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Differential Cross-Section for Excitation of Helium by Electron from Field Theoretic View Point.

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A N N A L E N D E R PHYSIK
7. Folge. Band 33. 1976. Heft 1, S. 1-80
Differential Cross-Section for Excitation of Helium by Electron
from Field Theoretic View Point
By T. ROYand S. BHATTACHAHYYA
Jadavpur University, Department of Physics, Calcutta (India)
A b s t r a c t . Cross-section for excitation of lieliuin from ground state to % a singlet state is
computed by field theoretic methods. Comparisons are made with experimental and Born crosssectione. Field theoretic croee-section is found t o be in good agreement with experimental results
for all ejection angles.
Differentieller Wirkungsquerschiiitt der Elcktroiieii-diireRiiii~voii Helium vom
aesiehtspunkte der Feldtheorie
I n h a l t s i i b e r s i c h t . Der Wirkungsquerschnitt fur die hnrrgung c\cs Heliums vom Grundzustand zum 2s-Singulett-Niveau wird mittels feldtheoretischer JIrtlioden berechnet und mit
den experimentellen und den Bornschen \VirkungsquerscIiiiitte!i vergliclirn. Die feldtheoretischen
Wirkungsquerschnitte sind fur alle Strenainkel in giiter ~'bereimtimmnng mit den experimentellen Ergebnissen.
1. Introduction
The angular dependence o i electron scattering from thc Iieliiiiii atom for small
scattering angles in the 31-100 cV impact energy range is esplnincd i n terms of the
first Born approxiination nnd tlie polarised Born apl~roxi~iitit
ion by TRUHLAR
e t al.
[I]. For impact energy 44 eV and for excitation of lieliuiii froin 1 S t o 2'5 state, the
above theory fits witli the esperiniental curve for nnglcs Icss than 40°. Beyond this
angle there is a great discrepancy between tlicory and rs1)criliicnt. \Ye have tried
t o remove this discrepancy by t i field theoretic approach.
I n our theory both statistics and exchange effect collie nutoilintically vin corninntators and anticommutntors of the field operators ["I.
2. Mathematical Fornialisrn
The t.otal Haiiiiltonian of tlie systein is given by
where He and HN are the free Hnniiltoninns of the elec*tron and of tlic niicleiis of the
helium atom respectively. p ( x ) a n d a(z)are the charge densities of liroton nnd electron field, and @ ( x ) and Y ( x ) the wave functions for them.
Now H is broken np into two parts ns follon-s
H = H,
1
+ 11,.
Ann. Physik. 7. Folge, Bd. 33
(2)
2
T. ROY and S. BRATTACHARYYA
If, is given by
and H , contains all other terms in (1).
The 8-matrix is given by
S = 1+ H ,
HTHT
+ higher order term
(3)
Here the interaction is broken up into the Coulomb HG and transverse H p parts.
We have kept all terms up t o the order 3.
There is no contribution from ‘I’in (3) because the initial and final state vectors
are orthogonal. Contribution from transverse part is of the order e2/c2, while contrihution from HG is of the order e2. So we neglect E l , term contribation compared to H c .
-
& 2 - ‘.-0.
.
44 e V
\
0
20
40
Hel7 ‘S-2 ‘S)
60 80
6 ldegl
TOO 720
Pig. 1
Fig. 2
Fig. 1 Differential cross-section vs. scattering angle 0 for excitation of the 2lS state of helium
at 44 eV impact energy. The solid curves represent the Born approximation. The other curves
are calculated using the polarised Born approximation (see ref. [l] Fig.4). Open circles represent experiments at 44 eh7
Fig. 2 Same as Fig. 1, buh from present calculations
3
Differential Cross-Sectionfor Excitation of Helium by Electron
The amplitude for the process is the matrix element,
Mfi = <yf181ul,>,
which becomes in our case
Mfi
=
<YfI&
yiui>
According to ROY[2] the initial and final state vectors can be written as
j lui>= J g%(P;P;PjP;) a,'tp;, as' (Pk).,t (Pa B+(Pk)10)
d3Pk d3P6d3Pk
j Yf)= s d$ (PlP2P3P4) 4-(Pl) a t (P2) a$ (PSI B+ (P4) dSPld3P2d3P3dSP4.
(5)
(6)
Here g$$ (p1p2p3p4)
and g$' (pipipjpk)are the Fourier transforms of the eigen solutions of the Schrodinger equation. It is implied that before scattering two electrons
are in the ground state of helium atom and one in the free state and after scattering
helium is in 215 excited state and one electron is in free state.
