Equilibrium deformations and excitation energies for non-collective states in odd-Z rare-earth nuclei

-q

Nuclear Physics A245 (1975) 376-396; ~ ) North-Holland PublishinB Co., Amsterdam Hot to be reproduced by photoprint or microfilm without written permission from the publisher

FOR

EQUILIBRIUM DEFORMATIONS AND EXCITATION ENERGIES NON-COLLECTIVE STATES IN ODD-Z RARE-EARTH NUCLEI t B. S. NIELSEN** and M. E. BUNKER

Los .4lamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87544 Received 25 October 1974 (Revised 25 February 1975) Abstract: Nuclear potential energy surfaces have been calculated for one-quasiparticle states in the odd-Z rare-earth nuclei. The calculations are based on the Nilsson model with inclusion of nuclear pairing effects. A renormalization of the energy surface to that of a liquid drop is made using the Strutinsky method. The lowest minima in the potential energy surfaces yield deformations (e2, e4) and excitation energies of the one-quasiparticle states,which can be compared with the experimental values after appropriate corrections, including a contribution from zero-point rotational motion. Many of the empirical energy trends among the one-quasiparticle states are well reproduced by these calculations. Some of the observed trends are attributable to variation in the hexadecapole (ca) deformation.

1. I n t r o d u c t i o n

F o r the odd-A deformed nuclei with 150 < A < 190, the ground-state assignments and the sequence of low-lying one-quasiparticle excited states are well u n d e r s t o o d in terms of existing single-particle models 1 - 6). However, because of various perturbing effects, the pure single-particle models do not give realistic predictions for the particle excitation energies in individual nuclides. If one examines the energy systematics of the one-quasiparticle states 4) it is apparent that certain orbitals exhibit definite local trends as a function of N or Z (cf. figs. 10b, 10c), for which one can hope to find a consistent theoretical interpretation. A m o n g the phen o m e n a k n o w n to exert an i m p o r t a n t influence on the single-particle energies are pairing correlations, quasiparticle-phonon interactions, and changes in nuclear shape. Calculations 3, 4, 6) in which these and other effects have been taken into a c c o u n t have been moderately successful in reproducing the state energies, but there is still m u c h r o o m for improvement. In the present w o r k we give a c o m p a r i s o n between calculated and experimental excitation energies for the deformed o d d - Z rare-earth nuclei. The calculations are based on the Nilsson model 7), with inclusion o f nuclear pairing effects. O n e of our main

t Work supported by the US Atomic Energy Commission. tt On leave from the Physics Laboratory, Royal Veterinary and Agricultural University, Copenhagen, Denmark. 376

EQUILIBRIUM DEFORMATIONS

377

interests was in determining to what extent the observed systematic energy trends can be related to changes in nuclear deformation, using theoretically calculated deformation values for individual states. Variation in the nuclear pairing strength, deduced from oddeven mass differences, was also taken into account. The odd-Z nuclei were chosen for study because in these nuclides the energies of the low-lying one-quasiparticle states are not strongly affected by the quasiparticle-phonon interaction, which, for simplicity, was not included in the system Hamiltonian. 2. Calculations 2.1. POTENTIAL ENERGY CALCULATION

The gross features of the present calculation follow the formalism outlined in refs. 7, a). A major part of the computer program was supplied by P. M¢ller. Changes in the program due to the fact that we are dealing with odd-A nuclei are discussed below. The total potential energy for a given odd-A nucleus is calculated from the following terms: Epot(/32, 84) = E l . d . + ~ E s . p . - ( ~ Es.p.). . . . ageWEpair(N, Z)'q-Eodd, N,Z

(1)

N,Z

with ELd. being the liquid-drop energy 9) and ~N,Z Es.p. being the summation of singleparticle energies for occupied states (the summation taken over both proton and neutron states). The term (~N. z Es.p.)..... ge is the total single-particle energy calculated after a Strutinsky smearing-out of the level scheme 1o). In addition, the total potential energy includes the pairing energy Epair(N, Z) and the quasiparticle energy of the odd particle, Eodd.

