Low-lying magnetic and electric dipole transitions in odd-mass deformed nuclei: A microscopic approach

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 613 (1997) 45-68

Low-lying magnetic and electric dipole transitions in odd-mass deformed nuclei: A microscopic approach V.G. Soloviev a, A.V. Sushkov a, N.Yu. Shirikova a, N. Lo Iudice b a Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow region, Russia b Dipartimento di Scienze Fisiche, Universitd di Napoli "Federico 11" and lstituto Nazionale di Fisica Nucleare, Mostra d'Oltremare Pad. 19, 1-80125 Naples, Italy

Received 10 June 1996; revised 9 October 1996

Abstract A contextual study of low-lying magnetic and electric dipole excitations is carried out for several deformed odd-mass nuclei using a separable Hamiltonian within the quasiparticle-phonon nuclear model. The M 1 and E 1 core states are composed of one plus two-phonon components, those having a different multipolarity are simply one-phonon states. All phonons are computed microscopically in the random-phase approximation. The resulting M1 and E1 spectra are compared with the available experimental data. PACS: 21.10.Re; 21.60.Ev Keywords: M1 E1 low-lying transitions, deformed odd-mass nuclei, quasiparticle-RPA (QPNM) phonons

1. Introduction A large body of information is now available on the magnetic dipole excitations, known as the scissors mode [ 1], first discovered in inelastic electron scattering experiments carried out on 158Gd [2] and, since then, detected in most deformed even-even nuclei [ 3 ]. We know from combined (e, e ~) and ( y , y ~ ) experiments [4] that the strength of the mode is fragmented into few peaks, mainly of orbital nature [ 5 - 7 ] , around -~ 3 MeV. We learned more recently from experiments carried out on a chain of Sm isotopes [8,9] and subsequently on a Nd chain [10] that the mode is closely correlated with nuclear deformation. 0375-9474/97/$17.00 Copyright (~) 1997 Published by Elsevier Science B.V. All rights reserved PII S0375-9474(96)00426-5

46

V.G. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

The measurement of the linear polarization of the scattered photons in nuclear resonance fluorescence (NRF) experiments carried out on some even Dy and Gd isotopes have enabled parity assignment for these nuclei [ 11 ]. It was possible in this way to identify electric dipole transitions in the same energy region of the scissors mode [11,12] and thereby to select the M1 transitions out of the E1 transitions. The M1 mode was studied in a large variety of phenomenological as well as microscopic approaches [ 13 ]. Only the latter, however, are suitable to describe the fragmentation of the mode. The most recent calculations of such nature [ 14] have been carried out in the Tamm-Dancoff approximation [15] and, more systematically, in random-phase approximation (RPA) [ 16-18] calculations. The low-lying M1 transitions were studied recently within the quasiparticle-phonon nuclear model (QPNM) [ 19]. This is an extension of RPA where a separable Hamiltonian is brought in a diagonal form in a space spanned by one plus two RPA phonon states [20]. It was already used to study most of the properties of all non-rotational states up to 2.5 MeV in deformed nuclei, including all E,~ and M,t transitions [21-25]. Its application to the M1 mode enabled us to investigate how the M1 strength distribution is affected by the coupling with two RPA phonons. It was found that such a coupling, though affecting appreciably the M1 distribution, does not alter substantially the RPA picture [ 19]. The question of whether the scissors mode survives as we move from even- to oddmass nuclei was posed some time ago in studies carried out within the interacting Boson Fermion model (IBFM) [26,27], in schematic RPA [28] and within the generalized coherent state model (GCSM) [29]. The first experimental search for M1 transitions in odd-mass nuclei was carried out in an inelastic electron scattering on 165Ho [30]. No strong M1 transition around 3 MeV was found in this nucleus. Subsequent NRF experiments on 163Dy have detected a sizeable MI strength around 3 MeV which, though more fragmented, fits nicely into the systematic of the scissors mode in the neighboring even-even Dy isotopes [ 31 ]. A concentration of dipole strength with the same properties has been found also in other rare-earth nuclei [12] and more recently in 167Er [32]. In this latter experiment, which covered a wider energy range, 1.9-4.3 MeV, peaks around and above 4 MeV have been detected. The interpretation of these excitations as a manifestation of a scissor-like oscillation mode found support in a theoretical analysis carried out within the IBFM [31,32]. Though appealing, such a response cannot be considered conclusive. In IBFM, in fact, as in all other schematic approaches [28,29], the problem of fragmentation, of crucial importance in odd nuclei, is overlooked. An additional source of uncertainty is inherent in the experimental analysis. In fact, the polarization techniques, adopted for parity assignment in doubly even nuclei, are ineffective when applied to odd-mass nuclei. Because of this limitation, the presence of E1 excitations intermixed with M1 transitions cannot be ruled out. One may hope to gain a more clarifying response from microscopic calculations. These may display the detailed mechanism which produces fragmentation and may give

VG. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

47

some insight into the actual structure of the observed dipole strength around 3 MeV by estimating the relative weight of the E1 transitions versus the M1 transitions in this region. The purpose of the work reported in this paper is to give a contribution in this direction. Low-lying M1 and E1 excitations in odd-mass nuclei have been contextually studied within the QPNM. This approach provides a microscopic particle-core scheme particularly suitable to these nuclei. It generates, in fact, a microscopic particle-core Hamiltonian which can be put in a diagonal form using a microscopic quasiparticlephonon basis. Such a basis embodies not only M1 (or E l ) core states but also all low-lying excitations of other multipolarities entering either in the M1 (or E l ) channels. We will carry out two series of calculations. In one case all core states are computed in RPA. A scheme of this nature was already adopted to describe the M1 transitions in 163Dy [33]. In the second case, the M1 ( E l ) modes are computed within the QPNM which accounts also for two RPA phonon admixture, while the core states of different multipolarity are simply treated in RPA. The method adopted here is outlined very shortly in Section 2 for doubly even nuclei and more extensively in Section 3 for odd-mass nuclei. The numerical procedure is illustrated in Section 4. The results are discussed in Section 5. The final conclusions are drawn in Section 6.

2. Q P N M excitations in doubly even nuclei A description of the QPNM can be found elsewhere [20]. We briefly remind here that one starts with an intrinsic Hamiltonian having the following composite structure: H = Hsp + Hpair -}'-Hph -'~/-/pp.

