Cascade-exciton model of nuclear reactions

Nuclear 0

Physics

North-Holland

A401 (1983) 329-361 Publishing

Company

CASCADE-EXCITON K. K. GUDIMA+,

MODEL

OF NUCLEAR

S. G. MASHNIK+

REACTIONS

and V. D. TONEEV

Joini Institute for Nuclear Research, Laboratory of Theoretical Physics, Dubna, USSR Received

17 September

1981

Abstract: An approach combining essential features of the exciton and intranuclear cascade models is developed. The cascade-exciton model predictions for the energy spectra, angular distributions and double differential cross sections of nucleons and complex particles as well as for the excitation functions are analyzed at incident.nucleon energies To 5 100 MeV and in a large range of nuclear target masses. We discuss the relative role of different interaction mechanisms and their possible experimental identification.

1. Introduction Nucleon-nucleus reactions in the medium-energy range T, 5 100 MeV are still attracting much attention because of the opportunity to investigate the preequilibrium particle emission. The mechanism of particle emission during the attainment of statistical equilibrium in an excited nuclear system is somewhat intermediate between direct reactions and decays through the states of a compound nucleus, and is not reduced to their simple combination. The development of the pre-equilibrium concept of the nuclear reactions has allowed one to understand the importance of this mechanism and its relation to the intermediate nuclear structure, and to explain a number of interesting physical effects [see review articles l* ‘) and references therein]. Among the available pre-equilibrium emission models similar in their physical assumptions, preference is given to those which, being internally self-consistent, describe the largest set of experimental data. The majority of the exciton models claim only to describe the shape of angle-integrated energy spectra of secondaries, mainly of nucleons. Some models are used to investigate the excitation functions and more rarely the angular distributions of particles. The latter stems from the difficulties of the statistical formulation of pre-equilibrium decay models assuming quasi-equilibrium and from the necessity of taking into account direct mechanisms of particle production. On the other hand, at higher energies many features of nuclear reactions are fairly well reproduced within the intranuclear cascade model 3). However, a direct + Institute

of Applied

Physics

of the Moldavian

Academy 329

of Sciences,

Kishinev,

USSR.

330

K. K. Gudima et al. / Cascade-exciron

model

extension of the cascade-evaporation model to the energy region To 5 100 MeV reveals essential quantitative discrepancies with experiment in the doubledifferential cross sections of secondary nucleons 3-6). In this paper we suggest a model marrying specific features of the cascade model with the exciton version of the pre-equilibrium decay model. We shall start with considering the physical grounds of these models and ascertaining interrelations between these approaches ; then we shall give an account of our cascade-exciton model of nuclear reactions. The subsequent sections are devoted to a detailed analysis of various characteristics of nucleon-nucleus interactions at energies To 5 100 MeV. Some conclusions are drawn in the closing section of this paper.

2. Pre-equilibrium

decay of nuclei and intranuclear cascade

To understand interrelations between models of pre-equilibrium nuclear decay and intranuclear cascade, we trace briefly the main physical assumptions resulting in different equations to describe relaxation phenomena. Let us specify an excited nuclear system by the hamiltonian I? = 8, + P, where Z?, is referred to undisturbed nuclear constituents. Let us choose the representation in which the undisturbed energy E is diagonal, E?,lEa) = E(Ea). Here, all indices of a nuclear state are included in CI except for energy. Assume that the disturbing residual interaction I/ is small enough to neglect changing of the system energy E. Then, starting with the dynamical Liouville equation and using statistical mechanics methods, one can show that the diagonal elements of the density matrix P(E,u,t), treated as a probability of finding a system at the time moment t in the Ea state, will satisfy the master equation 6, ‘) +

“(;;” t, = c

[A(Ecx, Ed)P(E, d, t) - i(Ed, Ea)P(E, ct, t)].

*‘+a

Here, the energy-conserving time-dependent perturbation

probability theory,

J.(Ecc, Ed) = ;

rate

is defined

in the first order

of the

I(Ec#IELT’)~%o,(E),

(2)

where the matrix element (EcrlPlEor’) is believed to be rather a smooth function in energy, and w,(E) is the density of final states of the system. One should note that

+ We do not discuss here the problem of initial and irreversibility in eq. (1); see the original papers [refs. ‘*“)I.

boundary

conditions

related

to arising

of

K. K. Gudima et al.

eq. (1) is derived

provided

that

I Cascade-exciton model

the “memory”

time r,,,,,

331

of the system

is small

compared to the characteristic time for intranuclear transitions - h/1(&, Ecc’), but, on the other hand, eq. (1) itself is applicable for the time moments t x- h/i(Ecc, Ed). Due to the condition r,,,,, > h/?.(Ecc, Ed), being described by eq. (1) the random process is the markovian one. In physics applications an inverse approach is frequently used: First the markovian property of a process is postulated and then eq. (1) which is a form of the well-known Smoluchowski equation is written down. The validity of these equations for describing such a finite system as an atomic nucleus has been proved in refs. 9, lo). The master equation (1) is the mathematical basis of a large class of preequilibrium decay models + known as exciton models. Within these models, an excited nuclear state is completely defined by an excitation energy and a number of excited particles p and holes h (n = p + h is a number of excitons), that is CIE n. A further simplification is achieved by assuming that the sum ca, + o1in the right-hand side of (1) is contributed only by the terms representing the exciton-exciton scattering, that gives rise to the selection rule An = 0, f2 under intranuclear transitions. With the hypothesis of a priori equal probability for all available nuclear states, the main problem is then reduced to the evaluation of the statistical factor o,(E) in eq. (2) the averaged matrix element M,,. = (E,ljPIEcc’) being estimated phenomenologically. Passing over in eq. (1) to the representation of occupation numbers CI+ vi, vz,. . ., vA in the case of a dilute system (or short-range potential for constituents), we get the Boltzmann equation in the Uehling-Uhlenbeck form when simultaneous collisions of more than two particles and correlations between particles are neglected (see appendix) : w,) ~

at

=

4;

-A

[Aij+kl(Vi)
kl-ij(Vk)
+

+

o(vk))(l

o)(l

+

+

o(vL))

@Cvj>)].

(3)

This form of the master equation for average occupation numbers (v& is a starting point for the pre-equilibrium decay model of Harp-Miller-Bern 13). In the classical limit (0 = 0) the single-particle states can be specified by a position vector Y and a linear momentum p, and one may introduce the singleparticle distribution function

(vk) = f(rky pk>t)drdP,

= hd”dP,.

+ Assuming a continuous spectrum of nuclear states a, we arrive at the diffusion equation by expanding the right-hand side of eq. (1) up to second order in (M-U’). This equation is widely applied for describing mass, charge and angular momentum transfer in deep inelastic collisions of heavy ions l’,‘*).

