Microscopic theory of the isovector dipole resonance at high angular momenta

Nuclear Physics A419 (1984) 261-294

0 North-Holland Publishing Company

MICROSCOPIC THEORY OF THE ISOVECTOR DIPOLE RESONANCE

AT HIGH ANGULAR MOMENTA+

P. RING

Physik-Department der Technischen Universitiit Miinchen D-8046 Garching, W. Germany

L. M. ROBLEDO and J. L. EGIDO

Departamento de Fisica TPoriea, Universidad Autonoma Madrid 34, Spain

M. FABER

lnstirut ,jiir Kernphysik der Technischen Unicersitiit Wien, A-1020 Vienna, Austria

Received 22 September 1983

Abstract: The giant dipole resonance at large angular velocities and finite temperatures is studied within the framwork of temperature-dependent linear response theory for superfluid Fermi liquids. The peak energy of the resonance and its splitting is discussed as a function of angular momentum and temperature. The influence of the shape and gap parameters on the fine structure is investigated.

1. Introduction

For more than thirty years experimental and theoretical efforts have been

devoted to the study of giant resonances in nuclei [for a recent review, see ref. ‘)I.

The best-known and the most pronounced of these resonances is the giant dipole

resonance (GDR) which is observed in nearly all nuclei at a resonance energy of

roughly 78A - ‘I3 and corresponds to an isovector mode, where protons and

neutrons move out of phase. Within the last ten years other types of giant

resonances have been studied in great detail.

Most of these investigations, however, have been concentrated on the giant

resonances based on the ground state. Morinaga, however, had already pointed out in

1956 [ref. ‘)J that there should be giant resonances based on each nuclear state.

Only recently has experimental evidence been found for giant resonances based on

higly excited rotating compound states, which are built after heavy-ion fusion

reactions 3* 4). A pronounced peak in the region between 10 and 20 MeV was

found.

’ Supported in part by the Bundesministerium fiir Forschung und Technologie and by CAICT, Spain.

261

262 P. Ring et al. 1 Microscopic theory

The study of giant resonances in hot rotating systems is interesting for several

reasons: Temperature and angular velocity provide two new degrees of freedom

which give us, in general, additional insight, into nuclear interactions and nuclear

structure. An example are shape changes and phase transitions at high excitation

energies or at large spins which could show up eventually in the fine structure of

the GDR. Another example is the temperature dependence of the effective mass.

We should be aware, however. that at the moment experimental study of such

resonances is just beginning. Their resolution cannot yet be compared with the

resolution of experiments on conventional giant resonances, as for instance electron

scattering experiments, and it will probably require a large additional experimental

effort in the future to obtain all the interesting information on nuclear structure at

high excitation energies and high spins which one could expect from the theory.

In the meantime it is a challenge to study those resonances in highly excited

systems at least theoretically. The theory of conventional giant resonances has been

pushed very much in the last twenty years. Very different phenomenological

models have been proposed and a number of microscopic methods have been

used ‘). There is general agreement that the random phase approximation (RPA)

with effective (possibly energy-dependent) forces and mass parameters provides a

numerical tool to investigate the peak energies and their splitting. For a

quantitative description of the width, however, one has to go beyond the RPA and

one must include more complicated configurations 5).

In the theoretical description of giant resonances at high spins so far two types

of calculations have been performed:

(i) The harmonic oscillator model introduced by Brink6) for the GDR built on

the ground state has been extended to the rotating scheme by several

authors ‘-lo).

(ii) Linear response theory was used by Egido and Ring l’) for the study of the

nucleus 164Er at zero temperature within a small configuration space. These

calculations have extended to finite temperatures 12) and to a realistic Woods-

Saxon potential 13). It was found that there is some splitting of the giant dipole

state caused by the rotation directly. It is, however, not very large at realistic

velocities in heavy nuclei. The main part of the splitting is caused by the

deformation, which itself can change at high spins due to the stretching effect.

So far, most of the investigations have neglected pairing correlations, which are

very important in heavy deformed nuclei, at least for the vicinity of the ground

state. In the calculations of ref. ” ) there were some indications that they can also

have an influence on the GDR. In order to study this more systematically in

connection with realistic potentials and finite temperatures we use in this paper

the most general form of linear response theory including temperature and pairing

correlations.

In sect. 2 we present the theory using a generalized quasiparticle formalism

which simplifies the equations considerably. In sect. 3 we discuss in detail special

P. Ring et al. 1 Microscopic theory 263

features which occur in the description of rotating resonances. As an example we

study in sect. 4 the rotating harmonic oscillator with pairing at finite temperatures.

Sect. 5 is devoted to realistic investigations in two heavy deformed nuclei: “‘Er

and “‘Er. In particular we study phase transitions such as pairing collapse and

shape changes as a function of the angular momentum and the temperature and

their influence on the fine structure of the GDR. In sect. 6 we summarize our

results.

2. Temperature-dependent linear response theory of superfluid nuclei

It is well known that linear response theory is a very powerful technique for

describing giant resonance phenomena in a microscopic framework 14). It is

equivalent to the random phase approximation, but it allows directly the

calculation of the y-absorption cross section as a function of the y-energy. It has

been used by the Soloviev group ’ 5, for the description of collective excitations

based on the ground state and can easily be extended to the description of giant

resonances based on excited states, such as high-spin states along the yrast line l1 )

or on compound states described by a temperature-dependent mean-field

theory 16. 13).

The basic idea is that one investigates the behavior of the nuclear many-body

system in an oscillating external field P with small amplitude in the framework of

time-dependent mean-field theory. The response of the system is obtained in linear

order in E. Details of the derivation for systems without pairing correlations and at

zero temperature are given in ref. “). In the following we give a short derivation

for the general case with pairing and at finite temperatures using a very elegant

formalism which avoids the lengthy formulas showing up in other derivations 18).

