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Giant dipole and quadrupole modes in highly rotating nuclei
Nuclear Physics @ North-Holland
A435 (1985) Publishing
GIANT
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DIPOLE AND QUADRUPOLE MODES IN HIGHLY ROTATING NUCLEI
M. DI TORO, Zstituto Dipartimentale
U. LOMBARD0
and
G. RUSSO
di Fish, Universitci di Catania, Zstituto Nazionale Catania C. so Ztalia, 57-Z 95129 Catania, Italy
di Fisica Nucleare,
Sez. di
Received 22 June 1984 (Revised 22 August 1984) Abstract: We discuss isovector CD and isoscalar GQ resonances built on high-spin states observed in heavy-ion reactions within a microscopic fluid-dynamical theory. We reproduce the main features of the experimental data from the analysis of scaling solutions for small oscillations of two Vlasov fluids in a rotating frame. Results are shown for 16sEr. We discuss the possibility of studying excited nuclear states by just looking at the properties of the giant resonances built on top of them.
1. Introduction
A great deal of progress has been made during the last years in the investigation of GR’s built on high-spin states observed in high-energy y-ray spectra from the deexcitation of medium and heavy nuclei produced in fusion and deep-inelastic reactions I). The appealing physical aspects of such a study lie in the properties of the GR strength function in systems far from the ground state and its dependence on properties of highly-excited nuclei such as excitation energy, spin and nuclear deformation. Although some experimental discrepancies still exist, three main features come out from the available experimental data: (i) the strength function does not depend much on the ground-state deformation; (ii) the centroid of the resonance shifts to lower energies with higher angular momenta (A&/AL-0.05-0.1 MeV/A), (iii) the GR overall widths are much broader than typical ground-state GR widths. Several theoretical attempts to solve this problem have been proposed recently 2), ranging from simple solvable models to quite huge cranked RPA calculations. The approach proposed in this paper is fluid-dynamical in that the collective nuclear dynamics is developed in a phase space as a semiclassical limit of the self-consistent TDHF equations 3,4). C onsequently our results, which are quite easy to work out, can be obtained using realistic interactions and can be directly compared with full cranked RPA calculations. We can say that with respect to other theoretical approaches the most interesting feature of our method is that we get analytical and physically transparent results while retaining the fully microscopic foundations with 173
M. Di Toro et
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~1. /
Giant dipole and quadrupole modes
the consequent use of realistic interactions and the possibility of a simple analysis of the dynamical effects of each term of the effective force. A semiclassical approach to this problem was also recently introduced by Winter and Schuck2) but with a model residual interaction related to the particular collective motion studied. The key point of our approach is to assume that for a giant collective state all the strength is concentrated on only one level. This ansatz is largely justified from RPA calculations as well as from variational fluid-dynamical approaches and it corresponds to taking into account only the lowest multipole distortions of the momentum distribution during the vibration, which is described as a scaling mode. Our general philosophy is to use this simplifying assumption to study giant resonances in a wider context, such as in a rotating nucleus or as doorway states in particular reactions. In this paper we are able to show the effects of dynamical deformations and Coriolis coupling on the frequencies and strengths of GDR’s and GQR’s built’on high-spin states. In particular an agreement with the above-mentioned experimental features is found. 2. Scaling solutions of the cranked Vlasov equation The dynamics of a two-fluid system will be derived from the Vlasov equation in a frame rotating with angular frequency w (assumed to be along the z-axis):
(2-l) where h’ = h -or * L is the Wigner transform of the self-consistent cranked hamiltonian with a Skyrme interaction. The label q is the isospin coordinate and quantities without any isospin labels are understood to denote total values. Eq. (2.1) can be obtained as a semiclassical limit of the cranked TDHF equation for the Wigner transform f,( r, p, t) of the one-body density matrix. Transforming in the intrinsic coordinates (r’ = r, k = p - mo x r) the equation of motion becomes “(z,”
‘)={hb(r,
k, t),f,(r,
k, t)}+2mw
* (V&h XV&),
(2.2)
where from now on, the Poisson brackets are defined with respect to r and k. For Galilei-invariant Skyrme forces 5, the transformed cranked hamiltonian assumes the form +B,(r).
k+A,(r)-imlwxr12,
(2.3)
where the explicit expressions for m *, A and B are given in the appendix. In the following we shall focus our attention on the SKM force which reproduces very
M. Di Toro et al, / Giant dipole and quadruple
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175
well the nuclear compressibilities and the symmetry energy near the nuclear surface “). It is worth noticing that, as in the classical case, the centrifugal and the Coriolis forces appear explicitly in the hamiltonian and in the equation of motion, respectively, as an effect of the rotation. The rotating states of the nucleus are solutions of the stationary Vlasov equation in the rotating frame, (2.4) In general, the function ff is not isotropic in the intrinsic momentum space and the Coriolis force gives rise to a current flow in the rotating frame. We restrict our analysis to isotropic stationary solutions for which the Coriolis term in eq. (2.4) vanishes exactly. This kind of solution exists for Galilei-invariant non-local interactions and implies rigid currents and rigid moments of inertia for the rotating nucleus ‘). As discussed in ref. 3), this solution corresponds to a classical equilibrium condition for the stationary rotating system. In this sense we can say that in our results shown in sects. 3, 4 there is some kind of temperature effect, in the reference state, which is acting in the direction of a classical description of the rotating nucleus. Giant resonances are described as small-amplitude vibrations on the rotating nucleus. We introduce a variation with respect to the stationary rotating distribution function discussed before,
and then we linearize the cranked Vlasov equation (2.2). In this respect our procedure is completely equivalent to a RPA theory in phase space. However, we interpret as giant resonances only simplified solutions corresponding to scaling modes. This assumption has been proven to be quite successful in describing giant resonances in the static case, i.e. for nuclei in the ground state ‘). It corresponds to a maximum collectivity condition for the studied motion in so far it is equivalent to take into account only the lowest multipole (I = 0, 1,2) distortion of the distribution function in momentum space. Consistently the equation of motion can be reduced to a harmonic form which means that all the strength is concentrated on only one level. Of course in the RPA dynamics there are explicit quantum terms which are neglected in the Vlasov approach. In this respect we would like to remark the foliowing: (i) Quantum-dynamical contributions are small for the particular solutions, scaling oscillations, we associate with giant resonances ‘*12). (ii) In principle we include all the quantum effects in the stationary state on top of which we build the giant mode. The numerical results shown in this paper are limited to a classical stationary rotating distribution function just because we do not have available fully self-consistent cranked Hartree-Fock solutions.
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The extension of the scaling ansatz for giant modes on rotating nuclei is straightforward in the limit of rigid current assumption for then the only effect of the rotation on the momentum distribution is a shift of the Fermi sphere by the amount moxr. We assume for the variation of the distribution function
%=K$x,+k%,
(2.6)
where S,(r, t) is the displacement vector and x4(r, t) is a velocity field, assumed to be local. Eq. (2.6) implies second-order distortions of the Fermi surface, as in the static case. The most intuitive scaling motion is obtained from an irrotational displacement field S,(r, t) = V&(r, t) ,
(2.7)
where &(r, t) gives the shape of the deformation associated with the studied vibration and the corresponding time-dependent amplitude. At this point we are left with the problem of finding a connection between x and S and this will be worked out within the framework of the fluid-dynamical equations. From eq. (2.2) we can generate an infinite chain of fluid-dynamical equations for the k-moments of the distribution function. If we are able, with some physical ansatz, to truncate the chain at the lowest k-moments level, instead of the complicated cranked Vlasov equation, we would be able to solve a set of differential coupled equations for relatively few unknown (r, t) functions, as density, current, kinetic energy tensor and so on. These equations are completely classical, quantum effects being in the initial conditions and in the truncation procedure. For scaling modes we can exactly close the fluid-dynamical chain at the lowest two /c-moments, continuity and Euler equations “). The equation for the zeroth k-moment assumes the form
with pq =
I
d3kfq(r,k t) ,
Jq
=I tf,(r,
Jt“’=; ; t+[,04,Jq - p,J,,l ,
S t) d3k9
t+= t,+ t* .
As is well known the current is not conserved locally due to the presence of the neutron-proton exchange term in the non-local potential. The total number of nucleons of a given isospin type is, of course, conserved. For isoscalar modes the source term disappears, being Jt’.+ Jtj. = 0. Obviously, the total density is locally conserved in any case.
