ISIBSIR 78-1523

Numerical Solution of the

Electron Slowing-Down,

Equation for Molecular

Hydrogen

L. V. Spencer

Center for Radiation Research

National Measurement Laboratory

National Bureau of Standards

Washington, D.C. 20234

R. Pal

Howard University

Washington, D.C. 20059

December 1978

I.S. DEPARTMENT OF COMMERCE

100

U56

78-1523

lATIONAL BUREAU OF STANDARDS

MatJsnat itsrsaa oJ StanderiiS?. MAY 1 4 iS7’i

/

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'j n

NBSIR 78-1523 ' IV.

NUMERICAL SOLUTION OF THE

ELECTRON SLOWING-DOWN

EQUATION FOR MOLECULAR

HYDROGEN

L V. Spencer

Center for Radiation Research

National Measurement Laboratory

National Bureau of Standards

Washington, D.C. 20234

and

R. Pal

Howard University

Washington, D.C. 20059

December 1978

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Seciretary

Dr. Sidney Harman, Under Secretary

Jordan J. Baruch. Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

NUMERICAL SOLUTION OF THE ELECTRON SLOWING-DOWN EQUATION FOR MOLECULAR HYDROGEN

L. V. Spencer

National Bureau of Standards

Washington, D.C. 20234

and

R. Pal

Howard University

Washington, D. C. 20059

ABSTRACT

Procedures are outlined for solution of the slowing-down equation for

fast electrons- by direct numerical integration. Spectra for H^ are computed

by use of Gerhart's cross section set, suitably modified, and with inclusion

of vibrational excitations. The flexibility and stability of the approach is

demonstrated by solutions for source energies ranging from 1 MeV down to 24 eV

with the spectrum extended below 2 eV. Energy and number conservation

are checked in detail, and the ionization yield is computed as a function of

energy. Spectrum and ionization yield so obtained are compared with previous

calculations; general agreement is found with some significant differences.

Agreement with Combecher's measured ionization yields at low energies is

rather satisfactory overall, but with significant deviations of trend. Some

exploration of secondary and tertiary electron contributions is included.

Key words Electron slowing-down; energy degradation spectrum; inelastic

cross section set; ionization yield; molecular hydrogen; W-value.

NUMERICAL SOLUTION OF THE ELECTRON SLOWING-DOWN EQUATION FOR MOLECULAR HYDROGEN*

L. V. Spencer and R. Pal

I . Introduction

Electron energy degradation or "si owl ng-down" is due to inelastic interactions with the surrounding medium. The process generates a spectrum y

which expresses the relative likelihood that a fast electron is detected in

different energy intervals dT. More specifically, ydT can be considered as

the expected value of the pathlength traveled by a source electron, together

with that by secondary electrons which it generates during the slowing down

process, while the one or any of the others have kinetic energy within dT.

The importance of y derives largely from the fact that one need only

multiply y_ by the (suitably normalized) cross section for some process, and

integrate over all electron kinetic energies, to obtain the yield for that

process, whether excitation or ionization. But given a correct spectrum y,

the resulting yields are no more accurate than the cross sections which are

used to evaluate such integrals. Errors in evaluation of the slowing-down can

accentuate and greatly complicate the errors and uncertainty of any yields so

calculated.

There are two difficulties associated with determining the energy degradation spectrum. First, the inelastic scattering cross sections are complex

and hard to obtain. Secondly, because of the complexity of the cross sections

the numerical solution of the transport equation is not a trivial task. As a

result there have been nemerous investigations with various types of approximations which will be briefly reviewed below. But since the basic integral

equation of which y is the solution can in principle be solved by standard

numerical techniques, we feel that it is appropriate to make an

•k Supported in part by the Office of Naval Research.
attempt to do this. This paper describes a successful calculation of y for

molecular hydrogen by application of well-known numerical integration techniques

to the standard form of the slowing-down equation. Because the cross sections

utilized are current the resulting data is of general interest.

Before proceeding to describe this effort, we identify some of the other

methods which have been applied in the following section, two of the most

recent (for H

2

) being due to Douthat [1] and to Garvey, Porter, and Green [2].

Our comparisons with results from these and other calculations are of interest

perhaps more because different cross section sets have been applied than

because of contrasting computational methods. We use the cross section

compilation of D. E. Gerhart [3]; but we must modify Gerhart's forms to render

them more readily applicable to our procedures, as well as to make extensions

to relativistic energies. In addition, we take account of vibrational modes

in order to extend our results well below the threshold for electronic

excitations, i.e. down to about 1.5 eV.

We believe that the calculations described here are significantly more

accurate than previous evaluations of y for molecular hydrogen. Comparison

with recent measurements of the ionization yield as a function of electron

source energy can be construed as pointing out the importance of more accurate

ionization and excitation cross sections in the range from 15 eV to perhaps

30 eV for future improvement of both slowing-down spectrum and yield results.

New aspects explored here include careful and complete checks of energy

conservation and conservation of the number of primary particles. In addition,

the spectral structure near the source energy is resolved in its most

important features; and there is some discussion of general character! sties of

spectral structure, particularly as it results from those generations of

secondary electrons which are its most important components.

2
1 1 . Comments on Computational Methods

If electrons are generated with initial kinetic energy T^, the slowingdown equation of which y is the solution is

K(T) y(T,Tj = ^ dr k(T',T) y(TM^) + 6(T-TJ , (1)

where K(T) is the total inelastic interaction probability per unit pathlength

at kinetic energy T; and k(T',T)dT is the differential probability, per

unit path length, for an electron of energy T' to undergo an interaction

resulting in an electron having kinetic energy in the interval dT containing

T. This electron can be a new secondary, or the result of an energy loss

(T'-T). The Dirac delta function in eq (1) describes a monoenergetic source

of unit strength with kinetic energy T^; and the spectral function y(T,T^)

has dimensions path length per unit energy.

Direct numerical integration of eq (1) has not been thought very feasible

because of the general structure of k(T',T) for T' near T. A large and

complex concentration of interaction probabilities occupies a narrow region

11 eV < T'-T < 18 eV, as seen in Fig. 1. Further, for whatever value of i

may be selected, energies within a few tens of eV of T^ can only be reached

if no more than one, two, or at most a few interactions have occurred. This

means that the structure for T' near T in the cross sections also appears

for T near T^ in y(T,T^). Thus the integrand of eq (1) has complicated

features for T' near T, and for T' near T^; and these two regions of

structure both make significant contributions to the integral. Direct

numerical integration must therefore deal with the necessity to use integration

steps perhaps less than 1 eV wide to resolve structure, while at the same

time covering an energy range below T^ that can be of the order of 10^ eV

or greater.

3
Most calculations with above a few hundred eV have attempted to bypass the complexities of y(T 5 T^) for T near T^. Incidentally, this structure was first described by H. W. Lewis, and therefore has been called

the "Lewis effect" [4]. The various computational methods fall into distinct

classes as follows;

(1) Numerical evaluation of an integral form of eg (1 ). If one integrates all terms of eq (1) over the variable T, from a value (say) , to the source energy ‘’'q, one obtains an equation of balance for the number

0

-^ electrons with kinetic energy above T. ^ A similar integral over al'' terms., in which an additional weighting factor T is used in the resulting

integrands, gives a corresponding equation of balance or "conservation" for energy flowing into, and out of, the reservoir of electrons with kinetic energies between T and These conservation equations exhibit the significant advantage of having kernel functions with less irregular and singular trends than eq (1). Further, conservation of number or energy is usually more important than a detailed description of cross section structure,

when one attempts to solve for y at most spectral energies. A number of authors have successfully used the "number balance" equation (eq (7)) to obtain solutions for y [1,5-10].

(2) A more elementary and older method, which is sometimes used to obtain approximate solutions to the slowing down equation employs the con -

t j nuous slowing down approximation (CSDA) [11 -1 5]. While the CSDA is more

suited to the slowing down of heavy charged particles, it also describes in a rough way the consequence of the frequent, small-loss interactions of electrons

with the medium through which the electrons move. One can readily derive this

’See eqs (7) and (8).

4
approximation from eq (1). Any electron, once generated, contributes a term

1/S(T) to the slowing-down spectrum below the energy at which generated, where S(T) is the stopping power; and the number of secondary electrons generated can be calculated from a much simplified integral equation. Calcu- lations of this type give a good general description of y(T,T^) which can be sufficiently accurate for some purposes. They are easily developed and inexpensive, but they are not highly accurate.

(3) Monte Carlo methods can be used to evaluate the slowing-down spectrum

as an average over a few, or perhaps a few hundred or thousand case histories

[16-19]. All that is required for good statistics is many collisions in each energy interval specified. Since each source electron not only remains a part of the calculation to very low energies, but also gives rise to many sec- ondaries, the statistics actually improve as the calculation progresses to low energies. Energy conservation can be readily imposed by insuring that all energy resulting from each interaction is properly accounted for. The Lewis effect region can be described if and as desired. Hence results obtained by use of this method can be impressive. Perhaps its main drawback is the time required to calculate each case history, which makes computations of many case histories with high source energies comparatively expensive.

(4) Use of a discrete analog , in which electrons progress from higher to lower energy "bins" according to computed transition probabilities between

pairs of bins. Mostly, such calculations have been used with rather low source energies (e.g., < 1 keV) [20-24]. A recent publication gives results

of systematic calculations of this type to much higher energies [2]. The authors of this last paper chose an interval near the source energy suffi- ciently wide that the Lewis effect was wiped out; but they indicate that the method can be extended to give a more detailed description if desired.

5
(5) Use of methods which solve an equation adjoint to eq (1) [3,24-28].

Such equations have a form similar to eq (1); but the solution progresses

upward in energy rather than downward. Difficulties with the integration

process are not removed, because the complexities of the function k(T',T)

remain. These calculations usually evaluate yields directly, rather than

y(T,T^). Most such calculations have been performed at low energies (up to

1 keV); and the development of a wide variety of methods to solve these

equations is still in process.

(6) The application of numerical integration to eq (1), which is the

subject of this paper, and which has greatest resemblance to method (4), that

is, use of a discrete analog. Here one attempts to apply standard techniques

of numerical integration to the evaluation of the scattering integral, for T

values which progress stepwise to lower energies, some use of this method at

low energies is reported by Douthat [1].

We know of two attempts to make comparisons of the different methods when

the same, or closely similar, cross sections are being used [23,24]. The more

recent study features W, the reciprocal of the ionization yield, as an

integral comparison parameter [24]. It gives little information on relative

accuracies of different types of computations of y(T). The earlier study

[23] compared CSDA results for He and N

2 with results obtained by the

discrete analog method. Only electron sources of 500 eV and below were used.

Differences are significant; but qualitative and rough quantitative agreement

are found. One reason for attempting to apply direct numerical integration

systematically is the possibility that accuracy for a given cross section set

can thereby approach ~ 1%. This would effectively eliminate computational

methods as a source of significant error in evaluation of si owi ng-de-wn spectra

or yields. Such accuracies probably require a more detailed evaluation of the

Lewis effect region; but this can be desirable on its own merits.

6
A certain amount of structure occurs at low energies in the si owi ng-down

spectrum also, because the thresholds for different types of transition can

give rise to irregularities. We think that this structure can be more easily

evaluated by direct integration of eq (1) than by use, e.g., of integral forms

of the transport equation (method (1) preceding).

Successful methods for direct numerical integration also can in principle

be used to evaluate space and directional aspects of electron transport by

methods not heretofore applied with realistic cross section data to this type

of particle transport.

Because the systematics of slowing-down spectra are important for obtaining

similar spectral information about materials and source energies not otherwise

studied in the same detail, we have organized the calculations to permit

identification of the primary electrons as well as the 1st, 2nd, and higher

generations of secondary electrons. The different generations play rather

different roles; and their spectral components require a somewhat different

description for each generation.

One problem avoided in our calculations for ^2 is the problem of inner

shells. While hydrogen has only one shell, most other materials of interest

have inner shells. In the kernel function k(T',T) different atomic or

molecular shells can contribute terms describing electrons with the same

kinetic energy T; so one must in principle include calculations for all

contributing shells whenever k is required. These calculations must be

performed many times, because k appears as a factor at each step of each

integration. We do not see any fundamental difficulty in this requirement,

although it surely increases the computational time of calculations by any

method.

7
A related problem which likewise appears only in materials with Z > 2

is that of including Auger electron components. These have energies characteristic of transitions between different shells. They must be treated as an

additional source whose inclusion is necessary for conservation of energy.

Again, there appears to be no fundamental difficulty in accounting for this

component; but our current task is simplified by their absence from

spectra.

III. Cross Sections and Stopping Powers

The shape of the differential cross sections has a dominant influence on

integration procedures appropriate for this and other similar problems.

Section IV, which follows, illustrates this in detail. In this section we

give general information about the set of differential cross sections with

which we work, and the total cross section and stopping power data derived

from it. More details about these cross sections are given in Appendix I.

We have adopted a well-known set of cross sections for our calculations,

namely the cross section set developed by Donald E. Gerhart in his PhD

thesis [3]. Gerhart followed Platzman's use of the Weiszacker-Wil 1 iams

approach in which collisions are classified as either hard or soft, but not

both. This involves specifying cut-offs to both soft ("glancing") and hard

("knock-on") collisions.

In the process of adapting these cross sections to our use, we have made

a number of modifications. A brief summary of these changes is as follows:

1) We have extended Gerhart's cross sections into the relativistic range in a

manner which gives stopping powers of the Bethe theory type, with reasonable
values for the "mean excitation potential", here designated (I). 2) In

addition, we have modified the form of Gerhart's cut-off function for small

energy losses due to hard collisions; for our purpose it is more advantageous

to use an abrupt, rather than gradual cut-off of this kind. 3) At low electron

kinetic energies, where the differential spectrum of energy losses is poorly

known, extend the use high energy cross section forms by ad hoc adjustments

of the cut-off for momentum transfers, which is a constant at high energies

in cross section formulae for glancing collisions. This replaces an abrupt

transition at 350 eV to low energy forms which Gerhart used, while preserving

good agreement with total ionization cross sections. 4) We have added vibrational excitation cross sections which were omitted by Gerhart, and which play

a dominant role for kinetic energies below about 12 eV. 5) Finally, we

symmetrized the glancing collision cross sections. These matters are given

in greater detail in Appendix I, which also contains additional numerical

information omitted from Gerhart's publication, such as a set of transition

energies for the discrete part of the spectrum as given by Namioka [29].

In the process of satisfying sum rules, Gerhart adopted an oscillator

strength distribution (OSD) which gives a value for the ionization dipole

strength ~ 1 OX lower than data from other sources, with OSD values for the

excitation region correspondingly a little higher than other data. We doubt

that this is desirable, and one of us (RP) has developed an OSD safisfying

the sum rules without such adjustments. But the importance of Gerhart's cross

sections is such that we have used them -- with modifications as outlined

above — in the extensive numerical explorations reported here.

9
Figures 1 and 2 present differential cross section data as calculated by

our version of Gerhart's cross sections. Figure 1 displays the variation with

energy loss. Figure 2 shows that our version of Gerhart's cross sections

agrees rather well in shape with the Opal, Beaty, and Peterson [30] experimental data for the spectrum of secondary electrons at T = 500 eV. One

should note, however, that the experimental data have been divided by a scale

factor of 1.65.^ A similar comparison at T = 100 eV with recent experimental

data due to Rudd and Dubois [31] shows somewhat poorer agreement as to shape,

but better agreement in magnitude. In both cases, the experimental data

appear to drop below the theoretical data at very low secondary electron

energies.

Figures 3 and 4 present stopping power and total cross section results.

Values for the stopping power obtained using Bethe's expression are also given

in fig. 3, as are some recent unpublished data due to Green [32]. Gerhart's

OSD leads to a computed value for <I) of about 19.2 eV as stated in his

paper [3]; and this value was used in calculating the curve labeled "Bethe".

On the other hand, our set of cross sections leads to an equivalent constant

with the value <I)p = 18.6 eV even though utilizing Gerhart's OSD. While it

is possible to make adjustments which would increase the latter value, we have

not attempted to do so. See Table 1.6 for other data on <I>, both

experimental and theoretical.

^This means that the Opal, Beaty, Peterson data extrapolated in a reasonable

way give an integrated value significantly larger than data due to Rapp and

Englander-Golden [33], to which Gerhart's cross sections are normalized.

