NUCLEAR
PHYSICS A
Nuclear Physics A560 (199312.53-273
North-Holland
On the calculation of transition probabilities
with correlated angular-momentum-projected
wave functions
and realistic forces *
J.L. Egido, L.M. Robledo and Y. Sun
Departamento de Fisica Tebrica C-XI, Vniversidad Autdnoma de Madrid, E-28049 Madrid, Spain
Abstract: In this paper we propose the use of angular-momentum-projected
generator coordinator
method (GCM) wave functions for the evaluation of transition probabilities in heavy nuclei. We
derive the relevant equations and discuss ways to cope with the technical difficulties which
appear in the application of the theory. We show the feasibility of the method by applying it to
the calculation of B(E3) transition probabilities in light nuclei within the GCM, in the gaussian
overlap approximation (GOA). In the calculations we use the density-dependent Gogny force.
The theoretical projected results are in much better agreement with experiment than the
unprojected ones.
1. Introduction
The calculation of transition probabilities in nuclear physics is a problem whose
magnitude has grown considerably in the last decade. This is because we are
continually challenged by new experimental results in very soft nuclei where the
traditional random phase approximation (RPA) loses its validity. The actual
limitation is even larger as it can be inferred by the fact that the only RPA
calculations of transition probabilities available in the literature, with Skyrme or
Gogny forces, are for closed shell nuclei. A typical example which illustrates the
situation is the evaluation of the transition probability between states in two wells
of different deformation. The usual case, however, is the calculation of transition
probabilities when anharmonicities are present. This is the situation we have
chosen in order to illustrate our theory, specifically we consider the case of
octupole correlations. We stress, however, that our theory is completely general
and can be applied to different cases.
Correspondence to: Dr. J.L. Egido, Departamento de Fisica Tedrica C-XI, Univ. Autonoma de
Madrid, E-28049 Madrid, Spain.
* Dedicated to Prof. Dr. Hans A. Weidenmilller on the occasion of his 60th birthday. Work supported
in part by DGICyT, Spain under project PB88-0177 and PB91-0006.
03759474/93/$06.00
0 1993 - Elsevier Science Publishers B.V. All rights reserved
254
J.L. Egido et al. / Transition probabilities
In the last years much progress has been achieved in understanding the role of
the octupole degree of freedom in nuclei. It has turned out in a variety of
situations that one must explicitly consider the octupole correlations to fully
understand the underlying physics. One has found out, in particular, that even if
the concept of permanent octupole deformation does not apply in most nuclei,
fluctuations in this degree of freedom are very important.
On the theoretical side, a large part of our knowledge on this topic is based on
energy considerations based on potential energy curves. From these curves we
have learned about the height of the barrier between the symmetry-breaking
minimum and the symmetry-conserving one. The calculations are usually done in
the mean-field approach by constraining the nucleus to have a given octupole
moment (breaking thereby the spatial parity symmetry). There have been several
approaches at this level: the phenomenological
Nilsson (folded Yukawa or
Woods-Saxon) plus Strutinsky 1-5) as well as the self-consistent q,-constrained
calculations (HFB or HFBCS) with Skyrme III plus monopole pairing 6> and with
Gogny forces 7-9) . In these types of approximations other static properties such as
dipole moments can be evaluated, too. All these calculations predict, at least,
octupole instability for nuclei around 222Th and 144Ba.
To describe the energy splitting Of-l(3-j, however, one has to take into
account correlations beyond those included in the mean-field approximation. Such
calculations have been done in refs. l”,ll) within a many-body theory with separable
forces and in the ATDHF + ZPE (GCM + GOA) with the realistic Gogny force
12*13).The latter one, which describes all the bulk nuclear properties and has no
adjustable parameters, has the advantage of making reliable predictions. In this
approach we have done extended calculations in the light actinides ? and in the
lanthanides 12). Since the collective hamiltonian obtained in the ATDHF + ZPE
(GCM + GOA) approach and the parity operator commute, the energy eigenstates
have good parity. This allows us to calculate the mentioned splitting as well as
transition probabilities. The overall results we have obtained in these calculations
are very satisfactory.
While the results for the energy splitting are valid under very general assumptions, the formula used to calculate the transition probabilities is based on the
rotational model approximation; hence, its use is restricted to well-deformed
nuclei. This condition is satisfied by most of the nuclei of the extended calculations
of refs . ‘,i2) . There are many nuclei, however, where this condition does not apply;
namely, for spherical and transitional nuclei and also for those deformed nuclei
that are soft towards small deformations. It is important to realize that in GCM
type of calculations the transition probabilities are calculated as averages over all
constrained values. That means, there will always be some points in the calculation
for which the rotational model approximation is not a good one. The larger the
weight of these points is in the wave function, the worse the approximation will be.