(2.1)

The first is a one-body Hamiltonian which includes a deformed axially symmetric Woods-Saxon potential. The second is a proton (neutron) monopole pairing interaction with strengths Gp (Gn). The other two terms, Hph and Hpp, are separable interactions acting, respectively, in the particle-hole (ph) and particle-particle (pp) channels. The ph interaction is composed of spin-independent and spin-dependent pieces. Their strengths . IA,u will be denoted, respectively, with K~u and ~v , where T labels the isoscalar (T = 0) and isovector (T = 1) terms. The pp interaction is spin independent with strengths G a•. The first step of the QPNM consists in expressing the Hamiltonian in terms of quasiparticle operators aqo- and aqt,~ through the use of the Bogoliubov canonical transformation. The symbol q stands for the single-particle asymptotic quantum numbers q = N n z A T for K = A + I and q = N n za+ f o r K = A 1, w h i l e o - = ± l relates time reversal states. In the second step one construct the RPA phonon operators

Q~ = I Z qlq2

{~bq,q2a~c~( q'q2) - qS~J'q2a~z-'r(qlq2) } '

(2.2)

V.G. Soloviev et aL/Nuclear Physics A 613 (1997) 45-68

48

where v, stands for the quantum numbers v -- {/A/x}, i = 1,2 . . . . labeling the RPA roots with a given ,~/x. A~(qlq2) and A~(qlq2) are, respectively, pairs of creation and annihilation quasiparticle operators. Their actual structure can be found in Refs. [20,21 ]. The Hamiltonian (2.1) is then expressed in terms of Q~x~,~and Qx,~, thereby assuming the quasiparticle phonon form HQPNM = Hqp q- H,, + Hvq.

(2.3)

The first term is the quasiparticle Hamiltonian of the simple form

EqOl';trOlqcr

Hqp = Z

(2.4)

q(r

The other two, Hq and H,,q are, respectively, the RPA phonon and the quasiparticlephonon coupling terms. The transformed Hamiltonian is finally brought into diagonal form by resorting to the variational principle with a trial wave function

-=

~ #* ~ 0 ,

(2.5)

where ~ ~ {7~/2}, vj = ilAj/xl, v2 = i2a2/x2 and - -

S~U°-'"2°-z=6°-"~l+'r2'~2'~;*

(

1 "~ ~it,~1/,/,1,i23,~/~ 2

1 ~--~,o ~ --- 6 ~ o )

(2.6)

The eigenvalues E~ are the roots of the secular equation (m~ det

C(uI~' ~rr0 i7o'

- E~)(3i,i, - Z

UI~U20)UI

+

21tJt'IL'z~'qt'2

(A)U2

+ Aw(vlv2) + A(vlv2) -- E~

=

0,

(2.7)

where

C(VlV2) -

1+

](~(UI, U2)

1 + 6~,,,~,2

(2.8)

The quantity w,, denotes the RPA energies, UC,',l,,2describes the coupling between oneand two-phonon states, K~is a term introduced to take into full account the Pauli principle in the two-phonon components of the wave function (Eq. (2.5)). This term induces the energy shift 2xw (vl v2) in the eigenvalue equation. The other energy shift zX(c,i u2) is due to the coupling with three-phonon states not included explicitly into the calculation. Its value is approximately A(vlv2) ~-- --0.2AgO(UlV2).

49

V.G. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

3. Excitations in odd-mass deformed nuclei 3.1. With RPA core states

Equations for describing the nonrotational states in odd-mass deformed nuclei using RPA core state were derived in [ 34]. These states have the form TO

n

7/'O

11~11 (O'oK; ) =

t

r0

CqoO~qo00 -Ju Z k

qo

qlO-I

}

Z

I~°'lgl+°-2]'l'2'°'°K°Oq IU20~ IO'1 0~20"2

]/P0.

(3•1)

,'2~'2

The normalization condition yields T0 7"0 n )2 [ l + £ X ° ( q ~ ' ~ ( Cqo ) 2 -[- ~ - ~ ( D qlv2 qo ql v2

(3.2)

1;v2)1=1'

where the factor £K°(ql;V2)

=

_ ~ v2 2 ~..,(~Pqlq2)

(3.3)

q2

comes from antisymmetrizing the quasiparticle-phonon components of (3.1). This has been done for the first time in Ref. [35]• Using the above wave function one deduces energies and eigenvectors from the variational principle. They can be found in Ref. [20]. 3.2. With Q P N M core states

As shown in [19,23,36], at energies above 2.5 MeV the coupling with the twophonon configurations induces fragmentation of the K ~ = 1+ and K ~ = 0 - , 1- modes in doubly even nuclei. Being interested in the distribution of the magnetic and electric transitions falling in the energy range 2.5-3.5 MeV, we need to take such a two-phonon coupling into account also in odd-mass nuclei. We followed the mathematical procedure developed in [37]. The basic idea is to use the already fragmented phonons in the wave function of excited states according to the procedure described by Eqs. (4.86)-(4.90) in [20]. To this purpose we chose the following trial wave function: T0

~0

[S-'T°C" ,~t qo

ql(TI

'2~r2

+Z'°Z q3gr3

1

(3.4)

hfr

From the normalization condition we got TO

~--~(Cq"o) 2 + qo

Z

_ ]2= "°(D".--q,,,2. ~211• + c K ° ( q , , • U2)] "~- ~--~r°(~)n L..~ "--q, afzh*

ql r2 a2,~2 ~ Aft

q3fl

1.

(3.5)

50

V.G. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

The QPNM core states have the quantum numbers {h, ,~ = 2, /2 = 1} for the M1 excitations and {h, ~ = 3, /2 = 0} or {h, ,~ = 3, /2 = 1} for the AK = 0 or AK = 1 E1 transitions, respectively. Under the assumption

=0,

36)

which is a valid approximation when the energy of the core states is above 2.5 MeV, we obtained after a variational calculation the secular equation V ~r° det (Eq° --En)~qoq'o -- ~4~.a Z ql

ro

4 Z Z Ei' -± q3 h

V~,2 (qoql)V[,2,,(q~oq,)[1 + £XO(q,, Xo ro . U2 ) ] SKi# 2

'2 a2/'2 * a~

Eql -}- g'Ov2 q- A K ° ( q l ' U2) -- En

vro_ t - - ~ R~ v-', vro e_l_ ~ R ~ SXO = i'Aft ~'qOq3) i ~--.~i' i,,]/2 ~'q0q3) i" K3fz

) ~ ~ _ - - ~ - _-- g

"

O.

(3.7)

The rank of such a determinant equals the number of one-quasiparticle states of the wave function (3.4). The quantities eq, o9v and E ~ n are, respectively, the quasiparticle, RPAphonon and QPNM energies, AX°(ql, v2) is the energy shift due to the Pauli principle and SxX~,= ~$1g~lzl,g0( 1 -- ~5#,0) + a~,,06g, go ( 1 + ~3~,0).

