Mean-field approximations for deformed odd-mass nuclei

L YSICS

uclear Physics A543 (1992) 469-494 North-Holland

tons %r de%rmed odd-mass nuclei Larsson and S. Pittel artol Research Institute, University of Delaware, Newark, DE 19716, USA Yker

R.J. van de Graaff Laboratory, University of Utrecht, P.O. Box 80000, 3508 TA Utrecht, The Netherlands Received 2 December 1991 As

ct: The Hartree-Bose-Fermi, Tamm-Dancoff and random-phase approximations are developed for deformed odd-mass nuclei described by the neutron-proton IBFM . Both axially symmetric and triaxial solutions are considered. The methods are applied to the negativeparity bands of 165 Ho, which are described by coupling an odd proton (hole) in the I h 11/2 orbit to the even-even core nucleus 166 Er. We also consider variations in the occupation number of the odd-proton orbit and find that there is a window of values for this parameter for which the axial symmetry of the core is destroyed by its interaction with the odd proton and triaxial solutions are obtained. 1 . Introductio

The interacting boson model (IBM) ') and the interacting boson fermion model (IBFM ) 2 ) have been very successful in describing a wide variety of collective properties exhibited by medium-mass and heavy nuclei 3 ) . These models are ideally suited for exploiting dynamical symmetries and for using effective hamiltonians that describe rotational, vi rational and transitional features in nuclei 4,5 ) . This, together with the small number of parameters in the hamiltonian- and in many cases relatively small matrices that are 5agonahzed, makes these models very appealing. However, even though the matrices are much smaller than those that arise in shell-model calculations, they can still be very large. This is particularly true in the version of the IBFM called IBFM2, in which the neutron and proton degrees of freedom are distinguished in a description of odd-mass collective nuclei . In this model, a full diagonalization of the hamiltonian is impossible without severe truncation of the basis. Furthermore, even if we could diagonalize the hamiltonian in the full basis, we would not be able to draw much understanding from the very complicated wave functions. Mean-field techniques provide an attractive way of truncating the full basis in a physically meaningful manner. can-field techniques have been shown to be useful for obtaining approximate solutions to the Schr6dinger equation in the IBM 1 6 ) and the IBFM 1 7,8), where no distinction between neutron and proton bosons is made. In these models, an approxi mation to the intrinsic ground state can be obtained using the Hartree-Bose (HB) or 0375-9474/92/$05 .00 @ 1992-Elsevier Science Publishers B.V. All rights reserved

47 0

R. Larssort et al. / Meant-field approximations

the artree-Bose-Fer i (HBF) approximation, respectively, and low-lying intrinsic states can be calculated using the Tamm-Dancoff approximation (TDA) and/or the random-phase approximation (RfA) . The TDA gives a very simple description of the low-lying intrinsic states in terms of one-particle-one-hole (1 pl h) excitations of the HB or HBF ground states, and is adequate in many cases. However, the TDA does not decouple spurious excitations (associated with the spontaneously broken symmetries) from physical excitations. Furthermore, it is less useful when two-particle-two-hole (2p2h) excitations admix appreciably into the intrinsic ground state. These shortcomings are avoided in the RPA, which allows excitations to be built on correlated intrinsic ground states that include p-h admixtures. Spurious states decouple and appear at zero excitation energy in the RPA. Also, the RPA matrix is closely related to the stability matrix, which provides important information as to whether the HB or HBF solution is a good approximation to the correlated intrinsic ground state of the system . These properties make the RPA very useful . All three many-body techniques have been used 7 in IBM 1 6 ) and the HBF method and the TDA have been used in IBFM 1 .8), in general with impressive success. In most cases, simplified hamiltonians with dynamical SU (3) symmetries have been used to test the methods. The energies calculated using mean-field techniques were typically within a few percent of their exact values. In this paper, we consider the extension of these mean-field many-body methods to the IBFM2, in which an odd-nucleon degree of freedom (either neutron or proton) is coupled to an even-even core built up from distinct proton and neutron bosons . This is needed if we wish to investigate excited states of odd-mass nuclei in which the proton and neutron degrees of freedom are not fully symmetric, an importait example being the M 1 scissors mode in deformed nuclei 9 ) . Obviously, it is essential in such cases to distinguish between protons and neutrons. Well deformed even-even nuclei are typically axially symmetric, and this will still be the case for odd-A systems. In most cases, coupling an extra fermion to the core will not destroy its axial symmetry. For this reason, we first develop the HBF method, TDA and RPA for axially symmetric deformed odd-mass systems. The relevant formalism is presented in sect. 3. For certain hamiltonians, two or more of the lowest quasi-particle states that arise in axial HBF cû%,alations may lie very close in energy. In such cases, solutions with different IC-values are almost degenerate and will mix when we relax the assumption of axial symmetry . For this reason, we extend the mean-field formalism to the case of triaxial symmetry . More specifically, we consider the coupling of the odd nucleon to a triaxial even-even core. Triaxiality of the core is induced by its interaction with the odd nucleon, and can occur even if the neighboring even-even nucleus is axially symmetric . The triaxial formalism is developed in sect . 4. The HBF method and TDA for the IBFM2 are straightforward extensions of their respective IBFM 1 versions. The RPA in non-scalar systems (i.e. systems in which the ground state has a non-zero angular momentum) has been investigated before using tensor couplings of states and excitation operators ") . Our approach is related to that

R. Larsson et al. / Mean-field approximations

47 1

one but is somewhat simpler since no angular momentum coupling coefficients enter in our deformed m-scheme formalism . Some properties of the RPA equations are summarized in sect . 5. In sect. 6, we report some schematic calculations for the negative-parity bands of 16s Ho. We assume that these bands can be described in terms of an odd-proton hole in the l h 11 /2 orbit coupled to the neighboring even-even core nucleus 166 Fr, and obtain their intrinsic energies using the formalism developed in sect . 3. In these calculations, we employ the axially symmetric mean-field formalism and arso discuss results for the stability matrix. We find that there is a window in the space of parameters of the hamiltonian that we use in which the axially symmetric HBF solutions are not stable. In sect. 7, we apply the triaxial formalism to the same model hamiltonian and show that it leads to stable solutions in the region in which the axial solutions did not. Finally, in sect . 8, we summarize the principal conclusions of this work . 2. The IBFM2 hamiltonian In the IBFM2, an odd-A nucleus is treated as a system of N,' proton bosons, N neutron bosons and a single fermion relative to an inert doubly magic core. The bosons and the fermion represent either particles or holes, depending on whether the active shells are less than or more than half filled, respectively . The bosons are restricted to 1 = 0 states (s-bosons) and 1 = 2 states (d-bosons). The corresponding boson creation operators are denoted by 7rt,n, t and v t,,,, for the proton and neutron bosons, respectively, where the indices denote the angular momentum and its projection. Similarly, the fermion creation operators are denoted by ask . The boson operators obey the usual commutation relations and similarly the fermion operators satisfy anticommutation relations . The IBFM2 hamiltonian is of the form H=H, +A,+V,,, +V,,f+V,f,

where Hp =

am,

Vn nßyb

Hf VPf

Çt Ptm,t P(tin,, + 4 E

E

ttßy8 mmßrn y rna

m mßrnyinj

Umß ngrn yina (P )lPtrn  l~ßtQ PÔtnj Pyrn 7 ,

Vn,'nQ,n y rna 7 tßrnßU7rn7 VSm, , rtm,t 7

= E Ejatk al k , jk

(tß.1J' rn,,rnßkk'

Wn,'Omßkk 1 (P)Pern,,Pßinaajkaj'k' ,

(2 .2)

and p = 7c, v. The matrix elements in (2.2) are properly symmetrized . This is the most general number-conserving two-body hamiltonian for a system with only one fermion.

