SOLUTIONS OF THE LANDAU-VLASOV EQUATION IN NUCLEAR PHYSICS

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SOLUTIONS OF THE LANDAU-VLASOV

EQUATION IN NUCLEAR PHYSICS

B. Remaud, F. Sebille, C. Gregoire, L. Vinet

To cite this version:

SOLUTIONS OF THE LANDAU-VLASOV EQUATION IN NUCLEAR PHYSICS

B . REMAUD* , " * , F . SEBILLE" , C . GREGOIRE* and L . VINET' * * '~nstitut de Physique, Universitk de Nantes, 2, rue de la Houssinigre, F-44072 Nantes Cedex, France

"IRESTE, 3 , rue du Marbchal Joffre, Universite de Nantes, F-44041 Nantes Cedex 01, France

' * * G A N I L , BP 5027, F-14021 Caen Cedex, France

RBsume: Nous discutons les propribtks des solutions de J'Bquation de Vlasov ohtenues p a r projection s u r u n e base d'Btats cohbrents. Elles v e r i f i e n t des conditions de stationnarite e t permettent de dBcrire l a diffusivit6 moyenne de l'espace de phase nuclbaire e t de reproduire les caractbristiques moyennes des noyaux Les methodes d'bchantillonnage e t l e u r s effets s u r l a dynamique sont discutBs pour l'etude des reactions nucleaires aux Bnergies intermbdiaires. Enfin, I'utilisation de la force n o n locale de Gogny est directe ce q u i donne I'opportunitB de l'utiliser e n dynamique nucl6ait-e

Abstract: The properties of Vlasov equation solutions obtained by projection o n coherent state basis a r e discussed. Such solutions satisfy stationarity conditions and satisfactorily describe t h e average diffusivity of nuclear phase space and reproduce t h e bulk properties of nuclei. Sampling methods and t h e i r effects on dynamics a r e discussed f o r the study of heavy ion reactions a t intermediate energies. The non-local Gogny force i s easily computable o n t h i s basis w h i c h allows to use i t f o r dynamical nuclear studies

I

-

INTRODUCTION.

The n e w experimental data obtained i n t h e intermediate e n e r g y r a n g e (10 to 100 A-MeV) have b r o u g h t forward t h e Vlasov equation ( a n d its associates t h e Landau-Vlasov.(ll o r V.U.U. equation (2)) i n the field of nuclear studies. It gives t h e time evolution of t h e nucleus one-body phase space u n d e r t h e influence of t h e n u c l e a r mean-field :

a

f D f = - +

[ f , H ]

= 0 t (1.1) at --++

f = f (

r ,

p ;

t )

i s the one-body density distribution ; t h e Poisson brackets ( ) a r e defined as usual from t h e Hamiltonian

H(

We shall n o t discuss h e r e t h e virtues and drawbacks of such an equation which constitute a lively subject of discussion between theoreticians. We shall take i t as i t is : a semi-classical approximation to more sophisticated self-consistent quantum evolution

C2-196 JOURNAL DE PHYSIQUE

equations ; i n particular i t can be found a s t h e h + 0 limit of

TDHF

equation ( 3 1. Then f ( 7

a

is a smoothed, positive approximation to the exact Wigner-transform of t h e one body nuclear density matrix.

The appealing features of Vlasov equation i n nuclear physics a r e threefold

-4 -+

i ) since i t i s positive, f (

r,

p ; t ) can be interpreted as a probability distribution of a n assembly of (fictious) pseudo-particles : i i ) its semi-classical character provides easier

interpretations of its properties ; i i i ) i t can be coupled to a collision term to take into

account t h e residual interaction. One of t h e unpleasant features of t h e Vlasov equation lies i n the f a c t t.hat i t is n o t easier to solve t h a n the TDHF equation from which i t c a n be derived. However, t h e Vlasov equation t u r n s out to be a - still unchallenged - method to studying t h e interplay between one - a n d two-body interactions i n heavy-ion collisions.

