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A variational procedure for improving the born approximation as applied to the scattering of electrons by hydrogen atoms

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A variational procedure for improving the born approximation as applied to the scattering of electrons by hydrogen atoms
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Jones, June Grimm, 1918-
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Atoms ( jstor )
Born approximation ( jstor )
Elastic scattering ( jstor )
Electrons ( jstor )
Hydrogen ( jstor )
Mathematical independent variables ( jstor )
Scattering amplitude ( jstor )
Scattering coefficients ( jstor )
Variational methods ( jstor )
Wave functions ( jstor )
Dissertations, Academic -- Physics -- UF
Electrons ( lcsh )
Physics thesis Ph. D
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bibliography ( marcgt )
non-fiction ( marcgt )

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Dissertation (Ph.D.)--University of Florida, 1953.
Bibliography:
Bibliography: leaf 52.
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Manuscript copy.
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Vita.

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A Variational Procedure For Improving The Born Approximation As Applied To The Scattering Of Electrons By Hydrogen Atoms


7
V)


Si


By
JUNE GRIMM JONES











A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY











UNIVERSITY OF FLORIDA
June, 1953


















ACKNOWLEDGEMENT



The author wishes to acknowledge her great debt to

Dr. M. M. Gordon, co-chairman of the supervisory committee, without whose guidance and assistance the work could not have been carried out and to Dr. R. C. Williamson, chairman of the committee, for helpful suggestions concerning the dissertation and guidance throughout her graduate work. She also wishes to thank the other members of the supervisory committee: Dr. J. W. Flowers, Dr. H. P. Hanson, Dr. A. G. Smith, and Dr. C. B. Smith.
















TABLE OF CONTENTS


LIST OF TABLES. . . . . . . . .

LIST OF ILLUSTRATIONS . . . . . . . .

Part
I. INTRODUCTIOR . . . . . . . .

11. THE VARIATIONAL METHOD . . . . .

III. THE CALCULATIONS . . . . . . .

IV. COMPARISON OF THEORETICAL RESULTS
WITH THE EXPERIMENTAL DATA. . . . V. CONCLUSIONS . . . . . . .

BIBLIOGRAPHY . . . . . . .


A


I


iii


Page

iv

v


1

13

22 36

50 52


1A,














LIST OF TABLES


Page


1. Elastic Scattering Coefficients and Crass
Sections Calculated by Born Approximation
with and without Allowance for Exchange
Effects . . . . . . .

Z. Parameters Determined by Linear Method
for 350 Volt Electrons . . . . .

3. Elastic Scattering Coefficients and Cross
Sections Dettrmined by Linear Variational Method Compared to Born Approximation.


4, Parameters Determined by Quadratic Method
for 350 Volt Electrons . . . .. 33

5. Cross Sections Determined by Quadratic
Variational Method Compared to Born
Approximation 350 Volt Electrons . 35


F--'-


* I0


I.


'-a


'


iv


Table


9


27



28


. .


. .



. .


I .














LIST OF ILLUSTRATIONS


Figure Page

1. The theoretical cross sections, obtained by the
linear (Kohn) and quadratic (Hulthin) variational procedures, for the elastic scattering of 350 volt electrons by hydrogen atoms are compared to the
cross sections calculated by the Born approximation
and to the relative scattered intensities measured
by Webb. . . . . . . . . . 30

2. The theoretical cross sections for the elastic
scattering of 350 volt electrons by hydrogen atoms
calculated by the linear (Kohn) variational
procedure employing the trial functions (44). (45)
with the parameter b (A, c a 0), with the parameter
c(A, b c 0). and with parameters b. c (A a 0) are
compared to those obtained using the parameters
A, b, c in the same trial function . . . . 43

3. The theoretical cross sections for the elastic
scattering of 350 volt electrons by hydrogen atoms
calculated by the linear variational method
employing the trial functions (44). (45) with the
parameter A (b. c a 0). with the parameters
A, b (c 0), and with the parameters A, c (b = 0) compared with those obtained using the parameters
A, b, c in the same trial function . . . . 46

4. The cross sections for the elastic scattering of
350 volt electrons by hydrogen atoms calculated
by the linear variational procedure employing the
trial function (57) with the parameters A. b, B
(c, C 0) and with the parameters A, B, C (b. c 0) are compared to those obtained employing the same trial functions with the
parameters A,b,c (BC = 0). . . . . 48


v











I. INTRODUCTION


In t h e application of wave mechanics to the calculation of the scattering of electrons by atoms, the simplest problem which can be treated is the collision of electrons with hydrogen atoms since the complete set of hydrogen eigenfunctions is available. Hence, any method proposed for treating electron-atom scattering problems must first be shown to be successful when applied to the hydrogen atom.

The wave function Y(f rk) which describes the scattering of an electron incident on a hydrogen atom in the ground state must satisfy the wave equation


A 2/Zm)(V t V )+ +(e 2 /rl) + (e2 /Y (2/r12){ 0. (1) where r1, r2, and r12 are the respective distances of the two electrons from the nucleus and each other. The mass of the electron is small enough compared to that of the proton so that the motion of the proton in the collision may be neglected. The total energy E of the system is the sum of the energy E of the atomic electron in the ground state and of the kinetic energy mv2/2 of the incident electron. The


IN. F. Mott and H. S. W. Massey. Theory of Atomic
Collisions (Clarendon Press, Oxford, 1949), second edition: Chap. VIII, section 2. This treatise on scattering problems gives a complete up-to-date account of most of the work done on this problem.


I


I











wave function r(I ,72) must remain finite as rl and r2 approach mero, and for large r, and r2 it must have the asymptotic form


+ (ri) exp(ikol. E2) + (1 /r2) I V.; ()f


exp(ikmr2 )


as r2-* D (Za)


o --7 (L/r1) 2) gm(n*;n ) exp(ikmr ),
mm


as r-P c. (2b)


where is a member, with energy Em of the orthonormal set

of hydrogen eigenfunctions; k2 = (2m/62) (E Em) ; is a unit vector in the direction of the incident electron and of unspecified orientation; II and i2 are unit vectors having the spatial orientations of Vo and r2 respectively; i and go are respectively the scattering amplitude and "exchange" scattering amplitude associated with the o -+ m transition.

The a priori symmetry condition imposed on the total wave function for particles with half integral spin requires that the space wave function be either symmetric or antisymmetric in the coordinates of the two electrons to correspond, respectively, with aitisymmetric (spin z 0) or symmetric (spin = 1) spin functions such that the total wave function is always antiymmetric. The total space wave functions


. I









3

are then


S( ) 2 t) (3)


which have the asymptotic forms for large r2


P(V",)exp(ik,A-. Y2) + (1 /r2) M


)exp(ikr) (r) (4a)

and for large r


T i(2)exp(k0 ) (1/r,)


f ij n) exp(ikor ) (r2. (4b)


For calculation. purposes, it is assumed that the intensity of the beam of electrons falling on a hydrogen atom initially in the normal state is such that one electron crosses unit area per unit time. The scattering of the electrons from a beam of unit intensity is measured in terms of the differential cross section I(ef) dw, defined as the number of electrons which fall per unit time on an area dS (dS = r2dw) placed at a large distance r from the scattering atom. For the o-+m transition, the differential cross section Imef)dw is the sum of the number of electrons which, after exciting the state m in the atom, are scattered into the solid angle dw located at (0, 4) in unit time and the number of electrons which, after the incident electrons have been









4

captured into the state m of the atom, are ejected into the same solid angle dw in unit time. Thus from either Eq. (4a) or Eq. (4b) we have


I (40) dw = (km/k ) 3/41f gJ2 + l/4(fm gm dw (5)


since the probabilities of the electrons having parallel spins and of the electrons having anti-parallel spins are in a ratioz. of 3 to 1. 2In order to obtain the scattering cross sections, the values of the scattering amplitudes imoand gmo must be obtained.

Returning to the wave equation (1), we rearrange the equation and multiply through by W("/( ): (12/2m) (V2 + k2) e(i*)Y(?, --) z Yro'[) [(-2/2m)2 0 o2 0 ~


-(02/r y .(r12]), 2 /r.) (e2/r2j *Ww If (rItr (6)


Since


(-2/2m)V (eZ/r,)] ( E0 ,(r) (7)


the integration of Eq. (6) reduces via Green's Theorem to


2lbid. Chap. V.









5


2) t k2


- (02/r 12 ItY1 2 d1.*


If we define a function


F (r2 021


it follows from (8) that


(A22/2M) V


+ k] F.(4) -


2/r2) (*Zr1


zt(,. E


The solution Vf\(7) of this integro-differential equation has the asymptotic form


F( 2) exp(ik 1. 2) + I,,exp(ikor2)/r2


(11)


as follows from Eqs. (9), (2). The required solution is3


F(i72) = exp(ikon- r) (2m/4frfh2


f xp(ikolt2 -II)/ j?2 -7'0


x


(12)


3_bid., Chap. VI.


(8)


(9)


x


I


(10)


( ) (fi2)dt, =


0 '


i


l(r' Y ?,. -rp2d


(OP',A)dt',


- (e2/r2









6

with, as a result,

S n, A' =0 ( /) x-k,'') (, )dt',(3
000

as follows from (1 ). Eq. (10) is only one of a set of simultaneous integro-differential equations for the functions


Fn 2 n' 12 l

By the same method, an integral equation of the form (13) can be found for the scattering amplitude f corresponding io each function F nW2):


f it2,i"') z (2m/02) exp(-ik -'*I') (-*I ~ t' (
no ~~~n r)t Ad




Similarly, for the set of functions r r r )d (16)
Gn('dz 2 1 2'

we have a set of simultaneous integro-differential equations corresponding to (10)

2 /2m) (V 2 + k 1



(e/r12jj Y' 2


r r 'n), (17)









7


whose solutions yield [Cf. (13)]


gnoKi 311) 2 (2m/rk2 ) jexP -ikniI'. 1') n(1', 31)dT'. (18) These integral equations (15), (18) for the scattering amplitudes f. and g are exact, and since the exact solutions cannot be carried out by any reasonable method, various approximate solutions have been based on them.

Before proceeding further we simplify our considerations by restricting the problem to elastic scattering at relatively high velocities of impact.

The simplest approximation is that due to Born. For high velocities of impact, the perturbation of the incident wave by its interaction with the atom will be small. Under these circumstances, Born takes as a zero-order approximation for Y, [cf. (2)] Y(F l2) r exp(ik,2 it- i ?) II


which assumes that the incident electron is unaffected by the atomic field. Substituting (19) into the right-hand side of (10). and following the calculations through Eq. (13), the Born formula for the elastic scattering amplitude is given by




4M. Born, Zeits. f. Physik 38, 803 (1926).






W

8

fg Zae(8 + K2a)/(4 + K2a2) (20)


where K r Zkosin(9/2) and a. z E2/me2 is the radius of the first Bohr orbit. Since ko = ZTmv/h, the scattering amplitude is a function of the product of the velocity of the incident electron and the sin (0/2). where 0 is the angle through which the scattering takes place.

The "exchange" scattering amplitude g), for the first Born approximation, obtained by substituting (19) into (I$) via (17), is given by Massey and Mohr5 as


goo = 4ao n 0 n(cos e) (21) where the gn are functio ns of ka 0 and the Pn(cos G) are Legendre polynomials. The actual numerical calculations of go were not carried out for the high energies of interest to us by Massey and Mohr. We have therefore calculated goo for an incident electron energy of 350 volts for values of koa0sin(8/2) from 0 to 5. which corresponds to an angular range of 00 to 1600. These values are compared with the corresponding fo0 values in Table 1. The table also contains the values of the elastic scattering cross sections (cf. Eq. (5)] with and without the contribution of "exchange" (i. e.. goo) to the scattering. It is


5H. S. W. Massey and C. B. 0. Mohr. Proc. Roy. Soc. A 132, 608 (1931).










9


TABLE I

ELASTIC SCATTERING COEFFICIENTS AND CROSS SECTIONS
CALCULATED BY BORN APPROXIMATION WITH AND WITHOUT
ALLOWANCE FOR EXCHANGE EFFECTS



kaoasine/2 f9/)a goo/) I2 G ):

3/41f-gl2. 2/4ig, 2


0.00 0.03 0.05 0.10 0.20

0.30
0.40 0.50 0.60 0.70

0.80 0.90 1.0
1.2 1.4

1.6 1.8 2.0
2.5 3.0


3.5
4.0 4.5 5.0


1.0000 .999 .996 .985
.943

.880 .803 .720 .638 .561

.491 .429 .375 .289 .226

.180
.146 .120 .0773 .0550

.0406 .0311
.0246 .0200


-0.0288
-0.0288
- .029
- .029
- .029

- .028
- .028
-.027
-.026
-.025

-.024
- .023
- .022
- .019
- .017

- .014
- .010
- .007
- .0011
+0.0024

t .0026 t .0001
-0.0018
+0.0034


27.9 27.9 27.7 27.1
24.8

21.6 18.0 14. 5 11.4 8.78

6.73
5.14 3.93 2.33
1.43

0.904 0.593
0.402 0.172
0.084

0.046 0.027 0.017 0.011


28.7 28.7
28.5 27.7 25. 6

22.3 18. 6 15.0 11.8 9.18

7.06
5.41 4.18 2.49 1.54

0.979 0.636
0.427 0.169 0.081

0.043 0.027 0.018 0.0092


S.


'


p










10

apparent that the "exchange" effect can be neglected at such high energies and the cross section for elastic scattering of electrons by hydrogen atoms can then be written simply


( f 2 (22)


as follows from Eq. (5) with goo w 0.

The adequacy of the Born approximation is determined by a comparison of the elastic scattering cross sections calculated from

(22) and (20) with observed scattered intensities. Such a comparison was made by Webb6 in his report of the experimental determinations of cross sections for the elastic scattering by hydrogen7 of electrons with incident energies varying from 30 to 912 volts over the angular range from 5 to 1500 In accordance with the assumption that the interaction of electron and atom is very small, the agreement between the Born approximation and experimental curves is best at the


6Glenn M. Webb, Phys. Rev. 47, 384 (1935).
7Molecular hydrogen was used. For the comparison of theoretical and experimental results, the cross sections calculated for atomic hydrogen by the Born formula were transformed by a suitable factor to equivalent cross sections for scattering by molecular hydrogen. See part IV A.
8The intensities of the scattered electrons at different angles are measured only relative to each other in scattering experiments. Hence the cross sections are reported in arbitrary units and the curves are compared with the corresponding theoretical curves by adjusting the ordinates of the experimental curves so as to obtain the best fit to the theoretical curves.









11

highest energy. 912 volts. At this energy, the experimental points lie on the calculated curve from 100 to 600. The angular range of the agreement decreases with decreasing energy. The experimental curves rise much more rapidly at small angles than the theoretical curves do. At large angles the observed points lie consistently above the calculated curves. At the lower voltages (30-100 volts) there is scarcely any agreement. At 350 volts, the Born formula seems adequate from 400 to 600; the departure of the experimental curve from the theoretical curve at the small angles is very pronounced; although the agreement is not good at the large angles, the deviations are small compared to those at the small angles. Thus, the scattering of 350 volt electrons can be taken as a good test case of the adequacy of any theoretical calculation which attempts to improve to some extent the Born approximation. It is for this reason that we adopted this particular energy for the work described in the following pages.

The second Born approximation involves the substitution of

the expression (12) for F0(i2), obtained by the first Born approximation, as part :of the function Y(r,2) in Eq. (10) and the integrations of the equations a second time. This procedure is so involved that the calculation has never been carried through. It would be preferable to begin with a more accurate approximation of Y(.i'*, ) than to obtain successive Born approximations by iteration.





12


Another method, due to Massey and Mohr, 9 uses the second Born approximation only to obtain a "polarization" potential. With this potential added to the static potential of the atom, the cross sections are recalculated by the method of the first Born approximation. Although the method cannot be Justified rigorously (in the mathematical sense), the agreement of the theoretical and observed cross section curves is considerably better than that afforded by the first Born approximation at small angles. 6

There is still considerable room for improvement of the

first Born approximation by some simple analytical means. In search of such a method we have investigated variational techniques.


r
I


*,.
* .....>







**.
S.'
I-


9H. S. W. Massey and C. B. 0. Mohr, Proc. Roy. Soc. A 146, 880 (1934).


T . .lI-











II. THE VARIATIONAL METHOD



A. Me thods of Hulthbn and Kohn- -The variational

procedures introduced by Hulthan10 and Kohni treat the one-body problem consisting of the scattering of a particle by a potential Y(r). They involve the construction of an integral


L( _k -i2 YZ(E H)ld. (23)


where (E H) = v2 t k2 V(r)J. When Y is a proper solution of H r E Y, satisfying the boundary condition


--+ex(nq- I) + f(k",)exp(ikr)/r (i = 1, 2), (24)

"o


the integral (23) vanishes. In the event that an exact solution is not known, but a trial function Y = + SY can be set up, 4
t

which has the same asymptotic form as of (24) but with f replaced by f + S f, then it can be shown that to first order,


S L(klg, -k2) = -4 Tr 6f(k -E2). 25


Hence a variational procedure is available for calculating the unknown


10L. Hulthen, Kungl. Fysio. Stllskapets Lund Forhand. 14.
1 (1944).

W. Kohn. Phys. Rev. 74. 1763 (1948).
13









14

f. If a trial function is


S Y (. c 3 '(26)


where the c and are parameters, numerical values of these parameters may be obtained by solving the simultaneous equations derived from (25):


'aL/ aci =0, (27a)
according to Kohn's linear method aL/ f -4W (27b)

or L a 0, (28a)
according to Hulthen's quadratic L/ 3ci 0 (28b)

method.

Kohn's method then gives


41Tf = Lt + 41Fft


while Hulth~n's method gives f f ft (30)


because of the stipulation that L = 0. It is apparent that the better the trial function, the better the agreement between the scattering amplitude given by Kohn's method and that given by Hulthin'u method since in the limit of Yt Lt -- 0 and the two methods give the same results.









15


Variational techniques have been applied successfully to the elastic scattering of low energy (< 13 volts) electrons by hydrogen atoms. Huang. 12 using a modified Hulthin variational procedure, calculated the phase shifts for S-wave scattering. 13 He employed a trial function


(r1,r2) =[exp(-rI)/rZ [sinkr2 + u(rir2)coskr2


u z L exp(-r,)] [a + (b + cr12)exp(-r2)'


where a, b. and c are variational parameters. His results were inconclusive, however, due to a mistake in his method. 14

Massey and Moiseiwitsch applied the variational methods of Kohn and Hulthen to the elastic scattering of slow electrons by H-atoms. They used Huang's trial function and combinations of it to allow for exchange in the determination of the S-wave phase shifts.


12Su-Shu Huang. Phys. Rev. 76. 477 (1949).

1The scattered wave can be considered as a sum of partial waves of phase In (See, e.g., Mott and Massey, op. cit., Chap. IL) The S-wave scattering is that part of the scattering which is spherically symmetric (not dependent on the angle G) and the total S-wave cross section is given by Q = (4 r/kZ) sin Yt. S-wave scattering predominates at low energies.

14L. Hulthen and P. 0. Olsson, Phys. Rev. 79, 531 (1950).

15H. S. W. Massey and H. L. Moiseiwitsch, Proc. Roy. Soc. A 205, 483 (1951).









16

The fact that both (Kohn and Hulthen) variational methods gave very nearly the same results indicates that the trial function (31) is a fairly good approximation to the exact wave function for the scattering of slow electrons. In the absence of experimental data for e-H scattering at low energies, the scattering amplitudes were checked with the previous results obtained by direct numerical integrations of the differential equation -- the agreement was found excellent.

On the basis of the good results obtained for the elastic scattering of slow electrons, it should be possible to apply a variational technique to investigate the elastic scattering of fast electrons by hydrogen atoms. For this purpose the variational equations of Huang or Massey and Moiseiwitsch cannot be applied since they are formulated on the assumption that only S-wave scattering takes place.


