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Excited rotational bands in the high-spin region of deformed nuclei
1.D.2 : 1.E.1
Nuclear Physics A399 (1980) 390-414; © North-Solla~rd Ptrbliahina Co., Mtaterdam Not to be reproduced by photoprlnt os microfilm without wrlttan p~mladom ßrom the pnblisha
EXCITED ROTATIONAL BANDS IN THE HIGH-SPIN REGION OF DEFORMED NUCLEI J. L. F.GIDO t, H. 3. MANG and P. RING
Physik-Department, Technische Unioersit6t Mistehen, D-8046 Gorching
Received 28 August 1979 (Revised 8 November 1979) Abetrad : Rotational bands of deformed nuclei in the rare-earth region in the vicinity of the yrast line are investigated within the framework of the random phase approximation based on a selfconsistent solution of the Hartlee-Fork-Bogolyubov equations in the rotating systexo . Fôr low angular momeuta one fords the well-knows . ß, S and y vibtational bands and non~collective two-quasiparticle bands . In the high-spin region additional aligned bands develop. They are dramatically lowered in energy by the Curious interaction. Several band crossings occur. The calculated spectra are in fair agreement with experimental measurements.
1. Introdac8on In recent years a large amount of experimental information has been accumulated in the high-spin domain of deformed nuclei [for a .review see ref. 1)]. Besides the yrast band one has also observed the so-called "second" band (or Stockholm band which causes backhanding in many nuclei and in addition a series of other excited rotational bands Z). For heavy nuclei most of the microscopic attempts to describe the states along the yrast line s) are based on the mean field approach with constraints, the sculled self~onsistent cranking model (SCC) based on Harlxee-Fock-Bogolyubov (HFB) wave functions, and eventually improved . by a particle number projection before variation 4). This model is at present the only computationally feasible method for a realistic microscopic description, which allows simultaneous changes of shape, pairing correlations and alignment processes with increasing angular moments. One should, however, keep in mind that the wave functions thus obtained are only intrinsic wave functions. They have only a small overlap with the exact many-body wave function in the laboratory system, which is obtained by a projection onto good angular momentum 6. . '). Several authors') raised serious doubts as to the validity of the cranking model in the neighborhood of level crossings, i.e. in the backhanding region. These doubts have their origin in the application of the cranking approach to highly unrealistic s Work supported by the BMFr (Bundesministerium für Forschung und Technologie) . 390
EXCTTED ROTATIONAL BANDS
391
models involving only a few particles. We do not agree completely with these arguments and think that the cranking model is an approximation which works only in systems with many particles. It must fail in few-body systems. Therefore we feel no final decision about the validity of this method has been reached and it goes beyond the scope ofthis paper to enter into details. Eventually we have to regard our results in the backhanding region with some care. For the calculation of the low-lying excited rotational bands several authors [refs, is. s, aa)] proposed to use the random phase approximation (RPA) based on the self-consistent cranking model This method is the small amplitude limit of the time-dependent counterpart, of the static mean field approach- in the SCC model It has several advantages : (i) F.or low angular momentum it goes over into the well-known RPA based on an axially symmetric deformed product wave function, which has been used with great success in the literature for the calculation of vibrational states in deformed nuclei [see ref. 9) and references given there] . These levels are the heads of the bands investigated in this paper. (ü) The self~onsistency of the underlying single-particle basis guarantees that the spurious solutions corresponding to the violation of rotational symmetry and gauge invariance decouple completely from the other éxcitations. (iü) The sculled rotation-vibrational coupling is fully taken into account because for each angular momentum a new basis is used for the calculation of the vibrational levels. (iv) Since the RPA is also valid for non-oolleetive excitations such as pure twoquasiparticle states, it allows a consistent microscopic description -of a very large class of rotational bands, namely those with intrinsic wave functions that can be represented as superpositions of two-quasiparticle states in the rotating basis. (v) The method is the starting point for a theory going beyond the mean field approach : By. a requantization of the corresponding normal modes a ground state (the RPA-boson vacuum) can be introduced which contains additional correlations. In this way one is able to improve the energies of the yrast levels 1 ~ by including four-quasiparticle admixtures to the optimized product wave function of the selfoonsistent cranking model Within this paper, however, we do not use this latter possibility, but stay within themean field approach and restrict ourselves to the calculation of excitation energies, i.e. we use the SCC wave functions for the description of the yrast line. The main reason for this restriction is. that we use effective residual interactions (the pairing plus, quadrupole force) which is adjusted for the SCC energies of the ground state and of the yrast line. A part of the additional ground-state correlations is therefore akeady taken into account by the. ~ective force parameter. A detailed discussion of the problems connected with the definition of a correlated RPA ground state as an intrinsic wave function in a theory with angular momentum projection will be . given in a forthcoming paper' 1).
392
J . L EGIDO et al.
After having discussed the advantages of the suggested approach we should not forget to mention the diffculties connected with the RPA in the region of high spins: (i) A technical difficulty is caused by the rather strong violation of axial symmetry in the high-spin region There is no longer a K-selection rule to reduce the size of RPA matrices . In the case of rare-earth nuclei the dimension of the matrices to be diagonalized is 103-104. Only for separable forces can this be done using a reasonable amount of computer time. (ü) It is well known that the RPA breaks down if one of the frequencies goes to zero or turns imaginary, as for instance in the case of a phase transition In the highspin region such cases can occur. Examples are the regions of level crossings as in the backbending phenomenon, or at the breakdown of the pairing correlations at very high angular moments. In principle one knows well how to deal with this problem : One has to consider the ItPA as the lowest order of a suitable boson expansion of the hamiltonian. Higher-order terms in this expansion correct for the violation ofthe Pauli principle in the RPA, when one of the excitation energies goes to zero. They can be taken into account for instance in a self~onsistent version of the RPA 12). In the calculations presented in this paper it never happened that the RPA has imaginary eigenvalues, and we therefore stayed within the simple RPA approach. For some angular moments, however, we found very low-lying excitations. We therefore have to be aware that the results obtained at such I-values are less reliable. Investigations going beyond the quadratic order within the selfconsistent RPA approach are in progress . The method presented in this paper deviates. from the method proposed in ref. 2a) in so far as we do not take into account the angular momentum carried by the bosons for the calculation of excitation energies. A more detailed discussion of this point will be given in ref. li ), . In sect 2 of this paper we derive the RPA in the rotating basis as the smallamplitude limit of a time-dependent Hartree-Bogolyubov theory with variation after approximate projection onto good angular momentum . The simplifications resulting from the usç of separable forces are discussed in sect 3. In sect 4 we present calculations for the nucleus ie4Er and compare them with experimental data . The results are summarized in sect 5. 2. Time-dependent projected HsrIreo-Fock (TDPHF~ theory The projected Hartree-Fock (HF) or Hartree-Fock-Bogolyubov (HFB) theory has turned out to be a very powerful tool for the.description of static problems in nuclear physics. It is based on the fact that a large part of the many-body correlations in the nucleus can be taken into account by a simple symmetry-violating product wave function ~~~. Ifit is properly chosen, it can give rather reliable information on ground-state energies, r.m.s. radii or . deformation parameters. The quality of the approximation increases with the degree of symmetry violation The wave functions
IXCITED ROTATIONAL BANDS
393
themselves, however, are only a very poor approximation to the exact eigenstates of the many-body hamiltonian A. They have. to be understood as wave functions in a kind of intrinsic coordinate system . Only after projection onto eigenstates of the symmetry operator does one obtain a reliable approximation to the exact wave function in the laboratory frame: I~> = P'I~>~
( 1)
For simplicity we investigate in the following only one-dimensional rotations around a fixed axis. PI is the projection onto eigenstates of the angular momentum J.with respect to this axis. In realistic applications one has to deal with three-dimensional rotations and gauge transformations corresponding to the violation ofparticle number in superfluid systems. We will come back later on to these more general problems. The operator P' has the form 1 ax -nd~. Pi e`~U_ 2n ,~0 and the intrinsic wave function ~~) is determined by a variation of thé projected energy . For large symmetry violations the overlap function ~(w) _ <~I~~I~>
(3)
is sharply peaked at ~P = 0, and the overlap functions, for a general few-particle operator d can be expanded in derivations of the norm n(~p) [ref. i s)] Q(~P)
=
Cao+al
~ . .~ n(~P). i â~ +
(5)
For heavy systems a(~p) has â shape similar to n(~P) and one can stop this expansion in fast order. One then .fords for the projected expectation value of d
with eJ = J-<~~JI~> . A variation of the projected enemy obtained in this way with respect to the intrinsic wave function ~~~ yields the equations of the self-consistent cranking model with `~
<~I(aJlZl~>
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J. L. EGIDO et al.
and the subsidiary condition After this brief reaninder of static projected HFB theory we now turn to a timedependent description. We start out from the time-dependent wave function . (10) where I~(t)i is a time-dependent symmetry violating product wave functions and the projector .P' guarantees that the wave packet I ~P(t)~ in the laboratory frame has the proper angular momentum. In analogy to the usual dérivation of the TDHF equations from the variation of a classical action 14) we now determine the time dependence of the intrinsic wave function I~(t)i ûom the variational principle or
The exact TDPI-1F equations resulting from this variational principle will be discussed elsewhere. Within this paper we use two further .approximations : (i) We restrict ourselves in the intrinsic wave functions I~(t)i to harmonic vibrations with small amplitudes around a static time-independent solution I~oi of eq. (12). This means I~oi is determined from the variation principle and I~(t)i has the form I~(t)>
= eAcr>I~o>,
(14)
with an antihermitian single-particle operator A(t) e is a small parameter and B+ can be expressed by quasiparticle operators respect to I~ o ~ (akl~o) = 0) B+ _ ~
k
Xktak ai - Yktaiak.
ak
with (16)
Inserting eqs. (14}-(1~ into eq. (12) and varying with respect to the amplitudes X,~ and Y~ we obtain a set of projected 1tPA equations for the eigenmodes B~; of the system
EXCITED ROTATIONAL BANDS
39 5
We have to emphasize that I~ho) is a self~nsistent solution of eq. (13) and therefore depends on .the angular momentum . For each value of 1 we have a new quasiparticle basis âw a; . (ü) Since eqs. (1~ are still rather difficult to solve for realistic nuclei, we now introduce a second approximation for the projection. In analogy to eq. (6) we restrict ourselves to strông symmetry violations and stop after first order in the expansion (5).. Using the operator iSt -A instead of Û in eq . (6) we obtain from eq. (11) (lg) <~I(4~21~) Representing again I~(t)) in the form (14), we obtain from (13) for the statiç solution I~o) the self-consistent cranking approach of eq. (~ for the wave function I~o) with the cranking frequency w determined from the subsidiary condition <~olJl~o) = I.
