Electron Scattering from Hydrogen.pdf

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J. %'. M OTZ AN D G. M I SSON I

approximately 0.25. On the other ha,nd, for 660-kev

photons, the total Compton cross section appears to be

independent of the electron binding energy. It is in-

teresting to note that at 660 kev, the small- and large-

angle behavior of d|7~ has a compensating effect inwhich the total cross-section ratio, ore/a. r, is approxi-

mately equal to unity for both tin and gold.

ACKNOWLEDGMENTS

We wish to thank Professor Mare Ross for helpful

discussions. Also one of us (JWM) wishes to thank

Professor Mario Ageno for his suggestions during the

course of the work and for his kind hospitality in makingavailable the facilities of the Physics Laboratory of the

Istituto Superiore di Sanita.

PHYSI CAL REVIEW VOLUME 124, NUMBER 5 DECEMBER 1, 1961

Electron Scattering from Hydrogen*

CHARLEs SGHwARTz

Department of Physics, U'nzoerszty of Calzfornza, Berkeley, Calzfornza

(Received June 12, 1961)

Kohn's variational principle has been used to calculate S-wave elasticscattering

of electrons from atomichydrogen, using up to 50 trial functions of the type introduced by Hylleraas to describe the bound states oftwo-electron atoms. The phase shifts calculated at several energies up to 10 ev appear to have converged well,

leaving residual uncertainties mostly less than one thousandth of a radian. Taking extra pains to include theeffect of the long-range force at zero energy, we have also determined very accurate values for the scatteringlengths.

INTRODUCTION

HE scattering of electrons from hydrogen atoms

has been the subject of a great many calculations

since it presents what is probably the simplest non-

trivial real problem in scattering theory. We undertook

a program of computing definitive values of the S-wave

elastic phase shifts for this system, making no approxi-

mations other than those imposed by the 6nite speed

and capacity of modern computing machines. Our useof the variational method' for this scattering problem

completes, in a sense, the famous work on the bound

states of two-electron atoms begun more than thirty

years ago by Hylleraas.

Probably the most interesting, and quite unexpected,

result of this program has been the realization of the

extraordinary nature of the convergence of the "sta-tionary" phase shift. It has been recognized for some

time' that, in contrast with bound-state problems, the

addition of more variational parameters in a scatteringcalculation does not necessarily lead to a better answer.

This behavior is blamed on the nonexistence of anyminimum (or maximum) principle. The error in a varia-

tional calculation may be represented by

where 6 is the unknown error in the trial wave function.

Only for systems where one knows the (finite) number

of eigenvalues of II below the value E can one possibly

state that the expression (1) must be negative. ' Forscattering at any 6nite energy it is clearly impossible

to make any such statement. We have discussed else-

where4 how, by taking a great deal of numerical data,one can draw smooth curves and see an eGectively

regular convergence for the general scattering problemat any energy. This paper will present the results of

this treatment for the e-II problem.We use Kohn's variational principle,

ftan8/k7= tan8/k+ (2nz/ks) f(E—H)fdridrs, (2)

where

(2rtz/Pt') (E H) =k' 1+—1'+Vs'—

+ (2/ri)+ (2/rs) —(2/rrs),

with lengths in units of )tt'/ztze'; and our trial wave

function is (for singlet or triplet states) f= p+g,io=

(1&P)s)2e"zLsinkri/kri

+tan8 coskri/kri(1 — t"ts)"')7/4zrv2, (3a)

5 (E—H)Adre—a/2) (rz+rz)r, t

gl

t,m, n&0

*Supported in part by the Advanced Research Projects Ad-ministration through the U. S. Once of Naval Research.' A calculation identical to ours but limited to zero energy and

using only a small number of parameters has recently been re-ported by Y. Hara, T. Ohmura, and T. Yamanouchi, Progr.Theoret. Phys. (Kyoto) 25, 46t (1961).

