Ghosh, A S

Sil, N C

English

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By A. S. Ghosh and N. C. Sil Department of Theoretical Physics,Indian Association for the Cultivation of Science, Jadavpur, Calc'Utta~‘ti2 {Received 24 December 19G9)Glauber approximation has beon applied to calculate the total and the differential cross sections for ls-2a and la-2p excitations in e-H scattering. In the intermediate energy region, our results for the total cross section of 1®—2p excitation are in better agreement with the experimental observations than other existing theoretical results. In the case of U~2p excitation the total cross section curve almost coincides with the experimental findings. The differential cross sections for both the cases are more sharply peaked m the forward direction than those m Born approximation.I n tr o d u c tio nWith the recent developments in the experimental techniques, consider­able theoretical interests have been focussed upon electron-atom collisions. Electron-hydrogen system is theoretically the simplest one and as such has been most extensively studied. In the case of inelastic electron-hydi'ogen (e-H) collision process, there are long standing marked differences between the experi­mental results and the theoretical findings. There was no appreciable improve­ment in the theoretical results in spite of repeated attempts. Akerib & Boro- wibz (1961) have applied the impulse approximation to the inelastic e-H scattering. A new method, which explicitly takes into account the repulsion between the atomic and incident electrons in the choice of the total wave function, has been introduced by Vainshtein, Presnyakov & Sobelman (1963). Ochkur (1963) has given a modified form of Born-Oppenheimer approximation allowing for exchange interaction. Recently (1969), he has presented an improved version of this previous work. Sloan & Moore (1968) have given a theoretical formulation for both the elastic and inelastic processes based on Foddeev equation (1961) for throe particle scattering. This approximation amounts to an unitarized Born approximation with the exchange effect taken into account. Several workers (Hamburg & Peterkof 1962, Burke, Schey & Smith 1963) have applied the close coupling approximation to the o-H scattering problem. This approximation, though theoretically sound, is laborious in practice and the results obtained in the case of the inelastic processes are not upto the expectation.The purpose of the present paper is to make an analysis of the inelastic (1s“ 25 and \s-2p) electron-hydrogen scattering processes. The dynamical basis of our calculation is the multiple scattering model proposed by Glauber (1969). This approximation is extensively used in nuclear and particle physics BesselIndian J. Phya. 44, 163-161, (1970)G la u b e r  a p p ro x im a t io n  in in e la stic  e -H  s ca tte r in g153154 A. S. Ghosh and N. C. Sil& Willsin 1968, Harrington 1968). In atomic physics, it has been applied to clastic eloctron-hydrogon scattering (Franco 1968, Tai et al 1969). Glauber approximation is based on Eikoual approximation. The latter applies to the scattering by a fixed potential and is thereby restricted to single scattering where- as Glauber approximation takes account of multiple scattering. Contrary to the method of impulse approximation the interaction between the incident electron and the proton has been taken into account in the present method.Wo have calculated the 15-26- and ls-2p excitation cross sections in e-H collision covering the energy region 10-6 eV to 200 eV.T h e o b yWe consider the target proton to be infinitely heavy and the origin of the co-oT'diuate to be placed at the position of the j)roton. Let r denote the position vector of the atomic electron and b be the