10
IV. General Method of Solution

A. Comments on the integration

In solving integral equations such as eq (1), the usual approach involves

replacement of the integral by a weighted sum as follows:

dT k(T',TJ y(T',Tjj) - ^ w- k(T.,T^) y(T.,T^) . (2)

If all values y(T.,T^) are known except for y(T ,T ), and if all values

for w^. have been specified, eq (1) gives a value for

When both factors of the integrand vary smoothly and slowly from one

value of T^. to its neighboring values, weights w. can be adopted from

one of the standard formulae, such as Simpson's rule or even the trapezoidal

rule. This method of calculation is suitable for regions in which k(T',T)

and y(T,T^) vary with similar rapidity and irregulari ty , because the many

values for X i^equired to describe such functions correspond rather well in

spacing and number to the mesh of values required to evaluate the scattering

integral accurately.

An altogether different circumstance can occur, however. One of the

factors in the integrand can be rapidly varying while the other is smooth,

slowly varying, and predictable. Thus for large T^, at most kinetic energies

T', T' < T^, one has a slow and predictable function y(T',T^) combined

with very rapid and irregular variations of k(T',T^) whenever T' is within

perhaps 30 eV of T^. This is wel 1 -i 1 1 ustrated in fig. 1, where the cross

section, which is proportional to k, is given as a function of the difference

11
between the variables. The irregular behavior for (T^ - T) or (T' - T^)

between 11 eV and 25 eV occurs for virtually all T^, regardless of

trends of y.

Under these circumstances eg (2) is still appropriate; but it is no

longer desirable to use elementary numerical integration weights. Instead,

one can use integrals over k, usually moments, to construct an approximation

to k as a sum of Dirac delta functions. The integral is then readily

performed and gives an expression of the type shown by eq (2). We refer here

to such "6 - function approximations" by the general designation "Gauss

quadrature" (GQ) .

The €imple numerical integration algorithms for the solution of eq (1) on

the basis of expressions of the type illustrated by eq (2) require the T^ to

have the same values for different T^. Other algorithms are possible; but

this is preferable on the basis of simplicity. We thus choose T^. values for

the complete calculation at the start, with T^ ' s which march down this

preselected list. To achieve accuracy comparable with Simpson's rule in

eq (2), we evaluate three energy moments of k for the combined pair of

intervals T^_^ < T' < T^. The resulting three conditions^ fix 6-function

weights at T^, , T^ These weights can be cumulated with other

weights, since all play the same role in the numerical integration at various

T^. values of the same set.

^We here include the option of pre-specifying some or all x. values. This

results in a unified view of numerical integration procedures of many types.

The usual Gauss quadrature can be considered as an approximation of the type

N w(x) == a^6(x-x^. )> where the a^. and x^ are fixed by 2N jrioment

integrals. In the classic case, the weight function w(x) is unity for

-1 < X < 1. For another well-known case w(x) = e~^, with 0 < x <

12
By restricting our approach to integration formulae which yield sums of

the type shown in eq (2), we can determine w_j values by two or more different

procedures of this type, each appropriate to a sub-region of the domain of

i ntegration.

B. The integration mesh

Description of the Lewis effect, i.e. resolution of complex spectral

details near the source energy, requires that intervals for numerical

integration be ^ 1 eV, as already noted (see fig. 1). Similarly, at low

energies, near and below the ionization potential, trends of y(T,T^) with T

are again steep, with possible structure. Small intervals are therefore also

appropriate for solution of the integral equation at low energies. But it is

quite clear that most of the energy range, that between, say, 100 eV and

(T^ - 100) eV, does not require such a detailed description because y(T,T^)

has trends which are smooth and not particularly rapidly varying.

Our choice of the energies constituting the solution mesh was

therefore as follows: We used the tabulation intervals established for the

OSD to develop a set of values in the energy range from down a total

of 40 to 80 eV lower in energy. The interval size was then increased abruptly

from AT ~ 2 eV to AT = 20 eV. Following this, the intervals were increased

in geometric progression until the energy loss amounted to T^/2. Typically,

then, the energies T^, such that T^ < T^/2, were equated to the energy

1 osses (T^ - T^). Thus the solution energies were approximately twice in

number those required to carry the solution from T^ to T^/2.

The reason for an abrupt change in interval size as referred to above is

that intervals smaller than 2 eV can resolve at least moderately well the

main features of the glancing collision region, even between 13 and 17 eV.

On the other hand, intervals greater than 20 eV are large enough so that the

13
whole band of irregularities in k(T',T) is contained within a single

interval; and fine resolution is then rendered unnecessary except in regards

to the calculation of GQ weights, which is accomplished by interpolation on basic tables of cumulative integrals.

We did find it convenient to buffer the transition to large intervals with two intervals of size 10 eV, at the same time ensuring that the GQ region extended beyond one such interval. Table 1 gives an illustrative set of

T values, for a 1 keV source,

n From the attention given to such details, it should be evident that setting up a suitable integration mesh constitutes a significant part of our calculation. Typically, the fine spacing for high and low energy regions

requires about 40-45 points each, with up to 90 additional points assigned

to the remaining (usually large) energy region. We used up to ~ 200 integration points, so that instruction and data memory components together

occupied just under 64 K cells of the memory.

C. The integration procedure

Both glancing and knock-on cross section components have a rather simple

analytic structure; thus k(T',T), with T' = T + E, has the form

k(T+E,T) = E fyi+E) gyi.E) . (3)

i

The functions f^. (T + E) are slowly varying, and the functions g^-(T,E) can either be integrated analytically, or depend only on E, or both. We have already mentioned tabulations of cumulative integrals of the (rapidly varying)

g^.(E) for glancing collisions. Corresponding functions g^-(T,E) for knockon collisions can be integrated analytically.

14
Because the functions E) are slowly varying, if combined with

the function y(T + E,T^) the product tends to vary slowly or rapidly according

to the characteri sties of y. This leads us to pretabulate the f^. at the solution energies T^, with integration based on variables G^. defined by

dG3r=9i(T>E) • (9)

Then

k(T',T) y(T',iy dT' E JdG, f,(T,G,

1 1 y(T,G.,T^) (4')

This approach should be contrasted with the more direct procedure of simply evaluating k(T',T) as necessary for all combinations (Tj,T^) which appear in the set of integrations. We have ordinarily used 5 or 6 terms, i.e. values of i, so that the right side of eq (4') has ~ 6 terms

in the sum. On the other hand, pre-tabulation of the f^. is quick and cheap, because it need be done only once to obtain a list of values for all solution energies T.. By contrast, direct evaluation of k(T',T) requires calculations of all component factors of k for a quadratic, rather than linear

array, more or less cancelling out any extra computer time required by the multiplicity of i values. In addition, by differencing G^. we readily obtain the lowest moment required for the GQ; repetitious tabulations to evaluate GQ moments are avoided by use of several such cumulative functions.

15
We believe that it would be about equally feasible to use tabulations of

cumulative integrals for the Gauss quadrature, with direct evaluation of

k(T',T) reserved for evaluation of the remainder of the scattering integral;

but we have thus far chosen not to proceed in this way.

We always use GQ for the final interval, T^_.| > T' > T^, and we use a

version of Simpson's rule for the remainder of the scattering integral. Our

GQ contribution to eq (2) is as follows:‘"

T.

dT' k(TM„) y(TVy =» W,y(Tj^,,Ty + -^

0

) ' ( = )

n

where j_ is commonly, but not always, equal to n-1 . Since k(T',T) vanishes

for (f - T) < 11.2 eV, we find it advantageous to extend the GQ interval

to fully cover this gap, because other integration procedures are rendered •

inaccurate when extended into a region in which the integrand is zero. The

rule adopted for choice of ^ is therefore that it equal the largest m

value such that (T^ - T^) > 11.2 eV.

Six cumulative integrals over the OSD giving data for three energy

moments (see Appendix I), together with corresponding analytic integrals over

components of the knock-on cross section, give moments of the different

g^. (T,E) from which integration weights can be systematically developed for

all j_ values by combining contributions from adjacent integration intervals,

each covering two intervals of the tabulation.

“^Note that T. need not be included between T and T-.

J- 1 n j

/

16
We should comment that in integrals of the type shown on the right of

eq (4‘), for all ranges except those where 6Q is applied we systematically

use

dG.

dG.

du

du , ( 6 )

where for each three consecutive T.j values the corresponding u_ values are

-1, 0, and 1, thus permitting application of the elementary form of

Simpson's rule. We determine dG^./du by fitting each three consecutive G^.

values to a quadratic in u. Note that only two of the six cumulative integrals

over the g^- are required for this process; we use the other information

available in Table 1.2 only for the GQ.

D. Energy and number conservation

In order to monitor the calculations very closely we wrote a special

subroutine (ECHEK) to perform energy conservation checks at selected energies

(see Appendix II). The integral forms of the slowing-down equation which

were referred to in the introductory discussion are (for energy conservation)

Tq

dT' S(T',T) y(T',Tj = Tq , (7)

T

where S(T',T) is a limited stopping power which excludes energy transfers to

secondary electrons with kinetic energies exceeding T. For number conservation the equation

2T+I T^

J

dT' Kp(T',T) y(TM^) = 1 + r dT' K^T’ ,T) y(T' ,T^) (8)

' 2T+I

17
holds, where Kp(T',T) and K^(T',T) are, respectively , the probabilities

per unit path length for removal and addition of an electron to the kinetic

energy interval between T and T' [5]. Accurate evaluation of the integrals

in eqs (7) and (8) is of comparable difficulty to the main problem of evaluating eq (1), although one can use devices that have previously proven

satisfactory [5-10]. The accuracy of the numerical integration remains a

problem because S(T',T) and K (T',T) give important contributions for

P (T' - T) small, while retaining some irregularity.

For evaluation of eqs (7) and (8) we wrote a general stopping power

subroutine (STN) and an integrated cross section subroutine (TCX) according to

a pattern which permitted rapid and simple cumulative integration over

transition energies up to an arbitrary limit (see Appendix II). These

routines were essential to the evaluation of the kernel functions of eqs (7)

and (8). In preparing the stopping power subroutine, all contributions due

to the energy of ionization must be sometimes, though not always, included.

This requires a special term when fast (i.e. above cut-off) secondary

electrons were generated.

In the use of both stopping power and integrated cross section subroutines, one must bear in mind that for T' < 2T^ + I both primary and

secondary electrons from an interaction can lie below the energy cut-off T^.

For such interactions the total energy of the electron being scattered is

"dissipated.

The exercise of taking account of energy conservation has proved^ necessary

and very instructive. By this means we found that in the transition from

finely spaced integration intervals to coarse integration intervals, some of

the energy (typically ~ 4%) can be lost. This may be partly due -to the fact

that the changeover necessarily occurs sufficiently near the source energy to

18
be affected by the Lewis effect irregularities. The first few points with

coarse spacing are uncertain by a few percent for this reason; and any resulting

error is then "locked" in, because the major contribution to the scattering

integral is always due to a few points thus affected.

To correct such errors, we have calculated energy conservation as well as

a value for y^. at the first five T^. values with the coarse spacing. We

adjusted these y. values immediately as if any deficiency of energy conservation were solely due to the y^- value being determined. This worked well.

In general, we find that energy-conservation results give a good check on

the accuracy of the computations. The number conservation data are in

numerical agreement with energy conservation data above T^/2. Below T^/2

the number conservation data simply record the total number of fast electrons

generated by all processes. But the number conservation check can be extended

to very low energies by applying it to the primary electron component only.

E. Special features

At the time the computer program was being written, we felt that it might

be very important to distinguish between primary electrons, the first generation

of secondary electrons, and higher generations of secondaries. Hence in

setting up the basic table of integration weights, we arranged for two types --

i) those including, and ii) those not including the secondary electron

components of the scattering integral. The former are used to compute the

full slowing-down spectrum; the latter, together with the difference between

the two types, are necessary for computation of the different generations of

secondaries. To economize memory storage space, these two triangular arrays

of integration weights are both stored in the same nearly square array

(called WT).

19
Table II.l gives a summary description of the computer program which we

developed, and which we call ELSPEL, for E^nergy l^oss ^ectrum of Electrons.

Discussion of each of the eighteen routines is presented in Appendix II.

IV . Discussion of Results

A. Energy Conservation

We begin our discussion of results with a table showing typical energy

and number conservation data, i.e., table 2. A value of unity is the desired

result, and departures from unity indicate fractional differences from source

energies T^. Such departures are seen to approach a few percent, although

for most energies the agreement is ~ 1%. Since the integrals required for

these checks feature a large contribution from the region of T' near T,

the departures from unity of table 1 are partly due to evaluation of the energy

conservation integral, in addition to errors of y. But for the most part, we

believe that the energy conservation integral is a reliable measure. The

results shown are sufficient to demonstrate a general feasibility of. the

approach which we have developed. Further improvements in computational

accuracy can no doubt improve this check. One should note that the rate of

change of the energy conservation data is probably a better indication of the

accuracy of y in any given part of the spectrum than are the values shown.

Figure 5 gives the fractional contribution to the energy deposition by

primaries, secondaries, and all other components combined. Here, energy

deposition includes energy losses of all types except the kinetic energies of

secondary electrons still capable of ionizing, i.e. kinetic energies > 16 eV.

There is evident a rather remarkable persistence of the importance of the

primaries and first generation secondaries, which together are respcfnsible

for more than 90% up to source energies greater than 300 keV. Since both

20
components can be represented rather well with fairly simple functional forms,

one can hope to obtain analytic representations in this manner which are

usable for many types of yield calculations.

B. The Lewis effect

Figure 6 demonstrates the Lewis effect and its mai-^ features at a single

(high) source energy (1 MeV). The abscissa measures the energy loss in eV.

The Lyman and Werner bands appear as peaks, with multiple interactions producing

a similar structure which decreases in prominence. The maintenance of the

similarity of structure for multiple interactions is very likely due to the

dominance of the Lyman peak: probably all but one of the interactions combining

to give prominent secondary and tertiary peaks are Lyman interactions.

While not evident, the peaks of fig. 6 are in reality composed of many

individual lines which are not resolved. It is possible to resolve these

lines with finer spacing, birt we have not seen fit to attempt it.

The rather arbitrary cut-off of low-energy knock-on interactions leads to

a rather small discontinuity, labled "Miller," which is identified on the

figure. The form chosen for this cut-off is evidently not very important.

Also displayed is the energy-loss region for which energy conservation is used

to correct the solution. The smoothness apparent in this region is partly an

artifact of the method used, and partly a consequence of increased interval

size, both of which act to wash out the physical irregularities.

The ordinate has been chosen so that at large energy losses the curve in

fig. 6 should approach unity, corresponding to the increasing validity of the

reciprocal stopping power approximation for y(T,T^). This result is apparent

although the asymptotic trend develops more slowly at larger source energies

than we would have expected.

21
C. Properties of si owing-down spectra

Figure 7 presents spectra for four source energies. The ordinate has

been chosen so that the curve is fairly flat in the region of dominance by

secondary electrons.^ For the 30 keV case, contributions by primaries and

f i rst-generation secondaries are indicated by dashed lines; and in the case of

the primaries, the dashed line has approximately the functional form T/T^.

According to the CSDA approximation, the first generation secondary

electrons due to a monoenergetic source at are described by

y(l)s

= f

dT K^(T',T)/S , (9)

"2T

where is the kernel function of eq (8). If we write the stopping power

in the form S = k(T) B(T), where B(T) is the stopping number, then the

lead term of K^/S is [T~^ + (T' - T)~"' ]/B(T' ) . We assume >> T and

make use of the usual slow rate of variation of B(T') in evaluating this

factor at T' = T^/2, which is approximately the midpoint (T^ + 2T)/2 of

the integration interval. Then,

S(T) y*'’(T,T 1 oT ^ T . ^o‘^

1 - 2— + — log

0

T.0 T T < T^/2

1/B(y2), T « T^. (9')

The minima of the weighted spectra shown in fig. 7 are thus expected to

have roughly the value l/B(T^/2), due to 1 st-generation secondaries which

^Since the product S • y is a rough measure of the number of electrons,

(primaries plus secondaries) per primary, at any given kinetic energy T,

the ordinate variable can be interpreted as a rough measure of the fraction

of the input energy T^ which these electrons represent.

22
dominate in this region. Below ^ 1 keV the minimum tends to disappear.

But the tendency of the spectra to converge keeps a value ~ B”''(.5 keV)

significant to still lower energies. At the right of the figure, some of

these values have been indicated.

