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Nuclear intrinsic vorticity and its coupling to global rotations
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A 627 (1997) 259-304
Nuclear intrinsic vorticity and its coupling to global rotations* I.N. Mikhailov a,b, R Quentin c, D. Samsoen c a Bogoliubov LaboratoryJbr Theoretical Physics (JINR), Dubna, Russia b Centre de Spectromitrie Nucldaire et de Spectromdtrie de Masse (IN2P3-CNRS), Orsay, France c Centre d'Etudes Nucld aires de Bordeaux-Gradignan (IN2P3-CNRS and Univ. Bordeaux 1), Gradignan, France
Received 4 April 1997
Abstract Important collective modes which are generally neglected within current descriptions of nuclear excitations in terms of fluid dynamics, are studied here. The intrinsic vortical modes are defined in a general way from which a specific mode, both simple and versatile enough, is particularly discussed. In this paper the main emphasis is made on the coupling of the chosen intrinsic mode to the rotation of the nuclear principal axes frame with respect to the laboratory system. A semi-quantal description of such excitations is proposed which is a generalization of the so-called routhian approach of global rotations. The results of a semiclassical treatment of the corresponding variational problem are presented. A simple mean field approach where the one-body potential is mocked up by a harmonic oscillator is discussed in a somewhat detailed fashion. The broad range of validity of a quadratic approximation for the collective energy in terms of the relevant angular velocities, is hinted from the previous simple model approach. Some general consequences of the latter are then drawn which have bearing on some possible fingerprints for the existence of such excitations, as the staggering phenomenon observed in gamma transition energies in some superdeformed states and the occurrence of identical rotational bands in neighbouring nuclei. @ 1997 Elsevier Science B.V.
1. Introduction S i n c e the p i o n e e r i n g w o r k s o f B o h r and W h e e l e r [ 1] on nuclear fission as well as o f B o h r and M o t t e l s o n on nuclear q u a d r u p o l e collective m o t i o n at low excitation * Work partially supported by the IN2P3-JINR Collaboration grant #91-9. 0375-9474/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PH S 0 3 7 5 - 9 4 7 4 ( 9 7 ) 00401-6
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energy [2,3], it is a usual approximation to describe large amplitude nuclear collective mode in terms of the dynamics of an equivalent incompressible nuclear fluid. In nuclear physics, however, an important part of the possible modes for a corresponding selfbounded classical fluid drop has not received a sufficient attention, even though their existence has already been recognized in a more or less direct fashion. These modes are the intrinsic vortical modes. Among their many manifestations, it appears that they strongly couple, in particular, with the global rotations of the fluid drop. In this paper we want first to define these vortical modes as precisely as possible and then within a generic model case, widely studied in other parts of the physics, we intend to discuss the above mentioned coupling. Indeed this seems to us of some relevance in view of the important current experimental effort devoted to understand very high spin and/or superdeformed nuclear states. In a fluid dynamical approach (within the so-called eulerian description), the essential concept is the velocity field u ( r ) with respect to the laboratory frame [4]. It may be split into an intrinsic part U i n t ( r ) and a rotational part u R ( r ) u ( r ) =Uint(r) + UR(r) ,
uR(r) = g~ × r ,
(1)
where g2 is the angular velocity associated with the rotation from the laboratory frame to the body-fixed or intrinsic frame (i.e. where the mass tensor is diagonal). The intrinsic field Uint(r), as any vector field under very general conditions, may be split into two parts, as well known Helmholtz's theorem) (2)
//int = US -]- UV ,
with us = - g r a d V ; uv = rot A ;
AV = -div//int , A A = - r o t Uint
,
(3)
whose first part us is irrotational and defined by a scalar field V whereas its second part uv is divergence-free and defined by a divergence-free vector field A. The uv field is defined as the intrinsic vortical velocity field. We picture the incompressible nuclear fluid as a self-bounded finite system having a constant density within a sharp-edged boundary, called the surface. The equation lbr the latter writes in terms of some shape time-dependent parameters labelled generically, ei as
F(r, ei) = 0.
(4)
Since our fluid is self-bounded, writing the condition of no surface-crossing in the intrinsic frame, yields OF
Uint • g r a d F + ~ - - e i i Oei
= 0,
(5)
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providing thus a boundary condition of the type Uint" n = qg(ei,r) ,
(6)
where n is the outward unit normal vector and qb(ei, r ) is given by ~.-~i aee'
(lh(ei, r ) = --
(7)
~/grad F - grad F " To specify the vortical modes under consideration here, we make a further restriction on the intrinsic velocity field by assuming that all the variation of the intrinsic shape is carried only by the field us deriving from a scalar potential. This assumption stems from our choice to make the minimal generalization of the Bohr-Mottelson approach [2,3] where this is obviously the case. To do that, we impose that the intrinsic vortical field Uv is purely tangential (imposing thus the boundary condition (6) on the field Us only) as we will see below. We define, as usual, the current in the intrinsic frame by Jint = pUint ,
(8)
in terms of the relevant density p. The latter is written here as
p = poO(F),
(9)
where P0 is the value of the constant density into the nuclear volume and O the Heavyside step function. As a result, the contribution (Op/Ot)v of the field Uv to the time change of the density, which is defined according to the continuity equation c~p d i v j + ~ - = 0,
(10)
as
(OP) -
-~-
=pdivuv+uv.gradp
(11)
v
is vanishing due to the divergence-free character of uv and to its tangential nature, since n is proportional to grad F and thus, at the surface, to grad p. The resulting equation for the field us deriving from a scalar potential is therefore of the Laplace form inside the volume, with a boundary condition of the von Neumann type at the surface (which is simply the Eq. (6) with us in lieu of Uint). Let us now set up a possible classification scheme of the uv fields, assuming that their components are homogeneous polynomials in the coordinates r of order p, which can always be done upon adding as many components as necessary to retrieve any given analytical field. For p = 0, the tangential character of Uv over a closed surface clearly implies that this field is vanishing, The collective motion is thus merely a combination of an irrotational shape-changing field Us and of a global rotation UR. It is thus consistent - here for the most general us mode though - with the Bohr-Mottelson ansatz [2,3]. The p = 1 case has been widely studied when the surface is defined in Eq. (4) by
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a quadratic function F (which implies for a bounded surface that the corresponding nuclear volume is ellipsoidal). Riemann and many others [4], among which one must single out Chandrasekhar [5], have studied the general problem of a total velocity field linear in the components of r whose domain is bounded by such an ellipsoid. There, the scalar potential V defining us, as well as the components of the vector potential A defining Uv, must obviously be quadratic. Taking into account the boundary condition for Uv, one finds ( u v ) i . ~. e t j.k c o. l x.g C I , ),k
(12)
where e~ik stands for the totally antisymmetric third rank tensor while x~ is the kcomponent of r, (.Oj is the j-component of a coordinate-independent vector oJ and ci, ck the semi-axes of the ellipsoid in the i and k directions (obviously all indices i, j, k refer to principal axis, i.e. correspond to the frame attached to the body's inertia tensor). This mode is the only intrinsic vortical motion possible for an ellipsoid having a uniform vorticity. It will be the one considered here. Some ways to extend the study to nuclear shapes other than ellipsoidal or to higher (p ~> 2) polynomial orders for the velocity components will be, however, proposed below. Even though, in the previous discussions, we have separated the global rotation UR and the intrinsic vortical Uv velocity fields, they are well known to be strongly coupled in classical systems [4]. In nuclei, the existence of such a coupling is strongly suggested by the well known fact that the moments of inertia characterizing the spectra of nuclear collective rotational bands are significantly different from their rigid body value [6]. This is particularly the case for superdeformed nuclei. In 152Dy for instance, the experimental dynamical moment of inertia has been fitted in Ref. [7] as being equal to 0.75ORB ÷ 0.25Om, where the subscripts "RB" and "IR" label the rigid body and irrotational moments of inertia respectively. As proposed years ago by Bohr and Mottelson [6], such a renormalization is generally understood as resulting from residual interactions, for instance of the pairing type. This interpretation is of course not in contradiction with the introduction of intrinsic vortical modes. The latter may be seen as a translation of the effect of these residual interactions in terms of collective modes. As a matter of fact, the microscopic calculations of Ref. [8], introducing rather schematically some effects of the pairing residual interactions, clearly exhibit current line patterns which strongly deviate from the ones obtained in a rigid rotation. In addition to the pairing phenomenon, there are of course many other sources for intrinsic vortical motion, among which one may mention the rotational equivalent of the Landau diamagnetism, resulting in a retardation of the fermions near the surface as compared to the centre of the nucleus (see e.g. Ref. [9] and references quoted therein). For integral quantities like the moments of inertia, the quantitative analysis of Ref. [9] has demonstrated that this diamagnetic property is roughly compensated by a Pauli paramagnetic alignment of the spins. Yet, the currents are significantly affected by this surface behaviour.
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To illustrate the importance of the coupling of a global rotation with an intrinsic vortical motion, we have chosen here the simplest intrinsic vortical field in a somewhat generic yet crude description of vortical modes in deformed nuclei. As a matter of fact, the introduction of the Riemann linear velocity field in an ellipsoidally bounded fluid into Nuclear Physics has a long history since it has been proposed long ago by Cusson [10] as well as widely studied, classically and quantally, mostly by group theoretical methods, by various authors [ 11-16]. Most of the previous works, however, were lacking a proper balance between a quantal treatment of the collective flow properties on one hand of its coupling to the multitude of intrinsic degrees of freedom on the other hand. At the same time recent experimental advances, in particular with the operation of the new generation of gamma multidetector arrays such as EUROBALL or GAMMASPHERE, have revealed the crucial importance of the latter couplings which provides an incentive to push forward a microscopic study of the intrinsic vortical modes. Recently the concept of a vortical field linear in the coordinates has provided a theoretical frame for a possible explanation [17,18] of two rather surprising experimental facts found with EUROGAM and GAMMASPHERE, namely the AI = 2h staggering [ 19,20] and the identical bands [21 ] phenomena. If confirmed, the former tentative explanation has been shown to yield a nuclear analog of the Aharonov-Bohm phase quantization [22]. The nuclear modes considered here, are described within a rather simple model for the nuclear density. Clearly, one cannot ignore the importance of the existence of a smooth surface for the collective motions under study. As an example of this, let us just recall that, just above, we have alluded to one of its effects when dealing with an analog of the Landau diamagnetism. On the other hand, the well known leptodermous character of the nuclear density may justify a semiquantal treatment of these modes. It is the purpose of this paper to examine the coupling of global rotations with intrinsic vortical modes h la Chandrasekhar (S-type ellipsoids) in nuclei within an approximate semiquantal approach similar in spirit with the widely used routhian description of global rotations. To do so, the content will be organized as follows. The geometrical and kinematical significance of the velocity field of Eq. (12) and its coupling to global rotations will be explicited in Section 2, together with some hints on its possible generalization. Section 3 will present a semiquantal description of the coupling between the global rotation and the uniform intrinsic vortical motion in one particular coupling scheme, the so-called S-type Riemann ellipsoid case. Section 4 will be devoted to a self-consistent semiclassical solution of the corresponding generalized routhian problem and the results of a simple Harmonic Oscillator model will be presented in Sections 5 and 6. A quadratic approximation of the collective energy and some of its consequences will be discussed in Section 7. Finally some conclusions and directions lbr further investigations will be sketched in Section 8.
2. Some geometrical and kinematical considerations In what follows we will assume that the nucleus has an ellipsoidal shape. Let us consider now the velocity field of the uniform rotation of a sphere, defined by an
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angular velocity ~ . Let us scale both the distances and the velocity from the sphere to the considered ellipsoidai shape [5]. It is easy to check that such a field is nothing but the uniform intrinsic vortical fields of Eq. (12). We will now study the flow pattern associated with this field. It is, quite generally, defined by the current lines whose equations are dxl . .
.
HI
dx2 . .
dx3
H2
U3
(13)
The solutions of the above equation are ellipses. They result from the intersection of the plane 3
coigi
(14)
= 7],
i=1
with an ellipsoid homothetical with the boundary surface of equation 3
(15) i=1
where e and 77 are constants and the scaled coordinates are defined as Xi = x i / c i . Upon coupling the uniform vorticity field of Eq. (12) with a global rotation defined by an angular velocity vector g2, it has been shown (see e.g. Ref. [5] ) that the classically stable equilibrium solutions fall into two categories: o~ and g2 are collinear (the so-called S-type ellipsoids); ~ and g2 belong to the same plane of principal axis (the so-called P-type ellipsoids). As already pointed out [ 17], in the context of fastly rotating nuclei the S-type ellipsoids could, Ibr instance, correspond to a situation where the intrinsic and collective rotations are aligned (see Eq. ( 1 2 ) ) , whereas the P-type ellipsoids might rather describe tilted cranking situations studied in Ref. [23]. In what follows, we will limit our discussion to S-type ellipsoids, assuming that ~o and g2 are perpendicular to the (x, y) plane. To simplify the presentation we will also assume here the existence of an axial symmetry around the y-axis (generalizing to triaxial cases however, would be rather straightforward). In such a case, the divergence-free field u~ + Uv has the following components: -
-
Ux=-(s2+co(1/q))y,
u~. = ( s 2 + c o q ) x ,
ur=0,
(16)
where co and s2 are the projection on the z-axis of the corresponding vectors and q = (Cy/Cx) with ca: = cz. It is worth noting that the fields defined by Eq. (16) encompass quite a number of different flow patterns. Calculating as in Eq. (13) the lines of current, one finds the general quadratic curves (y2 + x 2 = r/,
(17)
where :7 is an integration constant and ~: is a dimensionless quantity, characteristic of the motion and given by
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1.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304 ax u 1 +-- -IGx
g=
-@
~ -
ay 2 ax
~
Qy
~
Q]z ]
~ =-
~-- -100
%= -1
~ = +1
~ -
2
l + a ~al'l n-w a× Y~_ g=
0
2
ax 2 ay
'~ = +0~
Fig. 1. Flow patterns corresponding to various values of the dimensionless parameter ( describing the relative importance of global rotation versus intrinsic vortical motion, whose angular velocities are respectively ~Oand to, for a given prolate ellipsoidal shape. In this figure located in the (x,y) plane, J2 and to are aligned on the z-axis. The quantities ax and ay are semi-axis values on the x-axis and y-axis respectively,with ax > ay. (=
l+qh
l+qh
.
(18)
The vorticity of this field, i.e. rot ( u ) , is aligned with the common rotation axis and given by (rot ( u ) ) 3 = (s2 + qco) ( 1 + sc) .
(19)
As seen in Fig. 1 representing the flow pattern in a plane perpendicular to ~o and g2, the cases where ( is infinite or vanishing correspond to shear mode situations (without discontinuity of the velocity). For ~ going from - ~ to 0, the modes go from a shear mode in the plane defined by the collinear angular velocities and the symmetry axis to a mode where the corresponding plane is equatorial, through a purely irrotational mode for ( = - 1. The global or rigid body rotation corresponds to co = 0 or ( = 1. The purely intrinsic vortical mode occurs when 12 = 0 or ( = 1/q2. One may associate to any velocity field u ( r ) , a coordinate transformation upon summing infinitesimal displacements for all fluid elements located around r with respect to the time t, such that 6r = u ( r ) & .
(20)
Whenever u ( r ) is a linear function of some parameters generically called ~i, the infinitesimal displacement fields are thus obtained by replacing these ~i parameters by the infinitesimal quantities 6q~i = ~ i ~ t . In the particular case of the velocity field considered in Eq. ( 1 6 ) , the corresponding coordinate transformation T may be factorized [24] as
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I.N. Mikhailov et aL/Nuclear Physics A 627 (1997) 259-304 T = Rz (O)A(q)iCz ( O ) A - l q ,
(21)
where A ( q ) is the volume conserving scaling transformation from a sphere to the considered ellipsoidal shape defined by the axis ratio q, Rz (O) is the rotation of an angle O = s2t around the z-axis and Rz (O) is the rotation of an angle O = wt around the same axis of the corresponding scaled sphere. The above point (i.e. local) transformation T clearly conserves the intrinsic shape. It is possible to generalize the uniform intrinsic vortical field hypothesis of Eq. (16) in two different directions. One would like, indeed, to be able to deal with shapes differing from ellipsoids and also to deal with intrinsic vortical modes more general than what results from the rotation Rz (O) in the corresponding scaled sphere. There are, a priori, many ways of doing so. For the sake of illustration, we will briefly sketch here one possible approach. The main point is that we have to define divergencefree fields tangential to a given boundary surface. This may be easily formulated in some cases, in terms of curvilinear coordinates (i. Using the notation of Ref. [25] and assuming that the equation of the surface may be expressed in terms of the first curvilinear coordinate ~:j as scj = C ,
(22)
where C is a constant, a possible solution for such a velocity field could be
u(r)
= rot(aj~)
-
a2 0 - hlh3 ~(3
(hl~)
a3 (hl~), hlh2 0~2
(23)
where the three vectors a~ constitute the set of unit orthogonal vectors associated to the chosen curvilinear coordinates, ~ is a scalar function of the position r and the quantities hi a r e scale factors as defined in Ref. [25]. As a particular realization of the above, it is easy to generalize the transformation Rz (O) considered in Eq. (21). Indeed, noting that for a sphere of radius R, the Eq. (22) is simply r = R,
(24)
with the norm r of r playing the role of (l, the velocity field considered in Eq. (23) becomes u ( r ) = ~7~ × r ,
(25)
where ~ is, as above, a scalar function of the position r. Including in Eq. (16) this generic intrinsic vortical field Uv in lieu of the one defined in Eq. ( 1 2 ) , would clearly enrich the coupling scheme under discussion here. Releasing the assumption of an ellipsoidal shape now, would imply to introduce an equivalent of the volume conserving transformation A ( q ) associated with a given family of nuclear shapes.
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267
3. F r o m a classical to a s e m i q u a n t a l description
In this section we will present and discuss a generalization of the Routhian approach widely used for a microscopic description of nuclear rotations [26]. By generalization we mean an extension of this framework to the most general vortical velocity field. However, here we will restrict ourselves to the modes defined in Eq. (16). We will sketch here the main ideas sustaining such an approach leaving a complete discussion of this approach to a forthcoming publication. A discussion within a purely classical framework will serve as an introduction for a semi-quantal description of the considered modes. Let us denote by E(J2, o2) the total energy for equilibrium classical states at given values of the angular velocities /2 and o2. Let us also define two single-valued functions I and J, of .(2 and w: l=l(O,w),
J = J(O,w),
(26)
allowing an inverse representation: £2= 12(I, J ) ,
w = ¢0(1, J ) ,
(27)
and satisfying the following property: 0 ~2= ~ ( E ( O ( l , J ) , o 2 ( I , J )
)) ,
0 o2= ~ ( E( g2(1, J),o2( l , J ) ) ) .
(28)
In classical mechanics Eqs. (28) define the time derivatives of generalized coordinates when 1 and J are the corresponding momenta. In our case, /2 and o2 are the angular velocities measuring the rates of change of the angles O and 0 introduced in the T matrix of Eq. (21) determining the displacements of fluid elements. Thus the quantities satisfying Eqs. (26) may be taken as conjugated momenta of these angles. The energy may thus be written as H ( I , J) = E(s2(I, J ) , co(l, J) ) .
(29)
To find the functions l(s'2, w) and J(s2, o2), we notice that d H ( l , J) = ~0(I, J) dl 4- o2(I, J) d J , dR(O, w) =
l(f2, w ) d O - J ( J 2 ,
w)dw,
(30)
where R(J2, co) is the generalized Routhian (associated to the Lagrangian for the two considered angular variables): R( ~ , w ) = H ( l ( O, co),J( $2, o2) ) - l ( f2, w ) 1 2 - J(12, w ) w .
(3l)
Differentiating Eq. (31) and using Eqs. (32) and 30, one finds a system of two equations:
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I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304 Ol c~J OE 0-an + (0T6 = gfi '
O ~I- + o ~ -a-J= - - .0E c~(0 &o a(0
(32)
The solution to Eqs. (32) may be given in an analytical form by expanding the collective energy in powers of the angular velocities: E( 0, (0) = ~
Im .,~ ,m2om~ °°"12 •
(33)
nil ,m2
Expanding also the functions I ( O , (0) and J ( O , (0) as I ( ~2, (0) = ~
• ~,'2.vtl i (O/712 t,,,,,m2
II11,1112
" ~(~nq Jm~,,,2
J ( O, oo ) =
O)"112
(34)
,
1111,m2
and equating the terms with the same power o f / 2 and (0 in Eqs. (32), one obtains the following equations for the coefficients imp,m2 and jml,m2: mliml,m2 + (ml + l )jm,+l,m2-1 = ( m l + l)Imml+l,m2 .
(m2 + 1)in,t-l.m2+1 + m2jm,,m2 = (m2 + l)Imm,..,2+l •
(35)
Upon taking into account the classical time-odd character of both I and J. Eq. (35) yields 1 ( 0, (0) = 2Im 2,0s2 + Im
4
3
2
2
1
3
(0 -}- 5Im 4,0 a'2 -'}-Im 3,1 02o0 + ~Im 2 , 2 0 ( 0 + 5Im 1,3(0 -- • • 2 2 + .{Im 3,1 ° - + . . J(O,(0) = 2 I m o . 2 ( . o + I m l , l O + 4Im0,4(03 + I m l , 3 ~ o 2 0 + ~Im2,2(00 1,1
(36) The Hamiltonian in Eq. (29) does not depend on the angular coordinates O and O7 and consequently the conjugated momenta I and J are conserved during the motion: {I,H} = {J,H} =0.
(37)
Therefore the evolution of the nucleus consists merely in a time-change of the angular variables governed by the following equations of motion: O={O,H}
= o,
0 = {O, H} = (0.
(38)
This situation changes when the nucleus is perturbed b y a quadrupole electric field. The latter adds to the Hamiltonian H(I, J) defined in Eq. ( 2 9 ) a perturbative term: /'/pert = Z aijQ(j, ij
(39)
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where Qij is the usual inertia tensor (for the charge distribution) whose rotationally irreducible part is proportional to the electric quadrupole moment of the nucleus. In the presence of such a perturbation one obtains [ = {1, npert}
=
- -
Z ij
"~ij oOi)
,9(9
(40)
The perturbative part of the Hamiltonian does not depend on the angle O, and thus the conservation of the quantity J is not affected by the interaction of the nucleus with the electric field (in the lowest order approximation on r / a , where ,t is the electromagnetic wavelength): J = {J, Hpert} = 0 .
(41)
The last equation formulates a very important result, showing indeed that the quadrupole electric interaction does not change the quantity J. The routhian defined in Eq. (31) may be regarded as the hamiltonian resulting from a transformation into a coordinate frame moving with fluid elements [27], as we will discuss below. The reason for considering such a coordinate transformation is indeed, to introduce a reference frame moving with respect to the laboratory frame according to some velocity pattern. This allows us to build a family of solutions exhibiting the type of collective currents which we want to study. Let us proceed now towards a quantal description from what we may learn from classical mechanics. In this case, performing a change of coordinates, one should add a so-called generating function G to the Hamiltonian, so as to preserve the mathematical structure of the Hamilton equations of motion [27]. In the particular case of a local transformation (i.e. where the new coordinates do not depend on the momenta) this generating function G writes as G = a .p,
(42)
where ~e is a (local) vector field which is equal to minus the collective velocity associated with the considered transformation, as for instance those discussed in Section 2. In quantum mechanics an equivalent of the above considered canonical transformation, may be obtained by applying to the wavefunction of the system a unitary transformation. Restricting the discussion here to the Hartree-Fock approximation, a solution p of a TDHF equation is transformed into /5 by the so-called Thouless unitary transformation defined by a hermitian matrix S as /5 = eiS p e - i s .
