- Email: [email protected]
Self-consistent calculation of nuclear rotations: The complete yrast line of 24Mg
Nuclear Physics A467 (1987) 115-135 North-Holland, Amsterdam
SELF-CONSISTENT THE
CALCULATION COMPLETE
OF NUCLEAR
YRAST
LINE
ROTATIONS:
OF =Mg
P. BONCHE Service de Physique 7’ht!orique, CEN Saclay, 91191 Gif sur Yvette Cedex, France H. FLOCARD Division de Physique 7’ht!orique*, Institut de Physique Nucliaire, P.H. Physique Nu&aire
BP1 91406 Orsay, France
HEENEN’
The’orique, Universite Libre de Bruxelles, CP229, B-1050 Bruxelles, Belgium Received
22 October
1986
Abstract: We present a method of solution on a three-dimensional mesh of the self-consistent cranked Hartree-Fock+ BCS equations. Using several parametrizations of the Skyrme interaction we apply our method to the study of the complete yrast line of 24Mg. We find that 5=26 is the limiting angular momentum for this nucleus. We have also studied examples of collective rotations along the y = 60” axis involving hexadecapole deformations. Our work corroborates earlier calculations using the Nilsson-Strutinsky method and extends them to higher spins.
1. Introduction As demonstrated by numerous calculations, the microscopic description of lowspin properties of nuclei by means of an effective interaction like the Skyrme force is remarkably successful ly2). In view of the importance taken in the last decade by nuclear high-spin physics it appears worthwhile to test how well this microscopic approach can help understanding the presently accumulating data. In this work we present a method of solution of the three-dimensional (3D) coupled Hartree-Fock (HF) + BCS equations using a discretization of the wave functions on a rectangular mesh. In this way we can describe any type of even multipole** deformation generated by the interplay of the nuclear mean-field and the constraining rotational field. We can also study with similar good accuracy the low as well as the high-spin configurations, irrespective of the magnitude or shape of their deformation. As compared to earlier 3D HF+ BCS calculations ‘) the breaking of time reversal symmetry induced by the cranking field generates two types of technical difficulties. First all degeneracies of the single-nucleon wavefunctions have to be removed, doubling therefore the size of the calculation. Second the structure of the mean-field ass0ci.G au CNRS. FNRS. l * The method can easily be extended l
Laboratoire
’ Maitre de Recherches
to include
0375-9474/87/%03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
a description B.V
of odd multipole
deformations
-‘),
P. Bonche et al. / 24Mg Yrast line
116
itself becomes
more complicated:
in addition
to the usual
effective mass operator,
the scalar and spin-orbit potentials, there appears contributions from the interaction to the spin-vector and current potentials. These terms usually neglected in semimicroscopic equations
approaches including
have
a treatment
been
included
of pairing
in the present
and temperature
work.
The relevant
effects are given in sect.
2 and in the appendices. As a first application of this method we choose to study the rotational properties of a light nucleus 24Mg from the ground state up to the fission limiting angular momentum. The low-spin properties of this nucleus are well known experimentally 4-9). They extend up to the yrast 12+ and possibly 14+. Recent theoretical studies of rotational properties have been made by shell-model calculations lo) (up to J = 12) and by means of the Nilsson-Strutinsky (NS) method 11V12).Results from cranked HF calculations (without spin-orbit interaction [refs. 13,14)] and from the cranked a-cluster model “) are also available. They are compared with our findings in sect. 3. Sect. 4 contains our conclusions.
2. Solution of the self-consistent The cranking
approximation
cranking equations
is based on the assumption
that a nucleus
with spin
J can be described in terms of an intrinsic state at rest in a frame rotating with some angular velocity w around an axis. The optimal intrinsic state 1P) is determined by minimization of the routhian 8 E=E-wJ,,
(I)
where E and _I, are respectively the expectation the third component of the angular momentum*
values
of the hamiltonian
J, =(?P]@). The angular
velocity
w which
acts as a lagrange
and of
(2) parameter
is determined
by the
condition** J,=J.
(3)
In this work the energy E is calculated within the HF+ BCS approximation, using a functional derived from a Skyrme interaction and a constant strength pairing interaction. We also added a treatment of temperature effects to study the incoherent excitations of the nucleus. According to thermodynamics, one should then replace the energy E by the free energy F = E - TS in the expression (1) of the routhian to describe the isothermal rotation of a system at temperature T. l Our choice of the z-axis instead of the customary structure of the spin Pauli matrix mz which simplifies particle wavefunctions (see appendix A). ** For correction to formula (3) see ref. 16).
choice of the x-axis is motivated by the diagonal the formulation of the symmetries of the single-
P. Bonche et al. / 24Mg Yrast line 2.1. NUCLEAR
AND
The mean-field
COULOMB
energy
(EC). It is well known be written spin
ENERGY
is the sum of a nuclear that for a Skyrme
as the integral
degrees
MEAN-FIELD
of freedom
(EN) and a Coulomb
interaction
of a local functional are taken
into
117
the nuclear
contribution
of a set of densities
account,
the breaking
contribution “). When
of time
can the
reversal
symmetry induced by the constraint wJ,, leads to a rather complicated form for the functional i8) which we give in appendix A. The direct component of the coulomb energy, is calculated by solving the Poisson equation while the exchange contribution is evaluated by the Slater approximation. The single-particle routhian h’ = h - w_!, defined by
(h’@k)(Ca) = ,g; takes the following
a)
k
form for a state k of isospin
,
p
with the effective mass rnz, the spin scalar U,, spin vector V,, current C, and spin-orbit W, potentials given in appendix B. It can be noted that the spin vector (B.3), (B.7) ayd current (B.4), (B.8) potentials are not generated by the cranking constraint -wJ, only. They incorporate additional contributions from the two-body interaction involving the nucleon currents jr and the spin densities p7.
2.2. PAIRING
ENERGY
For a system as light as 24Mg, pairing is not expected to play an important role. For this reason we choose to work with the simple BCS method. As compared to the Hartree-Fock-Bogoliubov (HFB) method, HF+BCS relies on two approximations. First, the contributions to h’ (4) resulting from the variation of the pairing energy with respect to the wavefunctions 02 are neglected. Second, one assumes that the basis within which the pairing tensor takes the simple canonical form, is identical to that diagonalizaing the one-body routhian h’ [ref. ‘“)I. In other terms if cl is the creation operator associated with the eigenstate 0,‘ of h’, the pairing hamiltonian for a constant pairing interaction with strength G, which couples time-reversed states, can be approximated as 20):
ci,= c
G,,C:C;CjC,,
k,l>O
where the sum over k and 1 is restricted to the eigenstates ,+ A signature. On the other hand the states k and 1 are eigenstates
(6) of h’ with positive of h’ with negative
118
P. Bonche et al. / 24Mg Yrast line
signature,
that we associate
described
below.
state E of k and
reversed
an eigenstate the partner coefficient
with k and 1 respectively,
When time reversal
symmetry
Gkl = G. For a rotating
of h’. Following
the prescription
k^ of k, as the eigenstate Gkl is then related
according
is not broken nucleus
suggested
of h’ whose
to the prescription (w = 0), k^is the time
the state E is no longer
by Marshalek
overlap
*‘), we define
with f is maximal.
The
to G by: Gkr = GFkFi,
(7)
Fk = l(El k^)l.
(8)
with
For non-zero values of the angular velocity, the coefficients Fk are smaller than unity, accounting in this way for coriolis anti-pairing. In fact this prescription has been shown to overestimate the anti-pairing effects, and a renormalization prescription of the coefficients Fk has been proposed 2’). In the case of 24Mg we have checked that for reasonable values of the pairing strength G, only the ground state and first 2+ are affected by pairing correlations. As this work is more concerned with higher values of spin which are not affected by pairing, we have decided to use the simpler Marshalek prescription for calculating the matrix elements Gkl. As in ref. ‘) we also introduce a cut-off factor to ensure that pairing acts only on states whose energy lies within
5 MeV of the Fermi
level.
2.3. TEMPERATURE
To describe incoherent excitations a formalism involving temperature
above the yrast line, it is customary to introduce and to consider that collective decays proceed
along isentropic lines. In the mean-field approximation temperature is described by means of a density operator, 6 = exp (-i/ where
l? is a non-interacting
T)/Tr
quasiparticle ff=
1
(exp (-R/T))
a nucleus
,
at non-zero
(9)
operator
&@;@kfi?~~@k^.
(10)
and pz are related
to the c’,+“s and C~“S by the
k>O
In eq. (10) the quasiparticle usual
p:
BCS relations:
(11) pk^= u&-,&c;. C we give the equations and the Ei’s are the quasiparticle energies. In appendix resulting from a minimization of the grand potential with respect to the Ek’s, Et’s, vk’s and &‘s.
P. Bonehe
2.4. NUMERICAL
et al. /
z4Mg Yrast line
119
METHODS
The numerical methods of the present work involve a discretization of the wavefun~ions @Joon a three dimensional mesh. They have been presented in refs. 22,2, where tests of accuracy are also given. We have also explored defo~ation energy surfaces around the yrast intrinsic state and fission barriers using the constraint on the quadrupole operator described in ref. 2). 3. Results 3.1. DEFORMATION
PROPERTIES
OF NON-CRANKED
24Mg
The deformation energy surface of 24Mg calculated with the Skyrme SIII interaction 23) is shown in fig. la. It displays a single minimum for a prolate defo~ation of Q = 107 fm2. This result is in marked difference with that obtained in an earlier calculation I”) with the BKN force “*). Indeed the BKN interaction, which does not incorporate a spin-orbit term leads to a triaxial ground state. The obliteration of triaxiality seems to be common to microscopic calculations using more refined effective interactions. For instance, we checked that the ground state of 24Mg calculated with the SGTI [ref. 25)], T6 [ref. ““)I or the SkP [ref. “‘)] interactions is also prolate. A similar rest& was found in calculations performed with the Dl interaction ‘“). The self-consistent minimum is slightly underbound [E = 196.73 MeV versus E ex.= - 198.3 MeV, see ref, ““)I. This difference is compatible with the magnitude of the correction expected from the projection of the intrinsic state on a O+ state. An estimate of this correction can be obtained by means of the simple formula AE = (~l”/2~)(J*) where (J2) is the expectation value of the total angular momentum. With our calculated values of (J’) (19.46) and the Valatin moment of inertia* fi2/2Yr =0.28 MeV, we find a correction of 5.4 MeV. The charge radius of the
1
0
1
Q(b)
Fig. 1. (a) HF+ BCS deformation energy surface of 24Mg calculated with the $111 interaction pairing strength G = 1.22 MeV. The contour lines are drawn every 0.5 MeV. The deformations given in barn. (b) Same as (a) but without pairing interaction (pure I-IF).
and a Q are
* In principle
Q(b)
the Peierls-Yoccoz
moment
of inertia
should
be used i6).