~ See, for example, H. S. W. Massey in Irundbgch der Physikedited by S. Flugge (Springer-Verlag, Berlin, 1956),Vol. 36.

X (ri rs "&ri "rs )/4)rv2, (3b)' L.Rosenberg, L. Spruch, and T.F.O'Malley, Phys. Rev. 119,

164 (1960), have in this manner established minimum principlesfor scattering at zero energy. These authors have recently ex-tended their method to positive energies, but only by mutilatingthe potentials.

4 C. Schwartz, Ann. Phys, (to be published).



I-, LEcTRON CATTERI NG F ROM H

exchanges coordinates ri an d x and ~is thehere Ei2 exc anh' h

'important in study-ariable scale parameter wnicu is so im

.Th actual variation of parametersng the convergence. e ac uayields the standard linear problem

(i, j=1, 2, , /If),

Mz ' (2'/8 ) Xz(E H)Xjdridrs)2

E,= (2ns/Ii') X (Li' H) pd—,dr2)

lb.O--

15.5—

15.0—

14.5—

14.0—Ian Q

k

13.5—

13,0—

003d I0h

, -- 1.940.003roclloh

1.92

1,90

1.88

1.86

'I.84

tan81.82

I F80

le78

'tion of tanb is also a trivial inear

ameters. The numbersumber E of paramtif the order of50 in the following results identi y e o

in'

(3b) through totalncludin al/ terms in

2 to 7 fo th t i 1 t t t dowers, (1+m+I), from 2 to or e ri

1 to 6 for the singlet state, respectively.

-tan3

k

2.90

2.89-

2.88—

2.87—

IP

~ I

lI:.Il:Il

Il .(1;

)ls

I

t3

I

III

I I

l /

I /

/

2.85—

2.84—

2.83—S =14 =0.4

I I I

20 24 28 32 340.4 0.8 ).2 l.b

ri let bases ita'ft t k=0.4: the most thoroughly150 oints represented.tudies case, over. po'

RESULTS

. ~,tan6 ki ure 1 shows the stationary values o

for the tnp e

2 ev). This is our most thoroug y s u ie

entin some 150 points in t isurves shown representing som

so a reat deal of structure is no s o

erin roblem see re ereharacteristic of the scattering pcertainl do exist or a). Similar singularities cer

'y

ut the are so narrow as a unc ionarger matrices, but .ey ar unc ion

0 K) a h ~ ~ and then come up again rom

12.5—

12.0—

11.5—

0.6 1,0 1,4

0 Sk =ox k

I I

1,8 2.2 OA 1.0K

1.76

1.74

1.72

=0= 0.7 —1.70

I I I

1.4 1.8 2.2

(a) (b)

hase shift at k=0.4; typical well-behavedes. h S' glet phase shift at k=0.7;esults using only five //: values. b ing e p

typica resuesult with wild values at ~=1.4.

ariational calculations of S-wave e-IXI. esu s o varia ona e 7scattering. ehe numbers in parentheses give e uthe last digit quoted.

0.10,20.30.40.50.60.70.80.866

S=OLtans/kg k (radians)

—.68 (2) 2.553(1)—.23 (2) 2.0673 (9)—6.4(1) 1.6964(5)+15.87 (4) 1.4146{4)

5.17(2) 1.202 (1)2.845 (8) 1.041(1)1.917(5) 0.930(1)1.530(5) 0.886(1)

~ ~ ~~ ~

5=1I tans/kg S (radians)

—.056(4) 2.9388(4)—.260 (3) 2.71/1(5)—.492 (4) 2.4996(8)—.833(2) 2.2938(4)—.384(3) 2.1046(4)—.40(1) 1.9329(8)—.74(2) 1.7797 (6)—7.2 (6) 1.643{3)

+150(20) 1.563 (1)

fina 1 results (see Ta e isl I is derived from such curves

answer and assigned our uncertainty w ic s ou

h ht that a reasonable measure o e

leadtoa rossoveresay as a unc ion overestimate of the accuracy;asses through zero, an i s m