M. Di Toro et al. / Giant dipole and quadrupole modes
177
For the first k-moment we get the Euler equation
+i Tr (Tq)V, --& (
1
+Jqv;B,+(B,-v,)Jq++(J,~v,)B,+J,x(v,xJ,)=O. (2.9)
The last four terms in the second line do not give any contribution after linearization in the limit of rigid current. The kinetic energy tensor d, is given by
(Tq)ij=
I
3fq(
r, k, t) d3k.
3. Isovector dipole mode
Isovector giant dipole resonances are described as out-of-phase small-amplitude collective oscillations Sf,(r, k, t) = -Sf,,(r, k, t) of the neutron/proton distribution function around the stationary value ff(r, k, t). Scaling deformations correspond to a flow pattern given by an irrotational dipole velocity field. Using a generalized scaling generator 4), a,(r, k, r) =x4(*, t)+k.
S,(r, r),
(3.1)
we have Mq = CC& cq4).
(3.2)
This involves second-order distortions of the momentum distributions while taking into account the shift of the Fermi sphere to the rotation. Assuming the nucleus to undergo rigid rotations (,I:* = 0) and in the limit of the irrotational displacement field S, = Vrqhq,one obtains from the continuity equation the simple relation xq=-
“6 1+2mqp
’
77=2
t+
(3.3)
which allows us to close the fluid-dynamical chain at the lowest two k-moments. In fact, the first k-moments of the transition distribution function can be expressed only in terms of the field S,(r, t): 6P,=Vr*
J, = - 1
9
(3.4)
$:qp,
(3.5)
(P$sj)
(3.6)
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M. Di Toro et al. / Giant dipole and quadrupole modes
The linearized Euler equation becomes the equation of motion for the displacement field: P”,‘st,
-1+2m7)p +pv;
-2pS,‘wxS,+6p,ox(oxr)+V; V D”’
[+f”,‘6p,]+spqA
+4pV,[Sp,
m
+P:
[
s;i, *st
mq 1
VJD, m
Tr ;i”,‘+p”,‘Tr a?,]
+l)[GpqV,(Tr?‘)+TrG;iqV,p”‘]=O,
(3.7)
where Dq is the local part of the Skyrme HF potential and p = - t_/4fi2, t_ = tl - tz (see appendix). It describes the dynamics of the collective motion and, in principle, can be solved by imposing suitable boundary conditions. As far as we are interested in gross properties of the giant modes such as centroid energies and transition strengths in terms of the angular frequency, we follow a simplified procedure. Let us assume for the scalar field the usual Tassie-Bohr form
d,k t) = &3t)x+
ay4(t)y
Sq=(aZ,
+ a,4(t)z,
a;, cxg.
(3.8)
By taking the scalar product of eq. (3.7) with each component of the real scaling field (3.8) and by integrating over the spatial coordinates we end up with three differential coupIed equations for the (Y&,=(t) unknown functions. AS Vi(Sq)j = OV,j, the gradient and divergence terms in the linearized Euler equation give no contribution after projection. Actually, from a direct evaluation of the collective kinetic energy, it is possible to show how this procedure gives the right collective mass. For rotations about the z-axis, the z-mode is not affected by the presence of centrifugal and Coriolis forces but only indirectly “feels” the rotation through dynamical deformations in the stationary density. For a stationary distribution function which is invariant with respect to rotation of 7~ about each of the three spatial orthogonal axes (good signature), the equation for the z-mode is
while the x, y modes, coupled only through the Coriolis term, satisfy M,Gtj-2mwZqai,4+(Cx-mo2Zq)a’:=0, M,&,4+2moZ,ci~+(C,,-mo2Zq)a,4=0,
where Mc=m
I
Pq
st
1 + 2m7jpsf
d3r
(3.10)
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Giant dipole and ~~ad~pole
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179
is the collective inertial parameter, and 2, = f d3rp, ;
c&Y,Z
J +J
= rrzq
d3r[(Tr P)a&,
p;+ (Tr
%$J&,,p”‘]
d3r([D”,‘+ft,(l+2xo)p”,‘-t(ti+tt-)V2p”,’]a:,,p”,’
+~t3(1+2~3)(pSf)“-‘[(a-l)p~~-(tr+l)ps,’](~,,,p”,‘)2}
(3.11)
are the kinetic energy and interaction contributions, respectively. In the harmonic approximation, eq. (3.9) for the z-mode gives n, = (c;/WJ”2.
(3.12)
The motion in the x -y plane, eqs. (3.10), has two independent solutions of the form CY,= A(*) cos (.f-?*)t + t$‘*‘) ,
ay = I#*’ sin (@*‘r + 4”‘)
with J-p’=
C+- mo2Zq+2
I
Mc
mwZ, 2 ( M, )
*[(~)2+(~)2(c+-~~)+4(~)4]“2}“2,
#*>/A(*t)
2mwZ&!(*)
=
-M c f2’*‘”+ C, - mw”Zq .
(3.13)
(3.14)
It should be noticed that the major cont~butions to the integrals of eq. (3.11) come from the nuclear surface. Thus, it is not the detailed behavior of the local density inside the nucleus which is important here, but rather the shape at the surface. Transforming to the laboratory system, ff:=a,cos&-f+,sinwt, ak=ffXsinwt+a;coswt,
ffzI- - %,
(3.15)
we get, in general, a splitting into four components: CY:=~(A(*)-B(*tf)cos[(n’*‘-w)t+#(*)] +$(A’*‘+B’*‘)
cos[(~(*‘+u)t+c$‘*‘],
a~=~(B(*‘)-A(*‘)sin[(~t(*)-w)ti-#f*)] +~(A(*~+~(*))sin[(~(*)+~)~+#(~)].
(3.16)
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M. Di Toro et al. / Giant dipole and quadrupole modes
Apart from the splitting of the GDRinto two components analogous to that displayed by static triaxial deformations, the Coriolis force gives rise, in the laboratory frame, to a further splitting into four frequencies. The resulting modes can be classically interpreted as linear superpositions of two circular motions in opposite senses for each component. In particular, for axial deformed nuclei along the z-axis we get from eq. (3.14) B (*) = TA(*) and only two frequencies a’*-‘~ w are expected to occur in the laboratory frame. The expressions for the dipole frequencies, obtained so far, are very general and allow fully self-consistent calculations starting from a finite-temperature cranked HF code to get the stationary solution 8). As a preliminary calculation we specialize our method to a classical rotating nucleus about the z-symmetry axis. A very nice model calculation was discussed by Hilton ‘) just studying the normal oscillations of a system of 2 protons and N neutrons interacting via a dipoie-dipole two-body force, moving in a deformed rotating harmonic-oscillator well. In our microscopic picture we could get similar results using a local Hartree-Fock mean field. Indeed if we specialize our frequencies (3.12), (3.13) to the local case we easily get the results eq. (13) of the Hilton work. Of course in this model, contrary to our approach, there is no way to get a self-consistent analysis of the dynamical effects of the interaction on the rotational as well as on the vibrational motion. EVALUATION
OF THE DIPOLE ENERGIES
Although we have developed the formalism for a general case, in order to simplify the discussion and to emphasize the main effects of the rotation on the dipole frequencies, we shall assume a proportional density distribution of neutrons and protons and very close values of the Fermi energies for the two systems. All that means (3.17) We also neglect Coulomb effects in the calculation of the restoring parameters. In the Thomas-Fermi approximation for the kinetic energy density, integrating by parts eq. (3.11) we get Ci = Z&/A, where F=,YJ=
with
I d3cf(
$%&,y,epst12
(3.18)
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M. Di Toro et al. / Giant dipole and quadrupole modes
For axially-symmetric
dynamical
deformations
along the z-axis (F” = F,) we get (3.19)
(3.20) with M
For the cranked
stationary
s d3r
= m
density
‘“’
1 +2m?7pst
we assume
SmA.
an oblate
Psf(r’ ‘)= 1 +exp [(r: R(tl))/a]
Wood-Saxon
’ (3.21)
R(e)=Ro(~)[1+~(3cos2e-I)] with p. = 0.145 fmm3, a = 0.5 fm and R,(p)
shape:
fixed by the condition
5
d3rpSf(r, p) = A.
