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TDHF WITH COLLISIONS IN HEAVY ION DYNAMICS
H. Köhler
To cite this version:
H. Köhler. TDHF WITH COLLISIONS IN HEAVY ION DYNAMICS. Journal de Physique Colloques,
1984, 45 (C6), pp.C6-389-C6-394. �10.1051/jphyscol:1984646�. �jpa-00224248�
Colloque C6, suppl6ment a u n06, Tome 45, juin 1984 page C6-389
TDHF WITH C O L L I S I O N S I N HEAVY I O N DYNAMICS
H.S. ~ ~ h l e r *
Physics Department, University o f Arizona, Tucson, Arizona 85721, U.S.A.
RQsumQ
-
La c o r r e c t i o n des c o l l i s i o n s B deux c o r p s B l ' a p p r o x i m a t i o n champ moyen dans l a dynamique n u c l Q a i r e e s t d i s c u t d e . Le r g s u l t a t de l ' a p p r o x i m a t i o n du temps de r e l a x a t i o n p o u r l e noyau de l a c o l l i s i o n e s t compard aux c a l c u l s "exacts" de Danielwicz. C e t t e approximation s l a v S r e Qtonnamment bonne l o r s q u e l e s c a l c u l s e x a c t s comprennent l e s e f f e t s q u a n t i q u e s . L ' Q l a r g i s s e m e n t des Q t a t s ( 2 cause d e s e f f e t s de durde de v i e ) permet l a p o p u l a t i o n d l Q t a t s i n a c e s s i b l e s p a r des moyens c l a s s i q u e s . Abstract-
The two-body collision correction t o t h e mean-field approximation in nuclear dynamics is discussed. The result of t h e relaxation-time approximation f o r t h e collision kernel is compared with the "exact1' calculations of Danielwicz. The relaxation-time approximation is found t o be surprisingly good when t h e e x a c t calculations include t h e quanta1 effects. The broadening of energy s t a t e s (because of lifetime e f f e c t s ) allows classically unreachable s t a t e s t o be populated.We aim a t a microscopic description of nuclear dynamics; in particular, of collisions between nuclei. The resolving power of our "microscope" changes with time. Maybe quarks will be the ultimate fundamental (microscopic) particles. In nuclear structure and dynamics, calculations on nuclei a r e still conventionally t r e a t e d a s a many-body system of nucleons.
There is, however, good evidence t h a t already ground-state nuclear m a t t e r is not purely nucleonic. Other degrees of freedom manifest themselves in three-body forces t h a t a f f e c t quantities such a s binding energies in a quantitative way.
In t h e microscopic many-body theory of heavy ion collisions t h a t w e shall present here, the particles are assumed t o be nucleons, however. The transport properties of t h e hot nucleus a r e therefore assumed t o be purely due t o nucleonic collisions. I t appears, however, t h a t one ultimate goal of the theory and experiments would be t o look f o r nucleonic degrees of freedom t h a t might show up in hot nuclei.
There a r e several possible avenues of approach t o arrive a t t h e transport equation t h a t we shall use t o describe heavy-ion collisions. To follow t h e historical development, w e s t a r t with the TDt-F equation, which w e write here in t e r m s of t h e one-body density m a t r i x
(la) op(f<;t1 = 0
with
where U is a mean field t o be defined later.
This is t h e quantum mechanical analogue of t h e classical Vlassov equation. This can be considered as an approximation for t h e Boltzmann transport equation by including only t h e particle-hole excitations by neglecting the two-particle two-hole excitations.
*supported in part by NSF G r a n t Number PHY-8100141
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984646
C6-390 JOURNAL DE PHYSIQUE
In a low-temperature fermi gas this is a consistent approximation. I t is consistent wih a long mean-free path due to Pauli blocking, if the particles a r e considered t o be quasi- particles. In an only slightly different language: I t is consistent with t h e Brueckner theory of effective interactions
if
t h e interaction between t h e particles is the Brueckner reaction matrix.A t higher temperatures, the Pauli blocking decreases, however, and "two-body collisions"
become important.
The role of two-body nucleon collisions in nuclear dynamics and especially in heavy ion collisions is a challenging and often-addressed problem. The problem is, however, not always well defined, so this should be t h e f i r s t topic of any discussion. Nuclei a r e highly correlated systems. The nucleon f o r c e is strong and of short range. I t h a s long been known t h a t a perturbation theory in t e r m s of t h e nucleon force is inadequate t o low order even if efforts have often been made t o construct "smooth" nucleon forces in order t o improve t h e convergence. But these smooth f o r c e s a r e very unrealistic by not fitting high-energy data.
Because of t h e strong binding of t h e nuclear system, t h e propagation is so f a r off t h e energy shell t h a t the hiqh-enerqy d a t a a r e important as an input. As a consequence, one can make the following ~ e ~ r n i n ~ l y ~ c o n t r a d i c t o r ~ statement: Low-energy nuclear physics is a high- energy problem.
