Bhullar, A S

Sud, K K

IACS Institutional Repository

Indian J. Phys. 73B (2), 2 9 9 -3 0 4  (1999 )I J P  B— an in ternationa l jou rnalC a lc u la tio n  o f  tr ip le  d if fe r e n t ia l A -sh e ll io n iz a t io n  c r o s s  sec tio n s  o f  a to m sA S Bhullar and K K SudD epartm ent o f Physics, C ollege o f  Science C am pus, M L Sukhaciia University, U da ip u r-313 001, R ajasthan, IndiaA b s tr a c t  In th is c o m m u n ica tio n , we rep o rt our c a lc u la tio n s  o f  tr ip le  d iffe ren tia lA '-shell io n iza tio n  c ross sec tio n s (d iffe ren tia l in energy  and ang les ol the sca tte red  and e je c te d  e le c tro n s)  o f  a tom s by re la tiv is tic  e lec tro n  im pact u sin g  one pho to n  ex ch an g e  approxim ation In the present investigation, we use Dirac plane waves foi both incident and scattered electrons and Darwin and relativistic Som m eifeld-M aue wave functions to represent the bound and ejected electrons, respectively It is dem onstrated that the shift in the peak of the b inary m axim a and form ation  o f  the recoil peak in (e , 2e)  co llision  occurs due to the in te r fe r e n c e  b e tw e e n  th e  a to m ic  tr a n s i t io n  c h a rg e  d e n s ity  and  a to m ic  t r a n s it io n  curren t density  co n tribu tion  to the trip le  d ifferen tia l cross section  (TD C S) in asym m etric  g eo m etryK e y w o rd s  A -shell ionization , T D C S ,  u \  2e)  p r o c e s s  P A C S  N o. 3 4  8 0  D p1. IntroductionThe electron impact inner-shell ionization cross sections of atoms (differential in energy and angles of the ejected and scattered electrons) are sensitive to the details of the atomic wavefunctions as well as to the details of the influence ol the atomic potential on the incoming and outgoing electrons and thus their study offers a sensitive test of theories. Since the first coincident measurement of (e% 2e) process on atoms by Ehrhardt etal [ 1) and Amaldi et al [2] extensive theoretical and experimental investigations have been done. Wc refer the reader, for details on (e% 2e) process for atoms, molecules and solids to the review work of Ehrhardt et al [3], Byron and Joachain [4| and McCarthy and Weigold 15, 6J. The absolute triple differential cross section (TDCS) for relativistic electron impact AT-shell ionization of different targets ranging from copper (Z= 29) to uranium (Z= 92) have been measured by Nakel and collaborators 17-12]. Theoretical calculations of the differential cross section of/f-shcll ionization by electron impact have been performed by a number of workers [13-191. Fuss et al[ 13) and Bell [14) have calculated triply differential /('-shell ionization cross section of atoms in plane wave impulse approximation whereas Cavalli and Avaldi [15] have calculated in semi-classical impulse0 1 9 9 9 1ACS300 A  S  B h u lla r  a n d  K  K  S u dapproxim ation. The theoretical lorm alism  o fT D C S  has been developed by D as and K onar 116] using the sem i-relativ istic  Som m crfeld-M aue wave function for the slow  electron  w hereas Jakubassa-A m undsen [171 used non-relativistic C oulom b w ave function  m ultip lied  by a free Dirac spinor for both outgoing electrons while W alters c t  a l l  1 0 ] used sem i-relativistic  Coulom b wave function for the slow electron and plane wave for the fast electron. Jakubassa-A m undsen  117,18) has calculated TDCS in Coulomb-Born approxim ation in which the continuum  eigenstates o f  both electrons are described by C oulom b w aves to an effective  charge. D re i/lc r  e t  a l  119] have com puted TD CS in relativistic d istorted wave Born approxim ation  in w hich incident and outgoing electrons arc described in the field o f atom  or ion w hile the ion ization  is described  in term  o f a first order electron-electron  interaction. We present in Section 2, an ou tline  ot the theoretical form alism  used by us to com pute the triple d ifferential A'-shell ion ization  cross section. In Section 3. the results ol the present calculation  for copper and silver targets is p resented .2. TheoryWe present in this section, the theoretical form alism  in w hich we have usetl one photon exchange approxim ation to calculate triple differential X-shcll ionization cross section o f atoms Feynm