B. Variational Procedure for Improving the Born Approximation for e-H Scattering--A solution of the wave equation (1) for the scattering of an electron by a hydrogen atom has the asymptotic form (cf. Eq. (2))


+()--, W(- )exp(ik # -2


+( /r2 ) I Ym 2r-- f)exp(ikmr2) (32a)


as r;









17


y (1/r,) 7- (2m(n);)exp(ikmrl), (32b)





In order to derive a variational equation for calculating f0 specifically another solution to the same wave equation with the asymptotic form


2) (g1 es,. o2),-+


+r(1/r2) m E fmo*( );n2)exp(ikmr ) (33a)


as r2 epo;

-( /r,)2) 8o )n )ep r ),(33b)



as r,- ;


is introduced. The integral [(cf. (23)]


L00(d; 2)) J H) Y1) drdt2 (34)


where H = (-A2/2m)(Vz + V) (e2/r) (e2/r)+ (e2/r2), is constructed and its first-order variation is obtained by replacing each Y by Y + S where the Yt eS Y have the same asymptotic form as but with each f and g replaced by f + f and g S Sg. Making use of Green's theorem, we then have









18











Substituting into (35) th' asymptotic forms (33) and the variation of (32), and performing the volume integrations, we find that the first of these integrals vanishes so that (35) then reduces to


2L Eoa%4exp(-ikrcos a ) Sf00(2 ;;n)exp(ik~r2)r 2



ikcosoc + i] r 2T sindad., (36)


where cos oC = ~~ -2. An integration by parts then gives

Ef. (25))
00 00














Eq. (37) provides a variational equation for calculating fe. A similar equation is obtained for g if the roles of r and are interchanged



We try out this variational equation with the simple trial

functions (the one used for the Born approximation -- cf. Eq. (19)): [io, a OW +) exp(ik i 7i) (38a)









19


(38b)


and (ri7 o 2'r).


Then since we have ft z 0, it follows from (37) that


if ();(2)) a 1(4 7rE a ) L (i ;P) ) 00 t


(39)


Substituting (38a), (38b) into (34). we find


L =8 t (6 + KZalZ)/(4 + Ka) 2
t 0%'&' 0


(40)


Substituting (40) into (39). we obtain


f,( 9) z 2a(8 + K a )/(4 + K2 2 2


(41)


which L is exactly the scattering amplitude (20) obtained for the first Born approximation. 39ence, it appears that Eq. (37) provides a variational method for improving the Born approximation.


C. Selection of Trial Function--The scattering process we are considering may be broken down into the following components:

1. Scattering by the field of the atom, considering the

atom undisturbed by the incident electron and the incident

electron only slightly affected by the collision;

2. The distortion of the incident and scattered electron

waves by the potential field of the atom (the latter field being taken as the undisturbed atomic field);









20

3. The disturbance or polarizationn" of the atomic field

by the incident and scattered electron waves;

4. The exchange of electrons between the incident electron

wave and the atom.

The trial function adopted should allow for as many of these component processes as possible. The number of parameters in our trial function was limited to three in order that the calculations should not become undauy cumbersome. (Massey and Moiseiwitsch 5 found three parameters sufficient to give excellent results--see above.) Adapting Huang's trial function (31), which gave such good results at low energies, for use at high energies, we obtain a trial function


Yt Z 100 *p oko r 2)1 1 + bexp(-r2/a%) + (crl2/a ) x


exp(.r2/ao) + A [exp(ikorZ)/r2 [ exp(-r2/a 4)


The first term in the bracket gives the Born approximation and therefore allows for the first process in the list above. The second term allows for effects on the incident wave close to the hydrogen atom since exp(-r2/a0) falls off rapidly for values of r2 greater than as, the radius of the first Bohr orbit; this term attempts to take care of the distortion of the incident wave by the field of the atom listed as part of process 2. The third term (cr12/a0)exp(-r2/a%) allows for the polarization of the atomic field by its explicit dependence on



's.42









21

the distance between the incident and atomic electrons. This term will have a peak value in the vicinity of the atom but will also be zero when the two electrons' positions coincide; it attempts to take care of process 3. The effect of an exchange of the Incident and atomic electrons on the scattered electron wave (process 4) has been discussed in the Iptroduction (p. 8). At the energy considered (350 volts), the effect of exchange Is not significant (see Table 1). In the last term, A is the trial scattering coefficient. The factor

I exp(-r2/a)] is included so that the wave function behaves properly when r2 approaches zero. and this factor is appreciably different from unity only in the neighborhood of the atom. A very important reason for choosing the trial function in the particular form of (42) is to make the required integration involving the function feasible (see III A). The asymptotic form of (42) is


~I~t( -190 -*P o. j2) + A exp(ikor2)/r2 (43)
nor7)/ (3






Z 00










III. THE CALCULATIONS


A. The Variational Integral Lg--To obtain the scattering coefficient foo for the elastic scattering of electrons by hydrogen atoms in the ground state, the integration involved in L of (34) must be carried out. In addition to the trial function


I) exp(ik in"- 1 + b exp (-r/a)


+ (cr iz/aaexp(-rz/a + A exp(ikor2)/r I


exp(-r0/a)] i()) (44) selected above [cf. (42)] an auxiliary trial function P. (33)]


y(?i)exp(-ikO2)' rZ) [ + b exp(-r2/a0)


+(cr12/a )exp(-rZ/ao) + A exp(ikor2)/r 1


exp(-r2/a )]) (45)


is required. Then


L (i ,i )J E H] (y ,)d i


4 TrE a2 (D + Dlb + D2b2 + D3 + 2


+ DsA + D6A2 + D7bc + D8bA t D cA) (46)

22









23


where, with K = Zkosin 0/2, D, z Za (8 + K2a2)/(4 + Ka2)2
0 00

Di : 4a [-K2/(1 + K2a2 2 + (15 + K a )/(9 + K a )l


D2 C 2[ -2(1 + K2a2)/(4 + K2a2)2 +(24 + K2a )/(16 + K2 2 D3 = 4a (K2a2 1)/(1 + K 2 )2
3 0 0

a (-132 1OK2a)/(9 + K2a,)J(3/K) [-tan-'(/Ka0)


tan I(3/Ka )J


D 2(64 14Ka 3K a)/(4 + K2 2 )4


4a(4 + K 2a)2

2 2 -1I 2 )-1
D5 : -1 + (1 + k0a) 4(9 + 4k a )


+(2/ka%) f-tan I(1/koa%) + tan- (3/2ka) i k a (1

2 2-1 2 2 -1 2 2
+k~a) 8(27 + 12k-a) + (I /kOa) I n(9 + 9k a )


ln(9 + 4koao)








24

D6 z (I/ao) (2 t Z2k;) 12(9 t 4k a ) + 2z i k
2 2a 2 2 2 2t k0
In(4 + 4koa%) + 21n(9 t 4k;a) In(16 t 4k a )

r a-1 2 2-I 2z-2
f i [k {(2 + 2ka ) (9 + 4ka) (4 + -k ao)

I -L Iil
(2/a ) -tan- (1/koa%) t Ztan- (3/2koa%) tan- (2/k a0)

2 2 22 2 2 2 222
D7 = 2a0(-2-K ao)/(4 t K a) t 4a0(-64-3K ao)/(16 t K a)


+(4/K)-tan'(2/Ka%) + tan-I(4/Ka%)

D8 = -1 + (2 + 2k a0) + 4(9 + 4k a0) (4 + k 0a)


+(2/koa) I-tanI(3/Zkoa) t tan- (2/k a )l


+ ka [(2 + 2ka2)- f 8(27 t 12k2)- (8 2ka 0)j


+(I/koa0) in(36 t 26k 2a in(36 t 9ka )

2 2-1 2 2 -1 22 2 2
D 9z -901 + 4ka) -(4ka 0 + 9) t (5 2koa0)/(4 + 4koa0)


11/12 (1/2)ln(8I + 36k a2) (1/2)ln(16 + 64koa%)


(1/ka%) 1(5/2)tanI (I/2koa0) + 3tan1(I/koa%)


+ ( /2)tan1 (3/2koao ) i koaof-l9(i t 4k2a a)2 2)- 2 2 -1
+7(4 4koa%) 2(27 + 12k a.) t (I /k a ) x









25


(5/4)In(l t 4koa ) + (3/2)n(1 +


+(1/4)ln(9 4k~a$) (1/2)In3j + tan- (3/2kea )


tan- (1/2koa,)].


The coefficient Do is the same as the Born approximation scattering amplitude of Eq. (20). Some of the coefficients Di depend only on the kinetic energy of the incident electron (i. e. k), while other coefficients depend both on the kinetic energy of the incident electron and the scattering angle 9 (i.e., K).


B. Linear Determination of Parameters--If the variational method of Kohn is paralleled (cf. (27)]. we have from Eq. (37) and Eq. (46) the following set of linear equations


L/ ab z DI + 2D2b + D7c + D8A a 0 (47a)


L/ cc a D3 + D7b + ZD4c + DA a 0 (47b)


(4TE a2 )-1 L/ DA D5 + D8b + Dc + 2D6A -1
(47c)


for determining b. c, and A. Whence,


L00(b, c, A) -4 T Eoa2 (A fo) (48a)









26


determines I


D + A + D5A t D6A2 + D b + D2b2 t D3c


+ D c2 + D7bc + D8bA + DgcA (48b)


Numerical values of the coefficients Di have been calculated for electrons of 350 volt energy over a range of 0 from 00 to 1600. Using these numerical values, Eqs. (47) were solved simultaneously for the parameters b, c, and A. The values of these parameters are listed in Table 2. Numerical values of fo and the cross sections


I( e ) = |f0( e )1 2 (49)


were then obtained via (48b). These values are tabulated along with the cross sections given by Born's formula (i. e.. Eq. (20)) in Table 3. A comparison is made graphically in Figure 1.

It in seen from Eq. (48b) that the linear variational method gives the scattering coefficient as the sum of the Born approximation scattering coefficient (i. e., Do) and additional contributions from the trial scattering amplitude (A) and from the parame-ters (b. c) which were chosen to allow for the effects of distortion and polarisation. Figure I shows that allowance for these additional contributions to the scattering amplitude results in a sharp rise in the cross section curve at small angles well beyond the Born approximation









27

TABLE 2


PARAMETERS DETERMINED BY LINEAR METHOD
FOR 350 VOLT ELECTRONS


k0aosin e/2 b c. A/a,


4.31 4.21 4.25 4.02 2.77

0. 812 .358 .587 618 540

.454 382 .308 .203 .129


0
0.03 .05
1 2

.3
.4
.5
.6
.7

.8
.9
1.0 1.2
1.4

1. 6 1.8 2.0 2.5 3.0

3. 5 4.0 4.5 5.0


-1. 231
-1. 151
-1. 291
-1. 291
-1. 381

-1. 241 +0. 2611 + 6481 + 712i + 5841

. 5291 + 4901 +.4411 +. 3601 +. 328i


-8.56
-8.42
-8. 33
-7. 52
-4. 78

-0. 358
-1.22
-2. 25
-2. 40
-2.05

-1.78
-1.56
-1.32
-0.965
- .750

- .580
- .455
- 360
- .203
- .109


- .0819 +. 1611
- .0856 +.1521
-0932 +.1431
- .105 +.1321


0.921 .892 900 .771
.464

-0. 0874
- .0432 0.139
.244 .218

.209
194 .168 .122
.0914


.0673
.0502 .0383 .0209 .01283

.00819 .00578
.00435 .00344


-0. 5961
- .5851
- .6211
- 6781
- .7241

- .6011
-.1151
- .003261 +0. 03761 + .03021


1
+
+
+
+


. 0422i 05211 .05251 .04441 .04141


+ 03301
-. 02571 + 02051 +.01151 +.007201

+. 004981 4.003141 +.002551 +.00202 I


.0768 0397 .0103
-0.0344
- .0621


+. 2911 +. 2621
2421 +.2021 +.1781


- .0410- .0156+0.0186+ .0439-


+8. 691 +8. 381 + 8. 751 + 8. 171 +7. 061

+4.241 + 1091
-0. 5241
- .6491
- .4541


.4371 .4421 .4211 3631 .3811

3601 3401 3281 .2911 .2721

. 258i .2371 .2301 .2191


7-












-









28

TABLE 3

ELASTIC SCATTERING COEFFICIENTS AND CROSS SECTIONS
DETERMINED BY LINEAR VARIATIONAL METHOD
COMPARED TO BORN APPROXIMATION



k0&03i3 0/2 fLinear Var. Born

Method Approximatio a

0 -9.91 +5. 541 3595 x 10 cm2 27.9 x 10-1cm
0.03 -9.54 +5.201 3295 27.9
.05 -9.31 t5. 42i 3239 27.7
1 -7.65 + 4.561 2213 27.1
.2 -2.64 +2. 44i 362 24.8

.3 0.456 +0.5831 15.3 21.6
.4 .900 -0.08381 22.8 18.0
.5 1.12 +0.100i 35.3 14.5
6 1. 36 + 212i 37. 2 11. 4
.7 0.975 + .1811 27.4 8.78

.8 .808 + .1631 19.0 6.73
.9 .663 +.1431 12.8 5.14
1.0 .535 +. 113i 8.34 3.93
1. 2 .363 + 04941 3.75 2.33
1.4 .261 + .04541 1.95 1.43

1.6 .196 +.02921 1.09 0.904
1.8 .153 +.01911 0.661 .593
2.0 .123 +.01311 .425 .402
2.5 .0771 4.00565i .167 .172
3.0 .0544 +.00292i .0829 .084

3.5 .0400 +.00167i .0448 .046
4.0 .0308 +.00104i .0265 .027
4.5 .0244 +.00074i .0166 .017
























Fig. 1. The theoretical cross sections, obtained by the

linear (Kohn) and quadratic (Hulthln) variational procedures. for the elastic scattering of 350 volt electrons by hydrogen atoms are compared to the cross sections calculated by the Born approximation and to the relative scattered intensities measured by Webb. Ordinate, Elastic scattering cross sections in units of 10-cm2 Abscissa, Angle 0 through which the scattering takes place.









30


I I1

Linear var. method Quadratic var. method

-10 Born approx.
0 Experimental values
0







2 0





-lo\


INN











0


0






ice 200 300 400 0








31

curve. At large angles, however, the two curves merge.


C. Quadratic Determination of Parameters--If the variational method of Hulthhn is paralleled [cf. (28)). the restriction that the trial function be such that the variational integral L vanishes is imposed. Then, from (37),


6 Loo = -4 Tr E & 4 fa= 0 (50)
00 00

and

L 0 (5la)


3L/ 3b 0 (51b)


aL/ ic = 0 (51c)


are the equations for determining b, c, and A. Note that (51b) and (Sic) are the same as (47a) and (47b) and hence linear in the parameters. but that (51a) sets the expression for Lo given by (46) equal to zero and is therefore quadratic in the parameters while its counterpart in the Kohn method, (47c), is linear. Eq. (50) also gives


010= A (52)

as opposed to the expression (48b) for f 0 given by the linear (Kohn) method.

The values of the parameters b and c were found in terms of









32

the parameter A by a simultaneous solution of Eqs. (51b) and (5 1c). Substituting these values into Eq. (51a), we obtained an equation of the second degree in A. Using the calculated values of Di (see above). the two possible values of A were then calculated from this equation over the angular range considered--these two sets of values being equivalent to two sets of fo values via (52). A method for discriminating between these two sets of f, values was 'ecessary and a test similar to that used by Massey and Moiseiwitsch was employed.

Eq. (13) is an exact integral equation for the scattering coefficient f,0. The integral of Eq. (13) was evaluated using the

of (44) corresponding to each of the two sets of parameters

resulting from the quadratic equation in A. The set of parameters for which the integral was more nearly equal to the value of A ( foo) tested was then the set of parameters selected.

The selected set of parameters b, c, and A calculated

for 350 volt electrons by the quadratic variational method is given in Table 4. The corresponding values of the integral in (13) are also listed for comparison with the values of the scattering coefficient A. It is apparent from the table that the agreement between the integral equation values of foe and the values of A is not very good except at large angles. But it in still sufficiently good to discriminate between the two sets of f00 values. The cross sections calculated by this method are compared to those calculated by the Born approximation




TABLE 4
PARAMETERS DETERMINED BY QUADRATIC METHOD FOR 350 VOLT ELECTRONS


k aesin 6/2 b c A/a Integral/a

9 -21.6 +16. 91 4.26 -3. 56i 2. 66 -0. 6341 -11. 1 +10. 11
0.05 -20.9 + 17. 11 4.23 -3. 701 2.62 6881 -10.6 +10. 11
1 -18.7 + 16.91 4.01 -3. 871 2.43 788i 9.04 + 9. 601
.2 -10.6 + 14. 71 2.66 -4. 021 1.59 -1. 011 3.77 + 7.021
.3 + 2.43 +10. 01 -1.65 -4. 00i 0.111 -1. 04i 0.178 + 3. 341
.4 1.83 2. 92i 0.278 +3. 65i .172 -0. 2291 .309 + 0. 282i
.5 3.07 1. 66i .499 42. 62i .463 1791 .0171 + .8251
.6 3. 16 1. 50i .454 +2. 341 .580 1341 -0.0348 + 9141
.7 2.70 0. 9371 .365 +1.881 .552 .1301 0.0251 + .8411
.8 2.31 .8651 .268 +1. 641 .502 .0892i .0430 + 692i
.9 1.99 8591 .190 + 1. 471 .444- 05)81 .0505 +- 553i
1.0 1.68 825i .129 +1. 32i .376 .031.1 .0657 + 4351
1.2 1.23 7241 .0524+1.1091 .270 .0143i .0797 + 2781
1.4 0.962- .7931 .0114+0.9941 .199- .001311 .0782 +. 1811
1. 6 .773- .7891 -0.0152+ .9171 .150 .00082i .0739 4 125i
1.8 .644- .7921 .0322 +.8681 .117 .00214i .0671 + .08991
2.0 .548- .8161 .0524 +.8451 .0942- .00259i .0592 + .06671
2.5 .392- .8401 .0826 +.7951 .0588- .00329i .0426 t-. 0353i
3.0 .312- .8931 .0996 +.7901 .0409- .003221 .0326 + 0212i
3.5 .251- .9721 116 +. 8091 .0298- .00254i .0252 + .01351
4.0 .241-- .9681 .109 '.7931 .0229- .00284i .0198 +.0o9661
4.5 .174- .9521 137 1.7561 .0174. .001421 .0161 +.006541
5.0 .175- .8891 .114 +.7161 .0140. .001601 .0137 +.005271


-




Wi


34

in Table 5 and graphically in Figure 1. It is evident that the quadratic (Hulthen) method does not provide any significant change in cross section values from those of the Born approximation.

It should be noted that Eq. (13) is a necessary, but not

sufficient, condition for the trial wave function to be a good approximation to the true wave function and that the linear (Kohn) variational procedure is such that this integral equation is satisfied identically. On the other hand, it should be noted that Lo z 0 is also a necessary, but not sufficient, condition for the trial wave function to be a good approximation to the true wave function and that the quadratic (Hulthon) method is such that this condition is satisfied identically.














-.









35

TABLE 5

CROSS SECTIONS DETERMINED BY QUADRATIC VARIATIONAL METHOD COMPARED TO BORN APPROXIMATION---350 VOLT ELECTRONS



k a0.in 9/2 e 1(0)
Quad. Var. Method Born Approx.


00
1
2
4
6


9
11 13
15 Is

20 22 27 32 36

41 46 .59 72 87

104 125 160


0'
8
16 32
44

0
18 34 58
4

28
46 22
4 48

36 28
6
34 18


10
8 56


209 x 10l1crm2
204 182 99.1 30. 5

2.29 6.87 9.89 8.97 7.27

5.58 3.97
2.04 1.10 0.629

.381
.248 .0968
.0469 .0249

.0149 .00851 .00550


27.9 x 27.7 27.1
24.8 21.6

18.0
14.5 11.4 8.78 6.73

5.14 3.93 2.33
1.43 0.904

.593 402 172
.084 .046

.027 .017
.011


10-18cm 2


0
0.05 .1
.2
.3

.4
.5
.6
.7
.8

.9
1.0 1.2
1.4 1. 6

1.8 2. 0 2. 5 3.0 3.5


4.0 4.5 5.0











IV. COMPARISON OF THEORETICAL CALCULATIONS WITH EXPERIMENTAL DATA



A. Adjustment of Experimental Data for Scattering by Atomic Hydrogen--The bulk of the experimental data for electronhydrogen scattering has been taken for molecular hydrogen. 16 Harnwell, 17 however, carried out investigations of the scattering of electrons by both atomic and molecular hydrogen and reported that the scattered intensities when atomic hydrogen is present are not much different than when molecular hydrogen alone is present. Since a difference is detectable, some allowance was made for it in the comparison of our theoretical calculations and the experimental data.

