Relativistic description of high spin states

Pergamon

frog. Part Nucl. Phys., Vol. 38, pp. 137-146. 1997 0 1997 Elsevier Science Lid

Printed in Great Britain. All rights reserved 0146-6410/97 $32.00 + 0.00

SOl46-6410(97)00018-S

Relativistic Description of High Spin States

P. RING and A. V. AFANASJEV

Physik-Department der Technischen Universirtit Miinchen, D-85748 Garching, Germany

Abstract

Cranked Relativistic Mean Field Theory with a non-linear coupling of the u-meson field is used for a microscopic description of rotational bands in superdeformed nuclei. I systematic investigations of all measured yrast superdeformed bands in the .A = 140 ~ 150 mass region is presented. The agreement with experimental data is excellent.

1 Introduction.

Since the discovery of the first discret.e superdeformed rotational hand (SD band) in the nucleus “*Dy

[l] in 19S6, the investigation of superdeformation at high angular moments remains one of the most

interesting and challenging topics of nuclear structure. The detailed experimental investigation of

superdeformation relreals many interesting properties of superdeformed bands like the phenomenon of

identical bands (see for example Ref. [2]) and suggested manifestation of &-symmetry (see [3] and

references therein).

By now, it is commonly accepted that the detailed properties of the SD bands like the dependence

of the dynamic moment of inertia .J(*) on the rotational frequency and the absolute values of charge

quadrupole moments Qo depend sensitively on the nunlber of occupied high-Y intruder orbit&. This

feature of the SD bands was investigated in detail for the first time in Ref. [4] using the Cranked

Silsson (CX) approach. Later, it was confirmed within the Cranked IIbods-Saxon (CWS) approach [5].

Systematic investigations of superdeformed rotational bands in different mass regions have so far been

performed only within the CWS or CN approaches (see, for example, Refs. [6, 7, 5, 8, 91). Moreover,

many experimental manuscripts contain applications of these models to the latest experimental data.

These investigations are based on a semi-phenomenological approach and contain many parameters

describing the underlying liquid drop as well as the single particle shell model. By treating the bulk

and the single particle properties separately they have the advantage to fit many details of the actual

nuclei under investigations directly to the appropriate region under investigation. The rich variety of

physical phenomena in the region of superdeformed shapes is based on a complicated and rather subtle

137

138 P. Ring and A. V. Afanasjev

interplay of collective and single particle properties. Therrforc it is (‘iI>>. to untlcrstantl that thcsc >c*nii-

phc~~o~iienological models can be very successful to reproduce many of the espcrimental details of these

superdeformed bands.

On the other side there are microscopic models, such as density dependent Hartree-Fock calcula-

tions with zero range forces of the Skyrme type or with finite range forces of the Gogny type. In these

models bulk properties and single particle properties are treated consistently based on the concept of

the variational principle. The parameterizations of these models are adjusted to a few spherical nuclei

only and therefore they have no freedom to introduce new parameters in the deformed region. Using

a non-relativistic concept there parameterization is however flexible enough to allow for a independent

adjustment of the spin-orbit splitting, which is turns out to be very crucial for our understanding of

shell effects. It is one of the great successes of these self-consistent models, that they do not only repro-

duce nuclei with normal deformations properly [lo] but they can also been used in the superdeformed

regions. The price to be paid for these microscopic models is a higher nunlerical effort. Therefore there

are so far not so many applications available yet. These are essentially applications to superdeformed

nuclei close to the doubly-magic superdeformed configurations “‘Dy and “‘Hg, as for instance Skyrme

calculations in the Hg-region at zero [ll, 12, 131 and finite angular velocity [14]. in the %-region [15]

and in the Dy-region [lG] and Gogny calculations [17, lS] in the Hg and the Dy region.