332

K. K. Gudima et al. / Cascade-exciton

Taking system be space-homogeneous. laws, we rewrite (3) for the two-particle

Let the conservation

afk -_= at

model

explicitly scattering

into account as follows:

the

sfs

dPidP,dPl’ij-kl(fifj-f,f,)b(pi+Pj-Pk-P,)6(T,+Tj_T,-IT;)

cross section da/da is where v,,, = Ipi -pjl/m, q = p,‘/2m and the differential defined in the first Born approximation. In the derivation of the last equality the distance between subsequent collisions was assumed to be large enough, so the plane waves can be used as the wave functions IEa) in evaluating the matrix elements Aij_kl (see (2)). If the system considered is not space-homogeneous, some additional flows are arised due to the density gradients and external forces. In the classical approach they are taken into consideration by substituting the “substantive” derivative with the local one d/at -+ a/& + (p/m). V, + F. V,, where F is an external force. Therefore, we have arrived at the Boltzmann equation +

(4)

The intranuclear cascade model is based on this equation but it is preliminarily linearized in the following manner. The fast (cascade) particles and the targetnucleus nucleons which have not yet been involved in the interaction are considered as two different types of particles, and the collisions between particles of the same type are neglected. The nuclear consituents are believed to be described by the equilibrium (maxwellian) distribution function fT(r,p). Then, for the distribution function of cascade particles, fcas(r,p, t) we have from eq. (4) the following equation :

g+5. According

V,+ F. V, fYr,p, >

to the normalization

t) = pT(r)<~vrellfFaS(r, p,t) + Q(r, p, t).

of the single-particle

s

#(r) = dpfT(rTp)

distribution

(5)

function,

(6)

+ For the case of nuclear reactions, the Boltzmann-like equation is derived more strictly in ref. 14) in the time-dependent Hartree-Fock approximation. The validity of eq. (4) for the intranuclear cascading is also discussed by Bunakov et al. lo, Is).

K. K. Gudima et al. / Cascade-exe&m

is the local distribution

particle function

number

density.

of the target

Averaging

nucleus

model

333

in (5) is carried

out

over

the

nucleons

(7) and the cross section ~(v,,i) allows for the exclusion in the right-hand side of (5) is

Q(Y, p, t) =

dp,dnv,,,

‘~

f’(,,

principle.

The source function

pi)fcas(y, pj, t).

C-3)

The integro-differential equation (5) can be transformed into the integral one. In particular, if the fast particle flux collides with the semi-infinite slab of nuclear matter, we have for the cascade particles (neglecting recoil nucleons and assuming for simplicity pT = const and F = 0)

where pcas(y, t) is related to fcas(r,p, t) by an equation of the type (6). The probabilistic interpretation of this equation is quite obvious: A number of particles at the given point Y having momentum within the interval dp around the value p is built of the incident beam N, reaching this point (with an exponential damped factor) and rescatterings of every kind resulting in particles of interest. Relation (9) and its probabilistic interpretation are grounds for the cascade model proceeding from an analogy between the interaction of fast particles with nuclei and highenergy radiation transport through matter 3*16). In a real case one needs to solve the related system of integral equations like (9) with a distributed source function and complex initial and boundary conditions. It turns out to be more effective to use the analogy mentioned above and on this basis to simulate the fate of every particle inside a nucleus by the Monte Carlo technique. Therefore, both exciton and cascade models describing non-equilibrium processes have common assumptions. The main assumptions are “weak coupling limits” (that is smallness of perturbation V) and account for the two-particle interactions only. But in some points these models supplement each other. The cascade model takes explicitly into account the reaction geometry keeping all the

334

K. K. Gudima et al. 1 Cascade-exciton model

information on kinematic characteristics of fast particles, though neglecting residual interactions between cascade particles, the behaviour of the cascade particles being described by the equation of the state for the ideal gas. On the other hand, the exciton model considers an excited nucleus as a gas of quasiparticles (that is “particle-hole” degree of freedom is included) taking into account residual hh, ph and pp interactions. But the collision kinematics and nuclear geometry (in particular, peculiarity of peripheral collisions) are lost, making it difficult to consider the angular distributions of emitted particles. It should be noted that the applicabi!ity regions of these models are somewhat different. The cascade model conditions are fulfilled better at higher energies when the kinetic energy of particles exceeds essentially the nucleon binding energy that causes an additional smallness of disturbance. In fact, it is just a condition for using the impulse approximation and for treating classically the fast particle motion inside a nucleus. It is of great interest to combine these two models for explaining an anisotropy of angular distributions of emitted particles in a large region of incident energy and for improving an overall description of nuclear reaction features as compared to the conventional cascade-evaporation model. As an attempt to join the advantages of the cascade and exciton models, one may consider the hybrid model by Blann 17) and its extension to the description of angular distributions 18) though not all the points of this unification are indisputable 19). We shall follow here the scheme proposed in our earlier papers ‘O).

3. Formulation

of the cascade-exciton

model

The physical picture underlying our model is rather natural. A particle entering a nucleus can suffer one or several intranuclear collisions, that gives rise to the formation of an excited many-quasiparticle state like a “doorway state”. Due to residual interaction this state will evolve towards a more complicated one up to the formation of a compound nucleus. At each stage of this process a particle can be emitted. The behaviour of a primary particle and of those of the second and subsequent generations (if any) up to their capture or emergence from a nucleus is treated in the framework of the intranuclear cascade model. The number of captured nucleons and of “holes” produced due to the intranuclear collisions gives us the initial particle-hole configuration of the remaining excited nucleus, the excitation energy of which is defined by the conservation laws. A further destiny of the nucleus is traced in terms of the exciton model of pre-equilibrium decay which includes in a natural way the particle decay at the equilibrium stage too. Thus, the proposed cascade-exciton model (CEM) considers the nuclear reaction as proceeding through three stages - cascade, pre-equilibrium and equilibrium (or compound nucleus) - unlike the two-stage Serber mechanism r6). So, in a general case three components will contribute to each experimentally measured value. In

335

K. K. Gudima et al. 1 Cascade-exciton model

particular,

for inclusive

particle

o(p)dp

spectrum

to be widely discussed

later, we have

= gin[NCas(p) + Np’“(p) + Neq(p)]dp.

