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A relativistic theory of superdeformations in rapidly rotating nuclei
Nuclear Physics North-Holland
A51 1 (1990) 279-300
A RELATIVISTIC
THEORY
RAPIDLY
OF SUPERDEFORMATIONS
ROTATING
W. KOEPF Physikdepartment
der Technischen
Universitiit
IN
NUCLEI*
and P. RING Miinchen,
D-8046 Garching,
Fed. Rep. Germany
Received 7 August 1989 (Revised 27 September 1989) Abstract: Superdeformed shapes in rapidly rotating nuclei are investigated in the framework of a cranked relativistic mean field theory. A well established parameter set containing a non-linear self-coupling between the cr-mesons is used. Excellent agreement with experimental data in the nucleus “*Dy is achieved and new superdeformed bands in “Sr are predicted. It turns out that the spatial components of the vector w-mesons (nuclear magnetism), which are usually neglected in applications of this theory for small angular momenta, play an important role for a quantitative understanding of the moments of inertia in these superdeformed shapes.
1.
Introduction
Since the theoretical prediction ‘) and the experimental discovery ‘) of superdeformed bands in the nucleus ‘52Dy a large number of experimental 3-5) and have been devoted to this interesting phenomenon in theoretical 6-‘o) investigations fast rotating nuclei. On the experimental side similar bands have been found in a number of neighbouring nuclei in the same region and in other regions of the periodic table. Quadrupole moments have been measured and it has been verified that these bands indeed correspond to shapes with an axis ratio of roughly 2: 1. As Strutinsky has already pointed out more than twenty years ago “3’2), these shapes produce large negative contributions to the shell correction energy. In the actinides the Coulomb and surface energies nearly balance each other, and one therefore In lighter nuclei the observes the fission isomers 13*14)with large deformations. Coulomb energy is weaker and the surface energy dominates at small angular momenta thereby not allowing these large deformations. Only at very large angular velocities the deformation driving rotational energy takes over, and allows the minima in the shell corrections to appear. On the theoretical side nearly all of the investigations have been carried out in the semi-phenomenological framework proposed by Strutinsky “,‘*) and extended to rotating nuclei by several groups in the seventies 15-“): The bulk properties of the nucleus are treated in the liquid drop model, to which shell corrections, calculated l
Work supported
0375-9474/90/$03.50 (North-Holland)
in part by the Bundesministerium 0
Elsevier
Science
Publishers
fiir Forschung B.V.
und Technologie.
W. Koepf
280
P. Ring / S~perdef~~alions
in a rotating single particle potential, are added. At angular been shown ‘*) that this method is an excellent approximation
momentum zero it has to a full Hartree-Fock
or Hartree-Fock-Bogoliubov calculation with density dependent forces. In the rotating frame such calculations are extremely complicated *9-23), because time reversal
symmetry
is broken
and spin saturation
self-consistent descriptions of superdeformed framework of cranked Hartree-Fock calculations in the strontium region 2’-23).
is no longer
valid. As far as fully
bands of heavy nuclei in the there exist to date only calcutations
On the other hand it has become fashionable to describe nuclear structure in the framework of a relativistic quantum field theory. In analogy with early attempts of Teller and collaborators 24,25), Walecka 26,27) introduced a model, in which the nucleons were treated as Dirac particles interacting by the exchange of mesons. In the simplest approximation this field theory is treated at the mean field level. Only very few parameters are introduced and adjusted to experimental data of nuclear matter and a few finite spherical doubly-magic nuclei. In recent years it has been shown that without changing these parameters one can also describe nuclear deformations in light 28-3’) and heavy nuclei 32*33 ). As compared to density dependent Hat-tree-Fock calculations with Skyrme 34+35)or Gogny 36) forces these relativistic theories start at the meson level and are therefore more fundamental. In addition they include spin properties from the beginning in a consistent way and - dealing only with a few local fields - they are technically considerably simpler to handle and require less effort as density dependent Hartree-Fock calculations. These theories have recently been extended to the rotating frame in ref. 37). First applications to the nucleus ‘“Ne show that the results are similar to Skyrme calculations. This light nucleus, however, is not a very good rotor and a mean field approximation is only possible with restrictions. In this paper we therefore present the first applications of cranked relativistic mean field theory to heavy nuclei. We go directly to the region of large angular momenta - as typical for superdeformed nuclei - because to date there is no relativistic description of pairing which plays an important role in the low spin region of deformed nuclei. Theoretical investigations of pairing properties in superdeformed bands show that they are considerably quenched
9,38). As a first stage it should
be a good approximation
to neglect
them
completely. In sect. 2 we briefly present the theory and give the basic de~nitions. In sect. 3 we investigate the nucleus ‘“Sr and predict superdeformed bands in agreement with earlier non-relativistic calculations using Skyrme forces. Sect. 4 is devoted to the first experimentally discovered superdeformed band in the nucleus 152Dy. The calculated deformation is in excellent agreement with experiment. In order to understand the proper value of the moment of inertia, however, it turns out to be important to inciuCe the spatial contributions of vector mesons - nuclear magnetism - which do not show up in nuclear structure calculations at spin zero, where time reversal invariance is valid. In sect. 5 we draw conclusions about our results.
W. Koepf; P. Ring / Superdeformations
2. The theoretical
281
formalism
In relativistic mean field theory 27) the nucleus is described as a system of point-like nucleons, Dirac particles coupled to mesons and to the photon. There are essentially three
types
of mesons:
between
the nucleons,
a nearly
as strong
the scalar
u-meson,
which
and the two vector mesons,
repulsion
and the isovector
provides
a strong
attraction
the isoscalar
w-meson
providing
takes
care of the
p-meson,
which
symmetry energy and which is important in heavy nuclei. We therefore start with the well known effective lagrangian L= q( y”(ia,
-gww,)-(m+g,a))~+-C(a,aac”a-
-&,w,-a,w,)(a”w”
-d”wp)+$?12,w,w~.
density
rn~(~~)-~b~c~-~b~u~
(1)
In our actual calculations we also took into account the isovector p-mesons and the photons, i.e. the Coulomb field. As both of them are vector mesons, they are described in analogy to the o-meson. In order to keep the equations short their contributions are not explicitly written out in eq. (1) or in following equations. Apart from the linear coupling between the mesons and the nucleons the lagrangian (I) contains a non-linear coupling between the cr-mesons, which has been introduced by Boguta and Bodmer 39) and which turns out to be crucial for a quantitative description of many nuclear structure data, as for example for the surface properties or the deformations *“). This lagrangian contains as parameters the masses of the mesons m,, m, and m,, the coupling constants g,, g, and g,,, and the non-linear terms b2 and b,. The mesons are not necessarily considered as bare mesons as observed in scattering experiments, but they are considered as effective particles carrying the most important quantum numbers and generating the interaction in the corresponding channels in a Lorentz invariant manner by a local coupling to the nucleons. In that sense the lagrangian (1) is an effective lagrangian constructed for the mean field approximation. Its parameters are adjusted to experimental data. Therefore, we do not include the v-meson explicitly, because it carries negative parity and its contribution would vanish in the mean field approximation. It would show up in two-meson-exchange processes and is in that sense represented by the a-meson. In a sense the lagrangian (1) is a relativistic analogy to the well-known multipole expansions of the effective nuclear interaction, as for instance the pairing plus quadrupole model 40). where one also takes into account only the lowest quantum numbers in the appropriate channels, and where the parameters are fitted to experimental data. In this paper we use the well established parameter set NLI of ref. 41). It was determined by fitting to the nuclear matter characteristics and to the ground state properties of a few spherical nuclei. It has been used recently in a number of investigations over the entire periodic table 33342343) producing excellent results for binding energies, nuclear radii, quadrupole and hexadecupole deformations, and
282
W. Koepf, P. Ring / Superdeformarions
also for the nuclear turned
densities
out that its results
in spherical
Skyrme forces. The corresponding listed in table
and deformed
nuclei.
are very close and sometimes
even better
values for the masses and coupling
In particular
it
than those of constants
are
1.
As already outlined in a previous paper 37), the rotation in the relativistic mean field theory using the assumptions
of nuclei can be described of the Cranking model 44)
by transforming the effective lagrangian density (1) which acts in the laboratory system into a frame which rotates with constant angular velocity L! around a fixed axis (in our cast the z-axis). We then obtain the following covariant generalization of the lagrangian density (1) using the techniques of general relativity: ~=~(~~(i(~,+~~)-g,~,)-(m+g,~))~+~(g~’”d,~~,~-m~cT2)-~b2~‘-~b3~4 I M I -~g~““g”P(a,3,-a”;,)(a,;,-a,~,)+~m~g~~;,w,.
(2)
This lagrangian contains the mesonic fields, 6 and GIL, and the baryonic spinors 4 in the rotating frame. They are found by simply transforming the fields and spinors in the laboratory frame, a, wF and $, using the standard formulas already known from special relativity, cT=e
iR.U
u,
(3a)
-0
( >(-(Ax;)y)(“d::,w,“)? w
=
&
6
=
,iiE.Jl
g
,
(3b) (3c)
where L and J = L+ S are the operators for the orbital and for the total angular momentum. The spin operators S are either 3 x 3-matrices for the vector mesons or the Dirac spin matrices for the nucleons. In addition the lagrangian (2) also contains the non-diagonal riemannian metric tensor g Py, i.e. the generalization of the flat TABLE 1
The parameters of the lagrangian non-linear set NLl 4’)
for the
NLl
m,, (MeV) m, ( MeV) g, &J & b, (fm-‘) b,
492.25 795.36 10.138 13.285 4.976 -12.172 -36.265
The nucleon mass is taken to be mN = 938 MeV, and the mass of the p-meson is M,, = 763 MeV.
W. Koepi
Minkowsky
metric,
of the baryonic
the modified
spinors
ing transformations The variational
P. Ring / Sueerdeformatio~s
gamma
matrices
283
7” and the covariant
derivative
fW. All the details are given in ref. 37), where the correspond-
are described explicitly. principle gives the equations
of motion,
the Dirac equation
for
the baryons (-iaV+g,(w*and the Klein-Gordon
ff.O)+p(m+g,a)-n.J)f/,=&*JI,,
equations
for the mesons
(-A-(Q4)2+m2,)cr=-g<,p,-b2u2-b3d,
(5)
(-A-(WL)2+m~)wo=gwp,, (-A
-(a
(4)
. #+
(64
m:)o
=
g,,,j,
f6b)
.
For the sake of convenience the tilde sign is suppressed and a simplifying substitution [see eq. (22) in ref. “‘)I is used in the case of the w-field: G’*=SwP as @fiF
w”
( >( w
ZE
-0
w
ij+(nxi)o*
>.
