Spreading width of giant resonances in fluid-dynamical approximation

Volume 138B, number 5,6

PHYSICS LETTERS

26 April 1984

SPREADING WIDTH OF GIANT RESONANCES IN FLUID-DYNAMICAL APPROXIMATION ~

K. ANDO 1, A. HAYASHI and G. HOLZWARTH Universitiit Siegen, FB 7-Physik, 59 Siegen 21, Fed. Rep. Germany Received 30 December 1983 Revised manuscript received 7 February 1984

On the basis of a semiclassical argument a fluid-dynamical expression is derived for the coupling matrix element between a giant multipole state and a two-phonon state responsible for the spreading width. Significance of effective threebody forces originating from the density-dependent potential energy functional is pointed out in the description of the coupling.

After having shown that nuclear fluid dynamics [ 1 ] is able to reproduce escape widths for Giant Multipole Resonances (GMR) obtained from corresponding RPA equations [2], it seems desirable to have also a simple expression for the spreading width within that formalism. The spreading width F + of GMR describes the decay of the simple giant "doorway" state into nuclear configurations with a more complicated structure [3]. According to the golden rule of perturbation theory F ~ is given by F ~ = 2 r r2t~. ~ t I12N2x, g ( W L - ~ x - W x , ).

(1)

Here, the giant L-pole resonance is represented by IL>, while Nxx' IXX'> indicates the "complicated" normalized two-phonon final state, and the g-function takes care of energy conservation. Of course, the XX' sum extends over physically different states IXX'> only. If we consider X and X' as solutions of RPA equations then I XX'> will, of course, include also almost pure (dressed) two-particle-two-hole, as well as one-phonon plus l p - l h configurations. If we take V as a two-body force and sum up all possible (lowest order) diagrams by taking into account the presence of the g-function, the matrix element in (1) can be cast into the form * Supported by Deutsche Forschungsgemeinschaft. 1 Present address: Department of Physics, Kyoto University, Kyoto, Japan. 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

+ (qg01[Ox, [gutx,,B[]] leP0>,

(2)

where lee0> stands for the HF ground state, O~ the creation operator of the X-phonon, and gU x is the transition potential <~lgUxlt3> = ~ ] ~ ) ~ v g p x ( ~ ' g ) , 75

(3)

associated with the transition density matrix. In the microscopic description the operator B~ in (2) is defined through <~lB[lt3> = /(w - ( e ~ - @)) i.e. wB~ = g UL + [H0, B [ ] ,

(4)

H 0 being the static single-particle hamiltonian (Ho)a> = e~la>). Obviously, eq. (4) uniquely determines the operator B~, i.e. all the matrix elements of B~ that are required in the evaluation of (2). This is in contrast to the RPA equations: {~o0~ - (8UL + [H O, OTLI)}Iq)0> = 0 , (qb01 {6oO~ -- (gU L + [H0, O~])) = 0 ,

(5)

which specify only ~ - h (and h - p ) components of the creation operator O L. 333

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In the semiclassical approximation the Wigner transform 8n(x, p) of the transition density matrix is given by

resonances based on the generalized scaling description with a further assumption of irrotational flow, the Wigner transform Ow is approximated by

6n(x, p) ~- -i{no(x,p), Ow(x, p)*}

oFD (x, p) * = -(M6o/2) 1/2 mdp(x)

(6)

in terms of the Poisson bracket between the Wigner transform n O of the ground state density matrix p0(xl, x2) and that of the creation operator Ot (here and in the following we suppress the subscript L and leave out spin-isospin variables). If we further assume that nO(x, p) is a function of the Wigner transform cO(X, P) of riO

no(x, p) = F (eo(x, p ) ) ,

(7)

i.e., the classical equilibrium condition used, for example, in a study of semiclassical fluid [4], then eq. (6) reduces to

8n(x,p) "~ [--Ono(x,p)/OeO(x,p)] u(x,p) ,

(8)

with

(12)

in terms of the collective velocity potential ¢, the associated velocity field u = V q~, and the collective mass M:

M -1 =

mfpo(Ixl) I,(x)[ 2 d3x.

