Z. Phys. A 348, 273-279 (1994)
ZEITSCHRIFT
FORPHYSIKA
9 Springer-Verlag 1994
Novel approach to angular distributions in precompound reactions
Does the Bohr hypothesis always work?
S.Yu. Kun
Theoretical Physics Department, RSPhysSE, Institute of Advanced Studies, ANU, Canberra ACT 0200, Australia (Fax: + 61-6249 4676)
Received: 23 February 1994
Abstraet. We modify the standard statistical model for
precompound reactions (exciton model) by taking into
account the correlations between fluctuating S-matrix elements with different J (total spin) values. This is done in
the framework of the statistical approach to nuclear reactions. While angle-integrated cross-sections are not affected by our modification, differential cross-sections become asymmetric about 90 ~ c.m. This asymmetry weakens with time and with increasing complexity of the decaying nuclear system, but need not disappear even for
the compound (thermalized) system. We present a comparison with data showing such an asymmetry.
PACS: 24.10.Cn; 24.60.Dr
In the modern approach to precompound reaction
theory, it is customary to distinguish the multistep-direct
and the multistep-compound processes [1]. The former
yields forward-peaked angular distributions, and the latter cross-sections which are symmetric about 90 ~ c.m. The
distinction is based on the difference between a continuum and a bound-state shell-model wave function for
the nucleon in intermediate states of the collision process
[1-3]. While intuitively appealing and quite successful in
general, this approach suffers from two shortcomings.
First, the transition from one type of reaction to the other
is abrupt: Entirely different statistical assumptions are
used in the theory [1-3] for nucleons in continuum and
in bound-state orbitals. In the first case, direct amplitudes
dominate the process; in a statistical description [4], such
direct reactions imply strong correlations between reaction amplitudes carrying different total spins J and, hence,
give asymmetric angular distributions. In the second case,
such correlations are from the outset assumed to be absent; this implies symmetry about 90 ~ c.m. A second
shortcoming becomes apparent upon inspection of a
number of experimental angular distributions of neutrons
of low energy < 1-2 MeV (see Figs. 1 and 2 as examples)
and sub Coulomb charged particles emitted in nucleoninduced precompound reactions with a few to several tens
of MeV incident energy. Although the above-mentioned
theories predict these low-energy particles to originate
overwhelmingly from the multistep-compound process, a
considerable fraction of them (up to 20-30%) displays
forward peaking outside of experimental errors reported.
This is not consistent with present-day theory.
It is the purpose of this work to overcome these difficulties by a suitable modification of the statistical model.
Here we briefly present the main sheme of our formulation, the results and simple applications to the analysis
of data. The detailed derivations and discussion will be
given elsewhere.
The main idea is this. We relinquish distinguishing
continuum and bound-state orbitals and classify intermediate configurations by their particle-hole numbers (m)
only. (In this respect, our approach is a kin to the exciton
model [5]). For each class of states with fixed m and
J, we introduce the usual statistical assumptions. Spin
correlations of S-matrix elements result from the nonvanishing of the square of a certain matrix element
(see (7)). As a result, we describe as a smooth process
the transition from the strongly forward-peaked angular
distribution typical for small partical-hole m values to the
nearly isotropic distribution typical for m-values corresponding to the compound-nucleus situation. We do so
at the expense of introducing an additional parameter,
the relaxation time r,~ of spin correlations in class m, or
the corresponding correlation width Bm = h/rm. Our approach is physically meaningful if reactions on different
targets and/or at different bombarding energies can be
described by similar relaxation times.
To be quantitative, we introduce two complete sets of
states in class with m excitons. (i) The set of Slater determinants Xm/ of single-particle states, with j a running
index. We observe that the Xm/ are not eigenfunctions
of the total spin J. Each Xmj has as many particles
(holes) as corresponds to the definition of the class m.