It has to be mentioned that the above equations of motion are to be understood for fields and spinors, which are stationary in the rotating frame, and they contain extra terms due to the rotation, namely the Coriolis term --a . J for the baryons, which is already well known from non-relativistic cranking 44), and in addition terms proportional to a2 for the mesons. The Coriolis term explicitly breaks time reversal invariance. Therefore, and in contrast to the usual applications of relativistic mean field theory for the ground state of even-even-nuclei, it can here no longer be expected that the spatial components of the baryon currents j, vanish. As these baryonic currents are a consequence of the rotation, we expect that their feedback on the rotation is small for the angular velocities under consideration. In order to limit the numerical effort we therefore used in a first step for the self-consistent the following approximation,
solution
fLJ*+ (fOO,0) i.e. we neglected
eq. (6b) and the term -g,a
of the above equations
(8)
- w in eq. (4).
As a second step we took into account the contributions of the spatial components of the vector fields, i.e. the nuclear magnetism, in a semi-classical approximation. As will be outlined in the appendix, the spatial components of the w-field give in leading order corrections to the total angular momentum J,(O) = ($]Jz I$) and to the total energy in the laboratory frame E [see eq. (27) in ref. “‘)I of the form,
W. Koepf; P. Ring / Superdeformations
284
where 9rig = rn(~‘+ y’) is the rigid body moment nucleonic
mass,
and
pho,,, = l/&r:
of inertia,
= 0.138 frnm3 (for
m = 938 MeV being the
r, = 1.2 fm)
represents
the
homogeneous density in the nuclear interior. As we will see in sect. 3, the correction terms in eqs. (9a) and (9b) yield a positive contribution to the moment of inertia and therefore for a given angular momentum I a lowering of the corresponding level in the rotational band. The set of coupled partial differential equations [eqs. (4)-(6)] is solved by expanding the Dirac spinors and the mesonic fields in terms of the complete set of eigenfun~tions of the three-dimensional harmonic oscillator. We use the same techniques and simplifying assumptions concerning the symmetry of the mesonic fields as in our previous publication of ref. 37), where all the numerical details are given. In the cases where the constants of the present calculation deviate from the ones in ref. 37), as for example in the case of the cut-off parameters for the baryonic and mesonic basis, the parameters will be explicitly given.
3. Superdeformation
in the nucleus *“Sr
3.1. INTRODUCTION
The region of the nucleonic periodic table around N = 2 = 40 has recently been the subject of a number of investigations, both theoretical and experimental 4s). This big interest originates in the wide variety of nuclear phenomena that occurs in this region and the strong variation of these phenomena with both particle number and spin. Furthermore it was shown in actual experiments that this region contains some of the most deformed nuclei known 46). The interest in the neutron deficient Sr isotopes is due to the remarkable diversity in their spectroscopic behavior and the very rapid changes of shape that occur, when the number of neutrons is varied by a few units 21,22). We present in this article some results obtained for the ground state band and a few superdeformed bands of *‘Sr. As has already been mentioned, we have used the parameter set NLl given in table 1 and neglected pairing correlations. Thus our calculations are not fully realistic for low spins, say I G 20R. In our calculations we work with an axially symmetric deformed oscillator basis with the deformation parameter PO = 0.25 which we truncate by taking into account only a certain number of those oscillator states which have the lowest energy in the corresponding oscillator potential. For the large components of the Dirac spinors the basis contains 120 states which would correspond to N, = 7 shells in the case of spherical symmetry. For the expansion of the bosonic fields the number of basis states corresponds to Ns = 10.
W. KoepA P. Ring / Superdeformarions
3.2. GROUND-STATE
PROPERTIES
We find the ground
OF
“Sr
state of this nucleus
with a mass-quadrupole
moment
of Q0 = 3.53 e - b. Using
the ansatz
to be triaxial
R,=
(y = ll.!?)
of Q = 7.44 b and an electrical
J(r’ Y&= + 2( r* YZ2)== 2 with
285
and well deformed quadrupole
AR:@ ,
moment
(10)
1.2A1’3 (fm) this corresponds
to @ = 0.41. The ground-state energy is of the total energy surface at an axially symmetric prolate deformation with the same p-value occurring at an energy difference of only 0.792 MeV higher. This is in good agreement with the results obtained in non-relativistic calculations by Bonche et al. *‘), where the effective Skyrme III interaction was used. In these calculations a mass quadrupole moment of Q = 7.4 b was obtained, however the absolute minimum of the total energy surface was found to be axially symmetric and prolate. On the other hand the nucleus turned out to be extremely soft against of rotational y-deformation and the authors of ref. *O) conclude that a restoration
E = -681.79 MeV, and we also find a second minimum
invariance would earlier applications
probably lead to a triaxial ground state, as it has been seen in of angular momentum projection in the rare-earth region 47). A
ground-state energy of E = -680.74 MeV was found by Bonche et al. ‘*). In another non-relativistic calculation with Skyrme interactions but extending the constrained Hartree-Fock approach through the use of the generator coordinate method, Flocard 23) finds the ground state of “Sr to be triaxial (y= lo”), and also observes a local minimum at an axially symmetric prolate deformation. This is in full agreement with our findings.
3.3. THE
GROUND-STATE
We calculated
BAND
the properties
IN
*(%I
of the ground-state
in the four parity-signature-blocks, namely ll-+, lo+-, and ll-fortheneutrons
band assuming
with the occupation and9++, lo-+,9+-
fixed occupation numbers lO+ +, and IO--
for the protons. The numbers given above are the numbers of occupied single-particle states of the parity and signature which follows. These occupation numbers lead to a rotational band of positive parity and of positive signature. They are also valid for the ground state and they were found to be related to the ground-state band in cranked Hartree-Fock calculations in refs. 2’-23) and also in cranked NilssonStrutinsky calculations of ref. 48). We find a regular spectrum extending up to I= 36h which corresponds to a collective rotation of a nearly prolate shape around one of its small axes. In fig. 1 we compare the theoretical spectrum with and without the contributions of the nuclear magnetism (9) with the experimental spectrum.
W. Koepf P. Ring / Superdeformations
286
E’(MeV)
aoSr
\
15
I~----
1.
26+
-
10 -’
5-
oNLl
NLl(cor.)
Expt.
Fig. 1. Experimental and calculated rotational spectra for the ground-state band in ‘%. NLl stands for our relativistic calculation without nuclear magnetism and NLl(cor.) stands for our calculation including this correction.
For low spins, say I s lOh, both theoretical spectra are too compressed. This type of discrepancy also occurs in the other non-relativistic calculations 21~23*48),and it can be explained as due to the neglecting of pairing correlations. These correlations induce a gap in the quasi-particle spectrum and reduce therefore the possibility to excite virtual two-quasiparticle states, which leads to a considerable reduction of the moment of inertia. In non-relativistic theories it is possible to reproduce the proper size of this quantity by taking into account pairing correlations. So far, however, there is no consistent treatment of pairing in the framework of a relativistic field theory and a full understanding of the rotational spectra at low spins in such a scheme is therefore a task for the future. For higher spins, say Ia lOh, the theoretical spectrum without the spatial contributions of the o-mesons is too extended, while the corrected one remains too compressed. The reason for this type of deviation can be seen far more clearly from the following figures. Fig. 2 shows the component of the total angular momentum along the rotational axis J,(a) as a function of the angular velocity 0, fig. 3a shows the kinematical
moment
of inertia
and fig. 3b shows the dynamical
2, calculated
moment
as
of inertia
2”‘,
calculated
as
(12) as functions of the total angular momentum. The experimentaf moments of inertia (the squares in fig. 3) vary much more strongly with spin than their theoretical counterparts (the solid and dashed lines in fig. 3). The experimental R-dependence of the angular momentum (J,) in fig. 2 is far from being a straight line, while the theoretical values definitely exhibit such a
W. Koepf, P. Ring / Superdeformations
287
JZlA 30
20
10
J
0.5
1.5
1.0 R(MeV)
Fig. 2. The component of the angular momentum in i-direction as a function of the angular velocity R for the ground-state band in “Sr. The full line corresponds to the uncorrected calculation, the dashed line corresponds to the corrected calculation and the squares refer to the experimental values.
, J(nrev-‘)
/
30
20
I!
80Sr
: _________________________________-----------_+_*-cd-9 &_A-4 ..u/ ) +_/ (
.A
0
J___~___~____~____~____~_____
,
0
30
20
10 I/h
J(*)(MeV-‘)
0 /\ d ,‘?\, t: :,, :t,t, 30 / a: ; ;’ ,’ ~_______+______:~____~___+-----------. 1: / /
*OSr
Fig. 3. The kinematic (upper figure) and the dynamical (lower figure) moment of inertia as a function of the total angular momentum for the ground state band in ‘“Sr. The meaning of the different lines and of the squares is identical to fig. 2.
288
W. Koepx P. Ring / Superdeformations
behavior, and therefore in our calculation the two moments of inertia 3a and 3b are much more similar than the experimental ones.
shown in figs.
This indicates that the nucleus in reality does not behave as a rigid rotor, while in our calculations it practically behaves like that. This means that in our calculations the details
of the intrinsic
field approximation
single-particle
and the feedback
structures of the rotation
are smeared
out in the mean
on the intrinsic
structure
is
underestimated. This can be understood partly by the fact that we did not take into account the spatial components of the vector fields self-ConsistentIy, which would contribute in the case of broken time reversal invariance, i.e. for 0 # 0 or I # 0. Although our semi-classical approximation for the nuclear magnetism (9) has just the right order of magnitude, as will be discussed later on, it cannot simulate this crucial self-consistency. As in the case of two parallel currents in electromagnetism, the spatial components of the vector fields are attractive, and so their contribution would lead to a smaller increase of the energy with spin and thus to a larger moment of inertia. This increase in the moments of inertia can be seen in fig. 3 as the difference of the calculation with (the dashed lines) and without (the solid lines) the contribution of the nuclear magnetism. The fact, that the consideration of the spatial components of the vector fields, i.e. the nuclear magnetism, really leads to a considerable improvement can be seen most clearly in fig. 3a. The experimental moment of inertia nearly reaches the theoretical value including the corrections (the dashed line) and the decreasing difference between both can be explained as due to a slow vanishing of the pairing correlations as the spin gets higher ‘*). Fig. 4 finally shows the quadrupole moment Q (the solid line) and the hexadecupole
moment
Q4” (the dashed
line)
as a function
of the total
angular
Fig. 4. The quadrupole moment (solid line, left scale) calculated as Q =gF4rZ Y&‘+ 2{t* Yzz)’ and the hexadecupcle moment (dashed line, right scale) calculated as Qlo = (r” Y,,) as a function of the total angular momentum for the ground-state band in “Sr. The calculation includes nuclear magnetism in the approximation (9).
W. Koepf, P. Ring / Superdeformations
289
momentum. We observe a steady decrease of the deformation This is in agreement with the non-relativistic results. Taking
into account
that we have neglected
pairing
with increasing
correlations,
spin.
and that we did
not treat the spatial components of the vector fields self-consistently, we find quite good overall agreement with both experiment and with other non-relativistic calculations. This would seem to make it appropriate to apply this model under extreme conditions, namely to the case of superdeformation.
