the compound nucleus at high excitation energy

Nuclear Physics A519 (1990) 3c-16c 3c North-Holland

THE COMPOUND NUCLEUS AT HIGH EXCITATION ENERGY

David BRINK

Department of Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK.

Heavy ion reactions produce compound nuclei with high excitation energies and large angular momentum. Under these conditions many of the concepts used in compound nucleus theory lose their validity. Energy levels become broad and overlap and the level density is not properley defined. The life time of a compound nucleus becomes very short and it may not reach equilibrium before it decays. Ways of thinking about the compound nucleus under these conditions are discussed in this lecture.

1. INTRODUCTION

The compound nucleus model was developed by Bohr [1] to describe neutron resonance

reactions. A compound nucleus produced in such a reaction has an excitation energy of

about 8 MeV and a low angular momentum. The central idea of Bohr's theory is that

a well defined compound nucleus exists as an intermediate state in a reaction, and that

its mode of decay depends on its energy and on other conserved quantities such as angular

momentum but not on the specific way it was produced. Bohr's assumption implies that the

compound nucleus reaches some kind of equilibrium before it decays. Very highly excited

nuclear systems are produced in heavy ion reactions. At high excitation energies the life

time of the compound nucleus is very short and there may not be time for the internal

degrees of freedom of the nucleus to reach thermal equilibrium. In that case theories based

on Bohr's compound nucleus hypothesis might break down. The purpose of the present talk

is to discuss non-equilibrium effects and ways of thinking about the compound nucleus at

high excitation energy.

Statistical compound nucleus theories based on the Bohr hypothesis were developed by

Weisskopf [2], Weisskopf and Ewing [3] and by I-Ianser and Feshbach [4]. The nuclear level

density plays an important role in such theories. For example Friedman and Lynch [5] have

a formula based on ref [2] for the decay rate of a compound nucleus C into a channel A + a

where A is the residual nucleus and a is a neutron, a proton, an alpha particle or something

similar. It is based on detailed balance and relates the decay rate for the process C --+ A +

a to an average cross-section tr(A + a --o C) for forming the compound state C in the inverse

process. The formula of Friedman and Lynch is

d2Na (2I, + 1)k~a(a + A --, C,~pa(l~,) dEd t - 2 r z h - " p c ( E ~ ) (1)

It gives the partial width for making a transition to a particular state of the final nucleus A

with excitation energy E~. In equation (1) p(E~) is the density of states at the excitation

energy E~ of the compound nucleus, p(E~t ) is the density of states in A at energy E~t in

0375-9474/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)

4c D. Brink / The compound nucleus

the final nucleus, ka is the wave-number of relative motion of A + a and I~ is the spin of

the fragment a. The total transition rate is obtained by summing over final states.

Near the ground state a nucleus has discrete quantum states. The spacing D of levels

in the compound nucleus depends on the excitation energy, the angular momentum and the

mass number. For a heavy nucleus with energy near the threshold for neutron evaporation it

is typically a few electron volts. As the excitation energy increases above neutron threshold

the compound nucleus states can decay by neutron emission and acquire a width I" related

to the mean life-time r for neutron decay by F = h / r . At first the widths are small

(F < < D) but as the excitation energy of the compound nucleus increases the levels become

wider and soon they overlap. At higher energies the decay rates increase, the level widths

become much larger than the spacings (1" > > D) and it is no longer possible to distinguish

individual states. In the region of overlapping resonances it is not clear how the density of

states factors in equation (1) should be defined.

At low excitation energies where level widths are much less than their spacings the com-

pound nucleus survives for long enough to reach a state of complete equilibrium before it

decays. The Bohr hypothesis of independence of formation and decay is valid. At higher

excitation energies when resonances overlap there are interference effects between the con-

tributions of several compound states and the decay of the compound nucleus will depend

on the state in which it was formed. In this situation statistical formulae like eq.(1) might

no longer be valid. J

When resonances overlap the statistical model can still be valid if the compound nucleus

survives for long enough to reach a state of thermal equilibrium before it decays. Then the

densities of states Pc and PA are measures of the phase space available in the initial and

final states and have a meaning even if they cannot be measured by counting levels.

The sequential decay hypothesis is important in statistical compound nucleus theories.

When several decays take place it is assumed that one follows another and that the com-

pound nucleus has time to reach thermal equilibrium between successive decays. There is

evidence that this hypothesis breaks down for slow decay processes like fission. A nucleus

with a high excitation energy may emit several neutrons while fission is taking place.