The fascinating thing is, however, t h a t t h e problem of the strong interactions t o a very good approximation can be simplified by introducing an effective two-body interaction. This is t h e so-called Brueckner reaction
(K-
or G-) m a t r i xwhere v is t h e nucleon-nucleon interaction and Q is t h e Pauli ooerator. The enerav denominator in the propagator Q / e can be chosen so a s t o manipulate t h e perturbation series (diaarams) effectively summed by t h e definition of t h e K-matrix. The K-matrix was defined for t h e purpose of summing ground-state energy diagrams for nuclear matter. For t h e purpose of summing other diagrams, say single-particle or effective interactions in open-shell nuclei, the K-matrix still serves as a first-order effective interaction t o be used in a diagrammatic expression, but higher-order t e r m s a r e often important.
We a r e now able t o go back t o t h e opening s t a t e m e n t regarding t h e role of two-body collisions. Two-body collisions in nuclei a r e always important. If by t h a t we mean collisions between actual nucleons interacting via t h e two-body f o r c e v in Eq. (1). If, however, w e talk about the effective interaction, t h e K-matrix, the answer will be different. We can now regard t h e nucleons a s quasi-nucleons interacting via a K-matrix r a t h e r than a v-interaction.
Hartree-Fock theory can now be formally applied if the v-interaction is replaced by a K- interaction, even if t h e definition of single-particle energies prompts us t o include some other t e r m s (so-called Brueckner rearrangement t e r m s ) in t h e definition of the B W field (see above). In this theory we can therefore say t h a t t h e problem is reduced t o a one-body problem; t h a t of nucleons moving in a one-body potential, t h e mean or Hartree-Fock field.
So in this sense, two-body collisions a r e eliminated from t h e theory; t h e two-body collision (the K-matrix) only serves to define t h e mean field. The diagrams defining t h e one-body field a r e shown in Fig. 1. These a r e particle-hole diagrams. A nucleon collides (is excited) by the collision with another nucleon t h a t s t a y s in i t s orbit (remains unexcited). This also implies t h a t t h e r e a r e no scatterings in nuclear m a t t e r other than forward scatterings.
Scatterings only t a k e place a t t h e nuclear surface where VU f 0 ; U being t h e mean field.
This
W
theory of t h e ground s t a t e of nuclei has been taken over in a time-dependent description in T D W , where the W - f i e l d is changing with time. In this theory, t h e one-body density m a t r i x then statisfies Eq. (1).In principle, one would like t o calculate t h e mean field U in Eq. (1) and defined by Fig. 1 from a K-matrix. In practice, t h e K-matrix is replaced by some approximation of t y p e Skyrme f o r c e or similar.
produced i n a nucleus, say by a p-2p reaction, t h i s hole has a l i f e t i m e t h a t w i l l decrease sharply w i t h the depth o f the hole-state. This is known experimentally, and calculations also show this. F o r a large nucleus, the w i d t h o f a 1s state is -10 MeV, i.e., a l i f e t i m e o f - 3 x 1 0 - ~ ~ sec. The l i f e t i m e o f a p a r t i c l e o f -20 MeV above the f e r m i surface is f r o m the optical nucleus potential of about the same magnitude.
Fig. 1
-
Diagrams defining the one-body field.I n an ion-ion collision, such states are easily produced.' W i t h l i f e t i m e s shorter than the i n t e r a c t i o n t i m e o f a nuclear collision, we see t h a t the mean-field approximation would break down. A n interaction w i t h the other nucleons t h a t results i n decays o f hole and/or particle states is what we r e f e r t o as a two-body collision i n this context. They are typically represented by the diagram shown i n Fig. 2. The 10-MeV widths given above are just some average value t o get an idea about what l i f e t i m e s a nucleon m i g h t encounter i n a heavy-ion collision. Just as important is the strong momentum dependence being roughly quadratic around the f e r m i energy. Although the nuclear f l u i d i n several respects can be treated by classical or semi-clasical methods, this is an e f f e c t t h a t is specifically quantal.
Fig. 2
-
(a) The imaginary p a r t o f this diagram represents a decay. (b) A two-body collision.It differs f r o m the mean-field contribution i n Fig. 1 i n t h a t both particles are excited.
I n r e l a t i o n t o heavy-ion collisions, it is actually more relevant t o discuss deformations and, i n fact, strong deformations of the f e r m i surface. I f one does TDHF and looks at the phase-space representation (Wigner function), then the only change i n the momentum distribution (deformation o f the f e r m i surface) a t any one point would come f r o m the f l o w i n phase space and collisions w i t h the one-body field. Calculations have shown t h a t these effects tend t o increase the deformations (except perhaps i n a coarse-grained picture).'
Calculations have been made following the time-development o f a deformation o f the F e r m i surface a f t e r an interaction is switched on. These calculations were made by Pawel
~anielewicz.' H e treats three d i f f e r e n t cases. We are especially interested i n t w o of these.
