This Letter discusses inner-shell excitation in collisions of very heavy ions (Z1+Z2&#8819;140) in the framework of the quasimolecular model. The importance of multistep excitations and of coupling between continuum states is demonstrated. The 1s&#963; vacancy probabilities resulting from coupled-channels calculations exceed perturbation theory by a factor 3-5, thus giving good agreement with recent experimental results

Greiner, Walter

Müller, Berndt

Reinhardt, Joachim

Soff, Gerhard

English

Hochschulschriftenserver - Universität Frankfurt am Main

VOLUME  43, NUMBER  18  PHYSICAL REVIEW LETTERS  29 OCTOBER  1979 Role of Multistep Processes in Heavy-Ion Inner-Shell Excitations J. Reinhardt, B. Müller,  and W.  Greiner Institut für  Theoretische Physik  der Johann  Wolfgafzg Goethe Universität,  GO00  Frankfurt arn Main,  German?) and G. Soff Gesellschaft  für Schwerionenforschung,  6100 Darrnstadt,  Gerrnany (Received 9  July 1979) This Letter discusses inner-shell excitation in collisions of  very heavy ions (2,  -2, 2 140) in the framework of  the quasimolecular model.  The  importance of  multistep exci- tations and of  coupling between continuum states is demonstrated.  The lso vacancy prob- abilities resulting from coupled-channels calculations exceed perturbation theory by  a factor 3-5,  thus giving good agreement with recent experimental results. The excitation of  inner shells in collisions of heavy ions at energies up to the nuclear Coulomb barrier is best described in the quasimolecular model.  Here one starts from the solutions of the stationary two-center  problem and treats the excitations as being caused by the explicit time dependence of  the electronic Hamiltonian. Calculations along these linesl have predicted large vacancy-formation  rates for superheavy systems.  These results have subsequently been confirmed in several e~periments.'-~  Whereas the measured impact-parameter (bombarding energy) and charge dependence was found in good agreement with the predictions of  first-order perturbation theory (in powers of  B/at), there remained an unexplained  discrepancy in absolute magnitude:  Experimental data lay above theory by a factor of  3-5. In this Letter we point out that it is imprtant to go beyond perturbation theory7 to obtain quan- titative agreement.  We demonstrate the impr- tance of  multistep processes, where an electron is excited via various intermediate states, by the first coupled-channels  calculations performed for collision systems with relativistic binding. In the following, we will first discuss the treat- ment of  excitation of  a single electron and  then show that the effect of  the Pauli principle can be easily implemented.  In the second part of  the Letter, numerical results will be presented. We start from the complete set of  stationary solutions of  the two-center  Dirac equation which is parametrized by  the internuclear dis- tance R.  The time dependence R(t) is prescribed by  the nuclear trajectory.  To solve the time- dependent Dirac equation H~~~[R(~)]  qi(+)  (t)  = i(a/a  t)  qi"'(t)  (2) with the initial condition cp,(+)(-  W)  = cp,  we expand the wave function idij(t)=  -Ckaik(t)(~jIa/atI(P,) X  ex~[-  i(~k-~B)I  (3) with the phase xj  = I:odt'~j@(t')).  Projection on a particular basis state yields the coupled dif- f erential equations bij(t)  = -Ckain(l)(<pjI'/'tIq,) subject to the initial condition aij(- W)  = 6,,.  Here and in the following the sum implies an integral over the continuous spectrum.  The squared am- plitude laij  l2  denotes the probability that the elec- tron has been excited from state i to state  j  dur- ing the Course of  the collision.  The amplitudes ai, completely describe a collision where a single electron is present.  Normally,  however,  the projectile ion is only partially stripped and the target atom is neutral so that several bound states are occupied. If  pair creation is to be included in the descrip- tion,  one even has, in principle,  to deal with in- finitely many particles.  