120
minimum
P. Bonche et al. / 24Mg Yrast line
(3.12 fm) is in good agreement
ref. ‘)I. The proton than the value
quadrupole
deduced
from measurements
62.3 f 4.5 fm*, ref. “)I. The calculated operator*
(e)
with the experimental
moment
which
value [3.08 f 0.05 fm,
we find (Q, = 54.2 fm*), is smaller
of the 2+ quadrupole
expectation
= 51.6 fm4 agrees with the available
value of the proton
moment
[Qy=
hexadecapole
data [41 f 14 fm4, ref. “)I. When
the pairing interaction is switched off the deformation energy surface is slightly modified (fig. lb). The minimum is still prolate (Q = 109 fm2) and its energy (E = -196.62 MeV) is almost unaffected by the pairing. However, the energy difference between the spherical configuration and the ground state increases from 1 to more than 10 MeV. 3.2. ROTATIONAL
PROPERTIES
OF =Mg
The self-consistent cranked yrast and yrare bands of 24Mg are shown in fig. 2. We have restricted our study to positive-parity bands which, except for the band labelled VIII, have been obtained with identical fillings of neutron and proton orbitals (see appendix D). In the upper part of the figure we compare the experimental and calculated spectra. The latter is too compressed, a result generally found in microscopic HF [ref. i3) and references therein] calculations of light nuclei. The pairing correlations, which are rapidly destroyed by Coriolis anti-pairing, affect very little the calculated spectrum. A measure of the Coriolis effect is given by the neutron and proton gaps (eq. (C.3)) whose values are respectively 1.07 and 1.18 MeV for (J,) = 0 and drop to 0 for (J=) = 2. The trajectory of the ground state band in the Q, y plane is drawn on fig. 3. Up to J = 6 the deformation of the intrinsic state stays almost constant. From J =6 to 10, the quadrupole moment decreases while the nucleus remains prolate. The trajectory reaches the oblate axis for the critical value J = 12 (i.e. the maximal angular momentum of a pure sd configuration). From then on, the nucleus flattens continuously along the oblate axis up to the angular momentum 20 beyond which we could not follow this configuration (it is anyhow no longer yrast). The rather puzzling existence of a cranking state evolving smoothly from J = 12 to J = 20 as an apparently oblate solution rotating around its symmetry axis can be understood from the results shown in fig. 4. This figure displays the evolution versus (JZ) of the expectation values (r4Yr) (quantization along the rotation axis 0,). For values of J larger than 12 the moment (r”Yi) cancels exactly as it should be expected for states lying on the y = 60” axis. On the other hand the moment (r4Yi) is non-zero so that a kind of collective rotation** is still allowed. This phenomenon had already been noticed in an earlier calculation of *‘Ne [ref. ‘“)I in which nuclear densities with squarish shapes have been found for values of J between 8 and 14. On fig. 5 we have plotted the single-particle routhian for three * In contrast with the rest of this article, fl is defined here with respect to the axis of the nucleus and not the rotation axis. ** It differs markedly however from a purely classical evolution in which hexadecapole modes are not necessary to achieve a continuous trajectory along the y = 60” axis.
P. Bonche et al. / “Mg
E
121
Yrast line
10+ 12+
20
__
70 9 P 21
50
IO-
8'
8+ 6+
4+ _ 2+
30 o-
vThe
0
IO
20 (JZ)h
Fig. 2. Excitation energies (in MeV) versus angular momentum (in fi) of the several bands studied in this work. The arrows point to the critical point of the bands indicated in the parentheses. The left-up part of the figure shows a comparison of the calculated and experimental spectra of the ground state band.
values of (J=). For (J,) = 12 and 20 where 24Mg is almost axially symmetric, each state is labeled not only by its parity but also by the expectation value of J_ in that particular state. While the occupation scheme at (JZ) = 12 is compatible with a pure (sd) shell filling, it is no longer the case at 20 where one of the occupied state (of negative parity) brings a s contribution to (JZ). In these two cases one can notice empty states in the Fermi sea. The reorganization of the occupation levels which remove these holes leads to different configurations (and different spins) studied later in this section. Comparing our results with those obtained by Ragnarsson, Aberg and Sheline 12) with the NS method using a modified oscillator potential, we find many similarities and some differences. Their ground state is triaxial until J = 6, as opposed to our prolate solution. From J = 6 to the band termination at J = 12
Fig. 3. Trajectories
in the Q, y half-plane
of the several bands (with the exception of the fission band IV) studied deformation plane has been drawn on the right side of the figure.
(24)
0 l
in this work. The lower part of the
XXIII
4’
3Xt
P
II m
A
AlP
I
0
123
- -5o__-. I
0
< r4Y42) < r4 Y44) II
f Ll’
I
I
I
I
IO (J,??
Fig. 4. Expectation
I
I
I
20
0
I
I
I
(J,bi
values versus angular momentum (in h) of the hexadecapole moments fm4) for band I and II. The quantization axis is the rotation axis.
_
IO r’Yy
(in
the path of the NS trajectory follows closely that found in our microscopic calculation. As in the present work, they find a spectrum too compressed compared with experiment. In contrast with our results, their ground-state trajectory stops at .I = 12. We think that this is due to the neglect of Yi components (with respect to the rotation axis) in the potential. In fact many (if not all) of the qualitative differences between their results and ours stem apparently from the restricted parametrization of the potential adopted in ref. “). Another example of collective rotation allowed by hexadecapole deformations is given by the band associated with the yrast 8+ state at 11.86 MeV. It has been established experimentally ‘**) and confirmed by shell-model calculations lo) that this state does not belong to the ground state band. In our calculation (see fig. 5) it results from the transfer of one neutron and one proton from the d=,,2,-3,2 to the “) determined this cranked state to be d 5,2,5,2 orbitals. Similarly, NS calculations on the y = -120” axis: a prolate nucleus rotating around its symmetry axis. In ref. 12) the 8+ is an isolated state. The authors propose to correlate this 8+ with an excited 2+ state obtained by the promotion of one particle from the [211, -51 level to the i]. On the other hand we could follow down to lower spins a band (band II) terminating on the yrast 8+ state. This band connects with the ground state of 24Mg located on the y = -120” axis. From J = 0 to J = 8, the prolate deformation of the nucleus decreases while the principal axis of symmetry remains aligned with the rotation axis. Again this smooth transition is made possible by the non-zero expectation value of r4Yi (right side of fig. 4). As for the 2+ state studied in ref. I*), we find [Zll,
124
P. Bonche et al. / z4Mg Yrast line -r
TI
I
r-
-+
-10
1/2’
-
__1/2+ -+1/Z+
--+
--cY2+
0
Y
$
-+
-z W
-
_5/2+
-
n/2+
-
Y.?
-0
312’
_
,,2+@G-E:
F
--+ -
0 3,2+ -
1/z+
+
-
+ G3c
:
-=+ +
t/2+
--__
,,2-
--+ --3/a-
_--
--+
--3/2-
3/Z+
_
a/*-
-
i/27;;:
,,2-
-
5/2+
-
_ 3/2+ -_3/-2-
i/2-
---
--3R-
-
,,2-
1/2-
--f1/2-
-
-
=; ---
5/z+
3/2-
---I/*-
-5/2+
-+
-_,,2-
-
-
__--
__--
/
--+
__t3,,-
,-p:rrj , -+-
_,’ r/2-
--
T/2-
=:
---i/2-
-. ,’
-_1,2-
_-i/2-
0 =_ +
0 0 =___= --- -+ -:4
-i1,2+
--t/2-
-30
7:;:
----_,,2-
-+
-20
___. 5/2+
=-::; -
5/2 -
-_
-_3/2+
0
-+ -+ --
--1/z+
0
-_:+
-
1,2-
--
1/2-
-+ --
1/2-
A___--__-_*
--
,-+-
/’ -
I’
Y2-
-
--
3/2-
--f -4o--+
-*1/z-
_
J,llil
0
,,2+
12
-
-.
-.___
-+
--“Z+
-_,,2+
=,y:
3/2-
-.1/2+ 1/2+
-+
t/2+
-t/2+
0
20
--l/2+
8
14
16
-+ 20
22
Fig. 5. Single particle routhians of several bands studied in this work. The energies are given in MeV and the spins (J,) indicated at the bottom in h. The circled number 12 points to the position of the fermi level. The dashed lines indicate empty states below the Fermi level. The + and - signs refer to the parity. For nearly axial configurations and when the principal axis of symmetry is identical to the rotation axis, the parity quantum number is preceded by the expectation value of J, in the state. For band IV the signs indicate
the signature
and the parity
of the level.
that this state does not belong to band II. By considering either the promotion of a neutron or a proton orbital, we could create two almost degenerate excited bands (band VIII) which are also located on the y = -120” axis but are not related to the yrast 8+ state.
26
I? Bonche et al. / z4Mg Yrast line
The removal
of the g- hole in the occupation
125
scheme of the ground-state
configur-
ation at J = 12 can be achieved by emptying either the 3’ or the +’ states. In the former case, one obtains the band III at spin J = 16. We could follow this band up to J = 26 and down to J = 8. From J = 14 to 20-22 (with the exception is the Yrast band.
