Download Particle emission from a hot, deformed, and rotating nucleus

Z. Phys. A 351, 397-404 (1995)
ZEITSCHRIFT
FOR PHYSlKA
9 Springer-Verlag 1995
Particle emission from a hot, deformed, and rotating nucleus
K. Dietrich ~, K. Pomorski 2, J. Richert 2'*
Physik Department, TU Munich, James Franck Strasse, D-85748 Garching, Germany
2Department of Theoretical Physics, M. Curie Sktodowska University, ul. Radziszewskiego 10, 20-031 Lublin, Poland
Received: 11 May 1994/Revised version: 14 December 1994
Abstract. The emission of nucleons from a hot, deformed
and rotating nucleus is treated within the Thomas-Fermi
approximation. We study in particular the dependence of
the transmission coefficient on the deformation and the
rotational frequency of the emitting nucleus. A tractable
form of the transmission coefficient is given.
N and Z is given by [2]
c2 (E*, I, N, z ) =
2Sv+ 1
2~zhp(E*, li
f dew~(e,l~;r)pR(E~,IR) "
(1)
IR--lI--lt~ ] a~-- Ae
PACS: 25.85.-w; 24.75.+i
1. Introduction
Whenever the excitation energy of a compound nucleus
exceeds considerably the binding energy of a nucleon or
a light composite particle (especially the c~-particle), the
emission of this particle competes favorably with the Yemission. Therefore, a detailed understanding of the particle emission is of great importance in the early stage of
the decay of highly excited nuclei. Such highly excited
nuclei are generally deformed and in rapid collective rotation. Consequently, the dependence of the emission probability of neutrons, protons, and ~-particles on the shape
and the angular momentum of the mother nucleus is of
great physical interest. A specific example is the competition between particle emission and nuclear fission in the
decay of highly excited medium heavy and heavy nuclei,
where the neutrons, protons, and ~-particles emitted prior
to fission carry interesting information on the fission
process [1].
The famous formula by Weisskopf describing the
decay widths FIB of a nucleus of excitation energy E*, total
angular momentum I, and neutron and proton numbers
This work is partly supported by the Polish State Committee for
Scientific Research and by the European Economic Community
(EEC) under contract No. ERBCIPACT93 1576. K.D. acknowledges support by the BMFT
*Permanent address: Physique Th6orique, CRN, BP20, F-67037
Strasbourg C~dex 2, France
The quantities e~,tp, S, denote the energy, the orbitM
angular momentum, and the intrinsic spin of the emitted
particle of type v 1. The quantities p and PR represent the
level densities of the emitting and of the residual nucleus.
The arguments E*, [R are the excitation energy and the
angular momentum of the residual nucleus. Finally, the
transmission coefficient ~'(e, l~; r) represents, for each type
v of particles, the fraction of the flux penetrating the
barrier with an angular momentum l~ and a final energy e.
The argument r denotes all further parameters upon
which the transmission coefficient depends. For the emission from a deformed, rapidly rotating nucleus, the transmission coefficient depends on the deformation and the
rotational frequency, i.e. the argument r represents a deformation parameter and a rotational frequency. It is the
purpose of this paper to investigate this dependence of the
emission width within the Thomas-Fermi approximation.
The transmission coefficients are either determined
empirically from elastic scattering on spherical nuclei [3]
or calculated on the basis of various assumptions, for
instance the WKB expression for the penetrability of
a barrier [4].
If the mother and daughter nucleus are deformed, not
only the level densities p(E*, 1) and pe(E~, IR) depend on
the deformation, but also the transmission coefficient
#"(e,l~; r). The dependence of the level densities on the
deformation of the nucleus is usually taken into account in
an approximate way (see for instance ref. [5]).
The dependence of the emission width on the deformation of the nuclear surface was considered in different
1The subscripts ~ and /3 are introduced, because in practice we
calculate the emission probability for a finite set of discrete values for
and I. ~ and fi denote a "cell" in a two-dimensional grid for e and I
398
approximations by various authors [-6-10]. In none of
these publications the distribution of the neutrons and
protons prior to emission was described by a Fermi-Dirac
distribution as we shall do in the following chapter, but
rather by classical approximations. A detailed discussion
on the relation between our theory and the one by Aleshin
will be given in Sect. 5.
