Sum rules for compression modes

Volume 108B, number 4,5

PHYSICS LETTERS

28 January 1982

SUM RULES FOR COMPRESSION MODES S. STRINGARI Dipartimento di Fisica, Libera Universitd di Trento, Povo, Italy Received 6 October 1981

Compression modes in nuclei are investigated by means of a sum rule approach. The link with the hydrodynamical and generalized scaling models is discussed. It is shown that the relevant physical parameters are the nuclear incompressibility and the Fermi energy. The predictions for the relative location of the isoscalar giant dipole and monopole energies are in agreement with the first recent experimental data in 2°apb. Transition densities are also analyzed.

During the last few years much theoretical effort has been devoted to the study of the breathing mode [ 1 - 3 ] . In particular the relation between the incompressibility modulus K of nuclear matter and the frequency of the collective monopole vibration in nuclei has been widely discussed. Due to the increase of experimental data on the giant monopole mode, it now seems possible to extract rather precisely the value of K (K = 220 -+20 MeV). Evidence for a possible excitation o f the giant isoscalar dipole resonance at co = 21.3 MeV has been recently found [4] in 208pb. The availability of further data on this excitation, which is associated with a volume oscillation, will evidently increase our knowledge of the compression mechanisms in nuclei. The purpose of this letter is to analyze compression modes using a sum rule approach in which the role of the nuclear incompressibility is pointed out. We find it convenient to derive explicit expressions for sum rules in the limit of large nuclei, thereby neglecting surface effects. Though a realistic calculation should take care of these effects, important properties of these modes can be understood in this limit. In particular the relative location of the monopole and dipole energies can be easily discussed. In the following we shall evaluate three different sum rules for the isoscalar excitation operator F

= ~,if(ri): (i) the polarizability sum rule:

232

I(0lFIn)l 2 S-l=n~J 0 En-E 0 '

(1)

(ii) the energy-weighted sum rule S1 = ~ n

I(0lFln)12(En - E o )

(2)

and (iii) the cubic energy-weighted sum rule

S 3 = ~ I(OIFIn)12(En - E0)3.

(3)

n

Let us firstly consider the sum rule S 1 which is easily evaluated in terms of a double commutator:

1(

S 1 =½(01[f, [H,F]] 10)=~--~

- A-l(01 ~.. 1,1

(01/~

Vf(ri)Vf(rl.)lO) ~. I

IVf(ri)12lO) (4)

The second term on the right hand side of eq. (4) comes from the center of mass motion and is crucial in the isoscalar dipole mode. In order to evaluate the sum rules S 1 and S 3 we consider two different models for the description of the nuclear motion: the hydrodynamical model [5] and the generalized scaling (fluid-dynamical) model [ 6 - 8 ] . In both cases the relevant quantity for the present discussion is the collective potential energy V(o (r, t)) associated with the displacement field o(r, t). In the generalized scaling (GS) approach V is de-

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PHYSICS LETTERS

rived by evaluating the change of the nuclear energy due to the unitary transformation iq~(t)) = e x p [ ½ i / ~ (t~(rit).Pi +pft~(rit)] ]10)

(5)

applied to the ground state. Using Skyrme-type energy functionals E(p, r) one only needs the changes 5p and 5r of the nuclear density p = Y.it~7~i and of the kinetic energy density r = Z i V~TV qJi induced by transformation (5) [7]. A simptified expression for the energy change can be derived if one takes into accoutn only volume effects. Retaining terms up to second order in t~ 1 the following result is then found * i.

VGS=aF(v)= ~ eFf [(VkVl +VtUk)2 4(VO)2] fl dv _

28 January 1982

Both methods can be regarded [9,10] as approximations to a quantum treatment of the equations of motion as described, for example, in the time dependent Hartree-Fock theory or, equivalently, in the RPA. In fact they do not account for all the high-multipolarity distortions of the Fermi surface that are expected to occur during the collective motion and that are predicted, for example, by Landau's theory of zero sound. However, the two models can be used to evaluate correctly the sum rules S 3 and S_ 1 in the framework of the RP,~. In fact it has been shown [7] that the generalized scaling approach exactly reproduces the sum rule S 3, when evaluated in the RPA. For isoscalar operators F = Y.if(ri) the following identity holds [11] :

S 3 = (O2/Ov2)(vlHlv)lv=O

(8)

with

+ &Kf (vo)2p do.