2.1. DERIVATION OF THE LINEARIZED BETHE-SALPETER EQUATION

We start with the assumption that the system can be described at each time as a

statistical ensemble of many-quasiparticle configurations

In) = Jm, . . . m,) = cl;, . . . c$$j> (2.1)

characterized by the quantum numbers m, . . .m,. 14) is the vacuum to these

quasiparticle operators :

%l4> = 0 for all m = 1 . . . M. (2.2)

We assume that the system is always in thermal equilibrium with a heat bath of

temperature T, i.e. we can calculate the expectation value of an arbitrary operator

264

6 by averaging

P. Ring et al. / Microscopic theory

<@, = CP,<&>, (2.3) n

where the probabilities are given by

P, - ew[ - PbGI + Em2 + . + -kJl (2.4)

and normalized to unity, E, are the quasiparticle energies in the intrinsic frame

and the parameters i and o are determined by the constraints

(JIJT = .$(I+ l), (&), = N. (2.6)

Since we assume that the states In> are generalized Slater determinants we can

use a generalized Wick theorem 19, 20) and express the thermal averages of any

operator by the single-particle density matrix

(2.7)

where the operators c,,c: represent an arbitrary basis. It is convenient to

introduce a set of operators a,, which combine the creation and annihilation

operators :

a, = a,

a, = a,’ m = 1 . ..M. p = 1 . . . M, -1, -2 ,..., -M

They obey the commutation relations

{a,, a,.,\ = 6,,,. (2.9)

An arbitrary single-particle operator E given in its quasiparticle representation 1 7,

P = F0 + 1 F;;a,‘a,, + c (F;;a,‘cr,‘, +F;;ama,), (2.10) mm’ m -c m’

can be written as

P. Ring et 01. / Microscopic theor) 265

with

i

F” F20 .F =

_FO’ >

__FLP ’

.F” = F”f~Tr,(F”). 2 (2.12)

Using (2.8) we find

.$$.@ = - .Q,. (2.13)

In particular, we can express the expectation value of any single-particle operator E by

<p> r = 5’ +~Tr,,(.~.?Z). (2.14)

Within the same formalism a general Bogoliubov transformation

LY,’ = c u,,r: + QmCk (2.15) k

can be represented through the matrix

yjlI‘= u v* ( ! v u*’

or by the unitary operator exp(i2) given in analogy to (2.11) by

(2.16)

(2.17)

with the matrix

+#‘ = exp (i?Y). (2.18)

The density matrix of the form (2.7) is diagonal in the quasiparticle representation

(2.19)

with the Fermi occupation numbers

f,L!---. efiEm + 1

(2.20)


Assuming the operators up to be time dependent,

a,(t) = eiHta,(0)e-iH’,

and using the Wick theorem again we derive the equations of motion.

(2.2 1)

with

The compound

solution do

i.2 = [.x(a), a], (2.22)

-x,,. = <{a,,Hl,at)),. (2.23)

state with vanishing external field corresponds to the static

[-X(@), @] = 0. (2.24)

These are the temperature-dependent HFB equations ‘l) which can also be

obtained from a minimization of the free energy. The solution of this equations

determines the self-consistent HFB basis. In this basis 1’ as well as X(.2’) are

diagonal. The eigenvalues are

r; = .L j;i = l-f,

E,= E,, E,= -E, 11 > 0. (2.25)

In the presence of a time-dependent oscillating external field

E(t) = fle-‘E’+h.c. (2.26)

Eq. (2.22) has the form

i&2 = [ST++,@]. (2.27)

Linearizing this equation in the external field we obtain

a(t) = 9’ + (69 eeiEt + h.c.) (2.28)

and 6W is determined by the linear response equation

(2.29)

P. Ring et al. / Microscopic theor)

with the effective interaction

261

We now define the response function [w by

S~,,~ = f c R,,. ,,,, .(E).F,,,.. (2.31) VY’

From (2.29) we see that the response function [w” for vanishing interaction W is

given by

(2.32)

and the full response function is determined by the linearized Bethe-Salpeter

equation

[w = lw”+Iwo~vv~~, (2.33)

where the dots mean ~~,,~.

The matrix elements VV can easily be derived from (2.30). All the indices pp’, vv’

can run over creation and annihilation operators for quasiparticles (p >< 0.. .);

there are 16 combinations. In contrast to the quasiparticle RPA matrix, which

contains only the H22 and the H4' part of the hamiltonian in the quasiparticle

representation [see, for instance, ref. l’), p. 3441, we have in the temperature-

dependent quasiparticle RPA also H3’ matrix elements.

2.2. LINEAR RESPONSE THEORY WITH SEPARABLE FORCES

It is evident that in the general case eq. (2.33) will have an extremely large

dimension, in particular if there are not further symmetries, as in the case of

rotating giant resonances, where the quasiparticles c(+ correspond to a rotating

basis.

We therefore use in the following separable interactions of the form

H = Ho +)x 1 D;D,, (2.34) D

where Ho is a single-particle hamiltonian and p runs over a set of single-particle

operators D,, : which are either hermitian or antihermitian (D+ 5 f D) as for instance

Q, (quadrupole-quadrupole-interaction), P I!I P+ (pairing interactions) or dipole

operators. xp are the corresponding strength parameters. (In the following we replace

the sum over p by a dot.)

268 P. Ring et al. / Microscopic theory

Using the quasiparticle representation (2.11) for the operators 0 and neglecting

exchange terms we find the simple expression

We can introduce the matrix

(2.36)

R,,,.(E) measures the change in the expectation value of the operator B, if we bring

our system into an external field of type DPS, which oscillates with the frequency E. The dimension of this matrix corresponds to the number of separable terms in our

hamiltonian; it is therefore very small.

From eq. (2.33) we derive the linear response equation for the matrix R

R(E) = R”+xRo.R, (2.37)

where the response without interaction is given by

R(E) is obtained by matrix inversion

R(E) = R’(E)

1 -xR’(E)’

(2.38)

(2.39)

The only numerical work remains in the transformation of the operators D, into

the quasiparticle representation and the subsequent sum in eq. (2.38).

The expression in eq. (2.38) is rather general. For the convenience of the reader

we therefore give a number of special cases:

(i) For systems without temperature and pairing we have the well-known result

RpDp,(E) = c (ilD,‘I~XW,W _ <Ml lWWph) mi E-E,+Ei+iq E+E,--Ei+iq ’

The sum runs over particles (m) and holes (i) only.