In this paper, we shall discuss two questions which a r e connected with the use of Vlasov equation i n nuclear physics i ) how c a n we define its initial conditions ii)

what a r e t h e methods to solve i t taking advantage of its considerable redundancy. The first point i s of major importance a t lower energies since the close-to-equilibrium properties cannot be safely studied unless one is able to provide descriptions of nuclear ground states which a r e stationary solutions of Vlasov equation. The second one is essential to the possibility of making simulations of r e a l nuclear reactions.

The n e x t section presents t h e basis t h a t we use ; t h e third section gives some results f o r nuclear statics. Dynamics of nuclear reactions is discussed i n section 4 .

I1

-

STATIC BASIS OF THE PHASE SPACE.

The c o h e r e n t states a r e gaussian wave packets labelled by a continuous complex parameter

IZ>,

connected with the position of t h e i r center (x0, po) , we firstly discuss h e r e t h e one-dimension case since the extension of t h e formalism to t h r e e dimensions i s straightforward. The c o h e r e n t state properties a r e well known (see r e f 4

f o r a review). Let u s briefly recall t h a t i) according to t h e metric dp(Z) = dxodp0/h,

they obey t h e closure relationship : Jdp(2) Z > 21 =

1

(2.1)

i i ) they f o r m a n overcomplete basis of t h e Hilbert space ; i.e one can develop a n y density operator as

b =

sdk(Z)

~ F ( z ' )

p Z , * l ~ > <z'I (2.2)

which determines a mapping of t h e quantum phase space onto t h e space spanned by the continuous parameters

z.

The c o h e r e n t states a r e quantum objects, particularly well fitted to t h e semi-classical approximations. The Wigner transform of < > is :

X - X , p - PO ; A A A e x

(2.3

Although t h e y originates from minimum uncertainty wave packets (ApAx =

h / 2 )

; o n e can use reduced c o h e r e n t states (i.e.

h

-+ 0) while preserving t h e above

"classical quasi particles" 6(x

-

xo,

p - p,) if

YI

= 0 .

The projection of a given phase space distribution t h e n reads :

~ ( x , P ) = W ( X , P )

* ~ ( x , P

; AX, AP) ; ( 2 . 4 )

it is a folding product of t h e coherent states basis on a weight function w(x.pl

f(x,p) = (4nhAxhp)-' Jdxodp0

w ( x ~ ,

P O )

~ ( x - x , ,

p- PO ; hx, h p ) (2.5)

I n principle, o n e could deduce s u c h expressions f r o m

PZZ

i n e q . 2.2. However, i n this case t h e resulting f(x,p) is not fitted to o u r goals since i t keeps track of all quantum effects (such as shell effects) and presents wide oscillations. The general method to deduce a smooth approximation f(x,p) from eq. 2.4 i s still to be performed. We h a v e used

a v e r y simple a n s a t t ( 1 ) guided by t h e Thomas-Fermi limit t h a t we must recover a t t h e

h

+ 0 limit and by quantitation methods of semi classical orbits i 5 ) .

f ( x , p ) = @(EF - X ( X , P))* d(x,p, ; Ax, Apj (2.6)

The symbole

R

indicates t h a t o n e takes t h e expectation value of t h e classical harniltonian o n t h e c o h e r e n t state, w h i c h i s imposed by energy-conservation conditions i n dynamical equations (see below). Eq. 2.6 extends t h e existence domain of f(x,p) beyond the classical allowed region-while n o t inducing a n y coupling to t h e continuum. As In most smoothing procedures, we a r e l e f t w i t h t h e choice of t h e couple

(a,

Ap! which is fixed by requiring

t h a t t h e average diffusivity of t h e real phase space is reproduced. We notice t h a t i n e q . 2 . 5 ,

the step function plays t h e role of a n occupation number f o r a continuous set of semi-classical trajectories. I n fig. 1, one can see t h e comparison between e q . 2.6 f o r fermions i n a harmonic field a n d exact o r smoothed quantum results.

Fig.1 : Phase density distributions i n

( ~ n l f i ) ~

units f o r 224 fermions i n a harmonic well The exact (dashed c u r v e ) and smoothed (dashed and dotted c u r v e ) solutions a r e from ref.10. Our result i s given by t h e full line.

