Excitations of the atomic shell in heavy-ion collisions are influenced by the presence of a nuclear reaction. In the present Rapid Communication we point out the equivalence between a semiclassical description based on the nuclear autocorrelation function with an earlier model which employs a distribution of reaction times f(T). For the example of U+U collisions, results of coupled-channel calculations for positron creation and K-hole excitations are discussed for two schematic reaction models

Greiner, Walter

Müller, Berndt

Müller, Udo

Reinhardt, Joachim

English

Hochschulschriftenserver - Universität Frankfurt am Main

PHYSICAL REVIEW A  VOLUME 28, NUMBER 4  OCTOBER  1983 Description of atomic excitations in heavy-ion reactions J. Reinhardt, B. Müller, and W. Greiner Institut für  Theoretische Physik der Johann  Wolfgang Goethe-Universität, Robert-Mayer-Strasse 8-1  0, Postfach 111  932, 0-6000  Franwurt-um-Main.  West Germany U. Müller Gesellschaft  für  Schwerionenforschung mbH, Planckstrasse I, Postfach 110 541, D-6100 Darmstadt 11,  West Germany (Received 23 June 1983) Excitations of  the atomic shell in  heavy-ion collisions are influenced by  the presence of  a nuclear reac- tion.  In  the present Rapid  Communication we  point  out the equivalence between a semiclassical descrip- tion  based  on the nuclear autocorrelation function with  an  earlier model which  employs a distribution of reaction times f ( T).  For the example of  U +U collisions, results of coupled-channel calculations for posi- tron creation and K-hole excitations are discussed for two  schematic reaction models. Recently,  several  suggestions  have  attracted  attention which aim at the determination of nuclear reaction  parame- ters  by  measuring  atomic  excitations  (cf.  Refs.  1-11  and references  therein).  Among  other  effects,  variations  of inner-shell vacancy formation probabi~ities~.~  and oscillations in  the spectra  of  emitted  6  electrons4 are expected to arise from interference of excitation  taking place before and after the nuclear  reaction.  Furthermore, the emission of quasi- molecular (united atom)  X  raysSe6  and spontaneous positron production7,' in supercritical collisions will be enhanced dur- ing the reaction. These effects are most easily  described in a semiclassical model.  Here the internuclear motion consists of an incom- ing  and outgoing branch  of  a  classical  (usually  hyperbolic) trajectory, while the nuclear reaction acts to keep the nuclei close  together  for  an  interval  of  time  T.  If  the adiabatic basis  is  employed the amplitude of  an electronic transition i -+  f is (cf. Refs. 3 and 8) where (we Set fi = 1  ) a„(O)  and änif(=) are the amplitudes  for one branch  of the  trajectory  ( -  = +  0  or 0-  +CO)  calculated  with  the energy and angular momentum of the nuclear motion in the ingoing (outgoing) channel.  To obtain, e.g., the number of holes  in a  state q, the squared amplitudes  (1)  have to be summed over all  initial  states above the Fermi level  Fand averaged  with  a  distribution  function  of  reaction  times f ( T) yet to be specified: with the normalization condition summation was  sugge~ted,~  leading  to an obvious generali- zation of  (3). Recently,  Tomoda  and  Weidenm~ller'~,'~  applied  the semiclassical  approximation  to the full  electronic plus  nu- clear  quantum  problem  assuming  that  the  excitations  of both Systems take place  in  separate regions of  space.  They obtain  a  formula  very  similar  to  (1).  Suppressing angular momentum indices the electronic transition amplitude is with  the nuclear  S matrix  SoN for a transition between the states  0-  N.  If  the  S  matrix  is  a  randomly  fluctuating function  of  energy,  the atomic excitation  probability,  after averaging over the beam energy, is found to beI3 a(6) is a measure for the ratio of  "direct"  and "delayed" reactions  and  the  matrix  A  is  the  nuclear  autocorrelation function of the fluctuating part of the S matrix: In standard compound-nucleus theory, with  the  compound-nuclear  width  r,.  