u

1Olc

Nuclear physics A428 (1984) 101~118~ North-Holland, Amsterdam

FAST NUCLEON EMISSION 100 MeV/u

: A PROBE OF THE HEAVV ION REACTIONS AT ENERGIES BELOW

Bernard REMAUO*,Francois

SEBILLE*,

* Institut de Physique,

Universite

Christian GREGOIRE+

and Franz SCHElJTER+

de Nantes, 44072 Nantes Cedex, France

+ GANIL, BP 5027, 14021 Caen Cedex, France.

After a review of the various theoretical models which describe the fast nucleon emission in heavy-ion reactions at bombarding energies lower than 100 MeV/u, the Boltzmann equation is discussed as a tool for studying the reaction mechanisms in this energy range. The Boltzmann equation allows to analyze the balance between one - and two - body processes. A linearized version is derived to study the transport in nuclei of fast nucleon flows generated by the reaction dynamics. Results are presented showing that the high energy tails of the spectras can be interpreted without reference to statistical emissions from hot nuclear fragments.

1. INTRODUCTION Free nucleons are a quasi-permanent least bound nucleons

in

output of the heavy-ion collisions

nuclei have a binding energy of approximately

; the

8 MeV,

which is small when compared with the relative kinetic energy of the ions. Even at low energies, the excitation

nucleon emission

energy, the linear and angular momenta of the interacting

At very low bombarding accepted

energies

that the nucleon emission

fragments energies

is then an open channel for the dissipation

or from the compound (typically

tablished

evidences

occurs statistically

system (see refs. 1-4).

for non-statistical character,

of the processes

neutron5-8

from either the emerging At higher bombarding there are well es-

and proton'-I1

these emissions

system.

it is generally

higher than 8 MeV per incident nucleon),

to their non-equilibrated signatures

(a few MeV per nucleon),

of

emissions.

Due

have been thought to bear

from which they originate and, by the way, to pro-

vide probes of the reaction dynamics. This work will follow such a philosophy the bombarding

energies of which do not exceed 100 MeV per nucleon. This energy

range is presently data12

; it will be specialized in reactions

under focus owing to the recent availability

of experimental

; it is interesting too from a theoretical point of view, since one ex-

pects to observe governed

the transition

between a regime where the reactions

are mainly

by the nuclear mean field and a regime where, the binding energies of

the nucleons

becoming progressively

the individual

negligible,

collisions.

0375-9474/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

the reactions

are dominated

by

102c

B. Remaud et al. /Fast nucleon emission

This paper is organized approaches

currently

as follows.

developed

In section 2, we sketch the various

in the literature and we precise how our work

places itself among this wide diversity have not yet discriminated. Boltzmann

equation

to the nuclear reaction

how this general equation emission. devoted

Specific

of theories that the experimental

In section 3 we develop the application

context, and in section 4 we discuss

can be used (and simplified)

calculations

to our (provisional)

are presented

APPROACHES

When an ion impinges on another onethere

the time-dependent

are three basic mechanisms

of single nucleons

mean field

and iii) de-excitation

to treat the nucleon

in section 5 and section 6 is

conclusions.

2. SKETCH OF THE VARIOUS THEORETICAL

can lead to the emission

ii) individual

all the currently

which

: i) interaction of a nucleon with collisions

between two nucleons

of heated pieces of nuclear matter. Although

imagine other processes,

data

of the

developed

one can

theories are connected

with one (or more) of the above mentioned mechanisms. The time-dependent

Hartree-Fock

(TDHF) theory is a fundamental

which allows to treat the self-consistent during a heavy-ion

collisionI

adjustement

; calculations14s15,

approach

of the nuclear mean field

at bombarding

energies

higher than 10 MeV per nucleon, clearly show the emission of fast nucleons as localized

"jets" of low density nuclear matter which occur on a time scale

approximately

equal to the characteristic

target without atic

interacting.

However

time for the projectile

to pass the

the TDHF method does not provideanysystem-

way to compute angular distributions.