(43)
Defining the Hartree-Fock Hamiltonian ,~ as obtained with/5 from the two-body Hamiltonian, one gets the following transformed TDHF equation [[~ - S , / 5 ]
= ih~,
where the "cranking" potential S writes
(44)
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I.N. Mikhailov et aL/Nuclear Physics A 627 (1997) 259-304 S = ihe i s d (e -is) •
(45)
We consider now the particular case where the time dependence of S may be factorized out; - the hermitian operator S depends linearly on the momentum operator p. One has thus -
S =/3(r.p
+ p. r)/2,
(46)
where/3 is a real function of the time and F is a (constant in time) local vector field. Both restrictions allow one to consider e is a s a quantal analog of a local transformation associated with a velocity field depending linearly on some generalized velocity parameter as considered in Section 2. This analogy is substantiated by the fact that the transformed operator of r does not imply in this case the momentum operator p and also by the fact that the Wigner transform of S writes Sw = / ~ / F - p .
(47)
If one wants to approximately describe in a microscopic fashion, the quantal analog of a motion whose classical analog is a transtbrmation associated with the velocity field oe = - / ~ F , one may solve the constrained TDHF equation (44) with the cranking potential S defined in Eq. (47). The stationary solutions of the TDHF equations of the type of Eq. (44) amount therefore to solving the generalized Routhian stationary problem 8(H+G)
=0,
(48)
where G is the quantal generalization of what has been defined in Eq. (42). In the case considered here, one takes for the vector field ~ minus the collective velocity of Eq. (16). Its solutions are thus functions of the two angular velocities g2 and o~. This holds in particular for the energy. This quantity depends explicitly on the two angular velocities as a quadratic function, but also implicitly through the coupling of the one-body potential coming from the generating function with the twobody hamiltonian in the Hartree-Fock self-consistent solution process (e.g. through deformation adjustments). In this context, it is worth emphasizing a consequence of the vanishing contribution to the time-derivative of the diagonal part p of the density matrix from the intrinsic vortical current, as demonstrated above from Eq. (11). As a result, the modes associated to the field uv should not contribute to the definition of the equilibrium shape, at least in the sharp-edged density limit where this result has been obtained. In realistic leptodermous cases, their influence on the deformation should be small. One therefore expects the w-dependence of the energy to be a priori closer to a quadratic expression, than it is the case for the O-dependence. We will now, describe briefly some properties of the quantization of the energy in Eq. (29). Our present discussion will be mostly qualitative. However, this subject has
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
271
been treated in great details in Refs. [ 12,13,16], for instance. First of al, the fact that the mass parameter entering the purely kinematical energy of Eq. (29) does not imply the associated variables (the angles O and O) frees us from any ambiguity in the quantization process. As usual, a quantal description of the collective motion involved in such a model may be obtained by substituting the Poisson bracket relations by the commutator relations to yield h [Z,o] = [ L O ] = _ , 1
[Lo] = [),o] =o, [ [, J] = 0.
(49)
Substituting the classical momenta I and J by their quantal counterparts [ and ] in the energy (29) one obtains the quantal analogue of the classical model for S-type Riemann ellipsoids. The conservation of the classical collective momenta [ and J established in Eq. (37) yields the following commutation relations. [ B , [ ] = [/4, J ] 0 .
(50)
Taking into account the last commutation relation in Eqs. (49), one sees that the quantal states so obtained may be labelled by the eigenvalues of the operators squared which we shall specify by I and J (with eigenvalues I(1 + h) and J ( J + h) as usual). Finally, from the first commutation relation in Eqs. (49), one obtains the standard commutation relation boy [1, f ( O ) ] - i 30
(51)
showing that the operator f is to be interpreted as the z-projection of the orbital angular momentum operator of the nucleus. The corresponding eigenvalue problem may thus be formulated as the periodicity condition of the wavefunction. ~ ( I , J) ~ exp ( i ( l O + JO) )
(52)
at the endpoints of the definition interval of the angle variables O and O. This interval depends on the symmetry of the mean field. Let us consider here the case of even-even nuclei (the generalization to other nuclei has been considered in Ref. [18]). In our ellipsoidal case, the mean field is C2-symmetrical (i.e. with respect to the rotation by an angle 7r around the collective rotation axis). Thus, neither of the two relations Rz (Tr) and Rz (Tr) in Eq. (21) changes the physical state of the system. Thus the periodicity condition is formulated in this case as: exp(ilTr) = exp(iJTr) = 1 ,
(53)
from where it follows that both 1 and J quantum numbers are even. We notice that the commutation relations (50) constitute a part of the commutator algebra of the so-called O ( N - 1)-invariant model [ 16]. In the latter, the J-quantum number is unequivocally
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
272
defined as a quantum number of an intrinsic state. Having this in mind, one may call the present approach as a spontaneously broken O ( N - 1) symmetry model.
4. A semiclassical approximation In this section we will give close tbrm expressions for various quantities related to the solutions of the generalized routhian or cranking variational problem of Eq. (48), in a semiclassical limit. By semiclassical we mean here an approximation h la Wigner [28] corresponding to a truncated h expansion (up to second order terms). More specifically we will work within the so-called Extended Thomas-Fermi approach [29]. It yields functional relations between various local densities carrying informations on the nonlocal part of the density matrix and its diagonal (in r) part, the local density function p(r). These densities are the kinetic energy densities ~-o(r), the so-called spin-orbit densities J,,(r), the current densities j r ( r ) , the spin vector densities pv(r), where v stands for the considered charge state. They are sufficient, when using a Skyrme type effective force, to express the expectation value of the energy of any Slater determinant as an integral over the whole space of an energy density H in the lab frame. (see for instance, Ref. [30] where one may also find the exact definitions of these densities and of 7-{). General expressions for such functional relations have been explicitly derived in Ref. [31 ], in the case of a routhian approach restricted to the global rotation. In another paper [32], they have been recently extended to the most general cranking field, or generating function, linear in p, as in Eq. (42). Therefore we will merely sketch here the results obtained for the mass parameters in the general case and apply them to the particular vortical velocity field uR 4- uv of Eq. (16), which is the quantity ol = - / ~ / ' , where F and/3 have been defined in Eq. (46) and the vector potential a in Eq. (42). Let us first ignore the spin degrees of freedom. We will decompose the integral expression of the Slater determinant energy, when using a Skyrme effective force. Namely, to obtain the generalized inertia parameters, one identifies 7~z, the part of which 7-/ is quadratic in or. As a matter of fact, one may follow exactly the same path of calculations as in Section 3.1 of Ref. [31] to obtain
(54) u
where the quantity 6(2)(rv) is the second order correction in h of r,,(r) and f,, is the usual effective mass form factor associated with a Skyrme effective force. One notices that the first (Thomas-Fermi) term corresponds, as expected, to the classical collective energy. We have shown in Ref. [32] that
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
6(2)('/'v) = (37.r2)-2/3m2
{
1/3 [(rOt~)2 [-- ~
273
~ 3 I } ~ . (rotrotae) .
Pv"
(55)
2h 2
+Z. [rot × V( vp!./3)l Jv t
Let us assume now that the constraining field o~ results from several external fields a = ~-~' a i .
(56)
i
As a trivial consequence of the quadratic character of ~2 in a, one may write (with an obvious notation and i, j, labelling the various modes) 7~2 = ~ 2 ( / )
+ ~2(i,j).
i
(57)
i
The uncoupled terms are readily obtained upon replacing a by ai in Eqs. (54), (55). The coupling terms are given by ( rnp~ai . a j + B']-{2(v; i , j ) ) .
7~2( i, j ) = Z
(58)
q
with (3~ 2) -2/3 m b'7-C2(v; i , j ) =
4 [ fvpJo/3 (rot ~i rot aej - ~fi rot rot a j - aei • rot rot ~lfi) "
"
+ V ( f v p l v / 3 ) • ( a i . r o t a j + a j • rot ~i) .
(59)
One thus gets inertia parameters lot the coupling of several modes whenever o~ results from their addition, upon integrating over r the 7{2(i) terms and dividing the results by fl~/2 where/3i is the collective mode velocity (for the coupling terms one would of course divide instead the result of the integral of 7J2 (i, j ) by/~i/3j). When taking the spin degrees of freedom into account, one should evaluate first the spin-vector densities p , , ( r ) (as defined for instance in Eq. ( [ 11] ) of Ref. [31] ). They are solutions of the following linear system of equations p,,
l+~p,,
(X+Y)
+p~,\T~,
r,,,
-
+Z,,rota~
, (60)
where ~, stands for the other charge state with respect to the one defined by v. In Eq. (60) also, the constants X, Y, Z,, and /_t are defined from Skyrme force parameters and from the densities Pv (note that p = ~-~,,p,,), as: X=Bio + peBl2, Y=BI1 + pYBj3, Zt,
=--
mB9 .
h2 ( P + P~,),
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
274
/z =
6m ( 37re) - 2/3 h2 ,
(61)
(see e.g. Ref. [30] for the definition of the exponent y of the density dependence of the force and of the Bi coefficients). In Eq. (60), the term in g2 comes from the global rotation in the lab frame which has been assumed, so that a spinconstraining field -S2. s has been added to ¢e • p. In our case where ~ is aligned on rotce spin densities for neutron and protons are also aligned on g2. Including the spin degrees of freedom, one gets a supplementary part (7-/2)s into 7-[2 namely: h
(']-{2) s = - - ~
Z Z,,rot a
(62)
'P,,
1)
exhibiting a coupling term between space and spin degrees of freedom. Let us now apply the previous formalism to our S-type ellipsoidal modes. We will ignore from now on the spin degrees of freedom. It is worth noting that, by construction 7"(2 being quadratic in the aei's, is also quadratic in $2 and ¢o, so that one can define the collective rotational energy as E = ~i A w 2 + B w a Q + 1 Cf22 .
(63)
Of course, due to the coupling in the variational Eq. (48) between the time-odd part of the density matrix and its time-even part, the latter and thus the non-collective energy is also dependent on ,(2 and w, as well as the moments of inertia A, B and C. The density will be schematically described as being constant and bounded by a sharp edged ellipsoid whose semi axes are given by a,. = aoq 2/3 ,
(64)
ax = a~.. = aoq -U3 ,
where a0 is given in terms of the total number of particles N and of the usual size parameter r0, by a0 = ro Nj/3, whereas q is now the ratio of ax and ay. We will also use below the following unit for the moments of inertia 2 ~,2 Ar5/'3~l/3 r / = ~ .... 0'" '~ ,
(65)
which is nothing but the rigid body moment of inertia tbr a sphere multiplied by the shape factor ql/3. The various inertia parameters are given below to order h 2 (the first terms being the Thomas-Fermi estimates): A=~70[1-(D)
oI,
B='r/[l
(D)
o 1,
C=T/O[I-(ff-~)
l , (66)
where O is a geometrical factor given by O = 1/2(q + l / q ) and the dimensionless semiclassical expansion factor D/~7 is given for this mode by D=-(37r2)-2/3mZ//./fv(r)pl//3(r)d3r, tl
(67)
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
275
which is approximated within our simplified model for the density (assuming furthermore that p,, = pp) in terms of the isoscalar effective mass in nuclear matter (m*/m)NM of the Skyrme force by -D- =
( 8 "~2/3 5
(
i
~~m*m / )~MN-23/).
(68)
n
It is important to note that its inverse surface dependence ( N -2/3) makes D / ~ small with respect to 1 in heavy nuclei. In our case as in the classical case (see Ref. [ 13] ), the kinematic moment of inertia of S-type Riemann ellipsoids is given, in terms of the rigid body and irrotational moments of inertia, with the notation used previously in Section 1, by -
I = 0 = rORB q- ( 1 -- r)OlR , /2
(69)
-
where w C r= 1 + /2B'
B2 OIR=C
C '
ORB
C.
(70)
Assuming for instance r = 0.75, as it has been fitted by Swiatecki [7] for the first discovered super deformed band in lS2Dy and q to be equal to two, one gets a ratio c o / ~ of about - 2 0 % . Of course, the previous estimate relies on the assumption that the reduction of the moment of inertia from its rigid body value can indeed be all explained in terms of an intrinsic vortical motion.
5. A s i m p l e m o d e l case: the h a r m o n i c oscillator m e a n field
In this section, we will illustrate the above discussed semiquantal approach, by considering its application in a (non-self-consistent) simple model where the mean field is described by a harmonic oscillator potential. In spirit, the following is very close to what has been done in Ref. [ 14] by Rosensteel, excepted for two important points. Namely, we take here into account properly the volume conservation as well as the quantal character of J. In the following, we will assume that the two rotation axes (for global and intrinsic rotations) are collinear with the x-axis. The one-body generalized Routhian h(g2, w) whose eigenfunctions will be used to build the requested Slater determinant solution is written as: p2 m h( g2, w ) = ~m + -2 (wxx2 + wyy2 + w z z - ) - g2L,. - w J x ,
(71)
where Lx and Jx are the first components of the orbital angular momentum L and of the operator J introduced in Section 3, also called, in Ref. [14] for instance, the Kelvin circulation. In this particular case, as it may be easily interred from our previous presentation of the intrinsic vortical mode, the Kelvin circulation is nothing but the
276
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
operator L after a double stretching in positions and momenta. One has thus, in the r-representation J,=-ih
qY~z
q-Jz
(72)
,
with q = c / b , where the semi-axis of the ellipsoid are called a, b and c respectively. We express now h(.O, 0)) in terms of the usual boson operators ( a / , a i ) as i=3
h0)i(a+ai + 1/2) - i h S ( a ] a 3 - a~a2) - i h D ( a ~ a f
h(.O, 0)) = ~
- a2a3),
(73)
i=1
with 8---
D =
.O(0)3 -}- 0)2) ÷ c0(q0)3 + 1 (1/q)0)2) 2 WV/~7~ ~2( 0)3 - 0)2) ÷ 0)(q0)3 - ( I / q ) 0 ) 2 )
(74)
2 vmT 3
Since this operator depends quadratically on the creation and annihilation operators (a +, ai), one can perform a Bogoliubov transformation restricted to the boson operators in the (y, z) plane i=3 + % =
Vk = 2, 3;
~-~/&i a+ + #kiai),
(75)
i=2
such that one has Vk = 2,3;
~+k = [h(.O, 0)),~k],+
(76)
which corresponds to an homogeneous system of linear equations such that o92 -- -ok -iS 0 iD iS w3 - -ok iD 0 0 -iD 0) 2 + .Ok iS -iD 0 -iS 0)3 + .Ok
ak2 "~ ,~k3 I
(77)
]'Zk2 ]
IZk3 /
This system admits non-trivial solutions only when the determinant of the above matrix is vanishing. This gives a second order equation in -O2 yielding thus two positive eigenfrequencies -O2 and ,'23. The former is given by -
-
+ 2
"
+ ($2 -- D 2 ) + A2,
(78)
where the quantity A2 is given as =
~
÷ ( S 2 -- D 2 ) ( 0 ) 2 ÷ w 3 ) -r- 20)20)3(S 2 ÷ D 2)
.
(79)
The eigenfrequency -O3, is deduced from the above eigenfrequency g22 upon replacing A2 by - A 2 in Eq. (78).
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
277
Such a transformation allows us to write h(s2, w) in a canonical form i=3
i=2
Some further details about this transformation are given in Appendix A. In what follows we will calculate the equilibrium solution for a given harmonic oscillator configuration of N particles (i.e. a set of spin, isospin and eigenvalues of the particle number operator ni in each directions, for each fermion) at a given value of I and J (dimensioned as h) as it will be defined below. There are five variables in the problem: the three harmonic oscillator frequencies w i and the two angular velocities 12 and w. For each set of these variables and for a given configuration we may calculate trivially the value of the total generalized routhian, and evaluate (see Appendix B), the expectation values of L x and J~, from which one deduces readily the corresponding energy. We can also calculate the three expectation values ( x ~ / N } whose positive square roots are identified with the three characteristic lengths a, b and c. Some details about the analytical expressions for these expectation values are given in Appendix B. We are thus left with a variational problem implying the above listed five variables which are subject to the following three constraints: the volume conservation enforced in a quite natural way whatever the values of the deformations and of the angular velocities g2 and w by imposing -
47"r
--abc
3
-
(V being a constant fixed once for all by scale arguments or by the actual radius value if known for the considered nucleus) ; the value of the angular momentum given (for small values of the projection K of the total angular momentum on the symmetry axis, as expected for the rotationally aligned superdeformed states, and at any rate only in average) by (Lx} = ~ +
-
(81)
= V,
h) ;
(82)
the value of the Kelvin circulation quantum number again given in average by (J~> =
v/J(J + h).
(83)
In the absence of any time-odd constraint, the question of volume conservation in the harmonic oscillator model is readily seen [3], to be expressed in terms of a constant w0 (fitted so as to provide a good scale) as o91o92o93 = o93 .
(84)
The volume conservation condition has attracted a lot of attention when used in usual Routhian-type calculations [33,34]. Basically two prescriptions have been used. The first is in use here (see Eq. (81)), another results merely from taking into account in the routhian case the above Eq. (84). This has been used for instance in Ref. [14]. It
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
278
has been shown in Ref. [34] that some attention should be paid when using a Lagrange multiplier technique to enforce the volume conservation. It has to be stressed that we are not concerned here by this problem since we do not use Lagrange multiplier to guarantee the volume conservation, but rather perform the routhian minimization within a space of volume-conserved (according to our prescription) density matrices, as we will specify below. Indeed, to provide a solution of our variational problem for a given harmonic oscillator configuration, we proceed for each quantized value of I and J in the following iterative way. Starting from some fixed values s2~n) and w ~') of the angular velocities, we minimize the generalized routhian in the two dimensional space of triaxial shapes defined by a, b and c, upon taking into account the volume conservation condition of Eq. (81 ). Having thus fully specified the variational solution, we compute (see Appendix B) the expectation values 1 (n) and j(n) of Lx and Jx at this stage of the iteration, from which we infer by the gradient method new values s2~n+l~ and w ~n+j) for the angular velocities. We then re-iterate the above process until a convergence is reached (namely whenever both (I - 1 ~')) and ( J - j~n)) falls below 10-sh). As a result, the Lagrange multiplier associated with the quantity (Lx) which is conserved during the variation, is undoubtedly the angular velocity corresponding to the global rotation from the lab system to the body-fixed system, in the usual routhian case. This conclusion may be generalized to the coupling case between the two vortical modes as considered here. One may further note that it seems hopeless in such a coupled mode, to yield an analytical expression for the variational condition as it was the case for the usual routhian. This is due to the rather involved character of the expectation value (Jx) since the Kelvin circulation operator Jx, itself, depends on the variational quantities (wj, w2, w3, w, s2) through the b and c deformation parameters in quite a complicated fashion (see Appendix B). We still have to give the rationale behind the choice which we have made of the prescription for the volume conservation given in Eq. (81). Indeed this equation is a constraint on the nuclear density which is grounded on the high value (relatively to similar quantities for other bulk or surface modes) of the incompressibility modulus in nuclear matter. Eq. (84) corresponds to the conservation of the volume enclosed by some equipotential surfaces, more specifically as far as that part of the potential which is momentum-independent is concerned. While such a choice entails clearly substantial simplifications in the numerical treatment, its physical motivation is not completely clear to us. It is worth noting that the expectation values of the squared components of r and of p, as well as those of Lx and Yx, depend exclusively on the harmonic oscillator configuration through the following sums (see Appendix B)
j= 1
where the sum runs over all the N occupied fermion states labelled by j and n~j) is the
279
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
number of quanta in the /-direction for the j-fermion. The moments of inertia A, B and C, are clearly very important quantities to determine out of our microscopic collective model. In general they can be obtained by making a local quadratic fit of the collective energy as a function of the moments I and J, according to the following formula which will be justified below in Section 6 E ( I , J) - ~A C
B
12 - B I J +
j2
(86)
,
or as a function of the angular velocities s2 and 0) (see Eq. (63)). Equivalently one may deduce A, B and C from the numerical calculation of second order derivatives of the energy with respect to the 1 and J, or J2 and w, variables. Approximate moments may also be evaluated within the Inglis cranking formula [35] which may be viewed as a perturbative approach in two different ways. In general it corresponds to neglecting the effect of the time-odd part of the density matrix (or equivalently of the currents) on the Hartree-Fock mean field, thus ignoring the socalled Thouless-Valatin corrective terms [36]. Clearly such time-odd potentials are not taken into account anyway in such phenomenological mean field calculations. On the other hand, this approach neglects also second and higher order effects, in the angular velocities, affecting the time-even part of the local density, as e,g. rotation induced stretching deformations. Indeed it provides the linear response of the time-even part of the variational solutions at given values of the angular velocities ~ and 0), keeping it fixed under the action of the time-odd constraining field. We will label these moments as Aio BIC and Clo Their analytical expressions are given in Appendix C. numerically. Around vanishing values of the two angular velocities, the resulting limits of the moments are labelled A°o B°c and Cl°c. They must of course be identical to the values A, B and C yielded by the fit in that angular velocity domain. Extending what is known [37] in the pure global rotation case to our coupling case, one finds (see Appendix C) for A°c (upon assuming that the single particle states are such that the ensembles of the number of quanta in each direction varies from 0 to a maximum value without gaps) ]Z
A°c
-
20)20)3
[(0)2 -- 0)3) 2 (~'2 -~- ~'3) -- (0)2 q- 093)2 (,--~3 -- --~2)] L O92 -~- 0)3 O92 -- 0)3 J
(87)
m
Similarly one gets for C°c cO_
h 20)20)3
- b
_ c
2
(c0)2 ~0)3) (,--~2 -}- ,--~3) -~ 0)2 "+" 0)3
( 0)2 +
0)3) 2
]
( 2 3 -- ~--~2) •
O92 -- (03
(88)
Finally the coupling moment B°c is written as h B°lC- 20)20)3 %
X
(~0)2-
~0)3)(0920)2 + o93
0)3) (,~2 -~- ~"3) q-
0)2 +
0)3) (0)2 + 0)3) 0)2 -- 0)3
(,-~3 -- 2 2 ) ]
-
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
280
(89) In the limit of vanishing values of the angular velocities 82 and w, one searches for the linear response of the static solution which is well known to be determined by the self-consistency relation (90)
~-~l¢01 -----~ ' 2 W 2 = ~ ' 3 0 ) 3 ,
and the following relation for the semi-axis value a
a2 = 5h
(91)
~'1,
m¢ol and similar values for b and c. One deduces from the above that in this limiting case b o~3 . . . .
~.
C
-~3
~'2
402
(92)
Using these results one finds for the moments Ai°o B°c and C°c A°c=CI°c=~z
~2-2+~7
,
B°c=~
~22+~33
.
(93)
Finally one deduces in particular that B°c
co
- -
-
2
q+ l/q
- O,
(94)
which is exactly the Thomas-Fermi result found in the previous section (see Eq. (66)). It may be noted that the equivalence of Inglis cranking masses and rigid body masses upon making use of the self-consistency condition, was known already in the pure global rotation case [3].
