Structure of the ground and excited states of odd-mass deformed nuclei in the region 153 ≦ A ≦ 187

1 l.D.2

1

Nuclear Physics A92 (1967) 449-414; Not to be reproduced

STRUCTURE

by photoprint

North-Holland

or microfilm

OF THE GROUND

OF ODD-MASS

without

AND

DEFORMED

IN THE REGION V. G. SOLOVIEV Laboratory

@

written

Publishing permission

EXCITED

Co.,

Amsterdtrm

from the publisher

STATES

NUCLEI

153 5 A s 187 and

P. VOGEL

of Theoretical Physics, Joint Institute for Nuclear Research, Dubna Received

12 April

1966

Abstract: The interactions of quasi-particles with quadrupole and octupole phonons in deformed odd-mass nuclei are considered in the framework of the method of approximate second quantization. The properties of phonons are determined in investigating even-mass nuclei. In considering the interaction of quasi-particles with phonons in odd-mass nuclei, there is therefore not a single free parameter. The peculiarities of the secular equations are studied and it is shown that for many states, besides the interaction of quasi-particles with gamma-vibrational phonons, the interactions of quasi-particles with octupole and beta-vibrational phonons are very important. The energies and wave functions for the majority of odd-mass nuclei in the range 151 5 A 5 187 are found. It is shown that the interaction of quasi-particles with phonons leads to the appearance of admixtures to the one-quasi-particle states and improves the description of these states as compared to the independent quasi-particle model. It is shown that this interaction leads to the formation of the collective non-rotational states and of the complex structure states in odd-mass nuclei. The calculated energies of the excited states, values of B(E2), the decoupling parameters a and the spectroscopic factors agree rather well with the corresponding experimental data. It is concluded that the interaction of quasi-particles with phonons gives a correct description of the structure of the ground and excited states of deformed odd-mass nuclei.

1. Introduction The structure of the ground and excited states of odd-mass deformed nuclei is very simple in the independent quasi-particle model I). The ground state and some excited states have a single-quasi-particle structure. Further three-quasi-particle states of two types appear: (i) (2n, p) and (2p, n) when there are two neutron quasi-particles and one proton quasi-particle or when there are two proton quasi-particles and one neutron quasi-particle, (ii) (3~) and (3n) when the three quasi-particles are protons or neutrons. Then, there must be five-quasi-particle states, etc. As is known, the interactions of quasi-particles in the nucleus play an important role. In even-mass nuclei the interactions between quasi-particles cause formation of collective non-rotational states which are described as one-phonon excitations. In the odd-mass nucleus there is one quasi-particle in addition to the phonons and quasiparticles of the even-mass nucleus. The interaction of quasi-particles with the phonons in odd-mass nuclei was considered in refs. 2, “). The energies of the ground and excited states of odd-mass nuclei are the roots of the 449

450

V.G.

S O L O V I E V A N D P. V O G E L

secular equations obtained in refs. 2.3). There the wave functions were also found and it was shown that the interaction between quasi-particles and phonons leads to the appearance of admixtures to the one-quasi-particle states as well as to the formation of collective non-rotational states and complex structure states. A secular equation was studied and some general conclusions were presented about the pecularities of the collective non-rotational states in odd-mass deformed nuclei. In the present paper a further investigation of the secular equations is performed. The energies of the non-rotational states of odd-mass deformed nuclei in the range 153 _< A _< 187 are calculated. The structure of these states is studied. The reduced electromagnetic transition probabilities B(E2), the decoupling parameters a and the spectroscopic factors in direct nuclear reactions are calculated.

2. Secular equation investigation The structure of the collective non-rotational states of even-mass deformed nuclei is rather well investigated 4-9). As is known, the wave function of the collective state is a superposition of the wave functions of two-quasi-particle and two-quasi-hole states. The more strongly collective the state is, the larger the number of two-quasiparticle states which gives a noticeable contribution to the wave function of this state. In the ranges 150 < A < 190 and 226 < A < 256, the first K ~ = 0 +, 2 + , 0 - states of even-mass deformed nuclei are strongly collective in the overwhelming majority of cases. The two-quasi-particle component corresponding to the first pole is about (80-98) ~ for the first K ~ = 1- and 2 - states, therefore the latter are weakly collective. The energy and the structure of K ~ = 0 +, 2 +, 0 - , 1 - and 2 - states for the first and second roots of the secular equation are given in ref. 9). In the odd-mass deformed nucleus there is one quasi-particle in addition to the quasi-particles and the phonons of the even-mass nucleus. Let us take into account the interaction between quasi-particles and phonons describing the even-nucleus collective states without going beyond the framework of the method of approximate second quantization in the model with pairing and multipole-multipole interactions. The Hamiltonian of the system is written in the form:

"=

4,)B(ss)+ E 1

--¼ Z . . . . . . . .

+0 , ( ; 4 0

{[ Z (fz"(ss')B(ss') +Y'U(ss')B(ss'))vss '

+ Z (f~'(vv')B(vv') +f~U(vv')B(vv'))v,¢](Qi(21~) + + Q,(2/0) + e.c. }.

(l)

vv"

The phonon operator Qi()4t) of multipolarity 2 with the projection # is of the form

Q,(;.~) = ~{ y~ (~.,,(2~)A(ss')- ~p~,(;4,)A(ss') + + ~ , , ( ~ ) A ( s s ' ) - ,~,,(;~)A(ss') +) +

++

451

ODD-MASS DEFORMED NUCLEI

where

A(vv') = ½~/72~ aa~,,,a,_~,

A(vv') = ½x/2 ~ ~ , ~ , ,

a

U(vv')

Z

-=-

+

B(vv')

O{vagv, a ,

Z

=

+

O'O{v_aO~v, ~ ,

~ being the quasi-particle operator. The sum s(v) is over the one-particle levels o f the average field o f the neutron (proton) system. The a = ___1 states are time-conjugate states. The quantitiesf~"(ss ') andfa~'(ss ') are the matrix elements o f the multipole operator 2~l, in this case f z° = raY;.o, f~u = ½,/~2ra(y~u+(_l)uya_~)" In investigating t h e / z ve 0 states, in addition to the matrix element fa'(ss'), where K~ + # = K2 we take into account the matrix element f~U(ss') with K 1 + K 2 = +_i~, where K~ and K 2 a r e the projections o f the m o m e n t u m on the nuclear symmetry axis. By C. we denote the correlation function, by 2~ the chemical potential, and E(s) are the one-particle level energies in the neutron system, e(s) = x/C 2 + { E ( s ) - 2 n } 2 (g(S) is the quasi-particle energy) llss,

=

I)ss,

M s U s , ~ - U s, U s ,

2

u~ =½

( 1+

1

,~(s) / '

=

II s II s, - - V s l) s , ,

2

lJs

2 =

I--U

s

.

F o r all the cases, except ). = 2 and # = 0

I

~,~.(~#) = ~,/~

1 ,/y,(;.#)

fZ'(ss')Gs, f~U(ss')u~, ~(s) + ~(s') + ~ot "'

in the functions ~ , ( 2 # ) and ~p~¢(2#)i we havefZU(ss ') instead off~U(ss'). By o~" we denote the energy o f the collective non-rotational state of an even-mass nucleus with A - 1 nucleons. The quantity y i ( 2 # ) is o f the form

Y'(2#) = L (faU(ss')2 +f;~(ss')Z)uZ~" w~(~(s) + e(s')) ss,

((~(s) + ~(s')) 2 - (~t")2) 2

+ • (faU(vv')2 +f;'U(vv') 2)u~, co~"(e(v) + ~(v')). ,~, ((~(v) + ~(¢))2 - (~ot")2) ~

(2)

It is comparatively small for the collective states and Yi(2g) ~ ~ when o9~" o ~ ( v ) + e(v'), i.e. it is very large for states close to two-quasi-particle states. It should be noted that in writing the Hamiltonian (1) we have used the secular equations determining the energies w~" of the collective state in the even-mass nucleus. F o r K(~) = ~:p (a) = ,,~p . ( x ) _ to(x) (K. , (a~, ~z), ,.(z) ,,~p are the multipole-multipole inter-

452

V. G.

S O L O V I E V A N D P. V O G E L

action constants) this equation is of the form 1

2~c(a)

E

(f;"(ss')2+faU(ss')2)u2v+(co~,,)2 Z

•

2 # 1jVt 2 ~ 2 2 (s-(~,) ~+f~,2 p (,,,, ! )uv~,] , x,,,,2 t

~(~)+ ~(s')

•

(3)

~(,,) +~(v ) J

The summation in eq. (1) over i is the summation over the roots ofeq. (3). In studying the excited states with 2 = 2 and It = 0, the spurious state in even-mass nuclei was excluded. To this end one took into account an additional interaction between quasi-particles. In investigating the interaction of quasi-particles with 2 = 2, p = 0 phonons we should take into consideration those terms of the Hamiltonian which give a similar effect as the terms used in excluding the spurious state. Therefore, two terms H ' ( n ) and H ' ( p ) where =--X/2 ss'

i

{(~(20)QI(20)

+~p,~s(20)Qi(20))

x B(s's') + B(s 's ')(~,s~(20)Q~(20)+~p~,~(20)Qi(20) ~ ~ )},

(4)

should be added to the Hamiltonian (1) (for an elaboration of the notation see refs. s, 9)). We obtain an equation for the energies and wave functions of the odd-mass nuclei. To this end let us take into account the interaction of quasi-particles with phonons having different 2pi values. The wave function for e.g. an odd-proton nucleus which describes the states with the projection of the momentum on the nuclear symmetry axis K and parity ~ is written in the form

~,(K =) = O(K=)+~,o,

(5)

Q,()VO~o = o,

(6)

where

= Co

.,/2

o

;,,i w

Op~,ev~,Q,(2#) },

(7)

where p (or pq) stands for the average field level with a given K = value. The normalization condition for the wave function 7S(K =) is written as follows: 2 1 Z Z (D:~Ui'12"~ C,{1+~ \--pv~/ ) 2~i

=

1.

