Hyperfme Interactions 89 (1994) 469-476 469

Electron and positron impact excitation of hydrogen from the initially excited 2 2S state to 3 2S and 3 2p states

Surbhi Verma and Rajesh Srivastava

Department of Physics, University of Roorkee, Roorkee 247 667, India

The total and differential cross sections for the electron and positron impact excitation of hydrogen from its initially excited metastable 2 2S state to 3 2S and 3 2p states are reported. A distorted-wave approximation theory is used for the calculation. The angular correlation parameters A, (Ly) and alignment angle 7 are also reported for 2 2S--3 2p excitation. For com- parison, similar first Born results are also obtained. Interesting features are noted on compar- ing our distorted wave results with the first Born results as well as with the other available results for n 2S and n 2p (n = 2, 3) excitations from the ground 12S state of hydrogen.

1. I n t r o d u c t i o n

The study of collisions involving electron impact excitation of atoms from an initially excited state is of fundamental importance. This has several useful applica- tions in high density gas discharges, laboratory and astrophysical plasmas and gas lasers. Although demand for such studies trace back several decades, it is only recently that systematic efforts have begun [ I-3].

Even for the simplest systems like hydrogen and helium, where excitation from the ground state to higher states is becoming fairly well understood, not much attention has been paid to the study of excitation from excited states. However, for helium some theoretical and experimental works have been reported. These deal with the excitation from excited 2 IS, 2 3S states to higher states by electrons and (or) positrons giving differential cross sections (DCS), total cross sections (TCS) as well as angular correlation parameters (ACP) for S-P excitations [1,3-7]. In con- trast, for hydrogen, only TCS results for electron excitation from the 2 2S and other higher states using the first Born approximation (FBA) were available [8]. It is only recently that Scott and Burke [2] have reported TCS results using R-matrix theory for n = 2 to n = 3 excitations in hydrogen in the I Se mode. Burke has also

reemphasized the need of such studies [9]. In the present work, which is to some extent a continuation of our previous

work on electron and positron excitation of hydrogen from the ground state to n = 3 states [10,1 I], we take up a systematic theoretical study of excitations from

�9 J.C. Baltzer AG, Science Publishers

470 $. Verma, R. Srivastava / Electron andpositron impact excitation of hydrogen

n -- 2 states to n = 3 and 4 states in hydrogen by impact of electrons and posi- trons. For the purpose of describing collision dynamics a distorted-wave approxi- mat ion (DWA) theory is used and calculations are performed to obtain DCS, TCS and, where applicable, different ACP results. In the present paper, however, the results for a part of our work will only be presented, viz. the results for 2 ~S--3 2S and 2 2S--3 2p excitations in hydrogen.

2. Theoretical calculation ( D W A theory)

The T matrix for electron and positron impact excitation of hydrogen f rom initial state i to final state f in singlet and triplet modes can be expressed as (atomic units are used throughout)

T s (i -,- f) = T(direct) + 6z,.-1 r(exchange)

and

with

Tt(i --+ f) = T(direct) - ~z~,_l T(exchange)

T(direct) = (F - (2 , kf)uf(1)[ V(1,2) - U(2)lui(1)F+(2, ki))

and

T(exchange) = (F - (2 , kf)uf(1)l V(1,2) - U(2)iui(2)F+(1, ki ) ) .

Here , ki and kf are the wave vectors of the incident and scattered electron (posi- tron), V(1,2) = Z' (1 /n - 1/n2) is the interaction potential between projectile and target hydrogen a tom and ui(f) is the initial (final) state wavefnnction of the target atom. The projectile charge Z' = - 1 (+1) for electron (positron). The projectile distorted wavefnnction F ~:(r, k) satisfies the following differential equation along with theusual - (+) in (out)-goingbonndarycondit ions:

(V 2 + ~ - 2Uj)F+(r, kj) = 0 with j = i , f .