The Fermi jet16 and PEP17 models,

although much more phenomenological,

ranked in the same family of approaches, sion to the coupling of the internal motion of the heavy ions nucleon wave functions superimposition

since they attribute

can be

the nucleon emis-

(Fermi) motion of nucleons to the relative

; this amounts to take rough approximations "

for the

(plane waves) and for the nuclear mean field (mere

of the equilibrium potential wells). However, TDHF and PEP model 18 nucleon emission characteristics which present remarkable

calculations

display

similarities

; this demonstrates the underlying relationship between the two

models.

It is then not possible

sions without questioning ion reactions.

to exclude such mechanisms

the bulk of our knowledge about the low energy heavy-

The PEP model calculations

for the energy spectra but generally tions7'17

usually yield

fail in reproducing

(see refs. 18, 19 for discussions

The mechanism

for the prompt emis-

reasonable

aboutthesediscrepancies).

ii) above is taken into account in knock-out

for both low - and high - energy collisions.

agreement

the angular distribu-

calculations

The low energy model"

11,20

is usually

103c

B. Remaud et al. / Fast nucleon emission

specialized undergoes

in the peripheral

a quasi-free

of the knock-out

collisions

scattering

contributions

to the spectra provides

nent which lacks in the evaporation factors

being free parameters,

out processes

where a nucleon of the projectile

with a nucleon from the target. Calculations

code outputs

it is difficult

(see ref. 22 for a discussion

the high velocity

21

compo-

; however, the normalization

to weight the role of the knock-

at high energy).

Since the nuclear fragments laxation of their excitation

of a nuclear reaction are observed after the re23,24 have been developed energy, various methods

to study the decay chains of excited nuclear matter with the purpose to proceed up to the primary fragment

It is precisely

characteristics.

dures that has been proven the need for other mechanisms tation of thermally explain

equilibrated

fragments

or compound

systems, in order to

the single nucleon yields.

Higher temperatures ly successful

mean

in fitting

that the considerable the interacting

higher escape velocities,

the experimental

excitation

energy remains localized in a small region of 25,26 suggests that the friction

kinetic energy

the two nuclei. Very high temperatures carried out using an equilibrium In the lab reference

into heat in the overlap region of

are obtained and the calculations

statistical

with this intermediate

of which are intermediate

and that of the compound

sources have been very efficient

system. Calculations in reproducing

results. The moving source model seems to extrapolate

energies the spectator-participant collisions,

of the compound

data at around 20 MeV per nucleon 27 have

shown the need for thermal sources the velocities between that of the projectile

the participant

are

treatment of the heated region.

system, the hot spot has the velocity

system. The fits to experimental

experimental

models - which are usual-

results - have been derived assuming

nuclei. The hot spot model

forces convert the relative

nuclei

using these proce-

than the only de-exci-

models

nucleons

the

at lower

; in such pictures, during peripheral

belong to the overlapping

regions of the

; they are pulled out of their mother nuclei then constituting a piece

of very hot nuclear matter which travels with a velocity

roughly equal to the

mean of the two nucleus velocities. The models

based on the hot spot or hot sources suppose that emissions

place on a very restricted

time scale, since Tocal equilibration

ved in order to use the statistical

hypotheses

for the emission.

take

must be achieOn the other

hand, this time scale must not be too long since the hot nuclear matter has to remain confined

; this latter time scale is difficultly reconcilable with a long nucleon mean free path ( > 2.5 fermis in our energy range). Another family of approaches

cascade calculations

consists

in precompound

which result from the extension

excitation

28,30 ,,31,32

to the heavy-ion

reactions

104c

B. Remaud

of studies of nucleon-nucleus cally are Monte-Carlo

et al. I Fest nucleon emission

reactions. The nuclear intracascade

simulations

models basi-

which follow the nucleon trajectories

nucleon inside the target, individually

treating each nucleon-nucleon

of each

collision;

by definition,

they are not well adapted to treat the collective effects. Even 33 , it seems that their use at energies lower 30 than 250 MeV per nucleon is questionable. In the prec~pound calculations , in the case of light projectiles

the philosophy

is comparable

but a great simdlification

results when the nuclear,

cascade is followed in the momentum

space instead of the laboratory

system. The approach to equilibrium

is governed

cribes the time evolution ties. Recently,

clude the angle-energy tributions

of the particle-hole

the preequilibrium

orthogonal

viewpoints,

the above theore-

- but often partial - successes

in fitting

the

data, as if each model told its part of a more complex story. This

situation may reveal the signature and two - body processes the b~barding

energies

gies are available

two-body processes

of the often-claimed

transition

between one -

that one should observe in the heavy ion reactions at that we consider, To tackle this problem,

