. The pressure dependence of an excitation function would have to be determined experimentally to permit an exact correction or to determine the safe pressure range corresponding to some maximum error. Most of the experimental papers men60
Fig. 75. The relative he- lium 5016- A (3'P-*2l S)

excitation functions of Lees

(1932), Curve 1, and of Yakhontova (1959), Curve

2 which include our cor- rections applied for polari- zation and cascading (optically thick case) , and

the excitation function of Moustafa Moussa (1967)

corrected by that author,

Curve 3, compared with

the shape of the cross sec- tion for direct excitation of the 3

1 P level calculated by Ochkur and Bratsev

(1965b), Curve 4.

tioned below include linearity checks of one form or another. However, in every case the linearity data contains scatter much larger than a few percent in the

10~5 to 10-4 Torr range. It must be presumed that all of

the absolute measurements could have been seriously

affected by imprisonment of resonance radiation.

Measurements of the absolute value of the 3

1 P—>2 l S

excitation function at very low pressures were accomplished by St. John, Bronco, and Fowler (1960), and

later by St. John, Miller, and Lin (1964). The latter published a peak value (at ~100 eV) for the apparent

excitation cross section of the 3

l P level of 350X10-20 cm2 per atom. This value was obtained by absolute

measurement of the intensity in the 3

1 P—*21 S line

“at pressures as low as 1CF4 Torr.” Transition probabilities calculated by Dalgarno, Lynn, and Stewart

[from tables published by Gabriel and Heddle (1960)

who referenced these to Dalgarno, Lynn, and Stewart

(1959, private communication)]] were used to calculate

the apparent excitation cross section for the 3

l P level, on the assumption that imprisonment of resonance

radiation is negligible at these pressures. A curve of pres- sure dependence of the excitation is given in Miller’s

thesis but the data shown in the low-pressure range are not sufficient to support that assumption. Equation

(176) suggests that imprisonment may have been

responsible for 10 or 20% of the observed intensity. No

estimate of the experimental error in the absolute meas- urement of the excitation function is given.

This “apparent” excitation cross section is uncorrected for polarization and cascading effects. The authors did not measure the polarization of the lines observed in their measurements. To obtain a correction

they applied Eq. ( 158) using the data of McFarland and

Soltysik ( 1962) . Their correction was —4.4% at 100 eV.

An additional correction for cascading of 4 percent at 100 eV then led to a 3

X P level-excitation cross section

of ~322X 10~20 cm2 per atom.

Another low-pressure measurement was carried out

by Zapesochnyi and Feltsan (1965). They worked at pressures <4XlO~5 Torr, where imprisonment effects could be of the order of a very few percent. They give a

Fig. 76. Estimates of the effect of imprisonment of resonance

radiation on the fraction /3 _2 of helium atoms excited to the 3

1 P

state which contribute to observed 5016-A emission. The two

curves are based on imprisonment coefficients gp and Sr given by

Phelps (1958) in his Fig. 4. The pressure p is in Torr and the

effective radius of the interaction chamber p is in centimeters.

61
peak value of 17. 4X 10-20 cm2 for the absolute apparent

excitation function of the 3

X P—+2X S line. This corres- ponds to a cross section for excitation of the 3

X P level of

about 750X10~20 cm2

. They estimated the error as being 50-100% from all causes. Corrected for cascading

and polarization, this becomes 675X1CV20 cm2

. V. E. Yakhontova (1959) measured the absolute

value of the excitation cross section for the 3

X P level by

working at a pressure of 0.1 Torr where imprisonment of resonance radiation is essentially complete so that all excitations to the 3

l P level give rise to 3X P—>2X S radiation. Under these circumstances no polarization cor- rection is needed since pressure depolarization is essentially complete. A magnitude of 3.5X10-20 cm2 at 100

eV was obtained for the excitation of the line, uncorrected for cascading. This magnitude corresponds to a

cross section for excitation of the 3

X P level of about

150X 10-20 cm2

. The author refers to an auxiliary investigation of secondary processes as indicating that these

are small.

Gabriel and Heddle (1960) measured the 3

l P—>2X S

excitation function over a range of pressures using

photographic detection. By applying Phelps’ trapping

analysis to data taken at 0.011, 0.03, 0.06, and 0.1 Torr

they obtained a value for the 3

X P level-excitation cross section of 457X10-20 cm2

, at 108 eV, corrected for cas- cading. The radiation is so nearly depolarized at these

pressures that correction for angular anisotropy is negligible. The method would seem to be subject to considerable uncertainty since there are several free parameters; e.g., the effective radius (~0.5 cm) and

the collisional transfer cross sections as well as the excitation cross section are simultaneously determined

from the four measurements.

Moustafa Moussa (1967), working at Leyden, has

published absolute excitation cross sections for a number

of helium lines over the energy range from 50 eV to 6

keV. For the 3

X P cross section at 10-4 Torr he obtained

260X1CU20 cm2 at 100 eV after correction for polarization (7%) and cascading (4.1%). The pressure is high

enough, according to Fig. 76, so that a significant imprisonment error could have occurred.

Other absolute measurements, by Lees (1932), by

Thieme (1932) , and by Stewart and Gabathuler (1959) were carried out at pressures for which imprisonment of

the resonance line is severe. Phelps has shown that the

results of Lees and of Thieme are consistent with the

theory of imprisonment of resonance radiation if an

effective radius of 0.75 cm is assumed for their some- what similar excitation chambers.

The range of results, from 150 to 750 X 10~20 cm2 for

the 3

X P level excitation cross section, certainly is unsatisfactory. If it were necessary to narrow the choice

we would disregard the results obtained by Yakhontova

because of the errors introduced by excitation transfer at the high pressures used. The remaining results differ

within a factor of three. Zapesochnyi and Feltsan admit

to the possibility of their result being 1.5-2 times too

large so that the two results are apparently not contradictory, but by inference they are allowing for trapping of resonance radiation. They worked at lower pres- sures than did St. John, Miller, and Lin. Obviously

there were gross inaccuracies in several of these measurements. It is impossible to make a valid judgment

as to where the correct answer might lie. These absolute

values and those for the higher n

x P excitations are

listed in Table LXIV(a), as are absolute values calculated at 100 eV by Ochkur and Bratsev (1965b). It is in- teresting to note that there is rather good agreement in the measured ratios of the 3

X P and 4X P excitation cross J ; sections at 5016 and 3965 k. The work at a given ai laboratory tends to be internally consistent, but the 11 basic radiometric, pressure, and other calibrations are not consistent from one laboratory to another. Improvement of radiometric and density standards is needed. \ In Table LXIV a-f we have applied branching ratios t to convert the photon-excitation cross sections to cross sections for populating the upper levels. Zapesochnyi

(1966) , in Astron. Zh., has summarized results obtained J in his laboratory over several prior years. His results are

f presented as direct excitation cross sections for the c levels corrected for cascading. Because we have not in- cluded cascade corrections the values we have obtained

from Zapesochnyi and Feltsan (1965) do not agree for

all levels with the values presented by Zapesochnyi

(1966). A level by level comparison suggests that

Zapesochnyi may have introduced corrections other

than for cascading or may have included some results

not otherwise published. For example, Zapesochnyi

( 1966) lists the peak 3

X P level excitation cross section as 530 X 10~20 cm2

. In the text above we have given a value

of 675X10-20 cm2 based on Zapesochnyi and Feltsan

(1965). Since the source of Zapesochnyi’s numbers is obscure we have not included them in our table. Zapesochnyi found that his peak level-excitation cross sections for a given series lie along a straight line on a log-log plot against principle quantum number. The

peak cross sections for each series of levels can be

represented by the expression

Qxjmax) =c/w“, (177)

where c and a are constants for each value of orbital

angular momentum L, and n is the principal quantum

number. He finds a = 3 for the X P, X D, and 3 D levels, a = 4 for the X S and 3 S levels and a = 9 for the 3 P levels. Values for the 2

X S, 2X P, and 2

3 S cross sections are obtained by extrapolation. No physical justification is given for formula (177).

The same general physical considerations apply to excitation of the higher n

x P levels as apply to the 3

X P

level, with the exception that collisional transfer of

excitation may be more severe for the higher levels. In particular, imprisonment of resonance radiation

presents essentially the same problem for each of the

62
I;

Fig. 77. Relative excitation functions for the helium 4

l P and

5

1P—*2l S transitions [St. John, Miller, and Lin (1964) , 2 p, Curve

1; Yakhontova (1959), Curve 2; Lees (1932), 44 p, Curves 3 and 5J compared with direct excitation cross section for 4

l P

and 5

lP levels calculated by Ochkur and Bratsev (1966), Curves 4 and 6. The experimental data are uncorrected for polarization or cascading. The Ochkur and Bratsev 4

l P and 5l P cross sections are shown with the proper relative magnitudes. The data obtained

by Moustafa Moussa ( 1967) for 4

1 P excitation closely correspond

to Curve 4 above 100 eV, falling below it at lower energies; and

for 5

l P they fit Curve 6 at all energies.

lines n

1 P—*2x S. In the low-pressure range the fraction

fn—2 of n

x P atoms which give rise to nx P—>2x S radiation can be represented by

/^2 = jB_2+(0.04^„_1/^3-i)p(/>X103 ), (178)

where An~m is the transition probability for the line n

x P—*ml S. The first term on the right, P„_2 , is the

branching ratio for the n

l P—*21 S line, which does not

vary greatly within this series. The pressure-dependent

term is approximately proportional to the transition

probability of the appropriate resonance line, and decreases by a factor of two or less with each higher level. The pressure p is in Torr and the effective radius p is in centimeters.

All these n

x P lines are 100% polarized at threshold

according to Percival and Seaton, and presumably

behave qualitatively in the same way as the 3

X P—>2X S

line with respect to energy dependence and pressure

dependence of the polarization.

Some experimentally determined relative excitation

functions for the 4

1P—>21 S and 5X P—+2X S lines are shown in Fig. 77. Many of these results were obtained

at pressures high enough so that pressure depolarization and imprisonment of resonance radiation are very

effective. Zapesochnyi and Feltsan (1965) carried out

measurements of excitation functions for the TP, 5

X P

and 6

1P—+21 S lines, not included in the figures, which

have much the same characteristics as their 3

1 P—>21 S

function shown in Fig. 71. Thieme’s data for TP excitation is also omitted. They are characterized by a steep

slope at the higher energies.

Moustafa Moussa used an ultraviolet spectrometer

to obtain the first measurements of the 2

l P excitation

cross section. He measured the energy dependence but

did not obtain an absolute value.

16.2. Helium n

l S—+21 P Transitions

The situation for the n

x S—>2x P series of excitation

functions is usually presumed to be simpler than for n

x P excitation. The nx S state has spherical symmetry

and the lines are not polarized so that there is no question of instrumental polarization effects or of angular

distribution. Furthermore, since the n

x S state does not

optically connect to the ground state the problem of

imprisonment of resonance radiation enters only

through the population contributed by cascading from

the X P levels. Therefore, the poor agreement between

the available measurements has been regarded as surprising. However, because of the cascading contribution

these lines are much more pressure sensitive than the

corresponding triplet series, for example. Figure 78

shows estimates of the cascade contribution to the

T.S—>2l P line at several pressures. The extreme sensitivity of the cascading fraction to pressure in the range

where most of the measurements were carried out has

not been recognized.

The available measured excitation functions are

clearly of limited value, in view of this pressure dependence. Figure 79 displays the uncorrected data for the

5048- A (41 5'-^21 P) . line. Here we have normalized

the experimental results at 60 eV and shown them

in comparison with the shape of the theoretical curve by Fox (1965). It should be noted that the cascading

fraction is quite significant at l-/z pressure and varies

sharply with pressure. Furthermore, the pressures

are not explicitly stated for two of the curves. The comparison with theory should not be carried too far here.

Data for the 4438- A (51 S—*2l P) excitation function

is displayed in Fig. 80. The comparison with theory

looks much the same as for the 4

1 ^'—»21 .P excitation

function and should be regarded as only qualitative

since the measurements were carried out at pressures

for which cascading should be large.

A few results are available for the 3

X S (St. John et al., Zapesochnyi and Feltsan, and Moustafa Moussa),

Fig. 78. The cascade fractions for the helium 5048-A line (41 S^21

JP) are estimated for several values of the product pp, where p is pressure in Torr and p is effective radius of the collision volume. The basis of the calculations is the same as for Fig. 72. In addition the imprisonment curves of Fig. 76 and data from the

text are used to estimate the degree of suppression of transitions

to the ground state from higher l P states.

63
Fig. 79. Relative excita- tion functions for the helium

4

1 S—>2l P transition [St. John, Miller, and Lin

(1964), 1.5 n, Curve 1; Heddle and Lucas (1963),

1.2 n, Curve 2; Thieme

(1932), Curve 3; Yakhontova (1959), Curve 4; and

Zapesochnyi and Feltsan

(1965), Curve 5J are compared with the 4

1 5 level- excitation cross section cal- culated by Fox (1965),

Curve 6.

6

J 5 (Thieme, St. John et al., Zapesochnyi and Feltsan,

and Moustafa Moussa), and 7

l S (Zapesochnyi and

Feltsan) excitation functions. These results are similar to the results for the 4*5 and functions. The

Zapesochnyi and Feltsan results are limited to the

energy range below 40 eV.

Measurements of a large number of helium excitation

functions by H. R. Moustafa Moussa became available

too recently to be considered in planning the figures for

this review article. We have added them where possible

to the already drafted figures. Moustafa Moussa’s data

extends from 50 eV to about 6 keV. Our figures were

designed to emphasize the lower energy range, and

extend typically to 250 eV so that the slope and curvature of excitation functions are displayed at energies

Fig. 80. Relative excitation functions for the 5

1 5-^21 F helium

transition [St. John, Miller, and Lin (1964), 5.5 ju, Curve 1; and Yakhontova (1959), Curve 2J are compared with the 5*5

level excitation cross section calculated by Fox (1965), Curve 3.

higher than the energy of maximum excitation. Moustafa Moussa’s data for the 1 S, l D, 3 Sy 3P, and 3D excitation functions do not contain information about the

shape of the peak and threshold regions, but show the

shape beyond thf peak. In comparing Moustafa

Moussa’s data with other results we have used data

given in his thesis to reintroduce cascading and polarization effects. For 4

1 5 excitation the negative slope of the Moustafa

Moussa curve is slightly less than that of the curve

representing the work of St. John, Miller, and Lin. The

two would be indistinguishable in our Fig. 79. For 5*S

excitation, shown in Fig. 80, Moustafa Moussa’s re- sults again have a slightly smaller negative slope in the

range from 75 to 250 eV than that of the results by

St. John et al. In Table LXIVb are listed the absolute values of the

n'S levels measured by a number of workers. Most of

these absolute values are measured at the peak of the

excitation function. This practice of measuring absolute

values at the peaks may not be the best in every case, particularly for the triplet lines because of the very

sharp peaking near threshold. In the first place, variations in the widths of electron energy distribution

from one measurement to another may well make comparisons between peak values rather meaningless. In the

second place the entire question of polarization near

threshold seems to be quite undecided so that corrections to peak values for angular distribution and instrumental effects may not be meaningful. In the third

place, the question of pressure sensitivity near thresh64
Fig. 81. Relative excitation functions for the helium 4

1 /)—

transition [St. John, Miller, and Lin (1964), 1.7 n, Curve 1; Heddle and Lucas (1963), 1.2 n, Curve 2; Yakhontova (1959),

Curve 3; McFarland (1967), 10~2 n (crossed-beam measurement),

Curve 5; and Moustafa Moussa (1967) , ~1 n, Curve 6] compared

with the 4

l D excitation cross section calculated by Ochkur and

Bratsev (1965), Curve 4.

old, raised by Bogdanova and Geitsi, leads to some uncertainty about the validity of peak measurements. We

have extrapolated the n

l S absolute measurements to 100 eV, and compared them in Table LXIVb with the

absolute values calculated by Fox.

16.3. Helium n

l D^>2l P Transitions

The l D excitation process results in radiation which is rather strongly polarized (see Fig. 73). Therefore, all the earlier discussion of distortion due to polarization

is applicable here. The possibility of Larmor precession

causing depolarization is especially significant here

because of the longer lifetimes involved (see Figs. 61

and 62) . Also, as discussed by Lin and St. John (1964)

,

these excitation functions are subject to distortion by

excitation transfer, presumably from the high n

l P

states to n

l F states followed by cascading. The pressure

dependence curve for 4

1 Z7—>2l P radiation of Heddle and

Lucas ( 1963) does not indicate a very great sensitivity

to pressure, but it was made at 35 eV, well below the

peak of the n

l P cross sections. The distortion should be

greatest at 100 eV where the n

l P cross sections have

their maxima.

We have not calculated the cascading fraction and its pressure dependence for lack of reliable parameters and

quantitative information about excitation transfer at higher levels. The l D excitation functions are of the same order of magnitude as the l S functions, while the n

1 P-^ml D transition probabilities are typically a third

or a fourth of the corresponding n

l P^nnl S transition

probabilities. Therefore, the n

l P^>ml D cascading is relatively smaller than the n

l P-^m1 S cascading. Cascading fractions for l D excitation functions should be

smaller than for l S excitation functions. Cascading

also occurs from l F states but St. John et al. estimate the

cascading from 'F levels to be about a tenth of the cas- cading from l P levels. They give the total cascading

fraction as a maximum percentage (5 to 8%) at 450 eV.

Figure 81 shows some of the measurements of the

4922 A (41 D—>2l P) excitation function. The polarization of this line is 60% at threshold and appears to be

significant out to several hundred volts. Since the

pressures at which Yakhontova, and Zapesochnyi and

Feltsan made their measurements are not explicitly

stated it is not possible to apply polarization corrections

to the different excitation functions. The most complete

treatment covering the region of the maximum is the

work of Heddle and Lucas, explicitly corrected for

instrumental polarization, at an explicitly stated pres- sure of 1.2 ju, coupled with their polarization measure- ment at 1.3 u shown in Fig. 73. McFarland (1967)

measured the 4

l D excitation and polarization using a helium beam. His objective was to study threshold

effects. The beam pressure in this work was low enough

so that secondary processes would not have been signifi- cant. His polarization curve is presented in Fig. 73.

McFarland’s 4

l D excitation function, shown in Fig. 81,

and that obtained by Heddle and Lucas are in rather

good agreement. Also shown are Moustafa Moussa’s

results, including cascading, and without polarization

correction, for 4

l D excitation. From 140 to 250 eV

Moustafa Moussa, Heddle and Lucas, and McFarland

have essentially identical slopes. At lower energies

Moustafa Moussa’s results are slightly higher than the

others.

These results could possibly be reconciled with Yakhontova’s results, for example, through depolarization

if she worked at higher pressures than Heddle and

Lucas, or through some instrumental polarization,

which might not have shown up in the 1 P work where

pressure depolarization was more nearly complete. The

Zapesochnyi and Feltsan curve is again very different

from those of other authors.

Figure 82 shows the 4387-A (5 1 Z)-^2 1 F

> ) excitation

functions of Yakhontova, St. John et al ., and Moustafa

Moussa uncorrected for cascading or polarization, in

comparison with the 5

1 /? excitation cross section calculated by Ochkur and Bratsev. The theoretical result

characteristically falls below the experimental results at the peak for all the l D lines of helium.

Additional measurements have been carried out for

Fig. 82. Relative excitation functions for the helium 5

1 Z?-^2 1 F

> transition [St. John, Miller, and Lin (1964), 1.1 /u, Curve 1; Yakhontova (1959), Curve 2; and Moustafa Moussa (1967),

1 ix, Curve 4J compared with the 5

l D excitation cross section cal- culated by Ochkur and Bratsev (1965b), Curve 3.

65
3

1 D, 6l D, 7

1 T>, and 8

1 D excitation (St. John et al., Yakhontova, Zapesochnyi and Feltsan, and Moustafa

Moussa). St. John and Jobe (1967) have measured a helium excitation cross section at 18 695 A. The bandwidth of the monochromator is not given and the paper

does not specify whether the measurement includes both

singlet and triplet 4F—>3D transitions at 18 697 and

18 686 A, respectively. They obtained a direct 4 F

excitation cross section of 17 X 10-20 cm2 at 100 eV.

Measured absolute values for 1 D excitation are compared in Table LXIVc.

16.4. Helium n

3 S—*23 P Transitions

The helium 3 5 excitation functions would appear to be as nearly experimentally fool-proof as any excitation

function. Since the orbital angular momentum is zero

there is no polarization in Russell-Saunders coupling

so that all the problems of instrumental polarization,

pressure depolarization, and anisotropy do not occur. According to Heddle and Lucas (Fig. 58) the 3 S and 3 P

excitation functions are not particularly pressure sensitive. The triplet system is free of direct influence of

imprisonment of resonance radiation, but is subject to it rather indirectly through collisional transfer of excitation from the singlet system.

The fraction of excitation due to cascading is given

as 20 to 30% for n

3 S excitation by St. John, Miller,

and Lin. This rather large contribution reflects the

inaccessibility of the ground state. (The 2

3 S state exal- tation is a special case and will be discussed separately.)

Since 3 P state excitation function is not pressure sensitive below 7 or 8 n the component of 3 5 excitation due

to cascading should not be pressure-dependent.

The 3 S excitation functions are displayed in Figs. 83

and 84, and the measured absolute values in Table

LXIV(d)

.

An intercomparison of the shapes of the excitation

functions, and a comparison with theoretical results

reveals an interesting difference which seems to be

characteristic of the triplet cross sections. The theoretical results are based essentially on an expansion of

Fig. 83. Relative excitation functions for the helium 4

3 5—>23P

transition [St. John, Miller, and Lin (1964), 2.0 n, Curve 1; Heddle and Lucas (1963), 0.5 m, Curve 2; and Yakhontova

(1959), Curve 3], compared with the 4

3 S excitation cross sec- tion calculated by Ochkur and Bratsev (1965b), Curve 4.

E(eV)

Fig. 84. Relative excitation functions for the helium 5*5—»23 P

transition [St. John, Miller, and Lin (1964), 5.3 n, Curve 1; and

Yakhontova (1959), Curve 2] compared with the 5

3 S excitation ; cross section calculated by Ochkur and Bratsev (1965b) , Curve 3. The data obtained by Moustafa Moussa (1967), if normalized t . the other curves at 60 eV, would correspond closely to Curve 1 ' at higher energies.

the cross section as a series in the electron energy, E.

At high energies the leading term, E~3

, presumably

dominates. The experimental results, without exception, fall off much more slowly with energy.

The question arises of whether the slow fall off of

the experimental results is instrumental, due to (1) a

high background of stray light emitted by the monochromator or (2) excitation by slow electrons produced

from the primary beam by scattering at slit edges.

Gabriel and Heddle (1960) give a total excitation cross section of 9.6X10-18 cm2 for all “observed states,”

and 3 5 cross sections ranging from 1 to 15X10-20 cm2 at 100 eV. Therefore a transmission of 1% in light scat- tered throughout the spectrum would give the observed

high energy tail. This question is not discussed in detail

in the papers but there are indications that the background level just off the line in question was checked in

most of the work. In this case the theoretician is left with an interesting problem of accounting for a dependence weaker than E~3

. Perhaps the slow decay may be

due to a large coefficient for the term of next higher

order in the expansion of the cross sections in powers of

E. Possibly the slow fall off may arise through cascading

from higher levels where a weakening of the RussellSaunders coupling scheme may result in the excitation

cross sections having some of the character of singlet

excitation.

In Fig. 83 we intercompare some shapes of the 4713-

(43 S-+23 P) function. The result is disappointing in

view of the absence of polarization effects. The pressure

at which Yakhontova worked is unspecified. The rela- tively low peak-to-tail ratio would be consistent with a helium pressure of 10 u or higher, where according to

Fig. 58, the excitation function is enhanced and pre- sumably distorted by excitation transfer. The possibility that space charge effects would account for the

difference would not seem likely.

Some further careful work at low pressure, low elec- tron current density, and high electron energy resolution over a broad energy range on the 4713- A excitation

66
Fig. 85. Relative excitation functions for the helium 3

3 P—>23 S

transition [St. John, Miller, and Lin (1964), 1.5 n, Curve 1 and Q;

Heddle and Lucas (1963), 0.25 n, Curve 2; and Yakhontova

(1959), Curve 3] compared with the 3

3 P excitation cross sec- tion calculated by Ochkur and Bratsev ( 1965b) , Curve 4. Data by

Moustafa Moussa (1967), normalized to the other curves at 60

eV, closely correspond to Curve 1 at the higher energies.

function would be very useful. This line provides a good oppurtunity for a highly precise and reliable

measurement which would permit a definitive comparison with the theoretical results for triplet excitation, a comparison which may have considerable physical

significance.

16.5.

Helium n

3 P—>23 S Transitions

The 3P excitation functions appear to be as insensitive

to pressure as are the 3 S functions. Here, there is the

added complication of polarization. This polarization is as high as 25% at low energies and falls off slowly to zero above 200 eV. Therefore, some distortion of the

excitation function is possible. The measured excitation

functions at 3889 A (33 P—>23 S) are compared in Fig. 85

and the absolute values of n

3 P functions in Table

LXIVe.

The comparison between the data by Heddle and

Lucas and by Yakhontova is much like that of the 3 S

case. The lower peak-to-tail ratio obtained by Yakhontova could be accounted for in terms of polarization

only through the assumption that there was nearly

100% instrumental polarization.

The 2

3P—>23 S excitation measurement at 10 829-30

oA, by Zapesochnyi and Feltsan (1965) is worth special

notice. They obtained a value of 2510X10-20 cm2 at

26.5 eV. Since all 2

3 P excitation decays to the 2

3 S

state, the 2

3 S excitation function is necessarily larger

than the 2

3 P excitation function. Measurements of 2

3 P

excitation functions (Fleming and Higginson, 1964;

and Schulz and Fox, 1967) yield values at the peak just

above 20 eV, a factor of ten smaller. The Zapesochnyi

and Feltsan result implies a rapid increase in the 2

3 P

cross section and therefore in metastable production in the range from 20 to 26.5 eV. The cascade contribution

would be quite significant at 26.5 eV. Since the measure- ments were made at pressures of 8 to 25X10-3 Torr

there may also have been significant excitation transfer

into the triplet system, much of which would cascade

into the 2

3 P state. 16.6. Helium n

3D—+23 P Transitions

The 4

3Z)—>23 P excitation function at 4471- A has been

shown to be quite pressure sensitive, presumably due to excitation transfer into the higher 3 F states followed by

cascading. This has been discussed in an earlier section.

Quite low pressures are required to obtain reliable

absolute values, and some slight distortion of the excitation function should be evident at pressures of a few

times 10~3 Torr. Figure 86 displays the excitation function for the 4471- A line and measured absolute values

are listed in Table LXIVf.

Figure 86 illustrates again the remarkable difference

between the behavior of the experimental and theoretical results at higher energies. It also shows that the

peak-to-tail ratio in Yakhontova’s results is still smaller

relative to the results of Heddle and Lucas, for this

very pressure sensitive line.

16.7.

Excitation of the Helium n = 2 Levels

Measurement of helium n= 2 level excitation presents

a special set of problems which have led to a variety

of measurement techniques. The lowest excited states are listed below with energies and lifetimes:

State Energy (eV) Mean life (sec)

3

3 S 22.72 3.6X10-8 TP 21.22 5.6XKF-10

2

3 P 20.96 1.05X10-7 2

1 S 20.61 0.14

2

3 5 19.82 105

.

