Kushwaha, M S

Saksena, Vandana

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Indian 1  Phys. 70B (2), 97-101 (1996)I IP B— an intemtlional journalBethe collisional parameters for the ionizing collisions of molecules by relativistic electronsVandana Saksena*+ and M S Kushwaha*"Department of Applied Physics, Madhav Institute of Technology and Science, Gwalior-474 005, India+Department of Physics, Chaudhary Charan Singh University.Meerut-250 004, IndiaReceived 15 December 1995, accepted 27 December 1995Abstract : A method based on the fin’ Bom approximation which includes relativistic effects, longitudinal as well as transverse interactions of the virtual photons with the molecules but uses only the optical oscillator strengths as its input, has been employed to obtain the Bethe collisional parameters for H2, Nj. Oj. CO, FfeS. COj, NHj. CS2 and SFfc molecules ionized by relativistic electrons. For most of the molecules, our values of the parameters are in good agreement with the experimental data.Keywords : Ionization, moleculesPACSNo. : 34.50 GbIonization cross sections of atoms and molecules find applications in a number of fields such as upper atmospheric physics, space physics, plasma physics, fusion, laser, radiation physics, mass spectrometry, chemical reactions and even in biological sciences [1]. Hence a large number of investigations for the experimental determination and theoretical evaluation of the ionization cross sections have been carried out. More than twenty years ago Rieke and Prepejchal [2] measured the total ionization cross sections of a number of molecules by electron impact in the energy range 0.1 to 2.7 MeV. They fitted their measured cross sections Op to the following relation, extracted from the Bethe theory,where m, ft and c are the rest mass of the electron, the Planck's constant divided by 2n and the velocity of light, respectively and /J « v/c, v being the velocity pf the projectile,©19961ACS98 Vandana Saksena and M  S KushwahaA/2 and cRP arc iwo Bethe parameters. M2 can be evaluated with the help of the optical oscillator strength ^  but lhc parameter cRP depends on the distribution of thegeneralized oscillator strengths of the involved transition. Due to non-availability of the generalized oscillator strengths for the ionizing collisions of the molecules, there seems to be no calculation for cRP. In this paper, we propose a new method to calculate the above parameter which uses only optical oscillator strengths as input. We report our calculated' values of cRP for a number of molecules namely H2,.N2, O2, CO, H2O, H2S, C 0 2, NH$, CSj and SF6 ionized by relativistic electrons in the energy range 0.1 to 10? MeV.In the first Born approximation, we have [3]aT = a, + <x,, (2)where G\ is the ionization cross section of the molecule due to those virtual photons which have their polarization vectors parallel to K_ (longitudinal interaction) and is|given by (3]8;za$R r ,n( ^ ) mitt R d f (W ,K 2a$)° x * mv2 h  J in (*^)mili W dWx Fck (E \ K 2a2) d  (In K2a 2) dW, (3)where a0 is the first Bohr radius, E ' is equal to j  mv1 and / is the ionization potential.df(W ,K  Oq) gencra|jze(j oscillator strength for the energy loss W and momentum dW  *transfer hK. The exchange factor FCX(E', K 2ciq) has been evaluated in the Ochkurapproximation, Indistinguishability of the electrons leads to EmM = ( £ + / ) /  2, £  being incident energy. Relativistic expressions for v and (Ka0f max min are utilised [3].The cross section <7, due to those virtual photons which have their polarization vectors perpendicular to (transverse interaction) is significant only at relativistic velocities and in this region, it is given by [3]8m l R  , , „<7, = -  -----M 2 In (I -  P2) + P 2]. (4)m r  1 JNow we put eq. (4) in eq. (2) and equate the result to eq. (1) to obtain1r p  ~_E 'a,4 m%RM 2 In P2. (5)It is evident from eq. (5) that to calculate cRP, we require the values of 0\ which in-tum, requires the distribution of the generalized oscillator strength for its evaluation. In general, the values of the generalized oscillator strengths are not available for molecules. Only for the H2 molecule, Liu [4] has calculated inelastic scattering factors by using Cl waveBethe collisional parameters for the ionizing collisions etc 99functions. Hence, we have adopted an alternative approach. We express the generalized oscillator strengths as a sum of two terms (Mayol and Salvat [5]>df(W ,Q ) dWdf(W.  0)dWO (W -Q ) + h(Q )8(W -Q ), (6)where 0  ( W -  Q) is a step function and 8 (W -  Q) is the Dirac delta function, h (Q) is given byKQ )  = r df(W,  0) dW (7)and Q = ( t o ,,)2R is the recoil energy.The above equations show that in the present approach, the soft collisions (Born- Bethe term) contribute to the ionization only for the recoil energy Q varying from / to W and the contribution of the hard collisions (Mott-Moller term) is only at Bethe ridge Le. Q = W. Using Ochkur exchange approximation, putting eqs. (6) and (7) in eq. (3) and integrating over Q we get<Ti = Ok + aM.where Ob% the Born-Bethe cross section is given by4 TtalR f<£+/)/2 R df(W,0)  dWag/? HP J/ ~W‘ ( c ) - R M - 55-1 - - " dWwith ^minI2 me2y E ( E  + 2mc2) -  -y/(E- W ) ( b -  W + 2mc2)}2.Similarly, the