The distort ion potential Ui(f) is taken to be the same in both the initial and final channels and is given by the following expression as the sum of the static potential (Vstatic) of the final state of the target and the semiclassical exchange potential (Ve~) of Furness and McCarthy [12] corresponding to the final state of target,

Ui(f)(r) = Vstatic(r) + 6z,,-I V~e(f)(r)

with

Vstatic(2) = <uf(1) lV(1,2)[uf(1))

and

S. Verma, R. Srivastava / Electron and positron impact excitation of hydrogen 471

_ 1 1

- - [(lk~i(f ) - Vstatic) 2 q- 8 /~Tp] l / 2} .

Here p is the spherically averaged charge density of the final state of the target and r is taken to be +1 for scattering in the singlet mode of the system while - 1 for the triplet mode [13]. We know that the choice of the distortion potential Ui(0 is arbitrary but should be chosen such that the DWA model works the best. Thus the choice taken here has been found to be the most appropriate as suggested by Bartschat and Madison [14]. This is also the definition of the potential U in the direct and exchange T matrices.

After choosing a particular transition and using the partial wave expansion of the distorted waves F • the T matrices are evaluated. The differential and total cross sections are then obtained in the usual manner from the T matrices.

In case of 2 2S-3 2p excitation, the ACP results are also obtained. Since the electron-hydrogen collision is explicitly spin dependent, we consider the spin aver- age of the amplitudes and define different parameters as [ 15]

A = ([a(O)Zl) /a ,

R = R e ( a ( 1 ) a ( O ) * ) / a ,

I = Im(a(1)a(O)*) / rr ,

where a(m) refers to the scattering amplitude for magnetic sublevel m of the excited final P state, Re and Im refer respectively to the real and imaginary parts and

a = (lal02) + 2(lal2).

Here ( ) means the spin average defined by

(a(m' )a*(m) ) = [aS(m')d*(m) + 3at(m')at*(m)]/4.

The superscript s (t) refers to singlet (triplet) amplitudes. The parameters, angular momentum transfer (Ly) and alignment angle 7 can be given by

(Ly) = - 2 x / 2 I ,

tan2 7 = - 2 x / 2 R / ( 2 A - 1).

Similar calculations for ACP are also done in the case of positron-hydrogen colli- sions but due to the absence of exchange the spin averages are not required.

3. Resul t s and discuss ion

We have calculated differential and total cross sections for 22S-32S and

472 S. Verma, R. Srivastava / Electron and positron impact excitation of hydrogen

2 2S--3 2 p excitations as well as ACP for 2 2S--3 2p excitation for incident electron and positron energies ranging from near threshold energy (1.889 eV) to 40 eV (i.e. nearly 20 times the threshold). However, DCS and ACP results are presented only for two selected incident energies, viz. 10 and 20 eV which are respectively close to 5 and 10 times the threshold energy, and our DWA calculations should be quite reliable for these energies. Since there were no other calculations except the TCS results at selected energies by Vainshtein [8] using FBA, we repeated our calcula- tion using FBA to compare with our various distorted wave results. We also found that our FBA results for TCS agree very well with those of Vainshtein [8]. Our dif- ferent electron and positron impact results for both the transitions are displayed through figs. 1-4. It should be remembered that there is only a single set of results for FBA as this approximation does not distinguish between electron and positron impact results.

Fig. 1 displays the DCS results at 10 and 20 eV incident electron (positron) ener- gies for 2 2S-3 2S excitation. From the figure we see that at these energies the elec- tron and positron DWA results are quite close and both differ to a great extent as compared to the FBA results, at almost all angles except near the forward direc- tion. Also, structures present in the FBA results are absent in the distorted wave results. Interestingly, the FBA results for 12S-n 2S and 12S-n 2p (n = 2, 3) excita- tions in hydrogen do not show any such structures. These structures in FBA may be an artifact of the projectile wavefunctions which cause these to appear in the FBA and to disappear in the DWA due to distortion effects of the projectile.