(20)
In a second step we vary eq. (18) with respect to the amplitudes X~, and Ykl in the wave packet ~(t) of eq. (14). In the lowest order in the smallparameter e we now obtain instead of eq. (1~ the RPA equations in the rotating frame ~~OICak}al , ~ vB v
-
C~ ~, By ]]I~0) - ~,
(21)
The same equations have been dérived earlier is) by boson expansion techniques in the rotating basis. The determination of the cranking frequency co by condition (20), however; seems to be somewhat arbitrary in this case. One of the advantages of the Kamlah approximate projection technique is that it yields this condition automatically. In the case of three-dimensional rotations we have to make further assumptions: (ink We restrict ourselves towave functions I~(t)j whichstay close to axial symmetry for all times, ie. the width dK of essentially admixed K-values in the wave function I~(t)) is small compared to I: <~I(d~s)21~) ~ I(I + 1).
(22)
We then again obtain the cranking model with respect to the x-axis and only the subsidiary condition (20) has to be changed into (23) The derivation follows closely the time-independent case [for details see refs. i 3, is)] . Well-deformed nuclei in the middle of the rare=earth region are axially symmetric in the ground state. We can therefore neglect <~olJz too) to a good apprôximation.
396
J. L. EGIDO et al. T~ 1
Quantum numbers of the single-particle states and their energies . in MeV Protons
Neutrons
BtatC
enel'gj~
dg9/z lg .,/z 2d,/2 lhlln 2d3/z 3s 1/2 lh 9/3
-4.9811 =0.6741 0.0000 0.9513 2 .4119 2.6965 6.1795
~
statG
energf
~11/2 ~~/z 189/z 1113/2 3P3/z ~s/z 3P1/z 2gs/z
-5.9548 -1 .5730 -0.7865 0.4400 1.4606 1.9100 2.9212 6.5072
For increasing angular moments the axial symmetry is sometimes lost. For instance, we fmd m 164Er (see table 3) y > 1Q° for I z 24. Nevertheless the condition (22) is always satisfied with an accuracy of a few percent for the yrast band and for the excited bands with K _ 0. The only exception are bands with K $ 0 for smâll Ivalues, for instance the y-band with K = 2. In this case the K quantum number is nearly a good quantum number, which has to be treated on a more quantummechatücal footing than we did in the time-dependent wave packet ~(t). In principle one can derive theRPA in the rotating frame also in a purely quantummechanical static picture by. the variational ansatz ~~P~ = PIBy ~ItPA),
(24)
where. ~ItPA~ ~is a suitable vacuum to the boson operator B and is determined by variation after projection . For the details of this derivation see re£ t i~ In the case of rather pure K-values (dK ~ I) we again obtain the cranking condition (23) where one has to replace ~~oi by Bq ~RPA). For the y-band we now have <.is ) = 4. That means in practice thatwe have to calculate the y-band at the angular momentum I with a slightly modified cranking frequency. The corresponding correction in the energy is of no importance for I-values larger than 8. 3. N~erkal solation of the RPA egaations The Coriolis term a~?s vïolates axial symmétry and time reversal For heavy nuclei the size of the RPA matrix is extremely large. A truncation of this matrix does not seem to be reasonable, since in such a case .the spurious solutions would no longer be .separate from the other normal modes. One is therefore restricted to the use of simple separable forces . The pairing plus quadrupole force has proven to be a very reasonable approximation to a realistic residual interaction to describe the interplay bet~öveen single.particle structures and collective quadrupole and pairing degrees of
EXCTTED ROTATIONAL BANDS
397
freedom. The basic blacks of the oonesponding hamiltonian aré a spherical singleparticle field, AO = L, EkCk k
Cb
(~)
the five quadrupole operators,, pw = ~~k ~ k,l
and the two pair operators,;
P + _ ~, Ck C~+ ,
P- _ ~
I~c k =
P+,
(2~
which describe the creation or the annihilation of a Cooper pair. The oonesponding hamiltonian is of the form
For calculations in the rotational frame the symmetry operation etc= has turned out to be very important We therefore introduce the symmetrized quadrupole operators Qo = rz~z~, . Qf~=,~(rZpz~frZ~z_~ (h = 1, 2),
which have definite signature
etx1~Q}~e~-txlx
= tQtw eif~txii e-if[.t,~ = Qo~ ~/o
We then get far the hamiltonian
2
(3O)
0 11 ~ -~ ~ ~ BP , P P
where we have introduced the seven operators Isv =
~Qw
15t = ,~~t,
(31)
and p runs over the values -2, . . ., +2 and +, -. Introducing a general 13ögolyubov transformation with thé quasiparticle operators ak+
_~ + V,kci , i ,UuFi .
(32)
398
J. L. EGmO et al.
these operators. can be expressed as Rnih
UP = DP
ßP l
_- ~ DP~ak ah k!
+ ßP l +ßPO,
ßPU
- ~,,(DPr~iak al ±DPkta~ak)~ k
(33)
Dp is the expectation value of the operator 1SP in the IiFB vacuum ~ ) to the ciuasiparticles ak : Dô = ~~ r2Y2o)~
Di =
V `~~~Y22)~
(34)
The symmetry with respect to the operation e;~= requires D_0 1 =D_0 2 =0,
(3~
and D° 1 vanishes automatically because of the antihermiticity of ~i+l . Inserting eq. (33) into the hamiltonian without normal ordering yields where ~°° is a simple constant : ~Po
= ~Ho) - i ~ i
P
_ ~Ho)-i~
(3~
.~1 is a bilinear form in the operators a, a+
with
Hl1 - HÔl ~ ~ ~ßP)DP1' P
H2o = Hôo - ~ `ßP)DPO~ P
where Ho, Hô1 and Hô° are defined in analogy to eq. (33) for the operator A° in a+a+ eq. (25) and ~2 is a bilinear form in the pair operators as and ~2 -
2 ~ ßP0+ßP0 P
_ -2 ~ (aktat.t k<1 k, «,
~~fDPO ~ 02 . alak) L .~ j D20+ / (DP . P. \ P /k!
a~.ak, 20 DP ~~~~ Ca+a+~ . k !
(39)
~' finally contains terms of the form a +aa + a+, a +aa + a which are of higher order in a perturbative boson expansion 21) . A normal ordering of these terms would
EXCITED ROTATIONAL BANDS
39 9
certainly give contributions to the RPA matrix. It is, however, within the framework of the pairing plus quadrupole model to neglect them. We now come back to the variational problem in the rotating frame as discussed in sect. 2. In this case we have to replace 8° in eq. (24) by A° -coJ~. Neglecting the exchange terms which come from the operator ~a in the variational equation (~ we end up with the well-known HB equations in the rotating frame. Their solution gives a diagonal operator ~ 1 ~i = ~ Ekak aw k
where Ek are the quasiparticle energies in the rotating frame. The vacuum defined by these operators is the intrinsic wave function of the yrast level at the corresponding angular momentum. Its energy is given by
with the exchaangé contributions
<~~i = -i~ ~ ~Dp2,°,Iz p kl
z
= -~ ~ ~dQa dQai - z~ ~ ~ dP~ dPti p=-2 s=t
(42)
which are determined by the fluctuations of the collective operators Q~ and P~ : These aretaken into account in the calculation ofthe energy, but not in thevariational equation for the determination of the intrinsic wave function. In the next step one has to solve the 1tPA equations (21). Here the terms ~' are neglected. One then ends up with a separable RPA problem
(43)
for the determination of the amplitudes X~ and Y,~. Introducing the matrix elements _
the linear eigenvalue problem (43~ whose dimension is given by the number of pairs kl (k ~ I), can be reduced to a homogeneous system of equations non-linear in éa~, whose dimension is equal to the number of collective operators 1Sp:
J. L. EGIDO et al.
400
with
DozDoz " w ~~-Ek-Ft
DzoDzo " Sl+Ek+Et)
It has non-trivial solutions tD only for discrete values Sa~, where the corresponding determinant vanishes. They are determined numerically. The symmetry e'~= reduces this 7 x 7 determinant to a 5 x 5 determinant containing the operators ¢o, ~+~, ~+ z~ p+ and ~- with Positive signature and a 2 x 2 determinant containing the operators ¢ _ .i and Q _ z with negative signature. Using the RPA equations (43) we can express the amplitudes X~ and Y,~ by the matrix elements v, Xr
_
- ~ Doz "t P
Pki
r P
Slr -Ek-Ei '
~
~rr -_
~
P
Dzo "tr Pkt
P
Slr+Ek+Ei'
(4~
and obtain in this way the norm of the elements tp. They measure the excitation probability of the one-boson state [see eq. (24)] Iv> = Bv I~A> by the collective operator ßv . We thus get for the quadrupole matrix elements ~RPAIQ~,Iv~ = X -~t~,
p = -2, . . ., +2,
(49)
ând for the pair transfer matrix elements
164Er Because of the large amount of experimental data gathered for the nucleus [ref. z)] we used this nucleus to give a unified microscopic description of the different bands within the theory discussed so far. In this paper we concentrate on states with positive parity, because the pairing plus quadrupole interaction (2~ does not introduce oorrelations between two quasiparticle states with negative parity {at least as long as we neglect the exchange terms as we did in sect. 3).. On the other hand, the negative parity bands can be described to a good approximation within_ the general-. ized single-particle picture, as has been shown in ref ~1 ').
40 1
EXCITED ROTATIONAL BANDS 4 .1 . FORCE PARAMETERS AND THE YRAST LINE
As discussed in sect. 2, we used the self~ons>stent cranx~ng model for the description of the yrast line. The wave functions ~~~ are determined by the variation of the energy (37) with the constraints on an angular momentum <1x>, proton number . <1Gpi and neutron number
j(I+1)-< =i,
I <~pi = Z~
<~ni
= A - Z.