Knowledge of  the single- particle amplitudes aij,  however, is still suffi- cient if one neglects the two-body  part of  the electron-electron interaction in the Hamiltonian HTcD. The Pauli principle does not lead to com- plications since the "dynamic"  wave functions yi(')(t) which contain the time development of single-electron states are mutually orthogonal at every instant of  time because of  the unitarity of  the time development operator.  The set  cpi(+), O 1979 The American Physical Society  1307 VOLUME  43, NUMBER  18  PHYSICAL REVIEW LETTERS  29 OCTOBER  1979 therefore, can be used as a basis for describing the many-particle problern.' We can expand the (time-dependent) field oper- ator in the basis qi")  so that where bi, 6,+  and di,  dit are particle and hole operators, respectively, which satisfy the usual anticommutation relations.  The operators bi and di are defined such that they annihilate the (Hei- senberg) state /F):  ~,IF)=o  for i>F  anddilF)=O for i <F. To determine the asymptoticaily ob- served particle excitation we use the alternative expansion where Y,(-)  are the solutions to Eq.  (2) satisfying the condition qi(-)(+  m) ='Pi.  Equating (5) with (6) leads to a canonicai transformatioc  between the two sets of  operators  and b"i,di. The nurnber of  particles excited into an initially vacant level <pi  (irrespective of  the occupation of  the other states) then reads and the number of  holes in an initiaily occupied state qj  is Thus the particle (hole) excitation probability in a particular state is simply given by  an incoher- ent sum of  single-electron transition probabilities as shown by Reading (Ref, 8)-  The result is in- fluenced by the number of  electrons present sole- ly via the position of  the Fermi surface F.  The number of  correlated pairs, however, is not just the product of  single excitation numbers.  For particle-hole pairs (i >F,  j  < F)  we find This formula, which was also derived independ- ently by Ford and Reading,grlo  should be applied to analyze experiments like those presently per- formed for coincidences between electrons and characteristic X  rays.  Equation (9) holds also for particle-particle  or hole-hole correlations if  the sign of  the second term is changed.  If  the calcu- lation involves channels which are treated inco- herently, e.g.,  spin-up and spin-down states when neglecting rotationai coupling and magnetic effects, Eq. (9) has to be supplemented by inco- herent terms to account for random coincidences. The mentioned case of  spin degeneracy leads to an enhancement by more than a factor of  2, namely N~,~=~N~,~O+  2Ni0~,0, where qO,  NjO, and  are calculated from Eqs. (7)-(9) for one spin state and NiSj  is  summed over all combinations. Having established the relevante of  the single- particle amplitudes, we must turn to the solution of  the coupled differential Eqs. (4).  For any practical use the basis has to be restricted to a finite and not too large set.  In particular, the continuuin nlust be discretized into finite seg- ments.  This is most simply achieved by replac- ing the integral J;,~E f  (E)  by a finite sum C,AE,  f(~,).  Since the matrix elements vary smoothly with energy,  the allowed spacing AEi is limited by the oscillations of  the phase x,(t): Ax = (AE)(;lt),&  T.  If  we assume a mean excita- tion region of  AR= 300 fm and collision velocity v/c= 0.1,  this condition gives roughly AE«  0.4nzc2 Numerical calculations are  feasible down to AE = 0.1nzc2 at low electron kinetic energies.  The upper energy limit for states to be included is determined by the Fourier spectrum of  the colli- sion and by the energy dependence of  the coupling matrix elements.  We  have tested the latter by applying the completeness relation and checking the identity We found that it is satisfied sufficiently if  states up to 2 MeV kinetic energy are included.  About half  of  the coupling strength goes into the contin- uum. The coupling matrix elements were obtained  in monopole approximation,"  which was proven to be remarkably good in very heavy collision sys- tems.  Only nsu states are included in the present calculation.  