- which explains

putting more one-body effects

extending

(exciton) probabili34 to in-

and then to get access to the angular dis35 reactions .

they start fran apparently

tical approaches get reasonable experimental

excitation

exciton model has been generalized

correlations

in the nucleon-nucleus

Although

coordinate

by a master equation which des-

the varieties

at higher energies

of models -, one consists in

in theories which have been deviced

in the high energy

two strate-

limit. The other strategy

to treat the

consists in

the mean field theories which have been success-

fully worked out in heavy-ion

reactions

at less than 10 MeV-per-nucleon

bombar-

ding energies. This latter strategy

does not seem presently

but it will allow to describe low energy, collective and deep inelastic

the vanishing,

phenomena

reactions.

so well developed

as the former,

with increasing energies,

such as fusion,

of the

fast fission, quasi fusion

In the following, we shall discuss a theoretical

approach based on our present knowledge of low-energy reactions ; this line is 36 ; at 80 MeV per nucleon, events are still observed supported by recent results with large linear manentum fragments.

This indicates

transfers and high temperature

emitters

that, even at these high energies,

field is strong enough to sanetimes melt the two colliding

of light

the nuclear mean

ions in a kind of

compound system. 3. THE BOLTZMANN Weaddress

EQUATION

IN HEAVY-ION

REACTIONS

ourselves with the problem of finding

cleus phase space distribution

the time evolution

during a violent collision

of a dinu-

; this phase space

105c

B. Remaud et al. /Fast nucleon emission distribution

evolving

in time under the influence

of i) the interaction

particles with the nuclear mean field and ii) the direct interaction particles

by individual

collisions.

ble in its full generality

during the last years discuss

This general problem of physics is unsolva-

and particularly

We can step forwardsanapproximate

of the

of the

in the context of nuclear physics.

solution, using the knowledge

accunulated

; this implies a hierarchy of approximations that we shall

:

3.1. &uantal-versus-semklassicaZ

mean field

TDHF theory is a well developed method cribes in a quanta1

framework

of nucleons scattering a oarticular

the reorganization

37,38

; it des-

of the N-body wave functions

on the walls of their mean field. The starting point is

exoression

of the Liouville-Von

density operator defined

Neumann eouation L

J

&N) isthe .

approaches.

in heavy ion collisions

: (3.1)

J

in terms of Nsingle

particle wave functions

P(N) (F',F",t) = "z ~i*(f,t)~i(~',t)' i

(3.2)

is the Hartree Fock Hamiltonian in which the one-body potential U(F',t) de'HF In the right-hand side of eq. (3.1) pends self-consistently on the density A. V is the residual dual collisions.

two-body

interaction

which includes the effects of the indivi-

In the low energy reactions, this term is neglected since most

collisions

are inhibited

babilities

induced by the Pauli exclusion ih ap at

by the restrictions

R2 + 5

They are now standard

2 (v; -v&b

procedures

on the phase space occupation

principle. The TDHF equation

(N) -(u(F,t)

13

-u(P,t))&N)=

to derive semi-classical

of the single-particle

density matrix

f'(?,F;) = /& The same transformation equation

equations

equi-

transform

:

exp(iS;ffi)P(N)(? + $ ,? - 5).

(3.4)

made on eq. (3.3) gives the semi-classical

TDHF

:

afW + + fi ?+fW - nio(g)2n E

-&

($2n+1

U(?,t)(;@+Ifw

=o.

(3.5)

Starting from eqs (3.4) and (3.5), we can derive a sequence of approximations to get formally

simpler equations which retain as much as possible the physical

characteristics

of the fully quanta1 one. The most celebrated

truncating

the expansion 13 Vlasov equation

at the order two in powers of%.