(8)

va

Let us find the mean value of H i n the state 7*(K=) and determine C; and DRy ,a~'i using" the variational principle in the form

5{<(2(K'~)HO(K'~)+>--~Ij[C2(1 +½ Z E (Dpw) a~,i 2)--1]} = 0, kpi

(9)

va

where rb is the Lagrangian multiplier. After some calculation we get the following

ODD-MASS

DEFORMED

453

NUCLEI

secular equation 2):

P(q) =- ¼ Z ~

2

v°,

faU(Pv)z+ faU(PV)Z -(~(P)-rlJ) = 0.

(10)

The poles in eq. (10) are the sum o f the quasi-particle energy e(v) and the p h o n o n energy ~o/ . The solutions for eq. (10) ql determine the energies o f the non-rotational states in odd-mass nuclei, i.e. if eq. (9) is taken into account then ([2(K~)HI2(K~)+> = tlj. The factor v2~ indicates that the particle-particle and hole-hole interactions are preferred to the particle-hole interactions. The terms in eq. (10) with 2 > 3 and i > 2 give a very small contribution since the quantity Y~(2/t) -a tends to zero when the corresponding even-nucleus state approaches the two-quasi-particle state.

/

P (7)

t

// //

~o /' j / /

/'// 0 250

7

/

f

i- f f /

/ / / J

//,2.

Fig. 1. The behaviour of P(r]) (in units h~30) of the left-hand side ofeq. (10) for p = 521'~ (the dashand-dot line) and for p -- 512~ (the dashed line). The behaviour of P(~) (in units ( / ~ 0 ) 2, the scale is increased ten times for clarity) of the left-hand side of eq. (16) for P l - } - p 2 = 521~+512d¢ (the continuous carve). The first pole of eqs. (10) and (16) equals to e(521 ~) +co122 = 0.324/~tfi 0 and the second pole to e(521~)+~% 22 = 0.340/'~?b-

The behaviour o f P(r/) for K ~ = ~-- states in iT1yb is given in fig. 1, where the dot-and-dash line denotes t P(t/) for p = 521]`, e (521]') = 0.317h&o and the dashed t By NnzA'~ we denote the K = Ad-~' state of the Nilsson potential, by NnzAl, the K = A - - -r state hdJo-- 41A-k MeV.

454

V. G . SOLOVIEV A N D P. V O G E L

line is P ( q ) for p = 512~,, e(512~) = 405hob o. The first pole corresponds to e(521J,)+ 0 9 22 1 = 0.324fi&o and the second to e(5215)+~o 22 0.340h& 0. It is seen in fig. 1 that POD has one root up to the first pole, the second root between the first and the second poles, etc. Using the normalization condition (8), the functions C~ and/)au~ --o~a can be written in the f o r m

v~ f~U(pv)2+faU(pv) 2

f2U(pv)-af24'(pv) 2 x: yi()4, ) e,(v)+~o~'-q~

Da,,i = 1

-"~

t,,2,

(ll) (12')

The quantity C~ determines the contribution of the one-quasi-particle state with a given p to the state under consideration (see eq. (7)). Consequently if C~, ~ 1, the state is a one-quasi-particle state, if C,2 is somewhat smaller than unity, the state possesses a complicated structure and if C 2 << 1 the state is collective as a rule. The quantity of the c o m p o n e n t with a quasi-( -7 2p1' ',/-)-A p v' u i }J 2 determines the contribution particle in the v-state plus p h o n o n 2Hi to the wave function 7ffK~). Here 2

(D;.uh 2 V,(l);.,,i]z = 1 vp~ f~"(pv)2+f~"(pv)2 ,--0~, = ½ ~ ,--o .... 4 Y;(2/~) ( c ( v ) + , . ~ u - qj)-~

(12)

The secular equation is derived under the a s s u m p t i o n that [:~,, Qi(2tz)] = 0, i.e. it is believed that the p h o n o n is a boson and the fact that Qi()./0 is a superposition of the operators e,.,~¢,, and ~,,,~v'~' + + is neglected. The p h o n o n - p h o n o n scattering and the Coriolis forces which are of importance in some cases are neglected as well. The secular equation (10) can be improved in some respects. Thus the ~(p)wdues should be calculated using the blocking effect. This implies the use of the correlation function C(p) and the chemical potential ).(p) for a given ? state of the system consisting of an odd n u m b e r of particles N + 1. Tile other possibility is to replace in eq. (10) e(p) by the difference d " ( p ) - c 5 o of the energies (reckoned from the corresponding 2, 2(0)) of the system consisting on N + 1 particles with a quasi-particle in the p-state and of the ground state of the N-particle system. Further we could take into account the influence of the exclusion principle on the phonons, i.e. calculate the quantities ~o~" and Y~(2/) utilizing the fact, that the wave function of the ground state is ~(,~7' o and the wave function of the excited state is Qi(2~O+y.,+fl*o . Due to the Pauli principle, in this case, the p h o n o n operator Q/(2H) contains no c o m p o n e n t with a quasi-particle :~+ r i f t and the quantities ~of~ and Y~()41) weakly depend on v> Corrections of this type are taken into a consideration in ref. ~o ). We take into account the terms (4) which are related to the non-conservation of the

ODD-MASS

DEFORMED

455

NUCLEI

particle number and obtain the secular equation in the form:

v~ fa"(pv)2 +f~"(pv) 2 ~(~) + ~2"- '?~

~(p) - ,Tj- + 4.,Z -vZ YOp)

1

Z

m ~

1

1 [f2O(pp)(E(p)_2) e(P)2

Y yI(20) 8(p)+m 2 ° - q d

f Z°(vv) , E(v)~ E(v') 12 ~,~ ~ ~(~)(44v)~-(<4°)~) ~(v)(~(od°)~)J = o;

4C~

(13)

all notations are from refs. 8,9). However, if the contribution of terms with 2 = 2, # = 0 in eq. (13) is small, then eq. (10) can be used. This is just the case of the considered region 153 < A < 187. We make a further improvement of the secular equation (10). We bear in mind, that there are several one-particle states with the same K ~ and take the wave function 7S(K ~) in the form

7J(K~) = Q(K=px ...

p,)+Tto,

(14)

where

a(~%.., •

"

"

((7' ~+

l

p,)+

,,/2 N(p,... v,) Z~ ~-,~, o,o..-

I + Co, :+p+ ~ + Z n

pi211i[n ~ v a

~P'I

"

"

"

p,)~+ Q,(2p) + }•

(14')

2,uiv

This general case is considered in ref. 2). In the present paper we study in more detail a particular case when only two one-particle states Pl and P2 with a given K ~ are taken into account• By analogy with eq. (5) the wave function is written as follows: ~ ( K =) =

(2(K'~plP2)+7Jo,

(l 5)

where

Q(K~plpz)

+

~

1N(plp2) Z{Cm%,~+Cp~¢~+ ZD~(p, p2)o:+~Qi(21z)+}. t

x/2

+

.

p

+

2#i

(15')

;..ui~

Using the variational principle we obtain a secular equation in the form:

P'(rl) = i!

Vi(P'P')-(e(Pl)-tlJ) I/)(PaP2)

VJ(P'P2)

Vj(PzPz)--(8(Pz)--rlJ)

= 0,

(16)

"

(17)

where

Vi(PqP") = ¼,~ vp~vPJ';U ' (pqv)fa~('P"V)~;"'(Pqv)faU(P"V)~ e ( v ) ~ ~

As can be seen, eq. (16) contains only poles of the first order and the coefficients of ¢ ¢ the second order poles are cancelled. The quantities Co, and Cp2 can be taken in a

456

V. G. SOLOVIEV AND

P. VOGEL

symmetric form:

c~,, = 1 - V ) ( p , pv~(p,p:) ,)-(~.(p,)-,O' C'p~ = 1

(18')

vj(pl pz)

Vj(R2D2)

(18)

- (~:(P2) - ~'j))

then for D,.,, ~#i(PIPE) we have ~.ui D,,(p,p2)

N(p, p2)-2 =

=

(7'

F) a'ui 4-(7'

--P! --Pl

0 "a'ui

VO" - - - - p 2 - - p 2 v o "

(19)

C'R](1 + ~V'(p'P')IOq~ / + Cpa'2(1 + aVj(p2P2)]~?,i/ + 2C'o, C'p~c?Vj(p,pa) &b"

(20)

In this case the contribution of the one-particle state p~ to the wave function (15) is