Coupling to other states, in particu- lar rotational coupling, was found to be weak in high-Z near-symmetric systems.  For more de- tails on the basis used, See Soff  etal.12 All  radi- al matrix elements were extrapolated to distances R >ZOO fm by a power law and  cut off  smoothly with a Gaussian factor at R > 1500 fm to simulate the effect of  translation factors.  The integration VOLUME  43, NUMBER  18  PHYSICAL REVIEW LETTERS  29 OCTOBER  1979 FIG. 1.  (a)  Probability for lsu  excitation as  a function of  impact parameter b  for Xe-Pb  collisions at 4.6  MeV/ N laboratory energy.  Full line, coupled-channels  re- sult assuming a Fermi level above F=  3su;  dashed line, perturbation theory.  Also shown are experimental re- sults of  Greenberg et  al. (Ref. 3) (circles) and Anholt et  al. (Ref. 14) itriangles).  (b) Energy dependence of P„,(b)  at various fixed impact parameters. was performed along Rutherford trajectories up to R = 3000 fm with use of  a standard predictor- corrector Hamming routine. The most remarkable new result of  the coupled- channels treatment is an increase of  vacancy- formation rates by up to a factor of  5 due to multistep excitations compared to previous per- turbative calc~lations,~'~~  the effect being largest for very heavy systems.  To demonstrate the in- fluence of  excitation via intermediate states the Table I shows the lsa hole probability (normal- ized to 2 because of  spin degeneracy) for Pb-Pb collision at b = 0 and  2a = 20 f m (E%/A=  4.65 MeV/ N).  The results of  several calculations including a varying number of  nsa bound-state  channels TABLE I.  n  designates the highest bound state includ- ed; F is  the last occupied state.  Thus the left column (F  = X) includes excitations to the continuum only while in the median (right) column the Fermi surface was ta- ken above the 3su (lsa) state.  All calculations include 16 continuum states.  Doubling of this number results in a less than 1Wh  change of  the excitation rates. 32  F=X  F=  3so  F  = lsa lso  2.96%  .  es  ... 2su  4.80%  ...  19.92% 3s~  6.84%  .*.  21.40% 4so  8.08%  9.24%  22.12% 6so  9.06%  11.00Y0  22.56% 8s  U  9.42%  11.34%  22.68% FIG.  2.  Same as Fig. 1  for Pb-Cm collisions at 4.7 MeV/N  laboratory energy.  The various curves in (a) belong to different positions of the assumed Fermi lev- el F  as indicated in the figure. are given. The coupling between continuum states mainly leads to a shift of  the spectrumr  The emitted electrons are rescattered to higher energy,  The inclusion of  intermediate bound  states increases the ionization by  supplying mainly low-energy electrons so that the net effect of  multiple exci- tations is a rise of  the electron spectrum without change of  shape. Fig, l(a)  shows the impact-parameter depen- dence of  K-vacancy production for Xe-Pb colli- sions at E /A = 4.6  MeV/N  caiculated in perturba- tion theory (dashed line) and in the coupled- channels approach (full lines) for the (last oc- cupied) Fermi level F = 3sa (X  = 8sa in ail caicula- tions).  The multistep excitation is Seen to lead to an overall increase of  P(b). Also shown are recent experimental results obtained by  two groups in Darmstadt and Stanf~rd.~"~  Theory is in remarkable agreement with these data and also with measurements in other systems like Xe-Au (Ref, 3).  Similar agreement is found for the dependence on bombarding energy,15 which is shown in Fig. l(b), Our theory, however, does not show a peak at small impact parameter,  This special feature was observed in one of  the ex- periments and was tentatively explained in the Briggs modelL6  with use of  perturbation theory in the united-atom basis. Fig. 2 gives the predicted lsa vacancy prob- ability for the very heavy system Pb-Cm (Z,+Zz = 178) for various positions of  the Fermi surface. Note,  in particular, the strong impact-param- VOLUME  43, NUMBER  18  PHYSICAL REVIEW LETTERS  29 OCTOBER  1979 eter dependence and  the steep increase with bom- barding energy. In Summary, we have demonstrated the impor- tance of  multistep processes in the quasimolecu- lar description of  heavy-ion inner-shell excita- tions.  