(ii) For systems with temperature, but without pairing we find

RO tEI = c C’W;lO(W,4k) tf _f.j PP’

kk’ E-Ek+ck..+iq k’ ”

(2.40)

(2.41)

P. Ring et al. 1 Microscopic theor) 269

The sum runs now over all single-particle states k and k’. (iii) For systems with pairing, but without temperature we have

q,.(E) = 2 D:“,: 02,. D;;: Dxm,

E-Em-E,.+iq -E+E,+E,.+irj . > (2.42)

m < In’

(iv) The general case with pairing and temperature eq. (2.38) can be written as

R$(E) = 2, E_F;:+iq (f;,-fM) m m’

+c m < m’

E_$~~~y+iq (I-fm.-fm)

+c m < rn’

(2.43)

2.3. THE STRENGTH FUNCTION AT FINITE TEMPERATURE

The strength function of an operator g in the hot system is defined as

S,(E) = C pil(flPli)l’S(E - E,+ Ei). (2.44) if

Using detailed balance2’) it can be rewritten as

I(flFli)12 I(flW” E-Ef+Ei+ir] - E+E,-E,+ir]

(2.45)

As in the case of zero temperature this expression is related to the response

function. Starting from the definition (2.31) of’the response function we find by

simple perturbation theory an exact expression for R

(ila~a,lf)(fla~a,~li>. (ild~,~lOOl+2,li> E-E,+E,+iq - E+Ef-Ei+iq

(2.46)

and the strength function (2.44) can be written as

1 S#) = - k l_e-BE Im RF,@). (2.47)

This equation deviates from the usual equation at zero temperature “) only by the


factor (1 - e-PE)-1, which is for the relevant temperatures and resonance energies

very close to one.

From the strength function we can easily derive the absorption cross section.

2.4. PHENOMENOLOGICAL DESCRIPTION OF THE WIDTH

So far we have calculated the response function in RPA approximation. In a

truncated oscillator space it therefore has poles at the discrete RPA

eigenfrequencies. For an infinitesimal parameter q in eqs. (2.32) or (2.38) the

function R would show up a complicated structure, because there are a large

number of two-quasiparticle configurations in the rotating scheme. The resonance

is then fragmented over many such configurations (fragmentation width).

In reality we have to include couplings to many more complex configurations

(decay width) and also to the continuum (escape width). In the experiment we

observe therefore a much larger width. Within the RPA approximation, where we

include only two-quasiparticle configurations, we are not able to calculate the

width on a microscopic basis. Going beyond RPA and using e.g. Mori theory 22)

one can show that the coupling to more complicated configurations can be taken

into account by replacing the parameter ‘1 in eq. (2.32) by an energy-dependent

operator.

We therefore use the phenomenological approximation and replace 9 by a finite

constant *I-. As a reasonable approximation we have chosen 23) f = 1 MeV. We

are aware, however, of the fact that at high temperatures one expect changes of the

width, which we are not able to calcu!ate micrqscopically so far. The investigation

in this paper concentrate therefore on the position and on the splitting of the

resonance maxima only.

3. Giant resonances in rotating nuclei

In the description of the giant dipole resonance based on the nuclear ground

state it has turned out to be rather useful to calculate the absorption cross section +

o(E) = 4n2crC(E,,-E,)I(v((D110)12G(E-E,+E,), (3.1) 1’

where M is the fine structure constant and D is the dipole operator

(3.2)

’ The following considerations can be easily extended to other giant resonances, as for instance, the giant quadrupole resonance.

P. Ring et al. / Microscopic theor) 271

IO) is the nuclear ground state and the sum runs over all excited states IV) with

angular momentum I,, = 1 and negative parity..

a(E) describes the absorption cross section of a dipole photon with energy E at

the ground state. Since we are in this paper dealing with giant resonances based on

excited states we have to calculate the absorption cross section starting from an

initial state ii) and to average over all initial states (thermal averaging and

averaging over initial orientations) :

OtE) = 4n2x~Pi(E[pE’)ZI: I(fllDl(i)12. 6(E-E,+Ei). (3.3) if 1

In subsect. 2.3 we have seen that this expression is closely related to the strength

function and can be derived from the response function (2.42). a(E) is the

absorption cross section in the laboratory scheme, i.e. the states Ii) and If, carry

good angular momentum. As long as we deal with spherical nuclei, the two-

quasiparticle states showing up in the random phase approximation for R can be

coupled to good angular momentum and we obtain directly the absorption cross

section in the laboratory frame.

In deformed and rotating nuclei rotational invariance is broken. We describe the

states by (possibly time-dependent) Slater determinants lb(t)) in the intrinsic

frame. The wave functions in the laboratory system can be obtained through

angular momentum projection :

IV = m#Q. (3.4)

where the projection operators P$ are given by “)

(3.5)

Q = (CI, p, 7) are Euler angles and

R(Q) = exp(iaJ,)exp(igJ,)exp(iyJ,)

describes a rotation in these variables.

Using projected states of the type (3.4) in the definition of the densities (2.7) or

in the response function (2.42) destroys the whole beauty and simplicity ofJinear

response theory. We therefore have first to transform into the intrinsic frame.

It can be shown that for deformed nuclei with large particle numbers it is

enough to carry out the projection only in a first approximation 24,17), the so-

called cranking approximation. In this approach one has to solve the equations of

motion for the density (2.22) in the rotating frame.

id = [Yq9) - mYa,, a]. (3.6)

272 P. Ring er al. / Microscopic theory

The quasistatic solution is given by the self-consistent cranking density aCO

[It (a,) - w.Yx., n,] = 0 (3.7)

and the angular velocity is determined through the constraint

(J,), = &1+ 1). (3.8)

The giant resonances in the intrinsic scheme are obtained as vibrations with small

amplitude around this rotating equilibrium in the sense of the cranked RPA “. “).

We could, of course, solve the temperature-dependent cranked RPA equations. As

discussed in sect. 2, it is, however, much easier and more elegant to use the linear

response formalism. We therefore start again with eq. (3.3) which is still an exact

expression and transform in this equation the energies and the matrix elements of

the dipole operator to the intrinsic system. In this way we can express cr(E) by the

response function in the deformed and rotating frame.

For simplicity we discuss in the following only the case of zero temperature. It is

trivial to extend it to finite temperatures. We therefore assume that we start at an

initial state Ii) with angular momentum I at the yrast line (fig. 1). Since the dipole

photon carries angular momentum one, we then have three possibilities: the final

state can have either angular momentum 1 or I If: 1. In the first case we have no

Fig. 1. Schematic representation of the absorption of a dipole photon starting at a level with

angular momentum I at the yrast line.

P. Ring et al. / Microscopic theory 273

change in signature [see ref. l’), p. 4761; in the other two cases the signature

changes.