:

(3.3)

0.

valent to the TDHF one. This can be achieved by taking the Weyl-Wigner

pro-

is then

one consists

The result is the

in

B. Remaud et al. / Fast nucleon emission

106~

afw 72

+

i;

{

-W

.v;f

- ($.($)

which is formally similar to the Liouville

= 0,

(3.6)

equation

$;" + {H,fWl = 0, well-known

in classical mechanics.

conservation

The Liouville

of the phase space densities,

the Vlasov equation

is contained

equation mainly guarantees

such as the only quanta1

in its initial conditions.

the

character of

If the initial value

of fw at t = o does not violate the quantum requirements,

it will not violate

them at any time. The relationship

to the Liouville

implies the remarkable solution

properties

through the constants

of the Vlasov equation

of motion associated with H. It is easy to see

that if I(?,$,t) is a constant of motion, the Vlasov equation. motionandthen

In the static

any function

will be a solution of eq. 3.6

+ D(F'))

(3.8)

; this contains the following cases : (3.9a)

fW(F',i;,t)= (1 + exp( !++,-I,

(3.9b)

the Thomas-Fermi

distribution

and the Fermi-Dirac

; E being a Fermi energy,6 a temperature-like of the Vlasov equation

automatically

parameter.

for the quanta1 problems

the validity of the local harmonic nic potential

of I is a solution of

= 0), H is itself a constant of

fw(f,S;,t) = O(E - H),

in which one recognizes tribution

any functional

case ($

:

f'(H) z fw(g

of validity

one

that one can derive a very general class of

approximation

extinguishes

in eq.

dis-

The limit

is connected with

to U(f), since a purely harmo-

3.5 the powers of ii larger than

two. The close-to-the-classical the phenomenological

limit character

illustration

ding the phase space distribution

of the Vlasov equation makes easy

of its solutions. This is achieved in elementary

cells w'(;,$

in divi-

; r;,p;) centered

; as a consequence of the Liouville equation (3.7), their mean positionsaresolutions of the Newton equations :

on (rT,pT)

drt -& = pT/m

d P;

,

Such cells (called quasiparticles5*) cal particles

r

must not be confused with true classi-

since they are only elementary

they can be deformed evolves according

(3.10)

= - VFi V($).

probability

densities

and since

in the phase space in the same time as their mean position

to eqs. 3.10.

107c

3. Remaud et al. f Fast m&eon emission

The same kind of approximations

as for eq. 3.5. can be studied on the Wigner itself3' ; we write the density operator as

transform of the density operator

:

(3.11) where the inverse Laplace transfo~

is computed with B = it/# and c(B) is the

single particle propagator. One can then perform a Wigner-Kirkwood

a semi classical Vlasov equation Numerical

distribution

of c in powers of II and 3.11 and 3.4 yield

with the same order of approximations

(see ref.4;94ior

calculations

expansion

; the successive transformations

troncate it at the order Bz

-

as the

more details).

have shown that the solutions

of the Vlasov

equation are comparable in many respects with the TDHF results. It has been 41 argued that the Vlasov equation presents no advantage since it is not simpler to solve numerically.

As far as we

using rough, time-independent obtained

know,numerical

endeavours

have been done

bases ; we think that large improvements

by looking for time-dependent

could be

bases in better link with the symmetry

of the equation.

3.2. The BoZtzmann equation Many efforts

are currently

TDHF theories43-45,

developed

to include the two-body collisions

The more straightfo~ardwayis

tion,a master equation which governs the occupation particle

to couple,to

in

the TDHF equa-

probabilities of the single is43344 :

states. The general form of this master equation

&

"i =ji,

Vijkl Gijkl{iiiFjnknl - i$,iilninj1,

Which includes the matrix elements of the two-body which takes care of the energy conservation. single-particle

(3.12)

interactions

Itisonlyat

Wanda

term G

the limit of vanishing

widths that Gijkl tends towards a mere s-function.

