Elastic scattering of electrons and positrons by the Ca atom

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J. Phys. B: At. Mol. Phys. 18 (1985) 1827-1840. Printed in Great Britain

Elastic scattering of electrons and positrons by the Ca atom

S P Kharet, Ashok Kumar$§ and Vijayshrit t Department of Physics, Institute of Advanced Studies, Meerut University, Meerut-250005, UP, India $ Department of Chemistry, Faculty of Science, The University of Western Ontario, London, Ontario, Canada N6A 3K7

Received 10 September 1984

Abstract. Differential and total elastic cross sections for the elastic scattering of electrons and positrons by calcium atoms in the intermediate energy regions 10-500 eV and 10-75 eV, respectively, are evaluated using a real optical potential and are compared with available theoretical results. Various parameters associated with electron spin polarisation in the elastic scattering processes, namely S( e ) , T( e ) , I f l , lg/ and the relative phase 4re, between f and g, are also reported. Three sets of critical points are obtained for elastic e--Ca scattering. The differential cross section results agree very well with the theoretical values of Gregory and Fink; however, the spin polarisation factors overestimate their values by a factor of about two wherever comparison is possible.

1. Introduction

Mott (1929, 1932) in his two historic papers gave for the first time a comprehensive theory of the spin polarisation of electrons elastically scattered by atoms and thus opened up altogether new dimensions of investigations in atomic and molecular collision theory. The earlier studies on the theoretical and experimental aspects of electron spin polarisation in various collision processes have been reviewed exhaus- tively by various authors (Tolhoek 1956, Massey and Burhop 1969, Kessler 1969, 1976, Walker 1971, Kieffer 1971). There are two aspects of electron spin polarisation in elastic scattering by atoms that we shall be investigating in this paper with the calcium atom as the target. Firstly, we shall study the structures of the differential cross sections. An interesting feature of these structures is the presence of the critical points (E,, 6,) defined as the energy and the angle at which the DCS attains a minimum with respect to both (Buhring 1974). The interest in the critical points ensues from the fact that if an unpolarised beam of electrons with incident energy E, is scattered elasticaily at the angles close to e,, the beam of the scattered electrons is highly spin polarised. The second aspect relates to the determination of all the parameters that are to date considered necessary to describe theoretically the elastic electron scattering from spinless targets (Berger et al 1983), namely the direct and spin-flip scattering amplitudes f and g (i.e. their magnitude and relative phase) as well as the spin polarisation parameters S, T, U (to be defined in § 2). In view of the numerous applications of polarised electrons (Kessler 1982) their study becomes exceedingly important par- ticularly in their scattering from heavier targets.

8 Present address: Department of Physics, Institute of Advanced Studies, Meerut University, Meerut-250005, UP, India.

0022-3700/85/091827 -I- 14$02.25 @ 1985 The Institute of Physics 1827

1828 S P Khare, Ashok Kumar and Vijayshri

It is noticed that out of the various atomic and molecular targets the heavier atomic targets like Xe and Hg have been studied more often. A detailed review of these aspects has been given by Raj (1981). From the later work, we would like to mention the three ‘perfect’ elastic e- triple scattering experiments on Xe and Hg which give all the spin polarisation parameters mentioned above-these are by Berger er a1 (1981), Wubker et al (1982) and Berger et a1 (1983).

In the present investigation we have determined the differential and total elastic cross sections, the critical points, the spin polarisation parameters f and g along with their moduli and relative phase and the functions S and T for elastic e- scattering from Ca atoms in the energy region 10-500 eV. Differential and total cross sections for e+-Ca elastic scattering are also reported for the first time, in the energy region 10-75 eV. The method of partial waves along with the optical model potential formalism has been employed. For electron scattering, the spherically symmetric optical potential comprises of the static field, polarisation, exchange and spin-orbit interactions. For positron scattering only the static and polarisation potentials are used. The only other investigation on elastic e--Ca scattering is that of Gregory and Fink (1974) who have solved the Dirac equation numerically taking only the static field and the spin-orbit interaction into account for energies varying from 100 eV to 2 keV. They have neglected the effects of exchange of the incoming electron with the atomic electrons as well as the atomic charge cloud polarisation. There are no experimental data for elastic e--Ca scattering for comparison. Similarly no other investigation, theoretical or experimental, is available for comparison for elastic e+ scattering from Ca in this energy region.

