The direct (Coulomb) formation of electron vacancies in the 1s&#963; state of superheavy quasimolecules is investigated for the first time. Its dependence on the impact parameter, projectile energy, and its contribution from excitations into the continum and higher bound states are determined

Betz, Wilfried

Greiner, Walter

Müller, Berndt

Soff, Gerhard

English

The direct (Coulomb) formation of electron vacancies in the 1sσ state of superheavy quasimolecules is investigated for the first time. Its dependence on the impact parameter, projectile energy, and its contribution from excitations into the continum and higher bound states are determined

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Vorr ME  17. NK~MBLR  16  PHYSICAL REVIER' LETTERS  18 OCTOUER  10-h Direct Formation of Quasimolecular lso Vacancies in Uranium-Uranium Collisions* Wilfried Betz, Gerhard Soff,  Berndt Müller, and Walter Greiner Institut für  TIzeoretisclze Physik der Jolzann  Tvolfgang Goethe -LTniz~ersität,  Frnnkfurt nnz  IlIai~z,  Gernla~iy (Received 14 June 1976) The direct (Coulomb) formation of  electron vacancies in the lsc state of  superheavy quasimolecules is investigated for the first time.  Its dependence on the impact parame- ter, projectile energy, and its contribution from excitations into the continuum and high- er  bound states are determined. The fundamental interest in the formation of quasimolecular electron states in heavy-ion  col- lisions centers around two points:  (1) It has been possible to observe radiative transitions between the quasimolecular states; particularly the mo- lecular K  X  rays.'  With further improvement of the experimental methods and their extension to heavier systems, a systematic spectroscopy of superheavy quasimolecules and even superheavy elements can be envi~aged.~  (2) For Z  =Z,  +Z, > 172, the conceptually new phenomenon of  spon- taneous pair creation due to the decay of  the neu- tral electron-positron vacuum in electric fields has been predi~ted.~.~  If  such overcritical fields occur during the collision of,  e.g.,  U with U,  the "decay  of  the neutral vacuum"  furnishes a new fundamental test of  quantum electrodynamics which also serves as  a prototype for similar  I processes in all areas of  the physics of  very strong fields and strong binding. The common feature of  these two classes of  ex- periments is the necessary vacancy production in the lsu  quasimolecular states.  Unfortunately, very little is  known about the formation of  lso vacancies,  because the well-understood  promo- tion mechanism is  not applicable and the contin- uum is expected to play an importarit role.  Ex- isting calculations of  continuum ionization5  are limited to H +H+  (nonrelativistic); their numeri- cal accuracy and agreement with experimental heavy-ion data6  are  poor.  EstimatesGDR  for the vacancy formation in U-U  collisions of  Ca.  1500 MeV lab energy range from 4 x10-"0  10". To calculate the quasimolecular lsa  vacancy formation amplitude during and after the colli- sion, we start from the instantaneous,  station- ary solutions of  the two-center Dirac equation9 We then expand the solution of  the time-dependent  Schrödinger equation into these basis states and the correspondent occupation amplitudes: In first-order perturbation theory the amplitudes a,(t) are  given by Here anP  is  a constant amplitude (mostly 0 or 1) that shows to what extent the transition 1s -X is allowed by the Pauli principle.  In perturbation theory,  unitarity is violated to the order Ap =C,a:,  where 11  runs over all excited states. In an actual U + U collision the 2sa and 3so states will be filled,  so  that in general u„,~  = aSsoP  = 0. Under these conditions we find  Ap -  0.1,  which is a measure of  the relative error we make using Eq.  (3) to solve the scattering equation.  If  we set, e.g,,  azSoP=  1 in a fictitious experiment,  we find P -  0.5 which shows that a true coupled-chan- nel calculation would be necessary in such an  ac- ademic situation.  It should be stressed that Eq. (3) is  not a kind of  "never  come back" approxima- tion because it allows for oscillations in the ex- citation amplitude a(t)  which indicates a "jump- ing" of  the vacancy between the Iso  and the iz state until it approaches the final value cz(t = a) after the collision, In the corotating coordinate system one can split the translation matrix element into a radial and rotational part: aiat-ka/a~  -G.;.  (4) For symmetric collisions the rotational part couples to the 3dr state as lowest state and \'OLI'ME  37. hTc  MBER 16  PHYSICAL REVIEW LETTERS  18 &TOBER  1976 -most  important-the  corresponding matrix element vanishes for R -  0.  On the other hand, the radial part couples to all nso, izdu,  etc. states with large matrix elements (17saj aIaR1 lso) at close distances.  We  have therefore limited our basis to the lso, 2so,  3s0,  4su bound  states and the Eso  continuum states.  For bound state wave functions1° we have taken the solutions of  the one- electron two-center  Dirac equation (1). The cal- culation of  the radial matrix elements can be ob- tained by use of (5)  =R(E,-E,)-'(w,  I  ~v„/~RI  Y,), Rlfnl l0  '$  ,b  iio  2io  560  ioia  2000  50  100  200  500 Rlfrnl  .  . from the two-center potential  V„,  provided the FIG. 1.  (a) Radial coupling matrix elements of  the  two wave functions are  orthogonal and eigenfunc-  lso to the 2so, 3su, 4su levels in U +U as a function of tions  of the  Hamiltonian.  The matrix ele-  internuclear separation.  The dashed lines are without ments between bound states as  obtained from Eq.  the translation factor, the full lines are with it.  The (5) are shown in Fig.  l(a) (dashed lines).  Between  dash-dotted  line shows a nonrelativistic ~su-2sa  ma- neighboring states, e.g.,  1s-2s,  2s-3s,  3s-4s, etc. they are  almost equal.  Hence only the matrix elements with the lsa  state are shown.  They do not vanish for large separations R but approach a constant value.  This is known to be due to the fact that the molecular states p,(~)  are  not as- ymptotic solutions of  the scattering equation, but rather states S,(R) = v,(R) exp[-(i/2)nze  .E] where  denotes the molecular state of  nega- tive parity (e.g.,  2p1,,o)  which in the separated atomic limit corresponds to the positive parity state  (e.g.,  1s~).  z  is the intrinsic coordin- ate along the molecular axis.  The matrix ele- ments (6) are  shown in Fig. l(a)  by solid lines, They vanish rapidly beyond R = 2000 fm,  indicat- ing the point where the two K  shells just begin to influence each other. The most noteworthy feature of  the matrix ele- ments is, however,  the steep increase even at very small R.  For comparison we have scaled a nonrelativistic matrix element to U + U  [dash-dot- ted line in Fig.  l(a)]. The difference is due to the fact that for 2,  +Z,  > 137 the relativistic izsa and npl,,o  dave functions are extremely sensitive on R,  especially for small separati~ns~'~'~  (this is the lack of  a "runway"  in the correlation dia- gram).  The smooth behavior of  the matrix ele- ments suggested calculating them also in the monopole approximation,  substituting a blown-up nucleus of  radius SR for the two nuclei.  The re- trix element scaled to U +U.  (b) The radial coupling elements of  the lsu level to the continuum states (ener- gies in electron masses).  The continuum states have been calculated in the monopole approximation. should be used."  Including this "translation fac- tor" in first order in the projectile velocity one finds that the matrix element (5) must be re- placed by12 '  sults are in agreement within 2% of  the exact curves in the range 20 fm -'R  e400 fm.  This jus- tifies one to calculate the radial matrix elements to the Eso  continuum states in the monopole ap- proximation [for both states in order for Eq.  (5) to remain valid] as  shown in Fig. l(b).  The total transition strength into the continuum, is larger than into the 2sa bound state,  1 LJ„,1'. For the bound states we find  1 D„,/~  1z4 -  const, i.e.,  the summation is rapidly convergent.  This behavior implies that the energy density of  ns states is clE/dn -n-4 in the Z = 184.  Indeed we find from numerical calculations that E„ -  12-  Y with y -  2.7  (E„-Iz-'  in the hydrogen atom).  We conclude that the radial coupling does not exhibit an anomalous threshold behavior toward the con- tinuum. The complex amplitudes a,(t) are  now  obtained by numerical integration of  Eq.  (3) along classi- Vo~r  MF 17, NI  MHIK 16  PHYSICAL REVII\\  LETTERS  1 8  Ocl OFJI  K  I()-() FIG.  2,  Total probability for Couloml~  excitation of a lsu electron in U + U into a racant nsu lerel and into the continuum (Esu) at zero impact parameter.  One- step and multistep channels are included.  