Progresses

have been made towards the solution way, towards the understanding

of eq. (3.12) in sample cases and, by the 45-48,66 of the role of two-body dissipation

However as far as we know, the present calculations tory, either because they crudely describe they would imply a costly expansion

are not completely satisfac49 or because

the two-body collisions

of the bases currently used in TDHF calcula-

tions5'. The inclusion

of collision

rallel way, recalling mations

on the discrete

grals over the momenta. equivalent

terms in the Vlasov equation

that it corresponds single particle

form of the density matrix

:

can be done in a pa-

limit

; then, the sum-

levels are replaced by continuous

In this classical

to the occupation

to a classical

probabilities

limit, the probabilistic are deduced

inte-

functions

from the Wigner trans-

108c

B. Remaud et al. /Fast nucleon emission

= n3/gf"(f,i;,t),

f(;,d,t) where g is the spin-isospin

degeneracy

unity when the Wigner distribution The collision

(1)coll

(3.13)

of the nuclear matter

function

term is then obtained

; f(?,d,t) is

takes its maximum value.

in its semi-classical

limit51y48 --

=g N-ISds'pd; di;4[ v(p*-p*) 31 *+I+E~-E~-E~) 2 3 1

1

: (3.14)

,

x a(PT+P;-P;-P;;) with fi 4 f(?,p;,t) This term describes two nucleons

the changes

;Ti =l -fi ;ci =p:/Zm

in the phase space distribution

interact through the residual

in eq. 3.14 through its Fourier Pauli blocking favors collisions one can evaluate

; N = 64n5h7

interaction

(3.15)

functions

when

; this last one appears

transform v(s) in momentum

space. Since the

between nucleons with high relative momentum,

this Fourier transform with the Born approximation

such as

(3.16)

{V(p)12 = (16m2h4/m2)do/dn In this approximation, cross section expressed

the main input is the free nucleon-nucleon in the center-of-mass

:

reference

differential

system of the colliding

nucleons. The Boltzmann

equation

is found in matching

body (eq. 3.14) contributions bution

(eq. 3.13)

(eq. 3.6) and two-

of the phase space distri-

: af+ at

r; G m.

= (1),,,,

- ,~,.,~,

As shown in ref. 48, the Fermi-Dirac

distribution

already a solution of the Vlasov equation, the collision

the one-body

to the time-evolution

(3.17)

of eq. 3.9b, which is

is also the equilibrium

solution of

term.

In its general form (3.17), the Boltzmann equation is tremendously difficult to solve directly, six-dimension

since it is a highly non-linear

phase space.

version to be used for the transport 4. A LINEARIZED

BOLTZMANN

in a

EQUATION

A standard issue of statistical

mechanics

52

equation,

when the total distribution

by assuming

lies in the search for linearized

to 53 one. We shall derive another class of linear kinetic equations

versions of the Boltzmann

equilibrated

equation

of fast nucleons through nuclei.

4.1. Forma2 bases

the equilibrium

differential

In the next section, we shall derive a linearized

that the solution

is close

function can be formally split into two parts : an

part fe and a low-density,

unequilibrated

one ft

:

B.begun

et al. / Fast nucleon

emission

109C

f(F,$) = fe(f,$) t ft(F',$). By low density,

we mean that the occupation

3.13, are small

: ft(F$)

f?F;)

in eq.

(4.2)

consequences

term vanishes for fe(‘t;): there are no collision distribution

as defined

<< I for all ifls.

From eqs. 4.1 and 4.2, we can draw the i~ediate

the equilibrium

(4.1)

probabilities,

: i) the collision

between nucleons belonging

to

; ii) the collision term is vanishingly small for

: the collisions are scarce between nucleons belonging to the far-from-

equilibrium

part of the distributions

to the states occupated

; iii) the Pauli blocking is mainly due

in the saturated

nuclear matter. Formal algebraic mani-

pulations 53 then lead to a simple form for the collision

term (3.14)

:

(1) co71 = ..*~Z(l-f~)f~f~ -(1-f~)(l-f~)f~f~~.

(4.3)

The factor 2 in the gain term arises from the relabelling

symmetry of the

indexes 3 and 4. We can make use of the linearity of (I)c,l, in terms of f, the Boltzmann

equation appears now as a set of two coupled differential

tions. The main physical

assumption,

equa-

which will further simplify the problem,

will consist in taking fe as in eqs 3.9 and in solving independently

eq. 4.4

:

t

F % t m

.9;ft

- $U."i; ft = (I)coll.

