Symmetry conserving Hartree-Fock-Bogoliubov theory

Nuclear Physics A3B~i (1982) 189-204 ® North-Holland Publishing Company

SYMMETRY CONSERVING HARTREE-FOCK-BOGOLIUBOV THEORY (n . On the solution of variatiooal equations t J. L. EGIDO rr

Max-Planck-Institut für Kernphysik, 69 Heidelberg, West-Germany

and P. RING

Phystk-Department der Technischen Unioerstt6t München, 8046 Garching, West-Germmty

Received 16 November 1981 (Revised 18 January 1982) Aóetraá : A new method is presented for the general solution of variational equations after projection onto eigtnspaces of symmetry operators. The minimum of the projected energy surface is found following a path in the direction of the steepest descent. The method is discussed for the case of particle number projection in connoction with cranked Hartree-Fock-BogoGubov theory and for three~imensional angular momentum projection . The applicability of the method is shown for number projection before variation in the high-spin region of realistic heavy nuclei.

1. Introduction An essential tool for treating strong oorrelations in many-body systems in cases where perturbation theory and even a summation of certain subseries of diagrams, such as the random phase approximation, break down, is the mean field approach. It is widely used in nuclear physics in the form of the shell model, the deformed Nilsson model, or the superfluid BCS models . The most general mean field theory of this type which includes pp as well as ph correlations is the Hartree-Fock-Bogoliubov (HFB) theory based on generalized Slater determinants . The validity of ttlis approach is. strongly connected with phase transitions and symmetry violations [see for instance ref. t)]. In many cases, in particular~if the correlations are very strong, such symmetry violations are of minor importance . An example is the violation of translational symmetry within the shell model. It is usually connected with correction of the order of 1 /A which can be neglected in heavy nuctei . The same is more or leas true for the violation of rotational symmetry for strongly deformed nuclei and with less accuracy in the case of gauge invariance violation in superfluid nuclei . f Work supported in part by the BundeamilSisterium für Forschung und Techndlogy. rt permanent ~~ ;Division de Fiskus, Universidad Autónoma de Madrid, Spain. 189 July 1962

19 0

J. L. Egido, P . Ring / Symmetry

conserving HFB (I)

There are, however, two reasons for going beyond the mean field approach and treating the symmetries properly by projection methods: (i) One is often interested in special quantities such as electromagnetic transition probabilities or moments, which can hafdly be calculated from symmetryviolating HFB wave functions. (ü) In transitional regions one observes often gradual changes from symmetry conservation to weak violation and eventually to strong breaking of the symmetry. In such cases the mean field approach without symmetry conservation breaks down . For the first problem it is sufficient to project from the deformed wave function after the variation. However, in the second case one needs a method to determine the mean field from a symmetry~onserving theory. Several authors have derived variational equations after projection z-e ) . Because of the complex structure of the projection operators, for heavy nuclei, where the mean field approach should be better justified, so far these equations have been solved only with dramatic restrictions . Only in the case of particle number projection in the simple BCS model was one able to carry out a general variation after projection [FBSC theory z)]. In the much more complicated case of HFB wave functions and cranked HFB wave functions, so far the projected energy surface has been mapped only as a function of a few collective degrees of freedom of the mean field such as quadrupole deformations ß, y or pairing gap parameters dP and do for protons and neutrons, and the variation has been carried out only within these few parameters'" e . za). In the case of angular momentum projection most of the authors restricted themselves to axially symmetric intrinsic states s " ~ only in light nuclei, a variation after three-dimensional angular momentum projection has been performed [ref. lo)] . In this paper we present a method for solving the projected variational equations without any restriction on the intrinsic HFB functions. It is based on a steepest descent method in the projected energy surface, which has been used with great success for the solution of unprojected variational problems i i . i z) and which can be extended without essential difficulties to the symmetry conserving cases. In sect . 2 we introduce the general method. The case of particle number projection is treated in sect. 3. Special problems showing up in the case of angular momentum projection are discussed in sect. 4. In sect . 5 essential matrix elements are evaluated and in sect. 6 we show the applicability of the method in the case of particle number projected cranked HFB theory for realistic heavy nuclei in the rare earth region .