In recent years it has been found that relat.ivistic mean field (RMF) tl leery based on effective La-

grangians [19,20,21] provides a very simple and very elegant way to describe ground state properties in

realistic nuclei [22]. The nucleons are t,reated as point-like Dirac particles interacting in phenomenolog-

ical way by the exchange of mesons, such as the a-meson responsible for the large scalar attraction at

intermediate distances, the w-meson for the vector repulsion at short distances and the p-meson for the

asymmetry properties of nuclei with large neutron or proton excess. Fine tuning the surface properties

by nonlinear terms in the a-field [23] one is able nowadays to describe with only six or seven parameters

in an astonishingly accurate way many essential ground state properties of nuclei all over the periodic

table [22, 211. This theory has been extended also to the rotating frame [23. 251 as Cranked Relativistic

Mean Field (further on CRMF) theory. It has been successfully applied for the description of lowest

SD band in “‘Dy [26, 271, for t,he identical bands in the pair of nuclei “*Dy and “‘Tb [2i] and the

superdeformed band observed in 83Sr (283. Moreover, preliminary CRMF calculations without pairing

have been performed for 1Q2,1Q4Hg and lg4Pb nuclei [29]. H owever, the discrepancy of the latter results

wit.h the experimental data indicates the necessity to include pairing correlations in the Hg-region,

u;here the superdeformed bands start already at rather low values for the angular momentum.

Cp to now the application of the CRMF approach was restricted to these few nuclei only. A more

systematic application of this method over an entire region in the periodic table is therefore highly

desirable, because only a systematic investigation can reveal the power and the deficiencies of this

method and/or the present mean field parameterization. Here we discuss the application of CRMF

theory for a systematic investigat.ion of all the experimentally known SD bands in the .-l N 140 - 150

mass region. This mass region has been selected for such a investigation since both the experimental

data on SD bands are very rich in this region and pairing correlations do not play a major role at the

high rotational frequencies observed here. In addition, this mass region is believed to be characterized

by many changes in the number of occupied intruder orbitals, while the .4 - 190 and A - 130 SD

bands are expected to differ mainly in the number of non-intruder orbitals [30. 311. Moreover, this is

a region of rather low level density at superdefornlatioll~ whew the mean field approach is expected to

Relativistic Description of High Spin States 139

\wrk n-~11. This rt$on is sul”rd~formatioll in some scnw similar to thca wgion of nllclci arolmrl “‘“Pb

at zero tlt~forination. the classical example where the eiuglc particle shrll motlrl is relatively successful.

2 Relativistic Mean Field Theory in the Rotating Frame

In relati\.istic mean field (RRIF) theory the nucleus is described as a system of point-like nucleons,

Dirac spinors, coupled to mesons and to the photons. The nucleons interact by the exchange of several

mesons. namely a scalar meson 0 and three vector particles J. p and the photon. The isoscalar-scalar

c-mesons provide a strong intermediate range attraction betlwen the nucleons. For the three vector

particles we have to distinguish the time-like components and the spatial components. For the photons

this means the Coulomb field and possible magnetic field in the case IThere currents play a role. For the

isoscalar-vector d-meson the time-like component provides a \ery strong repulsion at short distances for

all combination of of particles, pp, nn and pn. For the isovector-vector p-meson the time-likecomponents

gi\.r rise to a short range repulsion for like particles (pp and nn) and a short range attraction for unlike

particles (np). Thry also take care of the symmetry energy. In addition the spatial components of

the d and p-mesons lead to an interaction between possible currents, for the d-meson attractivcx for

all combinations (l>~~. nn and IIn-currents) and for the p-meson attracti1.e for ~JJ and nwcurrents but

rrpulsive for ~~trcurwnts. l\e ha\-e to keep in mind, however that within meiln field theory these currents

to only occur in cases of time-rexversal braking mean fields as for instance in the case of Coriolis fields

at high spin.

The starting point of relativistic mean field theory is the well known local Lagrangian densit>-

n-here the non-linear self-coupling of the a-field. which is important for an adequate description of

nuclear surface properties and the deformations of finite nuclei, is taken into account according to Ref.

[23]. The field tensors for the vector mesons and the photon field are:

In the present status of the art the relativistic mean field theory, the meson and photon fields are treated

as classical fields.

The Lagrangian (1) contains as parameters the masses of the mesons m,. m, and n?,,, the coupling

constant,s ga, gw and g,, and the non-linear terms g2 and 5~3.

The mesons 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 this sense, the Lagrangian (1) is an e,fectl:lre Lagmnginn constructed for

the mean field approximation.