A similar
result was found
by the authors
of J = 18) it
of ref. 12) except that
the positions of the 12+ states of band I and III are interchanged. They also conclude that the band III terminates at J = 16. However, we suggest that the secondary triaxial minimum appearing in their J = 20+ map (fig. 14 in ref. “)) and that they consider as leading to fission could in fact belong to band III. Indeed we shall see that the trajectory of our fission band lies very close to the y = 0 axis, while the states 20
value of the angular
momentum
(J = 26). In the left-down
corner
we have
drawn the contour lines of the total density at the minimum. The preformed 12C are nearly oblate and their principal axes are perpendicular to the rotation axis 0,. This preformation appears also on the spectra drawn on fig. 5. As (J,) increases from 22 to 26, one can observe a grouping of levels in pairs with opposite parity and signature. Our results differ in many respects from those found with HF using the BKN force 13) and the cranked a-model 15). Some of our bands (III, VI) cannot even be studied when the spin degree of freedom is neglected (see appendix D). The beginning of the yrast line looks similar in both LS and spin-orbit schemes and the critical point is reached for J = 12. However, the trajectories in the Q, y plane are somewhat different. The configuration D which was found yrast in ref. 13) seems to correspond to our band VII which is never Yrast (although never very
126
P. Bonche et al. / 24Mg Yrast line
-133
-135
0
5
Q(b)
Fig. 6. The upper part shows the deformation energy curve at J = 22. The 0.5 MeV. The minimum is indicated by the dot. The lower part shows the quadrupole moment for the limiting angular momentum J=26. On the sections of the total density for the yrast state (0; is the rotation axis). every 0.02 fmm3.
contour lines are drawn every fission path as function of the left side we have drawn two The contour lines are drawn
excited). The high spin yrast band IV corresponds to the band A3 of ref. 13) which is either yrast or slightly excited. The values of the scission limit angular momentum as well as of the maximal excitation energy are however comparable: (24 h, 60 MeV) versus (26 h, 73 MeV), reflecting probably the decreasing importance of spin-orbit effects near the scission limit angular momentum. 3.3. INFLUENCE
OF TEMPERATURE
On fig. 7 we show the isotherm T = 1.5 MeV, which exhibits two bands only. At finite temperature all the configurations favored at lower spins merge into a single
P. Bonche el al. / “Mg
127
Yrast line
IO m
5
t
2
50
H *-
w
0
1 0
.
I
I
IO
I
20
I
.
CJ;Pfi
Fig. 7. Isotherm T= 1.5 MeV versus fission band is drawn as the dashed
angular momentum in h. The dots indicate the Yrast states. The curve. The upper part displays the total entropy versus angular momentum.
one (the deformation
surface
energy
at J = 0 becomes
very flat with a spherical
minimum and the band II disappears). However the fission configuration curve), which corresponds to noticeably larger values of the quadrupole
(dashed moment,
remains separate. The excitation energy is always of the order of few MeV and the limiting angular momentum is still 26. The upper part of the figure shows that the isotherm does not differ very much from an isentropic: the value of the total entropy (e.g. (C.6))
remains
between
7 and 10. 4. Conclusions
This article presents the first study of very high-spin nuclear properties, based on a purely microscopic description using a standard effective
which is nucleon-
nucleon interaction. Since the domain of validity of such interactions extends in principle over the entire chart of elements, we expect the method exposed in this work to allow a microscopic description of nuclear rotations for all nuclei. Within the present cranked self-consistent formalism, and in contrast with more phenomenological models, there is no a priori assumption on the deformation of the mean-field. In some methods of solution of the HF+BCS equations the unavoidable truncation of the orbital basis introduces implicit constraints on the shape
128
P. Bonche et al. / 24Mg Yrast line
of the nucleus.
Here they are avoided
by a discretization
of the orbitals
on a regular
3D mesh. Actually we have found new examples of collective (although non-Yrast) rotations which are made possible by hexadecapole deformations of the routhian mean-field. We have studied Several effective
eight bands
interactions
of 24Mg. Four of them contribute
with incompressibilities
ranging
to the Yrast line.
from 220 to 350 MeV
and effective mass between 0.75 and 1 have been shown to give similar results, both qualitatively and quantitatively. This supports the widely accepted notion that rotational properties are primarily determined by properties of the interaction connected with multipole deformations for which 12 2. Our results corroborate in several respects earlier Nilsson-Strutinski calculations I’), which they complete and extend to the end of the Yrast line. In particular we were able to study all the bands beyond their critical point. The end part of the Yrast line (22cJ~26) is found to correspond to a very elongated system which ultimately fissions in two oblate 12C nuclei. The limiting angular momentum (J = 26) appears compatible with classical estimates 31). The study of the evolution of the fission barrier as a function of the angular momentum reveals that the optimal configuration as well as the saddle point are always slightly triaxial. It appears also that the fission valley in the Q, y plane is remarkably narrow in the y-direction. This could contribute to an increase of the stability of these fast rotating configurations.
We would like to thank F. Naulin for his help in calculating the deformation energy surface of 24Mg, and M. VCnCroni for discussions and a critical reading of the manuscript. This work was made possible by extensive computing facilities provided by the Centre de Calcul Vectoriel pour la Recherche. P.H. Heenen expresses also his gratitude to the FNRS for a computational grant. This work was partly supported
by the NATO
grant
RG85-0195.
Appendix A
NUCLEAR
MEAN-FIELD
ENERGY
In this appendix, we summarize results already presented in ref. 18) within the context of time-dependent Hartree-Fock and we recast them in a form suitable for threedimensional cranking calculations. With a simple effective interaction like the Skyrme force “) the energy EN can be written as the space integral of a local energy density: EN=
H(r)
d3r,
(A.1)
P. Bonche
et al. /
“Mg
Yrast line
129
with* H(r)=(h2/2m)7+B,p2+Bz(p~+p~)+B,(p7-j2)+B,(p,7,-j%+p,7,-j~) + Bs PAP + Bd~nh,
+ pp&,)
+ BTP’+~ + B,P” (d + P;)
+B,(pV.J+j.Vxp+p,V-J,+j;VXp,+p,V.J,+j,.VXp,) (A.21
+B,~P~+B~,(P~+P~)+B~~P~P*+B~~~~(P~+P~).
The densities p,, T,, j,, V . J, and pT (T = {n, p}) which appear in (A.2) can in turn be expressed in terms of the single-particle wavefunctions 4&( r, a) and their occupation factors
nk (defined
dr)
=
below
(C.5)) as
c nkiV@k(r, kc7
V . J(r) = -i
c
a)i29
r&(V@r(r,
a) xV@k(r,
a’)) * (c+~~‘),
km+
64.3) Using a generalized t,, x3, a and W:
Skyrme
interaction
2, with parameters
to, x,,,
t, ,
x1, t2, x2,
- kxsk’,
(A-4)
o= t,(1+x,P,)S+~t,(1+x,P,J(k26+6k’*) + t2(1+xzP,)k*
Gk’+~t,(l+x,P,)p”G+iW(a,+~~)
we can express the coefficients
Bi as:
B,=~t,(l+~x,)
B, = +t,(++xx,)
B2 = -$t&+x,)
B,=-;W
B3=+(t,(l+;x,)+t2(1++xZ))
Blo =&,x0
B4=-+(t,(;+xl)-t&+x2))
B,, = -&,
B5 = -&(3t,(l
++x,) - t2(1 +$x2)) B12 =$t,x,
B,=&(3tI(;+x,)+ B,=&t,(l++x,)
t&+x2))
BIj=-&t3. (A.5)
The complete Skyrme functional contains other terms resulting from a tensor coupling between spin and gradiant. The contribution of these terms which cancels exactly for spherical time reversal invariant wavefunctions is always an order of magnitude smaller than the spin-orbit contribution. Since furthermore they are generally not taken into account in the determination of the Skyrme parametrizations, they have been neglected in the present work. l
130
P. Bonche et al. / *4A4g Yrast line
In this calculation
we impose
the following
two symmetries
on the individual
wavefunctions: (i) parity &Jr,
a) = @A-r,
u) =pk@k(r, a),
(‘4.6)
pk=*l,
(ii) z-signature e ir(‘z-1’2)@k (x, y, z, u) = u@k (-x,
-y, z, u) (A-7)
= The complete
description
r]k@k(X,
of a wavefunction
y,
5
fl)
requires
,
vk=fl.
the knowledge
of four real
functions !Pku ( r) ( CY= 1 - 4) corresponding to the real (Re), imaginary (Im) spin-up and spin-down parts of @k. For positive and negative signature wavefunctions we adopted a different numbering of these four components:
This choice ensures that the parities of the components ‘P,,., with respect to plane symmetries about x, y and z planes depend only on the label (Y and the parity pk as indicated in table 1. The action of the time reversal operator on a function @k generates a state @n with same parity and opposite signature such that
,
The @‘E’Sare used to calculate k^ and the coriolis anti-pairing
the matrix coefficients
vz=--vk.
(El f) from which we determine Fk (eq. (8)).
TABLE 1 Parities of the components ?Pk,, of a wavefunction ok of parity pk, with respect to the x=0, y=O and z=O planes. a
x
Y
Z
1 2 3 4
+ _
+ _
+
+ _
Pk Pk -Pk -Pk
(A.9)
the state
131
I? Bonche et al. / 24Mg Yrast line
The densities following
V - J once
p, 7, j and
expressed
in terms
of the
Vka take the
forms: (A.lO)
T(r)=c nk i k
-
nktqk,lvwk,2
j(r)=1
(A.ll)
(vpk,a)2,
a=,
%Qv
vk,,
+
qk,,v
pk,4
-
qk,4v
vk,3)
,
(A.12)
k
tensor and where the coefficients a( I), p( 1), where Q,,,,, is the fully antisymmetric y(I) and 6( 1) are those given in table 2. Similarly the vector density p can be written: 2 ( wk, 1w;, + p(r)
=;
i Parity
and signature
induce
qk.2
2r]k(pk,l~k,4-
nk vk(
?&
definite
+
pk,4
qk,2pk,3) %,Z
-
?&
-
symmetries
. ?&I)
(A.14)
1
for each of the densities
(A.3)
with respect to the planes x, y, z = 0. They are listed in table 3. The direct part of the Coulomb interaction E&direct), 3 ,p,(r)p,(r’) d r lr-r’l is calculated the Poisson
’
(A.15)
in two steps. We determine first the Coulomb potential V,(r) by solving equation, then we evaluate the integral ij d3rp, V, . The exchange part TABLE 2 Indices
of the components Pk_ for V * J (formula
entering A.13)
the formula
a
P
Y
s
vk=+l 1
1
2
3
3
1
4 1 2
3
2
4 4
2 3
3 1 3
2 3 4
1 2 2
4 4 1
1
TJk=-l 1 2 3
The upper (lower) table corresponds positive (negative) signature vk.
