Bhattacharjee, Babulu

Deb, N C

Roberts, M J

IACS Institutional Repository

liliJmn J. Phys. 67h  (5 ). 3 9 5 - 4 0 1  (1 9 9 3 )E q u iv a le n c e  o f  t h e  f i r s t - o r d e r  t a r g e t - c o n t i n u u m  d i s t o r t e d - w a v e  a n d  a  s im p l i f ie d  f o r m  o f  t h e  s e c o n d - o r d e r  F a d d e e v - W a t s o n  a p p r o x i m a t i o n sB;thlu BhallacharjccBangdbasi College School, K/l, Sarilosli Mura Square, Cak iitiu  7(X) 012, IndiaN C  DebnepiirtmctU of Theoretical Physics, Iruhari Association loi the C ultivation of Science,Jadavpur, Calcutta 700 012, West Bengal.India anilM J RobertsC’enlre loi Copnitive and Coiiipiilatmnal Neumsucnce. IX'paiUncnl of Computing Science and Mathematics. Stilling Univeisiiy, Stilling F K ‘> IL A . l 'KA'n rnciJ IJ  J iwuary  I W JA b s tra c t : I he first ordei laigei-conimuum disioiled wave ( IC 'ItW l ) appioximation has been shown to be equivalent to a simplified lorm of the second oider Faddeev Watson (FW2) approximation when the projectile-Uiiget inieiaction is weak Ihe 'lC n W I is then compared with a peaking approximation to the FWI and fA\'2 total and diMeicntial cross sections lo i the lormation of positronium by the impact ol positrons on heliumK e y w o rd s  : C’haige transici, ion atom Scallcnng, Fadeev Watson appioaih F A C S  Nos. ; 4^ 70 -h e1. I n t r o d u c t i o nIn tlic htgh energy ion-atom rearrangement collisions, many theoretical approaches aic shown ii> be interconnected after some iuither approximations are made. DubctI | established such connections among the strong poicntial Born (SP B ) approximation 12], the impulse iipproximalmn 13| (lA),tbe peaked impulse approximalionl4| (P IA ) and the continuum clisiurted wave fC D W ) approximalion |5 |. Starling from the SPB approximation Briggs etui were able to derive several simplilicalions such as IA,continuurn intermediate stale (C IS ) ^tpproxirnalion [7] and eikonal approximation |8 ]. Recently, Datta et al 19) established a sonneciion between the first order boundary corrected Born IIO] ( B IB )  and the Coulomb- l>^ »in 11 11 (C B ) approximations A ll these efforts were made in order to bridge the gap among file well-known theoretical methods for ion-atom collisions, hi the present investigation, we I^mII stiow the equivalence t>f the first oidef target continuum distorted wave (T C D W l)(f) 1993 TAGSb7B (.S)43 9 6 Bahlu Bhattacharjee, N C Deh and M J Robertsapproximation [ 12), |13) and a sim plified form o f  the second order haddeev-W atson (FW2) approxim ation [14], [15] in which the projcclile-targel-nucicus and projcclile-clcciron  interactions are considered weak and taken only uplo first order m the respective 7’-matrices. The interaction between the target-nucleus and the bound electron is considered to be the strongest and taken all orders into the corresponding T'-mairix through the two-hody Green's lunctions. W e first establish the equivalence o f the two methods mathematically and then for a numerical check we apply these inethtKls to posilroniurn fonnalion from atomic helium.2. ^Equivalence of TCDWl and FW2 approachesWc consider the charge transfer process I + ( 2 ,3 ) - ^  ( 1 ,2 ) +  3with arbitrary masses ///, and charges The reduced masses are defined by(I)m, ni_______with the usual cyclic permutation o f  11,2,3) and the mass ratios are given by/;/,  ^ ///ia --f m fi = ///| 4Wc use the Jacobi spacc-cocirdinate r,, wilh conjugale momenta g,, in which r, IS the posilion coordinate o f / relative to the centre ol mass o f / and k and is the posilioii vector of j  relative to k.  