Barua, A K

Dastidar, K Rai

IACS Institutional Repository

Indian J, Phya. 49, 18-27 (1975)Angular distribution o f the fragments produced from collisionally dissociated HeH*^ ion using Slater type orbitalsK. R ai D abtidar  a n d  A. K. B a r u aDepartment of General Physios & X-rays,Indian Assooiaiion for the Cultivation of Science, Calcutta-100 032.(Received 2 September 1974)Tho angular distribution of the fragments produced from the electron impact dissociation of HoH;'' molecular ion has boon studied in the energy range from 30 ev to 1000 ev by applying the Born approxi­mation. For this purpose the electronic excitation coupled with rotational-vibrational transitioax has been considered. The electronic wave functions have been constructed from Slater ty[)o orbitals using two coiifigurations for tho state and a single configuration for the state of HcH^ ion. I t is found that there is no significant effect on the angular distribution of fragments duo to the Ixoteroxiuclearity of the molecular ion.1 . I ntro ductio n •The collision induced dissociation of HeH'*" ion has been the subject of several recent experiments (Stearns et at 1971, Schopman et al 1971). The most interest­ing feature of those studies is the observed asymmetry in the intensities of the dissociation fragments in the forward and backward directions for zero-angle scattering (Schopman et al 1971). Such asymmetry has also been observed in the case of the hoteronuolear HD+ ion (Dong & Dump 1970). Attempts have been made to interpret the asymmetry by including t)xe electron-dipole interaction toim in the Hamiltonian for HD++e system (Barua et al 1971, Saha et al 1972). I t  has further been suggested that the asymmetry may bo due to the effect of electric field of the ion source on the dipole moment of the heteronuclear mole­cular ion.Important details of the processes involved in the collision induced dissocia­tion of molecular ions can be obtained by a detailed study of the angular distri­bution of the dissociation fragments. Such studios have been extensively done for Ha+ ion both experimentally and theoietcally. For HeH+ ion experimental studios of tho angular distribution of the dissociation fragments by collision with inert gas atoms have boon performed (Schopman et al 1971). Theoretically, however, an attempt has been made to study in detail the angular distribution18Dissociation of HeH'^ ion 19of^he diasooiation fragments. Here, another approach has been made to study the angular distribution of the fragments produced from the electron impact dissociation of ion by the transition. The calculationshave been performed by using the Born approximation and Slater type orbitals to construct the molecular orbitals.2 . F o rm ulatio n  a n d  REStrLTsIn the collision of an electron with HeH^ molecular ioxi, the electronic and niioloar coordinates are specified as follows. The nuclei He and H of HoH+ are denoted by a and h respectively, the iuternuclear distance by jR, the coordi­nates of bound electrons 1 and 2 by r i and and the coordinate of the im pinging  electron 3 by (figure 1). kt and kf are the wave vectors of the incoming and outgoing electron respectively. TJien =- kf^^2/iAE, where ^E  is the energy required to raise the molecular ion at the equilibrium internuclear separation of the ground X^Z state to the first excited ^^2 state and fi is the reduced mass of the system.HRFig. 