NBS

^ecUnical ^iote

LECTURES ON ION-ATOM COLLISIONS

m. r. c. Mcdowell

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NATIONAL BUREAU OF STANDARDS

^ecknical <7(ote 185

ISSUED MARCH 15, 1963

LECTURES ON ION-ATOM COLLISIONS

M. R. C. McDowell

Royal Holloway College

University of London

Lectures delivered at the Joint Institute for Laboratory Astrophysics, University of Colorado, during September, 1962.

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C. Price 40 cents

LECTURES ON ION-ATOM COLLISIONS

By

M. R. C. McDowell*

PREFACE

This report contains lectures delivered at the Joint Institute

for Laboratory Astrophysics, University of Colorado, during September,

1962. The material of the first four is largely taken from recent

books by D. R. Bates, and was presented for pedagogic purposes only.

It does not purport, in any sense, to be a record of original work

by the author. Any new material is confined to the last two lectures.

The numerical results quoted are taken, in part, from as yet

unpublished work by M. B. McElroy (Belfast), Rudd and Jorgenson

(University of Nebraska), and the author. Reference to this work

should be made to the appropriate publication.

The author takes this opportunity of thanking the Bureau of

Standards and the University of Colorado for their hospitality

during this summer.

* Permanent address: Royal Holloway College,

University of London

III
Contents

Page

Lecture I. The Impact Parameter Formulation of the Theory

of Atom-Atom Collisions 1

Lecture II. Applications of the First Born Approximation. . . 12

Lecture III. Slow Charge Transfer 22

Lecture IV. Charge Transfer in Other Cases 34

Lecture V . (i) Further Remarks on Charge Transfer

(ii) Curve - Crossing 44

Lecture VI. Ionization of Atomic Systems by Fast Protons. . . 56

IV
LECTURE I

The Impact Parameter Formulation Of The Theory

Of Atom-Atom Collisions

We will describe in this lecture the treatment of atom-atom

(including ion-atom collisions) developed by Bates and his coworkers^ * paying particular attention to the sequence of Born

approximations and to the so called "distortion" approximation, but

will leave to a later lecture any specific treatment of rearrangement

collisions

The theory to be described makes two basic assumptions.

(i) The relative motion of the colliding systems is

sufficiently fast that the usual Born-Oppenheimer

separation of electronic and nuclear motion is valid,

and further, that the nuclear orbits are rectilinear,

the velocity of relative motion v being constant

throughout the collision.

(ii) The velocity of relative motion |v I is sufficiently

low that nuclear and relativistic effects may be ignored.

Assumption (i) (a), the Born-Oppenheimer separation, is valid at all

energies of interest if both the systems are in S states, but may

lead to serious errors at thermal velocities in other cases.

Assumption (ib), is equivalent to supposing that the de Broglie

wave length \ of the incident particle, in the system in which the

target is at rest, is small compared with atomic dimensions.

T^t is y, -J^T « q-0 (1*1)

wf|V

1
2

For incident protons this is satisfied if E£<10ev, and for

heavier atoms at correspondingly higher energies. However, even

for atoms as heavy as Mercury, \ is ~ 0.1 a^ at 300° K, so one

should not suppose that cross sections calculated by the theory to

be described are necessarily adequate at thermal energies, or that

estimates of rate coefficients so obtained are applicable to upper

atmospheric conditions

Assumption (ii) is valid, insofar as relativistic effects

are concerned, for heavy particle impact, up to several tens of

Mev, provided inner shell electrons are ignored; but modification

of the cross sections by the nuclear forces becomes important

when \ ~ 10 cm. (If much of the contribution comes from small

impact parameters), which is 10 Mev for H

1" impact, but correspondingly lower for heavier atoms. For example, \ (A+

) is 10~l2

Cm at

100 kev. We can suppose that the limits of applicability of the

theory are approximately

10 ev < Ej < 10 Mev

for H* impact.

We consider an unperturbed system, Hamiltonian Hq , in an

eigenstate $ s supposed known, with energy e s Hofs = *s$s (12)

and suppose that the effect of the collision is to introduce a

perturbation V, which is an implicit function of the time. Then

the state of the system at time t may be described by a suitable
solution of

where

(h-'&.)£=° (1.3)

H = H„+V d.4)

It is customary to expand Yn

in terms of the S (all S)

5

*~s u=)4>s

^

(1.5)

where aRS (t) are the occupation numbers of state s at time t, the

system being originally in state n, and the collision taking place

at impact parameter p. In particular one supposes the target to

be .at rest at the origin of co-ordinates in a frame xyz, and that

the projectile moves with velocity v parallel to and at a distance

p from the z axis, the trace of its path in the xy plane at P

being such that OP makes an angle $ with the X axis. (Fig. I).

Then

X- fc<*>§ j Y- ^sv^% j 2* vt (1 6)

where Z (p) = o.

FIGURE I.

(See Separate Sheet)
4

The probability that such a passage causes a transition from

state n to state m is clearly | arm

(-+co)| 2 and integrating over all

impact parameters (supposing V independent of $) the cross section

is 00

VtVCA O

(1.7)

Our aim, therefore, is to obtain an expression for | a^ (<»)| 2

} and

to evaluate it, for the particular transition of interest.

Let us substitute (1.5) in (1.3), making use of (1.2) to

obtain , . [a*. Ct) 4>

s

«• "

S = L f«V* (-t) c^>s

e

S

s

s and multiplying on the left by m#e ^• emt

, integrate over all the ^m

electronic co-ordinates. We obtain

(

where

^s^vws

^

a u^lrt (18)

s V^s ^ i4>^VCt> s

Arc

This is a set of coupled integro-dif ferential equn's, and does not

admit of a general sol n

.