C6-392 JOURNAL DE PHYSIQUE
I n one he uses a Boltzmann (classical) collision term, i n the other a quantum-mechanical version. The difference between these t w o is t h a t i n the Boltzmann term, the energy- conservation is imposed i n each collision. B u t the energies are, i n fact,
=
sharp. O n the contrary, the f i n i t e l i f e t i m e s o f t h e states, i.e., the widths o f t h e energies are precisely what we are interested i n calculating. So this is a self-consistency problem. Danielwicz finds this self-consistency t o be quite important. A t t i m e t=O, the distribution is t h a t o f t w o f e r m i spheres, each of +=1.29 f m - l . They are separated by a distance o f 4.39 f m - l between the centers. The equilibrated temperature o f this system corresponds t o 70 MeV. As t-, we should therefore expect t o f i n d a f e r m i distribution o f this temperature. The momentum distribution at an angle o f 90° w i t h the symmetry axis is shown a t a f e w successive times i n Fig. 3. Table 1 shows t h e temperatures (in MeV) read f r o m t h e slopes of the curves. I n the Boltmann case, the distribution does reach this temperature, but we see t h a t the temperature is much l o w e r initially. The energy conservation allows particles t o reach higher momentums state only by repeated scatterings. I n the quantum case, t h e temperature is, on t h e other hand, reached much faster. (That the temperature is larger than 70 MeV here is probably because of numerical uncertainties.TABLE 1.
T i m e ( 1 0 - ~ l s ) 0.0033
Boltzmann 32
Quantum 86
I think these results are important. I think, in fact, t h a t it shows t h a t a quasi-elastic scattering or cascade model is inadequate. These models assume energy-conserving (free) scatterings just as i n t h e Boltzmann collision kernel. Danielewicz's result shows, however, t h a t the many-body effects are important. The distribution is instantaneously a F e r m i distribution w i t h the temperature being the equilibrated temperature.
The question we are concerned w i t h here is: How are t h e states populated as a result o f the collisions? (Or what is the f o r m of the Master equation?) The answer is t h a t they are populated as a function o f t i m e according t o
where fo is the f e r m i distribution. The angle 90° i n Fig. 3 does not include the i n i t i a l l y occupied states. Otherwise there would also be a t e r m describing the depopulation o f these states. According t o this result, we should thus always expect t o see particles e m i t t e d w i t h a temperature t h a t r e f l e c t s the energy-density o f t h e source irrespective o f i f they are "pre- equilibrium" particles or particles e m i t t e d later. We emphasize again t h a t his i s the result of l i f e t i m e effects
not
present in the classical Boltzmann equation.The reason we point t h i s out is t h a t the quantum case agrees w i t h our approximation o f a collision A general modification o f the T D W equation (1) is t o w r i t e
where D c is a collision operator. I n our (relaxation-time) approximation we w r i t e
Po (actually equivalent t o f o i n Eq. 3) is here a f e r m i distribution o f a temperature T obtained f r o m (the energy density o f ) p. I n the classical case (Boltzmann equation), this Eq.
( 5 ) i s an often-used ap roximation. I n the quantum case it may f i r s t have been used by
Karplus and Schwinger.
B
function of energy p 2 / 2 ~ in MeV and a t indicated t i m e in seconds. The dotted ( ~ o l t z r n a n n ) and the short-dashed (quantum) lines a r e from Danielewicz's paper.2 The long-dashed lines a r e from our relaxation-time approximation (see below).
F o r a homogeneous medium, t h e solution of Eq. (5) is
The l a s t t e r m describes t h e deppyulation of s t a t e s initially occupied. The f i r s t t e r m has t h e form of Eq. (3) with F ( t ) = 1-e-
'.
A plot of the anisotropy of t h e distribution a s a function of time, Fig. 4 shows some deviation from t h e exponential decay. The decay slows down a s the anisotropy decreases. F o r small excitations of the fermi sphere, ~ e r t s c h ~ found t h e relaxation t o depend mainly on t h e energy input. F o r larger deformations (excitations) this may not b e t r u e a s indicated by Fig. 4. If the energy goes into deformation r a t h e r than heating, t h e decay r a t e is larger. I t still needs t o be investigated if this is t h e correct interpretation of this result, however. The relaxation t i m e approximation (5) a s we have used i t in several calculations then is very much supported by Danielewicz's results. A possible refinement would be t o allow the relaxation t i m e . t o depend on t h e anisotropy of the deformation in momentum space.JOURNAL DE PHYSIQUE
Fig.4
-
Anisotropy as a function o f time. The t w o lines f r o m Danielewicz's work are drawn as i n Fig. 3. The t w o straight lines are the results using relaxation times T N and TR i n our work.',REFERENCES
1. K ~ L E R , H. S. and NILSSON, B. S., Nuclear Physics, t o be published.
2. DANIELEWICZ, P., Ann. o f Phys., t o be published.
3. KARPLUS, R. and SCHWINGER, J., Phys. Rev.
77
(1948) 1020.4. K ~ L E R , H. S., Nucl. Phys. A343 (1980) 315; Nucl. Phys.
A378
(1982) 159; Nucl. Phys.A400 (1983) 233c.
5. BERTSCH, G., 2. Phys.