(51)
The energy is. obtained as expeçtation value of the hamiltonian (27) with respect to these wave functions. It contains direct terms (37) and exchange terms (42). The single-particle configuration space was the same as used earlier 18). Single-- . particle' energies are given in table 1. The core consists of 40 protons and 70 neutrons, and a moment of inertia of the core was introduced as .~~ = 6 MeV- t . In order to reproduce the expérimental data as far as possible, the levels of the yrast line (see fig. 9) were used to adjust the strengths constants of the quadrupole-quadrupole force (~ and of the pairing force (G) t (see table 2). T~s~g 2 Strength constants of the pairing plus gnadrupole interaction X~ = 0.038 Xm = 0.038 Xvo = 0.100
G~ = 0.200 G, + , + = 0 .180 G,_ n _ = 0.157
The quadrupole strength parameters g (m MeV) haÎe3to be multiplied by dimensionless matrix elements hlce (n ~I(~/b~ Yzo In'l'j~m') where b r (41 .454/film) ni the oscillator length, G. t, t is the pairing force constant (in MeV) acting between neutron atabea with positive (negative) parity and G, +,_ _ (G.+ .+G.- .-)
The force constants determined in this way differ slightly from the parameters of ref. te). There were two reasons for this readjustment : (i) we wanted to use (as far as, possible) the spherical single-particle energies given in ref. t~ (only the ni b. level was shifted by 440 keV) ; (ü? we . also wanted to use a separable pairing interaction for the neutrons, i.e. Go-o+ _ (Gn+o+Ga-o-)~ .in order to simplify the solution of the RPA equation. Since the quadrupole force is no"t completely separable [Q~ ~ (Q~Q~~], and since the pairing strength for protons and neutrons is diûerent, we had in fact to generalize the dispersion relation (45) a little bit and to search for the zeros of a 10 x 10 det~±?inAnt for the states with positive signature (¢o, ~+v ~+z~ ~+ and ~for protons and neutrons) and of a 4 x 4 determinant for the states with negative signature (Q ~ 1 , ~1 : z for .~, protons and neutrons). . f For this adjustment we took into account the exchange terms because they were easy to calculate and improved the numerical agreement slightly .
. L. F.GIDO et al.
402J .
The moment of inertia of the inert, çore was used to modify the expectation value of the angular momentum ~Js% _ ~~~Js~~% +~core
and the energy where the angular frequency a~ is given by co = dE~/d(Jx ~.
(54)
In this way the inner shells N. 5 3 for protons and N 5 4 for neutrons are taken into account for the calculation of HFB expectation values. We neglect the interaction matrix elements between the valence shells and the côre shells in the RPA matrix and do not therefore allow for vibrational excitations of this core. Table 3 shows properties of the yrast line. lß column two and three we give the experimental spectrum and the calculated energies (53). Between I = 14 and I = 16 one observes back-bending. It is connected with a change in the intrinsic structure of the yrast line caused by the crossing of the ground-state band, which forms the yrast band up to I = 14 and an aligned band of two i~ neutrons, which forms the T~siB 3 grast line is 16aEr for the different Ivalues Properties of the
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
$e~ ~
~
0.000 0.091 0.299 0.614 1 .024 1 .518 2 .082 2 .70~2 3 .263 3 .768 4.345 4.999 5.728
0.000 0.106 0:345 0.701 1 .149 1 .664 2.215 2 .769 3 .317 3 .876 4.453 5.073 5.753 6.489 7277 8.105
°~ 0.0000 0.0759 0.1328 0.1798 0.2151 02388 . 0.2514 0.2528 0.2475 0.2434 0.2636 0.2921 0.3185 0.3417 0.3616 0.3842
dp
do
0.728 0.719 0.701 0.676 0.648 0.621 0.600 0.592 0.992 0.992 0.560 0.501 0.423 0.320 0.169 0.015
0.957 0.945 0.918 0.875 0.817 0.751 0.689 0.643 0.603 0.544 0.486 0.437 0.381 0.312 0214 0.031
~
ß 0.300 0.300 0.301 0.303 0.305 0.307 0.309 0.308 0.303 0.295 . 0.291 0.287 0.287 0.285 0.282 0.279
Y 0.00 0.20 0.76 1 .59 2 .55 3 .44 4.11 4.76 5.78 7.33 8 .62 9.40 10.05 10.65 11 .26 11 .57
The theoretical enegies F~ (Me~,are calculated by eq . (53) . They include the exchange terms (42). cv (Mew is the cranking frequency as defined in eq. (8). dp aad .d o (Mew are average gap parameters for protons and neutrons . The deformation parameters ß and y (degrees) are obtained from the expxtation values of ¢o and ¢z in eq. (28) : ß ~ (<¢o~~+<¢s)~r~2, tgy s<¢ s ~/<¢ oi~ ß ie normalized to 0.3 at I=O. ~ O . C. I~isfner et al. ~.
,EXCITED ROTATIONAL BANDS .
403
2 .0 1 :MéV) t5
Q5
0 Fig. 1 . (a) The spectrum of two~uasiparticle states with positive signature above the yrast line (y) in the nucleus 1s4Er as a function of the angular momentum I. (b) The spectrum of RPA frequencies with pôsitive sig(18tULC In 164 ~ a function of angular moment~n I. The calculation has beg carried out only for integea I-values. To guide the eye the Corresponding points are connected by hill lines. This connection is in general carried out according to the energy. On3y in cases where the miaingis very. small have we connected states with the same structure . We thus obtain sharp levd crossings, which would not occur in a tabulation with continuous I-vah~es.
J. L. EGII~O et al.
great band for I > 14. Only at very high spins at I ~ 30 a second alignment of two h,~ protons 1 takes place (see fig. 1aß The gap parameters for protons and neutrons show a continuous antipairing effect with increasing angular momentum . Although the neutron gap is larger for I = 0, it drops more rapidly in the region between I = 12 and I = 18 becausé of .the alignment of the i~. neutrons.. The pairing correlations vanish in both cases at I - 30. Thé deformation parameter ß does not change very much ovér the entire region of angular momenta under consideration ; it shows only a slight stretching effect for I .5 14 and afterwards a slight antistretching. The deförmation parameter y increases .continuously up to a value of 11° at I = 30. Nevertheless the essential condition in the derivation of the cranking model
In the following we discuss the structure of the spectrum with positive signature . (etwx = +1) in the region above the yrast.line as obtained from the diagonalization of the RPA equations (21). Its main features are determined by the two-quasiparticle energies E~ in the rotating frame.. The lowest are shown in fig. la. At I = 0 one observes a large gap of ~ 1.5 MeV between the great level (y) and a number of twoquasiparticle states. Some of them are labeled by the latin indices. For increasing angular momentum they decrease roughly parallel to each other. The only exception , (a) is one pair of i,~ neutrons, which aligns along the rotational axis and is therefore drastically lowered in energy . Between I = 14 and 16 it crosses the ground-state band and forms afterwards the great levels. Between I = 12 and I = 18 the increment of angular momentum in th,e great line is mainly caused by the alignment process, i.e. the collective frequency.does not change very much in this region - it even goes back a little bit (see table 3). As a consequence, the two~uasiparticle energies stay roughly constant between I = 12 and I = 18, and one observes a kind of plateau in the general behav;our of many. two~uasiparticle energies. Only for I values larger than 20, where the collective frequency increasës again, the two-quasiparticle energies start to decrease furthermore. In this region we also observe a pair (c) of two aligned hy, ,protons, which cross the great band at I ~ 30. Fig. lb shows the lowest eigenvalues t2, of the ItI'A equations -in the rotating frame. Many. levels are very close to the uncorrelatted two-quasiparticle energies ; an example is thé pair of levels f which corresponds to two combinations ak+ a; and arai which are degenerate in energy at I = 0 and show only a very smà11 splitting for increasing angular. velocity. In fig. 6 we will find the corresponding combinations ax+a~ and arai with negative signature. ~.
.
EXCTfED ROTATIONAL BANDS
~ 405
On the other hand, there are a series of important differences between fig. la and fig. lb caused by the two-body cortelations : (i) We did not draw the spurious solutions dl d1PP and dRn for positive signature . which lie exactly at zero energy. Since the other wave functions are orthogonal to these states, there occur drastic differences in the wave functions of the levels in fig. la and fig. lb, even in cases where the behaviour of the energy looks rather similar. (ü) The RPA correlations produce collective states which are drastically lowered in energy. The most important example is the y-band (y) which has no analogon in fig. la. Fig. lb contains the members with even 1-values (ie. positive signature of this band). Its excitation energy above the yrast band is rather constant up to I ~ 16. In this region the . yrast levels change their character. The ground-state band has now a positive excitation energy above the yrast level The y-band stays roughly parallel to the ground-state band and therefore increases for I > 16 in the excitation energy above the yrast line: In this region it crosses other non-collective two-quasiparticle states. One observes a large mixing and the y-strength is split over some levels. (iü) In addition to the y-bands we find less pronounced collective bands : A proton pairing vibrational band (S) which at low spin values is essentially a superposition of two proton quasiparticle states (d and c in fig. la) and a ß-band (ß) which is less collective than the y-band. (iv) The aligned band (a) of two i~. neutrons is not very much affected by the RPA correlations. Only the crossing of the ground-state band is less pronounced in fig. lb than in fig. la. We have to emphasize that the indices y, ß, b and so on are only labels which characterize the structuue ofthese bands at low angular moments. For larger I-values there sometimes occur drastic mixtures and the character of a band may change completely. To get more detailed information about the structure of the wave functions we investigate in the following the collective strength of the operators ¢~ in eq. (28) and ~t in eq. (2~ for some of the bands in fig. lb. In figs . 2~ we show the quantities SY(ß). =
KRPAIOIv)I
E
z
I<~Alß l~> h
(5~
for the five operators 0 = ¢ o, .Q+1, Q+z~ p+ and P- ~~ Positive signature. They measure the degree to which the state v exhausts the sum rule of the operator 0, i.e. the overlap of the state Ivy with the normalized wave function O + IRPA). In the actual calculation the sum over v' in the denominator of eq. (5~ includes only the 801owest solutions of the RPA equations. There are certainly also contributions from higher states to this sum. In particular, we do not take into account the giant resonances . Nevertheless .the values of S,(0) calculated in this way allow us a .clear . interpretation of the collective structure of our wave functions.
Fig. 2. Zhe percentage of thecollective strength foithe operators ¢F and ~ t ddmed in eqa. (2~ and (28) exhausted by the y-baild of tsaEr ae a function of she angular momentum. Further details and a precise definition of the quantities Sr are given in the text.
Ob
Pairing - band (5)
a6 aa
- salP)proton
___sa~ P l~t~~,
0.2 0 ~.
.
-'
_______-
--~~__________________
x.