Moller cross section is given by<*M =II R 1 ME+DI2 (—  J, ^  b  | |*(IV) —=■-------- ;1 W2 WE' r'2dW,(8)(9)( 10)(M )where h(W) = J'w d f ( W \ 0) dW’dW'. ( 12)Wc have taken the values of the optical oscillator strengthsd f(W t 0) dWfrom Brion andThomson [6], Backx el al [7] and Zeiss et al [8,9]. Calculations have also been done withFej|(E ', K20q) = 1 neglecting the effect of exchange. In this case is equaltoE. The present values of A#2 obtained by integrating over W from / to oo,W dWgiven in terms of p  = c r f / M 2 in Table 1, are found to be within 15% of the experimental values of Rieke and Prepejchal [2] for the first eight molecules. Thus, the agreement is quite satisfactory. No experimental values of Af2 for CS2 and SF6 molecules seem to be available.I(X) Vandana Saksena and M  S KushwahaIn Table 1, we have also listed the present values of cRP as obtained from eq. (5) with the help of eqs. (8) lo (12) for impact energy ranging from 0.1 to I02 MeV. At higher impact energies, the values of cRP do not change and hence are not shown. The theoreticalTabic 1. Variation of the parameter t RP with (he incident energy £  and the asymptotic values of the ratio p -  i RP!M2 tor the ionization of rokcu les due to electron impact.< RP l<RP)dexp PMole/£( MeV) ic r 1 10° I01 I02H2 7 85 7.60 7 56 7 56 8.115 II (X)N2 35.76 34 56 34 37 34.37 34 84 10.60° 2 42 85 41.41 41 18 41 18 38 84 10 567CO 42.18 40 77 40 55 40.55 35 14 10 67h 2o 30 51 29 49 29.33 29.33 32.26 10 70h 2s 62 69 60.68 60.37 60 36 42.19 ijl 11c o 2 56 (X) 54 11 53 82 53 81 55 92 ll)57NHi 38 90 37 62 37.42 37 42 34.86 1(^84n Si to 85 25 82 46 82 03 82 02 - 1091SFft 160 35 154 88 154 03 154 01 - 10 41A shows the experimental values of Rieke and Prcpejchal [2] determined from eq. (I) in the eneigy range 0 I to 2 7 MeV'values of ( HP at energies higher than I MeV are shown to indicate the convergence of the results. The table shows that the present values initially decrease with the increase of the impact energy hut attain their asymptotic value at about I MeV. It is also noticed that in general, the values of cRP increase with the number of electrons in the molecule and also with its molecular weight. However, isoelectronic molecules like H20  and NH  ^ have quite different values of cRP. The same is found to be true for the isoelectronic molecule N: and CO.A comparison of the present values of cRP at 1 MeV with those obtained by Rieke and Prcpejchal [2] shows that for H2, N2, a ,  CO, H20, C0 2  and NH, the two sets of values arc in good agreement with each other, the difference between them being less than about l(K/f. However tor H2S molecule, the difference between the two values is quite large. The reason Ipr this large difference is not understood at present. It may also be noted that Of of Rieke and Prcpejchal [2] includes contributions of the inner shell also, but the present cross sections exclude them. However, it is expected to be small. For the remaining two molecules CS2 and SF6, no other values of cRP seem to be available for comparison. In our energy range, the calculations with Fn (E \  K 2a2) = 1 also give the same values o fc* ,. This is according to the expectation and is in agreement with the experimental observation of Rieke and Prcpejchal that the ionization cross sections due to electron and positron impacts are the same.Bethe collisionai param eters fo r  the ionizing collisions etc 101We conclude with the remark that the theoretical values of cRP for the above molecules are evaluated for the first time. A good agreement between the present and the experimental values of c RP for most of the molecules, shows that the present method which uses only optical oscillator strength as input should be useful in the evaluation of the ionization cross sections of complex target due to electron (positron) impact for incident energies greater thdn a few times of the ionization potential of the target.AcknowledgmentsThe authors are grateful to Professor S P Khare for useful discussions and to Dr. P D Marathe for encouragement. Thanks are due to Professor W J Meath for supplying the tabulated values of the optical oscillator strengths. Financial assistance by the Council of Scientific and Industrial Research (India), in the form of a Senior Research Fellowship to one of us (VS) is gratefully acknowledged.References[I] T D Mark and G H Dunn Electron Impact Ionization, Ch. 9, Ch. 7. (Berlin : Springer) (1985) 12] F F Rieke and W Prepejchal Phys Rev A6 1507 (1972)[31 S P Khare and J M Wadehrn Phys. Lett. A198 212 (1995)[4[ J W Liu Phys. Rev. A7 103 (1973)15j R Mayol and F Sal vat J Phys. B23 2 117 ( 1990)[6] C E Brion and J P Thomson J. Electron Spectrom. Rel. Phen. 33 301 (1984)[71 C Backx, G R Wight and M J van Der Wiel J Phys. B9 315 (1976)[81 G D Zeiss, W J Meath, J C F MacDonald and D J Dawson Radiat. Res. 63 64 (1975)(91 G D Zeiss, W J Meath, J C F MacDonald and D J Dawson Can J. Phys. 55 2080 (1977)

Bethe Collisional Parameters for the Ionizing Collisions of Molecules by Relativistic Electrons

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Bethe Collisional Parameters for the Ionizing Collisions of Molecules by Relativistic Electrons

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