2. Emission process in the Thomas-Fermi
approximation (TFA)
In the TFA the nucleus is described as a gas of free
neutrons and protons which are confined in the finite
nuclear volume and satisfy the Fermi-Dirac statistics. The
volume f2 with the surface Z may have any shape. In
addition we assume that the deformed volume is in rigid
rotation. For simplification we restrict ourselves to an
axially symmetric nucleus which rotates perpendicular to
its symmetry axis with a rotational frequency co.
We denote coordinates referring to the body-fixed
reference frame K ' by a prime' and we orient the 3'-axis in
the direction of the symmetry axis. The nucleus is assumed
to rotate around the Y-axis. The excited nucleus is described by a temperature T.
In the TFA the distribution of the neutrons and protons in the phase space is described by single-particle Wigner functions of the following form
2
00(~')
f,(2',~'; T ) = ~5. 1 + exp(-~(~-~ - Vo - coll- m))
2
(2)
In fact, strictly speaking, the "Thomas-Fermi approximation" doesn't imply this self-consistent procedure. Replacing the self-consistent potential ((2)) by a square well of
given form means that we replace the Thomas-Fermi
model by the simple Fermi-gas model. Since we want to
describe the emission from deformed nuclei, a self-consistent treatment would have to include shell corrections, i.e.
we would have to go beyond the TFM. Therefore, we
choose to stay with the simpler Fermi gas picture to start
with.
The chemical potentials #v for neutrons and protons
are determined from the conservation of the average neutron number N and proton number Z
jf~(2', fi') d3x ' d3p ' = N,
(6)
~fnO~:, ~') dax ' d3p ' = Z.
(7)
The quantity
. . .3. -- x 3 p 2
l'1 := Xzp
represents the orbital angular momentum of the nucleon
along the axis of rotation. The rotational frequency co is
related to the total angular momentum Ih by the condition
[h = ~ d 3 x ' 5
(3)
d3p'll. (f,(2', ~') + fp(2', ~'))
E*= f
d3x'~ d3p '[(fi'2-L\2mVo)(L(2',P',T)-f,,(Y',F';0))
+ \ ~ m - Vo + Vcb (fp(21,f; T ) - f p ( 2 ' , f i ' ; 0 ) )
where the step function 0o(2') is equal to 1 inside the
nuclear volume 0 and 0 outside of it
0o(2')
{10 f~ 2 ' ~ O
=
otherwise"
(4)
Henceforth, we leave away the argument T except in
Eq. (10) where it is important. The parameter V0
( ~ 50 MeV) represents the depth of the nuclear well. The
term - V o could be absorbed in the chemical potentials
g~,/~p. Since we shall discuss emission processes, it is more
convenient to incorporate - V o explicitly.
The term Vcb(2') represents the average Coulomb
potential felt by a proton at point Y' (co = elementary
charge)
Vcb(-~') =
y dBp ' y
d3 . . . . .
eg
Y f p ( Y , P ) i.~, ~ Y'I"
(5)
We shall see that this term influences the emission probability for protons. Of course, instead of replacing the
nuclear potential by a finite square well, we could also
determine the average potential V, acting on the neutron
selfconsistently on the basis of some effective nucleonnucleon interaction
V,(2') = ~ d~p'~ d3x'v(2', y ') [ f , ( y ' , ~') +fp(y',,F')]
(9)
implying that the average contribution from the intrinsic
spin is negligible.
Finally, the temperature T is determined by the average excitation energy E* of the total nucleus by the
relation
0o(2')
1 + e x p ( ~ ( ~ -- Vo + Vcb(x ) --col~ -- #~,))
(8)
(10)
We notice that the integration over the space coordinates
becomes trivial whenever we go to the limit co = 0 and
neglect Vcb(2'). In this case, there is no influence of the
deformation left given the fact that the volume f2 of the
nucleus should not depend on its shape.
In this limit the relation (10) takes approximately the
form
E* = (a,, + ap) T 2,
(11)
where a, and ap are the "level density parameters" for the
neutrons and protons (see f.i. ref. I-5]).
Let us emphasize at this point that, apart from fulfilling the requirements of the Pauli principle, the TFA treats
the dynamics classically.