J.L. Egido et al. / Transition probabilities
255
In this paper we propose a method to remedy this problem; i.e. angularmomentum projection applied to GCM-type wave functions. It is important to
notice that with this method we are able to calculate transition probabilities in
many nuclei where the RPA is not a good approximation or it is very difficult to
apply as in all open shell nuclei. Calculations with GCM wave functions but
without angular-momentum projection have been done in ref. 14).
On the experimental side there are many data which are not at all understood.
As an example, we would like to mention a very intriguing problem: the experimental results on the ,sPt, soHg, and s2Pb isotopes. These three isotopic chains
differing one from each other in two protons have B(E3) transition probabilities
which vary from 4 to 40 W.U. These unexpected results have been analyzed by
several authors “-l’) without success. In this paper we do not investigate this
problem because the main interest is to present the theory as well as the different
methods to cope with several difficulties which appear in its practical application.
However, and in order to illustrate the method, we shall apply the theory to
compute the B(E3) transition probabilities in the nuclei 160 180 and 14C, which in
a small scale display similar features to the Pt, Hg and Pb isotopes. The size of the
calculations (CPU time) is, however, one order of magnitude smaller.
The paper is organized as follows: In sect. 2 we discuss in some detail the
theoretical approaches. The configuration space, forces etc. are explained in sect.
3; in this section, furthermore, the main results are discussed and compared with
the available experimental data. Our conclusions are presented in sect. 4.
2. Theory
2.1. DERIVATION
GOA
OF THE
COLLECTIVE
HAMILTONIAN
WITHIN
THE
GCM
WITH
THE
In this section we briefly review the derivation of the collective hamiltonian
within the GCM formalism. Our main intention is to set the notation in order to
derive, later on, the formulae for calculating projected transition probabilities. For
a detailed description of the GCM, we refer the reader to ref. 18) which we closely
follow.
In the GCM the trial function I !P> is written as a continuous superposition of
the generating functions I q(a)), which are labeled by the continuous parameter a,
the generator “coordinate” *, i.e.
(1)
l
To make the derivation
simpler
we restrict
ourselves
to the case of just one coordinate.
J.L. Egido et al. / Transitionprobabilities
256
The wave functions ) cp(a>>can be calculated, for example, by solving the appropriate constrained HFB equations. The weights f(a) are determined by minimizing
the energy. This variational principle leads, as it is we11known, to the Hill-Wheeler
equation
Xf=df
(2)
with the overlap functions
zyu, a’) = (cp(Q>
I A 19@‘)),
M(a,
a’) =
(3)
<P@>lew-
(4)
Eq. (2) is rather difficuIt to solve. One way out to find the weights f(u), at least
in an appro~mative way, can be found when the norm overlap &u, a’) can be
written down in the form
a’)
N(U,
=~(q
+
is,q -
3s) = exp( -@(q)s2)
with q and s being the center-of-mass and relative coordinates,
s = a - a’. G(q) is related to the operator P = - (~/~)~/~~ by
(5)
4 = ;(a + a’) and
(6)
In the case of a gaussian overlap one can derive easily an equation of motion for
the coordinate 9 because the matrix JL”‘f2 can be calculated analytically, i.e. one
can write
&a,
a’) = /dg
fi~@/~(u,
q)H”2(q,
a’)
(7)
The expectation value of the hamiltonian with the GCM wave function eq. cl),
is given by
=
I
da da’ f*{a)Z(a,
= /-da da’
f *(a)/dq
u’)f(a’)
G/@/2(
a, q)h(a,
+@‘2(4,
a’)f(a’)
(8)
in an obvious notation. Under the assumption ~@“‘/~(a, 4) is sharply peaked at
q,
we
can expand h(u, a’) around a = u’ = 4 up to second order in the
u =
J.L. Egido et al. / Transitionprobabilities
differences (a -
q)
and (a’ -
q).
25-l
One obtains
(9 I HI ‘J’) = /dq 6b*(q)zma
(9)
with
d(q) = j-d” Jy”2(q, 4fW
(10)
and
B(q) is the mass parameter associated with the collection motion along q,
is the HFB energy V(q) = (q(q) ( I-? I (p(q)) and E,Jq) is the zero point energy
(ZPE) correction. The expression for B(q) and EO(q) can be found in ref. 12>.