In non-relativistic nuclear physics the cranking model [32] plays an important role in the descrip-

tion of rotating iiuc1G. It is the symmetry breaking mean ficltl version of a Yariational theory with fixed

angular momcwtum and can be derived as an approximate \.ariation after angular momentum projection


[33]. In the self-consistent version [3-l] it. allows to iucllitlc i~ligum<ut (+fccts [3j] as ncll as polarization

effects induced by the rotation, such as Coriolis-anti-Pairing or chaugcs of the tleformatiou. But alrcatly

in the simplified version of the Roto.ting Shll Model with fixed meau fields [3G] it is able to describe

successfully an extremely large amount of data in the high spiu region of deformed nuclei.

The cranking idea can be used for a relativistic description too [24: 251: one simply transforms the

coordinabe system to a frame rotating with coust.a.nt angular velocity fz around a fixed axis in space

assuming - as in non-relativistic nuclear physics - that this axis is perpendicular to the symmetry axis

of the nucleus in its ground state. Such a transformation in Mnkouski space is given in text books (3i]:

(3)

According to the cranking prescription the absolute value of the angular velocity ]fl] will be determined

after the self-consistent solution of the equations of motion in the rotating frame by the Inglis condition

[32]:

WJ), = IQId%%j (4)

Using the transformation properties of scalars, vectors, spinors. etr., we obtaiu in the rotating frame

the following quantities

(51

(6)

where Z = -i(r x p) is the orbital and J = Z + S is the total angular momentum containing the

4 x Cmat,rices S for the spinor fields with spin f and a 3 x S-matrices S for vector fields with spin I.

For details see Ref. [24].

Using these quantities we obtain the following La.grangiau in the rotating frame

E = 3(ic(~,+s,;;i+~~~+l),‘,~)~

+&&? - U(5) (3)

-$w - c?V)(@,, - &$) + ;n?$%~

-fJ . where D 1s the covariant derivative with respect to the rotating metric. Neglecting in the following

the tilde sign, we derive the classical equations of motion. In the quasi-static limit they have the form:

(a(p - sww) + g,wo + P(rn~ + g”g) - f2J)h = 4 (9)

{-A + (f2Z)Z) 0 + V’(0) = -gcp. (10)

{--A + (nzy + m:} 2 = ga/?” (11)

(-A+(flJ)’ + m:}u = g”j (12)

These equations are very similar to the RMF equations iu the nowrotating frame. There are only three

essential differences:


1. Thr Dirac equation (9) contains a Coriolis term 0J in full analogy to non-relativistic crauking.

2. The Klein-Gordon equations for the mesons contain terms proportional to the square of the

corresponding Coriolis terms. It turns out. howe\-er, that they can be neglected completely for all

realistic cranking frequencies, because (i) they are quadratic in Q and (ii) mesons being bosons

are to a large extend in the lowest s-states with only small d-admixtures

3. The Coriolis operator in the Dirac equation breaks time-reversal invariance. Currents j are

induced, which form the source of magnetic potentials in the Dirac equation (nuclear magnetism).

In this way the charge current j, is the source of the normal magnetic potential A, the isoscalar

barpon current jB is the source of the spatial components w of the w-mesons and the isovector

baryon current j, is the source of the spatial component p3 of the p-mesons. In contrast to the

Maxwellian magnetic field A having a small electromagnetic coupling? the large coupling constants

of the strong interaction causes the fields w and p to be important in all cases, where they are

not forbidden by symmetries, such as time reversal. They ha\-e a strong influence on the magnetic

moments [3S] in odd mass nuclei, where time reversal is broken by the odd particle, as well as on

the moment of inertia in rotating nuclei, where time reversal is broken by the Coriolis field [27].

