MICROSCOPIC CALCULATION OF THE NUCLEAR MEAN FIELD

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MICROSCOPIC CALCULATION OF THE NUCLEAR

MEAN FIELD

J. Cugnon, P. Grange, A. Lejeune

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C2, suppl6ment au n o 6, Tome 48, j u i n 1987

J. CUGNON, P. GRANGE* and A. LEJEUNE

UniversitB de Liege, Physique NuclBaire Thdorique, I n s t i t u t de Physique au Sart Tilman, B5, 8-4000 Liege 1 , Belgium ' c e n t r e de Recherches NuclBaires, BP 20 CRO.

F-67037 Strasbourg Cedex, France

R6sum6 - Pour un systhme de matiPre nuclCaire 3 densite normale, nous Ctu- dions la contribution, au champ nucleaire moyen, des corr6lations de 1'Ctat fondamental et ce, pour des tempCratures nulle et finie. Nous travaillons dans le cadre d'un d6veloppement en nombre de lignes trou en utilisant le po- tentiel nucl6on-nucl6on de Paris. Nous mettons en exergue la relation exis- tante avec le terme de collision de 116quation de Landau-Vlassov.

Abstract

-

Zero and finite temperature contribution of ground state correlations to the nuclear mean field is studied for nuclear matter at normal density within the framework o f the hole line expansion using the Paris potential. The connection with the collision term of the Landau-Vlassov equation is exhibited.

I. INTRODUCTION

The knowledge of the nuclear mean field is very important to explain the sta- tic as well as dynamic properties of nuclear systems. It 1s then Highly desirable to understand the properties of this mean field from first principles. By this term, we here mean nuclear forces described by potentials in the frame of nonrela- tivistic quantum mechanics. A very intensive effort has been devoted to the ground state nuclear matter mean field in the past. But, despite o f this effort, the mean field is known in first order only, not in the bare interaction, but in the renormalized interaction which accounts for medium effects. Here, we present the first microscopic calculation, based on a realistic NN interaction (namely the Paris potential), which goes up to second order in the renormalized interaction and so ta- kes the ground state correlations into account. The calculation is done for zero and finite temperatures.

2. THE FORMALISM

The framework is the Brueckner approach to nuclear matter (see ref. [ I ] for a review), extended to finite temperature according to the method o f ref. [ Z ] . The Green function, which describes at the same time the propagation of a particle ad- ded to the medium and the one of a hole punched into the medium (of momentum

z

and energy E ) , can be written as

(6

= 1 )

E -

X

-

M(k,E)

2m

The quantity M(k,E), known as the mass operator, can be expanded in series of the Brueckner reaction matrix g

.

The latter is related to the bare NN potential V

(l)~upported by the NATO grant No 025.81

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through the Bethe-Goldstone equation

Q being the Pauli operator for the intermediate states. Up to second order in g, one has :

with

where e(k) is the single-partlcle energy for state k . It has to be determined self-consistently as

In this scheme,

represents the momentum dependent complex mean field in nuclear matter (up to second order), identified to the optical-model potential. Eqs. (2.2)-(2.4) corres- pond to the diagrams contained in Fig. 1 . The diagram on the left is called pola-

risation diagram and the other one the correlation diagram. These denominations remind of the physical meaning of the diagrams for particle states. The first one describes the reaction of the medium on the propagation of a particle due to the polarisation produced by this particle. The correlation diagram describes the effect of the correlations present in the medlum prior to the interaction of the particle. In Eqs. (2.3)-(2.7), the quantities n(k) are the occupation probabi- lities in the unperturbed ground state

First order con- tributions to par-

ticle (upper row) where )J and the temperature T are related to the nucleon and to hole (lower p

row) self-energy.

p = C n(k)

.

(2.9)

[Fig. I ]

2

3. THE MEAN FIELD

In Fig. 2, we present our results for the real part of the correlation and the polarisation potentials for normal density po : 0.17 fmT3. We also present

~b:)

,

the mean field calculated in the first order approximation. We see that the polarisation field deepens when going to second order. However, the addition of the re- pulsive correlation field makes that globally, the mean field is less attractive than in first order approximation. This is an interesting result, which brings a better agreement with experiment, provided the deeplying single-particle states in- side heavy nuclei can be considered as indicative of nuclear matter properties.

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Vpo&, in nuclear matter at T-0

LO

+

Polarisation (Vpo), correlation ( V o ) and

20 total

(V)

single-particle potentials in cold nuclear matter at o ma1 density.

0 _{The short-dashed line}_:f_Y! _{gives the po-}

larisation potential when the usual con-

-

20 hoice is used. The short-dashed

%

s

y1:9

::::0

represents the first iteration

---LO of the correlation potential.

[Fig. 21

-60

-80

VpoYco in nuciear matter

-100

klfm-'I

Polarisation (Vpo), correlation

(Vco)

and total ( V ) slngle-particle potentla1 at

T

= 7 and 10 MeV.