2. Theory

In the optical model potential approach the basic idea is to analyse the elastic scattering of a particle from a complex target by replacing the complicated interactions between the projectile and the target particles by an optical potential or a pseudopotential in which the incident particle moves. Thus once the optical potential is determined, the many-body problem is reduced to an equivalent one-body problem. Then the differen- tial equation for the scattered electron, in atomic units, which we shall employ unless stated otherwise, is given by

( V 2 + k 2 - 2 V a p t ( r ) ) F ( r ) = O .

However, the equivalent optical potential, in general, is a complex, non-local, non- spherically symmetric and energy-dependent potential and its exact determination is not possible at present. One is required to take Vapt(r) in an approximate form. In this investigation Vapt( r ) is represented by a localised, real and spin-dependent potential which is spherically symmetric so that equation ( 1 ) can be solved using the method of partial waves. We have replaced Vopt(r) in ( 1 ) by different potentials corresponding to different physical effects, resulting in the following three approximations.

( I ) Static-field approximation (SF). In this approximation we have

Vopt(r) = Vodr) where the static field

Q 0 ( r , , r,, . . . , r , ) is the ground-state wavefunction of the target atom having 2 electrons

Elastic e - -& and e i -Ca scattering 1829

( Z = 20 for Ca) and V(r,, r,, . . .‘, r,, r ) is the interaction potential due to the target and the incident particle given as

Q = -1 for electron and +1 for positron.

tion we take (11) Static-field-polarisation-exchange approximation (SFPE). In this approxima-

(5) Vopt(r) = Voo(r)+ Vex(r)+ Vpol(r)

where Vpol(r) is the polarisation potential and Vex(r) is the electron exchange term which is obviously excluded for positron scattering. The non-local exchange term is converted into an equivalent local exchange potential following Vanderpoorten ( 1975) and is given as

Vex(r) =f{(;k’- VD) -[(ik2- VD)2+8.rrp(r)]”2} (6)

with VD= Voo(r)+ Vpol(r). p ( r ) is the spherical charge density of the target atom. Hartree-Fock wavefunctions for the ground state of the Ca atom as given by Clementi and Roetti (1974) have been employed to evaluate Voo( r ) and p ( r ) . The semiempirical energy-dependent Buckingham-type potential proposed by Jhanwar and Khare ( 1976) has been used as the polarisation potential:

aqr4 3k with d = -.

2 ( r 2 + d213 2 ( r 2 + d215 8A - f fdr2

Vp0l(r) = - (7)

a,,, aq and A are the dipole polarisability, quadrupole polarisability and the mean excitation energy, respectively. A is given as

where the dipole sums L(-1) and S(-1) are determined from the oscillator strength distributions of the target atom (Inokuti et a1 1967). Using the values of L(-1) and S(-1) given by Inokuti et a1 (1981), we obtain A = 0.14635 au for Ca.’ The experimental measurement of Miller and Bederson (1976) provides the value of a d = 168.71. aq = 3356.96 has been taken from the theoretical coupled Hartree-Fock calculations of Sen and Schmidt (1981). Since the polarisation potential (7) has been found not to yield satisfactory results for incident electron and positron energies E < 3.56A (Kaushik et al 1983 and references therein), hence following Raj (1981), for such energies the parameter d is replaced by an energy-independent cut-off parameter d, given as

d,=i(1.78/A)1’2. (9)

(111) Static-field-polarisation-exhange-spin-orbit approximation (SFPESO). In this approximation we have also included the spin-orbit interaction so that

Vopt(r) = Voo(r) + Vex(r) + Vpo,(r) + Vso(r) (10)

with

1830 S P Khare, Ashok Kumar and Vijayshri

where cy is the fine-structure constant = & and

5(j, 0 = 1/2 f o r j = l + $

((j, I ) = --;(/+ 1) for j = I - $ .

As expected Vso(r) also arises when the Dirac equation for a central field is recast as an equivalent SchrSdinger equation for the large components of the relativistic wavefunction (Schiff 1968).