N(plP2) Col. 2

~2

Let us investigate the peculiarities of eq. (16) and compare them with the secular equation (10). Consider the function Vj(pqp,~) for the r/-values which are noticeably smaller than the first pole in eq. (16). The function Vj(pq, p,,) is a sum of a large number of terms taken over 21~i and v, however, the main contribution is given by several terms posessing ihe following properties: their poles are not so far removed, their (~ , ")41 matrix el~mems/(~,,.) are large, the quantity Vp,,is about unity and the Yi(2tt) values are small. The diaganal functions Vj(pq, pq) are the sum of only positive terms. The non-diagonal functions Vj(p~p2) are much smaller than the diagonal functions because, first I / i ( p l p 2 ) is a sum changing the sign, second conditions under which the term in Vj(p~p2) is large are not simultaneously fulfilled for Pt and P2. Therefore the determinant in eq. (16) transforms actually into the product of diagonal terms alone. Thus, instead cf cq. (16), we are led to the product of eqs. (10), one for Pl and the other for P2. If the ~li wdue is rather close to the value of the first pole then in the sum r(pq, f),,) the predominant role is played by the pole term. In this case the secondorder poles in eq. (16) are cancelled and at the point ~(v)+e)~ ~, the function P(~I)has a pole of the first order. Then we are led again to an equation of type (10). But eq. (16) does not transform into the product of eqs. (10) for large qi" Besides, it is necessary to analyse the solutions of eqs. (16) in order to exclude false solutions, which appear in solving eqs. (10) for p, and Pc. In fig. 1 we have, as an example, the function P(tl) for t>~ +P2 = 521T +512{. The behaviour of the function P(rl) for 521T+512~, essentially differs from both curves P(q) for 521T and POD for 512j,, but the roots of eq. (16) actually coincide with those of eq. (10). However, there are two false solutions for 5125 when the r/values are close to those of the first and second poles. We find the structure of the first three states: for the first state q~ = 0.275hc5 o, the wave function contains 70 % of the one-quasi-particle 521]" state, 25 % of the

ODD-MASS DEFORMED NUCLEI

457

5 2 1 5 + Q 1 ( 2 2 ) state, 2 ~o of the 6 5 I T + Q 1 ( 3 0 ) state, etc., for the second state ~/2 -0.334 h&o and the wave function contains 67 ~ o f the 512+ state, 20 ~ o f the 514+ + Q1(22) state, 5 ~o o f the 510T ÷ Q1(22) state, etc., and for the third state q3 = 0.338 h&0 and the wave function contains 11 ~ of the 521T state, 6 ~ o f the 5125 state, 49 ~ of the 521J, + Q1(22) state, 30 o/ /o of the 521J, + Q2(22) state, 2 ~ o f the 514+ + Q~(22) state, etc. Therefore it is seen that both one-quasi-particle states Px and Pz contribute to the wave function corresponding to the third root. The solutions o f equations such as (16) with the determinant of higher order have similar structure. The lowest roots of eq. (16) coincide usually with the roots o f eq. (10). Besides, eq. (10) has superfluous roots as c o m p a r e d to eq. (16) which have no physical meaning. We formulate rules for excluding the superfluous roots in eq. (10) without solving eq. (16). E.g. the solution o f e q . (10) for p~ is false if the root is very close to the pole ~(v) + co{u and if the matrix elementfZ"(pl, v) is small. At the same time there must be a large matrix element f~"(Pz, v), which connects the given pole with another oneparticle state P2 having the same K ~ as Pl- The solutions for eqs. (10) somewhat rem o v e d from the poles are not usually false. Thus, the main role of eq. (16) is the exclusion o f false solutions o f eqs. (10) and the determination of the structure of higher states with a given K ". The secular equations (10) and (16) do not have a single free parameter and the arbitrariness consists only in how m a n y values of 2, /~ and roots i should be taken into account. We obtained the lowest roots (as a rule, the first two of three roots) of eqs. (10) and (16) for nuclei in the range 153 < A < 187. For the quantity a(p) we take the difference of the energies of an odd-mass nucleus in the p-state (calculated taking into account the blocking effect) and o f an even-mass nucleus. The first and second roots (i = 1, 2) of the quadrupole K ~ = 0 + and 2 + and o f the octupole K ~ = 0 - , 1- and 2 - states were taken into account in the calculations. The investigations showed that the lowering o f the energies with respect to the onequasi-particle values o f e(p) as well as with respect to the first poles ~(v)+co~" is mainly due to the terms o f eqs. (10) and (16) with 2 = 2, # = 2, i = 1 and 2 = 3, Ft = 0, i = 1. The terms with 2 = 2, # = 2, i = 1 are o f importance in almost all the nuclei, the terms 2 = 3, # = 0, i = 1 play an essential role in the beginning of the region of deformation and in the region o f the isotopes Y b - H f . Thus in some cases, (e.g. for K ~ = ~~+ , p = 660]` states, the 3+, p = 651]" state and the ~ , p = 512T state in a61Dy and in other nuclei) the terms with 2 = 3, tL = 0, i = 1 in eqs. (10) and (16) play a predominant role. Sometimes the terms in eqs. (10) and (16) with 2 = 2, # = 2, i = 2 a n d with 2 = 2, # = O, i = 1 are important. Thus the terms with 2 = 2, # = 2, i = 2 are of importance for the first K ~ = ~ - , ½- states in *TSHf and the terms with 2 = 2, ~ = 0, i = 1 in K = = z~+ states in 16IDy and 155Sm, etc. The positive-parity states in 157Tb and lS9Tb up to 1.6 MeV are shown in table 1. As example o f the solution for eq. (10) the energies, the structure and the lowering o f energies of these states with respect to the one-quasi-particle values e(p) as well as with respect to the first poles e(v) + col" (if their energies are not higher than e(p) or

458

V. G. SOLOVIEV AND

e(v)+o]')

are given. The experimental

that for a number

P. VOGEL

e n e r g y v a l u e s a r e f r o m refs. ~ - t 3 ) .

o f s t a t e s t h e t e r m s w i t h 2 = 2 , / ~ = 0, i =

It should be noted that for odd-proton

It is s e e n

1 are very important.

nuclei, the positive parity states are usually

more strongly collective than those with negative parity. For odd-neutron

nuclei, on

the contrary, negative parity states are more strongly collective, on the average, than those with positive parity. TABLE

1

Positive-parity states in XSTTb and ~SUTb Nucleus

,SrTb

K=

Energy (keV) exp.

calc.

0

0 300 530 1050 1100 1100 1100 1300

a~ •,:' ~

½+ ~+ 2z+ s~ ~ l+

597 990

9~ 7~ ½~ 7,.+ a+

1300 1400 1550 1550 1660

2

~agTb

.a.E ~+ ~ ~~ ~ 2 ~

0 348 580

~,~. ,7 ~ ~'~ 2' e:~._.a~

1270

0 330 430 1100 1100 1250 1150

971

1150 1350 1400 1500 1600 1650

Lowering with respect to the ~(p) I pole (keV) 200 80 400 420 420 270

1100 770 770

411'190 ~;; 411 ~,-f Q, (22)6 °,/, I o,o 4 1 3v96, 411,~65 o~,; 411~' ~ Q~(22)25 ',!0 411" ~ Q~(20)100 o~ 404],81 ~'£, 404v+Q~(20)lO '~; 413,'~ q- Q~(20) 100 ~ 404*70 °,o 404,* +Qa(20)25 ~'~ 413~ i Q~(22)75 o~; 411],q Q1(22) 20 ?o 413,1Qt(22)95 '~;,,, 404~, 4 ",;; 411~'4 Q~(22)100 3o 420"]'70 ~';, 4221 ~ Q~(22)15 ",; 523~'-{ Q,(30)90 ,q,,/,, 404~<3 ~i; 422,,75 "

260 0 200

520 400 260 120 570 420 290

Structure

1600 900 700 60 10 0 0

411,~.90 ~ , 411~, i Q~(22)8 i!,; 413,~94 ~/o, 411 !,q Q~(22)4 '!, 411,~60 ~,~, 411,~'q-Q~(22)30 !',,~ 404~66 ',I~,~,523~'q Q~(30)30 !!;i 413~ !Q1(22)95" ,, , 404 ~ ",, 5321 }-Q~(30)100 3~ 4 3'* ~ Q~(22)80°~;411! , ! Q~(221 5 i';i 4111 ! Q~(22)1007~ 523~, i-Q~(30)70 '~, 404,125 %; 404]'90 ~,;, 404'~ I Qz(20)4 ~,~i~ 420)65 ~(,, 422,~q Q1(22)20 ",, 422169 '?,i 411" I -! Q~(20)I00 '}ii

0

650 4_S0 0

Thus, on the basis of the present investigations, we may prove the conclusion d r a w n e a r l i e r i n ref. 3) t h a t t h o s e a p p r o x i m a t i o n s which take into account (although more

accurately

J o) t h a n

neglect the remaining n u c l e i in t h e r a n g e

in o u r c a s e ) o n l y p h o n o n s

with 2 -

2, tz

: 2, i =

1 and

(first o f all w i t h 2 = 3, IL = 0, i -- 1 ) a r e r a t h e r r o u g h f o r m o s t 153 < A =< 187. N e v e r t h e l e s s ,

2 = 2, I~ = 2, i = 1 p l a y a m a i n r o l e , a n a c c o u n t

e v e n in c a s e s w h e n

terms

with

o f t e r m s w i t h o t h e r 21~i v a l u e s is

necessary since in some cases the lowest pole corresponding

to a phonon

2t~i

which

ODD-MASS

DEFORMED

NUCLEI

459

differ f r o m 2 = 2, # = 2, i --- 1 can noticeably change the energy of the calculated state. In eqs. (10) and (16) it is not necessary to take into account terms with 2 > 3 and i > 2 since the total contribution of these terms is very small. 3. Odd-mass nucleus states close to the one-quasi-particle states