Let us suppose

(i) V„| -- o ; k

j-

- )

i

" ' iT "

" }

and impose the boundary conditions (1.10)

;
5

In virtue of (i), (1.8) reduces to

JV it) -_ a. ttf V

«1YW ***». V^.\AuA£ t

V»V) (1.11)

If m = n, we may readily integrate this and obtain

Z

* *)ft} ^- I - i ( V <*-*' (1.12) -ifv.j**'

- c» where we have changed the variable for convenience. If now Vnn is

small for all Z, or if v is very large, we may taice ann ^ 1 (all z),

and integrate (1.11) for n ^ m to obtain

' -co

Thus, to this approximation, the cross section is

o V

and is referred to as the 1st Born approximation since the assumptions made are mathematically equivalent to the usual wave formulation of the first Born approximation, when the nucleii are assumed

to move as classical particles. A rigorous proof of the equivalence

is provided by Frame (')*

This approximation is justified only if the sum of the

transition probabilities remains small throughout the encounter

\A 2- |^p U)|* « | (1.15)
6

If this is not the case, improvements can be made in two directions

We can take specific account of those anp judged to be most

important, to a high order of approximation ("Distortion approximation"), or partial account of all of them (at least in principle)

by a "2nd Born Approximation". In the latter, we start from (1.8)

and use (1.12) and (1.13) on the R.H.S. This yields

^s £/vrp ^*.s*'/v- ,

'SH.

v

g*^

*[l ~i? v^'u'] y

0© (1.16)

which may be written as

i*r &**» r. (V, or- VA/ \ £

***

(i.i7)

where ^

As v-^oo, the exponentials can be replaced by unity and

which is of the second order in the interactions.

(1.18)

If we integrate (1.17), we have

±. r,

-oo
giving for the cross section

«L. - a] I fcL^.).*-** |-

r -ra.»)

where the subscript 4 indicates that all terms of the 4th order in

the interactions should be dropped, since further terms of this

order arise from products of 3rd order and 1st order terms and these

/•g\

have not been included. in principle (1.20) takes proper

account, to the third order in the interactions, of all the matrix

elements, but in practice Wmn is approximated by a sum of what are

considered the two or three most important matrix elements, including say Vmm and Vmn and terms differing from these only in magnetic

quantum number

The distortion approximation, on the other hand, begins by

noting that of all the terms in (1.8), the only one whose magnitude

is not affected by oscillatory factors is that involving V^^ the

so-called "secular" term. The distortion approximation takes this

term almost completely into account.

We first of all remove the secular terms by writing

t

C^~- «^**T {-$**>**] (1 . 2l)

so that (1.8) is replaced by

b
with

*~s= ( e~+N/—)-(Es* vss)

(1.23)

so that the resultant formulae are analagous to those of the 1st

Born approximation, except that the unperturbed eigenenergies c

are replaced by the 1st order perturbed eigenenergies

\~- es* VSS (1.24)

Higher approximations may be obtained by retaining more than two

terms in the initial expansions.

A very simple alternative derivation of (1.22) has been given

by Bates (9), We start from (1.8) and the 1st Born solution for

&n

. We have

and suppose a

g = (s ^ n) to obtain

"32: -o- which integrates to give

O

Now to obtain a^ ( z ) we retain only am

and an , so that (1.81 )

becomes

'" *•«. ^ 1 fa^V^ +• <*„ CB V^ ? (1 , 27) 34 irl I
so using (1.26),

- I ft VU V -v MAMA.

T)i

4 -,

J (1.28)

which may readily be integrated to yield

t+«o.-if>uM-i^*-i/^r* (i.29) \4A

IT

where £

rw>^ 3

' (i.3o)

o

Thus the cross section can be written

o with Ymn given by (1.23)

(1.31)
10

REFERENCES

1. Bates, D. R. , Quantum Theory (Ed. Bates), Ch. 8 [N. Y.

,

Academic Press] (1961).

2. Bates, D. R., "Atomic and Molecular Processes " (Ed. Bates),

Ch. 14, [N. Y. Academic Press], (1962).

3. Bates, D. R. and McCarrol, R., Adv. Phys . 11_, 1. (1962).

4. Wu, T. Y. and Bhatia, A., J. Chem. Phys. , 24, 48. (1956).

5. Dalgarno, A., and McCarrol, R., Proc . Roy. Sco. , A, 237 , 383

(1956).

6. Dalgarno, A. and McCarrol, R., Proc. Roy. Soc

A, 239 , 413

(1957).

7. Frame, W. A., P. Comb. Phil Soc. -27, 511., (1931).

8. Moisiewitsch, B. L., Atomic and Molecular Processes (loc. cit.),

Ch. 9, (1962).

9. Bates, D. R., Proc. Phys. Soc. 73, 227. (1959).
11

FIG. 1

Coordinates used in the impact parameter formulation.
12

LECTURE II

Applications Of The First Born Approximation

§ (1). The 1S-2S Excitation of H by H

1" : Impact Parameter Form

90

t***>cr. _

b

The cross section is given by

(2.1)

o

where we have seen (1.13) that

+00

-00

Following Bates we have that the interaction potential is

Ut -c JL — -I—

ft \t-ffl (2.3)

where R is the internuclear separation, and £ the electron's

position vector. Taking

*«

fir

J T« 4iS <2>4)

we have

Using this in (2.2) and noting that it is an even function of Z, we

have(l)

' o
13

which is the sum of two standard Founer cosine transforms, and is

equal to

*,t* L- \ts/ J

< 2 - 7 >

where Kg (x) is the 2nd order moditied Bessel function of the

first kind, and the cross section may be readily evaluated

numerically. The results are shown in Fig. (2.1). It may readily

be seen that when v is large cos \^/ ~ 1, so that the cross section

is proportional to v~

2

. Further, when v is small the rapid

oscillations of the cosine factor make the cross section go rapidly

to zero with v. If we take V

igJ3S to have range a, then |alSj2s | 2

"is a maximum when

*

=f S»s,zs °* (2.8)

X

which is the usual adiabatic criteria.

Similar calculations have been carried out for transitions to

the 2p and 2p±' states, and for these transitions when the

incident particle is a hydrogen atom in the ground state, which

remains unexcited^ 2 ^ The cross sections obtained are identical

to these obtained earlier^3 ) by the usual wave formulation. The

collected results have been presented by Bates^'* However, in

the impact parameter formulation, the values of the transition

probabilities allow one to make some estimate of .their sum £' |a_ I 2

at impact parameter p, and yield some information on the range at
14

validity of the Born approximation as a function of impact

parameter. Since most of the contribution comes from small impact

parameter. Bates (4) has evaluated

P' - U*. SV^I1, (2.9)

and considers the criterion P'«l satisfied if P

1 < 0.15, but

grossly violated if P* > 0.5. For [H*,H (IS)] collisions he finds

the Born approximation satisfactory (provided distortion is also

unimportant) for E

± > 200 kev, and completely unreliable for E± < 50

kev.

