The components of the nuclear inertia tensor, functions of the separation distance R and of the radius of the light fragment R2, BRR(R,R2), BRR2(R,R2), and BR2R2(R,R2) are calculated within the Werner-Wheeler approximation, by using the parametrization of two intersected symmetric or asymmetric spheres. Analytical relationships are derived. When projected to a path R2=R2(R), the reduced mass is obtained at the touching point. The two one-dimensional parametrizations with R2=const, and the volume V2=const previously studied, are found to be particular cases of the present more general approach. Illustrations for the cold fission, cluster radioactivity, and &#945; decay of 252Cf are given

Gherghescu, Radu A.

Greiner, Walter

Poenaru, Dorin N.

English

Hochschulschriftenserver - Universität Frankfurt am Main

PHYSICAL REVIEW C  VOLUME 52, NUMBER 5  NOVEMBER  1995 Two-dimensional nuclear inertia: Analytical relationships R. A. ~her~hescu,'.~  W.  reiner.' and D. N. ~oenaru',~ '~nstitut  für  Theoretische Phy.rik der Universitüt, Frankfirrt arn Main, Postfach 111932, 0-60054 Frankfurt um  Main, Germany 'lnstitute of Atomic Physics, PO. Box MG-6, RO-76900 Bucharest, Rornarzia (Received 24 May  1995) The components of  the nuclear inertia tensor, functions of  the separation distance R and of  the radius of  the light  fragment R2,  BRR(R,R2),  BRR2(R,R2),  and  BR2R2(R,R2)  are  calculated within  the  Werner-Wheeler approximation, by  using the parametrization of  two intersected symmetric or asymmetric spheres. Analytical relationships are derived. When projected to a path R, =  R,(R), the reduced mass is obtained at the touching point. The two one-dimensional parametrizations with R2  =  const, and the volume V2  =  const previously stud- ied, are found to be  particular cases of  the present more general approach. Illustrations for the cold fission, cluster radioactivity, and  cu decay of  252~f  are given. PACS  number(s): 21.60.Ev, 23.70. fj,  25.85.Ca, 27.90.+b I. INTRODUCTION One group of methods frequently used to solve quantum dynamical problems in many branches of physics and chem- istry  (e.g., tunneling phenomena  in  solid-state and nuclear physics,  mass  transfer  in  nuclear  reactions,  mass  distribu- tions  in  fission,  scattering  reactions,  molecular  collisions. chemical  reaction-rate theory,  etc.)  [I-31,  relies  on  quasi- classical  approximation, in which  an iniportant quantity is the  inertia  tensor  [4].  The components  of  this  tensor  are strongly dependent on the arbitrarily chosen set of n  gener- alized coordinates {q  ,q2,  . . . ,  q,). In the present paper we have in  mind possible applica- tions  for  quantum  mechanical  tunneling  in  nuclear  decay processes  like  cold  fission,  cluster  radioactivities,  and  a emission [5], for which it was repeatedly stressed that a num- ber  of  collective  degrees  of  freedom  as  low  as  possible should be chosen in  order to  represent on an  axis or in  a plane the main physical quantities determining the basic fea- tures of these phenomena. Glas  and  Mosel  [6]  have  used  the  distance R  and  the angle q5  to express the kinetic energy in heavy ion collisions. Within fragmentation theory  of  binary  Systems [7]  the best suited deformation coordinates are the fragment  separation distance  ql=R  and  the  mass  asymmetry  Parameter q2=7=(V1-V2)/(Vl+V2),  where  V,  (i=l,  2) are  the volumes of  the  fragments AIZ, (which finally  will  be the daughter nucleus AdZd) and A2Z2  (which becomes the emit- ted cluster A,Z,  at the touching point configuration).  Such pairs of collective variables have been used to calculate the nuclear  inertia for the  mass  transfer  [8-101  in  heavy  ion collisions, by using either the cranking approximation [3] or the hydrodynamical approach [10]. As  a  consequence  of  the  assumed  incompressibility  of nuclear  matter,  the  total  volume  of  the  fragineilts  is  con- served  during the  deformation. Also,  we  are interested to study a given exit channel, hence the final value of the mass asymmetry, 7,jt =  (Ad- A,)IA,  is known. The parametrization of  two intersected  spheres with  radii  R1 and R2 has  beeil adopted  [11]  to  generate  two different  sequences  of  such shapes for a given mass asymmetry, with  an additional con- straint  of  R, =  Re  =  const  ["clusterlike"  (CL)  shapes]  or V2 =  Vr =  const ["more  compact" (MC) shapes]. In this way, by taking into account the total