Assuming the validity of the Born approximation. Massey and Mohr I8 have obtained the ratio of the scattering by molecular hydrogen to the scattering by atomic hydrogen:


itThe work of Webb6 is so comprehensive that his results alone are sufficient to test the adequacy of the theoretical values of scattering cross sections. Webb measured the scattered intensity over a wide range of angles and voltages with a single apparatus and found, moreover, that his results agreed closely with those obtained by other investigators in the overlapping regions.

17G. P. Harnwell, Phys. Rev. 34. 661 (1929).
18H. S. W. Massey and C. B. 0. Mohr, Proc. Roy. Soc. A 135, 258 (1932).


36


.


aim"





TI


37

2 H (1 + sinx/x)9(y), (53)


where x Zk d sin (G /2), d being the equilibrium nuclear separation of the molecule, and y : ka sin( 0 /2). This ratio performs a damped oscillation about unity as a function of (v)sin( e /2). For an energy of 350 volts over the range of Webb's experimental points, its maximum value is 1. 4 at 0 z So and its minimum value is

0. 78 at 0 a 300. This relationship was used to convert Webb's experimental data at 350 volts for molecular hydrogen to equivalent data for atomic hydrogen. 6


B. Comparison of Calculated Scattering Intensities with

Experimental Data--The experimental values of the cross sections are given in arbitrary units. Hence, for a comparison to the results of this investigation, the adjusted experimental data (cf. section A above) were fitted to coincide at 250 with the theoretical values obtained by the linear (Kohn) variational procedure as shown in Figure 1. In addition to these curves, the results from the Born formula are also given in Figure 1.

An examination of this figure reveals that at small angles ( e < 300), where the deviation of the Born approximation curve from the experimental points is very large, the curve obtained by the linear variational method fits the experimental curve very well except near 70. The dip in the theoretical curve at around 70 is probably not









38

significant. At less than 70, we found that the contributions to the scattering amplitude [cf. (48b)] of the terms which are functions of the parameters b and c were very large and negative compared to the scattering amplitude of the Born approximation. (The part of the scattering amplitude due to the terms which are functions of the parameter A was negligibly small over the entire range of 0. ) At angles greater than 7o, the Born term was largest; at angles greater than 460, terms other than the Born term were negligibly small. The dip in the theoretical curve appears at about 70 where the terms in b and c had become negative but were not yet larger than the Born term. Hence the dip appears in a transition region between the very small angular region where the (b and c) terms allowing for polarization and distortion effects contributed most to the scattering and the angular region ( V > 120) where the Born term predominated. It seems that three parameters are too few to make the transition smoothly.

The cross section values obtained from the quadratic

(Hulthan) variational procedure do not provide any improvement of the Born approximation. In Figure 1. the curve obtained by the quadratic method lies below the Born curve at the larger angles where the experimental points lie higher than the Born curve while, at small angles, it rises somewhat above the Born curve but not to such an extent that it can be considered to agree better with the experimental









39


curve.

It is interesting to note that the shapes of the curves in

Figure 1 obtained by the two variational procedures are quite similar even to the dip appearing around 70, but that the curves are displaced relative to one another. It follows from Eqs. (48a) and (52) that the closer the trial function is to the exact wave function, the closer the two cross section curves will agree. Hence, the displacement of the two curves demonstrates that the trial wave function is not a very good approximation to the exact wave function. The fact that the trial wave function is not a good approximation to the true wave function, however, does not necessarily mean the calculated coefficients are not good approximations to the true scattering coefficients. It is a peculiarity of quantum mechanical variational procedures that satisfactory results can be obtained even though the simple trial functions employed are relatively poor approximations. This is not surprising, however, since in our case, for example, the theory provides a stationary expression for the scattering coefficient, not the wave function. The criterion for adequacy of the theoretical scattering cross sections is still "goodness of fit" to the experimental



19For example, excellent results for the calculation of the binding energy of the hydrogen molecule by a variational method employing a very simple trial function were obtained by: W. Heitler and F. London, Zeits. f. Physik. 44, 455 (1927).








4.

dta ad. ab.. reut of ou le,...riati.. p..,.edre d.

provide & reasem"bMy gsod I1.


C. Results Obtained y Linear Variatieos Procedure with Various Other Trial F1atioss--The tutal wave IactMi ether than Eq. (44) whbik we have sed to deterslae the 1&.tO Ocattsviag creas section by the linear varIasesal pweo.dre* are als woth easiderties. Zopecially Interesting are the reosuts we obtained with the trial feactians.


o ,(Pi) 'mpei,'"9 Ya) L+ besp(r3/a*)

+ (euaai%)eu*"i''4.3/4)J (54) sed


y() ~(r) p(-. i (2)) [ + besp(-u3/a,)





whik are The saMe as (44)and (45) wIth A 0 0. ror this G00


to a Do + (DIDg + .DD DID3D7)/ 4D4L&).


RAfee to Eq. (44) *hews that the DI appearing (5) ae all fsitio of Xa, r 3,akosi( G /3). Three. the ecaneving amp*Utd* for the b sad c paramoSr case. (54). Ube these obtained by the









41

Born approximation (20), are functions of (v)sin( 9 /2) only.

In Figure 2. the results for the b. c parameter trial

functions (54. (55) are compared to those previously derived for the A, b, c parameter trial functions (44), (45). The curves are very similar. It seems then that the good fit obtained with the linear method and trial functions (44), (45) to the experimental data, as shown in Figure 1, at the small angles is provided predominantly by the contributions of parameters b and c to the scattering amplitude. For the b, c parameter case, the parameter A is sero, and the quadratic variational procedure gives a sero scattering coefficient. From these considerations, it is understandable that the linear variational method has yielded better results than the quadratic variational method for the trial functions (44), (45) as discussed above.

The linear variational procedure using the trial functions

considered so far provides no improvement of the Born approximation at large angles. At large angles, the experimental curve of the scattering cross section becomes almost independent of angle instead of falling off uniformly with increase of angle as predicted by the Born approximation. In Eq. (46) the coefficients Di of the terms involving the parameter A are independent of angle. From this, it seems that parameters in the scattered wave part of the trial function may allow for the improvement of the theoretical curves at large angles. Considering a trial wave function of the general form




Pr-'- ~" ',. -~ ', Pr a~..-,.* ..~ uu,~








'4' 4'14.,.
'~1




.4 4 .


I '4 '.4


9--


I :7

4"
''P
'p ~'






'~-"4?'*






'U


Fig. 2. The theoretical cross sections for the elastic


scattering of 350 volt electrons by hydrogen atoms calculated by the linear (Kohn) variational procedure employing the trial functions (44Y, (45) with the parameter b (A, c x 0), with the parameter c (A, b 0). and with parameters b. c (A x 0) are compared to those obtained using the parameters A, b, c in the same trial function. Ordinate, Elastic scattering cross sections in units of 10-18cm, 2 Abscissa, Angle & through which the scattering takes place.


'4, lr







43


A,b,c



0 Experimental values







.12 0
-10









-10
















100 200 30* 40* 50*

I I I N












tex(ik o 2i + bexp(-r2/a) + (crl2/a%)exp(-r/a.)


t(I/r2 or2) f exp( -r 2/a ) x


{A + B exp(-r21/%) + (Cr 12/a ,) exp(-r2/a )) (57)

we have examined the contributions of the various terms to the scattering by carrying out the cross section calculations with trial functions using various combinations of the five parameters in (57). In Figures 2, 3, and 4, the results of the various calculations are graphed. It is evident that the parameter b or c alone (Fig. 2) is not sufficient. Both b and c are required to give a good fit to the experimental data at small angles. The effect of the parameters A. B, and C is to raise the cross section curve at large angles (Figs. 3, 4). When A alone is used, the curve (Fig. 3) is raised at large angles; when A is used in combination with either b or c, or with b and c, no effect of A at large angles is evident. As the number of parameters in the scattered wave part of the trial functiott is increased, the rise of the scattering curve af large angles-also increases (Fig. 4). Since the parameters in the incident wave part of the trial function serve not only to improve the theoretical results at small angles but also to cut down the effectiveness of the parameters in the scattered wave part for raising the theoretical curve at large angles, we feel that a trial function employing all five parameters will provide improvement





Fig. 3. The theoretical cross sections for the elastic

scattering of 350 volt electrons by hydrogen atoms calculated by the linear variational method employing the trial functions (44), (45) with the parameter A (b, c z 0), with the parameters A, b (c = 0), and with the parameters A, c (b a 0) compared with those obtained using the parameters A. b, c in the same trial function. Ordinate, Elastic scattering cross section in units of 10i cm. 2 Abscissa, Angle 9 through which the scattering takes place.


-


- ~ ~ ~?~w~ ~y!I3~~ ~ ~ T -~




44



., /
I.'. .





.








46


A,b,c AJ b

A
0 0 Experimental values






-10 0









40



\\ 0











100 200 300 400 500













.4...,''


.9 -


Fig. 4. The cross sections for the elastic scattering of 350 volt electrons by hydrogen atoms calculated by the linear variational procedure employing the trial function (57) with the parameters A, b. B (c, C x 0) and with the parameters A. B, C (b, c = 0) are compared to those obtained employing the same trial functions with the parameters A, b, c (B, C a 0). Ordinate, Elastic scattering cross sections in units of 10-18cm. 2 Abscissa. Angle e through which the scattering takes place.


0


t









'9,
,., 4..









48




A,b,c
---- A,B,C t- - A b B
o Experimental values
0








102 0










0





00


00








100 200 30* 400 50*


L~. ~









49

of the Born approximation over the entire angular range of the scattering.


EM -











V. CONCLUSIONS



The success of wave mechanics in explaining phenomena

beyond the realm of classical physics has been unqualified. A major theoretical problem is resolving the frustration of not knowing the wave function which applies to a particular problem. In the event that a relatively simple asymptotic form of the wave function is known, however, a variational procedure may sometimes be employed to provide a solution to the problem without possessing the exact wave function. Since the complete set of hydrogen eigenfunctions is known. the variational method outlined in Part II should be useful in obtaining inelastic, as well as elastic, cross sections for the scattering of electrons by hydrogen without going into partial wave analyses as
20
is usually done. Only the elastic scattering at high velocities of impact has been considered here to illustrate the method. The simple form of our trial function accounts for the distortion and polarization effects expected to become important as the impact velocity or scattering angle decreases to where the Born approximation is inadequate. The deviation of our theoretical curves from the observed scattering cross sections at large angles remains when the number



2dA variational method, using partial waves, for obtaining inelastic cross sections has been derived by: B. L. Moiseiwitsch, Phys. Rev. 82, 753 (1951).
50





51

of adjustable parameters is limited to three. If the number of parameters could be increased to perhaps five, thereby greatly increasing the labor of calculations, an extension of the improvement of the Born approximation in the large angle direction could probably be realized. Since the departures 6f theoretical from experimental curves are of much larger magnitude in .the small angle region than in the large angle region. the three parameter linear variational procedure given here is considered to be a satisfactory improvement of the Born approximation in providing a theoretical basis for the elastic scattering of electrons by hydrogen atoms at high velocities of impact.


"IF Y,









- t *


Sf


I


S.
t. -


1':


, J -. *. .. : ,


i











BIBLIOGRAPHY



Born, M., Zeits. f. Physik 38, 803 (1926). Harnwell, G. P., Phys. Rev. 34, 661 (1929). Heitler, W. and F. London, Zeits. f. Physik 44, 455 (1927). Huang, Su-Shm, Phys. Rev. 76, 477 (1949). Hulthin, L., Kungl. Fysio. Sallskapets Lund Forhknd. 14. 1 (1944). Hulthin, L. and P. 0. Olsson, Phys. Rev. 79, 531 (1950). Kohn, W., Phys. Rev. 74. 1763 (1948). Massey, H. S. W. and C. B. 0. Mohr, Proc. Roy. Soc. A 132, 608
(1931).

Massey, H. S. W. and C. B. 0. Mohr, Proc. Roy. Soc. A 135, 258
(1932).

Massey. H. S. W. and C. B. 0. Mohr, Proc. Roy. Soc. A 146, 880
(1934).

Massey. H. S. W. and B. L. Moiseiwitach, Proc. Roy. Soc. A 205,
483 (1951).

Moiseiwitsch, B. L., Phys. Rev. 82. 753 (1951). Mott, N. F. and H. S. W. Massey, Theory of Atomic Collisions
(Clarendon Press. Oxford, 1949), second edition. Webb, Glenn M., Phys. Rev. 47, 384 (1935).









52











BIOGRAPHICAL SKETCH



June Grimm Jones was born In Wheeling, West Virginia, on October 16, 1918. She was awarded the A. B. degree in mathematics and physics by West Virginia University in 1938. After her undergraduate work, she was on the actuarial staff of Acacia Mutual Life Insurance Company located in Washington. D. C. until December, 1940 when she resigned to accompany her husband. Mark Wallon Jones, on an assignment in South America. Upon her return to the United States, she enrolled in the graduate school of the University of Florida in the summer of 1947. She was granted the Master of Science degree in June, 1948, by the Department of Physical. The following two years. she was an assistant member of the research staff of the Geophysical Institute located in College, Alaska. In September, 1950. she returned to the University of Florida to pursue studies in theoretical physics leading to the degree of Doctor of Philosophy. She is a member of Phi Beta Kappa. Phi Kappa Phi, Sigma Pi Sigma, Sigma Xi, and Kappa Kappa Gamma social fraternity.






53

53












This dissertation was prepared under the direction of the co-chairman of the candidate's supervisory committee and has been approved by all members of the committee. It was submitted to the Dean of the College of Arts and Sciences and to the Graduate Council and was approved as partial fulfilment of the requirements for the degree of Doctor of Philosophy.


June 8, 1953


Dean, College of Arts and Sciences Deea, Graduate School


SUPERVISORY COMMITTEE:



hairman

I I A/
Co-chairman




Full Text
A
o Experimental values


TABLE OF CONTENTS
Page
LIST OF TABLES iv
LIST OF ILLUSTRATIONS v
Part
I. INTRODUCTION I
II. THE VARIATIONAL METHOD 13
HI. THE CALCULATIONS 22
IV. COMPARISON OF THEORETICAL RESULTS
WITH THE EXPERIMENTAL DATA 36
V. CONCLUSIONS 50
BIBLIOGRAPHY 52
iii


Fig. 4. The cross sections for the elastic scattering of
350 volt electrons by hydrogen atoms calculated by the linear
variational procedure employing the trial function (57) with the param-
XI f- '
eters A, b, B (c,C 0) and with the parameters A, B, C (b, c = 0)
t ; 4 V- ; :
**
are compared to those obtained employing the same trial functions
: [T-it'l
with the parameters A, b, c (B, C = 0). Ordinate, Elastic scattering
18
cross sections in units of 10" cm. 2 Abscissa, Angle through
which the scattering takes place.


BIOGRAPHICAL SKETCH
June Grimm Jones was born in Wheeling, West Virginia,
on October 16, 1918. She was awarded the A. B. degree in
mathematics and physics by West Virginia University in 1938.
After her under graduate work, she was on the actuarial staff of
Acacia Mutual Life Insurance Company located in Washington, D. C.
until December, 1940 when she resigned to accompany her husband,
Mark Wallon Jones, on an assignment in South America. Upon her
return to the United States, she enrolled in the graduate school of
the University of Florida in the summer of 1947. She was granted
the Master of Science degree in June, 1948, by the Department of
Physics. The following two years, she was an assistant member of
the research staff of the Geophysical Institute located in College,
i i
Alaska. In September, 1950, she returned to the University of
Florida to pursue studies in theoretical physics leading to the degree
of Doctor of Philosophy. She is a member of Phi Beta Kappa, Phi

Kappa Phi, Sigma Pi Sigma, Sigma Xi, and Kappa Kappa Gamma
social fraternity.
53


12
Another method due to Massey and Mohr, ^ uses the second
Born approximation only to obtain a "polarization" potential. With
this potential added to the static potential of the atom the cross sections
are recalculated by the method of the first Born approximation.
Although the method cannot be justified rigorously (in the mathematical
sense) the agreement of the theoretical and observed cross section
curves is considerably better than that afforded by the first Born
6
approximation at small angles.
There is still considerable room for improvement of the
first Born approximation by some simple analytical means. In search
of such a method we have investigated variational techniques.

9
H. S. W. Massey and C. B. O. Mohr, Proc. Roy. Soc.
A 146, 880 (1934).


20
3. The disturbance or "polarization" of the atomic field
by the incident and scattered electron waves;
4. The exchange of electrons between the incident electron
wave and the atom.
The trial function adopted should allow for as many of these
component processes as possible. The number of parameters in our
trial function was limited to three in order that the calculations
15
should not become unduly cumbersome. (Massey and Moiseiwitsch
found three parameters sufficient to give excellent results--see above0
Adapting Huang's trial function (31)( which gave such good results at
low energies, for use at high energies, we obtain a trial function
The first term in the bracket gives the Born approximation and
therefore allows for the first process in the list above. The second
term allows for effects on the incident wave close to the hydrogen
atom since exp(-r2/aQ) falls off rapidly for values of r£ greater than
aQ, the radius of the first Bohr orbit; this term attempts to take care
of the distortion of the incident wave by the field of the atom listed as
part of process 2. The third term (cr j^/a^eapi-r^/e^) ^ow8 *or
the polarization of the atomic field by its explicit dependence on


25
{(5/4)ln(l t 4l£4> + (J/2)Xn(l + 44
+ (l/4)ln<9 t 4k*4) The coefficient DQ is the same as the Born approximation scattering
amplitude of Eq. (20). Some of the coefficients depend only on
the kinetic energy of the incident electron (i. e., kQ), while other
coefficients depend both on the kinetic energy of the incident electron
and the scattering angle O (i. e., K).
B. Linear Determination of Parameters--!! the variational
method of Kohn is paralleled [cf. (27)] we have from Eq. (37) and
Eq. (46) the following set of linear equations
W 3b = Di + 2D2b + D?c + DgA 0 (47a)
3c = D3 + D?b + 2D4c + D9A 0 (47b)
(4irE a*)-1 L/ 3A D_ + Db + Dgc + 2DA -1
(47c)
for determining b, c, and A. Whence,
L00(b.c.A) = -4 7TE0a2(A f^)
(48a)


37
In /Ijj = (1 + sinx/x) 2
where x s 2kQd sin ( & /2), d being the equilibrium nuclear separation
of the molecule, and y = koaQ8in( 0 /Z). This ratio performs a
damped oscillation about unity as a function of (v)sin( & ¡Z). For an
energy of 350 volts over the range of Webb's experimental points,
its maximum value is 1. 4 at 3 = 5 and its minimum value is
0. 78 at 0 30. This relationship was used to convert Webb's
experimental data at 350 volts for molecular hydrogen to equivalent
L
data for atomic hydrogen.
B. Comparison of Calculated Scattering Intensities with
Experimental Data--The experimental values of the cross sections are
given in arbitrary waits. Hence, for a comparison to the results of
this investigation, the adjusted experimental data (cf. section A
above) were fitted to coincide at 25 with the theoretical values ob
tained by the linear (Kohn) variational procedure as shown in Figure 1.
In addition to these curves, the results from the Born formula are
also given in Figure 1.
An examination of this figure reveals that at small angles
( 0 < 30), where the deviation of the Born approximation curve
from the experimental points is very large, the curve obtained by the
linear variational method fits the experimental curve very well except
near 7. The dip in the theoretical curve at around 7 is probably not


II. THE VARIATIONAL METHOD
A. Methods of Hulthen and Kohn--The variational
procedures introduced by Hulthn^ and Kohn* treat the one-body
problem consisting of the scattering of a particle by a potential V(r).
They involve the construction of an integral
M*i. -iT2) ^ J Y2(e Hj^dr. (23)
where (E H) = [ +- k^ V(r)j When Y is a proper solution
of H Y = BY, satisfying the boundary condition
Yj ^eaq>(iSj rj + fflc^l^expikrl/r (i 1, 2), (24)
v+cp
the integral (23) vanishes. In the event that an exact solution llT
is not known but a trial function ^ Y + £ Y can be set up,
which has the same asymptotic form as Y of (24) but with f replaced
by f + & f, then it can be shown that to first order, 11
*144,-1^) = -47T SfflTj.-T^). (25)
Hence a variational procedure is available for calculating the unknown
1L. Hulthen, Kungl. Fysio. Skllskapets Lund FOrhand. 14,
UW. Kohn, Phys. Rev. 74, 1763 (1948).
13
1 (1944).


