# Consistent nuclear model for compound and precompound reactions with conservation of angular momentum. [14. 6 MeV] Page: 1 of 5

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A CONSISTENT NUCLEAR MODEL FOR COMPOUND AND PRECOMPOUND REACT lO’.3

MASTER

WITH CONSERVATION OF ANGULAR MOMENTUM

C. V. Fu

Oak Ridge National Laboratory

Oak Ridge, Tennessee 37830, USA

CO/i' f~ -

The exciton model is modified such that it automatically reduces to the usual

evaporation formula after ecuil ibrium has been reached. The result is further

modified to conser\e annular monen'um in a form compatible with the Hauser-Feshbacn

formula. This allows a consistent description of intermediate excitations from

vihich tertiary reaction cross sections can be calculated for transitions to discrete

residual levels with known spins and parities. Level densities used for the compound

component of reaction cross sections are derived from direct summation of the

particle-hole state densities used for the precompound component.

[Nuclear reactions ?7A1, l‘6,1‘sTi, 51V, 57’5zCr, .s" ,55Fe,

E = 14.6 MeV. Calculated a(n,xn), (n,xp), (n,xi), a(E ),

and precompound analysts.]

V’ (Ea}'

55Cu, 53fib,

Hauser-Feshbach

IjPtrqduction j !c(E)

Development of fusion energy calls for substan- 1 j

tial improvement in the knowledge of neutron cross

sections in the energy range from a few MeV to about

40 MeV. 1 In tin’s energy range, the multi-step Hauser-

Feshbach model with precompound effects is the most

versatile and is considered an indispensible theoreti-

cal tool for cross-section evaluations.'’ In analyzing

cross sections such as hydrogen and helium productions

from 14-MeV neutron-induced reactions, wo showed5 that

spin and parity effects are more important in the

second step of the calculation than in the first step.

However, it is not xtraighUorward to conserve anguljr

momentum even in the first step because the presently

j available models for precomoound reactions do not con-

■ serve or even recognize angular momentum. In addition

the compound and precompound components are generally

calculated in the first step from tvio physically dif- .

ferent models, thus lacking a common basis for

carrying out the calculation to the second step. j

In this paper we develop a model capable of cal- ;

culating the compound and precompound cross sections ;

consistently and conserving angular momentum in both

compound and precompound reactions. The model becomes,

that of Hauser-Feshbach" at low energies where the

precompound effects are negligible. Level densities

used for calculating the compound component are made

consistent with those used for the precompound com-

ponent.

Theory

A full derivation of the formula given below has

been written up for publication elsewhere.5 Here we

only have enough space for a summary. We shall first

write down the final formula and then explain the

physical implications of the various components lead-

ing to its derivation.

The final formula is

ob(E,e)de

X!

l 9j I

Jtt j Si

£l_ y

DJtt sV

tj

T S ' i '

where ’

°Jit = ^b

-,E,U)

ti,,L £

p Ls'V 's'i'

f;b(i.E,u)

I Db(p,E) wb(p~l,h,I,LT) +■ C(E)

Db(p.E) = /

Pb(P,h,t)dt/uj(p,h,E)

nb(i,E,u),

(la)

(lb)

ob(!,U')

(lc)

(ld)

/ P(p,h,t)dt/m(p,h,E)

T

(le)

(If)

Pb(I>U’) = Ip ub(p-l,h,I,U‘)

and the various components and notations are explained

below. j

The compound and precompound components are con-

sistent if the precompound model automatically reduces

to the compound model after equilibrium has been

‘reached. This is achieved by introducing a set of

master equations containing time-dependent particle-

type distributions. Defining P,(p,h,t), the occupation

probability, as the probabi1ity°that the system will be

found in a state with p particles and h holes of type b

i at time t, the master equations which describe the I

•approach of the nucleus to statistical equilibrium are

j given by i

d Pb(p,h,t)

;fpb(P-i.h-i,t) n_! , yp)

|LP(p-l,h-I,t)“ p P

P(p-1,h-l ,t) X+(p-l,h-l,E)

t pyp+i.h+i.t)

+ LP(p-H ,h+l ,t)

P(p+l,h+l,t) A_(p+T,h+l,E)

■Emax

- Pb(p,h,t) [x+(p,h,E) + A_(p,h,E) + / maX Xb(p,h,c)dc]

wi th

P(P,h,t) = lb Pb(p,h,t)

I

(2a)

(2b)

and

h fb(p)

- i

(2c)

The transition rates A+ and X are given by Ribansky 1

et al.5 which contain an empirical residual two-body !

matrix element determined by Kalbach.6 E is the total:

energy of the reacting system. j

In the first term in Eq. (11), P(p-l,h-l,t) i

X (p-l,h-l,E) represents the total transition rates

I from (p-l,h-l) states to (p,h) states. Among the p

particles in the (p,h) states, (p-1) of them retain the

I old particle-type distribution P.(p-1,h-l ,t)/P(p-l ,h-l,

i t), and the newly created particle may have a different

I particle-type distribution given by fb(p), which will

j be determined analytically. Thus the°compositions of

| particle types in the new (p,h) states are given by the

^quantity in the brackets. j

FV; i

OISTMIUTUM OP THIS DOCUMENT IS UNLIMITED ^

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Fu, C.Y. Consistent nuclear model for compound and precompound reactions with conservation of angular momentum. [14. 6 MeV], article, January 1, 1979; Tennessee. (digital.library.unt.edu/ark:/67531/metadc1093950/m1/1/: accessed September 20, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.