- Sa(P. )protm
x
16 Fig. 3. 1be strength funcdona of fig. 2 for the pairing v~brational bead (b) in's4Er .
407
EXCTTED ROTATIONAL BANDS
An example in which the basic structure of the wave function is well preserved up to large I-values is the so~alled y-band (fig. 2). At small angular moments S,~(~+z) . is close to unity; ie. in this region the y-band nearly exhausts all the low-lying ~z strength. With increasing angular momentum one observes more and more mixing, which results in a generally diminishing value of Sr(¢+Z). At I = 20 this band contains only 40 ~ ofthe low-lying ¢ Z strength. However, the Qz strength is dominant for all angular moments under consideration. We only observe a small dip at I = 6. This corresponds in fig. lb to the crossing with band (a) of two aligned neutrons. It also shows up in the curves for S~,(¢ +1 ) and Sy(P_)°~ut~. For l-values larger than 18 we find even more mixing with other bands, as it is expected from fig. lb . Fig. 3 shows the corresponding quantities for the so~alled pairing vibrational band S of fig. lb. For low angular moments it is essentially a mixture of two two-quasiparticle (proton) states with K = 0, namely the states (Z + [411])2 and (i - [523])Z. At high angular moments I > 20 it changes its structure and . the protons in the h,~ shell align along the rotational axis . This gives drastic lowering of the energy and an increasing of the
0
-
"
ne ~.6 0
ß - bsnd -
- ~ SßIQpI
\
0.4
.~ "'
4
/
.'"' 8
~ 12
16
~ \.
_ _ _ _ g~ lp~t) - Sß 10,2) ~ - ~ .-
20
24
)
_ 28
Fig. 4. The strength functions of fig. 2 for the ß_vt~rational bend (~ in 's4 Er.
408
J . L. EGIDO et al.
I7X~)
Fig . 5. flue difference (1s') in the expectation valuup of 1s .between levels in the aligned bands (a) and t3zu; yrast band (y) as a function of I : (a) for positive aigoatuit+e ; (b) for negative aignatuure.
EXCITED ROTATIONAL BANDS
The band (a) (aligned band) in fig. lb is a rather pure band of two neutrons in the i~. shell. At I = 0 it corresponds mainly to the state (~+[642 2 with K = 0. With increasing angular momentum one observes a dramatic alignment along the rotational axis. This can be seen in fig. Sa, where we show the difference in the expectation value of Jx between the yrast line ~0) and the band (a) (Jx)w - w~Jzw~-~~~Jx~~i "
Between I = 12 and I = 18 this band changes its character because of the crossing with the ground-state band, i.e. ~O~.1x~0) increases drastically and w~,xw~ decreases. At I = 20 the band (a) is more or less the original ground-state band, whose angular momentum is smaller than that ofthe aligned yrast level at this I-value. This explains the negative values for (?xl)ev in this region. 4.3 . EXCITED STATES WITH NEGATIVE SIGNATURE
So far we havé investigated only vibrations of positive signature, i .e. the density distribution of the classical wave packet ~~(t)i in eq. (14) is at all times symmetric with respect to a rotation of 180° around the x-axis
This means that the rotational x-axis at all .times remains a prinçipal axis of the intrinsic density distribution . This is no longer the case for the wobbling motions which involve vibrations of the principal axis of ~the density distribution around the x-axis of steady rotation 2°). This motion is characterized by oscillating values of <~(t)~Q-i~~(t)i or <~(t)~Q_2~~(t)>, i.e. negative signature and odd I values. As discussed in sect. ~3, the 1tPA equations for negative signature are determined in the pairing plus quadrupole model by the operators Q_ 1 and Q_ 2. In figs . 6a and b we show in analogy to figs. la and b the two-quasiparticle spectrum EZ ~ and the itPA frequences ~q ~ for negative signature. Again the basic structure is formed by the two-quasiparticle excitations; again we observe an increasing level density at the Fermi surface with increasing angular momentum . Thé spurious wobbling motion, which corresponds to a steady rotation around a somewhat rotated axis and which has the excitation energy of the cranking frequency w is not shown in fig. 6b. The most drastic difference between fig. 6a and fig. 6b is therefore the low-lying collective y vibration. For small angular momenta it has precisely the same energy as the y-band with positive signature. For higher I-values we observe only small deviations. In the region of I ~ 11 an aligned band (a) of two i.,~ neutrons with negative signature approaches this band and produces some mixing (see figs. 7 and 8). This aligned band would correspond to the occupation of the i,~ levels with mx = ~ and . ms =. i in the decoupled limit. Its energy is somewhat higher than
Fig. 6. (a) Zhe spectrum of two-quasiparticle states with negative signattae above the yraet line (y) in the nucleus issue as a function of the angular mome~uum I. (b) Zhe spechvm of 1tPA fregtltncies with negative signature in 1~Br as a function of I.
F.XCIT® ROTATIONAL BANDS
411
0.5
1 5 . 9 13 17 21 25 l Fig. 7. The strength function for the operators Qa as defined in the text for the y-band with negative signature in fig. 6b .
0.5
Fig. 8. The strength function for the operators QF as defined in the text for the aligned band (a) with negative signature in fig. 6b.
the completely aligned band with positive signature (mx = ~ and mx = ~ in the completely decoupled limit) which corresponds to the band (a) in figs. la and b. To give more information about the structure of the corresponding wave functions, we show the quantities S(Q _ 1 ) and S(Q _ Z ) of e4. (5~ for the y-band in fig. 7 and for the aligned band in fig. 8 and the angular momentum (Jx~)r for the band (a)!in fig. 6b. Besides the y-band and the aligned band we see in the spectrum of fig. 6b a series of rather pure two~uasiparticle hands. An example are the two f-bands. They coincide~+ery precisely with the corresponding bands of positive signature in fig. lb. One reason for this little mixing is certainly the. schematic pairing plus quadrupole force, which provides an interaction only the Q _ t and Q_ a matrix elements in this channel. With a more realistic interaction one would expect more configuration mixing. 4.4. COMI'ARLSON wITH EXPERIIVIENT
Some of the experimental data of Kistner ei al. s) are given in fig. 9 and compared with our theoretical results.
412
J . L . P.(3iD0 et al. ~s4
.f 4
Er
ie'
3
g" 8" rs" 5'R' â
r
ss
gs
rx4.PoG
y
ss
g. s.
neg pn~
y
Fig . 9. A comparison of experimental aad theoretical spectra for several rotational bands in'~Er. 11îe dashed line indicates the yrast levels.
Thè yrast band is indicated by a dashed line. It consists of the ground-state (g.s.) up to I = 14 and of the aligned band of two neutrons (s.s.) for I >_- 16. These levels aré calculated within .the statïc self-consistent cranking model, as discussed in subsect. 4.1. The other levels with positive parity are obtained from the ItPA calculation presented in sûbsects. 4.2 and 4.3. For the low-lying members of the y-band the correction 'of the energy dE discussed in sect. 2 was introduced t
dE = a~dl = co(~I(I+ 1)-4- I(I + 1)):
This correction is of course only for small I-values of importance. It guarantees, however, that the band head of the y-band- comes closer to its experimental value. In fig. 9 we also show results for a band with negative parity : It is calculated within r Por a detailed discussion of this correction see ref. ii).
EXCITED. ROTATIONAL BANDS
413
the generalized single-particle model by a blocked HFB calculation, which has been discussed in more detail in ruf. i') : The overall agreement of our theôretical .calculations with experimental results is excellent in particular in view of the fact that nnly~ a few fit parameters (force constants and a moment of inertia of. the. core) were used to adjust the yrast line: All other levels were obtained with the same residual interaction and the same configur~tion space. In detail one observes small differences between the experimental and the theoretical spectrum, as for instance minor shifts in the band heads of the y and the negative parity band, which depend sensitively on the underlying single-particle scheme, or too strong a staggeringinthey-band, which indicates too strong aCoriolis interaction. It remains to be investigated whether this is a general feature of cranking calculations. S. Conclaslon
The time-dependent Hartree-Fock-Bogolyubov theory after projection onto states with good angular momentum in the limit of vibrations with small amplitudes was . used to investigate the . structure of rotational bands in the high-spin region of deformed nuclei . In this approximation the intrinsic wave function of the yrast line is obtained as stationary solution in the rotating frame. For each angular momentum the energies of the excited levels with the same spin are found as solutions of the random phase equations in a rotating basis. In the nucleus' 64Er extensive calculations show that one is thus able to understand a large number of experimental levels within a consistent microscopiç picture with a high degree' of accuracy. . Since the mean field. approach gives us only an intrinsic wave function, the calculation of transition amplitudes requires angular momentum and particle number projection. We observe an increasing level density at the Ferrai surface for increasing angular moments. This reduces the applicability of the ItPA at very high angular moments. It regtüres further investigation, if this fact also restricts the validity Qf the mean field approach in general for thedescription ofnuclei in the region of very high spins. References 1) R M. Lieder and H. Ryde, Adv. NucL Phys. 10 (1978) 1 2) I. Y. Lce et al., Phys. Rev. Leü. 37 (1976) 420; O. C. Kistner, A. W. Sunyar and E. der Mateosian, Phys . Rev. C17 (1978) 1417 3) P. Ring, R Beck and H. J. Mang, Z. Phys . 231 (1970) 10; B. Banerjee, K J. Mang and P. Ring, Nucl. Phys. A21S (1973) 366; A. L Goodman, Nucl. Phys. A266 (1976) 113; J. Fleckner, U. Mosel, K J. Mang and P. Ring, to be published . 4) A. Facades, K R SandhyarDevi, F. Graumer, K W. Schmid aad R Hilton, NucL Phys. A256 . (1976) 106
