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GIANT RESONANCES IN A POTENTIAL WELL : A SEMICLASSICAL DESCRIPTION
X. Viñas, A. Guirao
To cite this version:
X. Viñas, A. Guirao. GIANT RESONANCES IN A POTENTIAL WELL : A SEMI- CLASSICAL DESCRIPTION. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-19-C2-26.
�10.1051/jphyscol:1987203�. �jpa-00226468�
GIANT RESONANCES IN A POTENTIAL WELL : A SEMICLASSICAL DESCRIPTION
X. VIGAS and A. GUIRAO
Departament dfEstructura i Constituents de la Materia. Facultat de Fisica, Universitat de Barcelona, Diagonal 645,
SP-08028 Barcelona, Spain
Abstract: A semiclassica! method based on the sum-rule approach toge- ther with the Wigner-Xirkwood expansion in powers of h is used in order t o describe Giant Isoscalar Resonances at zero and finite tem- perature for an ensemble of fermions moving freely in a potential well.
Introduction
One of the methods to study Giant Resonances is the random phase approximation (RPA), for which the Hartree-Fock (HF) calculations provide the natural single-particle basis /1-4/. T o get the energy and width of the peak is not necessary to perform the complete solu- tion of RPA equations, it is enough to look for some moments of the strength. Sum rules give less detailed information than RPA solu- tions, but they are much easier to calculate and easier to interpret / 5 / .
I n recent years heavy-ion reactions have given preliminary expe- rimental evidence for the existence of collective states of giant resonance type sustained on excited nuclear states. Some theoretical effort has been made to describe such situation /6-9/.
I n this contribution we want to study the collective excitations of an ensemble of fermions moving freely in a given potential well using the sum rule approach and the Wigner-Kirkwood expansion in powers of % for the magnitudes which appear in our calculation.
Our goal is, rather than to reproduce the experimental results, to study the giant isoscalar resonances at zero and finite temperatu- re and for several multipolarities in a model potential well where the full quantum mechanics results are known. We want to check the ability of our semiclassical approach to reproduce these collective motions. Previous results about the giant isoscalar monopole resonan- ce in a harmonic oscillator well at zero temperature have been re- cently reported /lo/.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987203
C2-20 JOURNAL DE PHYSIQUE
1. T h e Giant Isoscnlar Mononole Resonance at zero temuerature
In order to derive the m l and mg sum rules w e proceed us fol- lows: First, we calculate the full quantum mechanics sum rule and in the last step we replace the quanta1 expectation values by the semi- classical ones.
T h e m l sum rule can be written as
T o calculate the m3 sum rule we use the identity /11/
where -2t'/m G=[H,F] =[K,F], ( , Y is the kinetic energy).
After llttle algebra
J Z V
I n order to evaluate ( 1 ) and ( 3 ) we use the semiclassical den- sity and kinetic energy density given in terms of the Wigner-Xirkwood expansion.
where w e assume A = 2 N = 2 Z fermions l n a central potential well V(r).
W e can also recover ( 3 ) from a scaling ( P ( r ) +a 3 P ( a r ) , 't ( r ) -)a5z ( a r ) ) of our semiclassical equilibrium densities ( 4 ) and (5).
This i s d u e to the fact that the virial theorem is fulfilled by these semiclassical densities /15/ (see reference 10 f o r details).
T h e static polarizability i s proportional to the m,l sum rule and i t can be calculated in the RPA approximation using the solution of a constrained Hartree-Fock calculation ( H + M F ) in t h e limit of
small p-values I
O u r semiclassical approximation is to do a constrained Thomas- Fermi ( T F ) + h ' calculation assuming that the potential is not pola- rized.
T a k i n g into account (41, ( 7 ) and ( 8 ) w e finally get the follow- ing analytical expression f o r thp c., sum rille
W e c a n a l s o check that the second equality in ( 6 ) i s fulfilled in o u r constrained Thomas-Fermi calculation /lo/.
F r o m t h e s e sum rules w e c a n estimate the e n e r g y o f the giant .esonance defining the energies /5/ m3/mg dnd E l = m i / m - l . R e - ults f o r t h e harmonic oscillator ue;?=Lave e e n reportid elsewhere /lo/. T a b l e 1 s h o w s sum rules and energies for a W o o d s - S a x o n poten- tial. T F and T F + k' results a r e llsted in c o m p a r i s o n with the exact o n e s f o r several number of fermions in t h e potential we1 I . T h e h1
corrections i m p r o v e t h e TF results and reproduce q u i t e well the quan- tal calculations.