(2 ). The Wave Formulation

Following Mott and Masseyt-') we have

®-^-^.jH^«^ 3*U* J' ' (2.10)

where

K c U^ tew (2.11)

and we may take

U . S O-AHA \ | 1- SlWh -V • • ' ' \ (2.11)

ir v MMMl

3|l*u- J and K as infinite. The interaction potential and the wave

max r

functions are identical to those given in §1. Substituting, and

making use of Bethe's integral

) *1 - cU - 4^ e, (2 . 12)

| y - ft \ K

2,
15

one obtains

GL^vIa \ (Tl^l1 K

5^

C««o

4

) (2-13)

where 4 ia»

(2.14)

and Xn» Xm are tne initial and final atomic orbitals. For the Is -»

2s transition this becomes,

*** g* J —"' ( 2 « 15 )

s 1i Ca- r>r * SuU(.iCr)c(rr (2.15)

iK J, which is readily evaluated by a suitable sequence of differentiations of

to yield

t = a i£_ (2.17)

and the cross section may then be evaluated from (2.13) by

numerical integration. The method may readily be generalized to

collisions of a bare ion of charge Z, incident on an ion isoelectronic with H of charge Za

.
16

The resultant expression reduces' -^' at high energies to

* ia

*

f (2.18)

£(3)« If the incident projectile is not a bare ion but another

atomic system, a similar analysis shows that 1^ is replaced by a

product of two factors, one being a matrix element for transitions

of the target and one of the incident atom, these factors being

identical to those in the single excitation case.

For

HOs*f v-nis) -> h(»£) * H(ye.j

) (2#19)

we have ^>o

SO J (2.20)

k

At sufficiently high energies K^^ -* o, and sum rules may be

applied to the matrix elements, to deduce the total cross section

for all inelastic collisions' -^' (E^ in kev.),

($h*i4V»**S) ~^ (*%*) (2.2D £ •v where the major contribution comes from double ionization

processes. Some representative results are shown in Fig. (2.2),

and a table of the transitions for which results are available

follows:
17

REFERENCES

1. Bates, D. R., Quantum Theory (loc. cit.), (1961).

2. Bates, D. R., Proc . Roy. Soc. A, 425, 299, (1958).

3. Bates, D. R. and Griffing, G. W., (1953), Proc. Phys

Soc . A 66, 961, (1954), Proc. Phys. Soc. A 67, 663.

4. Bates, D. R., Atomic and Molecular Processes (loc. cit.),

(1962).

5. Mott, N. F. and Massey, H. S. W., Theory of Atomic Collisions , 2nd Ed. (O.TT.P.), (1948).
18

FIG. 2.1. (From Bates 1961)

BortN t 5*StO«T«OM A-PPfcOIMMftf ICrUi To CAorSS S€C,Tt©K& R?*5 H(1s^* H* H(2S) + H"

BfKCS, *S<»1.

t

-2«oo

>**&

JBoAN A«>RoX

-I- \'(xt> 1-xS v^° Vl<f fc'oo 2-z$" *So

fc t.oa t impact fer»ieee»Y >ke>0
19

FIG. 2.2

Fi«ST 6O0.M APPftoMMPmo* ToTKfe CGjOS^ SfcCTLOM, !.- ... - — iM^ftOT fcKi£€,C?Y GofiufcS FOR.^"

M ps)-t Hlis)-» H (>S) * W (."* )

unegfe ->\4 «fePigfcS6NTs 2s t 2-pj 3sj 3>|> ) 3^ or C

VS" io i*ff 5-0 S-5 4-0

*. log t6N«2<i»Y ikc- *ro*^k.e*/3
20

TABLE 2.1

A list of processes for which calculations have been carried out

to the 1st Born approximation (not including charge transfer).

I. Proton Excitation

1) H+ + H (Is) -^ H+ + H (2s, 2p, 3s, 3p, 3d, C).

2) H+ + He (Is2

)

-* H

1" + He (ls2p or ls3p' p).

3) H+ + Na (3s) -» H+ + Na (3p).

II. Atom - Atom Excitation

4) H (Is) + H (Is) -» H (Is, 2s, 2p, C) + H (2s, 2p, 3s, 3p, 3d, C)

5) H (Is) + He (Is2)

-* H (E) + He (ls2p' p)

5

1 ) He+ (Is) + He (Is2

)

-» He+ (E) + He (ls2p' p)

6) H (Is) + He (2s3 S) -» H (E) + He (2p or 3p3 P)

Ne (Is) + He (2s3 S) - Ne (E) + He (2p or 3p3 P)

III. Ionization

7) H* (or He++) + He -* H+ (or He++

) + He+ (Is) + e

8) H+ + He (Is2

) - H+ + He+ (2s, 2p, 3p, 3d) + e

9) H+ + Li (Is2 2s) -> H

1" + Li+ (Is2

) + e

10) tf" + Ne - H

4" + Ne+ + e

IV. Double Ionization

11) He+ (Is) + H (Is) - He++ + H (E) + e

12) He (Is2

) + H (Is) -» He (E) + H+ + e
21

TABLE 2.1 (Con't.)

V. Electron Loss

13) H (Is) + H" (Is2

) - H (Is or 2p) + H (Is or 2p) + e

14) He (Is2

) + H" (Is2

) - He (Is2

) + H (Is) + e

REFERENCES

1) Bates and Griffing, Proc. Phys. Soc, A, 66, 961 (1953).

2) Moisiewitisch and Stewart, Proc. Phys. Soc. A, 67 , 1069, (1954),

3) Bell and Skinner, In Press. (1962).

4) Bates and Griffing, Proc. Phys. Soc. A, 6]_, 663. (1954)

5) Moisiewitisch and Stewart, (loc. cit) (1954)

6) Adler and Moisiewitsch (1957) Proc. Phys. Soc. A, 70 , 117.

7) Erskine, Proc. Roy. Soc. A, 224, 362. (1954).

Mapleton, Phys. Rev., 109 , 1166 , (1958).

Grospun and V. d Waale, Nuovo Cimento 19 , 872 (1961).

8) Dalgarno and McDowell, Airglow and Aurorae , p. 341 (1956).

Mapleton, (loc. cit) (1958).