J. L. Egido, P. Ring / Symmetry conserving HFB (I)

19 1

2. Projected Hartree-Fock-Bogoliubov theory We are interested in finding an approximate solution to the enact eigenstates of the two-body hamiltonian H = ~ EaßCáCß + 4 ~ Ua0Y8CáCÁCdCY. ad

aBYó

(l)

It contains a single-particle part eaa and a residual two-body force vaayd = -vadeY . The operators are fermion creation operators for some arbitrary basis, as for instance a spherical harmonic oscillator . Hartree-Fock-Bogoliubov theory is based on the mean field approximation, i.e. one assumes that the nuclear many-body system can be described as consisting of quasiparticles independently moving in a suitably chosen mean field. In order to be able to describe strong correlations amongst the original particles, this mean field has . to be allowed to break essential symmetries of the hamiltonian (1), such as particle number conservation (gauge symmetry) and rotational invariance. Within the framework of HFB theory one furthermore assumes a linear relation between the original particle ct, c and the quasiparticles ak = ~ UakCa + VatcCa~ a

The HFB wave function Iii is defined as vacuum with respect to these quasiparticles, akl~> _ ~,

(3)

and is characterized by the HFB coefficients U~ and Vim. The mean field approach is based on the fact that in many-body systems with strong oorrelations one often observes a kind of transition to a phase which is described by rather simple wave functions Iii of independently moving quasiparticles in a symmetry breaking ("deformed") mean field. In the following we call this wave function an "intrinsic" wave function, emphasizing the fact that this wave function is not meant as an approximation to the exact eigenstate of the hamiltonian (1) in the laboratory frame, which has to be an eigenfunction of the symmetry operators in the system. The word "intrinsic" has at this point nothing to do with a point transformation from the laboratory frame to some body-fixed system . So far we have neither specified how to determine the intrinsic function Iii, nor how to relate it to the exact eigenstate of the hamiltonian (1). It is clear that in order to compare Iii with an exact eigenstate, we have to restore the broken symmetries in Idà), i.e. we have to project onto an appropriate eigenspace of the symmetry operators. The projected normalized wave function

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J. L. Egido, P. Ring / Symmetry conserving HFB (IJ

is then an approximation to the exact eigenstate. ß is a projection operator which removes from Iii all components with, for instance, a particle number different from N and an angular momentum different from I. The detailed structure of these operators will be given in the following sections. Although the wave function Iii being a generalized Slater determinant has a very simple structure, the structure of I~ is in general very complicated, i.e. I~ contains many correlations . The transformation from the intrinsic system to the laboratory frame is achieved by the rather complicated procedure of projection. Now it is clear how we have to determine the intrinsic function I~~ . We only have to use the variation principle of Ritz and obtain the equation s) S <~Ixl~i = 0.

<`~I~>

This is variation after projection, i.e. the solutions of this equation are given by the minima in the projected energy surface Er {~} _

<~I~rxll~>

The dimension of this energy surface is given by the number of parameters, which characterize a Slater determinant Iii . In most of the applications so far one has used only a very restricted subset of all Slater determinants, as for instance Nilsson + BCS wave functions Iß(ß, y, dp, d o)~ depending on two quadrupole deformations ß, y and two gap parameters dP and do for protons and neutrons e . 9). Depending on the size of the configuration space or on the number ofmesh points the general HFB wave functions are characterized by a much larger number of parameters . Before we discuss in more detail, what kind of parameters we use, let us sketch the basic idea how to solve the variational problem (6). A very simple and elegant method to determine the minima in the energy surface is the gradient method ' 1 " 12) . One starts with an initial wave function I~oi~ It represents a point on this energy surface and one is able to determine the gradient of the energy on this point, i.e. the direction of the steepest descent. In order to find an improved wave function one follows the direction of steepest descent and calculates a wave function Iii. The size of this step is characterized by a positive parameter q which is somewhat arbitrary and which is changed during the interaction. For sufficiently small values of n we have EP {~i} < E~{~o },

i.e. Iii represents a better approximation to the exact solution than i~o~. Now one determines again the gradient in the point I~~i and repeates this game until the gradient vanishes and one has reached a minimum on the energy surface. For a practical application of this method we have to choose a general parametrization of the HFB wave function Iii . The HFB coefficients U,k and V,,~ are