In the calculation of the single-particle level scheme we have used the modified harmonic oscillator potential 7) V = ½h(oo p2(l - ~e2 Pz + 2e,, P,,)- rhcoo[21, • s + #(! 2 - (It2))],

(2)

which includes possibilities for both quadrupole and hexadecapole distortions. Values for the potential parameters K and/~ are adopted from ref. 7) and are given in table 1. TABLE 1 Nilssonmodel parameters Z

Kp

~p

Kn

~n

63455 67~9 71-73 75

0.0648 0.0637 0.0628 0.0620

0.591 0.600 0.608 0.614

0.0637 0.0637 0.0636 0.0636

0.438 0.420 0.405 0.393

378

B.S. NIELSEN A N D M. E. B U N K E R

The quasiparticle energy and the occupation factors are calculated on the basis of the usual pairing formalism without inclusion of blocking. The relation between the number of particles n, the pairing strength G, and the pairing parameters ,t and A (2 = Fermi level, A = pairing gap) is given by the usual BCS equations n = ~

1- ~/(E~.v._2)Z+A2/,

2 1 G = ~ ( x ~ . P . - 2 ) z + Az'

(3)

(4)

with the summations extended over a specified number of the single-particle states that lie nearest to the Fermi surface. In the case of formula (3) all levels below the lowest level in the basis are assumed occupied by a pair of nucleons and all levels above the highest level in the basis are assumed to be empty. By an iterative process 2 and A can be calculated from eqs. (3) and (4) on the basis of a given single-particle level scheme an d given values for n and G. Values for G were obtained from the following expression: N-Z GA = go +91 - - ,

A

(5)

which represents an isopin-dependent pairing strength, as introduced in ref. 7). The values adopted for the pairing constants are go = 19.2 MeV and 91 = ---7.4 MeV [ + for protons and - for neutrons], as recommended by Nilsson et al,7). Following ref. 7),we have used for the number of levels in the summations ofeqs. (3) and (4) 2 lx/~Z for protons and 2x/q5~' for neutrons. Using the quoted values for go, g l , and the number of levels, one obtains good average agreement between calculated values for the pairing gap and the odd-even mass difference 11. i 2) determined from experimental nuclear masses 13), which should be equivalent 11). On the other hand, rather large discrepancies are found in some cases [see, e.g., fig. 5 of ref. 14) or fig. 1 of this work]. In the present work we have chosen to calculate the position of the Fermi surface 2 on the basis of formula (3), with A equal to the empirical odd-even mass difference, the assumption being that more realistic values of 2 would thus be obtained. Having calculated the single-particle level scheme and the pairing parameters 2 and A, one can obtain the occupation factors U and V from the well known expression V?

= 2 1+ x/~E~.p._2)2+A2j.

(6)

The theoretical quasiparticle energy can now be calculated as

(7) where v refers to a given single-particle orbital.

379

EQUILIBRIUM DEFORMATIONS

In our computer program, the total potential energy is calculated for a set of grid points in the (e2, e4) plane. Then by use of spline interpolation, a so-called potential energy surface is obtained in a region of the (/32,/34) space. The calculated total potential energy at the absolute minimum of the energy surface is taken as the nuclear equilibrium energy, and the corresponding values of (e2,134) as the equilibrium deformation. In the approximation that dynamical effects may be ignored, the excitation energy for a specific state then corresponds to the difference between the nuclear equilibrium energy of the state and that of the ground state. 2.2. ZERO-POINT ROTATIONAL ENERGY AND OTHER ENERGY PERTURBATIONS As mentioned above, there are other factors that contribute to the energy of a nuclear state besides the total potential energy. In some instances, e.g., particle-phonon interactions may be quite important; however, according to the calculations of ref. 6) all of the odd-Z states considered here are predominantly (> 85 ~) one-quasiparticle in character, which implies that the energy perturbations associated with phonon admixtures are relatively small. There are also energy perturbations due to nuclear rotation, which we consider below in some detail. In the case of an axially symmetric odd-A nucleus, the rotational energy of the state II, K, ~) can be written as [ref. 15), eqs. (10}-(15a)] h2 Erot(I, K, ~) = ~ - ~I(I + 1 ) - 2 K 2 + ERpC(I, K, ct)+ (I, K, ~lj21I, K, 0t)].