(3.8)

The remaining term is

( K ~ " + p K 1 )UqlqlX~(U ) - a

V~r,(qlq2)=fa~(qlq2)

Uqlq2X[,(u )

,

(3.9) where f a ~ ( q l q 2 ) are the single-particle matrix elements of the multipole operator of rank A/z, while the other quantities are r

+

r

X~,(u ) = Z f qlq2

a,u

U(+)

v

v

(qlq2) qlq2(~llqlq2 "~ ~ q l q 2 ) ,

xr(v-)=~"~rf~lz(qlq2

( - ) ( ~llqlq2 v ~- ~ qvl q 2 ) ) Uqlq2

(3.10)

q]q2

and

u(+) qlq2 :

Uql Uq2 + l'lq2Uql

v qlq2 (±)

Uql Uq2 J: Uql Uq2"

=

(3.11)

In the above equations Uq and vq are the Bogoliubov transformation coefficients, ¢'u~q2 a n d qSqlq2 are the forward and backward amplitudes of the RPA phonons (2.2).

V.G. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

51

3.3. E1 and M1 transitions In computing the E1 and M1 transition probabilities we used the total wave function

1 [D~KgrK~,~ + (_I),+XD / KgGK'-'~ ]

g'/Mx~,r

V

(3.12)

-

The reduced E1 transition probabilities from the ground to the excited states, with 7rf quantum numbers {no = 1 l~°K0} (K0 va ½) and {nflf KT}, respectively, have the following form:

B~°(E1;noI~°Ko ---+nfl;I K f , # = [Kf - Kol) rO

1 ~ (El) = (loKoltz]IfKf)212.~pl ~ (Toqoqlo)C qo~q; "°c'"I qoq; TO

+½(1 + 8u,O) Z

Cqo (+) "o Z p(El) 1~, (Tqlq2)Uqlq2

qo

rqlq2

(t/t i'g# + d~i'~f'] 2, D qoh3[x hI- R~[1 +12XJ(qo;3[xi')].~.qlq2 l "rql q2 ~

x

(3.13)

fii'

where

(El) /~'qlq2) =(ql ] .Ad~(EI) ]q2) = (ql ] %ff PI~ _(l) ('r)rYl~lq2)

(3.14)

are the single-particle matrix elements of the El operator with effective charge

(1).

1 (

eeff trz) = - ~ e

re

N--Z)

A

(I + x).

(3.15)

The factor X is a fitting parameter introduced to quench the too large E1 transition probabilities obtained with the standard expression (X = 0). The M1 reduced strength is B~°(Ml; nolg°Ko --~ n f I ; ' K f , #

= IKf - Kol)

]@'p(M1)__ 1~ , ~ouo,

= tz~(logollxllfKf)2lqoZ._~

(~_. . . . , ~ , ~ + ) . ~ o . ~ . ~

qoq;t-'qo t..q;

"I"0

+½~3u,' ~_, Cqro° ~'~p~Ml) (rqlq2)u(q~X(qlq2). qo ×

rqlq2

~-'~Dnfqon2--R~' 1 1, [ -}- £ K r ( q o ; 2 1 i t ) ] ' T q (tit l q 2 i'21

_

d)i'21 ] 12 v-qlq2"[

where u}z~q~X(qlq2) = U(q~qlX(q2ql) with x2(qlq2) = 1 and

,

(3.16)

EG. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

52

(M1) Plu (~'qlqz) =(ql

IMr(M1)

I q2)

= ~/-~ (q, l [gt(r)l~, + geff(T)SF,] l q2)tzN

(3.17)

are the single-particle matrix elements of the MI operator. We used a bare orbital gyromagnetic factor and an effective spin factor g~ff = u./gs , . free• The terms in C"[D ~° do not appear in the equations giving the reduced probabilq0 q~'A~ ities. Indeed, the contribution of the quasiparticle-phonon configurations to the normalization of the ground-state wave function in odd-mass well-deformed nuclei is small. Other negligible terms have also been ignored. Given the impossibility of distinguishing experimentally between E1 and M1 transitions in odd-mass nuclei, we could appropriately compute the widths, multiplied by the statistical factor g = (21f + 1 )/(2•0 + 1). These quantities are in fact parity and spin independent and are related to the reduced strengths according to g F 0 ( E 1 ) = 1.0467(E~,[MeV])3B(E1)

T [ e2fmzl0-3] meV,

g F 0 ( M 1 ) = I1.547(Ez,[MeV] )3B(M1) ~ [/z~] meV.

(3.18)

We computed also the reduced widths gF~)ed(E1) = 1.0467B(E1) Y [ e2fm210-3] m e V ( M e V ) -3, g l~red o (MI)=

l l . 5 4 7 B ( M 1 ) T [/x2] m e V ( M e V ) -3

(3.19)

4. Numerical procedure Single-particle energies and wave functions were computed using a deformed axially symmetric Woods-Saxon potential following a procedure whose details are given in Refs. [20,38]. Such a procedure fixes the parameters of the potential as well as the quadrupole and hexadecapole deformation parameters /32 and /34. The values of all parameters can be found in Refs. [ 19,38]. The single-particle spectrum was taken from the bottom of the well up to +5 MeV. Two-quasiparticle configurations up to an excitation energy of 30 MeV were taken into account. Monopole and quadrupole pairing were included in the calculation of the quasiparticle energies and amplitudes. The quadrupole pairing was extracted from the A# = 20 pp interaction. Its strength was chosen to be G 2° = K~°. For this value of G 2°, the strength of the monopole pairing was fixed so as to reproduce the experimental odd-even mass differences. Blocking effect and the Gallagher-Moszkowski corrections [39] were taken into account in computing the two-quasiparticle energies. We used G a~ = 0.8K0au for all other multipolarities (A/z 4= 20). As shown below, by so reducing the constant G 21, we got a better agreement with the experimental M1 strength in doubly even nuclei. The separable interaction includes spin-independent particle-hole multipole terms with coupling constants Koa~ , tqa~ and spin-dependent components with strengths Kla/, 0 , Kl~ 1

VG. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

53

The K ~ = 0 - and K ~ = 1- RPA phonons were calculated in Refs. [22,40] using ph and pp isoscalar and isovector octupole and ph isovector dipole interactions. The K ~ = 1+ RPA phonons were computed in [19]. The Hamiltonian used in this case embodies isoscalar and isovector quadrupole interactions acting on both ph and pp channels as well as a ph spin-spin interaction. The redundant rotational state [41] was excluded by a proper choice of the isoscalar quadrupole coupling constant K~]]. We determined a critical value by imposing that the energy of the lowest RPA K = = 1+ state vanishes. We then searched for a slightly larger value for which the overlap of all RPA K ~ = 1+ states with the spurious rotational mode resulted to be practically vanishing. The phonons of different multipolarity were calculated using isoscalar and isovector interactions embodying the appropriate multipole fields. Our phonon basis consists of ten (i --- 1,2 . . . . 10) phonons of a given multipolarity: ,~/z = 2 0 , 2 2 , 3 2 , 3 3 , 4 3 , 4 4 , 5 4 and 55. We used 25 (i = 1,2 . . . . . 25) phonons of ,~/x = 21 and ,~/z = 30 and 31 multipolarities. The strengths of the spin-independent isoscalar particle-hole (ph) interaction terms K~u were fixed so as to reproduce the lowest experimental energy level for each K ~" l + [21-24]. The parameters so determined were also used to compute the M1 QPNM states in doubly even nuclei. The only parameters left were the isovector multipole constants and the spin strengths. The first were fixed according to the relation K~u = --1.5K~ u in substantial agreement with other choices [42]. The spin isoscalar coupling constant was taken to be ten times smaller than the isovector one consistently with the estimates obtained in a sum rule description of spin excitations in heavy spherical nuclei [43]. All parameters used in the study of the M1 modes can be found in Ref. [ 19]. The same phonon basis was used for doubly even and odd-mass nuclei. For the calculation of nonrotational states in odd-mass nuclei with mass number A + 1, we used a phonon basis for doubly even nuclei with mass number equal to A.