Figs. 2 and 3 present the results for DCS and ACP respectively at 10 and 20 eV incident energies for 22S-3 2p excitation. For ACP we present results only for A, <Ly> and -y while through their relations with R and I the latter can be easily

I0 §

10 * ~ E = lOeV

\

p .--

10 0 I } 1 I

Scattreing angle (deg)

I I \ / I , 1111 i I I I J I I

6O

1C

1(

-lo ~ Q

120 180 10-'0

4 I

E = 20eV

g : 1 g

" 1 I I i I I [ I I I I [ | i J t 60 120 180

Scattering angle (deg)

Fig. 1. Differential cross sections for 22S-32S excitation in hydrogen at incident energy of 10 and 20 eV. ( ) Electron impact D W A results, (- - -) positron impact D W A results, ( - - - ) FBA

results.

S. Verma, R. Srivastava I Electron andpositron impact excitation of hydrogen 4'73

90

60

30

0 E 0 E 0

-30

-60

-9C

\ . -" i - - " -

I! 1 1 1

60 120 180 Scattering angle (deg)

1.0

0.5

b -o.o V

-0.5

-1.0

/ / i / / ~t\

/ /

I //

~ i ~ U l t l l l l l l l l i l i l l 0 60 120 180

Scattering angle(deg)

I0 =

10 ~

1 0 - 2 \

1(? "% -'lrlLI

Scattering

I i I i ! I * I I i I * I I 60 120 180

angle(deg)

--J

1.0

0 . 8

0.6

0.4

/ 0.2 -"

0.0 J i t i I , , I 0 60

/ / \ ',, / / ' 1

/

J I I L t I I

120 180 Scattering angle(deg)

Fig. 2. Differential cross sections, alignment angle 7, angular momentum transfer (Ly) and A para- meter for 22S-32p excitation in hydrogen at incident energy of 10 eV. The notations of theoretical

calculations are the same as in fig. 1.

obtained if required. On comparing the DCS results we find the overall situation is almost similar to that for 22S-32S excitation except that the DWA results for positron and electron differ at larger angles. The differences in the two results at larger angles are due to the differences in the distortion potentials and electron exchange effect for this transition.

It is quite interesting, however, to compare the ACP results for 22S--32p excita- tion with those for 12S--22p and 12S-32p excitations [11]. For A we see from the figures that both the electron and positron DWA results show similar structures while the FBA results do not show any structure. In contrast, the results for 12S- 22p and 12S-32p excitations [11] show that only the electron impact results have structures while the FBA and positron impact DWA results do not show any struc- ture. This could be attributed to the fact (as noted by Mansky [16] for excitation

474 S. V e r m a , I L S r i v a s t a v a / E l e c t r o n a n d p o s i t r o n i m p a c t e x c i t a t i o n o f h y d r o g e n

9~ I 60

F

:i~ -90~-

7

/ I # / I /

i I

! I 1 I 0

" -1.0 60 120 t80 Scattering angle (deg)

t.0

0.5

-o.o

- 0 . 5

~ J~q i J J I i ~ J i i I i f '~ I I

0 60 120 180 Scattering angle (deg)

10

10 ~

10 0 j

\1 " ' ~ t I ~- -~

I i I 1 I i i i l i I i 1 I i I

60 120 180 Scattering angle (deg)

. . J

1.0

0.8

0.6

0.4

0.2

0.0 0

, ,, - / 7 " 1

, , [

I I I I I I I I I I I l 1 I 1 I I

60 120 180 Sca~edng angle (deg)

Fig. 3. Same as fig. 2 except at an incident energy of 20 eV.

in helium from the metastable states) that the role of exchange is not as strong in excitations from the metastable states as compared to the excitation from the ground state. This leads to the similar behaviour of the electron and positron results. This feature is also reflected to some extent in the DCS results.

