Volume 66B, number 1 PHYSICS LETTERS 3 January 1977

S T A T I C A N D D Y N A M I C A L T H O M A S F E R M I T H E O R Y F O R N U C L E I

Gottfried HOLZWARTH Physik-Department, Technische Umversitat Munchen, 8046 Garching, Germany

Received 22 July 1976

The Thomas 1: erml approximation together with a correction term for the kinetic energy and zero-range effec- tive mternucleon forces lead to static and dynamical results for nuclei In slab geometry which show strong sLmilarities to recent TDHF calculations.

In the following note we want to outline a fluid-dy- namical approach to large amplitude collective motion in nuclei and fusion-fission phenomena in heavy-ion collisions. It has been demonstrated recently [1-3] that the time-dependent Hartree-Fock (TDHF) method is capable of describing a wide range of dynamical phe- nomena by basically changing only initial conditions for the nuclei involved. A basic disadvantage of TDHF is that it requires the determination of the complete single-particle density matrix p(r, r', t) although one is mainly interested in its diagonal part only. The TDHF- method further restricts the density matrix by impos- ing p2 = p which might well be a condition too strin- gent for actual many-body dynamics including dissipa- tion of energy and temperature changes.

Considering the hmit r ~ r ' in the equation of mo- tion for the density matrix p(r, r', t) fluid dynamical equations are obtained [4, 5] for the local density p(r, t)and the current density ! (r, t). Further simphfi- cation arises from statistical assumptions about the energy expression, leading to general results about nu- clei of sufficient size, rather than to information about specific nuclei.

For the static case, Bethe [6] has suggested a dif- ferential equation for the local density, using the Thomas-Fermi (TF) approximation for the kinetic en- ergy and deriving the surface term from the long range part of the internucleon forces. With the success of the zero-range two- and three-body effective interactions of the Skyrme form (even without its velocity-depen- dent parts [1-3] ) it seems feasible to obtain the sur- face term from corrections to the TF kinetic energy expression instead. Of course both methods are sub- ject to criticism, because on the one hand the range of the force, on the other hand the surface thickness is

of the order of the internucleon distance. A simple cor- rection term to the TF approximation for the kinetic energy has been suggested by von Welzsacker [7] and subsequently discussed by many authors [8]. We shall use this correction term in the following without any attenuation factor [9].

For the purpose of comparison with the recent TDHF results we consider in this note the same (slab-) geometry discussed in ref. [1 ] and use the Skyrme pa- rameters t o = -1100 MeV fm ÷3 and t 3 = 17000 MeV fm +6. In this geometry the total energy is given by

E = \10 m PO] PO + 2 p dr

(1)

+ mJh2 ;(Vp)2dr + 3 p to fp2 + t3 fp3 at. The first term is the TF expression for the kinetic en- ergy in slab geometry where P0 stands for nuclear mat- ter density. The second term is the yon Weizsacker cor. rection while the last two terms are the Skyrme poten- tial energy without the small velocity dependent parts.

Static slabs: The differential equation determining the density distribution of static slabs is obtained from 8E/Sp = 0 subject to fixed particle number. De- noting the Lagrange parameter E F (Fermi energy) we have 8E/Sp = E F or

,, 1 (p')2 + p { a + ~ p +~/p2}, (2) P - 2 p

with

4/qrr2 \2/3 m m a = ~ ~:~-- P0) +4~-~ IEFI, /~= --3 ~-~ It01,

6 /3rr2~ 2/3 3 m 7 = 5 ~2p2] +-4-~ t3"

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V o l u m e 66B, n u m b e r 1 PHYSICS LETTERS 3 January 1977

fm -3

015

015 " 015t 015

010"

005

5=0 6 =10 ~5

° . ~:°, ~:°°~'-''x~.~.~~\ \ 2 4 6 8

Fig. 1. D e ns i t y profi les for static slabs.