The equilibrium deformation p(w) is constructed by the virial tensor method [see Mikhailov ‘)I which consists in imposing a balance among the forces acting on the nucleus, namely pressure, surface, centrifugal, Coulomb and nuclear forces. We solve analytically the integrals (3.18), assuming to a very good approximation
0aP=
Pi -c6(
G
In this way, explicit
.n(“‘(&)
=
expressions
MO
@(‘) M(p)
R,(
r-R(B)).
for the dipole
[ R,(O) 1
frequencies
are obtained:
2[1+~~+~p2]+
TtlA
*M(B)@’
(3.22)
where
is the dipole centroid energy for spherical nuclei in the non-rotating provides a good estimate of the experimental values ‘).
case, which
M. Di Toro ef al. / Giant dipole and quadrupole modes
182
10
I
50
100
150
200
Lo-l)
24hQZ 209 8 w
16;
(hR)
-
to"
hw(MeV)
Fig. 1. Shift and splitting, in the laboratory frame, of the isovector GDR energies in terms of ho for 16sEr (SKM force).
Transforming to the laboratory system, we get two frequencies, R, - o and R- + o, in addition to the unaffected 0,. This means that Coriolis effects are almost negligible. In the limit of local potentials (M(p) = mA), the first two frequencies become equal. In this case the rotation does not give any effect on the dipole mode apart from dynamical deformation. The expected effects, in the laboratory frame, are shown in fig. 1 for 16*Er. We predict a clear increasing of the width mainly due to a splitting of the giant level. The dipole absorption cross sections, easily computed in our approach 4), are used as weights in order to get the average dipole energy (h0). In fig. 1 we relate an angular momentum L to the angular velocity through a rigid moment of inertia for the corresponding oblate shape. The unphysically high values of L are due to the oblate spheroid constraint for the stationary rotating solution which prevents the nucleus to undergo fission. We obtain a small shift of the centroid of the resonance to lower energies of the order of 1.OMeV for angular velocities which can be reached in realistic heavy-ion reactions (ho G 1 MeV). Axially-symmetric oblate shapes along the yrast line should correspond to the final regime of shape changing with increasing angular momentum. At this stage the rotation is no longer collective being the angular momentum due to single-particle alignment. Nevertheless, a cranking evaluation of the average moment of inertia gives the rigid-body value, i.e. the average “single-particle” rotation around the symmetry axis shows quite a similar behavior to collective rotation ‘O).Our model calculation corresponds to a limit case somehow opposite to that analysed in other theoretical approaches 2, where rotations of prolate nuclei are considered. We would like to stress again that our results eqs. (3.1 l)-(3.16)
183
M. Di Toro et al. / Giant dipole and quadrupole modes
are completely general and can be used with any self-consistent solution.
stationary rotating
4. Giant quadrupole resonances We describe isoscalar giant modes as small-amplitude collective oscillations in which neutrons and protons undergo an in-phase irrotational flow. Now the Euler equation, eq. (2.9), after linearization becomes
(4.1)
In the case of quadrupole modes we assume for the corresponding field the form S = V4 (r, t) with 9(r, r) = ?j [ak(f)+k(r)+ &(r)=Gr2(Yzk+ &(r)
displacement
dt)@dr)l,
Y$)=zAr’XiXI, ij
= 4 r2( Yzk - Y&k)= C Af'XiXj ij
.
(4.2)
For total quantities (neutrons plus protons) using the same procedure outlined in sect. 3, from eq. (2.6) we get the same expressions (eqs. (3.4) and (3.6)) for the first two even k-moments of the transition distribution function. For the total current we have the simplified structure J = -p,$
(4.3)
simply because exchange terms are not playing any role if we sum up the currents. Following still the same method discussed for isovector modes, we take the scalar product of eq. (4.1) with each component of the displacement field. Integrating over the space coordinates, we reduce it to a set of coupled equations Mk~k(f)+Bk~~(t)+Ck(Yk(t)=O, M~~r(l)+B~dlk(f)+C~(YE(t)=O,
(4.4)
where, for rotations around the z-axis, one has Mk = ML = C JAf’&’
,
ij
Bk
=
-BE= 20 C I,(A’,5’A~~‘-AI:‘AI~))
Ck=C,-=2C(Ar’)2 ij
I
,
d3r-$fSl(r)+m2CIi(AljL))2(8i3+Sj3-2) ij
(4.5)
184
M. Di Toro et al. / Giant dipole and quadrupole modes
for the collective mass and Coriolis coupling and restoring parameters, respectively. The coefficients Ii = M j p,,( r)Xf d3r are the inertial parameters. We remark that, as in the state case 4Y7),th e only details of the interaction are in the effective mass. This coupling only between the k- and 6components of a given angular momentum projection is actually valid if the nucleus undergoes an axially symmetrical deformation with the rotation. In the general case of a triaxial shape we get a much more complicated set of five coupled equations. We will restrict our analysis to dynamical axial symmetric deformations. Therefore, we expect to see a splitting into three levels due to the effect of the deformations and a further splitting into five levels due to the Coriolis coupling. Assuming a harmonic time evolution, we can easily get the eigenfrequencies
iI@‘=
dR*&,
k=O, 1,2,
(4.6)
where, for k = 0, Bk = 0. Everything can be evaluated once we know the cranked self-consistent stationary solutions p,,(o) and ;i,Jw). As discussed in the previous section in this work we show some first, not self-consistent, results obtained from a stationary Thomas-Fermi distribution in an oblate cranked shape. In order to compare with the experimental values we need to transform the quadrupole motion to the laboratory system. By using the rotation properties of the spherical harmonic functions with respect to the z-axis, we get ff:=ffkcoskWf-~~sinkwt, ~~;=a~sinkot+~~~;coskOf, and, consequently,
(4.7)
for the frequencies O(,“(lab) = a’,“‘* kw .
(4.8)
In fig. 2 we report GQR energies, in the laboratory frame, for 16’Er as a function of the angular frequency and classical angular momentum L = I(o), I(w) being the rigid moment of inertia along the deformation axis. The pattern of the energy branches results from the competition between dynamical deformation and centrifugal force. For the k = 1 case the Coriolis splitting, in the rotating frame, is strongly reduced for high angular velocity. This is related to the oblate deformation we are only allowing in our calculation. The scaling velocity fields V+l,i become more and more aligned to the rotation axis and the Coriolis term of the Euler eq. (4.1) has no effect. This implies that, contrary to the dipole case, how we get a much wider level splitting in the laboratory frame. All together the five curves spread out with increasing angular momentum and this can partially explain the observed increasing of the width of the resonance ‘>. The average frequency, computed using as weights the quadruple absorption cross
M. Di Toro et al. / Giant dipole and quadrupole modes
10
I
50
100
150 200
185
L(h)
hw(MeV) Fig. 2. Shift and splitting of the isoscalar GQR energies, in the laboratory frame, in terms of the angular frequency for 16sEr.
sections, is almost independent interest.