472

R. Larsson et al. / Mean-field approximations

It is rotationally invariant (i.e. an angular momentum scalar) and also time-reversal invariant . The time-reversal invariance guarantees that for every state that emerges from the many-body calculations (HBF, TDA and RPA), there will be a degenerate state that is simply related to it by the time-reversal operation. 3. Mean-field equations - axial symmetry

For well-deformed nuclei it is natural to consider axially symmetric mean-field solutions . In the case of axial symmetry, the basis states all have good values of the angular momentum projection K on the symmetry axis, and therefore the number of variational parameters describing the intrinsic states of the system is greatly reduced. Furthermore, projection of states with definite angular momenta from intrinsic states with definite K-values is much easier . Thus, we first develop the mean-field formalism under the assumption of axial symmetry. 3.1 . THE AXIAL HBF EQUATIONS

The axial HBF approximation is based on the assumption that the ground band can be represented by a deformed intrinsic state Ig.s.;k) =

J(rno)N")I(r"o) "")Irk)

"ô),v"1-,kIO)

1

(3.1)

involving condensates of deformed K = 0 neutron and K = 0 proton bosons and a single deformed fermion with projection k. The collective bosons and fermions are linear combinations of the spherical boson and fermions introduced above, rt = ~ tm nm 1r

E R

v

t 1l(trnVL

Ft k

Y?~ka~k .

(3 .2)

These operators satisfy the same (anti) commutation relations as the spherical operators . The structure coefficients q,'o, rl :o and rl Jk are considered as variational parameters, chosen to minimize the expectation value of the hamiltonian for the trial state (3.1) . More specifically, we consider the variational condition 6 ( (g.s.;k 1HIg.s.;k) -

p=7r,v

AP((Tpo) N°~(rpo) w°) - af(rklrk)

(3 .3)

where the variations are carried out with respect to the parameters ~lno, J,"0* and n~k of the deformed annihilation operators. This yields the following set of coupled matrix

R. Larsson et al. / Mean-field approximations

473

equations Eh,':

-

ilâo

Â>t I

0,

ß

Q

hY~rlao - ~v flog

~hf/ 17f'k

(3 .4)

- Afr7fk,

where n

n

h~ta = e 4tß + z

+N v

h' ß

7S

7a

+ ~ WOOkk ' ( 70 17jk* ~%f'k,

VI OOd ;770r1S0

v

= E.t a~tQ + i z (N

ff~

~t7aß

E Uoooo

- 1)

7s

V600'?1rô1 lâo + f

hfl

.

f

- Ef (5ff'

( r) n70-'il aon

~t7bß (Ng - 1) E U0oo0 7

(v )

E W Okk r

*v

~ 70

v .6o

(v )~1lk1 ~'k,

+ E Np E Wö;kk (P )l tOIQO . p=Jt,v

(3 .5)

nfl

These equations are solved iteratively until convergence is achieved . In practice, we obtain solutions for all positive (or negative) k-values and then take the lowest solution, with projection k = K, as the HBF intrinsic ground state. Because of the time-reversal invariance of the hamiltonian, the solutions with projections fK are degenerate. We will denote the doubly degenerate ground-state solutions by 1g.s. ; K) and 1g.s.; -K) . The HBF equations of (3.4) yield in addition to the deformed bosons and fermions that define the lowest intrinsic state, also a set of deformed bosons and fermions orthogonal to them. We will denote the creation operators for these excited deformed bosons and fermions as Tso , r,s o and Tf They too can be expanded in terms of the spherical boson and fermion creation operators according to

k.

rsm

=

rsm

=

y~>ts

t

'I~trrlWrlrl , rt

vs t E fl"rrnv~rrn , t

rk f

klika;k ,

(3.6)

and likewise satisfy appropriate (anti)commutation relations. These excited boson and fermion operators will enter in the TDA and RPA descriptions of excited states to follow.

R. Lajsson et al. / Mean-field approxiinations

47 4 3.?. AXIAL T A

The TDA for odd-mass systems is based on the assumption that the lowest bands can be built up in terms of the lowest HBF solution (s) and one-particle-one-hole (1 pl h ) excitations built upon them. In the IBFM2, there are in general five different types of 1 p 1 h excitations with a given total projection K,

rfK TK Ig.s.; K),

rtpsK~Ti rpo lg .s.; +K) ,

(3.7)

with p = n, v . Here, the creation operators are the excited HBF bosons and fermions of (3.6), whereas the annihilation operators represent the correlated bosons and fermions of the HBF ground state of (3.1). The first term is a single-particle (fermion) excitation of the HBF solution with projection +K, and the second term denotes the collective (boson) excitations of the degenerate HBF ground-state solutions with projection +K and -K. Note that, in contrast with the single-particle excitation, for the collective excitations all degenerate HBF solutions have to be included. The HBF intrinsic state Ig.s.; K) does not couple to the excitations of Ig.s.; K) due to the usual HBF minimization condition that the terms H2o and Hoe of the quasiparticle hamiltonian are identically zero " ) . However, it can couple to the boson excitations of Ig.s .; -K) through the terms H3, and H, 3 of the quasiparticle hamiltonian, which are not necessarily zero '2 ). Thus, at the level of TDA, the HBF ground-state solutions can be modified by 1 p 1 h correlations. The most general TDA states with projection K can therefore be expressed as ITDA; K) _ 6K-,T Xh Ig.s .; K) + 6K.-T X-h Ig.s . ; -K) XXrhrk +

p=It,v

1 N

r

s

Xps.K-K rtps,K-Trpo

'P _FT .s.; s.K+K ps,K+K p nIg

-K)

Ig.s. ;K) (3 .8)

To obtain the energies and wave functions in TDA, we diagonalize the hamiltonian in she basis (3.7), including the HBF state (g .s .; ±K) if K = fK. The resulting eigenvalue equation can be written in the compact form AT XT = ET NT XT ,

(3 .9)

where 4T is the hamiltonian matrix in the basis mentioned above, the vector XT contains the amplitudes X in (3 .8), ET is the energy eigenvalue, and NT is a metric matrix from (KI K) that is equal to the identity matrix if the TDA basis is properly normalized and not overcomplete . The lowest ET will in general be lower than the BF energy due to the 1 p 1 h correlations that may be present in non-scalar TDA. The TDA gives a very simple description of low-lying excitations of the system. owever, it relies on the assumption that the only correlations beyond those of HBF are the 1 p 1 h correlations described above. Thus, it is unable to incorporate higher p-h correlations in the intrinsic ground state when they are important . Another limitation

R. Larsson et al. / Mean-field approximations

475

of the TDA is that spurious states (associated with the fact that the above meanfield solutions break rotational invariance) can mix with physical excitations . These limitations can be overcome by using the RPA to describe the low-lying excitations . 3.3. AXIAL RPA

The RPA permits the possibility that the intrinsic ground state of the system is not given exactly by the HBF solutions, but can also contain additional p-h correlations . For example, when acting on a state with 2p2h correlations, we can annihilate one of the particles in the excited pair and bring it down to its lowest single particle state, thus leaving only one particle above the Fermi level. To accommodate this, we must add to the excitation operators of (3.8) terms that take these de-excitations into consideration . To obtain excited intrinsic states with total projection K, we introduce two excitation operators OtK-K and QK+ t K , such that IRPA;K)

= Qt

K-K IVK)_

+ Pt

_IV_I) K+K

.