Since the optical potential is central in all the three approximations, the differential equation ( 1 ) reduces to the following radial differential equation which has been solved numerically by the Numerov method:

1(1+1) r2

($+ k2---

Phaseshifts are obtained under appropriate boundary conditions and the scattering amplitudes are then evaluated in the above mentioned three approximations from the following expressions, respectively

(14) l M

21k / = o f , ( e ) =- 1 (21+ l ) [exp(2i~,) - l]Pi(cos e)

l M 1 M 21k i=o k i=o

f , I ( 6 ) = - 1 (21+1)[exp(2i6,)-1]P,(cos e ) +f:,(e)-- (2l+l)Sf3P,(cos e ) (15)

In approximation I11 we have the spin-flip scattering amplitude g,,, given by

(17) 1 M

g&) = -z Zo [exp(2i~1+’”) - e x p ( 2 i ~ ] - ~ ! ~ ) ] ~ f ( c o s e) .

We note that since calcium is a closed-shell atom, in the elastic scattering, exchange cannot produce spin flip. In equations (15) and (16), f& is the scattering, amplitude in the Born approximation due to the long range polarisation ( 7 ) and Sf3 are the corresponding Born phaseshifts given as

S 7 = - 2 k Iom r2 V,,, ( r ) ( j, ( kr ) ) d r.

For higher partial waves, the evaluation of the spherical Bessel functions in the integrand of (18) becomes excessively time consuming. Since it has already been demonstrated by Raj (1981) that the semiclassical phaseshift 6: given by LaBahn and Callaway (1969) may be used without affecting the accuracy for several targets, therefore we have also replaced Sf3 by S s given as

with ro = ( I + 0.91 k.

Elastic e--Ca and e+-Ca scattering 1831

In (16) and (17) Sj+'/' is taken equal to Si-' / ' when the percentage difference

The differential cross sections in the three approximations are given by between them is less than or equal to 0.01 Oh.

respectively. We have also calculated the total cross sections using the optical theorem:

The spin polarisation parameters S ( e), T ( e ) and U ( e ) are defined as

with S 2 + T 2 + U 2 = 1. The function S ( O ) , known as the Sherman function, gives the values of the degree

of polarisation of an unpolarised beam after scattering, the direction of polarisation being perpendicular to the scattering plane. It also describes the left-right asymmetry in the scattering of a polarised beam. T( e ) and U ( e ) together give the change in the direction of the polarisation vector in the plane of scattering due to scattering.

Finally we describe the method of determining the critical points used for elastic e--Ca scattering. An earlier method proposed by Khare and Raj (1980) and employed successfully to obtain critical points for atoms with 6 2 6 10 and 2 = 4 (Kaushik et al 1983) could not be used for e--Ca (2 = 20) scattering as in more than one case it was found that the real and imaginary parts of the scattering amplitude did not become zero simultaneously, and the spin-flip scattering amplitude could not be taken to be zero either. These two approximation were the basic approximations on which the earlier method was based. Hence we have devised an alternative method in which we start from the basic definition of the critical points and consider

Then we apply a technique of numerical analysis, namely the two-way Newton- Raphson method, to this problem to obtain solutions of the equations

I.E(Ec, e,) = 0 Ie(Ec, 0,) = 0 (24)

where IE and Ie are partial derivatives of the differential cross section I ( E , e ) with respect to E and 8. Then a Taylor series expansion about the points E ' , e ' , up to the first order yields

1832 S P Khare, Ashok Kumar and Vijayshri

Solution of (25) for E, and 6, yields

IE, I,, I€€, IO€, l e e are first- and second-order partial derivatives with respect to E and 6. For the evaluation of E , and e,, we take three energies close to E, and at each energy calculate I ( E , 6) at three angles close to 6,. Thus we have nine values of I ( E , 6) corresponding to these energies and angles. Of these, ( E ', 6 ' ) is the point in the data set at which out of the nine values of I ( E , e) , the value of I ( E ' , 0') is the smallest. The first- and second-order partial derivatives are then obtained from this data set using numerical differentiation formulae. E , and 6, are then determined from (26).

3. Results and discussion

We shall first discuss the results for the elastic scattering of electrons from the Ca atom in the energy region 10-500eV. In figure 1 we present the differential cross section results in the approximations I and I1 at the energies indicated in the figure. It is found that the results in approximation 111 are practically the same as those obtained in approximation I1 except very near E , and 6,.