The true quantum-mechanical Wigner function is not
a probability and is not positive-definite as a consequence
of the fact that momentum and position cannot simultaneously have sharp values. If the Wigner functions occurs
in integrals over space and (or) momentum coordinates,
its deviation from the classical distribution function often
turns out to be less relevant than if we were to consider it
locally in phase space. It is with this optimistic expectation
in mind that we are now going to describe the emission of
399
particles within the TFA. As we shall see, it has the great
advantage that the dependence of the transmission coefficient on the deformation and the rotation of the nucleus
can be described in a simple and transparent way.
For simplicity, we only consider neutron emission in
what follows. The treatment of the proton emission is
completely analogous and slightly complicated by the
presence of the Coulomb field.
In the spirit of the simple Fermi gas model the nuclear
potential V,~d(Y') is assumed to have a constant depth
Vo in the nuclear interior and to be zero outside
The body fixed frame K' rotates with frequency
co around the (common) 1-axis of the laboratory frame
K (unit vectors 0'i).
The intuitively simple classical relation (14) between
the normal velocities of the incident and emitted particle
can be derived from the energy conservation during emission
htl
_ ~t2
__ - -
2m
__
V 0 --
(o(x;2p'
3 -
X'o3Pt2)
-
Vnucl(~') = -- VoOo(~',)
as we anticipated already in the form (2), (3) of the distribution functions.
A neutron which hits the surface Z of the nucleus at
a point 2; with a normal velocity gi(Y;) is classically
emitted if the kinetic energy of its motion perpendicular to
the surface exceeds the well depth V0
m
> Vo
03)
and otherwise elastically reflected. We denote velocity
components of the emitted particle by a tilde (~).
After the emission the velocity perpendicular to the
surface v-+i(2;) is reduced due to the loss of kinetic energy
v• tXo) = 5 v• txo) - Vo,
(14)
whereas the velocity parallel to the surface is unchanged.
If ~'(Y;) is a unit surface vector at the surface point
2; pointing outward, the velocity components of the neutron before emission are given by
< (~;):= ~'-~'(~;),
1)j_(.XT0) .=
:=
..
v•
n (Xo),
. . . . . .
-
(15)
(16)
(17)
and after emission by
ff'(Y~a) = v, (Xo) g'(N;) + vjl(Xo),
(18)
g,i(Y;) = glT(~;).
(19)
The velocity ~' = ~' and the canonical momentum fi! are
related by the equation
--~/
--+!
~' = m~' + moo x x
(20)
and correspondingly for the emitted neutron
~' = mg--'+ mc~' x Y'.
~
(12)
(21)
~ ' --- cog = toe'1.
(22)
Note that g' is the velocity relative to the body-fixed
reference frame K ' spanned by the unit vectors e-~[
3
-g'= ~ 2jell
i=1
and equally for v'.
(23)
t
~t
= 2m - (o(x0~p3 -
!
~i
(24)
x o 3 P 2 ).
Assuming that the tangential momentum P~I!is unchanged
in analogy to (19):
=
(25)
The normal and tangential momentum vectors P~,Pll,
•
are defined in complete analogy to the Eqs.
(i5)-(18).
We define the local classical transmission factor
-(2
= 00
)
(Xo) - Vo ,
(26)
where 0o is the Heaviside function
{~
00(~) =
for ~ > 0
for ~ < 0'
(27)
The total number n of neutrons emitted per time unit is
then given by
l
3
l
~i
,l
~t
.cl
l
-+t
n = ~z da 5 d p fn(X0, fi') • (xo) . wo [o l (xo) 3,
(28)
where do-' is the infinitesimal surface element, and where
the canonical momentum fi' and the velocity g' are related
by (20). Since the integration variable is fi', one has to
express the normal velocity vl in (28) in terms of the
normal momentum.
The reason why we introduced the velocity g' in addition to the canonical momentum ~" is that the classical
emission probability (26) can be formulated more simply
in terms of the normal velocity.
We now evaluate the probability per unit time that
a neutron is emitted with given final energy
g =
i=1
Here, the angular velocity vector o3' is given by
,2
2m'
where Pl, P2, PB are the components of the final neutron
momentum in the laboratory frame K. Once the neutron
is emitted and thus beyond the range of the nuclear
potential, its momentum components in the laboratory
frame are constants of motion. Thus the neutron assumes
the final values of its momentum immediately after emission contrary to the proton which is still subject to the
long range Coulomb field.
