INIS-mf—8669
ATOMKI PREPRINT HU ISSN 0231-2468
ATOMKI Preprint B/5 (1983)
MULTIPLE IONISATION EFFECTS DUE TO RECOIL IN ATOMIC COLLISIONS*
L. Vegh Institute of Nuclear Research of the Hung. Acad. of Sci. H-4001, Debrecen P.O.Box 51, Hungary
ABSTRACT A description of atomic recoil excitation caused by the projectile nucleus is given using the two-potential formula. The relation of the direct and recoil ionisation amplitudes are discussed demonstrating that their inter- ference in dipol approximation may be destructive only approximately. The recoil mechanism can produce strong multiple ionisation and the strength of the sate]lite lines follow strictly the binomial distribution. The recoil ionisation in neutron induced reactions, the life-time of the inner-shell vacancy in recoiled ions and the recoil effects in the electron conversion coeffizients are discussed. 1. Introduction In the last few years there has been an increasing interest in atomic ioni- sation phenomena influenced by the recoil of target nucleus. The recoil of the nucleus of the atom may "shake off" some of the electrons (Migdal 1939, Levinger 1953, Ciochetti and Molinari 1965) and it is known to be important at very small values of the impact parameter b where it can give large contribution to the ionisation. The recoil matrix element (Migdal 1939, Levinger 1953, Andersen et al 1976, Feagin et al 1979) obtained in sudden approximation has the following form: where * Submitted to Journal of Physics B., May 6 1983. In the case of the Coulomb interaction the dominant part of the recoil momentum is transferred at nucleus-nucleus distances close to the perihelion because this is the part of the classical orbit where the dominant part of the* change in the direction of the nuclei momenta occurs. Therefore it is a good estimate for the duration of the recoil momentum transfer the time needed to take the twice the distance of the closest approach at the given impact parameter b. This time denoted by tN is equal to tN- 2a [<• 0 •*/<*)]/« ,. a.OWil*. LEj.MeV where Ztis the atomic number of the incident nucleus, E andtf are the CM energy and velocity of the nuclei, respectively. If tN is much less than the periodic time of the investigated electron orbit with principial quantum number n, then the application of formula (1) for recoil calcula- tions is justified. This condition can be expressed as CO The sudden approximation condition (4) is fullfilled if b is small enough and if the incident particle is not very slow. The common feature of the most recoil calculations (see for example Andersen et al 1976, Amundsen 1978, Kleber and Unterseer 1979, Jakubassa and Amundsen 1979, Rosel et al 1982, Anholt and Amundsen 1982, Blair and Ariholt 1982, Reading et al 1982, Ford and Reading 1983) that they are essentially based on formula (1) if we retain only the first two terms in the expansion of the exponent operators in (1), that is e.xp • Tf and we neglect the second order terms ~Kmf\.'ri (Feagin 1982). Since for dif- ferent initial and final states the matrix element of the identity operator gives zero, the 8,,-body^ operator exp(ifc£ T$) disintegrates to the sum of one-body operators iS fer^ and the description of the recoil effects can be easily incorporated Into different treatments of the collision induced ionisation. The neglection of the higher order terms in expansion of the exponent operator in (1) can be justified if their contribution is small enough and this requirement can be formulated roughly for electrons with principial quantum number «. as loft .feJG^*- « A (5) where a, is the Bohr-radius, an is the radius of the Bohr orbit with n. It is possible that the condition (5) is fulfilled for inner shell e- lectrons but it is not true for electrons m higher shell. At very large recoil of the target condition (3) is not fulfilled already for the K-shell electrons. So it is an interesting problem to discuss the recoil effect in terms of formula (1) without any approximation. The discussion of large recoil effects for one electron target atom using the impulse approximation has been recently published by Feagin (1982). Using some elements of his work we discuss the scattering amplitude of the ion-atom collisions involving the recoil for many-electron atoms. 2. The scattering amplitude We discuss ion-atom processes where the incident ion is fully ionised. For the description of the collision in laboratory system we have the following coordinates. Let R» • • •R* be the coordinates of the N electron and ftrand" R#the coordinates of the target and incident nuclei, respectively. The notions of the corresponding momenta are ?„ ..