(10)

The cascade stage of interaction is described by the Dubna version of the of the system of intranuclear cascade model 3, 21 ). The Monte Carlo solution integral equations like (9) gives the single-particle distribution function fcas(r, p, t) through which all needed characteristics can be expressed. For example, for N”‘(p) from (10) we have

where the integration is carried out over all accessible impact parameters b for particles leaving a nucleus of radius R by the end of the cascade stage t,,,. It is noteworthy that, being defined by the size of the nucleus and its transparency (see the first term in eq. (9)), the reaction cross section gin is calculated within the cascade model itself. All the cascade calculations are carried out in the three-dimensional geometry. We take account of the diffusivity of the nuclear boundary and nuclear potential as well as exclusion principle effect on intranuclear collisions of nucleons. The subsequent interaction stages are considered in the framework of the modified exciton model proposed in our earlier paper 22). The model uses effectively the relationship of the master equation (1) with the markovian random processes. Indeed, an attainment of the statistical equilibrium described by eq. (1) is an example of the discontinuous markovtan process: The temporal variable changes continuously and at a random time moment the state of the system changes by a discontinuous jump, the behaviour of the system at the next moment being completely defined by its state at present. As long as the transition probabilities A(Ecc,Ecr’) are time independent, the waiting time for the system in the Ea state has the exponential distribution (the Poisson flow) with the average lifetime h/A(a, E) = h/~,,I(Ea, Ea’) [ref. ‘,)I. This fact prompts a simple method of solving the related system of eq. (1): Simulation of the random process by the Monte Carlo technique. In this treatment it is possible to generalize the exciton model to all nuclear transitions with An = 0, f2, and the multiple emission of particles and to depletion of nuclear states due to the particle emission. In this case the system (1) is as follows :

WE, n, t)

at

= -A(n,E)P(E,n,t)+l+(n-2,E)P(E,n-2,t) + 2,(n, E)P(E,

n, t) + a_ (n + 2, E)P(E,

+ c dT dE’lf(n, j s s

E, T)P(E’,

n + 2, t)

n + nj, t)d(E’ -E

- Bj - T).

(11)

336

K. K. Gudima et al. / Cascade-exciton model

with II = p+h

The lifetime of the excited system at the state different p-, h-composition) + is given by h

4,E)

h

-

NP,kE)

=

hC’+(P~h~E)+AO(P~h~E)+A-(p?hJ)+~rj(p~h~E)]-’~ j

where according to (2) the partial number by dn are

UP,

transition

h, E) = $

probabilities

changing

IMdn12W&, h, E),

and the emission rate of nucleon of the type j into according to detailed balance principle

rj(P, h, E) =

excitons

(but with

(12) the exciton

(13)

the continuum

is estimated

E-B~Ai(p,h, E, T)dT, s v;

Af(p, h, E, T) = T 2sj+ ’ P@j(P, h)

o(p-l,h,E-Bj-T) NP,

h, El

T"inv(T)3

(14)

where sj, B,, Vjc,pj are the spin, binding energy, Coulomb barrier and reduced mass of the emitted particle, respectively. A modification of (14) for the complex particle emission will be discussed in subsect. 5.2. The factor 9iTj(p,h) ensures the condition for the exciton chosen to be the nucleon of type j. This factor can easily be calculated by the Monte Carlo technique. Assuming an equidistant level scheme with the single-particle density g, we have the level density of the n-exciton state as 25, 26)

O(P,

This expression should one needs the number

h, E)

=

g@E)p+h-l p!h!(p+h-l)!

.

(15)

be substituted into eq. (14). For the transition rates (13), of states by taking into account the selection rules for

’ One should note that the particle-hole configuration is fixed by n and E with an accuracy of the position of the particle-hole pair with respect to the Fermi level. The summing over all possible positions of the pair gives rise to the mutual cancellation of the terms with I,(n,E) in eq. (11). In this case the transitions with An = 0 affect only the average lifetime ri/A(n,E) which has to do with the normalization of P(E, n, t). Such a cancellation does not occur when we add an index to specify the direction of the momentum brought into the system. This is necessary for the consideration of angular distributions of particles (see subsect. 5.2). In ref. 24) the third term in the right-hand side of eq. (11) is omitted by mistake, though the term La(@) in eq. (12) is kept.

337

K. K. Gudima et al. / Cascade-exciton model

intranuclear exciton-exciton scattering. The appropriate formulae have for the exclusion principle derived by Williams 27) and later corrected indistinguishability of identical excitons in ref. ‘“) : o+(p,h,E)

WJP,

[gE-.r$(p+l,h+1)]2

=$g

h, E) = &I

o-(P,kE)

gE-.LiQ+l,h+l)

n+l

C4E- .d(P>h)l n

[

been and

1’ n-l

.qE-.@f(p,h)

[P(P - 1) + 4Ph + W - 1 )I,

= &7Ph-2),

l(16)

where .d’(p, h) = a(p’ + h2 + p - h) - $I. By neglecting the difference of matrix elements with different An, M, = M _ = M, = M, we estimate the value of M for a given nuclear state by association of the A+ (p, h, E) transition with the probability for quasi-free scattering of a nucleon, which is above the Fermi level, on a nucleon of the target nucleus. Therefore, we have

where F,, is the interaction volume, and the averaging in the left-hand side of (17) is carried out over all excited states taking into account the exclusion principle. Combining (13), (15) and (17), we get finally for the transition rates :

l,(p

h

E)

=

(4%d%l)

n+1

gE - .d(P, h)

[ gE-&(p+l,h+l) A_,,.h.,,

_,,~;;...,,

,~L-j~yJ:,,~+l

“+l 1 p(p-

1)+4ph+h(h.9E--d(P,

p&l + l)(n - 2) [$7~-NPJ)12.

I”,

h)

1) ’

(18)

Thus, the initial conditions for the system of eqs. (11) t, = t,,,, n,, E,, are calculated within the cascade model. The Monte Carlo solution of (11) gives the population probabilities for the n-exciton states P(E,n, t). By the pre-equilibrium particles we call those which have been emitted before achieving of the statistical equilibrium appropriating to the time moment t,,. This moment is fixed by the condition A+(neo, E) = A_(nea, E) from which we get n,, 2: ,,/@. The preequilibrium component in (10) for the inclusive spectrum of particles of the type j can be represented as

JPeq(p)dp

=

l&g c Af(n, E, T)P(E, n, t) 8 s f,,, ”

F(Q)dTdGL ,

(19)

338

K. K. Gudima et al. I Cascade-exciton model

We shall return to the discussion of the angular anisotropy of the preequilibrium particle emission in subsect. 5.2. Here we note only that the angular function F(Q) is normalized to unity, IdQF(Q) = 1. Similarly component

to (19) we can write down :

Neq(p)dp =

the expression

for the equilibrium

d(P>Q)

o_ dt 1 nf(n, E, T)P(E, n, t) ~ F(Q)dTdQ, a(T, a) s L, n

(n 2 neq)

(20)

where the time moment t + a corresponds to the complete deexcitation of a nucleus due to the particle emission. As the nuclear states with different n are equally probable in the statistical equilibrium, the right-hand side of (20) is simplified to

Neq(p)dp = 1 If@, E, T) n

o(n - 1, E-5-T) ;W

G

E)

F(Q)dTdQ ’

CCO(~-~,E-B~-T) - T%,,,(T)\”

dTdQ.

(21)

; w(n’, E)

By summing

over n the total density

of excited states o(E) = &a(n,

to an exponential form o(E) - exp2m. Thus, for can use the conventional “evaporative” approximation o(p, h, E) + o(E) - exp2fi with the level density a.