9) McDowell and Peach, Phys. Rev. 121 , 1383.

10) Bates, McDowell and Omholt, J.A.T.P. 10, 51, (1957).

11) Boyd et al. , Proc. Phys. Soc. A 70, 110, (1957).

12) Bates and Williams, Proc. Phys. Soc. A, 70, 306 (1959).

13) McDowell and Peach, Proc. Phys. Soc. 74, 463 (1959).

14) Sida, Proc. Phys. Soc. A 68, 240. (1955).
22

LECTURE III

Slow Charge Transfer

£1. The Adiabatic Approximation

We suppose that below the maximum predicted by the adiabatic

hypothesis that the projectile and the target particle may be considered as forming a quasi-molecule, the velocity of relative motion

being such that the electron, considered as a classical particle,

would complete several orbits while the nuclear separation changed

inappreciably. Then we consider that the quasi-molecule makes

transitions (infrequently; ans «1, all s) under the perturbation

of the relative motion.

Let the total Hamiltonian be

H = ^ + V (r, R) (3#1)

so the system is described by a state function Yn satisfying

(H - i |t) Yn (R> r) = o (3.2)

Now at some fixed R, the quasi-molecule has eigen states y^s (R,r_)

with eigen energies e

s (R) such that

H Xs (*>D = E

s Xs (R>£).

(3 ' 3)

Then expand
23

and substitute (4) in (2). We obtain

(3.5)

S

Multiply on the left by

and integrate over the electron coordinates, assuming that

to obtain

(3.6)

s

J

Now

^^CSjH ) _ ^OLs . b£ (3.8)

so (3.7) may be written

s which is the P.S.S, approximation in its simplest form.
24

[Choosing the (arbitrary)^ ' phase of xs (r, R) such that

* **- xs _, -fix) only

and using the result v r f

)(s *v„ dr = o, we have

<& |?« |s> = o <3 - 10>

Now suppose that

as our boundary condition, and assume that we can put C

s (t) =

C

ng

(-00) on the r.h.s. of (3.9). Then we obtain the uncoupled

(2) equation, v '

which may be integrated to yield

and

Q -- W(Kh. (•*«•) I 2

*<*? (3-14)

o

These equations are appropriate only in the absence of coupling,

and even the coupled equations (3.9) are inappropriate if cond.

(3.6) is not satisfied. In the limit of high velocities we may

obtain the impact parameter version of the first Born from this by
25

using 1st order perturbed eigenfunctions on (3.13) and the

unperturbed eigenenergies.

8 2. Charge Transfer

We now consider the process

A + (B+e)n - (A + e)m

+ B (3.2.1)

in which an electron is transferred from state n of B to state m

of A. We deal first with the symmetric resonance case in which

A = B, m = n. (3.2.2)

If the atom (B+e) is in state S as R -» » the corresponding

molecular eigenfunctions are Xs

+ ,~, even and odd with respect to

interchange of the nuclei. The corresponding L.C.A.O. approximations to the total state function are

1A JT * s J U (3.2.3)

where Yi corresponds to the electron on nucleus B as R -* oo,

Y2 to nucleus A. Adopting the co-ords. shown in Fig. 3.1
26

FIG. 3.1

Co-ordinates For The Exchange Problem
27

and letting Z = vt determine the origin of t, we havew)

We now choose an expansion^' *'

s

where

(3.2.4)

(3.2.5)

(3.2.6)

Then

' **i "Sir. J

**r W &* +4

^

)«**']

(3.2.7)

such that

<K uTUcaa & -> *0 £** cjj£ V"

[To prove this note that

+ >\,± H JC

S

" - cs

~ X-g"

and that

<5

tuk^i
28

We now impose the boundary conditions

C

" ("CO * -L • C ±

i- OC ) r^ O S ^ VA

and make the assumption that transitions are infrequent

«f*^ =?

°

(3 - 2 - 8)

Then from

(H - i^

) 4> - ° (3.2.9)

we have

c f

^

+

1

(

f

)* ** ^" "l f 4"f(x* f X; 4j

4

l

fr (3 - 2 - 10 >

where

T*^V)*tM-a:Kj45 Qt J —

(3.2.11)

Clearly

JlV)*(.xs~)<£r = o , ^s

(3.2.12)

and if we further assume (incorrectly) [equiv. to assuming

C

s+ = o, s#]

J V

<> * — JS '^S (3.2.13)

we obtain

« 4» * « (3.2.14)
29

Now

•V • 1+ °$ + i'jwV- v*^}'"***5 (3 - 2 - 15)

The last term of this equation vanishes by symmetry arguments to

leave the Bates - McCarroll result

J*

X.: }coav^ dues. 2. 16)

Retaining only the symmetric part of T.+ one may then integrate

this to get the transfer probability in the form.

(3.2.17)

where

-too

(3.2.18)

(3.2.19)

and

If v is small, so is (3 - g+

), and the result reduces to the

P.8.S. method. In general, the effect of the ((3~ - p+

) term is

to reduce the cross section below the P.S.S. value. For the

resonant H

1" on H case, the effect is noticeable at 2 kev, and

significant above 25 kev.^ ' Excellent agreement with the

experimental results of Fite et al .'°' is obtained by normalizing
34

Fite's result to the theoretical value at the lowest measured

energy. Above 15 kev Ferguson's calculations give a result decreasing much more rapidly than the experimental values, but inclusion

of coupling to the 2p state ^ ' gives a marked improvement (Fig. 3.2)
31

REFERENCES

1. Bates, D. R., (1962), Atomic and Molecular Processes , loc. cit.

2. Bates, D. R., (1961), Quantum Theory , [Ed. Bates] Ch. 8.

3. Bates, D. R., and McCarroll, R., (1958), Proc . Roy . Soc . A 245,

175, (1962) Adv. Phys . U., 1.

4. Ferguson, A., (1961), Proc. Roy . Soc . A, 246, 540.

5. McCarroll, R., (1961), Proc . Roy . Soc . A, 246, 547.

6. Fite, W. L., et al., (1960) Phys . Rev . 119 , 663.

7. McElroy, M. B., (1962), To be published.
32

CAPTION TO FIGURE 3.2

The solid curve shows Ferguson's calculated values of Q rans

for

H

1" in H, the dashed curve the effect of including coupling to 2p.