The analysis of the secular equations (10) and (16) shows that the structure of the states is mainly determined by the position of r/1 with respect to ~(p) and the first pole. I f t/1 is very close to ~(p) then the state will be actually of the one-quasi-particle type. If r/1 noticeably differs from e(p) and from the first pole e ( v ) + eo~u, the structure of this state is very complicated since the contribution to the wave function is given not only by the one-quasi-particle states but also by many states with different quasiparticles and phonons. I f ql is close to the first pole of the secular equation then the state is collective. Thus if ql approaches the pole i.e. =

(21)

than this state is a g a m m a - v i b r a t i o n a l state, if 2 = 2, p = 2 or an octupole state if 2= 3,#=0, etc. Let us consider the states in odd-mass deformed nuclei which in their structure are close to the one-quasi-particle states. The contribution of the p-state to the wave functions is predominant, and the quantity C 2 is somewhat smaller than unity. The state possesses a structure similar to the one-quasi-particle state, if the following condition is fulfilled: e(p) << min {e(v) + ~o~"}, (22) i.e. when the quasi-particle energy is much lower than the value of the secular equation first pole. H o w e v e r in some cases there are comparatively high-lying states close to the one-quasi-particle states when r/is close to s(p) and condition (22) is not fulfilled. If the condition (22) is fulfilled then the K = ~-,1 1 9 states and partially the 27-states are close to the one-quasi-particle states and the contribution of the p-state to their wave function is, as a rule a) (95-99) %. F o r example, the contribution of the 5231" one-quasi-particle state to the K = = } - state is 99 % in 16~Tb, 98 o/ o / i n lS9Tb and in ~SVTb and 97 ~o in ~SSTb. The K = ~,~ 3, { states are rather close to the one-quasiparticle states and the contribution of the p-states is (90-97) %. This means that the quasi-particle plus p h o n o n admixtures are more important and the energies are more strongly' lowered with respect to e(p) for the K = {, { states than for K - - ~1 2,2 states. For example, as is seen from table 1 the contribution to the ground K= -- 73+ state of the one-quasi-particle 411]" state in tSVTb is 90 % and e ( p ) - r / t = 200 keV. Calculations of the levels of odd-mass deformed nuclei in the range 153 < A < 187 showed that the interactions of quasi-particle with phonons lead to a different decrease with respect to g(p), of the energies of the states close to the one-quasi-particle

460

v . G . SOLOVIEV AND P. VOGEL

states. Therefore in a n u m b e r of nuclei the calculated succession of the excited states differs from the succession of the Nilsson level schemes. Thus, in the Lu and Ta isotopes, the K = = ~ - state is very close to the one-particle 514T state (the contribution o f the state is larger than 99 ~(,) and the lowering e ( 5 1 4 T ) - q l ~ (10-20)keV. At the same time, in these nuclei the admixtures in the K ~ = ~7+ state close to the 404~, state are somewhat more important, since the contribution of p is of the order o f 97 o7~ o and the lowering is ~(404J,)-r/l ~ 50-100 keV. Therefore in all the Lu isotopes and in the Ta isotopes with A = 177, 179 and 181~ the -}+ 404 4 state is a g r o u n d state and the ~ - 514T state an excited state. In nuclei with N = 91, according to the Nilsson scheme, the ~zL-505]" state should be the ground state. According to our calculation, in lS3Sm, the 3+ 6511" state is the g r o u n d state and in ~SSGd the 3 - 521T state; this agrees with experiment. Thus, the ~J--505]" state is not a ground state in nuclei with N = 91-93, although ~(505"f) - - t h ~ 100 keV. Thus, the interaction of quasi-particles with p h o n o n s weakly affects the K - ~ , ~_ ~, - " states close to the one-quasi-particle states and more strongly affects states with smaller K. As a result the K = ~21-,~- states are not ground states in odd-mass deformed nuclei if in the average field level scheme there are levels with smaller K, near these states. We have calculated the energies o f the states close to the one-quasi-particle states and their structure for a large number o f odd-mass nuclei in the range 151 < A __< 187. The average field level scheme for ~5 = 0.3 was used 9) in which the following changes are introduced: in the neutron system the 5051" state is raised by 0.15h& o, the 6511" state is lowered by 0.05h&0 and the 660T state by 0.10 h&o; in the proton system, the 404]" and 422J, states as well as the 4044 and 514T states interchanged places and the 541 + state is lowered by 0.13 fi&0. The values o f oo~'~ and Yi(.).tO are recalculated according to the modified scheme, but in most cases they are close to those in ref. '~). The Re isotopes are calculated for the deformation r5 = 0.2 according to the level scheme and the values of~o~4' and Y~()4t) obtained in ref. ~4). Some calculation results are given in table 2, namely the experimental and calculated values o f the energies, the excitation energies in the independent quasi-particle model (taking into account the blocking effect) e ( p ) - g ( K 0 ) , (e(Ko) being related to the g r o u n d state) and the structure of these states. The experimental data are from refs. ~a, 1 5 - 2 2 ) . it is seen in table 2 that in some cases, even in comparatively strongly excited states, the admixtures are not so important, e.g. in the 612 keV state in ISJTa the contribution of the 411~, state is 95 ~ , in the K = = 77+ , 995 keV state in 17sYb the contribution o f the 633T state is 98 o/ /o, etc. The calculated energies o f the levels close to the one-quasi-particle levels agree somewhat better (especially highly excited ones) with experimental data than tho,~,e calculated in refs. 23,24) according to the independent quasi-particle model taking into account the blocking effect. However, this agreement is not sufficiently good since it depends on the position o f the average tield levels. In a number of cases the Coriolis interaction, which we neglected, could be important.

ODD-MASS

DEFORMED

461

NUCLEI

The change in the energies o f excited n o n - r o t a t i o n a l states in o d d - n e u t r o n nuclei in the t r a n s i t i o n f r o m one nucleus with a given N to a n o t h e r (or in different isotopes in o d d - p r o t o n nuclei) is due to m a n y causes, i.e. change in the average field levels, c h a n g e in the e q u i l i b r i u m d e f o r m a t i o n , interaction o f quasi-particles with p h o n o n s (due to the change in the values o f e)~", Y~) a n d others. In some cases the energy TABLE 2 States close to the one-quasi-particle states, in odd-mass nuclei Nucleus

K~

Energy (keV) exp.

calc.

Structure

e(p)--~(Ko)

l~aEu 16ITb a61Tb 161Ho 161Ho 18Silo 16~Tm l~3Lu ~s~Ya

7,+ ~-+ ~-½+ ~-~-+ ~z~½+

103 316 482 211 826 715 379 888 612

100 280 580 220 900 650 350 1300 540

180 200 450 240 920 850 330 1500 600

lS3Re 161Dy

~+ ~-

851 75

550 60

440 110

16aDy l~SDy X~SDy 167Er ~69yb l~Yb ~69Yb

~-~~~~-~~+

421 535 184 348 191 570 584

280 530 240 300 400 540 600

400 560 410 410 470 560 740

169yb mYb 171Yb 173yb ~r3Yb ~TaYb 17~yb 177Hf ~Hf

~-~ ~ ~+ ~-27~+ 7+ 9+ ,~-

657 835 95 1340 636 35l 995 32l 1060

820 1200 160 1560 450 530 1140 400 1200

1000 1300 120 •600 430 520 1100 350 1400

411 ~'94 °o~ 413vr96 o~; 411J,+Ql(22)2 532~96 ?Jo ~ 411 ~,96 532"~91.3% 404~,93 ~,; 402~,+Q~(22)5 5234,99 ~, 532~92 ~o 411vl95 °~;411~'+Q,(22)3 ~ ; 413,~-kQ1(22)2 404~98 521~95 ~; 651~ +Q~(30)2 ~;521~-[Q~(22)2 % 521 ~91 o/ /o 523~,94 512~'89 512~91 512~'95 523~97 642~'89 ~o; 523,~+Q~(30)4 ~ ; 642~+ -b Q1(20)4 521~'94 ~/~;651]'+Q~(30)4 514,~88 °/o 633~'99 % 512~90 514~99 %o 633"~98 o~ 633"!'98 ~/o 624~ 100 o~ 503~89 %

change in different isotopes or i s o b a r s m a y be due to the change o f the e q u i l i b r i u m d e f o r m a t i o n o f the nucleus in the excited state as c o m p a r e d to the g r o u n d state. T h u s the calculations o f ref. 25) showed that such a situation takes place for some states in a n u m b e r o f d o u b l y o d d nuclei in the t r a n s u r a n i u m region. The i n t e r a c t i o n o f quasiparticles with p h o n o n s is one o f these causes. Therefore, by using it one does n o t