(3.l0)

By definition, the annihilation operators QK _K and SDK+T satisfy

(3-11) _ DK+K IV±K) - 0' The states jyr±K) represent the doubly degenerate RPA intrinsic ground states of the system, including possible p-h correlations . The excitation operators of the RPA include both single-particle excitation terms (which also appear in the TDA) and single-particle de-excitation terms (which do rot). The excitation operators can be written as f t _ .Q Y,2K-K 2K_k IV±K) S

K-K

-

.KrfKrfs,2K-K

s

+E s

+ Z Pt _ K+K

P=7t .v

-

+

Krsh~K - Yf _K._Krt K

s~ - K

(X

1 NP

s

( Xs,K-Kps,K-K FPO

Z (Yf-KXFK~s .-K s

1 P=7c,v

NP

s

(XP

s

,K-K .0POrPS.K - K) '

.

+ Yf -2K-K.-Zf K fs .-2K-K

F" Yp r K _K.OPo P-s ,s,K+K ps,K+K~~O -

(3 .12)

The forward-going excitations (X-terms) are the same as in (3.8) . The backwardgoing excitations (Y-terms) represent the new physics of the RPA.. Now that the HBF solutions are not present in the RPA basis. The effect of the l p l h correlations that arise in TDA are included implicitly in the backward-going terms. The excitation operators given above yield the most general RPA state IRPA; K). However, depending on the specific values of K and K, not all of the terms necessarily contribute . For example, since the bosons have angular momentum 1 = 0 or 1 = 2, the boson excitation operators O~tt, .:FK only contribute if IK T 11 < 2.

. Leanson et al. / Meanfield c,)proximations

76

sing the more compact notation IGI) --_ Ig.s. ; K), IG2) _= I9.s4 -K), S2; = SDK -K- and A equations from the variational equation Q, E 92K+-E, we can generate the «G= l gà2i , H,

Q~ l jGj) -

(ER + Eo) (G®I (Qi , Q; l l Gj )) ) = 0 .

(3-13)

The double commutator is defined as [Q, H,

j]=2([

2([D`,[H,Q;1 ] +[[Qà,Hl,Q;1),

(3.14)

and ER is the RPA excitation energy relative to the correlated intrinsic ground state. The variations in (3.13) are with respect to the parameters X* and Y* that define the RPA de-excitation operators 0, and 522 . The RPA equations arising from (3.13) can be written in the compact matrix form ) (X ) = (Nx 0 ) (X~ Y Bt D Y ER 0 -Nr (,

(3.15)

ere, A, X and Ny are the submatrices and subvectors of AT, XT and NT that arise when the BF ground band solution Ig .s . ; K) is omitted in the TDA basis. Also, D is the backward-going TINA matrix, B is the matrix obtained from coupling 2p2h excitations to the I-IBF ground band, the vector Y contains the backward-going amplitudes and Ny is the corresponding part of the metric matrix. The matrices A,D,Nx and Ny are all hermitian; hence, the supermatrices on both sides of (3.15) are hermitian. Both A and D are square matrices; however, in RPA for non-scalar systems they have in general different dimensions. For the same reason, B is not necessarily a square matrix. Explicit expressions for the various matrices that comprise the full RPA matrix are given in the appendix . 3.4. THE RPA METRIC MATRIX

The component Nx of the metric matrix is diagonal and has its origin in the terms 12 E 1XpK-KI2 , (g.s .;KI52K - 52K .s . ;K) = IXs,K Ig

j

+

P

S

SP

(3.16)

XP` l s,K +K l

If the excitation operators are chosen so as to produce normalized states (as they are in (3.12) ), N v is simply the una matrix. The component N}- gets contributions from both diagonal and non-diagonal terms. The non-vanishing matrix elz~"nents are 2 + E l ~,p 12 , (g.s.; tTI52tKTK 52KI-k lg.s. ; ffC) s,t2n -K,±K 1 s,±! -K,0 (g .s . ;

±KI Qt

KfK 52KTKlg

.s . ,;

K) -

Yf* K .~K Yf -K.±K

SP

(3.17)

R. Larsson e,, il. / Mean-field approximations

477

The metric supermatrix of (3.15) is its own inverse. Thus, if both sides of (3.15) are multiplied by this matrix, we arrive at the eigenvalue equation A

B

) (X ) =

(X)

(3.18) Y - NrB i -NrD Y E where we have used the fact that, with our definition of the excitation operators, Nx is the unit matrix.

(

3.5. STABILITY OF THE HBF SOLUTIONS

To assess whether the HBF solutions provide a good first approximation to the correlated ground band of the system, we need to derive a stability condition similar to the one that exists for scalar systems 11 ) . The most general variations of the HBF ground-band solution Ig.s. ; K) can be expressed in the form

°t

e 1g.s.; K) ,

(3.19)

where Ot is a sum of forward-going 1 p1 h excitations only:

Ot-

oK-K-=cl Kft KFK + E CfKI'tfK~K

K

SK

+E

p-n,v

We now evaluate S «g.s.; KleoHe

°t

1 E Cps,K-K_rtps,K-K rpo . Np

(3.20)

sK

Ig.s.;7) - (Es + Eo) (g.s .; Kleoe

ot

Ig.s.; K) = 0,

(3.21)

up to second order in the coefficients C and C*, where the variations are with respect to these same coefficients. This yields a matrix equation that can be written in block diagonal form. In each block, the equation has the form As Bs

(Bs Ds

v*

(C

_ Es ( C. C

(3.22)

The matrices As, Bs and Ds are, respectively, submatrices of the matrices A, B and D that appear in (3.15) . As is a TDA-like matrix generated by the states OK-K Jg.s.; K), Ds is a TDA-like matrix generated by the states Ot_ -K Ig.s.; K), and Bs is a matrix that results from coupling Ig.s.; K) to 2p2h excitations of the type OK_ --OK-K Ig.s . ; K) . The HBF ground band solution 1g.s.; K) is stable if all of the eigenvalues Es of (3 .22) are non-negative . Stability of the degenerate solution 1g.s. ; -K) then follows directly from time-reversal invariance . Mean-field equations - triaxial symmetry As was mentioned in the introduction, the axial symmetry of the deformed eveneven core may sometimes be broken by the addition of an odd nucleon. In such cases, it is natural to consider triaxial solutions.