(McElroy 1962). The x's are the experimental points renormalized

(o's) to the P.S.S. calculation at 1 kev.
33

FIG. 3.2

1-o 1*5

to& CPaoTO*^ erttdox (nVCO/)
34

LECTURE IV

Charge Transfer In Other Cases

§1. The treatment given in Lecture III can readily be

generalized to take account of non- symmetric charge transfer.

Expanding the total wave function as

*»• jv°<<f-)

; **

f S^< If)

-i'.KJ/**

1 c

(4.1)

A B where , are proportional to eigenfunctions of the quasimolecule AB with the active electron on A or B respectively when

A B

R -* ». Noting that and are not in general orthogonal (in the

absence of exact wave functions) one can derive a set of differential

equations for the occupation coefficients of the form( '

21

<?o J

(4.2)
35

together with similar equations for the b. where

Mv- J%A* ^?*"**

4a

and . . 2 ,

(4.3)

)

«fe>V**'* °

(4.4)

*'

Clearly the coupling terms involving the M. are of considerable

importance. This set of equations has so far proved intractable.

A two state approximation which should be valid at somewhat higher

energies has been developed by several authors,^ »^» J »^-' and will

now be described. We expand (1)

(4.5)

where ^ (r, t) is orthogonal to both the a^ and b^ term and the

, A B

V-, \|f . are proportional to the initial and final unperturbed

atomic eigenfunctions. Now

/ m- ii \vl/ _ O

v. Tib/ i«-

'-fcfc y *^ (4.6)
36

so substituting (4.5) we get a set of coupled equations

^A.£..»»ti -w&£..fc -^A£fli£

with „ A a Iv-i , ^-.fr^Li^LTMi*^)*** \ (4 . 8)

Htf -_ Al.i^B/.vi^/vUv I

A B

and V , V are the initial and final unperturbed potentials.

We have taken the atomic eigenfunctions to be

A B

0j_ and 0* , where

(4.9)

J

Making the usual two state approximation

V « <T (4.10)

and assuming transitions are rare, gives „. ^ ^

J 'l»i+i

V

S

*i«'

U *Jt M' b

i*<.i

l\ f

^*- W~ (4.Ua)

- U 9 .b ^ — uAt»'t
37

Multiplying the lower equation by S..e J*- on the right, and

subtracting

and similarly

If we note that the imaginary parts of

tf; = Hd - i^ lt;& , g- = R^-S^K^

(4 13)

are anti-symmetric in t, we may remove the secular terms by putting

to obtain

*?* *'[ x%y] ^p[-: ^^u ^j

where

(4.14)

(4.15a)

(4.16)
38

and makes allowance for distortion. In the symmetric resonance case

. < 5 > 6^- = o, and the equations uncouple to give

and

However, we must generally make the simplifying assumption

with

These equations have been applied to

by McCarroll and McElroy^ ' and to

(4.17)

where -hi?«

|^l<v)|^^^(f VlOj'

^

j

(4.18)

(4.19)

b.(t) «a.(t) (all t), and we then obtain

3 x (^.(^OP -- |( M /. OLCJ

2 ("0)

K-/ - M,.

^

v (4 - 21)

(4.22)

(4.23)

by McElroy ' and the results are tabulated below. The agreement

with the results of Fite et al.^ 8 ) and the data taken in molecular

hydrogen' ' is shown in Fig. 4.1. It is far from clear whether
39

Bates and Lynn's' °' prediction of a rapid decrease in the cross

section for the accidental resonance case as v -• o is confirmed.

Probably coupling is as least as important an effect as distortion.

Equations (4.15) reduce to those of Bassel and Gerjnoy(-'-l)

if the |

i | 3 term is omitted.

|2. Comparison of the Various Treatments

If in the result (3.38) we ignor the (3± term, then the P.S.S,

approximation yields for the cross section for symmetric resonance

charge transfer

ee <% *. auit 5 *-1 ^)1 )^ CD

O

with

§- %o - ±

v f ^J_ft^d-« (2)

J> ca>

-*>)'*'

Similarly (4.18) yields

?= *,> X l«J- ^J^'cU (3)

If weignorethe e"

lv" term (v - o) and take |s..|s«i ) we have

DO

r a i. f

I Si -*jj) * <U

and

N - ^

^ <MvA |6>-<^lv6 |A>(5)

which differs from (2) only in that the molecular eigenenergies
40

are replaced by their L.C.A.O. approximations. Result * ' in the

limit e~ 1V! -» 1 has been obtained in a simple fashion by Gurnee

and Magee'^' and their method has recently been extended to the

asymmetric case by Rapp and FrancisTM), wno ignor the exponential

in (4.9). They show that in this approximation asymmetric resonance

charge transfer should show the typical

/Q = A log E + B

behavior down to some minimum energy E

Q which depends on the

energy to defect AE if in a complicated fashion. For E < E

Q

, the

usual adiabatic variation of the cross section is obtained.
41

REFERENCES

1. Bates, D. R. and McCarroll, R., (1962), Adv. Phys . 11 , 1.

2. Gurnee and Magee, (1957), J. Chem. Phys . 26, 1237.

3. Mittleman, M,, (1961), Phys . Rev. 122, 499.

4. Rapp and Francis, (1962), J. Chem. Phys . (In Press).

5. McCarroll, R., (1961), Proc . Roy. Soc . A. 246, 547.

6. McCarroll, R. and McElroy, M. B., (1962), Proc. Roy. Soc. A

(In Press)

7. McElroy, M. B., (1962). To be published.

8. Fite, W., et al . , (1960), Phys . Rev. 119 , 663.

9. Allison, S. K. , (1958), Phys. Rev. 110 , 670.

10. Bassel, R. and Gerjnoy, E., (1960), Phys . Rev. 117 , 749.
42

4*a;JP

OfI2o5

tfV\2

a!