15
Variational techniques have been applied succe86£uliy to
the elastic scattering of low energy (< 13 volts) electrons by
12
hydrogen atoms. Huang, using a modified Hulthen variational
13
procedure, calculated the phase shifts for S-wave scattering.
He employed a trial function
YrpTjj) = [expf-rj/rj jsinkr2 + u(rjr^coskrJ (31)
u = [l exp(-r2)J £ a + (b + cr. 2)exp(-r2)J ,
where a, b, and c are variational parameters. His results were
14
inconclusive, however, due to a mistake in his method.
15
Massey and Moiseiwitsch applied the variational methods
of Kohn and Hulthen to the elastic scattering of slow electrons by
H-atoms. They used Huang's trial function and combinations of it
to allow for exchange in the determination of the S-wave phase shifts.
i2Su-Shu Huang, Phys. Rev. Tj>, 477 (1949).
13
The scattered wave can be considered as a sum of partial
waves of phase r\n< (See, e. g., Mott and Massey, op. cit,, Chap.
II.) The S-wave scattering is that part of the scattering which is
spherically symmetric (not dependent on the angle e ) and the total
S-wave cross section is given by Q = (4 TT/k2) sin2 vjq. S-
scattering predominates at low energies.
-wave
14
(1950).
L. Hulthen and P. O. Olsson, Phys. Rev. 79, 531
15.
H. S. W. Massey and H. L. Moiseiwitsch, Proc. Roy.
Soc. A 205, 483 (1951).


41
Born approximation (20), are functions of (v)sin( 9/2) only.
In Figure 2, the results for the b, c parameter trial
functions (54, (55) are compared to those previously derived for the
A, b, c parameter trial functions (44), (45). The curves are very
similar. It seems then that the good fit obtained with the linear method
and trial functions (44), (45) to the experimental data, as shown in
Figure 1, at the small angles is provided predominantly by the
contributions of parameters b and c to the scattering amplitude. For
the b, c parameter case, the parameter A is zero, and the quadratic
variational procedure gives a zero scattering coefficient. From these
considerations, it is understandable that the linear variational method
has yielded better results than the quadratic variational method for
the trial functions (44), (45) as discussed above.
The linear variational procedure using the trial functions
considered so far provides no improvement of the Born approximation
at large angles. At large angles, the experimental curve of the
scattering cross section becomes almost independent of angle instead
of falling off uniformly with increase of angle as predicted by the Born
approximation. In Eq. (46) the coefficients of the terms involving
the parameter A are independent of angle. From this, it seems that
parameters in the scattered wave part of the trial function may allow
for the improvement of the theoretical curves at large angles. Con
sidering a trial wave function of the general form


V. CONCLUSIONS
The success of wave mechanics in explaining phenomena
beyond the realm of classical physics has been unqualified. A major
theoretical problem is resolving the frustration of not knowing the
wave function which applies to a particular problem. In the event
that a relatively simple asymptotic form of the wave function is
known, however, a variational procedure may sometimes be employed
to provide a solution to the problem without possessing the exact wave
function. Since the complete set of hydrogen eigenfunctions is known,
the variational method outlined in Part II should be useful in obtaining
inelastic, as well as elastic, cross sections for the scattering of
electrons by hydrogen without going into partial wave analyses as
is usually done. Only the elastic scattering at high velocities of
impact has been considered here to illustrate the method. The simple
form of our trial function accounts for the distortion and polarization
effects expected to become important as the impact velocity or
scattering angle decreases to where the Born approximation is inade
quate. The deviation of our theoretical curves from the observed
scattering cross sections at large angles remains when the number
l
>
20
A variational method, using partial waves, for obtaining
inelastic cross sections has been derived by: B. L. Moiseiwitsch,
Phys. Rev. 82, 753 (1951).
50


14
f. If a trial function is
c i c2* cy
(26)
where the c. and ffc are parameters, numerical values of these param
eters may be obtained by solving the simultaneous equations derived
from (25):
or
method.
^L/ dci
dL/ M
L = 0,
L/
0,
-4 IT
according to Kohn's linear method
according to Hulthen's quadratic
s 0
(27a)
(27b)
(28a)
(28b)
Kohn's method then gives
47Tf = + 47Tft
(29)
while Hulthen's method gives
f = ft (30)
because of the stipulation that L ~ 0. It is apparent that the better
the trial function, the better the agreement between the scattering
amplitude given by Kohn's method and that given by Hulthen's method
since in the limit of "y*, 0 and the two methods give the
same results.


27
TABLE 2
PARAMETERS DETERMINED BY LINEAR METHOD
FOR 350 VOLT ELECTRONS
koao,in 0/2
b
c
A/a0
0
-8.56 +8.691
4.31 -1.231
0.921
-0. 5961
0.03
-8.42 +8. 381
4.21 -1.151
.892
- .5851
.05
-8.33 +8.751
4.25 -1.291
.900
- .6211
.1
-7.52 +8.171
4.02 -1.291
.771
- .6781
.2
-4.78 +7.061
2,77 -1. 381
.464
- .7241
3
-0.358 +4.241
0.812 -1.241
-0.0874
- .6011
.4
-1.22 + 1091
.358 +0.2611
- .0432
- .1151
.5
-2. 25 -0. 5241
.587 + 6481
a 139
- 00326i
.6
-2.40 .6491
.618 + .7121
. 244
+ 0.03761
.7
-2. 05 4541
.540 + 5841
.218
+ .03021
.8
-1.78 .4371
.454 +.5291
.209
+ .04221
.9
-1.56 .4421
.382 + 4901
.194
+ .05211
1.0
-1.32 .4211
.308 + 4411
. 168
+ .05251
1.2
-0.965 .3631
.203 +. 3601
. 122
+ 0444i
1.4
- .750 3811
. 129 +.3281
.0914
+ .04141
1.6
- .580 3601
.0768 +.29U
.0673
+ .03301
1.8
- .455 .3401
.0397 +.2621
.0502
+ .02571
2.0
- .360 3281
.0103 +.2421
.0383
+ 02051
2.5
- .203 .2911
-0.0344 +.2021
.0209
+ .01151
3.0
- .109 .2721
- .0621 +. 1781
. 0128
+.007201
3.5
- .0410- .2581
- .0819 +. 1611
.00819 +.004981
4.0
- .0156- .2371
- .0856 +. 1521
.00578
+ .003141
4.5
+0.0186- .2301
- s0932 +.143
.00435
+. 002551
5.0
+ .0439- .2191
- 105 +. 1321
.00344
+.002021
*


Fig. 1. The theoretical cross sections, obtained by the
linear (Kohn) and quadratic (Hulthfcn) variational procedures, for the
elastic scattering of 350 volt electrons by hydrogen atoms are compared
to the cross sections calculated by the Born approximation and to the
relative scattered intensities measured by Webb. Ordinate, Elastic
scattering cross sections in units of 10 cm Abscissa, Angle
through which the scattering takes place.


IV. COMPARISON OF THEORETICAL CALCULATIONS
WITH EXPERIMENTAL DATA
A. Adjustment o Experimental Data for Scattering by
Atomic Hydrogen--The bulk of the experimental data for electron-
16
hydrogen scattering has been taken for molecular hydrogen.
Harnwell, however, carried out investigations of the scattering of
electrons by both atomic and molecular hydrogen and reported that
the scattered intensities when atomic hydrogen is present are not
much different than when molecular hydrogen alone is present. Since
a difference is detectable, some allowance was made for it in the
comparison of our theoretical calculations and the experimental data.
Assuming the validity of the Born approximation, Massey
18
and Mohr have obtained the ratio of the scattering by molecular
hydrogen to tire scattering by atomic hydrogen:
1 The work of Webb is so comprehensive that his results
alone are sufficient to test the adequacy of the theoretical values of
scattering cross sections. Webb measured the scattered intensity
over a wide range of angles and voltages with a single apparatus and
found, moreover, that his results agreed closely with those obtained
by other investigators in the overlapping regions.
17
G. P. Harnwell, Phys. Rev. 34, 66 i (1929).
18
H. S. W. Massey and C. B. O. Mohr, Proc. Roy. Soc.
A 135, 258 (1932).
36


17
Y1'-* I '^ as r1 > o .
In order to derive a variational equation for calculating fQO specif
ically another solution to the same wave equation with the asymptotic
form
+ <1/r2> m K> fmo*<2|;?2),!xP
as r > ;
H1/rl) \ ^ (*2) imo*^2);l) e*P as r ^ + oo j
is introduced. The integral[(cf. (23)]
Loo 1 / *12 (34)
where H = (-l¡Mv5 + V|) (2/rl> (e2/w2) + is constructed and its first-order variation is obtained by replacing
each Y by Y + S Y, where the Y + Y have the same
asymptotic form as Y but with each f and g replaced by f -til
and g -t- Sg. Making use of Green's theorem* we then have


26
determines £ j
oo
*, <-4 1 ) 1 % '; /
OO = r>o + A + D5A + D6a2 + Dlb + D2b2 + D3c
+ D4c2 + D?bc + DgbA + D9cA (48b)
Numerical values of the coefficients Dj have been calculated for
electrons of 350 volt energy over a range of from 0 to 160.
* 1 1 i i;* '
Using these numerical values, Eqs. (47) were solved simultaneously
for the parameters b, c, and A. The values of these parameters
are listed in Table 2. Numerical values of fQO and the cross sections
w) -- IW9>I2 <>
were then obtained via (48b). These values are tabulated along
with the cross sections given by Born's formula (i. e., Eq. (20))
in Table 3. A comparison ia made graphically in Figure 1.
It is seen from Eq. (48b) that the linear variational method
gives the scattering coefficient as the sum of the Born approximation
scattering coefficient (i. e., DQ) and additional contributions from
the trial scattering amplitude (A) and from the parameters (b, c)
which were chosen to allow for the effects of distortion and polari-
*>
%
zation. Figure 1 shows that allowance for these additional contri
butions to the scattering amplitude results in a sharp rise in the cross
section curve at small angles well beyond the Born approximation


48


LIST OF TABLES
Table
1. Elastic Scattering Coefficients and Cross
Sections Calculated by Born Approximation
with and without Allowance for Exchange
Effects .
2. Parameters Determined by Linear Method
for 350 Volt Electrons 27
* 4 ,-* j..I : t it'-' i i '* 4 ;* r V-- /. .* __ *
3. Elastic Scattering Coefficients and Cross
Sections Determined by Linear Variational
Method Compared to Born Approximation. ... 28
v ...
4. Parameters Determined by Quadratic Method
for 350 Volt Electrons 33
5. Cross Sections Determined by Quadratic
Variational Method Compared to Born
Approximation 350 Volt Electrons 35
iv


2
wave function) must remain finite as and r2 approach zero,
and for large r and r2 it mu8t have the asymptotic form
Y0-* y^i) exp(ik02*r2) + m 5 *2> x
exP as r^-^* o ; (2a)
as r^* (2b)
where i^f?) i a member, with energy Em, of the orthonormal set
of hydrogen eigenfunctions; = (2m/ii ) (E Em) ; n is a unit vector
in the direction of the incident electron and of unspecified orientation;
¡5^ and n2 are unit vectors having the spatial orientations of and ?2
respectively; f and gmQ are respectively the scattering amplitude
and "exchange" scattering amplitude associated with the o m
transition.
The a priori symmetry condition imposed on the total wave
function for particles with half integral spin requires that the space
wave function be either symmetric or antisymmetric in the coordinates
of the two electrons to correspond, respectively, with antisymmetric
(spin 0) or symmetric (spin 1) spin functions such that the total
wave function is always antisymmetric. The total space wave functions


43


4
captured into the state m of the atom, are ejected into the same solid
angle dw in unit time. Thus from either Eq. (4a) or Eq. (4b) we
have
+ 1/4|£mo *mo'2] dw (S>
since the probabilities of the electrons having parallel spins and of the
2
electrons having anti-parallel spins are in a ratio: of 3 to 1. In
order to obtain the scattering cross sections, the values of the scat
tering amplitudes fmoand g.nQ must be obtained.
Returning to the wave equation (1), we rearrange the
equation and multiply through by
(fi2/2m) (V + ko> i)Y?i,rz) '/^(rj) £(-fi2/2m)V^ Eq
-(e2/r1^Y(ri.r>2) ^2/rj) (2/r12>| (6)
Since
[(-E2/2m)V^ (e2/rt>j f0(?,) E<> ^(r^).
the integration of Eq. (6) reduces via Green's Theorem to
2Ibid., Chap. V.
(7)


31
curve. At large angles, however, the two curves merge.
C. Quadratic Determination of Parameters--If the varia-
tional method of Hulthen is paralleled [cf. (28)] the restriction that
the trial function be such that the variational integral L, vanishes is
imposed. Then, from (37),
SL00 -4lrEoaJ if00 = 0 (50)
and
L0o 8 0 <51*>
3L/ 3b 0 (51b)
L/ 3c 0 (51c)
are the equations for determining b, c, and A. Note that (5ib) and
(51c) are the same as (47a) and (47b) and hence linear in the parameters,
but that (51a) sets the expression for JLQO given by (46) equal to zero
and is therefore quadratic in the parameters while its counterpart in
the Kohn method, (47c), is linear. Eq. (50) also gives
(52)
as opposed to the expression (48b) for fQO given by the linear (Kohn)
method.
The values of the parameters b and c were found in terms of


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Fig. 2. The theoretical cross sections for the elastic
scattering of 350 volt electrons by hydrogen atoms calculated by the
linear (Kohn) variational procedure employing the trial functions
(44), (45) with the parameter b (A, c 0) with the parameter c
(A, b = 0), and with parameters b, c (A = 0) are compared to those
obtained using the parameters A, b, c in the same trial function.
Ordinate, Elastic scattering cross sections in units of 10~18cm.2
Abscissa, Angle $ through which the scattering takes place


34
in Table 5 and graphically in Figure 1. It is evident that the quadratic
(Hulthen) method does not provide any significant change in cross
%
section values from those of the Born approximation.
It should be noted that Eq. (13) is a necessary, but not
sufficient, condition for the trial wave function to be a good approxi>
mation to the true wave function and that the linear (Kohn) variational
procedure is such that this integral equation is satisfied identically.
On the other hand, it should be noted that L>00 0 is also a necessary,
but not sufficient, condition for the trial wave function to be a good
approximation to the true wave function and that the quadratic
(Hulthen) method is such that this condition is satisfied identically.


III. THE CALCULATIONS
A. The Variational Integral Lr>0--To obtain the scattering
coefficient f00 for the elastic scattering of electrons by hydrogen
atoms in the ground state, the integrations involved in LM of (34)
must be carried out. In addition to the trial function
Y't* > + b exp (-r2/aQ)
+ (cr12/aiexp*"r2/ao|I + A Jxp(ikDr2)/r2J |^1
- exp(-r2/a0)J Y^r^) (44)
selected above [cf. (42)] an auxiliary trial function jcf. (33)J
= V^^iJexpf-ik^2*- ?2)[l + b exp(-r2/a0)
+ cr12/ao )e*p(-r2/.0)] + A [.xp(ik0r2)/r2J[l
- exp(-r2/ao)J ^(r^) (45)
is required. Then
Loo J h] Yti)(?1.?2)dr1dt2
= 4 7TEQa2 (D0 t Dxb + D2b2 + D3c + D4c2
+ D5A + D6A2 + D7bc + DgbA + D9cA) (46)
22


I. INTRODUCTION
In the application of wave mechanics to the calculation of
the scattering of electrons by atoms, the simplest problem which can
be treated is the collision of electrons with hydrogen atoms since the
complete set of hydrogen eigenfunctions is available. Hence, any
method proposed for treating electron-atom scattering problems must
first be shown to be successful when applied to the hydrogen atom.
The wave function "If (^, rt,) which describes the scattering
of an electron incident on a hydrogen atom in the ground state must
satisfy the wave equation"
[(R2/2m)(V2 1-V2)<-E+(e2/r,)+(e2/r2)-(e2/ri2)]y 0. (1)
where rr^, and r^ are the respective distances of the two elec
trons from the nucleus and each other. The mass of the electron is
small enough compared to that of the proton so that the motion of the
proton in the collision may be neglected. The total energy £ of the
system is the sum of the energy E0 of the atomic electron in the ground
state and of the kinetic energy mv^/2 of the incident electron. The
*N. F. Mott and H. S. W. Massey, Theory of Atomic
Collisions (Clarendon Press, Oxford, 1949), second edition: Chap.
VIII, section 2. This treatise on scattering problems gives a complete
up-to-date account of most of the work done on this problem.
1


19
and (r J exp(-ik02^2^?2) (38b)
Then since we have f^ = 0, it follows from (37) that
ioo = 1(4TE0^| (39)
Substituting (38a), (38b) into (34), we find
Lt = an E0*03 (8 1- K22)/<4 + K2a2)2 (40)
Substituting (40) into (39), we obtain
f0o< a) 2a0(B + K2.2)/(4 + K2a2)2 (41)
which i is exactly the scattering amplitude (20) obtained for the first
Born approximation. Hence, it appears that Eq. (37) provides a
variational method for improving the Born approximation.
rt* .
C. Selection of Trial Function--The scattering process we
9
are considering may be broken down into the following components:
1. Scattering by the field of the atom, considering the
atom undisturbed by the incident electron and the incident
electron only slightly affected by the collision;
2. The distortion of the incident and scattered electron
waves by the potential field of the atom (the latter
field being taken as the undisturbed atomic field);


30


49
of the Born approximation over the entire angular range of the
scattering.


39
curve.
It is interesting to note that the shapes of the curves in
Figure 1 obtained by the two variational procedures are quite similar
even to the dip appearing around 7, but that the curves are dis
placed relative to one another. It follows from Eqs. (48a) and (52)
that the closer the trial function is to the exact wave function, the
closer the two cross section curves will agree. Hence, the displace
ment of the two curves demonstrates that the trial wave function is
not a very good approximation to the exact wave function. The fact
that the trial wave function is not a good approximation to the true
wave function, however, does not necessarily mean the calculated
coefficients are not good approximations to the true scattering coef
ficients. It is a peculiarity of quantum mechanical variational proce
dures that satisfactory results can be obtained even though the simple
19
trial functions employed are relatively poor approximations. This
is not surprising, however, since in our case, for example, the
theory provides a stationary expression for the scattering coefficient,
not the wave function. The criterion for adequacy of the theoretical
scattering cross sections is still "goodness of fit" to the experimental
19
For example, excellent results for the calculation of the
binding energy of the hydrogen molecule by a variational method
employing a very simple trial function were obtained by: W. Heitler
and F. London, Zeits. f. Physik. 44, 455 (1927).