The matrix elements (filDlli) h ave to be calculated with projected states. For

deformed nuclei and for large angular momenta (more precisely for Z-values

which are large compared to the K-values involved) we can use a technique

developed in ref. “) which is consistent with the cranking approximation and

express these projected matrix elements in a very good approximation by

unprojected matrix elements in the intrinsic frame, We obtain for the signature-

conserving transitions

(3.9)

and for the signature-changing transitions

with

D, = f&D&D,). (3.11)

In eq. (3.3) we also have to transform the energy difference Er - Ei from the laboratory

scheme to the intrinsic scheme. We do this within the cranking approximation. For

AZ = 0 (signature-conserving) transitions the initial and the final state belong to the

same angular velocity, i.e. the excitation energy stays unchanged. For Al = k lh we

have a change in the energy by the amount

AE=~Al=wAl dl

= fhw. (3.12)

We therefore can express the dipole absorption cross section (3.3) as

o(E) =: o,(E)+&,(E-hw)++,.(E+ho)+$o;(E-liw)++o,(E+ho), (3.13)

O,(E) = -4nzEIm RDpi(E), i = X, y, Z, (3.14)

are the absorption cross sections in the intrinsic frame. They can easily be

calculated in random-phase approximation. From eq. (3.14) we see clearly that the

transformation to the laboratory scheme induces an additional splitting. For the

214 P. Ring et al. / Microscopic theorj

GDR we obtain now in principle five peaks instead of three. We will see, however,

in sect. 5 that in realistic calculations this additional splitting can be hardly

resolved. It only increases the observed width.

Sumrnarizing the results of this section we find that we have to proceed in three

steps :

(i) For each angular momentum and each temperature we have to solve the

cranked temperature-dependent Hartree-Fock-Bogoliubov equations (3.7). The

solution of this equations defines a rotating mean field, which described the highly

excited compound state and which is used as a basis for the linear response theory

in the next step.

(ii) The giant resonances are calculated within linear response theory in the

intrinsic frame. The operators 6 are transformed to the quasiparticle representation

and for each energy E the response matrix R” of eq. (2.38) is evaluated. In the

dipole case it is a 3 x 3 matrix which can be decomposed into a 1 x 1 matrix for

positive signature describing AZ = 0 transitions and a 2 x 2 matrix for negative

signature describing AZ = + 1 transitions:

‘R;, 0 0 ’

R"= 0 R;! R;z ; (3.15)

\ 0 RfY RZz

for a finite value of q it is a complex matrix. By matrix inversion we obtain the full

response with interaction R(E) in eq. (2.39) and the cross sections ci (3.14) in the

intrinsic frame. They show maxima at the giant resonance energies.

(iii) The third step is the transformation back to the laboratory scheme in eq.

(3.13).

4.1. THE HARMONIC OSCILLATOR MODEL

In order to investigate the dependence of the giant dipole resonance and their

splitting on the deformation parameters we use in this section a model, which has

widely been used in the literature to describe the giant dipole state6). It has been

applied recently also in the rotating frame ‘-lo). We extend it and include pairing

and finite temperatures.

The model assumes a deformed mean field of oscillator type and includes a

dipole-dipole interaction :

H = Ho(w,, COP, CO,)+ C XiDTDi-O’$ i=x.y.z

(4.1)

The oscillator frequencies oi are related to the Hill-Wheeler deformation

P. Ring et al. 1 Microscopic rheoq

parameters /?, y by

275

, i = (1,2,3) = (x.I’,z). (4.2)

and the strength parameters xi depend in a deformed potential on the orientation

xi = 3&n”;. (4.3 1

The strength parameter xi is chosen in such a way, that we obtain for spherical

nuclei the GDR at

E ens = 2ho, = 82A-f (MeV), (4.4)

which is close to the experimental value 78A-f (MeV).

The hamiltonian (4.1) is quadratic in coordinates and momenta. Introducing

c.m. coordinates and intrinsic coordinates Brink6) could show that it can be

decomposed into (i) an intrinsic part, (ii) a part which describes the spurious c.m.

motion, and (iii) a part which describes the motion of protons against neutrons

and which depends only on the collective coordinates D,, D,, Dz. All coupling

terms vanish exactly. We are only interested in excitations of the dipole term. This

part has the form of a three-dimensional anisotropic rotating harmonic oscillator

and its deformations are given by

(4.5)

Valatin”) has shown that such a system has an exact solution. Its excitation

energies are given by

El = ho,,

E 2.3 = wo2 + co: f $o”, + 4wz 02 )t, (4.6)

with

They describe the splitting of the GDR caused by deformation and by rotation. In

fig. 2 we show the three peak energies 1, 2, 3 as a function of the cranking

frequency for fixed deformation j3 = 0.3 and for various triaxiality parameters

y =o, 10” ,. . .,60”. It is clearly seen that the deformation provides a considerable

276 P. Ring et ai, / Microscopic rheoq

y=60

E y=o

06 Cl06 012 Ol6 02L

hW

Fig. 2. The eigenfrequenties E,,EZ,E3 of the rotating harmonic oscillator as given in eq. (4.6) for /I = 0.3 and various y-values between 0’ and 60’. The units for the peak energies Ei and for the

cranking frequency hw are 78A- Is3 MeV i e. , . the energy of the GDR at vanishing deformation.

splitting which is very different for different y-values. On the other hand, the dependence on the cranking frequency o is rather weak for physically interesting w-values. In particular, we know that for heavy nuclei, even at the largest angular momenta (I - lOOh), w never exceeds 1 MeV, which for A = 164 is 0.06 in units of 78 A-” (MeV). This means for fixed deformation we hardly expect to find the rotational splitting in the experiment. As we will see in sect. 5 this results stays also true for realistic calculations.

in fig. 3 we see the IT-dependence of the splitting for all possible 7;-values ranging from 7 = -60” (a rotation of an oblate nucleus around the symmetry axis) over 1’ = O” (a rotation of a prolate nucleus around an axis perpendicular to the symmetry axis), though y = 60” (a rotation of an obiate nucleus perpendicular to the symmetry axis) to y = 120” (a rotation of prolate nucleus around the symmetry axis). Again the u-dependence is very weak. The physical region is limited by the second thin line close to the thick line at o = 0.