414
J. L. EGIDO et al.
~ R. R. Hilton, Contributed paper, Jülich Conf. Highly excited states m nuclei, Sept . 1975, vol. I, p. 33 ; I. Hamamoto, Nucl. Phys. AZ63 (1976) 315 ; Phys. Lett. 66B (1977) 222; E. R. Marshalek and A. L. Goodman, Nucl . Phys . A294 (1978) 92 ; F. Grnmmer, K. W. Schmid and A. Faesaler, Nuc1 Phys . A308 (1978) 77 ; S. Cwiok, J. Dudele and Z. Szymanski, Acta Phys. Pol. 139 (1978) 725 6) F. Grfimmer, K. W. Schmid and A. Faessler, Nucl . Phys . A306 (1978) 134 7) S. Islam, H. J. Mang and P. Ring, NucL Phys. A326 (1979) 161 8) B. L. Birbrair, Nucl. Phys. AZ57 (1976) 445 ; I. N. Mikhailov and D. Jensen, Phys . LetE . 72B (1978) 303 9) D. Zawisha, J. Speth and D. Pal, Nucl. Phys. A311 (1978) .445 10) S. Y. Chu, E. R Matshalek, P. Ring, J. Kramlinde and J. O. Rasmusaen, Phys. Rev. C12 (1975) 1017 11) J. L. Egido, H. J. Mang and P. Ring, Nucl . Phys . A to be publiahad 12) P, Schuck and S. Ethofer, Nucl . Phys . A212 (1973) 269; E . R. Marshalek, Phys . Left . 62B (1976) 5 13) A. Kamlah, Z. Phys . 216 (1968) 52 14) A. K. Kerman and S. É. Koonin, Ann. of Phys . 100 (1476) 332 15) E. R. Marshalek, Nucl. Phys . A275 (1977) 416 16) R. Beck, H. J. Mang and P. Ring, Z. Phys. 231 (1970) 26 ; P. Ring, H. J. Mang and B. Banerjee, Nucl . Phys. A225 (1974) 141 17) J. L. Egido, H. J. Mang aad P. Ring, Nucl . Phys . A334 {1980) 1 18) H. J. Mang, B. Samedi and P. Ring, Z. Phys. A279 (1976) 325' . 19) C. Gustafson, I. L. Lamm, B. Nilsaon and S. G. Nilsaon, Ark. Fys. 36 (1967) 613 20) A. Bohr and B. R Mottelson, Nuclear structure, vol. II (Benjamin, New. York, 1975); E. R. Marahalek, preprint 21) E. R. Marahalek, Nucl . Phys . A224 (1974) 221 ; 245 22) J. L. Egido, P. Ring and H. J. Maag, Phys. Lett. 77B (1978) 123
Nuclear Physics A399 (1980) 390-414; © North-Solla~rd Ptrbliahina Co., Mtaterdam Not to be reproduced by photoprlnt os microfilm without wrlttan p~mladom ßrom the pnblisha
EXCITED ROTATIONAL BANDS IN THE HIGH-SPIN REGION OF DEFORMED NUCLEI J. L. F.GIDO t, H. 3. MANG and P. RING
Physik-Department, Technische Unioersit6t Mistehen, D-8046 Gorching
Received 28 August 1979 (Revised 8 November 1979) Abetrad : Rotational bands of deformed nuclei in the rare-earth region in the vicinity of the yrast line are investigated within the framework of the random phase approximation based on a selfconsistent solution of the Hartlee-Fork-Bogolyubov equations in the rotating systexo . Fôr low angular momeuta one fords the well-knows . ß, S and y vibtational bands and non~collective two-quasiparticle bands . In the high-spin region additional aligned bands develop. They are dramatically lowered in energy by the Curious interaction. Several band crossings occur. The calculated spectra are in fair agreement with experimental measurements.
1. Introdac8on In recent years a large amount of experimental information has been accumulated in the high-spin domain of deformed nuclei [for a .review see ref. 1)]. Besides the yrast band one has also observed the so-called "second" band (or Stockholm band which causes backhanding in many nuclei and in addition a series of other excited rotational bands Z). For heavy nuclei most of the microscopic attempts to describe the states along the yrast line s) are based on the mean field approach with constraints, the sculled self~onsistent cranking model (SCC) based on Harlxee-Fock-Bogolyubov (HFB) wave functions, and eventually improved . by a particle number projection before variation 4). This model is at present the only computationally feasible method for a realistic microscopic description, which allows simultaneous changes of shape, pairing correlations and alignment processes with increasing angular moments. One should, however, keep in mind that the wave functions thus obtained are only intrinsic wave functions. They have only a small overlap with the exact many-body wave function in the laboratory system, which is obtained by a projection onto good angular momentum 6. . '). Several authors') raised serious doubts as to the validity of the cranking model in the neighborhood of level crossings, i.e. in the backhanding region. These doubts have their origin in the application of the cranking approach to highly unrealistic s Work supported by the BMFr (Bundesministerium für Forschung und Technologie) . 390
EXCTTED ROTATIONAL BANDS
391
models involving only a few particles. We do not agree completely with these arguments and think that the cranking model is an approximation which works only in systems with many particles. It must fail in few-body systems. Therefore we feel no final decision about the validity of this method has been reached and it goes beyond the scope ofthis paper to enter into details. Eventually we have to regard our results in the backhanding region with some care. For the calculation of the low-lying excited rotational bands several authors [refs, is. s, aa)] proposed to use the random phase approximation (RPA) based on the self-consistent cranking model This method is the small amplitude limit of the time-dependent counterpart, of the static mean field approach- in the SCC model It has several advantages : (i) F.or low angular momentum it goes over into the well-known RPA based on an axially symmetric deformed product wave function, which has been used with great success in the literature for the calculation of vibrational states in deformed nuclei [see ref. 9) and references given there] . These levels are the heads of the bands investigated in this paper. (ü) The self~onsistency of the underlying single-particle basis guarantees that the spurious solutions corresponding to the violation of rotational symmetry and gauge invariance decouple completely from the other éxcitations. (iü) The sculled rotation-vibrational coupling is fully taken into account because for each angular momentum a new basis is used for the calculation of the vibrational levels. (iv) Since the RPA is also valid for non-oolleetive excitations such as pure twoquasiparticle states, it allows a consistent microscopic description -of a very large class of rotational bands, namely those with intrinsic wave functions that can be represented as superpositions of two-quasiparticle states in the rotating basis. (v) The method is the starting point for a theory going beyond the mean field approach : By. a requantization of the corresponding normal modes a ground state (the RPA-boson vacuum) can be introduced which contains additional correlations. In this way one is able to improve the energies of the yrast levels 1 ~ by including four-quasiparticle admixtures to the optimized product wave function of the selfoonsistent cranking model Within this paper, however, we do not use this latter possibility, but stay within themean field approach and restrict ourselves to the calculation of excitation energies, i.e. we use the SCC wave functions for the description of the yrast line. The main reason for this restriction is. that we use effective residual interactions (the pairing plus, quadrupole force) which is adjusted for the SCC energies of the ground state and of the yrast line. A part of the additional ground-state correlations is therefore akeady taken into account by the. ~ective force parameter. A detailed discussion of the problems connected with the definition of a correlated RPA ground state as an intrinsic wave function in a theory with angular momentum projection will be . given in a forthcoming paper' 1).
392
J . L EGIDO et al.
After having discussed the advantages of the suggested approach we should not forget to mention the diffculties connected with the RPA in the region of high spins: (i) A technical difficulty is caused by the rather strong violation of axial symmetry in the high-spin region There is no longer a K-selection rule to reduce the size of RPA matrices . In the case of rare-earth nuclei the dimension of the matrices to be diagonalized is 103-104. Only for separable forces can this be done using a reasonable amount of computer time. (ü) It is well known that the RPA breaks down if one of the frequencies goes to zero or turns imaginary, as for instance in the case of a phase transition In the highspin region such cases can occur. Examples are the regions of level crossings as in the backbending phenomenon, or at the breakdown of the pairing correlations at very high angular moments. In principle one knows well how to deal with this problem : One has to consider the ItPA as the lowest order of a suitable boson expansion of the hamiltonian. Higher-order terms in this expansion correct for the violation ofthe Pauli principle in the RPA, when one of the excitation energies goes to zero. They can be taken into account for instance in a self~onsistent version of the RPA 12). In the calculations presented in this paper it never happened that the RPA has imaginary eigenvalues, and we therefore stayed within the simple RPA approach. For some angular moments, however, we found very low-lying excitations. We therefore have to be aware that the results obtained at such I-values are less reliable. Investigations going beyond the quadratic order within the selfconsistent RPA approach are in progress . The method presented in this paper deviates. from the method proposed in ref. 2a) in so far as we do not take into account the angular momentum carried by the bosons for the calculation of excitation energies. A more detailed discussion of this point will be given in ref. li ), . In sect 2 of this paper we derive the RPA in the rotating basis as the smallamplitude limit of a time-dependent Hartree-Bogolyubov theory with variation after approximate projection onto good angular momentum . The simplifications resulting from the usç of separable forces are discussed in sect 3. In sect 4 we present calculations for the nucleus ie4Er and compare them with experimental data . The results are summarized in sect 5. 2. Time-dependent projected HsrIreo-Fock (TDPHF~ theory The projected Hartree-Fock (HF) or Hartree-Fock-Bogolyubov (HFB) theory has turned out to be a very powerful tool for the.description of static problems in nuclear physics. It is based on the fact that a large part of the many-body correlations in the nucleus can be taken into account by a simple symmetry-violating product wave function ~~~. Ifit is properly chosen, it can give rather reliable information on ground-state energies, r.m.s. radii or . deformation parameters. The quality of the approximation increases with the degree of symmetry violation The wave functions
IXCITED ROTATIONAL BANDS
393
themselves, however, are only a very poor approximation to the exact eigenstates of the many-body hamiltonian A. They have. to be understood as wave functions in a kind of intrinsic coordinate system . Only after projection onto eigenstates of the symmetry operator does one obtain a reliable approximation to the exact wave function in the laboratory frame: I~> = P'I~>~
( 1)
For simplicity we investigate in the following only one-dimensional rotations around a fixed axis. PI is the projection onto eigenstates of the angular momentum J.with respect to this axis. In realistic applications one has to deal with three-dimensional rotations and gauge transformations corresponding to the violation ofparticle number in superfluid systems. We will come back later on to these more general problems. The operator P' has the form 1 ax -nd~. Pi e`~U_ 2n ,~0 and the intrinsic wave function ~~) is determined by a variation of thé projected energy . For large symmetry violations the overlap function ~(w) _ <~I~~I~>
(3)
is sharply peaked at ~P = 0, and the overlap functions, for a general few-particle operator d can be expanded in derivations of the norm n(~p) [ref. i s)] Q(~P)
=
Cao+al
~ . .~ n(~P). i â~ +
(5)
For heavy systems a(~p) has â shape similar to n(~P) and one can stop this expansion in fast order. One then .fords for the projected expectation value of d
with eJ = J-<~~JI~> . A variation of the projected enemy obtained in this way with respect to the intrinsic wave function ~~~ yields the equations of the self-consistent cranking model with `~
<~I(aJlZl~>
394
J. L. EGIDO et al.
and the subsidiary condition After this brief reaninder of static projected HFB theory we now turn to a timedependent description. We start out from the time-dependent wave function . (10) where I~(t)i is a time-dependent symmetry violating product wave functions and the projector .P' guarantees that the wave packet I ~P(t)~ in the laboratory frame has the proper angular momentum. In analogy to the usual dérivation of the TDHF equations from the variation of a classical action 14) we now determine the time dependence of the intrinsic wave function I~(t)i ûom the variational principle or
The exact TDPI-1F equations resulting from this variational principle will be discussed elsewhere. Within this paper we use two further .approximations : (i) We restrict ourselves in the intrinsic wave functions I~(t)i to harmonic vibrations with small amplitudes around a static time-independent solution I~oi of eq. (12). This means I~oi is determined from the variation principle and I~(t)i has the form I~(t)>
= eAcr>I~o>,
(14)
with an antihermitian single-particle operator A(t) e is a small parameter and B+ can be expressed by quasiparticle operators respect to I~ o ~ (akl~o) = 0) B+ _ ~
k
Xktak ai - Yktaiak.
ak
with (16)
Inserting eqs. (14}-(1~ into eq. (12) and varying with respect to the amplitudes X,~ and Y~ we obtain a set of projected 1tPA equations for the eigenmodes B~; of the system
EXCITED ROTATIONAL BANDS
39 5
We have to emphasize that I~ho) is a self~nsistent solution of eq. (13) and therefore depends on .the angular momentum . For each value of 1 we have a new quasiparticle basis âw a; . (ü) Since eqs. (1~ are still rather difficult to solve for realistic nuclei, we now introduce a second approximation for the projection. In analogy to eq. (6) we restrict ourselves to strông symmetry violations and stop after first order in the expansion (5).. Using the operator iSt -A instead of Û in eq . (6) we obtain from eq. (11) (lg) <~I(4~21~) Representing again I~(t)) in the form (14), we obtain from (13) for the statiç solution I~o) the self-consistent cranking approach of eq. (~ for the wave function I~o) with the cranking frequency w determined from the subsidiary condition <~olJl~o) = I.