Our final version of the linear Boltzmann

(4.4)

equation then reads

:

aft

2 +ff$

= &

Jd;4t2WG(p;,p;)f;

where WG and WL respectively

- WL(p;,p;)f;(I

- f;)l

are the gain and loss transition

;

(4.5)

rates :

G-t-, W (PIYP4) = W~(P~,P~),

(4.6)

G + -+ -+ z do 6(~1t5;2-if3-i;4)6(EI+"2-E3-E4)(i-f~)f~. W (PI,P~) = /dT2dp3 The loss term can be worked out independently

since it is independent

can be cast in terms of a local mean free path x($,)

(4.7)

on ft ;it

: (4.8)

The linearized

Boltzmann

nucleon distribution The underlying

equation

through a saturated

assumption

1 and 2 illustrate

the transport

nuclear matter

of a low-density

in thermal equilibrium.

is that one can neglect the excitation

clear matter by the particle-hole Figures

describes

pair creations

the equations

created

in nu-

induced by the collisions.

4.6 - 4.8 in the case of the cold infi-

IlCk?

3. Remaud et al. ,I Fast nucleon emission

nite nuclear matter loss transition

cal forms53. space

(fe is taken from eq. 3.9a). At this limit, the gain anti

rates and the local mean free path take fairly simple analyti-

In figure 1, we show the map of the transition rates for a phase

cell (labelled X) centered at p, = 1.5 in units of the Fermi momentum

The gain and loss zones are separated general s~etry momentum eq. 4.8

of the figure shows that the collisions

transfers+

pF.

by a circle with radius p/pf = 1.5. The favour the transverse

Figure 2 shows the local mean free path which follows from

; it is compared with the va'lues from ref. 56 ; both results are similar

since they are derived from comparable

approaches

somewhat higher since, to be consistent, nel in the nucleon-nucleon

; our high energy limit is

we have only included the elastic

chan-

cross section.

Collins

and Griffin

i 100

0 nucleon FIGURE 1 Map of the transition rates for a phase soace cell X. The contours are defined oh a linear scale with arbitrary units. The forbidden, toss and gain areas are respectively labelled F, L and G.

200

energY[ruleV]

FIGURE 2 Nucleon mean free path.

4.2. Past nucteon transport in infinite nuclear matter. To illustrate

the properties

case of the semi-infinite the transverse

of eq. 4.5, we have solved it in the idealized

nuclear matter. We neglect the space derivatives

directions.

to :

In a permanent

(P,/m)-&ft

where oz is the propagation

direction.

riant cross sections d30/d; = pft(c,z) ple situation. We take a gaussian with a full widthat

half maximum

regime, the Boltzmann

6.2) = (I)~~~,

equation

in

reduces (4.9)

In figure 3, the evolution of the invain the (px, p,)- plane is shown in a sam-

initial cross section centered at p, = 2.0pf equal to 0.4~~. Such conditions

are chosen in

order to mimic the flow through a thick target nucleus of a light projectile a head-on collision

at approximately

80 MeV per nucleon.

in

lllc

B. Remaud et al. /Fast nucleon emission

The figure 3.a shows the initial gaussian

in the py = 0 plane

circle marks the border of the Fermi sea where the occupation (not shown in the figure)

; the half

probability

is e(pF - p). Figure 3.b shows the situation after

that 8 fermis of nuclear matter have been flown through. The individual sions have creamed off the initial distribution distributed

almost isotropically

more effective

is sensible

in the energy range, a slowing

; it is due to the fact that the loss term is

on the faster particles

than on the slower ones.

the width is affected. To compare with any experimental have to introduce

the refraction

velocity

space distribution,

(4.5) transforms

around a velocity

some changes

of the initial conditions

one should

edges. Nevertheless, an initial

larger than the Fermi

into a sum of two isotropic velocity distributions.

keeps the memory

frame

isotropic

In the same way,

results,

on the nuclear potential

fig. 3 clearly shows that the transport equation

velocity,

colli-

; the scattered nucleons are

around the Fermi sphere. Since the mean free

path of eq. 4.8 is far from being a constant down of the centroid

fe

One distribution

with a shift of its centroid and

in its width. The other one is centered around zero in the rest

(i.e. the target reference

barely applicable an approximate

to realistic

frame). Direct calculations

are lengthy and

situations.