462

V . G. SOLOVIEV A N D P. V O G E L

succeed in explaining the change of the energies of the levels for different isotopes and isobars, e.g. the behaviour of the states close to the 541,l state in Lu isotopes. The fact that the states in odd-mass nuclei are not purely quasi-particle states is displayed in the beta-decay probabilities, in the magnitudes of the spectroscopic factors in direct nuclear reactions, in the values of the decoupling parameters a for K = ½ states, etc. Our investigations show that when a given average field level ? is near the Fermi surface then the admixtures in a state close to the one-quasi-particle state are as a rule the smallest and C 2 is close to unity. As the level p moves away fl'om the Fermi surface, i.e. as the excitation energy increases, the role of the admixtures in the state with a corresponding K = becomes greater and the quantity C~2, decreases. This peculiarity can be seen from the change of the decoupling parameter and spectroscopic factors. We now investigate the influence of the interaction of quasi-particles with phonons on the decoupling parameter a for the K = ½ states. Using the wave function of the state in the form (5) we get the decoupling parameter the following expression: 2

N

N ~20i~20i

vv'i

avv, Upv urn,,

vv'i

N ~ 3 0 i r , 30i) avv, LJov Upv, J,

(23)

N and a~v, N are the decoupling parameters calculated with the Nilsson wave where aoo functions is). For the states (21) close to the pole with tt ¢ 0 the quantity a is equal to zero. The role of the second and the third term ofeq. (23) is small in most cases. If we take into account that the phonon operator in the commutator [Qi()-l~), ~,.~] is a superposition of the quasi-particle operators then in the expression for a there appear other terms linear in the admixture amplitude, which will lead to an additional change of a. They are not taken into account here, Let us look how the quantity a changes for the K = ~ state close to the one-quasiparticle state 5215 in the transition from nuclei, where this state is a particle state to nuclei, where it is ground state and further hole excited one. Table 3 gives the experimental and calculated values of the energies of these states, the experimental 27) and calculated values of a as well as the quantity C 2 describing the contribution of the one-quasi-particle state. Table 3 gives also the difference of the energies ~ ( p ) - e ( K o) calculated in the independent quasi-particle model, taking into account the blocking effect (e(K0) is the value for the ground state). If the considered state is believed to be a pure one-quasi-particle 521], state, the values of a calculated with the Nilsson wave functions for 6 = 0.3 are a N = 0.89. It is seen in the table that when the 521~, state lies on the Fermi surface the admixtures are small ( c a = 0 . 9 6 - 0 . 9 9 ) and the a-values are close to a N'. In the cases, when the 521,~ state is a particle-excited state the quasi-particle plus phonon admixtures to it are large and the a-values are much srnaller than a N'. It is seen also that the account of the interaction of quasi-particles with phonons allowed the explanation of changes in the behaviour of a for states close to 5215 states in different nuclei.

ODD-MASSDEFORMEDNUCLEI

463

H o w e v e r , in s o m e cases the a c c o u n t o f the i n t e r a c t i o n s o f quasi-particles with p h o nons does not lead to the elimination of discrepancies between the calculated and TABLE 3 Energy, decoupling parameter and C2p quantities for a state close to the ½-521,~ state (calculation for 6 = 0.3, where for C2p = 1, aN = 0.89) Nucleus

15zSm91 1~5Sm93 161Dygz X6~Er95

Energy (keV) exp.

calc.

e(p)--e(Ko)

698 824 365 346

900 950 450 480 300 340 130 150 150 0 0 0 280 280 800 800 1000 1050 1000

1500 1350 920 920 530 530 180 180 180 0 0 0 290 290 850 850 1230 1230 1230

16ZDy97 165Er97 1~Dy99 167Er99 ~69Yb99 169Er101 171Ybio1 ~TaHf~ot ~7~Yb,o3 ~r~Hf193 175Yblo5 1~7Hflos l~TYb,o7 argHflo7 ~s~W~o7

Decoupling parameter a

297 108 208 24 0 0 0 400 126 913

746

exp. 0.334-0.36 0.324-0.28 0.44 0.47 0.56 0.58 0.71 0.79 0.83 0.85 0.82 0.74 0.75 0.71

0.59

C2 a

calc.

( × 100)

0.50 0.58 0.47 0.49 0.60 0.65 0.86 0.87 0.87 0.85 0.86 0.87 0.88 0.85 0.84 0.84 0.81 0.82 0.85

56 66 53 55 69 73 97 98 98 96 98 98 99 97 95 95 91 92 81

TABLE 4 Spectroscopic factor and the one-particle amplitude for states close to ½-510~' Nucleus

angYb99 lrlYblo 1 lvaYblo3 lrSYblos lrTYbxo7 175Hfloa 177Hflo5 argHf~or ~81Hf,o9

Energy(keV) exp.

calc.

805 950 1040 500 320

1300 1200 1160 660 103 1070 680 110 0

378 0

Spectroscopic factor

e(p)--e(K) 2300 1800 1340 800 300 1340 800 300 0

C2 o

exp.

calc.

0.39 0.51 0.65 0.90 0,89 0.75 0.90 0.90 0.96

0.40 0.52 0.76 0.85

0.38 0.49 0.63 0.85 0.77 0.73 0.85 0.79 0.70

e x p e r i m e n t a l v a l u e s o f t h e d e c o u p l i n g p a r a m e t e r a. E.g. f o r t h e v a l u e s are e q u a l t o - 0 . 2 ,

510T s t a t e

t h e a N_

w h i l e in ' 8 3 W w e h a v e a = 0.19. T h e a c c o u n t o f t h e i n t e r -

a c t i o n s o f q u a s i - p a r t i c l e s w i t h p h o n o n s d o e s n o t e l i m i n a t e t h i s d i s a g r e e m e n t . A s is

464

V . G . SOLOVIEV A N D P. V O G E L

shown in ref. 28), only a noticeable change of the Nilsson potential parameters leads to the elimination of this disagreement. In some other cases, e.g. for the 411 ~ state in o d d - p r o t o n nuclei, the values of a determined experimentally differ little from a N and the interaction of quasi-particles with p h o n o n s is not displayed as effectively as, e.g. for states close to the 521~ states. We consider the effect of the admixtures on the spectroscopic factors in direct nuclear reactions. Thus when the one-quasi-particle state p is excited in the (d, p) reaction on an even-mass target the spectroscopic factor is u,.2 If the admixtures, are taken into account i.e. the wave function is assumed to have the form o f e q . (5), then the spectroscopic factor is CT,"~u 2o. Table 4 gives the experimental (normalized to l vsYb) and calculated values of the spectroscopic factors for the excited K ~ = l - states close to 510~ states in the Yb and Hfisotopes, the values o f C~ as well as the energies 2 .2 o f these states. The calculated values of the spectroscopic lector C,,u~, correctly' reproduce the behaviour of the cross sections for the (d, p)reactions ~9), the decrease 2 o f C~2ou;"~ in the light Yb isotopes being due to the decrease o f C~,. Thus, the interaction of quasi-particles with phonons in some cases essentially affects states close to the one-quasi-particle states in odd-mass deformed nuclei and it should be taken into account in investigating excited states.

4. Collective non-rotational states Let us consider another limiting case, when the energy corresponding to the first pole (10) or (16) is m u c h lower than the one-quasi-particle energy, i.e. min {8(v) +,z)~u} < ~(p).

(24)

In this case, the term in (10) and (16), corresponding to the first pole plays a pred o m i n a n t role in K = @, ~ states, the latter have the structure quasi-particle plus p h o n o n . It is only in this case that we may use the terms gamma-vibrational, ocl upole, etc. for states in odd-mass deformed nuclei. The contribution o f the single-quasiparticle state p to these states is CZp = 0 . 0 0 1 - 0 . 0 5 0 . E.g. K s = l j_- state in 1('~t4o with an energy 687 keV is a gamma-vibrational state. The calculated energy is 850 keV, the lowering with respect to ~(523~')+ co~22 = 1700 keV is 3 keV, e(p = 505I") = 4500 keV, C~ = 0.001. The K = ½, ~ states has a somewhat more complicated nature, as compared to the K ¢) = -1~1, ~states since in eqs. (10) and (16) several terms with different 2 and tt play often an important role. The contribution of the one-quasi-particle state p is C~, = 0 . 0 1 - 0 . 1 0 . The K = ~, -} states occupy an intermediate position. The state is collective if its energy is very close to the pole. Highly excited states can be collective even if the condition (24) is not fulfilled. The most frequent is the intermediate case e(p) = (0.5 - 2.0) min {c(v)+m~'}.

(25)

465

ODD=MASS DEFORMED NUCLEI

Here the interaction between quasi-particles and phonons is most effective; it causes the strongest lowering of the roots of eqs. (10) and (16) both with respect to e(p) and to the first pole of the secular equation. The contribution of the one-quasi-particle state is C 2 = 0.3-0.8. The secular equation contains many terms with different 2/~i and v which are important. Thus, in the case (25) the interaction of quasi-particles with phonons leds to the formation in odd-mass deformed nuclei of non-rotational states having complex structure. The important quantity which characterizes the structure of an excited state is the reduced probability of the electromagnetic transition. The increase of the reduced probability B(E2) for the electric E2 transition as compared to the one-particle value indicates that the wave function of K -- K 0 - 2 or K = K o + 2 states (K o is related to the ground state) has an appreciable admixture of the component quasi-particle plus phonon 2 = 2, # = 2. An increase of B(E3) as compared to the one-particle value indicates a noticeable admixture of the component quasi-particle plus octupole phonon, etc. For the reduced probability of the electric E2 transition between the two states (7) we get: B(E2, IK ~ I'K') = ](I2KK'-KI I'K' )(K'lm(2, K ' - K ) I K )

+ ( I 2 K - g ' - K I I ' - K ' ) ( - 1)r-K'(K'R~ - Xlm(2, - g ' - g ) l g ) l 2,

(26)

where (K'lm(2v)lK) is the matrix element of the v-component of the E2 transition operator and Ri the operator of rotation at 180 ° around the axis 2. Though formally eq. (26) contains two terms, in all the cases of interest, only one of them operates and the other either is exactly zero or negligibly small. In tables 5-8 are the B(E2) values calculated with the Clebsch-Gordan coefficients equal to unity. These values are usually obtained in Coulomb excitation experiments (see, e.g. ref. 12)). The matrix element (K'lm(2v)lK) consists of four parts


+eeff

Z

~-'~ /-~2#i 2~tl 22 (vv)v¢~+ , /_ ~,p~+Op,,+f

2/ti vv'

221 • ZDpp,+M(2,)).