These notations are similar to that used by Macck and Shakeshafl | K>1 for charge transfer processes. We denoie by 17 > the milial target bound slate, wdiicli corresponds to energy e] and by I7*> the final projectile bound slate with energy If the initial relative momentum is Q, = ~h k , = JU|V, and the final relative momentum is Qf  = 1ik j ~ fj^vI  Then the energy conservation gives2 /'I+ f  , = Qi2^ i^+ i', -  EThe FW2 transition operator for the proLCss ( I ) is7|yy-i “  12 4  +  r^G[)T2 4 I 2G^)I ( 2 )Here V, denotes the interaction o f  the particle / and k with / spectating T, is liv corresponding transition operator which is given in terms of the two-body Green's opcratoi G / ( E)  by die Lippmann Schwinger equationr,  = V, + y , G r ^  (E )  V, (3)We shall now show how eq (2) gives rise naturally to the first order target continuum distorted wave approximation 112). Wc consider the case where the interactions lor the paiisEquivalence of the firs f-Order target-continuum distorted-wave etc 3971,2 V'l) and K3 ( i.e. V2) arc weak, hul that between the particle 2 and the target nucleus 3 ( i.e. V”^!) is strung. This would be the case lor smaller dellecUon angles U sing cq. (3) we can then replace 7 2  and by Vp and \ \  resj^cclivcly and retaining 7’j we have from eq (2)7'nv2 -  (' +  (V , 4 VO +  v , ( ; , :v , . (4)Taking the matrix clement o f (4) between the initial 1/ > and final 1/ > stales ol the entire system and introducing an olT-shell slate vcctorhave(v/‘/ 'I = (/I (1 + 7’,GO’) = (/I (1 + V,gO'),(71 7 ',w: |< ) -  ’ i + ' '0  I') + (71 V ,G ,VV , |/>.( 5 )(h)The lirsl term on the tight hand side ol cq. (b) is similar m structure to the TCDW I appioximation with an undistorted initial state and the intcrnuclear potential \ ' t ignored. To ^ce the relationship between the TCDW I and f'W2 approximations rmirc clearly, we have horn cq. (5)k ; - ’) -  (I + f'',' V,) | / )iiul the coriesponding position-space waverunction is(r„r; , |v / ;  ’) = I  ( r , ,k )I Icrc(7)(X)IS (he olT-shell C oulom b waverunction, with incom ing wave boundaiy condition at energy I , given byE, = E /^i q\ ^  g +iiul the momentum is given byP \  = f r k i  = ag .llic bars on the operators in eq. (8) denotes that they rcicr to their own centre-ol-m assliaiiics.If the main contribution to the integral in (7) is assumed to com e Irom the region ol ali/ation q = o o f  the final bound slate wavefunclion {g \P )  then39H Bahiu BhatUwharjcc, N T  Dch tuid M J Rohe nsl’\ ~ I■l2^  »•/ and fu-1^ 2^^ i'tThus lor R >> /1, V^ r,/| <-‘ncrgy shell. The on-shell limit of the oKshell Coulomh wave funelion is not denned in the sirict mathematical sense [2|. However, by lakiii” account ol ihe long range Coulomb distortion in Us del'inition (8 ) [cl. rcl'crcnccs 17,18] we can obtainInn w/' ' ( r„ J  ^ {27t ) c\[)(-Try/2)r{l~ir)(’I (  '7  • I - “  /  * " 2 ' ) 'if< Iwhere y A{/ \ jv^  so that etj (7) becomes( r i . r ^ l v ' /  ') -  ( r , , I/)) ( 2 ;r/t)‘ cxp( • ;ry/ 2 ) T( 1 - ly)f' '' |/■■| ( ry . \ ■ \ (‘^1Heie we ag*iin note that a sim ilar technujiie of ajipioxim ating the oK shell wa\el unction by its on-shell limit, has been used before by Briggs ct al |61 but starting Iroin the stiong |X)lential Born a[)pioximation ralhei than the I 'addeev-Watson approach Another nnpoitanl assumption in oui approach is llie leplacement td 7’ by V',, (/ ~ 2,3) in et| (2) In using ei|. (3). We also note that ee]. reduces to the position lepresenlation ol |/) uhen y -  0 when distoiUon is neglected.In the case of election cajUiiie by heavy ptirticles, -  1, / \  = //^ and using the nioiephysically ineainnglul Lo-oielinates ol Macek and Shakerhalt [ Ibj we haver\ -K,, and 7 -We now see that, apait Irom the momentum-space normali/ing factor (2nli) ec| (*-M IS pieusely the tmal state distorted wave used in the T C D W  approximation (c I a| (3) 111 lel 112|) We ihererore, Lonclude that a snn[nilied form ol the I ‘W2 approximation IS equivalent to the T C D W  1 approximation il the second term in the right-hand side nl C(.