7
whose solutions yield [cf. (13)]
g^(n.2') = (2m/E2)y'exp(-ikn'*?') Zn(?'.3)dT'.
(18)
These integral equations (15), (18) for the scattering amplitudes
fno and gno are exact, and since the exact solutions cannot be carried
out by any reasonable method, various approximate solutions have
been based on them.
Before proceeding further we simplify our considerations
by restricting the problem to elastic scattering at relatively high
velocities of impact.
The simplest approximation is that due to Born. 4 For high
velocities of impact, the perturbation of the incident wave by its
interaction with the atom will be small. Under these circumstances,
Born takes as a zero-order approximation for IP", [cf. (2)]
Y(?i*^2) = exp(iko& ?2) %(?!), (19)
which assumes that the incident electron is unaffected by the atomic
field. Substituting (19) into the right-hand side of (10), and following
the calculations through Eq. (13), the Born formula for the elastic
scattering amplitude is given by
4M. Born, Zeits. f. Physik 38. 803 (1926).


18
5L00 =0*0 it'd2] V;sY sY> v,d2] Jrf
+Vo [Y'oiV^Y^*1- SY^VJ^J d^dtj. (35)
Jrf~
Substituting into (35) the asymptotic forms (33) and the variation
of (32), and performing the volume integrations, we find that the
first of these integrals vanishes so that (35) then reduces to
Shoo s Eoao / *p(-ik0r2coset) ifoo(ir(1);i?2)exp(ikor2)/r2 x
£ik0C08 where cos oc = *2)- s2. An integration by parts then gives
|cf. (25)],
Loo> *
(37)
Eq. (37) provides a variational equation for calculating fOQ. A similar
equation is obtained for gQQ if the roles of iT, and ?2 are interchanged
We try out this variational equation with the simple trial
functions (the one used for the Born approximation -- cf. Eq. (19)):
Y^ (?1 *2) s V'ol?l) exp(ikon (38a)


8
00'2a0(* + kV)/(4 + K2.2)2
(20)
where K 2kosin(0/2) and aD = E2/me2 is the radius of the first
Bohr orbit. Since kQ = 2TTmv/h, the scattering amplitude is a
function of the product of the velocity of the incident electron and the
sin (0/2), where 0 is the angle through which the scattering takes
place.
The "exchange" scattring amplitude gQO for the first Born
approximation obtained by substituting (19) into (18) via (17), is given
by Massey and Mohr^ as
(21)
where the gn are functio ns of kQao and the Pn(cos 0) are Legendre
polynomials. The actual numerical calculations of gOQ were not
carried out for the high energies of interest to us by Massey and Mohr.
We have therefore calculated g for an incident electron energy of
350 volts for values of koaosin(0/2) from 0 to 5, which corresponds to
an angular rango of 0 to 160. These values are compared with the
corresponding fQo values in Table 1. The table also contains the values
of the elastic scattering cross sections [cf. Eq. (5)] with and without
the contribution of "exchange" (i. e., gQQ) to the scattering. It is
^H. S. W. Massey and C. B. O. Mohr, Proc. Roy. Soc.
A 132, 608 (1931)


A Variational Procedure For Improving The Born
Approximation As Applied To The Scattering
Of Electrons By Hydrogen Atoms
By
JUNE GRIMM JONES
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
June, 1953


Yt a Yo&i) j^exp(ik0n- r2) + bexp(-r2/a0) + (cr12/ao)exp(-r2/ao)
+ 0/r2)exp(ikor2) ^ 1 exp( -r2/aQ)j x
./
{a + B exp(-r2/a0) + (Cr l2/ao) exp(-r2/aQ)} J f (57)
we have examined the contributions of the various terms to the
scattering by carrying out the cross section calculations with trial
functions using various combinations of the five parameters in (57).
In Figures 2, 3, and 4, the results of the various calculations are
graphed. It is evident that the parameter b or c alone (Fig. 2) is not
sufficient. Both b and c are required to give a good fit to the experi-
t*
mental data at small angles. The effect of the parameters A, 3, and
C is to raise the cross section curve at large angles (Figs. 3, 4).
When A alone is used, the curve (Fig. 3) is raised at large angles;
when A is used in combination with either b or c, or with b and c,
no effect of A at large angles is evident. As the number of parameters
in the scattered wave part of the trial function is increased, the rise
/ '
of the scattering curve at large angles also increases (Fig. 4).
Since the parameters in the inoident wave part of the trial function
serve not only to improve the theoretical results at small angles but
also to cut down the effectiveness of the parameters in the scattered
wave part for raising the theoretical curve at large angles, we feel that
a trial function employing all five parameters will provide improvement


16
The fact that both (Kohn and Hulthn) variational methods gave very
nearly the same results indicates that the trial function (31) is a
fairly good approximation to the exact wave function for the scattering
of slow electrons. In the absence of experimental data for e-H
scattering at low energies, the scattering amplitudes were checked
with the previous results obtained by direct numerical integrations
of the differential equation -- the agreement was found excellent.
On the basis of the good results obtained for the elastic
scattering of slow electrons, it should be possible to apply a vari
ational technique to investigate the elastic scattering of fast electrons
by hydrogen atoms. For this purpose the variational equations of
Huang or Massey and Moiseiwitsch cannot be applied since they are
.* jp ,
formulated on the assumption that only S-wave scattering takes place.
B. Variational Procedure for Improving the Born Approx
imation for e-H Scattering--A solution of the wave equation (1) for
the scattering of an electron by a hydrogen atom has the asymptotic
form (cf. Eq. (2))
T(; yo(?i)e*p(ikos<1>.f2)
+ (*/r2> m £Tno f**i,i2)xP(ikmr2) <32)
as r<=*>;


9
TABLE 1
ELASTIC SCATTERING COEFFICIENTS AND CROSS SECTIONS
CALCULATED BY BORN APPROXIMATION WITH AND WITHOUT
ALLOWANCE FOR EXCHANGE EFFECTS
V0.me/2 £00/0 g<*>o IW* ll9)=
3/4l£-gl2+l/4|£ 0.00
1.0000
-0.0288
27.9
28.7
0.03
.999
-0.0288
27.9
28.7
0.05
.996
- .029
27.7
28.5
0. 10
.985
- .029
27.1
27.7
0.20
.943
- .029
24.8
25.6
0. 30
.880
- .028
21.6
22. 3
0.40
.803
- .028
18.0
18. 6
0.50
.720
- .027
14.5
15.0
0. 60
.638
- .026
11.4
11.8
0.70
. 561
- .025
8.78
9.18
0.80
.491
- .024
6.73
7.06
0.90
.429
- .023
5.14
5.41
1.0
. 375
- .022
3.93
4. 18
1.2
.289
- .019
2. 33
2.49
1.4
.226
- .017
1.43
1.54
1.6
. 180
- .014
0.904
0.979
1.8
. 146
- .010
0.593
0.636
2.0
. 120
- .007
0.402
0.427
2.5
.0773
- .0011
0.172
0.169
3.0
.0550
0.0024
0.084
0.081
3.5
.0406
+ .0026
0.046
0.043
4.0
.0311
.0001
0. 027
0.027
4.5
.0246
-0.0018
0.017
0.018
5.0
.0200
+0.0034
0.011
0.0092


32
the parameter A by a simultaneous solution of Eqs. (5lb) and (5lc).
Substituting these values into Eq. (51a) we obtained an equation of
the second degree in A. Using the calculated values of (see above)
the two possible values of A were then calculated from this equation
over the angular range considered--these two sets of values being
equivalent to two sets of fOQ values via (52). A method for discrim
inating between these two sets of f values was necessary and a test
15
similar to that used by Massey and Moiseiwitsch was employed.
Eq. (13) is an exact integral equation for the scattering
coefficient fOQ. The integral of Eq. (13) was evaluated using the
Yoi (44) corresponding to each of the two sets of parameters
resulting from the quadratic equation in A. The set of parameters
for which the integral was more nearly equal to the value of A
( fQO) tested was then the set of parameters selected.
The selected set of parameters b c, and A calculated
for 350 volt electrons by the quadratic variational method is given in
Table 4. The corresponding values of the integral in (13) are also
listed for comparison with the values of the scattering coefficient A.
It is apparent from the table that the agreement between the integral
equation values of fQO and the values of A is not very good except at
large angles. But it is still sufficiently good to discriminate between
the two sets of fQO values. The cross sections calculated by this
method are compared to those calculated by the Born approximation


u
highest energy, 912 volts. At this energy, the experimental points
lie on the calculated curve from 10 to 60. The angular range of
the agreement decreases with decreasing energy. The experimental
curves rise much more rapidly at small angles than the theoretical
curves do. At large angles the observed points lie consistently above
the calculated curves. At the lower voltages (30-100 volts) there is
scarcely any agreement. At 350 volts, the Born formula seems ade
quate from 40 to 60; the departure of the experimental curve from
the theoretical curve at the small angles is very pronounced; although
the agreement is not good at the large angles, the deviations are small
compared to those at the small angles. Thus, the scattering of 350
volt electrons can be taken as a good test case of the adequacy of any
theoretical calculation which attempts to improve to some extent the
Born approximation. It is for this reason that we adopted this partic
ular energy for the work described in the following pages.
The second Born approximation involves the substitution of
the expression (12) for FQ(r^), obtained by the first Born approximation,
as part of the function Y(ritl?^) in Eq. (10) and the integrations of the
equations a second time. This procedure is so involved that the calcu
lation has never been carried through. It would be preferable to begin
with a more accurate approximation of than to obtain
successive Born approximations by iteration.


35
TABLE 5
CROSS SECTIONS DETERMINED BY QUADRATIC VARIATIONAL
METHOD COMPARED TO
BORN APPROXIMATION350 VOLT ELECTRONS
: f : : .
koao8in 9/2
e
I(^)
Quad. Var. Method
Born Approx.
0
0
O'
209 x 10"18cm2
27.9 x 10i8cm2
0.05
1
8
204
27.7
. 1
2
16
182
27. 1
.2
4
32
99.1
24.8
.3
6
44
30.5
21.6
.4
9
0
2. 29
18.0
.5
11
18
6.87
14.5
.6
13
34
9.89
11.4
.7
15
58
8.97
8.78
.8
18
4
7.27
6.73
.9
20
'
28
5. 58
5. 14
1.0
22
46
3.97
3.93
1.2
27
22
2. 04
2.33
1.4
32
4
1.10
1.43
1.6
36
48
0.629
0.904
1.8
41
36
.381
.593
2.0
46
28
.248
.402
2.5
J59
6
.0968
. 172
3.0
72
34
.0469
.084
3.5
87
18
.0249
.046
4.0
104
10
.0149
.027
4.5
125
8
.00851
.017
5.0
160
56
.00550
.011


10
apparent that the "exchange" effect can be neglected at such high
energies and the cross section for elastic scattering of electrons by
hydrogen atoms can then be written simply
I00M lf00|2 <22>
as follows from Eq. (5) with gOQ 0.
The adequacy of the Born approximation is determined by a
comparison of the elastic scattering cross sections calculated from
(22) and (20) with observed scattered intensities. Such a comparison
was made by Webb6 in his report of the experimental determinations
n
of cross sections for the elastic scattering by hydrogen of electrons
with incident energies varying from 30 to 912 volts over the angular
range from 5 to 150. In accordance with the assumption that the
interaction of electron and atom is very small, the agreement between
g
the Born approximation and experimental curves is best at the
6Glenn M. Webb, Phys. Rev. 47, 384 (1935).
n
'Molecular hydrogen was used. For the comparison of theo
retical and experimental results, the cross sections calculated for
atomic hydrogen by the Born formula were transformed by a suitable
factor to equivalent cross sections for scattering by molecular
hydrogen. See part IV A.
g
The intensities of the scattered electrons at different an
gles are measured only relative to each other in scattering experiments.
Hence the cross sections are reported in arbitrary units and the curves
are compared with the corresponding theoretical curves by adjusting
the ordinates of the experimental curves so as to obtain the best fit to
the theoretical curves.


28
TABLE 3
ELASTIC SCATTERING COEFFICIENTS AND CROSS SECTIONS
DETERMINED BY LINEAR VARIATIONAL METHOD
COMPARED TO BORN APPROXIMATION
koaoflin diZ
*oo
$>
Linear Var.
Method
Born
Approximate n
0
*9.91 + 5.541
3595 x 10*Icm2 27. 9 x 10-18cm2
0.03
-9.54 5.20i
3295
27.9
.05
-9.31 +5. 42i
3239
27.7
.1
-7.65 + 4. 56i
2213
27. 1
.2
-2.64 + 2.441
362
24.8
.3
0.456 + 0.5831
15.3
21.6
.4
.900 -0.0838
22.8
18.0
.5
1.12 +0. 100
35. 3
14.5
.6
1. 36 + 212i
37.2
11.4
.7
0.975 + .1811
27*4
8. 78
4 r* ^
.8
.808 t .1631
19.0
6.73
.9
.663 + 1431
12.8
5.14
1.0
. 535 +. 113
8. 34
3.93
v 1.2
.363 + 04941
3.75
2. 33
1.4
.261 + 04541
1.95
1.43
1.6
.196 + .02921
1.09
0.904
1.8
.153 + .01911
0.661
.593
2.0
.123 +.01311
.425
.402
2.5
.0771 +.005651
. 167
.172
3.0
.0544 +.00292
.0829
.084
3.5
.0400 +.00167
.0448
.046
4.0
.0308 +.00104
.0265
.027
4.5
.0244 +.00074
.0166
.017
%


BIBLIOGRAPHY
Born, M., Zeits. £. Physik 38, 803 (1926).
Harnwell, G. P., Phys. Rev. 34. 661 (1929).
Heitler, W., and F. London, Zeits. £. Physik 44, 455 (1927).
Huang, Su-Shu, Phys. Rev. 76, 477 (1949).
Hulthen, L., Kungl. Fysio. Skllskapets Lund Forhhnd. L4, 1 (1944).
Hulthen, L. and P. O. Olsson, Phys. Rev. 79, 531 (1950).
Kohn, W., Phys. Rev. 74, 1763 (1948).
Massey, H. S. W. and C. B. O. Mohr, Proc. Roy. Soc. A 132, 608
(1931).
Massey, H. S. W. and C. B. O. Mohr, Proc. Roy. Soc. A 135, 258
(1932).
Massey, H. S. W. and C. B. O. Mohr, Proc. Roy. Soc. A 146, 880
(1934).
Massey, H. S. W. and B. L. Moiseiwitsch, Proc. Roy. Soc. A 205.
483 (1951).
Moiseiwitsch, B. L., Phys. Rev. 82, 753 (1951).
Mott, N. F. and H. S. W. Massey, Theory of Atomic Collisions
(Clarendon Press, Oxford, 1949), second edition.
Webb, Glenn M., Phys. Rev. 47, 384 (1935).
52


A Variational Procedure For Improving The Born
Approximation As Applied To The Scattering
Of Electrons By Hydrogen Atoms
By
JUNE GRIMM JONES
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
June, 1953

ACKNOWLEDGEMENT
The author wishes to acknowledge her great debt to
Dr. M. M. Gordon, co-chairman of the supervisory committee,
without whose guidance and assistance the work could not have been
carried out and to Dr. R. C. Williamson, chairman of the
committee, for helpful suggestions concerning the dissertation and
guidance throughout her graduate work. She also wishes to thank
the other members of the supervisory committee: Dr. J. W. Flowers,
Dr. H. P. Hanson, Dr. A, G. Smith, and Dr. C. B. Smith.
ii

TABLE OF CONTENTS
Page
LIST OF TABLES iv
LIST OF ILLUSTRATIONS v
Part
I. INTRODUCTION I
II. THE VARIATIONAL METHOD 13
HI. THE CALCULATIONS 22
IV. COMPARISON OF THEORETICAL RESULTS
WITH THE EXPERIMENTAL DATA 36
V. CONCLUSIONS 50
BIBLIOGRAPHY 52
iii

LIST OF TABLES
Table
1. Elastic Scattering Coefficients and Cross
Sections Calculated by Born Approximation
with and without Allowance for Exchange
Effects .
2. Parameters Determined by Linear Method
for 350 Volt Electrons 27
* 4 ,-* j..I : t it'-' i i '* 4 ;* r V-- /. .* __ *
3. Elastic Scattering Coefficients and Cross
Sections Determined by Linear Variational
Method Compared to Born Approximation. ... 28
v ...
4. Parameters Determined by Quadratic Method
for 350 Volt Electrons 33
5. Cross Sections Determined by Quadratic
Variational Method Compared to Born
Approximation 350 Volt Electrons 35
iv

LIST OF ILLUSTRATIONS
Figure Page
1. The theoretical cross sections, obtained by the
linear (Kohn) and quadratic (Hulthn) variational
procedures, for the elastic scattering of 350 volt
electrons by hydrogen atoms are compared to the
cross sections calculated by the Born approximation
and to the relative scattered intensities measured
by Webb 30
2. The theoretical cross sections for the elastic
scattering of 350 volt electrons by hydrogen atoms
calculated by the linear (Kohn) variational
procedure employing the trial functions (44), (45)
with the parameter b (A, c 0), with the parameter
c(A,b 0), and with parameters b, c (A = 0) are
compared to those obtained using the parameters
A, b, c in the same trial function 43
3. The theoretical cross sections for the elastic
scattering of 350 volt electrons by hydrogen atoms
calculated by the linear variational method
employing the trial functions (44), (45) with the
parameter A (b, c = 0), with the parameters
A, b (c 0), and with the parameters A, c (b = 0)
compared with those obtained using the parameters
A, b, c in the same trial function 46
4. The cross sections for the elastic scattering of
350 volt electrons by hydrogen atoms calculated
by the linear variational procedure employing the
trial function (57) with the parameters A, b, B
(c,C = 0) and with the parameters A, B, C
(b, c = 0) are compared to those obtained
employing the same trial functions with the
parameters A, b, c (B, C ~ 0). 48
v

I. INTRODUCTION
In the application of wave mechanics to the calculation of
the scattering of electrons by atoms, the simplest problem which can
be treated is the collision of electrons with hydrogen atoms since the
complete set of hydrogen eigenfunctions is available. Hence, any
method proposed for treating electron-atom scattering problems must
first be shown to be successful when applied to the hydrogen atom.
The wave function "If (^, rt,) which describes the scattering
of an electron incident on a hydrogen atom in the ground state must
satisfy the wave equation"
[(R2/2m)(V2 1-V2)<-E+(e2/r,)+(e2/r2)-(e2/ri2)]y 0. (1)
where rr^, and r^ are the respective distances of the two elec
trons from the nucleus and each other. The mass of the electron is
small enough compared to that of the proton so that the motion of the
proton in the collision may be neglected. The total energy £ of the
system is the sum of the energy E0 of the atomic electron in the ground
state and of the kinetic energy mv^/2 of the incident electron. The
*N. F. Mott and H. S. W. Massey, Theory of Atomic
Collisions (Clarendon Press, Oxford, 1949), second edition: Chap.
VIII, section 2. This treatise on scattering problems gives a complete
up-to-date account of most of the work done on this problem.
1

2
wave function) must remain finite as and r2 approach zero,
and for large r and r2 it mu8t have the asymptotic form
Y0-* y^i) exp(ik02*r2) + m 5 *2> x
exP as r^-^* o ; (2a)
as r^* (2b)
where i^f?) i a member, with energy Em, of the orthonormal set
of hydrogen eigenfunctions; = (2m/ii ) (E Em) ; n is a unit vector
in the direction of the incident electron and of unspecified orientation;
¡5^ and n2 are unit vectors having the spatial orientations of and ?2
respectively; f and gmQ are respectively the scattering amplitude
and "exchange" scattering amplitude associated with the o m
transition.
The a priori symmetry condition imposed on the total wave
function for particles with half integral spin requires that the space
wave function be either symmetric or antisymmetric in the coordinates
of the two electrons to correspond, respectively, with antisymmetric
(spin 0) or symmetric (spin 1) spin functions such that the total
wave function is always antisymmetric. The total space wave functions

3
are then
Yfil'fy t Y&.?,) <3>
which have the asymptotic forms for large
^(fyexpilc02.?2) + (l/r2) ^ [fmo(J;n2)
ilmotf^y] ** <4a)
and for large r ^
* ^<*>xP + m [mo^l)
mo(2|l)] e*P(i*ori> D*Z>* <4b>
For calculation purposes it is assumed that the intensity of the
beam of electrons falling on a hydrogen atom initially in the normal
state is such that one electron crosses unit area per unit time. The
scattering of the electrons from a beam of unit intensity is measured
in terms of the differential cross section dw, defined as the
number of electrons which fall per unit time on an area dS (dS = r2dw)
ft
placed at a large distance r from the scattering atom. For the o-*m
transition, the differential cross section 1 (e$dw is the sum of the
number of electrons which, after exciting the state m in the atom, are
scattered into the solid angle dw located at (4) in unit time and the
number of electrons which, after the incident electrons have been

4
captured into the state m of the atom, are ejected into the same solid
angle dw in unit time. Thus from either Eq. (4a) or Eq. (4b) we
have
+ 1/4|£mo *mo'2] dw (S>
since the probabilities of the electrons having parallel spins and of the
2
electrons having anti-parallel spins are in a ratio: of 3 to 1. In
order to obtain the scattering cross sections, the values of the scat
tering amplitudes fmoand g.nQ must be obtained.
Returning to the wave equation (1), we rearrange the
equation and multiply through by
(fi2/2m) (V + ko> i)Y?i,rz) '/^(rj) £(-fi2/2m)V^ Eq
-(e2/r1^Y(ri.r>2) ^2/rj) (2/r12>| (6)
Since
[(-E2/2m)V^ (e2/rt>j f0(?,) E<> ^(r^).
the integration of Eq. (6) reduces via Green's Theorem to
2Ibid., Chap. V.
(7)

5
(8)
If we define a function
Fo(?2> 51 /
(9)
it follows from (8) that
|(e2/r2) (2/ru)J
= $o (10)
The solution F0(r^) of this integro-differential equation has the
asymptotic form
Fo(?2^ exp(lko*?2) + fooexp(ikor2)/r2 U1)
as follows from Eqs. (9) (2). The required solution is3
FQ(?2) exP(*** *>) (2m/4tlR2) x
o 2
f r*p(ik0|?2 >/ |?2 -?'|] f0(?'.3)dt', (12)
3Ibid., Chap. VI.