[Fig. 31 4. THE EFFECTIVE MASS

This important quantity, defined as (m = I )

k =

de(k)

dk 1 2 3 . 1 2 3

klfm-'1 represents the true inertia of a par-

ticle embedded in the medium. It is given in Fig. 4 for various temperatures. The most important result is the peak appea- Effective mass m' in nuclear matter

. a ~ - r t k , ~ lHPVl ring around kF at zero temperature. The

-LO -20 0 20 LO 60 80 I

I d k l l ~ ~ v l ' I I presence of such a peak in an extended sys-

-60 -LO -20 o 20 LO 60 tem was surmised on the basis of the cal-

I

,

I , , , I

1.2-

-

culations up to second order in a semi-phe-

nomenological interaction done in ref. [3] and of extended model calculations for a hard sphere Fermi gas [ 4 ] and for a s-wave interaction [5]. The presence of a similar peak in finite nuclei was already con- jectured in ref. [6]. It was demonstrated

0.6- _L _I_/ _I _I - by Mahaux and Ng6 [7] and longly analysed I

0.5 1.0 1.5 2.0 2.5 in subsequent works (see ref. [8] for a

klfm-'1 list of references). It is also predicted

by semi-phenomenological calculations Nucleon effective mass at various tem- _{In nuclear matter, the peak appears} peratures. The curve with dots above kF

,

if only the polarisation con- and dashes indicates the effective mass _tribution_is_retained_(see_Fig._{5 ) .} _The calculated with the polarisation poten- correlation contribution drives the peak tial only.

[F.lg. 41 to the Fermi level (approximately). Phy- sically. this peak is due to the excitation of core-polarized (Ip-lh) states by a traveiling pe~ticle (or hole). These states are characteristic of the bulk matter. On the contrary, the origin of the peak in the finite nuclei is perhaps due to the excitation of surface vibrations [Ill although the coexistence of the two effects is more probable [12].

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5; THE MOMENTUM DISTRIBUTION

An expansion similar to eq. (2.3) exists for the momentum distribution ~ ( k )

,

which writes ~ ( k ) = p1(k) + p2(k)

,

with pl(k) = n(k)[2-ill

,

p2(k) = (l-n(k))i2

,

(5.3) and

i

-

K

M.(~,E)\

,

i = 1 , 2

.

(5.4)

-

E=e(k)

The results of our calculation at T = 0 are given in Fig. 5. The depletion below kF 1s due to the polarisation term and occupation probability at T=O the population above kF is due to the

correlation term. One can notice that the depletion of the Fermi sea is quite

,,

6 . CONNECTION WITH THE LANDAU-VLASSOV EQUATION

important. To illustrate this point, - - - _ _ _ _ _ _ _ _ _ _ we quote that the parameter K = I-~(k),

where the bar indicates an average over

-.-rL

;;,

Pllkj

1

the Fermi sea, is

-

0.25 in our calcu- 0'5 lation. In Fig. 6 , this value is com- compared with other predictions for p

=

PO

.

It turns out that the Paris

0.5 1.0 1.5 2.0 2.5 potential should be classified as a _{k ~fm-ll}

"hard" potential. This property ob-

viously comes from the momentum de- Momentum distribution p(k) for cold nu- pendence of the potential. cle'ar matter (full curve). The short-

dashed curve represents pl(k) when the

I t i s generally believed that the off-equilibrium behaviour o f nuclear systems should be governed by a Landau-Vlassov type o f equation :

A

0.2-

H.

O.'-

soft hard self-consistency is applied on the pola- 7 - risation potential only.

/"

"\

[Fig. 5 )

RHC

'.

Paris

0 Illustration of the variation parameter

K

with the "hardness" of the inner part of

0

various NN potentials : HT (Haftel and

v l ~ Tabakin), HH (Hamman-Ho Kim), RSC (Reid

7

R?C Soft Core), ~ 1 4 (Pandharipande)

,

RHC

HH

₀@ st. choice _{cont. choice} _{the standard choice of the Brueckner-}(Reid Hard Core). The full dots refer to

b

cont. choice (Ml+M2) Hartree-Fock approximation, the open dots

HI

0

variational to the continuous choice, the open squa- res to eqs. (2.3)-(2.6) and the lozenge >

"hardness" refers to variational calculations.

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+ +

where f(r,k,t) is the one-body Wigner distribution function and where fi = f(?,ei,t). It has been shown recently [I31 that both the mean field U and the transition matrix W have to be calculated from Brueckner g-matrix. O f course, in a non-quilibrium process, the g-matrix should be calculated with the instantaneous phase space occupation, but equilibrium calculations can be taken as giving the gross properties of U and W

Concerning U we have indicated that this field is nonlocal and depends upon the momentum. In first approximation (in

H),

a L?-dependence of the mean field U

would introduce the effective mass I n the 1.h.s. of eq. (6.1) which then would read

Fig. 4 indicates that this effect is non negligible. It has been studied phenomeno- logically in ref. [14].

As for the structure of the collision term, there is an interesting relation- ship with the mass operator. Indeed, one can write [I] from eq. (2.3)

_-

2

Im Ml(k,e(k)) = CCC n ( j ) ( l - n ( a ) ) ( l - n ( b ) ) l < a ~ g [ e ( k ) + e ( j ) ~ l % > l 6(e(k)+e(j)-

ttt

~ a b

and from eq. (2.4)

-

2

Im M (k,e(k)) = - 1 CCC n ( j ) n ( a ) ( l - n ( a ) ) [ < ~ [ g ~ e ( ~ ) + e ( ~ ) ] ( ~ > ( ~(e(k)+e(g)-

2 2

Assuming that the transition probability W can be identified to the g-matrix [13], one can write the collision term as

Strictly speaking, eq. (6.5) holds in a uniform medium, but it is expected to be correct locally in more general situations. The loss term is thus connected to the imaginary part of the polarisation potential and the gain term to the imaginary part of the correlation potential. At equilibrium, one should have

This relation is meaningful for T

+

0 only. For our case, these two ratios are given in Fig. 7 at two temperatures. We recall that M2 and MI are calculated independently. Fig. 7 shows that bhe relation (6.6) is very well fulfilled. The observation that the collision term has the structure (6.5) will help to have a better determination of the input data to be introduced in the collision term. Furthermore, it can be a good starting point for the estimation of equilibrium ti- mes and transport properties.

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C2-286 JOURNAL DE PHYSIQUE

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