(13)

For the sake of simplicity we assume that the underlying potential energy functional V[p] takes the form

Viol =fA (o(x)) d3x. Then,

cO(X, P) = p2/Zm + U(Po(r)) ,

v(x, p) = i{e0(x , p), Ow(x, p)*} •

(9)

Since we can repeat the above argument leading to (9) by using Bw in place of Ow m (6), we also have

v(x, p) ~- i {eo(x,p), Bw(x, p)*}

+ i(M/2w) 1/2 u(x). p

(10)

for non-vanishing Ono/Oe O. The comparison of (9) with (10) implies that in our semiclassical description v(x, p) is independent of p - p and h - h matrix elements o f B ?. Finally we manipulate (4) again by the semiclassical approximation as

¢oBw(x, p) * ~- ~ Uw(X, p) + i{eo(x, p), Bw(X, V)*), so that

ooBw(x, p)* "" 8 Uw(x, p) + ~'(x, p)

with r = Ix [ and U(p (r)) = 6 V[p ]/8 p (x). This simplification also implies a local residual interaction

a(Po(r)) = ~U(po(r))/3po(r) . The FD approximation for v of(10) now reads vFD(x, p) = -(M/2oo) 1/2 a(Po(r)) u(x) " VPO(r)

+ i(M¢o/2) 1/2 u(x)" p + (M/2co) 1/2 (O/Ok¢(X)) P}Pk/m.

= 6 Uw(x, p) + i {e0(x, p), Ow(x, p)*) = ~(x, p ) . In this way we have obtained a semiclassical expression for the operator B t. It is to be noticed that for long wavelength excitation in an infinite Fermi system the Fourier component v(q;p) of v(x, p) coincides with

(15)

The Wigner transform 6 UFwD(X,p) of the associated transition potential is ,1 <5UFwD(x,p)

(11)

(14)

=

a(Po(r))(M/2oo) 1/2 V" (PO(r) u(x)),

and similarly for the X-phonons

(16)

6Ux(x ) = a(Po(r)) 6Oh(X).

(17)

By substituting (15) and (16) into (11), the FD expression for Bw is obtained as

BFD (x, p)* = w -1 [(M/2w) 1/2 a(Po(r)) PO(r) V • u(x) + i(M6o/2) 1/2 u(x)" p

v (q; p) = ~ (1 + F l/( 2l + 1)) Vlm Ylm (P), + (1/m)'(M[26°) 1/2 (3/OkdP(x)) PiPk] , which plays a key role in the treatment of the collision integral [5]. In the fluid-dynamical (FD) formulation of giant 334

(18)

,1 Note that we are dealing with spin-isospin independent excitations.

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which contains terms up to quadratic in p. In view of Bw = co-l~, we see that eq. (18) is consistent with the fact that the generalized scaling approach takes account of momentum space deformations with multipolarity l ~< 2 [6]. It is interesting to note that the operator B~.D corresponding to (18) with q~= r 2 Y2 exactly solves the operator equation (4) for the harmonic oscillator plus the self-consistent quadrupole force model [7]. In a more realistic situation the FD operator B~:o cannot be the exact solution of(4), but is expected to provide a useful approximation for the solution in the case that the FD approach describes the giant vibration reasonably well. Given (18) the matrix element (2) is obtained in the form (XX' [ IA2) IL )=

-(M/2co) 1/2 ( ;Sp~,' u" V (a6p~) d3x

co2if(6j[,)j(O]

akO) Ok(a6p~O d3x + (X ~ X')), (19)

with the phonon transition density 6px and current

8ix: Spx(x) : , 8/x(x) : <%1V(x), otx] I%>, which may be taken from any feasible (microscopic) model for the phonons. The "l = 1" component (linear in the momentum p) in (18) yields the first term in (19), while the second term in (I 9) comprises both the "1 = 0" and "l = 2" contributions, as is evident from (18) and