(6)

In eq. (6) e v = K2F/2m* is the Fermi energy (m* is the effective mass) and K = K2 02(E/p)/OK 2 is the nuclear incompressibility. Eq. (6) is the classical expression for the potential energy of an elastic medium. The term proportional to eF (shear term) originates from the deformations of the Fermi surface induced by transformation (5) and vanishes only for special choices of the deformation field v. The potential energy in the hydrodynamical (HD) approach can be obtained using the Thomas-Fermi relation r = const, p5/3 to write the energy functional E(p;r) in terms of the only nuclear density P. This is equivalent to assume that the sphericity of the Fermi surface is preserved during the nuclear vibration. The change 5E = V H D iS then easily calculated:

: A x f ( w ? p dv

(7)

As follows from comparison of eq. (6) with eq. (7), the two models treat the kinetic energy term and the velocity dependent potentials in a different way. Let us now discuss the validity of the two approaches in the description of the nuclear motion. ,1 In deriving eq. (6) we have explicitly made use of the saturation condition for the ground state and neglected a term proportional to f(~TsVo)(~TsVo)pdo which contribution to eq. (6) is usually negligible, except in the case of large momentum transfer.

Iv) = exp(v [H, F] )]0)

Comparison with eq. (5) then yields the result S 3 = m -2 VGS(V = Vf(r)).

(9)

On the other hand, the hydrodynamical picture is well suited to study the static polarizability a (= 2.S_ 1) since it can be shown [12] that the Fermi surface preserves its sphericity in the presence of an external static field. In order to evaluate a one. has to minimize the expression E =

~Kf(v 0)20 do -

x f ;V(,a) dv

(10)

obtained adding the constraining term - ) , f f S p do to the hydrodynamical potential energy (7). A comment on the choice of the excitation operator F = )2if(ri) is in order here. Since we are interested in compression modes, the operator F should not excite surface vibrations that otherwise would exhaust the polarizability sum rule. In the limit of large systems this requirement is satisfied by excitation operators which vanish at the surface: f(R) = 0 where R is the radius of the nucleus. This condition, essential for multipolarities l 4= 0, permits one to write the constraining term in eq. (10) as - k f f V o p do. Minimization of eq. (10) then yields

V u = ~gf/K and

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Volume 108B, number 4,5

S_l=~X-l f f S o d v = ~ g K - l f f 2 p d v .

(11)

28 January 1982

the present experimental data in 208pb (coM = 13.5 M e V , co D = 21.3 MeV).

We shall now derive explicit expressions for the collective frequencies of the monopole ( f = r 2 - R 2) and dipole ( f = z ( r 2 - R 2) compression modes, labelled (M) and (D) respectively. Two different mean excitation energies can be defined in terms of the sum rules (4), (9) and (11). In the limit of large systems we find ((r2) -gRs 2)..

vTC

C031= S-l-

_

~r2)

(M), (12)

7K+~e

F

(D),

m(r 2)

The transition density 0 tr = V(op) of the collective vibration can be also evaluated. Of course the two approaches previously described yield two different forms for 0 tr. On one hand we have a scaling density [13] associated with the displacement field u = Vf. On the other hand, a constraining density given by the solution of eq. (12) (with the additional irrotationality assumption foro). In terms of sum rules these densities can be written as tr = ~n (n IFI0)(01 ~i 6 ( r Pscaling

- ri)ln)(En - EO)

and

and 0const r t r = n~ 0 (n IFI0)(01 6°1-1 =

_

m(r2 ) (13) =V~-~ K m(r 2)

K

m(r2 ) '

~z(100 + ( 3 r 2 - R 2 ) i r -1 0')

D =1/1.34 6°HD,n=I

K

m(r2 ) "

These energies turn out to be very close to the prediction (13) for co1-1" Concerning the applicability of eqs. (12) and (13) to actual nuclei we note that they have been derived in the limit of large systems and that consequently do not account for surface energy effects. However, we expect that the relative location of the monopole and dipole energies is less sensitive to these effects. From eqs. (12) and (13) we obtain co?_ 1/coM_1 = V/~-~ 1.46, D M =X/~(1 + ~~7 eF/K ') ,~ 1.66 6o31/w31 (having chosen eF = 38 MeV and K = 220 MeV). These predictions are in rather good agreement with

234

respectively. We find: ptsrcaling ~ 30 + ro'

(D).