As electron energy is increased the metastable states

2

3 S and 21 S are excited first, followed by the 23 P state

Fig. 86. Relative excitation functions for the helium 4

3 D—>23 P

transition [St. John, Miller, and Lin (1964) , 1.7 n, Curve 1 and

;

Heddle and Lucas (1963), 1 n, Curve 2; and Yakhontova (1959),

Curve 3J compared with the 4

3 D excitation cross section calculated by Ochkur and Bratsev ( 1965b) , Curve 4. Data by Moustafa

Moussa (1967), normalized to the other curves at 60 eV, closely

correspond to Curve 1 at the higher energies.

67
which decays entirely into the 2

3 S metastable state. At

21.22 eV the 2

l P state, which radiates primarily to the

ground state, is excited. The 2

1 P-^11 5 transition probability is a thousand times larger than the 2

l P—*2l S

transition probability so that the probability of exciting

the 2

l P-+2l S line must remain small in the energy

interval immediately above its threshold, even for a moderate amount of reabsorption. There is no evidence

or expectation of collisional excitation transfer in this

energy range, so that observations of excitation over the

three volt range from 20 to 23 eV have a simple interpretation in terms of a few processes. Good electron energy

resolution becomes of paramount importance in separating the effects of excitation of the different n= 2

levels. The oldest method of studying helium excitation in

this energy range is due to Maeir-Leibnitz (1935) and

was recently used again with refinements of technique

by Fleming and Higginson ( 1964) . An axial filament is used in a simple concentric cylindrical configuration of

electrodes and grids. Electrons were accelerated to the

excitation potential V and emitted through a narrow

slit into an excitation volume. An outer collector was

biased to receive only elastically scattered electrons.

The helium pressure was maintained at 0.47 Torr, at which electrons make a substantial number of collisions

before they reach the collector. Near threshold the drop

in collector current beyond the first excitation threshold

could be analyzed in terms of diffusion theory, taking

into account only elastic scattering and excitation of the

2

3 S state. A cross section for excitation of the 2

3 S state at the maximum just above 20 eV of 2.6X10-18 cm2 ±

17% was obtained.

The other absolute measurement of the 2

3 S peak that

is of interest was by Schulz and Fox (1957), using a method of detection of metastables by measurement of

the secondary electron current emitted from a metal

surface due to energy given up by the metastables. This

method was previously exploited by Dorrestein ( 1942)

and subsequently by Holt and Krotkov (1965) for

relative measurements.

Figure 87 compares the relative measurements of

metastable current from these last three papers. The

shapes obtained in the three measurements compare

quite well, with differences attributable to differences

in energy resolution. Schulz and Fox, using a retarding

potential difference method, had an effective energy

resolution of about 0.1 eV. They used an axially symmetric electrode configuration with the electron beam

projected down the axis along a confining magnetic

field. The metastable current was measured at an outer

cylinder. They put the measurement at the 2

3 S peak on

an absolute basis by assuming isotropic motion of the

metastables and using an efficiency for electron ejection by metastables at a gold surface obtained by

Stebbings (1957) . Their result was 4X 10-18 cm2 ±30%,

the stated error overlapping that given by Fleming and

Higginson,

The first peak in Fig. 87 is unambiguously attributable to the 2

3 S excitation in the Schulz and Fox experiment because it is well separated from the thresholds

for 2

1 S and 2

3 P excitation. Above 20.6 eV the metastable current includes the 2'S contribution and the 2

3 P

excitation through cascading into the 2

3 S. The interpretation of each of the humps and bumps in terms of

excitation of a different state has been proved inadequate through the work of Schulz and Philbrick ( 1964)

,

Chamberlain (1965) and Chamberlain and Heideman

(1965), these studies being based on the measurement

of components of the scattered electron current corres- ponding to specific inelastic processes. The Schulz and

Philbrick results for electrons scattered at 72° in an in- elastic process leaving the helium atom in the 2

3 S state

is included in Fig. 87. These results strongly suggest

that the maximum near 21 eV in the total metastable

production curve is due to a second peak in the 2

3 S

excitation curve rather than being due to 2

1 S or 23 P

production, and that the bump near 22.5 eV is due to resonance structure. In Fig. 88 results of the five-state

close coupling calculation (Burke, Cooper, and Ormonde 1966) discussed in Sec. 3.4 is compared with the

experimental results of Schulz and Fox. There is an

excellent correspondence between the measured total

metastable production cross section and that predicted

from the theory by summing the calculated contributions (see Fig. 33), in the range where cascading from

higher states does not contribute.

Chamberlain and Heideman obtained a spectacular

set of curves for the forward inelastically scattered components corresponding to 2

3 S, 21 S, 2l P, and 23 P excitation, which demonstrate that these curves are very

heavily structured. These are shown in Fig. 89. Both the

Schulz and Philbrick, and the Chamberlain and Heideman results were obtained with electron beams of about

0.06 eV half-width. A comparison of their results re- veals that the inelastic scattering corresponding to 2

3 S

excitation is anisotropic, so the results cannot be applied

to obtain a quantitatively correct subtraction of 2

3 S

excitation from the total metastable excitation curve. The work by Holt and Krotkov (1965) is another

very significant contribution to the analysis of the metastable production curve. Their total metastable production curve is included in Fig. 87. They capitalized on the fact that the 2

1 S metastables can be quenched

with 90% efficiency in a field of 226 kV/cm. The

quenched component of the production provided a cross section for the 2

1 S state excitation, which rises smoothly

from threshold to a broad maximum centered at 22 eV

and with about one-third the amplitude of the first 2

3 ,S maximum. The unquenched component gave a total cross section for excitation of the 2

3 S and 2

3 P states,

since all the 2

3 P atoms immediately decay into the 2

3 S

state, and no higher triplet states are excited in the

energy range up to 22.7 eV.

Holt and Krotkov also used an inhomogeneous magnetic field to spatially separate the three components of

68
Fig. 87. The total helium metastable excitation functions measured by Schulz and Fox (1957), Curve 1 £the data were

taken from Schulz and Philbrick (1964)]; Holt and Krotkov

(1966), Curve 2; and Dorrestein (1942), Curve 3 are intercompared on a relative basis. Curve 4 is the 2

3 5 excitation cross sec- tion measured at 72° by Schulz and Philbrick (1964).

Fig. 88. The total helium metastable production cross section obtained by Schulz and Fox (1957), Curve 2, is compared with the combined results of five-state close coupling calculations (Burke,

Cooper, and Ormonde, 1966), Curve 1. The 2

1 S production cross section obtained by Holt and Krotkov (1966), Curve 4, is also included, in comparison with the 2

1 S excitation cross section cal- culated by Burke, Cooper, and Ormonde, Curve 3.

20 21 22 23 24 INCIDENT ENERGt-eV

Fig. 89. Energy dependences of the components of forward in- elastically scattered electron current in helium corresponding to excitation to the 2

3 S, 21 S, 23 P, and 2l P states. The curves are smoothed tracings of the original data, and the width of the noise

is indicated by the error bars. (Chamberlain and Heideman, 1965)

.

the 3 S state. The metastables were allowed to pass

through an aperture out of the excitation region into a

drift space at lower background pressure, where the

magnetic and electric fields could be applied. At the

end of the drift space the metastables were detected by

counting the electrons ejected from a tungsten elec- trode placed in the path of the beam.

From the 2

3 S differential excitation cross sections

shown in Figs. 87 and 89 they inferred a total 2

3 P

excitation cross section which they used as the basis

for calculating the polarization of those 2

3 5 atoms

derived by cascading from the excitation of the 2

3 P

state. Cermak (1966) developed a method of analyzing

metastable production by energy analyzing the elec- trons, with a Lozier technique, produced by Penning

ionization as a metastable beam is passed through a gas

of lower ionization potential. For his study of helium

excitations he used argon, with an ionization potential

of 15.75 eV. The 2

3 S states (19.82 eV) result in elec- trons of 4.07 eV and the 2

1 5 states (20.61 eV) result in 4.86-eV electrons. Retarding potential analysis was

used to determine the electron currents in each energy

component. The behavior of these electron currents as a function of the energy of the exciting electron beam

yields excitation cross sections for the two states. The

results are relative cross sections but they could be re- lated to each other if the cross sections for Penning ionization were known.

Cermak extended his results up to 60 eV so that his 3 S

excitation function includes the cascading from all excitation into the triplet system. Similarly, the l S

excitation function necessarily includes a significant cas- cading contribution from higher X P states. The pressure

in the source, 5X 10-4

to 10~3 Torr, was high enough so that some modification of the singlet system cascading

pattern presumably occurred due to reabsorption of resonance radiation.

Dugan, Richards, and Muschlitz (1967), using a system somewhat similar to that of Holt and Krotkov,

studied the excitation functions of the 2

1 S and 23 S

states at electron energies of 25 to 135 eV. They used an inhomogeneous magnetic field to deflect the Lfs = ±l

components of the 2

3 S state out of the beam. It was

assumed that this reduces the 2

3 S population by twothirds, leaving the one-third in the Ms = 0 substate

undeflected. By this method they could determine the

separate excitation functions and their ratios. However,

the analysis depended on extrapolation to zero pres- sure from excitation over a helium pressure range of 40

to 130 juHg, in order to eliminate effects of imprisonment and differential scattering. For their geometry it seems certain that imprisonment is severe over the

whole range of pressures used, and an extrapolation to zero pressure cannot be reliable. As Phelps (1958)

has demonstrated, the cross section for excitation of the

5016 A (3! P—>2’.S) line is tremendously enhanced at helium pressures of 40 n, to the point that it alone is
comparable to the value of the 2

l S cross section esti- mated by Holt and Krotkov. The imprisonment process

is not linear at high pressures. An opposing error should

also occur due to enhancement of the 2

3 S state as a

result of excitation transfer into the triplet system. The

observations by Lin and St. John (see Fig. 56) show

that this would result in large contributions to the

triplet population at high pressure and at the higher

electron energies. This process would be quadratic in

pressure, not linear.

16.8. Thresholds of Helium Excitation Functions

The discussion of the n= 2 excitation functions has

centered on behavior near threshold, and has shown that

this is a region of special interest. This interest also

exists in thresholds for the excitation of higher levels

of helium. In the work which has been discussed on the

excitation of the visible helium lines thresholds have

not been emphasized but rather we have been con- cerned with the determination of the gross behavior of

the excitation functions. There has been a considerable

amount of threshold work, especially as it has related

to the polarization of the excited radiation near threshold.Perhaps the first experimental work to reveal the

general behavior of the thresholds was by Yakhontova

(1959). Detailed shapes were obtained by Smit, Heideman, and Smit (1963). Most of these results were obtained at pressures of 2 to 10 juHg. Zapesochnyi and

Feltsan (1965) measured the shapes of a large number

of thresholds at pressures from 0.3 to 3 u- None of these

workers considered effects due to the polarization of the

radiation. All used the thermal energy width of elec- trons from a hot cathode with no further energy selection. Energy resolution was about a third of an electron

volt for the two more recent works while Yakhontova,

using a more intense electron beam, had resolutions

ranging from 0.7 to 1.2 eV.

Zapesochnyi and Shpenik (1966), using a 127° elec- trostatic analyzer, studied threshold excitation for a number of atoms. They give results for the threshold

behavior of 3

3 S, 4

s

.S', 3

3 P, 33D, and 3

l D excitation obtained with electron beam half-widths of 0.3 to 0.5 eV,

showing pronounced structure immediately at threshold.Heddle and Keesing (1967a) have applied a hemispherical electron monochromator to the study of the

thresholds for 4

1 S and 4

3 S states. The energy spread of

electrons transmitted by the monochromator was nominally 0.070 eV. The 4

1 »S and 43 S functions had much the same behavior, each exhibiting a second maximum

beyond the threshold maximum, which Heddle and

Keesing suggest is a resonance with the configuration

(W)2 S1/2. To illustrate the threshold behavior Fig. 90 displays

the results obtained by Smit, Heideman, and Smit for one line from each series. General characteristics of

thresholds within a spectral series seem to be the same.

Fig. 90. Threshold behavior of the helium singlet and triplet

series excitation functions obtained by Smit, Heideman, and

Smit (1963).

An exception occurs for the 3D series, the structure in

the 3

3D-^>23 P excitation function being absent from the

4

3D and 5

3D functions.

Polarization of helium lines excited by electron impact

has been a controversial subject since Lamb and Maiman (1957) published a measurement of the polarization of the helium 3889- A (33 P—>23 S) line, showing a minimum at 28 eV. Earlier work by Skinner and

Appleyard (1927) showed that the polarization of elec- tron impact excited radiation for a number of mercury

lines drops toward zero as the electron energy approaches threshold from above.

In 1958 Percival and Seaton published the theoretical

study of polarization of impact radiation discussed in Sec. 9. In their paper Percival and Seaton reemphasized

the fact that for many lines conservation of angular

momentum dictates the value of polarization these

lines must have when excited by an electron carrying

only the energy of excitation and no more. Further,

they pointed out that the results of Skinner and Appleyard seemed to violate these essential requirements by

going toward zero at threshold, and they suggested that more careful experimental work seemed in order.

Heddle and Lucas (1961; 1962) and McFarland and

Soltysik (1961; 1962) carried out their studies of polarization using helium, and both studies showed polarization in helium going to zero at threshold. The results

have been reproduced in Fig. 73. 70
Hughes, Kay, and Weaver (1963) obtained similar

behaviors at threshold. Their work and the results of

McFarland and Soltysik showed in excellent detail the

structure of the 3

3 P—>23 S polarization near threshold

first observed by Lamb and Maiman. However, the

work of Hughes, Kay, and Weaver suffers from instrumental effects similar to but more severe than those

appearing in McFarland and Soltysik’s results. In both,

there are cases of negative polarization increasing with

pressure at fixed electron energy, which must be a nonphysical effect. The polarization observed (P0b8 ) at 90° to the elec- tron beam, through a monochromator, is related to the

true percentage polarization P of the line in equation

(169). We rewrite Eq. (169) to include depolarization

effects. Jb. (7P/ioo)+[(7-|i-r^)/(r,i+^)]

100 i+(Yi>/ioo)-[(rM -r±)/(rM +ri)]' 1 J

Here, 7 is a pressure depolarization factor which may

take values between zero and unity. At high pressures

depolarizing collisions and depolarization due to such

processes as imprisonment of resonance radiation and

collisional transfer of excitation, may occur. The factor

7 becomes small as such effects become dominant and

the observed polarization at sufficiently high pressure

is the instrumental polarization. Instrumental polarization is also observed at energies such that the true

polarization is zero. The observations of small values of polarization near

threshold have been the subject of continuing controversy. A simple integration sufficies to show that the

observed effects cannot be due merely to the finite energy distribution in the electron beam. McFarland

(1963, 1964) carried out an extensive series of tests to determine whether the observations might be the result

of radially scattered electrons. In the course of this

work he found low polarization values near threshold

for 3

3 P, 3

1 /*, 3

l D, and 4

l D excitation, but the polarization seemed to climb toward the theoretical value

again, very near to the threshold.

Heddle and Keesing (1964, 1965) have confirmed

this behavior for the 3

l P and 4

1 D excitations, and

Soltysik, Fournier, and Gray (1966) have reproduced

McFarland’s results for 3

X P, 3

3 P, and 4}D excitation.

More recently McFarland (1967) has carried out a

careful crossed beam measurement of the polarization

of the 4922- A (41 Z>2-^21 P1 ) line of helium. The results near threshold, shown in Fig. 91, confirm the earlier

measurements. Heddle and Keesing (1967b) have also

carried out very careful studies of 4

l D, 3

l P, 3

3 P, and 4

3 S

threshold polarization. The predicted zero polarization

of the 4

3 S case was confirmed. For the states with nonzero angular momentum, minima as much as several

volts wide near threshold appear to be established as physical properties of the excitation process, and the

indications are that the threshold polarizations required

by conservation of angular momentum are satisfied by

rapid increases in polarization within the last few tenths

of a volt above threshold.

17. Heavy Rare Gases

The first significant contribution on the excitation

functions of neon and argon came out of the work of Hanle (1930), at Jena, who studied the excitation

functions of neon at rather high pressures. He described some strong pressure dependences. Subsequently Fischer

(1933), also at Jena, studied the excitation functions

for a number of lines of neutral argon originating in the n = 5 shell and terminating in the n = 3 shell. He

also studied a number of lines of ionized argon excited

from the neutral ground state. Herrmann (1936) obtained excitation functions for a large number of argon

and neon lines. The pressures used were approximately

20 ju for argon and somewhat higher for neon. Maeir-Leibnitz (1935) obtained threshold measure- ments on neon and argon similar to those already described for helium, and Dorrestein (1942) studied

metastable production in neon. Milatz and Ornstein

(1935) observed absorption of the 6402- A line to determine the neon metastable concentration in an elec- tronic excitation measurement. Milatz and Woudenberg

( 1940) obtained absolute values for four neon excitation

functions, referring the measurements to a standard

tungsten lamp.

It would be impossible to critically evaluate these

data. The pressures used were in the range for which

excitation transfer and imprisonment of resonance

radiation are important for helium. There have been no

detailed quantitative studies of pressure-dependent

effects, analogous to those which have been carried out

for helium. The data were taken without regard to the

state of polarization of the observed radiation. Furthermore, there has been surprisingly little duplication of

E (eV)

Fig. 91. Threshold polarization, curve 2, and excitation, curve 1, of the helium 4*Dr-*21 Pi line obtained in a crossed beam measure- ment (McFarland, 1967).

71
measurements so that intercomparison of data is not

productive.

Data taken in the last decade are on a better basis

because lower pressures were used to reduce secondary

effects. However, there is still little duplication, and

there has been no recognition of the problems of polarization and imprisonment in this work on the heavy rare

gases. Devyatov and Kaptsov (1955) carried out relative

measurements of krypton excitation functions at about

10~3 Torr. Excitation functions were obtained for ten neutral krypton lines and four lines of singly ionized

krypton. Photographic detection was used. In 1957

Sin San Guk and A. M. Devyatov published further

relative measurements of krypton excitation functions,

also including some work on xenon. Pressures were again

about 10~3 Torr. These measurements were carried out

in sealed tubes, using a barium getter. Photoelectric

detection was used.

Volkova ( 1959) measured shapes of six argon excitation functions at pressures of 1.0X10-3 Torr, only one

of which is presented in her paper. Volkova and Devyatov (1959) measured the absolute values of these six lines and one other line at argon pressures of 4.68 X 10-3

Torr. They calibrated against a tungsten lamp, and

commented that their results for the 7 723-4- A argon

line agreed with the absolute value obtained by Herrmann in 1936. Volkova, Devyatov, and Kuralova

(1960) measured absolute values of several lines of

krypton and xenon. Zapesochnyi and Feltsan (1963),

measured relative excitation functions for three neon

lines. A number of these recent workers have commented on

the characteristic differences between helium excitation

and the excitation functions of the heavier rare gases. Russell-Saunders coupling holds for only a few levels for these heavier gases, and intercombination lines are common. Triplet excitation does not have the strongly

resonant character seen in helium. Zapesochnyi and

Feltsan (1966) presented shapes and absolute values for

excitation cross sections of the ten Paschen levels (2pn) of argon, and showed an interesting comparison of the

behavior of the 2p% levels of argon, krypton, and xenon,

which we show in Fig. 92. These excitation functions

peak more strongly toward threshold at higher atomic

masses.

18. Alkali Atoms

The properties of the light alkali metals are of much

interest from several points of view. They are, to a

reasonable approximation, one-electron systems and

have been investigated theoretically by a number of

workers (see Sec. 6). They are particularly interesting as components of stellar atmospheres and other plasmas,

in part because their low ionization potentials (~4-5

eV) lead to high ionization probabilities and put them

in the role of electron donors. Also, in contrast to hydrogen, the resonance lines occur in the visible or quartz

Fig. 92. Measured absolute excitation functions of the (Paschen) 2ps levels of argon (z = 18), Curve 1; krypton (z=36),

Curve 2; and xenon (z = 54), Curve 3. (Zapesochnyi and Feltsan,

1966).

( ultraviolet and may be prominent features of visible

stellar spectra.

Experimentally the alkalis present a unique set of

problems. Appreciable vapor pressures of these metals can be achieved at temperatures of a few hundred

degrees centigrade, and most of the excitation work has

been carried out in apparatus quite similar to that described for helium except that it is operated at elevated

temperatures. Typically, the reservoir of alkali metal is placed in a sidearm the temperature of which is controlled to achieve a desired vapor pressure. The rest of

the apparatus is usually operated at a somewhat higher

temperature to avoid condensation on the walls. At the

temperatures often used the experimental difficulties include the fact that glass becomes electrically con- ducting so that electrical measurements become diffi- cult. Alkali vapors are readily absorbed in glass causing

optical windows to deteriorate. This absorption also causes possible difficulties in attaining equilibrium.

Some attention has been given to the possibility of

dimer content in alkali atmospheres. Calculations based on thermal equilibrium indicate that the percentage of

atoms in the molecular form is typically a fraction of one percent at operating temperatures and increases

with temperature and vapor pressure. Serious contamination of atomic excitation functions would require

that the molecular excitation cross sections be large,

and would depend on the spectral resolution of the

detection equipment. Generally, it is probably not a

serious problem where high resolution spectrometers

are used but must be considered in each case. The effects of reabsorption of resonance radiation must also be

considered. Another complication is in the difficulty of

operating electron guns at energies down to one or two

volts in order to observe the excitation threshold. Some

of these experimental problems are circumvented for

measurements carried out with beams of alkali atoms,

but only a few such studies have been made.

For lithium the only available results were obtained

using lithium beams, because of the high temperature

(400°-500°C) required to obtain the necessary static pressure. The crossed beam results are all relative, no

absolute beam measurements having been attempted.

72
I

Fig. 93. Polarization (Curve 1) and relative intensity component 7|| (Curve 2) of the first resonance line (2p-*2s, 6708- A)

of lithium (natural isotopic composition) excited by electron impact, measured by Hafner and Kleinpoppen (1967). Curve 3, the total intensity observed at 90° to the electron beam, is calculated from Curves 1 and 2. Also shown on Curve 3 are the data of Hughes and Hendrickson (1964), 0.

Hughes and Hendrickson (1964) used a lithium beam

crossed by an electron beam to measure the relative

excitation cross section for the 2/>— line at 6708 A

(the 2p—*is line in terms of a hydrogenic scheme).

An interference filter was used so that the doublet was

not resolved. Beam density and the possibility of reabsorption of resonance radiation were not discussed.

In the course of studying the polarization of radiation excited by electron impact, Hafner and Kleinpoppen (1967) have obtained an excitation curve for

the 6708- A line over the range from threshold to about

80 eV, using an interference filter to isolate the line. Their published results show the intensity of the polarization component with electric vector parallel to the

electron beam. The measured function is easily corrected

using their measured polarization to obtain the total

intensity. This measurement and that of Hughes and

Hendrickson were carried out on lithium beams composed of natural abundances of the isotopes. The results

of both measurements are shown in Fig. 93. The agreement is quite good. In connection with their measure- ment Hafner and Kleinpoppen showed that depolarization of the radiation occurred as the beam density was

increased, due to reabsorption and reemission of the resonance radiation within the beam. Their results were

obtained at pressures low enough so that the effect was

negligible. The effect of such trapping on the observed

excitation curve for a resonance line would be threefold:

it would effectively increase the volume from which the

radiation was observed (although in a beam experiment convenient limits are provided by the size of the

beam) , it would depolarize the radiation, and it would

result in an increase in the cascading contribution due to an effective modification of branching ratios from the

higher p states. Hafner, Kleinpoppen, and Kruger (1965) measured

an excitation function for the 3

2D—>22 P (6103- A) line

of lithium and observed no depolarization as a function

of beam pressure, consistent with the fact that the 2rD

state does not optically connect to the ground state and

is affected only very indirectly through cascading with

imprisonment effects. Their results are shown in Fig. 94.

The most important result presented in these papers

by Kleinpoppen and his students is a measurement of

the threshold polarization of Li6

, Li7, and Na23

. Their

results for the resonance lines show that the polarizations decrease monotonically from threshold as a function of electron energy, and that the three threshold

polarizations obtained agree with the three values pre- dicted from the theory of Percival and Seaton, these

being different primarily because of differences in widths and separations of fine and hyperfine levels in the different isotopes. However, the polarization of the

non-resonant 3

2 D—>22 P (6103-A) line appears to fall below the theoretical value at threshold.

Feldman and Novick (1963) have carried out experiments which demonstrate the existence of long-lived

excited states in lithium, potassium, and rubidium.

These states are presumed to be associated with the

excitation of core electrons, rather than the valence

electron. The energies of such states are above the

ionization energy associated with the valence electron

but below the second ionization potential, and therefore

may be autoionizing states. By collecting either the

ions or electrons produced in the autoionization process, Feldman and Novick obtained excitation functions and,

by changing the distance from the excitation region to the detector, obtained measurements of the effective

lifetimes.

All measurements of excitation functions of sodium

and the heavier alkalis were carried out using static

systems rather than beams.

As in the case of lithium the reabsorption of reso- nance radiation in sodium should be significant down to quite low pressures. The effect of this on the first resoFig. 94. Polarization (Curve 1) and relative intensity component /|| (Curve 2) of the 6103-A (32 77—>22 P) line of lithium

(natural isotopic composition) excited by electron impact, measured by Hafner, Kleinpoppen, and Kruger (1965). Curve 3, the

total intensity observed at 90° to the electron beam, is calculated from Curves 1 and 2.

73
nance line would be to increase the source volume, to produce some modification of the cascading patterns,

and to depolarize the radiation.

Hafner, Kleinpoppen, and Kruger (1965) have measured the polarization of the sodium D lines and found

that it decreases monotonically with energy from about

15% at threshold. This results in an anisotropic radiation pattern as indicated by equation (158) but the

polarization is not large and the difference 7(90°)—/

is 5% or less of 7. The distortion due to instrumental

polarization could be 15% at most, and this only in the

case of nearly complete instrumental polarization.

The other sodium lines will be subject to greater

errors. Measurements of absolute values are subject

to modification of the effective branching ratios when

imprisonment of the higher resonance lines occurs. The

effects of excitation transfer would be apparent in dis- tortion of nonresonant lines, but most experimental

work has been carried out at pressures low enough so that this should not be important.

In general, the shapes of alkali excitation functions

determined experimentally should be somewhat less subject to imprisonment and polarization effects than

helium, while the measured absolute values should be

quite sensitive to the effects of imprisonment of reso- nance radiation. Unfortunately, there has not been

enough experimental work directed at evaluation of

these pressure and polarization effects. There has- not

been enough work for intercomparison to indicate

reproducibility. Much of the available work is confined

to a narrow energy range near threshold, where theoretical work is likely to be unreliable, so that comparisons with theoretical shapes over the narrow energy

range available are not very illuminating.