2. THE STATISTICAL MODEL

The basic formula of the statistical model of the compound nucleus is a relation between

the cross-section a(a + A --~ C) for formation of a compound nucleus state C from the

channel a + A and the partial width for the decay C --* a + A. It can be derived by using

a detailed balance argument. The cross-section for the formation of the compound state C

with excitation energy E and angular momentum J calculated from the Fermi Golden Rule

is 1

a ( a + A --* C) = va(2I, + 1)(2IA + 1 ,K(a + j A --* C ) p c ( E , J ) . (2.1)

D. Brink / The compound nucleus 5c

where v~ is the relative velocity in the initial state and Ia and IA are the total angular

momentum quantum numbers of a and A. The level density of compound nucleus states

with angular momentum J and excitation energy E is pc(E , J). The quantity K(a + A ---*

C) is proportional to the square of a transition matrix element summed over the magnetic

quantum numbers of a, A and C.

The corresponding Fermi Golden Rule for the partial decay width is

F~ - (2J + 1 - - - - ~ K(C --* a + A) \27r2va] . (2.2)

where k~ = I~v,~/h is the wave-number of relative motion of the fragments.

Detailed balance implies that the factor K(a + A ~ C) is the same for formation and

decay. Eliminating it from eqs.(2.1) and (2.2) gives a relation between the decay width and

the cross-section. This can be written in the simple form

F(C ---* a + A) - T(a + A --* C) (2.3) 2 rpc (E , J )

in terms of the transmission coefficient defined by

(2J + 1) a(a + A ---+ C) = ~ (2Ia + 1)(2/.4 + 1 'T(a) + A ---* C). (2.4)

Some statistical theories use simplified versions of the relations given above. Eq.(1.1)

in the introduction is obtained by summing over states A in the final nucleus in an energy

interval dE and averaging over all the angular momenta J of the compound nucleus. It is

necessary to assume that each state J of the compound nucleus is occupied with a probability

proportional to its statistical weight (2J + 1). The density of states p c ( E c ) in eq.(1.1) is

related to p c ( E c , J ) by

p c ( E c ) = ~ ( 2 J + 1)pc(Ec, J). (2.5) J

3. TIME SCALES

The assumptions of the statistical model are valid only if the time interval between the

formation and decay of a compound nucleus is long enough for it to reach a state of partial

equilibrium before it decays. The various degrees of freedom in a nucleus relax at different

rates. Some of the characteristic time scales are discussed in this section.

The relaxation time rn for the nucleonic degrees of freedom is one of the important

characteristic times. Each nucleon needs to make only a few collisions before its kinetic

energy is distributed amongst other nucleons in a more or less random way, so the time r,~

1 2 _= 37 MeV then the Fermi velocity is is short. If the Fermi energy is taken to be e¢ = ~mvf

6c D. Brink I The compound nucleus

given by v f / c =0.28. The mean free pa th of a nucleon between collisions can be est imated

from

)~n ~ 1/o'n

where ~ is the average nucleon-nucleon cross-section and n is the nucleon density. Following

Bertsch [5] we take ~ =40 m b = 4fm 2, and n = 0.16 fln -3. This gives ~,~ = 1.6 fm and a

mean t ime between collisions

r,~ = ~t,~lvf ~ 6fmlc . (3.1)

This es t imate is very crude. Pauli blocking effects increase rn and make it dependent on

the excitation energy.

Another collision t ime ~'opt can be es t imated from the nucleon-nucleus optical model.

The mean collision rate of a nucleon in a nucleus is related to the imaginary part of the

optical potential by 1/~op, = 2 W / h . In n u d e ~ mat t e r W ( E ) ~ ( E - ~S) 2 where ( g - ~/)

is the energy of the nucleon relative to the Fermi energy. This energy dependence is due to

Pauli blocking effects. Mahanx and Ngo [6] have analysed the systematics of nucleon optical

potentials and Brown and Rho [7] have fi t ted their results with a simple formula

W ( E ) ~ 1 2 2 ( E - e l ) 2 - 2MeV, (3.2) E 0 + ( E - ~s)

where E02 = 500 MeV 2. If (E - e l ) =20 MeV then W = 5.3 MeV and

Top~ = 19fm/c.