Most of our results were obtaine rom on y2 shows a typical case. Figure

~=1.4, ave results noticeably "out of line" wit itsir= . gneighbors, for the an siz



CHARLES SCH WARTZ

7,1—

7+0—

6,9-

6.8 "6.7—

6.6—

6.S-

6,4-

6.3—

6.2—

6.1—

r i T 1

3x37x 7

h 13 x 132.20

2.16

2.12

2.08

2.04

2,00

1.96

'1.92

—1.88

—1.84

Ot

atom is —A' 2ritrit) (n/r ), and it has beent at in such a circum t hcums ance the as m

a zero energyis

1—(a/r) —n/2r').

The sloww convergence noted above fork=0 „.b, bl, drne on the dificultu y of expanding a

ies o as slowl as1 r' '

At finite energies 'ty r in terms of r"e ".

at the long-rangegies it may be inferred th

is not hard to show that for the go

system with ground- tc arge rom a neutral s i-pin-zero atomic

h h

oun -state wave function

as t e followin ex1 of th oodi toor inate rp of the scattering particle.

6.0— S=O I( = 0

$,9 I I I

0.4 0.8 1.2 1 6 2,.0 0.4K

s =1

0.8 1.2

—1.80

1.6 2,0

8 0! ro'W+— +0(r(i—'),

ro 2mo' ro'

(a) (b)

Fio. 3(a) and (h'i.( ). irst zero-energy results.

where W&') is the olpo anzed target wave fe unction,

(Ep—Hp)e ' =DC(" . (6)

4mv2, (8)

d pe ldp-e—Iir 1

(n—1)!

t was

5L. Spruch, T. I'. O'Malle aey, and L. Rose berg, y .

' We have made a simimplified analysis of this "p 'o p

At l t fth se investigations we used o 1 teingth gorato t

ny e

resu s were essentially no better than in the

PROBLEM OF THE LONG-RARANGE FORCE

It is well known that, in the adiabatic a rthe pote tial seen at lar da arge distances from h dy rogen

data taken nearby, at ~=1.33 and 1.45 s

m 1 1 h i h

re

Quid dis- and is its staticn ese proceedings one sh

e ipo e moment o erator

an use only the smoothest

i s static polarizability,

R(g&(oi D.gy(t)

en a=1.8 and 2.4, but it

(&)

Thus an im proved choice for the

is seen.

function (3a)rom e next ehavior

is

Bavin sa'

g satisfied the requirements iven

a 225

sa men s given in reference

ci 1ie for the scattering length

n ee at =0 a strict minimf]

Figure 3(a) and (b) h

ooth b 1 bs ow our results w '

(r,+r22/2)

o as; ut it is also—os

ere is very poor. A

12 f3r'1

gests a convergence like 1

ces o successive minima of these curves sug-where the s are functions used to s

'

larities at r—0:

=595 = 7

f 1 h

in the next section.e o e much bettter results described

Another conver enc

of

her endence problem arises at the T

A ' ' ""'"""f th' this improvedrgy region. Abov

r o settin u al

e asymptotic art of

ve p was very reat.

a erm e' ", where k'=~k'—' &.

. , an t e scale parameter

= ( —.)

p e er rc in g was varied

is a y um er and the ese new ero-eo is a small imaginar numb

n y.

g , h ch ou

ero-energy results, shown

0~

c io 1 6 dimcult to repro rate approx t 1c ions wi 1 find it di

i i a much im rove

we were unable to makexima e y equal to .1 E4

ductions from d t'for th's'n let sbl'd' p't'd ur 6 1

t't"t k 0866 17686~0.0002

gy s more approachable, iscussion4 of these computations i



ELECTRON SCATTERI NG FROM H

6.20

0 3X30 7x7

6.16— g 13 x 13~ 22x 22~ 34x 34i 5px5p

1 808

S = 0 It = 0 5 =- 1

(improvedj

—1.800

TAsLz II. e+-H phase shifts.

kap 0.1 0.2 0.3 0.4 0.5 0.6 0.7

8 (radians) 0.151 0.188 0.168 0.120 0.062 0.007 —.054

6.12—

6.08—

6.04—

—1.794

—1.786

—1.778

numerical accuracy at each step. ' This is a very in-

teresting phenomenon for which we do not have anygeneral theoretical understanding, but it appears to bea rather general property of variational calculations.

