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, cochairman 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 electronatom 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 uptodate 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 rP 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 antiparallel 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 integrodifferential 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 ( /) xk,'') (, )dt',(3
000
as follows from (1 ). Eq. (10) is only one of a set of simultaneous integrodifferential 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 integrodifferential 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 zeroorder 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 righthand 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 ):
3/41fgl2. 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 (30100 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 onebody 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 Swave 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 Hatoms. They used Huang's trial function and combinations of it to allow for exchange in the determination of the Swave phase shifts.
12SuShu 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 Swave scattering is that part of the scattering which is spherically symmetric (not dependent on the angle G) and the total Swave cross section is given by Q = (4 r/kZ) sin Yt. Swave 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 eH 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 Swave scattering takes place.
B. Variational Procedure for Improving the Born Approximation for eH ScatteringA 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 firstorder 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 FunctionThe 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 resultssee 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 LgTo 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%) ftan I(1/koa%) + tan (3/2ka) i k a (1
2 21 2 2 1 2 2
+k~a) 8(27 + 12ka) + (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 a1 2 2I 2z2
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(2K ao)/(4 t K a) t 4a0(643K ao)/(16 t K a)
+(4/K)tan'(2/Ka%) + tanI(4/Ka%)
D8 = 1 + (2 + 2k a0) + 4(9 + 4k a0) (4 + k 0a)
+(2/koa) ItanI(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 21 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 koaofl9(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 ParametersIf 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 parameters (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 101cm
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 10cm2 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 ParametersIf 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 consideredthese 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 APPROXIMATION350 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
1018cm 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 HydrogenThe 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 DataThe 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 F1atiossThe 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 1018cm, 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 anglesalso 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 1018cm. 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, SuShm, 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 cochairman 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/
Cochairman