3.4. SUPERDEFORMED
BANDS
also to nuclei
IN “Sr
Here we shall focus our attention on two superdeformed bands (labeled j and z) in “Sr, which have also been observed in non-relativistic cranked Hat-tree-Fock calculations by Flocard 23) and by Bonche et al. 2’,22). As will be explicitly discussed in the following, the qualitative and sometimes even quantitative features of these bands are in good agreement with those of the corresponding superdeformed bands found in the non-relativistic calculations. The two bands have the same proton configuration, namely 104 +, 9 - +, IO+ -, 9- -, but differ in their neutron fillings, namely 11+ +, lo- +, 11-t -, lo- - for band j and 12+ + , 9 - + , 10-t - , 11 - - for band z. These occupation numbers are taken from ref. ‘*) and lead to rotational states of both positive parity and positive signature. In all the following results the contribution of the nuclear magnetism (9), i.e. the contribution of the spatial components of the vector fields, is included and it will not be stressed anymore. Fig. 5 shows the energy of the bands relative to an average yrast line, namely E” = E +670.5 - Z(Z + 1)/(2x 34.4) [MeV], as a function of the spin. Band j (the
-
E*(MeV)
I : /
2.0
: ,<*:
/
8F3r
f
_/*
‘,
__--
‘\
‘a : :
0.0 _._-----_------.“----_~-__--
-2.0
i-* I:
40
1
‘,
__-_ ‘\ \ ‘\ ‘.., .__c’
.’ .-.
I
80
120
flfl Fig. 5. The energy of the two superdeformed bands average yrast line E * = E +670.5 - I( I + 1)/(2x 34.4) approximation of eq. (9). The solid line corresponds the band z. Both bands differ
in ““Sr . The energies
are displayed relative to an [MeV] and nuclear magnetism is included in the to the band j and the dashed line corresponds to in their neutron fillings.
W. KoepA P. Ring / Superdeformations
290
solid line) can be followed from spin I = 55h to spin I = 120h, band z (the dashed line) from spin I = 40h to spin I = 125h. Above I = 102h band z is lower in energy than band j. In the non-relativistic to 80h after which it becomes this band taken
calculations
unstable
band j could be followed
is stable up to much higher (unrealistic)
too seriously,
because
from 30h
against fission. The fact that in our calculation
it is not clear, whether
angular
momenta
the fission
should
degree
not be
of freedom
can be appropriately described in the limited oscillator basis used here. Further in the non-relativistic calculations band z, which above spin I = 70h is lower in energy than band j, can be followed up to 92h. Thus our results are in qualitative agreement with the non-relativistic calculations, although quantitatively they are somehow shifted to higher spins. This again indicates the too small feedback of the rotation on the intrinsic structure. For the average moment
of inertia for the two superdeformed bands we obtain a value of g,, = 34 MeV’. This is much larger than the average moment of inertia of the ground state band (26 MeV-‘), and it is in good quantitative agreement with the average moment of inertia found for the superdeformed bands in the nonrelativistic calculations (37 MeV’). Fig. 6a shows the quadrupole moment of the bands as a function of spin. Their quadrupole moments are typically twice as large as the quadrupole moment of the ground state (744 fm*), although the ground state is already very deformed. The quadrupole moments correspond to values of /3 of about 0.65, which is equivalent to an axis ratio of 1.8 : 1. The quadrupole moments stay rather constant over large regions in spin or decrease slowly with spin, and then they increase at certain spins (at I = 55h, 90h, 120h for band z and at I = 115h for band j) in a nearly stepwise fashion to higher values. This fact, called the centrifugal stretching effect, reflects the filling of high lying, more aligned orbitals 4x), and in all cases it is connected to a smooth up-bending of the kinematical moment of inertia occuring at the same spins, as can be seen from fig. 7a. The drastic change in the quadrupole moment at Z = 78h for band z corresponds to a back-bending, caused by the crossing of two bands having rather different deformations 48). Between these stepwise stretchings Q decreases with spin. This is due to a smooth rotational alignment of the single-particle states, which 4x.4y). The qualitative behaviour of the quadleads to a dealignment in deformation rupole moment agrees very well with the nonrelativistic calculations and even the absolute values (Q 3 14 b is the average nonrelativistic value) are not very different. Fig. 6b shows the y-deformation as a function of spin. We observe a smooth increase of y with spin from y = 0” at I - 40h to y = 10” for the highest spins. The back-bending in band z leads to a stepwise increase in y. These upbendings and backbendings can be seen far more clearly in the moments of inertia. Fig. 7a therefore shows the kinematic moment of inertia (11) and fig. 7b the dynamical moment of inertia (12). There is one more up-bending occurring for I =94h in band j, which is not connected to a change of the nucleus shape.
W. KoepJ; P. Ring / Super~~formafi~ns
291
Y
/
10" ,I
,--\
._-
,’
.’
I’
I’
_--.._1
,’
I
*OSi5” I
,.-’ /. . .
_/* .-
cl”
__‘~-~-___--__---_--__--I--__----
40
80
120
Fig. 6. The quadrupole deformation as a function of the total angular momentum for the two superdeformed bands in “‘Sr. The deformation parameter y is determined from the expectation values of the intrinsic quadrupole moments CC&) = (r’Y,_) using the expression tan y = &(&)/(QJ, the quadrupole moment is calculated as described in tig. 4. The meaning of the different lines is identical to fig. 5, and again the semi-classicat correction for nuclear magnetism (9) is included.
4. Superdeformation
in the nucleus “*Dy
4.1. INTRODUCTION
The first experimental evidence for superdeformation in rotating nuclei has been found in the nucleus “’ Dy [ref. ‘)I. In May 1986 the Daresbury group around Twin resolved nineteen discrete y-ray-transitions in this nucleus with an almost constant energy separation corresponding to the moment of inertia of a superdeformed shape “). This was the first observation of a discrete line superdeformed band and it extended the spin, at which discrete states had been seen from 46h to 60h [ref. ‘)I. These fascinating experiments allow us to directly confront the results of recently developed advanced theoretical techniques of relativistic mean field theory in the rotating frame with experimental data in this extreme region of nuclear physics. We will therefore amplify here the properties of this nucleus in order to see, whether
292
I ~(2)(A~eV
-1
1
80 t 60
t
40
_c J ---. I! 40
I
80
120
Ilfi Fig. 7. The kinetmatic (upper figure) and the dynamical (lower figure) moment of inertia as a function of the total angular momentum for the superdeformed bands in ?Sr. The solid line refers to band j, the dashed line refers to band z.
relativistic mean field theory is able to describe quantitatively this phenomenon. Our calculation is also - apart from the correction term in eq. (9) - to the best of our knowledge the first fully self-consistent description of a heavy rotating nucleus in the superdeformed region, where not only the shell corrections but also the bulk properties are treated microscopically. We would like to stress again that we have no free parameter in our calculation. We use again the parameter set NLI from the literature 4’) given in table 1.
4.2. RESULTS
In contrast to our calculations for the much lighter nucleus 80Sr, which are described in sect. 3, in this case we used a more deformed (&-, = 0.50 instead of PO = 0.25) and larger basis ( NF = 8 major oscillator shells for the large components instead of NF = 7) for the expansion of the spinors and fields.
293
W. Koepf; P. Ring / Superdeformations
In the case of the superdeformed numbers for the four different necessary for the calculation frequencies
considered
occupation
of the lowest
(typically
bands
in “Sr we had to use fixed occupation
signature-parity blocks. It turns out that this is not of the superdeformed 15’Dy, since for all angular
the superdeformed lying single-particle
of a few MeV) between
the highest
state in this case simply states. occupied
Furthermore,
arises by the a gap appears
and the lowest unoccupied
single-particle states. This can be seen in fig. 8, where we show the single-particle orbitals around the Fermi surface for the neutrons (fig. 8a) and for the protons (fig. 8b) as a function of the angular velocity 0. It is also of interest to note that apart from some highly alignable states, the high-j states, whose energy decrease with angular velocity, the dependence of the single particle energies on the rotational frequency is very weak and there are pronounced shell gaps at the neutron number 86 and at the proton
86
-8 --
lszDy
+d.(MeV) -3
66
-5
152Dy
-7
.-.-.-.0.4
._I_.
I
_,_,_,_,
0.6 0.8 R(MeV)
1.0
Fig. 8. The energy of the single particle orbitals around the Fermi surface as a function of the angular “*Dy. The upper figure shows the neutron levels, the lower figure velocity R for the superdeformed shows the proton levels. The full line corresponds to positive parity and positive signature, the dashed line corresponds to positive parity and negative signature, the dashed and dotted line corresponds to negative parity and positive signature and finally the dotted line corresponds to negative parity and negative signature.
W. Koepf,
294
number deformed
66. This
is in perfect
shell closures
and Ragnarsson of the cranked
i? Ring / Superdefor~a~~ons
agreement
for this nucleus
7,52) using
a deformed
Nilsson-Strutinsky
with the observation
in non-relativistic Woods-Saxon
method.
of the so called
calculations potential
One of these intruder
by Dudek 5’)
in the framework states is responsible
for the vanishing of the superdeformed shell gap in the case of the protons $2 = 1.2 MeV, as can be seen from fig. 8b.
for
In fig. 9 we compare our results with the experimental spectrum ‘). It is obvious that the calculated spectrum without considering the contribution coming from nuclear magnetism - as given in eq. (9) - is too extended. The reason for this discrepancy is again the fact that we have neglected the spatial components of the vector fields in our first step. If we include them in the semi-classical correction, we achieve a much better agreement with the experimental data. The remaining deviations now go into the opposite direction and are very similar to those of the non-relativistic calculations ‘,“), where one also finds a somewhat too compressed spectrum. It is usually explained “) by the fact that there still remain residual pairing correlations in spite of the strong Coriolis-anti-pairing effect at these large angular velocities 53*54). A further comparison with experiment is possible within the moments of inertia. Fig. 10a shows the kinematic moment of inertia (11) and fig. lob the dynamical moment of inertia (12) as functions of the spin. The solid line corresponds to the calculation without the correction (9), the dashed line corresponds to the calculation including nuclear magnetism and the squares refer to experimental data ‘). It is interesting to note that the two moments of inertia are almost constant, have similar magnitudes, and are much more similar than those obtained from experiment. This tendency has also been observed in the non-relativistic calculations 7338).The fact that the dynamical moment of inertia $“’ varies very smoothly and slowly with
E(MeV) / -----Ii
*52Dy 0’
2’
I
NL1
NLl(cor.)
Expt.
Fig. 9. Experimental and calculated rotational spectra for the superdeformed band in “‘Dy. NLI refers to the relativistic calculation without the semi-classical correction for the spatial w-field and NLI (COT.) refers to the calculation including this correction. Both spectra are normalized at the lowest calculated levels, i.e. NLl at the 22+ and NLl (car.) at the 24+ state.