. Larsson et al. / Mean-field approxünations

47

Induced triaxiality occurs when two or more of the lowest axial I-IBF solutions with different k-values are very close in energy. If we permit mixing of m-values in the boson condensate (through the boson-fermion interaction), we may end up with a state that is lower in energy than the lowest axial I-IBF state. The mixing of in-values in the condensate should be small if the even-even core system has an axially symmetric FIB ground state. 4.1 . THE TRIAXIAL HBF FORMALISM

A wave function I1j`n) has triaxial symmetry ' 3 ) if it is invariant (up to a phase) under a rotation through an angle 7r about the x-, y- and z-axes (R, (7r), i = x, y, z), and reflection through each of the principal planes (a;, i = x, y, z ) . Only three of these six symmetry operations are independent and at least one must come from either rotations or reflections . One set of operators that can be used to generate triaxial symmetry is R, (7r), Rz (7r) and 6, ., which have the following effects on a single-particle state 11jin) :

Rr(7r)Iljm)

= (-1)1-"' j 1 j,

e-~~~~~Il jm)

Rz(7r)Ilj ;n)

q,IIjin)

-m)

Together, these operators set the following conditions on 1, j, in and the coefficients cl,, in a triaxial wave function E,,. c, j, I1jm) : (i) all terms should have the same parity, (ii) the m-values of the different terms should differ by an e".yen integer, and (iii) Cl,,_ , = f (-1),nc,,,. If these conditions are met, then the wave function is  triaxially symmetric . For (ii) and (iii) to hold simultaneously, states with + in and -in must differ by an even integer, which can only be realized for boson-like states. A fermion-like state can only be invariant under at most two of the three symmetry operations required for triaxiality . We will choose invariance under R--(7r) and the parity operator (which is equal to Rv(ar)Q,.) . Although the fermion states only satisfy two of the three conditions that define triaxiality, we will refer to these states as `triaxial' in the remainder of the text. The triaxial I-IBF approximation is based on the assumption that the lowest (doubly degenerate) intrinsic states can be expressed as IG I ) = Ig .s .;+j = Ice )

with

1 [N,!N,! ] -' Z(I~+)~~R(1;,+)Nvlf+l~), [N !j

r

' /2

+ _ 11ÔO SP + rl2ôd t

f+

jk

37~k a~~ ,

(r+)""n (r1,+)N.,rft Io),

+ '722

(d P2

k - ; even,

+

d n,_2)

,

(4.2)

R. Larsson et al. / Mean-field approximations

rt = 1:

47 9

k - z odd,

tlikaI ,

jk

(4.3)

where we have used the notation sP = pâo and dP, - P2 , . The indices + and refer to states with eigenvalues rz = f 1 of the operator R, (n) (for fermions, r_f exp (-i7r (k - i ) ) ). We note that there are other triaxial boson operators with good r=, but since our interest here is in induced triaxiality from an initially axial boson core, it is clear that the triaxial boson of the condensate must preserve the remnants of the deformed K = 0 axial boson. The coefficients that define the collective bosons and fermions of (4.2) can be obtained variationally by iterative matrix diagonalization, in much the same way as in sect . 3.1 . The HBF condition yields the bosons and fermions of the lowest triaxial intrinsic states. For the TDA and RPA analysis of excited states in the next section, we introduce the appropriate orthogonal boson and fermion operators, which can be expressed as t os t Ps t rps+ = fÎJpsp+~20d 00

Ps + p 2z

P3+ -

z (dt Z - drt.- z ) ,

t = rrP4_

2 (d t - dP. -, ) ,

rPt5_

~!2

rs+ rfs-

Pl

t t 1(dpz + dP,-~) ,

P'- 1

fs t i'l jkalk

k -

i even,

)7jkajk

k -

i odd .

jk _

s = 1, 2,

jk

(4.4)

4.2. TRIAXIAI. TDA AND RPA

The doubly degenerate HBF intrinsic states of sect . 4.1 were defined to have good values of the total r- = ±1 . As a result of time-reversal invariance, all excited bands will likewise be doubly degenerate, with one band having r_ = + 1 and the other rz = -1 . A triaxial RPA state with r-- = + 1 can be written as (4.5)

IRPA; +) = Q+ I V+) + Qt I w-) ,

where I y/f) denotes the correlated ground state with r_ = f l . The excitation operators in (4.5) are given by .Q+ _

s

(XS+rs+r+ 1

NP

YS+ . +r+rs+ - Yf

E (X+rs+FP+ s

.-rtrs-

/ - YS+FtF + ) ,

480

R. Larsson et al. / Mean-field approximations (Y'l.,-ir,t Fs+ + Yf_',i1,1 r,,-)

Prs- r.+ - YP r+rps_) .

(4.6)

As before, the annihilation operators satisfy Q+li/f)

= S2_1 yit) = 0,

(4.7)

As in the axial case, it is sometimes useful (e.g. in sect. 5) to use the notation S2, - 52+, V/+ and V/2 V/- . 2 =- Q- , V/1 The TDA equations are obtained as in sect . 3.2 by setting all the Y-coefficients to zero in (4 .6) and by including the I-IBF states lg.s.; +) and lg .s . ; -) in the basis. The RPA equations are calculated in the same way as in sect. 3.3, but now using the triaxial I-IBF solutions and excitation operators defined above. Both the TDA and RPA eigenvalue equations have the same general structure as in the axial case . The stability condition for the triaxial I-IBF solution lg.s .; +) is ot

a «g.s.; + leoHe

ig .s.; +) - (Es + Eo) (g.s.; + le oe

ot

lg.s.; +)) = 0,

(4.8)

with

®r =

Cf rft rf+ + sr-

sr,

sr_ r fsr_ 'f+

CP_rsr_rp+ .

(4.9)

Carrying out the variations in (4.8) through second order leads to the stability matrix for lg.s .; +) . 5.

ro erties of the

A

The excitation operators of sects. 3.3 and 4.2 have the same general structure . The RPA equations will therefore have a number of properties that depend only on the structure of the operators. The various properties that we consider below are very similar to those found for scalar RPA " ) . The differences come mainly from two sources : (i) in scalar RPA we have D = A* in the RPA-matrix - no such relation exists in non-scalar RPA; (ii) in non-scalar RPA the metric matrix may have off-diagonal matrix elements. 5.1 . STABILITY AND -.-,IPA ENERGIES It is convenient to write (3.15) in the more compact form Rv = ER NR V,

(5.1)

R. Larsson et al. / Mean-field approximations

48 1

or, if we collect all eigenvectors v in a matrix V and the eigenvalues ER in a diagonal matrix .6, RV = NRVE .

(5 .2)

Similarly, we can write the stability condition (3.22) as Sw = Esw .

(5 .3)

Note that the stability matrix S is not in general related in a simple way to the matrix R. This is contrary to scalar RPA where S = R. The lack of such a simple connection for non-scalar systems has some interesting consequences . Perhaps most importantly, complex energy solutions of (5.1) can arise even if the stability matrix is positive definite. It is easy to show, however, that a complex energy in (5.1) implies a vanishing norm, v t NRv = 0, so that such solutions represent unphysical excitations. 5.2. ORTHOGONALITY, NORMALIZATION AND CLOSURE These properties have been in part investigated before 10 ), so only a few further comments are needed . Any pair of solutions, vp and vq , of the RPA eigenvalue equation (5.1) satisfy the orthogonality relation (ERP - ERq )vgNRvp = 0 .