to a state with
132
P. Bonche et al. / 24Mg Yrast line TABLE
Parities
P. 7,
V .J
P.x pv P; Jx i, iz
of the Coulomb
energy
3
of the nucleon densities thex=O,y=Oandz=Oplanes x
Y
z
+ _ + + + _
+ + + + _
+ + + + -
is approximated
E,(exchange)
with respect
by means
to
of the Slater approximation
= -$e2(3/r)“3 I
d3rpp(r)4’3.
as: (A.16)
The center-of-mass recoil effect is approximately taken into account by a subtraction of the expectation value of (Ckpi)/2mA (A is the mass number) from the mean-field
EN.
energy
Appendix B HARTREE-FOCK
HAMILTONIAN
AND
ROUTHIAN
The variation of the nuclear energy EN with respect to the wavefunction @z( r, a) defines the one-body Hat-tree-Fock hamiltonian h. For a generalized Skyrme functional (13), its action following potentials:
h2
on a neutron
wavefunction
Ok takes the form (5) with the
h2
(B.1)
-=~~+B,P+&P,, 2m,* n
U,,=2Blp+2B2p,+B3(~+iV~j)+B4(~,+iV~jn) +2BSAp+2B6Ap,+(2+a)B,p’+” +B8(apa-‘(p;+p3+2papn)+ +
~P~-‘UGZP~+
B,,(P:+P;))
Bg(V - J+V
*J,)
,
(B.2)
V,,=B,(V~~+V~~,)+~B~~+~B,,P,+~P*(BIZP+BI~P~)
03.3)
C’,=2B3j+2B,j,-B,(Vxp+Vxp,),
054)
W, = - B,(Vp + VP,) .
(B-5)
133
P. Bonche et al. / z4Mg Yrast line
The Hartree-Fock identical,
hamiltonian
for a proton
except for the spin scalar potential
contribution
wavefunction
is mutatis
UP which is completed
mutandis
by the Coulomb
UC: U, = V, - e2(3p,/ 7r)“3.
03.6)
To obtain the routhian h’ the potentials V, and C, must be modified adjunction of terms generated by the constraint -oJ~,: V,(r) + V,(r) -$hwe,
by the
,
03.7)
C,(r)+C,(r)+hw(ezxr), where e, denotes
(B.8)
the unit vector along the z-axis.
Appendix C TEMPERATURE By virtue
FORMALISM
of the Wick’s theorem fit=
the expectation
C F~c:c~+E&cL-GC
by the density Tr(DfiO=
Z?
F,F,c;c$cjc,,
(C.1)
kl
k>O
in a state defined
value of the routhian
c
operator
(9), takes the simple
following
form:
(Eknk+&k^nk^-GF2knknk^)-d2/G,
(C-2)
k>O
where the q’s (i = k, f) are the eigenvalues and where the gap A is defined as A =-G
1
of h’ corresponding
to the operators
Fk(1-fk-f&&.
c’,
CC.31
k>O
In the above formulae
fk
fk=(l+exp
and fL denote
the usual
(l+exp
fc=
(EklT))-’
and the nk’s and n,-‘s are the occupation
Fermi coefficients:
factors
(EL/T)))l,
(C-4)
of the k and k^ orbitals,
nk=fk+(l-fk-fdd n&=f&+(l-fk-f.d&.
Similarly S=-
the entropy
S = -Tr (fi log 6)
(C.5)
can be expressed
C fklogfk+(1-fk)log(l-fk)+f~logf~+(l-f~)log(1-f~),
k>O
and the average
number
of particles A=Tr(fiA)=
in terms of the fk’s as (C.6)
as 1 k>O
(nk+ni).
(C.7)
P. Bonche et al. / 24Mg Yrast line
134
The variation
of the grand
potential
LZ=Tr(Dfi’)-TS-h,N-A&?, (2, N = proton, condition
neutron
number)
with respect
U: + V; = 1 gives the following E,‘-EL=Q
(C.8) to Ek, EL, uk and uk subject
equations
- EL+ GF;(n,
to the
*l) - nr) ,
Ek=+(Ek+~L), &=&+&Ek^), ii, =$(r&+nl;), ~~=&-h-GF~ii,J2+A2F~, ~;=i(l-(&-A-GF;&)/&).
(C.9)
Knowing the eigenvalues Q, sf; of h’ the solution of the equations (C.4), (C.5) and (C.9) gives the quasi-particles energies Ek and EC and the occupation probabilities nk, n,- used in the calculation of the field energy EN and the single-particle routhian h’ which in turn determines the &i’s* One can also check that in the limit where pairing vanishes, the quantities nk and nl; (and thus the field energy and the entropy) do not depend on the choice of which states k and k^ are being coupled. Appendix D CORRESPONDENCE
BETWEEN
LS AND
SPIN-ORBIT
(so.)
COUPLING
CONFIGURATIONS
In the LS coupling scheme the individual states are doubly (spin) degenerate and are labelled by two quantities P and R which denote respectively the quantum TABLE Correspondence
between
4
LS and so. configurations
(VP)
(J-1 ++
+-
-+
--
A,
2
2
1
1
A3
2
2
1
1
2
3
1
0
D
I II III IV V VI VII
++
+-
-+
-+
3 3 3 3 3 2 2
3 3 3 3 3 3 4
3 3 2 3 3 3 2
3 3 4 3 3 4 4
For the three LS configurations we have indicated their labeling as used in ref. 13) and the number of doubly-degenerate orbitals with quantum number (P, R). On the right side of the table we have reported for seven of the S.O. bands the number of orbitals belonging to each of the four blocks corresponding to a given signature and parity (7, p).
135
P. Bonche et al. J 24Mg Yrast line
numbers
associated
with parity with respect to a plane perpendicular
axis and a spatial rotation of rr around this axis. To a given correspond two S.O. coupling states with parity p = PR and Therefore
it is always possible
in ref. 13). The converse
to the rotation
LS coupling
signature
to assign a S.O. band to each of the LS bands
is not true since a correspondence
state IJ = *R. studied
can only exist for S.O.
bands with equal numbers of positive and negative signature states. In addition several LS fillings may correspond to the same S.O. filling. Table 4 indicates possible correspondences which take into account the considerations of symmetry exposed above and in addition make use of arguments of plausibility (similar deformation or excitation energy).
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31)
P. Quentin and H. Flocard, Ann. Rev. Nucl. Part. Sci. 28 (1978) 523 P. Bonche, H. Flocard, P.H. Heenen, S.J. Krieger and MS. Weiss, Nucl. Phys. A443 (1985) 39 P. Bonche, P.H. Heenen, H. Flocard and D. Vautherin, Phys. Lett. B175 (1986) 387 Y. Horikawa, Y. Torizuka, A. Nakada, S. Mitsunobu, Y. Kojima and M. Kimura, Phys. Lett. 36B (1971) 9 E.W. Lees, C.S. Curran, T.E. Drake, W.A. Gillespie, A. Johnston and R.P. Singhal, J. of Phys. G 2 (1976) 105 M.P. Fewel, S. Hinds, D.C. Kean and T.H. Zabel, Nucl. Phys. A319 (1979) 214 S.A. Wender, C.R. Gould, D.R. Tilley, D.G. Rickel and R.W. Zurmiihle, Phys. Rev. Cl7 (1978) 1365 J.L.C. Ford jr., T.P. Clear-y, J. Gomez de1 Campo, D.C. Hensley, D. Shapira and K.S. Toth, Phys. Rev. C21 (1980) 764 A. Szanto de Toledo, T.M. Cormier, M.M. Coimbra, N. Carlin Filho, P.M. Stwertka and N.G. Nicolis, Phys. Rev. C30 (1984) 1706 A. Watt, D. Kelwin and R.R. Whitehead, Phys. Lett. 63B (1976) 385 M. Diebel, D. Glas, U. Mosel and H. Chandra, Nucl. Phys. A333 (1980) 253 I. Ragnarsson, S. Aberg and R.K. Sheline, Phys. Ser. 24 (1981) 215 H. Flocard, P.H. Heenen, S.J. Krieger and M.S. Weiss, Prog. Theor. Phys. 72 (1984) 1000 D. Baye and P.H. Heenen, Phys. Rev. C29 (1984) 1056 S. Marsh and W.D.M. Rae, preprint Oxford University 45/86 (1986) P. Ring and P. Schuck, The nuclear many-body problem (Springer, Berlin, 1980) pp. 466-70 D. Vautherin and D.M. Brink, Phys. Rev. C5 (1972) 626 Y. Engel, D.M. Brink, K. Goeke, S.J. Krieger and D. Vautherin, Nucl. Phys. A249 (1975) 215 A.L. Goodman in Adv. Nucl. Phys. 11 (1979) 263, ed. Negele and Vogt (Plenum, New York) E.R. Marshalek, Phys. Rev. Cl5 (1977) 1574 0. Civitarese, A. Plastino and A. Faessler, J. Phys. G 9 (1983) 1063; M. Faber, A. Faessler and H. Markum, J. Phys. G 9 (1983) 1069 K.T.R. Davies, H. Flocard, S.J. Krieger and M.S. Weiss, Nucl. Phys. A342 (1980) 111 M. Beiner, H. Flocard, Nguyen Van Giai and P. Quentin, Nucl. Phys. A238 (1975) 29 P. Bonche, S. Koonin and J.W. Negele, Phys. Rev. Cl3 (1976) 1226 Nguyen Van Giai and H. Sagawa, Phys. Lett. 106B (1981) 379 F. Tondeur, M. Brack, M. Farine and J.M. Pearson, Nucl. Phys. A420 (1984) 297 J. Dobaczewski, H. Flocard and J. Treiner, Nucl. Phys. A422 (1984) 103 M. Girod and B. Grammaticos, Phys. Rev. C27 (1983) 2317 A.H. Wapstra and G. Audi, Nucl. Phys. A432 (1985) 1 H. Flocard, P.H. Heenen, S.J. Krieger and M.S. Weiss, Nucl. Phys. A391 (1982) 285 S. Cohen, F. Plasil and W.J. Swiatecki, Ann. of Phys. 82 (1974) 557
SELF-CONSISTENT THE
CALCULATION COMPLETE
OF NUCLEAR
YRAST
LINE
ROTATIONS:
OF =Mg
P. BONCHE Service de Physique 7’ht!orique, CEN Saclay, 91191 Gif sur Yvette Cedex, France H. FLOCARD Division de Physique 7’ht!orique*, Institut de Physique Nucliaire, P.H. Physique Nu&aire
BP1 91406 Orsay, France
HEENEN’
The’orique, Universite Libre de Bruxelles, CP229, B-1050 Bruxelles, Belgium Received
22 October
1986
Abstract: We present a method of solution on a three-dimensional mesh of the self-consistent cranked Hartree-Fock+ BCS equations. Using several parametrizations of the Skyrme interaction we apply our method to the study of the complete yrast line of 24Mg. We find that 5=26 is the limiting angular momentum for this nucleus. We have also studied examples of collective rotations along the y = 60” axis involving hexadecapole deformations. Our work corroborates earlier calculations using the Nilsson-Strutinsky method and extends them to higher spins.