T A B L E 1
3 4 4
( M e V f m ) ( M e V fm (tle~-'fm~) ( M e V ) ( H e V )
I
C2-22 JOURNAL DE PHYSIQUE
2. T h e G i a n t I s o s c a l * ~ r Mononole Resonance at finite temperat~lr+
T o c a l c u l a t e m and m l sum rules we follow t h e method of refe- rence / 9 / : we use 7 1 1 and ( 3 ) . but the expectation values a r e c a l c u - lated with the densities at flnite temperature g i v e n by:
where = ( - V ) / T 1 s the degeneracy parameter.
F o r the harmonic osclllator potential the m l and mg s u n r.1ies a t f i n i t e temperature read:
From ( 1 2 ) and ( 1 3 ) ~t follows that E 3 = 2 k w and thls value is independent of the temperature. T h e same 1s true i n the full quantal calculation a s it c a n b e easily seen from ( 1 ) and (3). I t is not surprising because the m g and ml,stt.n rules, f o r the harmonic oscllla- tor, a r e proportional t o < r a > ln quantal and semiclassical calcula- tions.
B e f o r e computing the m _ l sum rule, it 1s necessary t o point out t h e following: in HF calculations we can vary independently o c c u - pation numbers and the radlal part cf t h e wavefunctions. W h e n R P A is formulated the occupation numbers of each s i n g l e particle (s.p.) s t a t e a r e flxed and equal to those of t h e equilibrium HF state a t given T, and t h e collectlve m o t ~ o n affects only the radial part of the s.p. state. Consequently thls motion takes place, at thr s a m e time, a t c o n s t a n t temperature and a t constant entropy.
W e c a l c u l a t e t h e m - l sum rule performing a constrained calcula- tion. I n t h e quantal case, keeping flxed the occupation numbers, it i s e a s y t o s h o w that E1=2fiw.
I n t h e semiclassical approach based in TF method the situation i s d i f f e r e n t because w e have, a t the s a m e time, vzriations of the occupation numbers and t h e radial part of t h e wavefunctions. We can perform T F constrained calculations a t constant T o r a t constant S - O f course, in t h i s kind of calculations w e will h a v e spurious changes in S o r T respectively that a r e due to t h e u s e o f TF approximation and n o t t o any physical effect / 9 / .
F o r the harmonic osclllator in the pure T F approxination the e n t r o p y reads:
8 1
- - - Ai N
5(p)= ?, 2 ~ .
('7
A T - - T'It is not difficult to s h o w that
From ( 1 5 ) follows that E1=2%w. This result c a n be a l s o recovered
~f t ' corrections are taken lnto account.
We c-:. - 1 - 0 calculate the m - l sum rule ;r - n isothermal way, 1.e. keeping fixed the temperature. Following the s a m e strategy that
~n the T = O c a s e w e find f o r the harmonlc oscillator potentla1 and u p to % ' terms
T A B L E 2
( M e V fm 4 ) ( M ~ v - ' f m 4 ) ( ~ e v - l f m * )
- - - -. - --. -- - - -- . .-. . -- - . . - -. . - -- - -- . -
I I
T F .5951x10 6 T = O T F + % ~ .6014x10 6 ( M e V 1
Q M .5992x10 6
.3286x10 -- - - -. -- -- - - - - .- .- .- --. . - .. - . -. -
T F .6022x10 6
.3380x10 4
.3303x10 T = 2 T F + ~ ~ .6084x10 6
. 3 4 1 4 x 1 0 4
.3337x10 4 . 6 0 7 7 x 1 0 6
.3334x10 4
.6229x10 6
.3717x10 4
.3417x10 4
T = 4 .6293xlO 6
.3753x10 4
.3452x10 4 ( M e V
, 6 2 9 2 ~ 1 0 6
.3452xlO 4
.6563x10 6
. 4 2 4 4 x 1 0 4 .3600x104
T = 6 .6628x10 6
. 4 2 8 2 x 1 0 4
.3635x10 4 (?lev 1
.6627x10 6 .3286x10 4
T P .7007x10 6
.4929x10 4
.3844x10 1
T = 8 .7073x10 6
.4968x10 4
.3850x10 4 (?lev)
.7073x10 6
.3880x10 4
C2-24 J O U R N A L DE PHYSIQUE
T a b l e 2 s h o w s u s the dependence of the m l and m - ( a t constant T and c o n s t a n t S ) wlth temperature. T h e m l sum rule is 'not very sensi- tlve to T due to the fact that < r 2 i has a small d e p e n d e n c e on tempe- rature a s it is well k n o w n . T h e same 1 s true, in the harmonrc o s c i l - lator case, f o r the isentropic m - l sum rule whereas the isothermg.1 m - l s h o w s a more important dependence o n temperature d u e to the f a c t that t h e isothermal incompressibility decreases m o r e rapidly than t h e lsentroplc o n e when the temperature increases. a s rt has been pointed out in o t h e r semiclassical calculations /8/. W e can a l s o s e e that the isothermal m - l calculation gives a n overestimation of the width that i s m o r e important when the temperature increases.