1* Coordinate systom describing tho collision of an electron with HeH+ ion. 0  is the centre of mass of the ion and R is the internuclear distance.The wave function for the X^Z state has been chosen as ^o(ri, r*; JR)  -and that for the state asr,; B) =  (h tiirv n ;  B )+ 02^i(ri, r*; B)+  ( 7 2 )(1)(2)20 K. Rai Dastidar and A. K. BaruaIn eqs. (1) and (2) <f>’s are Slater type atomic orbitals and o’s are the ooefi&oients for the different configurations. The Slater type orbitals used may be writtenas^,w(r)== (-^ |* ex p (-j;ir)=" ( 9^ -)  ^r oxp (-z«r/2),(3a)(36)Zi is the nuclear charge for ith nucleus. and can he obtained by normalizing r-i', li) to unity and making -^h and orthogonal to each other.Since we are interested in calculating only the direct scattering amplitude by using Born approximation, the electron-dipole interaction term and electron- nuclei interaction terms in the interaction potential•Vo ra, ra, R)1 1, + r1________  _  _______I____  ' ____I ra+1 /5/J I I ra-4/5/? \  ^ | ra^r^ I ^ | ra-ri |will not contribute. The scattering amplitude for transition underthese conditions may be written avS/«(X, 0 , $) =  -  Ja  J J J {oxp (iK ri)-1-exp (iX ra))Ti, BHoiri, r^ ; B)xt*(R)Xo(R)^ridridR= - |a ( / . i + / « ) .  -  (5).wherefsi =  i i i  oxp(iK:.ri)^«*(ri, rai iO W n, ra; B)Xs*{R)Xo(R) r^  ^ dr, dR — J [Gi(Bi+R2+R,)+C2(R^+B^-{-Rf^+R^)]>^Xs*(R)Xo(RnR^ ... (6)where the R ’s are given below. A similar expression for fg, can be obtained by replacing by r,.Tn eqs. (5) and (6) K  “  ki^kf> the momentum transfer vector is taken along the 25-axis of the system, x^ kR) 8*^ 0 the nuclear wavefunctions for theinitial rotation-vibration state (v =  0, i7 =; 0) and continuum respectively.Dissociation of HeH* ion 21Tlie nuoloar wavefunctions used are similar to those for H2+ ion used by Zare(1967) which are given asandXo(R) —X»(/?) == S (2J'+ l)(-i)-^ 'exp(*ij<)J’»y P j '(k .B),... (7). . .  (8)where and F>j> (B) are the radial wave-functions; the latter has theform of a sine-wave with phase-shift Sj> at largo separations. The propagation vector k  is along the asymptote of the recoiling atoms and R  is the position vector which also coincides with the molecular axis. The polar angles for K and B are (ff, <j>) and ( 0 ,0 )  respectively and P r  (k, B) can be written asA  A. 4t7Pj '{k.B) =  2 ^  S (0. O),and henceX A R ) = ^ir I. S exp(idj’)F‘j iB )  Yj >m '(0, </>) Yj >m '(Q, *).Putting= ria—\R=  rih+tR>the Ri'n in eq. (6) are given as-  Li 2  (2»H-l)(-i)«j„(i KB)P„{oob(9 )(10)(11)16=  (4+^pp- ®^ P X R)=  S (2w+l)i«j„(| KB)P^{cob®), (12)B a =  ^ |r e x p ( - i K .^ J ? ) V ,=  S S La(l)(2n+l)(-i)»j„(i KB)P„{oob ©)P, (cos ©),n Iwherei?o ** exp ( - 2o»-jo)47r»'»(13)X^iaflj{R, ria, «6)dria+ f  Ji^^(Kr^a) exp (—ZaTja) JiXriafit(Jl.ria,Zi,)dria}], (14a)and22 K. Eai Dastidar and A. K. Barua00= i n j  exp (-a«r4„)/*(zj„ Tm ,0512 Z(l^  . • 1 r%\^  2V 6 (4za*+^“)®=  L, S (2n+l)(-i)nj„(i KH)P„(oos 0),n2K\2 s in [3 tan-i(-^^)lexp(*A. J E) -r i s  S (2n+l)i«j„(J KR)P„(coH 0 ),whereand=  S  S  Le(J)(2»+l)»"i„(|^JJ)P„(cos 0 )Pj(co8 ©),n I477®^*'® oo Hi'(2«+l)P,(oos©){J i-o 0Xoxp(—Z6rji,/2)rii,*/j8(P, r^ i,, Za)drjftCD+  J Ji+i(jK^»'i6)oxp(-Z6rii,/2)ri6*/ia(P>ri6,Zo)dri6}P? — “ i^gg  ®*P( ■ s R)Ia^i=  S S L,(J)(2w+l)(-i)«j„(|^P)P„(co8.0)Pi(oos©),n IOO7, =  4w J exp(-z„rj*)/„(2i,/2, r^ ft, P)rji4rji,.In the above expressione (1 1 ) to (20)r _  16V^  ~  (4zfl»+P®)* ’T 16® “  (4+iT*)* ’.. (146)... (15)(16)... (17)... (18)(19)(20)... (21a) ... (216)jy f^880Cl0ft%0th o f  io7i2 3M l)P l  (OOS 0 )  =. . .  (21o)Tj —2 7 ^ 5  (4 z „ a + ^ a ) a - ' . . .  (2 1 d )2  s in  [3  t t t n - i  J. . .  ( 2 l e )i a ( 0 P * ( o o s © )  =  ^ - ^ - ^  7 , /^ ,. . .  ( 2 1 /)i , ( l ) P , ( o o s 0 ) = ^ ^  7 ,7 ^ ,1 j- ^. . .  (2 1 y )> r.1 rM U , r^ a, *6) =  ifj.,.l(®6»-lo){/z+3/*(Z6i?)+/,_j(»6ie)}2" ■ *^+ (^*®-®H J^-i(2ii>'ia)+^i+3/2(z6rja)}], for 5  <  rjo .. . .  (22)Similarly, the expressions for / .  (R, r^, z,) can be obtained by  putting I =  0 6(|. (22). inTo avoid elaborate caloulations, the upper state radial wavefunotion has been replaced by a normalized ^-function. N S(R ~R ,)  whore R ,  is the intemuolear equilibrium distance for the ground state o f the H eH + ion. The axial-recoilapFoxim ation  has been assumed which results in  considerable simplifications in the calculations.For the num erical evaluation o f / , ,  the following «  and I values have beenused«  — 0 , 1  and 1 =  0 ,1 .Sample caloulations have shown th at the contributions o f the higher order terms to h i  Me small and m ay therefore be neglected. For a certain com bination of n and I values, only certain particular values o f  J ’ will contribute in the summa- lon over J ' ,  th e  final rotational quantum  number.Thorefore, fg  ^can be written as24 K. Rai Dastidar and A. K. Barua0 , ^  2  fnlfn<^o {«o(23)wherefnl =  {(2n+l)i^ exi){—iC)F'"vj(l^ Q)J^ ()^ YjM((^ > ^)^«o(©> ^)}[{L.,+L,)UiKRo)+ ( - 1  n L ^+ L ,)U ^K R ,)+ (2 l+1 )i» Fj„(0, ^){L^{l)j„(4l5KRo) +  ( - 1  n L ,( l ) + L M  jn i iK R M  ■•■ (24)ABSuming tho axial recoil approximtion to bo valid, the factor ( i)^' exp(t5j')equals exp(—iG ) wlvere 0  is a constant to order r(</'+^)/i<r]® and the sum over J and M' can be readily performed.Since = fs^ numerically, wo can put fs =  2/si- The Born differential scattering cross section may bo written as,W  ©,<t) =  ^  X IA (A .© ,0 ) r .  ■■■ (26)Kjwhere (0 , are the polar angles for R  which is the diroction of fragmentation according to tlu' axial recoil approximation (figure 2). Eq. (25) depends on the magnitude of the momentum transfer vector, whicli. is given by,A'2 =. fc,2-f-fc/^2fcifc/cOB w, ... (27)o) being the angle between ki and kf.Fig. 2. Courclinato system showing tho relation between scattering angles in the momen> turn transfer and electron beam frames. Dissociation fragments ejected along R  are described by tho polar angles (0 ,® ) and {6 ,0 ) in the two respective frames.In order to make the differentia] cross-section for ejected fragments inde­pendent of the angle of the scattered elo(;trons, the cross-section i ,  {K, 0 , $) should be integrated over sin co dw d<4„. Tims one can obtainT[27T maxI » { & ,  $ )  =  J S  1 U K ,  0 ,  $ )  I *A'dJ5: (27)where Kmt„ =  h - k f  ■ =  *«+*/•Using oqs. (23) to (27) one can obtainS|/»x(iT, 0 , 0 )P  m-\-g oos  ^ ©. (28)Th« values of tlve coefficionts in oq. (28) can be easily obtained from the preceding relations. So far, the angular distribution of fragments has been calculated in 7nomenlum transfer frame i.e., whore the momentum transfer vector K  is along Z-axJK. Tn order to compare tlieso results with experiments one must transfer the cross section to the electron beam frame, Avhore the electron beam directionDissociation of HeH* ion 25ueqf pe5!fig- 3. Angular distribution of dissociation fragments of HeH+ ion (XiS-+ A»£ transiton) fer different oleotron bombardin^nt energios.26 K. Rai Da&tidar and A. K. Baruaki is taken along 3-axis as shown in