When a nucleon enters a nucleus and collides with another nucleon then a 2-particle 1-hole

(2p- lh) s tate is produced. Each of these quasi particles has a lower energy than the original

nucleon and hence a longer mean life. Thus each successive stage of an equil ibration process

proceeds more slowly than the preceeding one.

Collective states in nuclei have their own characteristic relaxation rates. A giant reso-

nance is a superposit ion of l p - l h states and its relaxation t ime might be expected to be

related to the optical model damping of single particle and hole states. But coherence ef-

fects modify the relaxation rate and the damping widths of giant resonances are often much

less than would be expected from an uncorrelated particle-hole model. For example the

damping width of the giant dipole resonance in 2°spb is F =4 MeV. This corresponds to a

mean lifetime

~'aR = 50fm/c.

Low energy collective modes have a much longer relaxation t ime compared with giant

resonances. For example Grant6 et al [8] give an es t imate for the relaxation t ime of the

collective variable in nuclear fission as

"rfio°io,~ ~ 1600fm/c.


In order to judge the validity of the compound nucleus model it is necessary to compare

the various relaxation times with the life time of a compound nucleus. A compound nucleus

life time is determined by its most probable decay modes. In most cases this is neutron

emission. Neutron evaporation rates can be estimated from Weisskopf's formula (eq.5.1).

We give some numbers for the nucleus 2°Spb taking the neutron separation energy as B,~ =

7.4 MeV and the radius as R = 7.1 frn.

T = 1 MeV,

T = 2 MeV,

T = 5 MeV,

T = 10MeV,

~'e~ : 4.2 x 105fro/c,

re~ : 2550fm/c,

re~ : 45fro/c,

r ~ : 5fm/c.

where T is the nuclear temperature. It is related approximately to the excitation energy E

by the Fermi gas formula

E = aT 2 (3.3)

where a ~ A/8 MeV -1 is the level density parameter. A temperature of 2 MeV in Pb

corresponds to an excitation energy of about 100 MeV.

When the nuclear temperature is less than about 5 MeV the compound nucleus life time

is long enough for the single particle and giant resonance degrees of freedom to reach equilib-

rium before decay by neutron evaporation. On the other hand the equilibration time for low

energy collective degrees of freedom is much longer. When the temperature is higher than

about 2 MeV the mean time for neutron evaporation becomes shorter than the transition

time v~, between the saddle and scission configurations of a nucleus undergoing fission and

pre-equilibrium neutrons are emitted during the fission process. In these circumstances the

assumption of sequential decay is not valid.

Pre-fission neutron emission has been studied by several groups and the results have

been collected together in a review article by Newton [9]. Some recent results are given in

reference [10]. Hilscher [11] has given an estimate of the saddle to scission transition time

r~s ~ 104 fm/c for the fission of some nuclei with A ,~ 190 which is based on the analysis of

pre-fission neutrons.

4. LEVEL DENSITIES

The decay rates predicted by the statistical model depend strongly on level densities.

Just above neutron threshold individual levels can be identified and counted. Above about

8-10 MeV levels become broad and overlap and information about level densities is very

indirect. For this reason statistical codes use simple models whose parameters are adjusted

to give agreement with experiment. Such simple level density formulae are normally based

on the independent particle model with equal spacing for the single particle levels around

the Fermi level.


The level density formula from the equal spacing model for given angular momentum

and both parities is given by Bohr and Mottelson [12] as

(2J + 1)v:a ( h 2 ~ a/2 p(U,J) 12U 2 \ 2 ~ ] exp 2 v / - ~ (4.1)

where U = E* -Ero t is the excitation energy above the yrast line and E,ot = ( h 2 / 2 ~ ) J ( J + l )

is an approximation to the yrast line. The quantity ~ is the rigid body moment of inertia

for the nucleus. The level density parameter a is related to the single-particle level density g

by a = IrZg/6. When applied to a real nucleus, g should be taken as the sum of the neutron

and proton single-particle level densities at the Fermi surface. The Fermi gas model gives

a = ~r2A/4ef ~ A/15 MeV -1. In applications to heavy ion reactions the level density

parameter is usually taken to be a = A/8 MeV -1. The difference between the two values

has been ascribed to finite size effects [13].