(8)

Usually, terms independent of I are assumed included in the intrinsic state energy. However, since we have identified the calculated energy of the odd particle with the quasiparticle energy, eq. (7), it would appear that the total energy should include a contribution due to the zero-point rotational energy, Ezprt = Erot(K, K, 0t). In our analysis, we have calculated Erot(K, K, ~t) according to eq. (8), although it is possible that this equation yields an overestimate of the zero-point rotational energy, as discussed in a later section [see also Ogle et al. 3) and Stephens 36)]. In eq. (8), the term Expc, the Coriolis term, results from the coupling of the odd particle to the rotating core. The RPC matrix elements are given by, for K = K' = ½, (I, K, ctlHRpc[I, K', ct') = - ( K , alj-[K', a ' ) ( - 1)'+~(I +½)(UK Ux, + Vx Vr,),

(9)

and for all other cases by (I, K, c~lHRPc[I,K', ct') = - ( , K , alj-IK', a')~r(I-K)(I+K')(UK U~,+ V~VK,)

= -(K,a]j+IK',m')~(I+K)(I-K'XUKUr,+VKVK,),

(10) (11)

w h e r e K ' = K + I ~ ½. The RPC term has diagonal elements only for K = ½[cf. eq. (9)] given by ERPc(½, ½, ~) = --a = (½, ~[J-1½, ct),

(12)

B. S. NIELSEN AND M. E. BUNKER

380

with a being the so-called decoupling parameter 1). In evaluating ERpc, we have used only these elements [eq. (12)]. For the odd-Z states considered here, inclusion of the off-diagonal RPC terms would shift the band heads by no more than 25 keV. The results o f such a calculation for 165Ho are presented in fig. 11. The jz term is calculated using the relation

j2 = ~ + j_ +j_j+)+j2,

(13)

from which we get

(g, aljZlK, a ) --- 0.5[ ~ ((K, a ] j _ l g + 1, v)PK, K+I)z V

+ ~.,((K-I, vIj_IK, ct)PK,r_O2]+K 2,

(14)

V

f

I

[

I

I

Tm isotopes Odd-even mass diff.,Pp Colc. pairing gap, AO

I.I

>

~.0

:E

0.9-

&p

-

0.8-

pp

-

I o63

_

I i65

I i67

I J69

i iTi

_

A Fig. 1. Comparison of the empirical odd-even mass difference, Pp, and the calculated pairing gap zip [obtained from eqs. (3) and (4)], for the Tm isotopes.

EQUILIBRIUM DEFORMATIONS

381

where the sum extends over all available K + 1 states. In principle one should include off-diagonal terms of the form (K, ~lj2jK, ~'), but such terms have been neglected in the present analysis. Pairing has been taken into account using the well-known pairing factor PK, K±I = U K U K + I +

(15)

VKVK±I.

We thus obtain for EzpR Ezp R =

h2 ~ {g--fK,~a+0.5[ ~ ((g, ~ J j - i g + 1, V)PK, K+t) 2 V

+ ~ ( ( K - 1, vii-IK, cx)Pr, x- 1)2]}.

(16)

V

Fig. 2 shows the zero-point rotational energy (in units of h2/2J), calculated from eq. (16) for all N = 4 states, for the N = 5 "intruder" states originating from the h~ spherical state, and for the state ½- [541]. It can be seen that the zero-point rotational energy varies considerably and is especially important for low-K states which have large wave-function amplitudes of high-j spherical states, in which case the value of EZpRcan be several hundred keV. For the calculation shown, the model parameters used were those for 165Ho. Although one should, ideally, incorporate the zero-point rotational energy in the determination of the minimum of the total potential energy, test calculations have shown that EzpR is not very sensitive to the nuclear distortion. This fact justifies our procedure of adding the zero-point rotational energy tO the equilibrium energy after determining the potential energy minimum.

I

I

_ 40 550

541 "Lhg/2

I

I

I

I

Zero-point rotational energy for states in165Ho

~o

~13

431" - ~ - - - ' ~ ~

~

411~ ~ ~402 400t Sl/2 Id3/2 I/2 3/2

402 ~'d I 5/2 5/2

~,,~14

~ ~ 4 0~4

~ :

505

~7/2 I 7/2

I 9/2

I 11/2

K Fig. 2. The zero-point rotational energy, EzpR, for one-quasiparticle states in 165Ho,calculated from eq. (16). The points are identified by the Nilsson asymptotic quantum numbers [N, n,, A]. The lines connect states originating from the same shell-model state.