(5 .4)

Thus, if ERP i6 ERq then vgtNRvp = 0, which is the case for real and distinct eigenenergies. Complex energy states behave in a different way since ERp - ERp :A 0, making these states self-orthogonal . Complex conjugate solutions, ERp = ER q , are not orthogonal in general . A natural way to normalize the states vp is vp NRvp = sign (ERP ) , vp NRvq

= - 1,

if ERp real., if ERp = ERq and complex,

(5.5)

-+ VtNRV = 1J .

(5.7)

(5 .6)

or in matrix form vpN?v q = Npq

The matrix N, like the metric matrices, is its own inverse, _which means that Vt NR VN = I, 1 being the identity matrix. This shows that NR VN is the inverse of Vt and leads to the closure relation VNVt = NR .

(5 .8)

5.3. TRANSITIONS, SUM RULES AND SPURIOUS STATES There is an alternative way to derive the RPA metric matrix, that provides useful information on the properties of the RPA ground-state band . This in turn can be used

R. Larsson et al. / Mean-field appioxiinations

482

to derive an expression for the matrix elements of an arbitrary one-body operator ,M between the RPA ground state and excited states. Following ref. 14), we consider the scalar product (5.9) (y/ (P)Iy/(q,) (Y1'I'QPi-QgjIY1J) >, I [JQP>> 99JIIy/j) , ij

where y/, denote the degenerate RPA ground states, V/(P) is the pth excited RPA state and Qp, (i =1,2) are the excitation operators that create V ( P) . Expressing the commutators as a sum of one-body operators, one can show that the orthogonality condition for RPA solutions (y/ (p) I y/ (q)) = vp NRvq is satisfied if we take (Yl~I (~rirz~st - r t ~~nrn)Iy/j) = lârmajnêst

(5.10)

The indices m and n label the lowest HBF solutions, and s and t label excited HBF vectors . All operators in (5.10) represent one kind of particle (7r, v or f) . We now wish to evaluate the one-body matrix elements E, (Vi IM i I y/ (P )) . The nonzero contributions from .M, can be expressed in terms of the forward- and backwardgoing terms in (3.12) or (4.6), since these terms involve all possible excitations from the ground states to a state with the relevant K or r_ . Thus, we need to express Mi in this same basis. Clearly, the structure of M, and A42 will be the same as for Q, and S)2 , but with different coefficients: X --> M,Y, Y , My . Making use of the condition (5.10 ), we find that (y/,IM,SQpjIV/j) = MtNRvp, (5.11) i ij where the vector M contains the amplitudes Mx and My . We are now in a position to derive the following important sum rule satisfied by the matrix elements of .A4,: (+ll~I .MiI yl

jIy/(P))

(Pl) =

ERPNPq

~E(V/ (q J

'IMJIV/J)

= E(GjI[Mj,H,A4jjIGj) . ij

(5.12)

The matrix N is the norm defined in (5 .7) . To prove this sum rule, we separately consider the left- and right-hand sides of the equation . Using (5.11), we can express the right-hand side in matrix form as Pq

(v/,I .M,I1v/(P1)

I ERPNPq

z(VI(q)IM

jI VIj)

= MtNRVENVtNRM,

which can be rewritten, using (5.2) and (5 .8), as Mt NRVSNVtNRM = MtRVNT'tNRM = MtRNRNRM = MtRM .

(5.l3) (5.14)

Likewise, the right-hand side of (5 .12) can be readily expressed in matrix form as (G,I [M,, H, .Mj ]IGJ) = M t RM,

thereby confirming the sum rule.

(5.15)

R. Larsson et al. / Mean-field approximations

48 3

The sum rule (5 .12) can be used for any pairs of one-body operators . A particularly important application is to one-body operators that commute with the hamiltonian . For such operators, the left-hand side of (5.12) is identically zero. Thus, states VI(P) that have non-zero couplings to the ground states via such an operator must have ERp = 0. We can therefore identify spurious states (associated with spontaneous symmetry breaking by the mean-field solutions) as those that come out of RPA at zero excitation energy. 6. Axial calculations 6.1 . THE HAMILTONIAN

In this section, we apply the axial formalism of sect . 3 to the negative-parity bands in 165 Ho, which we describe as an odd-proton hole in the l h 11 /2 orbit coupled to an even-even 166Er IBM2 core. We also consider some simple variations in the parameters of the hamiltonian, so as to examine the structure of the mean-field solutions that may be expected in more general cases. The IBFM2 hamiltonian for 165 Ho can be expressed as H=HB+VBF,

where HB is the IBM2 hamiltonian of the 166 Er core and VBF is the interaction between the odd-proton hole and the IBM2 core. Note that since the odd proton is restricted to a single orbit it is not necessary to include its single-particle energy. The nucleus 166 Er is well deformed . For simplicity, we therefore assume a boson hamiltonian involving the SU (3) and SO (3) Casimir operators plus a Majorana 15 ) term, HB -

- aC2su(3) (n + v ) +

bC2SO(3) (7t + V ) +

CM7tv ,

(6 .2)

with C2SU(3)

(r

C2SO(3) (1c

+ V)

= 2(Q. + Qv) ' (Q. +

Qv) +

+ v ) = (L,, + Lu) ' (L., + Lv ) ,

4 (L T + Lv) ' (L., + Lv) ,

(2) t t - Sdn t t (2) (Snd M~w = (S~tdv ) ' Svd7c ) -2

E

(6.3)

(!) t j (/) ' (dndv ) (d ,du) (6 .4)

The parameter a is determined from the requirement that the experimental /3 and y band head energies are reproduced as well as possible by the relative eigenvalues of the (A, ,u) = QN, 0) and (2N - 4, 2) SU (3) representations. Similarly, the parameter b is determined from the moment of inertia of the ground-state band . The Majorana term acts on states that are not fully symmetric in the proton and neutron degrees of freedom . By choosing an appropriate value of c > 0, we can ensure that the states that are not fully symmetric are pushed up in energy in accord with their experimental positions. Reasonable values for the parameters of the IBM2 hamiltonian are a = 4.1 keV,

R. Larsson et al. / Mean-field approximations

484 E(MeV) 3.5

3.0

TDA

RPA

2+ 0+

3.339 3.289

2+ 0+

-- 3.339 3.289

1+

2.973

1+

2.972

2+ 1+ 0+

0.713 0.697 0.661

2+ 0+

0.713

0+

0.000

0+

0.000

1.0

0.661

0.5

0.0

Fig. 1 . Intrinsic energies from TDA and RPA calculations for 166Er . To the left of each intrinsic level is shown its KIT value. b = 13 keV and c = 175 keV . With this choice of c, the mixed-symmetry K = 1 + band head of 166 Er comes out at approximately 3 MeV. Exact diagonalization of the hamiltonian places the ß and y band heads at 0.713 and 0.791 MeV, respectively. The TDA and RPA intrinsic energies of '66 Er are shown in fig. 1 . Note that the intrinsic (3- and 7-energies are lower than the exact values . This reflects in part the fact that states of good angular momentum have not been projected from the intrinsic mean field states . It should be emphasized that better fits to the 166 Er spectrum could be achieved by permitting a breaking of the SU (3) symmetry ; however, for the qualitative purposes of this study, we do not feel that such improvements are warranted.