of the angular velocity in the region of physical
5. Conclusions
We describe giant resonances on high-spin states as small-amplitude vibrations of a system evolving through a cranked Vlasov equation, which is the semiclassical limit of a TDHF theory. Extending the notion of “shape scaling” in the rotating case, we study such particular scaling solutions which give the bulk of the response for giant modes on the ground state. The theory is fully microscopic in its foundations and .then we can use realistic interactions (Skyrme-type forces) and we can easily follow dynamics effects of each term of the force. Indeed the advantage of this approach is that while we retain the main features of a self-consistent RPA theory we finally get analytical results, with quite clear physical meaning, for giant frequencies and transition strengths. To get numerical results to compare with experiment we need a suitable description of the highly-rotating hot nucleus, and actually some knowledge of the density and kinetic energy density distributions for neutrons and protons. The results we discuss in this paper are obtained using a classical description of oblate spheroids with a deformation consistent with the angular velocity, rigid moment of inertia and rigid rotational current. Of course this stationary rotating solution must be improved, and we are planning to do so. At the moment our results could be related to the
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M. Di Tore et al. / Giant dipole and quadrupole modes
limit case of a fully aligned very-high-spin state. Bearing well in mind these limitations there are some main features we can extract from our calculations: (i) A splitting of the giant level essentially due to dynamical deformation eff e&s. We predict also a,further Coriolis splitting, which, however, is strongly reduced in the laboratory frame for the giant dipole resonance. All that is certainly one of the sources of the observed increasing widths of giant resonances. (ii) We get an overall decreasing of the main peak with increasing angular velocity. However this decrease is quite limited - smaller for giant quad~pole than for giant dipole, and hardly reaches 1 MeV in the physical region up to L- 50 &. (iii) Deformation effects could be extremely important. If the nucleus becomes very soft along some symmetry axis, the corresponding dipole frequency can be very small and we can reach some instability for this collective mode. Finally some physical points which are not included in our numerical results deserve a further comment: (i) Due to the classical description of the rotating solution we cannot study effects of deformations in the static ground state, which mainly correspond to a presence of a splitting already for u = 0 [ref. ‘)I. As already pointed out this drawback is not in the theory but in the approximate spheroidal rotating solution. A proper comparison with the data can be performed only in a vex-high-spin region after a transition to an oblate shape. (ii) There are three other sources of a possible increase of the width that we do not discuss in our present approach: (a) The spreading width due to a particle-phonon coupling that we do not include in our linearized theory. The effect of the rotation on this term is unknown but we believe that our phase-space approach could be very appropriate for studying this point. (b) The escape width due to the coupling to the continuum at the particle-emission threshold that should be enhanced in this high excitation energy region. This width has been recently analysed in a fluid-dynamical description of giant resonances on the ground state 13). (c) The damping coming from collision terms is also enhanced because the phase space available for explicit two-body collisions is increasing with the temperature. In principle the phase-space approach looks particularly suitable for studying such an effect starting from a Landau-Vlasov equation, i.e. with a Uehling-Uhlenbeck collision term 14). In conclusion in our present approach we can relate the “observed” width only to a splitting of the resonance which is perhaps the main effect within the present experimental resolution. A more appropriate study of the actual width is needed. This is still an open problem also for the giant resonances on the ground state “) and we think that our phase-space approach could be very useful in evaluating the relative importance of the three main damping sources discussed before, also with increasing spin and excitation energy.
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M. Di Toro et al. / Giant dipole and quadrupole modes
We would like to conclude with a very general statement. Apart from the problem of the description of the rotating solution, in this work we have shown in detail how the giant resonances are sensitive to the properties of the state on top of which the vibration is built. If we take the limit o + 0 all our equations can be used to study properties of giant resonances built on excited states, with arbitrary deformations. This is a quite exciting new spectroscopy: the idea is to study the dynamical structure of excited bands just looking at the properties of GR’s built on top of them. Some promising experiments have been already performed I’) and some more are also in progress using 47r y-ray detectors.
Appendix In the case of a spin-saturated system and neglecting spin-orbit and Coulomb effects, the HF potential, for a Skyrme interaction, in the phase space assumes the form ‘) V,(r,p)=A,(r)+C,(r)p’+B,(r).~,
(A-1)
where A, = Dq + Fq and Dq = to[( 1 -x,,)&
+$&[(l
+ (1 +AxO)p,,]-& t_V*p, -a(; t++ t_)V*p,,
-x3)(u+2)pzq+((1
-x&+2(2+x&$
+((2+XJ)(1+~)+(1--X3))2PqPq,l~a-19
Transforming
(A.3
F4=s[t+Triq.+(t+-&t_)TrtJ,
(A.3)
c,=S[f+P~,+(t+-ff-)p,l,
(A-4)
I$=-$+Jq.f(f+-ff_)JJ
(A.5)
to the intrinsic coordinates,
the cranked hamiltonian becomes
h;= ~+A,(r)-lrnlwxr,*+~~.k. 4
(A.@
It is understood that the quantities appearing in the transformed defined with respect to the intrinsic coordinates:
p(r) =
dkf(c k) ,
The velocity-dependent
+,=; I
hamiltonian
are
dk kikjf(r, k) .
part of the Skyrme interaction introduces an effective mass
M. Di Toro et al. / Giant dipole and quadrupole modes
188
given by
-$=;+ZC,(r)
(A.71
9
as in the non-rotating case, but it does not modify the effects of centrifugal and Coriolis force. Only in the case of the isovector dipole mode (Jq = -Jq.) does the neutron-proton exchange force produce an extra current (see eq. (2.9)) and then the Coriolis coupling term is modified by a factor (1 + 2mnp). Now we discuss in more detail than in the text the various contributions given by the Skyrme interaction to the GR energies. As is well known the potential gives no contribution to the restoring parameter for the case of isoscalar quadrupole, apart from the introduction of the effective mass and a negligible effect at the surface term. For the case of the isovector dipole, we have several contributions coming from the potential. They are
where w”,b,c
St
( > 2
=to(l+~Xg)-a(~t++t_)V2+~t3(a+1)
4’
x[a(l
-x3)+2(2+x3)]~“+$t3[2cr(2+x3)+(l
-x3)(2-o)]pU-‘pqpq,
( W,,. = W,., in agreement with the expected invariance of the Euler equation with respect to the exchange q-q’), (Cz)i=mq
+$I
= -mq
I
d3r[GpqViFS,‘+pS,‘Vi6Fq]
d”r[Tr(6r,)Vi(~)+Tr(i:)Vrs(~)]
I
d3r[Sq - V Tr (‘i”,‘)Vip”+ SpqVt Tr
(P)] .
According to the invariance of the stationary density with respect to a rotation of r about each of the three orthogonal axes, terms involving mixed second derivatives give no contribution to the integrals. References 1) J.E. Draper, J.O. Newton, L.C. Sobotka, H. Lindenberger, G.J. Wozniak, L.C. Moretto, F.S. Stephens, R.M. Diamond and R.J. McDonald, Phys. Rev. Lett. 49 (1982) 434; J.O. Newton, B. Herskind, R.M. Diamond, E.L. Dines, J.E. Draper, K.H. Lindenberger, C. Schtick, S. Shih and F.S. Stephens, Phys. Rev. Lett. 46 (1981) 1383;
M. Di Toro et ai. f Giant dipole and qnadmpole modes
189
A.M. Sandorfi, J. Barrette, M.T. Collins, D.H. Hoffmann, A.J. Kreiner, D. Branford, S.G. Steadman and J. Wiggins, Phys. Lett. 13OB (1983) 19; W. Hennerici, V. Metag, H.J. Hennrich, R:Repuow, W. Wahl, D. Habs, K. Hdmer, U.V. Helmolt, H.W. Heyng, B. Kolb, D. Pelte, D. Schwaim, R.S. Simon and R. Albrecht, Nucl. Phys. AS96 (1983) 329; B. Haas, D.C. Radford, F.A. Beck, T. Byrski, C. Gehringer, J.C. Merdinger, A. Nourredine, Y. Schutz and J.P. Vivien, Pbys. Lett. 120B (1983) 79; E.F. Garman, KA.A. Snover, S.H. Chew, S.K.B. Hesmondhalgh, W.N. Catford and P.M. Walker, The giant dipole resonances in the statistical decay of 63Cu, “%r and “‘Cs compound nuclei, Phys. Rev. C, in press 2) I.N. Mikhailov, in Nuclear collective dynamics, Proc. Brasov Int. Summer School (1982), ed. 0. Bucuresu, V. Ceausescu, N.V. Zamfir (World Scientific, Singapore, 1983) p. 263; K. Neergaard, Phys. Lett. 1lOB (1982) 7; R.R. Hilton, Z. Phys. A309 (1983) 233; J.L. Egido and P. Ring, Phys. Rev. C25 (1982) 3239: J. Winter and P. Schuck, Semiclassical approach to giant resonances of rotating nuclei, Proc. Topical Meeting on nuclear fluid dynamics, Trieste 1982, ed. M. Di Toro, M. Rosina and S. Stringari (IAEA, Vienna, 1983) p. 61; A.V. Ignatynk and I.N. Mikhailov, Sov. J. Nucl. Phys. 33 (1982) 483; 0. Civitarese, S. Furni, M. Ploszajczak and A. Faessler, Nucl. Phys. A408 (1983) 61; S.N. Fedotkin, I.N. Mikhailov and R.G. Nazmitdivov, Phys. Lett. I2IB (1983) 15; M. Faber, J.L. Egido and P. Ring, Phys. Lett. 127B (1983) 5; P. Ring and J.L. Egido, Nucl. PhysA419 (1984) 261 3) D.M. Brink and M. Di Toro, Nucl. Phys. A372 (1981) 151 4) M. Di Nardo, M. Di Toro, G. Giansiracusa, U. Lombard0 and G. Russo, Phys. Lett. 12SB (1983) 240; 132B (1983) 11; M. Di Toro, G. Giansiracusa, U. Lombard0 and G. Russo, J. de Phys. COB. C4 (1984) 297 5) Y.M. Engel, D.M. Brink, K. Goeke and D. Vautherin, Nucl. Phys. A249 (1975) 215 6) H. Krivine, J. Treiner and 0. Bohigas, Nucl. Phys. A336 (1980) 155 7) G.F. Bertsch, Nucl. Phys. A249 (1975) 253; and Nuclear physics with heavy ions and mesons, vol. I tNo~h-Holland, Amsterdam, 1978) p. 175; G. Holzwarth and G. Eckart, Z. Phys. AZ&l (1978) 291; Nucl. Phys. A325 (1978) 1; M. Di Nardo, M. Di Toro, G. Giansiracusa, U. Lombard0 and G. Russo, Lett. Nuovo Cim. 35 (1982) 113; Phys. Rev. C28 (1983) 929 8) A.L. Goodman, Finite temperature mean field theory of nuclei, Proc. Summer Work on nuclear theory, Santa Barbara 1981 World Scientific, Singapore p. 225 9) B.L. Berman and S.C. Fuhz, Rev. Mod, Phys. 47 (1975) 743 10) A. Bohr, Some aspectsof rotational motion, Proc. Scuola Inte~azione de Fisica “E. Fermi”, torso LXIX, Varenna 1976 (SIF, Botogna) 11) M.T. Collins, A.M. Sandor& D.H. Hoffmann and M.K. Saloma, Phys. Rev. Lett. 49 (1982) 1553; Z. Szeflinski, G. Szeflinska, Z. Wilhelmi, T. Rzaca-Urban, H.V. Klapdor, E. Andersen, K. Grotz and J. Metziger, Phys. Lett. 126B (1983) 159 12) M. Di Toro, Nuclear dynamics in phase space, Proc. Winter College on fundamental nuclear physics, Trieste 1984 (World Scientific, Singapore, in press); and preprint Ganil 84.07 13) G. Eckart and K. Ando, in Semiclassical methods in nuclear physics, Grenoble 1984, J. de Phys. C6 (1984) 285 14) E.A. Uehling and GE. Uhlenbeck, Phys. Rev. 43 (1933) 552 15) G.F. Bertsch, P.F. Bortignon and R.A. Broglia, Rev. Mod. Phys. 55 (1983) 287
A435 (1985) Publishing
GIANT
173- 189 Company
DIPOLE AND QUADRUPOLE MODES IN HIGHLY ROTATING NUCLEI
M. DI TORO, Zstituto Dipartimentale
U. LOMBARD0
and
G. RUSSO
di Fish, Universitci di Catania, Zstituto Nazionale Catania C. so Ztalia, 57-Z 95129 Catania, Italy
di Fisica Nucleare,
Sez. di
Received 22 June 1984 (Revised 22 August 1984) Abstract: We discuss isovector CD and isoscalar GQ resonances built on high-spin states observed in heavy-ion reactions within a microscopic fluid-dynamical theory. We reproduce the main features of the experimental data from the analysis of scaling solutions for small oscillations of two Vlasov fluids in a rotating frame. Results are shown for 16sEr. We discuss the possibility of studying excited nuclear states by just looking at the properties of the giant resonances built on top of them.
1. Introduction
A great deal of progress has been made during the last years in the investigation of GR’s built on high-spin states observed in high-energy y-ray spectra from the deexcitation of medium and heavy nuclei produced in fusion and deep-inelastic reactions I). The appealing physical aspects of such a study lie in the properties of the GR strength function in systems far from the ground state and its dependence on properties of highly-excited nuclei such as excitation energy, spin and nuclear deformation. Although some experimental discrepancies still exist, three main features come out from the available experimental data: (i) the strength function does not depend much on the ground-state deformation; (ii) the centroid of the resonance shifts to lower energies with higher angular momenta (A&/AL-0.05-0.1 MeV/A), (iii) the GR overall widths are much broader than typical ground-state GR widths. Several theoretical attempts to solve this problem have been proposed recently 2), ranging from simple solvable models to quite huge cranked RPA calculations. The approach proposed in this paper is fluid-dynamical in that the collective nuclear dynamics is developed in a phase space as a semiclassical limit of the self-consistent TDHF equations 3,4). C onsequently our results, which are quite easy to work out, can be obtained using realistic interactions and can be directly compared with full cranked RPA calculations. We can say that with respect to other theoretical approaches the most interesting feature of our method is that we get analytical and physically transparent results while retaining the fully microscopic foundations with 173
M. Di Toro et
174
~1. /
Giant dipole and quadrupole modes
the consequent use of realistic interactions and the possibility of a simple analysis of the dynamical effects of each term of the effective force. A semiclassical approach to this problem was also recently introduced by Winter and Schuck2) but with a model residual interaction related to the particular collective motion studied. The key point of our approach is to assume that for a giant collective state all the strength is concentrated on only one level. This ansatz is largely justified from RPA calculations as well as from variational fluid-dynamical approaches and it corresponds to taking into account only the lowest multipole distortions of the momentum distribution during the vibration, which is described as a scaling mode. Our general philosophy is to use this simplifying assumption to study giant resonances in a wider context, such as in a rotating nucleus or as doorway states in particular reactions. In this paper we are able to show the effects of dynamical deformations and Coriolis coupling on the frequencies and strengths of GDR’s and GQR’s built’on high-spin states. In particular an agreement with the above-mentioned experimental features is found. 2. Scaling solutions of the cranked Vlasov equation The dynamics of a two-fluid system will be derived from the Vlasov equation in a frame rotating with angular frequency w (assumed to be along the z-axis):
(2-l) where h’ = h -or * L is the Wigner transform of the self-consistent cranked hamiltonian with a Skyrme interaction. The label q is the isospin coordinate and quantities without any isospin labels are understood to denote total values. Eq. (2.1) can be obtained as a semiclassical limit of the cranked TDHF equation for the Wigner transform f,( r, p, t) of the one-body density matrix. Transforming in the intrinsic coordinates (r’ = r, k = p - mo x r) the equation of motion becomes “(z,”
‘)={hb(r,
k, t),f,(r,
k, t)}+2mw
* (V&h XV&),
(2.2)
where from now on, the Poisson brackets are defined with respect to r and k. For Galilei-invariant Skyrme forces 5, the transformed cranked hamiltonian assumes the form +B,(r).
k+A,(r)-imlwxr12,
(2.3)
where the explicit expressions for m *, A and B are given in the appendix. In the following we shall focus our attention on the SKM force which reproduces very
M. Di Toro et al, / Giant dipole and quadruple
modes
175
well the nuclear compressibilities and the symmetry energy near the nuclear surface “). It is worth noticing that, as in the classical case, the centrifugal and the Coriolis forces appear explicitly in the hamiltonian and in the equation of motion, respectively, as an effect of the rotation. The rotating states of the nucleus are solutions of the stationary Vlasov equation in the rotating frame, (2.4) In general, the function ff is not isotropic in the intrinsic momentum space and the Coriolis force gives rise to a current flow in the rotating frame. We restrict our analysis to isotropic stationary solutions for which the Coriolis term in eq. (2.4) vanishes exactly. This kind of solution exists for Galilei-invariant non-local interactions and implies rigid currents and rigid moments of inertia for the rotating nucleus ‘). As discussed in ref. 3), this solution corresponds to a classical equilibrium condition for the stationary rotating system. In this sense we can say that in our results shown in sects. 3, 4 there is some kind of temperature effect, in the reference state, which is acting in the direction of a classical description of the rotating nucleus. Giant resonances are described as small-amplitude vibrations on the rotating nucleus. We introduce a variation with respect to the stationary rotating distribution function discussed before,
and then we linearize the cranked Vlasov equation (2.2). In this respect our procedure is completely equivalent to a RPA theory in phase space. However, we interpret as giant resonances only simplified solutions corresponding to scaling modes. This assumption has been proven to be quite successful in describing giant resonances in the static case, i.e. for nuclei in the ground state ‘). It corresponds to a maximum collectivity condition for the studied motion in so far it is equivalent to take into account only the lowest multipole (I = 0, 1,2) distortion of the distribution function in momentum space. Consistently the equation of motion can be reduced to a harmonic form which means that all the strength is concentrated on only one level. Of course in the RPA dynamics there are explicit quantum terms which are neglected in the Vlasov approach. In this respect we would like to remark the foliowing: (i) Quantum-dynamical contributions are small for the particular solutions, scaling oscillations, we associate with giant resonances ‘*12). (ii) In principle we include all the quantum effects in the stationary state on top of which we build the giant mode. The numerical results shown in this paper are limited to a classical stationary rotating distribution function just because we do not have available fully self-consistent cranked Hartree-Fock solutions.