At the lowest incident energy, lOeV, it is seen that the inclusion of polarisation and exchange effects increases the number of minima in I ( 6 ) . Also the minima corresponding to both these approximations are considerably displaced from one another. However, as the energy increases, I ( 6 ) in approximation I1 have deeper minima and these are not far removed from those of I ( 6 ) in approximation I . As expected, the inclusion of polarisation and exchange effects leads to a noticeable increase in I ( 6 ) in the forward direction at all energies. With a further increase in energy, I ( 0 ) in the above mentioned approximations come closer and finally, say beyond 250 eV, they almost merge into each other. These features in the behaviour of I ( 6) are very much in accordance with the observations of other investigations (Khare and Kumar 1978, Khare et a1 1982, 1983). The sharpness of the structures in I ( 6 ) becomes more pronounced at energies close to the critical energies. The effect of spin-orbit coupling on I ( 6 ) is evident only at energies and angles very close to the critical points, where the direct and spin-flip amplitudes f r , , and g,,, respectively are found to be of the same order. The only other theoretical results available for comparison are those of Gregory and Fink (1974) at incident energies of 100, 250 and 500eV. A remarkable overall agreement exists in both the results except in the small-angle-scattering region. This is expected as Gregory and Fink (1974) have excluded the electron exchange and the long-range polarisation effects. No experi- mental data are available for comparison. However, keeping in view the general trend of the differential cross sections for the elastic electron scattering from complex atoms in the intermediate energy region (Khare and Kumar 1978, Khare e? al 1982, 1983), it is expected that the present results will yield satisfactory agreement with experimental data whenever available.

In table 1 we list the total elastic cross sections QT for e--Ca scattering obtained by the use of the optical theorem in approximations I and 11, QT in approximation 111 being again the same as those obtained in approximation 11. The total elastic cross sections in approximation I1 are always higher than those in 1. However, for E 2 200 eV they are practically the same in both the approximations.

Elastic e - - C a and e+- C a scattering

I C 1

1833

Id I L

Scattering ang le ldeg I 102

(e)

Figure 1. The differential cross sections (in a i sr-I) for the elastic scattering of electrons by calcium atomsat(a) lOeV, (b)4OeV, (c) 100eV,(d)250eV and ( e ) 500 eV. -, present results in approxima- tion I(sF); - - -, present results in approximation 4 20 i o 60 8'0 IO0 I20 I l O I60 is0

Scattering angle l deg I IUSFPE); - . - . -, results of Gregory and Fink (1974).

1834 S P Khare, Ashok Kumar and Vijayshri

Table 1. Total elastic cross sections (in a i ) for e--Ca scattering in approximations I and 11.

10 20 30 40 50 75

100 150 200 250 300 400 500

9 1.59 70.22 54.03 45.94 40.99 33.62 29.11 23.51 20.09 17.76 16.04 13.64 12.00

303.37 138.86 80.40 60.12 49.99 37.98 3 I .79 24.87 20.92 18.32 16.46 13.89 12.17

We now consider the results for the critical points. Our investigation has yielded three sets of critical points for the elastic e--Ca scattering, the first one in the low-energy small-scattering-angle region (set A), the second one in the low-energy large-scattering- angle region (set B) and the third one in the intermediate-energy large-scattering-angle region (set C). The results are listed in table 2 for both the approximations I and 11. The results are consistent with the observation that the regions where well defined minima in the cross sections occur depend on the atomic number 2, extending typically from a few eV up to many hundred eV and their number increases with increasing Z (Lucas 1979). For example, for the elastic scattering of electrons by Ar (2 = 18), Xe ( 2 = 5 4 ) and Hg ( Z = 8 0 ) 3, 8 and 12 sets of critical points respectively have been reported (Lucas 1979, Lucas and Liedtke 1975, Walker 1971).

Table 2. Critical energies ( E , ) and critical angles (e,) for elastic e--Ca scattering in approximations I and 11.

Approximation I Approximation I1

Set E, (ev) & (deg) E, (ev) 8. (deg)

A 43.7 69.07 37.3 72.24 B 41.9 140.62 39.7 141.38 C 137.4 119.84 137.8 120.99

The method of Khare and Raj (1980) for obtaining critical points, which worked so well for light atoms like C, N, 0, F and Ne is not found to be applicable in getting all the sets of critical points reported for Ca. However, our present results are in full agreement with the results obtained using their method wherever applicable. We also note that if one extrapolates the values of the critical points given by Khare and Raj (1980), Lucas and Liedtke (1975) and Lucas (1979) for the higher values of 2, just as Kaushik et al (1983) have done for the lower values of 2, a set of critical points for

Elastic e--Ca and e'-Ca scattering 1835

e--Ca is expected around the region where the third set has been obtained in the present investigation. Further, the results of the sets A and B are also consistent with the observation that the inclusion of polarisation and exchange effects leads to a decrease in E, and an increase in 0, in comparison with the values obtained in approximation I. However, in set C the value of E, increases by a small amount in going from approximation I to 11.