!. L. Egido, P. Ring / Symmetry conserving HFB (I)

193

not very well suited : They have to obey certain orthogonality relations [see ref. t) p. 246] and tío not represent a set of independent variables. Since we do need the energy surface only locally, i.e. in the vicinity of each point, a suitable parametrization is given by the Thouless theorem [ref. t), p. 615]. It states that any product function ~~) non-orthogonal to ~~o) can be written as ~~) _. ~4S(Z)) ac exp ( ~

k
Zkk,a,~a;,)~4so),

(~)

where the quasiparticle operators ak annihilate the vacuum ~4So) . The coefficients Z,~, with (k < k') represent a set of independent variables. The HFB coefficients U, V characterizing ~4s) can then be determined from the HFB coefficients Uo and Vo belonging to ~~o) in the following wayf U = Uo +YóZ*, V = Vo+UóZ*. The projected gradient is then given by the matrix elements ô

EP

ZI

~~~ak.ak(H-E~)ß~~)

,

and we obtain, following the direction of steepest descent by a step of the size rh I~t> a exp(-n ~

k
Ykk'a~ak')I~o)'

(lU)

The difference in the solution of unprojected HFB equations by the gradient method's' t2) and projected HFB equations (5) is therefore only given by the fact that in the latter case one has to calculate the projected gradient (9) instead of the unprojected gradient . In the following we have to distinguish the case of an ábelian symmetry group, which shall be discussed with the example of particle number projection, from the case of a non-abelian symmetry group, which shall be discussed with the example ofangular momentum projection . 3. Particle number projection before varlaüon

So far many authors have used the method of particle number projection before variation. However, most of them have restricted themselves to the BCS f The coefficients U, V obtained from eq. (g) are not necessarily orthogonal . A Gram-Schmidt orthogonaliza~ion is always possible without changing the vacuum .

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J. L. Eilido, P. Ring / Symnutry conserving HFB (I)

case z) or to the variation of a few parameters, such as the deformations or the gap parameters in the intrinsic Nilsson field 8). For general variation we need the matrix elements of the gradient (9) <~Iack.ak(FI -EN)ßNI ~~ (11)

<~I ~I~>

The projector onto the particle number N is given by z) - 1

zxe~~(A-x)d~P

2n ,lo

.

(12)

In the case where Iii has good number parity 13), the wave function Iii contains only even eigenvalues of A- N and we have - 1

zxe~(1P-NUzdW .

2n ,)o

Using an idea of Fomenko points

14,

ls) we replace the integral in e4 . (13) by a sum over L

_ 1 ~ ek~d~-N~ ,

ßN L -

(13)

L,= 1

~P,

= 2n 1. L

(14)

is a projector in the mathematical sense. It projects onto the subspace with particle numbers N, Nt 2L, Nt 4L, Nf 6L, . . . .For L going to infinity it converges to the exact projector (13) . Introducing the wave functions rotated in gauge space,

which are normalized to <~I~~`h = 1 with

we obtain for the projected wave function a sum of L Slater determinants ~il~i = ~ Y~I~~i, i_i with Ye

=

L

xil

~ xi, i=i

(17)

(18)

J. L. Egido, P. Ring / Symmetry conserving HFB (1)

195

and the projected matrix element of an arbitrary operator Ó is given by (19) The overlap matrix elements xi and <~I131~u h can be calculated from the Onishi formula (see sect . 5). It thus turns out to be possible to carry out a complete variation after particle number projection. In practical calculations in the rare earth region (see sect . 6) one needs L = 8 to remove all spurious components in the HFB wave function . From the relations (zo)