(ii) The set of (antisymmetric) eigenfunctions ~b~J, with
eigenvalues EmSu of the projection H,~ of the Hamil-
274
tonian H onto class m. Here, Ix is a running label, and
J the spin. Completeness implies that the expansion
=
C,~u ;j Xm~ exists, is unique, and has the inverse
J
Xmj -~-Z (C-1J
J
)j;mp ~)mu"
JU
We use statistical scattering theory in the form developed in [6]9 We label the wave functions in the entrance
(a) and exit (b) channels by Xe:
a and )~EJL respectively.
The channel labels a = { & l , , j a } and b = { b , lb,Jb} carry
the intrinsic state if(b) of the initial (residual) nucleus,
the orbital momentum l~(~) and the channel spin Ja(b)"
The shell-model multistep S-matrix (with a:#b) is given
by
S L (E) = - 2 iexp [i~o,~ (J)]
tL (E)
(1)
with
tL(E)= X
Vmi, nj Vm' i',n'j"
= ( O m m , Oii, Onn, ~ j j , + (~rnn, ~ i j , ~nm" ~ i.j )l)mn,2
- Eam~tJ ) - - Vaa,nvJ
c
I n ( l ) ~Oab ( J ) = 6 a ( J ) -q- (~b ( J ) , w h e r e d, (J) and d b (J) are
the potential phase shifts in the entrance and exit channels, respectively. In (1) and (2), E is the total energy of
the system and V is the residual interaction between classes, which does not depend on J. We assume that the
matrix elements of V vanish unless m and n label neighbouring exeiton classes, so that Im - n [ = 2. Also we restrict ourselves to the case when direct reactions are absent, i.e. <S~b)=6ab(Saa > where brackets <...> stand
for the energy averaging.
The partial width amplitudes ~,cz
m/2 have the form
r~1/2 <(as,, I VI Z ~:j>. The statistical model implies [6] that
cJ and V~u,~
J
__
Ymu
-({b,~J a ] V]~b~) are real Gaussian random variables with zero mean value.
It is our purpose to calculate the autocorrelation function of a pair of S-matrix elements
<S~b (E + 89e ) S~'* ( E - 1 a )>. Since we consider the random variables to be stationary, ergodicity is applicable
[7, 8] and we can change from the energy averaging to
the ensemble averaging [6]9
In order to perform the ensemble averaging, we expand the q~u wherever they appear in terms of the X,~j.
Accordingly the random variables ~-,u
~s and V ~ . ~ take
the form
~J ~ Z CJ
~rnl2
~mld ;i {~J
:mi,
i
(3)
q
C J
.v j
,
where
~cd'=.jT1/2 / ~r
mi -.'~rni
c
NIXEd),
Vm,,.+=<x,.,I VlX.+).
~e'S"
nj
t~mn6i] t~ge,
=
((~cJ)2
".'. miJ
(~c'J"
". mi
"~2~1/2 ,
] ]
(6)
(2)
+ i ~, ~ma
~: Ynv
~: 9
J
(5)
where, from the second (3) and the normalization condition (~bmu]q~,~u)=l,
:
s
2 In order to
we have Vm,
2 = V,~,.
further specify the correlation properties of the ('s we
postulate
~eJ
mi
[o~ (J)]rn~tnv=6mna~v(E
=Z
i
J
the (Omu.
We postulate that the second moment of the v's
equals
~n~,
-
mpnv
J
Since ?'ran
cJ and V~mu,nv
J
s
cd and
~mi
are real we take C~mu.i,
V~i,nj to be real also9 Random variables ~f~ and v,~,,j
are Gaussian with zero mean values since they can
be represented as linear combinations of the Gaussian
variables ~'mu
~J and Vm~.~.