Again, the present results for <Ly> and alignment angle 7 behave very differ- ently from the corresponding results for 12S-2 2p and 12S-3 2p excitations [11,17]. We note from the figs. 2 and 3 that the present positron results for both these parameters can be positive at certain angles rather than always being negative as seen for 1 2 S - 2 2p and 1 2S-3 2p excitations. This contradictory and striking feature is similar to that seen in the case of e+-Rb scattering [18]. In fact, all the previous calculations showed that the positron impact results for <Ly> and 7 were always negative. However, FBA results for 3' behave in a normal manner and show the usual behavior [11,15,17].

Finally, in fig. 4, we show our TCS results for both the 3 2S and 3 2p excitations

S. Verma, R. Srivastava / Electron and positron impact excitation of hydrogen 475

o r- o o

v) o o

0

40

20

65

I 1 1 I 1 1 I I , I I I l l I l l I I 10 20 30 40

Energy (eV)

No= o

0 ~

m 45

0

250

w

I i l i l l l i l i l l l l l O0 I 1 I I 1 0 2 0 3 0 4"0

Energy (eV)

Fig. 4. Total cross sections for 22S-32S and 22S-32p excitations in hydrogen. The notations of theo- retical calculations are the same as in fig. 1.

as calculated in FBA and using DWA by the impact of electrons and positrons. The inspection of the figure suggests that, as expected, all the results tend to merge at high energies. The difference, in the electron and positron results at lower ener- gies are basically due to exchange effects as well as due to the opposite nature of static distortion potential in the two cases. Although the FBA results as usual differ at lower energies with the DWA results for the 2 2S-3 2p excitation, surprisingly the two results are very close for the 2 2S--3 2S excitation by electron impact. On the other hand, if we consider fig. 1, the DCS results for the 2 2S-3 2S excitation look very different even at 10 and 20 eV energies and one tends to believe that the TCS results should also be different. Thus, this special feature of 2 2S-3 2S excitation that even at lower energies the FBA and DWA give quite close TCS results is quite interesting. In fact, the present situation is very similar to the electron excitation of the 21S and 21p states in helium [19], where DCS results in FBA decrease very steeply as the scattering angle increases and the main contribution to TCS comes from the smaller scattering angles. Hence it is not surprising that the FBA might give reasonable agreement with the DWA for integrated cross sections, and yet diverge from angular distribution by orders of magnitude at large scattering angles. Although we do not present a direct comparison of our results with the cor- responding TCS results for similar n 2S and n 2p (n = 2, 3) excitations in hydrogen from the ground 12S state a qualitative inter-comparison suggests that the present cross sections for excitations from excited 2 2S states are larger by more than an order ofmaguitude [2].

4. C o n c l u s i o n s

Cross sections for 3 2S and 3 2p excitations as well as ACP results for the 3 2p exci- tation from the initially excited 2 2S state were calculated for different electron

476 S. Verma, P~ Srivastava / Electron and positron impact excitation of hydrogen

and positron impact energies using DWA and FBA theories. The present results show interesting features when compared with the behaviour of 12S-n 2S and 12S- n 2p (n = 2, 3) excitations. Especially the existence of positive values of (Ly) and V parameters at certain angles shows a new feature. We hope our present work will initiate new theoretical and experimental work to be taken up to provide much more meaningful comparison and insight.

Acknowledgement

We are thankful to the Council of Scientific and Industrial Research (CSIR), New Delhi, India for financially supporting the present work. One of us (SV) is also thankful to CSIR, New Delhi, for the award of a Junior Research Fellowship.

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J.N. Bardsley (Plenum Press, NewYork, 1990) p. 349. [17] V.E. Bubelev and D.H. Madison, J. Phys. B 25 (1992) 3229. [18] K. Bartschat, R.P. McEachran and A.D. Stauffer, Proc. 17th Int. Conf. Phys. Elec. Atom. Coll.

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