In this way the problem of solving the Hill-Wheeler equation eq. (2) has been
reduced to solve the following Schrodinger equation for the collective wave
function:
where
V(q)
~4L(q)
The eigenfunctions
G(q)
4,(q)
(12)
=%4,(q)*
of eq. (12) are normalized
j-dq dG(q)4344&)
= k,,.
to one with the metric
(1%
The derivation we have presented is for a generator coordinate a(q), in this
paper we are mainly concerned with the coordinate associated with the octupole
moment so we shall constrain on q3. The formulae below are given for this case. If
the quadrupole fluctuations are important one should additionally constrain the
quadrupole moment q2. The corresponding formulae can be found in ref. 13).
It has to be mentioned that a collective Schrodinger equation can also be
obtained from the ATDHF theory 19-21) after quantization of the semiclassical
hamiltonian for the slow-moving collective degrees of freedom. The collective
hamiltonian obtained in this way has the same functional form as the GCM + GOA
one, but the expression of the collective parameters is different. The set of
parameters used in our calculations is an admixture of the two and it is known as
the ATDHF + ZPE set 22), It includes the mass parameter B(q,) coming out from
the semiclassical hamiltonian of the ATDHF theory, the metric of the GCM +
GOA and the ZPE correction that has the GCM + GOA form but uses the
258
J.L. Egido et al. / Transitionprobabilities
ATDHF mass instead, i.e.
(14)
This set of parameters was devised to put together the advantages of the ATDHF
set (time-odd components included in the mass term) and the ones of the
GCM + GOA (ZPE correction). This method can be somewhat justified in the
context of the extended generator coordinate method of Villars 23). Additional
justification for this choice can be found in ref. 9).
The calculation of the collective parameters involves the inversion of the HFB
stability matrix. At present, this is a formidable task and some approximation is
needed. The approximation used in this paper, the “cranking approximation”
24,25,22),neglects the off-diagonal terms of the stability matrix. This allows us to
invert it analytically at the cost of including the two-body interaction only through
the mean field. Using the cranking approximation, the ATDHF + ZPE parameters
are given by
M-*(q3)
G(q3)
= 2Ml,(q,)
where the quantities
M_Jq3)
(n =
Kn(q3)
’
B(q3)=
M-3(%)
M%q3)
’
(15)
1, 2, 3) are given by
=c
k,l
(Ek+El)n'
(16)
In the above expression, E, are the quasiparticle energies and <Q,‘“>,, =
( (p(q3) I a,ako3 I (p(q3)) are the matrix elements of the octupole operator Q3a in
the quasiparticle basis {(Ye, (Y:} of the HFB wave function 1(p(q3)) [ref. ‘91.
An interesting characteristic of the collective hamiltonian for the octupole
degree of freedom is that all the quantities entering into its definition (eq. (11)) are
even functions under the exchange q3 + -q3. Therefore, 2c0,, is invariant under
such exchange and it is possible to classify its eigenfunctions, +a(q3), according to
their parity under the q3 + -q3 exchange. It is easy to see that the parity of the
collective wave function under the q3 --, -q3 exchange corresponds to the standard spatial parity operation in the correlated wave function built up from 4,. The
inclusion of octupole correlations immediately restores the parity symmetry lost at
the mean-field level.
Therefore, the solution of the collective Schrodinger equation (CSE) (12) allows
the calculation of the 0+-1-(3-l
energy splitting. The corresponding B(E1) and
B(E3) transition probabilities connecting those states can be calculated in two
different approximations as we shall see in the next two sections.
J.L. Egido et al. / Transition probabilities
2.2. CALCULATION
259
OF TRANSITION PROBABILITIES WITHIN THE GCM WITH THE GOA
The solution of the collective Schrodinger equation (12) provides eigenvectors
42, where 7 denotes the parity (+ or -> and the subscript cr labels the different
states of a given parity. The inversion of eq. (10) (see next section) allows the
calculation of the weights f(4) of the GCM ansatz eq. (1) which fully determine
the wave function I Fz>. These wave functions, as indicated, do have good parity
but, in general, they are not eigenstates of the angular momentum operators. If we
restrict ourselves to constrain only the d3,, (and (izO when necessary) component
of the multipole operators, then the third component of the angular momentum is
conserved. We shall denote it by K. The corresponding wave function is given by
I qg.
In the limit of strong deformation, we can define an intrinsic system in which
the wave function ( Pi) can be interpreted as an intrinsic wave function. To
calculate transition probabilities in the lab system we shall transform the matrix
elements in order to express them in terms of the corresponding intrinsic analogues.
The wave function, in the lab system, with angular momentum I, A4 is given by
(17)
The multiple operator of order LA4 in the intrinsic system is related to the one in
the lab system by
with S_,&, being the Wigner function. The reduced matrix element entering the
expression for the transition probability is given by
X
Zf
L
Ii
-K,
M’
K,
In the special case K, = K, = 0 we obtain
(T&d“4% I%lK)*
( 19)
J. L. Egido et al. / Transition probabilities
260
Finally, for the B(EL) reduced transition probabilities to the ground state, we get
B(EL,Z"i+O+)=
with QLO= y’4P/(2L+‘YL,.