Zapesochnyi and Shimon ( 1962) studied the dependence of the shape of the excitation function for the first sodium resonance line on pressure. The results are shown in Fig. 95. The measurements at 2.5X10-4 and

5X10~4 Torr are in agreement, while the curve obtained at 10-3 Torr falls more rapidly with energy. This

Fig. 95. Behavior of the excitation function of the resonance

line of atomic sodium as a function of pressure: 2.5 X 10-4 Torr (), 5X16~4 Torr (A), and 1X10~3 Torr (X). The electron current density was 1.5 X10~4 A/cm2

. (Zapesochnyi and Shimon,

1962).

behavior is difficult to explain. Modification of cascading

patterns by imprisonment of resonance radiation would seem not to provide an explanation. The possibility of contamination of the spectrum, as from the sodium

dimer, does not seem likely. Under the experimental

conditions the dimer content is a few tenths of one per- cent and rises with temperature as is easily demonstrated from simple thermodynamic calculations. A

dimer excitation cross section peaked strongly at threshold would conceivably give the observed result. A more reasonable possibility is that the trapping of resonance radiation so extends the effective life of the

upper state at 10~3 Torr that chemical interactions between excited states of sodium provide an energy-dependent mechanism for depleting the population of the

upper state of the D line. In any case it is reasonable

to accept the shape obtained at 2.5 and 5X10~4 Torr

pending a more adequate understanding of the processes which cause the distortion.

Haft (1933) measured the excitation functions of a number of sodium lines, but worked at rather high pressures. Christoph (1935) measured the absolute . cross section for excitation of the sodium D lines using a tungsten strip lamp for radiometric calibrations, and working

at a side-arm temperature of 163.5°C. Volkova (1961)

measured absolute values of ten sodium lines at temperatures of 182° and 202°C (/>~5X 10-5 mm Hg) . Because of window darkening due to sodium absorption

the intensities were determined by comparison with

lines excited in helium at a pressure of 4X10-3 Torr.

Absolute measurements of helium excitation by Yakhontova (1959) were used to provide the calibration.

This indirect calibration process led to stated errors of

about 70%.

The most extensive set of measurements under conditions of reasonably low pressure have been carried out

by Zapesochnyi and Shimon (1965). They give errors

of 30-35% in absolute values and relative errors not

exceeding 10%.

The absolute value of the sodium D line excitation

function was measured by Zapesochnyi and Shimon

(1965) at 164°C, essentially the same temperature as was used by Christoph. In fact, Zapesochnyi and

Shimon used an absorption coefficient of 0.4 cm-1 determined by Christoph to correct for absorption of resonance radiation, so the two measurements, carried

out at the same pressure, are not completely independent. Figure 96 shows the absolute excitation functions

obtained by Zapesochnyi and Shimon for the principle

series of sodium. Christoph’s three absolute measure- ments for the D line, are shown there for comparison.

We have also included Born approximation results

(Vainshtein, 1965) and the results of a close coupling

calculation by Barnes, Lane, and Lin (1965).

Christoph gave his results in terms of cm2/cm3 per

electron at one millimeter pressure. He gave his working

pressure as 1.6X 10-5 Torr at 163.5°C, which is reason74
Fig. 96. Absolute excitation function of the resonance line of atomic sodium (5890-96 A, 3p—>3j). Included are three points by

Christoph (1935) () ; the excitation function measured by

Zapesochnyi and Shimon (Curve 1) ; the Born approximation

cross section for direct excitation of the 2>p state, not including

cascading, by Vainshtein (1965) (Curve 4) ; and a close coupling

result, not including cascading, by Barnes, Lane, and Lin (1965)

(Curve 5). Also shown are excitation functions for the sodium

\p—>4x and 5p—*5s transitions (Curves 2 and 3, respectively)

measured by Zapesochnyi and Shimon (1965).

ably consistent with more recent vapor pressure tables. From a summary of vapor pressure data by Hultgren,

Orr, Anderson, and Kelly (1963) we obtain a vapor

pressure of 1.7X10~5 Torr, and the difference is not

significant. Christoph’s statistical accuracy was of the

order of 25% to which must be added basic radiometric

errors and systematic pressure errors. He gave values of

105 (at 11 eV), 99 (at 21 eV), and 74 (at 32 eV) in

units of cm2/cm3. According to Kieffer and Dunn, gas

density for an ideal can be computed from

p= (3.535 XlOl6 )Pc ( 273/ Tc ) atoms-cnr3

, (180;

where Pc and Tc are the pressure in Torr and temperature in °K, respectively. Use of this expression permits

conversion of the data to cm2 per atom yielding 48, 45,

and 34X10-16 cm2

. Bates, Fundaminskv, Leech, and

Massey (1950) published Christoph’s results plotted in a figure at values of 46.6, 44.2, and 32.6X10-16 cm2

, which have been widely quoted. Zapesochnyi and

Shimon (1965) have erroneously remarked that values

in use in the literature, were a factor of almost two too

high, but they have evidently overlooked the factor

273/Tc in the density formula.

This being the case, it is possible that Zapesochnyi

and his colleagues have not correctly interpreted their

own experimental work on sodium and, perhaps, the

other alkali atoms. For excitation of the sodium D line their absolute values would be too low by the factor

273/437 = 0.625. As can be seen from Fig. 96, such an

adjustment would bring their excitation function into

better agreement with the theoretical results at higher

energies. The data for excitation of alkali atoms pre- sented in this review, however, are taken without modification from the papers by Zapesochnyi and his

colleagues.

Figures 97 and 98 give the absolute results obtained

by Zapesochnyi and Shimon for the sharp and diffuse

series of sodium. These results were obtained with pres- sures not greater than 4.5 X ICC4 Torr and with energy

spreads of not more than 0.8 eV. In Table LXV, taken

from Zapesochnyi and Shimon (1965), their absolute

values for the sodium photon-excitation cross sections,

taken at cross section maxima, are compared with

results of Christoph (1935) and of Volkova (1961)

(a corrected value is substituted for the last figure in the column, an error having occurred in the English

translation)

.

Zapesochnyi and Shimon (1965) investigated the

dependence of the absolute values, within a given spectral series, on the principal quantum number n of the

upper level. They found a good correspondence to the

relation Q(n) = c/w“, where for the diffuse series a= 5.25

and c=5.56X 10“4

; and for the sharp series a = 6.87 and

C= 1.26X 10-12. A departure is noted for the excitation

of the lowest D level, presumably because this lies in the

path of cascade transitions from higher F levels. The rather high values they obtain for a imply that

cascading contributions can be estimated on the basis

of only a few higher levels. For example, from Table

LXV and Figs. 96 and 98 it can be seen that the reso- nance line contains a contribution of about 25% due to cascading from the D states. Other cascade fractions are smaller.

The work on potassium has similarly all been carried

out in static vapor experiments rather than with beams.

Volkova (1959) measured excitation functions for two lines of the principal series of neutral potassium and

eleven lines of singly ionized potassium excited from

the neutral ground state, and measured the absolute

values at 60 eV. In another paper (Volkova, 1962) are presented absolute values measured at 5 eV for 19

lines of the diffuse and sharp series of neutral potassium.

These were all measured using a tungsten ribbon lamp

as a radiation standard, and presumably at pressures of

about 1.3Xl0~4 Torr, although no explicit information

Fig. 97. Absolute excitation functions for the sharp series of atomic sodium. 1. 6154-6161 A; 2. 5149-5154 A; 3. 4748-4751 A;

4. 4541-4646 A; 5. 4420-4423 A; 4343 A (Zapesochnyi and

Shimon, 1965).

75
to this effect is given in the second paper. The shapes of

three excitation functions were measured photographically using a retarding field to obtain simultaneous observations over a range of energies as a function of position in the electron beam.

Subsequently, Volkova, and Devyatov (1963) measured the excitation functions of the resonance lines at 7665 A and 7698 A. This measurement was carried out

at a vapor pressure of 2X 10-6 Torr (72°C), a pressure

at which some absorption of resonance radiation occurs. They built a special tube in which the distance from

the beam to the window was minimized. The absorption coefficient at the center of the line was calculated

and from this the total absorption for a Doppler line was estimated. The assumption of a Doppler line shape was verified by measurement with a Fabry-Perot

etalon. The shapes of the functions were also measured

at 2X10-6 Torr. The stated maximum error is 35%.

Another recent study of potassium was reported by

Zapesochnyi, Shimon, and Soshnikov (1965). They

measured excitation functions of 28 lines of potassium

at pressures between 9X10-5 and 5X10-4 Torr (122°-

150°C) and made an effort to achieve low electron

densities. The static vapor pressure method was used

but detection was photoelectric. The error quoted is 30-35% in the absolute values. These were measured

at the two extremes of the static pressure range except

for the principal series, and the results differed by 15%

or less. Excitation functions have been obtained by Zapesochnyi and Shimon (1966a) for cesium, and by Shimon

(1964) for rubidium.

Zapesochnyi and Shimon ( 1966b) designed an apparatus which included a movable window mounted on a

bellows, for the purpose of studying the resonance lines

of the alkalis. The window was driven by a micrometer

screw so that the position, and hence the thickness of

absorbing vapor, could be accurately controlled. They

showed that the intensity varied exponentially with the

thickness of the absorbing layer at sufficiently low pres- sure and distance in accordance with Beer’s law. An

extrapolation to zero thickness permits determination

of the absolute value. The cross sections for the reso- nance lines of potassium, rubidium, and cesium obtained

in this manner are shown in Figs. 99, 100, and 101.

Apparently the method was not applied to sodium, but

the results presented here have a similar form to that

shown in Fig. 96. Also shown in Fig. 99 are the results

obtained by Volkova and Devyatov (1963). Their

cross section is peaked near threshold in contrast to the

shape obtained by Zapesochnyi and Shimon. The latter authors point out that Volkova and Devyatov used a very high electron beam intensity. In their sodium work

Zapesochnyi and Shimon demonstrated that the excitation function was similarly distorted by such dense

electron beams.

The absolute values obtained for the resonance lines are shown in Table LXVI.

Fig. 98. Absolute excitation functions for the diffuse series of atomic sodium. 1. 8183-8194 A; 2. 5683-5688 A; 3. 4979-4983 A;

4. 4664-4669 A; 5. 4494-4498 A; 6. 4390-4393 A; 7. 4321-4324 A;

8. 4273-4276 A (Zapesochnyi and Shimon, 1965).

Fig. 99. Absolute excitation functions of the first resonance

lines of potassium. Curves 1 and 4 are the excitation functions

for the 7665-A and 7698- A lines, respectively, measured by Vol- kova and Devyatov (1963) . Curves 2 and 3 are the combined exci- tation function for the resonance line, with and without cascading,

obtained by Zapesochnyi and Shimon (1966b).

Fig. 100. Absolute excitation function of the rubidium resonance

lines (7948-7800 A) with and without cascading (Zapesochnyi

and Shimon, 1966b).

Fig. 101. Absolute excitation function of the cesium resonance

lines (8944-8521 A) with and without cascading (Zapesochnyi

and Shimon, 1966b).

76
19. Heavy Two-Electron Systems

The experimental study of helium excitation has been

discussed in considerable detail. It is the simplest of the

two-electron systems. It has been studied extensively,

is relatively tractable theoretically, and has important

applications in astrophysics and elsewhere. Of the other

two-electron systems, mercury has also been studied

extensively and some work has been carried out on cadmium and zinc. These atoms are more complicated,

have not received such extensive theoretical treatment,

and are not quite so interesting with respect to current

applications.

However, they provide some physically interesting contrasts with helium. In helium, L-S coupling holds

well and the spectrum is free of inter-system transitions.

In the heavier atoms spin-orbit interaction is strong and

inter-system transitions occur with competitive probabilities. The energy levels within a given shell are also

strongly modified by spin-orbit interaction energy so that there are quite significant differences in energy between states of different angular momenta. This strong

spin-orbit interaction, and the fact that the electrostatic interaction with the nucleus is less in the heavier

elements, results in a higher lying ground state and

larger intervals between some of the relevant upper

levels. The transition to the ground state is not so strongly favored over other transitions, energetically,

as in helium. One result appears to be that cascade

transitions make more significant contributions to the

observed excitation functions, as has been demonstrated quite effectively in the more recent experimental

work discussed below.

Another result is that trapping of resonance radiation

has relatively less influence on the effective lifetime of a given state of the heavy atom than for helium, and

therefore less effect on the observed intensity of lines not connecting to the ground state. In helium the

3

4 P—>2X S line may be enhanced by a factor of forty

through trapping of the 3'P—>1 ! A line. The factor is presumably lower for corresponding mercury, cadmium,

and zinc lines. The effects of trapping, however important, may be expected to occur at the same pressures, within an order of magnitude, as in the case of helium.

L. J. Kieffer’s “Bibliography of Low Energy Electron

Collision Cross Section Data” (1967) lists 53 papers

under the headings “Electronic Excitation, Experimental, Hg.” For helium there are 95, for cadmium 6,

and for zinc 3. For the case of helium, there is reason for desiring rather accurate information about the shapes of

the excitation functions as well as their absolute values.

Considerations of pressure-dependent secondary effects,

especially trapping of resonance radiation, have reduced

the number of helium papers considered in detail to a

relatively small number.

For the case of mercury, the very accurate determination of excitation functions is much less interesting

because the theory is not in a refined state and because

the applications are not so important. For the case of

mercury, zinc, and cadmium, it is the structure of the

excitation functions which provides the interest. Thus

the essential experimental problems are not so much the

refinement of spectroscopic techniques, to which some of the earlier workers may have been more alert than many

of the recent workers, but the refinement of photoelectric recording techniques and the techniques of obtaining high-energy resolutions in low-energy electron

beams, at which modern workers are beginning to excel. The earlier workers did not expect complex structure and most of them did not see it. Recent work has shown

the excitation functions to be rich in structure. Therefore, we will confine our attention to the more recent work. However, we note that authors in this field are

either blissfully unaware of, or find it easy to ignore,

the distortions which result from polarization through

instrumental polarization and as a result of anisotropic

radiation patterns.

The pressure dependence problem in mercury seems

less serious than in helium, although it should not be

ignored to the extent that it is in the literature, particularly where absolute measurements are involved. The

pressures used are not always stated explicitly. In most

cases the mercury pressures used were probably those

of mercury at room temperatures, for which the equilibrium vapor pressure is about 10-3 Torr, or lower

temperatures, for example that of melting ice, for which

the pressure is about 2X10~4 Torr. Zinc and cadmium

are used at temperatures of several hundred degrees

centigrade in order to achieve comparable vapor pressures. Evidence that excitation functions of certain lines are highly affected at mercury pressures characteristic of room temperature appear in the literature. Skinner and

Appleyard (1927) noted that it was necessary to work

at temperatures of —15° to — 20°C to avoid almost

complete depolarization of the 6

3 Pi—>61 6'

0 (253 7- A) line which occurs at room temperature. Federov (1965)

studied pressure dependence of the 6

3D2-j>61 Pi (5770- A)

polarization curve and found that the portion immediately adjacent to threshold where only direct 6

3 D2 excitation was effective was decreased by a factor of two in changing the pressure from 8X10-4 to 10-2 Torr, and

that significant depolarization occurred at 4X 10~3 Torr. His explanation was based on excitation by scattered

electrons but at these pressures this does not seem possible. Other secondary processes must be invoked. He

noted the appearance of a halo, reminiscient of the

effects observed in helium by Lee and others, and there

attributed to trapping of resonance radiation.

The mercury literature contains two papers which

should be of special interest to experimentalists working

in this or related fields, because they contain unusually

complete and interesting accounts of careful experimental investigations. The first of these, by Zapesochnyi (1954), marks the beginning of the modern high

resolution work which has revealed the extensive struc77
turing of the excitation functions. He attributed the structure primarily to cascading. The second paper is a

quite comprehensive report on experimental techniques

by Jongerius (1961). This paper also includes an excellent summary of the older literature. Both papers describe successful attempts to reproduce results of

earlier workers in order to account for the differences. In addition to papers by these workers and their colleagues, one other paper of particular significance by Anderson,

Lee, and Lin (1967) has appeared recently. This paper

reports a large number of experimental results. It in- cludes only the briefest possible account of experimental techniques, but it contains the most complete

analysis and discussion of the structure of the excitation available.

Figure 102 shows the experimental results for the

7

3 5i->6

3 P2 (5461- A) line of mercury published by

Frish and Zapesochnyi (1956) in support of their

original interpretation of the structure in terms of cas- cading. The lower curves taken from their paper show

their interpretation in terms of excitation functions of higher levels. The interpretation is qualitative with

respect to shapes and magnitudes but quantitative with

respect to intervals between excitation thresholds. It is a

quite convincing demonstration, indicating the great

importance of cascading in the development of the

excitation function. Zapesochnyi (1954) gives similar

interpretations of structure for other mercury lines. It is interesting to note that while most early papers

present smooth excitation functions for the 5461-A

line, Seibertz (1931) succeeded in obtaining most of

the essential features in his measurement of the 7

3 Sr~

6

3 Po (4047- A) line for which the excitation function

has the same shape as for the 5461-A line. Absolute values of a large number of lines have been

obtained by Jongerius (1961) and by Anderson, Lee,

and Lin (1967), with a fair degree of consistency between the results. The results by the latter group tend

to run higher than those by Jongerius by 15-50%. Older

results by Hanle and Schaffernicht (1930) are substantially higher than the recent values.

Of special interest are the relative values of excitation

functions with the same upper states. Several such sets have been studied. Jongerius gives relative transition

probabilities for lines with 7

3 5i, 83 »Si, and 6

3D2 upper

states. Anderson, Lee, and Lin present results for 7

3 5i,

7

1 vS0 , 8‘Pi, 9

1 Pi, and ltFPi upper states. These last three

involve comparisons between singlet-singlet and singlet-triplet transitions which yield information about

the mixing coefficients.

Except for some very early work, the only available

data on excitation functions for zinc and cadmium are from recent work by Zapesochnyi and his collaborators.

These data reveal structure similar in many respects to that found in mercury. We show on Figs. 103, 104, and

105 some results obtained by Zapesochnyi and Shpenik

(1966). This work was accomplished using a 127°

cylindrical electrostatic analyzer to obtain narrow

78

Fig. 102. Relative excitation function for the 7

3 Si—>63 P2 (5461-A) transition in mercury (Curve 1) showing an interpretation of structure in terms of excitation cross sections for higher

levels [(2) 7

3 P; (3) 7

3 S; (4) 8

3 P; (5) 103?; (6) 93 P; (7) ionization plus excitation] by Frish and Zapesochnyi (1956).

E (eV)

Fig. 103. Relative excitation functions for the corresponding 3 S

states in mercury (73 Si), Curve 1; cadmium ^Si), Curve 2; and zinc (53 Si), Curve 3, measured with an electrostatic electron monochromator to obtain energy spreads of 0.1, 0.2, and 0.3 eV,

respectively, for the three curves (Zapesochnyi and Shpenik,

1966).

Fig. 104. Relative excitation functions for the corresponding 3P

states in mercury (6*Pi), Curve 1; cadmium (53 Pi), Curve 2; and zinc (43 Pi) , Curve 3, measured with an electrostatic electron monochromator (Zapesochnyi and Shpenik, 1966).
Fig. 105. Highly resolved (~0.1 eV) excitation function for the mercury 3650- A line originating in the 6

Z D3 level (Zapesochnyi

and Shpenik, 1966)

.

E (eV)

Fig. 106. Polarization of the mercury triplet 7

3 Ai—>63 P2 ,i,o as a

function of electron energy, measured by Federov and Mezentsev

(1965) : Curve 1 , 4047 A; Curve 2, 5461 A; Curve 3, 4358 A. The

occurrence of finite polarization beginning 0.4r-0.5 eV above thres- hold is attributed to cascading from n

3 P levels.

Fig. 107. Polarization of the mercury lines at 4347 A (71 Di—>

6bPi), Curve 1; and at 5770 A (63 Z72—^EPi) , Curve 2, as a function of electron energy, measured by Federov and Mezentsev

(1965).

energy spreads in the electron beam. In Fig. 103 are compared the excitation functions obtained for lines

with the upper states 7

3 Sh 63 Sh and 53 Si in mercury,

cadmium, and zinc. The energy spreads used were given

as 0.1, 0.2, and 0.3 eV for the three cases. The fact that

the mercury structure appears to be narrower is due in part to the energy spread used.

In Fig. 104 are compared the excitation functions

corresponding to 6

s Pi, 53 Pj, and 43 PX levels in the three

respective atoms, and in Fig. 105 is a highly resolved

6

3Dz excitation for mercury. Zapesochnyi and Shpenik

relate the maxima to excitation of higher levels with

subsequent cascading.

Skinner and Appleyard (1927) investigated the

polarization of a number of the mercury lines excited

by electron impact. Their work has been the subject

of a great amount of interest because of their observation that the polarization of all lines studied went to zero at threshold. This conflicts with the requirements

of conservation of angular momentum at threshold.

Another main feature of their results was the highenergy trend to negative polarization of the 6

1 D2^61 Pi

and 7

1 A->61 Pi lines, and the observation of negative

maxima in polarization at lower energies for several of their lines. From the preceding discussion of the importance of

cascading in the formation of the mercury excitation

functions it is clear that the interpretation of the observed polarization should be quite complex, except near threshold below energies at which the cascading

mechanisms can be actuated. Furthermore, one would

expect a certain amount of structure in the polarization

which does not show up in the work of Skinner and

Appleyard. Federov and Mezentsev (1965) have re- cently studied the polarization of some mercury lines in the region near threshold. Using relatively narrow

energy spreads (~0.6 eV) they find structure in the

polarization curves related to cascading. They find the

lines S

1 ^—>6^1 and 71 S0—>63 Pi unpolarized near

threshold.

Their polarization curves for the 7

3 Si—>63 P2 .i,o lines are shown in Fig. 106. The polarization is zero near

threshold but becomes finite half a volt above threshold

because of cascading from the n

3 P levels. In Fig. 107

are shown the polarization curves for the 7

1 D2—+61 Pi

(4347-A) and 6

3Z)2—K^Pi (5770-A) transitions near

threshold. Skinner and Appleyard found the 4347-A

polarization to fall to zero near threshold, but with

better resolution Federov and Mezentsev show that

it rises again as threshold is approached from a higher

energy, in a manner not inconsistent with the requirements of angular momentum conservation. The behavior is similar to that now found in helium.

20. Experimental Summary

Experimental research in the electron impact excitation of atoms is entering a new phase. Many of the

79
important atomic excitation functions have been sur- veyed again and again in semi-quantitative fashion.

There is little point to more work of this type on helium.

The technology is available to support reliable measure- ments of relative excitation functions. The technology

must be applied with due attention to the physical

properties of the atom under study and to the effects of ambient fields on the observations. Hopefully, in this

article we have indicated the important physical properties and processes to be considered, at least for helium.

For atoms heavier than helium much more study is needed in order to determine the nature and importance

of secondary processes. The quality of absolute measurement depends on the

quality of radiometric and gas density standards, and on

the convenience with which they can be applied in the

laboratory. At the present time the reliability of absolute measurement cannot be much better than 10%.

The reproducibility with which the standards are applied from one laboratory to another is reflected in the

scatter of absolute measurements of a given excitation

function at the same energy, as in Tables LXIVd and

LXIVe. The scatter does not provide an indication of

the true reliability of the measured values, since certain

systematic effects may occur in all the measurements

and since the standards tend to be coupled through

intercomparison and through use of common supporting

data. The significance which may be attached to a comparison between theory and experiment is limited

accordingly. Other limitations include the uncertainty

in making proper allowance for cascading, polarization,

and imprisonment of resonance radiation.

Some indication of the significance of a shape comparison between experiment and theory can be gained

from considering the figures in which we have intercompared the experimentally determined helium excitation

functions. Some authors of the experimental results have

clearly been more thorough in their treatment of secondary and instrumental effects than others. However, more high quality work is necessary before a comparison between theoretical and experimental shapes

could be trusted in great detail. One serious limitation

clearly lies in the current confusion over the polarization of radiation excited by electron impact, apparent

in Fig. 73; but there are other limitations as may be

seen from consideration of Fig. 83 for the 4

3 S excitation

function and of Fig. 79 for the 4

1

S excitation function,

for which polarization should not be a factor. Accurate

allowance for cascading may be a serious limitation in

these and other cases. However, the comparison between theory and ex- periment has provided some extremely important re- sults:

(1)

In the range of energies below 50 or 60 eV the

close coupling calculations, involving as many states as it is presently practical to include but omitting higher

states and the continuum, yield results which rise much

higher than can be considered compatible with experi- t mental results, < (2) except in the immediate vicinity of threshold

where all open channels can be included in the close 1 coupling calculation. In the threshold case there has < been some remarkable agreement between theory and ! experiment, as is evident in Figs. 65, 67, and 88.

(3) The comparison between theory and experiment

for the helium triplet excitation functions shows that

there is a serious discrepancy in the high-energy behavior, the theoretical results falling off as E

~3 and the

experimental results falling much more slowly.

If it is difficult to make definitive statements about

the quality of helium excitation functions, it is even more difficult for the excitation functions of the heavier

atoms. The study of helium excitation gives a picture of

secondary and instrumental effects which must apply to work on other atoms. However, there is little evidence in

the literature that secondary effects for heavier atoms

have been properly considered, and it must be presumed

that the available data contains serious distortions.

The future of electron impact excitation work may

lie in the further development of methods of measuring

the inelastically scattered electron currents. This would

require measurement of scattered currents at all angles

and subsequent integration over the total solid angle.

It is not clear that accuracy can be achieved at the

lower electron energies. Nevertheless, the method is attractive because it would yield direct excitation

cross sections. There would be no cascade problem, no

imprisonment problem, and no collisional excitation

transfer problem. The angular distribution of scattered

electrons would certainly be nonisotropic, which would

present a problem fully as difficult as that presented by

the polarization of the radiation in the optical method.

However, the only one of the above problems which

cannot be avoided in the optical method is that of

cascading. It seems that the optical method has not

yet been properly exploited. With a proper consideration of the physics involved, more carefully planned

experimental work, and more adequate treatment in

publishing results, the full value of the optical method

could be realized.

ACKNOWLEDGMENTS

This review was supported by the National Bureau of

Standards through the National Standard Reference

Data Program. The authors are grateful for the use of

facilities of the JILA Information Center, which is supported by the Advanced Research Projects Agency

(Project defender), which is monitored by the U. S. Army Research Office—Durham, and the National

Bureau of Standards, through the National Standards

Reference Data Program. We are also grateful for the

many months of conscientious and careful work by the

Information Center staff, particularly by Elizabeth

Hosack, Mary Ann McCormick, and Ross Fraker, on

80
the preparation of data, figures, and references for this

article. We would like to express our thanks to M. J. Seaton,

D. W. O. Heddle, and L. J. Kieffer for very helpful

discussions of some aspects of this review. One of us,

S. J. Smith, would like to acknowledge the contribution

made by his scientific colleagues in JILA through in- numerable discussions of relevant experimental problems.

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Altshuler, S., 1952, Phys. Rev. 87, 992.

, 1953, Phys. Rev. 89, 1093. Anderson, R. J., E. T. P. Lee, and C. C. Lin, 1967, Phys. Rev.

157, 31. Barnes, L. L., N. F. Lane, and C. C. Lin, 1965, Phys. Rev. 137

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Spectry. USSR 10, 367 (1961)].

Volkova, L. M., 1959, Izv. Akad. Nauk SSSR Ser. Fiz. 23 , 968

[English transl.: Bull Acad. Sci. USSR Phys. Ser. 23, 957

(1959)].

, 1959, Opt. i Spektroskopiya 6, 273 [English transl.: Opt.

Spectry. USSR 6, 179 (1959)].

, 1961, Opt. i Spektroskopiya 11 , 775 [English transl.: Opt. Spectry USSR 1 1 , 420 ( 1961 ) ].