Since the transformation between the space-fixed
frame K and the rotating frame K' does not change the
400
absolute value of a vector, the total energy of the emitted
neutron is given by
&=
-- =
i=12m
~=~2m'
(29)
where the momentum components/~[ in K ' can be taken
at any time after the emission. Choosing the time t = 0
just after emission, the momentum components/~[(0) of
the neutron just after emission are related to its components Pl just before by the solutions (14)-(22).
dn
The number of neutrons G
Ae emitted per time unit
with an energy in the interval 8= - ~ < e < e~ + ~ is defined by the expression
dY/
_.=
5 d a ' 5 d s p f., ( .x o. , p. . ). v .l ( X. o. ) w o, D• , ~,
d&"
z
"618~'--~//~/(0)21".
2m ]
(30)
Using the energy conservation at the surface point Y; (Eq.
(24)) we may rewrite the argument of the a-function in the
form
e= -
~ 1)[(0),2
~ p[2
- - &+ 17o + (ll - ~1) co,
i=* 2m
i = 1 2ram
(31)
barrier region as compared to the neutron, and thus also
the value of the Wigner function.
The form of the angular momentum constraint in (33)
arises from the fact that the absolute value le of the
angular momentum of the emitted particle does not depend on the choice of the coordinate system.
As the chemical potentials for neutrons and protons
are roughly equal in not too exotic nuclei, the Wigner
functionfv of a proton is smaller than the Wigner function
f, of a neutron at the same given point of phase space:
.... p )
L(~o,
(34)
- f , ( x.o. ,. . p ) < O.
The difference (34) of the Wigner functions at the nuclear
surface represents the main difference between the emission probabilities for neutrons and protons. In general,
this reduces the emission probability of protons compared
to the one for neutrons. The reduction is largest around
the waist of a spheroidal nucleus and smallest in the
vicinity of the poles.
We denote the total number of emitted protons by ~t,
the number per time unit in the energy interval
4~
drt (e~) .
& - T < 8 < & + ~ by ~
a8 and the number per time
unit emitted into this energy interval with an angular
momentum 1~ by
d2Tg(&'la) AsAI.
de~dl~
with
(11 - 8) o = ~&(p; - ~;) - 2&(pl - Pl).
(32)
For small values of co, the last term in (31) can be neglected
which simplifies the evaluation of the integral (30) considerably.
The number
Then the quantities ~z,a7s
d~t and ~d2~ are given by
3
!
--+l
rc = [. d a ' 5 d p L ( x o ,
--+t
el
r
~') v; (Xo)Wo [v•
~t
(35)
t-
--=!
dzc
de~
da'
, ~, p- , )vl(Xo)~ofV~(xo)]
, ~,
ct , -+,
~d 3p'f,,(Xo,
d2n
- -
de, dl;;
AsAI
of neutrons emitted per time unit with an energy and
9
"
*
A~
angular momentum comprised between the hmlts (~= - T,
zi,s
zJl
Al
9
.
& + T) and (le - g, l~ + T) is obtained from the expression
d 2rc
de~dl~
(36)
[
t
_ ~ d a , I -u3 p:~,tXo,~')v•
. . . . . . . .
d2/,/
d&dl := ~da' ardS"'~c:2
......
wCot Fv,(Xo)]
' ~'
u:,,t o,p)vl(Xo)
wSt [v•, ~,
Z
9a 8~
- v ;3;2(-~176 a[;~ - IT( ootl].
~
2m J
(37)
,5
[
9a 8 ~ - ~ - m
j.
E;e-12gx~'(0)l],
(33)
where one ought to choose l: = 0, 1,2, ... ,h and Al = lb.
The form of the angular momentum constraint in (33)
arises from the fact that the absolute value of the angular
momentum l~ of the neutron remains constant after emission9
The classical treatment of proton emission proceeds
analogously. It is however slightly more complicated, because the protons continue to feel the long-range
Coulomb field of the deformed nucleus after emission.