W. Koepf 3(MeV-‘1
1
295
P. Ring / Superdeformations
_____________________-_____________________
80
40
ls2Dy
t
OT_;__-_------_;..____
_____
_ ____
152D Y
ot-;-----------;-----------;--20
100 $k
Fig. 10. The kinetmatic (upper figure) and the dynamical (lower figure) moment of inertia as a function to the of the total angular momentum for the superdeformed band in “‘Dy. The full line corresponds uncorrected calculation, the dashed line corresponds to the corrected calculation and the squares refer to the experimental values.
the angular velocity is in good agreement with experiment and with the nonrelativistic calculations and shows the difference between this superdeformed nucleus vary considerably with and normal deformed nuclei, where $J”’ can sometimes spin 2). In addition it shows that this superdeformed nucleus behaves as a nearly perfect rigid rotor, an assumption
we made for deriving
the semi-classical
approxima-
tion (9) for the nuclear magnetism. Furthermore, the calculation including nuclear magnetism (9) (the dashed line) agrees very well with experiment. As has already been stressed in the context of the spectra, our deviations are of about the same order of magnitude as those of the non-relativistic calculations and they can be explained by the missing pairing correlations. The fact that the calculation without the correction (9) (the solid line) underestimates the moments of inertia corresponds to the too extended spectrum of fig. 9, and can be understood as due to some lack of attraction due to the complete neglect of the spatial components of the vector fields.
W. Koepf, P. Ring / Superdeformations
296
Q40(fm4) 20500
20000
19500
20
60 Illi
100
Fig. 11. The quadrupole moment (solid line, left scale) calculated as Q = &d{r-’ YzO)*+ 2(rZ Y2Jz and the hexadecupofe moment (dashed line, right scale) calculated as QAo = (r4 I’.&) as a function of the total includes the semi-classical angular momentum for the superdeformed band in “‘Dy. The calculation correction for nuclear magnetism (9).
In fig. 11 we finally show the quadrupole moment (the solid line) and the hexadecupole moment (the dashed line) as a function of spin. There is a general tendency that both moments decrease with increasing spin. This is caused mainly by a genera1 increase of the rotational alignment of the single-particle states at the expense of the deformational alignment. The fact that the overall change of the quadrupole moment is below 4%, however, and that the energy of the single-particle states hardly changes with the angular velocity, as can be seen from fig. 8, shows the high degree of collectivity of the rotation of this superdeformed nucleus. In recent non-relativistic calculations “) a strong increase of the hexadecupole deformation with spin was predi&d. This cannot be supported by our results, since we observe a slightly decreasing or nearly constant hexadecupole moment. Concerning the absolute magnitude of the quadrupole moment we are in very good agreement with experiment and with the non-relativistic results. We obtain an average value for the electric quadrupole moment of 18.6 e * b, while the prediction from the non-relativistic calculations ‘*) is 18 e - b and the experimental value 3, is 19 e - b. The magnitude of the quadrupole moment corresponds to a deformation parameter of /3 = 0.72 and to an axis ratio of 1.9: 1, which is very close to the “classical” axis ratio of 2 : 1 arising from the studies of the fission isomers 12*13).The value for the y-deformation lies below 0.7” for all angular frequencies under consideration, and thus we observe the rotation of a nearly prolate highly deformed nucleus around one of its small axes. 5. Conclusion We use relativistic for an investigation
mean field theory in the rotating frame as proposed in ref. 37) of superdeformed shapes in rotating nuclei in two regions of
W. Koepf, P. Ring / Superdeformations
the periodic
table. We employ
in Hat-tree approximation, p-mesons,
the nowadays
neglecting
standard
291
version
vacuum-poIa~zation
of this theory:
and including
working u-, w- and
the Coulomb
field and a non-linear coupling between the u-mesons. have been In the relatively light nucleus *‘St=, where so far no superdeformations observed experimentally, we find - in agreement with earlier non-relativistic calculations of the Paris-Bruxelles group 2’) - two superdeformed bands with an axis ratio of 1.8: 1 and an average moment of inertia of roughly 34 MeV-‘. For the famous and now well known nucleus ‘52Dy, where superdeformation has been first discovered experimentally ‘>, we present the first fully self-consistent calculation using a parameter set from the literature, which is known to give an excellent description as well as deformed
of the ground-state properties of nuclear matter and of spherical finite nuclei over the entire periodic table j3).
Our calculated results for the superdeformed band in ls2Dy are in very good agreement with the experiment as well as with earlier semi-phenomenological calculations using the Strutinsky approach ‘). In detail the theoreticai moments of inertia are roughly 10% larger than the experimentally determined values. This can be however - at least in a nonrelativistic framework - understood as a lack of pairing correlations in the present calculation ‘*). For the moments of inertia this agreement can only be achieved if one includes the contributions of the spatial parts of the vector mesons (nuclear magnetism). Because of time reversal invariance they have no influence on the ground state of even-even nuclei, but they also play a crucial part in our understanding of the Dirac magnetic moments in odd nuclei 56)_ In the present work these contributions were - for reasons of numerical simplicity - taken into account only after the variation within a classical approximation.
We would like to express our gratitude of useful discussions.
to R.R. Hilton
and H.J. Mang for a number
Appendix SEMI-CLASSICAL
APPROXIMATION
FOR
THE
NUCLEAR
MAGNETISM
If we assume that the rotation of the nucleus with angular velocity +:R around the z-axis is that of a perfect rigid rotor, we find the following classical connection 57) between the spatial componentsj of the baryonic currentjp = ( p, j) in the laboratory frame and the density p:
P(fi
X
r) =
(A.11
W KoepA P. Ring / Superdeformations
298
Transforming
this baryonic
current
into the rotating
frame yields:
j” cos 0?+ j” sin a?+ j’ cos f2-j”
sin fli.Z J
@p
L?.fp
.
C-4.2)
Here x”‘ stands for the coordinates in the rotating frame. As discussed in detail in ref. 37) the spatial components w of the w-field, which appear in the Klein-Gordon equation (6b) and in the Dirac equation (4), arise not only by the usual transformation of a contravariant vector from the w-field in the laboratory frame, but there is this extra substitution in eq. (7) [and in eq. (22) of ref. “)I. Therefore the source of this field, j, in eq. (6b), is not just the spatial component J of the baryonic current in the rotating frame j’/* (A.2), but we find using eq. (17) of ref. 37): j,=J+(cnx This together
with eq. (A.2), eq. (A.l)
i)b.
(A.3)
and eq. (9) of ref. 37) finally -9 x’
j,=pfi
,
gives:
(A.4)
0 0 The Dirac equation (4) together with the total energy [eq. (27) in ref. “)I yields for the contribution of the spatial components of the w-field to the total energy: E nucl.magn. = -2
(A.9
Neglecting in the Klein-Gordon equation (6b) the Laplace operator, i.e. the kinetic energy of the meson, and the a-dependent terms with respect to the large w-mass,
we get, mto
= gw.iV
and further assuming that the rotating nucleus density p = phom, we finally find with eq. (A.4),
has a constant
and homogeneous
(A-6) where we used eq. (31) of ref. 37) for the rigid rotor moment of inertia of the nucleus. The quantity, which is stationary in our cranking calculation, is the energy E’ in the rotating frame: E’=E-OJ,.
(A.71
W. Koepf, l? Ring / Superdeformations
The variation
has to be carried out with fixed angular
we find the following relation for the component z-direction J, [see page 77 of ref. ‘*)I: J,=
velocity
299
0. With this assumption
of the angular
momentum
-g.
in
(A.81
In our actual mean field calculation we neglected the spatial components of the w-field. Eq. (A.6) yields a semi-classical approximation for their contribution to the total energy in the rotating frame. This leads to the following correction of E’: (A.9) and therefore
because
of eq. (A.8) to a correction
in J,: (A.lO)
and as due to eq. (A.7) also to a correction
of the total energy in the laboratory
frame: (A.ll)
The other vector fields, i.e. the p-meson and the Coulomb important, and so it is not necessary to take them into account
field, are far less in this context.
References 1) J. Dudek and W. Nazarewicz,
Phys. Rev. C31 (1985) 298 B.M. Nyako, A.H. Nelson, J. Simpson, M.A. Bentley, H.W. Cranmer-Gordon, P.D. Forsyth, D. Howe, A.R. Mokhtar, J.D. Morrison, J.F. Sharpey-Schafer and G. Sletten, Phys. Rev. Lett. 57 (1986) 811 P.J. Twin, Proc. of the Int. Conf. on nuclear shapes, Ahgia Pelaghai, Greece, June (1987) P.J. Twin, Contemporary Topics in nuclear structure physics, Cocoyoc, Mexico, June (1988) B. Haas, P. Taras, S. Filibotte, F. Banville, J. Ganson, S. Cournoyer, S. Monaro, N. Nadon, D. Prevost, D. Thibault, J.K. Johansson, D.M. Tucker, J.C. Waddington, H.R. Andrews, G.C. Ball, D. Horn, D.C. Radford, D. Ward, C. St.-Pierre and J. Dudek, Phys. Rev. Lett. 60 (1988) 503 T. Bengtsson and I. Ragnarsson, Nucl. Phys. A436 (1985) 14 I. Ragnarsson and S. Aberg, Phys. Lett. B180 (1986) 191 W.J. Swiatecki, Phys. Rev. Lett. 58 (1987) 1184 Y.R. Shimizu, E. Vigezzi and R.A. Broglia, Phys. Lett. B198 (1987) 33 T. Bengtsson, 1. Ragnarsson and S. Aberg, Phys. Lett. B208 (1988) 39 V.M. Strutinsky, Nucl. Phys. A95 (1967) 420 V.M. Strutinsky, Nucl. Phys. Al22 (1968) 1 SM. Polikanov, V.A. Druin, V.A. Karnaukhov, V.L. Mikheev, A.A. Pleve, N.K. Skobelev, V.G. Subbotin, G.M. Ter-Akopjan and V.A. Fomichev, Sov. Phys. JETP 15 (1962) 1016 H.J. Specht, J. Weber, E. Konecny and D. Heunemann; Phys. Lett. B41 (1972) 43 C.G. Andersson, S.E. Larson, G. Leander, P. Moller, S.G. Nilsson, 1. Ragnarsson, S.Aberg, R. Bengtsson, J. Dudek, B. Nerlo-Pomorska, K. Pomorski and Z. Szymanski, Nucl. Phys. A268 (1976) 205 K. Neergard, V.V. Pashkevich and S. Frauendorf, Nucl. Phys. A262 (1976) 61
2) P.J. Twin,
3) 4) 5)
6) 7) 8) 9) 10) 11) 12) 13) 14) 1%
16)
300 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48)