3 Applications in the A = 140 - 150 mass region.

In order to investigate the applicability of Relativistic Mean Field theory for the description of rotating

superdeformed nuclei, we have carried out a systematic investigation of all superdeformed yrast bands

in the A = 140 - 150 mass region using the parameter set XL1 [39]. All the experimental data presently

available for yrast and, in some cases, also for excited superdeformed rotational bands in this mass

region are compared with the results of these calculations. In Fig. 1 we summarize all the results

of the calculations for the yrast superdeformed bands in this region. We find large similarities Ivith

earlier phenomenological calculations of Nilsson-Strutinsky or Woods-Saxon-Strutinsky type. whose

parameters where obtained by fitting of single-particle levels in odd nuclei of deformed regions. Despite

this difference, the single-particle ordering in the superdeformed minima of the nuclei in the mass region

of interest obtained in the CRMF approach one side and in the CN and CWS approaches on the other

side reveals large similarities.

There are, however, differences in the level density close to the Fermi surface. It is lower in the

RMF model compared to the Nilsson and the M’oods-Saxon potentials. This is connected with the

low effective mass in relativistic theories [40]. Because of this low effective mass in the relativistic

theories one would not have expected such an good agreement of the moment of inertia J(‘) between

the relativistic theory on one side and the phenomenological theories and the experiment on the other

side. The phenomenological models have an effective mass one by definition and it is also known that

the effective masses of realistic nuclear descriptions taking into account higher order corrections should,

such as the virtual excitation of collective surface vibrations, lead to values close to one for the effective

mass at the Fermi surface of finite nuclei.

Although it is not understood why the low effective mass does not lead to a considerable dis-

agreement between the relativistic results and the experiment, we feel, that this fact could be in some

sense connected with a similar effect occurring in the description of isoscalar magnetic moments nuclei

differing by one nucleon form spin-saturated doubly magic nuclei. This quantities are well described


Figure 1: Systematic overview of the moments of inertia J t2) for the yrast superdeformed hands in the .4 _ 140 - 150 mass region. Cranked relati\-istic mean field calculations (full lines) are compared Gth

experimental values (dots)


in the extreme single particle model like the shell motlcl if one uses an effective mass on (Schinidt-

values). Relativistic theories with their small effccti\e masses show ver y large discrepancies from the

experimental values [al, 42, 431. They are connected with the violation of Galilean invariance in the

mean field approximation [44] and can be corrected by more sophisticated approximations treating

the symmetries properly, such as linear response or RPA theory, which takes into account polarization

effects in the nuclear medium. In fact, the disturbing problem of isoscalar magnetic moments in the

simple Wale&a model could be solved by taking into account polarizat.ion effects in the framework of

linear response theory [45, 46, 471. There it turned out that the spatial parts of the vector mesons

(m~clear magnetism) plays the essential role for this mechanism. On the other side. these polarization

effects can be taken into account also in the framework of pure mean field theory, if one considers that

the odd nucleon breaks time-reversal symmetry and causes polarization currents. If these currents and

the corresponding magnetic fields w, p and A are taken into account self-consistently one also obtains

good agreement with the experiment for the isoscalar magnetic moments [3S]. Voments of inertia are

related to the magnetic moments. They describe the response of the system to the external Coriolis

field. Therefore they should be calculated in linear response theory by the formula of Thouless and

1.alatin [4S], which corrects for the violation of symmetries in the mean field approximation by taking

into account polarization effects [49]. In fact. we suspect that a similar mechanism takes place in our

calculations, where the moments of inertia J(‘) are Thouless-\‘alatin moments of inertia, calculated

as derivatives of fully self-consistent solutions of the cranked RXIF equations. They yield the proper

values for these quantities although the effective masses are very different from one. This assumption

is supported by the fact, that neglecting the spatial parts of the vector mesons yields indeed very large

discrepancies between theory and experiment (see Fig. 1 and Ref. [26]) 1z’e admit, howevrer, that the

details of this mechanism are not fully understood so far and that further investigations in this direction

are necessary. A final remark concerns self-consistent calculations using Skyrme or Gogny forces, which

also show rather low values for the effective mass (- 0.i’). .%s mentioned in Ref. [50] one should expect

similar problems in these calculations too.