:
We postulate that
~ n i ~ m S = 6 i j "..rnie(:cS]2 and, from the first (3), find that
(g,~{)2 = (7~,)2 since Z (C~u;~)
s
2 = 1 due to normality of
(4)
i.e. we admit strong angular momentum correlations between the spins whenever they appear as indexes at the
channel (continuum) wave functions of the ingoing (outgoing) particle in the continuum. Thus (6) implies a correlation between ~'s with different orbital momenta
l c . l c , , channel spins Jc--kjc, a n d / o r total spins J . J '
(~,~s are matrix elements from the XceJ and not ~b~u)
provided the microstates of the residual nucleus are the
same ~ = Y. This is kin to the continuum-continuum coupling in the multistep direct reactions [ 1]. At the same
time (6) implies a noncorrelation condition with respect to the different microstates of the residual nucleus
~ . ~', which is the usual assumption of the statistical
model. It follows from (6) that we can consider the normalized ~,i = ~mi,~
cJ / f (cs)
m~.21~~/2 to be indepedent of l~, j~
and J due to continuum-continuum correlation-coupling
(6).
We
calculate
the
ensemble
averaging
S~b ( E + ~ o~ j~ ~J"
o ,. * ( E - - 89e ) and find that it is expressed in
terms of
-Jk
-J' k 2 k
(<~;,;#
I~bm; >) ,
(7)
where (...)k stands for the k-ensemble averaging, and the
~J,k=N1/2
mlt
-'rni
J
2
ui(m) X m i
(8)
i
9
9
- Jk
- J k
are normahzed
functions
<tkT&
ICm~,>
= 1. In (8) Uik( m ) are
J-independent orthogonal matrices U U r = I. The U-matrices arise from the diagonalization of the proper symmetrisied products Ca {b, ca(b)V and vv. For example, the
symmetric J-independent A-matrix obtained by symmetrization of the {a gb_matrix (which is associated with
one step contributions, i.e. formation of the m-exciton
class from the (a) entrance channel and decay from the
same m-class to the (b) exit channel) has the form
--2\
mi rnj -~
~,,,j ~mi) = A}7 )
(9)
275
The necessary condition for existence of the S-matrix spin
correlations is non-vanishing of the expression (7), i.e.
that different ~b2u~ within the same k-subspace are not
9
- J,k
- J',k
orthogonal. We note that while
(q~.u,
IqSmv
) k = ~::' Ou~,
-Jk
~-hJ',k',~2
((q~;g~]
v,~2 / : k does not necessarily vanish for J ~ J '
since the q~u~ are not eigenfunctions of the total spin.
Indeed using (8) and expanding X~i in terms of ~b~u we
obtain
-d,k__
1/2
~bmv-Nm
~, ~,
d
ll-(m){(~--l'~d"
CJm,u;i ~ik
\~
]i;mv
,,,hd"
Wmv
(10)
J'vi
i.e. (~mzt
- J,k a r e the linear combinations of the eigenfunctions
of total spin qS~'~ with different J'-values. We note that
the (q6m'
- ~uk Iq~mS'k'
; ) "~5~k,. Indeed, making use of the orthogonality of the U-matrices in (8) we can show that
-Jk
-J',k"
((~Om"
u I(bmv
) ) 2 ~-- 1/Nmi---~O for k . k ' , where N,,i>> 1 is
the dimension of the basis XmeOf the Slater determinants.
In order to support the non-vanishing of expression
(7) let us assume the opposite, i.e. that there exist at least
two orthogonal functions ~bmu
- s,k and ~)m,U..."