E) is reduced

t 2 t,, (or n 2 neq) we by substituting in (14)

4. Input data and condition to combine the models As we see from eqs. (5) (7) and (S), the input data for the cascade stage of the (CEM) are properties of the target-nucleus (that is the density pT(r) and energyfT(r,p) distributions of nucleons) and of the free nucleon interactions. The nuclear matter density is described by the Fermi distribution with two parameters taken from the analysis of the electron-nucleus scattering. The energy spectrum of nuclear nucleons is estimated in the perfect Fermi gas approximation with the local Fermi energy F or characteristics TF(r) = +(37r2)+h2/,[pT(r)]f. of the hadron-nucleon interactions, G, da/da, we employ the approximations given in refs. 3*21). A detailed comparison of the Dubna version of the cascade model with others is made in ref. 29).

K. K. Gudima et al. / Cascade-exiton

339

model

For the pre-equilibrium stage of the CEM, the average matrix elements and parameters of entrance nuclear states are considered as input information. As we have noted above, our approach allows one to define the entrance state for the preequilibrium decay in the framework of the cascade model. The value of M2 is estimated according to (17) in the approximation

(4%_l)c’,,l) ‘v (~>(~,,l>. In this approximation

where the exclusion

the effective cross section

principle

is allowed 1 -+x+%(2-

becomes

(22) n-dependent

22):

for by the factor l/x):,

if

x<2

if

x > 2,

q’(x) = 1 -zx,

and the average collision energy (T& is the sum of the kinetic energy of an excited particle averaged over all available configurations of the n-exciton state, Tr+ E/n, and the partner kinetic energy averaged over the spectrum of the perfect Fermi gas, $,. The interaction volume, entering into (17), can be presented as V,,, = $(2r, + ?)3 with the De Broglie wave length 4, the value of rc being of the order of the nucleon radius. It is noteworthy that the interaction radius is similarly defined in the intranuclear cascade models at high energies 30). Generally speaking, in the lefthand side of (I 7) an additional factor should be introduced to be found by comparing the results of calculations with experiment. The analysis carried out over many nuclei at incident neutron energy To = 15 MeY, when the influence of the cascade stage is negligible, has shown that this factor can be put to be unity, if we use rc = 0.6 fm [refs. 2. 3 ’ )]. Most of the proposed exciton models differ mainly by parametrization of M2. The most popular parametrization is that by Kalbach-Cline “): M2 = KE-‘Am3.

(23)

To achieve an agreement with experiment, however, the “optimal” value of K should vary from 95 to 7000 MeV with bombarding particle energy and reaction channel. In the recent paper by Kalbachj4), an attempt to take effectively into account the dependence of M2 on the exciton number has been made by introducing an excitation-energy dependent value of K. As one can see from fig. 1 this parametrization turns out to be rather close to our estimation of the squared matrix element (17), (22).

340

K. K. Gudima et al. / Cascade-exciton model

1

b

___-----c^

4

_____--_-_

&“-

/+

,‘,= I’ 5

n=3

..--

_=,_A =

75

2&d $ 2

1

I,

0

I

30 EXCITATION

,

I

,

60

90 ENERGY

,

,

I

,

120 E (MeV)

,

.-I

Fig. 1. Energy dependence of the average matrix element squared (a) and the intranuclear transition rates (b). The results of the present paper for different n are shown by solid lines. (a) The hatched area is the M2 estimation according to formula (23) with K = 95 MeV3 for the lower and K = 200 MeV3 for the upper boundary of this area; the dashed line is taken from the paper by Kalbach 34). (b) The hatched area is I+(n,E) calculated for n = 3, 5 and 7 according to Gadioli et al. j5). The dashed line is taken from the paper by Blann I’); the dot-dashed line is estimated from the optical model potential according to (24). The symbols 0 *), l 36), n 37) represent the measured values of /I+ (n. E) for the 3exciton state.

As long as extraction of M2 is connected with the assumption on n, E dependence of wAn(n, E), one can try to estimate directly the transition rate I. + in the line of (17). Using the results of classical cascade calculations Blann proposed the following approximation “) : 2+(T)

= 1.4x 1021(T+B,)-6x

10’s(T+&J2.

Gadioli et al. 33) obtained a certain expression for 1, from first principles by averaging over nuclear states with level density (15). Both these results are presented in fig. 1. As has been pointed out in ref. 3s) to bring the Gadioli results into agreement with the experimental data on energy spectra of particles, it is

K. K. Gudima et al. / Cascade-exciton model

necessary agreement through

341

to decrease these values of I, by a factor of 0.2-0.3, that results in with our estimation of A+. Finally, the transition rate can be expressed the imaginary

part of the optical

/I, =

potential:

-;w&.

(24)

Choosing W,,, from experiment, we get the values of I, close to ours. It is of interest that the absolute values of 2, for the state with n = 3 (p = 2,h = 1) can be experimentally obtained from the high-energy part of nucleon spectra. The analysis of more than three dozen neutron spectra in the (n, n’) reaction at To = 14.6. MeV of the mass number of has shown that A+ = (5.9f0.7) x 102’ ssl independently nuclei’). A similar study of the (n,p) reaction at T, = 14 MeV resulted in 2, = 4.9 x 1021 s-i [ref. ‘,)I. Close values for the transition rates were obtained from the excitation function analysis in the (p, n) reaction, A+ = 6 x 102’ s-l [ref. “‘)I. All these values agree fairly well with our estimation of M2 (see fig. 1). The parameter g entering into (14)-(16) is related to the level density parameter of single-particle states a = &c2g. We use a = &4 MeV-’ for both pre-equilibrium and equilibrium stages of interaction. The inverse cross sections ginv( T) in (14) are taken according to ref. 38) and the binding energies for particles Bj are from ref. 39). The pairing effects are neglected. An important point of the CEM is the condition for passing from the intranuclear cascade stage to pre-equilibrium emission. In the conventional cascade-evaporation model the fast particles are traced up to some minimal energy, the cut-off energy T,,,, being about 7 - 10 MeV. As is shown in ref. ‘), a reasonable variation of the value of Tcut does not change essentially the average number of particles in a nuclear collision. In other words, the matter is which particles should be called by cascade and which evaporative ones. As the zero-order approximation to our model, we shall also consider this “sharp cut-off’ method for passing to preequilibrium nuclear decay. In a real case a cut-off is expected to be somewhat smoothed. Moreover, when we move towards lower energies, the relative contributions of particles captured by periphery and interior region of a nucleus are changed. But this fact is completely outside the scope of the sharp cut-off approximation. Therefore, an attempt is natural to relate the condition for the fast (cascade) particle capture with the proximity of the imaginary part of the model optical potential to its experimental value obtained by analyzing the data on the particle-nucleus elastic scattering. This is inspired by the comparison of the classical equation (5) with its quantum-mechanical form in which the particle transport through nuclear matter is governed by the nuclear optical potential 4o, 41 ). In the “weak-coupling” approximation the imaginary part of the optical potential can be expressed through the cross section G for scattering of a particle

342

on nuclear

K. K. Gudima Ed al. / Cascade-exciton model

constituents

:

W,,,(r) = -~44o,,,)c,,,).