Tne strong increase sets in just after the point of  closest ap - proach. cal Coulomb trajectories for the heavy ions.  For excitation into the 2su and Eso  states this is straightforward, whereas for the 3so  and 4sa states a number of  channels add up coherently (e.g.,  the one-step excitation lso-3so and the two-step process lsu-2sa-3so). The total excitation probabilities  1 an/ and IdEla,l  are  shown in Fig.  2 as  a function of  R, under the assumption that the final state is va- cant.  As expected,  the curves show oscillations with a maximum twice as  high as  the final excita- tion probability ~(b).  The steep rise of  1 a(t)  1 ' just beyond the distance of  closest approach can easily be understood.  During the approach a ma- jor portion of  the lsa  electronic density flows in- ward following the motion of  the nuclei.  When this motion is suddenly reversed the electron is left behind and redistributes over other states. This acceleration process weakens quickly for high impact parameters, as  is Seen in Fig.  3(b). The excitations into the continuum, i.e.,  ioniza- tion,  vary also over the trajectory:  After the collision they are  comparable to excitation into the 4s state (which will be practically vacant from simple velocity arguments), whereas in the re- gion where the lsa  state of  U + U has dived into the positron continuum (R 6 35 fm) they are  dom- inant (see Fig.  3).  The projectile energy depend- ence of  excitation is  shown in Fig.  3(a), the im- pact parameter dependence in Fig.  3(b).  The fi- FIG. 3.  (a) Energy dependence of  the Coulomb ioniza- tion of  the U +U  lsu lcvcl at zcro impact parameter. Excitation into the 2sa, 3sn, 4s~  states and the contin- uum (E) is shown separately.  f  ({L) denotes the final atomic ionization probability; d(r~)  the average proba- bility over the region where the lsu state has clived  in- to the negative energy continuum.  (b) Impact parame- ter dependence of  the Coulomb ionization of  the lsa  lev- ei at Ei,, = 1600 MeV,  Notations are as in (a). nal excitation after the collision is denoted by f (n),  the average excitation over the diving re- gion by d(~).  To obtain cross sections,  the num- bers must be multiplied by a factor of  2 for the two-spin lso  states.  For example,  at 1600 MeV and Zero impact parameter the probability to have a  lso  vacancy in the diving region is pre- dicted to be larger than 0.08.  This number is crucial for the proposed positron experiment be- cause it is  bigger than the value 0.01 which was assumed in previous calculations of  the cross ~ections.~ With decreasing incident ion energy the excita- tion probabilities fall off  very steeply, until at tandem energies (E  c  200 M~v)  they are  much less than 10-',  The Same is true for increasing impact parameters:  Typically the probability shrinks to one half at h = 20 fm.  This will facili- tate coincidence experiments, because all the cross section goes into large angle collisions anyway.  Figure 3(b) allows also an estimate of the total cross section for atomic K-vacancy production oKdir  -  2 b, which is very close to the cross section expected from conversion of  Cou- lomb-excited nuclear states.13  We propose to separate the two mechanisms experimentally by looking at double-K-vacancy  production with !'<ILC  XIF  3-.  Xi  \II~I  R  10  PHYSICAL RE\ IE\\  LETTERS  18 C~TOBFR  1Y70 oMd" -  25 mb.  This cannot be caused by conver- sion because the lifetime of  nuclear states is  ca. 10-l1  sec  whereas a K  vacancy in U  lives only for about 10-l7  sec. We conclude that the 1so-vacancy production in U +U  by direct excitation is much larger than ex- pected.  The influence of  relativity on the behav- ior of  electrons in superheavy quasimolecules is doniinant and cannot be accounted for by small "relativistic  corrections."  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The combined effect of  interchannet and spin-orbit coupling is proposecl as the lilcely interaction niechanisn~. Photoionization of  the Xe 5s subshell by electric dipole interaction can  be  written scliematically as The orbital momentum 1 of the photoelectron is  restricted by angular momentum balance, -.  -. ~=5,+J,=5,+S+i.  (2  1 to the range  I  2.  Furthernlore, the parity of the ~liotoelectron  is a,  = - 1 (odd) because of parity 

Direct formation of quasimolecular 1sσ vacancies in uranium-uranium collisions

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