55s f00

w o

4-

CO

a*I X

n> \ — o

V) »

Oc oX I! * -4- 3^

-*» xo r : £/ i

or* i

Vf> j -

(0

X 3 7

<- 55

S3

q

to

i/1

Rf

00 > I *

w> I m

Doi

inX

n|OoxX

Ct OX

in - ! X -* ! ^, ^-

rsn

.9 1. I *to

Xo

tol iI

ft?I I X » O . £ X oXo

3*-

X

or. i oX

s>o

9S\ i * ; 41 I £To

"5- *x

4-

ToX i

*9 o OX

J*

X i J> j 4- I oX - j ^

-] iox

^>oO i X

—

x

CO

>0

X - \ + ! - o?

tAj

10ToXO

bo oX

Vn2a o I X i

VPx l tn i o i X

Si

GO

ToXn£DO

<i»

X

Oi

"it I I * i! ^ o

2,' ^

Rii or 

4

J) i o

ti- er !

S2T
43

[^**1°V».
44

LECTURE V

(i) Further Remarks On Charge Transfer

(ii) Curve - Crossing

|1. Elementary Treatments Of Some Validity At High Energies

The early work on charge transfer (H* on H) by Brinkmann

and Kramers^ * J and by Oppenheimer'-^ treated the process in the

first Born approximation and retained only the interaction between

the incident nucleus and the active electron (or the corresponding

past interaction) in the matrix element. That is they took

M = M = K.. (5.1} BK ji KJ ' }

which equivalent to the refined fast approximation (4.15) with

S.. - k.. = o. The cross section, to the first order in (m/M) is

then given by

% K * iiLS3{sfi +t*.***fi (

sc * a,

"^

) 1

(5 ' 2)

where S . 2 is the usual energy parameter and Z

l and Zg are the

nuclear charges. Similar expressions may be derived for capture

(4) into excited states v

. However, when

Qc = & Qls,ni (5.3)

for H* on H is compared with the experimental data'-3 ' taken on

Hg , (per H atom) it is found to overestimate by a factor of five

at energies up to 400 kev.
45

Bates and Dalgarno^>°' and independently Jackson and

Schiff^ ' suggested that

M__ TC = K. . + V (5.4) BDJS ij nn

where Vnn is the nuclear interaction

v**, » - f ^j,** 2

ji* 4> a

,e 4r (5.5)

was a better approximation in practice. However,^(8)' V vanishes

in the limit M-»°=. This approximation is equivalent to (4.15)

ignoring distortion, neglecting |

±

.

| 2 compared with unity and replacing the resulting Matrix element

^G = Hii " *ij

(5.6)

= K

ii + Vnn - < K. . + V > (5.61 ij nn ) ij nn v '

where < > indicates the initial static potential, by (5.4). The

resulting Q for H*" on H, and on He are in excellent agreement

with experiment over a wide energy range' '.

McDowell et al . are presently extending these calculations

to IT on H" and H* on L^+

. In these cases Coulomb forces play

some role and McDowell^ ' has recently shown that the correct

matrix element (to the 1st order) in that case is (5.1) where the

initial unperturbed eigenfunctions are now taken to be products

|U^ > of atomic orbitals and an incoming Coulomb wave. We have
46

incoming Coulomb wave. We have

where JUL > satisfies

[H *v fl l-c,)-<r vHJai>- B i |ui'>

Now the atomic functions used in Lecture (4) \wt > satisfy

(5.7)

(5.8)

(5.9)

so to the 1st order

i > = <fc

A > + (<»•* I v~ ' V> I <*.* > (5.10)

* E i- Ei

and substituting this in (5.7) we may obtain the correction due to

the Coulomb potential as

&K

J

m _ .. _ (5.11)

K ElC ~ *=k

That is Vnn scatters the incident system from state i into state k,

and capture proceeds from there to the final state. The important

virtual transitions are (i) the target is excited with A-L = ± 1,

and (ii) the target is unexcited but the relative momentum vector

is altered. Both these contributions can be evaluated by the

h

Femann technique, and calculations are in progress. The total (^

effect is small, for |m I 2 can be evaluated in closed form, and
47

it may be shown that if' '

£i_ « i (5.12)

then

CR

l M I 2 * [M

BK | 3

. (5.13)

Returning to (5.6) we may attempt to improve it by including

distortion, by noting that the most important effect is the

distortion of the unperturbed initial atomic eigenfunction by the

incoming particle, i.e. by V (£g ) . McDowell^ 8 ' has therefore

computed the cross section, replacing M^nTq by

W*st - <%* I V—V^J> | >0 > (5.14)

where

which is equivalent to replacing <K. . + V > in (5.61 ) by the sum

in (5.15). This modification greatly decreases the cross section

at low energies (<200 kev.). Calculations for He++ on H by this

approximation are now in progress.
48

(ii) Curve Crossing

If the Schrodlinger equation of the quasi-molecule formed

by the colliding systems is non-separable in the electron coordinates,

and the potential energy curves of states (ft and 9, are known, then

it is possible that \ (R) may equal V2 (R) for some R = R say. If

we then proceed to a higher approximation

%X ~~ C

l'

l<i>

> +

^^

(5.2-1)

Then it is found (Moisiewitsch, B. L. 1962) that the new potential

energies Vx

* (R), V2

' (R) are never equal unless Cf^ and

(fe belong

to different symmetry classes.

Landan*- ' and Zener(*-3) suggested that in such cases

transitions between the states corresponding to Ox (R~*°) and

<^2 (R - ») were likely only for R ± R^. if p is the probability of

such a transfer, on any one passage through R , then the transition

probability is

CP-. apo-p) (5 . 22)

and is small, both when P is small and when P is large. Bates' ^'

has recently criticized these assumptions, and their method of

approximating P, as follows :-

Consider the transition

(A + e) n + B - A + (B + e)

m

(5.2.3)

at some impact parameter p, under a perturbation H'
49

Let

be a zero order matrix element, where we take d> and <bm to be

appropriate linear combinations of the (exact) molecular eigenA functions and of the same symmetry class. If V describes the

state in which the electron is on nucleus A as R -» co, in the zero

th order approximation, then $ n

is the wave function describing

{(A + e) n + B} for R > R

Xj but {A +(B + e)m] for R < R and similarlyJ for o . Thus

m

H (R) 4 o (all R). (5.2.5)

nm

Further if H is the total Hamiltonian, (since in general _

I. A_- $; and in our case the sum runs only over j = n, m. ),

H<(V£>^ " H"-~t R )** t H^*.Cft.)<$vu

( 5 # 2.6)

and at R = R^

H

nn( V = WV (5.2.7)

while

AIWOSx) = 2| Hnm (Rx )|. (5.2.8)

We now proceed as in (4.5), taking a two state expansion

[Assumption 0]

fn-V^tn + Wtm < 5 ' 2 ' 9 >

with

t

lt> ^ d (L^P^ ki ~ J

J

C

**•"'*' 1 -^.^l *£' ? (5.2.10)
50

and substitute this into

\H-cd }q>^- o (5-2-11)

A typical term yields

(5.2.12)

We now make

Assumption I ; $ n

is a slowly varying function of Z, d<f> /z_l 4= o.