Thus the momentum components p~ in frame K continue
to be functions of time after emission. The dominant effect
of the Coulomb field is to change the potential in the
The formulae are analogous to the formulae (28), (30) and
(33) for neutrons and differ only by the explicit form of the
constraints. The final energy & and the final orbital angular momentum of the emitted proton are only attained at
time ~ = oo when the Coulomb potential has become
zero. The initial values for the trajectory calculation are
2; and ~'(0), where/~;(0) are related to the momentum
components Pl of the proton inside the nucleus in the
same way as for the neutron 9
The kinetic energy of the emitted proton at infinity
and just after emission differ by the Coulomb potential at
the surface point 2o:
/3,2(o0)
2m
i
;3?(o)
- ~-2m
+ Vcb(2'o).
(381
401
One can thus reformulate (36) in the form
67s
dc.~
-
z
~
f
da ~d3p'fp(xo, p )v•177
96 s~
~
2m
VCb(2;) ,
(39)
which shows that it is not necessary to perform a trajectory calculation at all.
The change of the orbital angular momentum of the
proton on the way from the nuclear surface to infinity is
due to the fact that the Coulomb potential is produced by
a deformed rather than a spherical nucleus. If the deformation is not too large, weexpect this change to be small.
In this case we may replace I(oQ) in Eq. (37) by the angular
momentum just after emission. In this approximation, (37)
may be given by the expression
dZrc
!
~!
(d~r ! ~d 3 pfp(xo,
p- + l )v2(2'o) w0cl D•, -~,
z
ds~d/~
(40,
Here, too, we need no evaluation of the trajectories of the
emitted protons.
Comparing the formulae (39) and (40) for the proton
emission with the formulae (30) and (33) for neutron
emission one sees that for given momentum ~'(0), the
proton ends up with a final energy e which is larger than
the one of the neutron by the Coulomb potential Vcb(2;),
an energy of the order of a couple of MeV. This implies
that the spectral distribution of the emitted protons is
shifted to higher energies by about this amount compared
to the spectral distribution of the neutrons. Quantum
effects tend to smoothen this effect.
W~
9
d2~t
~
~
~ d2~
ne expressmn ~ / 3 s ~ t
t~AsA1)
represents the
probability of emission per unit time of a neutron (proton)
with given final energy s~. and given final angular momentum I~. It is related to the Weisskopf formula in the
following way
Fig. I. Emission of a particle through the surface 2;. ~ symbolizes
the tangent plane at the emission point 2'o. The body-fixed coordinate frame is represented by its unit vectors (~"~,e"2,d'3). See further
explanations in the text
a plane wave by a potential step which coincides with the
tangential plane at the surface point 2;.
We introduce Cartesian coordinates (d,~7,~) with the
origin Ox at the surface point 2;, the d- an ~l-axis in the
tangential plane (g and (-axis coinciding with the surface
vector if'(2;) (see Fig. 1). If the rotational frequency co is
not t o o large, one may neglect the term -co[',. In this case
the hamiltonian for the neutron has the form
h 2
fi~ = - 2 5 Z ( 0 ~ + e , , , + o : : ) -
VoOo(-r
(43)
We consider a plane wave hitting the tangential barrier
from the side of the nuclear interior. Only the component
propagating perpendicular to the potential step is modified by the barrier:
@'~(~) = [e O'~/h + ~ e -~162 "0o(-~,) + J-e O;~/h 0o(0. (44)
Neglecting the influence of the rotation, the energy conservation implies
d2n
- -
ds~d/~
A~AI = F'~,
dZ~z
AsAI
de~dl~
-
(41)
P22
-.
- - - V o = -/~2z
2m
•
r~.
(42)
Let us note that in our derivation the spin degeneracy
factor (2S~ + 1) = 2 is contained in the definition of the
Wigner function (2) ((3)). The factor ~ in Eq. (1) enters the
Weisskopf formula as quantum unit of phase space similarly as in our Eq. (2) ((3)).
3. Quantum-mechanical correction of the classical
transmission factor
The classical transmission factor w~~cannot be expected to
be realistic whenever the energy of the nucleon is close to
the threshold of emission. We calculate a quantum-mechanical correction for the local transmission coefficient
wCol(26) by considering the transmission and reflection of
(45)
2m
The amplitudes .~ and -Y- of the reflected and transmitted
wave are obtained in the standard way. The ratio ,A,q,,,
,vo of
the transmitted current to the total incident current is
given by
q,, = ~l.y_12
W 0
p•
4p•
02 + ~,)2
0~o(/~i).