111 - STATIC DESCRIPTION OF NUCLEI.

C2-198 JOURNAL DE PHYSIQUE

+ ---+

~ ( T ~ = B ( E ~ - R ( ~ $ ) ) * d (

r

, p ; Ar, Ap) (3.1)

4 4

where we assume isotropy i n the

r

and i n the p directions. The Fermi e n e r g y EF is self-consistently deduced by constraining the norm of fc?) to the number of nucleons Till now, calculations h a v e been performed with simple Skyrme effective interactions :

~ ( p m )

-

a p m + b p V + ' 6 !

O < v <

l

(3.2)

which a r e easily combined with eq. 3.1, since the spatial density p m is :

p m =

Sf(T3dp

= f T F m dCi;); Ar) 13.3)

i.e. the Thomas-Fermi density (taking into account the zero-point e n e r g y ) folded with a

gaussian.

However, the structure of eq. 3.1 is particularly well fitted to t h e use of the Gogny effective interaction (6) which is non-local and allows good descriptions of ground state as well as fissionning nuclei. In the phase space notation, the one particle potenlial reads f o r

N=Z

nuclei (7) :

2

+

z

Si

f ( Z

S,

*

dtSi

d + h / p i )

( 3 . 4 ) i - I

The widths

pi

a n d depths

di,

a r e deduced from the D1 pararuetrization of the Gogny force. Since folding products between gaussians a r e straightforward

1 12

g(=, Q'") r

p(x,

p

= g(x, ( a ~ ) ' ' ~ ) ( 3 - 5 )

the Gogny force t u r n s out to be handy for semi-classical static and dynamic nuclear study ; i t provides a n effective way of studying t h e effects of non-locality and/or velocity dependence o n observables i n heavy-ion reactms. I n fig. 2, we show t h e nuclear profile obtained f o r a 4 0 ~ a nucleus. H.F. ( s k m * )

-

- - -

F V l a s o v (Gogny-GZ) 40 Ca

Fig.2 : Density profiles for protons, neutrons in a "ca

nucleus. We compare the results of Hartree-Fock with a Skyrme interaction and our calcutation with a G O B ~ Y

The main objective of t h e above considerations o n n u c l e a r statics is to get good initial conditions of t h e Vlasov ( o r V.U.U.) equation. If a n y functional of e n e r g y is solution of t h e static (time independent) Vlasov equation, one h a s to demonstrate t h a t i t is stationary before using i t i n dynamical calculations, i.e. t h a t all t h e observables a r e time-independent a t least o n a time-scale lager than t h e average duration of t h e nuclear reaction.

Solutions of e q . (1.1) h a v i n g t h e form of e q . (3.1) a r e given provided that

i )

R

i s a constant of motion and ii) t h e coherent states a r e solutions of t h e Vlasov

equation. The first condition :

d

,

,

--T(r,

p) = 0 (4.1

d t

i s exactly fulfilled if t h e gaussian centers follow the Ehrenfest equations :

d F

The second condition is fulfilled i n t h e sense that e q s ( 4 2) i n s u r e t h e minimal value of the action associated with t h e coherent state propagation i n g e n e r a l potential Our semi-classical approximation t h e n amounts to project t h e exact phase space on the class of solutions given by a swarm of c o h e r e n t states moving i n a n effective field

X.

The stationarity i s guaranteed since only t h e lower e n e r g y orbits a r e fully occupied These properties h a v e been checked with a v e r y h i g h degree of accuracy absolute fluctuations of less t h a n 1 % of t h e total e n e r g y , t h e r m s radius a n d quadrupole moments which a r e the low e n e r g y modes immediately excited b y small deviations f r o m t h e ground state

However, t h e phase space occupied by a nucleus is v e r y large w h e n compared to t h e actual extension of a c o h e r e n t states, t h e ratio scales r o u g h l y as

r

= (R/Ar13 (pp/hp13 ( 4 . 3 )

where R and p~ are respectively the radius and the Fermi momentum of the nucleus For nuclei with A >> 1. T is of t h e order of one thousand p e r nucleon ( i n o u r calculations. typical values f o r

Ar

a n d Ap a r e respectively 0 6 f e r m i a n d 0.15 fermi- l ) Real calculations can be performed i n two cases i) when symmetries reduce t h e d~mensionality of t h e problem, one can t r a c k t h e whole phase space (see r e f 8 f o r spherical symmetries)

ii) For r e a l 3-D problems, o n e h a s to reduce t h e complexity by a Monte-Carlo sampling of t h e step function i n e q 3.1 .