The result (4.6) has been obtained by a transformation to center of mass and intrinsic coordinates. One obtains the same result in linear response theory starting with a rotating harmonic oscillator as mean field and treating the dipole-dipole force as interaction. We than find the giant resonance energies as zeros in the denominator of eq. (2.39):

det(1 -xR”(E)) = 0. (4.7)


-& 2

-3

J; ) 1

p 2

--3

ct; -

@ --3

2

ct; -I &I -=:-3

2

Fig. 3. The eigenfrequencies E,, EZ, E, of eq. (4.6) for fixed /j’ = 0.3 as a function of the triaxiality 7. The thick lines corresponds to o = 0 and the other lines show increasing o-values in steps of 0.25.

The units are as in fig. 2 the peak energy of the GDR 78.4 “3 MeV.

For the signature-conserving transition D, we have (see eq. 2.41) in the case of

vanishing pairing correlations

R:,(W = ; E’T;D;‘,“‘)[i, (A, -.hJ. k k'

(4.8)

where Ik) = In,, nY, n,,s) are the oscillator quantum numbers. The operator D, connects only states with n; = n,k 1, n, = n;, n, = n: and s = s’. For these states

we have .sk-.sk, = +zZo, and we can bring the denominator in front of the sum.

We then can sum up and find

(4.9)

independent of the temperature T.

278 P. Ring et al. / Microscopic theor)

The resonance is therefore found at

(4.10)

Using the strength parameters (4.3) we obtain

E, = hti, (4.11)

in agreement with the first part of (4.6). The evaluation of the second part is more

tedious. It gives the same result. We therefore find that for the harmonic oscillator

model without pairing the peak energies do only depend on the deformation and

on the angular velocity o, but not on the temperature.

In realistic calculations this fact is no longer true in the exact sense. We will see,

however, in the next section. that the influence of the temperature on the splitting

for fixed deformation and gap parameters is indeed rather weak.

In the next step we add to the oscillator hamiltonian (4.1) a monopole pair field

A(P+ +P), (4.12)

where P+ = ‘&clct creates a Cooper pair. The sum over k is restricted to region

around the Fermi surface A, i.e. we allow only those levels Ik) in the deformed

nonrotating basis to participate at the pairing with

IQ-L1 5 l.lho,. (4.13)

Together with pairing the problem is no longer analytically soluble. We therefore

diagonalize

H&l,, co,, co;)-A(P+ +P)-wL,-AN (4.14)

in a spherical oscillator basis with N 2 10 major shells and solve the linear

response equations (2.37) for the dipole-dipole force in eq. (4.1).

First we investigate spherical nuclei at spin zero. In fig. 4 we show dipole

absorption cross section a(E) for two different temperatures (T = 0 and T = 1.5

MeV) and for various gap parameters A = 0,0.5,. . ., 2 (MeV).

As discussed above, there is no temperature effect for vanishing pairing

correlations. With increasing pairing correlations we observe a shift of the giant

resonance peak to higher energies. It is for zero temperature roughly proportional

to A as we see in fig. 5. Going from A = 0 to A = 1 MeV we find a shift of a little

more than 1 MeV. This shift is caused by two reasons as we can see from eq.

(2.42): (i) we have a dependence of the quasiparticle energies on A, i.e. already the

P. Ring et al. I Microscopic theor) 279

a(E)

4

3

2 --_-_-__;

l

0 IO 12 14 16 18 20

E (MeV)

Fig. 4. Dipole absorption cross sections a(E) in the harmonic oscillator model for a spherical nucleus with A = 164. Calculation with various gap parameters and two different temperatures are shown.

I I I

164Er ~zo,wz~

1.1

1.0

0.9

0.8

2

0 0.5 1 1.5 2 A,=A,=A (MeV)

Fig. 5. The dependence of the giant dipole resonance peaks on the gap parameter for vanishing angular velocity. For the spherical case (fi = 0) there is only one peak (three times degeneraie). For 1 = 0.3 we have two peaks, a vibration along the symmetry axis (z) and two generate vibrations per~ndicular

to it (x., y).

1.3

1.2


uncorrelated two-quasiparticle excitations are shifted to higher energies; (ii) for

increasing pairing we have no sharp Fermi surface any more. Apart from the ph

excitations we have pp and hh excitations. The collectivity is therefore increased.

For a repulsive interaction this causes a shift of the collective mode to higher

energies.

In fig. 5 we show also calculations at a temperature of T = 1.5 MeV and

calculations for a finite prolate deformation /l = 0.3. In the latter case we observe

the well-known splitting. The qualitative behavior is the same for all cases. In

detail, the temperature splitting is most pronounced for spherical nuclei. We have

to keep in mind, however. that large values of d are rather unphysical for large

temperatures. As we will see in the next section, pairing correlations collapse in

realistic nuclei already at rather moderate temperature. Our calculations show,

however, that changes in the pairing correlations should show up in shifts of

resonance peaks.

5. Realistic calculations in the nuclei “‘Er and 164Er

The harmonic oscillator model discussed in sect. 4 provides a very simple

description of the giant dipole resonance. It shows, however, some well-known

deficienties. Besides the fact that is does not give the proper level ordering in

realistic nuclei, it requires a very special residual interaction (4.1) which violates

rotational symmetry including three strength parameters xi, which are adjusted in

such a way as to reproduce the proper position and the proper splitting of the

GDR. In order to describe nuclear rotations properly one should at least in

principle start out with a rotational invariant residual interaction.

In the following we investigate the two nuclei “‘Er and 164Er. ls8Er shows for

large angular momenta a change from a prolate to an oblate shape which has

interesting consequences for the splitting of the GDR. The other nucleus ‘(j4Er

stays close to axial symmetry even for high spins. In this case we study the pairing

properties in more detail.

As discussed in sect. 3 the calculation has to proceed in two steps. In the first

step a temperature-dependent rotating mean field has to be determined. In the

second step dipole vibrations around this equilibrium shape are calculated in linear

response theory.