(46)
It is only non-vanishing, if the normal momentum/~2 of
the transmitted wave as obtained from (45) turns out to be
a real (positive) number. We indicate this explicitly by the
Heaviside function 0~o(/52).
If we are to take the rotational motion into account,
the quantum mechanical calculation becomes much more
complicated, because the eigenfunctions of the hamiltonian
/~" = h~ - co]'1
(47)
402
are not plane waves. Given the fact that we only aim at
a quantum correction of an otherwise classical theory, we
replace the operator l~ in (47) by its classical value. In this
approximation, the eigenstates of/~" continue to be plane
waves with the momentum fi' for the incident wave/~' for
the transmitted one. From the conservation of energy (24)
and the constancy of the momentum component Pll parallel to the tangent plane (see eqn. (25)) we find the following
relation between the normal momentum components
Pi and p•
--
2m
-
Vo -
c o p l b(~'o) = - -
2m
-
cof'• b ( X ; ) ,
(48)
where the quantity b is defined by
(49)
b(~;) = ~6~(~'. ~;) - x ; 3 ( ~ ' - ~ i ) .
g' is the unit vector perpendicular to the surface S at
the surface point Y6 and g[ are the unit vectors in
the direction of the axes of the rotating frame. From (48)
we obtain the normal momentum /~j_ of the emitted
neutron as a function of the normal momentum fi'~ of
the incident one
P'l = mco(2'o) +
,/(,Pi
- mcob) 2 - 2mVo.
(50)
The quantum-mechanically corrected form of the
transmission coefficient continues to be given by formula
(46).
The evaluation of a corresponding quantum correction for the proton is more complicated because of the
appearance of the Coulomb potential VCb in the hamiltonian h ~ of the proton
h2
h~ = ~m (~?~ + ann +acr - VoOo( - 0 + Vcb -- co'J1, (51)
where Vcb is to be calculated at the space point defined by
the coordinates (~, ~/, r which determine its position with
respect to the surface point Y6. As the Coulomb potential
is smooth and changes most strongly in the direction
normal to the surface, one may approximate Vcb in (51) by
its value for the coordinates (~ = 0, q = 0; ~):
(52)
v~ ~ v~(xo
+ Cn (xo)),
the function Vcb being given by Eq. (5).
Introducing again the classical approximation for the
term -col~ in (51), we arrive at the problem of a 1dimensional barrier penetration described by the effective
hamiltonian he~ff
h2
hC~ff -
2m ar162+ Vcb(Y'o + ~g')
+ [ - Vo
-
co(x62p'3 - X'o3P'2)] @o(
co(xo~p~ -
x o ~ p ~ ) 0o(~).
~)
(53)
The influence of the co-dependent terms is again expected
to be negligible for small enough co. One can use the WKB
for evaluating the transmission coefficient w~.
4. Averaging transmission coefficients over the surface
and results
The evaluation of the integrals (30) and (33) is technically
complicated, because it involves 4- and 3-dimensional
integrations. Therefore, as a first crude approximation, we
replaced the distribution functionf"(2~, fi') in eqn. (33) by
a constant. Since we used a deformed square well for the
nuclear potential, f" is by definition independent of )7'
apart from the small Coriolis term (see Eq. (2)). Neglecting
the momentum dependence off"(U, fi') is certainly a poor
approximation, which is expected to falsify the magnitude
of the average transmission factor #"(a~,lp). We hope,
however, that it does not influence appreciably the dependence of #" on the deformation.
The simple approximation for f, makes us lose the
temperature-dependence of the average transmission factor. Certainly, this implies that the present results are only
meaningful for large temperatures where the dependence
o f f . on the momentum ~' is smooth.