6
with, as a result,
as follows from (11). Eq. (10) is only one of a set of simultaneous
integro-differential equations for the functions
By the same method, an integral equation of the form (13) can be
< i *'
found for the scattering amplitude f^ corresponding to each function
^(3,5*) = (2m/fi2)^exp(-iknn'* r') $n(?',n)dt'.
Similarly, for the set of functions
(15)
Gn(i>> =
r*(r2)Y(rlt72)Z:
1* AZ*
(16)
we have a set of simultaneous integro-differential equations corre
sponding to (10)
(h2/2m)(Vf + k^)Gn(?)
- ^n(*i3) ,
(17)

7
whose solutions yield [cf. (13)]
g^(n.2') = (2m/E2)y'exp(-ikn'*?') Zn(?'.3)dT'.
(18)
These integral equations (15), (18) for the scattering amplitudes
fno and gno are exact, and since the exact solutions cannot be carried
out by any reasonable method, various approximate solutions have
been based on them.
Before proceeding further we simplify our considerations
by restricting the problem to elastic scattering at relatively high
velocities of impact.
The simplest approximation is that due to Born. 4 For high
velocities of impact, the perturbation of the incident wave by its
interaction with the atom will be small. Under these circumstances,
Born takes as a zero-order approximation for IP", [cf. (2)]
Y(?i*^2) = exp(iko& ?2) %(?!), (19)
which assumes that the incident electron is unaffected by the atomic
field. Substituting (19) into the right-hand side of (10), and following
the calculations through Eq. (13), the Born formula for the elastic
scattering amplitude is given by
4M. Born, Zeits. f. Physik 38. 803 (1926).

8
00'2a0(* + kV)/(4 + K2.2)2
(20)
where K 2kosin(0/2) and aD = E2/me2 is the radius of the first
Bohr orbit. Since kQ = 2TTmv/h, the scattering amplitude is a
function of the product of the velocity of the incident electron and the
sin (0/2), where 0 is the angle through which the scattering takes
place.
The "exchange" scattring amplitude gQO for the first Born
approximation obtained by substituting (19) into (18) via (17), is given
by Massey and Mohr^ as
(21)
where the gn are functio ns of kQao and the Pn(cos 0) are Legendre
polynomials. The actual numerical calculations of gOQ were not
carried out for the high energies of interest to us by Massey and Mohr.
We have therefore calculated g for an incident electron energy of
350 volts for values of koaosin(0/2) from 0 to 5, which corresponds to
an angular rango of 0 to 160. These values are compared with the
corresponding fQo values in Table 1. The table also contains the values
of the elastic scattering cross sections [cf. Eq. (5)] with and without
the contribution of "exchange" (i. e., gQQ) to the scattering. It is
^H. S. W. Massey and C. B. O. Mohr, Proc. Roy. Soc.
A 132, 608 (1931)

9
TABLE 1
ELASTIC SCATTERING COEFFICIENTS AND CROSS SECTIONS
CALCULATED BY BORN APPROXIMATION WITH AND WITHOUT
ALLOWANCE FOR EXCHANGE EFFECTS
V0.me/2 £00/0 g<*>o IW* ll9)=
3/4l£-gl2+l/4|£ 0.00
1.0000
-0.0288
27.9
28.7
0.03
.999
-0.0288
27.9
28.7
0.05
.996
- .029
27.7
28.5
0. 10
.985
- .029
27.1
27.7
0.20
.943
- .029
24.8
25.6
0. 30
.880
- .028
21.6
22. 3
0.40
.803
- .028
18.0
18. 6
0.50
.720
- .027
14.5
15.0
0. 60
.638
- .026
11.4
11.8
0.70
. 561
- .025
8.78
9.18
0.80
.491
- .024
6.73
7.06
0.90
.429
- .023
5.14
5.41
1.0
. 375
- .022
3.93
4. 18
1.2
.289
- .019
2. 33
2.49
1.4
.226
- .017
1.43
1.54
1.6
. 180
- .014
0.904
0.979
1.8
. 146
- .010
0.593
0.636
2.0
. 120
- .007
0.402
0.427
2.5
.0773
- .0011
0.172
0.169
3.0
.0550
0.0024
0.084
0.081
3.5
.0406
+ .0026
0.046
0.043
4.0
.0311
.0001
0. 027
0.027
4.5
.0246
-0.0018
0.017
0.018
5.0
.0200
+0.0034
0.011
0.0092

10
apparent that the "exchange" effect can be neglected at such high
energies and the cross section for elastic scattering of electrons by
hydrogen atoms can then be written simply
I00M lf00|2 <22>
as follows from Eq. (5) with gOQ 0.
The adequacy of the Born approximation is determined by a
comparison of the elastic scattering cross sections calculated from
(22) and (20) with observed scattered intensities. Such a comparison
was made by Webb6 in his report of the experimental determinations
n
of cross sections for the elastic scattering by hydrogen of electrons
with incident energies varying from 30 to 912 volts over the angular
range from 5 to 150. In accordance with the assumption that the
interaction of electron and atom is very small, the agreement between
g
the Born approximation and experimental curves is best at the
6Glenn M. Webb, Phys. Rev. 47, 384 (1935).
n
'Molecular hydrogen was used. For the comparison of theo
retical and experimental results, the cross sections calculated for
atomic hydrogen by the Born formula were transformed by a suitable
factor to equivalent cross sections for scattering by molecular
hydrogen. See part IV A.
g
The intensities of the scattered electrons at different an
gles are measured only relative to each other in scattering experiments.
Hence the cross sections are reported in arbitrary units and the curves
are compared with the corresponding theoretical curves by adjusting
the ordinates of the experimental curves so as to obtain the best fit to
the theoretical curves.

u
highest energy, 912 volts. At this energy, the experimental points
lie on the calculated curve from 10 to 60. The angular range of
the agreement decreases with decreasing energy. The experimental
curves rise much more rapidly at small angles than the theoretical
curves do. At large angles the observed points lie consistently above
the calculated curves. At the lower voltages (30-100 volts) there is
scarcely any agreement. At 350 volts, the Born formula seems ade
quate from 40 to 60; the departure of the experimental curve from
the theoretical curve at the small angles is very pronounced; although
the agreement is not good at the large angles, the deviations are small
compared to those at the small angles. Thus, the scattering of 350
volt electrons can be taken as a good test case of the adequacy of any
theoretical calculation which attempts to improve to some extent the
Born approximation. It is for this reason that we adopted this partic
ular energy for the work described in the following pages.
The second Born approximation involves the substitution of
the expression (12) for FQ(r^), obtained by the first Born approximation,
as part of the function Y(ritl?^) in Eq. (10) and the integrations of the
equations a second time. This procedure is so involved that the calcu
lation has never been carried through. It would be preferable to begin
with a more accurate approximation of than to obtain
successive Born approximations by iteration.

12
Another method due to Massey and Mohr, ^ uses the second
Born approximation only to obtain a "polarization" potential. With
this potential added to the static potential of the atom the cross sections
are recalculated by the method of the first Born approximation.
Although the method cannot be justified rigorously (in the mathematical
sense) the agreement of the theoretical and observed cross section
curves is considerably better than that afforded by the first Born
6
approximation at small angles.
There is still considerable room for improvement of the
first Born approximation by some simple analytical means. In search
of such a method we have investigated variational techniques.

9
H. S. W. Massey and C. B. O. Mohr, Proc. Roy. Soc.
A 146, 880 (1934).

II. THE VARIATIONAL METHOD
A. Methods of Hulthen and Kohn--The variational
procedures introduced by Hulthn^ and Kohn* treat the one-body
problem consisting of the scattering of a particle by a potential V(r).
They involve the construction of an integral
M*i. -iT2) ^ J Y2(e Hj^dr. (23)
where (E H) = [ +- k^ V(r)j When Y is a proper solution
of H Y = BY, satisfying the boundary condition
Yj ^eaq>(iSj rj + fflc^l^expikrl/r (i 1, 2), (24)
v+cp
the integral (23) vanishes. In the event that an exact solution llT
is not known but a trial function ^ Y + £ Y can be set up,
which has the same asymptotic form as Y of (24) but with f replaced
by f + & f, then it can be shown that to first order, 11
*144,-1^) = -47T SfflTj.-T^). (25)
Hence a variational procedure is available for calculating the unknown
1L. Hulthen, Kungl. Fysio. Skllskapets Lund FOrhand. 14,
UW. Kohn, Phys. Rev. 74, 1763 (1948).
13
1 (1944).

14
f. If a trial function is
c i c2* cy
(26)
where the c. and ffc are parameters, numerical values of these param
eters may be obtained by solving the simultaneous equations derived
from (25):
or
method.
^L/ dci
dL/ M
L = 0,
L/
0,
-4 IT
according to Kohn's linear method
according to Hulthen's quadratic
s 0
(27a)
(27b)
(28a)
(28b)
Kohn's method then gives
47Tf = + 47Tft
(29)
while Hulthen's method gives
f = ft (30)
because of the stipulation that L ~ 0. It is apparent that the better
the trial function, the better the agreement between the scattering
amplitude given by Kohn's method and that given by Hulthen's method
since in the limit of "y*, 0 and the two methods give the
same results.

15
Variational techniques have been applied succe86£uliy to
the elastic scattering of low energy (< 13 volts) electrons by
12
hydrogen atoms. Huang, using a modified Hulthen variational
13
procedure, calculated the phase shifts for S-wave scattering.
He employed a trial function
YrpTjj) = [expf-rj/rj jsinkr2 + u(rjr^coskrJ (31)
u = [l exp(-r2)J £ a + (b + cr. 2)exp(-r2)J ,
where a, b, and c are variational parameters. His results were
14
inconclusive, however, due to a mistake in his method.
15
Massey and Moiseiwitsch applied the variational methods
of Kohn and Hulthen to the elastic scattering of slow electrons by
H-atoms. They used Huang's trial function and combinations of it
to allow for exchange in the determination of the S-wave phase shifts.
i2Su-Shu Huang, Phys. Rev. Tj>, 477 (1949).
13
The scattered wave can be considered as a sum of partial
waves of phase r\n< (See, e. g., Mott and Massey, op. cit,, Chap.
II.) The S-wave scattering is that part of the scattering which is
spherically symmetric (not dependent on the angle e ) and the total
S-wave cross section is given by Q = (4 TT/k2) sin2 vjq. S-
scattering predominates at low energies.
-wave
14
(1950).
L. Hulthen and P. O. Olsson, Phys. Rev. 79, 531
15.
H. S. W. Massey and H. L. Moiseiwitsch, Proc. Roy.
Soc. A 205, 483 (1951).

16
The fact that both (Kohn and Hulthn) variational methods gave very
nearly the same results indicates that the trial function (31) is a
fairly good approximation to the exact wave function for the scattering
of slow electrons. In the absence of experimental data for e-H
scattering at low energies, the scattering amplitudes were checked
with the previous results obtained by direct numerical integrations
of the differential equation -- the agreement was found excellent.
On the basis of the good results obtained for the elastic
scattering of slow electrons, it should be possible to apply a vari
ational technique to investigate the elastic scattering of fast electrons
by hydrogen atoms. For this purpose the variational equations of
Huang or Massey and Moiseiwitsch cannot be applied since they are
.* jp ,
formulated on the assumption that only S-wave scattering takes place.
B. Variational Procedure for Improving the Born Approx
imation for e-H Scattering--A solution of the wave equation (1) for
the scattering of an electron by a hydrogen atom has the asymptotic
form (cf. Eq. (2))
T(; yo(?i)e*p(ikos<1>.f2)
+ (*/r2> m £Tno f**i,i2)xP(ikmr2) <32)
as r<=*>;

17
Y1'-* I '^ as r1 > o .
In order to derive a variational equation for calculating fQO specif
ically another solution to the same wave equation with the asymptotic
form
+ <1/r2> m K> fmo*<2|;?2),!xP
as r > ;
H1/rl) \ ^ (*2) imo*^2);l) e*P as r ^ + oo j
is introduced. The integral[(cf. (23)]
Loo 1 / *12 (34)
where H = (-l¡Mv5 + V|) (2/rl> (e2/w2) + is constructed and its first-order variation is obtained by replacing
each Y by Y + S Y, where the Y + Y have the same
asymptotic form as Y but with each f and g replaced by f -til
and g -t- Sg. Making use of Green's theorem* we then have

18
5L00 =0*0 it'd2] V;sY sY> v,d2] Jrf
+Vo [Y'oiV^Y^*1- SY^VJ^J d^dtj. (35)
Jrf~
Substituting into (35) the asymptotic forms (33) and the variation
of (32), and performing the volume integrations, we find that the
first of these integrals vanishes so that (35) then reduces to
Shoo s Eoao / *p(-ik0r2coset) ifoo(ir(1);i?2)exp(ikor2)/r2 x
£ik0C08 where cos oc = *2)- s2. An integration by parts then gives
|cf. (25)],
Loo> *
(37)
Eq. (37) provides a variational equation for calculating fOQ. A similar
equation is obtained for gQQ if the roles of iT, and ?2 are interchanged
We try out this variational equation with the simple trial
functions (the one used for the Born approximation -- cf. Eq. (19)):
Y^ (?1 *2) s V'ol?l) exp(ikon
(38a)

19
and (r J exp(-ik02^2^?2) (38b)
Then since we have f^ = 0, it follows from (37) that
ioo = 1(4TE0^| (39)
Substituting (38a), (38b) into (34), we find
Lt = an E0*03 (8 1- K22)/<4 + K2a2)2 (40)
Substituting (40) into (39), we obtain
f0o< a) 2a0(B + K2.2)/(4 + K2a2)2 (41)
which i is exactly the scattering amplitude (20) obtained for the first
Born approximation. Hence, it appears that Eq. (37) provides a
variational method for improving the Born approximation.
rt* .
C. Selection of Trial Function--The scattering process we
9
are considering may be broken down into the following components:
1. Scattering by the field of the atom, considering the
atom undisturbed by the incident electron and the incident
electron only slightly affected by the collision;
2. The distortion of the incident and scattered electron
waves by the potential field of the atom (the latter
field being taken as the undisturbed atomic field);

20
3. The disturbance or "polarization" of the atomic field
by the incident and scattered electron waves;
4. The exchange of electrons between the incident electron
wave and the atom.
The trial function adopted should allow for as many of these
component processes as possible. The number of parameters in our
trial function was limited to three in order that the calculations
15
should not become unduly cumbersome. (Massey and Moiseiwitsch
found three parameters sufficient to give excellent results--see above0
Adapting Huang's trial function (31)( which gave such good results at
low energies, for use at high energies, we obtain a trial function
The first term in the bracket gives the Born approximation and
therefore allows for the first process in the list above. The second
term allows for effects on the incident wave close to the hydrogen
atom since exp(-r2/aQ) falls off rapidly for values of r£ greater than
aQ, the radius of the first Bohr orbit; this term attempts to take care
of the distortion of the incident wave by the field of the atom listed as
part of process 2. The third term (cr j^/a^eapi-r^/e^) ^ow8 *or
the polarization of the atomic field by its explicit dependence on

21
the distance between the incident and atomic electrons. This term
will have a peak value in the vicinity of the atom but will also be
zero when the two electrons' positions coincide; it attempts to take
care o process 3. The effect of an exchange of the incident and
atomic electrons on the scattered electron wave (process 4) has been
discussed in the Introduction (p. 8). At the energy considered
(350 volts), the effect of exchange is not significant (see Table 1). In
\ ",
the last term, A is the trial scattering coefficient. The factor
1 exp(r2/a0)J is included so that the wave function behaves
properly when approaches zero, and this factor is appreciably
different from unity only in the neighborhood of the atom. A very
important reason for choosing the trial function in the particular form
of (42) is to make the required integrations involving the function
feasible (see III A). The asymptotic form of (42) is
(43)

III. THE CALCULATIONS
A. The Variational Integral Lr>0--To obtain the scattering
coefficient f00 for the elastic scattering of electrons by hydrogen
atoms in the ground state, the integrations involved in LM of (34)
must be carried out. In addition to the trial function
Y't* > + b exp (-r2/aQ)
+ (cr12/aiexp*"r2/ao|I + A Jxp(ikDr2)/r2J |^1
- exp(-r2/a0)J Y^r^) (44)
selected above [cf. (42)] an auxiliary trial function jcf. (33)J
= V^^iJexpf-ik^2*- ?2)[l + b exp(-r2/a0)
+ cr12/ao )e*p(-r2/.0)] + A [.xp(ik0r2)/r2J[l
- exp(-r2/ao)J ^(r^) (45)
is required. Then
Loo J h] Yti)(?1.?2)dr1dt2
= 4 7TEQa2 (D0 t Dxb + D2b2 + D3c + D4c2
+ D5A + D6A2 + D7bc + DgbA + D9cA) (46)
22

23
where, with K 2kQ8in 0/2,
Dq = 2ao(8 + K2a2)/<4 + K2a2)2
Dx = 4a0[-K2a2/(l + K2a2)2 + (15 + K2a2)/(9 + kV)2]
D2 r 2ao[-2(l + K2a2)/<4 + K2aJ)2
+ {24 + K2a2)/(l6 + K2a2)2]
D3 = 4ao(K2aJ 1)/(1 + K2aJ)2
+ aQ(-132 10K2a2)/(9 + K2aJ)J{3/K) [-tan'll/KaQ)
Xf. '
- tan_1(3/Kao)j *
D4 2a0(64 104K2aJ 6K4a4 3K6a*)/(4 + kV)4
- 4a0<4 + K22)2
D, a -I + (1 + k2a2)_i 4(9 + 4k2a2f
+<2/Vo>[-,im' + '(3/oaol] 1 [Vo {<1
+k2a2)- 8(27 + 12k2a2)-} + (l/k0a0> [ln(9 + 9k2,*)
J
- ln(9 + 4k2a2)]J

fM O
(4
fM O
M
fM
4
"t
(M O
4
T
M
(M O
V
(M
4
N O
4
V
O'
a
H
4
"rt
fM O
4
4
4 3
^ *
3
i#
sO
Q
fM*~0
t
*V
4-
+ *
--t
-v
CM O
4
4
O'
mm
oo