176
M. Di Toro et al. / Giant dipole and quadrupole modes
The extension of the scaling ansatz for giant modes on rotating nuclei is straightforward in the limit of rigid current assumption for then the only effect of the rotation on the momentum distribution is a shift of the Fermi sphere by the amount moxr. We assume for the variation of the distribution function
%=K$x,+k%,
(2.6)
where S,(r, t) is the displacement vector and x4(r, t) is a velocity field, assumed to be local. Eq. (2.6) implies second-order distortions of the Fermi surface, as in the static case. The most intuitive scaling motion is obtained from an irrotational displacement field S,(r, t) = V&(r, t) ,
(2.7)
where &(r, t) gives the shape of the deformation associated with the studied vibration and the corresponding time-dependent amplitude. At this point we are left with the problem of finding a connection between x and S and this will be worked out within the framework of the fluid-dynamical equations. From eq. (2.2) we can generate an infinite chain of fluid-dynamical equations for the k-moments of the distribution function. If we are able, with some physical ansatz, to truncate the chain at the lowest k-moments level, instead of the complicated cranked Vlasov equation, we would be able to solve a set of differential coupled equations for relatively few unknown (r, t) functions, as density, current, kinetic energy tensor and so on. These equations are completely classical, quantum effects being in the initial conditions and in the truncation procedure. For scaling modes we can exactly close the fluid-dynamical chain at the lowest two /c-moments, continuity and Euler equations “). The equation for the zeroth k-moment assumes the form
with pq =
I
d3kfq(r,k t) ,
Jq
=I tf,(r,
Jt“’=; ; t+[,04,Jq - p,J,,l ,
S t) d3k9
t+= t,+ t* .
As is well known the current is not conserved locally due to the presence of the neutron-proton exchange term in the non-local potential. The total number of nucleons of a given isospin type is, of course, conserved. For isoscalar modes the source term disappears, being Jt’.+ Jtj. = 0. Obviously, the total density is locally conserved in any case.
M. Di Toro et al. / Giant dipole and quadrupole modes
177
For the first k-moment we get the Euler equation
+i Tr (Tq)V, --& (
1
+Jqv;B,+(B,-v,)Jq++(J,~v,)B,+J,x(v,xJ,)=O. (2.9)
The last four terms in the second line do not give any contribution after linearization in the limit of rigid current. The kinetic energy tensor d, is given by
(Tq)ij=
I
3fq(
r, k, t) d3k.
3. Isovector dipole mode
Isovector giant dipole resonances are described as out-of-phase small-amplitude collective oscillations Sf,(r, k, t) = -Sf,,(r, k, t) of the neutron/proton distribution function around the stationary value ff(r, k, t). Scaling deformations correspond to a flow pattern given by an irrotational dipole velocity field. Using a generalized scaling generator 4), a,(r, k, r) =x4(*, t)+k.
S,(r, r),
(3.1)
we have Mq = CC& cq4).
(3.2)
This involves second-order distortions of the momentum distributions while taking into account the shift of the Fermi sphere to the rotation. Assuming the nucleus to undergo rigid rotations (,I:* = 0) and in the limit of the irrotational displacement field S, = Vrqhq,one obtains from the continuity equation the simple relation xq=-
“6 1+2mqp
’
77=2
t+
(3.3)
which allows us to close the fluid-dynamical chain at the lowest two k-moments. In fact, the first k-moments of the transition distribution function can be expressed only in terms of the field S,(r, t): 6P,=Vr*
J, = - 1
9
(3.4)
$:qp,
(3.5)
(P$sj)
(3.6)
178
M. Di Toro et al. / Giant dipole and quadrupole modes
The linearized Euler equation becomes the equation of motion for the displacement field: P”,‘st,
-1+2m7)p +pv;
-2pS,‘wxS,+6p,ox(oxr)+V; V D”’
[+f”,‘6p,]+spqA
+4pV,[Sp,
m
+P:
[
s;i, *st
mq 1
VJD, m
Tr ;i”,‘+p”,‘Tr a?,]
+l)[GpqV,(Tr?‘)+TrG;iqV,p”‘]=O,
(3.7)
where Dq is the local part of the Skyrme HF potential and p = - t_/4fi2, t_ = tl - tz (see appendix). It describes the dynamics of the collective motion and, in principle, can be solved by imposing suitable boundary conditions. As far as we are interested in gross properties of the giant modes such as centroid energies and transition strengths in terms of the angular frequency, we follow a simplified procedure. Let us assume for the scalar field the usual Tassie-Bohr form
d,k t) = &3t)x+
ay4(t)y
Sq=(aZ,
+ a,4(t)z,
a;, cxg.
(3.8)
By taking the scalar product of eq. (3.7) with each component of the real scaling field (3.8) and by integrating over the spatial coordinates we end up with three differential coupIed equations for the (Y&,=(t) unknown functions. AS Vi(Sq)j = OV,j, the gradient and divergence terms in the linearized Euler equation give no contribution after projection. Actually, from a direct evaluation of the collective kinetic energy, it is possible to show how this procedure gives the right collective mass. For rotations about the z-axis, the z-mode is not affected by the presence of centrifugal and Coriolis forces but only indirectly “feels” the rotation through dynamical deformations in the stationary density. For a stationary distribution function which is invariant with respect to rotation of 7~ about each of the three spatial orthogonal axes (good signature), the equation for the z-mode is
while the x, y modes, coupled only through the Coriolis term, satisfy M,Gtj-2mwZqai,4+(Cx-mo2Zq)a’:=0, M,&,4+2moZ,ci~+(C,,-mo2Zq)a,4=0,
where Mc=m
I
Pq
st
1 + 2m7jpsf
d3r
(3.10)
M. LX Toro
et al. /
Giant dipole and ~~ad~pole
modes
179
is the collective inertial parameter, and 2, = f d3rp, ;
c&Y,Z
J +J
= rrzq
d3r[(Tr P)a&,
p;+ (Tr
%$J&,,p”‘]
d3r([D”,‘+ft,(l+2xo)p”,‘-t(ti+tt-)V2p”,’]a:,,p”,’
+~t3(1+2~3)(pSf)“-‘[(a-l)p~~-(tr+l)ps,’](~,,,p”,‘)2}
(3.11)
are the kinetic energy and interaction contributions, respectively. In the harmonic approximation, eq. (3.9) for the z-mode gives n, = (c;/WJ”2.
(3.12)
The motion in the x -y plane, eqs. (3.10), has two independent solutions of the form CY,= A(*) cos (.f-?*)t + t$‘*‘) ,
ay = I#*’ sin (@*‘r + 4”‘)
with J-p’=
C+- mo2Zq+2
I
Mc
mwZ, 2 ( M, )
*[(~)2+(~)2(c+-~~)+4(~)4]“2}“2,
#*>/A(*t)
2mwZ&!(*)
=
-M c f2’*‘”+ C, - mw”Zq .
(3.13)
(3.14)
It should be noticed that the major cont~butions to the integrals of eq. (3.11) come from the nuclear surface. Thus, it is not the detailed behavior of the local density inside the nucleus which is important here, but rather the shape at the surface. Transforming to the laboratory system, ff:=a,cos&-f+,sinwt, ak=ffXsinwt+a;coswt,
ffzI- - %,
(3.15)
we get, in general, a splitting into four components: CY:=~(A(*)-B(*tf)cos[(n’*‘-w)t+#(*)] +$(A’*‘+B’*‘)
cos[(~(*‘+u)t+c$‘*‘],
a~=~(B(*‘)-A(*‘)sin[(~t(*)-w)ti-#f*)] +~(A(*~+~(*))sin[(~(*)+~)~+#(~)].