-•*«, PT and P, . We define a new set of coordinates as *»•«>- K CM Mr * Mm » Mi j, where M» is the mass of the incident nucleus andft Ai s the coordinate of the centre of mass of the target atom: (7) The conjugated mosnenta to the coordinates in (6) have the form: Nm This choose of the CM coordinates can be considered as generalization of the treatment of Feagin(1982)which has been constructed for a two nuclei- one electron three-body system. The Hamiltonian has the form: where TCM is the kinetic energy operator of the centre of mass and T** is that of the atom-incident nucleus relative motion. For our treatment of the ionisation process we split the Hamiltonian (9) into three part in the following way: H - Ho + V, • VH (13) with H. - TCM . T.. + H, ,^3 The inital and final asymptotic wave functions corresponding to the eigenfunctions of H# (14) have^the form: where u and Cf are the conjugated momenta (9). where the T operator depends on the nucleus-nucleus, electron-incident nucleus two-body interactions. We have calculated the matrix element (18) using the well-known two- potential formula of Gell-Mann, Goldberger and Watson (Gell Mann and Goldberger 1953)» Tfl (19) where the first term is the \matrix element of the Vj interaction which is assumed in this approximation to be well-known. The second term is the ^JjWRA matrix ^element of the Vu interaction with the scattering functions Vt$ and Vj* which are the solutions of the Schrodinger equation with Hamiltonian The first term of (19) can be calculated easily and accurately because the v*t interaction (15) depends only on the distance l«a-f$T|. Taking into account that the variable R in integral (18) can be expressed as see (7), and applying (15), (18) and neglecting in the nuclear amplitude the off -energy shell effects due to the binding of atomic electrons (impulse approximation), we have where we have denoted the first term of (19) by T4t . cj,* ^ -^ H is the momentum transfer to the target nucleus, fc is defined by (2), t Cep is the nucleus-nucleus scattering amplitude, Rfk(fc) is given by (1). For the calculation of the second, DWRA term of (19) we need the know- ledge of the fu and V/^* scattering wave functions. They can be obtained from the "^-matrix given in (21). Applying the general relations and neglecting the influence of the atomic electron energies on the nuclear motion we obtain - N » where *^(Rw)is the nuclear scattering wave function and the (-) motion in (flir relates to the continuum electron wave functions. Writing these scattering wave functions into the second term of (19) which we denote by T** • in the further we can write In order to calculate the 77^ matrix element we insert (23) with £ .iiij ^v /.#_ <-» I ^ i l£-> denotes nuclear scattering states and £ l^'X^lis the projection operator of the N electron states including sum" for the bound and integral for the scattering states, respectively. Taking into account that the Coulomb interactions in the second product can produce one-electron transi- tions only and for antisymmetrized N -electron wave function the N interac- tions in the sum give equal contributions, T^"' can be written as where <^(^ •?„) describes an atondc state differs from the initial ground state in one single-electron orbit and $ relates on the entity of -tie elec- tron states. Using (21) and (24), T+i has the form (25) Now we investigate the matrix-element where a K-electron goes into a positive energy state. In R,M(fc.) \n (25) we introduce the following approxima- tion * .. where pe is the momentum of the continuum electron and we can write with -*S(5> w ^ where q?"( jf4 ... ^denotes an N-< electron wave function which can be gener- ated by taking away one Is orbit in the initial state and the scattering orbit in the final state, respectively. If the recoil is very small (1*0), we can see from (26) that T^ is equal to the direct ionization T-matrix. If fc*0 the T^ amplitude (26) contains the recoil contributions to all order. In dipol approximation Tj,j, in (26) can be written as *. In the treatment of RSsel et al(1982)the term corresponding to our first term has the form: <*«)l|l^(*)><**)l*l<|i<»>> (28) In order to compare the factors of the two descriptions we assume that i**^) is equal to the Coulomb scattering amplitude. Taking into account that the vlq,} Coulomb amplitude differs only in a r)c phase factor from the first Born term of the V*(ft) Coulomb potential we obtain -J * (29) Since the matrix elements of f are conplex conjugated, their relative phase is equal to 2«*tjywhere*t$i3 the phase of <(^(r)if|^4jtT}). In our treatment T£* is calculated almost exactly in all order. We can see from (27-29) that the perturbative method followed by Rosel et al for the dipol term has introduced an approximation, namely which seems to be rather rough. We notice that in (2?) our second term is identical with the nonrecoil term of Rosel et al(1982) except for the mo- mentum of the final electron which in our treatment has the value fk+% at Rosel et al it is equal to %. In the treatment of R3sel et al(1982) in dipol approximation the recoil and direct terms have opposite sign. Taking into account the above conside- rations this may be valid only approximately. We can say that the accurate calculation of recoil-direct amplitude interference needs further efforts. At high recoil energies the recoil part of the ionisation amplitude can give much larger contribution to the ionisation cross section than the part corresponding to the direct ionisation. In the further we discuss such type of ion-atom collisions. 