, 1962, Opt. i Spektroskopiya 13, 849 [English transl.: Opt. Spectry. USSR 13, 482 (1962)].

, and A. M. Devyatov, 1959, Opt. i Spektroskopiya 7 , 819

[English transl.: Opt. Spectry. USSR 7 , 480 (1959)].

, and A. M. Devyatov, 1963, Izv. Akad. Nauk SSSR Ser. Fiz. 27, 1052 [English transl.: Bull. Acad. Sci. USSR Phys. Ser. 27,

1025 (1963)].

, A. M. Devyatov and A. V. Kuralova, 1960, Izv. Akad.

Nauk SSSR Ser. Fiz. 24, 950 [English transl.: Bull Acad. Sci. USSR Phys. Ser. 24, 952 (I960)].

Vriens, L., 1965, Physica 31 , 1081. Wheeler, J. A., 1933, Phys. Rev. 43 , 258. Weiss, A. W., 1963, Astrophys. J. 138, 1262. Wiese, W. L., M. W. Smith, and B. M. Glennon, 1966, “Atomic

Transition Probabilities—Hydrogen Through Neon,” National

Standard Reference Data Series, National Bureau of Standards

4, Vol. I. Wolf, R., and W. Maurer, 1940, Z. Physik 115, 410. Wu, T. Y., 1960, Can. J. Phys. 38, 1654. Yakhontova, V. E., 1959, Vestnik Leningr. Univ. Set. Fiz. i Khim. 14, 27. Yamanouchi, T., Y. Inui, and A. Amemiya, 1940, Proc. Phys.

Math. Soc. Japan 22, 847. Yavorskii, B. M., 1947, Zh. Eksperim. i Teor. Fiz. 17, 315. Zapesochnyi, I. P., 1954, Vestnik Leningr. Univ., No. 11, 67 [Translation available from American Meteorological Society,

AD 297 209 (1962)].

, 1966, Astron. Zh. 43 , 954 [English transl.: Soviet Astron.

AJ 10, 766 (1967)].

, and P. V. Feltsan, 1963, Izv. Akad. Nauk SSSR Ser. Fiz. 27, 1040 [English transl.: Bull. Acad. Sci. USSR Phys. Ser. 27, 1015 (1963)].

, and P. V. Feltsan, 1965, Ukr. Fiz. Zh. 11 , 1197.

, and P. V. Feltsan, 1966, Opt. i Spektroskopiya 20, 521 [English transl.: Opt. Spectry. USSR 20, 291 (1966)].

, and L. L. Shimon, 1962, Opt. i Spektroskopiya 13, 621 [English transl.: Opt. Spectry. USSR 13, 355 (1962)].

, and L. L. Shimon, 1965, Opt. i Spektroskopiya 19, 480

[English transl.: Opt. Spectry. USSR 19, 268 (1965)].

, and L. L. Shimon, 1966a, Opt. i Spektroskopiya 20, 753 [English transl.: Opt. Spectry. USSR 20, 421 (1966)].

, and L. L. Shimon, 1966b, Opt. i Spektroskopiya 21 , 261 [English transl.: Opt. Spectry. USSR 21 , 155 (1966)].

, L. L. Shimon, and A. K. Soshnikov, 1965, Opt. i Spektroskopiya 19, 864 [English transl.: Opt. Spectry. USSR 19, 480

(1965)].

, and O. B. Shpenik, 1966, Zh. Eksperim. i Teor. Fiz. 50, 890

[English transl.: Soviet Phys.—JETP 23 , 592 (1966) ].
Table I. Total cross sections for s—>s excitations of atomic hydrogen, calculated using the first Born approximation (Vainshtein, 1965)

.

Total cross sections in units of irao2 .“

Impact

energy e in threshold

units Is—*2s Is—>3s Is—>4s Is—>5s Is—>6s Is—>7s Is—>8s Is—>9s 2s—»3s

1.16 2. 16( — 1) 3. 93 ( — 2) 1.42( — 2) 6.79( — 3) 3. 79( — 3) 2. 33 ( — 3) 1 . 54 ( — 3) 1.07(— 3) 1.01(1) 1.64 2.44( — 1) 4.31 ( — 2) 1 . 54 ( — 2) 7.34(— 3) 4. 08 ( — 3) 2.51 ( — 3) 1 . 66( — 3) 1 . 15 ( — 3) 1.28(1)

2.44 1 . 94 ( — 1

)

3. 37 ( — 2) 1 . 20( — 2) 5. 70( — 3) 3. 1 7

( — 3) 1 .95 ( — 3) 1 . 28 ( — 3) 8.92 ( — 4) 1.11(1)

3.56 1 . 44 ( — 1

)

2. 48 ( — 2) 8.79 ( — 3) 4. 17 ( — 3) 2 . 3 1

( — 3) 1 ,42( — 3) 9. 37 ( — 4) 6.51 (—4) 8.66(0)

5.00 1 . 07 ( — 1) 1.83 ( — 2) 6.48( — 3) 3.07( — 3) 1.70( — 3) 1 .05 ( — 3) 6 . 89 ( — 4) 4. 79 ( — 4) 6.61(0)

6.76 8. 16 ( — 2) 1 . 38 ( — 2) 4. 89 ( — 3) 2.31 ( — 3) 1 . 28 ( — 3) 7. 89 ( — 4) 5. 20 ( — 4) 3. 61 ( — 4) 5.09(0)

11.24 5.04( — 2) 8.51 ( — 3) 3.01 ( — 3) 1.42(— 3) 7. 88 ( — 4) 4 . 84 ( — 4) 3. 19 ( — 4) 2.21 ( — 4) 3.20(0)

17.00 3.38( — 2) 5.69 (-3) 2.01 ( — 3) 9 . 49 ( — 4) 5.27( — 4) 3. 23 (—4) 2. 13 ( — 4) 1 . 48 ( — 4) 2.16(0)

24.04 2.41 ( — 2) 4.05( — 3) 1.43(— 3) 6. 75( — 4) 3. 75(— 4) 2. 29 ( — 4) 1 - 51 ( —4) 1 . 05 ( —4) 1.54(0)

Impact

energy e in threshold

units 2s—»4s 2s—>5s 2s—»6s 2s—>7s 2s—*8s 2s—*9s 3s—>4s 3s—>5s 3s—>6s

1.16 1.64(0) 5 . 82 ( — 1) 2 . 80 ( — 1

)

1 . 58 ( — 1) 9.91 ( — 2) 6. 65 ( — 2) 7.57(1) 1.07(1) 3.57(0)

1.64 2.05(0) 7 . 20 ( — 1

)

3 . 45 ( — 1

)

1 . 94 ( — 1

)

1.21(-1) 8 . 14 ( — 2) 1.24(2) 1.80(1) 6.07(0)

2.44 1.73(0) 6 . 00 ( — 1

)

2 . 86 ( — 1

)

1.61 ( — 1) 1.00(-1) 6. 70( — 2) 1.20(2) 1.68(1) 5.59(0)

3.56 1.31(0) 4.53( — 1) 2 . 1 5

( — 1

)

1.21(-1) 7.51 ( — 2) 5. 02 ( — 2) 9.82(1) 1.34(1) 4.39(0)

5.00 9.88( — 1) 3.39( — 1) 1.6K-1) 9 . 00 ( — 2

)

5. 60 ( — 2) 3. 74( — 2) 7.69(1) 1.03(1) 3.35(0)

6.76 7.54( — 1) 2 . 58 ( — 1

)

1 . 22 ( — 1

)

6. 84 ( — 2) 4.25( — 2) 2. 82 ( — 2) 6.02(1) 7.93(0) 2.57(0)

11.24 4.69( — 1) 1.60( — 1) 7 . 56 ( — 2) 4.23( — 2) 2. 63 ( — 2) 1 . 76 ( — 2) 3.85(1) 4.97(0) 1.61(0)

17.00 3 - 14 ( — 1

)

1 . 07 ( — 1

)

5.06( — 2) 2. 83 ( — 2) 1 . 76( — 2) 1 . 18 ( — 2) 2.62(1) 3.37(0) 1.08(0)

24.04 2 . 25 ( — 1) 7. 65 ( — 2) 3. 60 ( — 2) 2 .01 ( — 2) 1 .25 ( — 2) 8.37( — 3) 1.88(1) 2.41(0) 7 . 74 ( — 1)

Impact

energy e in threshold

units 3s—>7s 3s—>8s 3s—>9s 4s—»5s 4s—>6s 4s—>7s 4s-^8s 4s—»9s 5s—>6s

1.16 1.67(0) 9.30( — 1) 5 . 80 ( — 1

)

2.71(2) 3.28(1) 1.03(1) 4.57(0) 2.50(0) 6.40(2)

1.64 2.85(0) 1.60(0) 9 . 98 ( — 1

)

5.99(2) 8.07(1) 2.61(1) 1.20(1) 6.65(0) 1.95(3)

2.44 2.61(0) 1.45(0) 9.05( — 1) 6.48(2) 8.48(1) 2.70(1) 1.23(1) 6.74(0) 2.37(3)

3.56 2.03(0) 1.13(0) 7.02 ( — 1) 5.58(2) 7.05(1) 2.21(1) 9.99(0) 5.46(0) 2.15(3)

5.00 1.55(0) 8 . 57 ( — 1) 5.31 ( — 1) 4.50(2) 5.53(1) 1.72(1) 7.72(0) 4.21(0) 1.78(3)

6.76 1.19(0) 6.56( — 1) 4.06( — 1) 3.58(2) 4.33(1) 1.34(1) 5.98(0) 3.25(0) 1.44(3)

11.24 7 . 40( — 1) 4.09( — 1) 2 . 53 ( — 1) 2.32(2) 2.76(1) 8.44(0) 3.76(0) 2.04(0) 9.54(2)

17.00 4.98( — 1) 2 . 75 ( — 1) 1 . 70( — 1) 1.60(2) 1.87(1) 5.71(0) 2.54(0) 1.38(0) 6.60(2)

24.04 3 . 55 ( — 1

)

1 . 96 ( — 1

)

1.21(-1) 1.15(2) 1.34(1) 4.09(0) 1.82(0) 9 . 86 ( — 1

)

4.78(2)

Impact

energy t in threshold

units 5s—>7s 5s—>8s 5s—>9s 6s—>7s 6s—>8s 6s—>9s

1.16 6.51(1) 1.85(1) 7.98(0) 1.15(3) 9.06(1) 2.43(1)

1.64 2.47(2) 7.72(1) 3.47(1) 4.82(3) 5.82(2) 1.76(2)

2.44 2.93(2) 9.09(1) 4.05(1) 6.61(3) 8.45(2) 2.47(2)

3.56 2.58(2) 7.80(1) 3.44(1) 6.43(3) 7.92(2) 2.25(2)

5.00 2.08(2) 6.19(1) 2.71(1) 5.59(3) 6.44(2) 1.82(2)

6.76 1.64(2) 4.86(1) 2.12(1) 4.69(3) 5.04(2) 1.43(2)

11.24 1.06(2) 3.11(1) 1.35(1) 3.29(3) 3.10(2) 9.00(1)

17.00 7.22(1) 2.11(1) 9.14(0) 2.39(3) 2.06(2) 6.01(1)

24.04 5.19(1) 1.52(1) 6.56(0) 1.81(3) 1.49(2) 4.25(1)

a The numbers in parentheses denote powers of 10.

85
Table II. Total cross sections for s—>/> excitations of atomic hydrogen, calculated using the first Born approximation (Vainshtein,

1965) . Total cross sections in units of noo2 .*

Impact

energy e in threshold

units ls—>2p Is—>3/> Is—>4p Is—>5/> Is—>6/> Is—>7/> Is—>8/> Is—>9/> 2s—>3/>

1.16 7 . 70 ( — 1

)

1 - 41 ( — 1) 5. 08 ( — 2) 2. 43 ( — 2) 1 . 35 ( — 2) 8.35( — 3) 5.51 ( — 3) 3. 83 ( — 3) 2.75(0)

1.64 1.23(0) 2 . 12 ( — 1) 7 . 57 ( — 2) 3. 59 ( — 2) 1 .99 ( — 2) 1 .23 ( — 2) 8. 10 ( — 3) 5. 62 ( — 3) 1.10(1)

2.44 1.32(0) 2. 18 ( — 1) 7. 68 ( — 2) 3. 62 ( — 2) 2.01 ( — 2) 1.23( — 2) 8. 10(— 3) 5. 63 ( — 3) 1.70(1)

3.56 1.22(0) 1 . 96 ( — 1

)

6. 83 ( — 2) 3.21 ( — 2) 1 . 77 ( — 2) 1 .09 ( — 2) 7. 16 ( — 3) 4. 96 ( — 3) 1.91(1)

5.00 1.07(0) 1 . 69 ( — 1

)

5. 83 ( — 2) 2 . 73 ( — 2) 1.51 ( — 2) 9.23( — 3) 6.07(— 3) 4.21 (—3) 1.88(1)

6.76 9. 18 ( — 1) 1 - 43 ( — 1

)

4. 93 ( — 2) 2 . 31 ( — 2) 1 . 27 ( — 2) 7 . 78 ( — 3) 5. 12( — 3) 3. 55 ( — 3) 1.76(1)

11.24 6.88( — 1) 1 . 05 ( — 1) 3. 58 ( — 2) 1.67 ( — 2) 9.21 ( — 3) 5 . 63 ( — 3) 3.70 (—3) 2. 56 ( — 3) 1.43(1)

17.00 5.26( — 1) 7.91 ( — 2) 2. 70( — 2) 1 . 26 ( — 2) 6.91 ( — 3) 4. 22 ( — 3) 2.77 ( — 3) 1 . 92 ( — 3) 1.15(1)

24.04 4. 15(— 1) 6. 18 ( — 2) 2.09( — 2) 9. 76( — 3) 5.37( — 3) 3. 28 ( — 3) 2 . 15 ( — 3) 1 .49 ( — 3) 9.45(0)

Impact

energy e in threshold

units 2s—>4p 2s—*5p 2s—*6p 2s—>7/> 2s—*8p 2s—>9p 3s—Ap 3s—>5/> 3s—>6p

1.16 6.03( — 1) 2.43( — 1) 1 . 23 ( — 1) 7. 20 ( — 2) 4. 59 ( — 2) 3 . 12 ( —2) 3.25(1) 5.29(0) 1.83(0)

1.64 2.18(0) 8 . 29 ( — 1

)

4. 12 ( — 1) 2.38( — 1) 1.51(-1) 1 . 02 ( — 1) 4.88(1) 9.02(0) 3.37(0)

2.44 3.13(0) 1.16(0) 5.66( — 1) 3.24( — 1) 2.04( — 1) 1 . 37 ( — 1) 8.56(1) 1.55(1) 5.70(0)

3.56 3.31(0) 1.19(0) 5 . 82 ( — 1) 3.31 ( — 1) 2.07( — 1) 1 . 39 ( — 1) 1.10(2) 1.87(1) 6.67(0)

5.00 3.14(0) 1.12(0) 5 . 38 ( — 1) 3.04( — 1) 1 -91 ( — 1) 1 . 28 ( — 1) 1.18(2) 1.90(1) 6.65(0)

6.76 2.83(0) 9.99( — 1) 4.78( — 1) 2.70( — 1) 1 . 69 ( — 1

)

1. 13 ( — 1) 1.16(2) 1.80(1) 6.20(0)

11.24 2.23(0) 7 . 75 ( — 1) 3 . 69 ( — 1

)

2.07( — 1) 1 . 29 ( — 1) 8. 65 ( — 2) 1.02(2) 1.45(1) 5.05(0)

17.00 1.76(0) 6.05( — 1) 2.87 ( — 1) 1 .61 ( — 1) 1.00(-1) 6. 70 ( — 2) 8.55(1) 1.21(1) 4.05(0)

24.04 1.41(0) 4.83( — 1) 2 . 28 ( — 1) 1 . 28( — 1) 7. 96 ( — 2) 5. 32 ( — 2) 7.12(1) 9.91(0) 3.21(0)

Impact

energy e in threshold

units 3s—*7

p

3s—>8/> 3s—>9p 4s—>5p 4s—>6/> 4s—>7 p 4s—>8p 4s—>9/> 5s—>6p

1.16 8.69( — 1) 4.90( — 1) 3 . 07 ( — 1) 2.66(2) 3.81(1) 1.25(1) 5.74(0) 3.17(0) 1.15(3)

1.64 1.67(0) 9 . 72 ( — 1) 6. 21 ( — 1) 2.19(2) 3.41(1) 1.18(1) 5.64(0) 3.20(0) 9.33(2)

2.44 2.80(0) 1.61(0) 1.03(0) 3.01(2) 5.16(1) 1.85(1) 8.99(1) 5.15(0) 9.08(2)

3.56 3.23(0) 1.84(0) 1.16(0) 4.05(2) 6.66(1) 2.33(1) 1.12(1) 6.34(0) 1.16(3)

5.00 3.18(0) 1.80(0) 1.13(0) 4.62(2) 7.18(1) 2.45(1) 1.16(1) 6.52(0) 1.36(3)

6.76 2.94(0) 1.66(0) 1.04(0) 4.76(2) 7.06(1) 2.37(1) 1-11(1) 6.19(0) 1.45(3)

11.24 2.37(0) 1.33(0) 8.31 ( — 1) 4.41(2) 6.14(1) 2.00(1) 9.25(0) 5.13(0) 1.41(3)

17.00 1.89(0) 1.06(0) 6 . 59 ( — 1

)

3.84(2) 5.11(1) 1.64(1) 7.53(0) 4.16(0) 1.25(3)

24.04 1.51(0) 8 . 53 ( — 1) 5.31(-1) 3.28(2) 4.25(1) 1.35(1) 6.17(0) 3.39(0) 1.09(3)

Impact

energy e in threshold

units 5s—*7p 5s—»8/> 5s—>9p 6s—>7p 6s-+&p 6s—*9p 7s—>8/> 7s—>9p 8s—>9/>

1.16 1.52(2) 4.82(1) 2.16(1) 3.33(3) 3.93(2) 1.18(2) 4.23(3) 6.32(2) 5.75(3)

1.64 1.29(2) 4.12(1) 1.88(1) 3.19(3) 4.28(2) 1.29(2) 8.47(3) 1.05(3) 1.53(4)

2.44 1.42(2) 4.93(1) 2.34(1) 2.43(3) 3.64(2) 1.19(2) 9.16(3) 7.43(2) 2.11(4)

3.56 1.84(2) 6.32(1) 2.98(1) 2.59(3) 4.89(2) 1.52(2) 8.22(3) 7.04(2) 2.23(4)

5.00 2.06(2) 6.88(1) 3.21(1) 2.94(3) 6.23(2) 1.77(2) 6.96(3) 8.23(2) 2.12(4)

6.76 2.09(2) 6.82(1) 3.14(1) 3.13(3) 7.03(2) 1.84(2) 5.84(3) 9.39(2) 1.94(4)

11.24 1.90(2) 5.95(1) 2.70(1) 3.11(3) 7.28(2) 1.72(2) 4.17(3) 1.03(3) 1.53(4)

17.00 1.62(2) 4.95(1) 2.24(1) 2.82(3) 6.71(2) 1.50(2) 3.10(3) 9.87(2) 1.23(4)

24.04 1.37(2) 4.11(1) 1.85(1) 2.48(3) 5.96(2) 1.28(2) 2.38(3) 9.01(2) 9.90(3)

s The numbers in parentheses/lenote powers of 10. 86
Table III. Total cross sections for s—>d excitations of atomic hydrogen, calculated using the first Born approximation (Vainshtein, 1965)

.

Total cross sections in units of ttoo* *

Impact

energy « in threshold

units Is—3d Is—>4d Is—5d Is—6d Is—7d Is—8d Is—9d 2s—3d 2s—4d

1.16 1.50 ( — 2) 7. 07 ( — 3) 3. 73( — 3) 2.18(-3) 1 . 38( — 3) 9.29( — 4) 6. 54 ( — 4) 4.00(1) 5.64(0)

1.64 2. 14 ( — 2) 9. 96 ( — 3) 5. 21 ( — 3) 3.04(— 3) 1.92 ( — 3) 1 . 28 ( — 3) 9.07 (—4) 4.61(1) 5.63(0)

2.44 1 .99 ( — 2) 9 . 12 ( — 3) 4 . 75 ( — 3) 2.76 ( — 3) 1 . 74( — 3) 1 . 16 ( — 3) 8. 22 ( — 4) 3.79(1) 4.18(0)

3.56 1 .60 ( — 2) 7. 30 ( — 3) 3 . 79 ( — 3) 2 . 20( — 3) 1 . 39 ( — 3) 9. 30 ( — 4) 6. 53 ( — 4) 2.86(1) 2.98(0)

5.00 1 . 25 ( — 2) 5. 66 ( — 3) 2.93(— 3) 1 . 70(— 3) 1 .07 ( — 3) 7. 18 ( — 4) 5 . 04 ( — 4) 2.15(1) 2.16(0)

6.76 9. 73( — 3) 4.41 (—3) 2. 28 ( — 3) 1 .32 ( — 3) 8.33(— 4) 5. 57 (—4) 3.91 ( — 4) 1.64(1) 1.62(0)

11.24 6. 20 ( — 3) 2 . 80 ( — 3) 1.45 ( — 3) 8.38(— 4) 5. 28 ( — 4) 3 . 53 ( — 4) 2 . 48 ( — 4) 1.02(1) 9.82( — 1) 17.00 4 . 22 ( — 3) 1 . 90 ( — 3) 9.82(— 4) 5. 69 (—4) 3. 58 (—4) 2.59(— 4) 1 .68 ( — 4) 6.87(0) 6 . 52 ( — 1

)

24.04 3. 03 (—3) 1 .36 ( — 3) 7 . 04 ( — 4) 4. 08 (—4) 2.57 ( — 4) 1 . 72 ( — 4) 1 .21 (—4) 4.90(0) 4.63( — 1)

Impact

energy < in threshold

units 2s—5d 2s—6d 2s—7d 2s—>8d 2s—9d 3s—4d 3s—5d 3s—6d 3s—Id

1.16 1.88(0) 8 . 78 ( — 1

)

4.87 ( — 1) 3.01 ( — 1) 2.01 (— 1) 1.47(2) 2.67(1) 9.81(0) 4.80(0)

1.64 1.77(0) 8.03( — 1) 4.39( — 1) 2.69( — 1) 1 . 78 ( — 1) 2.50(2) 3.96(1) 1.36(1) 6.41(0)

2.44 1.28(0) 5.70( — 1) 3.09( — 1) 1 . 88 ( — 1) 1 - 24 ( — 1

)

2.44(2) 3.43(1) 1.12(1) 5.17(0)

3.56 8.95(— 1) 3 . 97 ( — 1) 2. 14 ( — 1) 1 . 30 ( — 1

)

8. 58 ( — 2) 2.00(2) 2.61(1) 8.30(0) 3.77(0)

5.00 6.44( — 1) 2.84( — 1) 1.53(— 1) 9. 32 ( — 2) 6. 13( — 2) 1.56(2) 1.95(1) 6.12(0) 2.76(0)

6.76 4 . 78 ( — 1) 2 . 1 1

( — 1) 1 . 14 ( — 1

)

6. 90 ( — 2) 4. 54 ( — 2) 1.22(2) 1.49(1) 4.61(0) 2.07(0)

11.24 2.89( — 1) 1 . 27 ( — 1) 6. 85 ( — 2) 4. 16 ( — 2) 2 . 73 ( — 2) 7.83(1) 9.18(0) 2.82(0) 1.26(0)

17.00 1.91(-1) 8. 42 ( — 2) 4. 53 ( — 2) 2 . 75 ( — 2) 1.81 ( — 2) 5.33(1) 6.14(0) 1.87(0) 8.39( — 1) 24.04 1 . 35 ( — 1) 5. 95 ( — 2) 3. 20 ( — 2) 1 .94 ( — 2) 1.28 ( — 2) 3.83(1) 4.37(0) 1.33(0) 5.95( — 1)

Impact

energy e in threshold

units 3s—8d 3s—9d 4s—5d 4s—6d 4s—7d 4s—8d 4s—9d 5s—6d 5s—Id

1.16 2.76(0) 1.75(0) 2.84(2) 5.37(1) 2.00(1) 9.93(0) 5.77(0) 4.95(2) 7.80(1)

1.64 3.59(0) 2.24(0) 7.98(2) 1.37(2) 4.84(1) 2.32(1) 1.31(1) 1.90(3) 3.32(2)

2.44 2.86(0) 1.77(0) 9.32(2) 1.41(2) 4.69(1) 2.18(1) 1.21(1) 2.66(3) 4.13(2)

3.56 2.07(0) 1.27(0) 8.30(2) 1.14(2) 3.68(1) 1.68(1) 9.20(0) 2.57(3) 3.62(2)

5.00 1.51(0) 9 . 27 ( — 1) 6.79(2) 8.85(1) 2.79(1) 1.26(1) 6.86(0) 2.20(3) 2.90(2)

6.76 1.13(0) 6.93( — 1) 5.46(2) 6.85(1) 2.13(1) 9.54(0) 5.18(0) 1.81(3) 2.29(2)

11.24 6.86( — 1) 4.20( — 1) 3.58(2) 4.31(1) 1.32(1) 5.88(0) 3.19(0) 1.22(3) 1.46(2)

17.00 4.56( — 1) 2.79( — 1) 2.47(2) 2.91(1) 8.86(0) 3.93(0) 2.12(0) 8.51(2) 9.97(1)

24.04 3.23( — 1) 1.98( — 1) 1.79(2) 2.08(1) 6.31(0) 2.80(0) 1.51(0) 6.21(2) 7.16(1)

Impact

energy e in threshold

units 5s—8d 5s—9d 6s—Id 6s—8d 6s—9d 7s—8d 7s—9d 00Co 1

1.16 2.71(1) 1.31(1) 1.16(3) 1.36(2) 2.18(0) 1.67(3) 3.10(2) 3.88(3)

1.64 1.19(2) 5.74(1) 3.61(3) 6.11(2) 2.37(0) 1.39(3) 7.40(2) 5.87(3)

2.44 1.38(2) 6.42(1) 5.78(3) 1.05(3) 1.77(0) 1.00(3) 1.73(3) 5.46(3)

3.56 1.16(2) 5.25(1) 5.97(3) 1.10(3) 1.26(0) 7.52(2) 2.13(3) 4.37(3)

5.00 9.01(1) 4.03(1) 5.28(3) 9.76(2) 9. 17 ( — 1) 5.78(2) 2.06(3) 3.40(3)

6.76 6.97(1) 3.18(1) 4.43(3) 8.19(2) 6.86( — 1) 4.51(2) 1.82(3) 2.65(3)

11.24 4.38(1) 1.93(1) 3.04(3) 5.61(2) 00 2.89(2) 1.32(3) 1.68(3)

17.00 2.95(1) 1.30(1) 2.15(3) 3.95(2) 2 . 78 ( — 1

)

1.98(2) 9.52(2) 1.13(3)

24.04 2.10(1) 9.25(0) 1.57(3) 2.89(2) \o 00 1 1.43(2) 7.07(2) 8.17(2)

The numbers in parentheses denote powers of 10.

87
Table IV. Total cross sections for the excitation of discrete levels of atomic hydrogen from the Is state, calculated using the first Born approximation (McCarroll, 1957; Omidvar, 1965). Total cross sections in units of too2

.