5. Numerical results Numerical calculations were carried out for

157Gd, 159Tb, 161Dy 163Dy and 167Er. The

core states entering into the quasiparticle-phonon basis are, respectively, the states of ]56Gd, 158Gd, 16°Dy, 162Dy, and 166Er. The spectra of !57Gd, because of their peculiar properties, will be discussed separately at the end of the section

5.1. M1 transitions A reliable study of the M1 spectra in odd-mass nuclei requires as preliminary condition an accurate description of the corresponding strength distributions in doubly even mass deformed nuclei. To this purpose we used for the A = 2 # = 1 quadrupole pairing strength the value G 21 = 0.8Kg ] instead of G 2j = K021 adopted in Ref. [ 19]. By this new choice a better

54

V.G. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

Table 1 Summed M1 strengths (in /z~) of doubly even nuclei in the 2.7-3.7 MeV energy range computed for two different quadrupole pairing strengths G21/K21

158Gd

16ODy

162Dy

166Er

1 0.8 Exp

4.67 3.41 3.39

3.25 2.46 2.42

3.28 2.60 2.49

3.07 2.51 2.67

Table 2 M I energies, strengths and structure of the K7r = 1+ states in 162Dy Experiment En (MeV)

B(M1)T (/zeN)

Calculation in QPNM E,, (MeV)

B(M1)]" (#~) gs = 0.7

Phonon structure ( A/z)i

(%)

0.14 0.09 0.76 0.07 0.90

(21)2 (21)3 (21)4 (21)6 (21)7 (31)1Q (32)2 (21)7 (2l)8 (32)1 • (33)t (21)7 (31)1 ® (32)1 (21)10 (21)H (31)2 @ (33)1 (21)11 (3l)2 ~ (32)1 (21)12 (31)~®(31)~ (32)1 ® (33)2

95 96 97 93 62 17 4 28 59 22 64 88 60 28 27 62 64 6 7

2.395 2.569 2.90

0.52 4- 0.03 0.13 4- 0.1 1.63 4- 0.10

2.35 2.40 2.52 2.74 2.83

2.965

0.10±0.001

3.01

0.07

3.06

0.86 4- 0.08

3.06

0.36

3.17 3.23

0.11 0.10

3.31

0.20

3.43

0.30

description o f the M1 transitions in all doubly even nuclei is achieved. Table 1 shows indeed that the resulting M I s u m m e d strengths in the energy range 2 . 7 - 3 . 7 M e V are smaller than the values obtained in the previous calculation [ 19] and are in better a g r e e m e n t with the experimental data [ 11,12]. A p p r e c i a b l e discrepancies between theory and experiments still remain as regard to the strength distributions (Table 2). In g o i n g f r o m even to odd nuclei the fragmentation o f the strength is dramatically enhanced. To illustrate this p h e n o m e n o n we show in Fig. 1 the M I spectra c o m p u t e d within R P A and Q P N M for 162Dy and 163Dy. The spectrum o f the odd-mass nucleus is m u c h richer than in the case o f the doubly even one. A further increase in fragmentation is observed w h e n the M1 core states are c o m p u t e d in Q P N M rather than RPA. That the strength should get strongly fragmented in g o i n g from doubly even to oddmass nuclei was largely expected. On one hand, the q u a s i p a r t i c l e ® ( ~ f i ) i c o m p o n e n t s

V.G. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

2

QPNM

lSSDy

~).5

55

RPA

16ZDy 1.5

1

~0.5

3.5

,I

Ii

QPNM

18aDy

3

,,I

(~n

x

I I

163Dy

RPA

#)

(~n

x

#)

z

~'Z'0.5

3.5

li[

.,. ,i,.i., ,i

0

163Dy

I

'I

,.. ,i,i i IBIlII

.'..,[,,

I

QPNM

I

63Dy

(X=2 #=1)

Ca

(X

1

i

't

i ,,

2.5

3 E (MeV)

i)

'

0 2

RPA

2 #

2

. 2.5

~1

3 E (MeV)

Fig. 1. QPNM and RPA M1 strength distributions in 162Dy and 163Dy. Full and dotted lines refer, respectively, to transitions to Kf = 3 and g f = } final states.

couple to several one-quasiparticle configurations. As a rule, the fragmentation so induced is rather weak due to the small number of one-quasiparticle states. On the other hand, the strength collected by each M1 state in a doubly even nucleus is distributed among four M1 levels in the neighboring odd-mass nucleus. In this latter system, in fact, the M1 operator can couple the {K0, I0 = K0} ground state to a multiplet of four excited states with quantum numbers { (Ko - 1,10 - 1 ), (K0 - 1, I0), (K0 - 1, Io + 1) } and { K o + 1 , 1 0 + 1}. Let us study this problem more quantitatively by analyzing the results obtained for ]63Dy. The strength of the strongest M1 transition occurring in ]62Dy at an excitation energy E = 2.90 MeV and estimated to be B ( M I ) Y-~ 0 . 9 0 # ~ (Table 2) is distributed

V.G. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

56

Table 3 Energies, widths and strengths of the ground-state ( K ~r = 5 - ) strongest M1 transitions in 163Dy. The summed widths and strengths are also given

Kf7r If

Ef (MeV) 2.59

2.89

3.12

I- ® (a~),}

(meV)

(meV (MeV) - 3 )

(#~)

2.42

0.21

17.98

1.04

0.09

62.93

3.63

0.32

82.87

3.45

0.30

35.52

1.48

0.13

32 - !2 _5 2

2.58

Structure

~ , , 8(M1) T

41.95

2

5_ 2

3.12

B (M 1 ) T

Eli 11,~a

5_ 2

32 - !2

3.1 I

Fred

~,~ r,,

3- 3 2

2.89

l'0

12 - 32

72 - 72 7 - 27

2

7 - 72

2

112.31

5.17

0.45

29.87

0.99

0.09

12.80

0.42

0.04

44.82

1.48

0.13

10.86

0.36

0.03

16.30

0.54

0.05

75.75

4.43

0.38

75.75

4.43

0.38

129.78

5.40

0.47

129.78

5.40

0.47

57.54

1.90

0.16

57.54

1.90

0.16

(%)