Level density formulae like eq.(4.1) are obtaind by relating the density of states to the

partit ion function (Bethe [14], Ericson [15]). In the simplest case

Z(~) = p(E)e-~EdE. (4.2)

where ~ = 1/T is the inverse of the nuclear temperature T. The free energy is defined as

F = -TlnZ so that

Z(~) = e -tn~(~). (4.3)

The inverse Laplace transform gives

1/o p(E) = ~r i e~(E-F(~))dfl (4.4)

where C is a Bromwich-type contour. Evaluating the integral by the saddle-point approx-

imation gives a relation between the level density and the entropy. The saddle point is

determined by the condition 0

~ ( ~ ( E - F ( ~ ) ) ) = 0. (4.5)

Using fl = 1 / T and the definition S = -OF/cgT of the entropy it is easy to show that

eq.(4.5) is equivalent to the thermodynamic relation between E and F

F = E - S T or /3(E - F(fl)) = S. (4.6)

Then the standard saddle point formula gives

p(Z) = N(E)e s(~), (4.7)


Where the normalization constant in eq.(4.7) is

~(E) = 1 / v ~ - ~

and 0 In Z 0 ~ = T2 OE (4.8)

D = ~ = 0--T"

The density of states given by eq(4.7) is much more sensitive to the entropy S in the

exponent than to the pre-exponential factor N ( E ) . To a good approximation the density of

states ratio in eq.(1) is

pA(EA)/pc(Ec) ~ exp[SA(EA) - Sc(Ec)] (4.9)

and is a function of the entropy 6S difference between the initial and final states. In the

case of gamma emission the initial and final states are in the same nucleus and the entropy

difference is

*S = (OS/OE),~E = -E , y /T .

When a neutron with energy e is evaporated then

where

6S = (It - e ) /T

OS ) - B,~ ,.m -B,~ It=-T -ONE

(4.10)

(4.11)

is the neutron chemical potential and B,~ is its binding energy. In this case the density of

states ratio is

p A ( E A ) / p c ( E c ) ,~ exp[--(e + B,,) /T]. (4.12)

The term depending on the derivative of the entropy in eq.(4.11) can usually be neglected

because in a Fermi gas

T(OS/ON)E ,,~ E / A

and it is small.

To complete the discussion we need a relation between the excitation energy and tem-

perature. A simple Fermi gas model gives eq.(3.3) where a is the level density parameter in

eq.(4.1). The thermodynamic relation OE/OT = T(OS/OT) then gives

S = 2aT = 2~/-a--aE. (4.13)

5. LIGHT PARTICLE EMISSION

1.1 Neutron evaporation Equation (1.1) can be used to estimate neutron evaporation rates. A simple result is

obtained by replacing the density of states ratio PA/RC by the Boltzmann approximation


(4.12) and the average cross-sectlon by the geometrical cross-section lrR~ where RA is the

radius of the target. For neutrons (2/¢ + 1) -- 2 and k~ = 2me/h ~ where e is the energy of

the evaporated neutron and m is its mass. Making these substitutions leads to a formula

for the neutron decay width (cf [19]

r~ 2mR2 -B. /T. [oo

- 7rh---- ~ e Jo

-- 2mR2 T2e -B~/T. (5.1) r h 2

The numerical values in section 3 were calculated with this formula.

1.2 Gamma emission

The decay rate for emitting gamma rays can also be estimated from eq.(1.1). In this

case PA/PC ~-, exp(-E~) and ka = E~,/hc. Substituting into eq.(1.1) gives

d2N 2 1__. Z. 7 .... E.r/T (5.2) dEdt - h (rhc) 2 a.yt~.y)e

The cross-section a~ for gamma absorption from the nuclear ground state is dominated

by the giant dipole resonance with mean energy ED and width FD. The giant dipole

resonance is a high frequency conective vibration. If it is described in terms of a damped

simple harmonic oscillator, the resonant cross-section can be calculated classically and can

be fitted with a Lorentzian

(FDE)2 (5.3) a~f(E) o¢ (E 2 _ E~)2 + (FDE)2

It is reasonable to assume that the structure of the dipole collective mode is not very

sensitive to the structure of the initial state so that if it were possible to perform a photo

absorption experiment on an excited state then the cross-section would still have an energy

dependence given approximatley by eq.(5.3) [16]. With these assumptions eq.(5.2) predicts

an enhancement of the gamma emission for gamma ray energies near the energy ED of the

giant dipole resonance. This enhancement was first observed by Newton et al [17] and has

been studied in many experiments. Some recent references are [18], [19].