Impact energy

(rydbergs) (eV) 01.2 01.3 0M 01,5 01.6 01.7 01.8 01.9 01.

1.00 13.60 1.2868 0.1787 0.0509 0.0199 0.0092 0.0050 0.0032 0.0018 0.0012

1.44 19.58 1.5354 0.2782 0.1000 0.0476 0.0265 0.0163 0.0104 0.0075 0.0054

1.96 26.66 1.4993 0.2798 0.1021 0.0490 0.0274 0.0169 0.0112 0.0078 0.0056

2.56 33.43 1.3886 0.2600 0.0951 0.0457 0.0256 0.0158 0.0104 0.0073 0.0053

3.24 44.06 1.2630 0.2358 0.0862 0.0413 0.0232 0.0143 0.0096 0.0066 0.0048

4.00 54.40 1.1424 0.2123 0.0775 0.0372 0.0208 0.0128 0.0088 0.0059 0.0043

6.25 85.00 0.8919 0.1637 0.0595 0.0285 0.0160 0.0098 0 0064 0.0045 0.0033

9.00 122.40 0.7101 0.1290 0.0468 0.0224 0.0125 0.0077 0.0048 0.0035 0.0026

12.25 166.60 0.5780 0.1041 0.0377 0.0180 0.0100

16.00 217.60 0.4797 0.0858 0.0310 0.0148 0.0083

20.25 275.40 0.4050 0.0721 0.0260 0.0124 0.0069

25.00 340.00 0.3468 0.0614 0.0221 0.0105 0.0059

36.00 489.60 0.2634 0.0463 0.0166 0.0079 0.0044

49.00 666.40 0.2075 0.0363 0.0130 0.0062 0.0034

72.25 989.40 0.1526 0.0265 0.0095 0.0045 0.0025

Table V. Total cross sections for the excitation of discrete levels of atomic hydrogen from the 2s state, calculated using the first Bom

approximation (Boyd, 1959) . Total cross sections in units of ttOo2

.

Impact energy

(rydbergs) 02a,

3

02a,02a, 5 02a.

8

02a,02a.

8

02a,

9

02a. 10

0.36 65.019 12.330 4.658 2.312 1.331 0.843 0.569 0.404

0.64 49.441 9.454 3.612 1.808 1.047 0.664 0.450 0.320

1.00 37.667 7.150 2.726 1.364 0.790 0.499 0.339 0.241

1.44 29.488 5.560 2.114 1.056 0.611 0.390 0.264 0.188

1.96 23.703 4.446 1.687 0.842 0.487 0.309 0.209 0.149

2.56 19.488 3.640 1.379 0.687 0.397 0.252 0.170 0.121

3.24 16.326 3.040 1.149 0.572 0.331 0.210 0.142 0.101

4.00 13.895 2.580 0.974 0.485 0.280 0.179 0.121 0.086

4.84 11.984 2.200 0.837 0.416 0.240 0.153 0.103 0.073

5.76 10.452 1.933 0.728 0.362 0.209 0.133 0.090 0.064

6.76 9.206 1.699 0.640 0.318 0.183 0.116 0.079 0.056

7.84 8.176 1.507 0.567 0.281 0.162 0.103 0.070 0.050

9.00 7.317 1.347 0.506 0.251 0.145 0.091 0.062 0.044

88
Table VI. Total cross sections for the excitation of discrete levels of atomic hydrogen from the 2p0 state, calculated using the first Born approximation (McCrea and McKirgen, 1960) . Total cross sections in units of nao

2

.

Impact energy

(rydbergs) 02pO.3 QipOA QipO.i Qipo.t (?2p0.7 QipOA QlpOA QipO.lO

0.36 74.837 14.426 5.259 2.499 1.409 0.892 0.601 0.424

0.64 64.529 12.323 4.749 2.347 1.348 0.861 0.586 0.417

1.00 52.482 9.826 3.790 1.859 1.072 0.684 0.461 0.328

1.44 42.803 oc 2.993 1.465 0.844 0.531 0.361 0.255

1.96 35.403 6.352 2.408 1.176 0.677 0.432 0.288 0.207

2.56 29.727 5.246 1.974 0.963 0.553 0.348 0.234 0.165

3.24 25.324 4.406 1.649 0.802 0.461 0.291 0.196 0.138

4.00 21.836 3.755 1.399 0.680 0.390 0.248 0.167 0.119

4.84 19.038 3.241 1.202 0.583 0.334 0.211 0.141 0.100

5.76 16.759 2.828 1.045 0.506 0.290 0.182 0.125 0.087

6.76 14.876s 2.491 0.918 0.444 0.254 0.160 0.108 0.077

7.84 13.304 2.213 0.807 0.393 0.225 0.143 0.095 0.066

9.00 11.976 1.988 0.726 0.350 0.200 0.125 0.084 0.059

Table VII. Total cross sections for the excitation of discrete levels of atomic hydrogen from the 2/>±l states, calculated using the

first Born approximation (McCrea and McKirgen, 1960) . Total cross sections in units of xao2

.

Impact energy

(rydbergs) (?2p±1.3 (?2p±1.4 (?2p±l.S (?2p±l.« (?2p±1.7 (?2p±1.8 (?2p±l,9 (?2p±I,10

0.36 76.515 13.346 4.995 2.512 1.414 0.890 0.600 0.424

0.64 57.44 10.692 4.052 2.040 1.166 0.740 0.505 0.358

1.00 45.051 8.181 3.077 1.537 0.883 0.560 0.378 0.270

1.44 35.937 6.381 2.380 1.190 0.679 0.426 0.288 0.202

1.96 29.278 5.102 1.891 0.935 0.537 0.342 0.229 0.160

2.56 24.313 4.173 1.538 0.758 0.435 0.268 0.180 0.128

3.24 20.535 3.480 1.276 0.625 0.359 0.224 0.151 0.108

4.00 17.587 2.949 1.077 0.528 0.302 0.190 0.129 0.091

4.84 15.248 2.533 0.922 0.451 0.258 0.164 0.110 0.078

5.76 13.360 2.202 0.799 0.391 0.223 0.140 0.094 0.067

6.76 11.812s 1.934 0.700 0.341 0.195 0.123 0.082 0.058

7.84 10.527 1.713 0.618 0.301 0.172 0.108 0.072 0.052

9.00 9.447s 1.534 0.551 0.272 0.153 0.096 0.064 0.046

89
Table VIII. Total cross sections for the excitation of discrete levels of atomic hydrogen from the n=3 level, calculated using

the first Bom approximation (Omidvar, 1965) . Total cross sections in units of 7rOo2-

Impact

energy

(rydbergs) (?3.4 (?3.5 (?3,6 (?3,7 Qi.8

0.07 657.1

0.08 709.2 83.37

0.111 735.3 126.98 42.92 19.16 9.94

0.16 676.9 125.33 47.26 23.56 13.67 0.36 460.9 83.69 31.86 16.08 9.44

0.64 322.3 56.35 21.16 10.62 6.21

1.00 237.4 40.33 14.98 7.47 4.35

Impact

energy

(rydbergs) (?3,4 Qi.i (?3,6 <23.7 @3,8

1.44 182.5 30.34 11.19 5.55 3.23

1.96 145.1 23.69 8.69 4.29 2.49

2.56 118.4 19.07 6.94 3.45 1.99

3.24 98.5 15.71 5.70 2.81 1.62

4.00 83.5 13.19 4.75 2.34 1.38 6.25 58.5 9.12 3.25 1.61 0.97

9.00 43.6 6.71 2.38 1.19 0.66

Table IX. Values of the constants Cn'i'.ni occurring in the Bethe total cross section formula (47) for transitions of atomic

hydrogen satisfying n — «'= 1 or 2 and l—l'= 1* (McCoyd and Milford, 1963).

V'

*'\ 1 2 3 4

n5-n'= 1

6 7 8 9 10

0 3.0(1) 1.70(2) 5.4(2) 1.32(3) 2.7(3) 5.0(3) 8.4(3) 1.34(4) 2.0(4) 3.0(4)

1 2.7(2) 6.9(2) 1.47(3) 2.8(3) 4.8(3) 7.8(3) 1.20(4) 1.76(4) 2.5(4)

2 1.14(3) 2.2(3) 3.8(3) 6.2(3) 9.5(3) 1.42(4) 2.0(4) 2.8(4)

3 3.3(3) 5.3(3) 8.2(3) 1.22(4) 1.76(4) 2.5(4) 3.4(4)

4 7.5(3) 1.10(4) 1.58(4) 2.2(4) 3.0(4) 4.1(4)

5 1.48(4) 2.1(4) 2-8(4) 3.7(4) 4.9(4)

6 2.7(4) 3.5(4) 4.6(4) 5.9(4)

7 4.4(4) 5.7(4) 7.2(4)

8 7.0(4) 8.7(4)

9 1.05(5)

n—n' = 2 v

*'\ 1 2 3 4 5 6 7 8 9 10

0 4.8 3.0(1) 9.3(1) 2.2(2) 4.3(2) 7.7(2) 1.27(3) 1.98(3) 3.0(3) 4.2(3)

1 3.5(1) 1.07(2) 2.3(2) 4.4(2) 7.4(2) 1.18(3) 1.79(3) 2.6(3) 3.6(3)

2 1.20(2) 2.9(2) 5.5(2) 9.1(2) 1.42(3) 2.1(3) 3.0(3) 4.1(3)

3 2.9(2) 6.4(2) 1.10(3) 1.71(3) 2.5(3) 3.5(3) 4.8(3)

4 5.6(2) 1.19(3) 1.96(3) 2.9(3) 4.1(3) 5.6(3)

5 9.7(2) 2.0(3) 3.2(3) 4.7(3) 6.4(3)

6 1.54(3) 3.1(3) 4.9(3) 7.0(3)

7 2.3(3) 4.7(3) 7.2(3)

8 3.3(3) 6.6(3)

9 4.5(3)

B The numbers in parentheses denote powers of 10.

90
Table X. Values of the constants Dn'i',ni occurring in the Bethe total cross section formula (47) for n'l'-^nl transitions of atomic

hydrogen satisfying n—n'= 1 or 2 and l—l'= 1 (McCoyd and Milford, 1963).“

\»'

/'\ 1 2 3 4

a 1 II

6 7 8 9 10

0 1 . 03 ( — 1

)

2 . 7

( — 1) 5.0( — 1) 8.0( — 1) 1.14 1.6 2 3 3 4

1 5 . 8 ( — 1) 7.6( — 1) 1.08 1.44 1.9 3 3 3 5

2 1.39 1.50 1.77 2 3 3 4 5

3 2.6 2.6 2 4 4 4 5 4 4.2 5 4 5 5 6

5 5 5 5 5 6

6 1.2(1) 8 6 8

7 l.KD 9 8

8 1.6(1) 1.2(1)

9 1-8(1)

n— n' — 2 V'

*'\ 1 2 3 4 5 6 7 8 9 10

0 1 - 41 ( — 1) 3(-l) 5.2(-l) 8.1(-1) 1.2 1.8 1.8 3 3 4

1 8( — 1) 9.7( — 1) 1.2 2 1.8 1.8 3 4 4

2 2.7 2 3 3 4 4 5 5 3 5.9 4 5 4 4 5 6

4 1.1(1) 8 7 8 7 6

5 2(1) 1.3(1) 8 7 9

6 3(1) 1.5(1) 1.6(1) 1.3(1)

7 5(1) 3(1) 1.3(1)

8 7(1) 3(1) 9 9(1)

a The numbers in parentheses denote powers of 10.

Table XI. Values of the constants C„',n and Dn\n occurring

in the Bethe total cross section formula (48) for n'—*n transitions

of atomic hydrogen satisfying n—n'= 1 or 2 (Kingston and Lauer,

1966a; 1966b) .“

n'

n—n'= 1 n—n'=

2

Cn’.n Dn'.n Cn'.n Dn-, n

1 3.02(1) 1 . 46 ( — 1

)

4.84(0) 2.24( — 1)

2 2.51(2) 8 . 61 ( — 1) 3.46(1) 2.35(0)

3 9.43(2) 2.27(0) 1.15(2) 9.48(0)

4 2.51(3) 4.36(0) 2.81(2) 2.42(1)

5 5.48(3) 7.08(0) 5.75(2) 4.79(1)

6 1.05(4) 1.04(1) 1.05(3) 8.13(1)

a The numbers in parentheses denote powers of 10.

Table XII. Total cross sections for the Is—>2s excitation of atomic hydrogen calculated using the first Bom approximation,

the Born-Oppenheimer approximation and the first-order ex- change approximation (Bell and Moiseiwitsch, 1963) . Total cross sections in units of irao2 .*

Impact energy

(rydbergs) B B.O. E,

1.00 0.248 1.648 0.447

2.25 0.167 0.204 0.148

4.00 0.102 0.0958 0.0905

9.00 0.0477 0.0451 0.0447

16.00 0.0273 0.0264 0.0263

a B=first Born approximation; B.O. = Born-Oppenheimer approximation; Ei =first-order exchange approximation.

91
Table XIII. Total cross sections for the Is—>2^0 and Is—>2/>±l excitations of atomic hydrogen calculated using the first Bom

approximation, the Born-Oppenheimer approximation and the first-order exchange approximation. The axis of quantization of the atomic wave functions is chosen along the direction of the incident electron. Total cross sections in units of vcrf.

Impact energy

(rydbergs)

t?2p0

8 (?2p±lb Qiv

Q2po

2Qtp±i

Bc B.O.® Ef B WoIII tq B B.O. Ei

1.00 0.873 1.498 1.209 0.0830 0.0811 1.039 1.660 1.371

2.25 0.732 0.641 0.630 0.275 0.243 1.282 1.126 1.115

4.00 0.475 0.442 0.441 0.283 0.265 1.041 0.973 0.972

9.00 0.231 0.227 0.227 0.215 0.211 0.662 0.648 0.648

16.00 0.134 0.133 0.133 0.159 0.158 0.453 0.449 0.449

8 Bell and Moiseiwitsch, 1963. c B=first Born approximation; B.O. = Born-Oppenheimer approximab Bell. 1965. tion; E\ =first-order exchange approximation.

Table XIV. Zero-order partial cross sections for the Is—»2s excitation calculated using the Born-Oppenheimer approximation, the

distorted waves approximation and the Is—>2s close coupling approximation.8

Impact

energy

(rydbergs)

Zero-order partial cross sections in units of 7nZo*

Exchange neglected Exchange included, singlet scattering

Q

Bb D.W.b C® B.O.b D.W.b D.W.d C*

1.00 0.198 0.239 0.204 0.287 0.714 0.0738 0.288

1.44 0.127 0.118 0.102 0.011 0.344 0.204 0.219

2.25 0.0585 0.0455 0.0450 0.014 0.127 0.069 0.0968

4.00 0.0194 0.0141 0.0i55 0.018 0.0255 0.0212 0.0290

Exchange included,

Exchange included, triplet scattering total scattering

Q~

B.O.b D.W.b D.W.d C® B.O.b D.W.b C®

1.00 2.02 0.0316 0.013 0.00274 1.59 0.178 0.0741

1.44 0.668 0.010 0.002 0.00814 0.503 0.094 0.0608

2.25 0.134 0.010 0.016 0.00974 0.104 0.035 0.0315

4.00 0.0205 0.006 ... 0.00624 0.020 0.011 0.0119

a B=first Born approximation; DW =distorted waves approximation; ® Bransden and McKee. 1956. B.O. = Born-Oppenheimer approximation; C = ls— 2s close coupling d Ochkur, 1958. approximation. ® Marriott, 1958.

D Erskine and Massey, 1952.

92
Table XV. Total cross sections for various excitations of atomic hydrogen calculated using the distorted waves method neglecting

exchange (Vainshtein, 1961). Total cross sections in units of iroo2 *

Impact energy

t in Is—>2s Is-—>3s Is—

>2p threshold

units Bb D.W.b B D.W. B D.W.

1.04 1 . 29 ( — 1

)

6.95( — 1) 2.37 ( — 2) 1 . 29 ( — 1

)

4. 12 ( — 1) 2 . 15 ( — 1) 1.16 2 . 14 ( — 1

)

7 . 09 ( — 1

)

3. 88 ( — 2) 1 .04( — 1) 7 . 79 ( — 1) 1.37(0) 1.36 2 . 45 ( — 1

)

5.20( — 1) 4. 39 ( — 2) 8. 64( — 2) 1.06(0) 1.47(0) 1.64 2 . 40 ( — 1

)

4 . 15 ( — 1

)

4. 24( — 2) 7. 02 ( — 2) 1.23(0) 1.39(0)

2.44 1 . 89 ( — 1

)

2.68( — 1) 3.28( — 2) 4.54( — 2) 1.31(0) 1.29(0)

3.56 1 . 40 ( — 1

)

1 . 82 ( — 1) 2. 40 ( — 2) 2 . 97 ( — 2) 1.21(0) 1.14(0)

5.00 1 .04( — 1) 1 . 27 ( — 1) 1 . 77 ( —2) 2 . 17 ( —2) 1.06(0) 9.90(— 1) 6.76 7. 85 ( — 2) 7. 95 ( — 2) 1 .34 ( — 2) 1 . 59 ( — 2) 9. 19 ( — 1

)

8.50( — 1)

Impact energy

e in threshold

IS—

i

*$P Is--»3d 2s—)3p

units B D.W. B D.W. B D.W.

1.04 7 . 57 ( — 2) 3.20( — 1) 8. 14 ( —3) 1 ,44( — 2) 4.88( — 1) 2 . 76 ( — 1) 1.16 1.41(-1) 3.44( — 1) 1 .51 ( — 2) 2 . 57 ( — 2) 2.82(0) 4.44(0)

1.36 1 . 87 ( — 1) 3. 19 ( — 1

)

1 .97 ( — 2) 2 . 47 ( — 2) 6.75(0) 2.06(1)

1.64 2. 12 ( — 1) 2 . 92 ( — 1

)

2. 16 ( — 2) 2 . 25 ( — 2) 1.10(1) 1.69(1)

2.44 2 . 18 ( — 1) 2. 28(— 1) 2.00( — 2) 1 . 72 ( — 2) 1.70(1) 1.93(1)

3.56 1 . 96 ( — 1

)

1 - 76 ( — 1) 1.61 ( — 2) 1 .48 ( — 2) 1.91(1) 1.89(1)

5.00 1.68( — 1) 1 . 49 ( — 1

)

1.26( — 2) 1 .25 ( — 2) 1.88(1) 1.77(1)

6.76 1 - 43 ( — 1

)

1 . 25 ( — 1) 9. 80 ( — 3) 1.04( — 2) ... . .

.

Impact energy

e in threshold

2P-+3s Is--4p Is—

4s

units B D.W. B D.W. B D.W.

1.04 1.00(0) 4.88( — 1) 2 . 75 ( — 2) 1 - 42 ( — 1

)

8. 62 ( — 3) 4 . 15 ( — 2) 1.16 1.52(0) 2.13(0) 5.07 ( — 2) 1 . 30 ( — 1

)

1.40(— 2) 3.60( — 2) 1.36 1.55(0) 2.18(0) 6.69(— 2) 1 . 20 ( — 1) 1 . 58 ( — 2) 3. 03 ( — 2) 1.64 1.34(0) 1.23(0) 7 . 55 ( — 2) 1 . 08 ( — 1

)

1.52( — 2) 2.46(— 2) 2.44 1.04(0) 1.02(0) 7. 65 ( — 2) 8.00( — 2) 1 . 17 ( — 2) 1.59( — 2) 3.56 8 . 57 ( — 1) 7.91 ( — 1) 6.81 ( — 2) 5.98( — 2) 8. 54 ( — 3) 1 .05 ( — 2) 5.00 7 . 33 ( — 1) 6. 71 ( — 1) 5.81 ( — 2) 4.93( — 2) 6.28(— 3) 7. 30 ( — 3) 6.76 . .

.

. .

.

4.92( — 2) 4. 38 ( — 2) 4.73( — 3) 5. 66 ( — 3)

a The numbers in parentheses denote powers of 10. b B=first Born approximation; D.W. = distorted waves approximation.

Table XVI. Rate coefficients for 2s--+2p transitions in atomic hydrogen (Seaton, 1955a). Rate coefficients W in cm3 sec *. a

T= 1 X104 deg K 2 X 104 deg K

I II Ill IV I II III VI

10*XW' 0.27 0.25 0.24 0.22 0.20 0.18 0.17 0.17

1(YXW" 0.45 0.42 0.38 0.35 0.34 0.31 0.29 0.27

8 W' for 2s—*2pi/i and W" for 2s-*2/>i/j. Approximations: I Bethe; II first Born; III semiclassical (Purcell. 1952); IV modified Bethe.

93
Table XVII. Values of XB+i.„ occurring in the formu’a (96) for the excitation cross section for the n—m+1 transition of atomic

hydrogen calculated using the semiclassical impact parameter

method (Saraph, 1964).

log £\

Values of Xn+l , n (Impact energy E in eV)

1 2 3 4 6 8

-0.8 1.12 1.47 -0.6 1.34 1.67 -0.4 0.850 1.55 1.82

-0.2 1.28 1.70 1.93 0.0 1.11 1.48 1.80 1.98

0.2 1.36 1.60 1.84 1.97 0.4 0.956 1.49 1.65 1.81 1.91 0.6 1.22 1.54 1.62 1.73 1.79

0.8 1.28 1.43 1.48 1.56 1.59

1.0 1.23 1.28 1.29 1.32 1.35

1.2 0.529 1.12 1.12 1.12 1.12 1.13

1.4 0.640 0.985 0.981 0.950 0.944 0.948

1.6 0.650 0.852 0.820 0.802 0.991 0.789

1.8 0.611 0.727 0.691 0.672 0.655 0.653

2.0 0.555 0.616 0.580 0.561

2.2 0.483 0.516 0.483 0.465

\n 10 15 20 25 30 40

log jE\

-2.6 0.99 1.31 -2.4 1.01 1.21 1.54

-2.2 1.23 1.44 1.79

-2.0 1.21 1.46 1.68 2.05

-1.8 1.08 1.44 1.70 1.92 2.32

-1.6 1.33 1.66 1.93 2.16 2.59

-1.4 1.03 1.55 1.88 2.16 2.41 2.86

-1.2 1.29 1.77 2.10 2.38 2.64 3.11 -1.0 1.52 1.96 2.29 2.57 2.84 3.33

-0.8 1.72 2.13 2.47 2.74 3.02 3.45

-0.6 1.88 2.28 2.60 2.88 3.14 3.58

-0.4 2.01 2.37 2.68 2.94 3.17 3.57

-0.2 2.09 2.42 2.69 2.92 3.13 3.46

0.0 2.12 2.40 2.64 2.81 2.99 3.30

0.2 2.09 2.31 2.50 2.69 2.82 2.99

0.4 1.99 2.20 2.34 2.44 2.52 2.89

0.6 1.89 2.00 2.08 2.15 2.20 2.29

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•*-> JD a T3

On 00 OO

00 <N 10*0

*-4 0\ OO N O -H

LO O 00

so

lo o lo OO

CN CO

Tt O 00

Csl oo «o lo

NO Qn ro O

O © O

O O

8 oo oo oo oo oo

•H <"0 co 88

8

o o o o

8 8 8 8 8 8

6 o do o o

§ § § §

^ ^

°

d o'd do o o'

Oc o © N

oo oo oo oo

m o O' —i CM 8 8 8 8 8 8

o’ o’ o’ © o’ o*

CS NO © CN ^ H N 88 S8

o’o o’o

CO CN <N 00 lO ^ §

«-< *-h M O co to .8 88 88 88

o' o' do’ o’ o' o o

O in

—l CM 88

o* o

o oo Q co

*“H CO CN co

r»^ —i r^. to oo O 88 88 8 o 8 o do oo do oo

O W)

o o o o

>0 rn CO CM 0\ r^vo OO -h o co f— OO cr> O Tfco ^ r-- cm O

8

—i H CM CM Q CM O OO O '-1 o oo oo oo 88 8 o

o o o o do o o’ d o o d

o o

CO

00 CO CM —

^ .—i -vj. u > V) O 00 CM O (O CM CO o o oo oo oo o_ __

s’ o o o' d o o' do o o

lO © 1C 00 lO CM 00 lO cC to -CO O N

S

io co ^ co IO in _ COCO -H —

O —< O-c OO OO OO do do o'd do oo oo

3 2 $ h N CN

vO vO

lO

CN

On CN nO vO 00

CN

toO

NO co 00 lO r^O

lOo

LO co LO On o

coO OO OO oo oo oo oo

<v JU

*3) a

.£

co H

1) 03 "&> "a

.£ ‘C

C/3 H

0) 'Si "H, .£

' c

c/i H

43 _43 "bC "a

.2 'C

co H

4) JU U Q. £ *C

c/) H

<D ^2

"ti "a

.2w £H

00o

3

9U

First

Born

approximation

values

are

given

in

parentheses.

b

Sum

column

is

the

total

of

all

significant

partial

wave

contributions.

Spin

weighting

factors

included.
Table XIX. Partial cross sections QtPL for the ls—+2p excitation of atomic hydrogen calculated according to the ls-2s-2p

Impact energy

(rydbergs) L 0 1 2 3 4 5 6 7

0.81 Singlet 0.0384 0.0754 0.1095 ^0.010 0.0030 0

Triplet 0.0007 0.0657 0.0083 0.0505 0.0013 0.0001

1.00 Singlet 0.0360 0.1105 0.2532 0.0352 0.0098 0.0025 0.0007 0.0002

Triplet 0.0033 0.0798 0.0458 0.1671 0.0438 0.0093 0.0020 0.0005

1.21 Singlet 0.0359 0.1105 0.3404 0.0863 0.0301 0.0112 0.0044 0.0016

Triplet 0.0068 0.0629 0.0549 0.1831 0.1046 0.0388 0.0133 0.0050

1.44 Singlet 0.0343 0.0815 0.02895 0.1256 0.0508 0.0229 0.0108 (0.0049)

Triplet 0.0095 0.0416 0.0539 0.1740 0.1404 0.0732 0.0347 (0.0147)

2.25 Singlet 0.0171 0.0176 0.0942 0.0999 0.0695 0.0451 0.0292 (0.0181)

Triplet 0.0106 0.0133 0.0357 0.1077 0.1342 0.1148 0.0838 (0.0542)

4.00 Singlet 0.0035 0.0024 0.0169 0.0302 0.0347 0.0329 0.0286 0.0237

Triplet 0.0052 0.0038 0.0139 0.0394 0.0624 0.0728 0.0719 0.0647

a First Born approximation values are given in parentheses. b Sum column is the total of all significant partial wave contributions

Table XX. Total cross sections for the Is—>2s excitation of atomic hydrogen calculated according to the \s-2s-2p close coupling approximation with exchange included (Burke, Schey,

and Smith, 1963)

.

Impact Total cross energy section in units (rydbergs) Of 7TOo2

0.81 0.211

1.00 0.3616

1.21 0.3395

1.44 0.2635

2.25 0.1604

4.00 0.1012

Table XXI. Total cross sections for the ls—>2p excitation of atomic hydrogen calculated according to the ls-2s-2p close coupling approximation with exchange included (Burke, Schey,

and Smith, 1963) . Total cross sections in units of 7roo2.