| (b) IS neglected This term, ( / 1 |/), is usually neglected m lon-atom coNision due tt> thepresence oflhe nucleus-nucleus interaction VS In the T C D W  I appo>ximation this (erin does not Lippear, noi in faU, does it appeal in the T C D W 2  approximation since higher order teiins in th'' latter aie accounted lor by a Cireen's operator with a connected Kernel (c.l. rel j I-I anti leleicnces theiein)Kelurnmg to etj (2 )  again, an alternative approach is to elfcLl a peaking approximation  dncctly to the /’- matrices The second order terms in (2) have been shown 119) to he behaved H regular ('oultmib / matrices, which are analogous to the regularized oll-'^hcHEquivalence of the first-order tarfiet-continuum distorted-wave etc 309Coulomb wavcfunctions, are used. The peaking approximation then results in a factoring out o f R utherlord scattering am plitude which for certain terms can be interpreted as corresponding to classical Thomas double scattering mechanisms. W e shall refer to this approach as the second order Faddeev-W atson peaking approximation (FW PA2). A similar peaking approximation applied to the first two terms in cq. (2) yields FWl^AI. Thus for electron capture by heavy particles, the term 7 ' , G o ' d o m i n a t e s  at higher angles and accounts quite w ell lor the observed Thomas peak 119,20). For capture by positrons two terms are important, viz. and which can interfere constructively for oddparity positronium states or destructively for even parity stales [21- 23) ,  although no experimental evidence is yet available to test these predictions.3. Numerical resultsBecause the FW PA2 docs not take account ol the cl feet ol distant collisions, such as target distortion, it might not be expected to be reliable for total cross sections. In contrast the TCDW approximation treats the target distortion explicitly and the resulting total cross-section arc in good agreement with experiment for electron capture by heavy particles [12,24] and by positrons [25] as shown in Figure 1 in this calculation. By contrast, m the latter case, the FW PA2 fails badly at these energies, although for capture by high energy protons, o f theE ( eV )Ffgiire 1. Total cross sections for positroniuiii formation in e'*' + He .scattering as a function of mci|Jent e"*” energy Solid circles , measurement of Fromme et al [26|, open circle . FWPAl [2.1], open triangles . TCDWl [25] and stars . IAVPA2 results [2^  Solid, dashed and dotted lines are drawn through TCDW t, FWFA1 and FWPA2 results respectively to guide the eye All theoretical results are for IS -  KS captureorder o f several M eV , the total cross sections are not bad [19]. From Figure 1 we also note that ihe T C D W l and F W P A l total cross sections are in good agreem ent w ith the4 0 0 Bahlu Bfiattacharjee, N C Deb and M J Robertsmeasurement of Fromme el al 126] at the higher energies. This is interesting since the FWPAl docs not account of the target distortion but includes the positron-target-core interaction explicitly via 7*2 while TCDWI