1523 + ®(21)4)

77

1660 T @(31),}

12

1523 I

@(21)7)

100

1523 .[ ®(21)9}

74

1660 T Q(31)3)

26

1523 I @(21)9)

26

1660 T @(31)3)

73

1523 i @(21)4}

100

1523 .L @(21)7)

100

[523 .[ @(21)9)

100

in 163Dy almost equally among the K y = 73 - and K7 = 7 - levels (Table 3), both having an intrinsic excitation energy 2.89 MeV. The two K7 states, indeed, collect, respectively, Z

B(M1)(KT=

I i~ = 5 - __+ K~~ = ~3 - ~I f ) ~-- 0.45/z~,

B ( M 1 ) (K7 = I F = 5 - ~ K~ = 17 = 27--) _~ 0.47tzr~. The K7 = 73 -

strength, however, is further distributed among the J7 = 73 - , 52 - , 7 -

states with strengths B ( M 1 ) ~_ 0.30/x 2, 0.13#~, 0.02/z~, respectively. The J f7"g ~ 3 - 2 of the strength. That the Jf~ = K7 states take ~ of the K f~ strength is state gets about .~ a general feature to be ascribed to angular momentum coupling. The states collecting appreciable amounts of M1 strength are dominated by a single quasiparticle-phonon configuration obtained by coupling the valence quasiparticle to the

V.G. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

57

Table 4 Energies, widths and strengths of the ground-state strongest El transitions in 163Dy E.f

K~fl f

(MeV) 2.25

2.50

2.85

~5+ 5 72

2.28

(meV)

(meV (MeV) -3)

B(E1)T ~ , I B ( E1 ) T (e2fm210-3)

85.94

7.53

7.19

3.01 10.54

2.88 10.07

~5+ 5 7_ 2

35.79

2.29

2.19

14.32 50.11

0.92 3.20

0.87 3.06

5+5 2 2

162.61

7.06

6.74

65.05 227.66

2.82 9.88

2.70 9.44

202.08

5.65

5.40

80.83 282.91

2.26 7.91

2.16 7.56

5.44

1.09

1.04

8.16

1.64

5_+ _5 2 2 72

1.71

Fred

~-~lfFo ~-~ifl'lwd 34.38 120.32

7_ 2 3.30

Fo

3+ ! 2

2

!2 + 23

12.39

1.04

0.99

17.59

1.55

1.29

Structure

1523 .L @(30)2)

96

1523 + @(30)3)

99

[523 ,[ @(30)4)

99

[523 ~ @(30)6) [512 T ®(30)3)

88 12

1523 .~ @(31)~) [521 T ®(30)1)

54 46

I523 .~ @(31)3)

100

(342) = 21 phonon. Only in few M1 transitions to the K 7 = ~3 - final states several other configurations like the [ 660 T ® ( 3 1 ) ) component play a significant role. The effect of the substantial purity o f the M I states is illustrated in the lower part of Fig. 1 showing the M1 strength distributions when only the (~42) are included in the calculation. This is what is done in schematic approaches [28]. The spectrum so obtained is quite similar to the corresponding one deduced from including all core states. Given the impossibility o f determining experimentally the spins of the final states, we have computed the spin-independent quantities g F 0 ( M 1 ) and gF(o red) ( M I ) in order to make a consistent analysis o f the experimental data. The spectra of the reduced widths of all nuclei but 157Gd, computed in QPNM, are compared with the experimental data in Figs. 2-5. The following points are noteworthy: (i) The computed M1 transitions fall in the region of the observed peaks in all nuclei. The discrepancies in the energy distribution with respect to experiments are of the same order as in the nearby doubly even nuclei. (ii) The K '~ ~ K ~ + l M1 transitions are fewer but in general much stronger than in the K ~ ~ K '~ - 1 case. As already said for 163Dy, the M I strength of the neighboring doubly even nuclei is equally shared by the two K~ states, but for K~ = K F - 1 the strength is further distributed mostly

58

V.G. Soloviev et a l . / N u c l e a r Physics A 613 (1997) 4 5 - 6 8

4

161Dy

Exp ? 3, > i (D

i v

,I ,

0 4

,JI

I

I

Ii

QPNM

co

;> ~3 >

~2

12 P~11 co

r

QPNM

'1

I

I

10 9 8 7 6

5 4

I I I I I II

£3 ! IJII

I

I

[

2.5

2

}

:1

,

3 E (MeV)

Fig. 2. Ground-state decay reduced width distribution in 161Dy. Full, dashed and dotted lines refer, respectively, to K f = 3 5, 7 final states. Table 5 Summed M1 and El reduced widths in odd-mass nuclei. The sum is given in meV (MeV) -3 gl'~)red )

159 Tb

161 Dy

up to 3.37 MeV M1 E1 Exp

16.7 7.5

163 Dy

J67 Er

up to 3.1 MeV 21.7 12.4 8.2

12.3 11.9 11.1

43.2 40.2

59

V.G. Soloviev et a l . / N u c l e a r Physics A 613 (1997) 4 5 - 6 8

163Dy

Exp ,

;>

3

3> 2

E

v

,ill

F..~ 1

oI 8 g-"7

, ILIll,

QPNM

>

>5 ¢0 v

E4

~'.3

r-..-~ 1

I il:

0

¢-7

i4 ~,

, =

I li

I QPNM

>a:

~_.,2 ~-

~"1 f 0 2

I I I I

I I I

L'

i',

2.5

[

,

,

3 E (MeV)

Fig. 3. Ground-state decay reduced width distribution in 163Dy. Full, dashed and dotted lines refer, respectively, to K I = 2' 2' 7 final states.

among two out of the three J f components of the K~ multiplet. (iii) The magnitude of the strongest M1 peaks is about twice the intensity of the corresponding observed transitions. Also the summed reduced widths are in general about twice the experimental values (Table 5). 167Er represents a remarkable exception. As shown in Fig. 6, the computed strength summed up to 4.3 MeV is only slightly larger than the experimental sum [32], if the latter includes also the transitions for which the experimental error is more than 50%. Computed and experimental running sums, however, increase with energy following different paths. This reflects the discrepancy in the energy distribution.

EG. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

60

159Tb

Exp

3 i

::> Gp

>~2i v

I i ,I,I, I,

I Ihll

IIIJllllI,II,l,II

I

I

QPNM

~"3 >. >

~2

L," 1

o

, i il,i,

°

0 2

2.5

3

3.5

E (MeV)

Fig. 4. Ground-state decay reduced width distributionin 159Tb. Full and dotted lines refer, respectively, to Kf = ½ and 25-final states. Indeed, the computed strength is on the whole shifted toward lower energies. In our case the spreading is a combined effect of the coupling o f the, already fragmented, K ~ = 1 + mode to the quasiparticle states and of the different rotational energies, of the order of 100 keV, carried out by the multiplet states, because of their different spins.