For the special case of L = 1, we have or0 =8,
with 6 being the dipole operator.
To calculate the matrix element <!PGf I QLOI ‘PGi) we proceed as with the
evaluation of the energy in eq. @), i.e.:
=
/ da
= /da
da’ fGf*(
a)(p( a) I Om14~‘) )fCi(a’)
&%~@‘~(a,
da’f’;‘*(a)/dq
q)&a,
a’).N”“(q,
a’)fli(a’).
(22)
Now we expand &a, a’) around a = a’ = q up to second order in the differences
(a - q) and (a’ - q>. It turns out 26) however, that for one-body operators which
behave smoothly as functions of a and a’ it is enough to stop at zero order l. The
result is
(23)
with
Q,(q) = (44
I &,, I cp(q))
(24)
being the expectation value of the multipole operator with the HFB wave function
satisfying the corresponding constraint. For example, in our calculations we constrain the octupole moment, in this case it must satisfy ((p(qJ I d3,, I qo(qJ) = q3.
In this way expectation values and transition probabilities are easily given in the
above formalism. For example, the quadrupole moment q2 of the nth excited state
is given by:
q2
=
/dq,
,/‘ml&z(q,)
12Q2(qd
(25)
with
Q,(s,>
l
= (cp(q,)
1020
1 P(d)-
(26)
We have checked this approximation
by calculating
numerically
the quantity &a, a’) in some nuclei.
The results obtained support the assumption
that &a, a’) behaves smoothly as a function of a and
a’.
J.L. Egido et al. / Transition probabilities
and the reduced transition probabilities
O+ ground state are given by
B(E1, l-*
0’) = ;
I/dq,
from the lowest l-
261
and 3- states to the
~~0*(4s)D(41)$:(4~)1*
(27)
for the El electric transition and
for the E3 one. In the above formulas II
and Q3(q3) are the mean-field
expectation values of the dipole and octupole operators, respectively, at the point
q3*
In the light of eqs. (27) and (28) we can now clearly see the unsuitability of these
formula to calculate transition probabilities in nuclei with a spherical minimum. In
these nuclei the GCM wave function is located around the minimum where the
multipole moments are zero. The main contribution to the transition probabilities
comes, therefore, from the tails of the wave function. We expect, therefore, to get
very undervalued B(EL) in the case of spherical nuclei.
2.3. CALCULATION OF PROJECTED TRANSITION PROBABILITIES WITH GCM WAVE
FUNCTIONS
In this section we shall develop the calculation of reduced transition probabilities with angular-momentum-projected
GCM-type wave functions.
We assume that the weights f,“(a) of the wave function eq. (l),
(29)
are known. It does not matter whether we have them from the Hill-Wheeler
equation (2) or from the solution of the collective hamiltonian equation (12). In the
latter case, however, we have to calculate the weights f,“(a) from +,“(a), what can
be done by inverting eq. (10). In this case one proceeds as follows:
Although the GCM is formulated by using continuous coordinates, in actual
calculations one always has to deal with a set of discrete points along the
generating coordinate. In this case _.&“(a, a’> is no longer a two-dimensional
function but a matrix, A”~,~,, and eq. (7) has to be written as
262
To determine
J.L. Egido et al. / Transition probabilities
J?;,{’ = JV,$~G:/~ we first diagonalize Jv,,,
NW=
Wn,
and then the eigenvalues and eigenvectors of J’ are used to write the square root
of the norm as
where the sum in s is restricted to those states with n, greater than a cutoff
parameter E. This parameter is introduced to deal with the problem of redundancy
in the I q(a)) basis. The discussion concerning how to determine the value of E
will be done in sect. 3.
In matrix form eq. (10) can be written as
which can be inverted to give
fl = C”4y’dk
k
where
Jl:k
‘I2 = C Wk,n;‘/2~s*.
We now turn to calculate the projected transition probabilities with the GCM
wave functions. The wave function (29) has good parity but it is not an eigenstate
of the angular-momentum operator. We can obtain such state by angular-momentum projection ‘*I, i.e.,
In the general case the coefficients g, are determined by the variational principle.