In the next section, we summarize 54 treatment which has been used for the fast nucleon ejection

(a)

(b)

FIGURE 3 Evolution of pft(p,z) in the(p,.p,) plane. The vertical axes are in logarithmic scale with arbitrary units. Fig 3a (right) : initial conditions. Fig 3b (left) situation at z = 8 fermis. 5. PRODUCTION

AND EJECTION OF FAST NUCLEONS

When two ions interact, the nucleons

:

the potential walls in the contact region co1 apse

can then be transferred

larger than the barrier height (classical escape) through the barrier

;

either because their kinetic energy is or because they tunnel

(quanta1 escape). This nucleon flux has been widely

nvesti-

gated

(see refs. 57-59 and ref. 60 for temperature effects). In the low bombar-

ding energy regime,

its characteristics

and drift, the one-body friction tial.

consistently

explain the mass diffusion

and the absorptive

part of the ion-ion poten-

If one keeps the underlying assumptions of these works (mainly, that the

transferred

nucleons

keep memory of their initial momentum),

part of the nucleon flux is unbound when the relative than a threshold

value

61

the most energetic

kinetic energy is higher

. We have restricted our calculations to this fast nu-

cleon flux which exists on a short time scale (see ref. 61 and lower} at the beginning

of the reaction

; this eases the treatment of the nuclear mean field

that we take in its sudden approximation linearized Boltzmann

equation.

However,

and allows to use consistently the one- and

play an even role all along the nucleon emission emission mechanism

and we have decomposed

inside the projectile particle exchanges

is negligible,

term for the nucleon flows

when compared with the contribution

between the two Fermi seas (1 particle

equation

reduces to a Vlasov equation

sketched by any phenomenological

model describing

forces of anion-ion

phase of nucleons

the dynamical

from one partner

processes

lisions in a quasi-homogeneous

medium described

iii) An ejection

less energetic

through the nuclear matter

by the Boltzmann

of the nucleons

of their trajectories

tribute to the one-body dissipation.

col-

integral redu-

are ejected to the laboratory

at the nuclear potential

This refraction

region of the nuclei where the collisions

We illustrate

the conser-

are given by the nucleon-nucleon

ones are trapped in the nuclear potential

5.1. Production

Of nU-

term.

phase, whereparts

after the refraction

can be

evolution

collision.

of the other one. The dominant

ced to its collision

of

-1 hole excitations).

whose solutions

cleans in the mean field of a nucleus, when this nucleus undergoes vative and frictional ii) A transport

the

into three steps (see refs 54 and 62).

i) an approach phase where the two-body collision

The Boltzmann

the

two- body processes do not

edges. The

well, where they con-

occurs in the low-density

are scarce.

phase. in figs 4, the characteristics

flux which crosses the potential

barrier at the

of the one-way fast nucleon border

of the two nuclei

;

only the nucleons which are unbound in the target referential frame are consi54 have been compared. In model A , the V?aSOV equation iS

dered. Two approaches

solved for the projectile field, each quasi-particle

quasi-particles

in the time-dependent

nuclear mean

which reaches the target nuclear medium contributes

to the flux. Model B63 relies on the window concept, the transferred flux is provided

by the shifted equilibrium

function modulated

nucleon

donor phase space distribution

by the influence of the barrier transparency.

Model 9 repre-

B. Remaud et al. / Fast nucleon emission

sents

approximation

a static

to the time-dependent

113c

treatment of model A

they yield very similar results. Fig. 4a shows that the production a property of the central collisions. velocity,

defined

as the difference

frame and the intrinsic velocity models

: but

is mainly

In fig. 4b, vu is the nucleon driving

between its total velocity

in the target

that it had initially in the projectile.

Both

predict a very short time scale (2-3~10-~~ set) for the process, which

subsequently

justifies

the use of the sudden approximation

to the mean potential.