6. FISSION

Fission decay rates are normally calculated by the Bohr-Wheeler [22] transition state

method. The argument used to derive the result is different from the one used in sections 2

and 5 to calculate neutron decay rates. In 1983 Swiatecki [22] compared the two approaches

and pointed out that the lack of symmetry between expressions for F,~ and Ff was already evident in equations (31) and (33) of the original Bohr-Wheeler paper. He went on to

analyse the origin of the differences.

In the Bohr-Wheeler theory the fission collective coordinate q is treated explicitly and

D. Brink / The compound nucleus 11 c

all the o ther internal degrees of freedom are t reated statistically. The internal and collective

degrees of f reedom are assumed to be in equilibrium inside the saddle point. The transi t ion

state formula is derived by focusing at tent ion on configurations near the saddle point. If p

is the m o m e n t u m conjugate to the collective coordinate q then the number of states with p

in the interval Ap and q in the interval Aq and energy in the interval A E is

p * ( X ) A E A h Aq, (6.1)

where p*(X) is the density of states of the internal degrees of freedom at an excitation

energy

X = E - B - K. (6.2)

In eq.(6.2) B is the potential energy at the saddle point and K = p2/2m is the collective

kinetic energy. Eq.(6.1) already contains simplifying assumptions because the density of

states p*(X) depends only on the energy in the internal degrees of freedom and has no

explicit dependence on the collective coordinate q. The only dependence on the collective

cordinate is through B and K. More general situations are possible. For example the level

density parameter could depend on the deformation and be different at the fission barrier as

compared with the equilibrium configuration [9]. There is also a weak coupling hypothesis

in eq.(6.2). The total energy is writ ten as a sum of an internal energy X and a collective

energy B + K and the interaction energy between the two is assumed to be small.

The fission decay rate is calculated from an ensemble of systems as the ratio

d N Number of decays per unit t ime

dt - Total number of systems[= p(E)AE]" (6.3)

The total excitation energy is E and in both the numerator and denominator of eq.(6.3) the

states considered lie in an energy range A E .

A fundamental assumption in the Bohr-Wheeler theory is that the nucleus fissions once

the collective coordinate q has passed the saddle point qB = 0. The systems which fission

in a t ime interval A t and have velocity v > 0 have the coordinate q in the range

o > q > - v A t = ( p / m ) h t . (6.4)

The max imum value of p is fixed by the condition that X = 0 or

K,na: = p~.: /2m = EB. (6.5)


The numerator of eq.(6.3) is obtained by integrating (6.1) over the appropriate region and

is

A I A E [P": -- h A vdpf(X)

h t h E L K'~'' - h d K p * ( E - B - K ) .

Substituting into eq.(6.3) gives the Bohr-Wheeler formula for the decay rate

dN 1 L E-B at - hp(E) p*(E - B - K ) d K . (6.6)

The total level density in eq.(6.6) is obtained by integrating (6.1) over the collective

coordinate q and momentum p

1 i p(E) = ~ p*(E - V(q) - K)dpdq. (6.7)

The main contribution to the integral over q comes from the region near the potential

minimum behind the fission barrier. Using the approximations (4.9) and (4.10)

p*(E - V - K ) ~ p*(E)e - ( v + K ) / T (6.s)

and evaluating the integrals (6.6) and (6.7) gives the Kramers formula [22,23] for the fission

rate d N _ o.'o e_B/T (6.9) dt 21r

where w0/27r is the classical frequency of the collective motion in the potential minimum

behind the fission barrier.

7. PRE-EQUILIBRIUM PROCESSES

There is an asymptotic relation between the density of states and the volume of phase

space

1 / / dNpdNq6(E__ H(p,q) ) (7.1) p ( E ) = -£W

for a system with a classical analogue. In eq.(7.1) N is the number of degrees of freedom

and H(p, q) is the hamiltonian of the system. Individual states are not well defined when

compound nucleus levels overlap but the density of states can still have a meaning as a

measure of the available phase space. In a particular system certain regions of phase space

might be more accessible from a given initial state than other regions. Then the phase

space can be divided into up into parts so that there is equilibrium in each region but not

necessarily between regions. This is the basic idea of a pre-equilibrium theory.


Different regions of phase space can be weakly connected for a variety of reasons. There

might be an approximate conservation law like isospin, so that regions of phase space can

be classified by isospin quantum numbers. Transitions from one region to another would be

weak but still possible because isospin is not exactly conserved. Another reason is related

to chaotic behaviour. The phase space of a classical chaotic system can be divide into

regular regions and chaotic regions so that trajectories remain in their respective regions.