Expression (46) gives the local value of the emission
probability at a given point of the surface X and a given
value of the momentum fi' of the particle. As expressed in
(1), we need the transmission probability for a particle
with fixed energy a and angular momentum I in the laboratory frame whereas the calculations have been performed
in the body fixed frame. Notice however the fact that
e = e' and ll = l] (e, I and the projection of-/on the 1-axis
correspond to the laboratory frame, the primed quantities
to the body-fixed frame). Hence we can calculate the
probabilities by proceeding in the following way. We fix e,
I and we take all values of l[ in the interval [ - l , +l]. For
each set {e, l,l~} and fixed point 26 on the surface we
determine ps (k = 1 to 3) compatible with the fixed energy
and angular momentum. Knowing (.~', 2;) we calculate
w through (46). The value of the transmission coefficient is
averaged over all allowed orientations of the angular
momentum of emitted particles. This calculation is repeated for fixed e and l over a dense mesh of points covering
the whole surface. The transmission coefficients u?~(a,l; r)
appearing in (1) are obtained by averaging the values
obtained for w over the whole surface over which this
quantity is different from zero (here r stands for the angular velocity co, the surface deformation parameters, the
total number of particles). In a similar way one may also
determine an average value <11> of the angular momentum projection and its square <l~> which must be
known for the calculation of the rotation energy of the
emitting nucleus on its fission path.
Expression (46) for fixed e, l and co has been used in
order to get the transmission coefficients for the emission
of neutrons from 126Ba. The shape of this nucleus has
been defined by using the Trentalange-Koonin parametrization of the surface [11]. Calculations performed in the
deformed case correspond to a ratio of the lengths of the
half-axes equal to 1.7. We present calculations corresponding to a nucleus at rest (co = 0) and a nucleus which
rotates with hco = 0.8 MeV which corresponds to an
angular momentum of approximately 60 h. The depth of
the mean potential Vo is fixed to 50 MeV.
Fig. 2 shows the behaviour of @ for an angular momentum l = 0 of the emitted particle. The transmission
403
1.0
1.0
, , , l l , , . ' , l , , , , i , , i ,
(Q)
0.8
j~
,~
0.6
0.6
j///
Ii/11
/
0.4
0.4
/
0.2
0.2
/
0.0
,
I=0
, , , . | 1 1 1 , t , , I , 1 . , , ,
0.0
5
10
15
20
~//'/
/
/
,
i
I
|
"
,
9
5
,
kll
,,I
I
(MeV)
10
]
15
I
"
I
20
1.0
(b)
0.8
t
0.8
0.6
~: 0.4
/
i"/
I
0.2
c0=O.8Me
v
/j
/P t
0.4
l//
0.2
0.0 ~
0
/
5
10
~: ( M e V )
15
20
Fig. 2a + b. Transmission coefficient u? as a function of the energy
e of the emitted neutron for l = 0 and angular velocity co of the
nucleus. Case (a) corresponds to co=0 and case (b) to
hco = 0.8 MeV. The dashed and fully drawn curves correspond to
emission fiom a spherical and deformed nucleus, respectively
coefficient increases with increasing energy of the particle.
The emission from the spherical nucleus is favored with
respect to the deformed one. This effect is particularly
strong in the energy range (2 10MeV). On the other
hand, the emission probability is not strongly affected by
the rotation of the nucleus. A close inspection shows that
for h e ) = 0.8 MeV the emission is slightly favored for
e < 4 MeV and the reverse is true for higher energy. The
situation changes quantitatively when l # 0, as can be
seen in Fig. 3 which shows cases corresponding to 1 = 4.
The spherical nucleus does not emit particles below some
particle emission threshold whereas the particle can escape from the nucleus for any energy of the particle in the
deformed case. These effects may have a sizable influence
on the calculation of the emission widths (1). The effect of
the nuclear rotation is quantitatively important (~ = 0 for
the spherical case and # ~- 0.27 for the deformed case at
= 10 MeV). The difference becomes smaller for increasing emission energy e.
In Fig. 4 we show the dependence of # on l. F o r an
emission energy e = 10 MeV the rotation of the nucleus
again does not affect the emission rate sizably. For both
//
I= 4 h
0.0
5
10
~: ( M e V )
15
20
Fig. 3a + b. Same as Fig. 2 for I = 4. See discussion in the text
co = 0 and co r 0 the
higher I values in the
one. This is expected
gated along its axis
rotation axis.
tail of the emission rates extend to
deformed case than in the spherical
since the deformed nucleus is elonof symmetry perpendicular to the
5. Conclusions and discussion
We have worked out a simple semi-classical model
which allows to calculate the transmission coefficients
which enter the emission widths (1) of a particle from
a rotating nucleus with an arbitrarily deformed surface.