figure 2 . by substitutingThis transformation can be doneandcos 0 “  cos 0 cos 0 '+sin  9 sin 9' cos 0 , ® =  8in“^(sin 9 sin ^/sin 0), (29)where (©, and {9, <p) are the polar angles of R  in the two different frames and are the polar angles for ki with respoci to R . In this case <}>' — 0, since the Y-axis for the two frames have been made to coincide. Averaging over the angle 0 , the differential scattering cross section for the fragments in the -frame is given by,Therefore, tJu> angular distribution of the fragments is symmetrical in the forward and backward directions and the angl(5-dependont part is suj^orimposed on an angle-independent term.3. Disoxtssion of ResultsThe nature of 7(<9)/7(90®) curves for the angular distribution of the fragments produced from transition of HcH^ ion is similar to tl\e curves forthe homonuclear dissociation. In the lower energy range tli.e ratio is expected to increase considerably when exchange is taken into account. Tins has been shown by Saha et al (1972) for the dissociation of HD+ ion. The differential cross- section for fragments of the hetoronuclear ion is obtained as a sum of an angle- independent term with a combination of cosine square term and the square of a cosine square term.The experiments performed for the dissociation of HeH+ ion by electronic excitation indicate that both and X^ll —> transitions are energeti­cally possible (Schopman et al 1971). The latter transition although optically forbidden is possible when excitation takes place by collision with another particle. However, from the available data it is not possible to ascertain experi­mentally the relative importance of the two transitions mentioned above. I t  is not possible to obtain scattering amplitude for X^S —> transition using Bornapproximation. Therefore, exchange scattering should be considered for the singlet-triplet transition.For the ground X^S state of HeH"^ ion no ionic term has been included in the wavefiinction although this state is known to bo strongly ionio (Michels 1966). The ionio term will affect the magnitude of the differential cross-section of the fragments and the results reported by us as ratios arc not likely to be affected significantly by this approximation. I t  can also be shown that if a purely ionicDissociation of HeH'^ ion 2 7wave function is chosen for ground state, the resulting differential cross section has the similar angle dependence as that for the non-ionic wave function.B e f e b e n o e sBama A. K., Saha S. & Mullick K. 1971 J .  P hya . B .  4 ,1450.Dong P. & Dump J. 1970 Ghe.m. P hya . iM t.  5, 340.Miohfils H. H. 1966 J .  C hem . P h ya . 44, 3834.Mulliok K„ Rai Dastidar T. K. & Barua A. K. 1974 J .  P hya . B .  7, 288.Saha S., Mullick K. & Bama A. K. 1972 7. Phya. B .  5. 1369.Suhopinan J., Los ,T. & Maas J. 1971 P h ya ica  51, 113.iStoarns J. W„ Berkner K. H., Pyle R. V., Brieglels B. P. & Warren M. L. 1971 P hya , E ev ,A4. 1960.Zare R. N. 1967 J .  Ghmn. P hya . 47. 204.

Angular Distribution of the Fragments Produced from Collisionally Dissociated HeH+ Ion using Slater Type Orbitals

Barua, A. K.

Dastidar, K. Rai

3.	Angular Distribution of the Fragments Produced from Collisionally Dissociated HeH+ Ion using Slater Type Orbitals

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Angular Distribution of the Fragments Produced from Collisionally Dissociated HeH+ Ion using Slater Type Orbitals

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