Thus

|H-il ^H^J^^-Cj IH^^^joU^ u)

Using this, (5.2.11) becomes

= cc-w^+ cc**M%*

e

(5.2.14)

Multiply in turn by Y^*, Yn

* on the ie ft and integrate over the

electron co-ordinates, to obtain
51

(5.2.15)

and make

Assumption II:

ikz

e = 1, (v- o).

Then

(5.2.16)

It is true that near R

x

c!H jJE -o (5.2.17)

and thus in this region (t « 1) a power series expansion gives,

(t = o at R = Rx),

(H

mm - H

nn> * a fc (5.2.18)
52

where a is a constant. We now make

Assumption III ; Equation (5.2.18) is valid for all t.

We can now write (5.2.16) as iC^ 3 <-vc Hvvma e- L }

}

(5.2.19)

and eliminate C to obtain (f = H \\. n ' mn 1 '

C^-*- CoibC^+4 2 C^ - O (5-2.20)

This is a version of Weber's equation, and solving it by standard

methods subject to the boundary conditions

C (-co) = 1 C (—) S 11 m

we obtain

K (^)l

2 - 1 - e

_UJ (5 .2.21)

(5.2.22)

with

yielding

-O)

P = e . (5.2.23)

This is the Landau - Zener result. We now examine its validity.

Taking the simplest problem first we have assumption II, that

e £ 1 as v -* o. This will be sufficiently accurate if
53

sin (kz) < %, or kz < ^ say, so that if I is a typical linear

dimension of either of the atomic orbitals involved, we will have

assumption II valid if the impact energy e- < c where' ^

So TV

e' V (5.2.24)

where M is in a.m.u. and t in atomic units. Clearly this is not a

very severe restriction. Assumptions and I are much more

significant. They imply that both vfm and (ftn describe S states ,

for if say <pn

± s not, then o(J)n/Qzr

contains appreciable components

of states differing from <|> n

i n azimuthal quantum number alone.

For example, Bates (1962) points out that if (pm

is a p state an

s state transitions of the sequence S -» Pq^P+i ~*P -^S will considerably decrease the Landau - Zener P factor. Even if the states are

both s type, at sufficiently high velocity the assumption

3$ /di ~ ° must be false. The most serious failure, however, _' (14) was pointed out by Bates v ' and this is the falsity of assumption

III. He points out that transitions are in fact likely throughout

the region in which the exponential factors in (5.2.16) are

slowly varying. If the width of this zone is AZ, then we have,

by the method of stationary phase,

where we suppose Hnm

(Z) (Zx

-% AZ £Z£ZX+%AZ) is constant.

(5.2.25)
54

*n+i Making assumption III, 4 *

/-^ M

&v A2. -^

(5.2.26)

That is, AZ -* °° as v -» °° and this is clearly unacceptable; so III

cannot be valid. Even when v is small AZ is so large that it is

not in general sufficient to treat H^ as constant. When P is a

maximum (oi = 0.69), Bates shows that

AZ= 0.5 |Hnm (Rx

)| Zx

2

with H in ev., which may be large. The correct procedure in the

simple case of s - s transitions, is to solve (5.2.16) as it stands

to obtain (Cv « C ) m n'

P=a t K-«*?HfCh~" h-~J ***

} **\

*"

oo ' (5.2.27)

which is very different from (5.2.23). This has not as yet been

applied to any specific case.

To sum up, one concludes that the Landau - Zener formulation

may be satisfactory for S-S transitions when v is well below the

velocity U at which the maximum cross section is expected.
55

REFERENCES

1. Brinkmann and Kramers (1930), Proc . Acad. Sci. Amsterdam 33 , 973.

3. Oppenheimer, J. R. , (1928), Phys. Rev. 31, 349.

4. Bates and Dalgarno, (1952), Proc. Phys. Soc. A 65_, 919.

5. Barnett and Reynolds, (1958), Phys . Rev . 109 , 355.

6. Bates and Dalgarno, (1953), Proc . Phys . Soc . A 66, 972.

7. Jackson and Schiff, (1953), Phys . Rev. 89, 359.

8. McDowell, (1961), Proc . Roy . Soc . A, Vol. 264, 277.

9. Bassel and Gerjnoy, (1960), Phys . Rev . 117 , 749.

11. McDowell, (1962), To be published.

12. Landau, (1932), Phys . Z. Sowjetunion 2, 46.

13. Zener, C, (1932), Proc. Roy. Soc . A 137, 696.

14. Bates, D. R. , (I960), Proc. Roy . Soc. A 257, 22.
56

LECTURE VI

Ionization Of Atomic Systems By Fast Protons

This final lecture will describe what little work has been

done up to the present on the ionization of atomic systems by other

atomic systems, which are usually supposed bare ions.

Following (2.13) we have for the transition

X

+ +Y-X+ +Y+ + (6.1)

the electron being ejected with momentum K 3

($ (v^->c) i M 3RV1 cLk

K M4(.VV and

'V?

(6.2)

(6.3)

where N (K.fc) is the relevant matrix element, nm v inr/

Apart from some external numerical factor depending on

the atomic states involved, this will in general be of the form,

NM

^ = j^)? i:*-

T

> <k irp ^

(6 4)

where we are now working to Born's approximation.

Now Y (r, ) is of the form,

f —J

where P permutes all the r t (i + j) with r j

, and the j th

electron is ejected, and U is the wave function of the residual

ion. Here we have normalized the ejected electron wave function
57

to (r, ) such that

so that F (£j ) has asymptotic form

P (.^j) ~* fc (6.7)

where a is a phase factor.