R. Larsson et al. / Mean-field approximations

485

The boson-fermion (BF) interaction VBF should consist mainly of two terms, a quadrupole-quadrupole interaction and an exchange interaction 2 ) . The quadrupolequadrupole BF interaction has its origin in the dominant quadrupole interaction between neutrons and protons 16) ; we thus assume that it only takes place between the odd-proton hole and the neutron bosons. The exchange interaction, which reflects the fact that the bosons are made up of fermion pairs, is a bit more subtle. Phenomenological IBFM2 studies ") have traditionally assumed that the origin of the BF exchange interaction is likewise the n-p quadrupole force. This leads naturally 18) to the three-body BF exchange interaction that has been used extensively. We do not agree, however, that this is the dominant component of the exchange interaction. As has been discussed in ref. '9 ), the strength of the IBFM 1 exchange interaction that emerges from such an assumption, after appropriate truncation to states of maximal F-spin, is too weak compared to that required by phenomenological fits. In the same work, it was shown that a quadrupole pairing interaction between like nucleons yields an IBFM 1 exchange interaction of more or less the correct strength. They thus concluded, and we concur, that the predominant origin of the BF exchange interaction is the interaction between like nucleons. Such distinctions are not particularly important at the level of IBFM 1, since all lead to the same general form of the exchange interaction, with differences in the strength only. But they are much more important at the level of IBFM2. As noted above, an interaction between neutrons and protons gives rise to a three-body BF exchange interaction, whereas an interaction between like nucleons gives rise to a two-body exchange interaction, between the odd fermion and the bosons of the same type. Based on the above considerations, we assume for the BF interaction (j = 1 hi,/2 ) the following parametrized form VBF

= rjQ~ - (anja~~)t2~ + Aji

1

r (0 : f(d~ânj)ii>(a~jd )

:~

(6.5)

where

=

= (jlly2llj) - (6 .6) Here, vj' is the occupation probability of the l h11/2 proton orbit, and To and A 0 are constants related to the quadrupole-quadrupole interaction between bosons . Ideally, the parameters of the VBF would be chosen from an appropriate microscopic calculation . Unfortunately, despite some recent progress 20), there does not exist at present a well justified microscopic prescription for deriving the parameters of the IBFM2 in strongly deformed nuclei . Thus, our approach is to simply choose what we consider to be reasonable estimates for TO and A 0 and then to determine the remaining parameter v~ so as to optimally reproduce the low-lying band structure of ' 65 Ho within our mean-field framework . Based on earlier work, the values that we take are TO = 52.8 keV and A 0 = -121 keV. Fig. 2 shows the results of HBF calculations for k - i, . . . , 'Z as a function of the remaining parameter vj2 . The intrinsic energy levels change their relative positions as rj

rOQjj (u 2 - v.i)

Aij = 4 Ao(Qjjujvj) 2 ~

Qij

R. Larsson et al. 111ean-field approximations

486

HBF energies vs . occupation probabilities . - - k = 1l/2 k = 9/2 __ k=7/2 k = 5/2 k=3/2 k = 1/2

0 .0

ß

-5 -6

f

0 .0

I

0 .2

I

0 .4 0 .6 0 .8 Occupation probability

I

1

Fig. 2. The expectation value of H as a function of v~ for the axial HBF solutions with k -

z-z.

j changes. We see that k = -', is the lowest level from v~ = 0.44 to v~ = 0.63. Since experimentally -'' ), the ground-state band has K = -', , v~ should be chosen in this range. To pin it down more precisely, we note that there is an experimental K = i band whose band head lies at 680 keV. If we choose v~ = 0.56, we obtain the k = z HBF level roughly 680 keV above the k = -', level. For the same reasons as noted above, it is not essential that our intrinsic energies precisely reproduce the experimental band head energies. Nevertheless, we will assume that this is indeed the appropriate value of aye, thereby completing the determination of our hamiltonian . With this hamiltonian, we then carried out both TDA and RPA calculations, for which the resulting spectra are shown in fig. 3. The TDA and RPA yield almost identical intrinsic excitation spectra, except that the RPA puts the spurious K = 1 and K = 9 - intrinsic bands at zero excitation energy (they are not shown in the column labelled RPA ) . Note again we are only presenting intrinsic energies, since no projection to states of good angular momentum has been made. Nevertheless, it is reassuring that the lowest K = ; and K = '-,' intrinsic bands come out (from both TDA and RPA) in reasonable agreement with-the experimental K = i and K = band heads at 515 .5 and 688.5 keV, respectively -'' ) . Further comparisons are hard to make due both to the lack of experimental data and to the fact that we have not incorporated either angular momentum projection or cranking in our treatment. There is an extra K = 9 intrinsic state that comes out very low in energy in the RPA calculations . This is not the spurious state mentioned above. Rather, it is a peculiar consequence of the RPA as it has been formulated above. The physical 27

R. Larsson et al. / Mean-field approximations E(MeV) 3.5

3.0 1 .0

TDA

MTDA

RPA

11/2 - - 3.408 3/2'- - 3.356

11/2 - - 3.408 3/2- - 3.356

11/2 - - 3 .402

7/2°' - 3.289

7/2- - 3.289

7/2- - 3.289

5/2- - 3.077 9/2 - - 3.052

5/2 - ..- 3.073 9/2- - 3.051

- 11/2- - 1 .026

11/2 - - 1 .026

5/20.782 11/2 - _- 0.771 9/2 - ~~ 0.763 0.728 3/2 1/2- -- 0.680 7/2- - 0.661

0 .5

- 3/2- - 0.491

11/2 - - 0.771

3/2 - - 0.728 1/2- - 0.680 7/20 .661

3/2- - 0 .491 5/2- - 0 .416

48 7

3/2- -3 .351

5/2- - 3.065 9/2- - 3.049 11/2 - -1 .010 -

11/2 - - 0.754 3/2 - - 0.717 1/2- - 000 7/20., 661

3/2- - 0.468 5/2 - - 0.388

9/2- - 0 .290

9/2- - 0.284

7/2- - 0 .000

9/2 - - 0.022 7/2- - 0.000 --

5/2- - 0.202 9/2 - - 0.151

0 .0

-- 7/2- - 0.000

Fig. 3. Intrinsic energies from axial TDA and RPA calculations for 165Ho . The results labelled TDA refer to complete TDA calculations, whereas those labelled MTDA refer to modified TDA calculations in which the spurious K = 2 and K = z states have been removed from the basis. To the left of each intrinsic level is shown its K" value. significance of this (if any) is not understood at this moment. In an exact diagonalization of a similar IBFM I hamiltonian no such low-lying K = 2 band is found, although also in this case it does arise in RPA. As such, we believe that this extra intrinsic state has no physical significance and arises as an artifact of the approximations inherent in the non-scalar RPA; however, further work to clarify this point is required . Further insight into the nature of the various bands can be obtained by carrying out a modified TDA calculation, in which the spurious K = z and K = 2 states have