A remarkable feature noted previously by Berger [34] is a strong tendency

of the curves to converge at low energies, so that the .:.hape of the slowingdown spectrum is nearly independent of in this region. One sees in

eq (9‘) a tendency of this type in the y^^\ since B varies slowly for

large T^. In higher generations of secondaries the upper limit of the

integral on the right of eq (8) becomes largely irrelevant, so that one expects

these components to depend on only through B(T^/2) at most. But the

convergence seen in fig. 7 is more striking than this and must reflect further

restrictions imposed by energy conservation. Table 3 illustrates this strong

trend towards T^-independence of the low energy components. Convergence is

evidently according to a trend somewhat of the type

y(T,T^) ~ y(T,-) + a^/log(T^/a^) , (10)

where a^, a-j depend on T only, because equal factors of increase in source

energy improve convergence by comparable i ncrements .

D. The ionization yield

Because degradation spectra have extensive applications to the evaluation

of excitation and ionization yields, we turn to the discussion of the ionization

yield, usually given in terms of a quantity W, eV per ion pair,

= ty’ /

^i(T) y(T,y) , 0

where k^. is probability for ionization per unit path length by electrons

with kinetic energy T. At the bottom of fig. 7 a curve of K_j/S has been

23
included. It is instructive to consider the different contributions to the

value of W"^ for H^. The T/T^ trend of fig. 7 for the primaries, which

follows from CSDA, has the consequence that the contribution by primaries can

be approximated by evaluating (k^./S) at the midpoint energy T^/2, e.g.,

.018 for Tq = 30 keV. Correspondingly , the fact that both (k^./S) and

(T/T^)Sy are comparatively flat means that a rough evaluation of the contribution due to secondaries can be obtained with similar ease. One notes that

large contributions, say in the case of 30 keV, occur between ~ 20 eV, and

~ 2500 eV. The midpoint of this range (in the logarithm) is (20 x 2500)^ =

224 eV. Evaluation of both K^/S and (T/T^)Sy curves at this energy, with

the product multiplied by £n(2500/20) = 4.83 gives a value .0095. We thus

estimate that W 1/(.018 + .0095) 36.4 eV. This agrees rather well with

more accurate calculations.

One can see from these curves that the high energy gradual decrease of

k^/S will decrease the contribution to W~^ due to primaries, as the source

energy T^ is raised. On the other hand, secondary electron contributions

will increase. Unfortunately, the resulting rough cancellation of competing

effects provides no simple explanation for the impressive T^-independence

found in experimental investigations of W.^ But this approach could probably

be developed to give rough estimates fairly readily.

Pursuing the question of W further, we present values due to other

investigators in table 4, and for different source energies in table 5. A

minimum occurs in the vicinity of 2 keV, with a very slow rise with increasing

source energy. Comparisons with results from experiments and other recent

6 See refs (7) and (8) for an interesting approach to this problem which uses scaling of the si owing-down spectra.

24
computations are very interesting, and are displayed in figs. 8 and 9. In

fig. 9 one sees that Garvey, Porter, and Green (GPG) obtain a much larger rate

of increase in W with source energy above ~ 3 keV. On the other hand, in

fig. 8 the data of GPG show a more rapid increase of W as T is

decreased from 3 keV to below 200 eV. We have not attempted to analyze

these differences in detail; but analogous differences in the case of H

2 O

can apparently be traced to differences of cross section. One need not

assume that computational procedures are the cause of such contrasting trends.

The case is different in the comparison with Gerhart shown in fig. 8.

While the cross sections used in our calculations are not identical with those

of Gerhart, the differences would not seem to be sufficiently great to give

rise to the ~ 10% differences in W which occur.

In fig. 8, the dashed line gives values of W inferred from the following

formula, which depends on the ratio of total to ionization cross sections:

W

- ^tot |~^o ° ^Toin^^

( 11 )

As discussed by Jones [51], such an expression is valid when one ionization is

energetically possible, but two interactions of which at least one is an

ionization are not possible. This condition applies below about 26 eV for

the case of H^. The bump shown in the W curve is therefore a consequence

of cross section ratios and does not follow from multiple processes.

Experimental values due to Combecher [57] do not show this reversal clearly,

although they exhibit a shoulder at correspond! ng energies. Such reversals

have been reported in references [23,58,22], and also by D. Douthat [54]. We

do not know whether this is a real phenomenon or not. It is strongly

C dependent on cross sections which are not well known as yet.

25
Equation (11) furnishes a good monitor for the accuracy of the slowingdown computations, since corresponding values can be obtained by application

of the computer programs, which were designed for much greater source energies

and are therefore put to a hard test at low energies. Agreement shown in

fig. 8 thus constitutes a good check.

Figure 10 gives a different plot of some of the data of fig. 8. Below

~ 27 eV, the solid curve represents the cross section ratio of eq (11).

Below 20 eV Combecher's data differs by a factor ~ 1.2 in magnitude from the

solid curve; and by inference from eq (11), this difference measures differences

of cross sections. Since our calculations thus appear to permit too much

ionization at low energies our W values may be somewhat low at high energies.

But this same argument means that Douthat's two versions of cross sections

below 100 eV should give rather different W values at high energies.^ In

our experience, high energy W values are rather sensitive to low energy

cross sections, even those below 25 eV.

It may be that the rather reasonable values for W obtained here occur

because Gerhart's cross sections for ionization are too large at low energies,

as evidenced by Combecher, but are correspondingly too small at high energies,

as much other data suggests.

Figure 11 compares our slowing-down spectra for = 10 keV with

corresponding spectra due to Douthat [1] and GPG [2], as read from the

published curves. In a general way, our values lie between those of the other

authors, with the largest differences occurring at low energies, where factors

of 2 can be seen. For the region above ~ 300 eV, which is dominated by

y - S \ the agreement is generally good, as can be expected.

^Douthat's lower values, € in fig. 7, correspond to a cross section set

rather similar to ours. His higher values, +, correspond to about a 28%

reduction in excitation cross section with 16% increase in ionization

cross sections.

26
Table 6 gives a few comparisons with data for = 100 keV due to

Okazaki and Sato [15]. No agreement is evident even at high energies where

the y values depend essentially on the stopping power alone.

27
Appendix I. Cross Section Data

For purposes of constructing a computer program for the integration, we

have used a well-known set of cross sections, viz the cross section set

developed by Donald E. Gerhart for his PhD thesis [3], even though there are

some unsati sfactory aspects to them. In this 'appendix we describe necessary

modifications and adaptations.

Gerhart uses Platzman's approach, which depends on the Wil 1 iams-Wei szacker

approximation in which the passage of a fast charged particle is replaced by a

nearly equivalent spectrum of virtual photons, which are subject to photoelectric absorption [84]. Collisions are classified as either "glancing'' or

"knock-on." Glancing collisions, with small momentum transfer, are assumed to

follow expressions derived for zero momentum transfer. Knock-on collisions

are assumed to follow cross section forms based on collisions between free

el ectrons.

A. Glancing Collisions (Small momentum transfer, large impact parameter).

First, in regards to excitations due to glancing collisions we follow

Gerhart in the use of the Bethe excitation cross section formula as follows:

Aira^R f

0 n „

, 4T N T E^/R R ^set^ ’ (I.l)

where R = 13.6 eV and f^ is the oscillator strength for the n'th

transition of a set, which can be any group of transitions treated together.

The constant c^^^ in the logarithm, which in principle differs from transition

to transition, is assigned an average value appropriate to a set of related

lines such as those constituting the Lyman or Werner bands of ^

Ionizations due to glancing collisions are described by the analogous

cross section form

28
a

( 1 . 2 )

where is an "ionization efficiency" function, and b is essentially

the reciprocal of an impact parameter in a^ units, or a cut-off for the

square of the momentum transfer in glancing collisions. The variable E

is the transition energy, and df/dE is the differential oscillator strength

distribution (OSD).

The OSD developed by Gerhart is based largely on discrete excitation

data due to Allison and Dalgarno [59] and experimental data due to Cook and

Metzger [60], both modified to satisfy information from sum rules. The

important and complex far-ul traviol et region from 14.4 to 16.5 eV is described

by replacing the discrete Rydberg states with a carefully estimated continuous

extension of the continuous part of the OSD at higher energies.

The f^ data of Table I.l are from Gerhart, and represent his modification of the data due to Allison and Dalgarno. The corresponding E^ data

were not given nor referred to in [3]; they are due to Namioka [29].

The ionization efficiency function n.j used by Gerhart is reproduced

from his paper in fig I.l. A dash-dot line, corresponding to an analytic

approximating function

has been added; we used this curve for convenience, despite the fact that it

can clearly be improved.

The mathematical form of eq (I.l) is a discrete analog of eq (1.2). In

both cases, the expression can be written as the sum of two products, each

having a function f(T) multiplied by a function g(E) (or g^) of the

transition energy. We proceed to a unified treatment of excitations and

(1.3)

29
ionizations in which cumulative integrals (or sums), extend from zero transition

energy over excitations and then ionizations, up to a maximum energy of about

220 eV. Three such cumulative integrals (f^, ^ multiplied by factors

or e\ i = 0,1,2), are tabulated at intervals of about 0.4 eV for small

transition energies, with the interval size increasing geometrically with E.

A second set of integrals includes a factor £n(E^) or £n(E). The resulting

6 tabulations are given in Table 1.2. The first three columns {£) are energy

j moments of f^ and while the latter three columns (^ ) are energy

moments of ^^n(E), E^ and E in eV. Factors (1 - n^.)

and n.j are included in calculations for excitations and ionizations,

respectively. These tables form the numerical basis for all computations

involving glancing collisions.

The use of three integrals, yielding three moments of the transition

energy, has the following significance; The zeroth momenth is essential to

the evaluation of the total cross section; the first moment is required- for

stopping power calculations; and by use of three moments one may perform Gauss

quadrature calculations of quadratic accuracy.^

The constant b was assigned a value 2.3 by Gerhart. As will be

discussed in section C below, we found it convenient to make b a function

b(T) for electron kinetic energies T below 707 eV.

We note that eqs (I.l) and (1.2) are non-relativistic and require

modification to achieve the relativistic forms. This is essential because we

wish to do calculations at relativistic as well as non-relativistic e'nergies.

Hence we modified all glancing cross sections by replacing the logarithmic

factor ln(4T/R) as follows:

^The second moment is important in straggling theory.

30
-> £n (T + 2 )

* * T (T +2)

(T +1)^

(1.4)

•k 2 Where T = (T/mc ) and the term on the right is the square of the velocity,

2 2

3 = (v/c) . Likewise, the factor (R/T) appearing at the left of cross

2 2 section forms given in eqs (I.l) and (1.2) was replaced by 2(R/mc”)3 .

B. Knock-on Collisions (large momentum transfer, small impact parameter).

Gerhart also made use of non-relativistic forms for the knock-on cross

section. These he multiplied by a factor c{)(E) and by another factor C(T).

The latter factor, which was evidently introduced to improve agreement with

total cross sections below ~ 1000 eV, makes it impossible to derive the Bethe

stopping power formula without some change in Gerhart's formulae. Hence we

simply eliminate C(T) by equating it to unity. This leaves us with Gerhart's

problem of obtaining agreement with data on total cross sections below 1 keV,

and this is discussed in the next section.

Gerhart's function (f)(E) was introduced as a smooth cut-off to the

spectrum of transition energies resulting from the knock-on process. (1)(E)

is shown in fig. 1.2, together with a simple step-function approximation which

assumes zero cross section for E < 17.5 eV, which we decided to use. In

our opinion, the slowing-down spectra are not sensitive to such a change of

shape; and the sharp cut-off simplified both numerical and analytic procedures.

Consistent with the point of view already expressed, we use the fullyrelativistic M(|)ller cross section to represent the knock-on process, modified

in a simple way to take account of a mean energy I « 1 6 eV for ionization:

31
aira^K l r

3^(mc^) |e

a

ko 2 +

(1.5)

+ u(E-17.5) ,

where the unit function, u(x) =

1 , X > 1

0, X < 1 , expresses the cut-off shown in

fig. 1.2 for low energy excitations by this process.

C . Low energy ionization cross sections . _2

Gerhart chose to use a simple E cross section for all processes at

low energies, i.e., T < 300 eV. This function was normalized to give a value

for the total ionization cross section in agreement with experimental data due

to Rapp and Englander-Golden (REG) [33]. We have chosen a different approach

to the problem of the selection of differential cross sections at low energies:

we continue to use eq (1.2), but assign to b (for T < 707 eV) an energy

dependence b(T) adjusted to give total cross sections in agreement with

REG. The theoretical basis for the T-independence of b breaks down at low

energies. In fact, since there is no theoretical basis at low energies for

the asymptotic form which we use, this procedure should be considered only as

an ad hoc extrapolation guided by an experimental ly fixed integral.

Incidentally, Miller used b = b(E) [58], but there is some theoretical

justification for this [62]. Even so, as the glancing component decreases in

importance at low T, it is not unreasonable to assign this to a decreasing

range of contributing momentum transfers.

Use of values for b(T) as large as b = 3 at T = 1 keV could

accomplish essentially the same thing as Gerhart's C(T) although we have not

incorporated this adjustment; while for T < 500 eV use of b(T) results in

32
more gradual change of spectrum than that due to Gerhart's prescri ption

Table 1.3 gives £n[b(T)] values which we have used. Table 1.4 shows the

comparison with REG thus attained.

Our approach is no less arbitrary than that adopted by Gerhart and in

modified form by Douthat. Our value for b above 707 eV is 2.34. This

sacrifices close agreement with REG in the region 707 eV < T < 1000 eV, but

preserves the asymptotic agreement of stopping powers at high energy with the

Bethe stopping powers based on a value for the mean excitation potential

consistent with Gerhart's OSD.

D. Low energy excitation cross sections

Figures 9 and 10 of Gerhart's paper [3] present his "adopted" electron

impact cross sections. Since our purpose here is to perform electron slowingdown calculations with minimal changes in Gerhart's cross section set, we

have reduced these curves to tabular form for our use, with no significant

modifications. But one should note that accurate values in the important

energy region 13 eV < T < 25 eV cannot be obtained in this way, and detailed

features of the spectra in this region can be wholly spurious.

These total cross section data must be in some manner complemented with

information about associated transition energies. In the case of the Lyman

and Werner discrete excitations one has values for individual transition

energies [29]; but for higher energy excitations a range of energies from

approximately 14.4 eV to 17 eV is involved, with the use of a continuous OSD

function in place of the actual complex line structure as already noted. Our

first approach was to choose for each process separately a functional form of

the type

[1 - n,(E)] I ^^n(4lb(T) . (1.6)

33
But since the si owing-down calculation depends for the most part only on three

moments (integrals) over the transition energy, we carried use of this form a

step further and applied it to the combination of al

excitation cross

sections. This leads to slightly different values b.(T), j = 1,2,3, for the

vj three different moments, as required to agree with calculations having different

b(T) values for different processes as in eq (1.6). However, it turned out

that these b- differed in a regular manner, but only by ~ 1%. Hence we

<J eventually used only b-j(T), which was assigned values very slightly larger

than those calculated from total cross section data. Tabulated data for

resulting b-j function are given in Table 1.3; and a comparison of resulting

total cross section values to those of Gerhart is given in Table 1.4.

The full differential spectrum is important when single interactions are

significant. We calculated these spectra by use of eqs (1.2) and (1.6), using

values for £n(b) and £n(b-j) already described. Figures 1 and 2 give some

results of these calculations.

Vibrational effects become important below about 20 eV; and they

constitute the main part of the cross section below the lowest electronic

transition energy, here taken to be 11.213 eV. Though ignored by Gerhart,

we have included them on the basis of data by Ehrhardt et al . [63] and

Trajmar et al. [64]. We use only two excitation energies, 0.54 eV and

1.08 eV. For electrons with kinetic energy greater than 20 eV, we use the

following simple expressions for the (non-relativistic) differential cross

sections

k^/K [- .00457 + .00588 £n(4T/R)] 5(E - .54) , (1.7)

k 2 /K ^ [- .000309 + .000287 £n(4T/R)] 6(E - 1.08) , (1.8)

34
2 where k = 4iTa^R/T. Below 20 eV, we tabulated the total cross section data

and perform interpolations as required. Tabular data are recorded in

Table 1.5; these are a somewhat smoothed version of the data in Ehrhardt et al

and Trajmar et al

The experimental data do not e: fend beyond 80 eV; hence the use of

eqs (1.7) and (1.8) for much higher energies is not necessarily accurate. But

because the contribution is a small one, this extrapolation was used (with the

relativistic modifications.) Equations (1.7) and (1.8) are of the type which

can be readily incorporated into the subroutine OSCUM which evaluates

differential and integral total oscillator strengths. But we prefer to treat

vibrations by use of separate computations which could be used or not as

desi red.