(20)
In a second step we vary eq. (18) with respect to the amplitudes X~, and Ykl in the wave packet ~(t) of eq. (14). In the lowest order in the smallparameter e we now obtain instead of eq. (1~ the RPA equations in the rotating frame ~~OICak}al , ~ vB v
-
C~ ~, By ]]I~0) - ~,
(21)
The same equations have been dérived earlier is) by boson expansion techniques in the rotating basis. The determination of the cranking frequency co by condition (20), however; seems to be somewhat arbitrary in this case. One of the advantages of the Kamlah approximate projection technique is that it yields this condition automatically. In the case of three-dimensional rotations we have to make further assumptions: (ink We restrict ourselves towave functions I~(t)j whichstay close to axial symmetry for all times, ie. the width dK of essentially admixed K-values in the wave function I~(t)) is small compared to I: <~I(d~s)21~) ~ I(I + 1).
(22)
We then again obtain the cranking model with respect to the x-axis and only the subsidiary condition (20) has to be changed into (23) The derivation follows closely the time-independent case [for details see refs. i 3, is)] . Well-deformed nuclei in the middle of the rare=earth region are axially symmetric in the ground state. We can therefore neglect <~olJz too) to a good apprôximation.
396
J. L. EGIDO et al. T~ 1
Quantum numbers of the single-particle states and their energies . in MeV Protons
Neutrons
BtatC
enel'gj~
dg9/z lg .,/z 2d,/2 lhlln 2d3/z 3s 1/2 lh 9/3
-4.9811 =0.6741 0.0000 0.9513 2 .4119 2.6965 6.1795
~
statG
energf
~11/2 ~~/z 189/z 1113/2 3P3/z ~s/z 3P1/z 2gs/z
-5.9548 -1 .5730 -0.7865 0.4400 1.4606 1.9100 2.9212 6.5072
For increasing angular moments the axial symmetry is sometimes lost. For instance, we fmd m 164Er (see table 3) y > 1Q° for I z 24. Nevertheless the condition (22) is always satisfied with an accuracy of a few percent for the yrast band and for the excited bands with K _ 0. The only exception are bands with K $ 0 for smâll Ivalues, for instance the y-band with K = 2. In this case the K quantum number is nearly a good quantum number, which has to be treated on a more quantummechatücal footing than we did in the time-dependent wave packet ~(t). In principle one can derive theRPA in the rotating frame also in a purely quantummechanical static picture by. the variational ansatz ~~P~ = PIBy ~ItPA),
(24)
where. ~ItPA~ ~is a suitable vacuum to the boson operator B and is determined by variation after projection . For the details of this derivation see re£ t i~ In the case of rather pure K-values (dK ~ I) we again obtain the cranking condition (23) where one has to replace ~~oi by Bq ~RPA). For the y-band we now have <.is ) = 4. That means in practice thatwe have to calculate the y-band at the angular momentum I with a slightly modified cranking frequency. The corresponding correction in the energy is of no importance for I-values larger than 8. 3. N~erkal solation of the RPA egaations The Coriolis term a~?s vïolates axial symmétry and time reversal For heavy nuclei the size of the RPA matrix is extremely large. A truncation of this matrix does not seem to be reasonable, since in such a case .the spurious solutions would no longer be .separate from the other normal modes. One is therefore restricted to the use of simple separable forces . The pairing plus quadrupole force has proven to be a very reasonable approximation to a realistic residual interaction to describe the interplay bet~öveen single.particle structures and collective quadrupole and pairing degrees of
EXCTTED ROTATIONAL BANDS
397
freedom. The basic blacks of the oonesponding hamiltonian aré a spherical singleparticle field, AO = L, EkCk k
Cb
(~)
the five quadrupole operators,, pw = ~
and the two pair operators,;
P + _ ~, Ck C~+ ,
P- _ ~
I~c k =
P+,
(2~
which describe the creation or the annihilation of a Cooper pair. The oonesponding hamiltonian is of the form
For calculations in the rotational frame the symmetry operation etc= has turned out to be very important We therefore introduce the symmetrized quadrupole operators Qo = rz~z~, . Qf~=,~(rZpz~frZ~z_~ (h = 1, 2),
which have definite signature
etx1~Q}~e~-txlx
= tQtw eif~txii e-if[.t,~ = Qo~ ~/o
We then get far the hamiltonian
2
(3O)
0 11 ~ -~ ~ ~ BP , P P
where we have introduced the seven operators Isv =
~Qw
15t = ,~~t,
(31)
and p runs over the values -2, . . ., +2 and +, -. Introducing a general 13ögolyubov transformation with thé quasiparticle operators ak+
_~ + V,kci , i ,UuFi .
(32)
398
J. L. EGmO et al.
these operators. can be expressed as Rnih
UP = DP
ßP l
_- ~ DP~ak ah k!
+ ßP l +ßPO,
ßPU
- ~,,(DPr~iak al ±DPkta~ak)~ k
(33)
Dp is the expectation value of the operator 1SP in the IiFB vacuum ~ ) to the ciuasiparticles ak : Dô = ~~ r2Y2o)~
Di =
V `~~~Y22)~
(34)
The symmetry with respect to the operation e;~= requires D_0 1 =D_0 2 =0,
(3~
and D° 1 vanishes automatically because of the antihermiticity of ~i+l . Inserting eq. (33) into the hamiltonian without normal ordering yields where ~°° is a simple constant : ~Po
= ~Ho) - i ~ i
P
_ ~Ho)-i~
(3~
.~1 is a bilinear form in the operators a, a+
with
Hl1 - HÔl ~ ~ ~ßP)DP1' P
H2o = Hôo - ~ `ßP)DPO~ P
where Ho, Hô1 and Hô° are defined in analogy to eq. (33) for the operator A° in a+a+ eq. (25) and ~2 is a bilinear form in the pair operators as and ~2 -
2 ~ ßP0+ßP0 P
_ -2 ~ (aktat.t k<1 k, «,
~~fDPO ~ 02 . alak) L .~ j D20+ / (DP . P. \ P /k!
a~.ak, 20 DP ~~~~ Ca+a+~ . k !
(39)
~' finally contains terms of the form a +aa + a+, a +aa + a which are of higher order in a perturbative boson expansion 21) . A normal ordering of these terms would
EXCITED ROTATIONAL BANDS
39 9
certainly give contributions to the RPA matrix. It is, however, within the framework of the pairing plus quadrupole model to neglect them. We now come back to the variational problem in the rotating frame as discussed in sect. 2. In this case we have to replace 8° in eq. (24) by A° -coJ~. Neglecting the exchange terms which come from the operator ~a in the variational equation (~ we end up with the well-known HB equations in the rotating frame. Their solution gives a diagonal operator ~ 1 ~i = ~ Ekak aw k
where Ek are the quasiparticle energies in the rotating frame. The vacuum defined by these operators is the intrinsic wave function of the yrast level at the corresponding angular momentum. Its energy is given by
with the exchaangé contributions
<~~i = -i~ ~ ~Dp2,°,Iz p kl
z
= -~ ~ ~dQa dQai - z~ ~ ~ dP~ dPti p=-2 s=t
(42)
which are determined by the fluctuations of the collective operators Q~ and P~ : These aretaken into account in the calculation ofthe energy, but not in thevariational equation for the determination of the intrinsic wave function. In the next step one has to solve the 1tPA equations (21). Here the terms ~' are neglected. One then ends up with a separable RPA problem
(43)
for the determination of the amplitudes X~ and Y,~. Introducing the matrix elements _
the linear eigenvalue problem (43~ whose dimension is given by the number of pairs kl (k ~ I), can be reduced to a homogeneous system of equations non-linear in éa~, whose dimension is equal to the number of collective operators 1Sp:
J. L. EGIDO et al.
400
with
DozDoz " w ~~-Ek-Ft
DzoDzo " Sl+Ek+Et)
It has non-trivial solutions tD only for discrete values Sa~, where the corresponding determinant vanishes. They are determined numerically. The symmetry e'~= reduces this 7 x 7 determinant to a 5 x 5 determinant containing the operators ¢o, ~+~, ~+ z~ p+ and ~- with Positive signature and a 2 x 2 determinant containing the operators ¢ _ .i and Q _ z with negative signature. Using the RPA equations (43) we can express the amplitudes X~ and Y,~ by the matrix elements v, Xr
_
- ~ Doz "t P
Pki
r P
Slr -Ek-Ei '
~
~rr -_
~
P
Dzo "tr Pkt
P
Slr+Ek+Ei'
(4~
and obtain in this way the norm of the elements tp. They measure the excitation probability of the one-boson state [see eq. (24)] Iv> = Bv I~A> by the collective operator ßv . We thus get for the quadrupole matrix elements ~RPAIQ~,Iv~ = X -~t~,
p = -2, . . ., +2,
(49)
ând for the pair transfer matrix elements
164Er Because of the large amount of experimental data gathered for the nucleus [ref. z)] we used this nucleus to give a unified microscopic description of the different bands within the theory discussed so far. In this paper we concentrate on states with positive parity, because the pairing plus quadrupole interaction (2~ does not introduce oorrelations between two quasiparticle states with negative parity {at least as long as we neglect the exchange terms as we did in sect. 3).. On the other hand, the negative parity bands can be described to a good approximation within_ the general-. ized single-particle picture, as has been shown in ref ~1 ').