6. Discussion
of the results
of some
illustrative
calculations
In what follows, we will discuss the results obtained for the nucleus defined by Z = 80 and N = 110, i.e. for 19°Hg nucleus. Spin degrees of freedom are present to double the available space. Yet, within this simple model, they are not active in the dynamics. The parity and x-signature operators do commute with the generalized routhian R. As a matter of fact, one should in principle compute expectation values of R not for single particle states (using a straightforward standard notation) such as
li~vi) = Z
x":i nAInznA)l~vi)'
(95)
IT:nA
but rather for the good signature states i
lisi) =
O-n~na~X"-nA (In2na) lsi) + si(- )n=tnzn _ A) I _
~i))
(96)
281
LN. Mikhailov et aL/Nuclear Physics A 627 (1997) 259-304
Table 1 Definition at zero spin of the retained configurations. They correspond to three very different deformation regions (weak deformation, WD; super-deformation, SD; hyperdeformation, HD). The quantites ~ are defined in the text together with the three harmonic oscillator frequencies hwl (in MeV) and the three semi-axes (in fm)
WD SD HD
~n ~1
~"P ~1
~n ~2
~P ~2
~--'n ~3
~"P ~3
177 155 139
116 100 94
177 155 123
116 100 76
227 295 421
148 200 286
h,'~o I
ht02
]ZO)3
a
b
c
7 . 7 4 2 7 . 7 4 2 6.049 6.427 6.427 8.225 8.895 8.895 4.582 5.594 5.594 10.858 9.854 11.538 3.248 5.080 4.339 15.414
(si denoting the signature of the /-state). However, since the spin-up and spin-down components of a single particle state should not couple in our simple model (no explicit spin dependence in R), this choice does not matter since one obviously gets (iXilRliXi) = (isilRlisi) .
(97)
We have chosen to study three different deformation regions corresponding to weakly deformed ( W D ) shapes with q ~ 1.32, super-deformed ( S D ) shapes with q ~ 1.94 and hyper-deformed ( H D ) shapes with q ~ 3.55. For that purpose we have selected three configurations which minimizes the harmonic oscillator hamiltonian for vanishing angular velocities and satisfy the self-consistency condition (90), whose characteristics ~ ' , ~oi, a, b, c ( i = 1, 2, 3; v = n , p ) are given in Table 1 It is to be noted that by no means the selected solutions are the only possible ones. For instance between the values of q corresponding to the W D and SD solutions, there are three other prolate self-consistent solutions (with the following values of the q parameter: 1.36, 1.55, 1.78). The HD solution is significantly triaxial. It has been retained due to its resulting value for the ratio of the two moments of inertia B / C for reasons which will appear clearly later. Of course this arbitrary choice is merely made to provide a sampling of rather different type of solutions. Anyway, we are not intending to achieve completeness in such a crude model but rather to get qualitative trends. We then describe states with non-vanishing values of 1 and J, in particular on the so-called yrast line (lowest energy state with respect to J for a fixed value o f 1) by choosing as above described the appropriate angular velocity values. In this process, we may have to change the configuration to keep the total routhian at its minimum. In such a case, there is some ambiguity in choosing the configuration which will replace at spin 1 + 2h the configuration retained at spin 1. In this work, we have adopted at spin 1 ÷ 2h, the configuration that minimizes the routhian R when keeping the deformation parameters at the values they had at spin 1. However, the fully variational values of the parameters for the new configuration will be in general different from the previous ones. Then for this new deformation one should check that such a configuration is optimal and if not change again the configuration accordingly. While the configuration of the HD and SD solutions remain stable within the range of calculated /-values (from 0 to 80h), it is changing very much tbr the W D solutions when I > 40h. The retained configurations for the W D solution are listed in Table 2. The resulting values o f q = c / b
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
282
Table 2 Definition as a function of the spin of the retained WD configurations by their corresponding quantites ~'~' defined in the text 1/h
~n ~1
='p ~1
_--n ~2
~p ~2
~n ~3
~p ~3
0-40 42-46 48-56 58-60 62-72 74-80
185 189 189 189 183 185
ll8 ll8 124 122 122 120
169 161 157 157 151 145
114 ll4 106 100 100 96
227 233 239 239 257 265
148 148 152 164 164 174
~3.6
]
" HD F 3.2 ~ .J!
SD
1.8
/' /,,
1.4
WD
1.2
HD
1.1
WD
/
/
s
F
.'
s
/
!
/''
' SD 0
20
40
60
80
1/14 Fig. 2. Equilibrium deformation parameters along the yrast line as functions of the total angular mmnentum 1 for each of the three retained configurations (WD, SD, HD). Yrast is understood here and in the next figures, in a restricted sense, i.e. as the solution having the lowest energy at a given spin for a given configuration. The parameters q and q~ are defined in terms of the semi-axes a, b and c as q = c/b and q~ = a/b. a n d q / = b / a a l o n g t h e yrast line, are d i s p l a y e d , as f u n c t i o n s o f I, in Fig. 2. In Fig. 3 the y r a s t values o f the K e l v i n c i r c u l a t i o n J as a f u n c t i o n o f 1, d e n o t e d Jyrast(1), are s h o w n for the t h r e e s t u d i e d s o l u t i o n s . F o r the W D s o l u t i o n , o n e sees t h a t J ~ I w i t h o n e a c c i d e n t a r o u n d 1 = 2 0 h . A b o v e I ~ 50h, the s o m e w h a t r e g u l a r p a t t e r n o f Jyrast(1) is b l u r r e d by the p r e v i o u s l y d i s c u s s e d c h a n g e s o f c o n f i g u r a t i o n s . B e l o w this spin value, the o b s e r v e d b e h a v i o u r o f Jyrast(1) m a y be h i n t e d to r e s u l t f r o m its d e p e n d e n c e o n I, w h i c h w o u l d b e o f the type Jyrast(1) ~ [ x l ] ,
(98)
l.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
60
20
WD
I
283
~,~*
°°
.~
i
j ~
20
.~,'"
,
.o.~ •
40
i !
HD
.f 20
o
20
40
60
80
I/kl Fig. 3. Quantal yrast values of the Kelvin circulation as functions of the total angular momentum 1 for each of the three retained configurations (WD, SD, HD). where x is a real number slightly smaller than I and [y] stands for the integer part of y. This explanation will be further substantiated in the next section. The curves Jyrast (1) for the SD and HD solutions might be explained in a rather similar fashion, with a x-value, however, which is much smaller than 1. In the HD case, one finds a remarkable periodic dependence whose period would be very close to 4h, thus consistent with x ~ 1/2. The curves E ( 1 ) giving the total energy at a given spin value 1 for the three considered solutions are plotted on Fig. 4. From one deformation to another, they obviously exhibit different curvatures yt related to the trivial deformation dependence of the moments of inertia O, namely,
w h e r e / 3 denotes the usual axial quadrupole deformation parameter. It is also to be noted that the HD solution is very far indeed from the yrast line and is only considered here merely for the purpose of varying widely the deformation. Let us discuss now the spectra in both 1 and J associated with one configuration, e.g. the yrast HD configuration which is, as we have seen unambiguously defined. As shown in Fig. 5, the states sharing the same J-value fit very nicely into parabolic curves whose minima are located at higher and h i g h e r / - v a l u e s upon increasing J. This feature displayed in Fig. 5 at rather low values of 1 and J, is also present for larger values of the two moments as shown in Fig. 6. This seems to be a rather general phenomenon and is
284
1.N. M i k h a i l o v
et al./Nuclear
Physics
A 627 (1997)
259-304
120
HD ~o /
/
/
WD 40 ¸
•
y/l•
.-/ ~
SD
0 0
. . . . . . . . 2 .0 . . . .
4 ~0
. . . . . . . 6 .0 .
8'0
1/14 Fig. 4. Total energy curves as functions of the total angular momentum 1 for each of the three retained configurations (WD,SD,HD).
400
j/o/ 200
J = 8 ¢t
o
fz :'
J=61~
ii 2
J = 4 ]~ 0 [
t 4
1 6
o o
• 10
[
12
114 1/14
Fig. 5. Total energy curves corresponding to the HD configuration and given values of the Kelvin circulation as functions of the total angular momentum 1 in the 4-8 h range.
found also in light nuclei. In Fig. 7 we have plotted similar energy curves for the ground state configuration of ZONe, making contact thus with the work of Rosensteel [ 14] with the important difference, as already noted, of limiting ourselves to even values of the Kelvin circulation. Let us turn now to the discussion of inertia parameters A, B and C. For the three considered deformations we have made local quadratic fits of the energy yielding thus these parameters around the yrast line. The fits whose results are displayed in Figs. 8-10, have been made either assuming an energy quadratic in the moments (see Eq. ( 8 6 ) ) or in the angular velocities (see Eq. ( 6 3 ) ) . Both fits have been performed within intervals of the relevant quantities which are much smaller than what is needed to vary 1 or J by 2h. They both give exactly the same values for the inertia parameters. In Figs. 8-10 the Inglis cranking values (see Eqs. (87), (88), (89)) are also reported. As expected
285
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
,
i
".'. ,.
/'
.'
,' .;
'. ,
,' ".,
//
/ /'
,
,,'
.;
,
('
"'~\
/ \ J = 3 4 1~
-5
62
/' \ J=36 ~
/
/
J = 3 8 ]'l
J = 4 0 k~
68
74
I
80
Fig. 6. Same as Fig. 5 at higher/-values (namely in the 34-40 h range). The plotted energy results from the substraction of a continuous approximation of the yrast energy obtained by releasing the restriction to (even) integer values for J/h.
40l j = 6 k~ ~
/
/
J = 2 1{
.-I
.....
0
2
4
'6 l/lt
Fig. 7. Same as Fig. 5 for the 2°Ne nucleus in the 2-6 h range for the Kelvin circulation J. plotted energies are defined by difference with the ground state (1 = J = 0) energy.
they exactly coincide with the fitted values for vanishing angular velocities. One gets thus for A, B and C the relations previously discussed in Eqs. (93), (94) for A°¢, B°¢ and C°c. For finite values of S2 and co now, the linear response estimates ~ la Inglis, i.e. neglecting the rearrangement of the time-even part of the density matrix, are also plotted. One observes for the fitted values of A, B and C, a slight angular momentum dependence which is of course not to be understood as a quadrupole deformation effect (since it is constant with I in cases where the configuration is kept identical, see Fig. 2), for which we have not found yet any obvious explanation. However, the ratio B/C or B/A are found to be much more stable. For the HD solution for instance, one finds indeed that, when I increases from 0 to 80h, B/A does not change from its zero spin value of 0.42 up to some tenth of one per cent and B/C does not vary from the same value
286
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304 "7,
IO0
~.....//"~-./ "~ 5O
125
~
~"-~_~
230 [
"~.~
.
0
.
.
2o
.
.
.
.
4o
.
.
]
-
.
.
.
60
80
I/I~ Fig. 8. Inertia parameters A (in h2-MeV-1 units) as functions of the total angular momentum 1 along the yrast line for each of the three retained configurations (WD, SD, HD). The dots result from a quadratic fit of the collective kinetic energy. The other curves correspond to the Inglis cranking formula values defined in the text. At the limit of vanishing angular momenta, the two curves coincide and are given by the Eq. (93). These asymptotic values are represented as horizontal lines. more than by some percent. Kinetic moments of inertia, defined in terms o f the frequency S2(1), necessary to yield the spin value as Ot~ n = l / S 2 ( 1 )
(99)
are given for the three retained solutions in Fig. 11. On this figure the subscript "th" indicates that we have used the Lagrange multiplier value ~O(I) for the angular velocity, as opposed to the usual "experimental" estimate of s2(1) $'2exp(l ) = E , / ( 1 ) / 2 h ,
(100)
which is nothing but a consequence of the classical Hamilton relation between the variables 1 and s2 in Eqs. (28) and where E~,(1) is the transition energy to the yrast state of spin 1 from the yrast state of spin I + 2h. From the above "experimental" angular velocity, one may deduce "experimental" kinetic angular moments of inertia o kexp in =
i/f2exp(1)
( 101 )
which are displayed in Fig. 12 and found roughly similar to their "theoretical" counterparts of Fig. 11. This fact reflects the similarity in the general trend of the dependence
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
287
75 50
100 80 60 120
115 ,
0
I
20
,
,
,
L
,
,
,
40
I
60
80
I/t~ Fig. 9. Same as Fig. 8 for the inertia parameter B. of /2(1) and g2exp(l ) as functions of 1, as shown in Fig. 13. These kinetic moments are at first sight to be compared with the dynamical moments C. Indeed they are calculated to belong to the same range of values. Their detailed comparison however is rather delicate indeed. As seen from Eqs. (33), (34), the collective energy and thus the angular momentum depend in general on powers in S2 whose order go beyond f2 2 and therefore the two moments of inertia are different since they involve different orders of derivation of the energy for their evaluation. This is rather well known. What is new here, is the fact that the energy and its derivative depend also on powers of o) and therefore in the relation between 1 and /2 for instance, even in the linear case (i.e. for a quadratic energy) one would have beyond a term in /2, a constant term which may yield a collective alignment resulting from the intrinsic vortical motion. This point will be explicitly discussed in Section 7. In the context of the AI = 2h staggering phenomenon observed in some super deformed bands [ 19,20], one introduces a difference g E e ( l ) in gamma transition energies E7(1) from a "smooth" behaviour (assumed to be merely in this case [20] a polynomial in 1 of order 4 or less) ~E:,(1)=~
ET(1)-
-6Er(I
[4Ee(1-2h)+4Ee(l+2h)
- 4h) - 6 E e ( I + 4 h ) ] } .
(102)
288
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304 ~-. lOO
~
50 140
120
100 235
233
0
20
40
60
80 1/14
Fig. 10. Same as Fig. 8 for the inertia parameter C. The calculated energies ~ E v ( 1 ) are displayed on Fig. 14. It is to be noticed HD case one finds a rather well-marked periodic effect of staggering with a thus consistent with the observed period in Jyrast(1). This behaviour will discussed in the next section, together with the observed modulation of the of the energies 8E~, (1).
that in the 4h period, be further amplitude
7. The quadratic approximation The results of the previous section, clearly indicates that the quadratic approximation of Eq. (63) for the collective energy in terms of the angular velocities w and s2 (given in its most general form in Eq. (33)) should be valid in rather wide domains. In such a case the conjugated momenta I and J of the rotation angles O and 0, of which the time derivatives are /2 and w respectively, are merely related by linear expressions as special cases of Eqs. (34): 1 =CS2 + Bw,
(103)
J = Bg2 + A w ,
yielding upon inverting these relations ~2= ( A 1 - B J ) / ( A C
- B 2) ,
o) = ( C J -
BI)/(AC
- B 2) .
(104)
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
289
WD
~sop
I i i
=
100 i o
200 71"",, I
150 100 -
SD
,,
.,
l'~
ii
250
[l
HD
1', 1
200 j 0
;
,
20
40
60
t
80
1/14 Fig. 11. Theoretical kinetic moments of inertia, defined in the text (see Eq. (99)), as functions of the total angular momentum 1 along the yrast line for each of the three retained configurations (WD, SD, HD).
Inserting Eqs. (104) into the Eq. (33) for the energy, one finds readily the quadratic expression for the collective energy in terms of 1 and J, postulated in Eq. (86). Let us draw a first consequence of the preceding. Indeed, expressing I in terms of s2 and J one finds
AC - B 2
1= ~ - S 2 +
B
~-J.
(105)
One thus gets within a conserved J band a collective alignment proportional to J and a kinetic moment of inertia which is rather close to the irrotational ansatz, upon assuming A ~ C, as approximately obtained in both our semiclassical and harmonic oscillator estimates in a wide range of deformations. Another interesting property of the quadratic energy given in Eq. (63) has been widely exploited in a previous paper [17]. It will only briefly sketched here and further discussed. Defining the yrast state as having the lowest energy for a given value of I, one gets classically (that is in particular assuming J to be a continuous variable) from
3E[I = 0 ,
(106)
from which one obtains the following yrast-J value B
Jyra~t(/) = ~ / ,
(]O7)
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LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
"- . . . .
loo
,
,..~i
,
-
!-.
!
160 II / 140
~. ,
300
i
~"
"b
"~
?,%
~
!
!
/ II 275 250
, ~ 0
• 20
40
60
80
1/14 Fig. 12. Same as Fig. 12 for the experimental kinetic moments of inertia defined in the text (see Eqs. (]00), (t01)). and yrast energy 12 Eyrast(1) = 2 ~ "
(108)
The latter is nothing but the classical rotor expression whenever the intrinsic vortical currents are seen not to contribute to the collective energy. In general the quadratic collective energy of Eq. (86) may thus be written as E ( 1 , J) = E y r a s t ( / ) -t- f i E ( l , J ) , 1 C (j_jyrast(l))2 B E ( l , J) - 2 A C - B 2
'
(109)
where the part that varies with J at a fixed value of I, f i E ( I , J ) , is of the from of a rotational energy for the distance of J from its classical yrast value, involving a moment of inertia which is again very well approximated by the irrotational moment of inertia (assuming A ~ C ) . From the preceding discussion one clearly sees that the Kelvin circulation is not a direct measure of the intrinsic vortical motion content of the considered solution. The relevant quantity is indeed J - Jyrast(1), i.e. the deviation of J from its classical yrast value. To support that statement is it useful to note from the second of Eqs. (104) that whenever J assumes its classical yrast value one gets a vanishing (o value. Turning back now, to the collective alignment property, one sees that if one follows the yrast line and thus inserts the yrast value of J as a function of I into
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
"'"
I
wo
0.5 ~
SD
291
0.5
51" ~ "f~U
:
/
0
.... /
HD
~,~
r
0.2 t
~
[
y~::
I 0 i J_
0
.
.
.
20
.
.
.
.
40
k
60
•
.__
I !
80
1/]4 Fig. 13. Theoretical and experimental angular velocities as functions of the total angular momentum 1 along the yrast line for each of the three retained configurations (WD, SD, HD). The theoretical (experimental resp.) velocities correspond to the solid (dashed resp.) curves. The theoretical values is simply the relevant Lagrange multiplier, whereas the experimental one is defined in Eq. (100).
Eq. (105), one looses the alignment term and gets a kinetic moment identical to the dynamical one since in that case. 1 = C~.
(110)
Now, one should take into account the essential fact that J is quantized in the very same way, as we have seen, that an orbital angular momentum. Its yrast value is not in general equal to Jyrast(1). In other words, J (in units of h) should be, in our case, an even integer number. Consequently the yrast energy differs from Eyrast([) by a rotational-like energy similar to BE(I, J) (i.e. implying the same moment of inertia) and proportional to the square of the difference t~Jyrast(l) between the quantal and classical yrast J-value, namely t~Jyrast(/) = 2hE ~ - ~ where E[y] denotes the integer number which is closest to y, namely
E[y] = M i n { E [ y ]
y,E[y] + l - y } ,
with [y] being again the integer part of y. The energy 8Jyrast(1) of Eq. (111), is obviously a periodic function of I whose period T(1) is given by
292
1.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304 WD 2.5 / I, i I i '1I,
: : : : -
SD
20
I
i
,~
/
'1
I i
I ,
i~
'
i f
HD ii
,i
I,! II
I
'1 /
-2O
5
i'~
t', t C
5
,I '1
/
'i /
' '
/
.t.
?
a
20
40
60 1/14
Fig. 14. Rapidly varying part of the gamma transition energies as functions of the total angular momentum 1 along the yrast line for each of the three retained configurations (WD, SD, HD). These energies are defined by Eq. (102). T ( 1 ) = 2h C .
(112)
Such a quantization o f J in the yrast line generates therefore a structure in the yrast line energies which exhibits a regular pattern of kinks. This has allowed us to propose it as a possible explanation o f the staggering observed in some superdeformed states. Clearly for B / C = 0.5, one gets a A I = 2h staggering pattern (see however the extended discussion o f this case below). We refer to Ref. [ 17] for a complete discussion o f this matter. There are two points presented in Ref. [ 17] which we would like to discuss and complete in the present paper. First, since experimentalists do observe bands over large domains in 1, and thus in J, one is compelled to explain how much our yrast band is still a band, i.e. composed o f states connected by strong stretched E2 transitions. We previously rightly argue that despite the fact that the E2 operator does conserve J, the experimental bands might be explained by the existence of residual interactions beyond the simplified model presented here. These interactions lead to a J-mixing which should be maximal near the crossing o f the energy parabolas corresponding to two adjacent J values. An estimate of the quenching of the E2 operator shows that it is compatible with the present uncertainties on absolute measurements o f E2 matrix elements by D S A M techniques for instance [38]. Yet one should consider the cumulative effect in a full
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
293
cascade of such a quenching, l A quantitative answer to this point would necessitate a detailed study of the interband cross-transitions. For instance, one may notice that, if at angular momentum I one has an yrast state corresponding to a pseudo crossing (i.e. maximal mixing), its companion state (located just slightly above it) should decay to the next ( I - 2h) yrast state with an intensity which reduces by a factor two roughly, the lack of feeding (due to the J conservation) of that state through a direct yrast-yrast transition. Incidentally one may note that this side feeding would correspond to a gamma ray energy very close to the yrast-yrast gamma ray energy. The second point has to do with a possible improvement of the quadratic expression for the energy. Within our model of vortical and global rotation, which are assumed to be completely decoupled from other collective modes, the quantum mechanics analog of the classical energy of Eq. (86) is E(I,J) - A ~C-
B
"
"
For eigenstates of both the [2 and J~ operators, and assuming a full alignment of the operators [ and J, one gets the following quantal energy E(1, J) = A ( A I ( I
+ h) - B I j + C j ( j
+ h)) ,
(114)
with the constant A defined as A= 1/(AC-
B 2).
(115)
The analog of the classical yrast J-value is given also in this case by the variational Eq. (106) where the energy E is given by Eq. (114), leading to B Jyrast(/) = ~ - I
h 2 "
(116)
This yields the new following "classical" yrast energy Ia A ACh 2 Eyrast(1) = ~-~ + -~(A + B ) l h - ~
(117)
The total energy of Eq. (1 14) may thus be written upon Taylor-expanding it around J = Jyrast(1): E(], J) = Eyrast(1) q- 6(I, J) , ~( I, J) = 2 C x 2 ,
(1 18)
where x is defined as x = J - Jyrast(1) I We are indebted here to W. Andrejtscheffto have drawn our attention to this important question.
(119)
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
294
For quantal yrast states, one has therefore
x=2hE C2h
---d1+ 2"
(120)
For B / C equal to 1, which classically corresponds to spherical shapes, one gets x = - h / 2 for all even values of I, leading thus to an absence of staggering for the quantal yrast energy I? ,t Eyrast(1) = ~'~ q- ~ ( A + B)lh..