- :,,k within the same
J",k#
k-subspace. Further let us take one r176
from each
k" . k subspace. Since the total number of the different
k-subspaces is N,~ we obtain (Nm~+ 1) mutually orthogonal functions9 But these functions are linear combinations of the complete orthonormal set of Slater determinants whose number is Arm,-.Therefore it is not possible
to construct more than Arm; mutually orthogonal linear
combinations from the N,,~ Slater determinants (since
linear transformation can not increase the dimension of
space)9 Thus we come to a contradiction with the initial
- J,k
- J',k
assumption that qSm,
, and q~m,,
within the same k-subspace are orthogonal, which means that expression (7)
does not vanish. Another possible argument in favor of
the non-vanishing of expression (7) is the following. We
can show that the set qS~ with fixed J and running a, k
is overcomplete. That means that we can expand (although not in a unique way) any q~m;
- s, k" with j , :# J over
~b~'k with fixed J and running/~, k, Further since different
- or' , k '
k-subspaces are orthogonal all projections of q~m~
onto
~b~ with k . k '
vanish. Therefore ~bmv:',k and qSmuZk belonging to the same k-subspace can not be orthogonal
(otherwise we could not expand ~bmvs, ~, over ~ba;u_~ ~ whach"
would contradict the overcompleteness of the ~bmu-set
J'~
with fixed J and running ~, k) and thus expression (7)
does not vanish.
9 In order to model an explicit form of the expression
(7) we postulate for the neighbouring J and ( J + 1) spin
values
(<~b rn'#
we impose the condition
-:k
s , x ~ k < D ~: / A ~s,
((~bm'
u 2"r-~met"
:,'
(13)
where Am~I"
s
~s and F~s is the total average decay width
of the m-exciton configuration with spin value J. The
condition (13) is needed to reproduce the results [6] and
physically implies quick (as compared to the average lifetime) loss of memory of initial phase correlations in the
states with the same spin value9 This also implies the equal
population of all states with fixed spin value within each
exciton class [6]. Using (11-13) we find the unknown
normalization factor f ( J , J + 1 ;m) and derive an explicit
form for the expression (7) with arbitrary J # J '
((g J,k g, : ' , % V k
v- mt.t
=(D~
"r" m v
/:
s" I12
Din)
,a,,lJ-J
,
:
:"
I/zc[(E,~,-E,~v)
+
2
(14)
We would like to note that although the r.h.s, of (14)
does not vanish exactly it is extremely small provided the
Poincare recurrence time is much larger than the spin
relaxation time: h/D~>> rm, i.e. DJm/Bm--,O. Since from
the forthcoming analysis of data we find ,am > F ~ we
presume that the criterion for extremely small values of
the r.h.s, of (14) is the strong overlapping of resonances
J
J__+
of the intermediate nucleus D,,/F~
0. This is one of the
conditions for validity of the statistical approach [6]. Further we can show that the 1.h.s. of (14) equals the following fourth moment
cL.. cL. cA,;:
2 (c2
;Y
(15)
of the random C-coefficients with different J . J ' values.
The statistical model usually suggests that random variables have Gaussian distribution. We can show that if Ccoefficients are Gaussian variables then the fourth moment (15) vanishes (it is of order 1/Nmi~0 ). Therefore
the criterion Dm/F,
s ~---'0
:
(which physically implies that
the system will definitely decay before repeating its initial
configuration) is consistent with the asymptotically Gaussian distribution of C-coefficients. Altogether we presume
that the domain of the validity of the presented formulation coincides with that of the approach [6].
Restricting ourselves to purely internal mixing [6],
which only needs to be taken into account in practice,
and making use of (14) we obtain our central result
S~b(E + 89 Sfb* ( E - 8 9
I (]) m v
= f(j,j+
Further, for/~ . ~ ' ,
.
J
J+l ) 2 +,am].
2
1,rn)/[(E~mu-E~
(11)
= Sad, 588, exp [i~Oab(J)
- i(0a'b" (J')] ~ (T~:T~"r') 1/2
The physical meaning of the new ,am widths will be discussed later on. We assume that the random matrix elements (-~-mu,Tmv
g J,k Ig s+ 1,k,~
, are not correlated
(~ s,~ I-~ s+ 1,,,3 (2, s',k Ig s' +l,k 3 ~
rmlt
l "r m v
= a..
/ x'r" m l ~ " l "g r n v "
a... a.,
i
.
(12)
mtt
•
(e) (Tb:T,b'J') 1/2 .