impact parameter vector relative to the origin In Glauber ajiproximation, the amifiitudc of scattering Ffi(a) for the process in which the hydrogen atom undergoes a transition from an initial state i with wave function (j>i to a final state /  with wave function 0 / is given by (Franco 1968)'^ 'TT J 0/*(r)P(6, r)0i(r) exp {iq.b)d^b dr. ( 1)where q ----- kQ—kx, kn and ki being, respectively, the momenta of the incident and scattered electron The double integration with respect to d^ b is over the plane perpendicular to the incident beam direction. V{b r) has the form (Glauber 1959)r(b, r) 1 -e x p  (2/y(b)), (2)where ;\'(6) is the phase shift corresponding to the impact parameter b* Accord­ing to Glauber, the phase shift due to a number of scattering centres is just the sum of the individual phase shifts due to each, taken separately, of course, at the appropriate values of the impact parameters. Thus for the case of c-H scattering we may write (Franco 1968) _n b .r )  =  j( - ^ rwith r =  s + z , where s is the component of r  perpendicular to the incident beam and n =  v being the velocity of the incident electron.Next we calculate the scattering amplitudes for the two different cases under consideration,(A) (ls-2s) case :Here the initial and final states are, respectively, the Is and 2s states of hydrogen atom;/-e ., 0 j j ( r )  =  (Trao^)-! exp ( - r / u „ )  and (2 X ’*7r)-i( 1 ”  ) ^xp ( - r / 2 f l o ) ,whore Bohr radius.We take q to bo perpendicular to the incident momentum ko (figure 1), this assumption (Bessel & Wilkin 1968) is justified in Glauber's model which is applicable to small angle scattering at high energy.Olauher approximation in inelastic e^H scattering l5 5FigURB 1* C o o r d in a te  e y s to m  UBod fo r  in e la stic  e-H s c a tte r in g .Substituting the expressions for and V{b, r) and changingintegration variables from $4, X (=  X we can write equation (1) as ^1 _   ^ - - 1 ^  j ” j exp {iqb cos x)^  ^ X^256 A. S- Ghosh and N. C. SilIntegrating with respect to X have-  A  i i[ 1 _  ( 1 ° * * ^ ' ) ' ”  ] j„(g6) X bs db dz ds # ■where the iiitogratif)!! with respect to b is over the interval (0, co). perform the angular integration with respect to (p' and obtain'■/'W “ J t f  I[ "  (t ) ' '26t!?whore 0 [ij)  '” 2 ’^ i ( i+  1«». I +  2 »»= 1- y ‘ ) “ nd y  =  • s-f'il'cinpthe hyporgeometric fuuotion. Now we periorm the iutogratioii with respectto ri(—00, 00) and getX J q{(j b )bs d b d n  ... (4)whore and arc the Bessel functions of the third kind. The integration with respect to s is over the interval (0, co) Introducing the polar variables r, 0 given by b =  r cos 0 and s — r sin 0. to tlie integral (4) and evaluating the radial integration we haven 12911;^  rr r f 64 1J  ^ <?|^ sin^  ^  ^ J0X / sin  ^ '^^+^K j!7)“ cos2 0^XLI - ( 1  cos 201 /eos Ofn | cos 201 +  in, i  iw + l, 1; sin* 20)]d0 ... (5)(B) (U'~2p) case .Here the possible final states, with as the polar axis, are^ 2P>±ii '* ') == (2‘*7T«o‘^ )-ir exji(~/7-®o)  ^exp (i^0)0L>P)(O =  (2'’7r<')“ 0- exp (-r/2«,^) cos 0One can easily find that the factor r cos 0 that appears in the wave function for 2p0. makes the corre.sjionding matrix elements vanish. Further, it can beGlauher approximation in inelastic e-H scattering 157sliown that the cross sections for the othei two stales are the same. Thcro- loro we need calculate only a single scattering aniplitude. As in the previous case wo now substitute the expressions for and F(6, r) in equation (1)and obtainexp {iq b cos A^') oxp bs dh d,\ dz ds d '^Integrating with respect to A' we obtainW -  ! • = » «  { - I  V .H = - ) [ l - (X Ji{qh)b dh dz ds d(f>'Performing the integration with respect to 0' we getW  =-- f  «- “^P ( -  2^- ( ¥ ) ' “  ^  *wher-eandG[if) — 2; y~)2hsV = b“-\'S^Following the same procedure as in l5-2s, we finally get the scai tering amplitudePfj(a) — — — I ((/„i7)sin‘* 0 cos-'’ 0 (cosX(sin- 0— -  {%qy^  cos  ^0) (sin  ^0-\- cos  ^O yl  — J /??, 2; sin"20)dl7 . . .  (0)The dilforontial cross-section for a ])artioular transition is obtained by the relation/ ( a ) - | j W P . . .  (7)The total cross section for the process is given byQ =  27T J fF(a)] sin a dot0(9)158 A. S. Ghosh and N. C. SilR e s u l t s  a n d  D i s c u s s i o nWe have calculated the differential cross sections for l»-2s and l«-2 j. excitations with the help of the equations (5). (6) and (7). The integration in equations (5) and (6) have been done numerically using » Gatissianauadraturo. In figuie 2. wc have compared our results for the differential oross sections for l«-2« excitation at incident energies 50, 100. and 200 eV with the corresponding results of the first Bom approximation (FBA). Table 1 furnishesFiQUiiE 2. Iflxcitatinn difforontial crosH-Boefciona of hydrogen 2s Icvtil in the hist Born and Glauber approximations.Tabic 1. Difle.reniial croHS-sectious per unit solid angle in units of per steradian for excitation of 2p level o f hydrogen.ElectronEnergy(oV)Cosine of anglo o f scattering0.9999 0.990 0.985 0.939 0.868 0.75050 :i2.06 20 99 7 56 0.04 0.17 0.03100 83.51 22 56 4.22 0.25 0.03 0.004200 259.30 12.70 1.44 0.03 0.003 0.0004tho calculated values of the differential cross-section for \s-lp excitation at different values of the cosine of the angle of scattering for these incident cneigics.In the calculation of the total cross-section wc have carried out the integra­tion with respect to the cosine of tho angle of scattering numerically. Depend­ing upon the nature of the integrand we have divided tho total range of integra­tion (—1, 1) into suitable sub-intervals. In each of the sub-intervals a 16 point Claussian quadrature has been used. In table 2, wc have compared our valuesTable 2. Excitation total cross-sections 2s and 'Ip levels of Hydrogen (in units of in Bom, Ochkur and Glauber approximations.Glauber approximation in  inelastic e-H scattering 159ExcitedstatesE(eV) 30 40 60 100 2002sBornOchkurGlaubor0.1230.0800.1060.0810.0810.0720 057 0 053 0.05160.0290.0280.0282pBornOchkurGlauber1 12 0.7901.09 0 8430.950.7870.730.730.6370.47 0 48 0.462for the total cross-sections for both the cases at different incident energies with tho corresponding values obtained by using Ochkur (1969) and Born apiiroxi- mations. In figures 3 and 4, wo have shown our results for the total cross-3. Total oxcitation crosr^sections of the hydrogonio 2# lovol in cH scattoring. Solid lino B jm  ap proxinaation without exchange; small circles— close coupling approximation (Damburg e.t ul 1 9 0 2 ) with consideration of l«-2«-2p-lovels,- Chainod curve— Ochkiu- approxi- i^ natioii (1009); dotted lines—present calculation; circles with error indicated—experiment,160 A . S. Gho.sh an d  N . C. SilsoctioTiH o f  l-s-2,s’ and ].s-2^> processes, respectively , together w ith  the correspond- ing existing theoretical curves and com jiared  them  v it h  the cxperin ionta l findings (F ite H a} 11151), 1900, Lichten 1901).F ig t j iu s  4. T o t a l  o x c l t a t i o n  o ro a s-sn ctio n H  o f  t h e  h y r l m g o m o  2jj lo v o l  m  o - H  s c a t t e r in g .  