In the case of axial symmetry and even-even nuclei the ansatz above reduces to
I a, ZM) =
with the normalization
lv,-y2&&_g
(32)
constant N,,, given by
(33)
J.L. Egido et al. / Transition probabilities
The reduced transition probabilities
B(EL,
263
B(EL) are given by
e2
Zi + Zr) = ___
(yf, zr IIAL II(yiYIi)l’
21, + 1
‘(
(34)
and the reduced matrix elements by the relation
c
=
g,g,’
Ki,Kf
=2
c
K,,K,
g,g,!
c ( -)‘f-Kf(2zf+l)-’
;,,
M’,M”
To obtain this expression we have made use of some well-known properties of the
ghK functions, which can be found, for example, in sect. (4.6) of the book by
Edmonds 27). In case of axial symmetry (Ki = K, = 0) and for transitions to the
ground state of even-even nuclei CZr= Mf = O), the expression above reduces to
((yf, Zf 11
.AfL 11
(Yi,
li> = 2N&,142N~~,lj(2(
-)“C2’i
with Z?(p) = exp(--IpJ1).
If we now substitute the wave functions
we finally obtain
+
1)-1’2
I Pz> of eq. (29) into this expression,
J.L. Egido et al. / Transition probabilities
264
and the normalization
N,,=
constants are given by
/da da’ fb(a)[/dP
sinP dt&(P)(cp(a)
I ~(b-9
I4a’))].f&‘).
(38)
Substitution of these expressions in the formula for the reduced transition probability, eq. (34), provides the final result.
2.4. EVALUATION
OF MATRIX
ELEMENTS
In the last section we have seen that in order to compute projected
probabilities we must calculate the following matrix elements:
transition
(39)
where $ can be any one-body operator and I cpl>, I cpz> stand for any two different
HFB wave functions. To compute these quantities we will use the generalized
Wick theorem and the Onishi formula “1. The natural single-particle basis to
calculate such matrix elements is the one with spherical symmetry l. The corresponding formula are very well known in the literature 18V28P29).
In this paper, in order to get a better description of the mean-field wave
function, we are using an axially symmetric harmonic-oscillator basis with different
oscillator lengths. Because of this, even when our basis contains all the states
within a given number of major shells, the rotation operator takes us outside of
this subspace. The reason is that an axially symmetric basis with different oscillator
lengths including N, major shells is equivalent to an axially symmetric basis with
identical oscillator lengths but having states belong to major shells greater than
N, ** (see appendix A for details). It is possible, however, to establish a one to one
correspondence between an axially symmetric basis with identical oscillator lengths
and a spherical basis of the same size. Therefore, the rotation operator acting on
states of an axially symmetric basis with equal lengths and N major shells,
produces states which are linear combinations of states of the same basis.
Bearing this in mind it is possible to visualize clearly what happens when the
rotation operator is applied to an axially symmetric harmonic-oscillator
wave
function with different oscillator lengths: the components of this wave function expressed in the equal-length basis - belonging to shells greater than N, generate,
under rotation, states that are still in the same major shells but can not be mapped
back to any state in the original basis. As an example, suppose N, = 8 and a state
This is due to the fact that the rotation operator e-lpi’ acting on a state of a spherical basis with N,
major shells, produces states belonging to the same basis. This property is not valid for non-spherical
basis, as for example axially symmetric basis.
* This equivalence
preserves the j, quantum number R.
l
l
J.L. Egido et al. / Transition probabilities
265
with the maximum value of the j, quantum number R = f. When this state is
expanded in the equal-length basis it will have components belonging to major
shells greater than 8. Suppose that one of this components belongs to N = 10. This
component still has R = !$ but when rotated it will produce states with 0 = T
that is the maximum value of R for the N = 10 major shell. Obviously this 0 = q
component can not be mapped back to any state of the original basis as the
maximum value of 0 for the original basis is 0 = y. For this reason it is necessary
to extend the already known formulas of refs. 28*29).
In order to establish the notation we denote by CX;and /3: the quasiparticle
creation operators associated with 1cpl> and I (p2), respectively. These operators
are expressed in the particle basis as
a:= c upc:+ v,‘,“c,,
(40)
(41)
To apply the Wick theorem formula we have to establish the correspondence
between the set of quasiparticle operators CQ and the set of quasiparticle operators pk which annihilate the vacuum of the rotated HFB wave function &3> I (p2),
pk = I?(@&I?+(/I?) =
c U,‘k)*Z?(
/3)cJ?+(
/3) + V$)d(,+:I?+(P).
(42)
At this point it is convenient to introduce the rotation matrix of the single-particle
basis .9(p) which is defined as
where we have introduced the Greek index P Q.L= 1, 2,. . . , MO, MO + 1,. . . , Me> *
to include in the sum those states produced by the rotation operator that do not
belong to the original basis k (k = 1,. . , , MO). Looking at eq. (42) we see that the
rotated quasiparticle operator pk can not be expressed just in terms of the set (Ye.