1. Introduction As demonstrated by numerous calculations, the microscopic description of lowspin properties of nuclei by means of an effective interaction like the Skyrme force is remarkably successful ly2). In view of the importance taken in the last decade by nuclear high-spin physics it appears worthwhile to test how well this microscopic approach can help understanding the presently accumulating data. In this work we present a method of solution of the three-dimensional (3D) coupled Hartree-Fock (HF) + BCS equations using a discretization of the wave functions on a rectangular mesh. In this way we can describe any type of even multipole** deformation generated by the interplay of the nuclear mean-field and the constraining rotational field. We can also study with similar good accuracy the low as well as the high-spin configurations, irrespective of the magnitude or shape of their deformation. As compared to earlier 3D HF+ BCS calculations ‘) the breaking of time reversal symmetry induced by the cranking field generates two types of technical difficulties. First all degeneracies of the single-nucleon wavefunctions have to be removed, doubling therefore the size of the calculation. Second the structure of the mean-field ass0ci.G au CNRS. FNRS. l * The method can easily be extended l
Laboratoire
’ Maitre de Recherches
to include
0375-9474/87/%03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
a description B.V
of odd multipole
deformations
-‘),
P. Bonche et al. / 24Mg Yrast line
116
itself becomes
more complicated:
in addition
to the usual
effective mass operator,
the scalar and spin-orbit potentials, there appears contributions from the interaction to the spin-vector and current potentials. These terms usually neglected in semimicroscopic equations
approaches including
have
a treatment
been
included
of pairing
in the present
and temperature
work.
The relevant
effects are given in sect.
2 and in the appendices. As a first application of this method we choose to study the rotational properties of a light nucleus 24Mg from the ground state up to the fission limiting angular momentum. The low-spin properties of this nucleus are well known experimentally 4-9). They extend up to the yrast 12+ and possibly 14+. Recent theoretical studies of rotational properties have been made by shell-model calculations lo) (up to J = 12) and by means of the Nilsson-Strutinsky (NS) method 11V12).Results from cranked HF calculations (without spin-orbit interaction [refs. 13,14)] and from the cranked a-cluster model “) are also available. They are compared with our findings in sect. 3. Sect. 4 contains our conclusions.
2. Solution of the self-consistent The cranking
approximation
cranking equations
is based on the assumption
that a nucleus
with spin
J can be described in terms of an intrinsic state at rest in a frame rotating with some angular velocity w around an axis. The optimal intrinsic state 1P) is determined by minimization of the routhian 8 E=E-wJ,,
(I)
where E and _I, are respectively the expectation the third component of the angular momentum*
values
of the hamiltonian
J, =(?P]@). The angular
velocity
w which
acts as a lagrange
and of
(2) parameter
is determined
by the
condition** J,=J.
(3)
In this work the energy E is calculated within the HF+ BCS approximation, using a functional derived from a Skyrme interaction and a constant strength pairing interaction. We also added a treatment of temperature effects to study the incoherent excitations of the nucleus. According to thermodynamics, one should then replace the energy E by the free energy F = E - TS in the expression (1) of the routhian to describe the isothermal rotation of a system at temperature T. l Our choice of the z-axis instead of the customary structure of the spin Pauli matrix mz which simplifies particle wavefunctions (see appendix A). ** For correction to formula (3) see ref. 16).
choice of the x-axis is motivated by the diagonal the formulation of the symmetries of the single-
P. Bonche et al. / 24Mg Yrast line 2.1. NUCLEAR
AND
The mean-field
COULOMB
energy
(EC). It is well known be written spin
ENERGY
is the sum of a nuclear that for a Skyrme
as the integral
degrees
MEAN-FIELD
of freedom
(EN) and a Coulomb
interaction
of a local functional are taken
into
117
the nuclear
contribution
of a set of densities
account,
the breaking
contribution “). When
of time
can the
reversal
symmetry induced by the constraint wJ,, leads to a rather complicated form for the functional i8) which we give in appendix A. The direct component of the coulomb energy, is calculated by solving the Poisson equation while the exchange contribution is evaluated by the Slater approximation. The single-particle routhian h’ = h - w_!, defined by
(h’@k)(Ca) = ,g; takes the following
a)
k
form for a state k of isospin
,
p
with the effective mass rnz, the spin scalar U,, spin vector V,, current C, and spin-orbit W, potentials given in appendix B. It can be noted that the spin vector (B.3), (B.7) ayd current (B.4), (B.8) potentials are not generated by the cranking constraint -wJ, only. They incorporate additional contributions from the two-body interaction involving the nucleon currents jr and the spin densities p7.
2.2. PAIRING
ENERGY
For a system as light as 24Mg, pairing is not expected to play an important role. For this reason we choose to work with the simple BCS method. As compared to the Hartree-Fock-Bogoliubov (HFB) method, HF+BCS relies on two approximations. First, the contributions to h’ (4) resulting from the variation of the pairing energy with respect to the wavefunctions 02 are neglected. Second, one assumes that the basis within which the pairing tensor takes the simple canonical form, is identical to that diagonalizaing the one-body routhian h’ [ref. ‘“)I. In other terms if cl is the creation operator associated with the eigenstate 0,‘ of h’, the pairing hamiltonian for a constant pairing interaction with strength G, which couples time-reversed states, can be approximated as 20):
ci,= c
G,,C:C;CjC,,
k,l>O
where the sum over k and 1 is restricted to the eigenstates ,+ A signature. On the other hand the states k and 1 are eigenstates
(6) of h’ with positive of h’ with negative
118
P. Bonche et al. / 24Mg Yrast line
signature,
that we associate
described
below.
state E of k and
reversed
an eigenstate the partner coefficient
with k and 1 respectively,
When time reversal
symmetry
Gkl = G. For a rotating
of h’. Following
the prescription
k^ of k, as the eigenstate Gkl is then related
according
is not broken nucleus
suggested
of h’ whose
to the prescription (w = 0), k^is the time
the state E is no longer
by Marshalek
overlap
*‘), we define
with f is maximal.
The
to G by: Gkr = GFkFi,
(7)
Fk = l(El k^)l.
(8)
with
For non-zero values of the angular velocity, the coefficients Fk are smaller than unity, accounting in this way for coriolis anti-pairing. In fact this prescription has been shown to overestimate the anti-pairing effects, and a renormalization prescription of the coefficients Fk has been proposed 2’). In the case of 24Mg we have checked that for reasonable values of the pairing strength G, only the ground state and first 2+ are affected by pairing correlations. As this work is more concerned with higher values of spin which are not affected by pairing, we have decided to use the simpler Marshalek prescription for calculating the matrix elements Gkl. As in ref. ‘) we also introduce a cut-off factor to ensure that pairing acts only on states whose energy lies within
5 MeV of the Fermi
level.
2.3. TEMPERATURE
To describe incoherent excitations a formalism involving temperature
above the yrast line, it is customary to introduce and to consider that collective decays proceed
along isentropic lines. In the mean-field approximation temperature is described by means of a density operator, 6 = exp (-i/ where
l? is a non-interacting
T)/Tr
quasiparticle ff=
1
(exp (-R/T))
a nucleus
,
at non-zero
(9)
operator
&@;@kfi?~~@k^.
(10)
and pz are related
to the c’,+“s and C~“S by the
k>O
In eq. (10) the quasiparticle usual
p:
BCS relations:
(11) pk^= u&-,&c;. C we give the equations and the Ei’s are the quasiparticle energies. In appendix resulting from a minimization of the grand potential with respect to the Ek’s, Et’s, vk’s and &‘s.
P. Bonehe
2.4. NUMERICAL
et al. /
z4Mg Yrast line
119
METHODS
The numerical methods of the present work involve a discretization of the wavefun~ions @Joon a three dimensional mesh. They have been presented in refs. 22,2, where tests of accuracy are also given. We have also explored defo~ation energy surfaces around the yrast intrinsic state and fission barriers using the constraint on the quadrupole operator described in ref. 2). 3. Results 3.1. DEFORMATION
PROPERTIES
OF NON-CRANKED
24Mg
The deformation energy surface of 24Mg calculated with the Skyrme SIII interaction 23) is shown in fig. la. It displays a single minimum for a prolate defo~ation of Q = 107 fm2. This result is in marked difference with that obtained in an earlier calculation I”) with the BKN force “*). Indeed the BKN interaction, which does not incorporate a spin-orbit term leads to a triaxial ground state. The obliteration of triaxiality seems to be common to microscopic calculations using more refined effective interactions. For instance, we checked that the ground state of 24Mg calculated with the SGTI [ref. 25)], T6 [ref. ““)I or the SkP [ref. “‘)] interactions is also prolate. A similar rest& was found in calculations performed with the Dl interaction ‘“). The self-consistent minimum is slightly underbound [E = 196.73 MeV versus E ex.= - 198.3 MeV, see ref, ““)I. This difference is compatible with the magnitude of the correction expected from the projection of the intrinsic state on a O+ state. An estimate of this correction can be obtained by means of the simple formula AE = (~l”/2~)(J*) where (J2) is the expectation value of the total angular momentum. With our calculated values of (J’) (19.46) and the Valatin moment of inertia* fi2/2Yr =0.28 MeV, we find a correction of 5.4 MeV. The charge radius of the
1
0
1
Q(b)
Fig. 1. (a) HF+ BCS deformation energy surface of 24Mg calculated with the $111 interaction pairing strength G = 1.22 MeV. The contour lines are drawn every 0.5 MeV. The deformations given in barn. (b) Same as (a) but without pairing interaction (pure I-IF).
and a Q are
* In principle
Q(b)
the Peierls-Yoccoz
moment
of inertia
should
be used i6).