The expression (19) shows an unpleasant deficiency in the case of density-dependent forces: For constant velocity u (center of mass motion) the matrix element should vanish. However, for density-dependent forces where a(po) is not a constant, the " / = 1" contribution evidently is not zero. In that case we have to also include the contribution from the effective three-body force IA3)(1,2, 3) = 83 V[p]/6p (Xl) 8p (x2) 6p (x3)

= (Oa(PO)/Opo) 6 (x 1 -- x2) 6 (Xl -- x 3 ) , which adds a term of the form


26 April 1984

* * 3x 6px6px'SPLd

(20)

to the matrix element (2). Such a term is obviously of the "/-- l ' t y p e because only the u • p part of B~D contributes to 6PL = (M/2w) 1/2 V. (POU). Therefore we obtain the sum of the " / = l " terms in (19) and

(20) as -(M/2oo) 1/2 f d3x [u . (Sp%,V(aSp~0 + Sp~ v(aSp~,,))

-

(21)

Oa/Opo) Sp~,spl,v" O0,,)1

= (M/2oo)ll2jd3x[(a+Po3a/3PO)(V.r

u) 8Px6Px'] * * .

This expression now conserves the required symmetry. Furthermore it shows quite generally that for divergence-free flow fields the damping is due only to the " / = 2" term in B~D in the present simple FD description. The fact that the " / = 0" and " / = 1" terms do not contribute to the spreading width is familiar in the infinite matter case [8] where it is a general consequence of particle number and momentum conservation. It is interesting to notice that for divergence-free flow and zero-range forces the same holds also in finite systems if we consistently take account of the effective three-body coupling even in the case of densitydependent forces. It has been already implied by Adachi and Yoshida [9] that a consistent description of the density-dependent interaction matrix element between l p - l h and 2 p - 2 h states requires such a three-body coupling. Since the divergence-free Tassie ansatz u = V (r L YL) is expected to give a fair approximation to the flow field of Isoscalar Giant Multipole Resonances (ISGMR) such as the quadrupole vibration, a rough estimate of ISMGR spreading widths may be obtained with the simple expression (L :~ 0): (XX'I VI L ) = (2i/¢o)(M/2w)

1/2 (;(6f~t,)/

× Oj~k(r L YD) ~,(aS;~) d3x + (a ~ x') )

(22)

for the crucial matrix element in (1). Naturally the perturbative approach to the GMR width only makes sense if there does exist a strongly collective resonance structure. For cases where singleparticle effects and the coupling to 2 p - 2 h states wash 335

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out the whole resonance o t h e r m e t h o d s [10] m u s t be used for obtaining the strength distribution.

References [1] G.F. Bertseh, Nucl. Phys. A249 (1975) 253; G. Holzwarth and G. Eckart, NucL Phys. A325 (1979) 1; K. And~ and S. Nishizaki, Prog. Theor. Phys. 68 (1982) 1196. [2] K. And~ and G. Eckart, Siegen University preprint (1983). [3] G.F. Bertsch, P.F. Bortignon and R.A. Broglia, Rev. Mod. Phys. 55 (1983) 287. [4] D.M. Brink and M. Di Toro, Nucl. Phys. A371 (1981) 151.

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[5] D. Pines and P. Nozieres, The theory of quantum liquids (Benjamin, New York, 1966); G. Baym and C. Pethick, The physics of liquid and solid helium (Wiley, New York, 1978) Part II, Ch. 1. [6] T. Yukawa and G. Holzwarth, Nucl. Phys. A364 (1980) 29. [7] S. Stringari, Nucl. Phys. A325 (1979) 199. [8] E.g.C.J. Pethick, Phys. Rev. 185 (1969) 384; K. AndS, A. Ikeda and G. Holzwarth, Z. Phys. A310 (1983) 223. [9] S. Adachi and S. Yoshida, Phys. Lett. 81B (1979) 98. [10] G.F. Bertsch and I. Hamamoto, Phys. Rev. C26 (1982) 1323; B. Schwesinger and G. Wambach, SUNY preprint (1983).