Let us now consider the various spin polarisation factors, namely S( e), T ( e), I f l , (gl and the relative phase 4rel between f and g obtained under approximation 111. In figure 2 we have shown S ( e ) , T ( 0 ) and for all the three sets of critical energies and for energies lying close to them. The results are given in the angular regions close to 0,. We have taken 6 varying from 65-85" for the set A, 130-150" for the set B and 110- 130" for the set C. At other angles S( e) remains very small, T ( e) is practically one and 4re, shows a very smooth variation with little change in all the three sets. We notice variation'in all the three parameters S ( O ) , T ( 6 ) and 4rel in all the three sets with respect to E and 0 around critical points. However, for the set A the variations are much less pronounced in comparison with those observed for the sets B and C. We find that at 38 eV the positive peak of S( e) is bigger than the negative peak whereas at 39.7 eV both the peaks are almost equal and then at 40 eV the negative peak becomes bigger. Similarly T ( e) shows a very small variation at 38 eV, becomes sharply peaked to practically its minimum value (-1) at 39.7 eV and then again at 40 eV the peak becomes small. The phase factor 4rel also changes abruptly at E , and also at energies close to E,. Similar observations can be made regarding the set C. At 137.8 eV there are sharp variations in all the three parameters. U ( e) can be obtained from equation (22) of the text and are not shown here. These results can be interpreted following Kessler (1976) as follows. The magnitude of S ( e) gives the degree of spin polarisation. The sign of S ( 6 ) indicates whether the spin-up states are preferentially populated in which case it is positive or the spin-down states are more in number in which case it is negative. For example in the set B at 38 eV, S ( 9 ) is positive beyond the scattering angle 141" indicating the scattered electrons to be in spin-up orientation. At 40eV, the spin-down states are populated preferentially below the scattering angle 141.5". At 39.7 eV, i.e. the critical energy, at scattering angles greater than the critical angle, the spin-up states are more in number and at scattering angles less than 0, the spin-down states are more populated. Similar observations can be made regarding the set C where at 135 eV the degree of polarisation is peaked more towards the negative side beyond 121" and at 140 eV it is more positively peaked below 121".

Gregory and Fink (1974) have also determined the spin polarisation factors S ( e ) for elastic e--Ca scattering. The comparison of our results with those of Gregory and Fink for S ( 6) at incident energies of 100,250 and 500 eV shows that the present values overestimate their results at all angles by a factor of two. However, the magnitudes of S( 6) being very small ( at these energies, no conclusions can be diawn with any finality. However, the reason could probably be traced to the difference in the optical potential: equation (17) shows that the spin-flip amplitude g depends on the difference between Sf+l I2 and which in turn depend on Vop,(r). Thus the difference could be attributed to the different optical potentials chosen. T ( 0 ) and U ( e) together give the information that the component of the polarisation vector in the scattering plane rotates by an angle

to

1836 S P Khare, Ashok Kumar and Vijayshri

+I

0

-1 - +I E II,

0

-1 +I

0

- 1

+360

0

- 3 6 0 1360

- E o .E

E

8.

- -

-360 t 3 6 0

0

-360

A

36 eV

+ + +

37.3 eV

38 eV

75 85

6

38 eV

A

397 eV

-i.- 40 eV

T ) 140 150

Scatter ing angle ( d e g l

A

36 eV

f- 37.3 eV

r--

c 38 eV

I'-

75 85

4

B

38 eV

397 eV

-------

I

40 eV

1

v II

1 140 150

c 135 eV

?

4 137.8 eV I 140 e V

. I 120 130

c 135 eV

137 8 eV

140 eV

135 eV -7- ( b )

01 k h 65 75 85 '130 140 150

137.8 eV r 140 eV

T 0 120 130

Scatter ing angle i d e g l

( C )

- 1 120 130

Figure 2. The Sherman function S( e ) , T ( 0) and $rei at three sets (A, B, C) of energies near critical points:

Elastic e-- Ca and e+- Ca scattering 1837

when an initially polarised beam is scattered by the atom. It is observed that for sets B and C there is an abrupt change (-180") in a, at angles close to 6,.