488

R. Larsson et al. / Mean-field approximations

been projected out. To do this, we define a modified TDA basis which is orthogonal = f 1 spurious states. This new basis will have two states less - one to the K each from K = K + 1 and K = K - 1 - than the TDA basis discussed in sect. 3 .2. The results from this calculation are shown in the second column (labelled MTDA) in fig. 3. The lowest K = i and K = 2 states in MTDA are quite close to the PA states at 388 and 284 keV, respectively . This suggests that these are physically significant RPA states, and that the K = 2- RPA state at 22 keV is the one with no physical significance that should be discarded. In fig. 2, we see that the axial HBF solutions corresponding to k = i, i and i are very close in energy in the window 0.2 < vj2 < 0.3. The axial ground-state band changes from = 2 to K = z in this short interval. Even though this is outside the region of acceptable values for a description of 165 Ho, it is nevertheless interesting to explore this region further, so as to obtain insight into the kinds of effects that might arise in more general IBFM2 systems. As a side note, variations in v~ can be thought of as simulating a shift of the l h 11 /2 single-proton level relative to the other proton levels in the Z = 50-82 shell. When we evaluate the stability matrix associated with the axially symmetric HBF solution in this interval of v2 values, we find that it contains negative eigenvalues. Thus, for these values of vi , the axial HBF solutions are unstable. In contrast, for all other values of vj2, the stability matrix has positive eigenvalues only and the axial HBF solutions are stable. 7. Triaxial calculations In this section, we report triaxial mean-field calculations for the same IBFM2 hamiltonian as in sect. 6. Again, the calculations were performed as a function of vj2 , with all other parameters of the model fixed. We first carried out triaxial HBF calculations . In fig. 4, we compare the energies of the triaxial and axial HBF solutions. We see that precisely in the window in which the axial solutions were unstable, the triaxial solutions lead to a lowering in energy. The changes in the intrinsic ground-state energy are very small, typically less than 20 keV, but nevertheless they are enough to produce stable solutions for which all eigenvalues of the triaxial stability matrix are positive. The changes are more apparent in the wave functions than in the energies, as can be seen in tables 1-3. The m = t2 components in the boson condensate are small, as expected, and the bosons are mostly in an m = 0 state. The fermion has a strong mixing of k-values for most values of v2 in the triaxial region, with the largest amplitude in the fermion wave function gradually shifting from k - i to k - z as v; increases. The wave function of the triaxial ground-state band can be related to an axial wave function modified by the admixture of p-h excitations . To demonstrate this, we carry out a binomial expansion of a triaxial boson condensate wave function

R. Larsson et al. / Mean-field approximations -3 .3

_

489

HBF energies vs . occupation probabilities .

-3 .4

v

-3 .6

-3 .8 -3 .9 1 0 .18

1

1

i

1

I

1

0 .22 0 .24 0 .26 0 .28 Occupation probability

0 .2

0 .3

i

0 .32

Fig. 4. The expectation value of H as a function of v? in the window from v? ;zz 0.2-0.3. The dotted curves give the results of axial calculations for k = z-i and the solid curve gives the triaxial results .

I PT)

11OO S P

=

2 (dp-2 + d p2)

+ 1120 pO + 1122

Np NP n

)

(Y2P ) n

(Yôp

)Np-n

Np )

IO)

Np

L no (72p7op )n IPA)

,

n=0

where

P +

%Ôp - ilOOS

f%2pdp0

t P 72p - 1722

V-2'

(dtp-2

+

dt

p2)

TABLE 1 Coefficients glrn of the triaxial proton boson that enters in the HBF solution in the triaxial region v~ = 0.20 - 0.30. v2

7r

~00

n ~2-2

0.20 0.25

0.6410 0.6400

0.0165 0.0324

0.30

0.6372

0.0358

n ~20

0 .7672 0.7670 0 .7691

n ~22

0.0165 0 .0324 0.0358

(7.2)

R. l.af°ssort t al. / Mean-field approximations

491)

TABLE 2 Coefficients it,". of the triaxial neutron boson that enters in the same HBF solution as in table 1 . 17 2 a~ v v a~ ~12 _ ~120 ~l22 ~100 0.20

0.6301

0.0085

0.7764

0.0085

0.25

0.6301

0.0176

0.7761

0.0176

0.30

0.6293

0.0203

0.7766

0.0203

PA) =

1 NP.

(ypo)~'"10),

(7 .3)

denotes the axial boson condensate . Thus, the triaxial boson condensate, IPT), is equal to an axial condensate IPA), plus Np states that involve an increasing number of p-h

excitations built on IPA) . If rlp, is small (which it typically is), then only the first few terms in (7 .1) are important. This can easily be generalized to the full IBFM2 ground states, with the result that

we can consider any triaxial solution as being equal to an axial wave function (which is

not necessarily self-consistent) plus p-h correlations imposed on it . This is, of course, closely linked to the existence of negative eigenvalues of the stability matrix (3 .22) for

the axial HBF solutions. Some of the correlations that appear in the treatment above

will necessarily change the K-value of the axial wave function . Hence, the groundstate correlations that are incorporated in axial RPA are not sufficient to treat triaxial systems . g. Summary and conclusions We have developed Hartree-Bose-Fermi

(HBF), Tamm-Dancoff

(TDA)

and

random-phase (RPA) mean-field techniques for deformed odd-mass nuclei that are

described by the IBFM2. Our goal was to develop a practical approach to treat the coupling of an odd nucleon to an axially symmetric IBM2 even-even core . With this in mind, we first developed the mean-field formalism for axially symmetric odd-mass systems and applied it to the negative-parity bands in '65 Ho . Overall, we were able to achieve a reasonable description of the few negative-parity bands for which exper-

Coefficients 11 2

'I, I

TABLE 3

k of the correlated fermion that enters in the same HBF solution as in table 1 .

rlf

i

flii

3

'1i i ys i

1

1ii _5

_

7 "5

1ii -)