B.S. NIELSEN A N D M. E. B U N K E R

382

3. Results of calculations and comparison with experiment Figs. 3-5 display examples of calculated contour plots of the total potential energy surface, showing the onset of nuclear deformation in the Tb isotopes between N = 88 and 90. It is seen that a spherical shape is expected for 1s 1Tb (fig. 3), whereas two shallow potential minima- one prolate and one oblate- are calculated for 1s 3Tb (fig. 4). In the latter case, the difference in potential energy between the spherical configuration and the prolate equilibrium shape is only ~ 1.5 MeV, which is considered insufficient to

O.08

.

.-

I ~

. /- ~

o.o41~"

-

.

~

..--

.

.

.

i-,...

(

~+[4,]

", _.

oo° b.x.', \ ' , . . . - " ", \

-

.q-.".,.'.\ " ._ ~

~

"-.

!

.

1

/ ,'///t

"-..,'/'/,'/,,;I -

/,'/,'/,V/I

//,'////,4t

_~

~

-0?2

/ ; / ,I

~

~

-

-o.o~-,'..~ ~ . . ~ , -05

" - . "\.2

~

.

\ ' . . ,

,,~'b,

-_.__

>/.-~..Z.Z/,-t

-0.1

0 E2

0.1

0.~'

CI5

Fig. 3. Total potential energy surface in the (e2, e¢) plane for lSLTb, calculated for the ~+ 1411] state. The contour lines are drawn at 0.5 MeV intervals.

•

k

"

.

~

-

-

~

. . . . . . . . .

~3

__ " °,X

;',\t

\ \ \ -0.04~\ \ \

,X,

_~

-.

-._./

~,',,.',',..

-o.oe-,~,,,,,,,,',,,. -Q3

--a2

/

}/ /

/

\, \,\X \ \ ' X ' , ~ ~.

-

/

/

/

/

,",, ( i ,,

," A

.*"_-" /

//" -o,

Tb

/ //~.i~,,,] ",, I

'~

/

~-_

~ o

\

\

\

\

', \ \

",.~ ~

"'. N o.,

~ ~

-

-t-,"/,',' /

/

/// ///X

\---"///,?, Y . ./ // /~/ /2/ . "

"-

- 3-d/X~f0.2

03

E2

Fig. 4. Total potential energy surface for the 23-+ [411 ] ground state of 1s 3Tb" Contour lines are labeled in MeV.

EQUILIBRIUM DEFORMATIONS

o.oe,-~..',£,-'-,~-, ~

K

\'.."~ .."-~", ~ , o . o , ,"' x- - , "~~ -x. N"+.. _ ' .x° \ ,\ \ \ x

~v

\

k \\

-0.04~,

-" I

,, 6 ,/. >'J.----

, \ /, I/ , 'i ?1 1, ;/ .

\ I \

V

:.'5.:.~

I. /I i ' 1 1 1 1 ..-~ 'I.~'--~,

/

r'\',,\,,\k/! II Ill;lllll/l~+l,lll], I il!l,,l,'l,,I,-m,, \ \~lil!

',\\

\',,\ l..%.\\. ~ ' \ \ ~ ~,

\

383

~"'L \ \

'A!Illll I I I I I I

I

Iilil,'ll[lli4--, i I I I /, "

'

'

I

I

I

I

-02

I

I

,~

#l I

I

11

.I / ,"l I ; I /

I

-OI

0

I

:

i

i

I

~

I I I I I

"~\',~ \ \ OI

\ \

E2

".\

%., i

I

/ 1

.-ki)))iJ: , t t(t(t(ttt2,I

/

-008 -N\".'~.".. I

-05

I

02

/

~//~#~' 0.5

,

Fig. 5. Total potential energy surface for the 3+[411 ] ground state of i s5Tb" Contour lines are labeled in MeV.

0.04~//, / 6I1 // I1 / , / I

II1

I

I

I

I/

I

I,,'1~/? ,"~///I:/I/I/////165H0

,'1,,'/,,'/ i,/, I!1i1" /i/ill 'I

0'02~/1/

I

Ill -0.02~

-0.04[i 0.15

r

i

I

i

I

i/ /

I

7/"2-[5Z3]

I I

I

N\,

+

I

_

,/ - /

I I ;

[:1 ',/i \'\,:\

\

~\ X

\, ~

....