*M*o
(4
fM O
, ,>
fM
f4
"t
O
fM
4
4
1
d
MS
4

fM O
r4
IM
rt

CO
o
1

5
1
CO
o
t
f4
t
t
M
t
+J
fM
4
fM
4
fM*b
fM
a
4
fM O
(4
fM O
44
4
M*
"t
V
4
O'
4
fM O
(4
fM
-t

5^
w
44
fM
4
3
"7
fM
-*
4
fM
1
N

g
fM
>*
i
fM
2
w
o
1
i
4
44 ,
M
*o
t
?
t
H
w
fM
f
4
4-
II
3
M
i-
Q
oo
Q
+ (2/k0a0) -tan (3/2k_a ) + tan"*(2/k
fM O
fM O
t
t
fM O
fM O
44
M
fM
*
fM O
t
4
4
fM O
44
00
3
M1
M3
"(4
fM O
44
fM
r-
fM
I
I
<4
fM O
44
4
fM

>
i'Jf4
fM O
4
vO
CO
"fi
t
44
J ~T
O'
4
fM O
fM O
-4
I
O'
I
II
4
M3
4
't
<
=4 4<
fM O
4
v
co
00
'fl
t
fM O C
nj 44
fM 0
3 -
^34
3
CO
4
44
fM

I
4
fM
f4
44
3
fM
N.
oT
fM O
04
i
(4
fM'"o
fM O
4
t
in
CO
h
t
fM O
4
r-
4

25
{(5/4)ln(l t 4l£4> + (J/2)Xn(l + 44
+ (l/4)ln<9 t 4k*4) The coefficient DQ is the same as the Born approximation scattering
amplitude of Eq. (20). Some of the coefficients depend only on
the kinetic energy of the incident electron (i. e., kQ), while other
coefficients depend both on the kinetic energy of the incident electron
and the scattering angle O (i. e., K).
B. Linear Determination of Parameters--!! the variational
method of Kohn is paralleled [cf. (27)] we have from Eq. (37) and
Eq. (46) the following set of linear equations
W 3b = Di + 2D2b + D?c + DgA 0 (47a)
3c = D3 + D?b + 2D4c + D9A 0 (47b)
(4irE a*)-1 L/ 3A D_ + Db + Dgc + 2DA -1
(47c)
for determining b, c, and A. Whence,
L00(b.c.A) = -4 7TE0a2(A f^)
(48a)

26
determines £ j
oo
*, <-4 1 ) 1 % '; /
OO = r>o + A + D5A + D6a2 + Dlb + D2b2 + D3c
+ D4c2 + D?bc + DgbA + D9cA (48b)
Numerical values of the coefficients Dj have been calculated for
electrons of 350 volt energy over a range of from 0 to 160.
* 1 1 i i;* '
Using these numerical values, Eqs. (47) were solved simultaneously
for the parameters b, c, and A. The values of these parameters
are listed in Table 2. Numerical values of fQO and the cross sections
w) -- IW9>I2 <>
were then obtained via (48b). These values are tabulated along
with the cross sections given by Born's formula (i. e., Eq. (20))
in Table 3. A comparison ia made graphically in Figure 1.
It is seen from Eq. (48b) that the linear variational method
gives the scattering coefficient as the sum of the Born approximation
scattering coefficient (i. e., DQ) and additional contributions from
the trial scattering amplitude (A) and from the parameters (b, c)
which were chosen to allow for the effects of distortion and polari-
*>
%
zation. Figure 1 shows that allowance for these additional contri
butions to the scattering amplitude results in a sharp rise in the cross
section curve at small angles well beyond the Born approximation

27
TABLE 2
PARAMETERS DETERMINED BY LINEAR METHOD
FOR 350 VOLT ELECTRONS
koao,in 0/2
b
c
A/a0
0
-8.56 +8.691
4.31 -1.231
0.921
-0. 5961
0.03
-8.42 +8. 381
4.21 -1.151
.892
- .5851
.05
-8.33 +8.751
4.25 -1.291
.900
- .6211
.1
-7.52 +8.171
4.02 -1.291
.771
- .6781
.2
-4.78 +7.061
2,77 -1. 381
.464
- .7241
3
-0.358 +4.241
0.812 -1.241
-0.0874
- .6011
.4
-1.22 + 1091
.358 +0.2611
- .0432
- .1151
.5
-2. 25 -0. 5241
.587 + 6481
a 139
- 00326i
.6
-2.40 .6491
.618 + .7121
. 244
+ 0.03761
.7
-2. 05 4541
.540 + 5841
.218
+ .03021
.8
-1.78 .4371
.454 +.5291
.209
+ .04221
.9
-1.56 .4421
.382 + 4901
.194
+ .05211
1.0
-1.32 .4211
.308 + 4411
. 168
+ .05251
1.2
-0.965 .3631
.203 +. 3601
. 122
+ 0444i
1.4
- .750 3811
. 129 +.3281
.0914
+ .04141
1.6
- .580 3601
.0768 +.29U
.0673
+ .03301
1.8
- .455 .3401
.0397 +.2621
.0502
+ .02571
2.0
- .360 3281
.0103 +.2421
.0383
+ 02051
2.5
- .203 .2911
-0.0344 +.2021
.0209
+ .01151
3.0
- .109 .2721
- .0621 +. 1781
. 0128
+.007201
3.5
- .0410- .2581
- .0819 +. 1611
.00819 +.004981
4.0
- .0156- .2371
- .0856 +. 1521
.00578
+ .003141
4.5
+0.0186- .2301
- s0932 +.143
.00435
+. 002551
5.0
+ .0439- .2191
- 105 +. 1321
.00344
+.002021
*

28
TABLE 3
ELASTIC SCATTERING COEFFICIENTS AND CROSS SECTIONS
DETERMINED BY LINEAR VARIATIONAL METHOD
COMPARED TO BORN APPROXIMATION
koaoflin diZ
*oo
$>
Linear Var.
Method
Born
Approximate n
0
*9.91 + 5.541
3595 x 10*Icm2 27. 9 x 10-18cm2
0.03
-9.54 5.20i
3295
27.9
.05
-9.31 +5. 42i
3239
27.7
.1
-7.65 + 4. 56i
2213
27. 1
.2
-2.64 + 2.441
362
24.8
.3
0.456 + 0.5831
15.3
21.6
.4
.900 -0.0838
22.8
18.0
.5
1.12 +0. 100
35. 3
14.5
.6
1. 36 + 212i
37.2
11.4
.7
0.975 + .1811
27*4
8. 78
4 r* ^
.8
.808 t .1631
19.0
6.73
.9
.663 + 1431
12.8
5.14
1.0
. 535 +. 113
8. 34
3.93
v 1.2
.363 + 04941
3.75
2. 33
1.4
.261 + 04541
1.95
1.43
1.6
.196 + .02921
1.09
0.904
1.8
.153 + .01911
0.661
.593
2.0
.123 +.01311
.425
.402
2.5
.0771 +.005651
. 167
.172
3.0
.0544 +.00292
.0829
.084
3.5
.0400 +.00167
.0448
.046
4.0
.0308 +.00104
.0265
.027
4.5
.0244 +.00074
.0166
.017
%

Fig. 1. The theoretical cross sections, obtained by the
linear (Kohn) and quadratic (Hulthfcn) variational procedures, for the
elastic scattering of 350 volt electrons by hydrogen atoms are compared
to the cross sections calculated by the Born approximation and to the
relative scattered intensities measured by Webb. Ordinate, Elastic
scattering cross sections in units of 10 cm Abscissa, Angle
through which the scattering takes place.

30

31
curve. At large angles, however, the two curves merge.
C. Quadratic Determination of Parameters--If the varia-
tional method of Hulthen is paralleled [cf. (28)] the restriction that
the trial function be such that the variational integral L, vanishes is
imposed. Then, from (37),
SL00 -4lrEoaJ if00 = 0 (50)
and
L0o 8 0 <51*>
3L/ 3b 0 (51b)
L/ 3c 0 (51c)
are the equations for determining b, c, and A. Note that (5ib) and
(51c) are the same as (47a) and (47b) and hence linear in the parameters,
but that (51a) sets the expression for JLQO given by (46) equal to zero
and is therefore quadratic in the parameters while its counterpart in
the Kohn method, (47c), is linear. Eq. (50) also gives
(52)
as opposed to the expression (48b) for fQO given by the linear (Kohn)
method.
The values of the parameters b and c were found in terms of

32
the parameter A by a simultaneous solution of Eqs. (5lb) and (5lc).
Substituting these values into Eq. (51a) we obtained an equation of
the second degree in A. Using the calculated values of (see above)
the two possible values of A were then calculated from this equation
over the angular range considered--these two sets of values being
equivalent to two sets of fOQ values via (52). A method for discrim
inating between these two sets of f values was necessary and a test
15
similar to that used by Massey and Moiseiwitsch was employed.
Eq. (13) is an exact integral equation for the scattering
coefficient fOQ. The integral of Eq. (13) was evaluated using the
Yoi (44) corresponding to each of the two sets of parameters
resulting from the quadratic equation in A. The set of parameters
for which the integral was more nearly equal to the value of A
( fQO) tested was then the set of parameters selected.
The selected set of parameters b c, and A calculated
for 350 volt electrons by the quadratic variational method is given in
Table 4. The corresponding values of the integral in (13) are also
listed for comparison with the values of the scattering coefficient A.
It is apparent from the table that the agreement between the integral
equation values of fQO and the values of A is not very good except at
large angles. But it is still sufficiently good to discriminate between
the two sets of fQO values. The cross sections calculated by this
method are compared to those calculated by the Born approximation

TABLE 4
PARAMETERS DETERMINED BY QUADRATIC METHOD FOR 350 VOLT ELECTRONS
ko*o8in 6/2
b
c
A/ao
Integral/aQ
p
-21.6 +16.91
4.26
-3. 561
2.66 -0.
6341
-11.1 +10.11
0.05
-20.9 +17.11
4.23
-3. 701
2. 62 .
6881
-10.6 +10.11
. i
-18.7 +16.91
4.01
-3.871
2.43 .
788i
- 9.04 9.601
.2
-10.6 +14.71
2. 66
-4. 021
1.59 -i.
Oil
- 3. 77 7. 021
.3
+ 2.43 +10.01
-1.65
-4.001
0. Ill -1.
041
0. 178 + 3. 341
.4
- 1.83 -
2. 92i
0.278
+3. 651
. 172 -0.
2291
. 309 0.2821
.5
- 3.07 -
1.661
.499
+2. 62i
. 463 .
1791
.0171 + .8251
.6
- 3. 16 -
1.501
.454
+2. 341
. 580 .
1341
-0.0348 + .9141
.7
- 2.70 -
0. 9371
. 365
1.881
.552 .
130i
0.0251 + 8411
.8
-2.31 -
.8651
.268
+1.641
. 502 .
08921
.0430 .6921
.9
- 1.99 -
.8591
. 190
1.471
. 444 .
05) 8i
.0505 +.5531
1.0
1. 68
.8251
.129
+1. 32i
. 376 .
03121
.0657 +.4351
1.2
- 1.23 -
.7241
.0524+1. 109i
. 270 .
01431
.079? +.278
1.4
- 0.962-
.7931
.0114+0.9941
. 199 .
001311
.0782 +. 1811
1.6
- .773-
.7891
-0.0152+ 917i
. 150 .
00082
.0739 4.125i
1.8
- 644-
.7921
- .0322 + .868
.117 .
00214i
.0671 +.08991
2.0
- .548-
. 816i
- .0524 + 8451
.0942- .
002591
.0592 4.06671
2. 5
- .392-
.8401
- .0826 + .7951
.0588- .
00 3291
.0426 +.03531
3.0
- .312-
.8931
- .0996 + .7901
.0409- .
00322i
.0326 +.02121
3.5
- .251-
.9721
- 116
+. 8091
.0298- .
002541
.0252 +.01351
4.0
- .241--
.9681
- .109
+. 7931
.0229- .
002841
.0198 +.009661
4.5
- .174-
.9521
- .137
. 7561
.0174- .
001421
.0161 +. 006541
5.0
- .175-
.8891
- 114
+ .7161
.0140. .
001601
.0137 +.005271

34
in Table 5 and graphically in Figure 1. It is evident that the quadratic
(Hulthen) method does not provide any significant change in cross
%
section values from those of the Born approximation.
It should be noted that Eq. (13) is a necessary, but not
sufficient, condition for the trial wave function to be a good approxi>
mation to the true wave function and that the linear (Kohn) variational
procedure is such that this integral equation is satisfied identically.
On the other hand, it should be noted that L>00 0 is also a necessary,
but not sufficient, condition for the trial wave function to be a good
approximation to the true wave function and that the quadratic
(Hulthen) method is such that this condition is satisfied identically.

35
TABLE 5
CROSS SECTIONS DETERMINED BY QUADRATIC VARIATIONAL
METHOD COMPARED TO
BORN APPROXIMATION350 VOLT ELECTRONS
: f : : .
koao8in 9/2
e
I(^)
Quad. Var. Method
Born Approx.
0
0
O'
209 x 10"18cm2
27.9 x 10i8cm2
0.05
1
8
204
27.7
. 1
2
16
182
27. 1
.2
4
32
99.1
24.8
.3
6
44
30.5
21.6
.4
9
0
2. 29
18.0
.5
11
18
6.87
14.5
.6
13
34
9.89
11.4
.7
15
58
8.97
8.78
.8
18
4
7.27
6.73
.9
20
'
28
5. 58
5. 14
1.0
22
46
3.97
3.93
1.2
27
22
2. 04
2.33
1.4
32
4
1.10
1.43
1.6
36
48
0.629
0.904
1.8
41
36
.381
.593
2.0
46
28
.248
.402
2.5
J59
6
.0968
. 172
3.0
72
34
.0469
.084
3.5
87
18
.0249
.046
4.0
104
10
.0149
.027
4.5
125
8
.00851
.017
5.0
160
56
.00550
.011

IV. COMPARISON OF THEORETICAL CALCULATIONS
WITH EXPERIMENTAL DATA
A. Adjustment o Experimental Data for Scattering by
Atomic Hydrogen--The bulk of the experimental data for electron-
16
hydrogen scattering has been taken for molecular hydrogen.
Harnwell, however, carried out investigations of the scattering of
electrons by both atomic and molecular hydrogen and reported that
the scattered intensities when atomic hydrogen is present are not
much different than when molecular hydrogen alone is present. Since
a difference is detectable, some allowance was made for it in the
comparison of our theoretical calculations and the experimental data.
Assuming the validity of the Born approximation, Massey
18
and Mohr have obtained the ratio of the scattering by molecular
hydrogen to tire scattering by atomic hydrogen:
1 The work of Webb is so comprehensive that his results
alone are sufficient to test the adequacy of the theoretical values of
scattering cross sections. Webb measured the scattered intensity
over a wide range of angles and voltages with a single apparatus and
found, moreover, that his results agreed closely with those obtained
by other investigators in the overlapping regions.
17
G. P. Harnwell, Phys. Rev. 34, 66 i (1929).
18
H. S. W. Massey and C. B. O. Mohr, Proc. Roy. Soc.
A 135, 258 (1932).
36

37
In /Ijj = (1 + sinx/x) 2
where x s 2kQd sin ( & /2), d being the equilibrium nuclear separation
of the molecule, and y = koaQ8in( 0 /Z). This ratio performs a
damped oscillation about unity as a function of (v)sin( & ¡Z). For an
energy of 350 volts over the range of Webb's experimental points,
its maximum value is 1. 4 at 3 = 5 and its minimum value is
0. 78 at 0 30. This relationship was used to convert Webb's
experimental data at 350 volts for molecular hydrogen to equivalent
L
data for atomic hydrogen.
B. Comparison of Calculated Scattering Intensities with
Experimental Data--The experimental values of the cross sections are
given in arbitrary waits. Hence, for a comparison to the results of
this investigation, the adjusted experimental data (cf. section A
above) were fitted to coincide at 25 with the theoretical values ob
tained by the linear (Kohn) variational procedure as shown in Figure 1.
In addition to these curves, the results from the Born formula are
also given in Figure 1.
An examination of this figure reveals that at small angles
( 0 < 30), where the deviation of the Born approximation curve
from the experimental points is very large, the curve obtained by the
linear variational method fits the experimental curve very well except
near 7. The dip in the theoretical curve at around 7 is probably not

38
significant. At less than 7, we found that the contributions to the
scattering amplitude [cf. (48b)] of the terms which are functions of
the parameters b and c were very large and negative compared to the
scattering amplitude of the Born approximation. (The part of the
scattering amplitude due to the terms which are functions of the
parameter A was negligibly small over the entire range of 9 .)
At angles greater than 7, the Born term was largest; at angles
greater than 46, terms other than the Born term were negligibly
small. The dip in the theoretical curve appears at about 7 where
the terms in b and c had become negative but were not yet larger than
the Born term. Hence the dip appears in a transition region between
the very small angular region where the (b and c) terms allowing for
i
polarization and distortion effects contributed most to the scattering
and the angular region ( 9 > 12) where the Born term predominated.
It seems that three parameters are too few to make the transition
smoothly.
The cross section values obtained from the quadratic
(Hulthn) variational procedure do not provide any improvement of
the Born approximation. In Figure 1, the curve obtained by the quad*
ratic method lies below the Born curve at the larger angles where
the experimental points lie higher than the Born curve while, at small
angles, it rises somewhat above the Born curve but not to such an
extent that it can be considered to agree better with the experimental

39
curve.
It is interesting to note that the shapes of the curves in
Figure 1 obtained by the two variational procedures are quite similar
even to the dip appearing around 7, but that the curves are dis
placed relative to one another. It follows from Eqs. (48a) and (52)
that the closer the trial function is to the exact wave function, the
closer the two cross section curves will agree. Hence, the displace
ment of the two curves demonstrates that the trial wave function is
not a very good approximation to the exact wave function. The fact
that the trial wave function is not a good approximation to the true
wave function, however, does not necessarily mean the calculated
coefficients are not good approximations to the true scattering coef
ficients. It is a peculiarity of quantum mechanical variational proce
dures that satisfactory results can be obtained even though the simple
19
trial functions employed are relatively poor approximations. This
is not surprising, however, since in our case, for example, the
theory provides a stationary expression for the scattering coefficient,
not the wave function. The criterion for adequacy of the theoretical
scattering cross sections is still "goodness of fit" to the experimental
19
For example, excellent results for the calculation of the
binding energy of the hydrogen molecule by a variational method
employing a very simple trial function were obtained by: W. Heitler
and F. London, Zeits. f. Physik. 44, 455 (1927).