40 1
EXCITED ROTATIONAL BANDS 4 .1 . FORCE PARAMETERS AND THE YRAST LINE
As discussed in sect. 2, we used the self~ons>stent cranx~ng model for the description of the yrast line. The wave functions ~~~ are determined by the variation of the energy (37) with the constraints on an angular momentum <1x>, proton number . <1Gpi and neutron number
j(I+1)-< =i,
I <~pi = Z~
<~ni
= A - Z.
(51)
The energy is. obtained as expeçtation value of the hamiltonian (27) with respect to these wave functions. It contains direct terms (37) and exchange terms (42). The single-particle configuration space was the same as used earlier 18). Single-- . particle' energies are given in table 1. The core consists of 40 protons and 70 neutrons, and a moment of inertia of the core was introduced as .~~ = 6 MeV- t . In order to reproduce the expérimental data as far as possible, the levels of the yrast line (see fig. 9) were used to adjust the strengths constants of the quadrupole-quadrupole force (~ and of the pairing force (G) t (see table 2). T~s~g 2 Strength constants of the pairing plus gnadrupole interaction X~ = 0.038 Xm = 0.038 Xvo = 0.100
G~ = 0.200 G, + , + = 0 .180 G,_ n _ = 0.157
The quadrupole strength parameters g (m MeV) haÎe3to be multiplied by dimensionless matrix elements hlce (n ~I(~/b~ Yzo In'l'j~m') where b r (41 .454/film) ni the oscillator length, G. t, t is the pairing force constant (in MeV) acting between neutron atabea with positive (negative) parity and G, +,_ _ (G.+ .+G.- .-)
The force constants determined in this way differ slightly from the parameters of ref. te). There were two reasons for this readjustment : (i) we wanted to use (as far as, possible) the spherical single-particle energies given in ref. t~ (only the ni b. level was shifted by 440 keV) ; (ü? we . also wanted to use a separable pairing interaction for the neutrons, i.e. Go-o+ _ (Gn+o+Ga-o-)~ .in order to simplify the solution of the RPA equation. Since the quadrupole force is no"t completely separable [Q~ ~ (Q~Q~~], and since the pairing strength for protons and neutrons is diûerent, we had in fact to generalize the dispersion relation (45) a little bit and to search for the zeros of a 10 x 10 det~±?inAnt for the states with positive signature (¢o, ~+v ~+z~ ~+ and ~for protons and neutrons) and of a 4 x 4 determinant for the states with negative signature (Q ~ 1 , ~1 : z for .~, protons and neutrons). . f For this adjustment we took into account the exchange terms because they were easy to calculate and improved the numerical agreement slightly .
. L. F.GIDO et al.
402J .
The moment of inertia of the inert, çore was used to modify the expectation value of the angular momentum ~Js% _ ~~~Js~~% +~core
and the energy where the angular frequency a~ is given by co = dE~/d(Jx ~.
(54)
In this way the inner shells N. 5 3 for protons and N 5 4 for neutrons are taken into account for the calculation of HFB expectation values. We neglect the interaction matrix elements between the valence shells and the côre shells in the RPA matrix and do not therefore allow for vibrational excitations of this core. Table 3 shows properties of the yrast line. lß column two and three we give the experimental spectrum and the calculated energies (53). Between I = 14 and I = 16 one observes back-bending. It is connected with a change in the intrinsic structure of the yrast line caused by the crossing of the ground-state band, which forms the yrast band up to I = 14 and an aligned band of two i~ neutrons, which forms the T~siB 3 grast line is 16aEr for the different Ivalues Properties of the
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
$e~ ~
~
0.000 0.091 0.299 0.614 1 .024 1 .518 2 .082 2 .70~2 3 .263 3 .768 4.345 4.999 5.728
0.000 0.106 0:345 0.701 1 .149 1 .664 2.215 2 .769 3 .317 3 .876 4.453 5.073 5.753 6.489 7277 8.105
°~ 0.0000 0.0759 0.1328 0.1798 0.2151 02388 . 0.2514 0.2528 0.2475 0.2434 0.2636 0.2921 0.3185 0.3417 0.3616 0.3842
dp
do
0.728 0.719 0.701 0.676 0.648 0.621 0.600 0.592 0.992 0.992 0.560 0.501 0.423 0.320 0.169 0.015
0.957 0.945 0.918 0.875 0.817 0.751 0.689 0.643 0.603 0.544 0.486 0.437 0.381 0.312 0214 0.031
~
ß 0.300 0.300 0.301 0.303 0.305 0.307 0.309 0.308 0.303 0.295 . 0.291 0.287 0.287 0.285 0.282 0.279
Y 0.00 0.20 0.76 1 .59 2 .55 3 .44 4.11 4.76 5.78 7.33 8 .62 9.40 10.05 10.65 11 .26 11 .57
The theoretical enegies F~ (Me~,are calculated by eq . (53) . They include the exchange terms (42). cv (Mew is the cranking frequency as defined in eq. (8). dp aad .d o (Mew are average gap parameters for protons and neutrons . The deformation parameters ß and y (degrees) are obtained from the expxtation values of ¢o and ¢z in eq. (28) : ß ~ (<¢o~~+<¢s)~r~2, tgy s<¢ s ~/<¢ oi~ ß ie normalized to 0.3 at I=O. ~ O . C. I~isfner et al. ~.
,EXCITED ROTATIONAL BANDS .
403
2 .0 1 :MéV) t5
Q5
0 Fig. 1 . (a) The spectrum of two~uasiparticle states with positive signature above the yrast line (y) in the nucleus 1s4Er as a function of the angular momentum I. (b) The spectrum of RPA frequencies with pôsitive sig(18tULC In 164 ~ a function of angular moment~n I. The calculation has beg carried out only for integea I-values. To guide the eye the Corresponding points are connected by hill lines. This connection is in general carried out according to the energy. On3y in cases where the miaingis very. small have we connected states with the same structure . We thus obtain sharp levd crossings, which would not occur in a tabulation with continuous I-vah~es.
J. L. EGII~O et al.
great band for I > 14. Only at very high spins at I ~ 30 a second alignment of two h,~ protons 1 takes place (see fig. 1aß The gap parameters for protons and neutrons show a continuous antipairing effect with increasing angular momentum . Although the neutron gap is larger for I = 0, it drops more rapidly in the region between I = 12 and I = 18 becausé of .the alignment of the i~. neutrons.. The pairing correlations vanish in both cases at I - 30. Thé deformation parameter ß does not change very much ovér the entire region of angular momenta under consideration ; it shows only a slight stretching effect for I .5 14 and afterwards a slight antistretching. The deförmation parameter y increases .continuously up to a value of 11° at I = 30. Nevertheless the essential condition in the derivation of the cranking model
In the following we discuss the structure of the spectrum with positive signature . (etwx = +1) in the region above the yrast.line as obtained from the diagonalization of the RPA equations (21). Its main features are determined by the two-quasiparticle energies E~ in the rotating frame.. The lowest are shown in fig. la. At I = 0 one observes a large gap of ~ 1.5 MeV between the great level (y) and a number of twoquasiparticle states. Some of them are labeled by the latin indices. For increasing angular momentum they decrease roughly parallel to each other. The only exception , (a) is one pair of i,~ neutrons, which aligns along the rotational axis and is therefore drastically lowered in energy . Between I = 14 and 16 it crosses the ground-state band and forms afterwards the great levels. Between I = 12 and I = 18 the increment of angular momentum in th,e great line is mainly caused by the alignment process, i.e. the collective frequency.does not change very much in this region - it even goes back a little bit (see table 3). As a consequence, the two~uasiparticle energies stay roughly constant between I = 12 and I = 18, and one observes a kind of plateau in the general behav;our of many. two~uasiparticle energies. Only for I values larger than 20, where the collective frequency increasës again, the two-quasiparticle energies start to decrease furthermore. In this region we also observe a pair (c) of two aligned hy, ,protons, which cross the great band at I ~ 30. Fig. lb shows the lowest eigenvalues t2, of the ItI'A equations -in the rotating frame. Many. levels are very close to the uncorrelatted two-quasiparticle energies ; an example is thé pair of levels f which corresponds to two combinations ak+ a; and arai which are degenerate in energy at I = 0 and show only a very smà11 splitting for increasing angular. velocity. In fig. 6 we will find the corresponding combinations ax+a~ and arai with negative signature. ~.
.
EXCTfED ROTATIONAL BANDS
~ 405
On the other hand, there are a series of important differences between fig. la and fig. lb caused by the two-body cortelations : (i) We did not draw the spurious solutions dl d1PP and dRn for positive signature . which lie exactly at zero energy. Since the other wave functions are orthogonal to these states, there occur drastic differences in the wave functions of the levels in fig. la and fig. lb, even in cases where the behaviour of the energy looks rather similar. (ü) The RPA correlations produce collective states which are drastically lowered in energy. The most important example is the y-band (y) which has no analogon in fig. la. Fig. lb contains the members with even 1-values (ie. positive signature of this band). Its excitation energy above the yrast band is rather constant up to I ~ 16. In this region the . yrast levels change their character. The ground-state band has now a positive excitation energy above the yrast level The y-band stays roughly parallel to the ground-state band and therefore increases for I > 16 in the excitation energy above the yrast line: In this region it crosses other non-collective two-quasiparticle states. One observes a large mixing and the y-strength is split over some levels. (iü) In addition to the y-bands we find less pronounced collective bands : A proton pairing vibrational band (S) which at low spin values is essentially a superposition of two proton quasiparticle states (d and c in fig. la) and a ß-band (ß) which is less collective than the y-band. (iv) The aligned band (a) of two i~. neutrons is not very much affected by the RPA correlations. Only the crossing of the ground-state band is less pronounced in fig. lb than in fig. la. We have to emphasize that the indices y, ß, b and so on are only labels which characterize the structuue ofthese bands at low angular moments. For larger I-values there sometimes occur drastic mixtures and the character of a band may change completely. To get more detailed information about the structure of the wave functions we investigate in the following the collective strength of the operators ¢~ in eq. (28) and ~t in eq. (2~ for some of the bands in fig. lb. In figs . 2~ we show the quantities SY(ß). =
KRPAIOIv)I
E
z
I<~Alß l~> h
(5~
for the five operators 0 = ¢ o, .Q+1, Q+z~ p+ and P- ~~ Positive signature. They measure the degree to which the state v exhausts the sum rule of the operator 0, i.e. the overlap of the state Ivy with the normalized wave function O + IRPA). In the actual calculation the sum over v' in the denominator of eq. (5~ includes only the 801owest solutions of the RPA equations. There are certainly also contributions from higher states to this sum. In particular, we do not take into account the giant resonances . Nevertheless .the values of S,(0) calculated in this way allow us a .clear . interpretation of the collective structure of our wave functions.