Further, a,(pl) and B (p4)are annihilation operators for electrons and &-particles.
As in our previous paper [3] we ultimately get for the matrix element Mfi the
following expression. Transforming into centre-of-mass and relative coordinates we
get for the initial and final wave functions, respectively,
+ k' + P3 + p4 -Pz) tA @ldk)
@ldk')
yi(P3)
+ (&) (P3)
(7)
+ @lS(PS) yi(k')l
g g g 0, li' -k k - 1, P3, Pa)
S3 ( k + k' + p3 + - [D(@lS(l)
(k + li' - 0 +
( k + k' - 1 )
pf(P3)
$. E{@15(k + k' - z, @25(P3)+
$. k' - z)) yf(2)
f F
+ @lS(l)
YfCk$. k' - l)1. (8)
g:? (klc'P3P4) = 8 (k
[email protected]
==
~4
Qc)
@25
@15
@lS
yi(k)
@2d2))
@1S(p3)
@25(k
{@15(p3)@2S(l)
@25(?)3))
A , B, C , and D,E , F are combinations of Kronecker deltas obtained from the product of Pauli's spin symbols, such as
A = (6,6&, - da8t 6 b 8 ) Hence,
The matrix element after lengthy calculation and transforming back into coordinate
space becomes
M = 64 2 XrXeXr~Xa~BE d3(Pc- Qc) {I1- '2))
abc,def
where
1*
(10)
T.ROY and s. BHATTAOHARYYA
4
While deducing ( 1 0 )we found many terms make no contribution, such as
+
@,s(k')Y/,(p,)@lS(k k' - 0 @ 2 s ( 4 Y - ( P 3 ) ,
because !Pt(p3),@,,4p3),@,S(p3)are orthogonal t o one another being eigen functions
of the same Hamiltonian.
Again terms like
+
[email protected](k)@,s(k') Y'/,(P3) @ l d k
k' - 4 @zs(l) Y&3)
fail to contribute because they violates the principle of momentum conservation when
integrated over p,.
I n the matrix element Mf,the term Il is the Born term due t o direct scattering
while I , is due to exchange scattering.
Taking for
and @2shydrogen type wave functions and for Yt and Pfplane
waves with momentum p and q, respectively, we find
2nag
[
5 1 2 l / ] - 2nag
2 2
+ 913
4 = 7($2
I11
According t o CORINALDESI[4]
+
+
L, = 3x2Bl 1 6 x B 2 32 B3
M , = x 3 C , + 8XaCa + 128xC3 + 256 C4.
B'f and C't can be obtained from reference [4].
Finally, taking a,,= 1, e = 1, and x = 2 , the differential cross-section becomes
o ( 0 )= 14 (111 - 4,)2
(13)
1p1
in arbitrary units. Here we have omitted the constant multiplying factor arising
from spin and other things.
3. Conclusion
Expression (13) has been numerically computed for impact electron energy 44 eV
by, 1130 IBM computer of Kuljian. For angles less than 40", the error due t o rounding
up the operation in the computer propagates at a rate faster than the original value.
So i t is not possible t o compute a(@)numerically below 40". I n computer language
such problems are known as ill conditioned problems. Hence, for angles smaller than
40"' a(@)is computed analytically. I n this region t is less than one and
For the region above 40" the differential cross-section is computed numerically by
1130 IBM computer of Kuljian. We find a minimum in the field theoretic cross-section near about 50" and this agrees with experimental observation of TRUHLAR
e t al.
111. No other theoretical curve including Born's shows such a behaviour.
Differential Cross-Section for Excitation of Helium by Electron
5
References
[l] D. G. TRUHLAR,
J. K. RICE, S. TRAJMAN
and D. C. CARTCORIGHT,Chem. Phys. Lett. 9,
299-305 (1971).
[2] T. ROY,Nuovo Cimento 5, 1048-1051 (1972).
[3] T. ROY and S. BHATTACHARYYA,
Nuovo Cimento 9, 54-58 (1974); Nuovo Cimento 10,
499-503 (1974).
[4] E. CORINA,LDESIand L. TRAINOR,
Nuovo Cimento 2, 940-945 (1952).
Bei der Redaktion eingegangen am 7. Februar 1976.
Anschr. d. Verf.: Dr. T. ROYund Dr. S. BHATTACHARYYA
Jadavpur University, Dept. of Physics
Calcutta 32 (India)
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point, theoretical, field, differential, view, electro, section, cross, helium, excitation
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