W. Koepf, P. Ring / Superdeformations K. Neergard, H. Toki, M. Ploszajczak and A. Faessler, Nucl. Phys. A287 (1977) 48 M. Brack and P. Quentin; Phys. Lett. B56 (1975) 421 J. Fleckner, U. Mosel, P. Ring and N.J. Mang, Nucl. Phys. A331 (1979) 288 P. Bonche, H. Flocard, P.H. Heenen, S.J. Kriegec and MS. Weiss, Nucl. Phys. A443 (1985) 39 P. Bonche, FH. Flocard and P.H. Heenen; Proc. Int. Conf. on nucleus-nucleus collisions, Saint Malo, June (1988) P. Bonche, H. Flocard and P.H. Heenen, Proc. on the Workshop on novel nuclear shapes and high spin states, Argonne, April (1988) H. Flocard, Preprint DPhT IPN Orsay (1988), Microscopic description of the ground state and high-spin properties of light strontiums. M.H. Johnson and E. Teller, Phys. Rev. 98 (1955) 783 H.P. Duerr, Phys. Rev. 103 (1956) 469 J,D. Walecka, Ann. of Phys. 83 (1974) 491 B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1 W. Pannert, P. Ring and J. Boguta, Phys. Rev. Lett. 59 (1987) 2420 C.E. Price and G.E. Walker, Phys. Rev. C36 (1987) 354 S.J. Lee, J. Fink, A.B. Balantekin, M.R. Strayer, AS. Umar, P.G. Reinhard, J.A. Maruhn and W. Greiner, Phys. Rev. Lett. 59 (1987) 1171 W. Koepf and P. Ring, Phys. Lett. B212 (1988) 397 Y.K. Gambhir and P. Ring, Phys. Lett. B202 (1988) 5 Y.K. Gambhir, P. Ring and A. Thimet, preprint Technical University Munich (1989) D. Vautherin and D.M. Brink, Phys. Rev. C5 (1972) 626 D. Vautherin, Phys. Rev. C7 (1973) 296 J. Decharge, and D. Cogny, Phys. Rev. C21 (1980) 1568 W. Koepf and P. Ring, Nucl. Phys. A493 (1989) 61 J. Dudek, B. Herskind, W. Nazarewicz, Z. Szymanski and T. Werner, Phys. Rev. C38 (1988) 940 J. Boguta and A.R. Bodmer, Nucl. Phys. A292 (1977) 413 B. Baranger and K. Kumar, Nucl. Phys. All0 (1968) 490 P.G. Reinhard, M. Rufa, J. Maruhn, W. Greiner and J. Friedrich, Z. Phys. A323 (1986) 13 P. Ring, Proc. Workshop on relativistic many-body physics, Columbus, Ohio, June (1988) P. Ring, Proc. Workshop on nuclear structure with medium energy probes, Santa Fe, October (1988) D.R. Inglis, Phys. Rev. 96 (1954) 1059 J. Ebert, R.A. Meyer and K. Sistemich (eds.), Proc. Int. Workshop on nuclear structure in the zirconium region, Bad Honnef (1988), (Springer, Berlin, 1989) R.F. Davie, D. Sinclair, S.S.L. Ooi, N. Paffe, A.E. Smith, H.G. Price, C.J. Lister, B.J. Varley and 1.F. Wright, Nucl. Phys. A463 (1987) 683 A. Hayashi, K. Hara and P. Ring, Nucl. Phys. A385 (1982) 14 W. Nazarewicz, 5. Dudek, R. Bengtsson, T. Bengtsson and I. Ragnarsson, Nucl. Phys. A435 (1985)
397 49) B. Banerjee, H.J. Mang and P. Ring, Nucl. Phys. A215 (1973) 366 SO) N. Redon, J. Meyer, M. Meyer, P. Quentin, MS. Weiss, P. Bonche, H. Flocard and P.H. Heenen, Phys. Lett. Bl8l (1986) 185 51) J. Dudek, Proc. lnt. Conf. of nuclear shapes, Ahgia Pelaghai, Greece, June (1987) 52) I. Ragnarsson and S. Aberg, Preprint Lund MPh-88/01 (1988) 53) J.L. Egido and P. Ring, Nucl. Phys. A388 (1982) 19 54) U. Mutz and P. Ring, J. of Phys. GlO (1984) L39 5.5) Y.R. Shimizu, E. Vigezzi and R.A. Broglia, preprint Niels Bohr Institut (1989) 56) U. Hofmann and P. Ring, Phys. Lett. B214 (1988) 307 57) A. Bohr, B.R. Mottelson, Nuclear structure, vol. II (1975) $8) P. Ring, P. Schuck; The nuclear many body problem (Springer, Berlin, 1980)
A51 1 (1990) 279-300
A RELATIVISTIC
THEORY
RAPIDLY
OF SUPERDEFORMATIONS
ROTATING
W. KOEPF Physikdepartment
der Technischen
Universitiit
IN
NUCLEI*
and P. RING Miinchen,
D-8046 Garching,
Fed. Rep. Germany
Received 7 August 1989 (Revised 27 September 1989) Abstract: Superdeformed shapes in rapidly rotating nuclei are investigated in the framework of a cranked relativistic mean field theory. A well established parameter set containing a non-linear self-coupling between the cr-mesons is used. Excellent agreement with experimental data in the nucleus “*Dy is achieved and new superdeformed bands in “Sr are predicted. It turns out that the spatial components of the vector w-mesons (nuclear magnetism), which are usually neglected in applications of this theory for small angular momenta, play an important role for a quantitative understanding of the moments of inertia in these superdeformed shapes.
1.
Introduction
Since the theoretical prediction ‘) and the experimental discovery ‘) of superdeformed bands in the nucleus ‘52Dy a large number of experimental 3-5) and have been devoted to this interesting phenomenon in theoretical 6-‘o) investigations fast rotating nuclei. On the experimental side similar bands have been found in a number of neighbouring nuclei in the same region and in other regions of the periodic table. Quadrupole moments have been measured and it has been verified that these bands indeed correspond to shapes with an axis ratio of roughly 2: 1. As Strutinsky has already pointed out more than twenty years ago “3’2), these shapes produce large negative contributions to the shell correction energy. In the actinides the Coulomb and surface energies nearly balance each other, and one therefore In lighter nuclei the observes the fission isomers 13*14)with large deformations. Coulomb energy is weaker and the surface energy dominates at small angular momenta thereby not allowing these large deformations. Only at very large angular velocities the deformation driving rotational energy takes over, and allows the minima in the shell corrections to appear. On the theoretical side nearly all of the investigations have been carried out in the semi-phenomenological framework proposed by Strutinsky “,‘*) and extended to rotating nuclei by several groups in the seventies 15-“): The bulk properties of the nucleus are treated in the liquid drop model, to which shell corrections, calculated l
Work supported
0375-9474/90/$03.50 (North-Holland)
in part by the Bundesministerium 0
Elsevier
Science
Publishers
fiir Forschung B.V.
und Technologie.
W. Koepf
280
P. Ring / S~perdef~~alions
in a rotating single particle potential, are added. At angular been shown ‘*) that this method is an excellent approximation
momentum zero it has to a full Hartree-Fock
or Hartree-Fock-Bogoliubov calculation with density dependent forces. In the rotating frame such calculations are extremely complicated *9-23), because time reversal
symmetry
is broken
and spin saturation
self-consistent descriptions of superdeformed framework of cranked Hartree-Fock calculations in the strontium region 2’-23).
is no longer
valid. As far as fully
bands of heavy nuclei in the there exist to date only calcutations
On the other hand it has become fashionable to describe nuclear structure in the framework of a relativistic quantum field theory. In analogy with early attempts of Teller and collaborators 24,25), Walecka 26,27) introduced a model, in which the nucleons were treated as Dirac particles interacting by the exchange of mesons. In the simplest approximation this field theory is treated at the mean field level. Only very few parameters are introduced and adjusted to experimental data of nuclear matter and a few finite spherical doubly-magic nuclei. In recent years it has been shown that without changing these parameters one can also describe nuclear deformations in light 28-3’) and heavy nuclei 32*33 ). As compared to density dependent Hat-tree-Fock calculations with Skyrme 34+35)or Gogny 36) forces these relativistic theories start at the meson level and are therefore more fundamental. In addition they include spin properties from the beginning in a consistent way and - dealing only with a few local fields - they are technically considerably simpler to handle and require less effort as density dependent Hartree-Fock calculations. These theories have recently been extended to the rotating frame in ref. 37). First applications to the nucleus ‘“Ne show that the results are similar to Skyrme calculations. This light nucleus, however, is not a very good rotor and a mean field approximation is only possible with restrictions. In this paper we therefore present the first applications of cranked relativistic mean field theory to heavy nuclei. We go directly to the region of large angular momenta - as typical for superdeformed nuclei - because to date there is no relativistic description of pairing which plays an important role in the low spin region of deformed nuclei. Theoretical investigations of pairing properties in superdeformed bands show that they are considerably quenched
9,38). As a first stage it should
be a good approximation
to neglect
them
completely. In sect. 2 we briefly present the theory and give the basic de~nitions. In sect. 3 we investigate the nucleus ‘“Sr and predict superdeformed bands in agreement with earlier non-relativistic calculations using Skyrme forces. Sect. 4 is devoted to the first experimentally discovered superdeformed band in the nucleus 152Dy. The calculated deformation is in excellent agreement with experiment. In order to understand the proper value of the moment of inertia, however, it turns out to be important to inciuCe the spatial contributions of vector mesons - nuclear magnetism - which do not show up in nuclear structure calculations at spin zero, where time reversal invariance is valid. In sect. 5 we draw conclusions about our results.
W. Koepf; P. Ring / Superdeformations
2. The theoretical
281
formalism
In relativistic mean field theory 27) the nucleus is described as a system of point-like nucleons, Dirac particles coupled to mesons and to the photon. There are essentially three
types
of mesons:
between
the nucleons,
a nearly
as strong
the scalar
u-meson,
which
and the two vector mesons,
repulsion
and the isovector
provides
a strong
attraction
the isoscalar
w-meson
providing
takes
care of the
p-meson,
which
symmetry energy and which is important in heavy nuclei. We therefore start with the well known effective lagrangian L= q( y”(ia,
-gww,)-(m+g,a))~+-C(a,aac”a-
-&,w,-a,w,)(a”w”
-d”wp)+$?12,w,w~.
density
rn~(~~)-~b~c~-~b~u~
(1)
In our actual calculations we also took into account the isovector p-mesons and the photons, i.e. the Coulomb field. As both of them are vector mesons, they are described in analogy to the o-meson. In order to keep the equations short their contributions are not explicitly written out in eq. (1) or in following equations. Apart from the linear coupling between the mesons and the nucleons the lagrangian (I) contains a non-linear coupling between the cr-mesons, which has been introduced by Boguta and Bodmer 39) and which turns out to be crucial for a quantitative description of many nuclear structure data, as for example for the surface properties or the deformations *“). This lagrangian contains as parameters the masses of the mesons m,, m, and m,, the coupling constants g,, g, and g,,, and the non-linear terms b2 and b,. The mesons are not necessarily considered as bare mesons as observed in scattering experiments, but they are considered as effective particles carrying the most important quantum numbers and generating the interaction in the corresponding channels in a Lorentz invariant manner by a local coupling to the nucleons. In that sense the lagrangian (1) is an effective lagrangian constructed for the mean field approximation. Its parameters are adjusted to experimental data. Therefore, we do not include the v-meson explicitly, because it carries negative parity and its contribution would vanish in the mean field approximation. It would show up in two-meson-exchange processes and is in that sense represented by the a-meson. In a sense the lagrangian (1) is a relativistic analogy to the well-known multipole expansions of the effective nuclear interaction, as for instance the pairing plus quadrupole model 40). where one also takes into account only the lowest quantum numbers in the appropriate channels, and where the parameters are fitted to experimental data. In this paper we use the well established parameter set NLI of ref. 41). It was determined by fitting to the nuclear matter characteristics and to the ground state properties of a few spherical nuclei. It has been used recently in a number of investigations over the entire periodic table 33342343) producing excellent results for binding energies, nuclear radii, quadrupole and hexadecupole deformations, and
282
W. Koepf, P. Ring / Superdeformarions
also for the nuclear turned
densities
out that its results
in spherical
Skyrme forces. The corresponding listed in table
and deformed
nuclei.
are very close and sometimes
even better
values for the masses and coupling
In particular
it
than those of constants
are
1.