we see that we need only to calculate the cases l = 0, . . ., }L. A variation after particle number projection therefore seem to be at least a factor ~}L+ 1 more complicated than a variation without particle number projection. In practical calculations, however, it turns out that the convergence properties with particle number projection are in many cases much better than without projection. The necessary CPU time is therefore in general only increased by a factor of 1-2. One word remains to be said concerning the constraints. If the angular momentum is treated within the framework of the cranking model, the calculation has to be carried out with the constraint (21) This can easily be done by applying the gradient method to the hypersurface defined by eq. (19) [for details see ref. ia)] . Concerning the average particle number in the "intrinsic" state Iii a constraint does not seem to be necessary since we are projecting onto good particle number anyhow. In principle, the projected wave function Iii is invariant with respect to a rotation in the gauge space

The corresponding intrinsic function (23) however, is changed by such a rotation . This means there is a whole set of intrinsic functions I~~ which produce the same function I~ and the solution of eq . (2) is

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J. L. Egido, P. Ring / Symmetry concerning HFB (I)

not really a local minimum in the energy surface of all HFB functions, but a point on the bottom of a valley. To be unique one can fix the orientation in gauge space by requiring the corresponding gauge . angle a to vanish . Since the particle number has real matrix elements, this is achieved automatically if one calculates with real HFB coefficients . There remains, however, an additional ambiguity. A rotation of the type (23) with an imaginary angle a changes the average particle number in the intrinsic function ~~~. Although an exact projection from the new function ~~~ would not change the result, this is no longer true in the case of an approximate projection (14) with a finite Irvalue. To avoid numerical instabilities caused by this ambiguity we therefore carry out the variation (5) with a constraint on the average particle number in the intrinsic function <~~l~~~i = N.

(24)

Within the gradient method this additional constraint is rather trivial and it improves the convergence considerably . 4. Angalarmomentum projection before variation In the case of angular-momentum projection the underlying rotational group is no longer abelian. The projector is no longer defined uniquely 1). It depends on 2I+ 1 parameters gR ~r _ er -

~9g R

ßr~rg,

(25)

Wlth ßír~rR = 28n 1 ! Déta(~)k(~~~,

(26)

where rè are three angles characterizing a rotation in three~imensional space and k(i2) is the corresponding rotational operator. The parameters ga are determined by the variational principle and one therefore is left with a double variational method 16 " l'), (27) with

J. L. Egido, P. Rlng

/ Symmetry conserning HFB (I)

197

It can be solved again by an extended gradient method for the parameters Z,~. [eq. (7)], which determine ~8~) and the parameters ga. The gradient in these parameters is given by (29)

{YÎ = {Y'a. Yt'~k "h

with Y'a

-

a9á

~~~

<~~ az~. -

'~~)

t~~>

(30) (31)

Starting from an initial wave function ~~o) and the initial coefficients gá we obtain in the next step of the iteration 9á = 9á - nYí'~~

(32)

~~i) a exp(-n ~Yt'~x"a~at")I~o) " ,

(33)

kk

The matrix elements <~~ó~sa " I~) are again obtained by replacing the integral in eq. (26) by a sum over many mesh points in the Lt regime. Details are given in ref. ' e). As in the case of particle number the intrinsic function ~~) can be rotated and the rotation can be absorbed in a redefinition of the coefficients

with

~9k~nra~(~)~~) _ ~9á~~al~>, k a

(34)

9á = ~ Dá"a(~~a"" a"

(35)

<~I~il~) _ <~I~-ilk) _ <~I~-zl~) = 0,

(36)

This redundancy can be removed by constraining the intrinsic wave function to a fixed orientation, for instance which forbids rotations around real angles and to fined expectation values (37) (38)