61

V..G. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

167Er

Exp 4

b

~3 >

~2

Ii ,ILlllLI,ItI LIIJIll I QPNM

I

I

I

t--.4

e3

,> gJ

;>3 v

~-2 v

1

i,J

o

2

2.5

3

i,,liLLtt,ltl 3.5

4

E(MeV)

Fig. 5. Ground-state reduced decay width distribution in i67Er. Full and dotted lines refer, respectively, to Kf = 5 and 9 final states.

5.2. E1 t r a n s i t i o n s

In order to fix the effective charge we have computed within the QPNM the E l transition probabilities in 168Er, for which E1 transitions have been studied more thoroughly. Using for the effective charge Eq. (3.15), we fixed the parameter X by an overall fit of the experimental summed strength in the energy range 1.7-4.0 MeV [44]. We got from such a fit ( 1 + X) = 0x/-0-~.2,which is appreciably larger than the value ( 1 + X) = 0.3 adopted in other papers [45]. We have used this new effective charge also to compute

~uo~ls 'puv, q aoqlo oql uo 'SOA[~ 1I "sluomuodxo qa!A~ luatuOo.~e u! AoIN g ~OlO q uo!l!s -tll]Jl ]~ I]UOJ1S P, spla!X uo!lelnalea oql 'paapuI "uo!lnq!Jls!p £~'Jouo I~ oql JOl onJ1 lOLl s~ arums o q l "pa,'mpoJdaJ s! ql~uaJlS 1~ patutuns 0ql £luo leql 1no lu!od ol lUelJodtu! sl 1I "[17t7'?;I] mep lmuatupadxa oql ql!x~ 3:iluols!suoa [017] suo!lelnalea snmAaJd u! apetu ,(peaJle s e ~ uo!la!paJd s!qj. "suo!l!sueJl 1 = XV aql JOj ueql JagJe I sam!l 17-£ paapu! aJe sql8uaJls o%!puodsaJ.ma Oqd" 'sauo ]uv,u!tuop aql Jej ,(q aJe sUO!l!SUeJl l~l pamdtuoa 0 = XV aql !olanu qloq u I " tuj~a~ 01 × ~;g =.[ { d × + / ( l R ) 8 ~ [1717] anleA paJnseatu aql OnlgA oql ogu-eJ 3~gJouo AolAI 9"£-9"1 o41 ~OAO potutuns qlgumls oql ~ot gu!u!elqo .I~1991 llI tumlaods I~l oql

ql!A~ )tlOUlOO.l~l~OSOla 3~JOA U! ;tuj~a~: OI × g'gg" =.~ ( l ~ ) f i ~

".I~L91

U! sqI~uOJlS UO[llSUl~J1

IIAI JO s u l n s ~ u l u u n J

(ou!l [ l n j )

[lz~!l;)Jooql pu~

(aU!lpoqs~p) l~lUam.uadxB~'9 '~!d

(A~w) 3

jJ...J Y

.J

r-J

° 1-oo-~

f

M

ooo

I:i:1

A . j6" o ' " .I'

""J F

¢o+I

f....i

. .,.f( 'I~,W T

I

I

,

t

89-ffP (Z66I) YI9 V sa!s£Ztd JValanN/ 7° ta aa~aoloS "D'A

"~9

~G. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

63

transitions also in the energy range 2.2-3.2 MeV. Experimentally, only weak transitions have been measured in this region. In odd-mass nuclei the fragmentation mechanism of the AK = -4-1 El strength is exactly the same as for the MI transitions. Indeed, the K = 1 component of the E1 operator connects the {K0, I0 = K0} ground state to the states { ( K 0 - 1, I0 - 1), (K0 1,10), (K0 - 1,10 + 1)} and {K0 + 1,10 + 1} thereby distributing the strength of each E1 transition in a doubly even nucleus among four E1 levels in the neighboring oddmass nucleus. The strength of each AK = 0 E1 transition, instead, being shared by the { K f = Ko, If = Ko, Ko + 1} final states, splits into two E1 peaks. Exactly as in the case of the M1 transitions, the AK = 1 El strength is also equally shared by the K 7 = K7 - 1 and K 7 = / ( 7 + 1 levels and the J7 = K~~- states take ~2 of t h e / ( 7 strength. However, as in doubly even nuclei, the AK = 1 E1 strength represents a small fraction of the total transition probability. This is mostly concentrated in the AK = 0 transitions (Table 4) consistently with the results obtained for doubly even nuclei. These transitions are in general more than five times stronger. Even in this case the J f = Kf states are a factor two more strongly excited than the J f = KU + 1 states. As in the M1 case, the E1 states have a dominant configuration embodying the (A/2) phonon. In 163Dy for instance, the strongest transitions are dominated by the ] 523 ® ( 3 0 ) ) configuration, coupled in few cases to ] 512 Y ®(30)3) and other components (Table 4). Figs. 2 and 3 show the spectra of the El reduced widths computed for 16JDy and J63Dy, We can see that the M1 widths are much larger than the corresponding AK = 1 El widths. Indeed, as in the doubly even nuclei, the El strength is concentrated almost entirely in the AK = 0 transitions. These can be quite strong. In both nuclei, the strongest peaks are larger than the magnetic transitions and more than three times the experimental widths. I n 161Dy the E1 strength is concentrated in two peaks around 2.5 MeV. The spectrum overlaps little with the experimental one as well as with the computed M1 strength distribution. In J63Dy instead, no clear separation is noticeable, since the computed strongest peaks fall near the energy range of the observed widths. As shown in Table 4, these strongly excited states are composed of the (523 l) quasiparticle configuration coupled to octupole core states with excitation energies falling in the range 2.4-3.2 MeV. These core states are also strongly excited according to the model. Experimentally, however, no strong E1 transitions have been measured so far above 2 MeV neither in 16°Dy nor in 162Dy [46]. We could reduce the strength of these transitions by using a smaller effective charge, as done in Ref. [45]. This new effective charge, however, would quench also the low-lying E1 transitions thereby spoiling the agreement with experiments. This is an unsolved problem yet, which requires detailed specific studies. We now come to the analysis of 157Gd. Its observed dipole strength is known to be distributed over about 90 transitions covering almost uniformly the energy range 2 4 MeV [12]. As shown in Fig. 7, the computed MI strength, though quite fragmented, does not fill the full range of observation. It is in fact concentrated around 2.3 MeV and, to a much greater extent, in the interval 2.8-4. MeV. The gap is partially filled by

EG. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

64

Exp

l~TGd

~>3 >2 v

F..._,~ 1

0 6 QPNM

>

4

~3 v

~2 v

,idtL[ I QPNM

I

I

~3

I I

I

~2

, I i 0

,

f

~1 . . . .

i[. !i

I

I I

:I,

I

_!_

J

2.5

3

lii

'., 3.5

E (MeV)

Fig. 7. Ground-state decay reduced width distribution in ]57Gd. Full dashed and dotted lines refer, respectively, 3 5 to K f = ½, ~, 2 final states.

the E1 peaks. By including the E1 spectrum, we obtain a number of peaks comparable with the experimental data. Serious discrepancies still remain in the distribution and, to a greater extent, the magnitude of the strength. In some transitions, indeed, the peaks are five times higher. Moreover the two spectra overlap to a large extent.