Impact energy

(rydbergs) (?2p±l Qipo

Qip — 2 Qipm

m*=0,±l

0.81 «0.068 -0.223 «0.360

1.00 0.0997 0.6005 0.7999

1.21 0.1395 0.8149 1.0939

1.44 0.1631 0.8539 1.1801

2.25 0.2244 0.6423 1.0911

4.00 0.2394 0.3928 0.8716

95
close coupling approximation with exchange included (Burke, Schey, and Smith, 1963). Cross sections in units of irao2 .*

8 9 10 11 12 13 14 15 Sum6

*0.234

0.1265

0

0.0001

0.4481

0.3518

0.0006

0.0019

0.0002

0.0007

0.0001

0.0003

0.6212

0.4727

(0.0024)

(0.0072)

(0.0012)

(0.0035)

(0.0006)

(0.0017)

(0.0003)

(0.0008)

(0.0001)

(0.0004)

(0.0001)

(0.0002)

0.6243

0.5558

(0.0124)

(0.0371)

(0.0084)

(0.0251)

(0.0056)

(0.0169)

(0.0038)

(0.0113)

(0.0025)

(0.0075)

0.0016

0.0049

0.0011

0.0033

-0.0007

*0.0023

0.4275

0.6636

(0.0185)

(0.0556)

(0.0154)

(0.0463)

(0.0127)

(0.0381)

(0.0104)

(0.0311)

(0.0084)

(0.0252)

0.0066

0.0199

0.0054

0.0162

0.0044

0.0131

0.2640

0.6076

Spin weighting factors included.

Table XXII. Cross sections for the lx—»2s, \s—>2p, and Is—*ip excitations of atomic hydrogen calculated according to (a) the

ls-2s-2/>-3s-3/>, (b) the \s-2s-2p, and (c) the ls-3/> close coupling approximations with exchange included at 16.5-eV electron impact

energy (Burke, 1963) . Cross sections in units of jrao*.a

L—Q L= 1 L= 2 L=3 Total

Is—>2s Singlet (a) 0.387 0.0170 0.0351 0.0137

(b) 0.0588 0.0246 0.0645 0.0232 (a) 0.227

triplet (a) 0.0042 0.0663 0.0237 0.0038 (b) 0.340

(b) 0.0051 0.1000 0.0316 0.0069

Is*2p Singlet (a) 0.0368 0.0956 0.2303 0.06361

(b) 0.0359 0.1105 0.3404 0.0863 (a) 0.907

Triplet (a) 0.0064 0.0465 0.0525 0.1622 (b) 1.094

(b) 0.0068 0.0629 0.0549 0. 1831

j

1 s—>3p Singlet (a) 0.0066 0.0276 0.0740 0.0136]

(c) 0.0059 0.0086 0.1376 0.0468 (a) 0.250

Triplet (a) 0.0004 0.0216 0.0136 0.0431 (c) 0.319

(c) 0.0004 0.0209 0.0006 0.0485

a Spin weighting factors included.

96
Table XXIII. Partial cross sections for the Is—>2s and Is—>2/> excitations of atomic hydrogen calculated near the threshold energy according to the \s-2s-2p close coupling approximation

with exchange included (Damburg and Gailitis, 1963). Partial cross sections in units of ira02

.

L v QuL Q*PL' L+l QipL ' L

~l

fo 0.01 1 . 58 ( — I

)

1 . 16 ( — 1) ...

1 0.03 1 . 76 ( — 1) 1.40( — 1) ...

1 0.05 1 . 98 ( — 1

)

1 . 53 ( — 1

)

• • *

Singlet

1(1 0.01 7 . 8( — 2) 1 . 69 ( — 1

)

4. 29( — 1)

1 0.03 2.2( — 2) 7. 9 ( — 2) 2.51 ( — 1) 0.05 1 - 8 ( —2) 7 . 1

( —2) 2 . 28 ( — 1

)

(2 0.01 2 . 65 ( — 1

)

3 . 9

( — 2) 4 . 20 ( — 1

)

0.03 2 . 44 ( — 1

)

4.3( — 2) 3 . 57 ( — 1) 0.05 2.38( — 1) 49( — 2) 3.68( — 1)

'0 0.01 7. 62 (—4) 5.40( — 4) . .

.

0.03 1 . 12 ( —3) 6. 64(— 4) ... 0.05 1 .45 ( — 3) 6.59(— 4) * * •

Triplet- 1 0.01 6.47( — 2) 1 .80( — 2) 4.90( — 2) 0.03 7. 95 ( — 2) 2. 83 ( — 2) 4. 69 ( — 2) 0.05 8. 85 ( — 2) 3. 65 ( — 2) 4. 48 ( — 2)

2 0.01 2.84( — 5) 1 . 87 ( — 5) 2. 53 ( — 4) 0.03 6.77(— 4) 5. 88 (—4) 1.71(-3)

0.05 2.50( — 3) 2 . 21 ( — 3) 4. 75( — 3)

Table XXIV. Total cross sections for the Is—>2s and Is—>2/> excitations of atomic hydrogen calculated near the threshold energy according to the \s-2s-2p close coupling approximation

with exchange included (Damburg and Gailitis, 1963). Total

cross sections in units of irao2.

kip2 Is—>2s Is

*2p

Table XXV. Total cross sections for the 2

3 S—>23 P, 3

3 P, 33D,

and 4

3Z> excitations of helium calculated using the first Born

approximation and the modified Bethe approximation

(Moiseiwitsch, 1957). Total cross sections in units of 7rao2.

2

3 S—>23 P

Impact*

energy First Modified

(eV) Born Bethe 2

3 S—>33 P 23 S—*33D 2

3 S-^43 D

1.14 0 0

1.22 173 1.72 79 2.18 294 94

3.40 248 110 1.08 3.60

4.89 202 114 1.29 6.94 1.69 6.66 167 112 1.13 6.34 1.73 8.70 138 105 1.07 5.41 1.51 13.6 101 91 0.99 3.85 1.09 19.6 76 77 0.91 2.81 0.80

30.6 54 60 0.78 1.87 0.53

54.4 34 41 0.59 1.08 0.31

85.0 24 30 0.46 0.70 0.20

122 18 23 0.37 0.49 0.14

a The thresholds for the 2

35—*33P, 33 D, and

3.19, 3.25, and 3.92 eV, respectively. 43D excitations occur at

Table XXVI. Total cross sections for the excitation of the

2

3 S state of helium calculated using the Born-Oppenheimer

approximation and the first order exchange approximation (Bell, Eissa, and Moiseiwitsch, 1966). Total cross sections in units

of Trflfl2 .*

Impact energy

(rydbergs) BO Ef

1.44 9 . 52

(

— 1

)

1 . 04 ( — 1 ) 1.85 1.211(0) 1

43 ( — 1

)

2.25 8 . 48 ( — 1

)

9. 79 ( — 2) 2.56 6.23( — 1) 6. 95 ( — 2) 2.72 5 . 23 ( — 1 ) 5.81(-2)

3.24 3. 17 ( — 1) 3. 13 ( — 2) 3.69 2 . 08 ( — 1 ) 1 .89 ( — 2) 4.00 1 - 59 ( — 1 ) 1 .37 ( — 2) 9.00 6.45( — 3) 7. 35 (—4)

16.00 3. 63 (—4) 3. 29 (—4)

0.10 0.174 0.344

0.03 0.174 0.283

0.05 0.188 0.305

j* The numbers in parentheses denote powers of 10. b B.O. = Born-Oppenheimer approximation.

c Ei = first-order exchange approximation.

97
Table

XXVII.

Total

cross

sections

for

the

excitation

of

the 2

*P

states

of

helium

calculated

using

the

Born-Oppenheimer

approximation

and

the

first-order

exchange

approximation

(Bell,

Eissa,

and

Moiseiwitsch,

1966).

Total

cross

sections

in

units

of

rao

l

.“

a. Q»

a. o> C

5^ ^ III

ea

a.

«*O'

hj

I I I I I I I I I I rf X^- sO 1"- X^ CO O'

00 Tf Os Tf tO sO to

Os 00

vO co

t-h 00 O LO rO CM CN

(N(N-HaOO^tfOIN^(N

Tf CO

1 CO CO| COj CO

1 CO1 CO

1 Tf| to

1 1 1 1 1 1 1 1 O' 00 O' o 00 r-* 00 to o O' x^. CN O' o o sO x^ CN CO CN CN CN t-H CO o

HNNN(N(NMfO

I I I I I I I I iO rf O'

oo Tf co oo o

TH oo IT; IT) O O0 IO

rH CN X^ © O

-0o

>s

bo _

s a G 6?

<U 5

*J -Q U T3 S X

I I I I I I N N (S <N (S d 't

I I I

00 ?r> <T> O

N N -h O'

?r> 00 ^ N N

lO IO 'O O N if> oo cn in r—

Tf O'

CN vO

t^oo‘0'^^tNi'<$0''Oio

rt«NNN^«'fO''0

a

acd

t-3

-H-a

>

X

X

a

§H

98

£*,

toiO

fo

CJ

COTf

QCOCO

a,

COLO

a.

COrf

COCN

COto

Co

rh

<0

COCO

bo

<L> c .

,

v ,s , s ^ ^ ^ o o

# . o

1 sO sO

1 O x^ x^ x^

I 1 1 x^ 001 1 CO Os Qs t-H t-h

* * 1 1 1 1 1 1 1 1 1 1 1 1 1 1

x^. 00 x^ CN 't 't o —i so sO O -h O sO -1 t-H t-H O0 Tf (N t-H IT} T-H CN Os Tf

/-N ^ ^ ^ ^ <0> X^ vO

1 sOI sO

| sO| O N N1 I 1

X'— oo oO Os O' O' t-h

• 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 to CN CN o •>4< O CO rH co O CO V) H IO

CN CN CN -H to CO t-H SO N OO CS >-( If)

v y v ^ ^ v r —' ^ ^ Tf

1 Tf1 TfI to

1 to to to1 1 1 SO so1 1 O N N 00 001 1 1 1 1 • •

1 1 1 1 1 1 1 1 1 1 1 1 1 1 CN CN to O CN Os CN CN CN SO X^ Tf — ^H ^H 00 'O rO >-t OO Tf T-H to T-H SO CO

v , ^ ^ ^ ^ iO

. 1 Tf

1 Tf1 Tf1

1 ^ to to1 1 1 to so1 1 O N N N OO1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

• o CN O to H \c CO Tf CN T-H T—I X-— CN O

to CN CM CN H IO CO T-H X''- CN Os CN t-h sO

^ v , v , s ^ ^ ^ s ^ ^ ^

Tf Tf

1 Tf1 Tf1

1 Tf to to1 1 1 to to1 1 SO O N N N

• 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

* x^ *“• X— sO oo oo oo H< CN O O O

CN *f Tf CO CN —* Os to CN T-H co Tf CN T-H ^ ^ ^ ^ ^ ^ ^ rO

1 CO1 CO1 CO1 CO Tf Tf

I 1 1 Tf Tf1 1 to to so so X^1 1 1 1 1 • •

1 1 1 1 1 1 1 1 1 1 1 1 1 1 * sO O' 00 CN sO 00 co co CN if) Tf O' CN CN CN CN oo id CN t-h CO T-H Tf T-H Os

> _ ^

^ ^ ^ ^ ^ ^ CO

,

1 CO1 CO1 CO1 CO1 co co co

1 1 1 Tf Tf1 1 tO to O sO sO1 1 1 1 1 • 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

• 00 to O' 'O CO -h r^ o to co O' O' X"- X''- Os -1 to to to rO Tf CN O N 00 «) H

x v N s s s ^ ^ ^ ^ ^ ^ CO CN

1 CN1 CNI OJ

1 co fO co1 1 1 CO ^| | Tf IO LO O sO1 1 1 1 1 • 1 1 1 1 1 1 1 1 1 1 1 1 1 1

• O to to Tf o to to *“H LO so CS t-h x^ to o -1 — --1 Tf CN ^H tO -1 O CN 00 Tf

—V. ^

_

^ v ~v /—

V

^ ^ ^ ^

CN CN

|

1 CN1 CN1 CN1 CN1 CN CN CO

1 1 1 CO Tf

1 1 Tf to to to to

1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 CN CO CN o o to tO Tf t-H to oo T1 1*) 00 O' >f) Tt 'O o to to CO CN i-i 00 ro t-h tO CN sO CN t-h

v v ^ ^ ^ ^ . ^ ^ ^ ^

CO

| CO

1 COI CO1 CO Tf Tf

1 1 1 Tf Tf1 1 to to o o1 1 1 t 1 • *

1 1 1 1 1 1 1 1 1 1 I 1 * CO sO X^ sO CO CN co to Tf CN 00 Tf ro CN — 00 to CN t-h Tf -H IO (N rt

^

r— ^ . , , ^ ^ CO

.

1 CO

| CO

1 CO1 CO1 CO co co

1 1 1 Tf ^ iO to to sO so

• 1 1 1 1 1 1 1 1 1 1 1(111 1 1 1 1 1

* CN CO sO X^ CO X>> X>i T-H CN 00 CO oo T-H 00 Tf -1 CO CO CO CO CN T-H T-H to CN 00 CO T-H Tf CN

/^s _

^ ^ ^ ^ CO CO

| CN| CN

1 CO

|

.CO CO co

1 1 1 CO Tf

1 1 Tf Tf to to so

1 1 1 1 1 • 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

• to 00 rr> \o O Tf IO ro O O

sd o^ t-H Os N Tf CO T-H X^T CN T-H CO T-H so ^ / s /—v ^ ^ ^ ^ ^ s CN CN

1 1 CN1 CNI CN1 CN1 CN CN CN

1 1 1 CO CO1 1 CO Tf Tf Tf to1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 © x>- 00 to o 00 X^ CN Tf sO to t-h X^ Tf O' O

iO vO sd o SO Tf CO CN rH sd co T-H Tf rH to CO

CNTfs0OO©to©OQ©©©O©OO CNCNCNCN<O<Or^»O'O00OiOOOOO H rt N fO ^ lO

The

numbers

in

parentheses

denote

powers

of

10.
Table XXIX. Total cross sections for the \}S—+nl L excitations of helium calculated using the Ochkur approximation (Ochkur and

Bratsev, 1965). Total cross sections in units of irau

2 .“

Impact

energy

(eV) 2'P 3>P 4’P 5 >P 3'/) 4'D 5 'D 4>P 5

1 F

30 7 . 3 ( —2) 1 . 5

( — 2) 5. 3 (-3) 2. 5 ( — 3) 6. 3 (—4) 3. Of— 4) 1 . 6 f —4) 3.1 (-6) 2 A( — 6) 35 9 . 8 ( — 2) 2 . 2( — 2) 8.2( — 3) 4.0(— 3) 8.8(— 4) 4. 4 ( —4) 2 . 4 f — 4) 4. 3 ( — 6) 3 . 4 ( — 6) 40 l.l(-l) 2.6( — 2) 1 -0( — 2) 5 . 0 ( — 3) 1 .Of — 3) 5 . 2 ( —4) 2 . 8 ( — 4) 4. 8 ( — 6) 3 . 8 ( — 6) 50 1 - 3 ( — 1

)

3.2( — 2) 1 . 2

( — 2) 6 . 2( — 3) l.l(-3) 5 . 8 f —4) 3. 2 (—4) 5. Of — 6) 4. Of — 6) 60 1 4 ( — 1) 3 . 4 ( —2) 1 - 3 ( —2) 6. 7 ( — 3) l.l(-3) 5. 8 (—4) 3. 2 ( — 4) 4. 8 ( — 6) 3 . 8 f — 6) 80 1.4( — 1) 3.4( — 2) 1.4( — 2) 6.8( — 3) 1 .0( — 3) 5. 3 ( — 4) 3. Of— 4) 4. 1

f — 6) 3 . 3 ( — 6) 100 1.3(-1) 3 . 3 ( — 2) 1.3( — 2) 6.6 (-3) 9. Of— 4) 4.7 (—4) 2 . 6 ( —4) 3. 5 (-6) 2.8f — 6) 150 l.l(-l) 2 . 9 ( — 2) l.l(-2) 5. 7 ( — 3) 6. 8(— 4) 3. 6(— 4) 2. Of— 4) 2. 5 ( — 61 2 . 0 ( — 6) 200 9 . 9 ( — 2) 2.5( — 2) 1 - 0 ( — 2) 5.0 ( — 3) 5 . 4 ( —4) 2.9(—4) 1 . 6 ( —4) 1 . 9 ( — 6) 1 .6(— 6) 300 7 . 9 ( — 2) 2.0( — 2) 00 o 4.0( — 3) 3 . 8 f —4) 2 . 0 ( —4) 1.1 f-4) 1 . 3 ( — 6) l.l(-6)

400 6. 6 ( — 2) 1 . 7

( — 2) 6.7(— 3) 3 . 4 ( —3) 2. 9 ( — 4) 1 . 6 ( — 4) 8 . 6 ( — 5) 1 .Of — 6) 8. 1 f — 7) 500 5.8( — 2) 1.5 ( — 2) 5.8( — 3) 2 . 9 ( — 3) 2. 4 ( — 4) 1 . 3

( —4) 7 .Of — 5) 8. 1 ( — 7) 6. 5 f — 7)

a The numbers in parentheses denote powers of 10.

Table XXX. Total cross sections for the 2

3 S—>n3 L excitations of helium calculated using the Ochkur approximation (Ochkur and

Bratsev, 1966) . Cross sections in units of 7rao2 .*

Impact energy

(eV) 3

3 S 4

3 S 53 S 73 5 103 S 3

3 P 43 P 5

3 P 7

3 P 10*P

4 3.6(0) 3 . 7 ( — 1

)

• • • • • • • • • 1.6(0) 1 8 ( — 1) • • • • • • • • • 5 3.6(0) 5 . 8 ( — 1) 1 - 9 ( — 1

)

4 . 5 ( — 2) 1 - 2 ( — 2) 1.5(0) 3 . 4 ( — 1

)

1 . 3 ( — 1) 3. 3 ( — 21 8 . 9 ( — 3) 6 3.6(01 6. 1 ( — 1) 2.U-1) 5.6( — 2) 1 - 6 ( — 2) 1.2(0) 3 . 2 ( — 1

)

1 . 3 ( — 1) 3 . 8 ( — 2) 1 . 2

( —2) 8 3.2(0) 5 . 8 ( — 1

)

2.1(-1) 5.7 ( — 2) 1 - 7 ( — 2) 9. 8

( — 1

)

2 . 4 ( — 1

)

1.0(-1) 3. 1 ( — 2) 9. 6 ( — 3) 10 2.8(0) 5 . 2 ( — 1) 1 - 9 ( — 1

)

5 . 3 ( — 2) 1 - 6 ( — 2) 9. 5

( — 1) 2 . 4 ( — 1) 1.0(-1) 3 . 0 ( — 2) 9 . 4 ( —3) 15 2.1(0) 4 . 0 ( — 1

)

1 . 5

( — 1

)

4 . 2

( — 2) 1 . 2

( — 2) 8. 0( — 1) 2 . 3 ( — 1

)

1 ,0( — 11 3.1 (-2) 9. 9 ( — 3) 20 1.7(0) 3 . 2

( — 1

)

1 - 2 ( — 1

)

3 . 4 ( — 2) 1 - 0 ( — 2) 7 . 8 ( — 1) 2 . 3 ( — 1

)

1 .0( — 1) 3. 2

( — 2) 1 - 0 ( —2) 30 1.2(01 2 . 3 ( — 1

)

8.7 ( — 2) 2 . 4 ( — 2) 7 . 2 ( —3) 7 . 3 ( — 1

)

2. 2( — 1) 9. 5 ( — 2) 3. 0( —2) 9.66 — 3) 40 9. Of — 11 1 - 8 ( — 1

)

6. 7( — 2) 1 9 ( — 2) 5 . 6 ( — 3) 6 . 0 ( — 1

)

1 - 9 ( — 1) 8. 3 ( — 2) 2. 7 ( — 2) 8 . 5 ( —3) 50 7 . 4 ( — 1) 1 - 4 ( — 1) 5. 5 ( — 2) 1 - 5 ( — 2) 4 . 6 ( — 3) 5 . 5 ( — 1

)

1 - 7 ( — 1) 7.7( — 2) 2 . 4 ( — 2) 7.9(— 3) 70 5 - 4 ( — 1

)

l.l(-l) 4 . 0 ( — 2) i.K-2) 3.3( —3) 4. 8 ( — 1

)

1 • 5 ( — 1

)

6.7 ( — 2) 2. 1 (—2) 6.8( —3) 100 3 . 8 ( — 1

)

7 . 5 ( — 2) 2 . 8 ( — 2) 7 . 8 ( — 3) 2 . 4 ( — 3) 4. 3 ( — 1) 1 - 3 ( — 1) 5 . 8 ( — 2) 1 • 8 ( —2) 5 . 9 ( —3)

Impact energy

(eV) 3

3D ¥D 53D 7

3D lO3!) 4*F 53 P 7

3P 103F

4 5.7(0) 6 . 7

( — 1

)

... ... • • • l.l(-l) • • • • • • ...

5 6.4(0) 1.4(0) 5. 5

( — 1) 1 - 4 ( — 1

)

4. 1 ( — 2) 2 . 4 ( — 1) 1 - 3 ( — 1) 4. 1 ( — 2) 1.3 (—2)

6 6.5(0) 1.5(0) 6.0( — 1) 1 . 8 ( — 1

)

5 . 4 ( — 2) 2 . 5 ( — 1

)

1.4(— 1) 5. 1 ( —2) 1 - 7 ( —2) 8 6.0(0) 1.4(0) 5.7 ( — 1) 1 - 7 ( — 1) 5. 2 ( — 2) 2 . 4 ( — 1

)

1 • 4 ( — 1) 5.1 (—2) 1 - 7 ( —2) 10 5.5(0) 1.3(0) 5.1(-1) l.S(-l) 4. 7( — 2) 2.1(-1) 1 - 2 ( — 1) 4. 6 ( — 2) 1.6 ( — 2) 15 4.1(0) 9 . 8 ( — 1) 3 . 9 ( — 1

)

1 - 2 ( — 1) 3. 6 ( — 2) 1 - 6 ( — 1) 9.1(-2) 3. 4 ( —2) 1 • 2 ( — 2) 20 3.3(0) 7 . 8 ( — 1) 3 . 2 ( — 1) 9 .4 ( — 2) 2 . 9 ( — 2) 1 - 2 ( — 1) 7. 2 ( — 2) 2. 7 ( —2) 9.3( — 3) 30 2.3(0) 5.6( — 1) 2 - 3 ( — 1) 6.7 ( — 2) 2. 1 ( — 2) 8 . 9 ( — 2) 5. 1 ( — 2) 1 - 9 ( — 2) 6 . 5 ( —3) 40 1.8(0) 4.3(— 1) 1 - 7 ( — 1

)

5 . 2 ( — 2) 1 - 6 ( — 2) 6.6( — 2) 3. 8( — 2) 1 - 4 ( — 2) 5.0 ( — 3) 50 1.5(0) 3 . 5 ( — 1) 1 - 4 ( — 1

)

4. 2

( — 2) 1 • 3 ( — 2) 5 . 3 ( — 2) 3.3( — 2) 1 - 2 ( — 2) 4.0(— 3) 70 1.1(0) 2 . 6 ( — 1) 1.0(-1) 3. 1 ( — 2) 9 . 6 ( — 3) 3 . 9 ( — 2) 2 . 3 ( — 2) 8. 5 ( — 3) 2 . 9 ( —3) 100 7.6(-l) 1 - 8 ( — 1) 7. 3 ( — 2) 2 . 2

( — 2) 6. 8 ( —3) 2 . 8 ( — 2) 1 - 6( —2) 6.0( — 3) 2. 1 ( — 3)

The numbers in parentheses denote powers of 10.

99
Total

cross

sections

for

the 2z

S—>n lL

excitations

of

helium

calculated

using

the

Ochkur

approximation

(Ochkur

and

Bratsev,

1966).

Cross

sections

in

units

of

iro®*.' a3H

CM CM cm e? PO PO rt< to to to

ec. • • • 1 1 1 1 1 1 1 1 1 1 1 1 • to * O *-< CM T*< CM 00 t-H PO **

to CM PO t-H v-H Ov* PO

»—

cm CM CM <o PO Tt< to »o

ft* * • I 1 1 1 1 1 1 1 1 1 1 1 o CM Tt< sO 00 © Os to PO vo vO PO t-H CM to CM*

*—

CM ST <o PO rt< to

Q • • • 1 1 1 1 1 1 1 1 1 1 * to * * vO to CM T-H Ov «o vO

po CM rH to CM 00 s-H to ^ T-H —H CM CM PO <o 9 j * 1 cm © CM ^H CM 00 PO

so ID CM ^H PO* r*H Ov PO T-H

•»-—

v

t—

CM CM PO <o PO 1 1 l 1 1 1 1 1 Q # * © © o 1 1 1 1 1 1 1 1 1 po * CM © CN CM vO 00 PO CM vo PO CM to CM I'- PO o\ ^* CM CM* ^ 

»—

CM CM CM PO PO to • • • 1 1 1 1 1 1 1 1 1 1 1 * to * * ON ID r>» vO 00 © N

PO ^* t-H to PO -H PO

t-H t—H

H CM CM PO PO PO to • • 1 1 1 1 1 1 1 1 1 1 1 1 * vO \o CM ID 'O PO vO Os CM CM PO 00

cm PO r^. CM Os CM T-H vO* CM

t-H t-H T-i CM CM PO PO PO • • © © 1 1 1 1 1 1 1 1 1 1 PO * ID rH t-H PO © PO PO CM

CM T-H o^ rj«* CM s6 CM 00 PO VO CM H t-H CM CM CM PO PO 1 1 1 I 1 1 I tH T-H T—I © © © 1 1 1 1 1 1 1 CM Os © <D vO © © ©

Os CM T-H so CM tH to CM rHs c? CM CM CM CM CM PO PO ** • • • 1 1 1 1 1 1 1 1 1 1 1 • • • to * * * rD ID PO o Os o 00 00 00 00

SO <© to eH T-H PO & PO

H

CM CM CM PO PO <D Co • • • 1 1 1 1 1 1 1 1 1 1 1 • • TT * * ID Tt< CD Ov VO vO Os vO

t-H tH 00 PO CM* r>- PO t-H CM

*—h t-H ^H CM CM CM PO PO <o co 1 I 1 1 l 1 1 I • 1 © o © 1 1 1 1 1 1 1 1 1 po * CM O' CM CM 00 CM PO to to PO PO T_l to CM Os H vo PO rH 1s >

tJ «.

CM PO ID vO 00

©T-H to

t-H 20 30 5 50

© 8 aao

hH

100

The

numbers

in

parentheses

denote

powers

of

10.
Table XXXII. Partial and total cross sections for the PS—>2*5 excitation of helium calculated using the P5-21 5-2*5 close coupling

approximation (Marriott, 1964) . Cross sections in units of 7rao2 "

Partial cross sections 0

1 Total

cross (eV) (=0 1= 1 t=2 /=3 section

20.000b 1.00( — 2)

20.606 4. 02 ( — 2) 6. 86 (-2) 2. 98 ( — 5) ... 1 . 09 ( — 1

)

20.010 7. 68 ( — 2) 1.31 ( — 2) 8. 16 ( — 5) ... 8. 99 ( — 2)

21.400 8. 82 ( — 2) ... ... ... ...