The classification of SD bands in terms of the number of filled high-,V intruder orbitals [4], which

is commonly accepted by now, is supported also in the CRMF approach. The properties of the dynamic

moment of inertia Jt*) and the absolute values of the charge quadrupole moment Qu depend in this

mass region sensitively on the number of occupied :V = 7 neutron and N = 6,7 proton orbit,als. In most

cases, the configuration assignment in t.erms of the filling of high-,V intruder orbitals for the observed

bands agrees with that proposed within the CWS and the CN approaches. However, there are some

differences for the isotope chain 146-148Gd which are connected with the fact that at the self-consistent

deformation of the configurations of interest the energy difference between the v[642]5/2 and v[651]1/2

orbitals is underestimated in the present CRMF calculations. This leads to some difficulties with a

quantitative description of observed properties of the dynamic moment of inertia of the bands 1 and 2

in these nuclei. We cannot exclude that this difficulty is connected with the fact that we neglect pairing

correlations in the present calculations. It could also be connected with the small effective mass in

relativistic theories and the problems to describe properly the position of specific single particle levels

discussed abovve.

In Table 1 we show quadrupole moments. \Ve find that with an increasing number of occupied

high-.X’ intruder orbit,als, the charge quadrupole moment Q. and the mass hexadecupole moment QJo

increases The absolute values of (2s are reasonable close to the experimental data, and they are consistent


Table 1: Charge quadrupolc momnltq Q:“t.“F at 1 - 3Oh ;~nd 1 - GOfi of different configurations:

CRMF calculations arc compared with available cspcrimcntal values (27” and with results of cranked

\Voods-Saxon (Qs’“) and cranked S&son ( QcifS) calculations. The Qb’.” and Qg”’ values are givru for the configurations which have the same occupation of high-S intruder orbitals as in the CRMF calculations.

with the results of CWS and C’S calculations.

4 Conclusions

Summarizing this investigations we can conclude that it is indeed remarkable and by no means under-

stood in all details, t.hat a theory with only six or seven parameters is able to reproduce this complicated

and rather subtle interplay of collective and single particle degrees of freedom in the rotational bands of

superdeformed nuclei, The basic assumptions of this theory are extremely simple: Point-like nucleons

are mowing in simple classical fields. Up to a few parameters the structure of the couplings between the

nucleons and the fields is determined by the laws of relativity and principle of simplicity. A non-linear

self-interaction between the a-mesons takes care of the density dependence of this couplings. The ro-

tation is treated in the cranking approximation. Since the few parameters have been adjusted in the

literature more than ten year ago, long before most of these data described in this work have been

measured, we have here a paramet.er free description of superdeformation. The fact that this extremely

simple theory works so remarkably well not only in this field of superdeformations but also in many

ot.her areas of nuclear structure physics (see for instance [21]) g ives us confidence that this theory is

not only a simple phenomenological model but that it contains beyond that basic all the ingredients of

a proper theory of t,he nuclear many-body problem.

Before this confidence turns into solid knowledge, there is certainly still much work to be done,

in particular in the field of suI~erdeform;rtiolls. From investigations in the phenomenological models we


know (i) that pairing properties have to lie taken into accouut. in particular in the Hg rcgiou. wl~cr(.

superdeforniatioil occurs at lower iiugular momenta. but also in order to obtain a better uiitlcrstaiitliug

of many details in the .A=150 mass region. (ii) that collective phenomena such as octupole \ibratious

can have a strong influeuce on exited super&formed bands, (iii) that the description of the decay out

of the superdeformed bands requires a precise knowledge of the structure of the energy surface over

the entire deformation region. which depends in very subtle way on the parameters of the microscopic

models.

Last not least we have to emphasize: that relativistic mean field theory, which is definitively the

most microscopic theory among all of the models applied so far for the description of superdeformed

nuclei, and which contains also less parameters than all the other models, is still phenomenological in

nature. We need a deeper microscopic understanding of the connection of this theory to the low energy

domain of quantum-chrome-dynamics. the basic theory of the strong interaction. There is definitel\.

a long way to go before we can deri\.e the RMF-parameters from first principles, but RMF theor!

formulated as a relativistic framework in terms of Dirac particles and mesons uses at least the proper

terminology expected from such a connection.

Acknowledgments

This work is also supported in part by the Bundesministerium fiir Bildung und Forschung under the

project 06 TM 743 (6).

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Relativistic description of high spin states