an diagram  o f one photon exchange approxim ation  for ( < \  2 e )  inner shell ionization process is show n m Figure 1, where k {) and A represent the incident and scattered  electronwith four-m om enta k () =  { ^ ~ ^ () )and  ) in X-Z plane. T he scattered  e lectron  makesan angle w ith the incident electron beam  axis. The electron em its a virtual photon carryingF ig u re  1. Schem atic diagram  show ing Ihc four-m om enta of the incident e lectron , scattered electron and elected electron, respectively, indexed 0, I and 2 q  = kn - k , is the momentum tran sfe rm om en tum 0  (in Z-direction) and energy A E  which im pinges on the atom , resulting  in knocking out o f  one inner shell e lectron (ejected e lectron) and an ion. T he e jected  e lectron  has four-m om enta  as j is m oving in X -Z  plane m aking an angle 0 2 w ith the e lectron  beam  axis."The s  denotes the four-m om enta transfer. The present form alism  has been developedw ith the fo llow ing  assum ptions [2 0 ] :1. The incident and scattered electrons are described  by D irac p lane  w aves.Calculation o f  triple differential K -shell ionization cross sections etc 301A single inner shell bound electron in the process absorbs the virtual photon emitted by the incident electron.The bound electron is described by Darwin wavefunction in Coulomb-potentialV ( r )  =  -4. The ejected electron is described by relativistic Sommerfeld-Mauc wavefunction.In electron scattering process, the main work is to evaluate the transition matrix element. We define the transition matrix element a s :M , ( I )where./ is the atomic transition current given by J  ( i )  =  <t>. J  <p with J  representingthe atomic transition current operator whereas and 0* N respectively the bound andelected electron wavefunclions. is the lour potential generated by the electron current j  and is simply given in terms of the retarded Green’s function by/V r ,=  r i G(r<"r) j H{r' )(lr< ' <2)and 6 ( r . . r ) = - j ------- p .  (3)lrr " r |Here, j  is the electron current and is given in terms of the incident and scattered electron wave functions :V r< ) = < W r , , v ' r(r >.1’he triple differential cross section (TDCS) tor the { e ,  2 e )  process can be written asd ' < J ( c ,  2 c ) 2k IP(4)(5)where / n is the incoming electron flux U '  k f) / E {)). and P,. and P,s are the density of the finalslates have the same form lor the outgoing scattered and ejected electrons and is given by the I'crnn phase space isPr =-k  E .(2 K )  cX ~ r :with i  = 1,2. (6)On squaring the matrix clement (eq. I), summing and averaging over electron spins, we obtain4 k~ T .[/4g (.v,. mk ) +  A  j  ( s , , mk ) -  2 A QJ (.v,. mk )] (7)302 A S Bhullar and K K Sudand is the sum to be taken over the spin states of the bound (m* = ± 7 ) and the ejected\2.mkelectron (s2 = ± 2).Ag(s2.mk ).A)(sv  mk ) arc the atomic transition charge density and atomic transition currentdensity contribution and AQJ{sv mk ) is an interference term between the atomic transition charge density and the atomic transition current density contributions. The expressions for these quantities are as follows •( 4 f  e  t \ t i~A 2,(s2,mk) = ))(* ,.7 /((.v,.mk ) ) ] - ^  |J rM v mk :( 8). (9)and Qt = 2 Re r( E E \ - ~ k {)+ - y k  J • J  f f S2'mk )_ 1N_'( 10)Here Qfl(s1j n k) and J fl(s-),mk) arc atomic transition charge density and atomic transition current density, respectively.-160 -120 -go -40 0  40 go 120Angle in degree («2 )F ig u re  2. T riple different cross section (m b  /  s r 1 keV) for tf-shell ionization  o f  copper (Z  = 29) plotted against scattering angle 02 (in degrees) o f  the ejected electrons (E , =  71 keV) with incident electron energy E„ = 300 keV and scattered electron  (E, = 220 keV ) being detected ot 0, = - 9°. The dashed and dotted curves represent, respectively the T D C S by including the co n trib u tio n s o f  the atom ic transition  charge density  and  a tom ic  tran sitio n  charge  density  p lu s a to m ic  tran s itio n  cu rren t d ensity . T he so lid  cu rv e  re p re se n ts  th e  T D C S  by fu rther inc lud ing  the in terference  term .For brevity, the explicit expressions of AQ(s2,mk ), A j(s2,mk ) and Agj(s2ymk ) arc not presented here.3. Results and discussionIn Figure 2, we have