21.790 7. 34 ( — 2) 8.27 ( — 3) 2. 60 ( — 4) ... 8 . 20 ( — 2)

24.580 6 . 56 ( — 3) ... ... ... ...

27.365 1 . 14 ( — 2) 4. 16( —4) 1 .55 ( — 3) 8. 06 ( — 5) 1 . 35 ( — 2)

a The numbers in parentheses denote nowers of 10. b Calculated neglecting coupling with the singlet metastable state.

Table XXXIII. Partial and total cross sections for the l'S—>2' 5 excitation of helium calculated using the l

1 5-2 ‘5-2* 5 close coupling

approximation (Marriott, 1964) . Cross sections in units of iran

2 .“

Impact energy

(eV)

Partial cross sections Q

( Total

cross 1=0 (= l (=2 1= 3 sections

20.580 0.00 0.00 0.00 0.00 0.00

20.606 8. 38 ( — 4) 2.291 — 3) 1 .92 ( — 7) ... 3. 13 ( — 3)

20.010 8. 82 ( — 3) 1 . 72 ( — 2) 1 . 60 ( — 5) ... 2. 60 ( — 2)

21.400 1 . 31 ( — 2) ... ... ... ...

21.790 1.39( — 2) 2.01 ( — 2) 2. 09 ( — 4) ... 3. 42 ( — 2)

24.580 8. 69 (—4) ... ... ... ...

27.365 4. 13 ( — 4) 2. 57 ( — 2) 2. 58 ( — 3) 1 .09 (—4) 2.88( — 2)

a The numbers in parentheses denote powers of 10.

Table XXXIV. Partial and total cross sections for the superelastic <:onversion of the 2*5 state to the 2*5 state of helium calculated using the P5-21 5-2*5-3*5 close coupling approximation (Marriott, 1966). Cross sections in units of 7rao2 a

Partial cross sections Q

( Total

Impact energy — cross (eV) £=0 1= 1 1=2 (=3 sections

0.026 2.20( + l) 1.28(+2) 6. 08 ( — 3) 1.50(+2)

0.430 2.26(+0) 5.31( + 1) 6.08( — 1) ... 6 . 60 ( + 1

)

1.210 2 . 87 ( — 1

)

2 . 48 ( + 1

)

2.79(+0) ... 2.79( + l)

4.000 2. 54 ( — 2) 1.31(+0) 2.36(+0) ... 3.70(+0)

6.785 2.08( — 1) 1.23 ( — 3) 8.00( — 1) 2.61 ( — 1) 1.32(+0)

a The numbers in parentheses denote powers of 10.

101
Table XXXV. Partial and total cross sections for the l‘S—>3‘P excitation of helium calculated using the PS-^P close coupling ap- proximation with exchange neglected (Vainshtein and Dolgov, 1959). Cross sections in units of 10-2 toq2

.

Impact11 energy

Partial cross sections15 Total

cross (eV) Q

1o Q

01 Q2l Q12 Q52 Q23 sections

23.342 4.46(

1) 2.0 ( — 4) 1 . 72 ( — 1) 1.5 ( — 6) 4. 6( — 5) • • • 0.649

23.444 2 . 45 (

1) 4.3(— 3) 1.52(0) 3 . 5 ( — 6) 5.6( — 3) 3.7 (—6) 1.808

23.852 1 . 66 (

1) 2. 10 ( — 1

)

3.52(0) 1 . 2

( —3) 6 . 06 ( — 1

)

1 * 2 ( — 5) 4.503

25.484 1 . 32 (

1) 4 . 30 ( — 1

)

2.72(0) 2.6( — 3) 7 . 45 ( — 1

)

1 - 2 ( —3) 4.031

27.204 1 . 02 (

1) 3 . 66 ( — 1

)

2.12(0) 3. 2( — 3) 1.135(0) 4.7 ( — 3) 3.731

32.012 7.4( — 2) 2 . 98 ( — 1

)

1.52(0) 2. 8 ( — 3) 1.286(0) 9. 78(— 3) 3.191 36.908 5 . 03 ( — 2) 2 . 33 ( — 1

)

1.02(0) 1.7 ( — 3) 1.220(0) 1 .39 ( — 2) 2.539

s Threshold energy is 23.308 eV. b The numbers in parentheses denote powers of 10.

Table XXXVI. Total cross sections for the double excitation of helium calculated using the first Bom approximation (Massey and

Mohr, 1935)

.

Total cross sections in units of 7TOo2. a

Impact energy

(eV) (2s) 2 *5 ( 2s2pyP (2s3pyP (2s4/») 1 P (3s2p) 1 P

75 1.1 (-5) 6. 8 ( — 4) 3 . 3 ( —5) 2.0( — 5) 1-5 ( —4) 100 1 - 7 ( —5) 9.5(— 4) 5 . 2 ( — 5) 3. 5 ( — 5) 3 . 5

( —4) 200 2.4( — 5) 9.6(— 4) 6. 8 ( — 5) 4. 5 ( —5) 5- 3 (—4)

300 2.8(-5) 8. 2 (—4) 6.2( — 5) 4. 5

( —5) 5 - 8 ( —4) 400 3. 0(— 5) 7.3(—4) 5. 3 ( — 5) 4.0(— 5) 5 - 4 ( —4) 600 3.2( — 5) 6.0( — 4) 4. 4 ( — 5) 3- 4 ( — 5) 5 - 0 ( —4)

ft The numbers in parentheses denote powers of 10.

Table XXXVII. Total cross sections for the resonance transi- Table XXXVIII. Partial cross sections for the 3s—>3p transi- tions of the alkali atoms calculated using the first Born approxima- tion of sodium calculated using the 3s—3

p

close coupling approxition (Vainshtein, Opykhtin, and Presnyakov, 1964b) .“ Total mation with exchange neglected. (Barnes, Lane, and Lin, 1965)

.

cross sections in units of TTUfl2. Partial cross sections Q

l in units of irao2

.

Impact

energy e Li Na K Rb Cs

Impact energy

(eV) 4.210 7.364 10.520 16.832 23.144

in threshold

units 2s—>2p 3s—>3p 4s—>4/> 5s—>5p 6s—>6/> t0 0.01 0.72 0.94 0.54 0.33

1 0.87 1.66 1.49 0.65 0.31

1.02 41.4 30.7 54.7 58.5 75.0 2 15.96 6.02 2.62 0.99 0.52

1.04 57.1 42.6 75.7 81.1 104 3 12.31 10.06 6.22 2.44 1.16

1.08 77.1 57.8 103 110 141 4 7.94 9.52 7.16 3.72 2.03

1.16 99.8 75.6 135 146 186 5 4.58 7.90 6.82 4.23 2.63

1.32 120 93.2 166 180 230 6 2.48 6.13 5.98 4.21 2.88

1.64 131 104 187 203 260 7 1.30 4.59 5.01 3.92 2.88

2.28 124 103 186 202 260 8 3.37 4.11 3.53 2.74

3.56 103 88.4 159 176 226 9 2.45 3.33 3.12 2.53

6.12 75.3 66.6 120 133 172 10 2.68 2.73 2.30

11.24 50.1 45.4 82.1 91.3 117 11 2.15 2.39 2.08

21.48 31.2 28.7 52.0 58.0 74.6 12 1.73 2.08 1.88

41.96 18.6 17.3 31.4 35.1 45.2 13 1.81 1.69

14 1.57 1.52

(1966).

102
Table XXXIX. Total cross sections for the 3s—>3p transition of sodium calculated using the 3s-3p close coupling approximation

with exchange neglected (Barnes, Lane, and Lin, 1965).

Impact energy

(eV)

Total cross sections in units

of irao

2 Impact energy

(eV)

Total cross sections in units

Of Tdo2

4.210 46.8 16.832 48.9

7.364 58.7 23.144 42.1 10.520 57.2

Table XL. Total cross sections for the 6s—*6p resonance transition of cesium calculated using the modified Bethe approximation

(Hansen, 1964) . Total cross sections in units of ira^.

Impact energy

(eV) (?6p

Impact energy

(cV) (?6p

1.60 41.56 5.03 114.3

1.74 48.84 7.25 118.7 1.89 61.27 11.60 114.6 2.03 66.33 15.95 106.2 2.18 70.16 23.20 92.60

2.90 91.54 30.45 81.66

Table XLI. Total excitation cross sections for nonresonance transitions of lithium calculated using the first Born approximation

(Vainshtein, Opykhtin, and Presnyakov, 1964a) , a Total cross sections in units of xoo2

. b

Impact energy e in threshold units 2s—*3p 2s—>4p 2s—*5 p 2s—*3s 2s—>4s 2s—»5s 2s—>3d 2.v—>4<f 2s-+Sd

1.02 9 . 67 ( — 1

)

2 . 24 ( — 1

)

8. 95 ( — 2) 1.21(0) 2 . 1 3

( — 1

)

7.81 ( — 2) 1.38(0) 4 . 09 ( — 1

)

1 . 79 ( — 1

)

1.04 1.30(0) 3.01 ( — 1) 1 . 20 ( — 1

)

1.66(0) 2.91 ( — 1) 1 . 07 ( — 1

)

1.90(0) 5 . 62 ( — 1

)

2 . 45 ( — 1)

1.08 1.68(0) 3. 84( — 1) 1 . 52 ( — 1

)

2.21(0) 3 . 86 ( — 1

)

1 . 41 ( — 1) 2.56(0) 7 . 50 ( — 1

)

3 . 26 ( — 1

)

1.16 2.01(0) 4 . 49 ( — 1

)

1 . 76 ( — 1

)

2.79(0) 4 . 84 ( — 1

)

1 - 76 ( — 1

)

3.31(0) 9 . 53 ( — 1

)

4 . 1 0 ( — 1

)

1.32 2.11(0) 4.67( — 1) 1 . 77 ( — 1) 3.25(0) 5 . 55 ( — 1

)

2.01 ( — 1) 3.96(0) 1.11(0) 4. 73 ( — 1)

1.64 1.87(0) 3.91 ( — 1) 1 . 50 ( — 1

)

3.33(0) 5 . 58 ( — 1

)

2 . 00 ( — 1

)

4.19(0) 1.14(0) 4 . 78 ( — 1

)

2.28 1.39(0) 2 . 84 ( — 1

)

1 . 09 ( — 1

)

2.90(0) 4 . 76 ( — 1

)

1 . 60 ( — 1

)

3.77(0) 9 . 86 ( — 1

)

4.08 (— 1)

3.56 8.91 ( — 1) 1 . 84 ( — 1

)

7.21 ( — 2) 2.12(0) 3 . 42 ( — 1

)

1 . 21 ( — 1

)

2.82(0) 7 . 1 6 ( — 1

)

2 . 93 ( — 1

)

6.12 5. 20( — 1) 1 . 13 ( — 1) 4. 54 ( — 2) 1.34(0) 2. 13 ( — 1

)

7 . 50 ( — 2) 1.81(0) 4 . 50 ( — 1

)

1 . 82 f — 1

)

11.24 2 . 88 ( — 1

)

6. 67 ( — 2) 2 . 78 ( — 2) 7.68(0) 1 . 20 ( — 1

)

4. 23 ( — 2) 1.04(0) 2 . 56 ( — 1

)

1 . 03 ( — 1

)

21.48 1 . 55 ( — 1) 3. 85 ( — 2) 1.65 ( — 2) 4.12(0) 6.43( — 2) 2. 25 ( — 2) 5 . 62 ( — 1

)

1 - 3 7 ( — 1

)

5. 50 ( — 2)

41.96 8.23( — 2) 2. 19 ( — 2) 9. 63 ( — 3) 2.14(0) 3. 32 ( — 2) 1 . 16 ( — 2) 2 . 92 ( — 1

)

7. 08 ( — 2) 2. 85 ( — 2)

a Revised according to private communication from Professor Vainshtein (1966). b The numbers in parentheses denote powers of 10. 103
Table

XLII.

Total

excitation

cross

sections

for

nonresonance

transitions

of

sodium

calculated

using

the

first

Born

approximation

(Vainshtein,

Opykhtin,

and

Presnyakov,

1964a).'

Total

cross

sections

in

units

of

7rao 2b 1 3

^3T

<o cOT

•ft. IO

•ft.

M § c 12

<U

O

ti <n a m

»-h CM fVl t*j I I I I

r-H ro ^ ^ o o «0> © OO 00 to O

00 *+©

r->

© ON I I I I- o © CN »o I I I I

(NM fS I I o

fN (N ro r/)

IOO0

CN t-H

N ro (N

(N

N

NO ^O I I I I I I I I I I

co

On CN CO CN

t-h CN CN

O

IO On CN IO tO O to CN CN O

IO nO nO >0 fO CN . 3

\b <r> I 1 I o o o o I I I I

<NI IO © CN IO 3 © IO ,-H CN GO o ©

H*

© CN T*1 CNO

© tH. IO In. © T-H t-H tH t-H © IO CO rH od oooooooo N* tH. ON OM ^ *t

CN CN CN CN i i i i s: 5 ©

© i i 9 ^

IO IO 'O co CN t-h

CN Os nO nO ^ N

nO co

iO On

CN CN CN CN 04

r

O I I I I I I

CN

©©

©

IO <o

t-h CN CN

iO

NO

vl i.

IO CO ^O © © ©

ro iO 00

On iO

CN©

On CO © o o 00 CO

©

od - - od

© CO ^ ^ ^ 1 1 1 CN1 1 1 1 l 1 1 CO CO v-H 00 00 CO 1 CN CN T“H oo ^ 

1 1 1 1 1 1 1 1 1 IO o NO —

© IO

On o CN CO

IO •o NO iO CO CN o o o o o o © 00 IO CN

IO 1 CO

CN CO CO CO CN ^ T-H T“^1 CN1 1 1 1 1 1 1

Tt* i-H o CO

© r^. 00 o © IO O

CN

© ^ v 1 1 1 1 1 1 1 1 1 tH IO © IO NO

IO 00 o © o CN

CO CO CO CN CN i-H <^s ^ ^ I 1 1 1 1 1 1 1 IO s o CO 00

CO CN CO

oo On od iO CO o O n o o © o CO s © CN CN

CN NO o CN

CO CO CO CO* CN © CN 3

00 NO CM

T—(

CO CN IO

t—

i-H tH CN CO

©

t-H

© *h* NO CN

00

NO I I I

00 rf CN

CN CN CN CN

•O nO

nO On

CN CN CN

oo

©

CO CN

CN r-H IO

oo On cn

On 00

CO

© iO

00 ^ CN

00

©On

CN ^ 3OO

>>G

</i <u

u* ©Ga

rC

>,aO

-c

tn .s3

cuaaJGMo

PQ

tp

<u ^ f

tuO *+h G °

’an V] g .ti

-5

<D 3 Cfi - 55o O^ rrt

CJ n

’o o

1/2 £“< GOOHXan<H 3T

4PT

«o 3r

COT4PT©T

4P

»oTT£T

A,

IOT

>n

G

00 ^ G 2

<v o

Tj C/2 22 0/ aS - I 1 1 | 1 1 | CN CN CN 1 00© 1 ©00 1 ©IO 1 cOCN 1 ©CO 1 ©00 1

00 1 ©oo 1 00© 1 ©00 1 00© 1 ooIO

CN CN CO •3* CO CN IO CO 1 1 | 1 1 1 1 CN CN 1 1^. CN 1 IO 1 ©CO 1 © 1 1 CN

T—( 1 © 1 CO© 1 ©CO 1 to * CN 1 IO© 1 IO

CO

IO r^. oo CS oo NO CO CN

© CO o o o o o o o o

vH1 7 7 7 © rt< © 00 t-H CO © © 00 CN

CO oo CO 00 oo CN 00 oo IO CO

-H CN CN CO CN CN T-^ od CN r-H o ^7 ^7 T-H T-H iT' T-H 77 o o O o 3 3

00 oo

O NO CO

© tH- CO 1^. On O NO CN CN CN

T-H t-H T-H T-H T-H T-H CN T-H ^ 

^—V ^ V. /-v ,, V X—N. ^ -s 1

T-H1

T-H1 1 1 1 1T-H1 CN1 CN| CN1 1 1 1 1 1 1 1 1 1 1 1 1

t-H On tH. t-H 00 CN tH. H< CN o © t-H o CO 00 00 o NO On CN © iO On

t-H T-H ^H CN CN CO CN T-H T-H

© CO t-H

T-H1 1 | CN 1 1 1 1 1 1 1 1 1 1 1 1 o 8

00 CO NO CO 8

CO T-H 00

© CO On CN© IO IO o o IO

CN IO NO oo oo t^ »o CO CN TH IO o o S' & o o S O O o 7 7 © On s

NO tH.

© 00 CN On

© o CO

© CN On t^ o CO t-H CN 00 IO

T-H CN CO CO Th «o CO CN T-H

© CO 1 1 | 1 1

T-H1 I | CN1 CN1 1 1 1 1 1 1 1 1 1 1 1 1

t-H 8 i

CN

©

© o 00 CN tH T*« IO H< On tH IH. CN o CN

CN CO ,ri< to to LO CO CN T-H

© CO 7 7 o o o o o 7 7 7 7

CN1 CN©

to©

©©CN 3 o ©T-H CN

t-H CN©

©iO 3

IO od ^H t-H TH T-H T-H od IO CN TH 00 o S o o o o o o o 3 7 7 © CO t-H © fn. t+h CN

IO 00 CO

IO CN t^

CN o o o © CN CO CO On

CN CO IO IO IO ^ CO CN t-H tH. CO

CN© 3oo©

© CN

CO 3

OO

CN

©IO

CN CN

00

©

T-H rH -1 T-H CN CO

© CN

10U

Revised

according

to

private

communication

from

Professor

Vainshtein (

1966

). b

The

numbers

in

parentheses

denote

powers

of

10.
Table XLIV. Total excitation cross sections for nonresonance transitions of rubidium calculated using the first Born approximation

(Vainshtein, Opykhtin, and Presnyakov, 1964a) .• Total cross sections in units of ira^. h

Impact

energy e in threshold

units 5s—*6p 5s—* 7 p

5.9—>8/> 5.9—>6.9 5.9—>7.9 5.9—>8.9 5.9—>4/ 5.9—>5d 5s—*()d 5s—*7d

1.02 2.47(0) 6.48( — 1) 2 . 79 ( — 1

)

1.81(0) 2 . 78 ( — 1

)

1 .066 — 1) 1.04(1) 1.24(0) 3 . 38 ( — 1

)

1 . 23 ( — 1

)

1.04 3.37(0) 8 . 84 ( — 1

)

3 . 80 ( — 1

)

2.51(0) 3 . 85 ( — 1

)

1 - 47 ( — 1) 1.43(1) 1.67(0) 4 . 65 ( — 1

)

1 . 66 ( — 1

)

1.08 4.47(0) 1.17(0) 5.01 ( — 1) 3.38(0) 5 . 22 ( — 1

)

1 . 99 ( — 1

)

1 .88(1) 2.15(0) 5.83( — l) 2 . 12 ( — 1

)

1.16 3.62(0) 1.46(0) 6 . 23 ( — 1

)

4.38(0) 6.81 ( — 1) 2 . 58 ( — 1

)

2.35(1) 2.58(0) 6.89( — 1) 2.61 ( — 1)

1.32 6.47(0) 1.67(0) 7.081 — 1) 5.27(0) 8 . 25 ( — 1) 3 . 1 2

( — 1

)

2 .66(1) 2.71(0) 7. 1 1 ( — 1) 2.60( — 1)

1.64 6 .68(0) 1 .68(0) 7.06( — 1) 5.64(0) 8.81 ( — 1) 3.31 ( — 1) 2.62(1) 2.42(0) 6. 18 ( — 1

)

2 . 27 ( — 1

)

2.28 5.73(0) 1 .44(0) 6 . 00 ( — 1

)

5.10(0) 7 . 88 ( — 1

)

2 . 92 ( — 1

)

2.18(1) 1.81(0) 4.62( — 1) 1 . 38 ( — 1

)

.1 . 56 4.26(0) 1 .05(0) 4 . 33 ( — 1

)

3.84(0) 5 . 83 ( — 1

)

2. 146 — 1) 1.54(1) 1.17(0) 2.91 ( — 1) 1 .10(-1)

6.12 2.77(0) 6. 76( — 1

)

2 . 75 ( — 1

)

2.47(0) 3 . 69 ( — 1

)

1 . 35 ( — 1

)

9.52(0) 6 . 84 ( — 1

)

1 . 79( — 1) 6 .68( — 2)

11.24 1.65(0) 3 . 98 ( — 1

)

1 . 50 ( — 1

)

1 .42(0) 2. 10 ( — l) 7. 67 ( — 2) 5.36(0) 3 . 72 ( — 1

)

9. 29 ( — 2) 3.66( — 2)

21.48 9 . 29 ( — 1

)

2 . 22 ( — 1) 8. 86 ( — 2) 7 . 66 ( — 1

)

1 . 1 3 ( — 1

)

4. 106 — 2) 2 .86(0) 1 . 95 ( — 1

)

4. 88 ( — 2) 1.93 (-2)

41.96 5 . 08 ( — 1

)

1 . 20( — 1) 4.47 ( — 2) 3 . 98 ( — 1

)

5.85( — 2) 2 . 12 ( —2) 1.48(0) 9.98( — 2) 2.50( — 2) 9.97( — 3)

a Revised according to private communication from Professor Vainshtein (1966). h The numbers in parentheses denote powers of 10.

Table XLV. Total excitation cross sections for nonresonance transitions of cesium calculated using the first Born approximation

(Vainshtein, Opykhtin, And Presnyakov, 1964a) , a Total cross sections in units of jroo2. b

Impact

energy e in threshold

units 6s—*7 p 6s—*8p 69—>9/> 6s—*7s 69—>89 69—>99 6s—*5d 6s—>6d 6s—*7d 69—>8(7

1.02 2 . 88 ( 0 ) 7 . 64 ( — 1 ) 3 . 26 ( — 1 ) 1.83(0) 2 . 93 ( — 1 ) 1 . 02 ( — 1 ) 1.62(1) 3.44( — 1) 2. 76( — 2 ) 2. 18 ( — 2 )

1.04 3.95(0) 1.04(0) 4. 44( — 1) 2.54(0) 4. 08 ( — 1) 1 . 43 ( — 1 ) 2 . 21 ( 1 ) 4. 78 ( — 1

)

4.83( — 2) 3. 38 ( — 2)

1.08 5.25(0) 1.38(0) 5 . 84 ( — 1 ) 3.46(0) 5 . 60 ( — 1 )

1 . 98 ( — 1 ) 2.92(1) 6 . 53 ( — 1 ) 8. 98 ( — 2 ) 5. 46 ( — 2)

1.16 6 . 66 ( 0) 1.74(0) 7 . 25 ( — 1 ) 4.54(0) 7 . 43 ( — 1 ) 2 . 64 ( — 1

)

3.64(1) 8 . 53 ( — 1 ) 1.64( — 1) 8 . 74( — 2)

1.32 7.78(0) 2 . 00 ( 0 ) 8 . 20 ( — 1 ) 5.57(0) 9.21(-1) 3 . 29 ( — 1 ) 4.09(1) 1 . 01 ( 0) 2 . 67 ( — 1 ) 1 . 28 ( — 1 )

1.64 8.04(0) 2.03(0) 8. 13(— 1) 6.09(0) 1 . 01 ( 0 ) 3. 59 ( — 1) 3.97(1) 1.04(0) 3 . 54 ( — 1 ) 1 . 58( — 1 )

2.28 7.12(0) 1.75(0) 6 . 84 ( — 1 ) 5.61(0) 9 . 1 3 ( — 1 ) 3.24( — 1) 3.22(1) 9. 12( — 1) 3. 70( — 1) 1 - 57 ( — 1

)

3.56 5.36(0) 1.28(0) 4

.

88 ( — 1 ) 4.27(0) 6.81 ( — 1 ) 2 . 40 ( — 1 ) 2 . 22 ( 1 ) 6 . 89 ( — 1 ) 3 . 08 ( — 1 ) 1 . 28 ( — 1 )

6.12 3.53(0) 8. 24( — 1) 3.06( — 1) 2.76(0) 4. 34( — 1) 1 . 52 ( — 1 ) 1.35(1) 4. 55 ( — 1) 2 . 1

1

( — 1 ) 8. 69 ( — 2)

11.24 2 . 12 ( 0) 4.84( — 1) 1.76( — 1) 1.60(0) 2 . 48 ( — 1 ) 8. 69 ( — 2) 7.52(0) 2 . 68 ( — 1 ) 1 . 26( — 1 ) 5 . 1 7

( — 2 )

21.48 1 . 21 ( 0) 2 . 70 ( — 1 ) 9. 65 ( — 2) 8.62( — 1 ) 1 . 33 ( — 1 ) 4. 69 ( — 2) 3.98(0) 1 • 47 ( — 1 ) 6 . 93 ( — 2) 2. 83 ( — 2)

41.96 6 . 66 ( — 1 ) 1 . 46( — 1 ) 5 . 14 ( — 2 ) 4 . 49 ( — 1 ) 6. 92 ( — 2 ) 2.41( — 2) 2.05(0) 7. 70( — 2) 3.64( — 2) 1.48(-2)

B Revised according to private communication from Professor Vainshtein (1966). The numbers in parentheses denote powers of 10.

105
Table XLVI. Collision strengths for the 3 P—*1 D excitation of oxygen calculated for &i

2 = 0.3 (Seaton, 1953a).

Approximation i*(l,2) iV>{2, 1) S>(1,2)

1 0.48 0.48 0.48

II 0.81 0.62 0.71

III 0.98 0.88 0.93

IV 1.06 0.90 0.98

Table XLVII. Collision strengths for transitions connecting

the *P, l D, and 1 S terms of the lowest configuration of atomic

oxygen (Seaton, 1955b; 1956)

.

W H(l, 2) Si(l, 3) «(2, 3)

(a.u.) (« = 2) (« = 3) (« = 3)

0.000 0.00 0.000 0.000

0.025 0.15 0.017 0.009

0.050 0.34 0.050 0.024

0.075 0.53 0.075 0.040

0.100 0.69 0.100 0.057

0.150 0.97 0.144 0.088

0.163 1.04 ... ... 0.2 1.06 0.180 0.118

0.3 1.20 0.238 0.167

0.4 1.32 0.278 0.208

0.5 1.44 0.305 0.240

0.6 1.53 0.324 0.269

0.7 1.60 0.335 0.295

0.8 1.67 0.344 0.316

0.9 1.73 0.349 0.336

1.0 1.77 0.353 0.354

To obtain N collision strengths multiply

O strengths by 25/24 25/8 225/16

Table XLVIII. Deactivation coefficients for atomic oxygen and nitrogen (Seaton, 1955b; 1956). Deactivation coefficients for O and N (ann ' in units of cm3 sec

-1

)

.

Electron temperature T

O N

10*021 10*03. 10»O32 10*021 10*031 10*032

5X102 0.9 0.8 0.4 0.5 0.4 0.9

1X103 1.6 1.2 0.6 0.8 0.6 1 .5

5X103 5.0 3.5 1 .9 2.6 1.8 4.4

1X104 6.7 5.1 3.0 3.5 2.7 6.9

5X104 7.6 7.2 5.6 4.0 3.7 13

1X105 6.8 6.6 6.0 3.5 3.4 14

Table XLIX. Effective Kramers Gaunt factor g for the Is—>2p

excitation of hydrogenic ions (Burgess, 1961)

.