The division between regular and chaotic regions is not sharp for a quantum system but

transitions between the two kinds of regions might still be slow.

Pre-equillbrium theories have been developed by Kawai et al [24], by Feshbach [25] and

by other authors. They have been applied to the study of giant resonances by Dins et al [26]

and by Bracco et al [27]. I wili discuss this case as an example. The theory assumes that

the space of compound states can be divided into two regions. The first region C consists of

compound states where the Giant Dipole Resonance (GDR) is in its ground state and the

second D of compound states with the GDR in its first excited state. There would also be

regions where the GDR is in higher excited states but we assume that we are in an energy

domain where those states are not important. The results of refs.[26] and [27] are obtained

from conventional reaction theory and are expressed in terms of transmission coefficients.

In this section we derive equivalent results from a set of coupled rate equations.

Let PD(t) be the probability that a nucleus is in the region D at time t and Pc(t) be

the probability that it is in C. If the nucleus is in D it can decay with a rate FTD or it can

make a transition to C with a rate F~D . In the same way if the nucleus is in C it can decay

at a rate P~ or make a transition to D at a rate F~. The probabilities PD and Pc satisfy

the coupled equations

dPD _ (FTD + r~D)p. + r~Pc, (7.2) dt

dPc _ (F~ + F~)Pc + F~PD. (7.3) dt

Giant dipole photons can be emitted only if the nucleus is in the region D and the total

probability of emitting such a photon is

PD = 3'DID (7.4)

where

Im = Po(t)dt (7.5)

and 7D is a partial decay rate for GDR photon emission.

The probability PD of photon emission depends on the relative decay rates and on the

initial conditions. If the transition rates between the regions F~D and F~ are much larger

than the decay rates FTD and F~ the occupation probabilities of the two regions reach


thermal equilibrium and the decay is independent of the initial conditions. The transition

rates between the regions should be related by detailed balance so that in equilibrium

PC/PD 1 1 = r D / r v = Pc/PD (7.6)

where PD and Pc are the densities of states in C and D respectively. On the other hand

if the transition rates between C and D are small compared with the decay rates then the

resulting decay properties will be sensitive to the initial conditions. For example suppose

the nucleus starts off in a state in C and suppose the decay rate FTc is large compared with

F~. Then PD will be small because the nucleus can emit high energy gamma rays only from

states in D and the nucleus will have decayed before making a transition from C to D.

In order to make a comparison between a model based on rate equations and the theory

discussed in ref.[26] we look at a case where the nucleus can decay into a channel a from

D and also from C with partial decay rates 79 and 73. The total probability of decay into

the channel a is

p~ = 7~ID + 73Ic (7.7)

where ID is given in eq.(7.5) and Ic is given by a similar relation. The integrals 1D and Ic

can be expressed in terms of the initial values of PD and Pc by integrating equations (7.2)

and (7.3)

PD(O) = FDID - r~xo (7.s)

Po(O) = r j o - r~I~ (7.9)

where

r . = r L + r ~ and r o = r ~ + r ~ . (7.1o)

For the particular case where the initial state of the nucleus is produced in the region D by

absorption of a dipole photon (PD(O) = 1 and Pc(O) = 0) equations (7.8) and (7.9) can be

solved to give

---- +TCFD (7.11) pa 7~ Fc ~ I I I" FDFC - FDF c

This result car, be written in many ways. One of them is

7DFC~ 1 I

po ~ + ) r~ ---- l 1 + (7~ ~ FDF° - r D r °

(7.12)

where we have used eq.(7.6) and

~ / D ) ~ r~ (7.13) = (1 - ~ ) ~ + ~ ( ~ + FDDc FtDF~"


is the mixing parameter defined by Dias et al [26]. Equation (7.13) is equivalent to eq.(6) of ref.[26] ~ ~a + , ~ (7.14) Pa = ( 1 - ~ ) + / ~ + , , T D

when written in terms of transmission coefficients defined by eq.(2.3)

~'g = 2rTgpD , r~ = 2rT~pv

. ; = 2 rL. , . $ = 2 r .o

The limiting case where transitions from D to C are very weak corresponds to # ~ 0 and only the first term in eq.(7.14) to the decay into the channel a. The compound nucleus limit corresponds t o / m u = 1. Then there is complete mixing of C an D and the first term in eq.(7.14) vanishes.

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the compound nucleus at high excitation energy

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