E. Symmetry of glancing collision cross sections

One problem which must be solved to preserve energy conservation at low

energies is the following: In the case of glancing collisions, for which the

cross sections are not written symmetrical ly , secondary electrons with kinetic

energy (E - I) are generated with the same probability as that corresponding

to energy losses E. A type of symmetrizing is necessary in order to

guarantee this feature. We chose to accomplish this by terminating the cross

sections at an energy loss for which secondary and primary electrons had equal

kinetic energy, and then repeating the cross section structure below this

energy so that it has identical values for kinetic energies equally above and

below that value at which primary and secondary kinetic energies coincide.

This seems a little easier than the alternative approach of adding a

symmetrizing term. Where the difference between approaches is significant, we

35
feel that the effect on the spectrum is small, so that any improvement

would call for evaluation of more accurate and properly symmetrized cross

sections, i.e., a major effort.

F. The mean excitation potential

A mean excitation potential <I)g can be determined for the oscillator

strength distribution for use in Bethe's stopping power formula [65]:

<I)n = 2 R exp SZ' r

df £n (E/2R)]

(1.9)

Analogously, cross sections which we have adopted lead to a high energy

stopping power of Bethe's form; and if we determine an equivalent parameter

<I)p (P for Platzman) we find the following expression:

<I)p = 2R exp-jz‘

j

df £n

*-noniz. /bR

^ n

^ set'

sets n +

2 ( 4R

(I. 10)

where is a cut-off for low energy knock-on collisions, which was assigned

the value 17.5 eV to correspond to Gerhart's cut-off function j)(E).

Table 1.6 gives values for <I)g and <I)p, in comparison with values

by other authors. Consistency between <I)g and <I)p can be improved, but

it is interesting that (I)g agrees rather well with other theoretical

estimates, while <I)p is consistent with experimental values. Partly for

this reason we have not enforced agreement between <I)g and <I)p.

36
APPENDIX II. Description of the Subprograms constituting ELSPEL

The descriptions which follow are in the order shown in table II. 1,

proceeding from differential to integral cross sections, to the subprograms

guiding the solution.

1. OSCUM(E,ISET,ID)

The output from this subroutine consists either of cumulative integrals

over the oscillator strength (OS) distribution (ID =0), or the differential

distribution itself (ID =1). The six cumulative integrals (for ionization)

are as follows: (R = 13.6 eV)

where E is in eV in the factor £n(E).

For excitations the corresponding quantities are, (E^ e AE) and m = 0,1,2,

A. Elementary Cross Section Subroutines

(II. 1)

0

m-1 ^set

n (II. 2)

E E(E„/R)

sets n

m-1 ^set

n

37
where the distinct "sets" include Lyman, Werner, excitations with principle

quantum number n = 3, and the combination of all other excitations involving

transition energies between 14.2 eV and 16 eV. But we reduce these four

types of interaction to three in practice because the OS distribution, which

is taken to be effectively continuous in this energy region, is treated in

Gerhart's paper as identically the same in shape for all excitations which are

neither Lyman nor Werner [3].

Cumulative integrals over excitation oscillator strength appear in an

output array ASRE(I), where I = 1,2,3 correspond to m = 0,1,2 of the

integrals of eqs (II. 2), and I = 4,5,6 relate to the corresponding

integrals of eqs (II. 2). Similarly, for ionizations the output is in ASRI(I),

where index values I = 1,2,3 correspond to the first integrals of eqs (II.l);

and index values I = 4,5,6 correspond to the integrals of eqs (II.l)

containing the logarithmic factor.

For ID = 1 and E ^ 17 eV, the value of df/dE for ionizations

appears in ASRI(l), while £n(E)df/dE appears in ASRI(2). For E < 17 eV,

equivalent quantities are obtained by differencing the cumulative tabulations;

these appear in ASRE(l), ASRE(2), ASRI(l), and ASRI(2), the index 2

referring to the quantity with the logarithmic factor. (These equivalent

quantities are and divided by the width AE for any interval.)

Note that for the ID = 1 case, the output has units (eV)~"'.

Input tabulations for OSCUM are read from cards, when ISET f 1976.

One set of input data (EON, for ionizations) consists of cumulative 'integrals

tabulated at intervals from 15.176 eV to 17 eV. The other set (EXC, for

excitations) is a similar tabulation extending from 11.213 eV to 17 eV.

The two tabulations are listed in table 1.2. Correspond! ng energies are read

into EOS.

38
The method for computing ASRE and ASRI values is interpolation on the

prepared tables. For our calculations, based on Gerhart's set of cross sections,

the following forms were used to extend the tables to 222.7 eV, where E

values are in eV:

transition energy (list name: EOS):

En = E^^_^ (17/11. 213)^/^\ n . . . , 80 (II. 3)

excitations and ionizations, 14.2 eV to 17 eV (list name: EXC):

(II. 4)

ionization efficiency:

(II. 5)

ionizations (list name: EON), 17 - 35 eV:

^ ^ 275

(II. 6) dE ^2.77

for 35 eV to 222.7 eV:

df 1624

dE ^3.27

(II. 7)

for E > 222.7 eV

(II. 8)

39
Interpolation requires both positioning on the table, and an interpolation

formula. Positioning was accomplished as follows by determining the equivalent

index by the formula below, where N is the next integer below EX:

PY - loq(E/11.213)

(1/11 )log(17/ll .213)

(II. 9)

N 4 EX (II. 9')

The interpolation weights could then be calculated from (EX - N), with

integer separation between tabulated abscissa values. A simple 3-point rule

was used for this purpose.

The df/dE tabular values were obtained as previously mentioned by

differencing as follows in the region below 17 eV:

E0N(I,J+1) - E0N(I,J)

E0S(J+1) - EOS(J) ’

EXC(I,J+1) - EXC(I,J)

E0S(J+1) - EOS(J)

for 1=1,4 and 1 < J < 12. (II. 10)

No attempt was made to apply interpolation for E < 17 eV when df/dE was

required; a histogram structure was assumed. Interpolations were, however,

used for ionization above 17 eV.

2. VIB(T,ARGL,X1 ,X2)

Output for this subroutine are values for the total vibrational cross

sections at kinetic energy T, for 1st and 2nd excitation levels; these

values are designated XI and X2, respectively, for energies greater than

T = 20 eV, the values XI, X2 are given by formulae as follows:

40
XI .00457 + .00588 * ARGL

X2 = - .000309 + .000287 * ARGL, (1 1. 11)

where the constants .00588 • (.54/R) and .000287 • (1.08/R) are oscillator

strengths for the two transitions.

In adaption to the input parameter

ARGL = £n[2(T/R)(2 + T/mc^)] - 6^(T) , (11.12)

an i nput table VB for each level is read from cards, as is a corresponding

tabulation of the quantity iln(4T/R), which is stored in the list TVB.

For energies less than 20 eV, the method used is an interpolation, by

means of 3-point weights. Positioning on the TVB table for a calculation

at T = is accomplished by a search for the lowest tabular value exceeded

by U = X,n(4T,/R)

a Data on which the linear forms (for T > 20 eV) are based do not

extend above 80 eV; hence the constants used are probably not very

accurate at higher energies. But the cross sections are small at high

energies, so that this is not important.

3. BM0D(T,B1)

The output value of this function routine is B1 and

BMOD = X^n(bjb) , (11.13)

where b = b(T) is the momentum parameter which separates glancing from

knock-on collisions. For the tabulation we have taken b = b^ for

41
T > (1000)/2 eV, and have used ^ 3. For lower electron kinetic energies,

the value of b(T) has been chosen to render a total cross section for

ionization in agreement with experiment (Rapp, Englander-Golden [29]). But

in practice we modify this to £n(b^/b) = .25 for T > 707 eV, and b^

correspondi ngly equals 2.34.

The parameter B1 is defined similarly,

B1 = £n[b-| (°°)/b-| ] (11.14)

where b-j (°°) = 1, a value which applies for T > 350 eV, as shown in Table 1.3

This parameter corrects the total cross sections to agree with low energy

values, as discussed in section 3(D).

Input consists of a tabulation of values for £n(b^/b) and i^n[b.j (“)/b-| ]

,

which are read from cards into the lists BC0R(I,1) and BC0R(I,2), respectively. Data are given in Table 1.3.

Method of evaluation is interpolation on the table. Between 15.62 eV

and 707 eV, 2-point trapezoidal rule interpolation is used. Below 15.62 eV,

a 3-point rule is used instead. Positioning on the table is accomplished

as in OSCUM, viz. for N equaling the integer just less than X,

V _ loq(1000/T)

, , log/?

(II. 15)

N < X

An extra point has been

accuracy near threshold.

inserted at low energies in each list to give greater

, ^

42
4. DTU(T,B,TB)

Output of this function routine are two numbers, DTU and TB. Both

relate to maximum transition energy for glancing ionizations. For a given

value of T, and corresponding cut-off b(T), DTU is the maximum allowed

energy for glancing ionization transitions from higher kinetic energies T.

Correspondingly, TB is the least kinetic energy for electrons capable of

giving rise to the transition energy (T + I), where T is the kinetic

energy of secondary electrons and I = 16 eV is a mean ionization energy for

molecular hydrogen. Thus DTU relates to the energy loss of primaries,

and TB to the energy of secondaries; both express glancing collision limits.

In addition to T and B = £n(b^/b(T)), this subroutine requires BMOD

because both DTU and TB are evaluated by reverse interpolation to solve

the transcendental equations

* * / T

2T Rb(T ) \

2 + ^mc"^

(11.16)

* / T*

2T Rb(T ) 2 + -!-2

\ mc'^

The reverse interpolation is based on a list of energies

RAD(I,1) = 9(2^/^)^'^ eV, (11.17)

and a corresponding tabulation RAD(I,2) of values for the square root of the

left side of the transcendental equations above, when values of the energy

list RAD(I,1) are equated to T .

exp[-B^(T*)] = (T + I)^ , T* = TB

j

exp[-B^(T*)] = (DTU)^ T* = T + DTU

43
The positioning is accomplished by a search to locate both (T - T)

and (T + I) in the list RAD(I,2), treated as abscissa list. A 3-point

interpolation is then applied to RAD(I,1), treated as the ordinate list, to

•k -k find the value of T corresponding more exactly to (T - T) and (T + I)

•k than the neighboring tabulated values of RAD(I,2). This T then gives the

•k output, directly in the case of TB, and through use with (T - T) in the

case of DTU.

This rather lengthy positioning operation would probably take too much

time if DTU and TB values were required throughout, rather than for the

list of solution energies only, i.e., a linear set of no more than 100 to 200

values.

B. Combined Cross Section Functions

5. TCX(T,DT,IVB)

Output of this function routine is a quantity which can be considered as

2 me ^

2 2 either (for non-relativistic energies) total cross section times (T/Arra^R'^)

or total interaction probability divided by XKAP (as defined below), for all

processes at kinetic energy T, integrated over energy transitions up to DT.

For DT > (T + l)/2, the value of TCX is the total cross section. In

addition, this subroutine calculates certain partial cross sections, as follows

XKAP: This is the function which multiplies the stopping or cross

section number, to give interaction probabilities, in units

(cm^/g). That is, XKAP = j (N„/A) Z/3^, where "is

the Thomson cross section and 3

= v/c.

TGE: This is the glancing component due to electronic excitations

(i.e., not including vibrations.)

44
TBI: This is that part of the glancing component for ionizations

which has as a factor the cut-off parameter £n[b(T)].

2 Input to this function routine is the value of BRSQ = £n(b^R ), and.

through reference to BMOD, the correction term iin[b^/b(T)] . For IVB = 1,

vibrational components are included while for IVB = 0 they are ignored.

The basic method involves use of OSCUM to determine the total relevant

oscillator strength. To this component is added the knock-on contribution,

and, depending on IVB, the contribution due to vibrational excitations.

Note that for very small T, i.e., below I = 16 eV, the value of DT

can meaningfully exceed (T + I)/2, which becomes inoperable as a

limit. Note also that the contributing glancing transition energies are not

allowed to exceed a value TST, beyond which the glancing cross section is

negative.

Equations (I.l) to (1.8) in Appendix I give formulae by which the cross

sections are computed. We repeat below, using some of the computer terms, the

important expressions and definitions for the cross section functions.

Quantities used in calculations of glancing collisions are as follows, with

U = T/mc^, I = 1,2,3:

ARGL = logK (2 + U)

U(U+2)

(U+1)^

(11.18)

(11.19)

( 11 . 20 )

45
ASRE(I) = (1 ( 11 . 21 ) - nJ

all sets E <DT

n

ASRE(3+I) = X £ (R/E log(l/c ) (1 - nJ , (11.22)

all sets E^<DT

n n

where the summations over n are accomplished by integration for E > 14.2 eV.

Calculation of the knock-on contribution uses the following moments of the

Miller function:

(6^ = 16/mc^, €= E/mc^, U = T/mc^, = 17.5/mc^'

f.

DT

de 6 I-l +

1 2U+1

U(U+1 )' (U+1)'

. (11.23)

In the case of TCX and TCXI, 1=1, while for STN, I = 2, in

eqs (11.19) to (11.23).

6. TCXI(T,DT,GLI)

Output of this function routine is the interaction probability at kinetic

energy T for all ionizing processes, integrated over energy transitions up

to DT. For DT > (T + I)/2, the value of TCXI is the total cross section

for ionization. In the computation of this quantity, however, the glancing

and knock-on processes are separately calculated and can in principle 'be

referred to separately. The argument GLI gives the glancing component.

Input for this function routine, and the method of calculation are the

same as for TCX.

46
7. STN(T,DT,ISG)

Output of this function routine is the stopping "number" S/XKAP for all

processes at kinetic energy T, integrated over energy transitions up to the

limit DT. For DT > (T + I)/2, the value of STN is the total stopping

number. The index ISG is assigned either the value 0 or the value 1:

when DT < (T + I)/2, both glancing and knock-on ionizing transitions dissipate

a mean energy I = 16 eV to atomic systems with each such interaction which

occurs. In a general way, this additional dissipation should be included in

STN when DT is used to identify "energy loss"; but this dissipation component

should not be included in STN whenever DT is used to refer to (T + I),

i.e., the rate of energy loss due to formation of secondary electrons. In the

latter case DT is then usually associated with the maximum kinetic energy of

secondary electrons under consideration; hence electrons with greater kinetic

energies are not to be considered at all.

The value of ISG is irrelevant when total stopping number is being

computed, i.e., for DT > (T + I)/2. Otherwise ISG = 1 signals inclusion of

the extra component, while ISG = 0 refers to its exclusion.

Input for this function routine is the same as in the case of TCX, and

the method of computation is essentially the same, except that different

integrals (identified by I = 2) over the transition energy are used as indicated

in the notes for TCX given previously.

8. ONCE(TP,DT,XKD,IX)

Output of this subroutine is XKAP"^ times the differential cross

section for various processes for electrons of kinetic energy TP undergoing

energy loss DT. The output value is XKD. The value of IX relates to the

inclusion or exclusion of secondary electrons: IX = 1 means that secondaries

are included, while for IX = 0 or negative, secondaries are excluded.

47
Input into this subroutine is essentially the same as for TCX.

The calculation for glancing collisions utilizes OSCUM, in the mode

(ID = 1) in which the differential OS value for excitations or ionizations

below 17 eV is obtained by numerical differentiation. The differential OS

values for higher energy transitions are obtained in OSCUM by interpolation

on a predetermined list. Cross sections for knock-on collisions are given by

2 2 2 M(j)ller formula, with 6^ = 16/mc , e= E/mc , U = T/mc :

J_ ^ L 2U+1

U(U+1)2

One should note that the use of a limit (DT < TST) ensures evaluation

of glancing collision cross sections only when they are positive. Also, for

DT > (TP + I)/2, glancing collisions are calculated with (TP - DT) replacing

DT, corresponding to an energy loss totaling I plus the energy of the

secondary electron.

1 +

€ T+e

i_

m-"j

11.24'

(U+1)'

C. Subroutines for Evaluating the Scattering Integral

9. HARDP(N)

Output of this elementary tabulation subroutine is a list HKI(N,J)

which is used in the evaluation of the scattering integral. Setting T = T(N),

the n'th solution energy, the following values are currently included:

HKI(N,1

)

HKI(N,2)

HKI(N,3)

= ARGL = £n[2(T/R)(2 + T/mc^] - 3^

= 1 .