(121)
This is not so surprising since in this case the distinction between global and intrinsic rotations is completely disappearing. For B / C equal to vl now, which is the case mostly discussed in Ref. [17] (where however the full consequences of the quantal expression for the energy given in Eq. (114) have not been drawn), one gets x = qzh/2, where the plus (minus resp.) corresponds to an even (odd resp.) integer number I/2h. Since x enters quadratically the expression 114 tbr E(I, J), one retrieves in both cases the result valid for spherical deformations as given in Eq. (121), implying here too an absence of staggering. This happens because the AJ = hi2 shift in Kelvin circulation yrast values, with respect to the case studied in Ref. [17], is exactly suited so as to accidentally smooth out the kinks present in the yrast line made of pieces of parabolas intersecting with a AI = 4h period. This does not affect, of course, the general character of the advocated staggering phenomenon and its period in particular, but merely introduces a modulation of its amplitude as it will be discussed now. For that purpose, let us study now the yrast energy pattern as a lhnction of the distance of the ratio B / C from 1/2. Indeed introducing the parameter e by B C
-
l+e 2
(122)
'
one gets for I = 2ph
x=21~E
- (1 + e ) p h + ~ ,
(123)
and thus, x is clearly a periodic function of pe whose period is 2 or a periodic function of le whose period is 4l/. We will therefore restrict our study of the behaviour of x as a function of I, for a given value of B / C , i.e. of e, within one period, namely with the [ 0 , 4 h ] interval for Ie. One obtains i f p is even
x=h(½-
pe) ,
le < 3h,
x = h (5 - pe) ,
le > 3h.
(124)
In the case where le = 3h, the value of x is undetermined (since the function E[y] is not defined for y being half integer), but its absolute value is well defined as Ixj -= h ,
1~ = 3 h .
For odd values of p now, one gets
(125)
l.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
295
3
0 0
1
2
3
4
I~//4 Fig. 15. Rapidly varying part of the total energy 6y,.(l) over one period, as a function of the product of the total angular momentum 1 and the parameter e measuring the distance of the ratio B/C from ½; The quantities B, C and ~ have been defined in the text.
x= Y
=
h(½+pe) ,
le
h (\ 3z
Ie>h.
pe]/ ,
(126)
Similarly one gets for le = h
ix I = h ,
le=h.
(127)
From the above one may deduce the energy 6(I, J) on the yrast line, called 6yr(1) for even values o f p
t~yr(l )
=AC/8(Ie,- h) 2
6yr(1)=AC/8(l~-5h) 2
for
le < 3 h ,
(128a)
for
le > 3 h ,
(128b)
for
le < /7,
(129a)
for
le > 3h,
(129b)
and
6yr(1)
=AC/8(le-f-h)2
~ y r ( 1 ) = A C / 8 ( l e - 3h) 2
for odd values o f p. These energies which might be considered as envelops of the quantal 6yr(1) energies (i.e. for quantized values o f 1 and J ) are plotted in Fig. 15. In the limit where e is small with respect to 1, one clearly retrieves the harmonic oscillator result presented in the previous section, where the rapidly varying part of the yrast energy oscillates between two different values (corresponding to a sequence of odd, even, odd . . . . values for p ) and where the oscillation amplitude is modulated from 0 to a maximum difference of ½ACh 2. This translates into gamma transition energies showing a "beating-like" trend as shown in Fig. 14 for the HD solution for which the computed e is found to be slightly below 0.05. In this case the maximum staggering is expected to occur slightly below I --=20h and its vanishing slightly below I = 40h, which is roughly consistent with what
296
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
we have obtained in the microscopic calculation of Section 6 and as well with what has been discussed in our Ref. [17]. A similar staggering pattern would be apparent for somewhat larger values of e (as e.g. for e = 0.2), whereas it would be completed distorted for very large e-values (as e.g. for e = 0.5). In a recent paper [22], an analogy of this quantal effect possibly observed in some superdeformed states with an Aharonov-Bohm phase effect for some mesoscopic conductor [39] or semi-conductor [40] rings in a uniform and time-invariant magnetic field, was proposed. In those rings indeed, persistent electric currents have been observed whose intensity is a periodic function of the magnetic flux through the ring, with a period equal to the elementary quantum of flux e / h (e being the electron charge and h the Planck's constant). As demonstrated in Ref. [22] for the collective energy of Eq. (109), i.e. neglecting the beating modulation above discussed, our 3 E ( I , J) energy plays the role of the electron energy, ( B / C ) I is the analog of the magnetic flux, finding the yrast state at a given 1 value is equivalent to find the electron ground state and gamma transition energies (proportional to the derivatives of the energy as a function of 1) are the counterparts of the electric current (proportional to the derivative of the energy as a function of the flux). This analogy has been used to provide a possible explanation tbr the scarcity of the staggering phenomenon. Indeed, the staggering structure should not be damped by fluctuations in ( ( B / C ) I ) (i.e. in deformation) on a range of the order of larger than h, in very much the same fashion as the variation of the magnetic flux (i.e. of the ring radius) should be smaller than the elementary quantum of flux e / h . A sizeable increase of the vibrational axial quadrupole mass right in phase with the superdeformed well is of course producing an important squeeze of the collective wavefunction, reducing thus the deformation fluctuation which is crucial Jbr the staggering effect to be observed. It has been recently shown in a calculation [41] within an ATDHFB approach using the Gogny effective force that this mass parameter presents a sudden increase by more than one order of magnitude, at the right deformation, in 15°Gd and not for its neighbouring isotopes. This might explain why the phenomenon is observed around this nucleus and not elsewhere. Similar calculations for Hg and Ce isotopes near the nuclei claimed to exhibit possibly (but in a lesser clear, or widely accepted, fashion than in 149Gd) such a staggering, do not yield such a spectacular increase [41]. Finally let us briefly mention that the same quadratic expression for the collective energy of Eq. (109), has been used [18] to provide a tentative explanation for the intriguing phenomenon of identical bands observed in many superdeformed nuclei, as well as to a lesser extent in some normally deformed nuclei [21]. This phenomenon relates gamma transition energies in nucleus 1 at spin Ix to gamma transition energies in nucleus 2 at spin 12 through E ~ ( l l ) = Ez,(12) q-- x[Ez,(12) - E~(I2 - 2h)] ,
(130)
where x assumes the values 0, +¼, +1, ±¼, +1. We found, using exactly the quadratic energy of Eq. (63) as before, that Eq. (130) may hold. For that we consider now nuclear collective states as defined by two quantum numbers (I and J with our current
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
297
notation). Then it has been shown in Ref. [ 18] that Eq. (130) may be retrieved provided that (B/A)l
= (B/A)2 = Y.
(131)
This condition expresses essentially that the nuclei 1 and 2 have the same deformation (note that this quantity being a ratio of moments of inertia does not scale at all when changing the nucleon number). Moreover, the x-parameter entering Eq. (130) is defined in terms of Y defined in Eq. ( 131 ) and of the differences A I and A J between the values of the moments 1 and J in the nuclei 1 and 2, as x = ½(ai -raJ).
(132)
We have noted above that in some superdeformed states B / C ~ 0.5 yielded the right period for the almost elusive staggering phenomenon advocated there. It is interesting to note, upon assuming Y also to be close to 0.5 in these cases (i.e. C ~ A, that this l value leads here to the right "quantization" of x in multiples of ~.
8. C o n c l u s i o n s
The aim of this paper was to investigate a class of collective motions in atomic nuclei which, in our opinion, have not been the subject so far of the extensive study they may deserve. They are however present and well known at larger scales, like when considering self-gravitating stellar objects or fluids in a container as encountered in everyday life. Tea, for instance, when correctly stirred, may continue to rotate in its cup even when the latter is neither deforming nor rotating as a whole! The main stream descriptions of the nuclear collective excitations in terms of fluid dynamics have generally limited these studies to the coupling of scaling-like irrotational vibrations with global rotations, with some noticeable exceptions like the study of M2 giant resonances for instance. It is so for very good reasons including the fact that these modes are probably the most important as well as the most easy to decipher. Quite clearly it is more easy to "see" shape changing modes than the modes that merely affect the current distribution. Moreover the latter modes are strongly coupled with the Bohr-Mottelson types of modes as exemplified above for the global rotation. It is our opinion that it may be timely to investigate vortical modes in the intrinsic frame, and their couplings. Indeed recent experimental progresses have opened new pathways towards rather pure collective states (like superdeformed excited states in some part of the nucleidic chart), upon using very efficient spectroscopic tools, like 4 ~ high-granularity gamma detectors for instance. In order to proceed in this field which has not yet been thoroughly explored, we had to clarify some questions. First we had to define the relevant modes and to choose a way to treat them in a fashion which takes into account, at least approximately, the multitude of nuclear degrees of freedom and which is sufficiently quantal in the treatment of the
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1.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
collective properties. Finally, we had to examine possible observable signatures of such vortical modes in nuclei. As for the first problem we have decided to provide an answer which is both simple, flexible enough (see the variety of current patterns in Fig. 1) and well documented in classical physics as well as in algebraic collective dynamical description. This is why we chose Riemann-Chandrasekhar S-ellipsoids. The so-called "routhian" or "sell-consistent cranking" semiquantal approach is widely used in nuclear physics to describe global rotations. We deem it reasonable to assume that, if successful for one collective mode, it could also be useful for others, provided its foundation on Hamilton classical mechanics is well understood. We have taken full advantage of the physical relevance and relative simplicity of the harmonic oscillator model to validate some of the main simplifying ideas like the wide range of applicability of the quadratic approximation for the energy. Obviously a lot of theoretical work remains to be made. For instance, one would like to quantitatively estimate the close connexion between the intrinsic vortical velocity ~o (or equivalently the difference between the actual value of J and the classical yrast value Jyr~,st( I ) ) within the framework of constrained Hartree-Fock-Bogoliubov calculations. One also should investigate the amount of J-mixing as possibly resulting from a more comprehensive collective model approach, so as to assess the robustness of the present ansatz. At any rate, we feel that the results presently obtained are sufficiently suggestive to encourage further efforts to better understand the modes under scrutiny here. The last question concerning the measurable manifestations of such collective modes may be tackled in two different fashions. There are, indeed, both direct and indirect ways to see their effects. By "indirect" we mean here that upon ignoring them one has to renormalize the collective model parameters in use, in order to keep contact with the real world, while "direct" refers to observables implying the intrinsic vortical currents. To the first category clearly belongs the necessary renormalization of the moments of inertia away from rigid body inertia parameters. The preceding statement is both very useful to make a strong statement on the existence of intrinsic vortical modes and rather short to assess their properties. Pairing plays there a very strong role, so any quantitative evaluation should take into account these correlations. As for direct fingerprints of these modes, we came along with two possible candidates. The first physical phenomenon is the staggering observed in some superdeformed bands. It is a very tiny and almost elusive one, even though one has recently suggested that such a phenomenon could exist in the rotational spectra of some diatomic molecules [42]. If so explained, this staggering should indeed be viewed as a quantal zero-point fluctuation of the studied intrinsic mode, when following a rotational band whose properties are mainly given by the global rotation. To illustrate the latter point, one should keep in mind the relative magnitudes of the energies involved, a fraction of keV versus hundreds of keV. An analogy has been drawn between this phenomenon and another phenomenon which is, as the lbrmer, on the borderline of physical availability, namely persistent currents in mesoscopic rings. By no means this analogy is accidental, if one takes properly into account the well-known similarity between the motion of a massive particle in a rotating frame with the motion of a charged particle in a magnetic field.
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It is clearly, in our opinion, interesting by itself in embedding seemingly very different quantal properties in an unified framework. It has also been beneficial in allowing us to provide a frame of explanation of the scarcity of the observed nuclear phenomenon, and even of its actual occurrence in 149Gd,through the calculations of vibrational mass parameters of Ref. [41 ]. The other nuclear property which we have looked at (i.e. the phenomenon of identical bands, particularly in superdeformed nuclear states) deserves a more systematic study than what we have been able so far to perform in Ref. [ 18]. There we have been able to demonstrate that a priori our collective hamiltonian encompassed indeed such a property. We want to stress that not only we provide a coupling frame allowing this identity but also an explanation of why a scaling in the number of particle is not an issue of any relevance for that problem provided the two nuclei have the same deformations. There are many other places where the effect of such nuclear intrinsic vortical modes could be hunted for, as in magnetic excitations clearly and quite generally. Also, in excitation energy regimes where these collective modes are not too damped, they could affect significantly the nuclear level density. To conclude we should thus strongly advocate a systematical study of the role of intrinsic currents in low excitation energy excitations of nuclei.
Acknowledgements We are indebted to many physicists for their interest and/or their patience in numerous discussions on these ideas which might have appeared, at least at first sight, somewhat far-fetched. We would like to thank them all for their suggestions or criticisms, and particularly among them W. Andrejtscheff, J. Barrel, H. Doubre, B. Haas, F. Hannachi, J. Libert, W. Nazarewicz, J. Meyer, M. Meyer, N. Minkov and last but not least Ch. Brian ton. We are also grateful to E.Kh. Iouldashbaeva, J. Libert and M. Girod coauthors (with one of the authors of the present paper) of a paper not yet published, to have allowed us to make use prior to publication of some relevant results. Over the course of this work some of us have benefitted of the hospitality of some other's institutions. Therefore I.N.M. thanks the CENBG and EQ. the Bogoliubov Laboratory of the JINR, for the quality of their hospitality during many fruitful visits. This work has greatly taken advantage of a funding through the agreement between the I N 2 P 3 / C N R S and the JINR (Collaboration #91.9) which we thank very much for their continued support.
Appendix A. The boson Bogoliubov transformation associated with the generalized routhian in the harmonic oscillator mean field case We consider the boson Bogoliubov transformation (with k = 2, 3) +
o~k = ~
•kiaJ + tzkiai, i
l.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
300
satisfying the generic boson excitation equation %
+
= [h(.O,(0),a~-]
,
with the expression of h(s2,(0) given in Eq. (73). Calculating explicitly the above commutator yields the matrix equation given in Eq. (77). In view of the merely real or imaginary character of its matrix elements and defining Ak2 = lk2 ,
kk3 = i l k 3 ,
].Zk2 = ink2 ,
[.Zk3 = imk3 ,
one may rewrite the Eq. (77) in a real form as
0
0SD (03 - ,Ok
/
O
0
[ lk3
D
(02 @ ,(2k
-S
~ ink2
0
-S
(03 + s2~
\ mk3
= O.
The vanishing of the determinant associated to this homogeneous system of linear equations yields the following second order equation in .02 .(-24 - O~(O92 jr_ (023 - + - 2 ( S 2 - D 2 ) ) + ( 0 ~ ( 0 39 - 22( 0 2 ( 0 3 ( S 2 . q - D 2 ) ÷ ( S 2 - D 2 ) 2 = O ,
whose solutions have been explicited in Section 5. The transformation parameters now are given as 122 = (S'2~ - w~) (S22 + (o2) + c03(S 2 + D 2) - J22(S 2 - D 2) , 123 ----S ( ( f 2 2 + 093)(.(22 q- (02) -
( $2 - D 2 ) )
,
m22 = - 2 S D w 3 , m23 = D (($22 - ( 0 3 ) ( ~ 2
+ ~2) -
( $2 - D 2 ) )
132 = S ((.(23 q- (.o2)(f23 q- (03) - ( $2 - D 2 ) ) 133=(f22-(02)(.(23+(03)-+-(02(S2 //'/32---O
+ D 2)
, , .(23($2-D2),
((,(23 --w2)(S23 + co3) - (S 2 - O2)) ,
rn33 = 2SDo92.
The homogeneous character of such a linear system entails that these coefficients are only known up to a multiplicative constant which is fixed by the canonical condition k=3 k=2
The inverse transformation is given in the following form k=l * k=2
+
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301
Appendix B. Some expectation values of the generalized routhian solutions with a harmonic oscillator mean field
In this appendix we will evaluate the expectation values of some selected operators for the solution of the generalized routhian problem with a harmonic oscillator mean field. It is a Slater determinant made of A single particle wavefunctions characterized by the quanta numbers n~j), with i = 1,2, 3 associated with the frequencies eoj, /22 and .(23 respectively and j = 1 . . . . . A. It will appear useful to define the following sums j=A
1
_,= = Z ( n y
) + -i)
j=l
To compute the expectation values ponents of the position and/or the + boson operators a~, ak and then k = 2, 3, through the inverse boson One then finds readily
of one-body operators defined in terms of the comimpulsion, we have to rewrite them in terms of the + in terms of the boson operators a~, a~ whenever Bogoliubov transformation given in Appendix A.
= _Z_h z , , 2row1 and for i = 2, 3
mwi
"=
((lji-mji)2-~J) '
from where one deduces the values of the semi-axes of the associated ellipsoid as for instance for the first axis a= ~ , and similar formulae for b and c. To compute the expectation values of the x-component of the orbital angular momentum and of the Kelvin circulation, one needs "crossed" expectation values in x and p as
x2P3 = °)~ h / x3P2 ; = h /
k=3 k_~2 [ ( / k 2 _ - mk2)(lk3 + mk3)Z~] , k=3
09~ ~ _ [ (/~2 + mtz)(mk3 -- 1~3) Z k l •
One therefore gets x2P3 ~ _ x3P2 h~ ( h / '
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I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304 (~}
J~YO+l)=
C ( X2p3; - b f X 3 p 2 ~ "
=~
h /
7\
h /
The expectation value of the generalized routhian h(/2, 09) is simply
k=3 (h(sZ 09)) = h09,& + ~ h & ~ k , k=2 from where with the help of the above expectation values one can deduce the energy of the solution as
E= (h( a,09) ) + s~(L.) + 09(4}. One may also compute it directly from the definition of the energy operator
k=3 1 H = Z h09k(a~ak + ~ ) , k=l from where one gets
i=3 I k=3 2 ~, E=h091,_~l q-~-~ h 0 9 i E ( 1 2 i q - m k i ) _ ~ i=2 [ k=2 Appendix C. Moments of inertia in the Inglis cranking approximation The Inglis cranking approximation of Cic, for instance, leads to the following expression
Clc = 2 Z p,h
t(P I__Lxlh~l 2 ' ep -- eh
where ep and eh are the routhian mean values of the corresponding single particle states. Expressing Lx in terms of the original boson operators a +, a, one gets
_ 2//,2 Cic
[
4092(03 L(093 + 092)2 E
p,h
](pla~a3 - afa21h)l 2 ep -- eh
-t-(093 -- 092)2 Z p,h
+2(09~ - 092) ~ p,h
I(P[a~a~ - a3a2lh)12 ep -- eh
(Pla~af - a3a2lh)(hla2af - a3a~lP} el) -- eh
Using the Bogoliubov transformation, we can rewrite the particle-hole matrix elements appearing in the above expression in terms of the new boson operators a~+, o~, as
303
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304 ( p l a ~ a 3 -- a ~ a 2 1 h ) = i[(122m23 - m22123)(p]~2 + ~221h)
q-(/B2m33 - m32133) (Pla3 ~2 q- ~321 h) q-(/22m33 q- m23132 - m22133 - 123m32)(p1¢¢~¢e3~ if- ¢¢2~¢3]h) +(122133 -]-123132 - m22m33 - ma3m32)(PlO~¢3 q- ¢lt~t'alh)] , ( P l a ~ a ~ - aBaa]hl = i[(m22m23 - 122/23)(pit:c2I-2 -}- ¢¢221h/
+(mBam33 _ 132133) (p]~_2 -1- ¢232[h) +(m22m33 if- m23m32 -- 122133 --/23132) (pl%%q
+ ¢¢2o~31h)
+(m22133 q- 123m32 - /22m33 - m23132)(p1¢¢+¢e3 q- ~¢e21h)] • The calculation of C,c therefore involves the following sum rules (see e.g. Valatin's paper quoted in Ref. [ 3 3 ] ) Z p.h
Z p,h
(P1~2~0~3 + ° ~ 2 [ h ) [ 2 _ ep -- eh
h( 0 3 - 0 2) '
(Plee~a-~ + te2~e3lh)[ 2
2 2 '~ ~'3
ep -- eh (ploz +2 + oli2lh)
Z p,h
,-~2 -- ~'3
h( 0 2 + 0 3 ) ' 2
~'.
=2 -' ep -- e h
i = 2,3.
hJ'-2i'
One would get similar results for B,c upon replacing in the preceding expression of C~c one operator Lx by an operator Jx and similarly Arc, upon replacing both operators Lx of C~c by Jx. Of course the computation of the matrix elements of Jx implies the same matrix elements of products of two (a, a +) operators than those of Lx, up to a multiplication by appropriate scaling factors ( b / c or its inverse). The above equations provide linear response inertia parameters for finite values of the angular velocities. In the adiabatic limit, i.e. for small values of these velocities, the operators teu tend to the operators a u (where/x stands for 2 or 3). Using the previous sum rules where the energy denominators are now detemined in terms of static energies, and upon replacing O1,2 by wl,2 one gets the equations ( 8 7 ) - ( 8 9 ) for A°c , B°~c and C~,°.
References
[ I] [2] [ 3] [4] [ 51 [6]
N. Bohr and J.A. Wheeler, Phys. Rev. 56 (1939) 426. Aa. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26 (1952) 14. See e.g. Aa. Bohr and B. Mottelson, Nuclear Structure, Vol. 2 (Benjamin, New York, 1974). See e.g.H. Lamb, Hydrodynamics (Cambridge Univ. Press, Cambridge, 1975). S. Chandrasekhar, Ellipsoidal Figures of Equilibrium (Dover, New York, 1987). A. Bohr and B. Mottelson, Dan. Mat. Fys. Medd. 30 No. 1 (1955).