Z

Impact energy e in threshold units 2

1 0.155 0.21

4/3 0.21 0.25

8/3 0.37 0.38

4 0.45 0.46

16/3 0.53 0.54

106
Table

L.

Partial

cross

sections

for

the

Is—

>2s

excitation

of

He+

calculated

using

the

\s-2s-2p

close

coupling

approximation

(Burke,

McVicar,

and

Smith,

1964a).

Partial

cross

sections

Q

L

in

units

of

iroo 2

.*

VO rO

vO O' o o o

Ov Ov *-h ro OJ Cv its

o d o o o o

CO vO Ov O vo 00

ro CO to O 8 8

O -h O On tO CO ^ s s odd

vO iO

cm©

d

On00Oo

O —

00 ^< ^vo o ’'f PO

CN 00

CO O O N N

00 iO cc

CO *“H *-H o © © o o

M O' M r<J O0 On o o o

o o

O 0,0

o o o

o o o o o

1-H vo On H IO ^

o o o

o o o

O CS co N CO CM o o o

<u

bJD S3

ccJ 43

CJX

CJO

#5

<ubo

S3rt 43

bocd

43

<v 4> CJX

4—

>

4-> 4>

CJX

4-) 4-> <u 3, V 3 0> *3b 3. .9 *c o .s *c o .9 •c co H £ CO H 55 CO H

<L> bOa

cti 43

cjX

<uo&

<U <L> be b£ be C C (3 d d ctf

4-> 4J s 4-* 4-) 43O 4-> 4-* 43

CJ 4-> 4-> <u X V 1) X <U <u X u

”3) 3. <u a 3 <u 3> 3 0) ’bi 3

.2 •c 0 .g •c 0 .a •C 0 .9 •c

cn H C/3 H 55 C/3 H £ CO H

4>boa

d

43

bo a

.9

co £H

CM vO

107

Spin

weighting

factors

included.
Table

LI.

Partial

cross

sections

for

\s-*2p

excitation

of

He+

calculated

using

the

\s-2s-2p

close

coupling

approximation

(Burke,

McVicar,

and

Smith,

1964a).

Partial

cross

sections

QL

in

units

of

irao 2*

ro

PO On

PO iO

pn

Q v-i 8 8 8 odd

On PO iO

00 *-• PN Q PN 8 8 8 © © o’ oo o o o

On 0 00

O NO 0 O O O d d d

NO NO

rt4 PO PO

-H

O 00

ro ro 00

lo <r> 888 q o o

NO ^ o <r> < (n —< ro 888 odd

oo t-» ^ CN fD 00 ID -h ro 888 odd oo

NO

fDNO

<D N-I ID

On 00 O'

nO

—t §< 888 odd

50 NO 00

PN nO ro

NO ^ 888 odd

NO

fN) 00 'f

<r> o cs NO NO

O NO lO On

C4

O

O

O

O 0 0 O

00 to rf iO PO iO

PN NO On

r^- 8 § 8 8 § 8 8 O O o o o o o o o

On<DlOO NO

LO 00 04 O <~4 r-~ O 8 8 odd 3

00 On

PO

O 8

CN8LOOO 8 8 d d d d d

00 On 500

NO LO

cs to

PO s Oto 0 8 8 O 8 d d d d d 3

NO LO LO © On t'>» f"* »—I T+< 8 8 1 odd 2 oo oo l NO N 888 © © ©

t~» t'. O n* nO C4 O —

o o o

00 NO

N

N

N t> -H *D 8 8 8 odd

rf On PO

t* oo §

pn no . 8 8 o o o O PN

Q 00 oo ^ NO ^ O o 8 © dod

*-H (N

O0 n On ^ ^ © 8 o odd

'ON IO

NO NO 00 D- ID Q\ © 8 8 odd 8 8 2 ro no 3 8 8 odd

O' tH

O oo

NO

m

a tN (no 8 8 8 odd

»o o no

NO

J-J LO o 8 o odd —< LO CO

to o o

-t OO 04

t'. tD

© o © o dod

r-» <D

O 00 to 00 OO H to

I—i »—i © © © o odd

O' NO

O

z? oo 04 O 00 OO

05

© i-H

©

©© dod

NO O' 005 » o « O' O ON

LO -3i

O o S 55

o’ o o O i-» « (N Ol

<M

O

C4 S 08 doo

NO ro no NO NO ^ PN o po

po000^ doo

*-< lo ro

lO *-H OO

PO 00 PN 000PO Q d o o

00ooo

NO 00

l~-. LO

O' 04 o o

04 id

Tf CN NO

t"D <D <"4 <DO OO Od

•>* O'

O' 00 *-H N# to

tDOOOCN ©

odd

to LO

ts NO ID

no <n ~ 888 odd

O -H ro

0< O' O' 00 ^ “

C4o o

o’ o O 'tjt ON 00

On 10 00 04

C4o

O'0

LO

ON

On 1^- PO

On 10

PO 8 0

t-H O 8 8 8 8 8 8 d d d d O d d d

CM On PNlO PO ID ON 0 NOPN

PN 00 PO

PN Lf PO

PN od

O

D4 00 5 D 2 888 OOO 8 8

OOO 8 8

LO to

00

04

O

O o O O O O 0 0 0

00 00 00 0 PO

NO 00 PO PO 3: T—i PN OOO O O o

00

t-H PN *-H ^ Q

OOO

52 to O' o <0 00 LO o 8 8 1 odd

00 *0© 8 8 8 OOO

r* PO

PO PN

NO ^ O

OOO

PO On 00

PN

Q 8d

10 NO 00 00 N 8 8 § OOO

NO

O C4

04 LO u-> OOO

Lft to C4 PO

O

O o 1 /

<Db

EP

<L> a

<uas

<vtoa

aj 43uX

<uo

55

pn

boa

.Sa)

H

<DbDa

rto

t-i 0o£

44 _a> 13) a. c *C

to

H

<Vtoc

cj 43O <->

bX) C

c3 43O +->

bOCd

43

<J 4-> 4J

0)boc

rt 43O

0)boa

cti 430 +-> 4->

a>bOCa

43

X CL> ^5J

X 44 44

X JJ 44

X JJ 44

X <u 44

X

0) a; "3) "a 4> "3) "a <u "So ’S.

<u bo ’0, 0) bO

a 0 .S •E O .g •E 0 .s ‘E 0 .9 •E 0 .9 •E 0 £ c75 H 5? CD H a H £ c75 H 55 c/3 H oo

44

_ 4J 3) H S - c

/5

H

108

Spin

weighting

factors

included.
Table LII. Zero-order partial cross sections for the Is—>2s and Is—>2p excitations of He+ calculated using (a) the \s-2s-2p and

(b) the \s-2s-2p-3s-3p close coupling approximations at 68 eV (Burke, McVicker, and Smith, 1964a).

Transition

Cross sections in units of

irOo2 Transition

Cross sections in units of

7r<Zg2

Is—>2s Singlet (a) 0.00306 Is—>2p Singlet (a) 0.00421

(b) 0.00273 (b) 0.00409

Triplet (a) 0.00035 Triplet (a) 0.00073

(b) 0.00032 (b) 0.00071

Table LIII. Positions of the autoionizing levels of helium relative to the ground state of helium calculated using the \s-2s-2p

close coupling approximation (Burke, McVicker, and Smith, 1964b) and obtained experimentally (Madden and Codling, 1963);

Simpson, Mielczarek, and Cooper, 1964) . Positions of the autoionizing levels of helium relative to the ground state of helium given in eV.

L n

No exchange Singlet Triplet

Theory Theory Experiment Theory Experiment

0 2 58.151 57.860 57.

9

b no resonance calculated

3 62.850 62.916

4 64.139 64.179

5 64.653 64.673

1 2 58.658 58.352 58.

5

b

3 63.003 63.132

4 64.199 64.247

5 64.682 64.701

1 2 60.908 60.257 60.12“

3 63.626 63.683 63.65“

4 64.444 64.474 64.46“

5 64.801 64.817 64.81“

• Madden and Codling, 1963. b Simpson, Mielczarek. and Cooper, 1964.

109
-O3Os

i-H A

•§

C/3d

<uu>u oH

NO On rH On r— NO >o co co o co CO o tp oo v-H »—(

CO LO rj< Os v-H 00

Tt< 00 CN CO Os CO CO

O CO co CO 00

es v—

CN CN

O CN CN 8

CN CN

O o o o o o o o OO

O o o o o d o’ d d o d d o’ o d d o

bO££

ju

<jXW Ss

<ubO£

c3 -£

cjXW

as

<L> bJO £

c$ ^£uXW

<0

COCNoas

<u

b£) £d

,£

CJXW 8

CNO

Os

CNJ o©

vO

00Oo

II a£

bO£

ctf MVXW £

PQ£

.2*-£

cja

*xolxaad

bO .2a£O

cj

<1> CO0

13 *3? £ * 1 *s

ij’S 5 3

60 .s

.s ^ g

O*

in T3 3

4)

O

Cy cj 3 ”

ci $ “ 2 + CJ 4) , a 3 O ccj 3 * o o o o

2; ^ ro S <5 'O ill o o o

Os VO

oo lo VO

vO CN CN 8 8 8 d d o’ o o o

l^. M3

T-1 00 P'5 CN 8 8 8 o o o O cn lO CO O co co

v-H t-H O? o o o d o’ o’

rj< no *-• O O O

CN no

^1 O' t'* CN CN 8 8 8 d o’ o o o o ^ OO N O cn 8 o o o o o o o o O

On cn LO On LO o On (N CN

Os co lo co S

00 CNCO

Ot-H On 8 § § 8 8 § 8 8 8 o o o S

IO CN

co to

CO CN 8 8 8

o’d o’

noT

«o

<D J3

CN Tt< 00 CO ^ On O O OOn CN CN CN 00 o o o

On On co

t-H iO co o o o 3 2 o o o’

«-h u->

O

0\ rO O O O

ClJ 4-i O+-> "O£dd

Ph

o>bo£dACJX

<uO£

JD

bo£• H

C/3

<U "3. •cH

v

603

.3

cjKVoa

4> 13) .am

ju

•cH

<D603

oSOa

<uM

.am

JU

•c* H

o>603

cU .3

CJK4) o

ju

'60 .a

in

JZ

•cH

603d1 4-1 X <u <u

<v '60 'H. o g •c

55 in

H >HH

hJ9

ra<H

<o

00 o o

LO

110 I

Spin

weighting

factors

have

been

included.

The 1s—*3p

excitation

threshold

is

3.5

rydbergs.

The

total

column

includes

estimates

of

the

higher-order

partial

cross

sections.
Table LV. Total excitation cross sections for transitions of N4+ calculated using the Coulomb-Bom approximation (Burke, Tait,

and Lewis, 1966). Cross sections in units of iroo*.

Incident electron energy (rydbergs)“ 2s—>2p 2s—>3s 2s—*3p 2s->3d 2p-*3s 2p->3p 2p—>3d 3s—*3p 3s—*3d 3p—*3d

1.0 3.87

1.44 2.80

2.25 1.92 4.0

4.5455

1.20

76 5.0 1.01 0.0304 0.0097 0.0519 0.00316 0.0657 0.190 31.5 2.73 36.4

6.0 0.87 0.0257 0.0104 0.0462 0.00318 0.0551 0.172 17.0 1.25 15.4

8.0 0.69 0.0197 0.0112 0.0381 0.00311 0.0420 0.148 9.4 0.592 9.17

12 0.50 0.0134 0.0114 0.0286 0.00287 0.0287 0.120 5.23 0.283 4.19

16 0.42 0.0101 0.0110 0.0230 0.00263 0.0220 0.104 3.51 0.185 2.35

24 0.29 0.0068 0.0099 0.0164 0.00224 0.0150 0.084 2.12 0.106 1.35 32 0.23 0.0051 0.0089 0.0128 0.00196 0.0114 0.071 1.65 0.072 0.94

a The incident electron energy is given relative to the (Is)* 2s ground state.

Table LVI. Total excitation cross sections for the 4s—>4/>, 3d—*4p, and 4s—>3d transitions of Ca+ calculated using the Coulomb-Born

approximation and the unitarized Coulomb-Born approximation (Van Regemorter, 1960a; 1961). Total cross sections in units of ira<fl

4s—>4/>fl 3d—»4/>a 4s—

3<P> V 0 0.1 0 0.1 0.1062 0.2062

Coulomb-Bom 126.5 99.7 94.6 47.1 17. 35 16.38

Unitarized Coulomb- 58.2 52.0 58.3 31.5 10.8 10.2 Bom

a Van Regemorter, 1960a. b Van Regemorter, 1961.

Table LVII. Collision strengths for the excitation of forbidden Table LVIII. Values of the constants G, ho, hi, hi occurring

transitions between the terms of the lowest p

2

, p

3

, and p

4 con- in the polarization formulas for the helium fines LSJ—*J"

figurations of certain positive ions (Seaton, 1953b; 1955b 1956).“ (Percival and Seaton, 1958).

Configuration Ion 0(1,2) 0(1,3) 0(2, 3) SLJ J" G ho hi h2

2p

2 N+ 2.39 0.223 0.46 011 0 1 1 1

• • .

1 -1 1 3 ... Q2+ 1.73 0.195 0.61

2 1 7 13 ... p3+ 1.21 0.172 0.58 022 1 3 5 9 6 Ne4+ 0.84 0.157 0.53

2 -3 3 7 15 29

10

3 3 26

110 1 0 • • • • • • • • • 2p3 0+ 1.44 0.218 1.92

111 0 -1 1 3 . .

.

JT2+ 1.00 0.221 3.11 1 1 3 5

... 2 -1 13 27 • . . Ne3+ 0.68 0.234 3.51

112 1 21 47 73 • • • Na4+ 0.43 0.255 3.49 2 -7 11 29 ... 3 1 7 13 ...

2/>4 F+ 0.95 0.057 0.17 121 0 3 5 9 6

1 -3 7 15 18 Ne2+ 0.76 0.077 0.27 2 3 41 81 78 Na*+ 0.61 0.092 0.30 122 1 3 9 17 14

2 -3 7 15 18 Mg4+ 0.54 0.112 0.30 3 3 29 57 54

123 2 18 41 76 58 3p3 S+ 2.02 0.383 12.7 3 -9 11 25 34

4 3 15 29 26

a Values in italics were estimated by interpolation and extrapolation

111
Table LIX. Values of the constants G, ho, hi, fo occurring in the polarization formulas for the helium multiplets LS—*L"S

(Percival and Seaton, 1958)

.

SL L" G Ao h hi

01 0 1 1 1

• • • 2 1 7 13 • • •

02 1 3 5 9 6

3 3 15 29 26

11 0 15 41 67 • • • 2 3 73 143 • • •

12 1 213 671 1271 1058

3 213 2171 4271 4058

Table LX. Values of the constants G, ho, hi occurring in the

polarization formula for the 2P»/i—>2 S transition (Percival and

Seaton, 1958).

I G ho hi

0 3 5 7 * 15 37 59

1 33 161 289

§ 81 427 773

Table LXI. Values of the constants G, ho, hi occurring in the

polarization formula for the 2 P—+2 S multiplet (Percival and

Seaton, 1958).

1 G ho hi

0 3 7 11

12 15 53 91

1 33 236 439

§ 27 209 391

Table LXII. Threshold percentage polarization for resonance

lines of lithium and sodium.

Ion Calculated* Measured15

6Li 37.5 39.7±3.8

7Li 21.6 20.6±3.0

“Na 14.1 14.8=1=1.8

* Flower and Seaton, 1967.

b Hafner, Kleinpoppen, and Kruger, 1965.

Table LXIII. Percentage polarization of the 2p—»ls multiplet

of atomic hydrogen calculated using the ls-2s-2p close coupling

approximation Is—*2p excitation cross sections. w P(%)

0.01* 17.3

0.03* 15.7

0.05* 16.7

0.06*5 20.9

0.25b 28.35

0.46b 27.99

0.69b 26.67

1.50b 18.00

3.25b 8.55

* Damburg and Gailitis. 1963.

b Burke, Schey, and Smith, 1963.

112
Table LXIV. Measured Absolute Values of Helium Excitation Cross Sections.

n'P

Branching0

ratio B («—+2)

Q(n-*2)

(10-20 cm2

)

(a) Helium w'P—>2 1 5’

Q{nl P) Electron = Q(n-*2) /B(n—>2) energy

(10-20 cm2

) (eV) Reference Comments

3

l P 0.0231 3.5P 150 90-110 s c f g 457 108 u f k

350 100 V a d e g i 17.4 750 100-110 w adgi

260 100 X a h j

1 290 100 y Theoretical, m

4iP 0.0281 1.5 53 100-120 s c f g 210 108 u f k

159 100 V b d e g i 8.5 300 100-120 w adgi

89.2 100 X a h j1 114.5 100 y Theoretical, m

5'P 0.0283 84 108 u f k

3.9 140 100-120 w adgi

45.2 100 X a h j1 58.1 100 y Theoretical, m

6

l P 0.032 39 108 u f k

2.3 72 100-120 w adgi

7»P —0.025 -0.7 —30 100-120 w adgi

(b) Helium n

l S—>2l P

Qin'S)

Branching Q(n'S) Electron Translated'

ratio Q(»-»2) — Q(w—» 2)/B (n—>2) energy^ to 100 eV

n'S B(n—>2) (10-20 cm2

) (10-20 cm2

) (eV) (10-20 cm2

) Reference Comments

3'S 1.0 36 108 36.84 u bk

49 40 31.9 V b g

131.4 32-33 ... w bg

22.3 X bh

46.125 z Theoretical, m

4>5 0.59 12 20 33 15.25 s b g

16.5 108 16.88 u bk

24 43 15.58 V bg

15.1 25.4 33 15.41 w bg

7.67 X bh

17.81 z Theoretical, m

5

] S 0.471 3.75 8.0 33 5.21 s bg

7.0 108 7.16 u bk

9.2 45 6.37 V bg

5.8 12.3 33-34 ... w bg

3.64 X bh

8.78 z Theoretical, m

6'S 0.43 4.0 108 4.09 u bk

4.8 45 3.14 V bg

2.6 6.0 35-37 • • • w bg

1.98 X bh

4.93 z Theoretical, m

7'S —0.43 1.5 —3.5 35-38 • • • w b g

2.95 z Theoretical, m

8*5 —0.43 1.1 —2.5 33-36 . .

.

w b g 2.24 z Theoretical, m

113
Table LXIV ( Continued

)

n'D

Branching

ratio B (n—>2) Q(n~*2) (10-“ cm2

)

(c) Helium n

1D—*21 P

Q(nl D) Electron — Q(ji—*2)/B{n—>2) energy*1

( 10-20 cm2

) (eV)

Qin'D)

Translated1-

to 100 eV

(10-20 cm2

) Reference Comments

3

X D 1.0 25 108 26.94 u f k

42 46 24.09 V d e g i n

119.2 44-46 ... w d g i 14.2 X hjl

7.92 y Theoretical, m

4

l D 0.74 13 17.6 40-45 11.88 s d e g i n

12 108 12.93 u f k

17.6 53 13.16 V d g i 26 .7 36.0 42-45 27.22 w d g i 6.75 X hjl

4.13 y Theoretical, m

5

X D 0.652 5.5 8.5 44-48 5.70 s d e g i n

7.1 108 7.65 u f k

9.0 53 6.73 V d g i 7.9 12.1 40-45 ... w d g i 3.80 X hjl

2.29 y Theoretical, m

6

l D 0.617 3.0 4.9 42-46 2.99 s d e g i' n

3.0 108 3.24 u f k

4.7 53 3.53 V d g i 5.3 8.6 40-45 ... w d g i 2.00 X hjl

VD ~0.617 1.8 108 1.94 u f k

3.0 42-45 ... w d g i 8W ~0.617 1.4 108 1.51 u f k

1.6 ~2.5 41-46 ... w d g i

(d) Helium n*5—>2*P Q(n*5)

Branching Q(n3 S) Electron Translated1-

ratio Q(n~*2) = Q(n-*2)/B(n-*2) energy^ to 100 eV

n

3 S B{n-*2) ( 10-20 cm2

) ( 10-20 cm2) (eV) (10-20 cm2) Reference Comments

3*5 1.0 15 108 14.68 u k

107 35 11.06 V g

130.1 26.5 ... w g 3.24 X h

65.98 y Theoretical, m

4*5 0.62 15 24 26.5 2.51 s g n

21 .9 35 30 ... t g n

4.4 108 4.31 u k

35 35 3.6 V g 22 .0 35 27 4.95 w g

1.51 X h

2.46 y Theoretical, m

5*5 0.477 6.1 12.8 27 0.99 s g n

8.2 17.2 30 ... t g n

1.44 108 1.41 u k

12.3 35 1.27 V g

8.0 17.0 28 • • • w g

0.451 X h

1.23 y Theoretical, m

6*5 0.454 3.2 7.1 30 ... t g n 3.7 8.2 29 ... w g 7*5 ~0.45 1.4 3.1 30 ... t g n

1.06 108 1.04 u k

1.3 2.9 29 * * * w g

11h
Table LXIV ( Continued)

n

3P

Branching

ratio B («—>2) Q(n-+2) (10-20 cm2

)

(e) Helium n

3P—*23 S

Q(n*P) Electron = Q(n->2)/B(n-+2) energy*1 (10-20 cm2

) (eV)

Q(n3P)

Translated'

to 100 eV

(10-20 cm2

) Reference Comments

2

3P 1.0 2510 27 . .

.

w d g i 3

lP 0.898 75 84 28 11.21 s d e g i n 95 106 32 ... t d e g i n

11 108 13.41 u f k

97 37 17.13 V d g i 121 135 28 33.81 w d g i 5.01 X hjl

4.84 y Theoretical, m

4»p 0.779 10 3 13 32 ... t d e g i n 4 108 4.88 u f k

8 10 29 • • • w d g i 1.74 X hjl

2.02 y Theoretical, m

5

3 R 0.686 2 108 2.44 u f k

~0.5 0.75 30 ... w d g i 0.987 X hjl

1.06 y Theoretical, m

(f) Helium n

3D—>23P

Q(n3D)

Branching Q{n3D) Electron Translated'

ratio Q(n-+2) = Q{n-+2)/B{n-+2) energy*1 to 100 eV

n3D B («—>2) (10-20 cm2

) (10“20 cm2

) (eV) (10-20 cm2

) Reference Comments

3*D 1.0 36 26 9.04 s d e g i n

25 108 28.35 u f k

31 35 4.55 V d g i 35 27 • • • w d g i 1.61 X hjl

0.11 y Theoretical, m

4

3Z> 0.789 9.8 12.4 27 2.58 s d e g i n

4.6 108 5.22 u f k

12.0 35 1 .74 V d g i 11.6 14.7 27.5 5.44 w d g i 0.471 X hjl

0.063 y Theoretical, m

5

3D 0.713 6.1 8.6 30-32 3.78 s d e g i n

3.0 108 3.41 u f k

6.2 38 0.92 V d g i 6.4 9.0 29.0 ... w d g i 0.037 y Theoretical, m

6*D 0.657 1.5 108 1.70 u f k

3.9 35 0.56 V d g i 3.7 5.6 29.0 ... w d g i PD ~0.65 0.7 108 0.80 u f k

1.7 ~2.6 29-30 ... w d g i 8

3D ~0.65 0.8 ~0.01 29-30 • • • w d g i

“ Possible error due to imprisonment of resonance radiation. b Substantial error due to imprisonment of resonance radiation.

® Complete imprisonment of resonance radiation.

d Possible large instrumental polarization error.

® Some pressure depolarization likely.

' Essentially complete pressure depolarization.

* Uncorrected for cascading.

h Correction for cascading applied by author.

. Uncorrected for polarization.

1 Correction for polarization applied by author. k Extrapolated to zero pressure from data taken at several pressures ranging from about 0.01 to 0.1 torr.

* Correction for instrumental polarization applied by author. m Theoretical values do not include cascading contributions. “ Possible error due to excitation transfer.

° Branching ratios calculated from Wiese, Smith, and Glennon (1966). p Values presented in boldface are numbers which appear in the original Publication.

Q These are the energies at which the maximum excitation was observed

except for the data (Ref. 3) of Gabriel and Heddle for which observations were made at 108 eV.

' Data of Gabriel and Heddle (1960) was translated from 108 to 100 eV

using the excitation function from the appropriate series as measured by Heddle and Lucas (1963). Since the translation amounted to 8 eV a

negligible error is introduced. Other translations were based on curves

presented in the same papers as the original absolute values.

• Yakhontova (1959, 1960).

1 Stewart and Gabathuler (1959).

u Gabriel and Heddle (1960).

T St. John, Miller, and Lin (1964) ; Miller (1964). w Zapesochnyi and Feltsan (1965).

1 Moustafa Moussa (1967). y Ochkur and Bratsev (1965b).

* Fox (1965).

115
Table LXV. Absolute values for excitation of sodium lines.

QmaiXlO18 (cm2

)

Transition X(A)

' exo® (volts) MMI

(eV) Ref. b Ref. c Ref. d

3

2 S-32P 5890-5896 2.1 10-13 2180 (4800) ...

3

2 S-A2P 3302-3303 3.75 6.5 20.6 ... ...

3

2 S-52 P 2852.8-2853 4.34 7.0 0.16

3

2P-42 S 11 131-11 404 3.19 5 . 3 (?) ~90

3

2 P-52 S 6154-6161 4.12 6.2 20.2

3

2P-62 S 5149-5154 4.51 6.4 6.1

3

2 P-72 S 4748-4751 4.71 6.5 1.3

3

2P-82 S 4541-4546 4.81 6.8 0.70

3

2P-92 S 4420-4423 4.88 6.9 0.37

3

2P-1G2 S 4343 4.94 6.9 0.18

3

2P-32D 8183-8194 3.62 7-9 459

3

2P-42D 5683-5688 4.28 7-10 34.8 12.2

3

2P-52D 4979-4983 4.57 7-10 12.2 • • •

3

2P-62D 4664-4669 4.74 8-11 5.0 3.4

3

2P-72D 4494-4498 4.84 9-13 1.9 2.3

3

2P-82D 4390-4393 4.90 9-14 1.00 1.3

3

2P-92D 4321-4324 4.95 9-14 0.47 (0.7)

3

2P-102D 4273-4276 4.98 9-14 0.28 • • •

3

2P-ll2D 4239-4241 5.0 9-14 0.21 • • •

* Note: Vexe(volts) —excitation potential; EmaI (eV) —location of the ® Christoph (1935). maximum of the excitation function. “ Volkova (1961). b Zapesochnyi and Shimon (1965).

Table LXVI. Absolute excitation functions of the resonance lines of potassium, rubidium, and cesium; and of the resonance levels, obtained by summing doublets and applying a correction for cascading (Zapesochnyi and Shimon, 1966).

Gn.xXlO^cm*)

of resonance Atom Transition X( A) E,xa (eV) -Em«x(eV) of lines levels

K 4

25l/2—2Pi/2,3/2 7699-7665 1.61-1.62 12-16 6.8 6.3

( 5

2 Sl/2-52Pl/2 7948 1.56 1 1.56 1 Rb

l 5

25i/2-52Pj/2 7800 1.59 JH: 3 , Ir

( 6

2 Si/2-62Pi/2 8944 1.39

| f 3.27 1

I Csj

, 6

2 5i/2-62 Pj/2 8521 1.45 JM1 e, J

9.3

116
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