N

JOURNAL DE PHYSIQUE

The n u m b e r N is evidently a critical parameter, numerical tests show that complete stability i s obtained a s soon as N 3 50 A w h i c h is still large but much less

t h a n e q i4.3) would l e t expect.

We shall not show h e r e t h e variety of results t h a t one can extract from such calculations w h i c h go from the collective mode excitation of cold o r heated nuclei ( 8 ) to t h e intermediate e n e r g y reaction mechanism (9). These results show t h e ability of t h e formalism to follow the time evolution as well of relatively low e n e r g y modes as of reactions w h e r e considerable e n e r g y a n d momentum t r a n s f e r s occur.

I n fig. 3, we show calculations

f o r t h e l 2 C +

12c

reaction a t 84' A MeV with

t h e Gogny interaction ; i t is compared with calculations made with a simple local Skyrme interaction. I n this central collision, the bulk behavior looks similar, but we must remark t h a t the f i n a l relative velocities a r e different, this effect c a n be tracked back to the potential e n e r g y evolution w h i c h exhibits strong variations a t t h e b e g i n n i n g of t h e reaction due to t h e non-local character of t h e interaction. This p r e l i m i n a r y result shows t h e open field of research t h a t one can expect from t h e Landau-Vlasov equation used with such sophisticated interactions")

TDHF VLASOV, VLASOV2

r"'

Fig.3 : Projected nuclear densities on the reaction plane Ior the "C + "C (b4)A MeV) reaction the impact parameter i s 2.5 fm. The results labelled as TDHF and l VLASOVl are

found with a SKYRME force, VLASOV* gives the resuets with the GOGNY interaction.

V

-

CONCLUSIONS.

of n u c l e i . Taking use of t h e gaussian f o r m f a c t o r s of t h e non-local Gogny i n t e r a c t i o n , we h a v e s h o w n t h a t it c a n be used i n dynamical studies , t h i s i n t e r a c t i o n h a s been v e r y

successful i n t h e study of static n u c l e a r properties a n d of fission modes, o u r formalism provides a un,ique w a y of u s i n g i t i n t h e study of collective mode excitations a n d reaction mechanisms.

T h e a u t h o r s t h a n k M. PI, Y. RAFFRAY, P. SCHUCE a n d E. SCIRAUD f o r t h e i r collaborations.

REFERENCES

1) C. GREGOIRE. B. REMAUD, F. SEBILLE, L.

VINET.

Nucl. P h y s . A447 (1985) 55c a n d Nucl. P h y s . A465 (1987) 317.

2) J. AICHELIN, H. STOCKER, P h y s . Lett. 163B (1985) 59 a n d r e f e r e n c e s t h e r e i n .

3 ) P. CARRUTHERS. F. ZACHARIASEN, Rev. Mod. Phys. 55 ( 1983) 245

4 ) R.J. GLAUBER, i n Lecture i n Theoretical Physics, Vol. 1. ed. W.E. Brittin ( I n t e r s c i e n c e , New York, 1959) 315.

5 ) E.J. HELLER, J. Chem. Phys. 75 (1981) 2923.

6 ) J. DECHARGE. D. GOGNY, Phys. Rev. C21 (1980) 1568.

7) A. AYASHI, P. SCHUCK ( p r i v a t e communication)

8 ) L. VINFT, C. GREGOIRE, P . SCHUCK,%. REMAUD, F. SEBILLE, Nucl. P h y s . ( i n Press). 9 ) C. GREGOIRE, This Conference a n d r e f e r e n c e s therein:

10) M. PRAKASH. S. SHLOMO. V.M. KOLOMIETZ. Nucl. Phys. A370 ( 1981 ) 30.