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W.J. Swiatecki, Phys. Rev. Lett. 58 (1987) 1184. M. Durand, P. Schuck and J. Kunz, Nucl. Phys. A 439 (1985) 263. K. Bencheikh, 13 Quentin and J. Barrel, Nucl. Phys. A 571 (1994) 518. R.Y. Cusson, Nucl. Phys. A 114 (1968) 289. O.L. Weaver, R.Y. Cusson and L.C. Biedenharn, Ann. Phys. (N.Y.) 102 (1976) 493; B. Buck, L.C. Biedenham and R.Y. Cusson, Nucl. Phys. A 317 (1979) 205. 12] G. Rosensteel and D.J. Rowe, Phys. Rev. Lett. 46 (1981) 1119; and references therein. 131 G. Rosensteel, Ann. Phys. (N.Y.) 186 (1988) 230. 14] G. Rosensteel, Phys. Rev. C 46 (1992) 1818. 15] E.B. Balbutsev, I.N. Mikhailov and Z. Vaishvila, Nucl. Phys. A 457 (1986) 222. 16l M. Cerkaski and I.N. Mikhailov, Ann. Phys. (N.Y.) 223 (1993) 151. 17] I.N. Mikhailov and P. Quentin, Phys. Rev. Lett. 74 (1995) 3336. 18 ] I.N. Mikhailov and 13 Quentin, in Low Energy Nuclear Dynamics, ed. Yu. Oganessian, W. yon Oertzen and R. Kalpachieva (World Scientific, Singapore, 1995) pp. 512-521. 191 S. Flibotte et al., Phys. Rev. Lett. 71 (1993) 4299. [20] B. Cederwall et al., Phys. Rev. Lett. 72 (1994) 3150; A.T. Semple et al., Phys. Rev. Lett. 76 (1996) 3671. [21] C. Baktash, B. Haas and W. Nazarewicz, Ann. Rev. Nucl. Part. Sci. 45 (1995) 485. [ 22 J I.N. Mikhailov and E Quentin, submitted for publication; see also I.N. Mikhailov, Ch. Brianqon and 13 Quentin, Particles and Nuclei (JINR) 27 (1996) 303. [23] S. Frauendorf, Nucl. Phys. A 557 (1993) 259c. J 241 N.R. Lebovitz, "The Riemann ellipsoids", Lecture Notes, unpublished, quoted in Ref. [ 5 ]. 1251 P.M. Morse and H. Feshbach, Methods of Theoretical Physics (Mac Graw-Hill, New York, 1953), Sections 1.3 and 13.1. [261 See e.g., M.J.A. de Voigt, J. Dudek and Z. Szymanski, Rev. Mod. Phys. 55 (1983) 949. 1271 H. Goldstein, Classical Mechanics (Wesley, Reading, 1950). [28] E.13 Wigner, Phys. Rev. 40 (1932) 749. 1291 M. Brack, C. Guet and H.-B. Hfikansson, Phys. Rep. 123 (1985) 275. [301 P. Bonche, H. Flocard and P-H. Heenen, Nucl. Phys. A 467 (1987) 115. [31] K. Bencheikh, P. Quentin and J. Barrel, Nucl. Phys. A 571 (1994) 518. J 32 ] 1.N. Mikhailov and 13 Quentin, to be published. 1331 J.G. Valatin, Proc. Roy. Soc. (London) A 238 (1956) 132; V.G. Zelevinsky, Soy. J. Nucl. Phys. 22 (1975) 565; G. Ripka, J.E Blaizot and N. Kassis, in Heavy Ions, High-Spin States and Nuclear structure, Vol. 1 (IAEA, Vienna, 1975) p. 445; A.13 Stamp, Z. Phys. A 284 (1978) 305; D. Glas, U. Mosel and EG. Zint, Z. Phys. A285 (1978) 83. [34] T. Troudet and R. Arvieu, Phys. Lett. B 82 (1979) 308. [35] D.R. lnglis, Phys. Rev. 96 (1954) 1059; 103 (1956) 1796. [36] D.J. Thouless and J.G. Valentin, Nucl. Phys. A 31 (1962) 211. 1371 See e.g. 13 Ring and 13 Schuck, The Nuclear Many-Body Problem (Springer, Berlin, 1980) § 3.4.3.1. [38] H. Savajol et al., Phys. Rev. Lett. 76 (1996) 4480; D. Nisius et al., Phys. Lett. B 392 (1997) 18. [39] L.13 L6vy, G. Dolan, J. Dunsmuir and H. Bouchiat, Phys. Rev. Lett. 64 (1990) 2074. [401 D. Mailly, C. Chapelier and A. Benoit, Phys. Rev. Lett. 70 (1993) 2020. 141 ] E.Kh. Iouldashbaeva, J. Libert, E Quentin and M. Girod, to be published. 142] D. Bonatsos, C. Daskaloyannis, S.B. Drenska, G.A. Lalazissis, N. Minkov, 1313 Raychev and R.13 Roussev, Phys. Rev. A 54 (1996) R2533.
Nuclear Physics A 627 (1997) 259-304
Nuclear intrinsic vorticity and its coupling to global rotations* I.N. Mikhailov a,b, R Quentin c, D. Samsoen c a Bogoliubov LaboratoryJbr Theoretical Physics (JINR), Dubna, Russia b Centre de Spectromitrie Nucldaire et de Spectromdtrie de Masse (IN2P3-CNRS), Orsay, France c Centre d'Etudes Nucld aires de Bordeaux-Gradignan (IN2P3-CNRS and Univ. Bordeaux 1), Gradignan, France
Received 4 April 1997
Abstract Important collective modes which are generally neglected within current descriptions of nuclear excitations in terms of fluid dynamics, are studied here. The intrinsic vortical modes are defined in a general way from which a specific mode, both simple and versatile enough, is particularly discussed. In this paper the main emphasis is made on the coupling of the chosen intrinsic mode to the rotation of the nuclear principal axes frame with respect to the laboratory system. A semi-quantal description of such excitations is proposed which is a generalization of the so-called routhian approach of global rotations. The results of a semiclassical treatment of the corresponding variational problem are presented. A simple mean field approach where the one-body potential is mocked up by a harmonic oscillator is discussed in a somewhat detailed fashion. The broad range of validity of a quadratic approximation for the collective energy in terms of the relevant angular velocities, is hinted from the previous simple model approach. Some general consequences of the latter are then drawn which have bearing on some possible fingerprints for the existence of such excitations, as the staggering phenomenon observed in gamma transition energies in some superdeformed states and the occurrence of identical rotational bands in neighbouring nuclei. @ 1997 Elsevier Science B.V.
1. Introduction S i n c e the p i o n e e r i n g w o r k s o f B o h r and W h e e l e r [ 1] on nuclear fission as well as o f B o h r and M o t t e l s o n on nuclear q u a d r u p o l e collective m o t i o n at low excitation * Work partially supported by the IN2P3-JINR Collaboration grant #91-9. 0375-9474/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PH S 0 3 7 5 - 9 4 7 4 ( 9 7 ) 00401-6
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energy [2,3], it is a usual approximation to describe large amplitude nuclear collective mode in terms of the dynamics of an equivalent incompressible nuclear fluid. In nuclear physics, however, an important part of the possible modes for a corresponding selfbounded classical fluid drop has not received a sufficient attention, even though their existence has already been recognized in a more or less direct fashion. These modes are the intrinsic vortical modes. Among their many manifestations, it appears that they strongly couple, in particular, with the global rotations of the fluid drop. In this paper we want first to define these vortical modes as precisely as possible and then within a generic model case, widely studied in other parts of the physics, we intend to discuss the above mentioned coupling. Indeed this seems to us of some relevance in view of the important current experimental effort devoted to understand very high spin and/or superdeformed nuclear states. In a fluid dynamical approach (within the so-called eulerian description), the essential concept is the velocity field u ( r ) with respect to the laboratory frame [4]. It may be split into an intrinsic part U i n t ( r ) and a rotational part u R ( r ) u ( r ) =Uint(r) + UR(r) ,
uR(r) = g~ × r ,
(1)
where g2 is the angular velocity associated with the rotation from the laboratory frame to the body-fixed or intrinsic frame (i.e. where the mass tensor is diagonal). The intrinsic field Uint(r), as any vector field under very general conditions, may be split into two parts, as well known Helmholtz's theorem) (2)
//int = US -]- UV ,
with us = - g r a d V ; uv = rot A ;
AV = -div//int , A A = - r o t Uint
,
(3)
whose first part us is irrotational and defined by a scalar field V whereas its second part uv is divergence-free and defined by a divergence-free vector field A. The uv field is defined as the intrinsic vortical velocity field. We picture the incompressible nuclear fluid as a self-bounded finite system having a constant density within a sharp-edged boundary, called the surface. The equation lbr the latter writes in terms of some shape time-dependent parameters labelled generically, ei as
F(r, ei) = 0.
(4)
Since our fluid is self-bounded, writing the condition of no surface-crossing in the intrinsic frame, yields OF
Uint • g r a d F + ~ - - e i i Oei
= 0,
(5)
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
261
providing thus a boundary condition of the type Uint" n = qg(ei,r) ,
(6)
where n is the outward unit normal vector and qb(ei, r ) is given by ~.-~i aee'
(lh(ei, r ) = --
(7)
~/grad F - grad F " To specify the vortical modes under consideration here, we make a further restriction on the intrinsic velocity field by assuming that all the variation of the intrinsic shape is carried only by the field us deriving from a scalar potential. This assumption stems from our choice to make the minimal generalization of the Bohr-Mottelson approach [2,3] where this is obviously the case. To do that, we impose that the intrinsic vortical field Uv is purely tangential (imposing thus the boundary condition (6) on the field Us only) as we will see below. We define, as usual, the current in the intrinsic frame by Jint = pUint ,
(8)
in terms of the relevant density p. The latter is written here as
p = poO(F),
(9)
where P0 is the value of the constant density into the nuclear volume and O the Heavyside step function. As a result, the contribution (Op/Ot)v of the field Uv to the time change of the density, which is defined according to the continuity equation c~p d i v j + ~ - = 0,
(10)
as
(OP) -
-~-
=pdivuv+uv.gradp
(11)
v
is vanishing due to the divergence-free character of uv and to its tangential nature, since n is proportional to grad F and thus, at the surface, to grad p. The resulting equation for the field us deriving from a scalar potential is therefore of the Laplace form inside the volume, with a boundary condition of the von Neumann type at the surface (which is simply the Eq. (6) with us in lieu of Uint). Let us now set up a possible classification scheme of the uv fields, assuming that their components are homogeneous polynomials in the coordinates r of order p, which can always be done upon adding as many components as necessary to retrieve any given analytical field. For p = 0, the tangential character of Uv over a closed surface clearly implies that this field is vanishing, The collective motion is thus merely a combination of an irrotational shape-changing field Us and of a global rotation UR. It is thus consistent - here for the most general us mode though - with the Bohr-Mottelson ansatz [2,3]. The p = 1 case has been widely studied when the surface is defined in Eq. (4) by
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I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
a quadratic function F (which implies for a bounded surface that the corresponding nuclear volume is ellipsoidal). Riemann and many others [4], among which one must single out Chandrasekhar [5], have studied the general problem of a total velocity field linear in the components of r whose domain is bounded by such an ellipsoid. There, the scalar potential V defining us, as well as the components of the vector potential A defining Uv, must obviously be quadratic. Taking into account the boundary condition for Uv, one finds ( u v ) i . ~. e t j.k c o. l x.g C I , ),k
(12)
where e~ik stands for the totally antisymmetric third rank tensor while x~ is the kcomponent of r, (.Oj is the j-component of a coordinate-independent vector oJ and ci, ck the semi-axes of the ellipsoid in the i and k directions (obviously all indices i, j, k refer to principal axis, i.e. correspond to the frame attached to the body's inertia tensor). This mode is the only intrinsic vortical motion possible for an ellipsoid having a uniform vorticity. It will be the one considered here. Some ways to extend the study to nuclear shapes other than ellipsoidal or to higher (p ~> 2) polynomial orders for the velocity components will be, however, proposed below. Even though, in the previous discussions, we have separated the global rotation UR and the intrinsic vortical Uv velocity fields, they are well known to be strongly coupled in classical systems [4]. In nuclei, the existence of such a coupling is strongly suggested by the well known fact that the moments of inertia characterizing the spectra of nuclear collective rotational bands are significantly different from their rigid body value [6]. This is particularly the case for superdeformed nuclei. In 152Dy for instance, the experimental dynamical moment of inertia has been fitted in Ref. [7] as being equal to 0.75ORB ÷ 0.25Om, where the subscripts "RB" and "IR" label the rigid body and irrotational moments of inertia respectively. As proposed years ago by Bohr and Mottelson [6], such a renormalization is generally understood as resulting from residual interactions, for instance of the pairing type. This interpretation is of course not in contradiction with the introduction of intrinsic vortical modes. The latter may be seen as a translation of the effect of these residual interactions in terms of collective modes. As a matter of fact, the microscopic calculations of Ref. [8], introducing rather schematically some effects of the pairing residual interactions, clearly exhibit current line patterns which strongly deviate from the ones obtained in a rigid rotation. In addition to the pairing phenomenon, there are of course many other sources for intrinsic vortical motion, among which one may mention the rotational equivalent of the Landau diamagnetism, resulting in a retardation of the fermions near the surface as compared to the centre of the nucleus (see e.g. Ref. [9] and references quoted therein). For integral quantities like the moments of inertia, the quantitative analysis of Ref. [9] has demonstrated that this diamagnetic property is roughly compensated by a Pauli paramagnetic alignment of the spins. Yet, the currents are significantly affected by this surface behaviour.
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To illustrate the importance of the coupling of a global rotation with an intrinsic vortical motion, we have chosen here the simplest intrinsic vortical field in a somewhat generic yet crude description of vortical modes in deformed nuclei. As a matter of fact, the introduction of the Riemann linear velocity field in an ellipsoidally bounded fluid into Nuclear Physics has a long history since it has been proposed long ago by Cusson [10] as well as widely studied, classically and quantally, mostly by group theoretical methods, by various authors [ 11-16]. Most of the previous works, however, were lacking a proper balance between a quantal treatment of the collective flow properties on one hand of its coupling to the multitude of intrinsic degrees of freedom on the other hand. At the same time recent experimental advances, in particular with the operation of the new generation of gamma multidetector arrays such as EUROBALL or GAMMASPHERE, have revealed the crucial importance of the latter couplings which provides an incentive to push forward a microscopic study of the intrinsic vortical modes. Recently the concept of a vortical field linear in the coordinates has provided a theoretical frame for a possible explanation [17,18] of two rather surprising experimental facts found with EUROGAM and GAMMASPHERE, namely the AI = 2h staggering [ 19,20] and the identical bands [21 ] phenomena. If confirmed, the former tentative explanation has been shown to yield a nuclear analog of the Aharonov-Bohm phase quantization [22]. The nuclear modes considered here, are described within a rather simple model for the nuclear density. Clearly, one cannot ignore the importance of the existence of a smooth surface for the collective motions under study. As an example of this, let us just recall that, just above, we have alluded to one of its effects when dealing with an analog of the Landau diamagnetism. On the other hand, the well known leptodermous character of the nuclear density may justify a semiquantal treatment of these modes. It is the purpose of this paper to examine the coupling of global rotations with intrinsic vortical modes h la Chandrasekhar (S-type ellipsoids) in nuclei within an approximate semiquantal approach similar in spirit with the widely used routhian description of global rotations. To do so, the content will be organized as follows. The geometrical and kinematical significance of the velocity field of Eq. (12) and its coupling to global rotations will be explicited in Section 2, together with some hints on its possible generalization. Section 3 will present a semiquantal description of the coupling between the global rotation and the uniform intrinsic vortical motion in one particular coupling scheme, the so-called S-type Riemann ellipsoid case. Section 4 will be devoted to a self-consistent semiclassical solution of the corresponding generalized routhian problem and the results of a simple Harmonic Oscillator model will be presented in Sections 5 and 6. A quadratic approximation of the collective energy and some of its consequences will be discussed in Section 7. Finally some conclusions and directions lbr further investigations will be sketched in Section 8.
2. Some geometrical and kinematical considerations In what follows we will assume that the nucleus has an ellipsoidal shape. Let us consider now the velocity field of the uniform rotation of a sphere, defined by an
264
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angular velocity ~ . Let us scale both the distances and the velocity from the sphere to the considered ellipsoidai shape [5]. It is easy to check that such a field is nothing but the uniform intrinsic vortical fields of Eq. (12). We will now study the flow pattern associated with this field. It is, quite generally, defined by the current lines whose equations are dxl . .
.
HI
dx2 . .
dx3
H2
U3
(13)
The solutions of the above equation are ellipses. They result from the intersection of the plane 3
coigi
(14)
= 7],
i=1
with an ellipsoid homothetical with the boundary surface of equation 3
(15) i=1
where e and 77 are constants and the scaled coordinates are defined as Xi = x i / c i . Upon coupling the uniform vorticity field of Eq. (12) with a global rotation defined by an angular velocity vector g2, it has been shown (see e.g. Ref. [5] ) that the classically stable equilibrium solutions fall into two categories: o~ and g2 are collinear (the so-called S-type ellipsoids); ~ and g2 belong to the same plane of principal axis (the so-called P-type ellipsoids). As already pointed out [ 17], in the context of fastly rotating nuclei the S-type ellipsoids could, Ibr instance, correspond to a situation where the intrinsic and collective rotations are aligned (see Eq. ( 1 2 ) ) , whereas the P-type ellipsoids might rather describe tilted cranking situations studied in Ref. [23]. In what follows, we will limit our discussion to S-type ellipsoids, assuming that ~o and g2 are perpendicular to the (x, y) plane. To simplify the presentation we will also assume here the existence of an axial symmetry around the y-axis (generalizing to triaxial cases however, would be rather straightforward). In such a case, the divergence-free field u~ + Uv has the following components: -
-
Ux=-(s2+co(1/q))y,
u~. = ( s 2 + c o q ) x ,
ur=0,
(16)
where co and s2 are the projection on the z-axis of the corresponding vectors and q = (Cy/Cx) with ca: = cz. It is worth noting that the fields defined by Eq. (16) encompass quite a number of different flow patterns. Calculating as in Eq. (13) the lines of current, one finds the general quadratic curves (y2 + x 2 = r/,
(17)
where :7 is an integration constant and ~: is a dimensionless quantity, characteristic of the motion and given by
265
1.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304 ax u 1 +-- -IGx
g=
-@
~ -
ay 2 ax
~
Qy
~
Q]z ]
~ =-
~-- -100
%= -1
~ = +1
~ -
2
l + a ~al'l n-w a× Y~_ g=
0
2
ax 2 ay
'~ = +0~
Fig. 1. Flow patterns corresponding to various values of the dimensionless parameter ( describing the relative importance of global rotation versus intrinsic vortical motion, whose angular velocities are respectively ~Oand to, for a given prolate ellipsoidal shape. In this figure located in the (x,y) plane, J2 and to are aligned on the z-axis. The quantities ax and ay are semi-axis values on the x-axis and y-axis respectively,with ax > ay. (=
l+qh
l+qh
.
(18)
The vorticity of this field, i.e. rot ( u ) , is aligned with the common rotation axis and given by (rot ( u ) ) 3 = (s2 + qco) ( 1 + sc) .
(19)
As seen in Fig. 1 representing the flow pattern in a plane perpendicular to ~o and g2, the cases where ( is infinite or vanishing correspond to shear mode situations (without discontinuity of the velocity). For ~ going from - ~ to 0, the modes go from a shear mode in the plane defined by the collinear angular velocities and the symmetry axis to a mode where the corresponding plane is equatorial, through a purely irrotational mode for ( = - 1. The global or rigid body rotation corresponds to co = 0 or ( = 1. The purely intrinsic vortical mode occurs when 12 = 0 or ( = 1/q2. One may associate to any velocity field u ( r ) , a coordinate transformation upon summing infinitesimal displacements for all fluid elements located around r with respect to the time t, such that 6r = u ( r ) & .
(20)
Whenever u ( r ) is a linear function of some parameters generically called ~i, the infinitesimal displacement fields are thus obtained by replacing these ~i parameters by the infinitesimal quantities 6q~i = ~ i ~ t . In the particular case of the velocity field considered in Eq. ( 1 6 ) , the corresponding coordinate transformation T may be factorized [24] as
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I.N. Mikhailov et aL/Nuclear Physics A 627 (1997) 259-304 T = Rz (O)A(q)iCz ( O ) A - l q ,
(21)
where A ( q ) is the volume conserving scaling transformation from a sphere to the considered ellipsoidal shape defined by the axis ratio q, Rz (O) is the rotation of an angle O = s2t around the z-axis and Rz (O) is the rotation of an angle O = wt around the same axis of the corresponding scaled sphere. The above point (i.e. local) transformation T clearly conserves the intrinsic shape. It is possible to generalize the uniform intrinsic vortical field hypothesis of Eq. (16) in two different directions. One would like, indeed, to be able to deal with shapes differing from ellipsoids and also to deal with intrinsic vortical modes more general than what results from the rotation Rz (O) in the corresponding scaled sphere. There are, a priori, many ways of doing so. For the sake of illustration, we will briefly sketch here one possible approach. The main point is that we have to define divergencefree fields tangential to a given boundary surface. This may be easily formulated in some cases, in terms of curvilinear coordinates (i. Using the notation of Ref. [25] and assuming that the equation of the surface may be expressed in terms of the first curvilinear coordinate ~:j as scj = C ,
(22)
where C is a constant, a possible solution for such a velocity field could be
u(r)
= rot(aj~)
-
a2 0 - hlh3 ~(3
(hl~)
a3 (hl~), hlh2 0~2
(23)
where the three vectors a~ constitute the set of unit orthogonal vectors associated to the chosen curvilinear coordinates, ~ is a scalar function of the position r and the quantities hi a r e scale factors as defined in Ref. [25]. As a particular realization of the above, it is easy to generalize the transformation Rz (O) considered in Eq. (21). Indeed, noting that for a sphere of radius R, the Eq. (22) is simply r = R,
(24)
with the norm r of r playing the role of (l, the velocity field considered in Eq. (23) becomes u ( r ) = ~7~ × r ,
(25)
where ~ is, as above, a scalar function of the position r. Including in Eq. (16) this generic intrinsic vortical field Uv in lieu of the one defined in Eq. ( 1 2 ) , would clearly enrich the coupling scheme under discussion here. Releasing the assumption of an ellipsoidal shape now, would imply to introduce an equivalent of the volume conserving transformation A ( q ) associated with a given family of nuclear shapes.
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
267
3. F r o m a classical to a s e m i q u a n t a l description
In this section we will present and discuss a generalization of the Routhian approach widely used for a microscopic description of nuclear rotations [26]. By generalization we mean an extension of this framework to the most general vortical velocity field. However, here we will restrict ourselves to the modes defined in Eq. (16). We will sketch here the main ideas sustaining such an approach leaving a complete discussion of this approach to a forthcoming publication. A discussion within a purely classical framework will serve as an introduction for a semi-quantal description of the considered modes. Let us denote by E(J2, o2) the total energy for equilibrium classical states at given values of the angular velocities /2 and o2. Let us also define two single-valued functions I and J, of .(2 and w: l=l(O,w),
J = J(O,w),
(26)
allowing an inverse representation: £2= 12(I, J ) ,
w = ¢0(1, J ) ,
(27)
and satisfying the following property: 0 ~2= ~ ( E ( O ( l , J ) , o 2 ( I , J )
)) ,
0 o2= ~ ( E( g2(1, J),o2( l , J ) ) ) .
(28)
In classical mechanics Eqs. (28) define the time derivatives of generalized coordinates when 1 and J are the corresponding momenta. In our case, /2 and o2 are the angular velocities measuring the rates of change of the angles O and 0 introduced in the T matrix of Eq. (21) determining the displacements of fluid elements. Thus the quantities satisfying Eqs. (26) may be taken as conjugated momenta of these angles. The energy may thus be written as H ( I , J) = E(s2(I, J ) , co(l, J) ) .
(29)
To find the functions l(s'2, w) and J(s2, o2), we notice that d H ( l , J) = ~0(I, J) dl 4- o2(I, J) d J , dR(O, w) =
l(f2, w ) d O - J ( J 2 ,
w)dw,
(30)
where R(J2, co) is the generalized Routhian (associated to the Lagrangian for the two considered angular variables): R( ~ , w ) = H ( l ( O, co),J( $2, o2) ) - l ( f2, w ) 1 2 - J(12, w ) w .
(3l)
Differentiating Eq. (31) and using Eqs. (32) and 30, one finds a system of two equations:
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I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304 Ol c~J OE 0-an + (0T6 = gfi '
O ~I- + o ~ -a-J= - - .0E c~(0 &o a(0
(32)
The solution to Eqs. (32) may be given in an analytical form by expanding the collective energy in powers of the angular velocities: E( 0, (0) = ~
Im .,~ ,m2om~ °°"12 •
(33)
nil ,m2
Expanding also the functions I ( O , (0) and J ( O , (0) as I ( ~2, (0) = ~
• ~,'2.vtl i (O/712 t,,,,,m2
II11,1112
" ~(~nq Jm~,,,2
J ( O, oo ) =
O)"112
(34)
,
1111,m2
and equating the terms with the same power o f / 2 and (0 in Eqs. (32), one obtains the following equations for the coefficients imp,m2 and jml,m2: mliml,m2 + (ml + l )jm,+l,m2-1 = ( m l + l)Imml+l,m2 .