As already outlined in a previous paper 37), the rotation in the relativistic mean field theory using the assumptions
of nuclei can be described of the Cranking model 44)
by transforming the effective lagrangian density (1) which acts in the laboratory system into a frame which rotates with constant angular velocity L! around a fixed axis (in our cast the z-axis). We then obtain the following covariant generalization of the lagrangian density (1) using the techniques of general relativity: ~=~(~~(i(~,+~~)-g,~,)-(m+g,~))~+~(g~’”d,~~,~-m~cT2)-~b2~‘-~b3~4 I M I -~g~““g”P(a,3,-a”;,)(a,;,-a,~,)+~m~g~~;,w,.
(2)
This lagrangian contains the mesonic fields, 6 and GIL, and the baryonic spinors 4 in the rotating frame. They are found by simply transforming the fields and spinors in the laboratory frame, a, wF and $, using the standard formulas already known from special relativity, cT=e
iR.U
u,
(3a)
-0
( >(-(Ax;)y)(“d::,w,“)? w
=
&
6
=
,iiE.Jl
g
,
(3b) (3c)
where L and J = L+ S are the operators for the orbital and for the total angular momentum. The spin operators S are either 3 x 3-matrices for the vector mesons or the Dirac spin matrices for the nucleons. In addition the lagrangian (2) also contains the non-diagonal riemannian metric tensor g Py, i.e. the generalization of the flat TABLE 1
The parameters of the lagrangian non-linear set NLl 4’)
for the
NLl
m,, (MeV) m, ( MeV) g, &J & b, (fm-‘) b,
492.25 795.36 10.138 13.285 4.976 -12.172 -36.265
The nucleon mass is taken to be mN = 938 MeV, and the mass of the p-meson is M,, = 763 MeV.
W. Koepi
Minkowsky
metric,
of the baryonic
the modified
spinors
ing transformations The variational
P. Ring / Sueerdeformatio~s
gamma
matrices
283
7” and the covariant
derivative
fW. All the details are given in ref. 37), where the correspond-
are described explicitly. principle gives the equations
of motion,
the Dirac equation
for
the baryons (-iaV+g,(w*and the Klein-Gordon
ff.O)+p(m+g,a)-n.J)f/,=&*JI,,
equations
for the mesons
(-A-(Q4)2+m2,)cr=-g<,p,-b2u2-b3d,
(5)
(-A-(WL)2+m~)wo=gwp,, (-A
-(a
(4)
. #+
(64
m:)o
=
g,,,j,
f6b)
.
For the sake of convenience the tilde sign is suppressed and a simplifying substitution [see eq. (22) in ref. “‘)I is used in the case of the w-field: G’*=SwP as @fiF
w”
( >( w
ZE
-0
w
ij+(nxi)o*
>.
It has to be mentioned that the above equations of motion are to be understood for fields and spinors, which are stationary in the rotating frame, and they contain extra terms due to the rotation, namely the Coriolis term --a . J for the baryons, which is already well known from non-relativistic cranking 44), and in addition terms proportional to a2 for the mesons. The Coriolis term explicitly breaks time reversal invariance. Therefore, and in contrast to the usual applications of relativistic mean field theory for the ground state of even-even-nuclei, it can here no longer be expected that the spatial components of the baryon currents j, vanish. As these baryonic currents are a consequence of the rotation, we expect that their feedback on the rotation is small for the angular velocities under consideration. In order to limit the numerical effort we therefore used in a first step for the self-consistent the following approximation,
solution
fLJ*+ (fOO,0) i.e. we neglected
eq. (6b) and the term -g,a
of the above equations
(8)
- w in eq. (4).
As a second step we took into account the contributions of the spatial components of the vector fields, i.e. the nuclear magnetism, in a semi-classical approximation. As will be outlined in the appendix, the spatial components of the w-field give in leading order corrections to the total angular momentum J,(O) = ($]Jz I$) and to the total energy in the laboratory frame E [see eq. (27) in ref. “‘)I of the form,
W. Koepf; P. Ring / Superdeformations
284
where 9rig = rn(~‘+ y’) is the rigid body moment nucleonic
mass,
and
pho,,, = l/&r:
of inertia,
= 0.138 frnm3 (for
m = 938 MeV being the
r, = 1.2 fm)
represents
the
homogeneous density in the nuclear interior. As we will see in sect. 3, the correction terms in eqs. (9a) and (9b) yield a positive contribution to the moment of inertia and therefore for a given angular momentum I a lowering of the corresponding level in the rotational band. The set of coupled partial differential equations [eqs. (4)-(6)] is solved by expanding the Dirac spinors and the mesonic fields in terms of the complete set of eigenfun~tions of the three-dimensional harmonic oscillator. We use the same techniques and simplifying assumptions concerning the symmetry of the mesonic fields as in our previous publication of ref. 37), where all the numerical details are given. In the cases where the constants of the present calculation deviate from the ones in ref. 37), as for example in the case of the cut-off parameters for the baryonic and mesonic basis, the parameters will be explicitly given.
3. Superdeformation
in the nucleus *“Sr
3.1. INTRODUCTION
The region of the nucleonic periodic table around N = 2 = 40 has recently been the subject of a number of investigations, both theoretical and experimental 4s). This big interest originates in the wide variety of nuclear phenomena that occurs in this region and the strong variation of these phenomena with both particle number and spin. Furthermore it was shown in actual experiments that this region contains some of the most deformed nuclei known 46). The interest in the neutron deficient Sr isotopes is due to the remarkable diversity in their spectroscopic behavior and the very rapid changes of shape that occur, when the number of neutrons is varied by a few units 21,22). We present in this article some results obtained for the ground state band and a few superdeformed bands of *‘Sr. As has already been mentioned, we have used the parameter set NLl given in table 1 and neglected pairing correlations. Thus our calculations are not fully realistic for low spins, say I G 20R. In our calculations we work with an axially symmetric deformed oscillator basis with the deformation parameter PO = 0.25 which we truncate by taking into account only a certain number of those oscillator states which have the lowest energy in the corresponding oscillator potential. For the large components of the Dirac spinors the basis contains 120 states which would correspond to N, = 7 shells in the case of spherical symmetry. For the expansion of the bosonic fields the number of basis states corresponds to Ns = 10.
W. KoepA P. Ring / Superdeformarions
3.2. GROUND-STATE
PROPERTIES
We find the ground
OF
“Sr
state of this nucleus
with a mass-quadrupole
moment
of Q0 = 3.53 e - b. Using
the ansatz
to be triaxial
R,=
(y = ll.!?)
of Q = 7.44 b and an electrical
J(r’ Y&= + 2( r* YZ2)== 2 with
285
and well deformed quadrupole
AR:@ ,
moment
(10)
1.2A1’3 (fm) this corresponds
to @ = 0.41. The ground-state energy is of the total energy surface at an axially symmetric prolate deformation with the same p-value occurring at an energy difference of only 0.792 MeV higher. This is in good agreement with the results obtained in non-relativistic calculations by Bonche et al. *‘), where the effective Skyrme III interaction was used. In these calculations a mass quadrupole moment of Q = 7.4 b was obtained, however the absolute minimum of the total energy surface was found to be axially symmetric and prolate. On the other hand the nucleus turned out to be extremely soft against of rotational y-deformation and the authors of ref. *O) conclude that a restoration
E = -681.79 MeV, and we also find a second minimum
invariance would earlier applications
probably lead to a triaxial ground state, as it has been seen in of angular momentum projection in the rare-earth region 47). A
ground-state energy of E = -680.74 MeV was found by Bonche et al. ‘*). In another non-relativistic calculation with Skyrme interactions but extending the constrained Hartree-Fock approach through the use of the generator coordinate method, Flocard 23) finds the ground state of “Sr to be triaxial (y= lo”), and also observes a local minimum at an axially symmetric prolate deformation. This is in full agreement with our findings.
3.3. THE
GROUND-STATE
We calculated
BAND
the properties
IN
*(%I
of the ground-state
in the four parity-signature-blocks, namely ll-+, lo+-, and ll-fortheneutrons
band assuming
with the occupation and9++, lo-+,9+-
fixed occupation numbers lO+ +, and IO--
for the protons. The numbers given above are the numbers of occupied single-particle states of the parity and signature which follows. These occupation numbers lead to a rotational band of positive parity and of positive signature. They are also valid for the ground state and they were found to be related to the ground-state band in cranked Hartree-Fock calculations in refs. 2’-23) and also in cranked NilssonStrutinsky calculations of ref. 48). We find a regular spectrum extending up to I= 36h which corresponds to a collective rotation of a nearly prolate shape around one of its small axes. In fig. 1 we compare the theoretical spectrum with and without the contributions of the nuclear magnetism (9) with the experimental spectrum.
W. Koepf P. Ring / Superdeformations
286
E’(MeV)
aoSr
\
15
I~----
1.
26+
-
10 -’
5-
oNLl
NLl(cor.)
Expt.
Fig. 1. Experimental and calculated rotational spectra for the ground-state band in ‘%. NLl stands for our relativistic calculation without nuclear magnetism and NLl(cor.) stands for our calculation including this correction.
For low spins, say I s lOh, both theoretical spectra are too compressed. This type of discrepancy also occurs in the other non-relativistic calculations 21~23*48),and it can be explained as due to the neglecting of pairing correlations. These correlations induce a gap in the quasi-particle spectrum and reduce therefore the possibility to excite virtual two-quasiparticle states, which leads to a considerable reduction of the moment of inertia. In non-relativistic theories it is possible to reproduce the proper size of this quantity by taking into account pairing correlations. So far, however, there is no consistent treatment of pairing in the framework of a relativistic field theory and a full understanding of the rotational spectra at low spins in such a scheme is therefore a task for the future. For higher spins, say Ia lOh, the theoretical spectrum without the spatial contributions of the o-mesons is too extended, while the corrected one remains too compressed. The reason for this type of deviation can be seen far more clearly from the following figures. Fig. 2 shows the component of the total angular momentum along the rotational axis J,(a) as a function of the angular velocity 0, fig. 3a shows the kinematical
moment
of inertia
and fig. 3b shows the dynamical
2, calculated
moment
as
of inertia
2”‘,
calculated
as
(12) as functions of the total angular momentum. The experimentaf moments of inertia (the squares in fig. 3) vary much more strongly with spin than their theoretical counterparts (the solid and dashed lines in fig. 3). The experimental R-dependence of the angular momentum (J,) in fig. 2 is far from being a straight line, while the theoretical values definitely exhibit such a
W. Koepf, P. Ring / Superdeformations
287
JZlA 30
20
10
J
0.5
1.5
1.0 R(MeV)
Fig. 2. The component of the angular momentum in i-direction as a function of the angular velocity R for the ground-state band in “Sr. The full line corresponds to the uncorrected calculation, the dashed line corresponds to the corrected calculation and the squares refer to the experimental values.