120
minimum
P. Bonche et al. / 24Mg Yrast line
(3.12 fm) is in good agreement
ref. ‘)I. The proton than the value
quadrupole
deduced
from measurements
62.3 f 4.5 fm*, ref. “)I. The calculated operator*
(e)
with the experimental
moment
which
value [3.08 f 0.05 fm,
we find (Q, = 54.2 fm*), is smaller
of the 2+ quadrupole
expectation
= 51.6 fm4 agrees with the available
value of the proton
moment
[Qy=
hexadecapole
data [41 f 14 fm4, ref. “)I. When
the pairing interaction is switched off the deformation energy surface is slightly modified (fig. lb). The minimum is still prolate (Q = 109 fm2) and its energy (E = -196.62 MeV) is almost unaffected by the pairing. However, the energy difference between the spherical configuration and the ground state increases from 1 to more than 10 MeV. 3.2. ROTATIONAL
PROPERTIES
OF =Mg
The self-consistent cranked yrast and yrare bands of 24Mg are shown in fig. 2. We have restricted our study to positive-parity bands which, except for the band labelled VIII, have been obtained with identical fillings of neutron and proton orbitals (see appendix D). In the upper part of the figure we compare the experimental and calculated spectra. The latter is too compressed, a result generally found in microscopic HF [ref. i3) and references therein] calculations of light nuclei. The pairing correlations, which are rapidly destroyed by Coriolis anti-pairing, affect very little the calculated spectrum. A measure of the Coriolis effect is given by the neutron and proton gaps (eq. (C.3)) whose values are respectively 1.07 and 1.18 MeV for (J,) = 0 and drop to 0 for (J=) = 2. The trajectory of the ground state band in the Q, y plane is drawn on fig. 3. Up to J = 6 the deformation of the intrinsic state stays almost constant. From J =6 to 10, the quadrupole moment decreases while the nucleus remains prolate. The trajectory reaches the oblate axis for the critical value J = 12 (i.e. the maximal angular momentum of a pure sd configuration). From then on, the nucleus flattens continuously along the oblate axis up to the angular momentum 20 beyond which we could not follow this configuration (it is anyhow no longer yrast). The rather puzzling existence of a cranking state evolving smoothly from J = 12 to J = 20 as an apparently oblate solution rotating around its symmetry axis can be understood from the results shown in fig. 4. This figure displays the evolution versus (JZ) of the expectation values (r4Yr) (quantization along the rotation axis 0,). For values of J larger than 12 the moment (r”Yi) cancels exactly as it should be expected for states lying on the y = 60” axis. On the other hand the moment (r4Yi) is non-zero so that a kind of collective rotation** is still allowed. This phenomenon had already been noticed in an earlier calculation of *‘Ne [ref. ‘“)I in which nuclear densities with squarish shapes have been found for values of J between 8 and 14. On fig. 5 we have plotted the single-particle routhian for three * In contrast with the rest of this article, fl is defined here with respect to the axis of the nucleus and not the rotation axis. ** It differs markedly however from a purely classical evolution in which hexadecapole modes are not necessary to achieve a continuous trajectory along the y = 60” axis.
P. Bonche et al. / “Mg
E
121
Yrast line
10+ 12+
20
__
70 9 P 21
50
IO-
8'
8+ 6+
4+ _ 2+
30 o-
vThe
0
IO
20 (JZ)h
Fig. 2. Excitation energies (in MeV) versus angular momentum (in fi) of the several bands studied in this work. The arrows point to the critical point of the bands indicated in the parentheses. The left-up part of the figure shows a comparison of the calculated and experimental spectra of the ground state band.
values of (J=). For (J,) = 12 and 20 where 24Mg is almost axially symmetric, each state is labeled not only by its parity but also by the expectation value of J_ in that particular state. While the occupation scheme at (JZ) = 12 is compatible with a pure (sd) shell filling, it is no longer the case at 20 where one of the occupied state (of negative parity) brings a s contribution to (JZ). In these two cases one can notice empty states in the Fermi sea. The reorganization of the occupation levels which remove these holes leads to different configurations (and different spins) studied later in this section. Comparing our results with those obtained by Ragnarsson, Aberg and Sheline 12) with the NS method using a modified oscillator potential, we find many similarities and some differences. Their ground state is triaxial until J = 6, as opposed to our prolate solution. From J = 6 to the band termination at J = 12
Fig. 3. Trajectories
in the Q, y half-plane
of the several bands (with the exception of the fission band IV) studied deformation plane has been drawn on the right side of the figure.
(24)
0 l
in this work. The lower part of the
XXIII
4’
3Xt
P
II m
A
AlP
I
0
123
- -5o__-. I
0
f Ll’
I
I
I
I
IO (J,??
Fig. 4. Expectation
I
I
I
20
0
I
I
I
(J,bi
values versus angular momentum (in h) of the hexadecapole moments fm4) for band I and II. The quantization axis is the rotation axis.
_
IO r’Yy
(in
the path of the NS trajectory follows closely that found in our microscopic calculation. As in the present work, they find a spectrum too compressed compared with experiment. In contrast with our results, their ground-state trajectory stops at .I = 12. We think that this is due to the neglect of Yi components (with respect to the rotation axis) in the potential. In fact many (if not all) of the qualitative differences between their results and ours stem apparently from the restricted parametrization of the potential adopted in ref. “). Another example of collective rotation allowed by hexadecapole deformations is given by the band associated with the yrast 8+ state at 11.86 MeV. It has been established experimentally ‘**) and confirmed by shell-model calculations lo) that this state does not belong to the ground state band. In our calculation (see fig. 5) it results from the transfer of one neutron and one proton from the d=,,2,-3,2 to the “) determined this cranked state to be d 5,2,5,2 orbitals. Similarly, NS calculations on the y = -120” axis: a prolate nucleus rotating around its symmetry axis. In ref. 12) the 8+ is an isolated state. The authors propose to correlate this 8+ with an excited 2+ state obtained by the promotion of one particle from the [211, -51 level to the i]. On the other hand we could follow down to lower spins a band (band II) terminating on the yrast 8+ state. This band connects with the ground state of 24Mg located on the y = -120” axis. From J = 0 to J = 8, the prolate deformation of the nucleus decreases while the principal axis of symmetry remains aligned with the rotation axis. Again this smooth transition is made possible by the non-zero expectation value of r4Yi (right side of fig. 4). As for the 2+ state studied in ref. I*), we find [Zll,
124
P. Bonche et al. / z4Mg Yrast line -r
TI
I
r-
-+
-10
1/2’
-
__1/2+ -+1/Z+
--+
--cY2+
0
Y
$
-+
-z W
-
_5/2+
-
n/2+
-
Y.?
-0
312’
_
,,2+@G-E:
F
--+ -
0 3,2+ -
1/z+
+
-
+ G3c
:
-=+ +
t/2+
--__
,,2-
--+ --3/a-
_--
--+
--3/2-
3/Z+
_
a/*-
-
i/27;;:
,,2-
-
5/2+
-
_ 3/2+ -_3/-2-
i/2-
---
--3R-
-
,,2-
1/2-
--f1/2-
-
-
=; ---
5/z+
3/2-
---I/*-
-5/2+
-+
-_,,2-
-
-
__--
__--
/
--+
__t3,,-
,-p:rrj , -+-
_,’ r/2-
--
T/2-
=:
---i/2-
-. ,’
-_1,2-
_-i/2-
0 =_ +
0 0 =___= --- -+ -:4
-i1,2+
--t/2-
-30
7:;:
----_,,2-
-+
-20
___. 5/2+
=-::; -
5/2 -
-_
-_3/2+
0
-+ -+ --
--1/z+
0
-_:+
-
1,2-
--
1/2-
-+ --
1/2-
A___--__-_*
--
,-+-
/’ -
I’
Y2-
-
--
3/2-
--f -4o--+
-*1/z-
_
J,llil
0
,,2+
12
-
-.
-.___
-+
--“Z+
-_,,2+
=,y:
3/2-
-.1/2+ 1/2+
-+
t/2+
-t/2+
0
20
--l/2+
8
14
16
-+ 20
22
Fig. 5. Single particle routhians of several bands studied in this work. The energies are given in MeV and the spins (J,) indicated at the bottom in h. The circled number 12 points to the position of the fermi level. The dashed lines indicate empty states below the Fermi level. The + and - signs refer to the parity. For nearly axial configurations and when the principal axis of symmetry is identical to the rotation axis, the parity quantum number is preceded by the expectation value of J, in the state. For band IV the signs indicate
the signature
and the parity
of the level.
that this state does not belong to band II. By considering either the promotion of a neutron or a proton orbital, we could create two almost degenerate excited bands (band VIII) which are also located on the y = -120” axis but are not related to the yrast 8+ state.
26
I? Bonche et al. / z4Mg Yrast line
The removal
of the g- hole in the occupation
125
scheme of the ground-state
configur-
ation at J = 12 can be achieved by emptying either the 3’ or the +’ states. In the former case, one obtains the band III at spin J = 16. We could follow this band up to J = 26 and down to J = 8. From J = 14 to 20-22 (with the exception is the Yrast band.