(m2 + 1)in,t-l.m2+1 + m2jm,,m2 = (m2 + l)Imm,..,2+l •
(35)
Upon taking into account the classical time-odd character of both I and J. Eq. (35) yields 1 ( 0, (0) = 2Im 2,0s2 + Im
4
3
2
2
1
3
(0 -}- 5Im 4,0 a'2 -'}-Im 3,1 02o0 + ~Im 2 , 2 0 ( 0 + 5Im 1,3(0 -- • • 2 2 + .{Im 3,1 ° - + . . J(O,(0) = 2 I m o . 2 ( . o + I m l , l O + 4Im0,4(03 + I m l , 3 ~ o 2 0 + ~Im2,2(00 1,1
(36) The Hamiltonian in Eq. (29) does not depend on the angular coordinates O and O7 and consequently the conjugated momenta I and J are conserved during the motion: {I,H} = {J,H} =0.
(37)
Therefore the evolution of the nucleus consists merely in a time-change of the angular variables governed by the following equations of motion: O={O,H}
= o,
0 = {O, H} = (0.
(38)
This situation changes when the nucleus is perturbed b y a quadrupole electric field. The latter adds to the Hamiltonian H(I, J) defined in Eq. ( 2 9 ) a perturbative term: /'/pert = Z aijQ(j, ij
(39)
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
269
where Qij is the usual inertia tensor (for the charge distribution) whose rotationally irreducible part is proportional to the electric quadrupole moment of the nucleus. In the presence of such a perturbation one obtains [ = {1, npert}
=
- -
Z ij
"~ij oOi)
,9(9
(40)
The perturbative part of the Hamiltonian does not depend on the angle O, and thus the conservation of the quantity J is not affected by the interaction of the nucleus with the electric field (in the lowest order approximation on r / a , where ,t is the electromagnetic wavelength): J = {J, Hpert} = 0 .
(41)
The last equation formulates a very important result, showing indeed that the quadrupole electric interaction does not change the quantity J. The routhian defined in Eq. (31) may be regarded as the hamiltonian resulting from a transformation into a coordinate frame moving with fluid elements [27], as we will discuss below. The reason for considering such a coordinate transformation is indeed, to introduce a reference frame moving with respect to the laboratory frame according to some velocity pattern. This allows us to build a family of solutions exhibiting the type of collective currents which we want to study. Let us proceed now towards a quantal description from what we may learn from classical mechanics. In this case, performing a change of coordinates, one should add a so-called generating function G to the Hamiltonian, so as to preserve the mathematical structure of the Hamilton equations of motion [27]. In the particular case of a local transformation (i.e. where the new coordinates do not depend on the momenta) this generating function G writes as G = a .p,
(42)
where ~e is a (local) vector field which is equal to minus the collective velocity associated with the considered transformation, as for instance those discussed in Section 2. In quantum mechanics an equivalent of the above considered canonical transformation, may be obtained by applying to the wavefunction of the system a unitary transformation. Restricting the discussion here to the Hartree-Fock approximation, a solution p of a TDHF equation is transformed into /5 by the so-called Thouless unitary transformation defined by a hermitian matrix S as /5 = eiS p e - i s .
(43)
Defining the Hartree-Fock Hamiltonian ,~ as obtained with/5 from the two-body Hamiltonian, one gets the following transformed TDHF equation [[~ - S , / 5 ]
= ih~,
where the "cranking" potential S writes
(44)
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I.N. Mikhailov et aL/Nuclear Physics A 627 (1997) 259-304 S = ihe i s d (e -is) •
(45)
We consider now the particular case where the time dependence of S may be factorized out; - the hermitian operator S depends linearly on the momentum operator p. One has thus -
S =/3(r.p
+ p. r)/2,
(46)
where/3 is a real function of the time and F is a (constant in time) local vector field. Both restrictions allow one to consider e is a s a quantal analog of a local transformation associated with a velocity field depending linearly on some generalized velocity parameter as considered in Section 2. This analogy is substantiated by the fact that the transformed operator of r does not imply in this case the momentum operator p and also by the fact that the Wigner transform of S writes Sw = / ~ / F - p .
(47)
If one wants to approximately describe in a microscopic fashion, the quantal analog of a motion whose classical analog is a transtbrmation associated with the velocity field oe = - / ~ F , one may solve the constrained TDHF equation (44) with the cranking potential S defined in Eq. (47). The stationary solutions of the TDHF equations of the type of Eq. (44) amount therefore to solving the generalized Routhian stationary problem 8(H+G)
=0,
(48)
where G is the quantal generalization of what has been defined in Eq. (42). In the case considered here, one takes for the vector field ~ minus the collective velocity of Eq. (16). Its solutions are thus functions of the two angular velocities g2 and o~. This holds in particular for the energy. This quantity depends explicitly on the two angular velocities as a quadratic function, but also implicitly through the coupling of the one-body potential coming from the generating function with the twobody hamiltonian in the Hartree-Fock self-consistent solution process (e.g. through deformation adjustments). In this context, it is worth emphasizing a consequence of the vanishing contribution to the time-derivative of the diagonal part p of the density matrix from the intrinsic vortical current, as demonstrated above from Eq. (11). As a result, the modes associated to the field uv should not contribute to the definition of the equilibrium shape, at least in the sharp-edged density limit where this result has been obtained. In realistic leptodermous cases, their influence on the deformation should be small. One therefore expects the w-dependence of the energy to be a priori closer to a quadratic expression, than it is the case for the O-dependence. We will now, describe briefly some properties of the quantization of the energy in Eq. (29). Our present discussion will be mostly qualitative. However, this subject has
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
271
been treated in great details in Refs. [ 12,13,16], for instance. First of al, the fact that the mass parameter entering the purely kinematical energy of Eq. (29) does not imply the associated variables (the angles O and O) frees us from any ambiguity in the quantization process. As usual, a quantal description of the collective motion involved in such a model may be obtained by substituting the Poisson bracket relations by the commutator relations to yield h [Z,o] = [ L O ] = _ , 1
[Lo] = [),o] =o, [ [, J] = 0.
(49)
Substituting the classical momenta I and J by their quantal counterparts [ and ] in the energy (29) one obtains the quantal analogue of the classical model for S-type Riemann ellipsoids. The conservation of the classical collective momenta [ and J established in Eq. (37) yields the following commutation relations. [ B , [ ] = [/4, J ] 0 .
(50)
Taking into account the last commutation relation in Eqs. (49), one sees that the quantal states so obtained may be labelled by the eigenvalues of the operators squared which we shall specify by I and J (with eigenvalues I(1 + h) and J ( J + h) as usual). Finally, from the first commutation relation in Eqs. (49), one obtains the standard commutation relation boy [1, f ( O ) ] - i 30
(51)
showing that the operator f is to be interpreted as the z-projection of the orbital angular momentum operator of the nucleus. The corresponding eigenvalue problem may thus be formulated as the periodicity condition of the wavefunction. ~ ( I , J) ~ exp ( i ( l O + JO) )
(52)
at the endpoints of the definition interval of the angle variables O and O. This interval depends on the symmetry of the mean field. Let us consider here the case of even-even nuclei (the generalization to other nuclei has been considered in Ref. [18]). In our ellipsoidal case, the mean field is C2-symmetrical (i.e. with respect to the rotation by an angle 7r around the collective rotation axis). Thus, neither of the two relations Rz (Tr) and Rz (Tr) in Eq. (21) changes the physical state of the system. Thus the periodicity condition is formulated in this case as: exp(ilTr) = exp(iJTr) = 1 ,
(53)
from where it follows that both 1 and J quantum numbers are even. We notice that the commutation relations (50) constitute a part of the commutator algebra of the so-called O ( N - 1)-invariant model [ 16]. In the latter, the J-quantum number is unequivocally
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
272
defined as a quantum number of an intrinsic state. Having this in mind, one may call the present approach as a spontaneously broken O ( N - 1) symmetry model.
4. A semiclassical approximation In this section we will give close tbrm expressions for various quantities related to the solutions of the generalized routhian or cranking variational problem of Eq. (48), in a semiclassical limit. By semiclassical we mean here an approximation h la Wigner [28] corresponding to a truncated h expansion (up to second order terms). More specifically we will work within the so-called Extended Thomas-Fermi approach [29]. It yields functional relations between various local densities carrying informations on the nonlocal part of the density matrix and its diagonal (in r) part, the local density function p(r). These densities are the kinetic energy densities ~-o(r), the so-called spin-orbit densities J,,(r), the current densities j r ( r ) , the spin vector densities pv(r), where v stands for the considered charge state. They are sufficient, when using a Skyrme type effective force, to express the expectation value of the energy of any Slater determinant as an integral over the whole space of an energy density H in the lab frame. (see for instance, Ref. [30] where one may also find the exact definitions of these densities and of 7-{). General expressions for such functional relations have been explicitly derived in Ref. [31 ], in the case of a routhian approach restricted to the global rotation. In another paper [32], they have been recently extended to the most general cranking field, or generating function, linear in p, as in Eq. (42). Therefore we will merely sketch here the results obtained for the mass parameters in the general case and apply them to the particular vortical velocity field uR 4- uv of Eq. (16), which is the quantity ol = - / ~ / ' , where F and/3 have been defined in Eq. (46) and the vector potential a in Eq. (42). Let us first ignore the spin degrees of freedom. We will decompose the integral expression of the Slater determinant energy, when using a Skyrme effective force. Namely, to obtain the generalized inertia parameters, one identifies 7~z, the part of which 7-/ is quadratic in or. As a matter of fact, one may follow exactly the same path of calculations as in Section 3.1 of Ref. [31] to obtain
(54) u
where the quantity 6(2)(rv) is the second order correction in h of r,,(r) and f,, is the usual effective mass form factor associated with a Skyrme effective force. One notices that the first (Thomas-Fermi) term corresponds, as expected, to the classical collective energy. We have shown in Ref. [32] that
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
6(2)('/'v) = (37.r2)-2/3m2
{
1/3 [(rOt~)2 [-- ~
273
~ 3 I } ~ . (rotrotae) .
Pv"
(55)
2h 2
+Z. [rot × V( vp!./3)l Jv t
Let us assume now that the constraining field o~ results from several external fields a = ~-~' a i .
(56)
i
As a trivial consequence of the quadratic character of ~2 in a, one may write (with an obvious notation and i, j, labelling the various modes) 7~2 = ~ 2 ( / )
+ ~2(i,j).
i
(57)
i
The uncoupled terms are readily obtained upon replacing a by ai in Eqs. (54), (55). The coupling terms are given by ( rnp~ai . a j + B']-{2(v; i , j ) ) .
7~2( i, j ) = Z
(58)
q
with (3~ 2) -2/3 m b'7-C2(v; i , j ) =
4 [ fvpJo/3 (rot ~i rot aej - ~fi rot rot a j - aei • rot rot ~lfi) "
"
+ V ( f v p l v / 3 ) • ( a i . r o t a j + a j • rot ~i) .
(59)
One thus gets inertia parameters lot the coupling of several modes whenever o~ results from their addition, upon integrating over r the 7{2(i) terms and dividing the results by fl~/2 where/3i is the collective mode velocity (for the coupling terms one would of course divide instead the result of the integral of 7J2 (i, j ) by/~i/3j). When taking the spin degrees of freedom into account, one should evaluate first the spin-vector densities p , , ( r ) (as defined for instance in Eq. ( [ 11] ) of Ref. [31] ). They are solutions of the following linear system of equations p,,
l+~p,,
(X+Y)
+p~,\T~,
r,,,
-
+Z,,rota~
, (60)
where ~, stands for the other charge state with respect to the one defined by v. In Eq. (60) also, the constants X, Y, Z,, and /_t are defined from Skyrme force parameters and from the densities Pv (note that p = ~-~,,p,,), as: X=Bio + peBl2, Y=BI1 + pYBj3, Zt,
=--
mB9 .
h2 ( P + P~,),
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
274
/z =
6m ( 37re) - 2/3 h2 ,
(61)
(see e.g. Ref. [30] for the definition of the exponent y of the density dependence of the force and of the Bi coefficients). In Eq. (60), the term in g2 comes from the global rotation in the lab frame which has been assumed, so that a spinconstraining field -S2. s has been added to ¢e • p. In our case where ~ is aligned on rotce spin densities for neutron and protons are also aligned on g2. Including the spin degrees of freedom, one gets a supplementary part (7-/2)s into 7-[2 namely: h
(']-{2) s = - - ~
Z Z,,rot a
(62)
'P,,
1)
exhibiting a coupling term between space and spin degrees of freedom. Let us now apply the previous formalism to our S-type ellipsoidal modes. We will ignore from now on the spin degrees of freedom. It is worth noting that, by construction 7"(2 being quadratic in the aei's, is also quadratic in $2 and ¢o, so that one can define the collective rotational energy as E = ~i A w 2 + B w a Q + 1 Cf22 .
(63)
Of course, due to the coupling in the variational Eq. (48) between the time-odd part of the density matrix and its time-even part, the latter and thus the non-collective energy is also dependent on ,(2 and w, as well as the moments of inertia A, B and C. The density will be schematically described as being constant and bounded by a sharp edged ellipsoid whose semi axes are given by a,. = aoq 2/3 ,
(64)
ax = a~.. = aoq -U3 ,
where a0 is given in terms of the total number of particles N and of the usual size parameter r0, by a0 = ro Nj/3, whereas q is now the ratio of ax and ay. We will also use below the following unit for the moments of inertia 2 ~,2 Ar5/'3~l/3 r / = ~ .... 0'" '~ ,
(65)
which is nothing but the rigid body moment of inertia tbr a sphere multiplied by the shape factor ql/3. The various inertia parameters are given below to order h 2 (the first terms being the Thomas-Fermi estimates): A=~70[1-(D)
oI,
B='r/[l
(D)
o 1,
C=T/O[I-(ff-~)
l , (66)
where O is a geometrical factor given by O = 1/2(q + l / q ) and the dimensionless semiclassical expansion factor D/~7 is given for this mode by D=-(37r2)-2/3mZ//./fv(r)pl//3(r)d3r, tl
(67)
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
275
which is approximated within our simplified model for the density (assuming furthermore that p,, = pp) in terms of the isoscalar effective mass in nuclear matter (m*/m)NM of the Skyrme force by -D- =
( 8 "~2/3 5
(
i
~~m*m / )~MN-23/).
(68)
n
It is important to note that its inverse surface dependence ( N -2/3) makes D / ~ small with respect to 1 in heavy nuclei. In our case as in the classical case (see Ref. [ 13] ), the kinematic moment of inertia of S-type Riemann ellipsoids is given, in terms of the rigid body and irrotational moments of inertia, with the notation used previously in Section 1, by -
I = 0 = rORB q- ( 1 -- r)OlR , /2
(69)
-
where w C r= 1 + /2B'
B2 OIR=C
C '
ORB
C.
(70)
Assuming for instance r = 0.75, as it has been fitted by Swiatecki [7] for the first discovered super deformed band in lS2Dy and q to be equal to two, one gets a ratio c o / ~ of about - 2 0 % . Of course, the previous estimate relies on the assumption that the reduction of the moment of inertia from its rigid body value can indeed be all explained in terms of an intrinsic vortical motion.
5. A s i m p l e m o d e l case: the h a r m o n i c oscillator m e a n field
In this section, we will illustrate the above discussed semiquantal approach, by considering its application in a (non-self-consistent) simple model where the mean field is described by a harmonic oscillator potential. In spirit, the following is very close to what has been done in Ref. [ 14] by Rosensteel, excepted for two important points. Namely, we take here into account properly the volume conservation as well as the quantal character of J. In the following, we will assume that the two rotation axes (for global and intrinsic rotations) are collinear with the x-axis. The one-body generalized Routhian h(g2, w) whose eigenfunctions will be used to build the requested Slater determinant solution is written as: p2 m h( g2, w ) = ~m + -2 (wxx2 + wyy2 + w z z - ) - g2L,. - w J x ,
(71)
where Lx and Jx are the first components of the orbital angular momentum L and of the operator J introduced in Section 3, also called, in Ref. [14] for instance, the Kelvin circulation. In this particular case, as it may be easily interred from our previous presentation of the intrinsic vortical mode, the Kelvin circulation is nothing but the
276
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
operator L after a double stretching in positions and momenta. One has thus, in the r-representation J,=-ih
qY~z
q-Jz
(72)
,
with q = c / b , where the semi-axis of the ellipsoid are called a, b and c respectively. We express now h(.O, 0)) in terms of the usual boson operators ( a / , a i ) as i=3
h0)i(a+ai + 1/2) - i h S ( a ] a 3 - a~a2) - i h D ( a ~ a f
h(.O, 0)) = ~
- a2a3),
(73)
i=1
with 8---
D =
.O(0)3 -}- 0)2) ÷ c0(q0)3 + 1 (1/q)0)2) 2 WV/~7~ ~2( 0)3 - 0)2) ÷ 0)(q0)3 - ( I / q ) 0 ) 2 )
(74)
2 vmT 3
Since this operator depends quadratically on the creation and annihilation operators (a +, ai), one can perform a Bogoliubov transformation restricted to the boson operators in the (y, z) plane i=3 + % =
Vk = 2, 3;
~-~/&i a+ + #kiai),
(75)
i=2
such that one has Vk = 2,3;
~+k = [h(.O, 0)),~k],+
(76)
which corresponds to an homogeneous system of linear equations such that o92 -- -ok -iS 0 iD iS w3 - -ok iD 0 0 -iD 0) 2 + .Ok iS -iD 0 -iS 0)3 + .Ok
ak2 "~ ,~k3 I
(77)
]'Zk2 ]
IZk3 /
This system admits non-trivial solutions only when the determinant of the above matrix is vanishing. This gives a second order equation in -O2 yielding thus two positive eigenfrequencies -O2 and ,'23. The former is given by -
-
+ 2
"
+ ($2 -- D 2 ) + A2,
(78)
where the quantity A2 is given as =
~
÷ ( S 2 -- D 2 ) ( 0 ) 2 ÷ w 3 ) -r- 20)20)3(S 2 ÷ D 2)
.
(79)
The eigenfrequency -O3, is deduced from the above eigenfrequency g22 upon replacing A2 by - A 2 in Eq. (78).
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
277
Such a transformation allows us to write h(s2, w) in a canonical form i=3
i=2
Some further details about this transformation are given in Appendix A. In what follows we will calculate the equilibrium solution for a given harmonic oscillator configuration of N particles (i.e. a set of spin, isospin and eigenvalues of the particle number operator ni in each directions, for each fermion) at a given value of I and J (dimensioned as h) as it will be defined below. There are five variables in the problem: the three harmonic oscillator frequencies w i and the two angular velocities 12 and w. For each set of these variables and for a given configuration we may calculate trivially the value of the total generalized routhian, and evaluate (see Appendix B), the expectation values of L x and J~, from which one deduces readily the corresponding energy. We can also calculate the three expectation values ( x ~ / N } whose positive square roots are identified with the three characteristic lengths a, b and c. Some details about the analytical expressions for these expectation values are given in Appendix B. We are thus left with a variational problem implying the above listed five variables which are subject to the following three constraints: the volume conservation enforced in a quite natural way whatever the values of the deformations and of the angular velocities g2 and w by imposing -
47"r
--abc
3
-
(V being a constant fixed once for all by scale arguments or by the actual radius value if known for the considered nucleus) ; the value of the angular momentum given (for small values of the projection K of the total angular momentum on the symmetry axis, as expected for the rotationally aligned superdeformed states, and at any rate only in average) by (Lx} = ~ +
-
(81)
= V,
h) ;
(82)
the value of the Kelvin circulation quantum number again given in average by (J~> =
v/J(J + h).
(83)
In the absence of any time-odd constraint, the question of volume conservation in the harmonic oscillator model is readily seen [3], to be expressed in terms of a constant w0 (fitted so as to provide a good scale) as o91o92o93 = o93 .
(84)
The volume conservation condition has attracted a lot of attention when used in usual Routhian-type calculations [33,34]. Basically two prescriptions have been used. The first is in use here (see Eq. (81)), another results merely from taking into account in the routhian case the above Eq. (84). This has been used for instance in Ref. [14]. It
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
278
has been shown in Ref. [34] that some attention should be paid when using a Lagrange multiplier technique to enforce the volume conservation. It has to be stressed that we are not concerned here by this problem since we do not use Lagrange multiplier to guarantee the volume conservation, but rather perform the routhian minimization within a space of volume-conserved (according to our prescription) density matrices, as we will specify below. Indeed, to provide a solution of our variational problem for a given harmonic oscillator configuration, we proceed for each quantized value of I and J in the following iterative way. Starting from some fixed values s2~n) and w ~') of the angular velocities, we minimize the generalized routhian in the two dimensional space of triaxial shapes defined by a, b and c, upon taking into account the volume conservation condition of Eq. (81 ). Having thus fully specified the variational solution, we compute (see Appendix B) the expectation values 1 (n) and j(n) of Lx and Jx at this stage of the iteration, from which we infer by the gradient method new values s2~n+l~ and w ~n+j) for the angular velocities. We then re-iterate the above process until a convergence is reached (namely whenever both (I - 1 ~')) and ( J - j~n)) falls below 10-sh). As a result, the Lagrange multiplier associated with the quantity (Lx) which is conserved during the variation, is undoubtedly the angular velocity corresponding to the global rotation from the lab system to the body-fixed system, in the usual routhian case. This conclusion may be generalized to the coupling case between the two vortical modes as considered here. One may further note that it seems hopeless in such a coupled mode, to yield an analytical expression for the variational condition as it was the case for the usual routhian. This is due to the rather involved character of the expectation value (Jx) since the Kelvin circulation operator Jx, itself, depends on the variational quantities (wj, w2, w3, w, s2) through the b and c deformation parameters in quite a complicated fashion (see Appendix B). We still have to give the rationale behind the choice which we have made of the prescription for the volume conservation given in Eq. (81). Indeed this equation is a constraint on the nuclear density which is grounded on the high value (relatively to similar quantities for other bulk or surface modes) of the incompressibility modulus in nuclear matter. Eq. (84) corresponds to the conservation of the volume enclosed by some equipotential surfaces, more specifically as far as that part of the potential which is momentum-independent is concerned. While such a choice entails clearly substantial simplifications in the numerical treatment, its physical motivation is not completely clear to us. It is worth noting that the expectation values of the squared components of r and of p, as well as those of Lx and Yx, depend exclusively on the harmonic oscillator configuration through the following sums (see Appendix B)
j= 1
where the sum runs over all the N occupied fermion states labelled by j and n~j) is the
279
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
number of quanta in the /-direction for the j-fermion. The moments of inertia A, B and C, are clearly very important quantities to determine out of our microscopic collective model. In general they can be obtained by making a local quadratic fit of the collective energy as a function of the moments I and J, according to the following formula which will be justified below in Section 6 E ( I , J) - ~A C
B
12 - B I J +
j2
(86)
,
or as a function of the angular velocities s2 and 0) (see Eq. (63)). Equivalently one may deduce A, B and C from the numerical calculation of second order derivatives of the energy with respect to the 1 and J, or J2 and w, variables. Approximate moments may also be evaluated within the Inglis cranking formula [35] which may be viewed as a perturbative approach in two different ways. In general it corresponds to neglecting the effect of the time-odd part of the density matrix (or equivalently of the currents) on the Hartree-Fock mean field, thus ignoring the socalled Thouless-Valatin corrective terms [36]. Clearly such time-odd potentials are not taken into account anyway in such phenomenological mean field calculations. On the other hand, this approach neglects also second and higher order effects, in the angular velocities, affecting the time-even part of the local density, as e,g. rotation induced stretching deformations. Indeed it provides the linear response of the time-even part of the variational solutions at given values of the angular velocities ~ and 0), keeping it fixed under the action of the time-odd constraining field. We will label these moments as Aio BIC and Clo Their analytical expressions are given in Appendix C. numerically. Around vanishing values of the two angular velocities, the resulting limits of the moments are labelled A°o B°c and Cl°c. They must of course be identical to the values A, B and C yielded by the fit in that angular velocity domain. Extending what is known [37] in the pure global rotation case to our coupling case, one finds (see Appendix C) for A°c (upon assuming that the single particle states are such that the ensembles of the number of quanta in each direction varies from 0 to a maximum value without gaps) ]Z
A°c
-
20)20)3
[(0)2 -- 0)3) 2 (~'2 -~- ~'3) -- (0)2 q- 093)2 (,--~3 -- --~2)] L O92 -~- 0)3 O92 -- 0)3 J
(87)
m
Similarly one gets for C°c cO_
h 20)20)3
- b
_ c
2
(c0)2 ~0)3) (,--~2 -}- ,--~3) -~ 0)2 "+" 0)3
( 0)2 +
0)3) 2
]
( 2 3 -- ~--~2) •
O92 -- (03
(88)
Finally the coupling moment B°c is written as h B°lC- 20)20)3 %
X
(~0)2-
~0)3)(0920)2 + o93
0)3) (,~2 -~- ~"3) q-
0)2 +
0)3) (0)2 + 0)3) 0)2 -- 0)3
(,-~3 -- 2 2 ) ]
-
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
280
(89) In the limit of vanishing values of the angular velocities 82 and w, one searches for the linear response of the static solution which is well known to be determined by the self-consistency relation (90)
~-~l¢01 -----~ ' 2 W 2 = ~ ' 3 0 ) 3 ,
and the following relation for the semi-axis value a
a2 = 5h
(91)
~'1,
m¢ol and similar values for b and c. One deduces from the above that in this limiting case b o~3 . . . .