1

9 z

q iI _ ii i

0.20

0.9418

0.25

0.3305

0.0609

0.0028

-0 .0002

-0 .0000

0.7076

0.6804

0.1892

0.0212

0.30

0.5527

0.6036

0.5731

0 .0412

-0 .0012

-0 .0002

-0 .0040

-0 .0005

R. Larsson et al. / Mean-field approximations

491

imental data exist. We then considered variations in one of the parameters of the input hamiltonian and found that for certain values of this parameter the axial HBF solutions were not stable. Motivated by this observation, we extended tNe mean-field formalism to the case of triaxial symmetry and confirmed that triax.ality of the core can be induced by the presence of an odd fermion. Much of the analysis followed closely earlier work on mead-field ter hniques developed for even-even nuclei in the context of the IBM, both for on!: kind of bison (IBM 1) and for distinct neutron and proton bosons (IBM2) . However, sevcrA ûn :que features arise when treating odd-mass nuclei because of their non-scalar nature arid the associated degeneracies that occur. To generate a physical excitation, either in TDA or RPA, it is necessary to introduce a distinct excitation operator for each of the degenerate groundstate solutions and then to add them coherently. A particularly interesting consequence of this is that in non-scalar systems the TDA can directly couple l p l h excitations to the Hartree ground state. Thus, in contrast to the scalar case, ground-state correlations are already introduced at the level of TDA. An interesting outcome of our axial RPA treatment of '65Ho was the presence of an extra band differing from the ground band by I d K( = 1 and occurring at very low excitation energy . We believe that this band has no physical significance, although further work to clarify its origin is clearly needed. An attractive feature of the RPA formalism is that it decouples spurious states from physical excitations. This is not true in TDA. However, we have shown how the TDA formalism can be modified to project out all spurious excitations from its basis. The resulting modified TDA results are in very close agreement with the RPA results, except that the extra low-lying band dGI-S not arise. This suggests that the modified TDA formalism may be particularly useful for non-scalar odd-mass systems . In addition to the need for clarification of the extra JA K I = 1 band in odd-mass RPA, there are several other issues that still require further investigation . Most importantly, the formalism as presented only provides information on intrinsic energies. One possible direction for future study is the development of methods either to project states of good angular momentum from the IBFM2 mean-field solutions or to include cranking in the formalism . Alternatively, it would be interesting to derive equations analogous either to the Inglis " ) or the Thouless-Valatin 22 ) formulae for estimating moments of inertia from non-projected mean-field solutions, and to incorporate Coriolis decoupling in the analysis " ). Some work along these lines has been reported in the context of IBFM 1 8) ; their extension to IBFM2 is necessary if we wish to derive meaningful spectra using our mean-field methods. One of the authors (S.P.) wishes to acknowledge fruitful discussions with Jorge Dukelsky and Michael Kirson on several aspects of the triaxial mean-field formalism . This work was supported in part (R.L. and S.P .) by the National Science Foundation under grant Nos. PHY-8901558 and PHY-9108011, and in part (RB) by the Stichting

R. Larsson

492

a al.

It

Mean-field approximations

voor Fondementeel Onderzoek der Mate6e (MM) with financial support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). ppendfix A

A.1 . TDA AND RPA MATRIX ELEMENTS

We present here the TDA and RPA matrix elements for the case of axial symmetry . ALL TDB matrix elements. The (normalized) axially symmetric TDA basis for positive K is

Qjj&;T) ' Ifs, K)

= r.,fsK riK- 19* S« ;Y) 1 1

I Ps, K ::F K; ±Z) =

V1_V_7P

rt

psK :F

Folge ±T)

(A . 0

with p = 7r, v. The non-vanishing matrix elements of H in this basis, relative to the F ground-state energy Eo = (g.s. ; ±KIHIg .s. ; ±K), are given by (ps,K ::F Z; ±TI H 1 pt, K ::F T; ±7) - Jm Eo

E ,P P* P

+ (Np - 1

(Iß76

(k

+ bp, N,, (Q6

bp, X,

+

"flij,

a#Y6

'OS*_i,O "ß,à

,q.0 1&,0 1 7,K:Fi (jp )"(#SKTi 1P*

V" -8 L5 A :F,K,KTT,00 V076 -

7r*

7t

VS*

V1

00,K :F K,K :FT?n'r ,O ?110 011 7,K 1-.-,jq6,K:F -T 1

1 PS* pi f* f W4-2 (p)?i KTK,K:FT,±-T,±*K (t.K :F«Klß,K :FK'J,±jZ"1',±Z

Ssta p,

(f.s,KjHjft, K) - 6SjEo f f* f EI llj,K 17j,K + P

(g.s . ; :-F

K1H1ps, ::F

(K

P* P f* f - 45 S I W"flij, I OOKK (P)q,,,0 17fl,OUj,KqjI,K

NP kpjj

+ K) ; ±K)

f* f p* PS N--,7,= & « / jr 7(P)1(W oqq,=XI '±zg I , f _r III I -rIP__V aflij , (fs,KIHIpt, K =F 1; ±!) f* f p* PS = V7NP 37'% W"ß]J p a m 'P &-- - (P) 1 .ol ,zrli,KIIo-i-V 9 i . II_VIP-t, ßli l " , (7rs,K :F T; ±"KI HI vt, K ~ T; ±T)

= -v NN,, 7 V"075 nz n ns* n ff,t%n L'i* /' -r AT UU,rt :F (k, -r OPY85

V

V7

R. Larsson et al. / Mean-field approximations

493

(ps,K f K; ::FKIHl pt, K :F K; ±K) ~l , r~fll_

_ afij'

_

K±K,K~K,~K,±T

( p)r ps* _ ~pt

rf*

f

717 .±K' a,K±K ß,K~K J " :l'

(A.2)

Here, Af, A,, and A denote the lowest eigenvalues of the HBF equations for fermions, proton bosons and neutron bosons, respectively . A.1.2. RPA matrix elements In the RPA, there are, in addition to couplings between 1 p l h excitations (as described above), couplings between the HBF ground state and 2p2h excitations. The 2p2h excitations that couple to Ig.s.; ±K) in the RPA matrix are (p=7rorv)

Ips, K ::F K; pt, -KfK; ±K)

= _l' 1 t

17ts, K :F K; vt, -K f K; ±K) =

_rPOrt

.s.; fK)

psK~K

Pt-KtK rpolg

1

-rrtvi- K± - rvo i g .s.; fK) ,

Np

N,~ N

rt

nsK :K

1 Ifs, K; pi, -K f K; ±K) = Ifs (A ~g.s .; ±K) . NP KrfKp,-KtKFPO

.3)

The coupling matrix elements are then given by (g.s.; ±KIHI ps,K ::F K ; pt, -K f K; ±K) _ '(Np - 1) U"Î;la la _ rt f yi5

P*rP* Pl _ Ps OO,K :F K,-KfK ( p ) "0 POrg,-K± l ;,K :FK'

±K1 H17ts,K :F K; v t, -K f K; ±K)

(g . s . ;

aß7Ö (g . s . ;

076 :F O,K T,o,-KfK

ß .K :F

~

6,-KfK

±KIHIfs,K; pt, -K f K ; :f--Y) NP

f*

,,Pli'

fs

0,-K±K .±K,K(p)rp0rß`-K±KrJ .±Krl' .K

(A.4)

References

1) 2) 3)

4) 5) 6) 7) 8) 9)

A. Arima and F. Iachello, Phys. Rev. Lett . 35 (1974) 1069; Ann. of Phys. 99 (1976) L53; 111 (1978) 201 ; 123 (1979) 468 F. Iachello and O. Scholten, Phys. Rev . Lett. 43 (1979) 679 F. Iachello and A. Arima, The interacting bosun model (Cambridge Univ. Press, Cambridge, 1987); F. Iachello and P. Van Isacker, The interacting boson fermion model (Cambridge Univ. Press, Cambridge, 1991) O. Scholten, F. Iachello and A. Arima, Ann. Phys . 115 (1978) 325 ; O. Scholten, Ph . D. Thesis, University of Groningen, 1980 O. Castanos, P. Federman, A. Frank and S Pittel, Nucl. Phys. A379 (1982) 61 J . Dukelsky, G.G. Dussel, R .P.J . Perazzo and H.M. Sofia, Nucl . Phys. A425 (1984) 93 S. Pittel and A. Frank, Nucl . Phys . A454 (1986) 226 J . Dukelsky and C. Lima, Phys . Lett . B182 (1986) 116 D. Bohle, A. Richter, W. Steffen, A.E.L. Dieperink, N. Lo Iudice, F. Palumbo and O. Scholten, Phys . Lett . B137 (1984) 27 ;

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10) 11) 12) 13) 14) 15) 16) 17) 18) 19"' 20) 21) 22)

R. Larsson et al. /

approximations

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