VG. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

65

6. Conclusions A parameter-free, fully microscopic, quasiparticle-phonon scheme has been adopted to study the M1 and E1 low-lying levels in deformed odd-mass nuclei. In order to give an exhaustive account of these transitions the one-quasiparticle states were coupled to core states of several multipolarities (up to ,~ = 5). Those describing the E1 and M1 modes were composed of one plus two RPA phonons and were treated within the QPNM. The others were RPA pbonons only. The use of QPNM rather than RPA E1 and M1 core states has produced more fragmented spectra consistently with experiments. The effect is modest nonetheless. The enhanced fragmentation is almost entirely due to angular momentum coupling between valence quasiparticles and M1 or El QPNM core states. The dominance of these configurations and the geometrical origin of the fragmentation indicate that the scissors picture of the M1 mode remains substantially valid also in odd-mass nuclei. We have found that the strongest computed M1 transitions fall in the region of the observed widths, around 3 MeV. Also the quite pronounced observed fragmentation of the strength is qualitatively reproduced. In particular, the large spreading observed in 167Er is accounted for to a fair extent. All these results support the magnetic nature of the observed transitions. On the other hand, our calculation has produced an E1 strength comparable in magnitude to the M1 total strength. In 161Dy the strongest MI and E1 transitions fall in different energy regions. This could help, in principle, to discriminate between them. We are dealing however with energy differences of the order of a few keV. The computed distribution of the strength is not so accurate as to enable us to draw such a conclusion. It may be worth to point out that, in fact, even the problem of reproducing accurately the MI spectra in doubly even nuclei is unsolved yet [ 19]. In any case, such a criterion would not apply to 163Dy, where the two spectra overlap. It must be pointed out once again that in the present model, the strong E1 transitions predicted for odd-mass nuclei correspond to strong E1 excitations of collective octupole core states. However, no strong E1 transitions above 2 MeV have been observed in the doubly even nuclei entering in our calculation. Only once such a discrepancy will be understood and removed, it will be possible to draw reliable conclusions for odd-mass nuclei on the possible existence of strong E1 transitions in the same region of the MI excitations. Independently of this uncertainty, our calculation suggests a criterion for identifying the nature of the transitions. According to our findings, almost the whole El strength should be carried out by the AK = 0 transitions, while the MI strength is entirely promoted by the AK = 1 M1 operators. The nature of each observed transition could therefore be established by measuring of the spin of the final state. Unfortunately, an experiment which measures spins in odd-mass nuclei is not easily feasible. Alternatively, if the decay branching ratios to another low-lying state (different from the ground state) could be measured for each transition, one would hopefully deduce the K quantum numbers of the excited states from an analysis of the experimental data based on the

66

VG. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

Alaga rule. It would therefore be possible to reliably assess the nature of each transition in view of the different A K quantum numbers carried by the E1 and M1 operators. Our calculation poses one additional problem. The strengths of the strongest M I or E1 transitions are at least a factor two larger than the measured values. With the exception o f E67Er, the summed M I widths are also more than two times larger. These discrepancies imply that, on the theoretical side, more fragmentation and some overall quenching should be produced. On the other hand, we like to stress that no attempt has been made to adjust the parameters so as to improve the agreement with experiments. All parameters were in fact fixed in previous calculations by a best fit of some selected properties of the electric and magnetic levels below 2 MeV in doubly even nuclei. A fine-tuning adjustment might yield substantial improvements. The occurrence o f E1 transitions of appreciable strength in the observation region cannot be ruled out but cannot be assessed with certainty due to the discrepancies between theory and experiments found in the doubly even nuclei for these transitions. On the other hand, the M1 and E1 transitions are predicted to carry different A K quantum numbers. More refined experiments may hopefully exploit this fact in order to settle upon the exact nature of the observed widths. The above comments do not apply to 157Gd. Both M1 and E1 theoretical spectra are to be included in order to be able to account, though only qualitatively, for the extreme fragmentation of the observed dipole strength. On the other hand, the computed strength is exceedingly larger than the observed one. Before making any assessment on the nature of the observed spectrum, it is necessary to explore if it is possible to find a quenching mechanism for the transition which redistribute the M I and E1 strength. Investigations in this direction are under way.

Acknowledgements We wish to thank U. Kneissl for fruitful discussions and for having provided us with experimental data before publication. One o f the authors (N.L.) likes to thank A. Richter and P. yon Neumann-Cosel for useful suggestions and for having made available the data on 167Er before publication. This work was supported by the grant R F F R 95-01-05701 from the Russian Foundation for Fundamental Research.

References [ l] N. Lo ludice and E Palumbo, Phys. Rev. Lett. 41 (1978) 1532; G. De Franceschi, E Palumbo and N. Lo ludice, Phys. Rev. C 29 (1984) 1496. 12] D. Bohle, A. Richter, W. Steffen, A.E.L. Dieperink, N. Lo ludice, E Palumbo and O. Scholten, Phys. Lett. B 137 (1984) 27. [3] For a summary see A. Richter, Nucl. Phys. A 507 (1990) 99c; A 522 (1991) 139c. [41 D. Bohle, A. Richter, U.E.P. Berg, J. Drexler, R.D. Heil, U. Kneissl, H. Metzger, R. Stock, B. Fischer, H. Hollick and D. Kollewe, Nucl. Phys. A 458 (1986) 205.