5.1. CALCULATION OF THE ROTATING MEAN FIELD

In principle. we should use a realistic, possibly density-dependent two-body

interaction and in the first step we should solve the temperature-dependent HFB

equations in the rotating frame. This step would require a tremendous numerical

effort in heavy nuclei. So far, calculations of this type have been carried out either


without temperature 29) or without pairing 30). It is, however, well known that the

semiclassical method of Strutinski 31 ) provides a very simple and powerful

approximation to Hartree-Fock calculations with realistic forces. It has been

extended for superfluid nuclei 32), for finite temperatures33) and for rotating

nuclei 34). The version which we use in the following has been described in detail in

refs. 35-37): We assume that the nucleons move in average potential

H,(P, 7) = T + Vi,+ I’“-~ + b,,,,. (5.1)

where T is the kinetic energy; V& is the deformed Woods-Saxon potential in the

form proposed by Damgaard et al. 38):

VLs = Vl/(l +exp (@)/a,)). (5.2)

The length function I(v) defines an effective distance from the nuclear surface, given

through the deformation parameters /? and 7:

. i = 1,2.3 (5.3)

For a spherical nucleus with radius R; the function I(r) has the simple form

I(r) = r-R;.

The spin-orbit potential is defined by the gradient of a Woods-Saxon potential

P”. = - ; vvgs”+J x p) = -iKa(VVg x V). (5.4)

The Coulomb term Vcou, is given by the potential of a uniformly charged deformed

liquid drop with Z- 1 protons. The radii parameters R& Rs”. and Rcou, for the

central, spin-orbit and Coulomb terms, the surface thickness parameters a,, a_

and the potential depth parameters V$, riV,S.“. are determined according to the

droplet model 39) and the modifications suggested in ref. 40). They are listed in table

The hamiltonian Ho (5.1) is diagonalized in a deformed Cartesian basis of an

anisotropic oscillator with the frequencies

wf XI; : 1 1 1

d=(x2):@ypy

ho,O = ~(w,w+o,)~ = 55A-4 (MeV), (5.5)

where the expectation values (x’), 0.‘) and (2’) are determined from the liquid

282 P. Ring et al. / Microscopic theory

TABLE 1

Single-particle potential parameters of the nuclei “*Er and ‘%r (R and a are given in fm, v. in MeV)

R’, R‘,“. 0 R cod vo’ K Vo’.” a, a E.0.

protons 6.7480 6.0455 6.111 - 57.859 - 12.0 0.66 0.55 neutrons 6.7050 6.0455 -49.341 - 12.0 0.66 0.55

protons 6.8406 6.1320 6.170 - 58.755 - 12.0 0.66 0.55 neutrons 6.7885 6.1320 - 48.445 - 12.0 0.66 0.55

drop of the same shape. All basis states (II,. nY, n_, sx) are included which satisfy the

energy condition 41 )

(n,+3)ho,+(n,+4)ho,+(IZ;+~)hLO; 5 w,+t)h$ (5.6)

We use in this paper N, = 10.

Short-range correlations of the pairing type are taken into account by a residual

interaction

Hpair = - G C am+a~a,q,,,, mm’>0

(5.7)

where m runs over a limited configuration space defined by all the eigenstates of

H, in (5.1) whose distance from the Fermi surface is smaller than A:

IEm--pl I A. (5.8)

The cut-off energy n is chosen as l.lhw,. ’ The pairing strength G in eq. (5.7) is

determined by the average pairing-gap method 32)

G-r = g(X)ln(2n/j), (5.9)

where d” = 12/,6 (MeV) is an average gap parameter and g(x) is the average level

density at the Fermi energy 1 of the smoothed single-particle level distribution. As

discussed in ref. 32) the results are independent of the cut-off energy /i, if the

condition n % d” is satisfied.

So far, we have defined the deformed single-particle potential and the pairing

force. This hamiltonian should now be diagonalized in the rotating ,frame, i.e. we

should solve the BCS equation in the rotating frame. Since the Coriolis term


violates time-reversal invariance we then end up with cranked HFB equations4’).

Their solution is rather time consuming, because they require iterative

diagonalization of rather large matrices. Such calculations have been done in

ref. 43) for zero temperature.

At the present stage they are too complicated for finite

therefore use an approximation proposed by Marshalek 44). In a

is diagonalized.

H, - oJ, = c @‘b+bi + t$‘b+by),

and the pairing field is transformed to the rotating basis b:, b+ :

Hpair = -G 1 F;cFFijib’bj;‘bjbi. iji’j

temperatures. We

first step Ho - oJ,

(5.10)

(5.11)

Neglecting the off-diagonal elements of Fij and substituting EY and .$’ by their

mean values Ei = ~(e~+$‘) we end up with a modified temperature-dependent gap

equation,

A=+G c FiiA(l -h-f;)

i>o t/(Ei-2)2+A2Fi’ (5.12)

which is easy to solve. Neglecting the off-diagonal elements Fij(i # j) usually

means a too strong reduction of the gap. To avoid this strong reduction we use

renormalized effective matrix elements,

(5.13)

which give rise to a better agreement with exact HFB results45).

Having now defined the rotating mean field we are able to calculate the energy

E, the free energy E - TS, the energy in the rotating frame E -wJ, or the grand

canonical potential E -oJ, --AN - TS by the Strutinsky method. Details of those

calculations are given in the literature 35 - 37).

Each of these quantities depend on the deformation parameters /I and y and we

have to determine the self-consistent solution by minimization with respect to these

parameters. There are several ways to proceed: we can either vary the energy E with fixed values of the entropy S, the angular momentum I and the particle

number N, or we can vary the free energy at fixed values of the temperature T, the

angular momentum I and the particle number, or we can vary’the grand canonical

potential at fixed values of the temperature, the cranking frequency and the

chemical potential.


All these methods are equivalent and give us the same results for the stationary

points :

(5.14)

In figs. 6 and 7 we show for pedagogical reasons the energy surfaces E(/?, y)

obtained by the temperature-dependent Strutinsky method from the Woods-Saxon

potential without pairing for constant entropy and particle numbers and for

various angular momenta I = 0, 20, 40. 60h. The minima in these surfaces

correspond to equilibrium points. are therefore approximate solutions of the

cranked HF equations with temperature.

The energy surfaces in figs. 6 and 7 correspond only for zero entropy (at the

yrast line) to a constant temperature. For the plots with nonvanishing entropy we

have only an average temperature of roughly T = 1.5 for S = 58.5 and T = 2.5 for

s = 97.5.

For zero temperature the nuclei “*Er and “j4Er have rather different shapes :

with increasing angular momentum lssEr undergoes a transition to a prolate

shape for I > 40h, whereas 164Er stays close to axial symmetry, even for high spin

values. We therefore neglect for this second case in the following the y-degree of

freedom.

We also observe that the minima in the energy surface become flater and flater

for higher temperatures. We therefore expect in this case more and more

anharmonizities. The simple RPA will certainly become less reliable in this region.