3. G i a n t I s o s c a l a r Resonances of hlqh multipolarity
I n t h i s c a s e the c o l l e c t i v e operator i s given by F = r l y l O . F o B l o - winq the s a m e method described in the first s e c t i o n m , and m, read
3 4
w h e r e G = % V ~ ' Y ~ ~ . V and the k i n e t i c energy contribution i s taken from reference /9/.
T h e m - l sum rule i s g i v e n by a constrained TI? calculation. At f i n i t e t e m p e r a t u r e t h i s sum r u l e reads
T A B L E 3
I n T a b l e 3 w e g i v e o u r semiclassical result f o r Eg (obtained from ( 1 7 ) and ( 1 8 1 ) for different number of particles in a Woods-Sa- xon potential a t several temperatures and multipolarities ( e n e r g i e s
pure T F calculation g l v e s a n overestimation of the energy. T h e energy E3 is q u i t e insensitive to changes in temperature, however this chan- ge is m o r e important when multlpolarity increases and the number of particles in the potential well decreases.
F o r the harmonlc potential E g and E l energies a r e grven by
I n pure T F a p p r o a c h E3 and E l a r e constant with temperature, however when h 2 corrections a r e taken i n t o a c c o u n t a small dependence in t e m p e r a t u r e appears.
F o r the q u a d r u p o l e case ( 2 1 ) reproduces the ex'act result Ej=2tw ( t h a t c a n be obtained from /17/ and /18/ taking t h e quantal expecta- tion values). T h e situation is different f o r E l , t h e exact calcula- tion ( e q u i v a l e n t to a deformed oscillator potential) g i v e s u s E 1 = 2 k w whereas the semiclassical result ( 2 0 ) is E l n m .
D u e to the f a c t o f the e x a c t equivalence b e t w e e n the T F + ~ ' c a l - c u l a t i o n and the Strutinsky averaged o n e in t h e harmonic oscillator problem /13/ w e h a v e performed this last c a l c u l a t i o n i n order t o check o u r semiclassical result. We find E -1.43 h w f o r A = 4 0 when w e a r e in t h e p l a t e a u region. T h i s is the va1;e that w e c a n g e t from ( 2 0 ) i n t h e T=O limit.
T h e d i s c r e p a n c y between t h e quantal and semiclassical results can be understood in t h e following way: a s it h a s been pointeE out by J e n n i n g s 1 small d i s t o r t i o n s before the level crossing obtai- ned with a constraint o n r a y 2 i n the h a r m o n i c o s c i l l a t o r causes just t h e s a m e c h a n g e in the wavefunctions a s the s c a l i n g . Consequently a d e f o r m a t i o n o f t h e F e r m i s p h e r e a p p e a r s and it i s n o t t a k e n into a c - count i n o u r semiclassical approach. T h i s is n o t t h e c a s e of o u r m3 c a l c u l a t i o n w h e r e the semiclassical a p p r o a c h i s t a k e n i n the last s t e p a f t e r a quantal d e r i v a t i o n of the sum r u l e t h a t considers t h e F e r m i s e a distortions.
T h e c h a n g e of orbits a f t e r t h e crossing i s a b s o l u t e l y necessary i n o r d e r t o k e e p t h e momentum distribution roughly spherical /14/ and i t h a p p e n s in s l o w oscillating motion like f i s s i o n where semiclassi- cal methods based o n the T F approach h a v e been used successfully / 1 5 / .
4. C o n c l u s i o n s
First, w e h a v e checked t h a t the u s e o f o u r semiclassical sppro- x i m a t i o n reproduces we1 the giant monopole resonance a t zero and f i n i t e temperature. I n this latter case, w e h a v e s h o w n for the harmo-
C2-26 JOURNAL DE PHYSIQUE
nic oscillator that a constraines TF calculation keeping the entropy constant reproduces the e x a c t calculation whereas the constrained TF a t c o n s t a n t temperature underestimates the E l value, increasing thus the r e s o n a n c e width. T h i s result i s ln good a g r e e with other finite t e m p e r a t u r e calculations /8, 9/.
T h e second polnt is that o u r semiclassical calculation overesti- mates the m - l sum rule in the quadrupole case for the harmonic poten- tial because t h e deformation of the Fermi sea plays an important role and i t is n o t i n c l u d e s in o u r approach.
Ack7,-wledqements: T h e authors a r e inde~t2.3 t o M.Barranco, H.Xrivine, J . M a r t o r e l 1 , P.Schuck and J.Treiner f o r stimulating discussions.
T h e y a l s o thank CAICYT ( g r a n t P B 8 5 - 0 0 7 2 - C 0 2 - 0 0 ) for financial s u p - port.
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