depicted the result of our calculation of /(-shell ionization cross section of copper in coplanar asyfTunetric geometry at incident electron energy of E() = 300 keV for two outgoing electrons having energies £, = 220 keV and£ 0 = 71 keV at a scattering angle 0, = -9 °  (last electron) as a function of detection angle 6\ of the slow electron. In Figure 3 TDCS for silver target in coplanar asymmetric geometry is shown. Here the angle of the fast electron is fixed at -  7° and the energy values E} = 375 keV and £ ? = 100 keV for the incident electron impact energy £ () = 500 keV. In Figures 2 and 3 dashed curve represents the results of our calculation of TDCS by including the atomic transition charge density contribution only whereas the dotted curve depicts the contribution to the TDCS, due to atomic transition charge density as well as atomic transition current density. The solid curve in Figures 2 and 3 depicts the iesults of our calculation of TDCS by including the contributions of the atomic transition charge density, atomic transition current density as well as the interference between the atomic transition charge and current densities. The results presented in Figures 2 and 3 demonstrates that the shift in the peak of the binary maxima and the formation of recoil peak in (e. 2e) collision occurs due to the interference term to the TDCS in asymmetric geometry. We suggest here that there is a necessity of extensive computation on different incident electron energies at different targets to appreciate better the contribution to the binary and recoil peak due to the interference term. Extensive computation of the TDCS at relativistic incident electron energies and for diHerent targets for available experimental data is in progress.Calculation of triple differential K-shell ionization cross sections etc 303Figure 3. Same as in Figure 2 but for silver (Z = 47) with £0 = 500 keV, £, = 100 keV, Ex = 375 keV and 0, = - 7 .304 A S Bhullar and K K SlidAcknowledgmentsThe support from the Special Assistance Programme of the University Grants Commission (UGC), New Delhi to the Physics Department, Mohanlal Sukhadia University is gratefully acknowledged. One of us (ASB) acknowledges to the Council ol Scientific and Industrial Research (CSIR), New Delhi for senior research fellowship.R e fe re n c e s[1] H b h rhard t, M Schulz. T Tekaat and K W illmann / V i v a  Rev U ’tt 2 2  89 (196V)[ 2 1 U A m aldi, A Egidi. R M areonnero and G P i/ /e la  Rev Set lu s trum  40 1001 (1969)|3 |  H Ehrhardt, K Jung, G Knoth and P Schlcinincr Z /7/v.s. 1)1 3 (1986) f4 ] F W Byron and C J Joachain /V/v.s Rep  179 21 1 (19X9)[ 5 | I E M cCarthy and E Wcigold R ep  P u P h y s  51 299 (1988)(6 ) I E  M cC arthy and E W cigold Elei n o n -A to m  Coll is ions  (C am bridge C am bridge  U niversity  Press)(1 9 9 5 )(71 E Schule and W Nakel J P in s  B I5  L639 (1982)[81 H R uoff and W Nakel J P in s  B20 2299 (1987)[9] J Bonlert, H O ral and W Nakel ./ P in s  B24 1423 (1 9 9 |)[101 H R J W alters, H Ast, C T W helan, R M D re i/le r. H C.raf, C D Schro ter, J B onlert and W NakelZ  Phys  1)23 353 (1992)[ 1 1 J C D  Schroter, H-TH P n n /, N Keuler and W Nakel (c. 2e) a n d  R ela ted  Processes  eds C T W helan. H R J W alters, A Lahm am -B cnnam  and H E rhaid t (D ordrecht K luw er) p 4(H (1993)[121 K H Besch, M Sauter and W Nakel ./ Phvs  B30 E73 (1997)[13 ] 1 Fuss. J M itroy and B M Spicer ./ Pins  B IS  U21 (1982)[ 14 J F Bell ./ P i n s  B22 287 (1989)[ 1 5 1 A Cavalli and L Avaldi Nuovo Can  1)16 1 (1994)[ 16) J N Das and A N Konar J Phys  B7 2417 (1974)[17J D H Jakubassa-A m undsen Z. Phys D l l  305 (1989)[181 D H Jakubassa Am undsen J P in s  B25 1297 (1992)119) R M D rei/ler, H Ast, S Keller, C T W helan, L IJ Ancaram and H R J W alte r  ./ /V/v.s B30 L77 (1997)(20 ) A S Bhullar, C Selvarajii and K K Sud E lv i t ron  Im pact  t u p l e  D if feren tia l  K -S h e l l  Ioniza t ion  C io wSection of  Silvei  (Presented at the XI N C A M P held at IIT M adras. Dec 17-20) (1996)

Calculation of Triple Differential K-Shell Ionization Cross Sections of Atoms

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Calculation of Triple Differential K-Shell Ionization Cross Sections of Atoms

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