One expects very large fluctuations which should show up in particular in the

width of the resonance. In fact, experimental data seem to indicate a considerable

increase of the width of the GDR at higher excitation energies4).

Pairing correlations collapse at large angular momenta and at moderate

temperatures (T > 0.5 MeV). Therefore they are usually neglected for the region of

large excitation energies46) and they have not been taken into account in the

energy surfaces of figs. 6 and 7. They play, however, an important role in the

region close to the yrast line. In the following we take ino account pairing for

164Er as described above.

In fig. 8 we show the gap parameters obtained from the solution of eq. (5.12)

and how they depend on the angular momentum for various temperatures. The

Fig. 6. Energy surfaces for “‘Er in the shape parameters fi and 1’. They correspond to constant entropy S and constant angular momenta. The contour lines describe an energy difference of 2 MeV and absolute values for the energies are given in MeV. Pairing correlations are neglected. The entropy values

S = 58.5 and S = 97.5 correspond on the average to the temperatures T = 1.5 and T = 2.5 MeV.

P. Ring PI al. / Microscopic theory

0 P 0.6

0 0.6 0 0.6

0 P

0.6 0 P

0.6

0 0.6

0 13 0.6

285

0 P 0.6

Energy

“‘Er T=O

0 ’ 06 .

Energy

15*Er S=58.5

06 .

-60

Energy 15*Er

s=97.5


0 P

0.6 06 .

0 0.6 0 0.6

0 0.6

Energy

‘fib Er

T=O

0 P 0.6 0

P 0.6 0 ’ 06 .

Energy

‘% S 58.5

Energy

ts’Er

s=97.5

Fig. 7. Energy surfaces for ‘%r. The details are explained in fig. 6.

TG I

a

0.5


1 I I I I I I I

16’Er p = 0.3, y = 0

(4 t b) -

0 0 10 20 30 0 10 20 30 40 !

I 0-l)

287

Fig. 8. Gap parameters for neutrons (a) and protons (b) for the nucleus lb4Er as a function of the angular momentum at various temperatures. The deformation parameters are kept fixed : /I = 0.3.~ = 0”.

results for vanishing temperature are in good agreement with earlier calculations in

the framework self-consistent cranked HFB theory in the pairing-plus-quadrupole

model 47). The neutron gap breaks down already at I = 3Ofi. somewhat earlier than

the proton gap, which goes to zero only for I h 5Ot1. With increasing temperature

we observe in both cases a rapid collapse. For T = 0.5 all pairing correlations have

vanished. This is again in agreement with results obtained from the solution of the

full cranked temperature-dependent HFB equations in the quadrupole-pairing

model 48). We have to be a little careful, however. Both phase transitions (pairing

collapse with increasing angular momentum and with increasing temperature are

only so sharp in calculations of the mean-field type). Improved theories, such as

number projection 47), h s ow that in finite nuclei these transitions are always more

or less washed out.

5.2. THE GIANT DIPOLE RESONANCE IN REALISTIC NUCLEI

For the description of the giant dipole resonance we use a rotational invariant

dipole-dipole interaction

V=xD+.D. (5.16)

The constant x is adjusted the peak energy of the K = 1 vibration for I = 0. We


find that we need in the Woods-Saxon potential a strength somewhat reduced to

the corresponding oscillator model. We use for the case with pairing correlations a

reduction of 0.55 and in the case without pairing a reduction of 0.7, i.e. we have

I 0.55 X 3 $ wlo; for calculations with pairing, (5.17a)

%=

0.7x3&w: for calculations without pairing. (5.17b)

In the first step we concentrate on pairing correlations and use (5.17a). In the

harmonic oscillator model we have seen in sect. 3 that pairing correlations have an

influence on the position of the peak energies. This result is also true for realistic

calculations.

In fig. 9a we show the temperature dependence of the peak energies E,,, in

164Er for zero angular momentum. Pairing correlations are treated in three

different ways. In the full line we use the self-consistently determined pairing gaps

of fig. 8. In the dashed-dotted lines the gap parameters are kept fixed at their self-

consistent values at T = 0 (dp = 0.87, A, = 0.81 MeV) and in the dashed lines we

have used vanishing gap parameters from the beginning. We find that in the latter

two cases, where the gap parameters do not depend on the temperature, practically

16

2 p 15

E W

12

I I I I

(4 I -

164Er -A selfc.

I=0 _-__A=0

----A frozen _

---.-___ -_-.---______ _____________________--___-_

x, Y

l- I I I I I

0 0.1 0.2 0.3 0.4 0.5 T

Fig. 9a. The temperature dependence of the peak energies for Woods-Saxon calculations in 16“Er at zero angular momentum. Pairing correlations are treated in three different ways as described in the text. The deformation is j? = 0.3, y = O”. The dipole strength is 0.55 in units of the spherical oscillator

model.

15

3

F 11

E W

12

11

P. Ring et al. 1 Microscopic theory

(b) ' I I I I

164Er -Aselfc. ____A=0

T=O .--.-Afrozen /x -Li.-.-

289

0 lo 20 30 40 50 60

10)

Fig. 9b. The angular momentum dependence for zero temperature.

no change in the peak energies appears. There is again a shift in the peak energies,

if pairing is switched off. It is - 0.5 MeV for the z-mode and the self-consistent

calculation connects the two limits.

Similar results are obtained for the angular momentum dependence of the giant

resonance peaks at fixed temperature. In fig. 9b we see those results for

temperature zero. Again the pairing collapse produces a shift in the peaks. It is

again most pronounced for the z-mode. Calculations with fixed gap parameters

(either frozen at there values for angular momentum zero or at A = 0) show more

or less no angular momentum dependence, because the deformation stays fairly

constant for this region in 164Er.

Next we study the nucleus “‘Er with vanishing pairing and the force (5.17b). In

fig. lOa, b we show dipole absorption cross sections a,(E), gJ(E) and a,(E) in the

intrinsic frame. In order to see clearly the influence of the shape changes (see fig. 6)

we show results of a self-consistent calculation with angular-momentum- and

temperature-dependent shapes in fig. 10a and results of a non-self-consistent

calculation, where the deformations are kept fixed at their values in the ground

state (zero angular momentum and zero temperature) in fig. lob.