We note that the ratio 6o31/6o1_ 1 is always larger than 1. This means that also in the limit of large systems the sum rules are not exhausted by one state. The result 6o31/6Ol-1 = ~O-/7 for the monopole mode has been discussed in ref. [2]. It is also interesting to evaluate the energies of the lowest monopole and dipole states of the hydrodynamical model [5] : M =V0.66 COHD,n=I

~5(r-r.)ln)/(E n i -E°)

(M),

(14)

(D)

and Pconst rtr "(r 2

-R2)o+(~r 2 -]R2)ro '

(M),

-,- z[10(r 2 - R 2 ) 0

+~2s ~ r 2 - 3R 2 + ~R4/r2)ro

'] (D),(15)

where 0' = do~dr. In fig. 1 we have plotted the transition density for 208pb, assuming a Wood-Saxon shape for 0 (R = 6.5 fm, a = 0.54 fm). The figure shows that the scaling and the constrained densities have different behaviours in the interior region. In conclusion we have shown that the generalized scaring and the hydrodynamical models provide a very practical evaluation of sum rules in compression modes. In the limit of large systems the relevant parameters are the nuclear compressibility and the Fermi energy. Of course the inclusion of surface, Coulomb and symmetry effects should be taken into account for a realistic analysis in actual nuclei, as it has been recently done for the monopole mode [ 1 - 3 ] . Further availability of experimental data on the giant isoscalar dipole resonance would make the extension of the

Volume 108B, number 4,5

PHYSICS LETTERS

28 January 1982

lot, monopole

,,,'/~'~

8

I

I0''

/

dipole

'

10 r (fro)

I

I

8

1,0 r(,,)

',

/;" 0

'

l

,

!

/ ,

1I

2

........ ~"

6

Fig. 1. Transition density (in arbitrary units) of the isoscalar giant monopole and dipole resonances in 2°epb. The full line corresponds to the scaling model, the dashed line to the constrained calculation. 235

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PHYSICS LETTERS

analysis in this direction e x t r e m e l y interesting *2 I wish to acknowledge B.K. Jennings for some useful suggestions. *:~ A recent RPA calculation of compression modes has been performed by Nguyen Van Giai and H. Sagawa (Orsay preprint) using Skyrme forces. Their results for transition densities and for the relative location of the monopole and dipole energies are in agreement with the predictions of the present work.

References [1] J.P. Blaizot, Phys. Rep. 64 (1980) 171; J.P. Blaizot and B. Grammaticos, Nucl. Phys. A355 (1981) 115. [2] B.K. Jennings and A.D. Jackson, Nucl. Phys. A342 (1980) 23.

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28 January 1982

[3] J. Treiner, H. Krivine, O. Bohigas and J. MartoreU, Nucl. Phys. A371 (1981) 253. [4] H.P. Morsch, M. Rogge, P. Turek and C. Mayer-B6ricke, Phys. Rev. Lett. 45 (1980) 337. [5] A. Bohr and B. Mottelson, Nuclear structure, Vol. 2 (Benjamin, 1975). [6] G. Bertsch, Nucl. Phys. A249 (1975) 253. [7] S. Stringari et al., Nucl. Phys. A309 (1978) 189. [8] G. Holzwarth and G. Eckart, Nucl. Phys. A325 (1979)1. [9] B.K. Jennings and A.D. Jackson, Phys. Rep. 66 (1980) 141. [10] G. Eckart, G. Holzwarth and J. de Providencia, Nucl. Phys. A364 (1981) 1. [11] O. Bohigas, A.M. Lane and J. Martorell, Phys. Rep. 51 (1979) 267. [12] D. Pines and P. Nozi6res, The theory of quantum liquids (Benjamin, 1966). [13] T.J. Deal, Nucl. Phys. A217 (1973) 210; M.N. Harakeh and A.E.L. Dieperink, Phys. Rev. C23 (1981) 2329.