volume conservation and the matching condition in the separation plane, it was possible to arrive at a single independent shape variable which could be either the separation distance, R, of  the geometrical centers of  the spheres or, z,,-the  distance between their centers of mass. The Werner-Wheeler  approximatioil  [12,13]  allowed us to obtain analytical relationships for the nuclear inertia (a scalar in this case) in a wide range of mass asymmetry. Our aim at present is to relax the above-mentioned restric- tions, leaving both q  =  R and q2=  R2  to vary. We shall show that from the general  expressions obtained in this way  one can get the preceding ones by giving the corresponding law of variation R2  =  R2(R) in the overlapping region of the two fragments. Also, another local test is  provided  by  the  fact that the inertia scalar B(R), which is the result of projection on a general trajectory R2=  R2(R), is equal to the reduced mass, R(R,) =  p, at the touching point R,= R,+  Re  . 11.  MULTIDIMENSIONAL APPROACH During  the  decay  process,  leading  from  one  parent nucleus, AZ, to two different fragments (AdZd-the  daugh- ter or heavy fragment; AeZ,-the  emitted ion or light frag- ment), the  shape of  the  system changes  continuously.  The potential energy surface in a multidimensional hyperspace of deformation parameters, {q) =  q  ,q2,  . . . ,q, , gives the gen- eralized forces acting on the nucleus. Information concerning how  the  system  reacts  to  these  forces  is  contained in  the inertia tensor {B„).  The contribution of  a shape change to the kinetic energy of the system at any time, t, is expressed by The  inertia  tensor  with  components  B„=B,,(q)  corre- sponds to the variation  in time of  the nuclear shape. Their values depend on the particular choice of  deformation coor- dinates. On the other hand, for a system with axial symmetry relative to the z  axis, the kinetic energy of a nonviscous fluid is given by 0556-28 13/95/52(5)/2636(7)/$06.00  -  52  2636  O 1995 The American Physical Society 52  -  TWO-DIMENSIONAL NUCLEAR  INERTIA: ANALYTICAL  . . .  2637 Here  V  is  the  volume  considered  to  be  conserved, a=  3rnl(4.rrrO) is the mass density, rn  is the nucleon mass, V  is  the  velocity,  and  r, =  1.16 fm is  the  nuclear  radius constant. By  assuming irrotational motion  (V  X V= rot  V= 0),  the velocity field may be derived from a scalar velocity potential p, i.e.,  V= V  q.  From the continuity equation of  an incom- pressible  fluid  it  follows  that  the  Laplace  equation, V  *  =  h  <p =  0,  should be satisfied with kinematical boundary conditions. is different from Zero  if  the origin of z is not placed in the center of  mass. Here p, =  p,(z) is the nuclear surface equa- tion  in cylindrical coordinates, with  zmi, ,zm„ intercepts on the z  axis. For another Set of  deformation parameters {cu}  describing the same shape, expressing the need of  a vanishing normal component of the velocity at the surface. In this way there is no flow of matter through the  surface. The  surface equation for axially  sym- metric shapes in cylindrical coordinates (p,<p,z)  is written as F(r,t,q)=p-p,(z,t,q)=O  in whichp,  is thevalueofp on the  surface.  The  velocity  components,  i=  dpldz  and p = dpldp, are both functions of  z  and p. As  an  approximation  to  the  incompressible  irrotational flow, one can  use  the  Werner-Wheeler  assumption. In  this approximation the flow is considered to be a motion  of  cir- cular layers of fluid, z  is independent of p, and p is linear in P: The quantities X, are calculated separately for the left- and right-hand  side of  the body  by  requiring  a  vanishing  total (convective) time derivative of  the fluid volume to the left- hand side, or right-hand side of an arbitrary plane normal to the z  axis: In order to get a vanishing normal component of the velocity at the surface, one needs The functions X,  and Yi  are found as a sum of two terms for the left- (1) and right- (r) hand side of the shape. After Substitution in the relationship for the kinetic energy and comparison with the initial equation for E,  we find the following equations for the components of the inertia tensor: where the correction term B:,(q)  due to the center of  mass motion [11] Also, one can  define  a nuclear inertia scalar B(s)  along  a trajectory,  given parametrically by  the equations q,  =  q,(s)  , (i= 1,2,.  . . ,n): In this way the multidimensional tunneling penetrability can be reduced  to a one-dimensional problem. When s=R, or s =z,,  a good test  of  accuracy of  the computations  is ob- tained at the touching point configurations, where one should obtain the reduced mass B(R,)  = p. 111. TWO-DIMENSIONAL PARAMETRIZATION OF INTERSECTED SPHERES The surface equation, assuming a rhape of two intersected spheres, can be written as where R , R2 are the radii of the two overlapping fragments, z, is the position of the separation (intersection) plane, and R is the distance between the two Centers. By taking q,  =  R and q,  =  R2 as independent deformation parameters and placing the origin of coordinates in the center of  the left-hand side fragment, the limits of  integration are zm„=  -  R, and  z„„=  R +R2,  as  can  be  seen in Eq.  (11). The radius  of  the  heavy  fragment  will  be  considered  as  a function of these variables, R =  R (R,  R2).  The Xl1  quanti- ties defined above for the left-hand side fragment are and the corresponding Yli 2638  R. A. GHERGHESCU, W.  GREINER, AND D. N. POENARU  52  - Similarly, for the right-hand side fragment we have  i B„(R,R,)  B~~2(R,R2) X  r R 1--  p?,(x)dx=  1,  (16)  ,  B~2~(R,R2)  B~2~2(R>R2) a, has  the  nondiagonal  components  equal  to  each  other, BR2R=BRR2,  due to the symmetry property. By  taking into 2R2 (17)  account Eqs. (7),(8),  the inertia components have the follow- R2+i-R  ing form: and the corresponding Y,.,  B„(R,R*)=  B:,+  BK,+  B:, The tensor of inertia for two independent variables  where BRR:(R,Rz)=  B~PR?+  ~RR~+BKK~ R+R2 =  TU/::,  p$(xiRxm2  +  + ymyiR2)d:  +  VUJ  p:(xr~xfi2  + iyr~yf12)dz 7  7 (24) in which where 52  -  TWO-DIMENSIONAL NUCLEAR  INERTIA: ANALYTICAL  . . .  2639 The  following  geometrical  quantities  can  be  defined: D1=z,;  Hl=Rl-z,;  D2=R-z,,;  H2=R2-0,.  By substituting  Eqs.  (12)-(19)  into  Eqs.  (2 1)-(29),  and  per- forming the integrations we  obtain for the first component, BRR,  the contribution of the left-hand side fragment: For the right-hand side fragment and for the correction term Similar calculations for the mixing component, BRR2,  lead to for the heavy fragment contribution, and for the  light fragment  contribution. The correction  term  is expressed as For  the  last  diagonal  component  of  inertia,  BRZR2,  the corresponding terms are found to be Finally, by  summing up the contribution of  the two frag- ments and the correction term for every of  the three tensor components, and by  taking into account that all lengths are expressed  in  units  of  the  radius  of  the  parent  nucleus, R,=  roA  I",  we  obtain the  three components  of  the inertia tensor: 2640  R. A. GHERGHESCU, W.  GREINER, AND D. N. POENARU The volumes of  the two fragments are given by  the fol- lowing relationships: By using the total volume conservation, V, + V2=  V= const (where V=~TR;/~,  Ro is the radius of  the parent nucleus), and the matching condition in the intersection plane we  will calculate analytically the two partial  derivatives of R  ,(R,  R,),  according to the theory of implicit functions: The involved quantities are After performing the calculations, we obtain By choosing a trajectory in the plane of  the two indepen- dent  coordinates,  R  and  R2 given  in  a  parametric  form: R2=  R2(s):  R= R(s),  and by taking s=  R, we can write ac- cording to Eq. (10) expressing  an  inertia  scalar which  is  used  to  calculate the tunneling  penetrability along this path.  In the limit R-tR,, when  Hl+O  and H2+0,  the  diagonal  component, becomes  infinitely  large.  Consequently, for a  finite  inertia scalar B(R,), one has to choose a path R2(R)  fulfilling the condition of a vanishing derivative, R;(R,)  =  0, at the touch- ing point. IV. TWO ONE-DIMENSIONAL SEQUENCE OF SHAPES AS PARTICULAR CASES We  have  the  possibility  to  check  the  validity  of  the general  relationships  (39)-(41),  by  comparing,  in  the whole  range  of  R=(R„R,),  where  R,=R,-R„,  and R21  =  R2(RI)  is the initial  value of  R2,  the  results  for two particular one-dimensional configurations  mentioned in  the introduction  (MC and CL) with the similar equations previ- ously  published  [ll] assuming only  one independent vari- able. Also, at the touching  point, R  =  R„  the  inertia  scalar B(R)  equates the reduced mass. For  the particular pararnetnzation  with  compact shapes, there  is  a  second  restriction  besides  V=const,  narnely V2=const.  