The computations reported here were carried out onthe IBM 704 facility of the University of California

Computation Center.

6.00—~ / «e'

~Z~g

QX P ~o1 77P~$

5.96—

0.4

o,= 5.96540.003i i

0.8 1.2 1.6

a,= 1.768640.0002't i I l I

2.0 0.4 P.B 1.2 1.6k

—1.762

2.0

(a) (b)FIG. 4 (a) and (b). Improved zero-energy results.

mentioned that numerical inaccuracies Lround-off

errors accumulated in solving Eq. (4)] could be aserious problem. In particular, we could not determinethe stationary value from our 50+50 matrix at zeroenergy (the old way) to any better than about 1%%A.

However, with this improved calculation the numerical

uncertainties were much reduced. There thus seems tobe a correlation between good convergence and good

POSITRON SCATTERING

A few simple modiications of the programs allowed

us to calculate elastic S-wave phase shifts for thescattering of positrons by atomic hydrogen. Sincewithout the space symmetry we now need more terms

in (3b) for each total power, (l+nt+tt); the results donot converge as rapidly as for e . The results, shown

in Table II, have probable errors of about ~0.001radian. For the scattering length we find the upperbound u+~—.10; and from the apparent rate of con-

vergence we believe that a+ will not be as little' as2.11.

8 This behavior was also noted at the end of Appendix 2 ofreference 4.

8 Compare with previous results of L. Spruch and L. Rosenberg,Phys. Rev. 117, 143 (1960).

P H YSI CAL REVIEW VOI UME 124, NUMBER 5 DECEM BER 1, iggg

Continuous Photoelectric Absorption Cross Section of Helium*

D. J. BAKER, JR., D. E. BEDO, AND D. H. TGMBQULIANDepartment of Physics and Laboratory ofAtomic and Solid State Physics, Cornell University, Ithaca, Zero borh

(Received Juiy 28, 1961)

The continuous photoelectric absorption cross section of helium has been measured in the spectral regionextending from 180 to 600 A with greater accuracy and the observations are found to agree with the calcula-tions of Huang and Stewart and Wilkinson. A grazing incidence spectrometer with a photomultiplier wasused for a single measurement at 180A while the remaining measurements were carried out in a normalincidence spectrometer utilizing photographic techniques. Whereas in previous experiments the absorbing

gas sample was allowed to 611 the entire spectrometer chamber, in the current measurements the gas wasconfined to a small cell provided with suKciently transparent windows. The use of an absoprtion cell reducescontamination and facilitates the measurement of gas pressures. The results indicate that the cross section

varies from a value of 0.98&0.04 Mb at 180A to a value of 7.7~0.3 Mb at the absorption edge located at504 A.

I. INTRODUCTION~OLLOKING recent success in the development of

windows which are transparent to extreme ultra-violet radiation, we have carried out measurements ofthe continuous photoelectric absorption cross section of

*Supported in part by the Office of Naval Research.

helium. The use of such windows in the construction ofa gas cell permits coninement of the gas sample to arestricted volume. This technique has certain advan-

tages over earlier methods'' which used the entire

' P. Lee and G. L.Weissler, Phys. Rev. 99, 540 (1955}.2 N. Axelrod and M. P. Givens, Phys. Rev. 115,97 (1959).

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Electron Scattering from Hydrogen.pdf