3. The calculation of the cross section Now we calculate the ionisation cross section assuming that Tb* »Tf?^*" • T^11 using (21) we can write (3D Our initial state t is a two-body system, it contains the projectile nu- cleus and the atom in ground state. In the final state $ the electrons can exist in bound and scattering states, so. generally we have a many-body final state. If we have *v electron with positive energy in the final state, the general cross section formula in our normalization has the form A is the considered integration volume in the space of the final momenta, PA • • • £* are the momenta of the final positive energy electrons related to the recoiled nucleus, see (8), E^and Ej are the initial and final total energies, respectively. If we measure the Q( angle of the scattered nucleus and the momenta of the final electrons are unobserved, from (32) we obtain (33) The E4~ Ei. energy difference is equal to where fy is the binding energy of the -t-th electron before the collision and cf is the contribution of the electrons axcited into higher atomic bound states. The Tji matrix in (33) has the product form given by (31). Regarding that in our case the atomic kinetic and binding energies in (34) are many orders of magnitude less than the incident ion energies and that the nuclear cross sections (except ior the narrow resonance regions) does1not depend dras- tically on the projectile energy we can write using (31) and (33): Using tl) , (31), (33) and (35) the cross section has the forms *"<***' .. (36) where (&(**•«.. •?«> is the wave function of the bounded final state elec- trons, "tfjsjjl*1) are the continuum electron wave functions. For the calculation of the n-fold integral we apply the completness relation .< (37) where the sum is performed for the entity of the possible one-electron bound states. Writing' (37) n-times into (36) we obtain ' (38) .ff U - f \k > () <*>tt where tft. V<> denotes the k-th bound state of the K-th electron in the atom. For the calculation of the cross section (38) we have used nonrelativ- istic screened hydrogenic wave functions. We have followed the screening procedure of Slater(193O). In this case the calculations can be performed in analytical form. In the calculations we have done two important approximations. Though formally we have taken into account the Pauli principle using antisymmetri- sed wave function in the initial state, in the practical calculations we : • have neglected the exchange terms. This neglection may cause errors ~to%- Since for large recoils in the case of simultaneous ionisation of n electron in (38) the matrix elements of N-n .... N body operators appear, the calcu- lation of the vevy large number of exchange terms is cumbersome. Secondly, for the calculation of the ionisation probability according to (37) the projection operator for hydrogen-like atom has the form (39) P - A - f t?><3| In the case ot many-electron atoms one must take into account the screening effect of the inner and outer shell electrons. L'erioerin>7, trvm the outer electrons reduces the binding energy in the k-th shell. We can in- corporate this effect by defining an upper limit in the sum (39), Following the procedure of Kleber and Unterseerll979), if we have a vacancy in the shell with principial quantum number k, then the upper limit \ has the form where EB is the experimental binding energy in eV. £ has in general non- integer value and in our calculations 15. ~ 1-2 fe . We have approximated fe by its rounded up value. We have found that in our cases the contribution of i orbits other than % can be neglected. We have accepted the "k limit also in the case of multiple ionisation because we calculate in (38) matrix elements between states which contain only bound-state electron orbits. 