(27)

i

Here M(2i) is the matrix element of the collective E2 transition in an even mass nucleus between the ground state 5Uo and the ith phonon state Q~+(22)~o. All terms in eq. (27) have a direct physical meaning. The first term corresponds to absorption of the phonon in the 5U(K'~') state, the second to the one-particle transition between quasiparticles in both states, the third to the one-particle transition accompanied by absorption and creation of a phonon and the fourth connected with the phonon admixture in the 7~(K~) state. In purely one-quasi-particle states only the second term will operate. In the transition from a purely collective to purely one-quasi-particle state only the first will operate (and we get B(E2)oaa = ½B(E2) .... ). For many solutions the contribution is given by all terms in eq. (27). If ~/i is lower than the first pole then the first, second and fourth terms are summed up coherently.

466

V. G. SOLOVIEV AND P. VOGEL

In this case enhanced E2 transitions can occur, although the p h o n o n admixture is comparatively small. This takes place for the first K ~ = z1 + states in the Tb isotopes. If r/j is higher than the first pole different terms in eq. (27) have different signs and an additional enhancement does not take place. The quantities M(2i) in eq. (27) are from earlier 29) calculated B(E2) for E2 transitions between the ground and gamma-vibrational states o f even-mass nuclei. The values of the effective change are also from ref. 29). In computer calculations an error was comitted 29) which did not affect the B(E2) values but led to an incorrect determination o f eerr. In the present paper corrected values eeff = 1.2e for protons and eeff = 0.2e for neutrons were used. It should be also noted that in ref. 29), B(E2) for 172yb is rather small and therefore in ~73Lu and 173yb underestimated values for B(E2) were obtained. On the other hand, for the even-mass G d and D y isotopes in ref. 29) the B(E2) values were slightly overestimated; this should affect the B(E2) values in the corresponding odd-mass nuclei. The results for the complex and collective states are given in tables 5-9; tables 5 and 6 for the o d d - p r o t o n nuclei and tables 7 and 8 for the odd-neutron nuclei give the results for K = K 0 - 2 and K = K 0 + 2 states, where K o is related to the ground states o f odd-mass nuclei. The experimental and calculated values o f the energies, o f the decoupling parameters a and o f the reduced probabilities B(E2) as well as the contribution o f the one-quasi-particle state C 2 and the contribution -p,--pv(72(t922~)2of terms in eqs. (10) or (16) corresponding to the first pole with 2 = 2, # = 2 are given. In some cases there are several states with given K ~ which have different structure, therefore tables 5-8 give not only the first but sometimes the second and third states with those K ~ values. Some complex structure states which are most interesting are given in table 9. The experimental data are f r o m the reviews 13,20) and also from refs. 1 5 - 2 2 , 2 6 - - 3 7 ) . Let us consider the peculiarities o f some nuclei having an odd n u m b e r o f protons. In 153Eu and lSSEu the admixture o f the 4 1 3 5 + Q 1 ( 2 2 ) state to the first K ~ -- ±÷2 states is 11 and 10 ~o and to the second states, 80 ~ . The calculated values o f the decoupling parameter a disagree with the experimental values which are larger than a N = - 0 . 7 9 for b = 0.3 and a N = - 0 . 8 9 for 6 = 0.2. The contribution o f the gamma-vibrational state to the first K ~ = 9+ states is negligibly small while the second K,~ = ~o + states are pure gamma-vibrational states. In 155Eu, the K ~ = 3 - 1095 keV state is interesting; the structure is given in table 9. In the Tb isotopes the first K ~ = ~+ states contain ( 6 0 - 6 5 ) ~ o f the one-quasiparticle 411 ~ state and, nonetheless B(E2) is large and equals 1.6-2.8. Due to the large contribution of the 411 ~ state the disagreement with experiment in the a-values is large. In 159Tb the calculated deeoupling parameter for the first K ~ = ~+ state is closer to the experimental value for the second K ~ ~1 + state and on the other hand the calculated a-value for the second state is closer to the experimental a-value for the first state. As to K ~ = ~+ states the first states have a complex structure, the third states are octupole states to a large extent.

TABLE 5 K = /(0--2 states in odd-proton nuclei (Ko is related to the ground state) Nucleus

v for

K~

Energy (keV)

B(E2)/B(E2)s.p.

ground

I~Eu

½+

155~69~U

½+

4114

4114

exp.

4134

635

765

calc.

exp.

157Th 65--159Th 65 - v

½+

½+

½+

411~

411+

411+

411~

411~

411~

761

597

calc.

650

0.5

1800

1.1

650

0.6

1800 155Tb 65--~

½+

411+

2 221 ) 2 Cp(Dpv

( × 100)

( x 1oo)

500

2.2 0.2

530

2.1

1300

0.5

430

971

1150

calc.

--0.95

--0.61

72

11

--0.10

10

80

--0.63

74

10

-0.10

10

80

--0.50

64

28

-1.0

1.2

1200

580

exp.

1.5

0.1

0 0.04

--0.50 0

2.8

0.05

0.3

--0.81

--0.47 0

1600 161Th 65---

C2 p

parameter a

state

413+

Decoupling

411~

550

1.6

--0.50

1200

0.2

0

0.3 65 0.2 60 0.4

10 25 20 30 15

38

40

64

25

0.5

15

1611-1-67_.u

~-

541~

523~

593

1000

1.9

12

65

163--67tlO

3-

541~

523)

583

940

2.4

9

78

1651"1A67-.O

-~--

541~

523)

515

820

2.8

4

90

167~ 691m

~+ ~-

411~

411+

570

0.7

81

16

1050

2.3

15

80

169~ 69tm 171T-6 9 -Ill 173 71Lu.

~~+ ~+

Z+

411f 411~

411~

4114

411+

1.9

570

620

1.5

0.3

91

6

900

1200

0.3

2.0

6

90

650

0.1

95

2

1350

1.6

3

9O

675

404~

1050

96

1400

1

1

9O

175 71Lu.

3+

411~

404~

1000

0.2

82

16

177--71LU

~+

411~

404~

900

0.5

61

36

179T~ 73 - a

3+

411~

404~

1000

402+

1200

18173Ta.

3+

411~

404+

183 75R~

½+

400~

402~

4114

460 1103

~

75R~

½+

400~ 4114

647

400

872

850

400~ 187 75Rc

½+

400~ 4114 400~

0.5

511

400

618

850 1150

36

5

9O

87

5

94 2.5 3.6

1100 402~

0.34

900 1400

402~

53

62

1200

4oo I 185

1.1

0.5

~s0 0.38

1.8 3.1

0.6

0.32 ~0

0.38

0.31

-- 1.1 2.4

~0

2

95

79

7

92

0

4

9O

78

9

9O

0

3

9O

c~

.~

~

c~

Ix~

~

~

t-o

c~

o~

~

+

o~

o

+

~ o

c~

+

0

.<~_ __.>.