' i'0 frn

Setting the curly bracket equal to zero fixes the con- nection between the Fermi energy E F and the con- stant nuclear matter density P = P0" For the value of P0 = 0.15 fm -3 we obtain E F (P = P0) = E(O) = -16 . 7081673 MeV. The remarkable features of solu- tions for eq. (2) are shown in fig. 1. Devianons o f E F from E(F 0) by small amounts/5 yield density distribu- tions (starting the integration with p (x = 0) = 0.15, P' (x --- 0) --- 0) which differ only in the mean lengths

fro- 3

o~-

o

o1

0 -

o~--

oli 0 -

Ol

0

01

0

O1

0

O}

0

/ ~/A=IMeV ~ t=0

- ~ . ~ t =00.6

t=01

t=o16

t=022

t=04 \ ~ ~ =054

t=078

i i i I i i i i i 5 10 frn 15

Fig. 2. D e ns i t y profi les for two identical slabs co lhdmg w i t h e/A = 1 MeV energy per particle. (T ime unit = 10 -21 sec).

t=0

t = 002

t = 004

t=006

t = 008

t =012

t=02

t=03

t = 0 4

01 ~ t=05

~ ' I , I ' I 5 10 15 20 fm 25

Fig. 3. Dens i ty profi les as in fig. 2 w i th e/A = 30 MeV.

of the slab while the surface thickness of about 2 fm remains unaffected. The maximum of the slope occurs at half-density. One order of magnitude in 6 changes the mean length by about 0.95 fm. Of course, we could have alternatively kept E F fixed and changed p (x = 0) instead by small amounts. In a subsequent paper [10] we consider spherical nuclei with coupled proton and neutron fluid including Coulomb terms and find similar results.

Slab collisions: The fluid dynamical equations governing the motion of p and] in slab geometry are

~-P + 1 = 0 (3)

~-~I + bx \ p ] m Ox " (4)

In this form they are of the Euler type and conserve the total particle number f o dx and the total energy

m:7 W =-~ dx + E,

as cffn be easily verified by multiplying eq. (4) with the velocity ],/p, making use of the continuity eq. (3) and integrating over x. On the right hand side of eq.

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Volume 66B, number 1 PHYSICS LETTERS 3 January 1977

0.4

fm-3

03

02

01

I I I I 20 40 60 80 1 O0

elA (MeV)

Fig. 4. Central density of compound system for slabs colhdmg with e/A MeV per particle.

(4) the term originating from the von Weizsacker cor- rection can be rewritten in the form

P b - 2 + = m ~x m bx "

Therefore we can introduce the pressure P by writing the right hand side of eq. (4) as - ( 1 / m ) ( a / O x ) P to ob- tain the relation between pressure and density at tem- perature T = 0:

_3 2 2 {a t h2 (3,t r2 2/3 "

4 m P ~ •

For finite temperatures the TF-expression for the ki- netic energy can be modified in the familiar way [11 ] .

Integrating eqs. (3) and (4) with initial condit ions of two static slabs moving with uniform velocity to-

wards each other leads to fusion, oscillating compound nuclei and fission depending on the incident energy per particle e/A. We have used an explicit fourth order Runge-Kutta method which usually became unstable after the build-up of the compound system. For stabil- ization we had to introduce artificial viscosity to smooth out oscillations especially in the tail region of p (note the p -1 terms in 6E/6p!). This leads to non- conservation of W and to much less structure as com- pared to the TDHF results. Typical results are shown in figs. 2 and 3. Remarkably the density of the com- pound system shows the same dependence on the inci- dent energy as given by TDHF in ref. [1 ] (cf. fig. 4).

In a subsequent paper [12] we shall consider more closely the dissipation of energy and ItS connection to changes in the local temperature of the system.

References

[1] P. Bonche, S. Koonin and J.W. Negele, Phys. Rev. C 13 (1976) 1226.

[21 S.E. Koonin, Phys. Lett. 61 B (1976) 227. [3] R.Y. Cusson, R.K. Smith and J.A. Maruhn, preprint 1976;

R.Y. Cusson and J A. Maruhn, Phys. Lett. 62 B (1976) 134.

[4] N.N. Bogoliubov, Quantum statlitics, vol. 2 (Macdonald, London 1971).

[5] C.Y. Wong, J.A. Maruhn and T.A. Welton, Nucl. Phys. A253 (1975) 469.

[6] H.A. Bethe, Phys. Rev. 167 (1968) 879. [7] C.F.v. Weizsacker, Z. Phys. 96 (1935) 431. [8] N.H. March, Advan. Phys. 6 (1957) 1, and references

therein. [9] R. Berg and L. Wllets, Proc. Phys. Soc. 68 (1955) 229.

[10] G. Eckart and G. Holzwarth, to be published. [11] P. Gombas, Die statistische Theorle des Atoms und lhre

Anwendungen (Springer-Verlag, Wien, 1949) p. 8ff. [12] G. Holzwarth, to be published

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