Here  we  Want  to Stress that (6) is  completely equivalent to the results (1)-(3). Indeed, Eq.  (3) can be written in the form The time distribution function f ( T) and the nuclear  corre- (4)  lation  function  AEtE are  simply  related  by  a  Fourier More generally, at a given scattering angle 8 a superposition  transformation of  both  direct  (undelayed)  scattering  and  reaction  events  - may  be  present.  For  these  contributions  an  incoherent  Ar+u,r=S,  dTf(T)piuT . 28  2558  -  01983 The American Physical Society 28  -  DESCRIPTION OF ATOMIC EXCITATIONS IN HEAW-ION REACTIONS  2559 Here  we  have  implemented  causality  by  the  condition f (T)  =O for  T <  0. This is satisfied if  the autocorrelation function  is  an analytic function  in  the upper  energy  half- plane.  As is  well  known,14 the  assumption  of  stochastic fluctuations in the S matrix leads to an exponential decay in time.  Indeed,  the ansatz f ( T) =  T, exp( -Tc T) leads  to (8). It is envisaged to apply the theory to heavy-ion collisions close  to the Coulomb barrier  or to deep-inelastic  reactions. In  both  cases  it  is  not  clear  whether  the  stochastic argu- ments  leading  to  the form  (8)  are well  founded.  On the contrary, one expects time distributions more or less sharply peaked at a finite time T,.''  According to (10) this could be transformed to an autocorrelation function  exhibiting long- range  order.  Such  an order  might  be  established  by  the ponential  de~ay).~  From  precise  experimental data,  there- fore, information may be gained both on the mean time de- lay  (or width r,) and on the distribution f ( T), and thus on the nuclear autocorrelation function A„,. The  equivalence  also  holds  for  supercritical  col~isions'~ where the matrix  C,,,  of  Eq.  (2) acquires nondiagonal ele- ments8 due to the decay of the lsu  resonance into the adja- Cent  positron  continuum  E.  The modified  expression  for the excitation probability is In the case of an exponential time distribution the matrix M is given by dominant influence of a few collective modes in the nuclear  iT,/(E,  -  E,  +iTc)8nnImmI  , S matrix.  This question has direct  bearing  on experiments (12) looking  for  structure  in  the  8-electron  spectrum  since  in-  where the atomic 1s resonance enters with a complex energy terference  Patterns  are washed  out in  the case  of  (8)  (ex-  EI,-  iTo12.  Additional  nondiagonal  terms  arise  from  the I  decay to the positron continuum 1  s +  E,: Jra/27;  G M1sEplsE~= Ep- ElS+iro/2 Ei- Els-ir0/2 FIG. 1.  Spectra of positrons emitted in  the sl/z channel in  supercritical  U +Cm collisions at 6.2-MeVIu impact energy, calculated  under the assumption that a nuclear reaction takes place.  Predictions for an exponential (full lines) and a sharp (thin lines) distribution of reaction times, i.e., f (T)  = r, exp( -  T,T)  and f (T)  =  6( T -  l/r,), are compared.  (a) Nuclear widths T, =  400, 200, and 100 keV.  (b) Tc  = 50, 25, and 12.5 keV. 2560  J. REINHARDT,  B. MÜLLER, W. GREINER, AND U. MÜLLER FIG. 2.  Same as Fig.  1 for the subcritical pilz states.  For  the exponential time distribution  the spectrum is smooth and changes only weakly with T,. FIG. 3.  Probability  for exciting K vacancies in a U +Cm reaction as a function of the nuclear delay time (inverse width)  T,. DESCRIPTION OF ATOMIC EXCITATIONS IN HEAW-ION REACTIONS  2561 where T' =  Tc +  ro/2  and Mnnlmm'  =  M"'nmm,  = predictions  of  the sharp  (bold  lines)  and exponential  (thin lines)  distribution  functions  are  compared.  While  the shapes  of  the  corresponding  spectra  for  the  two  models  -. differ  quite distinctly it  is  not immediately clear how much In  (13)  the decay  matrix  element  was  approximated  by  a  information  the  distribution  function  ( T)  can  be constant  T(E,)  =  To, and coupling  between  different  posi-  gained  from  the  experiment.  Additional  broadening  and tron states was neglected.  