, J(nrev-‘)
/
30
20
I!
80Sr
: _________________________________-----------_+_*-cd-9 &_A-4 ..u/ ) +_/ (
.A
0
J___~___~____~____~____~_____
,
0
30
20
10 I/h
J(*)(MeV-‘)
0 /\ d ,‘?\, t: :,, :t,t, 30 / a: ; ;’ ,’ ~_______+______:~____~___+-----------. 1: / /
*OSr
Fig. 3. The kinematic (upper figure) and the dynamical (lower figure) moment of inertia as a function of the total angular momentum for the ground state band in ‘“Sr. The meaning of the different lines and of the squares is identical to fig. 2.
288
W. Koepx P. Ring / Superdeformations
behavior, and therefore in our calculation the two moments of inertia 3a and 3b are much more similar than the experimental ones.
shown in figs.
This indicates that the nucleus in reality does not behave as a rigid rotor, while in our calculations it practically behaves like that. This means that in our calculations the details
of the intrinsic
field approximation
single-particle
and the feedback
structures of the rotation
are smeared
out in the mean
on the intrinsic
structure
is
underestimated. This can be understood partly by the fact that we did not take into account the spatial components of the vector fields self-ConsistentIy, which would contribute in the case of broken time reversal invariance, i.e. for 0 # 0 or I # 0. Although our semi-classical approximation for the nuclear magnetism (9) has just the right order of magnitude, as will be discussed later on, it cannot simulate this crucial self-consistency. As in the case of two parallel currents in electromagnetism, the spatial components of the vector fields are attractive, and so their contribution would lead to a smaller increase of the energy with spin and thus to a larger moment of inertia. This increase in the moments of inertia can be seen in fig. 3 as the difference of the calculation with (the dashed lines) and without (the solid lines) the contribution of the nuclear magnetism. The fact, that the consideration of the spatial components of the vector fields, i.e. the nuclear magnetism, really leads to a considerable improvement can be seen most clearly in fig. 3a. The experimental moment of inertia nearly reaches the theoretical value including the corrections (the dashed line) and the decreasing difference between both can be explained as due to a slow vanishing of the pairing correlations as the spin gets higher ‘*). Fig. 4 finally shows the quadrupole moment Q (the solid line) and the hexadecupole
moment
Q4” (the dashed
line)
as a function
of the total
angular
Fig. 4. The quadrupole moment (solid line, left scale) calculated as Q =gF4rZ Y&‘+ 2{t* Yzz)’ and the hexadecupcle moment (dashed line, right scale) calculated as Qlo = (r” Y,,) as a function of the total angular momentum for the ground-state band in “Sr. The calculation includes nuclear magnetism in the approximation (9).
W. Koepf, P. Ring / Superdeformations
289
momentum. We observe a steady decrease of the deformation This is in agreement with the non-relativistic results. Taking
into account
that we have neglected
pairing
with increasing
correlations,
spin.
and that we did
not treat the spatial components of the vector fields self-consistently, we find quite good overall agreement with both experiment and with other non-relativistic calculations. This would seem to make it appropriate to apply this model under extreme conditions, namely to the case of superdeformation.
3.4. SUPERDEFORMED
BANDS
also to nuclei
IN “Sr
Here we shall focus our attention on two superdeformed bands (labeled j and z) in “Sr, which have also been observed in non-relativistic cranked Hat-tree-Fock calculations by Flocard 23) and by Bonche et al. 2’,22). As will be explicitly discussed in the following, the qualitative and sometimes even quantitative features of these bands are in good agreement with those of the corresponding superdeformed bands found in the non-relativistic calculations. The two bands have the same proton configuration, namely 104 +, 9 - +, IO+ -, 9- -, but differ in their neutron fillings, namely 11+ +, lo- +, 11-t -, lo- - for band j and 12+ + , 9 - + , 10-t - , 11 - - for band z. These occupation numbers are taken from ref. ‘*) and lead to rotational states of both positive parity and positive signature. In all the following results the contribution of the nuclear magnetism (9), i.e. the contribution of the spatial components of the vector fields, is included and it will not be stressed anymore. Fig. 5 shows the energy of the bands relative to an average yrast line, namely E” = E +670.5 - Z(Z + 1)/(2x 34.4) [MeV], as a function of the spin. Band j (the
-
E*(MeV)
I : /
2.0
: ,<*:
/
8F3r
f
_/*
‘,
__--
‘\
‘a : :
0.0 _._-----_------.“----_~-__--
-2.0
i-* I:
40
1
‘,
__-_ ‘\ \ ‘\ ‘.., .__c’
.’ .-.
I
80
120
flfl Fig. 5. The energy of the two superdeformed bands average yrast line E * = E +670.5 - I( I + 1)/(2x 34.4) approximation of eq. (9). The solid line corresponds the band z. Both bands differ
in ““Sr . The energies
are displayed relative to an [MeV] and nuclear magnetism is included in the to the band j and the dashed line corresponds to in their neutron fillings.
W. KoepA P. Ring / Superdeformations
290
solid line) can be followed from spin I = 55h to spin I = 120h, band z (the dashed line) from spin I = 40h to spin I = 125h. Above I = 102h band z is lower in energy than band j. In the non-relativistic to 80h after which it becomes this band taken
calculations
unstable
band j could be followed
is stable up to much higher (unrealistic)
too seriously,
because
from 30h
against fission. The fact that in our calculation
it is not clear, whether
angular
momenta
the fission
should
degree
not be
of freedom
can be appropriately described in the limited oscillator basis used here. Further in the non-relativistic calculations band z, which above spin I = 70h is lower in energy than band j, can be followed up to 92h. Thus our results are in qualitative agreement with the non-relativistic calculations, although quantitatively they are somehow shifted to higher spins. This again indicates the too small feedback of the rotation on the intrinsic structure. For the average moment
of inertia for the two superdeformed bands we obtain a value of g,, = 34 MeV’. This is much larger than the average moment of inertia of the ground state band (26 MeV-‘), and it is in good quantitative agreement with the average moment of inertia found for the superdeformed bands in the nonrelativistic calculations (37 MeV’). Fig. 6a shows the quadrupole moment of the bands as a function of spin. Their quadrupole moments are typically twice as large as the quadrupole moment of the ground state (744 fm*), although the ground state is already very deformed. The quadrupole moments correspond to values of /3 of about 0.65, which is equivalent to an axis ratio of 1.8 : 1. The quadrupole moments stay rather constant over large regions in spin or decrease slowly with spin, and then they increase at certain spins (at I = 55h, 90h, 120h for band z and at I = 115h for band j) in a nearly stepwise fashion to higher values. This fact, called the centrifugal stretching effect, reflects the filling of high lying, more aligned orbitals 4x), and in all cases it is connected to a smooth up-bending of the kinematical moment of inertia occuring at the same spins, as can be seen from fig. 7a. The drastic change in the quadrupole moment at Z = 78h for band z corresponds to a back-bending, caused by the crossing of two bands having rather different deformations 48). Between these stepwise stretchings Q decreases with spin. This is due to a smooth rotational alignment of the single-particle states, which 4x.4y). The qualitative behaviour of the quadleads to a dealignment in deformation rupole moment agrees very well with the nonrelativistic calculations and even the absolute values (Q 3 14 b is the average nonrelativistic value) are not very different. Fig. 6b shows the y-deformation as a function of spin. We observe a smooth increase of y with spin from y = 0” at I - 40h to y = 10” for the highest spins. The back-bending in band z leads to a stepwise increase in y. These upbendings and backbendings can be seen far more clearly in the moments of inertia. Fig. 7a therefore shows the kinematic moment of inertia (11) and fig. 7b the dynamical moment of inertia (12). There is one more up-bending occurring for I =94h in band j, which is not connected to a change of the nucleus shape.
W. KoepJ; P. Ring / Super~~formafi~ns
291
Y
/
10" ,I
,--\
._-
,’
.’
I’
I’
_--.._1
,’
I
*OSi5” I
,.-’ /. . .
_/* .-
cl”
__‘~-~-___--__---_--__--I--__----
40
80
120
Fig. 6. The quadrupole deformation as a function of the total angular momentum for the two superdeformed bands in “‘Sr. The deformation parameter y is determined from the expectation values of the intrinsic quadrupole moments CC&) = (r’Y,_) using the expression tan y = &(&)/(QJ, the quadrupole moment is calculated as described in tig. 4. The meaning of the different lines is identical to fig. 5, and again the semi-classicat correction for nuclear magnetism (9) is included.
4. Superdeformation
in the nucleus “*Dy
4.1. INTRODUCTION
The first experimental evidence for superdeformation in rotating nuclei has been found in the nucleus “’ Dy [ref. ‘)I. In May 1986 the Daresbury group around Twin resolved nineteen discrete y-ray-transitions in this nucleus with an almost constant energy separation corresponding to the moment of inertia of a superdeformed shape “). This was the first observation of a discrete line superdeformed band and it extended the spin, at which discrete states had been seen from 46h to 60h [ref. ‘)I. These fascinating experiments allow us to directly confront the results of recently developed advanced theoretical techniques of relativistic mean field theory in the rotating frame with experimental data in this extreme region of nuclear physics. We will therefore amplify here the properties of this nucleus in order to see, whether
292
I ~(2)(A~eV
-1
1
80 t 60
t
40
_c J ---. I! 40
I
80
120
Ilfi Fig. 7. The kinetmatic (upper figure) and the dynamical (lower figure) moment of inertia as a function of the total angular momentum for the superdeformed bands in ?Sr. The solid line refers to band j, the dashed line refers to band z.
relativistic mean field theory is able to describe quantitatively this phenomenon. Our calculation is also - apart from the correction term in eq. (9) - to the best of our knowledge the first fully self-consistent description of a heavy rotating nucleus in the superdeformed region, where not only the shell corrections but also the bulk properties are treated microscopically. We would like to stress again that we have no free parameter in our calculation. We use again the parameter set NLI from the literature 4’) given in table 1.
4.2. RESULTS
In contrast to our calculations for the much lighter nucleus 80Sr, which are described in sect. 3, in this case we used a more deformed (&-, = 0.50 instead of PO = 0.25) and larger basis ( NF = 8 major oscillator shells for the large components instead of NF = 7) for the expansion of the spinors and fields.
293
W. Koepf; P. Ring / Superdeformations
In the case of the superdeformed numbers for the four different necessary for the calculation frequencies
considered
occupation
of the lowest
(typically
bands
in “Sr we had to use fixed occupation
signature-parity blocks. It turns out that this is not of the superdeformed 15’Dy, since for all angular
the superdeformed lying single-particle
of a few MeV) between
the highest
state in this case simply states. occupied
Furthermore,
arises by the a gap appears
and the lowest unoccupied
single-particle states. This can be seen in fig. 8, where we show the single-particle orbitals around the Fermi surface for the neutrons (fig. 8a) and for the protons (fig. 8b) as a function of the angular velocity 0. It is also of interest to note that apart from some highly alignable states, the high-j states, whose energy decrease with angular velocity, the dependence of the single particle energies on the rotational frequency is very weak and there are pronounced shell gaps at the neutron number 86 and at the proton
86
-8 --
lszDy
+d.(MeV) -3
66
-5
152Dy
-7
.-.-.-.0.4
._I_.