(16)
Here, T,~"r is the usual transmission coefficient [6] for
population of states with m excitons and spin J from
channel a. The matrix H ~ ' (e) obeys the "probability
balance" equation
276
2~ (Fff s" + P, I J - J '
-- Z
I -ie)
H,~J J"k (e) Tsj,,r, = g,~,.
(17)
k
With F~ (J) = F ~ (J) + F~ (J) the total average decay
width [6] of states at energy E in the class n with spin J,
where F ~ (J) = 2 ~, (?~J)2 and F~ (J) = 2 n ~, V2n/DS~ is
c
/q
the spreading width, we have
D nJ J ' =
Fff'" = (In (J) + Fn (J'))/2,
T,,~Jff--(T
s TJ']I[ 2
-- ,.-mn -Inn;
(D.J D.J" ) 1 / 2 ,
(18)
9
The flux of probability between classes is determined
by the real and symmetric transition coefficients
J
J
--1
T~n=2rc(Dm)
(Vm,)22rc(DS) -1. For J : J ' , a = a ' ,
b = b ' and e =0, Eqs. (16) to (18) coincide with the results of [6]. The difference between the cases J = J' and
J . J ' in (16) to (18) is that for J . J ' we have additional
"decay widths" tim ] J - J ' l . In order to interpret these
new energy scales we consider the Fourier transform of
(16) assuming that only the simplest no-exciton configuration is populated from the entrance channel (a). We
obtain
(2nh) -1 ~ d e e x p ( - i e t / h ) S ~ b ( E + 8 9 1 8 9
~- fia~' g ~ ' exp [ i(Oab (J) - i~oa, b" (J" )]
2ff
X
Z
[GaltbF,n ( J ) a a H " l b ' , n ( J t ) l l / 2 p n ( t )
n~no
X exp (-- t l J - J ' l / ' C n ) ,
(19)
where a~b,n
He (J)= T~ s T,bS/~, T2 J and P, (t) is the solur
tion of the ordinary angle-independent exciton model
master equation (with average effective spin value), subject to the initial condition P,(t=0)"~0,no. In the derivation of (19) we have assumed: (i) fl,~tm>___flnt, for
m
m > n with tm = ~. (h/Q). This requirement ensures
n=no
that more complex configurat/ons produce less forward
peaking of the emitted particles; (ii) It m <_fl, for m > n,
which means that spin relaxation rate Bm is not increased
as the system approaches to an equilibrated, in phase
space, state. Conditions (i) and (ii) imply that fin is a
smooth function of n and B, can be considered to be
approximately independent of n for n ~ 10. The assumption of smoothness for ft, is consistent with the results
[9] of the analysis of the energy autocorrelation functions
in the 27A1(3He, p) reaction. From (19) one can see that
spin correlations relax not only with increase of the complexity of configurations but also within each single configuration with fixed number of excitons as time goes on.
Since the rate of spin relaxation is proportional to
IJ - J ' ] (i.e. the difference of impact parameters or the
difference in initial conditions), the relaxation times
z, = h/fln have the formal meaning of Lyapunov exponents.
From (19) for small times t ~ h/fi n the time dependent
spin off-diagonal contributions retain memory of the potential phase shifts, and the main contribution to the sum
over n comes from n-values close to n0. Therefore for
short times (19) yields forward-peaked angular distributions. On the other hand, for times t>>h/fi n, the spin
correlations in (19) are exponentially suppressed which
leads to the symmetry about 90 ~ c.m. in angular distributions. Moreover, the main contribution to (19) now
comes from classes near ~. This is the compound nucleus
limit. We can show that (19) (after substitution into the
expression for the double differential cross-section and
summation over angular momenta) formally results in
the hot spot model [10] with a time and angle-dependent
nuclear "temperature" which relaxes from its maximal
E / n o value for the forward scattering angles in the initial
reaction stage towards the compound nucleus angle-independent temperature E/~. A connection between the
exciton model and the hot spot model has been discussed
in [11].