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1. L. Egido, P . Ring / Symmetry conserving HFB (I)

which forbids rotations around imaginary angles . For real wave functions with a good signature 19), erR.r,l~) a

I~),

(39)

only the condition (3~ is left . 5. Evaluation of projected matrix elements In sects. 3 and 4 we saw that the essential quantities for the solution of the projected HFB equations are matrix elements of the form <~lak'akHl~"'),

<~Ik,l~), <~IHI~`"),

(~)

where ~~`~i is given by eq . (15) . For the calculation of these matrix elements we use the theorem of Thouless and express the rotated function ~~"h by ~~) and the quasiparticle operators ak belonging to the vacuum ~~)

which is in agreement with the normalization<~~~~`~) = 1. In order to determine coefficients ~B~ we express the quasiparticle operators ak belonging to ~~) as well as the rotated quasiparticle operators aknt belonging to ~~~`h in the original particle basis as a,~ _ ~U,,~cQ + Va,~ca, a

a(4, k _ ~ (D«~ U)akCa a

+

(D(i)~

V)akCa~

where D~`~ is the representation of the rotation f2, in the space of the basis states ~a) _ Cá IO). In the case of particle number projection Dc" is just a phase e'"~2 and in the case of angular momentum projection these are the Wigner functions with arguments él,. Using eq. (42) and the orthogonality relations for the HFB coefficients U, V, Uc'~ and Vc`~ we can express the operators aknt by the operators ak, ak, ak!) r _

~ ~k~kak' + ~iak'~ k'

J. L . Egido, P. Ring / Symmétry concerning HFB (I)

Wlth

~Plc'~

199

= UiDc'~U+VtD°~'V,

~~a~~ is defined as the vacuum of the operators ak'~ : From this condition we can derive the coefficients ~8~~ in eq. 41 [for details see ref. 1), p. 615], ~'~ _ ~a~"~Plcp"_,

(43)

The overlap functions of the norm x, in eq. (16) are given by the determinant

They are defined by this expression only up to a sign. Because of the signature symmetry this causes no problems in the case of number projection. In the case of three-dimensional angular-momentum projection, however, the sign has to be determined by analytic continuation from the point llt = 0, where the overlap is unity. This procedure eventually requires an increased numerical effort. For the evaluation of the matrix elements <~~Hl~ c'~ we use an extension of the theorem of Wick [ref. 1), p. 619] and find ~~~g~~c~~ = Tr(EPcn+ZrcoPu~_Zda~~c~'),

(45)

with

~é = ~~~cácp~~~i =

(VU~c~t )aß,

and Uc" = U+V*~ir, Vc'~ = V+U*~Bc'~' rárq -

~ vaßTAhbß, ßa

d~ - ~ ~ vaß7óK~~ ra

Y

L~~ 3

= ~ ~ vaßTóKá`~ aß

(47)

200

J. L. F .gido, P. Ring / Symmetry conserving HFB (I)

and we obtain for the projected energy (49) The gradient of the hamiltonian can be calculated from eqs. (9) and (19) with the matrix elements

where HZ°~`) is defined in analogy to the unprojected HZ° by Hz°c`) = U(t)'hp)Vp)~ - Vp)'hp)TU(q~+U~r~td~~~U~~~~-V~~~fdi~~V~~)~

(51)

with Of course, in the case of particle number projection the number operators for protons and neutrons have to be treated separately. In the theory of high-spin States of heavy nuclei one uses very often simple separable interactions of the form

~ is a single-particle term, Qa are the five quadrupole operators (or higher multipole operators of ph type) ~v

-_

~

Qw~cáca,

and Px are multipole operators of pp type ßz

O]

(53)

°sting

= ~Px .rcaca .

a

The simplest case is the monopole pairing force with Po.~ = Sad. Neglecting the exchange terms the overlap matrix elements have then a very simple form jít) ay - - ~ 4pgQtY.r ct) d~~ = P P'~.r ~

~~ =

( 55 ) Pct)Px.,~

J. L. Egido, P. Ring / Symmetry conserving HFB (IJ

201

with

Using the quasiparticle representation for any single-particle operator ~, it might be of pp or ph type

we obtain for the overlap and for the gradient

kk'

<~~b~~u>)

k
(Dk,°akak . +D,°~,~~.ak.ak~

= Do_jrTr(Doz~Bv>)

(57)

(58)

~~Iak.aklJl~`)) _ (DZ°+~`~(~~D~~~)+Dtt~`~+ß`)D117+~Bct)T°z~8u))~' .