We  have  to  take  into  account  the  Fact  that R,=Rl(R,R2),  in the general case when only the total vol- ume is conserved without any other constraint, so that The derivative  of  R2(R)  with  respect  to  R, is  calculated following the same prescription as above, The two partial derivatives of  R1 with respect to R and R2 are given by Eq. (48). We  find for the other terms in the above formula After  replacing  these  equations  in  (51)  and  performing calculations, we obtain the derivative of R2 with respect to R for the compact shape parametrization: which is identical to the corresponding Eq. (21) from [I I]. By substituting (55) in (39)-(41),  the three components of the inertia tensor may be written: 5 2  -  TWO-DIMENSIONAL NUCLEAR INERTIA: ANALYTICAL  . . .  2641 where and V2 is the volume of the emitted fragment [see Eq. (42)]. Now, in the general expression of  the total inertia  scalar (49) we  introduce  the  derivative  of  R,  with respect to R, (55), obtained for the compact shape parametrization. We get in this way the inertia for V2 =  const: 3 H~(R,+D,)(R~+D~)~  3  -- 4  R+Rl+R,  + 4%-  P  V?] The last term is the correction due to the Center of  mass motion.  One can  observe that, during  the  shape evolution, this  correction  remains  constant -  9A ~2/(4  T)?  =  -A;/A. The final result for the compact shape is where  ,UA  =A,AdIA  is the reduced mass number. This term is  derived  from  3AV2/(4%-)  -  ~AV~/(~%-)~=A,-A~/A =PA.  The equation  (62) reproduces the formula of  inertia obtained for MC shapes [ll]. Despite  the  differences  between  B  and  B„  shown  in Figs.  1-3  for cold  fission  with  12?cd light fragment,  46~r cluster radioactivity, and cr decay of  252~f,  the final result of the  projection  on  the  R2(R) path  coincides  with  the  one- dimensional B(R), the mixing term, 2Bm2R; being  nega- tive, due to the sign of the derivative R;.  Also, by multipli- cation  with  R;  and  (R;)?,  the  last  two  terms  give  no FIG.  1. The components  of the nuclear inertia tensor BKR  (top left), B„,  (top right), and BRiRZ  (bottom left), leading to a scalar B(R)  along a path R,(R) with a negative derivative (bottom right), for the cold fission of 252~f  with  lZ2cd  light fragment in the param- etri~ation  of  two  intersected  spheres  with  V2=const. The  one- dimensional  inertia  B  (top  left),  calculated  with  the  Eq.  (49), is exactl y reproduced. contribution  at  the  touching  point,  in  spite  of  the  general trend of BRZRZ  toward an infinite value when R+R,  . For clusterlike shapes, R,=const,  dR, ldR =  0. Unlike the preceding case, where all three components of inertia tensor contributed to B(R), now FIG. 2.  Same quantities as in Fig.  1, for the  "Ar  cluster emis- sion from 252~f. 2642  R. A. GHERGHESCU, W.  GREINER, AND D. N. POENARU  52 FIG. 3. Same quantities as in Fig. 1, for the a decay of  '52~f. which is again the Same as in Ref. [ll]. The  variations  with  R  of  B„  equating  B(R),  and  of BRR2  for CL shapes, and the Same decay modes like in Figs. 1-3  are plotted in Fig. 4. The other component BRZRZ  shows a similar behavior with that of MC shapes. By  analyzing the formula of  the total  inertia  B(R), and that of the derivative of R2  with respect to R, we can See that at the touching point where H2-+0, we have from Eq. (55) (dR2)l(dR)=0  when  R-+R,. In  this  way  the  two  last terms  of  B(R),  namely  2BRR2(R,R2)(dR2)I(dR) and B~,~,(R,R~)[(~R~)I(~R)]~  vanish at R =  R, . FIG. 4. The components of the nuclear inertia tensor BRR  equat- ing exactly the one-dimensional B  (left-hand side) and BRRi  [right- hand side, giving no contribution  to the scalar B(R) owing to the vanishing  derivative  R; =  01,  for  two-intersected  spheres  with R, =  const. The plots refer to the cold fission with  12'cd  light frag- ment  (top),  the  46~r  cluster  radioactivity  (middle),  and  a  decay (bottom) of  252~f. ACKNOWLEDGMENTS This work  was  supported  by  the Bundesministerium für Forschung  und  Technologie,  Bonn,  and  the  Institute  of Atomic  Physics, Bucharest. D.N.P. received  a donation  of Computer and copying equipment from the Soros Foundation for an Open Society. R.A.G. is grateful to the Konferenz der Deutschen  Akademien  der  Wissenschaften,  Mainz,  for  a grant of  a Volkswagenstfiftung fellowship. [I] P.  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Two-dimensional nuclear inertia : analytical relationships

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