~.
C
-~3
~'2
402
(92)
Using these results one finds for the moments Ai°o B°c and C°c A°c=CI°c=~z
~2-2+~7
,
B°c=~
~22+~33
.
(93)
Finally one deduces in particular that B°c
co
- -
-
2
q+ l/q
- O,
(94)
which is exactly the Thomas-Fermi result found in the previous section (see Eq. (66)). It may be noted that the equivalence of Inglis cranking masses and rigid body masses upon making use of the self-consistency condition, was known already in the pure global rotation case [3].
6. Discussion
of the results
of some
illustrative
calculations
In what follows, we will discuss the results obtained for the nucleus defined by Z = 80 and N = 110, i.e. for 19°Hg nucleus. Spin degrees of freedom are present to double the available space. Yet, within this simple model, they are not active in the dynamics. The parity and x-signature operators do commute with the generalized routhian R. As a matter of fact, one should in principle compute expectation values of R not for single particle states (using a straightforward standard notation) such as
li~vi) = Z
x":i nAInznA)l~vi)'
(95)
IT:nA
but rather for the good signature states i
lisi) =
O-n~na~X"-nA (In2na) lsi) + si(- )n=tnzn _ A) I _
~i))
(96)
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Table 1 Definition at zero spin of the retained configurations. They correspond to three very different deformation regions (weak deformation, WD; super-deformation, SD; hyperdeformation, HD). The quantites ~ are defined in the text together with the three harmonic oscillator frequencies hwl (in MeV) and the three semi-axes (in fm)
WD SD HD
~n ~1
~"P ~1
~n ~2
~P ~2
~--'n ~3
~"P ~3
177 155 139
116 100 94
177 155 123
116 100 76
227 295 421
148 200 286
h,'~o I
ht02
]ZO)3
a
b
c
7 . 7 4 2 7 . 7 4 2 6.049 6.427 6.427 8.225 8.895 8.895 4.582 5.594 5.594 10.858 9.854 11.538 3.248 5.080 4.339 15.414
(si denoting the signature of the /-state). However, since the spin-up and spin-down components of a single particle state should not couple in our simple model (no explicit spin dependence in R), this choice does not matter since one obviously gets (iXilRliXi) = (isilRlisi) .
(97)
We have chosen to study three different deformation regions corresponding to weakly deformed ( W D ) shapes with q ~ 1.32, super-deformed ( S D ) shapes with q ~ 1.94 and hyper-deformed ( H D ) shapes with q ~ 3.55. For that purpose we have selected three configurations which minimizes the harmonic oscillator hamiltonian for vanishing angular velocities and satisfy the self-consistency condition (90), whose characteristics ~ ' , ~oi, a, b, c ( i = 1, 2, 3; v = n , p ) are given in Table 1 It is to be noted that by no means the selected solutions are the only possible ones. For instance between the values of q corresponding to the W D and SD solutions, there are three other prolate self-consistent solutions (with the following values of the q parameter: 1.36, 1.55, 1.78). The HD solution is significantly triaxial. It has been retained due to its resulting value for the ratio of the two moments of inertia B / C for reasons which will appear clearly later. Of course this arbitrary choice is merely made to provide a sampling of rather different type of solutions. Anyway, we are not intending to achieve completeness in such a crude model but rather to get qualitative trends. We then describe states with non-vanishing values of 1 and J, in particular on the so-called yrast line (lowest energy state with respect to J for a fixed value o f 1) by choosing as above described the appropriate angular velocity values. In this process, we may have to change the configuration to keep the total routhian at its minimum. In such a case, there is some ambiguity in choosing the configuration which will replace at spin 1 + 2h the configuration retained at spin 1. In this work, we have adopted at spin 1 ÷ 2h, the configuration that minimizes the routhian R when keeping the deformation parameters at the values they had at spin 1. However, the fully variational values of the parameters for the new configuration will be in general different from the previous ones. Then for this new deformation one should check that such a configuration is optimal and if not change again the configuration accordingly. While the configuration of the HD and SD solutions remain stable within the range of calculated /-values (from 0 to 80h), it is changing very much tbr the W D solutions when I > 40h. The retained configurations for the W D solution are listed in Table 2. The resulting values o f q = c / b
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
282
Table 2 Definition as a function of the spin of the retained WD configurations by their corresponding quantites ~'~' defined in the text 1/h
~n ~1
='p ~1
_--n ~2
~p ~2
~n ~3
~p ~3
0-40 42-46 48-56 58-60 62-72 74-80
185 189 189 189 183 185
ll8 ll8 124 122 122 120
169 161 157 157 151 145
114 ll4 106 100 100 96
227 233 239 239 257 265
148 148 152 164 164 174
~3.6
]
" HD F 3.2 ~ .J!
SD
1.8
/' /,,
1.4
WD
1.2
HD
1.1
WD
/
/
s
F
.'
s
/
!
/''
' SD 0
20
40
60
80
1/14 Fig. 2. Equilibrium deformation parameters along the yrast line as functions of the total angular mmnentum 1 for each of the three retained configurations (WD, SD, HD). Yrast is understood here and in the next figures, in a restricted sense, i.e. as the solution having the lowest energy at a given spin for a given configuration. The parameters q and q~ are defined in terms of the semi-axes a, b and c as q = c/b and q~ = a/b. a n d q / = b / a a l o n g t h e yrast line, are d i s p l a y e d , as f u n c t i o n s o f I, in Fig. 2. In Fig. 3 the y r a s t values o f the K e l v i n c i r c u l a t i o n J as a f u n c t i o n o f 1, d e n o t e d Jyrast(1), are s h o w n for the t h r e e s t u d i e d s o l u t i o n s . F o r the W D s o l u t i o n , o n e sees t h a t J ~ I w i t h o n e a c c i d e n t a r o u n d 1 = 2 0 h . A b o v e I ~ 50h, the s o m e w h a t r e g u l a r p a t t e r n o f Jyrast(1) is b l u r r e d by the p r e v i o u s l y d i s c u s s e d c h a n g e s o f c o n f i g u r a t i o n s . B e l o w this spin value, the o b s e r v e d b e h a v i o u r o f Jyrast(1) m a y be h i n t e d to r e s u l t f r o m its d e p e n d e n c e o n I, w h i c h w o u l d b e o f the type Jyrast(1) ~ [ x l ] ,
(98)
l.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
60
20
WD
I
283
~,~*
°°
.~
i
j ~
20
.~,'"
,
.o.~ •
40
i !
HD
.f 20
o
20
40
60
80
I/kl Fig. 3. Quantal yrast values of the Kelvin circulation as functions of the total angular momentum 1 for each of the three retained configurations (WD, SD, HD). where x is a real number slightly smaller than I and [y] stands for the integer part of y. This explanation will be further substantiated in the next section. The curves Jyrast (1) for the SD and HD solutions might be explained in a rather similar fashion, with a x-value, however, which is much smaller than 1. In the HD case, one finds a remarkable periodic dependence whose period would be very close to 4h, thus consistent with x ~ 1/2. The curves E ( 1 ) giving the total energy at a given spin value 1 for the three considered solutions are plotted on Fig. 4. From one deformation to another, they obviously exhibit different curvatures yt related to the trivial deformation dependence of the moments of inertia O, namely,
w h e r e / 3 denotes the usual axial quadrupole deformation parameter. It is also to be noted that the HD solution is very far indeed from the yrast line and is only considered here merely for the purpose of varying widely the deformation. Let us discuss now the spectra in both 1 and J associated with one configuration, e.g. the yrast HD configuration which is, as we have seen unambiguously defined. As shown in Fig. 5, the states sharing the same J-value fit very nicely into parabolic curves whose minima are located at higher and h i g h e r / - v a l u e s upon increasing J. This feature displayed in Fig. 5 at rather low values of 1 and J, is also present for larger values of the two moments as shown in Fig. 6. This seems to be a rather general phenomenon and is
284
1.N. M i k h a i l o v
et al./Nuclear
Physics
A 627 (1997)
259-304
120
HD ~o /
/
/
WD 40 ¸
•
y/l•
.-/ ~
SD
0 0
. . . . . . . . 2 .0 . . . .
4 ~0
. . . . . . . 6 .0 .
8'0
1/14 Fig. 4. Total energy curves as functions of the total angular momentum 1 for each of the three retained configurations (WD,SD,HD).
400
j/o/ 200
J = 8 ¢t
o
fz :'
J=61~
ii 2
J = 4 ]~ 0 [
t 4
1 6
o o
• 10
[
12
114 1/14
Fig. 5. Total energy curves corresponding to the HD configuration and given values of the Kelvin circulation as functions of the total angular momentum 1 in the 4-8 h range.
found also in light nuclei. In Fig. 7 we have plotted similar energy curves for the ground state configuration of ZONe, making contact thus with the work of Rosensteel [ 14] with the important difference, as already noted, of limiting ourselves to even values of the Kelvin circulation. Let us turn now to the discussion of inertia parameters A, B and C. For the three considered deformations we have made local quadratic fits of the energy yielding thus these parameters around the yrast line. The fits whose results are displayed in Figs. 8-10, have been made either assuming an energy quadratic in the moments (see Eq. ( 8 6 ) ) or in the angular velocities (see Eq. ( 6 3 ) ) . Both fits have been performed within intervals of the relevant quantities which are much smaller than what is needed to vary 1 or J by 2h. They both give exactly the same values for the inertia parameters. In Figs. 8-10 the Inglis cranking values (see Eqs. (87), (88), (89)) are also reported. As expected
285
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
,
i
".'. ,.
/'
.'
,' .;
'. ,
,' ".,
//
/ /'
,
,,'
.;
,
('
"'~\
/ \ J = 3 4 1~
-5
62
/' \ J=36 ~
/
/
J = 3 8 ]'l
J = 4 0 k~
68
74
I
80
Fig. 6. Same as Fig. 5 at higher/-values (namely in the 34-40 h range). The plotted energy results from the substraction of a continuous approximation of the yrast energy obtained by releasing the restriction to (even) integer values for J/h.
40l j = 6 k~ ~
/
/
J = 2 1{
.-I
.....
0
2
4
'6 l/lt
Fig. 7. Same as Fig. 5 for the 2°Ne nucleus in the 2-6 h range for the Kelvin circulation J. plotted energies are defined by difference with the ground state (1 = J = 0) energy.
they exactly coincide with the fitted values for vanishing angular velocities. One gets thus for A, B and C the relations previously discussed in Eqs. (93), (94) for A°¢, B°¢ and C°c. For finite values of S2 and co now, the linear response estimates ~ la Inglis, i.e. neglecting the rearrangement of the time-even part of the density matrix, are also plotted. One observes for the fitted values of A, B and C, a slight angular momentum dependence which is of course not to be understood as a quadrupole deformation effect (since it is constant with I in cases where the configuration is kept identical, see Fig. 2), for which we have not found yet any obvious explanation. However, the ratio B/C or B/A are found to be much more stable. For the HD solution for instance, one finds indeed that, when I increases from 0 to 80h, B/A does not change from its zero spin value of 0.42 up to some tenth of one per cent and B/C does not vary from the same value
286
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304 "7,
IO0
~.....//"~-./ "~ 5O
125
~
~"-~_~
230 [
"~.~
.
0
.
.
2o
.
.
.
.
4o
.
.
]
-
.
.
.
60
80
I/I~ Fig. 8. Inertia parameters A (in h2-MeV-1 units) as functions of the total angular momentum 1 along the yrast line for each of the three retained configurations (WD, SD, HD). The dots result from a quadratic fit of the collective kinetic energy. The other curves correspond to the Inglis cranking formula values defined in the text. At the limit of vanishing angular momenta, the two curves coincide and are given by the Eq. (93). These asymptotic values are represented as horizontal lines. more than by some percent. Kinetic moments of inertia, defined in terms o f the frequency S2(1), necessary to yield the spin value as Ot~ n = l / S 2 ( 1 )
(99)
are given for the three retained solutions in Fig. 11. On this figure the subscript "th" indicates that we have used the Lagrange multiplier value ~O(I) for the angular velocity, as opposed to the usual "experimental" estimate of s2(1) $'2exp(l ) = E , / ( 1 ) / 2 h ,
(100)
which is nothing but a consequence of the classical Hamilton relation between the variables 1 and s2 in Eqs. (28) and where E~,(1) is the transition energy to the yrast state of spin 1 from the yrast state of spin I + 2h. From the above "experimental" angular velocity, one may deduce "experimental" kinetic angular moments of inertia o kexp in =
i/f2exp(1)
( 101 )
which are displayed in Fig. 12 and found roughly similar to their "theoretical" counterparts of Fig. 11. This fact reflects the similarity in the general trend of the dependence
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
287
75 50
100 80 60 120
115 ,
0
I
20
,
,
,
L
,
,
,
40
I
60
80
I/t~ Fig. 9. Same as Fig. 8 for the inertia parameter B. of /2(1) and g2exp(l ) as functions of 1, as shown in Fig. 13. These kinetic moments are at first sight to be compared with the dynamical moments C. Indeed they are calculated to belong to the same range of values. Their detailed comparison however is rather delicate indeed. As seen from Eqs. (33), (34), the collective energy and thus the angular momentum depend in general on powers in S2 whose order go beyond f2 2 and therefore the two moments of inertia are different since they involve different orders of derivation of the energy for their evaluation. This is rather well known. What is new here, is the fact that the energy and its derivative depend also on powers of o) and therefore in the relation between 1 and /2 for instance, even in the linear case (i.e. for a quadratic energy) one would have beyond a term in /2, a constant term which may yield a collective alignment resulting from the intrinsic vortical motion. This point will be explicitly discussed in Section 7. In the context of the AI = 2h staggering phenomenon observed in some super deformed bands [ 19,20], one introduces a difference g E e ( l ) in gamma transition energies E7(1) from a "smooth" behaviour (assumed to be merely in this case [20] a polynomial in 1 of order 4 or less) ~E:,(1)=~
ET(1)-
-6Er(I
[4Ee(1-2h)+4Ee(l+2h)
- 4h) - 6 E e ( I + 4 h ) ] } .
(102)
288
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304 ~-. lOO
~
50 140
120
100 235
233
0
20
40
60
80 1/14
Fig. 10. Same as Fig. 8 for the inertia parameter C. The calculated energies ~ E v ( 1 ) are displayed on Fig. 14. It is to be noticed HD case one finds a rather well-marked periodic effect of staggering with a thus consistent with the observed period in Jyrast(1). This behaviour will discussed in the next section, together with the observed modulation of the of the energies 8E~, (1).
that in the 4h period, be further amplitude
7. The quadratic approximation The results of the previous section, clearly indicates that the quadratic approximation of Eq. (63) for the collective energy in terms of the angular velocities w and s2 (given in its most general form in Eq. (33)) should be valid in rather wide domains. In such a case the conjugated momenta I and J of the rotation angles O and 0, of which the time derivatives are /2 and w respectively, are merely related by linear expressions as special cases of Eqs. (34): 1 =CS2 + Bw,
(103)
J = Bg2 + A w ,
yielding upon inverting these relations ~2= ( A 1 - B J ) / ( A C
- B 2) ,
o) = ( C J -
BI)/(AC
- B 2) .
(104)
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
289
WD
~sop
I i i
=
100 i o
200 71"",, I
150 100 -
SD
,,
.,
l'~
ii
250
[l
HD
1', 1
200 j 0
;
,
20
40
60
t
80
1/14 Fig. 11. Theoretical kinetic moments of inertia, defined in the text (see Eq. (99)), as functions of the total angular momentum 1 along the yrast line for each of the three retained configurations (WD, SD, HD).
Inserting Eqs. (104) into the Eq. (33) for the energy, one finds readily the quadratic expression for the collective energy in terms of 1 and J, postulated in Eq. (86). Let us draw a first consequence of the preceding. Indeed, expressing I in terms of s2 and J one finds
AC - B 2
1= ~ - S 2 +
B
~-J.
(105)
One thus gets within a conserved J band a collective alignment proportional to J and a kinetic moment of inertia which is rather close to the irrotational ansatz, upon assuming A ~ C, as approximately obtained in both our semiclassical and harmonic oscillator estimates in a wide range of deformations. Another interesting property of the quadratic energy given in Eq. (63) has been widely exploited in a previous paper [17]. It will only briefly sketched here and further discussed. Defining the yrast state as having the lowest energy for a given value of I, one gets classically (that is in particular assuming J to be a continuous variable) from
3E[I = 0 ,
(106)
from which one obtains the following yrast-J value B
Jyra~t(/) = ~ / ,
(]O7)
290
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
"- . . . .
loo
,
,..~i
,
-
!-.
!
160 II / 140
~. ,
300
i
~"
"b
"~
?,%
~
!
!
/ II 275 250
, ~ 0
• 20
40
60
80
1/14 Fig. 12. Same as Fig. 12 for the experimental kinetic moments of inertia defined in the text (see Eqs. (]00), (t01)). and yrast energy 12 Eyrast(1) = 2 ~ "
(108)
The latter is nothing but the classical rotor expression whenever the intrinsic vortical currents are seen not to contribute to the collective energy. In general the quadratic collective energy of Eq. (86) may thus be written as E ( 1 , J) = E y r a s t ( / ) -t- f i E ( l , J ) , 1 C (j_jyrast(l))2 B E ( l , J) - 2 A C - B 2
'
(109)
where the part that varies with J at a fixed value of I, f i E ( I , J ) , is of the from of a rotational energy for the distance of J from its classical yrast value, involving a moment of inertia which is again very well approximated by the irrotational moment of inertia (assuming A ~ C ) . From the preceding discussion one clearly sees that the Kelvin circulation is not a direct measure of the intrinsic vortical motion content of the considered solution. The relevant quantity is indeed J - Jyrast(1), i.e. the deviation of J from its classical yrast value. To support that statement is it useful to note from the second of Eqs. (104) that whenever J assumes its classical yrast value one gets a vanishing (o value. Turning back now, to the collective alignment property, one sees that if one follows the yrast line and thus inserts the yrast value of J as a function of I into
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
"'"
I
wo
0.5 ~
SD
291
0.5
51" ~ "f~U
:
/
0
.... /
HD
~,~
r
0.2 t
~
[
y~::
I 0 i J_
0
.
.
.
20
.
.
.
.
40
k
60
•
.__
I !
80
1/]4 Fig. 13. Theoretical and experimental angular velocities as functions of the total angular momentum 1 along the yrast line for each of the three retained configurations (WD, SD, HD). The theoretical (experimental resp.) velocities correspond to the solid (dashed resp.) curves. The theoretical values is simply the relevant Lagrange multiplier, whereas the experimental one is defined in Eq. (100).
Eq. (105), one looses the alignment term and gets a kinetic moment identical to the dynamical one since in that case. 1 = C~.
(110)
Now, one should take into account the essential fact that J is quantized in the very same way, as we have seen, that an orbital angular momentum. Its yrast value is not in general equal to Jyrast(1). In other words, J (in units of h) should be, in our case, an even integer number. Consequently the yrast energy differs from Eyrast([) by a rotational-like energy similar to BE(I, J) (i.e. implying the same moment of inertia) and proportional to the square of the difference t~Jyrast(l) between the quantal and classical yrast J-value, namely t~Jyrast(/) = 2hE ~ - ~ where E[y] denotes the integer number which is closest to y, namely
E[y] = M i n { E [ y ]
y,E[y] + l - y } ,
with [y] being again the integer part of y. The energy 8Jyrast(1) of Eq. (111), is obviously a periodic function of I whose period T(1) is given by
292
1.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304 WD 2.5 / I, i I i '1I,
: : : : -
SD
20
I
i
,~
/
'1
I i
I ,
i~
'
i f
HD ii
,i
I,! II
I
'1 /
-2O
5
i'~
t', t C
5
,I '1
/
'i /
' '
/
.t.
?
a
20
40
60 1/14
Fig. 14. Rapidly varying part of the gamma transition energies as functions of the total angular momentum 1 along the yrast line for each of the three retained configurations (WD, SD, HD). These energies are defined by Eq. (102). T ( 1 ) = 2h C .