Thus (19) provides us with a unified description of a
precompound and compound-nucleus reactions, with the
appropriate change from a strongly forward-peaked angular distribution to one which is symmetric about 90 ~
c.m. The relative contributions of the different reaction
mechanisms are completely determined by the relationship between the F,'s and ft,'s, i.e. by the relationship
between the relaxation time and the angular momentum
correlation loss time. It is obvious that the off-diagonal
spin contributions ( J . J ' ) disappear ever faster with increasing fin. It is also clear that monotonically increasing
with n quantity fin t, is the parameter which determines
the degree of angular momentum correlation in class n.
At the same time the forward-backward asymmetry in
angular distribution is consistent with the statistical assumption of equal population of all states within each nconfiguration with fixed spin value.
Our model does not necessary imply that the emission
of particles from classes n near ~ is symmetric about 90 ~
c.m. Whether or not this is the case depends on the value
fie te, i.e. on the competition of the two relaxation processes (increasing of the complexity of the nucleus configurations (thermalization) and the decay of the offdiagonal spin contributions). The model presented in this
work is also related to the model of the "leading particle"
[12]. This point will be elaborated in the future publication.
We must admit that we have not justified the
Lorentzian form in the r.h.s, of (11) where a higher
than second power of (E~u-E1m +1) can be taken.
Should we take the r.h.s, of (11) to be proportional to
e x p [ - ( F J -F,~+l)2/2fi~] we would obtain a result
similar to (19) but with exp ( - t21j - J ' l / 2 rZm) instead
of e x p ( - t] J - - J ' l / 2 " m ) .
As we have seen the angular distributions of particles emitted from class m are determined by the product
flmtm/h=lm/Z m. Therefore the analysis of the angular
distributions for different energy losses can allow in principle the extraction of the set of fl~n tm/h" Furthermore
the analysis is considerably simplified for: (i) energetic
emitted particles when the cross-section overwhelmingly
277
[
0.20
~
o9 0.19
"T
>(D 0.18
&
f
0.17
uLD 0.16
2~
~
184W(n,n'), En=14.1MeV
E n = 14.1 MeV
0.6-0.8 MeV
"5-
0.20
>
s
0.15
l~
"(3
-~ 0.15
%
%
0,14
0,10
I
0
I
[
30
I
I
I
60
I
i
I
I
90
I
I
120
I
I
I
150
I
i
180
OCM (~
Fig. 1. Centre-of-mass angular distribution in 2~ (n, n') inelastic
scattering at E,= 14.1 MeV and E,, =0.6-0.8 MeV outgoing energy. Data are from [14]. The curve is calculated with ,8e/I'cN= 10
originates from the simplest no configuration; and (ii)
low energy particles in the domain of the evaporation
spectrum where the main contribution comes from the
decay of nuclei with equilibrated numbers of excitons ft.
The second possibility has additional interest since, as
was mentioned above, the thermalization time te and the
one for loss of angular momentum correlation memory
may be similar, so that the emission of particles from the
compound nucleus need not be symmetric about 90 ~ c.m.
This is a novel and interesting modification of [ 13] Bohr's
compound-nucleus picture: the spectrum of emitted particles is thermalized and yet shows a forward-backward
asymmetry. This situation occurs when fl~ (t~+ h/1-'CN)/
h_<_ 1, i.e. when the total time
,
0
t*= ~, h/Fm needed
to reach the compound-nucleus a n d h~ve~ decay (h/FeN
with FCN=Fa) is not much longer than the decay time
of the spin correlations.
Our basic picture and the quantitative consequences
we deduce from it are supported by the data. As an example, we display in Fig. 1 an experimental angular distribution for the 2~
reaction at E ~ = 14.1 MeV
[14]. The data in Fig. 1 are transformed to the c.m. system. The energy of the outgoing neutrons is from 0.6 to
0.8 MeV. We observe that the small-angle cross-section
is about 25% larger than the back-angle cross-section.