(59)

For separable forces of the type bt ~ ß we have

~~~ak,a~t . 6~~n) _ <~~ak'akbt~~1') ~ <~~D~~n)

For a small number of separable terms it turns out to be much more economic concerning the numerical effort to use thé quasiparticle representation (57) based on the function ~~). 6. Test of convergence in the case of n®ber projection In order to test the convergence of the approximation projector ~ for L = 4, . . . we solve the number projected cranked HFB equations as discussed in sect . 3 for a typical realistic case of the well~eformed nucleus 16~Er. We use the pairing-plus-quadrupole hamiltonian of eq. (52). The configuration space and the single-particle energies are taken from ref. z1). The strength of the 0, 2,

J. L. Fgido, P. Ring / Symmetry construing HFB (IJ

202

quadrupole force is X = 70A

(62)

-t .a~

and the strengths of the pairing forces for protons and neutrons are Because of the particle number projection these values are slightly reduced as compared to the values given in ref. ~~) . We also use a small core moment of inertia J~ contributing to the energy ~Jc~z and to the angular momentum co.l~ with .!~ = 6.5 MeV -t :,The results of these calculations as concerned the physical properties of the yrast line are discussed in more detail in ref. ZZ). Here we only study the convergence with the number L of mesh points in the integration of e9 . (14) . Tesr e 1

Cranking frequency m, deformation parameters ß and y, gap parameters for protons and neutrons dP and do, and excitation energies E, for the yrast level with I ~ 106 in the nucleus' 6`Er I

w

ß

Y

do

do

Ef

0

0.000 0.000 0.000 0.000 0.066 0.071 0.075 0.075 0.115 0.124 0.130 0.130 0.157 0.170 0.175 0.176 0.188 0.196 0.206 0.207 0.208 0.213 0.223 0.224

0.313 0.313 0.311 0.311 0.313 0.313 0.312 0.312 0.314 0.314 0.313 0.313 0.315 0.315 0.314 0.314 0.317 0.317 0.316 0.316 0.318 0.318 0.316 0.316

0.000 0.000 0.000 0.000 0.099 0.082 0.082 0.077 0.366 0.360 0.330 0.326 0.887 0.830 0.707 0.708 1 .103 1 .089 1 .074 1 .091 1 .635 1 .336 1 .321 1 .331

0.819 0.984 0.977 0.976 0.810 0.889 0.970 0.970 0.794 0.876 0.956 0.956 0.774 0.858 0.936 0.936 0.754 0.840 0.916 0.916 0.741 0.827 0.903 0.903

0.712 0.814 0.840 0.848 0.699 0.807 0.840 0.840 0.674 0.789 0.819 0.818 0.635 0.759 0.786 0.786 0.584 0.705 0.743 0.744 0.519 0.657 0.696 0.696

0.000 0.000 0.000 0.000 0.082 0.087 0.092 0.092 0.266 0.285 0.300 0.301 0.542 0.581 0.609 0.609 0.890 0.950 0.993 0.994 1 .292 1 .356 1 .423 1 .425

2

4

6

8

10

The angular moments are given in 6, the quantities m, d and Ef in MeV and the angle in degrees. For each I we give the successive approximations for the projection : L = 1 (unprojected), L = 2, L = 4 and L = 8. The energies are normalized to 0.0 at I = 0. The absolute values are -287.404 (L = 1), -288.882 (L = 2), -289 .106 (L = 4) and -289 .109 (L = 8).

J. L. F.gido, P.

Ring / Symmetry conservixg HFB (I)

203

In table 1 we give the cranking frequency ~ determined by the constraint