This expression may be used in actual dynamical calculations which take care of prefission particle emission. Our
preliminary numerical results show that deformation
effects indeed strongly affect the transmission coefficients.
On the other hand, the effect of the nuclear rotation is
found to be weak, even for large rotation velocity.
Our evaporation model is similar in spirit to the
Richardson theory describing the emission of electrons
from a hot metal. In our case, an additional complication
is due to the deformed shape of the emitting body and to
its rotation in space.
404
1.0
'
9
'
~
I
"
9
9
*
I
'
'
9
9
I
'
~
'
"
!
(o)
-~
0.8
'
'
"
"
I
~
"
'
"
( ,p)=C.exp
~,=o
a=lOMeV
-
\~mm
V(2')-col;
,
(54)
where the normalisation factor C is given by the ratio of
the level densities PR and +plof the daughter and m o t h e r
nucleus times the factor ~
as in the Weisskopf formula
(see Eq. (1))
0.6
V
2S~ + 1 DR(E~, IR)
27rh p(E*, I)
0.4
c =
0.2
Using the Fermi gas formula for PR and p, Aleshin obtains
.... !,
0.0
0
1.0
....
2
''''
4
....
6
P~ = e -s/T,
8
i(h)
10
12
~=O.8MeV
l , , , , l , , ' ' l ' ' ' ' l ' ' ' ' i ' ' ' '
(b)
a=lOMeV
0.8
,
,
0
2
\
0.6
v
0.4
0.2
0.0
4
,
6
8
,
10 12
I
.
,
,
,
Fig. 4a + b. Transmission coefficient w as a function of the angular
momentum I of the emitted neutron for fixed e. Case (a) corresponds
to co = 0 and case (b) to he) = 0.8 MeV
So far, we only calculated the average transmission
factor #"(e~, la) for neutrons assuming a constant distribution of the neutrons prior to emission. In the next step, we
shall i m p r o v e this calculation by using a m o r e realistic
phase space distribution of the neutrons and p e r f o r m
analogous calculations for protons and c~-particles. We
also will calculate the angular distribution of the emitted
particles assuming that the decaying nuclei are partially or
completely polarized. Finally, it is tempting to investigate
the effect of the q u a n t u m corrections in a m o r e systematic
way than as done in chapt. 3, including the effect of shell
corrections.
Of course, there is also the question whether nucl e o n - n u c l e o n collisions will have a noticeable effect on the
emission rate. It is difficult to foresee the o u t c o m e of such
an investigation which one could base on the Bethe-Goldstone equation at finite temperature.
Finally, we would like to c o m m e n t on the relation
between our work and the one of Aleshin [10]:
In Aleshin's work the distribution of the nucleons
inside the nucleus is described as a classical gas of particles
m o v i n g in the average potential V(U) at a given temperature T. In our notation, Aleshin's distribution has the
form (see 1st of the references [10])
(55)
(56)
P
where S is the (empirical) separation energy of the emitted
particle from the m o t h e r nucleus in its ground state.
This has to be c o m p a r e d with the F e r m i - D i r a c distributions (2) and (3) in our theory:
In the classical limit, our distribution f ( 2 ' , fi') becomes
equivalent to the one of Aleshin with the difference that we
obtain the chemical potential ~t a t finite temperature T instead of the negative separation energy ( - S) (see eqs. (56)
and (54)). If the temperature T is small, the chemical
potential is almost equal to the negative separation energy
S from the ground state and, furthermore, the ratio ~T ~ S
is large c o m p a r e d to 1. In this case, our theory becomes
identical to the one of Aleshin. As the temperature rises, the
absolute value of the chemical potential becomes smaller.
When ~T becomes comparable to 1, the F e r m i - D i r a c
distribution will yield a different emission probability from
the classical one. In a later paper we shall investigate the
differences in detail as a function of the temperature.
One of us (J.R.) acknowledges a grant for three months from the
Commission of the European Community DG XII under contract
no. ERBCIPACT-93-176 which allowed him to stay and to perform
the present work at the Institute of Physics, M. Curie-Sktodowska
University in Lublin, Poland. K.D. acknowledges gratefully the
support during a two week stay in Lublin by the Marie Curie
Sktodowska University and support by the BMFT.
References
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1159