A similar
result was found
by the authors
of J = 18) it
of ref. 12) except that
the positions of the 12+ states of band I and III are interchanged. They also conclude that the band III terminates at J = 16. However, we suggest that the secondary triaxial minimum appearing in their J = 20+ map (fig. 14 in ref. “)) and that they consider as leading to fission could in fact belong to band III. Indeed we shall see that the trajectory of our fission band lies very close to the y = 0 axis, while the states 20
value of the angular
momentum
(J = 26). In the left-down
corner
we have
drawn the contour lines of the total density at the minimum. The preformed 12C are nearly oblate and their principal axes are perpendicular to the rotation axis 0,. This preformation appears also on the spectra drawn on fig. 5. As (J,) increases from 22 to 26, one can observe a grouping of levels in pairs with opposite parity and signature. Our results differ in many respects from those found with HF using the BKN force 13) and the cranked a-model 15). Some of our bands (III, VI) cannot even be studied when the spin degree of freedom is neglected (see appendix D). The beginning of the yrast line looks similar in both LS and spin-orbit schemes and the critical point is reached for J = 12. However, the trajectories in the Q, y plane are somewhat different. The configuration D which was found yrast in ref. 13) seems to correspond to our band VII which is never Yrast (although never very
126
P. Bonche et al. / 24Mg Yrast line
-133
-135
0
5
Q(b)
Fig. 6. The upper part shows the deformation energy curve at J = 22. The 0.5 MeV. The minimum is indicated by the dot. The lower part shows the quadrupole moment for the limiting angular momentum J=26. On the sections of the total density for the yrast state (0; is the rotation axis). every 0.02 fmm3.
contour lines are drawn every fission path as function of the left side we have drawn two The contour lines are drawn
excited). The high spin yrast band IV corresponds to the band A3 of ref. 13) which is either yrast or slightly excited. The values of the scission limit angular momentum as well as of the maximal excitation energy are however comparable: (24 h, 60 MeV) versus (26 h, 73 MeV), reflecting probably the decreasing importance of spin-orbit effects near the scission limit angular momentum. 3.3. INFLUENCE
OF TEMPERATURE
On fig. 7 we show the isotherm T = 1.5 MeV, which exhibits two bands only. At finite temperature all the configurations favored at lower spins merge into a single
P. Bonche el al. / “Mg
127
Yrast line
IO m
5
t
2
50
H *-
w
0
1 0
.
I
I
IO
I
20
I
.
CJ;Pfi
Fig. 7. Isotherm T= 1.5 MeV versus fission band is drawn as the dashed
angular momentum in h. The dots indicate the Yrast states. The curve. The upper part displays the total entropy versus angular momentum.
one (the deformation
surface
energy
at J = 0 becomes
very flat with a spherical
minimum and the band II disappears). However the fission configuration curve), which corresponds to noticeably larger values of the quadrupole
(dashed moment,
remains separate. The excitation energy is always of the order of few MeV and the limiting angular momentum is still 26. The upper part of the figure shows that the isotherm does not differ very much from an isentropic: the value of the total entropy (e.g. (C.6))
remains
between
7 and 10. 4. Conclusions
This article presents the first study of very high-spin nuclear properties, based on a purely microscopic description using a standard effective
which is nucleon-
nucleon interaction. Since the domain of validity of such interactions extends in principle over the entire chart of elements, we expect the method exposed in this work to allow a microscopic description of nuclear rotations for all nuclei. Within the present cranked self-consistent formalism, and in contrast with more phenomenological models, there is no a priori assumption on the deformation of the mean-field. In some methods of solution of the HF+BCS equations the unavoidable truncation of the orbital basis introduces implicit constraints on the shape
128
P. Bonche et al. / 24Mg Yrast line
of the nucleus.
Here they are avoided
by a discretization
of the orbitals
on a regular
3D mesh. Actually we have found new examples of collective (although non-Yrast) rotations which are made possible by hexadecapole deformations of the routhian mean-field. We have studied Several effective
eight bands
interactions
of 24Mg. Four of them contribute
with incompressibilities
ranging
to the Yrast line.
from 220 to 350 MeV
and effective mass between 0.75 and 1 have been shown to give similar results, both qualitatively and quantitatively. This supports the widely accepted notion that rotational properties are primarily determined by properties of the interaction connected with multipole deformations for which 12 2. Our results corroborate in several respects earlier Nilsson-Strutinski calculations I’), which they complete and extend to the end of the Yrast line. In particular we were able to study all the bands beyond their critical point. The end part of the Yrast line (22cJ~26) is found to correspond to a very elongated system which ultimately fissions in two oblate 12C nuclei. The limiting angular momentum (J = 26) appears compatible with classical estimates 31). The study of the evolution of the fission barrier as a function of the angular momentum reveals that the optimal configuration as well as the saddle point are always slightly triaxial. It appears also that the fission valley in the Q, y plane is remarkably narrow in the y-direction. This could contribute to an increase of the stability of these fast rotating configurations.
We would like to thank F. Naulin for his help in calculating the deformation energy surface of 24Mg, and M. VCnCroni for discussions and a critical reading of the manuscript. This work was made possible by extensive computing facilities provided by the Centre de Calcul Vectoriel pour la Recherche. P.H. Heenen expresses also his gratitude to the FNRS for a computational grant. This work was partly supported
by the NATO
grant
RG85-0195.
Appendix A
NUCLEAR
MEAN-FIELD
ENERGY
In this appendix, we summarize results already presented in ref. 18) within the context of time-dependent Hartree-Fock and we recast them in a form suitable for threedimensional cranking calculations. With a simple effective interaction like the Skyrme force “) the energy EN can be written as the space integral of a local energy density: EN=
H(r)
d3r,
(A.1)
P. Bonche
et al. /
“Mg
Yrast line
129
with* H(r)=(h2/2m)7+B,p2+Bz(p~+p~)+B,(p7-j2)+B,(p,7,-j%+p,7,-j~) + Bs PAP + Bd~nh,
+ pp&,)
+ BTP’+~ + B,P” (d + P;)
+B,(pV.J+j.Vxp+p,V-J,+j;VXp,+p,V.J,+j,.VXp,) (A.21
+B,~P~+B~,(P~+P~)+B~~P~P*+B~~~~(P~+P~).
The densities p,, T,, j,, V . J, and pT (T = {n, p}) which appear in (A.2) can in turn be expressed in terms of the single-particle wavefunctions 4&( r, a) and their occupation factors
nk (defined
dr)
=
below
(C.5)) as
c nkiV@k(r, kc7
V . J(r) = -i
c
a)i29
r&(V@r(r,
a) xV@k(r,
a’)) * (c+~~‘),
km+
64.3) Using a generalized t,, x3, a and W:
Skyrme
interaction
2, with parameters
to, x,,,
t, ,
x1, t2, x2,
- kxsk’,
(A-4)
o= t,(1+x,P,)S+~t,(1+x,P,J(k26+6k’*) + t2(1+xzP,)k*
Gk’+~t,(l+x,P,)p”G+iW(a,+~~)
we can express the coefficients
Bi as:
B,=~t,(l+~x,)
B, = +t,(++xx,)
B2 = -$t&+x,)
B,=-;W
B3=+(t,(l+;x,)+t2(1++xZ))
Blo =&,x0
B4=-+(t,(;+xl)-t&+x2))
B,, = -&,
B5 = -&(3t,(l
++x,) - t2(1 +$x2)) B12 =$t,x,
B,=&(3tI(;+x,)+ B,=&t,(l++x,)
t&+x2))
BIj=-&t3. (A.5)
The complete Skyrme functional contains other terms resulting from a tensor coupling between spin and gradiant. The contribution of these terms which cancels exactly for spherical time reversal invariant wavefunctions is always an order of magnitude smaller than the spin-orbit contribution. Since furthermore they are generally not taken into account in the determination of the Skyrme parametrizations, they have been neglected in the present work. l
130
P. Bonche et al. / *4A4g Yrast line
In this calculation
we impose
the following
two symmetries
on the individual
wavefunctions: (i) parity &Jr,
a) = @A-r,
u) =pk@k(r, a),
(‘4.6)
pk=*l,
(ii) z-signature e ir(‘z-1’2)@k (x, y, z, u) = u@k (-x,
-y, z, u) (A-7)
= The complete
description
r]k@k(X,
of a wavefunction
y,
5
fl)
requires
,
vk=fl.
the knowledge
of four real
functions !Pku ( r) ( CY= 1 - 4) corresponding to the real (Re), imaginary (Im) spin-up and spin-down parts of @k. For positive and negative signature wavefunctions we adopted a different numbering of these four components:
This choice ensures that the parities of the components ‘P,,., with respect to plane symmetries about x, y and z planes depend only on the label (Y and the parity pk as indicated in table 1. The action of the time reversal operator on a function @k generates a state @n with same parity and opposite signature such that
,
The @‘E’Sare used to calculate k^ and the coriolis anti-pairing
the matrix coefficients
vz=--vk.
(El f) from which we determine Fk (eq. (8)).
TABLE 1 Parities of the components ?Pk,, of a wavefunction ok of parity pk, with respect to the x=0, y=O and z=O planes. a
x
Y
Z
1 2 3 4
+ _
+ _
+
+ _
Pk Pk -Pk -Pk
(A.9)
the state
131
I? Bonche et al. / 24Mg Yrast line
The densities following
V - J once
p, 7, j and
expressed
in terms
of the
Vka take the
forms: (A.lO)
T(r)=c nk i k
-
nktqk,lvwk,2
j(r)=1
(A.ll)
(vpk,a)2,
a=,
%Qv
vk,,
+
qk,,v
pk,4
-
qk,4v
vk,3)
,
(A.12)
k
tensor and where the coefficients a( I), p( 1), where Q,,,,, is the fully antisymmetric y(I) and 6( 1) are those given in table 2. Similarly the vector density p can be written: 2 ( wk, 1w;, + p(r)
=;
i Parity
and signature
induce
qk.2
2r]k(pk,l~k,4-
nk vk(
?&
definite
+
pk,4
qk,2pk,3) %,Z
-
?&
-
symmetries
. ?&I)
(A.14)
1
for each of the densities
(A.3)
with respect to the planes x, y, z = 0. They are listed in table 3. The direct part of the Coulomb interaction E&direct), 3 ,p,(r)p,(r’) d r lr-r’l is calculated the Poisson
’
(A.15)
in two steps. We determine first the Coulomb potential V,(r) by solving equation, then we evaluate the integral ij d3rp, V, . The exchange part TABLE 2 Indices
of the components Pk_ for V * J (formula
entering A.13)
the formula
a
P
Y
s
vk=+l 1
1
2
3
3
1
4 1 2
3
2
4 4
2 3
3 1 3
2 3 4
1 2 2
4 4 1
1
TJk=-l 1 2 3
The upper (lower) table corresponds positive (negative) signature vk.