Fig. 2. Zhe percentage of thecollective strength foithe operators ¢F and ~ t ddmed in eqa. (2~ and (28) exhausted by the y-baild of tsaEr ae a function of she angular momentum. Further details and a precise definition of the quantities Sr are given in the text.
Ob
Pairing - band (5)
a6 aa
- salP)proton
___sa~ P l~t~~,
0.2 0 ~.
.
-'
_______-
--~~__________________
x.
- Sa(P. )protm
x
16 Fig. 3. 1be strength funcdona of fig. 2 for the pairing v~brational bead (b) in's4Er .
407
EXCTTED ROTATIONAL BANDS
An example in which the basic structure of the wave function is well preserved up to large I-values is the so~alled y-band (fig. 2). At small angular moments S,~(~+z) . is close to unity; ie. in this region the y-band nearly exhausts all the low-lying ~z strength. With increasing angular momentum one observes more and more mixing, which results in a generally diminishing value of Sr(¢+Z). At I = 20 this band contains only 40 ~ ofthe low-lying ¢ Z strength. However, the Qz strength is dominant for all angular moments under consideration. We only observe a small dip at I = 6. This corresponds in fig. lb to the crossing with band (a) of two aligned neutrons. It also shows up in the curves for S~,(¢ +1 ) and Sy(P_)°~ut~. For l-values larger than 18 we find even more mixing with other bands, as it is expected from fig. lb . Fig. 3 shows the corresponding quantities for the so~alled pairing vibrational band S of fig. lb. For low angular moments it is essentially a mixture of two two-quasiparticle (proton) states with K = 0, namely the states (Z + [411])2 and (i - [523])Z. At high angular moments I > 20 it changes its structure and . the protons in the h,~ shell align along the rotational axis . This gives drastic lowering of the energy and an increasing of the
0
-
"
ne ~.6 0
ß - bsnd -
- ~ SßIQpI
\
0.4
.~ "'
4
/
.'"' 8
~ 12
16
~ \.
_ _ _ _ g~ lp~t) - Sß 10,2) ~ - ~ .-
20
24
)
_ 28
Fig. 4. The strength functions of fig. 2 for the ß_vt~rational bend (~ in 's4 Er.
408
J . L. EGIDO et al.
I7X~)
Fig . 5. flue difference (1s') in the expectation valuup of 1s .between levels in the aligned bands (a) and t3zu; yrast band (y) as a function of I : (a) for positive aigoatuit+e ; (b) for negative aignatuure.
EXCITED ROTATIONAL BANDS
The band (a) (aligned band) in fig. lb is a rather pure band of two neutrons in the i~. shell. At I = 0 it corresponds mainly to the state (~+[642 2 with K = 0. With increasing angular momentum one observes a dramatic alignment along the rotational axis. This can be seen in fig. Sa, where we show the difference in the expectation value of Jx between the yrast line ~0) and the band (a) (Jx)w - w~Jzw~-~~~Jx~~i "
Between I = 12 and I = 18 this band changes its character because of the crossing with the ground-state band, i.e. ~O~.1x~0) increases drastically and w~,xw~ decreases. At I = 20 the band (a) is more or less the original ground-state band, whose angular momentum is smaller than that ofthe aligned yrast level at this I-value. This explains the negative values for (?xl)ev in this region. 4.3 . EXCITED STATES WITH NEGATIVE SIGNATURE
So far we havé investigated only vibrations of positive signature, i .e. the density distribution of the classical wave packet ~~(t)i in eq. (14) is at all times symmetric with respect to a rotation of 180° around the x-axis
This means that the rotational x-axis at all .times remains a prinçipal axis of the intrinsic density distribution . This is no longer the case for the wobbling motions which involve vibrations of the principal axis of ~the density distribution around the x-axis of steady rotation 2°). This motion is characterized by oscillating values of <~(t)~Q-i~~(t)i or <~(t)~Q_2~~(t)>, i.e. negative signature and odd I values. As discussed in sect. ~3, the 1tPA equations for negative signature are determined in the pairing plus quadrupole model by the operators Q_ 1 and Q_ 2. In figs . 6a and b we show in analogy to figs. la and b the two-quasiparticle spectrum EZ ~ and the itPA frequences ~q ~ for negative signature. Again the basic structure is formed by the two-quasiparticle excitations; again we observe an increasing level density at the Fermi surface with increasing angular momentum . Thé spurious wobbling motion, which corresponds to a steady rotation around a somewhat rotated axis and which has the excitation energy of the cranking frequency w is not shown in fig. 6b. The most drastic difference between fig. 6a and fig. 6b is therefore the low-lying collective y vibration. For small angular momenta it has precisely the same energy as the y-band with positive signature. For higher I-values we observe only small deviations. In the region of I ~ 11 an aligned band (a) of two i.,~ neutrons with negative signature approaches this band and produces some mixing (see figs. 7 and 8). This aligned band would correspond to the occupation of the i,~ levels with mx = ~ and . ms =. i in the decoupled limit. Its energy is somewhat higher than
Fig. 6. (a) Zhe spectrum of two-quasiparticle states with negative signattae above the yraet line (y) in the nucleus issue as a function of the angular mome~uum I. (b) Zhe spechvm of 1tPA fregtltncies with negative signature in 1~Br as a function of I.
F.XCIT® ROTATIONAL BANDS
411
0.5
1 5 . 9 13 17 21 25 l Fig. 7. The strength function for the operators Qa as defined in the text for the y-band with negative signature in fig. 6b .
0.5
Fig. 8. The strength function for the operators QF as defined in the text for the aligned band (a) with negative signature in fig. 6b.
the completely aligned band with positive signature (mx = ~ and mx = ~ in the completely decoupled limit) which corresponds to the band (a) in figs. la and b. To give more information about the structure of the corresponding wave functions, we show the quantities S(Q _ 1 ) and S(Q _ Z ) of e4. (5~ for the y-band in fig. 7 and for the aligned band in fig. 8 and the angular momentum (Jx~)r for the band (a)!in fig. 6b. Besides the y-band and the aligned band we see in the spectrum of fig. 6b a series of rather pure two~uasiparticle hands. An example are the two f-bands. They coincide~+ery precisely with the corresponding bands of positive signature in fig. lb. One reason for this little mixing is certainly the. schematic pairing plus quadrupole force, which provides an interaction only the Q _ t and Q_ a matrix elements in this channel. With a more realistic interaction one would expect more configuration mixing. 4.4. COMI'ARLSON wITH EXPERIIVIENT
Some of the experimental data of Kistner ei al. s) are given in fig. 9 and compared with our theoretical results.
412
J . L . P.(3iD0 et al. ~s4
.f 4
Er
ie'
3
g" 8" rs" 5'R' â
r
ss
gs
rx4.PoG
y
ss
g. s.
neg pn~
y
Fig . 9. A comparison of experimental aad theoretical spectra for several rotational bands in'~Er. 11îe dashed line indicates the yrast levels.
Thè yrast band is indicated by a dashed line. It consists of the ground-state (g.s.) up to I = 14 and of the aligned band of two neutrons (s.s.) for I >_- 16. These levels aré calculated within .the statïc self-consistent cranking model, as discussed in subsect. 4.1. The other levels with positive parity are obtained from the ItPA calculation presented in sûbsects. 4.2 and 4.3. For the low-lying members of the y-band the correction 'of the energy dE discussed in sect. 2 was introduced t
dE = a~dl = co(~I(I+ 1)-4- I(I + 1)):
This correction is of course only for small I-values of importance. It guarantees, however, that the band head of the y-band- comes closer to its experimental value. In fig. 9 we also show results for a band with negative parity : It is calculated within r Por a detailed discussion of this correction see ref. ii).
EXCITED. ROTATIONAL BANDS
413
the generalized single-particle model by a blocked HFB calculation, which has been discussed in more detail in ruf. i') : The overall agreement of our theôretical .calculations with experimental results is excellent in particular in view of the fact that nnly~ a few fit parameters (force constants and a moment of inertia of. the. core) were used to adjust the yrast line: All other levels were obtained with the same residual interaction and the same configur~tion space. In detail one observes small differences between the experimental and the theoretical spectrum, as for instance minor shifts in the band heads of the y and the negative parity band, which depend sensitively on the underlying single-particle scheme, or too strong a staggeringinthey-band, which indicates too strong aCoriolis interaction. It remains to be investigated whether this is a general feature of cranking calculations. S. Conclaslon
The time-dependent Hartree-Fock-Bogolyubov theory after projection onto states with good angular momentum in the limit of vibrations with small amplitudes was . used to investigate the . structure of rotational bands in the high-spin region of deformed nuclei . In this approximation the intrinsic wave function of the yrast line is obtained as stationary solution in the rotating frame. For each angular momentum the energies of the excited levels with the same spin are found as solutions of the random phase equations in a rotating basis. In the nucleus' 64Er extensive calculations show that one is thus able to understand a large number of experimental levels within a consistent microscopiç picture with a high degree' of accuracy. . Since the mean field. approach gives us only an intrinsic wave function, the calculation of transition amplitudes requires angular momentum and particle number projection. We observe an increasing level density at the Ferrai surface for increasing angular moments. This reduces the applicability of the ItPA at very high angular moments. It regtüres further investigation, if this fact also restricts the validity Qf the mean field approach in general for thedescription ofnuclei in the region of very high spins. References 1) R M. Lieder and H. Ryde, Adv. NucL Phys. 10 (1978) 1 2) I. Y. Lce et al., Phys. Rev. Leü. 37 (1976) 420; O. C. Kistner, A. W. Sunyar and E. der Mateosian, Phys . Rev. C17 (1978) 1417 3) P. Ring, R Beck and H. J. Mang, Z. Phys . 231 (1970) 10; B. Banerjee, K J. Mang and P. Ring, Nucl. Phys. A21S (1973) 366; A. L Goodman, Nucl. Phys. A266 (1976) 113; J. Fleckner, U. Mosel, K J. Mang and P. Ring, to be published . 4) A. Facades, K R SandhyarDevi, F. Graumer, K W. Schmid aad R Hilton, NucL Phys. A256 . (1976) 106
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