40
data and th abova result; >t oar linoar variational procedure do
prvida a reasonably good lit.
C, Results Obtained by Linoar Variational Procodaro with
Vartoua Qtksr Trial runctioas--Ths trial wave function other than
Ej. (44) which wo havo nod to determine the elastic scattering cross
action by the linear variational procodaro aro alto worth consider
ation. Especially interesting aro tito roonlte wo obtained with the trial
functions.
l^lrj) onpCik^^^rg) [i + be*p(-r2/*o)
+ (eri2/e0)esp(-r2/ao)] (54)
^(? ) eapt-ik^2*-?2) [] + bexp(-r2/a0)
+ (cri2/a0)ojip(-r2/ao)J (55)
which are the aamo aa (44) and (45) with A 0. For this casa
*0# Do + (54)
Reference to Eq. (4b) shows that the D& appearing in (54) aro all
fuacitienof Ka* n 2kQa0ia( e/2). Therefore the scattering ampli-
todee lor the b and c parameter cao# (54), like these obtained by the

41
Born approximation (20), are functions of (v)sin( 9/2) only.
In Figure 2, the results for the b, c parameter trial
functions (54, (55) are compared to those previously derived for the
A, b, c parameter trial functions (44), (45). The curves are very
similar. It seems then that the good fit obtained with the linear method
and trial functions (44), (45) to the experimental data, as shown in
Figure 1, at the small angles is provided predominantly by the
contributions of parameters b and c to the scattering amplitude. For
the b, c parameter case, the parameter A is zero, and the quadratic
variational procedure gives a zero scattering coefficient. From these
considerations, it is understandable that the linear variational method
has yielded better results than the quadratic variational method for
the trial functions (44), (45) as discussed above.
The linear variational procedure using the trial functions
considered so far provides no improvement of the Born approximation
at large angles. At large angles, the experimental curve of the
scattering cross section becomes almost independent of angle instead
of falling off uniformly with increase of angle as predicted by the Born
approximation. In Eq. (46) the coefficients of the terms involving
the parameter A are independent of angle. From this, it seems that
parameters in the scattered wave part of the trial function may allow
for the improvement of the theoretical curves at large angles. Con
sidering a trial wave function of the general form

Fig. 2. The theoretical cross sections for the elastic
scattering of 350 volt electrons by hydrogen atoms calculated by the
linear (Kohn) variational procedure employing the trial functions
(44), (45) with the parameter b (A, c 0) with the parameter c
(A, b = 0), and with parameters b, c (A = 0) are compared to those
obtained using the parameters A, b, c in the same trial function.
Ordinate, Elastic scattering cross sections in units of 10~18cm.2
Abscissa, Angle $ through which the scattering takes place

43

Yt a Yo&i) j^exp(ik0n- r2) + bexp(-r2/a0) + (cr12/ao)exp(-r2/ao)
+ 0/r2)exp(ikor2) ^ 1 exp( -r2/aQ)j x
./
{a + B exp(-r2/a0) + (Cr l2/ao) exp(-r2/aQ)} J f (57)
we have examined the contributions of the various terms to the
scattering by carrying out the cross section calculations with trial
functions using various combinations of the five parameters in (57).
In Figures 2, 3, and 4, the results of the various calculations are
graphed. It is evident that the parameter b or c alone (Fig. 2) is not
sufficient. Both b and c are required to give a good fit to the experi-
t*
mental data at small angles. The effect of the parameters A, 3, and
C is to raise the cross section curve at large angles (Figs. 3, 4).
When A alone is used, the curve (Fig. 3) is raised at large angles;
when A is used in combination with either b or c, or with b and c,
no effect of A at large angles is evident. As the number of parameters
in the scattered wave part of the trial function is increased, the rise
/ '
of the scattering curve at large angles also increases (Fig. 4).
Since the parameters in the inoident wave part of the trial function
serve not only to improve the theoretical results at small angles but
also to cut down the effectiveness of the parameters in the scattered
wave part for raising the theoretical curve at large angles, we feel that
a trial function employing all five parameters will provide improvement

Fig. 3. The theoretical cross sections for the elastic
scattering of 350 volt electrons by hydrogen atoms calculated by the
linear variational method employing the trial functions (44), (45) with
the parameter A (b,c 0), with the parameters A,b (c 0), and
with the parameters A, c (b 0) compared with those obtained
using the parameters A,b, c in the same trial function. Ordinate,
Elastic scattering cross section in units of 10" cm. Abscissa,
Angle 9 through which the scattering takes place.

A
o Experimental values

Fig. 4. The cross sections for the elastic scattering of
350 volt electrons by hydrogen atoms calculated by the linear
variational procedure employing the trial function (57) with the param-
XI f- '
eters A, b, B (c,C 0) and with the parameters A, B, C (b, c = 0)
t ; 4 V- ; :
**
are compared to those obtained employing the same trial functions
: [T-it'l
with the parameters A, b, c (B, C = 0). Ordinate, Elastic scattering
18
cross sections in units of 10" cm. 2 Abscissa, Angle through
which the scattering takes place.

48

49
of the Born approximation over the entire angular range of the
scattering.

V. CONCLUSIONS
The success of wave mechanics in explaining phenomena
beyond the realm of classical physics has been unqualified. A major
theoretical problem is resolving the frustration of not knowing the
wave function which applies to a particular problem. In the event
that a relatively simple asymptotic form of the wave function is
known, however, a variational procedure may sometimes be employed
to provide a solution to the problem without possessing the exact wave
function. Since the complete set of hydrogen eigenfunctions is known,
the variational method outlined in Part II should be useful in obtaining
inelastic, as well as elastic, cross sections for the scattering of
electrons by hydrogen without going into partial wave analyses as
is usually done. Only the elastic scattering at high velocities of
impact has been considered here to illustrate the method. The simple
form of our trial function accounts for the distortion and polarization
effects expected to become important as the impact velocity or
scattering angle decreases to where the Born approximation is inade
quate. The deviation of our theoretical curves from the observed
scattering cross sections at large angles remains when the number
l
>
20
A variational method, using partial waves, for obtaining
inelastic cross sections has been derived by: B. L. Moiseiwitsch,
Phys. Rev. 82, 753 (1951).
50

51
of adjustable parameters is limited to three. If the number of param
eters could be increased to perhaps five, thereby greatly increasing
the labor of calculations, an extension of the improvement of the Born
approximation in the large angle direction could probably be realized.
Since the departures of theoretical from experimental curves are of
much larger magnitude in the small angle region than in the large
angle region, the three parameter linear variational procedure given
here is considered to be a satisfactory improvement of the Born
approximation in providing a theoretical basis for the elastic scattering
of electrons by hydrogen atoms at high velocities of impact.

BIBLIOGRAPHY
Born, M., Zeits. £. Physik 38, 803 (1926).
Harnwell, G. P., Phys. Rev. 34. 661 (1929).
Heitler, W., and F. London, Zeits. £. Physik 44, 455 (1927).
Huang, Su-Shu, Phys. Rev. 76, 477 (1949).
Hulthen, L., Kungl. Fysio. Skllskapets Lund Forhhnd. L4, 1 (1944).
Hulthen, L. and P. O. Olsson, Phys. Rev. 79, 531 (1950).
Kohn, W., Phys. Rev. 74, 1763 (1948).
Massey, H. S. W. and C. B. O. Mohr, Proc. Roy. Soc. A 132, 608
(1931).
Massey, H. S. W. and C. B. O. Mohr, Proc. Roy. Soc. A 135, 258
(1932).
Massey, H. S. W. and C. B. O. Mohr, Proc. Roy. Soc. A 146, 880
(1934).
Massey, H. S. W. and B. L. Moiseiwitsch, Proc. Roy. Soc. A 205.
483 (1951).
Moiseiwitsch, B. L., Phys. Rev. 82, 753 (1951).
Mott, N. F. and H. S. W. Massey, Theory of Atomic Collisions
(Clarendon Press, Oxford, 1949), second edition.
Webb, Glenn M., Phys. Rev. 47, 384 (1935).
52

BIOGRAPHICAL SKETCH
June Grimm Jones was born in Wheeling, West Virginia,
on October 16, 1918. She was awarded the A. B. degree in
mathematics and physics by West Virginia University in 1938.
After her under graduate work, she was on the actuarial staff of
Acacia Mutual Life Insurance Company located in Washington, D. C.
until December, 1940 when she resigned to accompany her husband,
Mark Wallon Jones, on an assignment in South America. Upon her
return to the United States, she enrolled in the graduate school of
the University of Florida in the summer of 1947. She was granted
the Master of Science degree in June, 1948, by the Department of
Physics. The following two years, she was an assistant member of
the research staff of the Geophysical Institute located in College,
i i
Alaska. In September, 1950, she returned to the University of
Florida to pursue studies in theoretical physics leading to the degree
of Doctor of Philosophy. She is a member of Phi Beta Kappa, Phi

Kappa Phi, Sigma Pi Sigma, Sigma Xi, and Kappa Kappa Gamma
social fraternity.
53

This dissertation was prepared under the direction of the
co-chairman of the candidate's supervisory committee and has been
approved by ail members of the committee. It was submitted to the
Dean of the College of Arts and Sciences and to the Graduate Council
and was approved as partial fulfilment of the requirements for the
degree of Doctor of Philosophy.
June 8, 1953
SUPERVISORY COMMITTEE:



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oo
Q
+ (2/k0a0) -tan (3/2k_a ) + tan"*(2/k
fM O
fM O
t
t
fM O
fM O
44
M
fM
*
fM O
t
4
4
fM O
44
00
3
M1
M3
"(4
fM O
44
fM
r-
fM
I
I
<4
fM O
44
4
fM

>
i'Jf4
fM O
4
vO
CO
"fi
t
44
J ~T
O'
4
fM O
fM O
-4
I
O'
I
II
4
M3
4
't
<
=4 4<
fM O
4
v
co
00
'fl
t
fM O C
nj 44
fM 0
3 -
^34
3
CO
4
44
fM

I
4
fM
f4
44
3
fM
N.
oT
fM O
04
i
(4
fM'"o
fM O
4
t
in
CO
h
t
fM O
4
r-
4


TABLE 4
PARAMETERS DETERMINED BY QUADRATIC METHOD FOR 350 VOLT ELECTRONS
ko*o8in 6/2
b
c
A/ao
Integral/aQ
p
-21.6 +16.91
4.26
-3. 561
2.66 -0.
6341
-11.1 +10.11
0.05
-20.9 +17.11
4.23
-3. 701
2. 62 .
6881
-10.6 +10.11
. i
-18.7 +16.91
4.01
-3.871
2.43 .
788i
- 9.04 9.601
.2
-10.6 +14.71
2. 66
-4. 021
1.59 -i.
Oil
- 3. 77 7. 021
.3
+ 2.43 +10.01
-1.65
-4.001
0. Ill -1.
041
0. 178 + 3. 341
.4
- 1.83 -
2. 92i
0.278
+3. 651
. 172 -0.
2291
. 309 0.2821
.5
- 3.07 -
1.661
.499
+2. 62i
. 463 .
1791
.0171 + .8251
.6
- 3. 16 -
1.501
.454
+2. 341
. 580 .
1341
-0.0348 + .9141
.7
- 2.70 -
0. 9371
. 365
1.881
.552 .
130i
0.0251 + 8411
.8
-2.31 -
.8651
.268
+1.641
. 502 .
08921
.0430 .6921
.9
- 1.99 -
.8591
. 190
1.471
. 444 .
05) 8i
.0505 +.5531
1.0
1. 68
.8251
.129
+1. 32i
. 376 .
03121
.0657 +.4351
1.2
- 1.23 -
.7241
.0524+1. 109i
. 270 .
01431
.079? +.278
1.4
- 0.962-
.7931
.0114+0.9941
. 199 .
001311
.0782 +. 1811
1.6
- .773-
.7891
-0.0152+ 917i
. 150 .
00082
.0739 4.125i
1.8
- 644-
.7921
- .0322 + .868
.117 .
00214i
.0671 +.08991
2.0
- .548-
. 816i
- .0524 + 8451
.0942- .
002591
.0592 4.06671
2. 5
- .392-
.8401
- .0826 + .7951
.0588- .
00 3291
.0426 +.03531
3.0
- .312-
.8931
- .0996 + .7901
.0409- .
00322i
.0326 +.02121
3.5
- .251-
.9721
- 116
+. 8091
.0298- .
002541
.0252 +.01351
4.0
- .241--
.9681
- .109
+. 7931
.0229- .
002841
.0198 +.009661
4.5
- .174-
.9521
- .137
. 7561
.0174- .
001421
.0161 +. 006541
5.0
- .175-
.8891
- 114
+ .7161
.0140. .
001601
.0137 +.005271


ACKNOWLEDGEMENT
The author wishes to acknowledge her great debt to
Dr. M. M. Gordon, co-chairman of the supervisory committee,
without whose guidance and assistance the work could not have been
carried out and to Dr. R. C. Williamson, chairman of the
committee, for helpful suggestions concerning the dissertation and
guidance throughout her graduate work. She also wishes to thank
the other members of the supervisory committee: Dr. J. W. Flowers,
Dr. H. P. Hanson, Dr. A, G. Smith, and Dr. C. B. Smith.
ii


6
with, as a result,
as follows from (11). Eq. (10) is only one of a set of simultaneous
integro-differential equations for the functions
By the same method, an integral equation of the form (13) can be
< i *'
found for the scattering amplitude f^ corresponding to each function
^(3,5*) = (2m/fi2)^exp(-iknn'* r') $n(?',n)dt'.
Similarly, for the set of functions
(15)
Gn(i>> =
r*(r2)Y(rlt72)Z:
1* AZ*
(16)
we have a set of simultaneous integro-differential equations corre
sponding to (10)
(h2/2m)(Vf + k^)Gn(?)
- ^n(*i3) ,
(17)


This dissertation was prepared under the direction of the
co-chairman of the candidate's supervisory committee and has been
approved by ail members of the committee. It was submitted to the
Dean of the College of Arts and Sciences and to the Graduate Council
and was approved as partial fulfilment of the requirements for the
degree of Doctor of Philosophy.
June 8, 1953
SUPERVISORY COMMITTEE:


21
the distance between the incident and atomic electrons. This term
will have a peak value in the vicinity of the atom but will also be
zero when the two electrons' positions coincide; it attempts to take
care o process 3. The effect of an exchange of the incident and
atomic electrons on the scattered electron wave (process 4) has been
discussed in the Introduction (p. 8). At the energy considered
(350 volts), the effect of exchange is not significant (see Table 1). In
\ ",
the last term, A is the trial scattering coefficient. The factor
1 exp(r2/a0)J is included so that the wave function behaves
properly when approaches zero, and this factor is appreciably
different from unity only in the neighborhood of the atom. A very
important reason for choosing the trial function in the particular form
of (42) is to make the required integrations involving the function
feasible (see III A). The asymptotic form of (42) is
(43)


3
are then
Yfil'fy t Y&.?,) <3>
which have the asymptotic forms for large
^(fyexpilc02.?2) + (l/r2) ^ [fmo(J;n2)
ilmotf^y] ** <4a)
and for large r ^
* ^<*>xP + m [mo^l)
mo(2|l)] e*P(i*ori> D*Z>* <4b>
For calculation purposes it is assumed that the intensity of the
beam of electrons falling on a hydrogen atom initially in the normal
state is such that one electron crosses unit area per unit time. The
scattering of the electrons from a beam of unit intensity is measured
in terms of the differential cross section dw, defined as the
number of electrons which fall per unit time on an area dS (dS = r2dw)
ft
placed at a large distance r from the scattering atom. For the o-*m
transition, the differential cross section 1 (e$dw is the sum of the
number of electrons which, after exciting the state m in the atom, are
scattered into the solid angle dw located at (4) in unit time and the
number of electrons which, after the incident electrons have been


5
(8)
If we define a function
Fo(?2> 51 /
(9)
it follows from (8) that
|(e2/r2) (2/ru)J
= $o (10)
The solution F0(r^) of this integro-differential equation has the
asymptotic form
Fo(?2^ exp(lko*?2) + fooexp(ikor2)/r2 U1)
as follows from Eqs. (9) (2). The required solution is3
FQ(?2) exP(*** *>) (2m/4tlR2) x
o 2
f r*p(ik0|?2 >/ |?2 -?'|] f0(?'.3)dt', (12)
3Ibid., Chap. VI.


LIST OF ILLUSTRATIONS
Figure Page
1. The theoretical cross sections, obtained by the
linear (Kohn) and quadratic (Hulthn) variational
procedures, for the elastic scattering of 350 volt
electrons by hydrogen atoms are compared to the
cross sections calculated by the Born approximation
and to the relative scattered intensities measured
by Webb 30
2. The theoretical cross sections for the elastic
scattering of 350 volt electrons by hydrogen atoms
calculated by the linear (Kohn) variational
procedure employing the trial functions (44), (45)
with the parameter b (A, c 0), with the parameter
c(A,b 0), and with parameters b, c (A = 0) are
compared to those obtained using the parameters
A, b, c in the same trial function 43
3. The theoretical cross sections for the elastic
scattering of 350 volt electrons by hydrogen atoms
calculated by the linear variational method
employing the trial functions (44), (45) with the
parameter A (b, c = 0), with the parameters
A, b (c 0), and with the parameters A, c (b = 0)
compared with those obtained using the parameters
A, b, c in the same trial function 46
4. The cross sections for the elastic scattering of
350 volt electrons by hydrogen atoms calculated
by the linear variational procedure employing the
trial function (57) with the parameters A, b, B
(c,C = 0) and with the parameters A, B, C
(b, c = 0) are compared to those obtained
employing the same trial functions with the
parameters A, b, c (B, C ~ 0). 48
v


38
significant. At less than 7, we found that the contributions to the
scattering amplitude [cf. (48b)] of the terms which are functions of
the parameters b and c were very large and negative compared to the
scattering amplitude of the Born approximation. (The part of the
scattering amplitude due to the terms which are functions of the
parameter A was negligibly small over the entire range of 9 .)
At angles greater than 7, the Born term was largest; at angles
greater than 46, terms other than the Born term were negligibly
small. The dip in the theoretical curve appears at about 7 where
the terms in b and c had become negative but were not yet larger than
the Born term. Hence the dip appears in a transition region between
the very small angular region where the (b and c) terms allowing for
i
polarization and distortion effects contributed most to the scattering
and the angular region ( 9 > 12) where the Born term predominated.
It seems that three parameters are too few to make the transition
smoothly.
The cross section values obtained from the quadratic
(Hulthn) variational procedure do not provide any improvement of
the Born approximation. In Figure 1, the curve obtained by the quad*
ratic method lies below the Born curve at the larger angles where
the experimental points lie higher than the Born curve while, at small
angles, it rises somewhat above the Born curve but not to such an
extent that it can be considered to agree better with the experimental


23
where, with K 2kQ8in 0/2,
Dq = 2ao(8 + K2a2)/<4 + K2a2)2
Dx = 4a0[-K2a2/(l + K2a2)2 + (15 + K2a2)/(9 + kV)2]
D2 r 2ao[-2(l + K2a2)/<4 + K2aJ)2
+ {24 + K2a2)/(l6 + K2a2)2]
D3 = 4ao(K2aJ 1)/(1 + K2aJ)2
+ aQ(-132 10K2a2)/(9 + K2aJ)J{3/K) [-tan'll/KaQ)
Xf. '
- tan_1(3/Kao)j *
D4 2a0(64 104K2aJ 6K4a4 3K6a*)/(4 + kV)4
- 4a0<4 + K22)2
D, a -I + (1 + k2a2)_i 4(9 + 4k2a2f
+<2/Vo>[-,im' + '(3/oaol] 1 [Vo {<1
+k2a2)- 8(27 + 12k2a2)-} + (l/k0a0> [ln(9 + 9k2,*)
J
- ln(9 + 4k2a2)]J


51
of adjustable parameters is limited to three. If the number of param
eters could be increased to perhaps five, thereby greatly increasing
the labor of calculations, an extension of the improvement of the Born
approximation in the large angle direction could probably be realized.
Since the departures of theoretical from experimental curves are of
much larger magnitude in the small angle region than in the large
angle region, the three parameter linear variational procedure given
here is considered to be a satisfactory improvement of the Born
approximation in providing a theoretical basis for the elastic scattering
of electrons by hydrogen atoms at high velocities of impact.


Fig. 3. The theoretical cross sections for the elastic
scattering of 350 volt electrons by hydrogen atoms calculated by the
linear variational method employing the trial functions (44), (45) with
the parameter A (b,c 0), with the parameters A,b (c 0), and
with the parameters A, c (b 0) compared with those obtained
using the parameters A,b, c in the same trial function. Ordinate,
Elastic scattering cross section in units of 10" cm. Abscissa,
Angle 9 through which the scattering takes place.


40
data and th abova result; >t oar linoar variational procedure do
prvida a reasonably good lit.
C, Results Obtained by Linoar Variational Procodaro with
Vartoua Qtksr Trial runctioas--Ths trial wave function other than
Ej. (44) which wo havo nod to determine the elastic scattering cross
action by the linear variational procodaro aro alto worth consider
ation. Especially interesting aro tito roonlte wo obtained with the trial
functions.
l^lrj) onpCik^^^rg) [i + be*p(-r2/*o)
+ (eri2/e0)esp(-r2/ao)] (54)
^(? ) eapt-ik^2*-?2) [] + bexp(-r2/a0)
+ (cri2/a0)ojip(-r2/ao)J (55)
which are the aamo aa (44) and (45) with A 0. For this casa
*0# Do + (54)
Reference to Eq. (4b) shows that the D& appearing in (54) aro all
fuacitienof Ka* n 2kQa0ia( e/2). Therefore the scattering ampli-
todee lor the b and c parameter cao# (54), like these obtained by the