^ _ (2T/mc^+l

)

(T/mc^)(l+T/mc^)^

HKI(N,4) = 1/(1 + T/mc^)^ = (1 - 3^)

HKI(N,5) = £n(b^/b) = BMOD

HKI(N,6) = £n(l/b^)

HKI(N,7) = DTU

HKI(N,8) = TB, as given in FUNCTION DTU. (11.25)

48
These will all be recognized as functions of T required in cross section

formulae. The first six are required sequentially in the order given above,

in a DO- loop.

10. CUMK2(N,NP,A,B,C)

Output for this subroutine are integration weights A,B,C to be applied

to y(T^p_^ ,Tq) , respectively. These weights

take account of both the energy region T^p to Tj^ in the scattering integral

and the "scatter-out" term K(Tj^)y(Tj^,T^) usually placed on the left side as i

eq (1). Note that this region of the scattering integral is complicated and

difficult; the three weights contain the contribution by the kernel function

k(T',T), which has singular and irregular behavior for T' near T.

The weights are obtained by Gauss quadrature, using 3 moments and

3 Dirac delta functions except when NP = 1 , in which case only 2 moments

and Dirac delta functions are used. The general concept is described in

section IV, C in the text. Basical ly, 'one is using eq (4) and making the

approximation = T|.^p - T^^, etc.)

g^E) »= o.«(E - AT|^,p^, ) + B.«(E - AT,jp) + yy(E - AT^p_q , (11.26)

such that

dE E'^ g,(E) a.j ATj

NP+1

+ 3.jATj

NP

+ y.ATj

NP-1 (11.27)

One then cumulates all terms, with the appropriate factors;
WT(NP + 1,N) = Z + *T^p+l)

1

WT(NP,N) = Z 3 ^-f^-(TN + AT^p) , (11.28)

1

WT(NP-1,N) = Z Y,-f.(T„ + ATfjp ,) .

i

We found it desirable to include the term K(T|^)y (T|^,T^) as a separate

weight, WT(N,N), when NP + 1 7

^ N. But when NP + 1 = N, this term was

incorporated into the integrand of eq (11.27) by use of the representation

K(T^ + E) 6 (E).

11. CUMK3(T,DT,IX)

Output for this subroutine is a set of numbers GL(J), J = 1,...,6, on

which integration weights for the scattering integral are based. These numbers

correspond to the following integrals:

T+DT °° GL(J) = J

dE 9 j(E) - JdEgj(E) . (11.29)

T T

The final term in eq (11.29), which is independent of DT and is included

only for glancing interactions, always cancels because only differences

between GL values appear in the weights. It is used to control round-off

errors; note that for large DT the value of GL should vanish for glancing

interactions. Further, DT can be very large, with very many integration

points.

5C
The value of IX determines whether or not a contribution due to secondary

electrons is to be included. For IX = 1 they are included, while for IX = 0

secondary electrons are excluded. In the case of knock-on interactions, this

exclusion is effected by ensuring that for DT > (T + I), the upper limit in

the above integral is .^{2T +1). In the case of glancing interactions,

inclusion or exclusion of secondaries is a more complex problem because the

cross section does not explicitly have a symmetric structure. For DT < (T + I)

eq (11.29) gives the relevant (glancing interaction) integrals, while for

DT > (T + I), i.e., for glancing secondaries, we include a contribution

.T+DT

9j(T + I) I dE . (11.30)

^2T+I

One should note however that in all the above, the limits of integration

can be modified by the requirement that the cross section never be negative, a

condition expressed through limits to the value of E. Thus the upper limit of

the integral in eq (11.29) cannot exceed T + DTU, while the lower limit of

the integral in eq (11.30) must be at least TB, as evaluated with the DTU

function routine.

12. WATES(N,IX)

Output of this subroutine is a full set of weights for evaluation of the

scattering integral at the N'th value of T in the solution list. The

value IX = 1 means that these weights take account of secondary electrons,

while for IX < 0 secondary electrons are excluded.

Basic to this subroutine is the concept expressed in eq (9), the use of

Stieltjes integration: values of G^. are given in the list 6L(I) which

51
constitutes the output of CUMK3. These values must be turned into integration

weights, after which the output of CUMK2 can be included.

The procedure used (see section IV, C) is a bit indirect. The assumption

is made that the integrand, which is the product f^. (T + E) y(T + E^T^), is

no more sharply varying than quadratic in an arbitrary variable u such that

u = - 1,0,1 at three successive points T^, , and Tj+2' these same

three points the basic integration variables G^. have values G^. (1),

and G^. (3). We then write

(IT. 31)

We next fit G^. (u) by a quadratic function of u, from which (dG^/du) can

be approximately evaluated;

2

G^- (u ) = a + bu + cu ,

G. (-1 ) = a - b + c ,

etc. (11.32)

Three-point integration weights can then be applied for evaluation of the

integral. While this integration procedure may not appear so, it is exact

if the basic assumption is obeyed, that f^.y can be exactly represented by

a quadratic function of the variable u.

As in the case of CUMK2, the cumulation over i values with the f^.

factors, and combination of different weights for the same point must be

carried out.

52
We should note that general 3-point or Simpson rule calculations are

suitable to an odd number of points, whereas the scattering integral will

involve odd and even numbers of points alternately. For cases in which an

even number of points occurs, the interval from TY(1) to TY(2) is separately evaluated as follows: The procedure previously described is followed

for points TY(1), TY(2), and TY(3), except that the integration over the

variable u is taken only from -1 to 0. The resulting three weights are

included with the same cumulations and factors already described.

The subroutine closes by calling CUMK2 and including weights for the

final integral, including the "scatter-out" term.

13. C0RRX(X,UI2,UI1 ,UI0,A,B,C)

Output of this small correction subroutine are three values A, B, and

C, which are essentially differences between B-poi'nt integration weights based

on the assumption of a continuous integrand, and Gauss quadrature weights which

can take account of a discontinuity in the same interval.

The three points of the integration interval are UI2, UIl, and UIO,

in that order. The value of X falls somewhere in the interval, this being

the location of the discontinuity, with X = 0 corresponding to UI2. Size

_2

of the discontinuity is I , as shown in the sketch below:

The function sketched is a constant times a unit function.

53
Such a discontinuity occurs in the differential knock-on cross section;

and we take account only of the discontinuous part of the largest term,

7 7

E = I , evaluated at the discontinuity. Integration as in CUMK3 here

involves the variable (see eq. (4))

dG = 1 dX , (11.33)

and

The procedure used is that of constructing weights at the three points by

straightforward application of three point numerical integration as discussed

in WAXES. But we also calculate weights by applying Gauss quadrature (in the

variable X) to the step function by Dirac delta functions located at the

three points indicated.

Differences between the two sets of weights give the values of A, B,

and C.

The reason for this calculation is our impression that the Gauss quadrature approach is significantly more accurate for any interval containing the

discontinuity generated at the 17.5 eV cut-off of the knock-on transitions.

D. General Evaluation of the Slowinq-Down Spectra

14. TLIST(NPH)

Output of this subroutine is a solution list of energies, N3 in number,

which are stored in the array TY(I). A peculiarity of this list is that for

T > T^/2, the quantity TY ( I ) corresponds to (T^ - T), while for T < Tq/ 2,

TY(I) = T. The reason for this distinction is mostly the avoidance of roundoff error in the computations.

54
The Input constant NPH gives the factor by which T changes (in the

intermediate energy region) between successive values, viz

T - t

= <To - Vl) •

• (”-35)

The input constant N1 determines how many finely spaced intervals are

utilized. These finely spaced intervals are separated approximately as the

basic tabulations of OSCUM, and as given in an input list EOS. But the

TY(I) are located between the EOS values according to the formula

TY(I) = {EOS(I) • E0S(I + 1)}^^ . (11.36)

After N1 of these values, two buffer intervals of 10 eV are inserted

separating the fine spacing from the coarse spacing. The Gauss quadrature is

arranged to extend over an energy region exceed i ng 11.2 eV. When (T^ -j - T^)

corresponds to the first of these buffer intervals, fine spacing can resolve

the peak region of the oscillator strength distribution; while for (T^ .j - T^^)

corresponding to the second of these intervals, the peak region is covered by

the Gauss quadrature region, now 20 eV wide.

Coarse spacing begins with an interval of 20 eV and increases at the

rate already given.

AT ^t/AT = 2 n+1 n 1/NPH

(11.37)

Intervals increase in spacing until exceeded, at some value T^.

For this value, one writes N2 = n, and TY(N2) = Tq/ 2. The value

TY(N2 - 1) is then readjusted to give a more even sequence near Tq/2.

55
For n > N2, the values of TY(n) repeat in reverse order those which

precede, according to the rule

TY(n) = TY[2-N2-n] . (11.38)

This has the consequence that the intervals decrease in width progressively

towards low electron kinetic energy.

Finally, intervals may be added as necessary. In particular, 10 to 20

additional values are usually added below the lowest electronic transition

energy to cover the region in which vibrational transitions are dominant.

(No attempt is made in the interval near to perform corresponding

calculations.

)

One should note that for low source energies T^, the sequence just

described is interrupted whenever the tabulated value recorded in TY exceeds

T^/2. Below T^/2, however, the procedure is as already outlined.

Both N2 and N3 are output constants that are recorded in memory

locations available to other subroutines.

15. CXPRT(KX)

The name for this subroutine is short for "cross section printout."

Output determined by this routine includes the stopping number, the total

cross section and various components, including the total ionization cross

section, and the function XKAP which multiplies stopping number and^total

cross section "number" to give stopping powers and interaction probabilities

per unit path length.

For KX = 1 , the differential cross section is printed as wen,

normalized to be compared with a Platzman plot. For KX = 0 (or negative),

only the integral cross sections are recorded.

5b
16. SOLVE(IS)

Output of this subroutine is a solution list for the index IS; the

values are recorded in the array S0L(I,2-IS). Interpretation of the

different possible cases is as follows:

IS = 1 : The total slowing-down spectrum including secondaries of all

orders;

IS = 0: primary electrons only

IS = -1: 1st generation secondaries only;

IS = -J: J'th generation secondaries only.

For IS = 1, the integration weights required are those based on

inclusion of secondary electrons. For IS = 0, the integration weights

are zero for secondary electron components. For negative IS values, the

integration weights which are appropriate are di fferences between the above,

i.e., weights only for secondary electrons, which are used in tandem with

weights only for primary electrons.

For negative IS values, the two required sets of weights are put above

and below the diagonal element of the WT array, with the diagonal reserved

for WT(N,N).

The calculation proceeds by setting up an array of integration weights as

called for. However, the different generations of secondary electrons are

calculated in order; and all use the same array of weights, which is thus

determined only once, and used for all J < - 1.

57
Next, the calculation proceeds by evaluating the source contribution

(ONCE), the scattering integral contribution, and finally, by use of WT(N,N),

the value of y. Because the different generations of secondaries are computed

in chain fashion, the solution for the preceding generation is retained in a

special list, y(I), for further reference. (This is convenient rather than

essential, because this information is also in the general solution array SOL.)

17. ECHEK(N,IX)

Output is the fraction of the input energy which is dissipated in

the form of electrons of kinetic energy lower than the N'th solution energy

T, plus the energy deposited with molecules, due both to excitations and

ionizations. Also computed is the number of electrons making the transition

from kinetic energy above T|^ to kinetic energy below

Because all energy is dissipated regardless of the criterion used, energy

conservation requires a value of unity for the fraction mentioned above.

Similarly, number conservation requires a value of unity for the number of

electrons crossing the cut-off for T such that > (T - I)/2 or for

calculations based on the spectrum of primaries only.

The index IX merely selects the spectrum whose energy fraction is to be

computed. Note that energy and number dissipation can be computed not only

for the total slowing-down spectrum, but also for its various components,

viz., primaries only, or n'th generation secondaries.

Let us write k(T',6) for

^

^

where e= T' - T, where k is a

function earlier called XKAP,

k{1'

3^(T

(11.39)

58
We then define various derived quantities as follows:

JC(T',x) = fd6 k(T',6), i.e., TCX(T' ,x,IVB)

-'o

K(r ) =Jt(T' ,(T'+I)/2), also TCX(TMMVB)

^(T',x) = Jde ek(T',e), i.e., STN(T',x,0)

S(T') =^(T',(T'+I)/2), also STN(T',T',0) ,

J*(T',x) =^(r,x) + I[K(T') -J((T',x)], i.e., STN(T',x,l)

(11.40)

(11.41)

(11.42)

(11.43)

(11.44)

Then eq (7) is interpreted numerically with the following identifications:

ydM^) =ay(T',T^)/K(T') , (11.45)

S(T',T) = k(T') _/*(!', T+I), T’>2T+I , (11.46)

= k(T') {

T'[K(T') -JC(T' ,T'-T)]}, T- < 2T + I .

Similarly, eq (8) can be interpreted as follows:

Kp(T',T) = k(T')[K(T') - J{(T' ,T'-T)], T' < 2T + I and T' < , (11.47)

K3(T',T) = k(T')[K(T') - J((T',T+D], T' > 2T + I . (11.48)

59
In both cases, solution is for 0/(T',T^). But to check number conservation,

it is more useful to take = 0.

The integrands of eqs. (7) and (8) become very large for T' T. In

principle this suggests an application of Gauss quadrature. But we prefer

here simply to add and subtract integral quantities with two properties:

a) the modified integrands tend to vanish as T' -> T; and b) the subtracted

integral term can be accurately and readily evaluated. Our choice of supplementary term for evaluation of the energy conservation equation is the

following identity:

Correspondingly , we use the following term in the number conservation

equation.

(3T+I)/2

dT- yd,!^) T[K(T) -.X(T,T'-T)] = T y(T,T^) S(T) . (11.49)

T

(3T+I)/2

dT' y(T,T^) [K(T) -jf(T,T'-T)] = y(T,T^) S(T) . (11.50)

T

Equations (7) and (8) then take the forms

T

'' I0

T y{T,iy S(T) = Tg dT' y(T',Ty k(T') S(T',T) - , (11.51)

T

y(T,Tjj) S{T) = 1 +1 dT' y(T',Tg) k(T') Kj(T',T)

2T+I

2T+I

(11.52)

60
If one combines terms in braces into single integrals, one finds that the

integrands then approach zero in the limit of T' T. One remaining question

pertains to whether the approach to zero may not be so slow that the contribution of the final interval is, say, underestimated; this can in principle be a

significant problem because of an often sizable contribution due to this

interval. To keep track of this, an extrapolation of the integrand to a

finite value, rather than zero, has been made, with an associated estimate of

the corresponding modification to the integrated result. On the basis of this

estimate, the problem did not appear to be serious.

The above addition-subtraction is not carried out below the lowest

electronic transition energy, taken to be THR = 11.25 eV. In that low

energy region, evaluation of the integrands is carried out directly.

The outline of the computation is as follows: First, a list of integration

weights is computed, using the Stieltjes integration method described in

section II. 11. This method was not essential; but the intervals vary

in size, and this is a suitable way to take account of these variations.

Next, the numerical integration sum is evaluated; and evaluations of the

addition-subtraction and the source term follow. These three terms, their

sum, and the extrapolation fractional increment used to check the additionsubtraction procedure are recorded for possible output. A sixth quantity is

also evaluated, namely the number of electrons crossing the specified energy

cut-off T|,^.

In addition, three other quantities are routinely evaluated. One of

these is the value of 1/W(T), number of ions per eV generated by

y(T'), r > T. The second is the total yield of excitations. The third is

the factor by which the solution niust be modified to bring about

accurate energy conservation. These last three quantities are stored separately

from the other 6 quantities.

61
18. ELSPEL

Output of ELSPEL constitute the main results of the computation,

including stopping powers and cross sections, the value for W, the yield of

excitations of all types, the slowing-down spectrum and its components, and

the results of the energy-and number-conservation tests.

Input quantities are as follows:

NEX: the number of source energies,

IX: the number of solution lists. IX = 1 here corresponds to

the slowing-down spectrum alone. IX = 2 calls for a

calculation also of the primaries as a separate component.

IX = 3 calls for the 1 st-generation secondaries as well, etc.

SORS(I): the list of source energies, in eV.

BCOR(I): see BMOD

NO, N1 : see TLIST.

B: the value of b .

In addition, NPH determines the number of solution energies and their

separation, as discussed in TLIST.