Integrating over R and all the electrons except the jth,

the integral (6.4) turns out to be proportional to

i*--Ci

tTC (6.8)

Two alternative procedures are possible. If we are willing to

suppose that the ejected electron moves in a Coulomb field, of

some effective charge Z, we can write

and perform the integration by standard methods, provided

$ (nt,r_j ) has a reasonably simple analytic form. For hydrogen

(z = 1) this is of course exact, and the calculations for both

if and H

1" impact have been performed by Bates and Griffing*- » '

and are shown in Fig. (6.1).

The alternative procedure is to expand $ (x, r\j ) in

partial waves,

.£

la* $(*,*; ) * f(*U^ e fi^C-O P^e)

(6.10)
58

where R*»p (r) is the solution of the Schrodinger equation for an

electron in the field of Y* (n' '£''), having an electron of momentum it in the V wave of the continuum. The cross section then

becomes

(5. . ^±£ SCa^OQwCSt.)

with

s ,^) s

J la (Kii K olK

6^,00 = [p(^-r)P(^U',r)^lK-r)^

(6.12)

(6.13)

(6.14)

Here P(n£,r) = r R(n£,r), and j^ i s the usual £th order spherical

Bessel function.

§2. We now discuss the application of these procedures to

atoms other than hydrogen.

(i) Helium . This has been investigated by Erskine^ ', Dalgarno

and McDowell^4

), Mapleton^ 5 \ and Grosjean and V. d Waale^. All

the above authors represent the ground Is state by a simple

one parameter variational wave function

(j> (Hs,-^> - ^»__ e ;> £,- 1^5 (6.15)

ilT
59

Mapleton takes

^eV) = *(*>* P[*,,,jt1*" ilS " X

]

With this choice one must take Zg = Z

1 to ensure orthogonality,

otherwise one obtains a cross section tending to a finite limit as

Ej -» co. (Fig. 6.1). However, asymptotically one has a Coulomb

field with Zg = 1.0. A better choice therefore is to expand (6.16)

in partial waves and take

Zg = Z. I = o ^ - 1 a to) (6.17)

His results for this case are in excellent agreement with

( 7 8 ^ experiment^ ' ' (Fig. 6.2) and with the calculations of Erskine,

who ignored the -t = o contribution and took P (h, £ = 1, r) to be

the solution of the Hartree equation

d»P(*t,r ) . fax-, i r|+ tlvir) e

+*]_ «(£!> 7 p^-r) = o

d,-fi I T L v* J (618)

as did Grosjean and V. d Waale. Dalgarno and McDowell solved

(6.18) with a numerical potential term, and used this to obtain

Zg (x) for the p - wave, evaluating the other contributions with

the same Zg (k)-

(13) (ii) Lithium . McDowell and Peach v

' pointed out that in this

case their is no choice of Zg which will ensure orthogonality,

and that the cross section with Coulomb waves diverges for all Zg .

However, the dominant p - wave contribution is finite and was
60

computed for Z2 = 7^ and Zg = 1, (Fig. 6.3), and is clearly very

sensitive to the choice of Z2. They show that the corresponding

photoionization cross section varies as Z2

7C)(p{~^^2? Even for

Zg = 1, Q , is sensitive to the choice of bound state wave

function. (Table 6.1)

TABLE 6.1

Photoionization cross sections of Li (10~18 C*vt) with Coulomb

ien

(10)

(9) function Z=l and (a) 3 - parameter bound function (Holoien 1958)

(b) 9 - parameter bound function (Roothan et al . , 1960)

b c

0.513 2.50

0.547

0.634 2.40

0.726

0.779 2.20

0.772

"c" indicates the experimental results of Tunstead (1953).

This implies that the agreement between Maple ton 1 s calculations

and the experimental results may well be fortuitous.

McDowell and Peach are now recomputing Q.; on for H*" on Li

using Hartree - Fock wave functions for the ejected electron

and a 3-parameter Holoien function for the bound state.

H

2 (Rydbergs) a

0.769

0.01 0.819

0.02 0.869

0.03 0.915

0.04 0.956

0.05 0.982
61

(iii) Other Calculations .

Bates and Griffing^ 1 * 2 ) , McDowell and Peach^ 12 ' l3 \ and

Bates, McDowell and Omholt^ ' have calculates the energy distribution of the ejected electrons at energy E t

, from H, H~ , Li and Ne

respectively for either H* or H° impact. They find it depends

primarily on the initial angular momentum t, as would be expected

from consideration of the photoionization cross sections. Rudd

has extended this work to He using Mapleton's calculations, and

finds reasonable agreement with experiment.^ ' (Fig- 6.4).

(iv) Classical Calculations .

Rudd (loc. cit.) and Alsmiller' 5 ' have applied Grsinskis 1

modification of Thompson's method to calculate Q. for H

1

" v x on He ion

and Hg respectively. (Fig. 6.2 and 6.4). They obtain surprisingly

good agreement with the observations, considering the nature of the

approximation, [in this method one assumes that if the ion passes

at impact parameter p it will transfer energy AE (p,v) to an

electron of velocity v, and computes the cross section as

<S -.-i*(J_ S(v-v-)J j> olfc dUr (6.19)

where v is the expectation value of the velocity and p (v) is the

greatest impact parameter for which AE (p,v) exceeds the

ionization potential.]
62

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66

REFERENCES

1. Bates and Griffing, (1953), Proc. Phys. Soc. A, 66 , 961.

2. Bates and Griffing, (1954), Proc. Phys. Soc. A, 68, 90.

3. Erskine, G., (1954), Proc. Roy. Soc. A, 224, 362.

4. Dalgarno and McDowell, (1956), Airglow and Auroae . (loc. cit.)

5. Mapleton, (1958), Phys. Rev. 109 , 1166.

6. Grosjean and V. d. Waale, (1961), Nuovo. Ginento . 19 , 232.

7. Hooper, et al., (1961), Phys. Rev. 121 , 1123.

8. Rudd and Jorgenson, (1962), To be published.

9. Holoien, (1958), Proc. Phys. Soc. 72, 461.

10. Roothan, et al., (1960), Rev. Mod. Phys . 33, 221.

11. Tunstead, (1953), Proc . Phys. Soc. A, 66, 304.

12. McDowell and Peach, (1959), Proc. Phys. Soc. 74, 463.

13. McDowell and Peach, (1961), Phys. Rev . 121 , 1383.

14. Bates, et* al., (1957), J.A.T.P . _10, 51.

15. Alsmiller, (1961), Priv. Comm.

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