calculation of preequilibrium angular distributions with the exciton model code preang

39 Pages·2007·0.51 MB·English

MAY 1979
CALCULATION OF PREEQUILIBRIUM ANGULAR
DISTRIBUTIONS WITH THE EXCITON MODEL
CODE PREANG
BY
J.M. AKKERMANS
H. GRUPPELAAR
ECN does not assume any liability with respect
to the use of. or for damagrs resulting from
the use of any information, apparatus, method
or process disclosed in this document.
Netherlands Energy Research Foundation ECN
KO. Bo» »
1755 ZG Petten INH)
The Netherlands
Telephone [phone] - 6262
Telex 57211
ECN-60
MAY 1979
CALCULATION OF PREEOUILIBRIUM ANGULAR
DISTRIBUTIONS WITH THE EXCITON MODEL CODE PREANG
J.M. AKKERMANS*
H. ORUPPELAAR
Pm«nt addfM*: PONMnttKuto for Ptowiw Phytic» "RijnhuiMn", Nimmgcln, Tlw Netherlands.
- 3 -
Akkeroans, J.M. and H. Gruppelaar ECN-60
CALCULATION OF FREEQUILIBR1UM AW3JLAR DISTRIBUTIONS WITH THE EXCITOS
MODEL CODE PREANC.
Petten, Netherlands Energy Research Foundation ECN. «979,May.
45 pages, 4 figures, 4 tables.
A description is given of the computer code PREANC which calculates
emission spectra and angular distributions of particles emitted in
preequilibrium nuclear reactions. This code employs the generalized
master equation approach of the exciton model, as proposed by
Mantzouranis et al. The first part of this report expounds the theo
retical framework. It is demonstrated that the generalized master
equation can be transformed to a set of simple tridiagonal matrix
equations, directly yielding the Legendre coefficients of the angular
distributions. Also some useful closed-form expressions ate derived.
Matrix methods are shown to be very helpful in simplifying both theory
and numerical calculations. The second part of this report discusses
the code PREANG based on these methods. Structure of the program and
of input and output are described. A sample problem is given, viz.
the reaction 93Nb(n,n') at 14.6 MeV.
Keywords
INELASTIC SCATTERING MEV RANGE 10-100
COMPUTER CALCULATIONS EXCITONS
P CODES CROSS SECTIONS
ANGULAR DISTRIBUTION PARTICLE-HOLE MODEL
PRECOMPOUND-NUCLEUS EMISSION LEGENDRE POLYNOMIALS
NIOBIUM 93 PROGRAMMING
NEUTRON REACTIONS EMISSION SPECTRA
- 5 -
CONTENTS
P«ge
Abstract 3
1. INTRODUCTION 7
2. ANALYTICAL SOLUTION OF THE GENERALIZED MASTER EQUATION . . 10
2.1. Diagonalization of the integral operator 10
2.2. Solution of the Generalized majter equation II
2.3. Time integral of the solution 13
3. CLOSED-FORM EXPRESSIONS FOR THE AWGULAR DISTRIBUTIONS . . 15
4. EIGENVALUES AND INITIAL CONDITION 18
4.1. Determination of the eigenvalues 18
4.2. Initial condition 19
4.3. Remark on convergence 21
5. THE CODE PREANC 22
5.1. Description of the code ?2
5.2. Input and output description 24
5.3. Sample problem 26
6. REFERENCES 28
TABLES 30
FIG^TIES 43
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I. INTRODUCTION
Preequilibrium statistical theory has proved to be very useful for
calculating spectra of light particles emitted in nuclear reactions
at energies ranging from 10 up to about 100 '2V. An important problem
in preequilibrium theory is how to describe the angular distributions
of emitted particles. In this report we shall discuss this problem
in the framework of the exciton model, both in its closed form and
in its master equation formulation. Some results presented here have
already been reported in a short communication {I[.
For the analysis of preequilibrium angular distributions, Mantzouranis
et al. |2,3| have proposed a generalization of the master-equation
exciton model for fast nucleons as incoming particles. In this model,
the state of the composite nucleus is thought to be characterized at
each time t by the exciton number n and a direction Q , which corres
ponds to the direction of the projectile on its way inside the nucl.us.
In a series of binary collisions with the target nucleons the pro
jectile gradually loses its energy as well as its correlation with
the incident, forward, direction. It is assumed that, when emission
occurs from a nucleus in state (n,Q), the direction of the emitted
particle coincides with ft. Thus, the first few collisions (also cor
responding to the lowest exciton numbers) generate the forward peaking
in the emission cross sections, which distinguishes the preequilibrium
part from the evaporative part of the reaction.
The above-mentioned authors factorize the generalized internal trans
ition rates from state (n,f!) to state (m,ft') into the usual transition
rates between different exciton states and an angle-dependent part
that is supposed to be proportional to the differential free nucleon-
nucleon scattering cross section:
x (rc + a') - x xG(n,n')
n-*m n*m
.[/«£J
w» £V £ • ">
The direction of Q has to be specified in the center-of-mass system
of projectile and nucleus.
- 8 -
Let q(n,R,t) denote the occupation probability for the composite nu
cleus state (n,R) at time t. The generalized master equation then
reads:
^•q(n,n,t) - I x /dn,G(n,n,)q(m,n\t)
BMi
m
-q(n,n,t)|w(n)*]r X 1, (2)
where we have included an emission term with emirsion rate v(n).