(3.16)
180
M. Di Toro et al. / Giant dipole and quadrupole modes
Apart from the splitting of the GDRinto two components analogous to that displayed by static triaxial deformations, the Coriolis force gives rise, in the laboratory frame, to a further splitting into four frequencies. The resulting modes can be classically interpreted as linear superpositions of two circular motions in opposite senses for each component. In particular, for axial deformed nuclei along the z-axis we get from eq. (3.14) B (*) = TA(*) and only two frequencies a’*-‘~ w are expected to occur in the laboratory frame. The expressions for the dipole frequencies, obtained so far, are very general and allow fully self-consistent calculations starting from a finite-temperature cranked HF code to get the stationary solution 8). As a preliminary calculation we specialize our method to a classical rotating nucleus about the z-symmetry axis. A very nice model calculation was discussed by Hilton ‘) just studying the normal oscillations of a system of 2 protons and N neutrons interacting via a dipoie-dipole two-body force, moving in a deformed rotating harmonic-oscillator well. In our microscopic picture we could get similar results using a local Hartree-Fock mean field. Indeed if we specialize our frequencies (3.12), (3.13) to the local case we easily get the results eq. (13) of the Hilton work. Of course in this model, contrary to our approach, there is no way to get a self-consistent analysis of the dynamical effects of the interaction on the rotational as well as on the vibrational motion. EVALUATION
OF THE DIPOLE ENERGIES
Although we have developed the formalism for a general case, in order to simplify the discussion and to emphasize the main effects of the rotation on the dipole frequencies, we shall assume a proportional density distribution of neutrons and protons and very close values of the Fermi energies for the two systems. All that means (3.17) We also neglect Coulomb effects in the calculation of the restoring parameters. In the Thomas-Fermi approximation for the kinetic energy density, integrating by parts eq. (3.11) we get Ci = Z&/A, where F=,YJ=
with
I d3cf(
$%&,y,epst12
(3.18)
181
M. Di Toro et al. / Giant dipole and quadrupole modes
For axially-symmetric
dynamical
deformations
along the z-axis (F” = F,) we get (3.19)
(3.20) with M
For the cranked
stationary
s d3r
= m
density
‘“’
1 +2m?7pst
we assume
SmA.
an oblate
Psf(r’ ‘)= 1 +exp [(r: R(tl))/a]
Wood-Saxon
’ (3.21)
R(e)=Ro(~)[1+~(3cos2e-I)] with p. = 0.145 fmm3, a = 0.5 fm and R,(p)
shape:
fixed by the condition
5
d3rpSf(r, p) = A.
The equilibrium deformation p(w) is constructed by the virial tensor method [see Mikhailov ‘)I which consists in imposing a balance among the forces acting on the nucleus, namely pressure, surface, centrifugal, Coulomb and nuclear forces. We solve analytically the integrals (3.18), assuming to a very good approximation
0aP=
Pi -c6(
G
In this way, explicit
.n(“‘(&)
=
expressions
MO
@(‘) M(p)
R,(
r-R(B)).
for the dipole
[ R,(O) 1
frequencies
are obtained:
2[1+~~+~p2]+
TtlA
*M(B)@’
(3.22)
where
is the dipole centroid energy for spherical nuclei in the non-rotating provides a good estimate of the experimental values ‘).
case, which
M. Di Toro ef al. / Giant dipole and quadrupole modes
182
10
I
50
100
150
200
Lo-l)
24hQZ 209 8 w
16;
(hR)
-
to"
hw(MeV)
Fig. 1. Shift and splitting, in the laboratory frame, of the isovector GDR energies in terms of ho for 16sEr (SKM force).
Transforming to the laboratory system, we get two frequencies, R, - o and R- + o, in addition to the unaffected 0,. This means that Coriolis effects are almost negligible. In the limit of local potentials (M(p) = mA), the first two frequencies become equal. In this case the rotation does not give any effect on the dipole mode apart from dynamical deformation. The expected effects, in the laboratory frame, are shown in fig. 1 for 16*Er. We predict a clear increasing of the width mainly due to a splitting of the giant level. The dipole absorption cross sections, easily computed in our approach 4), are used as weights in order to get the average dipole energy (h0). In fig. 1 we relate an angular momentum L to the angular velocity through a rigid moment of inertia for the corresponding oblate shape. The unphysically high values of L are due to the oblate spheroid constraint for the stationary rotating solution which prevents the nucleus to undergo fission. We obtain a small shift of the centroid of the resonance to lower energies of the order of 1.OMeV for angular velocities which can be reached in realistic heavy-ion reactions (ho G 1 MeV). Axially-symmetric oblate shapes along the yrast line should correspond to the final regime of shape changing with increasing angular momentum. At this stage the rotation is no longer collective being the angular momentum due to single-particle alignment. Nevertheless, a cranking evaluation of the average moment of inertia gives the rigid-body value, i.e. the average “single-particle” rotation around the symmetry axis shows quite a similar behavior to collective rotation ‘O).Our model calculation corresponds to a limit case somehow opposite to that analysed in other theoretical approaches 2, where rotations of prolate nuclei are considered. We would like to stress again that our results eqs. (3.1 l)-(3.16)
183
M. Di Toro et al. / Giant dipole and quadrupole modes
are completely general and can be used with any self-consistent solution.
stationary rotating
4. Giant quadrupole resonances We describe isoscalar giant modes as small-amplitude collective oscillations in which neutrons and protons undergo an in-phase irrotational flow. Now the Euler equation, eq. (2.9), after linearization becomes
(4.1)
In the case of quadrupole modes we assume for the corresponding field the form S = V4 (r, t) with 9(r, r) = ?j [ak(f)+k(r)+ &(r)=Gr2(Yzk+ &(r)
displacement
dt)@dr)l,
Y$)=zAr’XiXI, ij
= 4 r2( Yzk - Y&k)= C Af'XiXj ij
.
(4.2)
For total quantities (neutrons plus protons) using the same procedure outlined in sect. 3, from eq. (2.6) we get the same expressions (eqs. (3.4) and (3.6)) for the first two even k-moments of the transition distribution function. For the total current we have the simplified structure J = -p,$
(4.3)
simply because exchange terms are not playing any role if we sum up the currents. Following still the same method discussed for isovector modes, we take the scalar product of eq. (4.1) with each component of the displacement field. Integrating over the space coordinates, we reduce it to a set of coupled equations Mk~k(f)+Bk~~(t)+Ck(Yk(t)=O, M~~r(l)+B~dlk(f)+C~(YE(t)=O,
(4.4)
where, for rotations around the z-axis, one has Mk = ML = C JAf’&’
,
ij
Bk
=
-BE= 20 C I,(A’,5’A~~‘-AI:‘AI~))
Ck=C,-=2C(Ar’)2 ij
I
,
d3r-$fSl(r)+m2CIi(AljL))2(8i3+Sj3-2) ij
(4.5)
184
M. Di Toro et al. / Giant dipole and quadrupole modes
for the collective mass and Coriolis coupling and restoring parameters, respectively. The coefficients Ii = M j p,,( r)Xf d3r are the inertial parameters. We remark that, as in the state case 4Y7),th e only details of the interaction are in the effective mass. This coupling only between the k- and 6components of a given angular momentum projection is actually valid if the nucleus undergoes an axially symmetrical deformation with the rotation. In the general case of a triaxial shape we get a much more complicated set of five coupled equations. We will restrict our analysis to dynamical axial symmetric deformations. Therefore, we expect to see a splitting into three levels due to the effect of the deformations and a further splitting into five levels due to the Coriolis coupling. Assuming a harmonic time evolution, we can easily get the eigenfrequencies
iI@‘=
dR*&,
k=O, 1,2,
(4.6)
where, for k = 0, Bk = 0. Everything can be evaluated once we know the cranked self-consistent stationary solutions p,,(o) and ;i,Jw). As discussed in the previous section in this work we show some first, not self-consistent, results obtained from a stationary Thomas-Fermi distribution in an oblate cranked shape. In order to compare with the experimental values we need to transform the quadrupole motion to the laboratory system. By using the rotation properties of the spherical harmonic functions with respect to the z-axis, we get ff:=ffkcoskWf-~~sinkwt, ~~;=a~sinkot+~~~;coskOf, and, consequently,
(4.7)
for the frequencies O(,“(lab) = a’,“‘* kw .
