Elastic scattering of electrons by helium atoms in high energy higher order Born approximation

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Pramin.a Vol. 17, No. 4, October 1981, pp. 309-314. ~) Pdnted in India

Elastic scattering of electrons by helium atoms in high energy higher order Born approximation

N S RAO and H S DESAI

Physics Department, Faculty of Science, M S University of Baroda, Baroda 390 002, India

MS received 3 January 1981 ; revised 27 August 1981

Abstract. The differential cross-sections for Y-helium elastic scattering are calculated by using Yates high-energy higher order Born approximations, through 0 (K~ 2) of the incident electron momentum, and comparisons have been made with the recent theoretical and experimental results.

Keywords. Elastic scattering; electrons; helium atoms.

1. Iu/roduction

The development of theoretical methods for intermediate energy collisions is gaining importance. Glauber's method, the modified Glauber's method, the eikonal Born series method, the fixed-scatterer approximation, the higher order Born approximation, are some of the methods which have been applied successfully (Glauber 1959;

Yates 1974; Byron and Joachain 1977; Ghosh 1977). Motivated to describe an alter- native approach for the high energy expansion of the differential scattering cross- section (DCS) in terms of reciprocal powers of Kl, Yates (1979) has proposed the theoretical method of high energy higher order Born (HnOa) approximation. This method has an advantage in that the computation of the higher-order Born approxi- n~tion terms is simpler. Further expressions for higher orders can be obtained in the closed form. The divergent integrals in the Glauber eikonal series (n~) method given by Yates (1974) are not present in this new treatment of HHOS. This method is very much like the modified Glauber approximation and thus has many of the attractive features of the Glauber approximation.

In this paper we examine the elastic scattering of electrons from the helium atom by the HHO8 approximation method and compare the results with other recent theoretical and experimental data. In § 2 the amplitude factors for the two terms in the nHOS approximation are calculated using the Hartree-Fock wavefunction for the ground state of the helium atom. The results for the Hylleraas wavefunction are obtained as a particular case from the general results. In § 3 we discuss the results of the present calculations with the other and the experimental data.

309

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310

N S Rao and H S Desai

2. Theory

In the ImOS approximation the scattering amplitudes are given by (throughout this paper atomic units are used).

if

~,~s¢'1' = - - ~ d% exp (iq'ro) VI, (ro), (1)

hJL=-- JHEA¢(~) ---- d p _f,rf~ ( q - - p - - Bl [; p + B , ~),

R . ¢ ( 2 ) _

~ ' J H E A

2 zr~ 0 K~ OB~

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oo

4zr 2 f f dp~

, , ~ , ( q _ _ p _ _ p , ~" pq_p: ~)

Kl ff~ dp (pz__Bl---~)~fl

- - 0 0 OD

~ f ap f dp"(P~+P:)rz'"'

(p~ - - B,) ~ " (q - - p - - p"~; p + P"~)' (3)

^ ^

- - O 0

where q -- K, - - K s is the momentum transfer to the target atom, Kt is the momentum o f the incident electron, B~ =

AE/K,,

is the average excitation energy of the target and the symbol @ is for the principal value o f the integration.

Vj., (r0) ---- ( % (h, r~)

I v[ w,

(rl, rz)~, (4) where V is the interaction between the incident electron and the target atom and is given by

V (r o, r~, r~ = - - 2 _q_ 1 . _~_ 1 (5)

,0

I,o-,11 t ro-,81'

where r 0, h and r~ are the position vectors o f the incident electron and the target

A

electrons with respect to the target nuclei. ~ is a unit vector in the z-direction. The general form o f rr~z~ is given in the appendix. Meanings of other symbols are same as in Yates (1979). We u s e t h e Hartree-Fock wavefunction for the ground state o f the helium atom as given by Byron and Joachain (1973).

T (h, r2) = ~I, (r0 ~1, (r~.),

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with ~ l s (r) : 1 [A exp

(--y'lr) + B

exp (--y'2r)],

P !

A = 2.60505, B = 2.08144, Yx = 1.41, y~ = 2-61.

The scattering amplitudes can be written as

,

k = l

(7)

(3)

where

Elastic scattering of e- by He atoms in HHOB

Im¢¢9.1- 1 ~ [ da(~ykylk )

--

~H~A =K, -- ×

k = 1 , 2 , 3 , 4 j = 2 , 3, 1,0

_ q2 I z ( B ~ ; 0 ) l +

I 2 11 (B~; y~) q2 + y~

~ Y YJ +* Y* Yj

1 ~ [ (~yy~y~)

~,~ ¢ ~ --~rg~r(lk=l,2,3, 4

A'~I d HEA A k X

j=2,3,1,0 q2

1212 (B~; Y~) -- q~ + y'---~ l~ (B~; O) 1

-- Bkk OY-k-OY~ Y~ Y~ I4 (B~; k

(o, 1)1 }

--

Bkj 0Y70Yj Y~ Y] ~

( i, Yk, Y~)

R,~ 1,t9,) 1 ~[ 0 [,4, (~yyk)

~2 J H E A

27r 2 K~

k = 1,2,3,4 ]=2,3,1,0

I I

(Bl; 0)

+Ia(Bl;y ~)

(By; y~)l

=i--'~ 2 --Is q + Yk Yk

Bk~( ~2 1 )

(B~. ,#. v2, ~

2 ~-~y~ y~ y~

-- 20Y-kbY, YfY] In (B~; y~; y~) ,

A~ = [q-- p[~ + B~ + fl; B~ =pZ + By + y~;

_k

(Ck'S, Ak's, Bk~'S, and Bkfs are constants).

A~.=[q--pl 2+B~+y];B, =p2+B~+y?.