To solve this problem it is necessary to enlarge the single-particle basis in such a
way that the rotated single-particle states of the original basis are included in the
enlarged basis. The minimal enlargement is to go up to a base size of M, (see
appendix A for details). We will then have a single-particle basis, which we shall
denote by {CL, k = 1,. . . , Me}, made of the original set {c:, I = 1,. . . , MO) and the
extended set (d:, 1 = M,, + 1,. . . , Me). It has to be mentioned that the rotated
d~(Z?(PM~Z?t(/3)) do not necessarily belong to the extended basis but this is not
* M, can be infinite.
In such cases some truncation
scheme
is needed.
266
J.L. Egido et al. / Transitionprobabilities
important as the dl states are not relevant from a dynamical point of view. The
Bogoliubov transformation of eq. (41) also has to be extended to accommodate the
new single-particle states. The easiest way to enlarge the Bogoliubov transformation without disturbing the already determined solution of the mean field is to
define new Bogoliubov matrices u and v of dimension A4, x M,
(44)
which preserve unitarity. In eq. (44) the U and I/ matrices are the M, x M,,
original matrices determined by the mean-field equations, and 1 represent the
unity matrix. We can now establish the relation between the rotated and extended
quasiparticle set BP and the extended (Ye set (from now on Greek indices are
meant to represent states belonging to the extended basis).
where
A,@)
=
(P)+i3(p)u(2)+ v(1)+sB*(p)v~2))p,p’
(46)
I?,&?)
=
(tF1’B*(p)v’2’+ ,(~)T~(,)D~2))p,p’
(47)
where 5 is a M, X Me square matrix standing for the rotation
extended particle basis. We can now use this relationship to write
matrix in the
(48)
where
Z(P) =WM-‘(P)
= -Z’(P)
(49)
and ‘Z is the normalization constant $F= (cpl le- @‘yI (p2) given by the Onishi
formula %F= (det(A(P)>>‘/2.
In order to compute the matrix element of eq. (39) the operator & has to be
written in the extended quasiparticle representation of I cpl>
J.L. Egido et al. / Transitionprobabilities
using this representation
261
it is easy to obtain
Taking into account
it is straightforward
to get the final result
3. Numerical results
The lowest state of negative parity in I60 is a 3-. This state has been studied by
several authors 30) and described as an octupole vibration. A 3- state has been
also observed in the neighboring nuclei 14C and 180. The state of 160 is the most
collective one, having a B(E3) to the ground state of 14.1 W.U., to be compared
with 8.3 W.U. in “0 and 2.4 W.U. in I46 [ref. 33)]. The last value, being that small,
make us forecast that this state is not a collective vibration. Although these
B(E3)‘s do not change as dramatically as the ones for the Pt, Hg and Pb isotopes
mentioned in the Introduction, the changes are large enough as to test our theory.
To describe these states in the frame of the theory described above we shall use
the octupole moment q30 as a generator coordinate. Since these nuclei are
spherical in the ground state we shall not take into account either the quadrupole
moment as a coordinate or triaxialities.
To perform the calculation we have used the Gogny force 31). This interaction
has turned out to be very successful in the description of many properties of both
spherical and deformed nuclei over the whole periodic table in spite of its
parameter-free
character 32,30~22
I. The finite range of the Gogny force is also
crucial to describe properly pairing correlations which are very important in a
quantitative study of any kind of spatial deformation, The explicit form of the force
and the numerical values of the parameters used (DSAl set) can be found
elsewhere 31722).When axial symmetry is imposed, we only have to cope with four
268
J.L. Egido et al. / Transition probabilities
constraints in our mean-field calculation; namely, the proton and neutron number,
the octupole operator o,,, and the center-of-mass operator d,,,. The quasiparticle
operators of the Bogoliubov transformation are expanded into an axially symmetric
harmonic-oscillator basis including eight major shells.
To generate the wave functions I (p(q3)) of eq. (1) we first solve the constrained
HFBCS equations for different q30 values. These calculations obviously break the
parity invariance and in order to keep the center-of-mass at zero we further
constrain the center of mass coordinate to take this value. Beside these constrains
all other degrees of freedom are free to vary in search for the self-consistent
solution in the minimization process. Once we have found the solutions I (p(q3))
along the self-consistent path we can calculate all the quantities entering into
cO,l,eq. (111, that means mass parameters, zero point energies, etc. This allows us
c%?
the diagonalization of the CSE, eq. (121, to determine the quantities 4a(q3). To
obtain the correlated wave functions I ?PJ> we have to use the inversion method of
sect. 2.3 to determine the weights fz(qf)
from the corresponding collective wave
functions. As it was mentioned in sect. 2.3 a cutoff parameter E has to be
introduced in the inversion procedure in order to deal with the redundancy
problem of the basis. If some of the eigenvalues of the norm are zero it implies
that some of the basis states are linearly dependent and have to be taken away.