1. K-shell and multiple ionisation due to recoil Let us consider the effect of recoil on the K-shell ionisation probabil- ity. Generally we can say that if the projectile velocity v0 is smaller than the K-shell velocity vt<, then the recoil part of the scattering amplitude even on its maximum gives smaller or equal contributions than the direct part, that is in (26) Tjf/TK"M • Increasing the projectile velocity to the K-shell velocity, the recoil and direct contributions are increasing together. In the region irB > wK with the increase of vB the recoil contribution T^T becomes dominant over the direct part T,t which decreases with vr0 here. In Fig. 1 we have plotted the recoil and direct K-shell ionisation prob- abilities for wAlU, w)wAl scattering at E^ =20 and HO MeV as a function of the CM scattering angle2. At large angles.the recoil ionisation proba- bilities exceeds in some cases by a factor of ten the direct ionisation probabilities interpolated from the tabulated SCA results of Hansteen et al 0.975). The figure shows that in this region the recoil ionisation proba- bility increases nearly linearly with the incident energy while the direct ionisation probability decreases. Now we consider the processes where Vs ^trK and 2$has a small value. In this case the contribution of the direct ionisation for the L and other higher shell electrons is snail. Then due to the smallness of the probabili- ty of L-shell ionisation, the probability of the KC multiple vacancy pro- duction (n=1...8) caused by the direct Coulomb interaction is very small. If we investigate the small inpact parameter region, a significant satellite structure can be found due to the recoil. Since the recoil is described by the matrix element of a many-body operator which is the product of one-body operators identical in form, the strength of satellite lines follow binomial distribution. If the ionisation probability of one electron on the i--th shell which contain N electron is p^, then the strength of the satellite line corresponding to k holes has the form CM) The values of the p• ionisation probabilities depend mainly on the binding energy of the shell that is on the principial quantum number of the shell. Figure 2 shows the calculated PK t P^ ) PUJJ one electron ionisation probabilities for the recoiled nucleus aftf "as a function of the recoil energy CR. The PM ionisation probabilities are equal to unity already at ER*>1 MeV. The range of the recoil energies presented here can be covered for example in alpha elastic scattering at E^ =22 MeV. At this projectile energy the PL and PM direct ionisation probabilities are by orders of mag- nitude smaller than those for recoil. The probability of the K'L'M vacancy production for Al can be calculated using (41) and the values from figure 2. 5. Recoil ionisation in neutron induced reactions If the incident particle is neutron, neglecting the interactions due to the magnetic moment of the neutron, in elastic and inelastic scatterings the only mechanism of ionisation is the recoil. In neutron reactions where in the outgoing channels charged particles come out, we must take into consideration the direct Coulomb ionisation in the final state. The total cross section of such reactions is -^10% of the total neutron cross section tS^ in the incident neutron energy region B^IO^O MeV. We consider the vacancy production in neutron reactions for general case. At the above energies the sum of the (n,n),(n,2n),(n,3n).. cross sections has about the same values as the elastic one. The CM inelastic cross sec- tions are approximately isotropic, the elastic cross section has a strong forward peak. For the calculation of the recoil ionisation probabilities the knowledge of the dW/elE* differential cross section (E^ is the labora- tory energy of the recoiled ion) is needed. The ionisation cross section corresponding to the K-shell and multiple vacancy production characterized bynholes has the form: 0 where EMis the kinematic limit of the recoil energy, and (V,(e») is the recoil ionisation probability of the vacancy production at E«. Neglecting the anisotropy of the elastic scattering and using the almost isotropy of the inelastic and reaction cross sections, we have that has a constant value: & - Jr. and. (44) The probability of the K-shell .vacancy production for Al calculated with 4=1.8 bam at Ef =m MeV is <£"* 24.3 mb. • The ionisation due to the charge state channels is characterized mainly by one-hole production. In general its value is by many orders of magnitude smaller than t^and comparing with the above value of The results are presented in Table 2. and they show clearly that the recoil ionisation can produce highly ionized 19F states with high percentage, therefore the interpretation of the Life-time experiment of Morita et al Q.978) can be established. We notice that the model of Morita et al (1978) has the assumption that the 10 ionisation probability does not depend on the recoil energy. Since the re- coil mechanism is very sensitive on the recoil energy the model of Marita et al (1978) seems to be inaccurate. Its revision can be found in Vegh (1983). The experimental study of the life time of the inner-shell vacancy in recoiled ion is desireable because it can give interesting information about the ionisation mechanism and environmental effects. 