The main DO-loop permits the computations to be repeated for different

NPH values (see also TLIST), to check effects due to size of integration

mesh. In this DO-loop the solution energies are first fixed, by calling

TLIST. Then CXPRT is called to record various stopping powers and cross

sections. Next HARDP performs the preliminary computations of necessary

energy-dependent parameters. SOLVE is then called to compute the slowingdown spectrum; and this computation is followed by print-out of results.

62
together with related quantities including the integrand for computation of

(1/W), Finally, ECHEK performs the energy- and number-conservation tests,

and values for W and any for excitation yields.

The whole computation can then be repeated at other value of

NEX >1, or possibly with a different set of integration points.

Other special cross section routines are sometimes inserted in ELSPEL.

63
4

Table 1

Table 2

Table 3

Table 4

Table 5

Table 6

Table Captions

Integration energies for a calculation with T^ = 1 keV.

Values for energy and number conservation integrals above

different T, expressed as fractions of the source energy T^,

for T^ = 30 keV. For the column on the far right, integration was carried out only over the primary component.

Ratios y (T,T^)/y (T,l MeV), for various T,T^.

Experimental and theoretical values for W, FE, F, and DS

refer to Fowler equation, (Fano) energy balance, and degradation

spectrum methods of calculation [55].

Volts per ion pair (W) as computed for different source electron

kinetic energies.

Some comparisons with data from reference [15].

64
Table

1.

Integration

energies

for

a

calculation

with

T

=1

keV.

cn cn <m m m m cn m cn m cn cn cn cn m cn m cn cTi'^o^'^cTi

oo^CTicoco t^r^ijDyDLn LD'^-^nco c^JC^J

o O r—C\J OO LD MD CO o^Oi— c\joo '^LDcDr^

I— CNJOvJCMCM CMCMCNJCNJ

O) CM CO CO

OO CM CM CM

cn uo CO '3- OO

OO uo cTi CO I— CTl CTi OO

1— >3- CO OO cocot^cr.':^-

CO O '3- CO OO (—0^0 00 1^

CO CM CO OO CO CO CO CO

CM CM CM CM CM

OO CM r— O CM CM CM CM CM O CTi CO

CM I— I— 1— I

COCOLO'^'^ OOOOCMCM

c

CM OO '3' CO CO

CO CO CO CO CO

CO C3^ O I

CO CO CO CO CO

CM OO 3- CO CO

(j^ <ys cx^ <Ti <y\ CO CO o I—

^

(0> CO CO o o

CM CO 3- CO CO o o o o o

O) 3- CO CM

3- CO CM CO CM O O I— 3-0

3- OO CM I— I

OO

OO I

CM CM CM O CO

CO CO CO CO

o o r-^ I

CO O CO CO I

CO OO I— CO CO

CO CO CO 3- 3-

00 I— CO CO CM

00 CO CO 00 CO CO 3 CM I

3 3 3 3 CO

3 CM CO CO CO

00 CO CO CM (O

CO CO CO 3 CM

CO CO CO 00 CO

CO CO r~. CO o^

CO CO CO CO CO O I— CM CO 3 CO CO CO CO CO

CO CO r-~ CO cr» CO CO CO CO CO O I— CM CO 3 r

CO CO CO CTi

I—^ r-^

cu 3 CO CO CO CM

cr> CM CO 3 CM CO CO ICO CO CO CTl CO

CO r— O O CO I— cr> cci CTi CO o o CTi I— CO O CM CO

I— CM 3 CO CO CO CO CO CO CO

CO o^ I— CM 3

(O CO 3 3

CO CO I— CO 3 3 3 cO CO

CO CO O CM CM

LO CO CO CO CM CM CO CM CO

CO o 3 I— O I r— CM CO

c

CO c^ o I— CM

CM CM CO CO CO

CO 3 CO CO

CO (O CO CO CO

CO O I— CM

CO CO 3 3 3

CO 3 CO CO 3 3 3 3

CO cn o r— CM 3 3 CO CO CO

cu CO CO O CD 3 CO CO CO CM

I— 3 D CO

CO CO CO 3 O

CO CO D D I

CO CO D CO 3

CO CO 3 CO CO

I— D CO 3

CO D CO CO

CO CM CM CM CO CMCMCO C033COCO CO CO D O O f— CM CO

CM CM CM CM CcJ 3 CO vO CO

CM CM CM CM CM

CM CO 3 CO CO CO D O CM (O 3 CO CO CO D O I— CM CO 3 cO

CM CM CM CM CM

COCM CM

65

29.42

53

404.4

80

31.74

107

11.43

30.56

54

500

81

30.56

108

10.93
c:o

-M QJ 03 CO CsJ oo (XJ 03 1 f“* t r-“ LO CO c JO > 03 03 03 o 03 CO o CO

cu E s- 03 03 (03 o (03 (03 (03 (03 (03 (03 o CO e O OJ o cr CO r—

CL c E o

hO C_J u

-O >3 c

ro fOE

•r“ o i- 1— CL co

OJ •r“ cr sz >,-l->

OJ 4-> cn fx3 f— CO r-v CM CO LO CO CM (03 CO CM

OJ s- > CO CO (03 o r*" r— r— r— CM CM CM r— 3 !_ OJ s_ 03 03 03 o o o o o O O o o -l-> > a sc a

Ol OJ > LlJ CO r— r** r— 1— f— r— f—

jO o co

CO o >1 CJ

fT3 oo cz

i. o

03 II <D -M

+-> o o C h- o CO CO CO > CD CD (03 CO 03

T3 QJ c O QJ *> LO oo

O CO CM CD CO C03 r— o M- •f— 1— CM LO o CO CM

•r^ S_ 03 CO CO CO

4-> t. CO CO

t13 >

S_

O)

00

c:oa

s_QJ

0) H-O

O

COco

00

O)s_CLX

O)

rOU

CO03

O) -M 4-> c

JO (J ro o E rO i- •r- D i. <03 s_ +j C (+- OJ OJ 03 r— CM CD CM o CO CM

JO > CO CO CO (03 o (03 (03 03

•O LO

C s_ (03 03 (03 <03 o 03 (03 (03 c 03 •r— o O) • • (T3 (/) a #3 c

>3 QJ 4-> o

LO -C (CJ

i.

03

LO 0) c

<D JZ, o

Z3 4-) •r— (— >>+->

03 SZ (03 (T3 r—

CM ID CM o CM (03 > o s- > CO CO CO (03 o (03 CO CO

QJ S- <03 CT> (03 (03 o (03 <03 cn

sz

C QJ • • • • • • • . • £ LlJ (/3 CM 3 Co

OJ

O (D

i.O

O)I o

I—

CO

m

C\J

CM

=dO

CD

CM

LO CO LO

CO uo o f—

CO LO LO

Ovi oo

66
Table 3 . Ratios y(T,T^ )/y(T,l MeV), for various T’ToTo,keV

T = 9.9 12.3 17.3 22.6

.10 1.376 1.470 1 .813 2.197

.30 1.132 1.163 1.279 1 .401

.50 1.112 1.134 1.211 1 .288

1.0 1.0775 1 .0922 1 .144 1.190

3.0 1.0564 1.0653 1 .0965 1 .122

5.0 1.0490 1.0564 1 .0824 1 .103

10. 1.0394 1.0453 1.0658 1.0822

30. 1.0269 1.0308 1 . 0448 1 .0558

100 1 .0159 1 .0181 1.0265 1 .0304

300 1 . 0080 1.0090 1 .0133 1 .0166

67
Table

4.

Experimental

and

theoretical

values

for W

(see

also

Whyte

[56]

and

Booz

and

Ebert

[21]).

FE,

F,

and

DS

refer

to

Fowler

equation,

(Fano)

energy

balance,

and

degradation

spectrum

methods

of

calculation

[55]. >OJ UJ 1

U- '

LlJ UJ

LL- LlJ LL- p' UJ UJ

(/) Ll- Ll. oo U- LU C <' Q o ( 1 •r“ 1—

1—

1

\ I 1 4-> LO CO CM CM i 1 CO •=d* ro 1 1 l_l 1 1 LO CM LO LO 1 1 1 I i 1 1 1 3 CNJ CM CO U r— LO O CO CTi CO

r—

• • • • • • • ro o CO LO LO LO LO O CO CO CO CO CO CO

-X I—I I—

1—

LO MD 1 1 ( 1 1 i 1 1 rr> 1 1 CO

*N

V 1—

CO

N. CO CO 1—*> 1—

LO CO 3 CO CT> o

ro >1 03 o 1 1 LO JC 1 J 03 s_ 4-) l_l +J to s_ o -—

• r- OJ X ^ a s- CO >—*. S s CO S CL >> CO CO CO u >1 o 03 >5 >3 >3

CO n3 3 Q- > s_ 03 03 03

-t-> +-> i. O- OJ s- S- !-

c: i. • a

OJ rO 3 LO

a 3 3

!= Q_ r— o 3 o • • Q- o o o

S_ >1 Q_ CO '—a. a. Q_

OJ > + 1 + 1 '>

—

' CL fO r— X OJ oo o o CM o r-“ CO

O

LU zi: lO r*^ XO 'xo LO

CO CO CO CO CO CO CO CO 1 1 1—1 1—1 03 CO CO CO

CO 1 1 CO L-^ o c 1 1 o r-n >s • t s_ LO CO 00

4-> CO >3 CO >3 •X u 1 1 r> 03 >3 03 CO p— 1 J 1

1 • 0) LO XO 03 s- e CXJ s_ CO CO 1 o 1 i <c

LU 1 1 1 1 X ca s~

+-> «s s_

_!3 -—s. -—

> > o CO -—>» o

+-> CO CO cu o O)

a >3 CO CO H-

•p“ >3 >3 s; CJ r— 03 >3 >3 s 03 03 o <D s- 03 03 CM

s_ i_ CM XO S_ s_ • CO -—i-' 

> ca XO

+J CQ. ca I OJ ca ca CXJ e CO 21 <c

O)

d or * zc JZ II E- CO CO + 1 CM CO CO oo

•r“ '-W' + 3 s_a CO 03 CO o o CO CM o o C

Q. • • o X UD LO xo r-- CO CO

LU CO CO CO CO CO CO CO CO CO -a

O)

(/) 03 CQX

68
Table 5. Volts per ion pair (W) as computed for

different source electron kinetic energies

T, keV w

.024 69.4

.027 73.2

.030 67.4

.035 56.5

.05 50.7

.07 47.1

.1 44.2

.3 39.27

.5 37.44

1

36.24

2. 36.03

3. 36.05

5. 36.08

10. 36.20

30. 36.40

100. 36.59

300. 36.75

1000. 36.90

69
Table 6. Some comparisons with data from reference [1].

T(eV) y (cm/eV) Okazaki -Sato

10 ,26 X 10

-3

.46 X 10

'

10

^ 23 X 10

‘ .43 X 10

'

10

^ 80 X 10

-3

1.22 X 10

‘
Table Captions - Appendix I

Table I.

Table I.

Table I.

Table I.

Table I.

Table I.

Table I

1 Oscillator strengths for discrete excitations [3], based on Allison

and Dalgarno [59] and Namioka [29].

2 Tabulations of moments of the OSD, as defined in eqs (II. 1) and

(II. 2).

, ""i 3 A tabulation of low energy correction functions for ionization

(£n b^/b) using b^ = 3, and excitation (£n 1/b.j).

4 Comparisons of total ionization and excitation cross sections for

electrons with kinetic energies below 1000 eV. See references

[3,33]. (The number of digits given is for comparisons; neither

type of cross section is known to any such accuracy.)

5 Vibrational cross sections for the two lowest excitations, used in

conjunction with eqs (8) and (9) and due to Trajmar, et al [64].

6 Values of <I), the mean excitation potential for molecular hydrogen.

Table Captions - Appendix II

I.l Description of ELSPEL subprograms. Abbreviations are as follows:

c.s. —cross section, c . c . s

. --combined cross section, sc.int--

scattering integral, eval . --evaluation

%

71
Table I.l Oscillator Strengths for discrete excitations [3],

based on Allison and Dalgarno [59], and Namioka [29].

E„. eV

Lyman

f

n

E , eV

n

Werner

f

n

11.213 .001709 12.317 .05246

11.377 .005859 12.605 .08027

11 .535 .01170 12.875 .07694

11.674 .01776 13.129 .06030

11.846 .02277 13.367 .04269

1 1 . 982 .02602 13.589 .02863

12.132 .02736 13.794 .01873

12.268 .02705 13.984 .01211

12.405 .02553 14.156 .007822

12.524 .02326 14.319 .005060

12.679 .02059 14.446 .003280

12.790 .01785 14.560 .002104

12.902 .01522 14.650 .001290

13.026 .01281 14.710 .000616

13.139 .01068

13.238 .008835

13.305 .007271

13.450 .005962

13.549 . 004878

13.648 .003986

13.735 .003258

13.822 .002664

13.909 .002180

13.996 .001787

14.071 .001467

14.150 .001207

14.220 .0009933

Total .3107 Total .J923

72
o C\J LD (03 Csj CM C\J o o CM o UO CJO O CO LD CO CO ID UD 03 o r—

o O oo o O CO CO CM LO LD cn r— o O o oo o O O o CM CM CM CM CM (TcJ CM o o CM

-o o CO CcJ CM CO o CO CM o CM o LO

c: e m o COa o CCJ to n—

(03 •1~ £ o LO CO CO

rO S

CM o o cn (03 CO ID ro CO CM s o CO r*^ to o o CD o C\J C\J CcJ CNJ CM CM CM o o f— CO -—^ • • •

H—

'

cncr

OJ o o o r— LO CO CM CO CO CM o 03 LO c o CO to r— r— o ro r—

m CO OO CM o o (—

CO

»r“ f— o o CO o o CD 03 (Ol LO LO LO o CO (03 o o o r— CM (XI CM CM CM CM CM CM o o r—

CM

a (/) (/) OJ c c c o o

•r“ 'r* 4- +J 4-) d) fO 03 TO 4-> IM

tn

u c

03

X o

UJ 1—4

r* o CM CM ro CO o o LO CO r—> CO o CO CO CM CO o CO LO LO CO (03 CO o LO 00 CO o f— CM LO CO (03 CO LO CO o r— LO o o o o CM LO CO CO CO CO CO o o o <—

OJ +J4-O

(/) +Jc

cu o ro CO to CM CO r— r— CO r— CvJ LO o o cn e m o 03 LO r— LO o (03 CD LO CO •r“ — o cn CO o CM o cn CO o CO r-- <03 CD o f— CO CM E o o CD CM CO CO CO CO CO (03 o o CD 1

o

c/1 eo

+j

033 o (03 CM CD LO (03 o CM CO CM OO o m CO <y^ JD o CM (03 r—

<03 CO CO CO CD LO CO CO o CO

03 r~ o (XJ (03 o CM LO o CM CM CD f— LO CD

1— o o CD (XJ LO CO CO <03 <03 (03 CD o o o o o o o o o o o o o O o o o o

CO LO LO CO o CO CO r— <03 o CO r— <03 o

(03 r— (03 CO '=d“ r-s. r— CD CO o CD LO o

S_d)

SO

CM COO LO CD LO o CO '— CO o '— CO o

r— CM CM CO CO LO LO CD LO LO CO LU 1 f— 1— r— r— 1— r* 1— 1 * 1— ' '— '

73
Table .3. A tabulation of low energy correction functions for ionization

(An b^/b) using b =3, and excitation (£n 1/b^ )

.

T, eV

£n (b /b) ' oo'

' £n (1/b-j )

1414 .00 .000

1000 .13 .000

707 . .25 .000

500 .40 .000

353.6 .62 .000

250 .90 .010

176.8 1.23 .175

125 1.54 .408

88.4 1.78 .614

62.5 1.93 .819

44.2 1.99 1.037

31.25 2.02 1.185

22.1 1.92 1 .210

18.58 1.79

15.625 2.40 1 .100

14.00 .98

11.05 3.20 .81

74
Table

1.4.

Comparisons

of

total

ionization

and

excitation

cross

sections

for

electrons

with

kinetic

energies

below

1000

eV.

See

refs.

[3,33].

(The

number

of

digits

given

is

for

comparisons;

neither

type

of

cross

section

is

known

to

any

such

accuracy.)