~ o

.
~-,

~ o

.~

~o~

0

ix~

~

~.~

~

~

~

t~

o~

I~

~o

b~

~,~

o

~o

i~

~o

~

~o

.~

0

+

×

t~

~

Z

0

r~

o<

~

r~

.<

C=

0

÷

c~

ODD-MASS DEFORMED NUCLEI

469

As is seen from table 1 comparatively high lying K ~ = ~+, 2 25+ states in 159T b strongly differ from those in 157Tb. According to the calculations, in ~SVYb the first excited K ~ = 3 + 1050 keV state is beta vibrational and the 1660 keV state is close to the 4225 state, in 159Tb the first K ~ = 3 + 1600 keV state is close to the 422~ state and the second state is a beta-vibrational state. The second excited K ~ = -~+ 1100 keV state in 157Tb is beta vibrational, and in ~59Tb with 1250 keV energy it is an octupole state. This is due to the increase ofco~ ° in a58Gd as compared to 156Gd. In the H o isotopes the calculated energy o f the K ~ = ~ - state is much higher than the experimental energy. The solutions o f eq. (16) with Pl = 541~', Pz = 532~, do not improve the situation. The } - 514 keV state in 165Ho has a c o m p o n e n t 523T + Q~(22) equal to 90 ~ . But the latter has a large three-quasi-particle c o m p o n e n t n521 + - n523+ +p523]'. That is why the beta transition from the 521~, state in 165Dy to the 23- state o f 165Ho is o f the au type with logJ~ = 5.2 (ref. 3z)). It is hindered about twice as compared with the single-particle transition 5235 ~ 523~. The K ~ a ~- states are g a m m a vibrational; the K ~ = 5+ 995 keV states in 165Ho for which l o g f t = 5.7 in the beta decay (ref. 32)) o f ~65Dy is possibly a three-quasi-particle state with the configuration n633T + n523 $ - p523 T. In the T m isotopes the first K ~ = a+ z states contain a large contribution o f the onequasi-particle 4111" states and the second states are mainly g a m m a vibrational. In the region 0.8-1.4 MeV there are three K ~ = ~+ states, two o f which contain a large contribution o f 411 $ + Q~(22) and 4135 states and the third state is close to the 402T state. In the Lu isotopes the first K ~ = ~+ states contain a large contribution o f the onequasi-particle 411T state, and the second state are mainly g a m m a vibrational; K ~ = J2~ + states are g a m m a vibrational. In 173Lu there is a K ~ = ~ - 888 keV state. According to our calculations the K ~ = 3 - state has an energy 1.3 MeV and it is rather close to the 5325 state, since C 2 = 0.9. This is due to the fact that in our level scheme ~(541]') = 1.6 MeV and Y~(22) = 7 x 103 becasue according to ref. 9) the first K ~ = 2 + state in 172yb is close to the two-quasi-particle state. The calculations o f the Re isotopes have a tentative character because o f some defects o f the average field level scheme. Thus s(4001") -- 0.210hObo is very small, this leads to underestimates o f the energy values of the first K ~ z1 + states and also to underestimates of B(E2) values due to a large contribution o f the 400]" state. The K ~ = ~-+ states o f the Re isotopes are mainly gamma-vibrational; this agrees well with experiment 33,34). Let us consider the peculiarities o f some odd-neutron nuclei. In a53Sm and lSSGd there is a K ~ --- ½+ state, which is rather close to the 400T state since CZo = 0.65-0.69; this state is lowered with respect to e(p) by 600 keV and with respect to the first pole by 400 keV, the K ~ = ½- state with an energy 600 keV has a complex structure. In nuclei with N = 93 K ~ = ½- and -}- states have a complex structure and it would be interesting to measure their B(E2) values.

470

V. G. SOLOVIEVAND P. VOGEL

I n nuclei w i t h N = 95 the first K = = ~-- state is g a m m a - v i b r a t i o n a l . I n 161Dy the K = = ½- 365 k e V state has a c o m p l e x s t r u c t u r e a c c o r d i n g to o u r c a l c u l a t i o n s ; t h e c o n t r i b u t i o n o f t h e 521~ state is 53 ~ . I n 163Dy t h e K ~ = 3+ state w i t h a c a l c u l a t e d e n e r g y o f 370 k e V s h o u l d h a v e a c o m p l e x s t r u c t u r e . I n t h e first K = = ½- state o f 163Dy t h e c a l c u l a t i o n s o v e r e s t i m a t e t h e c o n t r i b u t i o n o f t h e 521~ state. i n n u c l e i w i t h N = 99 t h e first K ~ = 3+ states in 165Dy a n d 167Er are g a m m a v i b r a t i o n a l as is seen f r o m t a b l e 7; t h e c a l c u l a t e d e n e r g i e s a n d B ( E 2 ) v a l u e s are in a r a t h e r g o o d a g r e e m e n t w i t h e x p e r i m e n t . H o w e v e r , in 169yb, a c c o r d i n g to the calc u l a t i o n s , t h e g a m m a - v i b r a t i o n a l state e n e r g y increases by 600 k e V as c o m p a r e d to TABLE 7 K -- K0--2 states in odd-neutron nuclei Nucleus

v for Energy (keY) B(E2)/B(E2)~.p. ground - - - state exp. calc. exp. calc.

K~

~S~Gd assSm tS";Gd *SgDy l~gGd ~lDy 16aDy

½ ½ ~ }½+ I +~

521,~ 521,~ 521,~ 521,~ 600~' 660~' 521 l,

521]` 521]` 844 521]' 521]' 642]" 642~' 523,~ 351

~';~Er

~

5211v 523~ 298

J65Dy l~:Er ~GgYb

!~ ~: ~+

651]' 633]` 539 651) 633I' 532 651~' 633~

*:~Yb mEr arayb x:sYb ~¢rHl" ~T:Yb ~saW

~-.{-~-~-½ :2+ ~-

521]` 510} 510]' 512,~ 512,~ 6421' 501]' 512~

5211v 902 512") 701 512} 1040 5141v 809 514,~ 624~" 510]' 458 209

600 950 700 750 250 420 300 900 340 950 700 750 1000 1350 1200 800 1200 800 800 1200 650 270

2.7

2.6 3.2

1.1 0.4 0.9 1.0 0.1 0.2 1.0 t.5 0.7 1.6 3.0 2.5 0.1 1.1 0.4 0.9

Decoupling parameter a exp. calc. 0.3:-0.3

0.25 ~0.65 0.56

--0.17 --0.22

0.5 0.5 1.3 0.3 0.2

165Dy a n d 167Er, w h i c h is d u e to the i n c r e a s e o f

0.39 0.59 0.45 0.42 4.3 4.3 0.60 0.07 0.65 0.09

0) 22

C"-o C2p(D2ell, v)e (:< 100) 44 66 51 47 70 70 68 8 73 10 9 7 72 3 70 48 65 77 77 0.4 86 93

(;< 100) 37 17 27 33 5 8 27 49 22 62 90 88 4 87 25 48 20 14 14 99 0 4

in 168yb as c o m p a r e d to l¢";Dy

a n d 166Er. T a b l e 9 gives t h e c o m p l i c a t e d s t r u c t u r e o f K = = 3 - a n d ½- states in l~'SDy a n d *67Er. I n l v t y b a n d ~ 7 3 y b the c a l c u l a t i o n s o v e r e s t i m a t e energies o f the K 0 - 2 a n d K 0 + 2 states d u e to t h e i n c r e a s e o f the values o f 0 22 a n d y 1 ( 2 2 ) in l v ° Y b a n d esp e c i a l l y in 1 7 2 y b as c o m p a r e d to ~ 6 s y b and 176yb. T h e K ~ -= 3 - a n d -I- states in 1 7 a y b has a c o m p l e x s t r u c t u r e , i n 1 7 3 y b t h e r e are t w o c o m p a r a t i v e l y h i g h - l y i n g K ~ = 3 - states, o n e o f 1340 k e V a n d close to the 512J, state a n d the o t h e r o f 1224 k e V energy. W e get t w o p o s s i b l e s o l u t i o n s for the latter, one w i t h a c o m p l e x s t r u c t u r e a n d t h e s e c o n d close to the 52l]" state. I n 1 7 s y b the first K ~ = 3 - 809 keV state is

471

ODD-MASS DEFORMED NUCLEI

relatively close to the 5125 state (Coz = 0.77), the second K ~ = 3 - 1616 keV state is mainly gamma-vibrational, and the contribution o f the 521T state is 18 ~ . The first K ~ = 3 - 709 keV state in x 7 7 y b is close to the 5123, state (Cap = 0.77), and the second K" = ~-- 1365 keV state has the contribution o f the one-quasi-particle 501T state o f 67 ~ . In calculating the levels o f 1 7 5 y b and a v Y y b and o f the H f and W isotopes we should take into account the decrease o f the equilibrium deformation as c o m p a r e d to 5 = 0.3. This has not been done. In refs. 2, 3) it was noted that the energies o f the states close to K = K o - 2 g a m m a vibrational states are lower than those for K = K o + 2 states: g ' ( K o - 2 ) < g ° ( K o + 2 ).

(28)

TABLE 8 K = K0-}-2 states in odd-neutron nuclei N

Nucleus

K~

p

r for Energy ground (keV) state calc.

