Different collective deformation coordinates for neutrons and protons are introduced to allow for both stretching and &#947; transitions consistent with experiments. The rotational actinide nuclei 234-238U and 232Th are successfully analyzed in this model. NUCLEAR STRUCTURE 232Th, 234-238U calculated B (E2) values, collective model

Greiner, Walter

Hess, Peter Otto

Maruhn, Joachim

Seiwert, Martin

English

Hochschulschriftenserver - Universität Frankfurt am Main

PHYSICAL REVIEW C  VOLUME 23, NUMRER 5  MAY 1981 Different deformations of proton and neutron distributions in nuclei M. Seiwert, P. 0.  Hess, J. A. Maruhn, and W. Greiner Institut für  Theoretische Physik, Johann  Wolfgong Goethe Universität. 6000 Frankfurt um Main, Germany (Received 3 November 1980) Different collective deformation coordinates for neutrons and protons are introduced to allow for both stretching and  y  transitions consistent  with expenments. The rotational actinide  nuclei  234~23aU  and 232Th  are successfully analyzed in this model. STRUCTURE  232~h,  234-238~  calculated B (E2)  values,  collective model.  1 In constructing the quadrupole operator of  col- lective models, one of  the basic assumption~~*~  is the homogeneous charge distribution (HCD) with a uniform ratio of  the neutron and proton densities. The qualitative systematic agreement but quanti- tative disagreement of  calculated B(E2) values with experimental data in the rare earth and acti- nide nuclei contradicts this assumption.  Further  - more, it is a well-known fact that the collective model in the standard formulation yields vanishing probabilities for M1 transitions among any col- lective nuclear ~tates.~'~  This, as well as the lowering of  the g, factor from  Z /A, has been re- medied in earlier investigations  .3'4  Therefore several attempts have been made to generalize collective models  in order to describe magnetic nuclear properties c~nsistently.~'~  In these works different shapes for the neutron and proton distri- bution are introduced to allow for an individual be- havior of  neutron and proton liquid: &P,  and a; are independent degrees of  freedom but they are coupled strongly via the symmetry energy which prevents very large differences in the spa- tial distributions of  protons and neutrons.  This suggests the introduction of  the center of  gravity coordinate ff,  and the difference coordinate 4,: t,=a!;  -.P,,  (2) where B,  and B,, are the proton and neutron mass pararneters. The Hamiltonian is now decomposed as H=H„,(ff)+H„(S)+H,(ff, 51,  (3) where H„„ describes the common collective mo- tion, where both the kinetic as well as  the potential term are rotational invariants, which may also be expressed in terms of  the Euler angles B, and in- trinsic variables U„  U,.  The principal axes are determined by  requiring that the products of  in- ertia of  the kinetic energy contained in Eq. (4) vanish.  H„(f,)  describes the surface oscillations of  protons against neutrons.  Its typical energy is of  the order of  giant resonance energy, i.e., 15  MeV (see  also estimates  by  Faessler5). Because of  the small amplitudes involved, a harmonic ap- proximation seems to be  appropriate for this pro- ton-neutron-asymmetry  degree of  freedom. The coupling Hamiltonian HI (ff ,  t;) in lowest order  , being also rotationally invariant, must be  of the form This Hamiltonian has ten degrees of  freedom, five for each kind of  nucleori.  With the assumption of  a  homogeneous charge distribution ap=  an=  ff or t = 0, the number of  coordinates is reduced to fjve and the Hamiltonian becomes the well-known H„,  of  Eq. (4).  It is our goal now  to find a more realistic condition to reduce the number of  degrees of  freedom.  We consider the potential energy Determining the  t;  value of  the potential minimum 2335  O 1981 The Amencan Physical Society 2336  M. SEIWERT, P. 0.  HESS, J. A. MARUHN, AND W. GREINER  23  - for fixed a!, we obtain which leads to Considering only excitations near the ground state, the core is forced to oscillate along the direction of  the vibration with a constant ratio (/a.  The ratio (8) is constant for the potential of  the form (7) because the potential parameters C,!  and C, are constants.  For more involved interactions be- tween the 5  and  a!  degrees of  freedom this is no longer so.  This constancy implies the introduction of  new coordinates (Y,  =U,  COS~, 5, =U,  sin6.  (9) The Hamiltonian is then reduced to five dimen- sions, H(u)=  (SB, cos26+  SB, sin26+  B„  cos6 sind) X  [EI X  + ($C, sin26  + C,,  cos6 sind) X [U  xxu]C+ v'(u) =SB,[;  X ;]Co1+ f(u),  (10) where we included all the rotationally invariant terms constructed out of  uCZ1  in the potential ?(U). The mathematical form of  the Hamiltonian cor- responds to the usual formulation of  collective modes in the coordinates uCZ1;  the energy levels and eigenfunctions will not be  modified compared to earlier calculations by  this procedure.  The quadrupole operator , defined as2 depends only on the proton deformation a!;  which can be  written,  as follows, in terms of  the co- ordinates uC2'  using Eqs. (2) and (9): ---- neutron surface -  proton surtace FIG.  1. Proton and neutron distribution for ß =0.26. FIG. 2.  M(E2) values plotted as a function of  spin I for 232~h,  2347  236~.  The experimental data are taken from Ref. 6.  The predictions of  the present work (solid line) are compared to the rotation-vibration model with homogeneous charge distribution (dashed-dotted lines) and the interacting boson approximation in the SU(3) limit (dashed lines). Hence, the quadrupole operator depends on the proton-neutron deformation difference 6: 6  = 0 corresponds to the old assumption of  equal proton and neutron deformation.  As an illustra- tion, the proton and  neutron distributions for 6=  30'  are shown in Fig. 1.  Obviously the neutrons oscillate with a larger amplitude than the protons . Considering rotational excitations the centrifugal stretching is mainly done by  the neutrons whereas the proton distribution is more or less constant. This effect is clearly Seen in Fig. 2.  There, for FIG. 3. Different branching ratios for different iso- topes.  The calculation of  the homogeneous charge dis- tribution (HCD) and the present work are compared to experiment (Ftef.  9). 23  -  DIFFERENT DEFORMATIONS OF PROTON  AND NEUTRON  ...  2337 the nucleus 232Th  and some U isotopes, the M(E2) values are  plotted as a function of  the spin I.  The experimental values6*' follow the rigid rotor as if  the proton distribution remains constant.  In our calculations we use the rotation-vibration modelL2  and the reduction of  the proton stretching according to the quadrupole operator of  Eq. (13) and compare the old and new  model predictions with various experiments (Figs. 2 and  3). The predictions6 of  the interacting boson ap- proximationS seems to be unrealistic for higher spins.  The proton-neutron deformation difference 6 in these calculations is fitted to the transition proves considerably the agreement of  the theory with the B(E2)  transitions within the ground-state band at high spins and also with various intraband- interband branching ratios.  In addition, there is the well-known  lowering of  the collective g,  fac- tors from Z /A and also the collective M1 transi- tions explained by  the Same idea (mechanism). We may thus conclude that there is considerable evidence for different proton and neutron deforma- tions in nuclei,  the  neutrons having the larger deformations.  This may be due to the smaller pairing force of  the neutrons compared to the one of  the ~rotons. from the P  to the ground-state band.  We  obtain values between 23"  and 28'.  In  Fig.  3 some  The authors would like to thank Professor Th. W. Elze and Dr. H. Ower from the Institut für Kern-  branching ratios are given.  The ratio of  an intra-  physik  Frankfurt for illuminating discussions,  band to an interband transition is sensitive to the suggestions, and comments.  We  also thank Dr.  charge distribution while this is not true for the ratio of  two interband or intraband transitions.  H. Ower for the permission to use some figures of Obviously the introduction of  different deforma-  his thesis. tions for the proton and neutron distributions im- 'J.  M.  Eisenberg and W.  Greiner, Nuclear Theory, 2nd  'A.  Faessler, Nucl. Phys. E, 653 (1966). ed. (North-Holland,  Amsterdam,  1975), Vol. 1.  6~.  Ower, thesis, University of  Frankfurt/Main,  1980. 'A.  Faessler, W. Greiner, and R. K. Sheline, Nucl.  ''Th.  W.  Elze and H. &er,  private communication. Phys. 0,  33  (1965).  8~.  Arima and F. Iachello, Phys. Rev. Lett. E,  1069 'W.  Greiner, Phys. Rev. Lett. o,  599  Q965).  (1975). 4~.  Maruhn,  W. Greiner, and J~A.  Manihn, Phys. Lett.  Qucl.  Data  Sheets  (1977);  (1977). 57B, 109 (1975). 

Different deformations of proton and neutron distributions in nuclei

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