(25)

Here the averaging is carried out over the spectrum of nuclear nucleons and includes the Pauli principle effect (see (7)). Formula (25) is valid only at sufficiently high energies and for the nuclear interior. As is seen from (25) the radial behaviour of the density pr(r) follows that of the optical potential. In a general case, the pr(r) function lags behind W,,,(r) due to the finite radius of particle interaction and nonlinear relation of W,,, to pT. Since at present these effects cannot be considered in a consistent manner, we define the imaginary part of the optical potential for cascade particles Wo’,“t”by relation (25). Describing pT(r) by the two-parameter Fermi distribution we use the parameter corresponding to the volume part of the imaginary optical potential taken from the analysis of the available experimental data, that indirectly allows for the effect of nonlinear relation between W,,, and pT. The results of the Monte Carlo calculation of WoCpa(s(y) are pictured in fig. 2, which also presents the experimental values of the imaginary part of the optical potential W:;:(r) obtained by two different groups 42, 43). One should note that for T, 2 30 though thereat the angular MeV the measured values of Wo’,:” diverge noticeably distributions and even polarization in elastic scattering coincide nearly in the x2 criterion. At low energy T, the calculated imaginary part of the optical potential does not reproduce the absorption bump occurring at the nuclear periphery. It should be added also that the conditions of the validity of the cascade and optical model do not coincide: The cascade model considers the scattering on bound nucleons rather than on a potential well as the optical model does. Thus, the agreement between Way,“;”and WoyF is treated within a certain accuracy which can be characterized by the proximity parameter

If the 60 MeV proton is assumed to obey the conditions of the validity of the cascade model then, as is clear from fig. 2, the parameter -9’ = 0.34.5. This value can be chosen more accurately by comparing the calculated characteristics with experiment. Fig. 3 shows spectra of secondary protons from the p + ,A + p’ + . . . reaction calculated within different versions of models. One can see that the sharp cut-off approximation results in the unphysical dip in the particle spectra around T,,,. Going to higher energy of an incident proton the dip is masked by the choice of the histogram step but at To N 15 MeV its effect distorts abruptly the general form of the spectrum. Being included the mechanism of pre-equilibrium particle emission smooths the theoretical curves and improves the overall agreement with experiment. It should be noted, however, that the angle-integrated spectrum is not a very sensitive characteristic of the reaction, therefore even the conventional

K. K. Gudima et al.

30MeV

t

-8

343

Cascade-exciton model

8

1

0

L&L-L_, 0

RADIUS

(fm)

Fig. 2. Imaginary part of the optical potential for proton scattering from the nucleus s4Fe. The model calculation results are shown by dots. The solid and dashed lines are experimental values from ref. 42) and ref. 43), respectively.

20 KINETIC

i_ LO 60 ENERGY

80 100 T IMeV)

Fig. 3. Kinetic energy spectra of protons. The continuous lines 1 and 2 are calculated within the CEM with .P = 0.3 and 0.1; the continuous histogram is the sharp cut-off version of the CEM (see text). The calculation results of the conventional cascade-evaporation model are shown by the dashed histogram. The experimental points are taken from refs. 44, 45).

results. As we shall see cascade-evaporation model provides reasonable later, the most sensitivity to the pre-equilibrium emission is revealed by fast backward emitted protons, that allows one to fix the value 9 = 0.3 to be used =p we employ the Becchetti-Greenlees everywhere in our calculations. For W,,, potential 42) for protons with T -=I30 MeV and neutrons with T < 20 MeV; at higher energies the optical potential by Menet ef al. 43) for protons and by Marshak et al. 46) for neutrons is used.

344

K. K. Gudima et al. 1 Cascade-exe&m

5. Comparison 5.1. ANGLE-INTEGRATED

NUCLEON

model

with experiment and discussion SPECTRA

The CEM calculation results for da/dT are presented in fig. 4, in which the contributions of different components are shown as well. The model reproduces well the change in the spectrum shape with increasing incident energy and in passing from light to heavy target nuclei, providing the right absolute values for the particle yield. One should stress that din is evaluated within our approach rather than being taken from the available experimental data or independent calculations in the optical model as one used to do in all other models of preequilibrium decay. Such absolute calculations turn out to be possible due to the employment of the intranuclear cascade model. The relative contributions of cascade and pre-equilibrium particles depend on both To and mass number A of a target nucleus, the contributions of these two mechanisms to do/dT being hardly localized in a certain narrow energy range of T. It is noteworthy the overall agreement of the CEM predictions with experiment for the angle-integrated spectra is similar to that provided by other versions of the exciton models; however, the parameters of the entrance states for pre-equilibrium quite different from usually postulated values n, = 3, decay are E, = E,, = To + B,. As is seen from fig. 5, there is a very large spectrum of initial states in the n, and- E, values, 99% of which gives rise to pre-equilibrium decay. The presented values of n, and E0 concern the total spectrum of post-cascade residual nuclei rather than some pair of A and 2. It is of interest that in this case a quarter of all interactions results in a capture of the incident proton, i.e. the nuclear excitation energy equals that for the compound nucleus formation, E, = ECN; however, the incident proton (and possibly recoil nucleon) suffers more than one intranuclear collision, that is n, # 3. At lower energies To the incident nucleon has a high probability of being captured by a nucleus even after the first collision. In particular, at T, = 15 MeV the cascade stage is nearly degenerated and the main contribution is due to the entrance state with n, = 3 and E, = Tf B,. The agreement with experiment at low energies is exemplified in fig. 6. 5.2. ANGULAR

DISTRIBUTIONS

The CEM predicts asymmetrical angular distributions for secondary nucleons. Firstly, this is due to high asymmetry of the cascade component. A possibility to have asymmetrical distributions for particles emitted during the pre-equilibrium interaction stage is related to keeping some memory of the direction of a projectile. It means that along with the energy conservation law we need to take into account the conservation law of linear momentum P, at each step when a nuclear state is getting complicated. In a phenomenological approach this can be realized in different ways.

K. K. Gudima et al. I Cascade-exciton model

-

- - -._ ‘;

N

0

0

-

-

$29

N

0

-

-

r-

K. K. Gudima et al. / Cascade-exciton

346

model

‘,“J

10-l

L’

-10-2

10-2

.

-1.510-

!n-3

0

-81~10-2

3

- 24G-2

0

-10-3

I

1

I

I

20

40 EXCITATION

60 ENERGY

E IMeVl

Fig. 5. Contour plot of constant probability for entrance states of pre-equilibrium nuclear decay in the s4Fe+p (62 MeV) reaction. The circles correspond to the states with entrance energy E, = Ec,. The distributions over exciton number n and excitation energy E which are the projections of this twodimensional distribution are shown in insertions.