accounts for target distortion but excludes positron-target-core inieraclion in (6). In Figure 2, we present differential cross sections in both the approximalions al 250 eV. In the forward angles the TCDWI (excluding c"^  -  Hc"^  interaction)to  p10MO 10ci■o10 ' r1010TO A Mo A «1— —^ ro : FWPA 1 A 1 TCDW 1 ; FWPA 21“IkO A *O t o li.O .1’^6 A * >t^6 , ^  ^ X ^ * ik n X30 60L_900 (d eg ree)120 160J1BCFigure 2. IS - IS differential cross sections tor the same system at 250 eV 'Ihe symbols repa'sent the results in the same tlieones as described in Figuic 1 mresults are a factor of two large compared to FWPAl (including c/*" -  He"^  interaction) results while in the backward direction they nearly match with each other. At this energy FWPA2 results arc loo large throughout (he entire angular range.4. SummaryWe have established a relationship between the TCDWI and the FW2 approximations when the projectile-target interaction is ignored. We also demonstrate that the TCDWI (excluding e*^ “ He'  ^ interaction) total cross sections gives energy dependence similar to the FWPAl (including e^ -  He"^  interaction) total cross sections.AcknowledgmentNCD acknowledges the financial support by the Council of the Scientific and Industrial Research, Govt, of India, New Delhi.Rcfcrence.sm  LJ Dube 1984 7 Phys B17 64112) J H Macek and S Alston 1982 Phys Rev. A26 250Equivalence of the first-order target-continuum distorted-wave etc 401[3] T Pradhan 1957 Phys. Rev. 105 1250 , M R C McDowell 1961 Proc Roy. Soc A264 277 . J S Briggs 1 9 7 7 / Phys. BIO 3075[4 ] B H Bransden and I M Cheshire 1961 Proc PIm. Soc. 81 820|5 J D S F Crothcrs 1981 7 Phy.s B14 1035 , D S F Crothers and J F McCann 1982 PIm Lett. 92A 170 .Dz Belkic, R Gayel and A Salin 1979 Ptm Re\r21^[61 J S Briggs, J Macck and K Taulbjerg 1982 Comm. At Mol Phv.s J2 I [71 Dz Belkic 1977 J Phy.s. BIO 3491[81 J Eichler and FT Chan 1979 P/m Per. A20 104 , F T Chan and J Eichlcr I W  Py/v.r Rev A2 0  1841 [9 ] S K Dalta, D S F Crothers and R McCarroll 1990 J Phys B23 479[101 DPDewangan and J Eichler 19867 Phys. B19 2939 , Dz Belkic, R Gayct, J Hanssen and A Salin 19«6 J Phy.\. B19 2945S Mukherjee, K Bhadra, N C Sil and D Basu 1978 Phy.\ I ctf 65A 285 , C Smha. S Mukherjee and N C Sil 1979 J Phy.s. B12 1259 , C Smha, S Mukherjee and N C Sil 1979 J  PIm B12 139 |1 12] D S F Crothers and K M Dunsealh 1987 J Phy.s B20 4115113] N C Deb 1988 Phy.s Rev A38 1202114| J C Y Chen 1974 Ca.se Studies in Atomic Physu .s' eds E W McDaniel and M R C McDowell (Am.stcrdam North-Holland) Vol.3 3051 1 51 C J Joachain 1983 Quantum Collision Theory (AiiLsierdam Nonh-Hollund) ch 19 116| J H Macck and R Shakeshaft 1980 Phys Rev A22 1441117] H Van Haeringen 1976 J Math. Phys 17 995118] M J Roberts 1985 J Phys B18 L7071191 M J Robert-s 1987 J Phys B20 55 1jjOj S Alston 1990 P/iy.v /fev A42 331 ^|JI] R Shakeshaft and J M Wadehra 1980 Phvs Rev A22 968122] N C Deb 1989 J PIm B22 3755 [21] M J Roberts 1989 J Phys. B22 3315[24] K M Dunseath, D S F C o^ihci'S and T Ishihara 1988 J Phys B21 L46l[25 | N C Deb. D S F Crothers and D Fromme 1990 J Phys B23 L483|26| D Froiiinic, G Kru.se. W Raith and G Sinapius 1986 Phys Rev Lett. 57 3031

Equivalence of the First-Order Target-Continuum Distorted-Wave and a Simplified form of the Second-Order Faddeev-Watson Approximations

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Equivalence of the First-Order Target-Continuum Distorted-Wave and a Simplified form of the Second-Order Faddeev-Watson Approximations

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