(112)
Such a quantization o f J in the yrast line generates therefore a structure in the yrast line energies which exhibits a regular pattern of kinks. This has allowed us to propose it as a possible explanation o f the staggering observed in some superdeformed states. Clearly for B / C = 0.5, one gets a A I = 2h staggering pattern (see however the extended discussion o f this case below). We refer to Ref. [ 17] for a complete discussion o f this matter. There are two points presented in Ref. [ 17] which we would like to discuss and complete in the present paper. First, since experimentalists do observe bands over large domains in 1, and thus in J, one is compelled to explain how much our yrast band is still a band, i.e. composed o f states connected by strong stretched E2 transitions. We previously rightly argue that despite the fact that the E2 operator does conserve J, the experimental bands might be explained by the existence of residual interactions beyond the simplified model presented here. These interactions lead to a J-mixing which should be maximal near the crossing o f the energy parabolas corresponding to two adjacent J values. An estimate of the quenching of the E2 operator shows that it is compatible with the present uncertainties on absolute measurements o f E2 matrix elements by D S A M techniques for instance [38]. Yet one should consider the cumulative effect in a full
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
293
cascade of such a quenching, l A quantitative answer to this point would necessitate a detailed study of the interband cross-transitions. For instance, one may notice that, if at angular momentum I one has an yrast state corresponding to a pseudo crossing (i.e. maximal mixing), its companion state (located just slightly above it) should decay to the next ( I - 2h) yrast state with an intensity which reduces by a factor two roughly, the lack of feeding (due to the J conservation) of that state through a direct yrast-yrast transition. Incidentally one may note that this side feeding would correspond to a gamma ray energy very close to the yrast-yrast gamma ray energy. The second point has to do with a possible improvement of the quadratic expression for the energy. Within our model of vortical and global rotation, which are assumed to be completely decoupled from other collective modes, the quantum mechanics analog of the classical energy of Eq. (86) is E(I,J) - A ~C-
B
"
"
For eigenstates of both the [2 and J~ operators, and assuming a full alignment of the operators [ and J, one gets the following quantal energy E(1, J) = A ( A I ( I
+ h) - B I j + C j ( j
+ h)) ,
(114)
with the constant A defined as A= 1/(AC-
B 2).
(115)
The analog of the classical yrast J-value is given also in this case by the variational Eq. (106) where the energy E is given by Eq. (114), leading to B Jyrast(/) = ~ - I
h 2 "
(116)
This yields the new following "classical" yrast energy Ia A ACh 2 Eyrast(1) = ~-~ + -~(A + B ) l h - ~
(117)
The total energy of Eq. (1 14) may thus be written upon Taylor-expanding it around J = Jyrast(1): E(], J) = Eyrast(1) q- 6(I, J) , ~( I, J) = 2 C x 2 ,
(1 18)
where x is defined as x = J - Jyrast(1) I We are indebted here to W. Andrejtscheffto have drawn our attention to this important question.
(119)
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
294
For quantal yrast states, one has therefore
x=2hE C2h
---d1+ 2"
(120)
For B / C equal to 1, which classically corresponds to spherical shapes, one gets x = - h / 2 for all even values of I, leading thus to an absence of staggering for the quantal yrast energy I? ,t Eyrast(1) = ~'~ q- ~ ( A + B)lh..
(121)
This is not so surprising since in this case the distinction between global and intrinsic rotations is completely disappearing. For B / C equal to vl now, which is the case mostly discussed in Ref. [17] (where however the full consequences of the quantal expression for the energy given in Eq. (114) have not been drawn), one gets x = qzh/2, where the plus (minus resp.) corresponds to an even (odd resp.) integer number I/2h. Since x enters quadratically the expression 114 tbr E(I, J), one retrieves in both cases the result valid for spherical deformations as given in Eq. (121), implying here too an absence of staggering. This happens because the AJ = hi2 shift in Kelvin circulation yrast values, with respect to the case studied in Ref. [17], is exactly suited so as to accidentally smooth out the kinks present in the yrast line made of pieces of parabolas intersecting with a AI = 4h period. This does not affect, of course, the general character of the advocated staggering phenomenon and its period in particular, but merely introduces a modulation of its amplitude as it will be discussed now. For that purpose, let us study now the yrast energy pattern as a lhnction of the distance of the ratio B / C from 1/2. Indeed introducing the parameter e by B C
-
l+e 2
(122)
'
one gets for I = 2ph
x=21~E
- (1 + e ) p h + ~ ,
(123)
and thus, x is clearly a periodic function of pe whose period is 2 or a periodic function of le whose period is 4l/. We will therefore restrict our study of the behaviour of x as a function of I, for a given value of B / C , i.e. of e, within one period, namely with the [ 0 , 4 h ] interval for Ie. One obtains i f p is even
x=h(½-
pe) ,
le < 3h,
x = h (5 - pe) ,
le > 3h.
(124)
In the case where le = 3h, the value of x is undetermined (since the function E[y] is not defined for y being half integer), but its absolute value is well defined as Ixj -= h ,
1~ = 3 h .
For odd values of p now, one gets
(125)
l.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
295
3
0 0
1
2
3
4
I~//4 Fig. 15. Rapidly varying part of the total energy 6y,.(l) over one period, as a function of the product of the total angular momentum 1 and the parameter e measuring the distance of the ratio B/C from ½; The quantities B, C and ~ have been defined in the text.
x= Y
=
h(½+pe) ,
le
h (\ 3z
Ie>h.
pe]/ ,
(126)
Similarly one gets for le = h
ix I = h ,
le=h.
(127)
From the above one may deduce the energy 6(I, J) on the yrast line, called 6yr(1) for even values o f p
t~yr(l )
=AC/8(Ie,- h) 2
6yr(1)=AC/8(l~-5h) 2
for
le < 3 h ,
(128a)
for
le > 3 h ,
(128b)
for
le < /7,
(129a)
for
le > 3h,
(129b)
and
6yr(1)
=AC/8(le-f-h)2
~ y r ( 1 ) = A C / 8 ( l e - 3h) 2
for odd values o f p. These energies which might be considered as envelops of the quantal 6yr(1) energies (i.e. for quantized values o f 1 and J ) are plotted in Fig. 15. In the limit where e is small with respect to 1, one clearly retrieves the harmonic oscillator result presented in the previous section, where the rapidly varying part of the yrast energy oscillates between two different values (corresponding to a sequence of odd, even, odd . . . . values for p ) and where the oscillation amplitude is modulated from 0 to a maximum difference of ½ACh 2. This translates into gamma transition energies showing a "beating-like" trend as shown in Fig. 14 for the HD solution for which the computed e is found to be slightly below 0.05. In this case the maximum staggering is expected to occur slightly below I --=20h and its vanishing slightly below I = 40h, which is roughly consistent with what
296
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
we have obtained in the microscopic calculation of Section 6 and as well with what has been discussed in our Ref. [17]. A similar staggering pattern would be apparent for somewhat larger values of e (as e.g. for e = 0.2), whereas it would be completed distorted for very large e-values (as e.g. for e = 0.5). In a recent paper [22], an analogy of this quantal effect possibly observed in some superdeformed states with an Aharonov-Bohm phase effect for some mesoscopic conductor [39] or semi-conductor [40] rings in a uniform and time-invariant magnetic field, was proposed. In those rings indeed, persistent electric currents have been observed whose intensity is a periodic function of the magnetic flux through the ring, with a period equal to the elementary quantum of flux e / h (e being the electron charge and h the Planck's constant). As demonstrated in Ref. [22] for the collective energy of Eq. (109), i.e. neglecting the beating modulation above discussed, our 3 E ( I , J) energy plays the role of the electron energy, ( B / C ) I is the analog of the magnetic flux, finding the yrast state at a given 1 value is equivalent to find the electron ground state and gamma transition energies (proportional to the derivatives of the energy as a function of 1) are the counterparts of the electric current (proportional to the derivative of the energy as a function of the flux). This analogy has been used to provide a possible explanation tbr the scarcity of the staggering phenomenon. Indeed, the staggering structure should not be damped by fluctuations in ( ( B / C ) I ) (i.e. in deformation) on a range of the order of larger than h, in very much the same fashion as the variation of the magnetic flux (i.e. of the ring radius) should be smaller than the elementary quantum of flux e / h . A sizeable increase of the vibrational axial quadrupole mass right in phase with the superdeformed well is of course producing an important squeeze of the collective wavefunction, reducing thus the deformation fluctuation which is crucial Jbr the staggering effect to be observed. It has been recently shown in a calculation [41] within an ATDHFB approach using the Gogny effective force that this mass parameter presents a sudden increase by more than one order of magnitude, at the right deformation, in 15°Gd and not for its neighbouring isotopes. This might explain why the phenomenon is observed around this nucleus and not elsewhere. Similar calculations for Hg and Ce isotopes near the nuclei claimed to exhibit possibly (but in a lesser clear, or widely accepted, fashion than in 149Gd) such a staggering, do not yield such a spectacular increase [41]. Finally let us briefly mention that the same quadratic expression for the collective energy of Eq. (109), has been used [18] to provide a tentative explanation for the intriguing phenomenon of identical bands observed in many superdeformed nuclei, as well as to a lesser extent in some normally deformed nuclei [21]. This phenomenon relates gamma transition energies in nucleus 1 at spin Ix to gamma transition energies in nucleus 2 at spin 12 through E ~ ( l l ) = Ez,(12) q-- x[Ez,(12) - E~(I2 - 2h)] ,
(130)
where x assumes the values 0, +¼, +1, ±¼, +1. We found, using exactly the quadratic energy of Eq. (63) as before, that Eq. (130) may hold. For that we consider now nuclear collective states as defined by two quantum numbers (I and J with our current
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
297
notation). Then it has been shown in Ref. [ 18] that Eq. (130) may be retrieved provided that (B/A)l
= (B/A)2 = Y.
(131)
This condition expresses essentially that the nuclei 1 and 2 have the same deformation (note that this quantity being a ratio of moments of inertia does not scale at all when changing the nucleon number). Moreover, the x-parameter entering Eq. (130) is defined in terms of Y defined in Eq. ( 131 ) and of the differences A I and A J between the values of the moments 1 and J in the nuclei 1 and 2, as x = ½(ai -raJ).
(132)
We have noted above that in some superdeformed states B / C ~ 0.5 yielded the right period for the almost elusive staggering phenomenon advocated there. It is interesting to note, upon assuming Y also to be close to 0.5 in these cases (i.e. C ~ A, that this l value leads here to the right "quantization" of x in multiples of ~.
8. C o n c l u s i o n s
The aim of this paper was to investigate a class of collective motions in atomic nuclei which, in our opinion, have not been the subject so far of the extensive study they may deserve. They are however present and well known at larger scales, like when considering self-gravitating stellar objects or fluids in a container as encountered in everyday life. Tea, for instance, when correctly stirred, may continue to rotate in its cup even when the latter is neither deforming nor rotating as a whole! The main stream descriptions of the nuclear collective excitations in terms of fluid dynamics have generally limited these studies to the coupling of scaling-like irrotational vibrations with global rotations, with some noticeable exceptions like the study of M2 giant resonances for instance. It is so for very good reasons including the fact that these modes are probably the most important as well as the most easy to decipher. Quite clearly it is more easy to "see" shape changing modes than the modes that merely affect the current distribution. Moreover the latter modes are strongly coupled with the Bohr-Mottelson types of modes as exemplified above for the global rotation. It is our opinion that it may be timely to investigate vortical modes in the intrinsic frame, and their couplings. Indeed recent experimental progresses have opened new pathways towards rather pure collective states (like superdeformed excited states in some part of the nucleidic chart), upon using very efficient spectroscopic tools, like 4 ~ high-granularity gamma detectors for instance. In order to proceed in this field which has not yet been thoroughly explored, we had to clarify some questions. First we had to define the relevant modes and to choose a way to treat them in a fashion which takes into account, at least approximately, the multitude of nuclear degrees of freedom and which is sufficiently quantal in the treatment of the
298
1.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
collective properties. Finally, we had to examine possible observable signatures of such vortical modes in nuclei. As for the first problem we have decided to provide an answer which is both simple, flexible enough (see the variety of current patterns in Fig. 1) and well documented in classical physics as well as in algebraic collective dynamical description. This is why we chose Riemann-Chandrasekhar S-ellipsoids. The so-called "routhian" or "sell-consistent cranking" semiquantal approach is widely used in nuclear physics to describe global rotations. We deem it reasonable to assume that, if successful for one collective mode, it could also be useful for others, provided its foundation on Hamilton classical mechanics is well understood. We have taken full advantage of the physical relevance and relative simplicity of the harmonic oscillator model to validate some of the main simplifying ideas like the wide range of applicability of the quadratic approximation for the energy. Obviously a lot of theoretical work remains to be made. For instance, one would like to quantitatively estimate the close connexion between the intrinsic vortical velocity ~o (or equivalently the difference between the actual value of J and the classical yrast value Jyr~,st( I ) ) within the framework of constrained Hartree-Fock-Bogoliubov calculations. One also should investigate the amount of J-mixing as possibly resulting from a more comprehensive collective model approach, so as to assess the robustness of the present ansatz. At any rate, we feel that the results presently obtained are sufficiently suggestive to encourage further efforts to better understand the modes under scrutiny here. The last question concerning the measurable manifestations of such collective modes may be tackled in two different fashions. There are, indeed, both direct and indirect ways to see their effects. By "indirect" we mean here that upon ignoring them one has to renormalize the collective model parameters in use, in order to keep contact with the real world, while "direct" refers to observables implying the intrinsic vortical currents. To the first category clearly belongs the necessary renormalization of the moments of inertia away from rigid body inertia parameters. The preceding statement is both very useful to make a strong statement on the existence of intrinsic vortical modes and rather short to assess their properties. Pairing plays there a very strong role, so any quantitative evaluation should take into account these correlations. As for direct fingerprints of these modes, we came along with two possible candidates. The first physical phenomenon is the staggering observed in some superdeformed bands. It is a very tiny and almost elusive one, even though one has recently suggested that such a phenomenon could exist in the rotational spectra of some diatomic molecules [42]. If so explained, this staggering should indeed be viewed as a quantal zero-point fluctuation of the studied intrinsic mode, when following a rotational band whose properties are mainly given by the global rotation. To illustrate the latter point, one should keep in mind the relative magnitudes of the energies involved, a fraction of keV versus hundreds of keV. An analogy has been drawn between this phenomenon and another phenomenon which is, as the lbrmer, on the borderline of physical availability, namely persistent currents in mesoscopic rings. By no means this analogy is accidental, if one takes properly into account the well-known similarity between the motion of a massive particle in a rotating frame with the motion of a charged particle in a magnetic field.
LN. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
299
It is clearly, in our opinion, interesting by itself in embedding seemingly very different quantal properties in an unified framework. It has also been beneficial in allowing us to provide a frame of explanation of the scarcity of the observed nuclear phenomenon, and even of its actual occurrence in 149Gd,through the calculations of vibrational mass parameters of Ref. [41 ]. The other nuclear property which we have looked at (i.e. the phenomenon of identical bands, particularly in superdeformed nuclear states) deserves a more systematic study than what we have been able so far to perform in Ref. [ 18]. There we have been able to demonstrate that a priori our collective hamiltonian encompassed indeed such a property. We want to stress that not only we provide a coupling frame allowing this identity but also an explanation of why a scaling in the number of particle is not an issue of any relevance for that problem provided the two nuclei have the same deformations. There are many other places where the effect of such nuclear intrinsic vortical modes could be hunted for, as in magnetic excitations clearly and quite generally. Also, in excitation energy regimes where these collective modes are not too damped, they could affect significantly the nuclear level density. To conclude we should thus strongly advocate a systematical study of the role of intrinsic currents in low excitation energy excitations of nuclei.
Acknowledgements We are indebted to many physicists for their interest and/or their patience in numerous discussions on these ideas which might have appeared, at least at first sight, somewhat far-fetched. We would like to thank them all for their suggestions or criticisms, and particularly among them W. Andrejtscheff, J. Barrel, H. Doubre, B. Haas, F. Hannachi, J. Libert, W. Nazarewicz, J. Meyer, M. Meyer, N. Minkov and last but not least Ch. Brian ton. We are also grateful to E.Kh. Iouldashbaeva, J. Libert and M. Girod coauthors (with one of the authors of the present paper) of a paper not yet published, to have allowed us to make use prior to publication of some relevant results. Over the course of this work some of us have benefitted of the hospitality of some other's institutions. Therefore I.N.M. thanks the CENBG and EQ. the Bogoliubov Laboratory of the JINR, for the quality of their hospitality during many fruitful visits. This work has greatly taken advantage of a funding through the agreement between the I N 2 P 3 / C N R S and the JINR (Collaboration #91.9) which we thank very much for their continued support.
Appendix A. The boson Bogoliubov transformation associated with the generalized routhian in the harmonic oscillator mean field case We consider the boson Bogoliubov transformation (with k = 2, 3) +
o~k = ~
•kiaJ + tzkiai, i
l.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
300
satisfying the generic boson excitation equation %
+
= [h(.O,(0),a~-]
,
with the expression of h(s2,(0) given in Eq. (73). Calculating explicitly the above commutator yields the matrix equation given in Eq. (77). In view of the merely real or imaginary character of its matrix elements and defining Ak2 = lk2 ,
kk3 = i l k 3 ,
].Zk2 = ink2 ,
[.Zk3 = imk3 ,
one may rewrite the Eq. (77) in a real form as
0
0SD (03 - ,Ok
/
O
0
[ lk3
D
(02 @ ,(2k
-S
~ ink2
0
-S
(03 + s2~
\ mk3
= O.
The vanishing of the determinant associated to this homogeneous system of linear equations yields the following second order equation in .02 .(-24 - O~(O92 jr_ (023 - + - 2 ( S 2 - D 2 ) ) + ( 0 ~ ( 0 39 - 22( 0 2 ( 0 3 ( S 2 . q - D 2 ) ÷ ( S 2 - D 2 ) 2 = O ,
whose solutions have been explicited in Section 5. The transformation parameters now are given as 122 = (S'2~ - w~) (S22 + (o2) + c03(S 2 + D 2) - J22(S 2 - D 2) , 123 ----S ( ( f 2 2 + 093)(.(22 q- (02) -
( $2 - D 2 ) )
,
m22 = - 2 S D w 3 , m23 = D (($22 - ( 0 3 ) ( ~ 2
+ ~2) -
( $2 - D 2 ) )
132 = S ((.(23 q- (.o2)(f23 q- (03) - ( $2 - D 2 ) ) 133=(f22-(02)(.(23+(03)-+-(02(S2 //'/32---O
+ D 2)
, , .(23($2-D2),
((,(23 --w2)(S23 + co3) - (S 2 - O2)) ,
rn33 = 2SDo92.
The homogeneous character of such a linear system entails that these coefficients are only known up to a multiplicative constant which is fixed by the canonical condition k=3 k=2
The inverse transformation is given in the following form k=l * k=2
+
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304
301
Appendix B. Some expectation values of the generalized routhian solutions with a harmonic oscillator mean field
In this appendix we will evaluate the expectation values of some selected operators for the solution of the generalized routhian problem with a harmonic oscillator mean field. It is a Slater determinant made of A single particle wavefunctions characterized by the quanta numbers n~j), with i = 1,2, 3 associated with the frequencies eoj, /22 and .(23 respectively and j = 1 . . . . . A. It will appear useful to define the following sums j=A
1
_,= = Z ( n y
) + -i)
j=l
To compute the expectation values ponents of the position and/or the + boson operators a~, ak and then k = 2, 3, through the inverse boson One then finds readily
of one-body operators defined in terms of the comimpulsion, we have to rewrite them in terms of the + in terms of the boson operators a~, a~ whenever Bogoliubov transformation given in Appendix A.
= _Z_h z , , 2row1 and for i = 2, 3
mwi
"=
((lji-mji)2-~J) '
from where one deduces the values of the semi-axes of the associated ellipsoid as for instance for the first axis a= ~ , and similar formulae for b and c. To compute the expectation values of the x-component of the orbital angular momentum and of the Kelvin circulation, one needs "crossed" expectation values in x and p as
x2P3 = °)~ h / x3P2 ; = h /
k=3 k_~2 [ ( / k 2 _ - mk2)(lk3 + mk3)Z~] , k=3
09~ ~ _ [ (/~2 + mtz)(mk3 -- 1~3) Z k l •
One therefore gets x2P3 ~ _ x3P2 h~ ( h / '
302
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304 (~}
J~YO+l)=
C ( X2p3; - b f X 3 p 2 ~ "
=~
h /
7\
h /
The expectation value of the generalized routhian h(/2, 09) is simply
k=3 (h(sZ 09)) = h09,& + ~ h & ~ k , k=2 from where with the help of the above expectation values one can deduce the energy of the solution as
E= (h( a,09) ) + s~(L.) + 09(4}. One may also compute it directly from the definition of the energy operator
k=3 1 H = Z h09k(a~ak + ~ ) , k=l from where one gets
i=3 I k=3 2 ~, E=h091,_~l q-~-~ h 0 9 i E ( 1 2 i q - m k i ) _ ~ i=2 [ k=2 Appendix C. Moments of inertia in the Inglis cranking approximation The Inglis cranking approximation of Cic, for instance, leads to the following expression
Clc = 2 Z p,h
t(P I__Lxlh~l 2 ' ep -- eh
where ep and eh are the routhian mean values of the corresponding single particle states. Expressing Lx in terms of the original boson operators a +, a, one gets
_ 2//,2 Cic
[
4092(03 L(093 + 092)2 E
p,h
](pla~a3 - afa21h)l 2 ep -- eh
-t-(093 -- 092)2 Z p,h
+2(09~ - 092) ~ p,h
I(P[a~a~ - a3a2lh)12 ep -- eh
(Pla~af - a3a2lh)(hla2af - a3a~lP} el) -- eh
Using the Bogoliubov transformation, we can rewrite the particle-hole matrix elements appearing in the above expression in terms of the new boson operators a~+, o~, as
303
I.N. Mikhailov et al./Nuclear Physics A 627 (1997) 259-304 ( p l a ~ a 3 -- a ~ a 2 1 h ) = i[(122m23 - m22123)(p]~2 + ~221h)
q-(/B2m33 - m32133) (Pla3 ~2 q- ~321 h) q-(/22m33 q- m23132 - m22133 - 123m32)(p1¢¢~¢e3~ if- ¢¢2~¢3]h) +(122133 -]-123132 - m22m33 - ma3m32)(PlO~¢3 q- ¢lt~t'alh)] , ( P l a ~ a ~ - aBaa]hl = i[(m22m23 - 122/23)(pit:c2I-2 -}- ¢¢221h/
+(mBam33 _ 132133) (p]~_2 -1- ¢232[h) +(m22m33 if- m23m32 -- 122133 --/23132) (pl%%q
+ ¢¢2o~31h)
+(m22133 q- 123m32 - /22m33 - m23132)(p1¢¢+¢e3 q- ~¢e21h)] • The calculation of C,c therefore involves the following sum rules (see e.g. Valatin's paper quoted in Ref. [ 3 3 ] ) Z p.h
Z p,h
(P1~2~0~3 + ° ~ 2 [ h ) [ 2 _ ep -- eh
h( 0 3 - 0 2) '
(Plee~a-~ + te2~e3lh)[ 2
2 2 '~ ~'3
ep -- eh (ploz +2 + oli2lh)
Z p,h
,-~2 -- ~'3
h( 0 2 + 0 3 ) ' 2
~'.
=2 -' ep -- e h
i = 2,3.
hJ'-2i'
One would get similar results for B,c upon replacing in the preceding expression of C~c one operator Lx by an operator Jx and similarly Arc, upon replacing both operators Lx of C~c by Jx. Of course the computation of the matrix elements of Jx implies the same matrix elements of products of two (a, a +) operators than those of Lx, up to a multiplication by appropriate scaling factors ( b / c or its inverse). The above equations provide linear response inertia parameters for finite values of the angular velocities. In the adiabatic limit, i.e. for small values of these velocities, the operators teu tend to the operators a u (where/x stands for 2 or 3). Using the previous sum rules where the energy denominators are now detemined in terms of static energies, and upon replacing O1,2 by wl,2 one gets the equations ( 8 7 ) - ( 8 9 ) for A°c , B°~c and C~,°.
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