The simplest way consists in sharing an incoming momentum P, (similarly to energy E,) between an ever-increasing number of excitons interacting in the course of equilibration of the nuclear system. In other words, the momentum P, should be attributed only to n excitons rather than to all A nucleons. Then, particle emission will be isotropic in the proper n-exciton system, but some anisotropy will arise in both the laboratory and center-of-mass reference frame. The nucleon angular distribution calculated in this approach is shown in fig. 7. A more detailed consideration needs taking into account the dependence on P,IP,in the level density. A step in this direction has been made in ref. 48). In another approach to the asymmetry effect for the pre-equilibrium component, the nuclear state with given excitation energy E should by specified not only by the exciton number n but also by the momentum direction 52, CI= {n,Q). Following Mantzouranis et al. ‘) the master equation (11) can be generalized for this case provided that the angular dependence for the transition rates A+, & and A_ is factorized. In accordance with (17) and (22), we assume

K. K. Gudima et al. / Cascade-exciton

model

341

Fe .p

162MeVI

------_________

+.

\\

\ \. w < c \

2 --_ ;1

0

8 10 12 2 L 6 NEUTRON KINETIC ENERGY T (klh’)

I 0

1L

I

30

60 EMISSION

90 ANGLE

120

150

.

1

P lf30

0

Fig. 7. Results of methodical calculations of angular distributions for protons. The histogram is the cascade component; the dashed line is the evaporative one. The pre-equilibrium component is calculated in two variants by using formula (26) (curve 1) and provided that the bringing-in moment concerns n excitons (curve 2) see text.

Fig. 6. Kinetic energy spectra of neutrons produced by 15 MeV neutron from different target nuclei. Notation is the same as in fig. 4. The experimental points are from ref. 47).

where F(Q)

=

da’“‘/dQ

s

(26)

dS2’daf”‘/dS2

The scattering cross section daf”‘/dO is believed to be isotropic in the reference fame of interacting excitons. This calculation scheme is easily realized by the Monte Carlo technique. The obtained angular distributions are demonstrated in

K. K. Gudima et al. 1 Cascade-exciton

348

model

7. We see that both methods give rise to similar distributions for preequilibrium particles. In comparing with experiment the details of these distributions become more obscured due to the contributions of cascade and evaporative (equilibrium) components. Everywhere below we shall use the second method to allow for the asymmetry of particles emitted at the pre-equilibrium interaction stage. The energy and A-dependences of the proton angular distribution for different components predicted by the CEM’are presented in fig. 8. fig.

5.3. INCLUSIVE

CROSS

SECTIONS

This characteristic implies a more detailed information. The above-mentioned regularities in the energy and A-dependence of separate components can also be followed in terms of inclusive distributions. We shall limit ourselves to a few examples. It is clear from figs. 9 and 10 that the cascade particle emission dominates at small angles though taking account of pre-equilibrium protons improves the agreement between theoretical and measured distributions [compare,

t

2?V +pi90MeVI /

5LFe+p139MeVI

0

60

120

180

PROTON

Fig. 8. Angular

distributions

0

60

EMISSION

120 ANGLE

c

0

5+e+p(29MeVI

60

120

180'

0

of protons. Notation is the same as in fig. 4. The curve marked corresponds to the cascade component.

by crosses

K. K. Gudima et al.

(AJW

JS/W)

1 Cascade-exciton

lPUP/9zP

model

350

K. K. Gudima et al. / Cascade-exciton

model

10

1'

> 10-l i L ln k 2

I 0

L

8

12 16 20 2'4 28

.

10

I-. z ? N" ' D j1

10-l

10-2

0 PROTON

ENERGY

TlMeVl

Fig. Il. Inclusive spectra of protons calculated in different versions of the CEM. Notation is the same as in fig. 3.

Fig. 12. Double differential cross sections of neutrons from the 93Nb+n reaction at TO = 15 MeV. Notation is the same as in fig. 4. The experimental points are from ref. 4’).

for example, with the results of the cascade model calculation from refs. “-“)I. It is of great importance that the fast proton yield at large angles is almost completely conditioned by the pre-equilibrium emission. It is a fact which allows one to fix the magnitude of the parameter 9. The influence of 9 on the inclusive spectra at large angles is shown in fig. 11. As follows from these results, the conventional cascade-evaporation model gives poor account of the hard part of nucleon spectra in the backward hemisphere. A similar agreement between theory and experiment is observed at lower energies (see fig. 12). An example of the nucleon exchange, p + lzoSn -+ n + . . ., is given in fig. 13. Again, we can note a reasonable agreement of the theoretical predictions with the measured values. A certain bump observed in neutron energy distributions near T N 30 MeV seems to be related to the excitation of the analogous resonance and needs a more sophisticated model by which it can be explained.

351

K. K. Gudima et al. 1 Cascade-exciton model

I

..1....1....1:...,.., 10 20 30 LO 50 NEUTRON ENERGY T I MeV)

0 Fig. 13. Inclusive

spectra of neutrons. thick continuous

The cascade component lines are the experimental

is shown by the dashed results from ref. 5).

histogram.

The

The results given in fig. 14 concern the highest energy under consideration. The CEM predictions agree quite satisfactorily with the experimental data especially as no protons with the energy T 5 20 MeV have been detected in this experiment49). The comparison of these results with those for lower energy T, shows that the relative contribution of the direct (cascade) mechanism increases when we move towards higher incident energies. Nevertheless, even at To = 156 MeV the preequilibrium decay contributes essentially to the high-energy part of double differential cross sections for secondaries emitted at the angle 8 > 90”. 5.4. COMPLEX

PARTICLE

EMISSION

In nucleon-nucleus reactions complex particles can be produced at different interaction stages and due to many mechanisms. These may be some fast processes like direct knocking-out +, pick-up reactions or final state interactions resulting in coalescence of nucleons into a complex particle. In the present version of the CEM we neglect all these processes at the cascade interaction stage. Therefore, fast aparticles, for example, can appear only due to pre-equilibrium processes. But the problem of complex particle emission introduces an additional uncertainty + An attempt

to take account

of direct knocking-out

effects has been undertaken

in refs. 50,s1).

352

K. K. Gudima et al.

1 Cascade-exciton

50 PROTON

ENERGY

100

model

0

50

im

T IMeV)

Fig. 14. Inclusive cross sections of protons produced by the 156 MeV protons from different targets. The dashed histogram shows the contribution of cascade particles; the dot-dashed line is the preequilibrium component. The contribution of all three (cascade, pre-equilibrium and evaporative) components is drawn by the solid histogram. The experiment is from ref. 49).

following

from the “origin”

of these particles.