The effect is clearly larger than the experimental errors
reported in [14]. The 25% increase of the angular distributions at the forward angles in the domain of the evaporation spectrum seems to be in agreement with other
measurements at E , = 14 MeV [15, 16] (as it is seen from
figures in these references). However we do not display
these data in Fig. 1 since their tables are not at present
available to us. In the theoretical analysis, we have taken
into account only l = 0, 1 values of the orbital momenta
of the emitted neutron. We neglect the phase-shift differences. The transmission coefficients are taken from
[17] and equal Tt=0=0.459, Tt= a =0.519 (we neglect the
l = 2 contribution since Tl= 2 = 0.132 [ 17]). The asymmetry around 90 ~ c.m. is due to the non-vanishing corre-
i
I
30
~
i
I
60
i
~
I
~
90
i
I
120
i
i
I
150
i
i
180
OcM (~
Fig. 2. Centre-of-mass angular distribution in 1 8 4 W ( n , n ' ) inelastic scattering at E,= 14.1 MeV and E,, =0.8- 1.0 MeV outgoing energy. Data are from [14]. The curve is calculated with
B,r/Fclv= 10
lations between the S-matrix elements with the neighbouring spins ] J - J ' [ =
1. The fit (see Fig. 1) of the
shape of the angular distribution is obtained with
Be(tS+h/FcN)/h=lO and reproduces the forwardpeaking. From Fig. 2.1 of [6], we have t*~0.01 h/FcN,
and FCN~--40keV, so that/~---0.4 MeV.
Having extracted fie~FeN from the 2~ (n, n') angular distribution, we use the same parameters (including
transmission coefficients) to describe angular distributions in the 184W(n,n') reaction at E , = 14.1 MeV [14]
and 0.8 to 1 MeV outgoing energy. The theoretical curve
qualitatively reproduces the increase of the cross-section
in the region of forward angles (see Fig. 2). We would
like to note that the data in Figs. 1 and 2 are also remarkable since, apart from the forward peaking, they
demonstrate angular anisotropy (about 20% in the forward half-sphere in the 2~ (n, n') angular distribution
and about 20-30% in the forward and backward halfspheres in the 184W(n,n') angular distribution). Such
relatively big values for anisotropies in the considered
reactions are not consistent with the prediction of the
Ericson-Strutinsky formula [18] which yields at most
1-2% anisotropy (for the rigid-body moment of inertia
of the residual nucleus) for heavy targets.
We would like to note that at E , = 14.1 MeV initial
energy and 0.6-1 MeV outgoing energy the multistep direct reaction contribution is expected to be very small.
This is supported by the analysis [19] of the 93Nb (r/, n')
inelastic scattering with E, = 14.1 MeV. It has been shown
that the relative contribution of direct multistep reactions
into the differential cross-section is about 2% for 1 MeV
outgoing energy. Therefore the 20-30% forward peaking
in the experimental angular distributions in Figs. 1 and
2 can not be explained, from our point of view, as due
to the direct multistep reaction effects.
To illustrate the applicability of our formulation to
relatively energetic emitted particles, which are expected
to originate from the simplest n o = 3 exciton configuration, we fit the shapes of angular distributions in the
278
Bi(n, xn)
X
~84W(n,n,)
En = 11.5 MeV
En= 14.3 MeV
101
E'n (MeV)
\ --~m
m\
\
(33 10o
6-8
(xl/2)
m
u- 10
>
o~
c3
E
En, = 5.5-6.5 MeV
~3
e m
t~
9
~
z3
m
'k%,"
S-lO
(•
%
E
n
'
= 8.5-9.5 MeV
g
t:)
E n = 25.7 MeV
I
10
10
I
I
30
I
I
50
i
I
70
I
I
I
I
1
I
I
[
I
I
90 110 130 150 170
OCM (~
~
9
12-16
Fig. 5. Centre-of-mass angular distributions in 184W(n, n') inelastic
scattering at E,=I1.5MeV, E,,=5.5-6.5MeV and E,,=8.59.5 MeV outgoing energies. Data are from [23]. The curves are
calculated with #,0=3/'n0= 3= 0.6
10o
0
I
I
30
60
I
I
I
90 120 150 180
ec M (o)
Fig. 3. Centre-of-mass angular distributions in 2~ (n, n ' ) inelastic
scattering at different ingoing and outgoing energies. Squares, data
from [20]; dots, data from [21]; triangles, data from [22]. The curves
are calculated with fl,,o=3/Fno = 0.6
10
I"l'
.+
Jtdt` {,~jtEn, ~1 i-15 MeV
"T
2~ (n, n ' ) and 184W(H, n ' ) reactions for different initial
energies (see Figs. 3, 4, 5). In the theoretical fit we take
for simplicity spinless reaction partners in the entrance
and exit channels. We neglect the phase-shift differences.