93 95 95 97 97 99 99 99 101

159Dy lS9Gd 161Dy 16aDy l~SEr 16~Dy l~TEr l~gYb 169Er

~~+ ~+ ~~.,t+~ '~ + ,sn+ ~

5145 624~ 624~ 514~ 514~ 615~ 615~ 615~, 5235

521~ 642~ 642~ 5235 5235 633~. 633~, 633~ 521~

101

mYb

~-~

5235

5215

101 103 105 109

~aHf ~rayb 17ayb 1saW

~~~11)-

5235 5055

521~ 512~ 514~ 510'~

505~ 512"~

1200 1000 950 750 800 730 830 1350 850 1250 1000 1600 960 1160 1700 1050 1400

B(E2)/B(E2)s.p" C2p calc.

2.7 2.8 2.7 3.1 2.6 3.3 2.8 2.1 1.3 0.8 0.1 1

1.0 0.7 1.2

(× 1 0 0 )

0.1 2 1 0.1 0.I 0.7 0.6 0.7 46 49 90 5 81 0 0 77 20

C-'v~D22~mj'-' (100)

85 94 95 87 99 95 98 99 47 30 6 90 99 99 20 80

The reasons for the fulfillment o f this relation are the following: first, for small K in eqs. (10) and (16) there are more terms in summing over v than for larger K. Second, in the average field level schemes there are less K = ~'-, ~2 states as c o m p a r e d to K = ~-, { states. Therefore for large K t h e r e are rarely cases when ~(p) is only somewhat higher than the energy o f the first pole and when the first non-rotational state energy is lowered very strongly. Eq. (28) must be fulfilled for states for which the energy o f the first gamma-vibrational pole is lower than e(p). Otherwise, these states have complex structure and eq. (28) m a y not be fulfilled. The calculations prove the validity o f eq. (28) when both states are close to the gamma-vibrational states; this is seen from tables 5-8. It is

472

V . G . S O L O V I E V A N D P. V O G E L

difficult to compare the calculation results with the experimental data on W(K o + 2) d~(K0- 2) since there are at present not m a n y experimental data on the splitting energies where both states are close to the gamma-vibrational states. The comparison o f the calculation results o f the characteristics o f the collective states and the complex structure state with the corresponding experimental data indicates that the interaction of quasi-particles with p h o n o n s gives a rather g o o d description of the energies o f these states and o f the quantities such as B(E2), the decoupling parameter a and others. It is necessary to stress that in m a n y cases states which were earlier interpreted e.g. in refs. 13,3o,37) as gamma-vibrational states, according to our calculation have a complex structure. TABLE 9 Complex structure states Nucleus

K~

a55Eu 155Gd lSgGd ~61Dy lr'3Dy ~6~Dy ~SDy 16~Er ~Er ~TEr ~gEr ~6~Yb

~ ~-' ~+ ~ :~ =3," 1½~ ~ ~ ~ .~

17tyb

~

17aYb ~7~Yb ~Tryb

~ ~ ~

Energy (keV)

Structure

exp.

calc.

1095

1060 850 340 450 370 800 640 630 750 800 850 1300 1200 1350 1700 900

365 574 570 508 745 915 805 945 1224 1616 1365

541]`70%, 541]`+Q~(20)19 }o; 411~'+Qx(30)5.5 i'£ 400]`69o/ /o 651]`77%; 532~,+Q~(30)6 %; 521]'+Q~(30)6 ~; 521~53~; 523++Q~(22)28 %; 521~+Q1(22)18 ','o 651]`71%; 521~+Q1(30)14 %; 532++Q~(30)6 7~; 521]'68~; 521++Q1(22)25 %; 651]`+Q1(30)4 ~,; 510]`32~; 512]`+Q1(22)63 o4; 512~,+Q~(22~4 i~; 660]`42o~; 642]`+Q~(22)43 %; 651]`~ Q1(22)7 '?,', 551~79%; 521Jc+Ql(22)15,o, 651]`+Qa(30)4% 510~32%; 512J'+Q~(22)54 %: 512~,+Qt(22)3 % 523~46%; 521~,+Q~(22)47 o~; 642]`+Q~(30)2 i',, 510]`39~o; 512]`+Q1(22)56 o£; 512+~ Q1(22)3 ", 510]`51%; 512~+Qx(22)41 o,~, 521~ Qt(20)3.5 ~'~ 521'I'7 %; 521+-t-Q1(22)90 % 521118 ~o; 521~+QI(22)80 ')o 501)67 %; 503]`+Q~(22)26 ~,;

5. Conclusion We calculated the properties o f the ground and excited states for 62 nuclei in the region 151 < A < 187. A total of 20-30 states was calculated for each nucleus. Thus, the obtained material is sufficiently large. Tables 1-9 give only a small part o f the results concerning the most interesting cases and the cases for which there are experimental data. The remaining material can be used when additional experimental data become available. The aim o f the present paper is to give a general picture o f the excited states for m a n y odd-mass nuclei. Therefore we do not analyse here each nucleus separately. In further development o f this topic, a detailed and careful calculation of the properties of the most interesting nuclei with improved Nilsson potential parameters, consideration o f the Coriolis interaction, etc. is needed. Using this approach, better agree-

O D D - M A S S DEFORIvlED N U C L E I

473

ment between the calculation results and experiment can be obtained and the predictions for the considered states can be improved; e.g. a small displacement of the average field one-particle levels 411 ~ and 4111" can noticeably improve the description of a number of odd-proton nuclei states. It should be noted that in investigating the interaction of quasi-particles with phonons there is no free parameter. The quantities ~o~" and Yi(2/0 are obtained in calculating the collective states of even-mass nuclei. Therefore in the cases when the agreement between theory and experiment was not sufficiently good in even-mass nuclei, this discrepancy should take place also in odd-mass nuclei. A general picture of the excited states of odd-mass nuclei is more complicated, and the descriptions are somewhat rough as compared to even-mass nuclei. The calculated position of the levels of deformed odd-mass nuclei is to a large extent determined by the position of the average field one-particle levels (Nilsson level scheme). Therefore the calculation accuracy of different characteristics of odd-mass nuclei is essentially restricted by a rough description of the energies and the wave functions with the Nilsson potential. Our investigation showed that the structure of the excited non-rotational states of deformed odd-mass nuclei is very different. Most low-lying and many more excited states in their structure are close to the single-quasi-particle states. The interaction of quasi-particle with phonons gives rise to admixtures to the single-quasi-particle states. The role of the admixtures increases, as a rule, with the excitation energy. The account of the interaction of quasi-particles with phonons has led to an improvement of the description of states close to the single-quasi-particles states as compared to the independent quasi-particle model. The interactions of quasi-particles with phonons have led to the formation of the collective non-rotational states and the complex structure states in odd-mass deformed nuclei. Comparing the calculated properties of these states with the experimental data it can be concluded that the model used rather well describes the energies and structure of these states. Many states which were earlier described as g a m m a vibrational have complex structure according to our calculations. For the further investigation of the structure of the states of odd-mass deformed nuclei, it is necessary to increase the amount of experimental data on the state energies, their spin and parity, the alpha-, beta- and gamma-transition probabilities, the spectroscopic factors in direct nuclear reactions etc. In conclusion we express our deep gratitude to Professors N. N. Bogolubov, A. Bohr, M. Bunker, D. B6s, D. Burke, B. Elbek, L. A. Malov, B. Mottelson and C. W. Reich for interesting discussions and to A. A. Korneichuk, K. M. Zheleznova and G. Jungklaussen for assistance in constructing the programme and machine calculation.

474

v.G. SOLOVIEVAND P. VOGEL

References I) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37)

V. G. Soloviev, ZhETF (USSR) 43 (1962) 246 v. G. Soloviev, Phys. Lett. 16 (1965) 308 v. G. Soloviev, Structure of complex nuclei (Atomizdat, Moscow, 1966) E. R. Marshalek and J. O. Rasmussen, Nuclear Physics 43 (1964) 438 V. G. Soloviev and P. Vogel, Phys. Lett. 6 (1963) 126; V. G. Soloviev, P. Vogel and A. A. Korneichuk, Izv. AN SSSR (ser. fiz.) 28 (1964) 1599 Lu Yuang, V. G. Soloviev and A. A. Korneichuk, ZhETF(USSR) 47 (1964) 252 D. B6s, Nuclear Physics 49 (1963) 544 V. G. Soloviev, Nuclear Physics 69 (1965) 1 V. G. Soloviev, Atom. Energ. Rev. 3 (1965) 117; K. M. Zhelesnova et al., preprint JINR D-2157 (1965) D. B6s and Cho Yi Chung, private communication L. Persson, H. Ryde and K. Oelsner-Ryde, Ark. Fys. 24 (1963) 451; H. Ryde, L. Persson and K. Oelsner-Ryde, Ark. Fys. 23 (1962) 195 R. M. Diamond, B. Elbek and F. S. Stephens, Nuclear Physics 43 (1963) 560 M. E. Bunker and C. W. Reich, private communication M. K. Kalpazhiu and P. Vogel, preprint, JINR E-2579 (1966) B. Mottelson and S. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 1, No. 8 (1959) K. Ja. Gromov, Zh. T. Zhelev, V. Zvolska and V. G. Kalinnikov, Yad. Fiz. 2 (1965) 783 L. Persson, R. Hardell and S. Nilsson, Ark. Fys. 23 (1962) 1 J. O. Newton, Phys. Rev. 117 (1960) 1820 D. Burke et al., Mat. Fys. Medd. Dam Vid. Selsk. 35, No. 2 (1966) B. Harmatz, T. Handley and J. Mihelich, Phys. Rev. 119 (1960) 1345 R. Wilson and M. Pool, Phys. Rev. 120 (1960) 1843 C. J. Orth, M. E. Bunker and J. W. Starner, Phys. Rev. 132 (1963) 355 Lu Yuang et al., ZhETF(USSR) 40 (1961) 1503 V. G. Soloviev, Mat. Fys. Medd. Dan. Vid. Selsk. 1, No. 11 (1961) L. A. Malov, C. M. Polikanov and V. G. Soloviev, Yad. Fiz. 4 (1966) 528 L. Funke et al., Nuclear Physics 55 (1964) 401 B. S. Dzhelepov and G. F. Dranischnina, Thesis XYI, Meetings on Nuclear Spectroscopy (1966~ Moscow Lu Yuang and Chou Lyang yuang, preprint JINR P-687 (1961) P. Vogel, Yad. fis. 1 (1965) 752 V. Hnatowicz and K. Gromov, Yad. Fiz. 3 (1966) 8 L. Funke et al., Nuclear Physics 70 (1965) 335 L. Perssun, R. Hardell and S. Nilsson, Ark. Fys. 23 (1962) 1 K. M. Bisg~rd, L. J. Kielsen, E. Stabell and P. (~)stergSrd, Nuclear Physics 71 (1965) 192; K. M. Bisghrd, private communication C. Sebille et al., Compt. Rend. 260 (1965) 3926 R. A. Kenefick and R. K. Sheline, Phys. Rev. 139 (1965) B1479 O. W. Schult, B. P. Meyer and U. Gruber, Z. Phys. 182 (1964) 171 R. K. Sheline and W. N. Shelton, Phys. Rev. 136 (1964) B351