It has been assumed

in the Milan

group papers [see the review article “)I that there exist preformed .a-clusters in nuclei with some probability cp, the cluster being treated as a single exciton with single-particle level density g, = &. The preformation factor cp is extracted by comparing the calculated and observed spectra of a-particles. Another approach developed in refs. 53-56) proceeds from the assumption that in the course of the reaction pj excited particles (excitons) are able to condense with probability yj forming a complex particle which can be emitted during the pre-equilibrium state. Below we shall follow the last approach and take into consideration the emission of the deuterium, tritium, helium-3, and helium-4 nuclei. Modification of the formula for the emission rate of complex particles reduces to the factor o(p- 1, h, E -BjT)/o(p, h, E) in (14) being changed by

Yj

o(P-P+ h, E-B,W(P,k E)

T) o(Pj, 0, Bj+ T) Sj

’

This substitution allows for all possible configurations of both the remaining nucleus and pi excited nucleons (the factor ‘w(pj, 0, Bj+ T)/gj). The chosen configuration of

K. K. Gudima et al. / Cascade-exciton model pj

nucleons

gj(p,h) integral particle volume

should

have a correct

isotopic

composition

guaranteed

353

by the factor

(14). The “condensation” probability yj can be found as overlapping the of the wave function of independent nucleons with that of the complex (cluster). Assuming that the wave functions are constant inside the nuclear I/ and vanish outside, we have the estimate 56)

(27)

It is rather a crude estimate. In the usual way the values yj are taken from fitting the theoretical pre-equilibrium spectra to the experimental ones, that gives rise to an additional, as compared to (27), dependence of the factor yj on pj and excitation energy [see e.g. refs. 55, “)I. In virtue of the above-mentioned variety of complex particle emission mechanisms, we do not see a deep physical sense in such a fitting procedure. Therefore, we define the condensation probability of pj excitons into the complex particle by the relation (27) in order to elucidate later the relative role of this statistical mechanism of complex particle emission. The single-particle density gj for complex particle states is found by assuming the complex particles to freely move in the uniform potential well whose depth is equal to the binding energy of this particle in a nucleus, g

,(T)

3

=

v(2sj

+

l )t2Pj)+

4X2h3

(T+Bj)*.

The angular distribution of complex particles is believed to be similar to that for the nucleons in each nuclear state. But the angular distributions summed up over all populated nuclear states (see (19)) will certainly differ, because the branching ratio for different particles depends essentially on the decaying nuclear state. The angle-integrated spectra of complex particles at To = 15 and 62 MeV are given in figs. 15-17. For comparison the proton spectra for the same reactions are shown as well. The inclusive particle distributions are contrasted with experiment at T = 90 MeV in rigs. 18-19. First of all, one should note that the CEM gives correctly account of the absolute yield of complex particles. This is achieved by using the relation (27) for yj while the values of yj needed for describing the complex particle spectra within the ordinary exciton model are lower by about two orders of magnitude 55). However, the a-particle yield is noticeably underestimated by our model in every reaction considered. This fact obviously shows that along with the statistical factor (27) one needs to allow for the nuclear structure effects which are of great importance just for the 4He nucleus. Due to the high probability of cl-cluster formation in nuclei, it is also important to take into account the processes of direct knocking-out of cr-particles. It can be seen that the CEM reproduces correctly the general shape of energy

354

K. K. Gudima et al. / Cascade-exciton

ImmJJ)

N9P

modef

K. K. Gudima et al. / Cascade-exciton

model

355

q3Nb+n (1SMeVI

2

6

10

IL

18

KINETIC ENERGY TiMeVl Fig. 17. Secondary

o

particle

20 LO

60 80

spectra

o

20

from the 93Nb+n (15 MeV) reaction. from ref. 58).

LO

60 80

0

20 LO

60

80

KINETIC ENERGY Fig. 18. Inclusive CEM calculation

0

20

LO

60

The experimental

80

data

are

0 20 LO 60 80 100

TIMeVI

cross sections for different secondaries from the 58Ni +p (90 MeV) reaction. The results are shown by the dashed lines; the experimental continuous lines are taken from ref. 45).

356

K. K. Gudima et al. / Cascade-exciton model

I&J,‘;-,

0

20

LO 60

80

0

20

10

60

80

0

KINETIC

20 LO

60

ENERGY

80

0

2C1 LO

,

60

80

,

0%‘0

!,

I,

60

80

I

100

T IMeVI

Fig. 19. The same as in fig. 18 but for the “‘Zr target

nucleus.

spectra of complex particles. The particle yield at T 2 15-20 MeV is caused almost completely by the pre-equilibrium emission. Nethertheless, fast particle yields at small angles are underestimated (see figs. 18 and 19) testifying to the direct mechanisms of complex particle production.

5.5. EXCITATION

FUNCTIONS

The excitation functions provide supplementary information to that from angleintegrated spectra. Within the CEM both these characteristics are evaluated on equal footing and under the same assumption. This is just the case for the reactions with many particles in a final state, in which the agreement with experiment is usually achieved by means of auxiliary suppositions as to the magnitude of the preequilibrium emission component and its energy dependence 37, 59). In fig. 20 we show the calculated excitation functions for the (p,n) reaction, which agree quite well with the experimental ones. The high-energy tail of excitation functions is mainly caused by the pre-equilibrium emission of a neutron. For the (p,xn) reactions the agreement is not so good (see fig. 21) though the general trends are well reproduced and discrepancies in absolute value are not catastrophic. On the other hand, the results obtained by various experimental groups differ by 30-50x.

6. Conclusions Thus,

combining

of the intranuclear

cascade

model

with

the pre-equilibrium

K. K. Cudima et al. / Cascade-exciton

model

357

i

If f

107~~~ (p, n ) lo7Cd

i

INCIDENT Fig. 20. Excitafion

function

PROTON

for the (p, n) reaction.

ENERGY

T, IMeW

The experimental

points

are from ref. “).

approach to nuclear reactions allows one to account for the bulk of the experimental data and to reveal the contribution of the pre-equilibrium mechanism of particle emission. The analysis presented above shows that the angle-integrated energy spectra, da/dT, of nucleons widely discussed in terms of the exciton models

18’Ta

INCIDENT Fig. 21. Excitation

function

PROTON

ENERGY

T,

I p ,Ln )178W

I MeV 1

for the reaction with many particles in the final state. points are taken from refs. 59,6’).