Finally, for elastic electron scattering we also show curves E, F, G in figure 3 giving the variation of I f 1 and lgl with energy, corresponding to the three critical angles 72.24, 141.38 and 120.94', respectively. It is found that at e,= 141.38' and 120.94' and at the corresponding critical energies /gl becomes even larger than (fl. Thus even though the effect of the inclusion of spin-orbit coupling is seen to be negligible at energies reported in figure 1, at the critical points, lg/ dominates over I f / . Since g arises solely from the inclusion of spin-orbit coupling, one can conclude that the spin-orbit interaction completely dominates the scattering at critical points and hence cannot be ignored in the regions close to the critical points. Another point that is noticed is that the set A of critical points do not exhibit any of the features common to the sets B and C of critical points. The reason for this may only be surmised-it could be because the minima in the differential cross sections are not very deep for set A and the direct scattering amplitude dominates over the spin-flip amplitude even at E, and 6,. Since all these features arise mainly due to the spin-orbit interaction and are included through the spin-flip scattering amplitude, they are not observed in the case of set A.

We now consider the elastic e+-Ca scattering. Figure 4 shows the differential cross sections at various energies up to 75 eV. A minimum is observed at 10 eV along with a dip in the lower scattering angle region when polarisation effects are included. This dip gradually decreases and shifts towards still lower angles, with increasing incident energy. Inclusion of absorption effects in the lower energy region is expected to

l o r E F G

11 I

1 I A i I I n 1 I 1 1

0 50 100" 10 50 100" 10 50 100 150

Energy ( e V 1

Figure 3. The direct and spin-flip scattering amplitude Ifl and /gl against incident electron energy corresponding to the scattering angles 72.24" (E) ; 141.38" (F); 120.94" (G).

1838 S P Khare, Ashok Kumar and Vijayshri

20 40 60 8 0 100 120 140 160 180 "- 'O 20 40 60 80 100 120 140 160 180 m vi Scattering angle ldeg I L

Figure 4. The differential cross sections ( a i sr-') for the elastic scattering of positrons by calcium atoms at ( a ) 10 eV, ( b ) 40 eV and (c ) 75 eV. -, present

in approximation II(sFP). 0 20 40 60 80 :Oo 120 140 160 '80 results in approximation I ( s F ) ; - - -, present results

1 0-1

Scottering angle l d e g l

increase the differential cross sections in the small-scattering-angle region (Khare er al 1982). Similarly at very low energies ( S 10 eV) positronium formation effects would also be very important.

Finally in table 3 we present the total elastic cross sections for e+-Ca scattering. At lOeV, the total cross section in approximation II(sFP) is higher than that in approximation I(sF). For 20 < E 6 100 eV, QT in I1 is lower than that in approximation I. For E > 100 eV both the approximations yield almost the same results. However, even at 500eV, the difference between & ( e - ) and &(e+) is noticeable. So far no theoretical or experimental investigation has been performed for elastic e+-Ca scatter- ing in this energy region; therefore, no comparison is possible.

Elastic e - - C a and et- Ca scattering 1839

Table 3. Total elastic cross sections (in a i ) for e+-Ca scattering in approximations I and 11.

10 20 30 40 50 75

100 200 300 400 500

58.26 47.45 41.13 36.78 33.53 28.04 24.53 17.45 14.16 12.16 10.77

141.36 42.68 29.19 27.9 1 27.41 25.15 22.84 17.12 14.05 12.12 10.75

Ours is the first study in recent times which attempts to determine all the parameters associated with the elastic scattering of an unpolarised electron beam from spinless atoms, mentioned above, albeit in an approximate formalism. Our formalism does have its own obvious limitations; for instance it is more suitable for describing scattering processes in the intermediate-energy region. Further, we have not accounted for absorption effects, the effect of which on all these parameters is yet to be investigated thoroughly. However, past experience (Khare and Kumar 1978) demonstrates that at least in the intermediate-energy region this method, even without the absorption effects, can be relied upon to describe the elastic e- scattering fairly well. Calcium being such an abundant element in nature a complete experimental study on the above lines is warranted and should be performed covering a wide energy region.

Acknowledgment

Financial assistance from UGC is acknowledged.

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1840 S P Khare, Ashok Kumar and Vijayshri

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Clarendon) ch 5