S o lid  l i n o -  B o m  a p p io x im a l i io n  w i t h o u t  e x c li a u g e ;  s m a ll  circlos^— c lo s e  c o u p l i n g  a p p r o x im a t io n  ( D a n ib u r g  ct al. 1 9 0 2 )  w i t h  c o n s id e r a t io n  o f  l - 2 s - 2 p -  l e v e l s ;  C h a in e d  c u r v e -  -O c h k u r  a p p r o x i -  m n t io n  ( 1 9 6 9 ) ,  d o t t e d  l in e — p ie s c n t  c a lc u la t io n , c ir c le s  w u th  e r r o r  in d io a t o d -  e x p e r im e n t .Tlie differential cross sections obtained in Cdauhcr approximation are more sharply peaked m the forward direction at cvcjy energy than those given fiy the first Born approximation. tAMth the increase of the incident energy, our results give closer agreement with those of the first Born approximation, This is in conformity with the obst?rvatioii of Tai et al. (1969)The calculated values for the total cross section for l.y-2,s* excitation agree more closol}" with the experimental findings than the results of other theoretical calculations in the intermediate energy range However, near the threshold energy, our results deviate considerably from the experimental findings, H does not reproduce the peak at the threshold. The theoretical curve for the total cross-section for l6-2p excitation in Glauber approximation almost coin­cides with the experimental observations even upto energies as low as 25 eV: below this energy, however, there is a slight discrepancy. For both the cases imdcr consideration, it appears from the table 2 that above 200 eV the theoretical results in first Born, Ochkur (1969) and Glauber approximations are almostideuiical. This has already been pointed out for the case of l{>-25 excitation in our previous note (1909). The better agreement of our theoretical results "with the experiments in the intermediate energy region may be attributed to the fact that contrary to the first Born and Ochkm’ approximations, the Glauber approximation takes account o f the interaction of the incident electron with the proton. Glauber approximation is a high energy approximation and is supposed to hold good for the small angle scattering. It is not generally expected to give good results in the low energy region Further, for the inelastic process it does not satisfy unitarity (Glaubei' 1959). In the present study, we have not take into acjcount the exchange effect, which is expected to play an important role in the low energy region. All those may bo responsible for the behaviour near the threshold. In addition to these, there are many doubtful assumptions in Glauber model. Jn order to come to a definite conclusion regarding the validity of this method, this method should be widely used in various problems and the calculated results should lie comjiared with exxierimental findings. An extension of Glaubei’ approximation to the clastic o-Ho scattering is under way.A c k n o w l e d g m e n tThe authors are thankful to Dr. S. C. Mukherjee for valuable discussions and to P. Sinha for helping with numerical calculation for Is— 2p excitation process.Glauber approximation in inelastic e-H scattering 161R e f e r e n c e sAkonb R  & Borowitz S-, 10<U Phys Itcv., 122, 1177.Bossjol K H & Wilkin C'., 19G8 Phys. Rev., 174, 1179.liurko P. G., Bdwy II M & Smith, K. 19«3 Phys Rev 129, 125S.Diimburg R. Ya. & Peturkop R, K. 1902 Proc Phys Soc. 80, 563.Kaddcjov, L. D 1961 Sov. Phys. JETP 12, 1014Pit.0 W. L., Stobbingg R. & Braclonan T. R . 1969 Phys. Rev. 116, 356.1060 Phys. Rev. 119, 1939.Pianco V. 1968 Phys. Rev. Letters 20, 700.Glaubor R, J. 1959 Lectures in ThcorcUcal Physics. Edited by Wesley E. Bnttm ct al. 1, 315. Ghosh A. S. & Sil N. C. [in course of publication). llarrmgton D. R. 1968 Phys. Rev. 176, 1982.Liebton W. L. 1961 Phya Rev. Letters 6, 12 Ochkixr V. T. 1964 Sov. Phys. JETP 18, 503.Oohkur, V. I. 1969 Opts, db Spcct. 25, 534.Sloan I. H. & Moore E. J. 1968 Proc. Phys. Soc. Sor 1, 1 ,414.Tai H., Toubonor P. J. & BohhoI R, H. 1969 Phys Rev. Letters 22, 1415.Vaiushtoin L., Presnyakov L. & Sobolmaii I. 1964 Sovl Phys. JETP 18, 1383.

Glauber Approximation in Inelastic e-H Scattering

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Glauber Approximation in Inelastic e-H Scattering

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