The experimental

358

K. K. Gudima et al. / Cascade-exciton model

turn out to be weakly sensitive to details of the interaction mechanism. The shape of the energy spectra points out the only fact that the interaction mechanism does not reduce to a single scattering of an incident nucleon or to the particle emission from particle-nucleus the compound nucleus state. In other words, the inelastic interaction in the medium-energy range has a multi-step character and at each reaction step a particle can be emitted. Allowing for the asymmetry of nucleon emission enlarges the class of characteristics discussed, but being mainly a reflection of general conservation laws, the study of inclusive distributions does not give much understanding of the relative role of different interaction mechanisms. A selection of separate components is possible by imposing essential kinematical restrictions on particles of interest. As an example we can consider the fast particle emission at large angles, where pre-equilibrium decay dominates. Another way to distinguish between different mechanisms is the study of correlations between secondaries ‘j2). The problem of complex particle emission is far from being solved. The complexity of this problem consists in the great variety of possible mechanisms and in their interrelation with nuclear structure effects. The isotopic dependence of the complex particle yield, which is an important problem, remained unnoticed until recent years. A detailed analysis 63) of the experimental data on nuclear fragmentation in high-energy nuclear reactions has shown that the yield of fragments (complex particles with charges 2 >= 2) and the shape of their energy spectra depend essentially on the isotopic composition of the target nucleus and emitted fragment. This requires an introduction of a nuclear structure factor into the “condensation” probability yY Accordingly, the measurements of complex particle spectra on separated isotopes are of particular interest. Exciting questions arise when we move to higher energies. The CEM predicts a noticeable yield of fast particles at large angles emitted due to the pre-equilibrium emission mechanism. As has been pointed out in our paper 64) particle emission during the equilibration stage in fast hadron induced reactions can be considered as a possible mechanism for the production of nucleons in the kinematical region forbidden for free particle scattering. This is very important in connection with recent discussions of the fast backward particle emission and the problem of nuclear scaling 62, 65). The proposed cascade-exciton model can be considered as a basis for studying the pre-equilibrium emission in high-energy heavy-ion interactions. The importance of the pre-equilibrium effect has been noted in ref. ‘j4), in which the signatures of the shock waves formed in high-energy collisions of two nuclei were analyzed in the framework of the cascade model. The authors

have appreciated

helpful discussions

with E. Betak.

K. K. Gudima et al. / Cascade-exciton

model

359

Appendix We shall

show

here under

what

assumptions

the Boltzmann

equation

follows

from the master equation (1). Let us rewrite eq. (1) and the probability rate (2) in occupation number representation c1+ vi, v2,. . ., vA 3 {vA}. Let the system change its state by changing simultaneously a few single-particle states. If it proceeds so that r particles are destroyed in {i,} states and appear in {k,} states, the number of particles in each state being changed only by unity, then expression (2) can be treated as a probability for r-particle scattering. In the case of dilute system or the system with short-range potential, we can limit ourselves to the two-particle scattering, that is r = 2. Then &!CX, Ea’) --f A(ij --t kl) = Aij_klvivj(l

+ Ov,)(l + Ov,),

where the values of 0 = + 1, - 1 and 0 correspond classical particles, respectively. Eq. (1) is reduced to aP(. .

.)

vi, . . vj, . . Vk,. . .,

at x[l+O(v,-l)]P(

ijl

fermions

and

.)

.)

-+c

to the bosons,

Vlr...Yt) = +;Aij,*r(vi+

l)(vj+

. ..) vi+1 )...) vi+1 )...) v,-l)...)

Akl+ijVkV1(l +ovi)(l

l)[l+O(v,-l)] v,-l)...)

+ OVj)P(. . .) vi,. . .,vj,.

t)

.) Vk,. . .) vl,. . .,t),

(A.11

where the factor 3 excludes identical states in summing over all values of i, j, 1. For simplicity, we assume also that all states are nondegenerate. Now we introduce the generating function =

G(zJt) =

for occupation

number

G(z,,

with the property

~2,.

. ., t)

(Vi...Vj...

1 z;‘z;‘. Y,Y*...

. P(v,, v2,. . .,t),

moments

zj;...z,$-G(zlt) , I

Then eq. (A.l) can be represented

Wzlt) ---_+ at

as

1 2*=22=...=1’

360

Hence

K. K. Gudima et al. / Cascade-exciton model

we

particular,

can

get

differentiation

equations

relating

moments

of (A.2) with respect

of

occupation

numbers.

In

to zk gives

An equation for (vivj) can be found by double differentiation of (A.2) but thirdorder correlation moments will enter it and so on. To get a closed equation from (A.3) the additional assumption of neglecting particle-particle correlations is necessary :

So, eq. (A.3) is reduced

to (3), the introduced

inaccuracy

being about

l/A.

References 1) M. Blann, Ann. Rev. Nucl. Sci. 25 (1975) 123 2) K. Seidel, D. Seeliger, R. Reif and V. D. Toneev, Particles and Nuclei 7 (1976) 499 3) V. S. Barashenkov and V. D. Toneev, Interaction of high energy particles and nuclei with atomic nuclei (Atomizdat, 1972), in Russian 4) H. W. Bertini, G. D. Harp and F. E. Bertrand, Phys. Rev. Cl0 (1974) 2472 5) A. Galonsky, R. R. Doering, D. M. Patterson and H. W. Bertini, Phys. Rev. Cl4 (1976) 748 6) J. Htifner and C. C. Chiang, Nucl. Phys. A349 (1980) 466 7) N. N. Bogolubov, Problems of dynamical theory in statistical physics (Gostekhizdat, 1946), in Russian 8) L. Van Hove, Physica 21 (1955) 517; 23 (1957) 441 9) G. Mantzouranis, H. A. Weidenmtiller and D. Agassi, 2. Phys. A276 (1976) 145 10) B. E. Bunakov, Yad. Fiz. 25 (1977) 505; 2. Phys. A297 (1980) 323 11) W. Nbrenberg, J. de Phys. C5 (1976) 141 12) H. A. Weidenmiiller, Prog. Part. Nucl. Phys. 3 (1980) 49 13) G. D. Harp, J. M. Miller and B. J. Berne, Phys. Rev. 165 (1968) 1166; G. D. Harp and J. M. Miller, Phys. Rev. C3 (1971) 1847 14) B. A. Rumjantsev and C. A. Kheifets, Yad. Fiz. 21 (1975) 510 15) V. E. Bunakov, M. M. Nesterov and N. A. Tarasov, Phys. Lett. 73B (1978) 267 16) R. Serber, Phys. Rev. 72 (1947) 1114 17) M. Blann, Phys. Rev. Lett. 27 (1971) 337 18) G. Mantzouranis, Phys. Lett. 63B (1976) 25 19) E. Gadioli, E. Gadioli-Erba and G. Tagliaferri, Phys. Rev. 17C (1978) 2238 20) K. K. Gudima and V. D. Toneev, Proc. 6th Int. Conf. on high energy physic and nuclear structure (Santa Fe, 1975), Abstr. of Contributed Papers, p. 262; Proc. 5th Int. Symp. on interactions of fast neutrons with nuceli (Gaussig, GDR, 1977) ZtK-324, p. 79 21) V. S. Barashenkov, K. K. Gudima and V. D. Toneev, Acta Phys. Pol. 36 (1969) 415 22) K. K. Gudima, G. A. Ososkov and V. D. Toneev, Yad. Fiz. 21 (1975) 260 23) C. Karlin, A. first course in stochastic processes (Academic, New York, 1968) ch. 8, 9 24) J. R. Wu and C. C. Chang, Phys. Lett. B60 (1976) 423 25) V. M. Strutinski, Comptes Rendus du Congres International de physique nucleaire (Paris, 1958) p. 617 26) T. Ericson, Adv. in Physics 9 (1960) 425

K. K. Gudima et al. / Cascade-exciton

model

361

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