I
_,_,_,_,
0.6 0.8 R(MeV)
1.0
Fig. 8. The energy of the single particle orbitals around the Fermi surface as a function of the angular “*Dy. The upper figure shows the neutron levels, the lower figure velocity R for the superdeformed shows the proton levels. The full line corresponds to positive parity and positive signature, the dashed line corresponds to positive parity and negative signature, the dashed and dotted line corresponds to negative parity and positive signature and finally the dotted line corresponds to negative parity and negative signature.
W. Koepf,
294
number deformed
66. This
is in perfect
shell closures
and Ragnarsson of the cranked
i? Ring / Superdefor~a~~ons
agreement
for this nucleus
7,52) using
a deformed
Nilsson-Strutinsky
with the observation
in non-relativistic Woods-Saxon
method.
of the so called
calculations potential
One of these intruder
by Dudek 5’)
in the framework states is responsible
for the vanishing of the superdeformed shell gap in the case of the protons $2 = 1.2 MeV, as can be seen from fig. 8b.
for
In fig. 9 we compare our results with the experimental spectrum ‘). It is obvious that the calculated spectrum without considering the contribution coming from nuclear magnetism - as given in eq. (9) - is too extended. The reason for this discrepancy is again the fact that we have neglected the spatial components of the vector fields in our first step. If we include them in the semi-classical correction, we achieve a much better agreement with the experimental data. The remaining deviations now go into the opposite direction and are very similar to those of the non-relativistic calculations ‘,“), where one also finds a somewhat too compressed spectrum. It is usually explained “) by the fact that there still remain residual pairing correlations in spite of the strong Coriolis-anti-pairing effect at these large angular velocities 53*54). A further comparison with experiment is possible within the moments of inertia. Fig. 10a shows the kinematic moment of inertia (11) and fig. lob the dynamical moment of inertia (12) as functions of the spin. The solid line corresponds to the calculation without the correction (9), the dashed line corresponds to the calculation including nuclear magnetism and the squares refer to experimental data ‘). It is interesting to note that the two moments of inertia are almost constant, have similar magnitudes, and are much more similar than those obtained from experiment. This tendency has also been observed in the non-relativistic calculations 7338).The fact that the dynamical moment of inertia $“’ varies very smoothly and slowly with
E(MeV) / -----Ii
*52Dy 0’
2’
I
NL1
NLl(cor.)
Expt.
Fig. 9. Experimental and calculated rotational spectra for the superdeformed band in “‘Dy. NLI refers to the relativistic calculation without the semi-classical correction for the spatial w-field and NLI (COT.) refers to the calculation including this correction. Both spectra are normalized at the lowest calculated levels, i.e. NLl at the 22+ and NLl (car.) at the 24+ state.
W. Koepf 3(MeV-‘1
1
295
P. Ring / Superdeformations
_____________________-_____________________
80
40
ls2Dy
t
OT_;__-_------_;..____
_____
_ ____
152D Y
ot-;-----------;-----------;--20
100 $k
Fig. 10. The kinetmatic (upper figure) and the dynamical (lower figure) moment of inertia as a function to the of the total angular momentum for the superdeformed band in “‘Dy. The full line corresponds uncorrected calculation, the dashed line corresponds to the corrected calculation and the squares refer to the experimental values.
the angular velocity is in good agreement with experiment and with the nonrelativistic calculations and shows the difference between this superdeformed nucleus vary considerably with and normal deformed nuclei, where $J”’ can sometimes spin 2). In addition it shows that this superdeformed nucleus behaves as a nearly perfect rigid rotor, an assumption
we made for deriving
the semi-classical
approxima-
tion (9) for the nuclear magnetism. Furthermore, the calculation including nuclear magnetism (9) (the dashed line) agrees very well with experiment. As has already been stressed in the context of the spectra, our deviations are of about the same order of magnitude as those of the non-relativistic calculations and they can be explained by the missing pairing correlations. The fact that the calculation without the correction (9) (the solid line) underestimates the moments of inertia corresponds to the too extended spectrum of fig. 9, and can be understood as due to some lack of attraction due to the complete neglect of the spatial components of the vector fields.
W. Koepf, P. Ring / Superdeformations
296
Q40(fm4) 20500
20000
19500
20
60 Illi
100
Fig. 11. The quadrupole moment (solid line, left scale) calculated as Q = &d{r-’ YzO)*+ 2(rZ Y2Jz and the hexadecupofe moment (dashed line, right scale) calculated as QAo = (r4 I’.&) as a function of the total includes the semi-classical angular momentum for the superdeformed band in “‘Dy. The calculation correction for nuclear magnetism (9).
In fig. 11 we finally show the quadrupole moment (the solid line) and the hexadecupole moment (the dashed line) as a function of spin. There is a general tendency that both moments decrease with increasing spin. This is caused mainly by a genera1 increase of the rotational alignment of the single-particle states at the expense of the deformational alignment. The fact that the overall change of the quadrupole moment is below 4%, however, and that the energy of the single-particle states hardly changes with the angular velocity, as can be seen from fig. 8, shows the high degree of collectivity of the rotation of this superdeformed nucleus. In recent non-relativistic calculations “) a strong increase of the hexadecupole deformation with spin was predi&d. This cannot be supported by our results, since we observe a slightly decreasing or nearly constant hexadecupole moment. Concerning the absolute magnitude of the quadrupole moment we are in very good agreement with experiment and with the non-relativistic results. We obtain an average value for the electric quadrupole moment of 18.6 e * b, while the prediction from the non-relativistic calculations ‘*) is 18 e - b and the experimental value 3, is 19 e - b. The magnitude of the quadrupole moment corresponds to a deformation parameter of /3 = 0.72 and to an axis ratio of 1.9: 1, which is very close to the “classical” axis ratio of 2 : 1 arising from the studies of the fission isomers 12*13).The value for the y-deformation lies below 0.7” for all angular frequencies under consideration, and thus we observe the rotation of a nearly prolate highly deformed nucleus around one of its small axes. 5. Conclusion We use relativistic for an investigation
mean field theory in the rotating frame as proposed in ref. 37) of superdeformed shapes in rotating nuclei in two regions of
W. Koepf, P. Ring / Superdeformations
the periodic
table. We employ
in Hat-tree approximation, p-mesons,
the nowadays
neglecting
standard
291
version
vacuum-poIa~zation
of this theory:
and including
working u-, w- and
the Coulomb
field and a non-linear coupling between the u-mesons. have been In the relatively light nucleus *‘St=, where so far no superdeformations observed experimentally, we find - in agreement with earlier non-relativistic calculations of the Paris-Bruxelles group 2’) - two superdeformed bands with an axis ratio of 1.8: 1 and an average moment of inertia of roughly 34 MeV-‘. For the famous and now well known nucleus ‘52Dy, where superdeformation has been first discovered experimentally ‘>, we present the first fully self-consistent calculation using a parameter set from the literature, which is known to give an excellent description as well as deformed
of the ground-state properties of nuclear matter and of spherical finite nuclei over the entire periodic table j3).
Our calculated results for the superdeformed band in ls2Dy are in very good agreement with the experiment as well as with earlier semi-phenomenological calculations using the Strutinsky approach ‘). In detail the theoreticai moments of inertia are roughly 10% larger than the experimentally determined values. This can be however - at least in a nonrelativistic framework - understood as a lack of pairing correlations in the present calculation ‘*). For the moments of inertia this agreement can only be achieved if one includes the contributions of the spatial parts of the vector mesons (nuclear magnetism). Because of time reversal invariance they have no influence on the ground state of even-even nuclei, but they also play a crucial part in our understanding of the Dirac magnetic moments in odd nuclei 56)_ In the present work these contributions were - for reasons of numerical simplicity - taken into account only after the variation within a classical approximation.
We would like to express our gratitude of useful discussions.
to R.R. Hilton
and H.J. Mang for a number
Appendix SEMI-CLASSICAL
APPROXIMATION
FOR
THE
NUCLEAR
MAGNETISM
If we assume that the rotation of the nucleus with angular velocity +:R around the z-axis is that of a perfect rigid rotor, we find the following classical connection 57) between the spatial componentsj of the baryonic currentjp = ( p, j) in the laboratory frame and the density p:
P(fi
X
r) =
(A.11
W KoepA P. Ring / Superdeformations
298
Transforming
this baryonic
current
into the rotating
frame yields:
j” cos 0?+ j” sin a?+ j’ cos f2-j”
sin fli.Z J
@p
L?.fp
.
C-4.2)
Here x”‘ stands for the coordinates in the rotating frame. As discussed in detail in ref. 37) the spatial components w of the w-field, which appear in the Klein-Gordon equation (6b) and in the Dirac equation (4), arise not only by the usual transformation of a contravariant vector from the w-field in the laboratory frame, but there is this extra substitution in eq. (7) [and in eq. (22) of ref. “)I. Therefore the source of this field, j, in eq. (6b), is not just the spatial component J of the baryonic current in the rotating frame j’/* (A.2), but we find using eq. (17) of ref. 37): j,=J+(cnx This together
with eq. (A.2), eq. (A.l)
i)b.
(A.3)
and eq. (9) of ref. 37) finally -9 x’
j,=pfi
,
gives:
(A.4)
0 0 The Dirac equation (4) together with the total energy [eq. (27) in ref. “)I yields for the contribution of the spatial components of the w-field to the total energy: E nucl.magn. = -2
(A.9
Neglecting in the Klein-Gordon equation (6b) the Laplace operator, i.e. the kinetic energy of the meson, and the a-dependent terms with respect to the large w-mass,
we get, mto
= gw.iV
and further assuming that the rotating nucleus density p = phom, we finally find with eq. (A.4),
has a constant
and homogeneous
(A-6) where we used eq. (31) of ref. 37) for the rigid rotor moment of inertia of the nucleus. The quantity, which is stationary in our cranking calculation, is the energy E’ in the rotating frame: E’=E-OJ,.
(A.71
W. Koepf, l? Ring / Superdeformations
The variation
has to be carried out with fixed angular
we find the following relation for the component z-direction J, [see page 77 of ref. ‘*)I: J,=
velocity
299
0. With this assumption
of the angular
momentum
-g.
in
(A.81
In our actual mean field calculation we neglected the spatial components of the w-field. Eq. (A.6) yields a semi-classical approximation for their contribution to the total energy in the rotating frame. This leads to the following correction of E’: (A.9) and therefore
because
of eq. (A.8) to a correction
in J,: (A.lO)
and as due to eq. (A.7) also to a correction
of the total energy in the laboratory
frame: (A.ll)
The other vector fields, i.e. the p-meson and the Coulomb important, and so it is not necessary to take them into account
field, are far less in this context.
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