OO LO LO CO CD LO o CM CTi CO CM

•r-» O "O- C\J C\J LO PO CM CO CTi a^ OO LO

CM .C 4- • • 1— C|_ LO LO o r—

CM CM CM CM CM o CM U Lul 1— I

'

<

r— r— o

exc +j

i. r-~ CO o OO CD <Ti CO LO

<o cn OO CM CD CM CO O r— CO CO o m -C • Q LO LO O 1 CM CM OO cn CM o CM

O) r— ' 1 ' ' 1 r—

c\j o o o o o o o o o o o o O LO r—

CO

OO

CM

OO

=d" CM

CD CM o CO CO LO cn CM r— r—

CO LO 1— o LO cn cn CO OO CM CO CO CO

•r* O CO CO LO CM CM CO CO o o CO CO LO o: q- CM CO cn cm OO o o I— f— o cn CO LO CM H- 4-

LU f— r—

CM o

CO

CT! c

1 CO CL 1 CO CM CO CM cn CM o o P*-. cn Q. S- CO LO CM CM CO LO CO o o CO CO LO LO CO

rO O) CM CD <n cn o o r— r— o cn CO LO CM r— o

q; -a

I, , r— ,

LO o o o o O O LO o LO o o o o 'd" =:j- cn CO CD o oLO oOO

LOCMLO CM o CO CO LO ro CM r75

073
Table 1.5, Vibrational cross sections for the two lowest excitations,

used in conjunction wi h eqs. (8) and (9) and due to Trajmar, et al [64],

Ta

4rra^R

0

T, eV 0.54 eV 1.08 eV

1.3 0 0

1.6 .0081 .000110

2.0 .0162 .000560

2.5 .0251 .00147

3.0 .0321 .00220

3.5 .0374 .00276

4.0 .0389 .00324

5.0 .0383 .00360

6.0 .0357 .00382

7.0 .0329 .00391

8.0 .0254 .00391

9.0 .0229 .00320

10.0 .0181 .00205

14.0 .00819 .000737

20.0 .00585 .000199

76
Table

1.6

Values

of

<I),

the

mean

excitation

potential

for

molecular

hydrogen.

00

O) 1—

1“^ (—1 |—

CTi o C\J CO CO CO E i_^ L_1 1 1 •r*

(/) LO

LU • + 1 S-. 03 O^J CO o CXJ E • • • £ CO CO CO 3 f^— r—

oo

oo

OJ

I 1 I—

oo ro>

r~i 1“^ 1 1 r-n r\ 00 LO CO 1—1 CO s_ c <r<-) 3

o 1 1 o •r-

+-) 03 LO CM —V. r— • • CO CL. C3a + 1 *4- 1 r— CO CM

03 LO OO CM C\J CM CM CO o • OO CO CO CTi CTO CTl OO CO

t“~ r— r-“ f— r— r— r—

00 4->

o;

s_

<DQ.

x; LU

I 1

1—r

<30 CO

A COCD

1 1

r—! 1 1 ro

CO o LO OJ

CO CO 1—J 1 i

’—1 U—

CO CO

-K <y\

C\J + 1 + 1 + 1

(T) CTl CO

CO o CO r-I

( f“~ CM 1— CM

c

O)CDo

i. a>5

-o

13cr

77
Table II. 1. Description of ELSPEL subprograms. Abbreviations are as follows: c.s. --cross section, c . c . s

. --combi ned cross

section, sc. int. --scattering integral, eval . --eval uation

SUBPROGRAM TYPE DEPENDENCE PURPOSE OR CONTEXT

OSCUM c.s. VIB c.s. BMOD c.s. DTU c.s. BMOD

TCX c.c.s. all c.s. TCXI c.c.s. all c.s. STN c.c.s. all c.s. ONCE c.c.s. all c.s.

HARDP sc. int. DTU

CUMK2 sc . int. all c. s.

,

HARDP, TCX

CUMK3 sc. int. all c.s. CORRX sc. int.

WATES sc .int. all other

sc. int.

TLIST eval

.

ECHEK eval all c.c.s. SOLVE eval

.

WATES, ONCE,

ECHEK

CXPRT eval

.

all c.c.s.

ELSPEL master al 1 eval

.

cumulative oscillator strength

vibrational cross sections

low energy glancing c.s. parameters

glancing c.s. limits

cumulative cross sections

cumulative ioniz. c.s. cumulative stopping power

differential cross sections

factors, parameters for integration

Gauss quadrature, final interval

cumulative integrations

correction for cut-off discontinuity

array of integration weights

prepare list of solution energies

evaluate energy, number conservation

evaluate slowing-down spectrum

evaluate total c.s., stopping powers

direct all computations, output

78
Figure Captions

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 1

Fig. 1

Differential cross sections for electrons of kinetic energy T as

a function of energy loss, T -T. Secondary electrons comprise that

part of the spectra below (T°-16)/2 eV.

Differential cross sections for secondary electrons of final energy

E due to 500 eV and 100 eV electrons in H . Solid curves are calculated from our cross section representations (R = 13.6 eV).

The 500 eV curve agrees closely with Gerhart's data from his Fig. 7,

for E

^ > .05. Circles are Opal, Beaty, and Peterson data [30]

divided by 1.65. The + values are unpublished data due to Dubois

and Rudd [31].

The stopping power for H^. .The dashed line gives results for the

Bethe stopping power formula using 19.22 eV for the mean excitation

potential. The circles give recent data due to Green [32].

Total cross section for inelastic interactions of fast electrons in

H^. Also given is the fraction due to ionizing interactions.

Relative contributions to the energy dissipated to kinetic energies

below 16 eV, as a function of the kinetic energy of the source electrons.

The Lewis effect for a 1 MeV source. S(T) is the stopping power.

Solution intervals in the Lewis effect region varied from ~ 0.4 eV

at (Tq-T) ~ 11 eV, to ~ 2 eV at (T -T) ~ 60 eV. The solution was modified to improve energy conservation in the region between the

arrows at the bottom.

Suitably scaled electron slowing-down spectra for source energies of

30 eV, 300 eV, 3 keV, and 30 keV. S(T) is the stopping power. For

the 30 keV source, primaries and 1st generation secondaries are indicated with dashed lines. The ratio k./S is also given (heavy

dashed line), as are several relevant reciprocal stopping numbers

(at the right). Below about 12 eV, data for the three higher

source energies are so close together that they are all represented

by a single curve.

Volts per ion pair (W) as a function of electron kinetic energy in

H,, . Comparison points are given for Combecher's experimental values

[57], and calculated values from references [2] and [3].

Volts per ion pair (W) in for electron kinetic energies greater

than 1 keV. The scale is enlarged so that small differences of

trend can be readily seen. At the right, a change of ]% is exhibited

The circles give values due to Garvey, Porter and Green [2], as read

from their curve. Green, Jackman, and Garvey (see [2]) give a few

numerical values which are in agreement to three significant figures.

Comparison of values for G = 100/W. The solid line represents our calculations. Circles give Combecher's experimental data [57].

Recent calculations due to Douthat are also shown [54].

1 The slowing-down spectrum of 10 keV electrons in H^, compared with

values due to Douthat [1,54] and due to Garvey, Porter, and Green [2]

79
=

lOkeV

80

O) M- ^£3 O r—I

+JU , != 2 3 O ; <+- r-0) ra J2

(/) (O fO i.

-(-1 : o

CXL) 1— a

c/1 >1 0) ! 01^,1

i. +J I <vc i OJ o ! O +J i •r- S. I +-> (0 O) a c < •I- -P i ^ ro 3 I 4- +J I O

o; I (/) C/l I C '

a; o !

O) c/l ^ O S. ! <+- +J I u ; u) ci; I C r- [ O <U ' 4J >1 u s- 0) (0 ! c/1 -O I C ; c/1 01

c/l u; O Ojj s- t/i! o

fO H-

•I- I -pC f- 0) Scl) <;

CD

•r“ U-
4TTao

Fig. 2 Differential cross sections for secondary electrons of final energy

E due to 500 eV and 100 eV electrons in . Solid curves are calculated from our cross section representations (R = 13.6 eV).

The 500 eV curve agrees closely with Gerhart's data from his Fig. 7,

for E~^ ^ .05. Circles are Opal, Beaty, and Peterson data [30]

divided by 1.65. The + values are unpublished data due to Dubois

and Rudd [31].

81
82

The

stopping

power

for H

.

The

dashed

line

gives

results

for

the

Bethe

stopping

power

formula

using

19.22

eV

for

the

mean

excitation

potential.

The

circles

give

recent

data

due

to

Green

[32].
CVJ

83

Total

cross

section

for

inelastic

interactions

of

fast

electrons

i

Also

given

is

the

fraction

due

to

ionizing

interactions.
CD

fO

>

(U

aaivdissia A9a3N3 Noiiovad

84

Fig.

5

Relative

contributions

to

the

energy

dissipated

to

kinetic

energies

below

16

eV,

as

function

of

the

kinetic

energy

of

the

source

electrons.
CfWTAN

CO

85 0) o

ITS I > cxu

S- h- JO o ' -M

4-) c O

•I— +J o

03 o o s o > 4-) •r- O o O Cvj JO

f— c o ? o

oo •t— o CD

4-1 o

s_ S- 'I o > o S O .o o

0.1—

4.J 1— c

CD •r* c

•r— c Q.^ o

0.1— *r“ O 1 4->

4-> 003

1/1 E” > > O o

OJ .c CO

4-> 4-> o

03 o

1/) o

•1- >o >3 1 oi 1 r— "Tf i- . O .. . o

h“ OO

O co 1 o O E > u o o

S- 5- <_ O 4- O £

CO "O •r— O > -I— o o s_ 21 03

4~) > -O

1— OJ C -I- 03 0 4-

i- cn-o O OJ o

4- S_ E

+->+-> 1/1 O O 03 (DOS 4- 4- 4- 4- C 0) 0)0

(/) </l +-> •>- -I- 13 0) 0)0 1 I 00 0 0)0 P ^ P

OO

CT)

arrows

at

the

bottom.
=

30eV

300

eV3

keV

30keV I I I i I I I I I I I I I I I I ! I L O T o Q g

CO

"O > c O) •o

fO 0) CM 4-1

rt o r—

C > c O) OJ O) o to

•r" (/) +-> O)

4-> 03 3 i- oo fO -o O CL

S- OJ O) >0 03 s- > sz >

O) O) 03 s r—

cn <D o r— o 03 o +-) 

p <v ro (/) CQ OJ f— £Z S9\ OJ 03 > T3 > • O) C r~ 5o

(X3 CTI +->

a o sz

oo (/) o cn +-> <U CO •I* H- •r“ s- 4-> O S_ fO 03

(T3 O) zz

(/) E CO JZ -M

O) •p— •r“ +-> •r“ s- S_ oi S2. -M O) s_ OO 03 z:

(D ^ 

+j

C <D •r“ QJ OJ O O) cn

S- S- o

O) 3 O) +-> o O o JZ

s_ OO *r“ E 0) 3 +J 3 CO o > 03 C O

1/) 0) S- I—

cn o

s- O) z o O o

4- OO 1— Q. CO Q.

(C OJO O)

i- JZ • 4-> s-

+-) 4-> tn to 03 u O)

O) S- c r— CO Q. O •P“ 03 0) O) U_ I— O T- o cn I sz -a s_ s_ ! 3 • (L) sz. q; O i- -Z •1- c 1 T3 O) CO a 0) 1 3 03 OJ cn O T3 Z OJ 1 c SZ. u

•I— z: +J z 3 cn +-> Z 3 O Z — 03 O 1 r- S > CO 1 CO CZ. su SZ.T3 r— Z 1 c O 0) (U cu 1 o +J -M Z JZ j S- CO 03 cn +J U I— 'r- i1 o 0) -r- 03 JZ 11 (D jz -a Z I -(-> z OJ 0) * ij O) •r" > 0) OJ CO O) z >

-o •1- O) CO JZ s- I O) s- +J z 1—

' 03 0) o

03 1

Z OJ ( CJ -— CO 03 JC 0) I, CO c/n OJ +J r— i' *r“ CO CD [ >> S- 03 Z C • 03 O I- |i s > "O »> C)_ CO 03 oi Z .—

j' 4-J .5^: o O) 03 03 *F“ o Z +J 1' 3 O O) •r- 03 >5 |l C/0 n CO r— -3 ^ i

CD

86
Fig. 8 Volts per ion pair (U) as a function of electron kinetic energy in

H . Comparison points are given for Combecher's experimental values

[57], and calculated values from references [2] and [3].
O) r— OJ

fO CD u s=

I/) ro "O

-C fC

QJ

O

x:u

oj x:2

fO

oo to

n fC 0) • 4-> 3 > x: " t—

QJ CT)I 1 fC T- C\J >

i_ I 1

O) c E x: 0)

fO +-> QJ ^ S- +J +-> o <s- *o

QJ E

+-> • fO

fO E

QJ QJ E E QJ QJ

C7) to +-> E

to >,

O

QJ I— Q_

OVt3 E fD

QJ QJ QJ E E >

QJ QJ 03 0X3 0

>, cn E I—I

CM >

QJ

-M E

QJ (T3 E O

-oE E QJ O E E -M

4->O t|- Q) O

QJ to

QJ

O 0)QJ

I/) QJ>_- E EO

4-

OE

QJE

QJ

cvA+- 3T 4-

•f— E -a ^ 03 S E '— to

+-> E 03

•r- x:

03 +-> Q- o E lO O

•r- -O

QJ E cn

QJ E

Q. 03

to E

+j OJ O tn

cn

cn

r— o

03> -oc

QJ fO > c

03 to £

QJ

X

rU O 03 E E

•r—O

XC

QJ 0) x: QJ 1— s- C5

to 0) ! • r— X+Jj o :

4- S- O 4- I

88
iT)

>

A3 001 /SdlVd NOI ‘ 9

89

Fig.

10

Col.iparison

of

values

for

G

=

100/W.

The

solid

line

represents

our

calculations.

Circles

give

Combecher's

experimental

data

[57].

Recent

calculations

due

to

Douthat

are

also

shown

[54].
,

cm/eV

Fig. 11 The si owi ng-down spectrum of 10 keV electrons in H , compared with

values due to Douthat [1,54] and due to Garvey, Porter, and Green [2].

90
Figure Captions - Appendix I

Fig. I.l The ionization efficiency function as reproduced from [3]. The

solid curve is due to Berkowitz [61]. The dashed line is a smoothed

form due to Gerhart. The dash-dot line is due to eg (1.3).

Fig. 1.2 Gerhart's $(E) (heavy curve) is reproduced from his Fig. 6 [3].

The light curve expresses a unit function with cut-off at 17.5 eV.

91
><v LU

CD

92

The

ionization

efficiency

function

as

reproduced

from

[3].

The

solid

curve

is

due

to

Berkowitz

[61].

The

dashed

line

is

a

smoothed

form

due

to

Gerhart.

The

dash-dot

line

is
>

QJ

UJ

{3)i>

cn

93

Gerhart's

4>(E)

(heavy

curve)

is

reproduced

from

his

Fig.

6

[3].

The

light

curve

expresses

a

unit

function

with

cut-off

at

17.5

eV.
REFERENCES

[1] D. A. Douthat, Electron energy degradation in molecular hydrogen.

Radiological and Environmental Research Division Annual Report, ANL-76-88

170-176 (1976) Argonne National Laboratory, Argonne, Illinois. See also

[54].

[2] R. H. Garvey, H. S. Porter, and A. E. S. Green, Relativistic yield

spectra for H

2

. J. Appl . Phys. 48, 4353-4359 (1977). See also

A. E. S. Green, C. H. Jackman, aM R. H. Garvey, Electron Inpact on Atmospheric Gases. J. Geophys. Res. 5104-5111 (Nov. 1 , 1977).

[3] Donald E. Gerhart, Comprehensive optical and collision data for

radiation action. I. H 2

. J. Chem. Phys. 6^, 821-832 (1975).

[4] H. W. Lewis, Straggling effects on resonant yields. Phys. Rev. 125,

937-940 (1962).

[5] L. V. Spencer and U. Fano, Energy spectrum resulting from electron

slowing down. Phys. Rev. 1172-1181 (1 954).

[6] Rosemary T. McGinnies, Energy spectrum resulting from electron slowing

down. Nat. Bur. Stand., Circ. 597, 16 p. (1959).

[71 Daryl A. Douthat, Electron degradation spectra in Helium. Radiat. Res.

61, 1-20 (1975).

[8] Li. Fano and L. V. Spencer, Quasi -seal ing of electron degradation

spectra. Int. J. Radiat. Phys. Chem. 7_, 63-76 (1975).