distortion  of  the line  will  result  from experimental effects Let us examine the consequences for the positron  spec-  (Doppler broadening).  Also, the position of the line is very trum  in  supercritical  collisions.  Leaving  aside  positron  sensitive  to  the nuclear  configuration  during the reaction. creation outside the reaction  region entirely  (which is not a  Therefore  realistic reaction model probably will produCe good approximation,  however, except for very long reaction  superposition of lines centered at various  (neighboring) en- times),  the combined  Eqs.  (11)  to  (13)  yield,  to  a  Breit-  ergies making the analysis more complex, Wigner-shaped  spectrum,  Figure  2,  in  addition,  shows the emission  of  p,12  posi- As a result of the presence of  large values of  T in the time integral  (3) the resulting "positron  line"  is clearly narrower compared  with  the  case  of  a  sharp  time  distribution f ( T)  =  6( T -  T,)  considered previously8 if  one identifies  Tc with  the inverse  nuclear  width  l/Tc.  While  the linewidth full width at half maximum, according to the simple analytic expression  (14), is  AE =2r1=2r„  the sharp time distribu- tion  leads to  AE =5.56rc.  The effect of  averaging  over a distribution of delay  times is demonstrated quantitatively by the coupled-channel calculations of  Fig. 1.  Here the spectra of positrons emitted in  the angular momentum channel sl/z have  been  calculated  for  U +Cm  collisions  at  6.2-MeVIu bombarding energy.  For several values of the nuclear width Tc (inverse  reaction  time)  between  400  and  12.5 keV  the trons.  Since these states are subcritical the time delay does not result in an enhancement of  the probability but only in interferences  which  are completely  smeared out by  an ex- ponential distribution f ( T). A similar effect is found for K-hole production.  Figure 3 shows  the  probability  of  lsg  and  2pl12a vacancies  as  a function  of  Tc  (or  l/rc). By  use of  the exponential  distri- bution the oscillatory  structure of  P(  Tc) is lost.  The sharp decrease  due  to  destructive  interference  at  rather  small values of  Tc, however,  survives  the folding  and should be observable experimentally. In Summary, we have shown the equivalence between two models for electronic excitations associated  with  a  nuclear reaction, one formulated in terms of  the nuclear autocorre- lation function, the other using a distribution of delay times. While the former  representation  is  more directly related to microscopic quantities,  i.e.,  the nuclear  S  matrix,  the latter has the advantage of a clearer intuitive interpretation. 'E. Merzbacher, in  X-ray and Atomic  Inner-Shell Physics-1982,  edit- ed  by  B.  Crasemann,  AIP  Conf. Proc. No. 94  (AIP, New  York, 1982), p.  1; W. Meyerhof,  ibid., p. 13. Anholt, Phys. Lett.  8&  262  (1979). 3U. Müller, J. Reinhardt, G. Soff, and W. Greiner, 2.  Phys. A 297, 357  (1980). 4G. Soff, J. Reinhardt, B.  Müller, and W. Greiner, Phys. Rev. Lett. 43,  1981 (1979). Anholt, Z. Phys.  292, 123 (1979). 6J. Kirsch, B. Müller, and W. Greiner, Phys. Lett. U,  151 (1983). 'J.  Rafelski, B. Müller, and W. Greiner, Z. Phys. A 285, 49  (1978). s~.  Reinhardt,  U.  Müller, B.  Müller,  and  W.  Greiner,  2. Phys.  A 303, 173 (1981).  - 9J. S. Blair and R. Anholt, Phys. Rev. A 25,  907  (1982). I0J. M. Feagin and L. Kocbach, J. Phys. B l4,  4349 (1981). "U. Heinz, B.  Müller, and W. Greiner, University of  Frankfurt  Re- Port No. 106/1983 [Ann. Phys. (N.Y.) (in press)]. 12T. Tomoda, Max  Planck  Institut Heidelberg Report No. MPI H- 1983-V8 (unpublished). I3T. Tomoda and H. A. Weidenmüller, Max  Planck Institut Heidel- berg Report No. MPI H-1983-V6 [Phys. Rev. C (in press)]. I4S. Yoshida, Annu. Rev. Nucl. Sci. 2,  1 (1974). 15~.  Gobbi and W. Nörenberg, in  Heavy Ion Collisions, Vol. 2, edit- ed by  R. Bock  (North-Holland, Amsterdam, 1980). I6J. Reinhardt,  B.  Müller, and  W. Greiner,  Phys. Rev. A 3,  103 (1981). 

Description of atomic excitations in heavy-ion reactions

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