to a state with
132
P. Bonche et al. / 24Mg Yrast line TABLE
Parities
P. 7,
V .J
P.x pv P; Jx i, iz
of the Coulomb
energy
3
of the nucleon densities thex=O,y=Oandz=Oplanes x
Y
z
+ _ + + + _
+ + + + _
+ + + + -
is approximated
E,(exchange)
with respect
by means
to
of the Slater approximation
= -$e2(3/r)“3 I
d3rpp(r)4’3.
as: (A.16)
The center-of-mass recoil effect is approximately taken into account by a subtraction of the expectation value of (Ckpi)/2mA (A is the mass number) from the mean-field
EN.
energy
Appendix B HARTREE-FOCK
HAMILTONIAN
AND
ROUTHIAN
The variation of the nuclear energy EN with respect to the wavefunction @z( r, a) defines the one-body Hat-tree-Fock hamiltonian h. For a generalized Skyrme functional (13), its action following potentials:
h2
on a neutron
wavefunction
Ok takes the form (5) with the
h2
(B.1)
-=~~+B,P+&P,, 2m,* n
U,,=2Blp+2B2p,+B3(~+iV~j)+B4(~,+iV~jn) +2BSAp+2B6Ap,+(2+a)B,p’+” +B8(apa-‘(p;+p3+2papn)+ +
~P~-‘UGZP~+
B,,(P:+P;))
Bg(V - J+V
*J,)
,
(B.2)
V,,=B,(V~~+V~~,)+~B~~+~B,,P,+~P*(BIZP+BI~P~)
03.3)
C’,=2B3j+2B,j,-B,(Vxp+Vxp,),
054)
W, = - B,(Vp + VP,) .
(B-5)
133
P. Bonche et al. / z4Mg Yrast line
The Hartree-Fock identical,
hamiltonian
for a proton
except for the spin scalar potential
contribution
wavefunction
is mutatis
UP which is completed
mutandis
by the Coulomb
UC: U, = V, - e2(3p,/ 7r)“3.
03.6)
To obtain the routhian h’ the potentials V, and C, must be modified adjunction of terms generated by the constraint -oJ~,: V,(r) + V,(r) -$hwe,
by the
,
03.7)
C,(r)+C,(r)+hw(ezxr), where e, denotes
(B.8)
the unit vector along the z-axis.
Appendix C TEMPERATURE By virtue
FORMALISM
of the Wick’s theorem fit=
the expectation
C F~c:c~+E&cL-GC
by the density Tr(DfiO=
Z?
F,F,c;c$cjc,,
(C.1)
kl
k>O
in a state defined
value of the routhian
c
operator
(9), takes the simple
following
form:
(Eknk+&k^nk^-GF2knknk^)-d2/G,
(C-2)
k>O
where the q’s (i = k, f) are the eigenvalues and where the gap A is defined as A =-G
1
of h’ corresponding
to the operators
Fk(1-fk-f&&.
c’,
CC.31
k>O
In the above formulae
fk
fk=(l+exp
and fL denote
the usual
(l+exp
fc=
(EklT))-’
and the nk’s and n,-‘s are the occupation
Fermi coefficients:
factors
(EL/T)))l,
(C-4)
of the k and k^ orbitals,
nk=fk+(l-fk-fdd n&=f&+(l-fk-f.d&.
Similarly S=-
the entropy
S = -Tr (fi log 6)
(C.5)
can be expressed
C fklogfk+(1-fk)log(l-fk)+f~logf~+(l-f~)log(1-f~),
k>O
and the average
number
of particles A=Tr(fiA)=
in terms of the fk’s as (C.6)
as 1 k>O
(nk+ni).
(C.7)
P. Bonche et al. / 24Mg Yrast line
134
The variation
of the grand
potential
LZ=Tr(Dfi’)-TS-h,N-A&?, (2, N = proton, condition
neutron
number)
with respect
U: + V; = 1 gives the following E,‘-EL=Q
(C.8) to Ek, EL, uk and uk subject
equations
- EL+ GF;(n,
to the
*l) - nr) ,
Ek=+(Ek+~L), &=&+&Ek^), ii, =$(r&+nl;), ~~=&-h-GF~ii,J2+A2F~, ~;=i(l-(&-A-GF;&)/&).
(C.9)
Knowing the eigenvalues Q, sf; of h’ the solution of the equations (C.4), (C.5) and (C.9) gives the quasi-particles energies Ek and EC and the occupation probabilities nk, n,- used in the calculation of the field energy EN and the single-particle routhian h’ which in turn determines the &i’s* One can also check that in the limit where pairing vanishes, the quantities nk and nl; (and thus the field energy and the entropy) do not depend on the choice of which states k and k^ are being coupled. Appendix D CORRESPONDENCE
BETWEEN
LS AND
SPIN-ORBIT
(so.)
COUPLING
CONFIGURATIONS
In the LS coupling scheme the individual states are doubly (spin) degenerate and are labelled by two quantities P and R which denote respectively the quantum TABLE Correspondence
between
4
LS and so. configurations
(VP)
(J-1 ++
+-
-+
--
A,
2
2
1
1
A3
2
2
1
1
2
3
1
0
D
I II III IV V VI VII
++
+-
-+
-+
3 3 3 3 3 2 2
3 3 3 3 3 3 4
3 3 2 3 3 3 2
3 3 4 3 3 4 4
For the three LS configurations we have indicated their labeling as used in ref. 13) and the number of doubly-degenerate orbitals with quantum number (P, R). On the right side of the table we have reported for seven of the S.O. bands the number of orbitals belonging to each of the four blocks corresponding to a given signature and parity (7, p).
135
P. Bonche et al. J 24Mg Yrast line
numbers
associated
with parity with respect to a plane perpendicular
axis and a spatial rotation of rr around this axis. To a given correspond two S.O. coupling states with parity p = PR and Therefore
it is always possible
in ref. 13). The converse
to the rotation
LS coupling
signature
to assign a S.O. band to each of the LS bands
is not true since a correspondence
state IJ = *R. studied
can only exist for S.O.
bands with equal numbers of positive and negative signature states. In addition several LS fillings may correspond to the same S.O. filling. Table 4 indicates possible correspondences which take into account the considerations of symmetry exposed above and in addition make use of arguments of plausibility (similar deformation or excitation energy).
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31)
P. Quentin and H. Flocard, Ann. Rev. Nucl. Part. Sci. 28 (1978) 523 P. Bonche, H. Flocard, P.H. Heenen, S.J. Krieger and MS. Weiss, Nucl. Phys. A443 (1985) 39 P. Bonche, P.H. Heenen, H. Flocard and D. Vautherin, Phys. Lett. B175 (1986) 387 Y. Horikawa, Y. Torizuka, A. Nakada, S. Mitsunobu, Y. Kojima and M. Kimura, Phys. Lett. 36B (1971) 9 E.W. Lees, C.S. Curran, T.E. Drake, W.A. Gillespie, A. Johnston and R.P. Singhal, J. of Phys. G 2 (1976) 105 M.P. Fewel, S. Hinds, D.C. Kean and T.H. Zabel, Nucl. Phys. A319 (1979) 214 S.A. Wender, C.R. Gould, D.R. Tilley, D.G. Rickel and R.W. Zurmiihle, Phys. Rev. Cl7 (1978) 1365 J.L.C. Ford jr., T.P. Clear-y, J. Gomez de1 Campo, D.C. Hensley, D. Shapira and K.S. Toth, Phys. Rev. C21 (1980) 764 A. Szanto de Toledo, T.M. Cormier, M.M. Coimbra, N. Carlin Filho, P.M. Stwertka and N.G. Nicolis, Phys. Rev. C30 (1984) 1706 A. Watt, D. Kelwin and R.R. Whitehead, Phys. Lett. 63B (1976) 385 M. Diebel, D. Glas, U. Mosel and H. Chandra, Nucl. Phys. A333 (1980) 253 I. Ragnarsson, S. Aberg and R.K. Sheline, Phys. Ser. 24 (1981) 215 H. Flocard, P.H. Heenen, S.J. Krieger and M.S. Weiss, Prog. Theor. Phys. 72 (1984) 1000 D. Baye and P.H. Heenen, Phys. Rev. C29 (1984) 1056 S. Marsh and W.D.M. Rae, preprint Oxford University 45/86 (1986) P. Ring and P. Schuck, The nuclear many-body problem (Springer, Berlin, 1980) pp. 466-70 D. Vautherin and D.M. Brink, Phys. Rev. C5 (1972) 626 Y. Engel, D.M. Brink, K. Goeke, S.J. Krieger and D. Vautherin, Nucl. Phys. A249 (1975) 215 A.L. Goodman in Adv. Nucl. Phys. 11 (1979) 263, ed. Negele and Vogt (Plenum, New York) E.R. Marshalek, Phys. Rev. Cl5 (1977) 1574 0. Civitarese, A. Plastino and A. Faessler, J. Phys. G 9 (1983) 1063; M. Faber, A. Faessler and H. Markum, J. Phys. G 9 (1983) 1069 K.T.R. Davies, H. Flocard, S.J. Krieger and M.S. Weiss, Nucl. Phys. A342 (1980) 111 M. Beiner, H. Flocard, Nguyen Van Giai and P. Quentin, Nucl. Phys. A238 (1975) 29 P. Bonche, S. Koonin and J.W. Negele, Phys. Rev. Cl3 (1976) 1226 Nguyen Van Giai and H. Sagawa, Phys. Lett. 106B (1981) 379 F. Tondeur, M. Brack, M. Farine and J.M. Pearson, Nucl. Phys. A420 (1984) 297 J. Dobaczewski, H. Flocard and J. Treiner, Nucl. Phys. A422 (1984) 103 M. Girod and B. Grammaticos, Phys. Rev. C27 (1983) 2317 A.H. Wapstra and G. Audi, Nucl. Phys. A432 (1985) 1 H. Flocard, P.H. Heenen, S.J. Krieger and M.S. Weiss, Nucl. Phys. A391 (1982) 285 S. Cohen, F. Plasil and W.J. Swiatecki, Ann. of Phys. 82 (1974) 557