LIQUID PARTICLE AND LIQUID GAS MODELS OF NUCLEAR DYNAMICS

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Submitted on 1 Jan 1984

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LIQUID PARTICLE AND LIQUID GAS MODELS OF NUCLEAR DYNAMICS

V. Strutinsky

To cite this version:

V. Strutinsky. LIQUID PARTICLE AND LIQUID GAS MODELS OF NUCLEAR DYNAMICS.

Journal de Physique Colloques, 1984, 45 (C6), pp.C6-281-C6-284. �10.1051/jphyscol:1984633�. �jpa- 00224235�

L I Q U I D P A R T I C L E AND L I Q U I D G A S MODELS O F NUCLEAR DYNAMICS*

V.M. Strutinsky

Institute for NucZear R e s e a r c h , Academy of S c i e n c e s o f t h e Ukr. S . S . R., Kiew, U . S . S. R.

Resum& - Nouspr6sentonsun modele dynamique du noyau qui tient compte d la fois des aspects macroscopiques (d'un liquide classique) et des composantes de quasi-particules (degres de libert6 microscopiques) .

Abstract - We present a model of nuclear dynamics which combines the macroscopic aspects of a classical liquid with the characteristic behaviour of a quanta1 gas of nearly independent quasi-particles.

This paper is to draw attention to a recent development of a dynamical theory of nuclear deformations in which the macroscopic behaviour of a classical liquid drop of almost incompressible condensed nuclear matter and the interaction with the quasi- particle components are given their balanced role. Since the first three parts of this theory are about to be published separately /1,2,3/, we content ourselves here with a short presentation of the basic ideas.

1. Liquid-Particle equations of motion / I / :

One starts from the selfconsistent Liouville equation for the single-particle density matrix p (3, , F2 ^,t)

where h is the selfconsistent single-particle Hamiltonian

and u i s some effective nucleon-nucleon interaction. By taking the Wigner transform of eq. ( I ) , one obtains the equation of motion for the Wigner function f (?,$,t) and performs the usual moment expansion /4/. Macroscopic quantities are introduced

+ + through a phase-space average of f(r,p,t):

which practically may be achieved using :he energy-averaging method used in static shell-correction calculations /5/. From f eq. (3) one obtains the average density

the average current density

and the average velocity field

The exact density matrix is written

*~anuscri~t presented and edited by M. Brack and R.W. Hasse.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984633

JOURNAL DE PHYSIQUE

where p l describes the dynamical single-particle or quasi-particle component of the nuclear fluid. The potential V in eq. ( 2 ) can be split according to (7) as

The above introduced quantities can be shown, up to second order terms in pl , ^to

fulfil the following equations of motion /I/:

'L " n,

where h = T

+

t r y

t r p

%

at the same time the continuity equation for p is fulfilled:

Equation (9) is a generalized Euler type of equation where V1 couples the liquid and the quasi-particle components.&[~] in eq. (9) is the macroscopic average local energy density which is assumed to be a functional of the local density :(see, e-g.

ref. /6/). The adiabatiz solution of eq. (10) corresponds to the cranking model with an external field V whose shape parameters can be considered as (slowly) time- dependent collective variables.

The conserved total energy is shown to be

E

C E + ^E,

(up to second order terms in pi), where the macroscopic part (including the collect- ive kinetic energy) is

and the quasi-particle correction

=

tr(Fp,)

-

is an implicit functional of P(r,t) and its time derivatives (through eq. (10)).

In the static case, _{E l} reduces to the familiar energy shell-correction / 5 / . The quasi-particle correction pi can be shown in perturbation theory to lead tp a correction B1 of the inertial parameter which in the adiabatic limit corresponds to the difference between the macroscopic (h~drodynamical) inertia, contained in eq.

(14), and its cranking-model value.

conditions matching the two regions involve such phenomenological parameters as the surfacetension and the compression modulus. The equations for the surface are most adequately formulated using a shape dependent curvilinear coordinate system / 7 / . The boundary conditions become particularly transparent for isoscalar density modes:

a) The average particle velocity component normal to the equivalent sharpe surface must be equal to the velocity of the normal displacement of the surface itself.

b) The normal component of the stress tensor caused by the density distortion should be compensated by the excess pressure due to the surface tension at the curved surface.

This approach to the nuclear surface dynamics becomes in the case of small distortions aroundaspherical equilibrium shape equivalent to the approach of Bohr and Mottelson as applied to the first-sound collective modes / 8 / . The general formalism developed in ref. /2/ will, however, also be applicable to large-amplitude processes such as fission or fusion and their coupling to intrinsic modes.

3. Landau Zero -Sound and Nuclear Giant Resonances /3/:

In the spirit of the shell correction method, nuclear dynamics of collective motion is split up into average behaviour and fluctuations. In this combined liquid + gas approach, the average behaviour of nuclear dynamics is treated within the framework of hydrodynamics and fluctuations by Landau theory of zero sound. Finite size effects from the nuclear surface are thus contained entirely in the hydrodynamic part so that Landau theory of nuclear matter can be used for the bulk. The boundary conditions then couple the liquid and the gas giving rise to zero sound admixtures to the otherwise first sound motion.

The resulting characteristic equation for the eigenvalues w = xsvF/Ro becomes

where the zero sound velocity svF has to be determined from the Landau equation,

Here ef is the Fermi energy, b is the surface energy coefficient, Fo, F1 are the Landau parameters and jR(xaUf$ the spherical Bessel function.

Eqs. (16,17) have been solved on a TI-59 desk calculator. By virtue of the coupling between zero and first sound, eigenenergies hwk of nuclei with masses 50 \< A 6 300 are not strictly proportional to ~ - l / 3 but exhibit small deviations from this law if the parameters 0.5 ,< Fo ,< 3, - 0.5 4 F1 d 1 .O, 2 6 eF/bsurf 4 3 are employed.

Resulting quadrupole and octupole giant resonance energies,

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are i n good a g r e e m e n t w i t h t h e e x p e r i m e n t .

R e f e r e n c e s

1. STRUTINSKY V.M., MAGNER A.G. a n d BRACK M . , 2. P h y s . 5 (1984) i n p r i n t

see a l s o Izv. A c a d . N a u k S S R ( 1 9 8 3 ) a n d P r o c . I n t . School o n Heavy Ion P h y s i c s A l u s h t a ( 1 9 8 3 )

2. STRUTINSKY V.M., MAGNER A.G.: t o be p u b l i s h e d

3 . STRUTINSKY V.M., MAGNER A.G. and DENISOV V . , 2. P h y s . A 3151984) 3 0 1 4 see, e - g . , MOYEL J . E . , P r o c . C a m b r . P h i l . Soc. 45 ( 1 9 4 9 ) 99

6. BRACK M., GUET C . , H ~ W S S O N H. -B., MAGNER A.G. and STRUTINSKY V.M.,

4 t h I n t . C o n f . o n N u c l e i f a r f r o m S t a b i l i t y , H e l s i n g b r (CERN 81-09, G e n e v a 1 9 8 1 ) p. 65

7. STRUTINSKY V.M. and TYAPIN A . S . , J. E x p . T h e o r . P h y s . (USSR) 2 ( 1 9 6 4 ) 6 6 4 8. BOHR A. and MOTTELSON B . , N u c l e a r S t r u c t u r e

( B e n j a m i n I n c . , 1 9 7 5 ) V o l . 11, Sect. 6 A - 3c