and the deformations ß and y of the self-consistent number projected solution fiu~°ßcosy = X<~PI¢ol~, ~°ßsiny = X~~Y~~z~~`i,

(65)

where itcv° = 41 .2 A_ f and ~~, are the quadrupole operators in units of the oscillator length and symmetrized with respect to e`~= as given in ref. z°). The pairing gap parameters for protons and neutrons dpand d are defined by the correlation energy of the pairing force d = ~J<~iptPi~,

which corresponds for unprojected wave functions just to the pairing gap of the intrinsic function given by d = G<~IPtl~i .

(67)

In a number projected theory the intrinsic wave function is no longer uniquely defined. We therefore use the definition (66) . In table 1 for each value of the spin I the first line corresponds to the unprojected value (L = 1) and the next three lines to the successive approximated projections L = 2, L = 4, and L = 8. L = 8 is practically exact and between L = 4 and L = 8 one has only minor changes, i.e. L = 4 seems to be already a very good approximation. As expected the expectation values ofthe quadrupole operators, i.e. the deformation parameters ß and y are not influenced very much by the particle number projection. For the gap parameters we observe larger values for particle number projection, because the projected wave function contains more pairing correlations . As a consequence the moment of inertia for the projected cases is smaller and we find larger level spacings for the projected case. We would like to thank R. R. Hilton and H. J. Mang for valuable discussions. References 1) 2) 3) 4) 5)

P . Ring and P . Schud, The nuclear many-body problem (Springer, New York, 1980) K. Dietrich, H . J . Mang and J . H . Pradal, Phys. Rev . 136 (1964) B22 H. D . Zeh, Z . Phys. 188 (1965) 361 H. Rouhaniejad and J . Yoocoz, Nucl. Phys. 78 (19C~ 353 J . Yoocoz, Varenna Lectures 36 (1966) 474

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1. L. Egido, P. Ring / Symmetry conserving HFB (I)

6) 7) 8) 9) 10) 11) 12) 13) 14) 1~ 16) 17) 18) 19) 20) 21) 22) 23)

N. Onishi, Prog . Theor. Phys. 40 (1968) 84 H. R. Dalafi, B. Banerjee, H. J. Mang and P. Ring, Phys . Lett. 44B (1973) 327 H. Gceke, J. Garcia and A. Faessler, Nucl . Phys . A208 (1973) 477 F. GrOmmer, K. Gceke, K. Allsem and A. Faessler, Nucl . Phys. A2ä (1974) 443 R. Y. Cutson and H. C. Lee, Nucl . Phys . A211 (1973) 429 H. J. Mang, H. Samedi and P. Ring, Z. Phys. A279 (1976) 325 J. L. Egido, H. J. Mang and P. Ring, Nucl. Phys . A334 (1980) 1 B. Hanerjee, H. J. Mang and P. Ring, Nucl . Phys. A215 (1973) 366 V. N. Fomenko, J. of Phys . A3 (1970) 8 R. A. Sorensen, Phys . Lett. 38B (1972) 376 H. D. Zeh, Z. Phys . 202 (1967) 38 A. K. Kermen and N. Onishi, Nucl . Phys. A281 (1977) 373 T. Hayashi, K. Hera and P. Ring, to be published P. Ring, R. Beck and H. J. Mang, Z. Phys. 131 (1970) 10 J. L. Egido, H. J. Mang and P. Ring, Nucl . Phys. A339 (1980) 390 K. Kurrar and M. Baranger, Nucl. Phys . A122 (1968) 273 J. L. Egido and P. Ring, to be published A. Faessler, K. R. SandhyaDevi, F. Grümmer, K. W. Schmid and R. R. Hilton, Nucl . Phys. A2S6 (1976) 106