‘2C(1032MaV)+‘*C F

Ca.u.1 (a)

” 2 -

mode,

A

-

mode,

A

.-

model

6

-

model

B

Apro,

1

20

40

t

60

0.6 V”(frn/

1.0 ld”JS~

FIGURES 4 Initial distributions for the nucleons at the entrance point of the target nucleus : 4a) cross section as a function of the initial orbital angular momentum. 4b) distribution of the displacement velocity vu (see text).

5.2. Inertia2 emission. For the more peripheral fast particle

emissions

collisions,

model A predicts an important source of

which escape to the laboratory without crossing the

target nucleus. These nucleons

travel along potential

valleys grazing the part-

ner nucleus and are directly ejected under the conjugate of the effective

potential

action of an up-stroke

and of the drift velocity given to them by their

mother nucleus. This kind of emission process was already predicted for the particles

by the cluster jet mode164

vents the passing of clusters,

u

, where the effective target potential pre-

belonging

to the projectile,

through the target

nuclear matter. As discussed

in ref. 54, such a mechanism

number of involved partial waves). terms, the thermal-like

Nucleon

tmnsport

In the transport

(large

behaviour of the spectra is observed resulting from a

memory of the initial momentum

5.3.

has large cross sections

In spite of the absence of two-body collision

distribution

width.

and emission in centml

coZZisions.

phase, the flux of nucleons which crosses the window between

the two ions serves as an input for the Boltzmann

equation.

In ref. 54, we give

114c

B. Remaud et al. /Fast

an approximate tial gradient correction

method

nucleon emission

to solve it in the bulk of the nucleus where the poten-

is small. The refraction

on the potential wall and the Coulomb

for the protons are also taken into account in the ejection phase.

In figure 5, we show the contour diagram of the velocity distribution cross section) of the emitted protons. cal about the mean drift velocity. V ,,

=

It presents

a dip near this value

0.6 fm/10-23 s and vA= 0 and a maximum fo‘r v,, = 1.2 fm/lO -23 s, i.e. around

the beam velocity

in the forward direction.

target travelling

through the nuclear matter of the projectile

V ,,

=

(invariant

It turns out that the plot is symmetri-

-

The contribution

of nucleons of the is centered at

0.2 fm/lO -23s and vL= 0. These shapes look like diagrams for a thermal

emission from a source with the velocity of the symmetrical half of the beam velocity). extreme assumption

Nevertheless

point (around the

we have to keep in mind that the

of cold nuclei was done for the.evaluation

of the collision

The effect of the two-body collisions

is mainly to

reduce the absolute cross sections for the highenergy tails of the spectra,

-

100

_.-._

modifying

60

the slopes and

--_

22

then increasing

-

10

temperature

the effective

parameter, which

turns out to be fairly comparable with the experimental ones (see ref. 54).

FIGURE 5 Invariant cross section of the emitted protons in the laboratory frame for the I2C(IO32 MeV)+12C

reaction.

6. CONCLUSIONS The Boltzmann

equation,

two-body collisions,

tion of the intermediate of proton emission,

including

as well the mean field properties

provides us with a general framework

as the

for a proper descrip-

energy heavy ion reactions. When applied to the problem

it is able to reproduce

ton cross sections. The thermal-like either by the two-body collisions

the main features

behaviour

of inclusive

pro-

of the energy spectra is obtained

or even by the dynamical

effects in the

B. Remaud et al. /Fast nucleon emission

115c

inertial emission. In some extents, have discussed mate methods

this approach

could reconcile

the various models

that we

in sect. 2 ; since the cascade models can be viewed as approxi-

to solve the Boltzmann

mean field self-consistency.

equation

The Boltzmann

in the limit when one neglects

equation provides

the

a better framework

to treat the Fermi motion and to test the sensitivity of nuclear observables to 65 . As developed in sect. 3-5, our model incorpothe nuclear equation of state rates most mean field effects in which we see the origin of the thermal-like behaviour

of the high energy spectrum tails

extension

of the PEP models.

It conflicts

nisms with the help of hot sources, on a relatively

; it appears then as a natural

the analysis of the reaction mecha-

providing an alternate description

long nucleon mean free path.

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