All curves in Figs. 3, 4 and 5 are identical (up to the
different normalization factors for the different angular
distributions), are obtained with fl,o = ~/F,o = 3 = 0.6, and
are in qualitative agreement with data. The transmission
coefficients have been taken in the form Tl = exp ( - I / d )
(see Fig. 7 in [1]). The angular distributions in Figs. 3, 4
and 5 are practically unchanged under the variation of d
from 3 to 8. Using the exciton model estimation (see
Fig. 2.2 from [6]) F n = s ~- 1 MeV for the 2~ composite
nucleus with an excitation energy about 20-25 MeV we
find/"no=3 1.5F~=5 ~- 1.5 MeV. Therefore since the extracted ratio is ]3,0=3//"n0=3=0.6 , we obtain ]~,0=3
"~ 0.9 MeV. Thus our analysis of low energy and energetic
particles results in ft,0 ~ 2 #~ which is consistent with the
discussed above condition of smoothness of B~ values
with respect to n (we estimate f i - 2 5 - 3 0 for the reactions
considered). Also the results of our analysis are consistent
with the assumption that B,-values should depend
smoothly on the target mass and the initial energy. However we should note that our analysis of the shapes of
the angular distributions in Figs. 3, 4 and 5 has a qualitative character since in the emission of energetic particles the multistep-direct process (originating from the
non-zero energy-averaged S-matrix [1]) may contribute. For example in the 184W(n, n t ) inelastic scattering
with E , = l l . 5 M e V
and outgoing energies E n , =
5 . 5 - 6 . 5 MeV and 8.5-9.5 MeV (see Fig. 5) the contribution of multistep direct reactions was calculated [23]
to be about 20-30% for the forward angles and 40-50%
for the intermediate and backward angles.
~-
10
E
, +X+§
LU
7D
%
1
0.1
I t I i I I I i I r I i I i I r I
10 30 50 70 90 110 130 150 170
OCM (~
Fig. 4. Centre-of-mass angular distributions in 184W(n, n') inelastic
scattering at E, = 26 MeV, E,,, = 14-15 MeV and E,, = 18-19 MeV
outgoing energies. Data are from [23]. The curves are calculated
with fl~o=3/Fno= 3 = 0.6
279
In conclusion, we have presented a novel a p p r o a c h to
angular distributions in p r e c o m p o u n d reactions, with the
essential feature o f a correlation between fluctuating Smatrix elements carrying different spins. This correlation
weakens with time and increasing complexity o f the nuclear states partaking in the reaction. W e have presented
theoretical arguments and analysis o f the data which support this approach.
I am grateful to Prof. H.A. Weidenmtiller for enlightening discussions and suggestions. I would like to express my deep gratitude to
the Max-Planck-Institut fiir Kernphysik, Heidelberg, where this
work was initiated, for their warm hospitality during my stay on
MPI and Alexander yon Humboldt Fellowships. I am grateful to
Dr. F.C. Barker for careful reading of the manuscript and useful
suggestions. Thanks are due to Prof. A. Takahashi for making
Ref. [14] available to me. The support by CPP is gratefully acknowledged.
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