Due to round-off errors, exactly zero eigenvalues are never found in actual
calculations. What is obtained, instead, is a set of very small eigenvalues indicating
that, within numerical uncertainties, linear dependency of the basis is present.
Therefore, the cutoff parameter E is introduced to decide whether a small
eigenvalue can be considered as zero or not. If E is too small, round-off errors and
numerical uncertainties will contaminate the physical results. On the other hand, if
E is too big most of the basis states will be eliminated along with the physical
information contained in them and the final results will also be affected. The
procedure followed in this paper has been to carry out a study of all the physical
quantities (in our case, B(E3) transition probabilities and wave function norms) as
a function of E and look for a plateau in the corresponding plot. The plateau gives
the value of the physical quantity as well as the range of reasonable value for E.
The lowest eigenstate I 9,“) with positive parity is interpreted as the intrinsic
state of the ground state and the lowest with negative parity as the octupole
vibration. These states, as mentioned before, are not eigenstates of the angular
momentum operator. To calculate the transition probabilities we have now two
possibilities. First, to assume the rotational model approach and the GOA, which
leads to eq. (28); the second possibility consists in projecting onto good angular
momentum, i.e., using eq. (34) with eq. (37) and eq. (38).
In fig. 1 we display the potential energy vC(q,) = (+4q3>I f? I (p(q3)) - EO(q3)
(solid line), taking as origin the spherical value, as a function of the octupole
moment q3 for the three nuclei 14C, 160 and 180. The potential energy curves rise
very fast showing the fact that these nuclei are not soft against octupole deforma-
J.L. Egido et al. / Transition probabilities
30
i
269
14c
220
e
‘;=
?I0
i\
I: :
I’.
I :
I :.,‘1
‘.....
03 ’
\
ra 0.2
0.0 L
0
1
2
q3 x 1 0m2(fm3)
Fig. 1. Collective potential energies (solid lines), f(q3)- f(q, = 0) with k&s) = (cp(qs)l fi 1cp&)) e&73) referred to the scale in the left, and amplitudes G ‘/*(q3)j d(qJ (* of the two lowest states in the
nuclei 14C, 160 and l*O, dotted lines represent the O+ amplitudes and the dashed ones the O- ones,
the scale is on the right-hand side. In the lower part of the figures the mass parameters B(q3) in units
of 10m4 h.frne6 MeV-’ are plotted.
tion, specially 14C. On the same picture we can see the amplitudes of the two
lowest states of the CSE, eq. (12); the dashed line represent the O- and the dotted
one the O+ state. Looking at the amplitudes of the Of states at q3 = 0 we realize
that the wave function of 14C is strongly peaked at this value, indicating the low
collectivity of this state. In the lower part of the same pictures we have plotted the
mass parameters B(q,) in units of 10e4 ZI- fmP6 MeV-l, the general behavior is to
decrease as q3 increases as expected 9), with the exception of the points where
level crossings take place.
16
14
=
m’
6
4
0'
14
16
18
A
Fig 2. B(E3) reduced transition probabilities, in W.U., for the nuclei 14C 160 and “0. Filled circles
represents the results of the theoretical predictions with the rotational model assumption and the GOA
for the evaluation of the matrix elements; empty circles are the theoretical results with projection onto
good angular momentum and the filled squares are the experimental measurements.
270
J.L. Egido et al. / Transition probabilities
In fig. 2 we have represented the reduced transition probabilities B(E3,3--,
O+)
(in W.u.> for the three mentioned nuclei. Filled circles represent the results of the
theoretical predictions with the rotational model assumption and the GOA for the
evaluation of the matrix elements; empty circles are the theoretical results with
projection onto good angular momentum and the filled squares are the experimental measurements. In this figure we clearly see the effect mentioned in the
Introduction, namely, that the rotational model approximation is not adequate for
calculations in not well deformed nuclei. We can also observe how the angularmomentum projection increases the unprojected values as to get a good agreement
with the experimental results. In the case of 160 and i*O the agreement of the
projected results with the experiment is excellent. We would like to point out that
our result for 160 agrees very well with the one obtained with the RPA approximation using the same type of forces 30). This result confirms the general believe that
this 3- state in 160 is an octupole vibration. In case of i4C the agreement between
the projected theory and the experiment is not as good as in the other two cases.
This was expected because the 3- state in this nucleus is probably of particle-hole
type. Obviously the GCM ansatz is not the most indicated to describe such states.