1

~ yk yjA2 B21 ]' 311

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0o)

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Ii... v @I - "o 1:} I0

\ \ % % \ % % "~° t t t I I I , I , I , 20 30 40 50 60 70 80 90 i00 110 120 130 Scattering ang[e (e) Figure 1. Differential cross-section for the elastic scattering of 200 eV electrons from the ground state of helium atom. Full curves; a and b present calculations using Hartree-Fock and Hylleraas wavefunctions respectively, chain curve; Byron and Joachain (1973, 1977), broken curve; Singh and Tripathi (1980), experimental data: • Jansen et al (1976), ORegister et al (1980).

- "1 I I D

I I I i I " I I I IllI in v Ol '10 b "0 10

'\

°~ 30 40 50 60 70 80 90 g)0 110 120 130 Scattering angle(e) Figure 2. Differential cross-sections for the elastic scattering of 400 eV electrons from the ground state of helium atom. O, experimental data Bromberg (1974). The remaining references are same as in figure 1.

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Elastic scattering ore- by He atoms in HHOB

313 The typical integrals I 1 (B2; y~), Is (B~; y~), and

I3(Bl; y~)

are analogous to Yates (1979). The results for the typical integrals

I~(B[; y~; y~), and Is(B[; y~; y~

are given in the appendix. In (7), (8), (9) and (10), if k = 4, j = 0, we will get the amplitude factors corresponding to the HyUeraas wavefunction. We can write (9) and (10) as

Re r ~ _{J H E A} = Re 1 + Re~, (11)

where Re x and Re~ are of the order Ki-1 and Ki -2 respectively, in (9) and (10). The differential cross-section through order Ki -~ for a fixed q can be approximated by

d~__ _-- [ f ~ , [2 + [T.._...,H~Ae'" 1, + IRe, 12 + 2fix':,

t-+.r ,tR-~ .,rtZ'HEA -k fG3),

(12) d ~

where

fG3

is the third Glauber eikonal term of Singh and Tripathi (1980).

3. Results and discussion

We have evaluated the integrals in (8) by reducing the two-dimensional integral to one-dimensional integral. Final results for the integrations were obtained by using the Gaussian quadrature method.

In figures 1 and 2 we exhibit our results for the Des for 200 eV and 400 eV incident energies. The wavefunctions used for these calculations are Hylleraas and the Hartree-Fock. A comparison with the recent theoretical calculations on the DCS and the experimental data is also made in these plots. It is observed that for small angles our results agree well with the experimental data, and the results of the theoretical calculations by other workers. This type of behaviour was observed by Joshipum (1981) for ~'-H~ DLS calculations. The results of the present calculations agree with the results of Byron and Joachain 0973). The use of the Hartree-Fock wavefunction improves the results for small angles. The improvement of the present approximation is significant at higher energies. Further it is observed that as B~ -> 0 the expression in (8) tends to the corresponding term given by Singh and Tripathi (1980).

In conclusion we expect that the ~nOB approximation give good results at large values of Kt.

Acknowledgements

The authors thank the referees for valuable and constructive comments. One of the authors (NSR) is thankful to MS University of Baroda for a research assistantship.

Appendix

In this appendix several results pertinent to § 2 are tabulated.

are done by standard techniques.

P.--2

All integrations

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314 N S Rao and H S Desai

[ I 1 s i n - l A ' } I 4 ( B ~ ; y ~ ; y g = - - 7r3 s g n ( y + q ~ ) ~ - - ~rel

sin -1 A " l - - s g n ( y - - q 2 ) [ 2 ~ 2 -~--~- ) ]

~ = (y + q2)~ + 4q2 (B~ + y~)

~a a = (y _ q2)2 q_ 4q2 (B~ + y~)

A' = 1 - - 2B~ (y + q~)~

4,,Zy ~ (B~ -t- y~) [(y q_ q2)2 _[_ "/

JJ

A " = 1 - - 2B~ (y - - q~)~

[(y _ q2)~ + 4q2y~] (B~ -~ y~)

15 (B~; y~; y~) = [I 3

(B,; y~) -- 14

(B~; y~; y~) (y~ + y~) + I 3 (B,; yjZ)]

(B, ; y~) = - - 7r 3 [ 1 - - 2 tan - 1 ~ ]

/3

L BiJ

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W e define Uz, ( q - - p - - x y ; pq-x~) ^

as U~2i )

=

(¢fl

^-V ( q - - p - - x y ; rl.r 2) ^ v ( p + x y ; rl.r ) A

I

)

where ~b~ = ~bf are the state functions as defined in (6), and V is the F o u r i e r t r a n s f o r m o f the interaction potential.

References

Bromberg J P 1974 J. Chem. Phys. 61 963

Byron F W Jr and Joachain C J 1973 Phys. Rev. A8 3266 Byron F W Jr and Joachain C J 1977a Phys. Rev. A15 128 Byron F W Jr and Joachain C J 1977b JPhys. B10 No. 2 Ghosh A S 1977 Phys. Rev. Lett. 38 1065

Glauber R J 1959 Lectures in theoretical physics, (ed.) Brittin W E and Duncan L G (New York:

Inter Science), p. 315

Jansen R H J, Deheer F J, Luyken H J, Wingarden B van and Blauw H J 1976 J. Phys. B9 185 Joshipura XII International Conference on the Physics of Electronic and Atomic Collision-1981

313

Register D F, Trajmar S and Srivastava S K 1980 Phys. Rev. A21 1134 Singh S N and Tripathi A N 1980, Phys. Rev. A21 105

Yates A C 1979 Phys. Rev. A19 1550 Yates A C 1974 Chem. Phys. Lett. 25 480

Elastic scattering of electrons by helium atoms in high energy higher order Born approximation