The nonadiabacity of  the electronic motion is the origin of  violent coupling between neigh- boring states which have the effect of  increasing the excitation rates.  The number of  vacancies depends monotonically [~q.  (8)] on the number of initially occupied states.  To obtain accurate re- sults for vacancy formation (and thus also for positron production) the initial distribution of electrons (i,e., the position of  the Fermi surface) must be known.  The characteristics of  the exci- tation process are changed  slightly by  multistep excitation, leading to a weaker dependence on impact parameter and energy.  Detailed studies of  the impact-parameter , energy, and charge dependence of  inner-shell excitation as a function of  the binding energies and  coupling strengths entering the coupled-channels calculations will be necessary for the envisaged spectroscopy of superheavy quasimolecules. This work was supported by the Bundesminis- terium für Forschung und  Technologie and by the Gesellschaft für Schwerionenforschung. 'W.  Betz,  G. Soff, B. Müller, and W.  Greiner, Phys. Rev. Lett. E,  1046 (1976). '5. S. Greenberg,  H.  Bokemeyer,  H.  Emling, E. Grosse, D. Schwalm, and F. Bosch, Phys. Rev. Lett. 39,  1404 (1977).  - 3~.  Liesen, P. Armbruster, H.-H.  Behnke, and S. Hag- mann, Z. Phys. A =,  417 (1978). 4~.  R. Macdonald,  P. Armbruster, H.  H.  Behncke, F. Folkmann,  S. Hagmann,  D.  Liesen, P. H.  Mokler, and A.  Warczak,  '2.  Phys. A 284,  57 (1978). 5~.-~.  Behncke,  D.  Liesen, S. Hagmann,  P. H. Mok- ler , and P. Armbruster,  Z. Phys. A 288,  35 (1978). 6~.  Anholt  and W.  E. Meyerhof,  Phys. Lett. @,  381 (1978). 'T.  H. Rihan,  B.  Müller, and W.  Greiner, J.  Phys. B 12, 761 (1979). *A. L. Reading,  Phys. Rev. A 8, 3262 (1973). 9~.  F. Ford and A.  L. Reading, to be published. ''J.  Reinhardt, B. Müller, and W.  Greiner, in Proceed- ings of the International  Workshop on Coherence und Correlation in Atomic Collisions, edited by H. Kleinpop- pen and J. F. Williams (Plenum, New York,  1979). "G.  Soff,  W.  Betz, J. Reinhardt, and J. Rafelski, Phys. Scr. 2,  417 (1978). "G.  Soff,  W.  Greiner, W.  Betz, and B. Müller, Phys. Rev. A a,  169 (1979). j3~.  Soff,  W. Betz, B. Müller, W. Greiner,  and E. Merzbacher,  Phys.Lett. s,  19 (1978); B. Müller, G. Soff,  W.  Greiner,  and V. Ceausescu,  Z. Phys. A 285, 27 (1978). 14~.  Anholt,  W.  E. Meyerhof and Ch. Stoller,  Z. Phys. A 291,  1287 (1979). l5T~iesen,  private comrnunication. I6p. A.  Amundsen, J. Phys. B X,  L737 (1978). Observations on the  V-0 --.  V =  5  Rotational-Vibrational Band of HD F.  W. ~alb~(~)  and J. Vigui! Laboratoire de Spectroscopie Hertzienne de 1'Ecole Nomnale Superieure,  Universite' Pierre et Marie  Curie, 75230 Paris Ce'dex 05, France, und Department  of  Physics,  University of British Columbia, Vancouver, British Columbia V6T  1 W5, Canada (Received 21 August 1979) The  = 0 -V  = 5 rotational-vibrational  spectrum of free HD has been observed by means of photoacoustic spectroscopy.  The electric dipole matrix elements are  found to depend significantly upon the rotational quantum number.  The dependence of the frequency of  the R(1) line upon density has been determined. Electric dipole vibrational spectra of  free H, molecules are rigorously forbidden by symmetry. For HD,  electric-dipole spectra are  permitted by  symmetry but are  very weak, being allowed only because of  nonadiabatic effects.  They are forbidden within the Born-Oppenheimer  approxi- mation.  The first observation of  a HD  dipole vi- brational spectrum was made by Herzberg.' Trefler and Gush2 have studied the pure rotation- al spectrum.  McKellar, Goetz,  and Ramsay3 have reported wavelength and intensity measure- ments of  the stronger lines in the 3-0,  4-0,  5-0, and 6-0 HD bands (we denote v = 0 -  vf  + 0 by V,-0 hereafter) and surveyed earlier work.  The spec- tral properties of  HD are  of  astronomical impor- tance as the vibrational spectrum of  HD  has been 1310  O 1979 The American Physical Society 

Role of multistep processes in heavy-ion inner-shell excitations

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