7. Recoil effects in electron conversion There are cases where the recoil ionisation can influence the measured value of conversion coeffizients of the electromagnetic decay of an excited nucleus. In in-beam experiments the investigated nuclei are produced in violent nucDear collisions where the residual nuclei can have quite large kinetic energy. Due to the recoil ionisation effect, the excited nuclei may have atomic shells with number of vacancies. Since the electron conversion coeffizient is proportional to the number of electrons in the given shell, the conversion coeffizient o^ for the i-th shell with holes has a de- creased value «!**• This effect is inportant in those cases when the f^ life-time of the inner-shell vacancy is larger than or equal to the characteristic nuclear life-time *CW. The life-time of a K-shell vacancy with intact outer shells is 16 IQ-l^-lO"" Sec and that of the L-shell is longer. Since the nuclear tran- sitions generally have the life-time 10~9-10~12 sec, the recoil effect can occur only in exceptional cases. For the transitions characterized by TM«%, we can make a simple estimate for the modification of the conversion coeffizients. We assume that the momentum of the projectile is transferred completely to the residual nucleus. This assumption is not too bad for example in the fl(«,xw) B* reactions where the angular distribution of the neutrons is nearly isotropic. Then the ki- netic energy of the residual nucleus has the approximate value Using this relation we have calculated for T^t. the ratio **&*)*•**— > 11 1 determined by the precise measurement of the conversion energies. The ef- fect can be checked in one experiment comparing the 3T-ray and conversion electron energies using calculated atomic binding energies. * 8. Conclusions Our formulation provides a convinient framework for the accurate des- cription of large recoil effects. The characteristic feature of the large recoil effects is the strong multiple ionisation which can influence a number of phenomena in different fields. Therefore we can say that the study of the recoil induced multiple ionisationsan interesting and important field and deserves attention. I would like to thank Laszlo Sarkadi for useful discussions and comments on the manuscript. References Andersen J U, Kocbach L, Laegsgaard E, Lund M and Moak C D 1976 J. Phys. B: At.Mol. Phys. 9_ 3247 Anholt R and Amundsen P A 1982 Phys. Rev. A25 169 Amundsen P A 1978 J. Phys. B: At.Mol.Phys. 11 3197 Bhalla C P, Folland N 0 and Hein M A 1973 Phys.Rev. A8 649 Blair J S and Anholt R 1982 Phys. Rev. A25 907 Ciochetti G and Molinari A 1965 Nuov.Cim. 40B 69 Feagin J M, Merzbacher E and Thompson W J 1979 Phys. Lett _81B 107 Feagin J M 1982 J. Phys. B: At.Mol.Phys. 15 3721 Ford A L and Reading J F 1983 IEEE Trans.Nucl.Sci. Hansteen J M, Johnsen 0 M and Kocbach L 1975 Atomic Data 15 307 Jakubassa D H and Amundsen P A 1979 J. Phys. B: At.Mol.Phys. 12 L725 Kleber M and Unterseer K 1979 Z. Phys. A292! 311 Levinger J S 1953 Phys. Rev. 90 11 Migdal A B 1939 see in Landau L D and Lifschitz Y M Quantum Mechanics (Pergamon London 1958) McGuire E J 1969 Phys. Rev. 185 1 Reading J F, Ford A L and Martlr M 1982 Nucl. Instrum. Meth. 192 1 RSsel F, Trautmann D and Baur G 1982 Nucl.Instrum.Meth. 192 43 Slater J C 1930 Phys. Rev. J36 57 Vegh L 1983 submitted to the 10th Int. Conf. on Atondc Collisions in Solids, Bad Iburg, 18-22 July 12 Table 1. The $KLn) cross section of the multiple production 2 of vacancies in the n+ ^Al process at neutron energy En=lU MeV. n 3 6 6 66 6 3 ' ' * °*12 1«2'1O" Table 2. The probability of the Ln ionisation in the case of a K-shell vacancy in the F ion recoiled by a N ion of projectile energy 1.75 MeV, n 0 1 2 3 4 5 6 7 P(KLn) 0. 168 0. 234 0 ..1.77 0 .13 0 .111 0.092 0.063 0. 024 PTKT Fig. 1. The recoil (- -) and direct ( ) K-shell ionisation probabilities in the ia+27Al scattering at projectile energies E«,=20 and 40 MeV as a function of the CM scattering angle X. The direct contributions are interpolated between the tabulated SCA results of Hansteen et al (1975). 13 1 0 1 2 3 4 5 6 7 8 9 » Fig. 2. The Pv, PT , PT recoil ionisation probabilities as a function K LQ_ •"23 27 of the recoil energy ER of the Al nucleus. Fig. 3. The Rj< and RJ_/RJ< correction factors for the «K conversion coeffizient andX-^/oC^ conversion ratio at projectile energy E^ =50 MeV as a function of the atomic number Z of the target. Kiadja a Magyar Tudomanyos Akad&nia Atommagkutat6 Intfizete A kiadasSrt Ss szerkeszt€s£rt felelSs dr.BerSnyi Dfines, az intSzet igazgatdja K^szUlt az ATOMKI nyomdajaban