[9] S. C. Soong, Inner-shell contributions to electron degradation spectra.

Radiat. Res. 67, 187-214 (1976).

[10] D. A. Douthat, Energy deposition by electrons and degradation spectra.

Radiat. Res. 141-152 (1975).

[11] N. Bohr, On the decrease of velocity of swiftly moving electrified

particles in passing through matter. Philos. Mag. and J. Sci . 30,

581-612 (1915).

[12] L. V. Spencer, and F. H. Attix, A theory of cavity ionization. Radiat.

Res. 3, 239-254 (1955).

[13] P. R. J. Burch, Cavity ion chamber theory. Radiat. Res. 361-378

(1955).

[14] Shin Sato, Kiyoshi Okazaki, and Shin-ichi Ohno, The estimation of the

G-values for the ionization and excitation of noble gases irradiated by

100 keV electrons. Bull. Chem. Soc. Jap. 2174-2180 (1974).

[15] Kiyoshi Okazaki and Shin Sato, Non-empirical calculation of the G-values

for the ionization and excitations in diatomic molecules irradiated by

100 keV electrons. Bull. Chem. Soc. Jap. 3523-3528 (1 975).

[16] Cornelius E. Klots and Harvel Wright, Aspects of degradation spectra.

Int. J. of Rad. Phys. and Chem. 191-200 (1970).

94
[17] Martin J. Berger, Monte Carlo calculation of the penetration and

diffusion of fast charged particles. Methods in Computational Physics , Academic Press, Inc., NY, 1 35-21 5 (1 963).

[18] D. 0. Schneider and D. V. Cormack, Monte Carlo calculations of

electron energy loss. Radiat. Res. 1J_, 418-429 (1959).

[19] Martin J. Berger, Spectra from energy degradation of fast electrons in

water. Nat. Bur. Stand. Interim Report NBSIR 77-1283, 16 pages (1977).

[20] M. Jura, Ph.D. thesis (Harvard University, 1971), unpublished.

[21] A. Dalgarno and G. Lejeune, The absorption of electrons in atomic oxygen.

Planet. Space Sci. J_9, 1 653-1667 (1 971 ).

[22] T. E. Cravens, G. A. Victor, and A. Dalgarno, The absorption of

energetic electrons bv molecular hydrogen gas. Plant. Space Sci.

1059-1070 (1 975).

[23] Lennart R. Peterson, Discrete deposition of energy by electrons in gases.

Phys. Rev. 1^, 105-11 1 (1969).

[24] R. H. Garvey and A.E.S. Green, Energy-apportionment techniques based

upon detailed atomic cross sections. Phys. Rev. 1 4A , 946-953 (1976).

[25] R. H. Fowler, Contributions to the theory of the motion of a-particles

through matter. Part II, Ionizations. Proc. Cambridge Philos. Society

531 -541 (1923).

[26] L. R. Peterson and A.E.S. Green, The relation between ionization yields,

cross sections and loss functions. J. Phys. B (Proc. Phys. Soc.) 1_, Ser. 2, 1131-1140 (1968).

[27] Mitio Inokuti, D. A. Douthat, and A.R.P. Rau, Degradation spectra and

ionization yields of electrons in gases. Proc. 5th Symp. Microdosimetry

,

Pallange, 977-1006 (22-26 September 1975).

[28] A.R.P. Rau, Ionization yields in gases on irradiation with charged

particles: A variational approach. Annual Contract Report to NSF,

Louisiana State University, Baton Rouge, LA, 23 pages (1976).

[29] Takeshi Namioka, Absorption spectra of H 2 in the vacuum-ultraviolet

region. III. Potential -energy curves for the +, C^n , B‘ +,

and D^n^ states. J. Chem. Phys. 1636-1644 (1965).

[30] C. B. Opal, E. C. Beaty, and W. K. Peterson, Tables of secondaryelectron-production cross sections. Atomic Data 209-253 (1972).

[31] R. D. DuBois and M. E. Rudd, Absolute doubly differential cross sections for ejection of secondary electrons from gases by electron

impact. II. 100-500-eV electrons on neon, argon, molecular hydrogen,

and molecular nitrogen. Phys. Rev. 1 7A , 843-848 (1978).

95
[32] A.E.S. Green, Jan. 23, 1978 letter, unpublished.

[33] Donald Rapp and Paula Englander-Golden, Total cross sections for ionization and attachment in gases by electron impact. I. Positive ionization.

J. Chem. Phys. 1464-1479 (1965).

[34] Martin J. Berger, Spectra from energy degradation of fast electrons in

water. NBSIR 77-1283, 15 pages (August 1977), NBS Interagency Report

(unpubl ished)

.

M. J. Berger and S. M. Seltzer, Calculation of energy and charge deposition

and of the electron flux in a water medium bombarded with 20-MeV electrons.

Annals of the New York Academy of Sciences , 1 61 , 8-23 (1969).

[35] William P. Jesse and John Sadaukis, Absolute energy to produce an ion

pair by beta particles from S^^. Phys. Rev. 1 07 , 766-771 (1957).

Ionization in pure gases and the average energy to make an ion pair for

alpha and beta particles. Phys. Rev. 1668-1670 (1955).

[36] G. G. Meisels, Gas-phase dosimetry by use of ionization measurements.

J. Chem. Phys. ^]_, 51 -56 (1964).

[37] Jerome Weiss and William Bernstein, Energy required to produce one ion

pair in several noble gases. Phys. Rev. 1 03 , 1253 (1956).

[38] Roger M. Leblanc and Jan A. Herman, Determination des valeurs Wo pour

les gaz et vapeurs organiques par la methode des courants de saturation.

J. Chim. Phys. 6^, 1055-1060 (1966).

[39] J. F. Lehmann, The absorption of slow cathode rays in various gases.

Proc. Roy. Soc. A115 , 624-639 (1927).

[40] W. C. Barber, Specific ionization by high-energy electrons. Phys. Rev.

1071-1077 (1955).

[41] J. M. Valentine, Energy per ion pair for electrons in gases and gas

mixtures. Proc. Royal Society of London, Series A. Mathematical and

Physical Sciences 211, 75-85 (Cambridge University Press, Bentley House,

N.W. 1 March, 195217“

[42] Leon M. Dorfman, Absorption of tritium beta particles in hydrogen and

other gases. Phys. Rev. 393-396 (1954).

[43] William P. Jesse and John Sadauskis, Alpha-particle ionization in pure

gases and the average energy to make an ion pair. Phys. Rev. 1120-1121

(1 953).

[44] T. E. Bortner and G. S. Hurst, Ionization of pure gases and mixtures of

gases by 5-MeV alpha particles. Phys. Rev. 1236-1241 (1954).

[45] C. Biber, P. Huber, and A. Muller, Arbeit pro lonenpaar von mehratomigen

gasen fur Po-a-Teilchen. Helvetica Physica Acta 503-521 (1955).

[46] C. J. Bakker and E. Segr^, Stopping power and energy loss for ion pair

production for 340-MeV protons. Phys. Rev. 8j_, 489-492 (1951 ).

96
[47] Cornelius E. Klots, Energy-partition parameters in radiation studies of

molecular gases. J. Chem. Phys. 44, 2715-2718 (1966).

[48] Georg Stetter, Die ionisation einzelner al pha-strahl en in verschiedenen

gasen. Z. Physik (H. Geiger, ed.) 1^, 639-651 (1943).

[49] W. Riezler and A. Rudloff, Ionisation und Energieverl ust von alphateilchen in verschiedenen gasen. Annalen der Physik. 1_5, 224-245 (1 955).

[50] J. M. Valentine and S. C. Curran, Energy Expenditure per ion pair for

electrons and a-particles. Philosophical Magazine 7th Series,

964-967 (Prof. M. F. Mott, ed., 1952).

[51] W. M. Jones, Some calculated quantities in the radiation chemistry of

molecular hydrogen: Average energy per ion pair and numbers of singlet

and triplet excitations per ion pair. J. Chem. Phys. 5688-5695 (1973).

[52] G. D. Alkhazov, Ionization fluctuations in gases. Soviet Physics-Technical

Physics 16, 1540-1545 (1972).

[53] A. Dalgarno and G. W. Griffing, Energy per ion pair for electron and

proton beams in atomic hydrogen. Proc. Roy. Soc. A248, 415-428 (1958).

[54] D. A. Douthat, Energy deposition and electron energy degradation in

molecular hydrogen. Submitted to J. Phys. B.

[55] R. L. Platzman, Total ionization in gases by high-energy particles: An

appraisal of our understanding. Int. J. Appl . Radiation and Isotopes,

1_0, 1 16-127 (1 961 ).

[56] G. N. Whyte, Energy per ion pair for charged particles in gases.

Radiat. Res. 1_8, 265-271 (1963).

[57] D. Combecher, private communication to M. Inokuti (Feb. 20, 1978).

[58] William Frederick Miller, A theoretical study of excitation and

ionization by electrons in helium and of the mean energy per ion pair.

Ph.D. Thesis, Purdue U., 166 pages (1956).

[59] A. C. Allison and A. Dalgarno, Band oscillator strengths and transition

probabilities for the Lyman and Werner systems of H 2 , HD, and D 2 • Atomic Data 1_, 289-304 (1970).

[60] G. R. Cook and P. H. Metzger, Photoionization and absorption cross sections of H 2 and D 2 in the vacuum ultraviolet region. J. Opt. Soc.

Am. M, 968-972 (1964).

[61] J. Berkowitz, unpublished.

J. Berkowitz and W. A. Chupka, Photoelectron spectroscopy of autoionization peaks. J. Chem. Phys. 2341 -2354 (1969).

W. A. Chupka and J. Berkowitz, High-resolution photoionization study of

the H 2 molecule near threshold. J. Chem. Phys. 4244-4268 (1969).

97
[62] Mitio Inokuti, Inelastic collisions of fast charged particles with atoms

and mol ecul es--The Bethe Theory revisited. Rev. Mod. Rhys. 297-347

(1971 ).

[63] H. Ehrhardt, L. Langhans, F. Linder, and H. S. Taylor, Resonance scattering

of slow electrons from H 2 and CO angular distributions. Rhys. Rev., 173-

222-230 (1968).

[64] S. Trajmar, D. G. Truhlar, J. K. Rice, and A. Kuppermann, Electron

scattering by H 2 with and without vibrational excitation. III. Experimental and theoretical study of inelastic scattering. J. Chem. Rhys.

4516-4533 (1970).

[65] H. Bethe, Quantenmechani k der Ein- and zwei-electronenprobleme. Handbuch

der Rhysik , 24 , Kapitel 3 (H. Geiger and Karl Scheel , Eds.) Miinchen,

Germany, 18-560 (see p. 519 ff) (1933).

[66] M. Bogaardt et B. Koudijs, Sur les potentiels moyens d 'excitation et

les relations energie-parcours dans les elements legers. Rhysica ]_8,

249-264 (1952).

G. Mano, Recherches sur I'absorption des rayons, Annales de Rhysique,

Series 11, 1, 407-531 (1934).

[67] F. W. Martin and L. C. Northcliffe, Energy loss and effective charge of

He, C, and Ar ions below 10 MeV/amu in gases. Rhys. Rev. 128, 1166-1174

(1 962).

[68] U. 'Fano, Renetration of protons, alpha particles, and messons. Annual

Reviews of Nuclear Science 2A, 1-66 (1963).

[69] Joseph F. Janni, Calculations of energy loss, range, pathlength,

straggling, multiple scattering, and the probability of inelastic

nuclear collisions for 0.1- to 1000-MeV protons. Air Force Weapons

Laboratory, Technical Report No. AFWL-TR-65-1 50, Kirtland Air Force

Base, New Mexico, 438 pages (Sept. 1966).

[70] R. H. Garbincius and L. G. Hyman, Range-energy relation in hydrogen.

Rhys. Rev. 1834-1838 (1970).

[71] Lucien Rages, Evelyne Bertel, Jemro Joffre, and Laodamas Sklavenitis,

Energy loss, range, and bremsstrahl ung yield for 10-keV to 100-MeV

electrons in various elements and chemical compounds. Atomic Data

1 -127 (1 972).

[72] T. J. Thompson, Effect of chemical structure on stopping powers of

high-energy protons, Univ. Calif. Radiation Laboratory Report UCRL-1910,

Univ. Calif. Radiat. Lab. Berkeley, pages (1952).

[73] G. D. Zeiss, William J. Meath, J.C.F. MacDonald, and D. J. Dawson,

Accurate evaluation of stopping and straggling mean excitation energies

for N, 0, H2, N2, O2, NO, NH3, H2O, and N2O using dipole oscillator

strength distributions. Radiat. Res. 7^, 284-303 (1977).

98
[74]

J. D. Garcia, Radiative corrections to the energies of atoms and

molecules. Phys, Rev. 147 , 66-68 (1966).

[75] 6. A. Victor and A. Dalgarno, Dipole properties of molecular hydrogen.

J. Chem. Phys. 2535-2539 (1969).

[76] P. W. Langhoff and A. C. Yates, Moment theory bounds on the mean energies of stopping, straggling, and molecular excitation. J. Phys.

B: Atom. Molec. Phys. 5, 1071-1081 (1972).

[77] A. Lewis Ford and J. C. Browne, Direct-resol vent-operator computations

on the hydrogen-molecule dynamic polarizability, Rayleigh, and Raman

scattering. Phys. Rev. 7^, 418-426 (1973).

[78] G. D. Zeiss, William J. Meath, J.C.F. MacDonald, and D. J. Dawson,

Dipole oscillator strength distributions, sums, and some related

properties for Li, N, 0, Ha, Na, Oa, NH 3 , HaO, NO, and NaO. Can. J. Phys. 2080-21 00 (1977).

[79] J. E. Turner, Patricia D. Roecklein, and R. B. Vora, Mean excitation

energies for chemical elements. Health Phys. 1_8, 1 59-160 (1970).

[80] H. H. Andersen and J. F. Ziegler, The Stopping and Ranges of Ions in

Matter , Hydrogen stopping powers and ranges in all elements, 317

pages (1977).

[81] Werner Brandt, Survey of stopping power. Health Phys. 1 1 -20 (1958).

[82] Hans Bichsel, Passage of charged particles through matter. American

Institute of Physics Handbook, 3rd Edition, 8_, 142-189 (1 972).

[83] Jochen Booz and Heinrich Georg Ebert, Mittlerer Energieaufwand W zur Bildung eines lonenpaares in Gasen durch Electronen, Beta-, Gamma- und

Rbntgenstrahl ung . Strahl entherapic 1 20 , 7-33 (1963).

[84] E. J. Williams, Correlation of certain collision problems with radiation

theory, Vid. Selsk. Math.-fys. Medd. XIII, 4, 50 pages (1935).

99
» V / .itTi.'i
NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. 2. Gov’t Accession

No.

3. Recipient’s Accession No.

4. TITLE AND SUBTITLE

Numerical Solution of the Electron Slowing-Down Equation for

Molecular Hydrogen

5. Publication Date

6. Performing Organization Code

7. AUTHOR(S)

I . V. Sppnr.pr and R. Pal

8. Performing Organ. Report No.

9. PERFORf^ING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 5331439

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address fS^reef, City, State, ZIP) 13. Type of Report & Period

Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information,

bibliography or literature survey, mention it here.)

If document includes a significant

Procedures are outlined for solution of the slowing-down equation for fast

electrons in differential form rather than in one of its integral forms. Data

for Hp are computed by use of Gerhart's cross section set, suitably modified,

and with inclusion of vibrations. The flexibility and stability of the

approach is demonstrated by solutions for source energies ranging from 1 .MeV down to 24 eV with the spectrum extended below 2 eV. Energy and number

conservation are checked in detail, and the ionization yield is computed as

a function of energy. Spectrum and ionization yield so obtained are compared

with previous calculations; general agreement is found with some significant

differences. Agreement with Combecher's measured ionization yields at low

energies is rather satisfactory overall, but with significant deviations of

trend. Some exploration of secondary and tertiary electron contributions is

included.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons

)

Electron slowing-down; energy degradation spectrum; inelastic cross section set;

ionization yield; molecular hydrogen; W-value.

18. AVAILABILITY Unlimited

I XI For Official Distribution. Do Not Release to NTIS

Order From Sup. of Doc., U.S. Government Printing Office Washington, D.C. 20402, SD Cat. No. C 1

3

I

! Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151

19. SECURITY CLASS

(THIS REPORT)

UNCL ASSIFIED

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

100

22. Price

SCOMM-DC 29042-
4

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