The exact figures that we obtain in the projected case are (6.8, 15.1 and 9.8
W.U.) to be compared with the rotational model prediction (1.56, 3.6 and 1.71
W.U.) and the experimental data (2.4, 14.1 and 8.3 W.u.> for the nuclei 14C, 160
and “0, respectively.
4. Conclusions
In this paper we have shown the unsuitability of the rotational model approximation, used in the calculation of B(EL) with wave functions of the GCM type,
for nuclei with a large component of their wave function close to the spherical
minimum. We propose to combine angular-momentum projection techniques together with wave functions of the GCM type. The corresponding theory has been
set up and the difficulties associated with such procedure have been pointed out
and solved. As an application of the theory we have performed realistic calculations using the density-dependent Gogny force to calculate the transition probabilities from the lowest 3- to their ground states in the nuclei 14C, 160 and “0. The
agreement with the experimental data increases dramatically when the projected
theory is used instead of the rotational model prescription.
We would like to express our gratitude to J.F. Berger, M. Girod and D. Gogny
of the Centre d’Etudes Nucleaires de Bruyeres le Chatel for their kindness in
allowing us to use their original Hartree-Fock
code and also to thank them for
helpful discussions. We also appreciate the careful reading of the manuscript by
J.L. Egido et al. / Transition probabilities
271
R.R. Chasman. One of us (Y.S.) wants to thank the Spanish Ministry of Education
and Science for a “Acciones de Formation de Personal Investigador” scholarship.
Appendix A
The single-particle basis used in this calculation is an axially symmetric harmonic-oscillator basis, characterized by two oscillator lengths b I and b,, with
quantum numbers L$ A, it i and n, and with the following properties:
j,Ifl,
A, II,,
n,> =finIfl,
A, IzI 9 n,),
(A-1)
(A.3)
(A.41
In the above expression 9 standard for the “simplex” operator @= 17e-ir’y, 9
for the time-reversal operator and fi means -0.
The natural basis to calculate the matrix elements of the rotation matrix is the
spherical one. However, and in order to reduce the computational time involved in
the mean-field calculations, we are forced to use in many cases axially deformed
harmonic-oscillator basis.
In order to obtain the rotation matrix the axially symmetric harmonic-oscillator
wave functions have to be transformed to a spherically symmetric basis. This
transformation is accommodated in two steps.
First the basis characterized by b I and 6, is transformed to an axially
symmetric basis with b I and b: = b I ,
lb,,
b,; LMn.n,>
= ~Cnz,,;lb,
4
7 b,;
RAn.n,L
(A.5)
where the transformation matrix C is given in ref. 34). In this transformation the
index n: runs from 0 to ~0 and therefore a truncation scheme is needed. Fortunately, for not very different oscillator lengths b, and bi the coefficients C
decrease very quickly as a function of the difference II, - nl and the truncation
converges very fast as a function of the number of additional n: included (in our
case we use two additional n:). It has to be mentioned that the maximum value of
the major-shell oscillator quantum number N = A + 2n I +n, in the original basis
N, increases to N, in the basis with identical oscillator lengths (the maximum
value of n: is now bigger than N,). The difference N, - N, obviously depends on
how large the truncation introduced in eq. (A.51 is.
272
J.L. Egido et al. / Transition probabilities
The states generated in the right-hand side of eq. (A.5) do not span all states
corresponding to the N, major shells. As the transformation of eq. (A.51 preserves
the 0 quantum number the states with 0 ranging from NO+ i to N, + i are
missing. When we apply the rotation operator to eq. (A.5), we will obtain, however,
linear combinations of all states belonging to the N, major shells.
As explained in sect. (2.4) we have to enlarge the original basis, used in the
mean-field calculations, as to include the relevant rotates states. The procedure we
have followed is to expand the original basis up to N, major shells (N, is chosen
big enough to have a good unitarity condition in the truncated transformation C,
usually two additional shells are included). This enlarged basis is then transformed
by eq. (A.5) to the axially symmetric basis with N, shells but identical oscillator
lengths. Once we have an axially symmetric basis with equal oscillator lengths we
can transform it to a spherical basis
Jbl,b,=bl;n,A,n.,nZ>=Cs~~4i;n,l,jIb=bl;nljm=~n),
nlj
(A.6)
where the transformation
coefficients are given in ref. 35) and the sum runs
through all the IZ, 1, j belonging to a given major shell N = 2n I +A + IZ,.
The extended basis can be written now in terms of the spherical basis with the
same number of shells N,
Ibl,b,;n,A,n.,n,>=CW,nl,~~;,,l,jIb=b.;nljm=R)
(A-7)
nlj
with
(A.8)
The rotation matrix expressed in the enlarged basis & can be written as
where dxn I is the standard Wigner rotation matrix.
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