V.G. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

67

[5] C. Djalali, N. Marty, M. Morlet, A. Willis, J.C. Jourdain, D. Bohle, U, Hartman, G. Kiichler, A. Richter, G. Caskey, G.M. Crawley, and A. Galonsky, Phys. Lett. B 164 (1985) 269. [6] C. Wesselborg, K. Schiffer, K.O. Zell, E von Brentano, D. Bohle, A. Richter, G.EA. Berg, B. Brinkmrller, J.G.M. R/Smer, E Osterfeld and M. Yabe, Z. Phys. A 323 (1986) 485. [7] D. Frekers, D. Bohle, A. Richter, A. Abegg, R.E. Azuma, A. Celler, C. Chan, T.E. Drake, K.E Jackson, J.D. King, C.A. Miller, R. Schubank, J. Watson and S. Yen, Phys. Lett. B 218 (1989) 439. [8 ] W. Ziegler, C. Rangacharyulu, A. Richter and C. Spieler, Phys. Rev. Lett. 65 (1990) 2515. [9] C. Rangacharyulu, A. Richter, H.J. Wrrtche, W. Ziegler and R.E Casten, Phys. Rev. C 43 ( 1991 ) R949. [10] J. Margraf, R.D. Heil, U. Kneissl, U. Maier, H.H. Pitz, H. Friedrichs, S. Lindenstruth, B. Schlitt, C. Wesselborg, E von Brentano, R.-D. Herzberg and A. Zilges, Phys. Rev. C 47 (1993) 1474. I 11 ] H. Friedrichs, B. Schlitt, J. Margraf, S. Lindenstruth, C. Wesselborg, R.D. Heil, H.H. Pitz, U. Kneissl, P. von Brentano, R.-D. Herzberg and A. Zilges, D. Htiger, G. Miiller and M. Schumacher, Phys. Rev. C 45 (1992) R892. [ 12] J. Margraf, T. Eckert, M. Rittner, 1. Bauske, O. Beck, U. Kneissl, H. Maser, H.H. Pitz, A. Schiller, P. von Brentano, R. Fischer, R.-D. Herzberg, N. Pietralla, A. Zilges and H. Friedrichs, Phys. Rev. C 52 (1995) 2429. 13] See for references, A. Richter, Prog. Part. Nucl. Phys. 34 (1995) 261. 14] For earlier references see C. De Coster and K. Heyde, Nucl. Phys. A 524 (1991) 441. 15] K. Heyde and C. De Coster, Phys. Rev. C 44 ( 1991 ) R2262. 16] I. Hamamoto and C. Magnusson, Phys. Lett. B 312 (1992) 267. 17] E Sarriguren, E. Moya de Guerra, R. Nojarov and A. Faessler, J. Phys. G 19 (1993) 291. 18] A.A. Raduta, N. Lo ludice and I.l. Ursu, Nucl. Phys. A 584 (1995) 84. 191 V.G. Soloviev, A.V. Sushkov, N.Yu. Shirikova, and N. Lo ludice Nucl. Phys. A 600 (1996). 201 V.G. Soloviev, Theory of Atomic Nuclei: Quasiparticles and Phonons (Institute of Physics, Bristol and Philadelphia, 1992). 121 I V.G. Soloviev, A.V. Sushkov and N.Yu. Shirikova, Nucl. Phys. A 568 (1994) 244. 122] V.G. Soloviev, A.V. Sushkov and N.Yu. Shirikova, Int. J. Mod. Phys. E 3 (1994) 1227. 1231 V.G. Soloviev, A.V. Sushkov and N.Yu. Shirikova, Phys. Rev. C 51 (1995) 551; Yad. Fiz. 59 (1996) 57. [241 J. Berzins, E Prokofjevs, R. Georgii, T. Hucke, T. yon Egidy, G. Hlawatsch, J. Klora, V.G. Soloviev, A.V. Sushkov and N.Yu. Shirikova, Nucl. Phys. A 584 (1995) 413. [25] V.G. Soloviev and N.Yu. Shirikova, Nucl. Phys. A 542 (1992) 410. [26] E Van Isacker and A. Frank, Phys. Lett. B 225 (1989) 1. [27] A. Frank, J.M. Arias and E Van Isacker, Nucl. Phys. A 531 (1991) 125. [28] A.A. Raduta and N. Lo ludice, Z. Phys. A 334 (1989) 403. [29] A.A. Raduta and D.S. Delion, Nucl. Phys. A 513 (1990) 11. [30] N. Huxel, W. Abner, H. Diesener, P. von Neumann-Cosel, C. Rangacharyulu, A. Richter, C. Spieler, W. Ziegler, C. De Coster and K. Heyde, Nucl. Phys. A 539 (1992) 478. [31] I. Bauske, J.M. Arias, E von Brentano, A. Frank, H. Friedrichs, R.D. Heil, R.D. Herzberg, E Hoyler, E van Isacker, U. Kneissl, J. Margraf, H.H. Pitz, C. Wesselborg, A. Zilges, Phys. Rev. Lett. 71 (1993) 975. [321 C. Schlegel, P. von Neumann-Cosel, A. Richter and E van lsacker, Phys. Lett. B 375 (1996) 21. [33] V.G. Soloviev, A.V. Sushkov and N.Yu. Shirikova, Phys. Rev. C 53 (1996) 1022. [34] V.G. Soloviev, Phys. Lett. 16 (1965) 308. [35] V.G. Soloviev, V.O. Nesterenko and S.I. Bastrukov, Z. Phys. A 309 (1983) 353. [36] V.G. Soloviev, A.V. Sushkov and N.Yu. Shirikova, J. Phys. G 21 (1995) 1217. [371 V.G. Soloviev, Yad. Fiz.40 (1984) 1163. [38] EA. Gareev, S.P. Ivanova, V.G. Soloviev and S.I. Fedotov, Sov. J. Part. Nucl. 4 (1973) 357. 1391 G.J. Gallagher and S.A. Moszkowski, Phys. Rev. 111 (1958) 1282. [40] V.G. Soloviev and A.V. Sushkov, Phys. At. Nucl. 57 (1994) 1304. [41 ] R. Nojarov and A. Faessler, Nucl. Phys. A 484 (1988) 1. [42] R. Nojarov, A. Faessler and M. Dingfelder, Phys. Rev. C 51 (1995) 2449. [43] E. Lipparini and A. Richter, Phys. Lett. B 144 (1984) 13. 144] M. Maser, S. Lindenstruth, I. Bauske, O. Beck, E von Brentano, T. Eckert, H. Friedrichs, R.D. Heil, R.D. Herzberg, A. Jung, U. Kneissl, J. Margraf, N. Pietralla, H.H. Pitz, C. Wesselborg and A. Zilges (to be published).

68

V.G. Soloviev et al./Nuclear Physics A 613 (1997) 45-68

[45] See for instance I. Hamamoto, Nucl. Phys. A 205 (1973) 225; A 557 (1993) 515c. [46] A. Zilges, E von Brentano, H. Friedrichs, R.D. Heil, U. Kneissl, S. Lindenstruth, H.H. Pitz and C. Wesselborg, Z. Phys. A 340 (1991) 155. [47] H. Friedrichs, D. Hfiger, P. von Brentano, R.D. Heil, R.-D. Herzberg, U. Kneissl, J. Margraf, G. Miiller, H.H. Pitz, B. Schlitt, M. Schumacher, C. Wesselborg and A. Zilges, Nucl. Phys. A 567 (1994) 266.