The positive terms at the right-hand side of Eq. (2) describe the
feeding to state (n,ft) from all possible states (m,ft'), whereas the
negative terms account for the losses of the system due to emission
and to transitions to other exciton states.
Strictly speaking, transitions with An»0 have to be included in the
generalized master equation. In contrast to the angle-integrated case,
these X -terms may not be dropped from the equation because of the
fact that each collision contributes to the loss of correlation with
the initial direction, irrespective whether the exciton number has
been changed after the collision or not. Carrying out a solid-angle
integration in Eq. (2), the An*0 terms cancel and the well-known
angle-integrated master equation is left |4j:
-j^q(n,t) - X+(n-2) q(n-2,t) + x'(n+2) q(n+2,t)
-q(n,t)[w(n)*X+(n) + X~(n)J, (3)
+
where X - X ,,.
n-*ni2
Comparing Eqs. (2) and (3), it is obvious that the solid-angle integ
rals in Eq. (2) spoil the simple mathematical structure of the master
equation (3). Mantzouranis et al. have solved the generalized master
equation by numerical methods. The results are in reasonably good
agreement with experimental data. However, these methods are very
time-consuming. Moreover, the calculations of spectra and of angular
distributions have to be performed in completely different ways.
- 9 -
In this report it is shown that it is possible to reduce the general
ized naster equation to a form siailar to that of the angle-integrated
•aster equation (3). Also, it will be deannstrated that the tin*
integral over the occupation probability, which is the quantity direct
ly needed for the computation of preequilibriua cross sections, can
be calculated at once, instead of using the iterative procedure often
eaployed. Further, we shall discuss the relatioa between the aaster-
equation and closcd-forst approaches and derive SOBK practical closed-
fora expressions for the angular distribution. Finally, a description
is given of the code PREANG based on the new methods |I|, discussed
in this report.
- II -
Next, we expand the generalized occupation probability q(n,Q,t) into
a Legendre polynomial series:
q(n,n,t) « I n. (n,t)P (cose). (7)
Making use of Eqs. (5) and (7), and taking advantage of the ortho
gonality of the Legendre polynomials, the generalized master equation
reduces to the following set of equations:
oT nl(n»t) " "l I Vn \(m^
m
-VM)t(n)»lJ. (8)
Thus, the integro-differential equation (2) has been transformed to a
set of linear differential equations, one for each Legendre polynomial
of order I. The form of these new master equations for the Legendre
coefficients of q(n,fi,t) is identical with that of the angle-integrated
master equation (3) (except that in Eq. (8) one has to take into account
transitions with m-n). Consequently, the calculation of preequilibrium
angular distributions and the determination of preequilibrium spectra
can be performed along exactly the same lines. In particular, the same
routines in a computer code can be used, see section 5.1.
2.2. Solution of the generalized master equation
It is possible to solve Eq. (8) by means of an iterative procedure.
This is the usual approach in solving the angle-integrated master
equation (3). However, more insight is gained by applying matrix-alge
braic methods. Let us write Eq. (8) in matrix form
(9
OTV^ - \ vi>- >
Here, 5 symbolizes the matrix of the transition rates and is given by
( w (,
Vi,7--<^v i>- -v*i'
(B„). . . - p. x X., and
I i-l i I i
f
(B ), . . • u x X., where the index i is related to the exciton
%, X • I f X X, X
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number n through i» l+(n-n )/2, n being the initial exciton number.
All remaining elements of B vanish. Hence, we are dealing with a tri-
diagonal matrix, a direct consequence of the postulated two-body na
ture of the intra-nuclear collisions.
Since ƒ dn'C(fi,n')» 1 according to Eq. (1), it is immediately seen
from Eq. (5) or Eq. (6) that u • • and |u | < 1 for l> 1. It may be
noted therefore that for 1*0 the A -transitions cancel and that B is
Q
identical with the transition rate matrix of the angle-integrated
master equation (3).
Because (B ). «."(B.).. . > 0, there exists a real similarity
X X y X J X X™1)1
transformation that transforms 6 to a real tridiagonal symmetrical
aatrix J5J. Furthermore, Gerschgorin's theorem (sometimes also named
after Lévy-Hadamard, e.g. [6[) shows that all eigenvalues of B lie
inside the domain consisting of the union of circles centered around
(Vi,iand with radii ri" l(Vi-i,i* (Vi+i,i|- Moreover» \ is
negative diagonal dominant because r. <IB ). .1.
(
As a consequence of the above properties, B can be diagonalized and
all eigenvalues are real and negative. Therefore, one can construct
the solution of the generalized master equation (9) in the same way
as was done recently by Luider for the angle-integrated case 171 .
The solution of Eq. (9) is
(,0)
Vo-Z ( « i W i ^ V i ' ]'
where (y.). and (v.). denote the eigenvectors and eigenvalues of B
rreessppeeccttiivveellyy,, aanndd ((cc )).. aarree iinnctoegg ration constants. These integration
constants can be determined from
n^t-0) - Y c , (II)
£ t
where Y. stands for the matrix of eigenvector (y.)> and n (t"0) is
x JL X Jt
the vector corresponding to the initial condition. The initial con
dition will be discussed in section 4.2. Eqs. (10) and (!]) with the
help of Eq. (7) constitute the analytical, time-dependent, solution
of the generalized master equation (2). It can be proved |)4| that
this solution becomes isotropic for large t.