In the ground state we observe the two degenerate modes x- and y- (K = + 1)

and the z-mode (K = 0) is shifted to lower energies because of the prolate

deformation (/I = 0.24). At higher temperature (T = 1.5 MeV, dashed-dotted lines

in fig. 10a) we find only minor changes, essentially only the width increases by a

small amount. Going to larger angular momenta we observe that the x-mode stays

more or less unchanged for temperature zero; for temperature 1.5 MeV it is shifted

upwards because of an increasing /?-deformation for high temperatures and large

290 P. Ring

I”““” ‘I

et al. / theory

QX- 158Er -T=O 7“ ., \\ -.- T:1.5

10 12 14 16 18 20 10 12 14 16 18 E (MeV) E (MeV)

Selfconsistent Not selfconsistent

1

0

GY

1

0

Fig. 10. The dipole absorption cronsections u,(E), o,(E) and ~J&E) in the intrinsic frame at T = 0 (full lines) and T = 1.5 MeV (dashed-dotted lines). The curves for different angular momenta are shifted against each other by a constant amount. The units are arbitrary. In part (a) we show self-consistent calculations carried out within the Strutinski approach in a rotating Woods-Saxon potential without pairing. In part (b) we show non-self-consistent calculations. Here the shape parameters are kept fixed

at the values of the ground state (T = 0. I = 0).

spins. The y- and z-modes show more pronounced shifts. From the energy surface in fig. 5 we see that the nucleus undergoes a shape change to oblate deformations. From the harmonic oscillator model we therefore should expect, that the y- and z- modes come close. Indeed we see this trend also in the realistic calculations. for zero temperature.

In the non-self-consistent calculations in fig. 10b the x-mode remains unchanged, the y-mode is shifted upwards and the z-mode slightly downwards as we will expect from the results of the harmonic oscillator (fig. 2). We also observe some mixing between the y- and z-modes. The calculations with ‘temperature are very similar.

In fig. 11 we show a comparison of the self-consistent calculation in the Woods- Saxon potential (full lines) and two applications of the harmonic oscillator model

P. Ring et al. / Microscopic theory 291

_ 158Er

16 - T=O

_*-- __-- *-

0 20 40 60 80

I (tl)

Fig. 11. The position of the peaks E,,, of the GDR in “‘Er as a function of the angular momentum.

Full lines correspond to the realistic self-consistent calculation in the Woods-Saxon potential. Dashed and dashed-dotted lines are obtained from the harmonic oscillator model of sect. 4. Details are given

in the text.

of sect. 4. The strength parameters of this model are reduced by a factor 0.8 as

compared to eq. (4.3) in order to find good agreement for zero angular momentum.

The dashed and the dashed-dotted lines are obtained from the formula (4.6) of the

rotating harmonic oscillator. As input we used the same deformation parameters as

in the self-consistent Woods-Saxon case. For the dashed lines we also used the

same angular velocities w, whereas for the dashed-dotted lines we neglected the

pure rotational splitting and calculated with o = 0. Apart from some deviations at

large angular momenta we find in all three cases qualitatively the same behaviour:

the y- and z-modes approach each other because of the change in shape at high

spins. At I = 80h we have 7 = -57”: the y- and z-modes are nearly degenerate in

the oscillator without rotational splitting (dashed-dotted lines). The same is the

case for the Woods-Saxon calculation (full lines). In the harmonic oscillator with

the Coriolis term we find some rotational splitting not observed for the rotating

Woods-Saxon potential.

Finally, we have to transform back to the laboratory scheme. This is done in fig.

12. For zero angular momentum and zero temperature we have the two well-

known peaks. As has been found in older calculations 49. 50) the lower resonance

(K = 0) has a larger width. For higher temperatures (T = 1.5 in fig. 12

corresponds roughly to an excitation energy of 45 MeV) the valley between the


10 12 14 16 18 20 E(MeV)

Fig. 12. The dipole absorption cross section u(E) in the laboratory frame (eq. (3.13)) for the self-consistent

calculation without pairing in ‘j8Er for two temperatures. The curves for different angular momenta are shifted against each other by a constant amount. the units are arbitrary. The lowest curve corresponds

to I = 80h. Below this curve the live components (1) a,(E). (2) u&E-k), (3) a,.(E+hw). (4) u,(E-hw) and (5) u,(E+hw) are given separately for angular momentum I = 8Oh and T = 0.

two peaks is washed out. The same occurs, if we go to higher angular momenta.

The additional splitting into five peaks finally washes out the whole structure. As

long as one cannot distinguish experimentally between the different resonance

peaks by a precise measurement of the angular distributions there is little hope that

the fine structure can be resolved.

6. Conclusion

We have derived a linear response theory for superfluid systems at finite

temperatures within the quasiparticle formalism. It is used for calculations in the

harmonic oscillator model and for realistic heavy nuclei in the basis of a rotating

Woods-Saxon potential to study the fine structure of the GDR for all angular

momenta and all excitation energies.

We found that the position and the splitting of the GDR is mainly characterized

by the deformation and gap parameters of the mean field. Deformation effects play

the most essential role, but changes of pairing cause additional shifts of 0.5 to 1

MeV. For a given shape and for given pairing parameters the two other degrees of

freedom temperature and angular velocity have only little influence.

This result has two advantages:

(i) the technical advantage, that one does not need to include temperature and

pairing into the solution of the linear response equation (if one has included these


degrees of freedom into the calculation of the self-consistent mean field, a simple

ph-response theory is enough.) ;

(ii) the advantage that the fine structure of the GDR, if we could measure it,

gives us even at finite angular velocities and temperatures information on the

nuclear shape at this angular momentum and at this excitation energy.

We did not investigate the behavior of the width. General arguments and

classical estimates 51) show that for higher temperatures one expects considerably

larger width parameters than at the ground state. This indicates, that the region of

high temperatures becomes less and less interesting. The fine structure can only be

observed at lower excitation energies not too far from the yrast line. This means,

however, that experiments which see at the moment mainly y’s from the high-

temperature region, are not able to obtain the information on nuclear structure

discussed in this paper.

The authors wish to thank R. R. Hilton and H. J. Mang for useful discussions

and valuable suggestions. Parts of this work were done during the stay of one of us

(J.L.E.) in the Max-Planck-Institut fur Kernphysik, Heidelberg; he gratefully

acknowledges discussions with H. A. Weidenmtiller and the support of the Max-

Planck-Gesellschaft.

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Microscopic theory of the isovector dipole resonance at high angular momenta