(4.8)
In fig. 2 we report GQR energies, in the laboratory frame, for 16’Er as a function of the angular frequency and classical angular momentum L = I(o), I(w) being the rigid moment of inertia along the deformation axis. The pattern of the energy branches results from the competition between dynamical deformation and centrifugal force. For the k = 1 case the Coriolis splitting, in the rotating frame, is strongly reduced for high angular velocity. This is related to the oblate deformation we are only allowing in our calculation. The scaling velocity fields V+l,i become more and more aligned to the rotation axis and the Coriolis term of the Euler eq. (4.1) has no effect. This implies that, contrary to the dipole case, how we get a much wider level splitting in the laboratory frame. All together the five curves spread out with increasing angular momentum and this can partially explain the observed increasing of the width of the resonance ‘>. The average frequency, computed using as weights the quadruple absorption cross
M. Di Toro et al. / Giant dipole and quadrupole modes
10
I
50
100
150 200
185
L(h)
hw(MeV) Fig. 2. Shift and splitting of the isoscalar GQR energies, in the laboratory frame, in terms of the angular frequency for 16sEr.
sections, is almost independent interest.
of the angular velocity in the region of physical
5. Conclusions
We describe giant resonances on high-spin states as small-amplitude vibrations of a system evolving through a cranked Vlasov equation, which is the semiclassical limit of a TDHF theory. Extending the notion of “shape scaling” in the rotating case, we study such particular scaling solutions which give the bulk of the response for giant modes on the ground state. The theory is fully microscopic in its foundations and .then we can use realistic interactions (Skyrme-type forces) and we can easily follow dynamics effects of each term of the force. Indeed the advantage of this approach is that while we retain the main features of a self-consistent RPA theory we finally get analytical results, with quite clear physical meaning, for giant frequencies and transition strengths. To get numerical results to compare with experiment we need a suitable description of the highly-rotating hot nucleus, and actually some knowledge of the density and kinetic energy density distributions for neutrons and protons. The results we discuss in this paper are obtained using a classical description of oblate spheroids with a deformation consistent with the angular velocity, rigid moment of inertia and rigid rotational current. Of course this stationary rotating solution must be improved, and we are planning to do so. At the moment our results could be related to the
186
M. Di Tore et al. / Giant dipole and quadrupole modes
limit case of a fully aligned very-high-spin state. Bearing well in mind these limitations there are some main features we can extract from our calculations: (i) A splitting of the giant level essentially due to dynamical deformation eff e&s. We predict also a,further Coriolis splitting, which, however, is strongly reduced in the laboratory frame for the giant dipole resonance. All that is certainly one of the sources of the observed increasing widths of giant resonances. (ii) We get an overall decreasing of the main peak with increasing angular velocity. However this decrease is quite limited - smaller for giant quad~pole than for giant dipole, and hardly reaches 1 MeV in the physical region up to L- 50 &. (iii) Deformation effects could be extremely important. If the nucleus becomes very soft along some symmetry axis, the corresponding dipole frequency can be very small and we can reach some instability for this collective mode. Finally some physical points which are not included in our numerical results deserve a further comment: (i) Due to the classical description of the rotating solution we cannot study effects of deformations in the static ground state, which mainly correspond to a presence of a splitting already for u = 0 [ref. ‘)I. As already pointed out this drawback is not in the theory but in the approximate spheroidal rotating solution. A proper comparison with the data can be performed only in a vex-high-spin region after a transition to an oblate shape. (ii) There are three other sources of a possible increase of the width that we do not discuss in our present approach: (a) The spreading width due to a particle-phonon coupling that we do not include in our linearized theory. The effect of the rotation on this term is unknown but we believe that our phase-space approach could be very appropriate for studying this point. (b) The escape width due to the coupling to the continuum at the particle-emission threshold that should be enhanced in this high excitation energy region. This width has been recently analysed in a fluid-dynamical description of giant resonances on the ground state 13). (c) The damping coming from collision terms is also enhanced because the phase space available for explicit two-body collisions is increasing with the temperature. In principle the phase-space approach looks particularly suitable for studying such an effect starting from a Landau-Vlasov equation, i.e. with a Uehling-Uhlenbeck collision term 14). In conclusion in our present approach we can relate the “observed” width only to a splitting of the resonance which is perhaps the main effect within the present experimental resolution. A more appropriate study of the actual width is needed. This is still an open problem also for the giant resonances on the ground state “) and we think that our phase-space approach could be very useful in evaluating the relative importance of the three main damping sources discussed before, also with increasing spin and excitation energy.
187
M. Di Toro et al. / Giant dipole and quadrupole modes
We would like to conclude with a very general statement. Apart from the problem of the description of the rotating solution, in this work we have shown in detail how the giant resonances are sensitive to the properties of the state on top of which the vibration is built. If we take the limit o + 0 all our equations can be used to study properties of giant resonances built on excited states, with arbitrary deformations. This is a quite exciting new spectroscopy: the idea is to study the dynamical structure of excited bands just looking at the properties of GR’s built on top of them. Some promising experiments have been already performed I’) and some more are also in progress using 47r y-ray detectors.
Appendix In the case of a spin-saturated system and neglecting spin-orbit and Coulomb effects, the HF potential, for a Skyrme interaction, in the phase space assumes the form ‘) V,(r,p)=A,(r)+C,(r)p’+B,(r).~,
(A-1)
where A, = Dq + Fq and Dq = to[( 1 -x,,)&
+$&[(l
+ (1 +AxO)p,,]-& t_V*p, -a(; t++ t_)V*p,,
-x3)(u+2)pzq+((1
-x&+2(2+x&$
+((2+XJ)(1+~)+(1--X3))2PqPq,l~a-19
Transforming
(A.3
F4=s[t+Triq.+(t+-&t_)TrtJ,
(A.3)
c,=S[f+P~,+(t+-ff-)p,l,
(A-4)
I$=-$+Jq.f(f+-ff_)JJ
(A.5)
to the intrinsic coordinates,
the cranked hamiltonian becomes
h;= ~+A,(r)-lrnlwxr,*+~~.k. 4
(A.@
It is understood that the quantities appearing in the transformed defined with respect to the intrinsic coordinates:
p(r) =
dkf(c k) ,
The velocity-dependent
+,=; I
hamiltonian
are
dk kikjf(r, k) .
part of the Skyrme interaction introduces an effective mass
M. Di Toro et al. / Giant dipole and quadrupole modes
188
given by
-$=;+ZC,(r)
(A.71
9
as in the non-rotating case, but it does not modify the effects of centrifugal and Coriolis force. Only in the case of the isovector dipole mode (Jq = -Jq.) does the neutron-proton exchange force produce an extra current (see eq. (2.9)) and then the Coriolis coupling term is modified by a factor (1 + 2mnp). Now we discuss in more detail than in the text the various contributions given by the Skyrme interaction to the GR energies. As is well known the potential gives no contribution to the restoring parameter for the case of isoscalar quadrupole, apart from the introduction of the effective mass and a negligible effect at the surface term. For the case of the isovector dipole, we have several contributions coming from the potential. They are
where w”,b,c
St
( > 2
=to(l+~Xg)-a(~t++t_)V2+~t3(a+1)
4’
x[a(l
-x3)+2(2+x3)]~“+$t3[2cr(2+x3)+(l
-x3)(2-o)]pU-‘pqpq,
( W,,. = W,., in agreement with the expected invariance of the Euler equation with respect to the exchange q-q’), (Cz)i=mq
+$I
= -mq
I
d3r[GpqViFS,‘+pS,‘Vi6Fq]
d”r[Tr(6r,)Vi(~)+Tr(i:)Vrs(~)]
I
d3r[Sq - V Tr (‘i”,‘)Vip”+ SpqVt Tr
(P)] .
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M. Di Toro et ai. f Giant dipole and qnadmpole modes
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