0.25

"

i',,,

/ ',,, \'-.____.--...2.. 0.20

/

0.50

E2

Fig. 6. Total potential energy surface for the 7- [523] ground state of 165Ho' Contour lines are labeled in MeV.

guarantee the existence of a stable deformation. In both of these calculations the odd particle is assumed to occupy the ~+ [411] orbital, which is the ground-state assignment for 155-161Tb" In the case of 15STb (fig. 5), as for all well-deformed nuclei in the rare-earth region, a deep prolate eflergy minimum is observed. Near the middle of the region (e.g., 165Ho; cf. fig. 6) the equilibrium energy for the ground state is typically favored by 15-20 MeV in comparison with the spherical configuration. Results of the excitation-energy and equilibrium-shape calculations for the deformed

16tTb

159Tb

157Tb

155Tb

lSSEu

153Eu

-

-

- -

97.4 199 0.235 0.038 104.3 196 0.245 -0.038 227.0 355 0.235 0.028 326.4 373 0.245 0.028 363.5 400 0.250 -0.024 480.0 402 0.260 -0.015

~-[532]

0 [0] 0.235 -- 0.038 0 [0] 0.245 -0.038 271 .2 103 0.240 - 0.028 327.6 ~') 107 0.250 - 0.028 348.2 111 0.255 -0.024 314.8 113 0.260 -0.015

-~+[413]

TABLE 2

103.2 20 0.235 -- 0.038 245.7 29 0.245 -0.038 0 [0] 0.235 - 0.028 0 [0] 0.250 - 0.028 0 [0] 0.255 -0.024 0 [0] 0.260 -0.015

~+[411]

376 0.255 -0.021 417.0 339 0.260 -0.011

545.3 a) 398 0.240 - 0.024 571.7 ") 388 0.250 - 0.024

7-[523]

½+[411]

466.9 ") 374 0.225 - 0.034 658.6 ~) 486 0.235 - 0.034 777 a) 580 0.245 -0.031 998 a) 696 0.250 -0.021

2÷[404]

9-[514]

Results of excitation energy and equilibrium shape calculations

499 ") 727 0.220 - 0.034 839 a) 883 0.235 - 0.038 1020 ") 995 0.245 -0.031 1255 a) 1122 0.250 -0.021

~+ [402]

½-[541]

- ~ - [505]

::

>. ~Z

t" r~

4~

~67Tm

~6STm

16SILO

t63Ho

t61Ho

159Ho

(1527.4)")

683 0.266 0.007

-

587 0.257 0.004

624.4 a) 425 0.230 -0.022 827.2 488 0.246 -0.016

(1580.8) a)

-

(649.3) a) 208 0.239 - 0.022 760.1 a) 209 0.252 -0.016 (876.0) ") 254 0.264 0.004 995.3 323 0.271 0.007

(470.7) a)

-

-

-38 0.240 0.020 298.6 a) -31 0.254 -0.014 360.4 ") -4 0.264 0.004 361.7 43 0.271 0.007

0

0.271 0.009 161.4 ~) 240 0.259 0.002 292.7 266 0.267 0.014

[o3

0.264 - 0.002 0

[o]

0.254 -0.012 0

Eo]

0.240 -0.019 0

[o]

-

-

0.270 0.017

[o1

0.261 0.006 0

O0 [03

205.8 19 0.241 0.020 211.2 11 0.255 -0.014 297.8 a) 2 0.265 0.004 429.4 16 0.271 0.007

7

9

.

4

216 0.261 0.01t

1

166.0 a) -42 0.229 - 0.025 252.6 a) 61 0.242 -0.020 439.9 a) 199 0.254 - 0.009 715.5 356 0.261 0.001 81.1 a) 130 0.254 0.001 (1055.9) a)

446.8 a)

- -

998 0.260 -- 0.025 423.8 938 0.266 0.020 471.2 ") 979 0.275 --0.011 680 a) 1119 0.281 0.001 182.2") 9O8 0.274 -0.001 171.7 a) 987 0.280 0.011

7~

>

,.r] ©

c

t'rl ,O

386

B . S . N I E L S E N A N D M. E. B U N K E R TABLE 2 (continued)

169Tm

171Tm

169Lu

171Lu

7 - [523]

½+ [41 l]

7+ [404]

379.3 324 0.275 0.025 424.9 371 0.279 0.037 493 520 0.256 0.013 662.0 608 0.265 0.024

0 [0] 0.276 0.029 0 [0] 0.280 0.038

316.2 314 0.269 0.022 635.5 415 0.272 0.033 0 [0] 0.259 0.016 0 [01 0.265 0.028 0 [0] 0.269 0.038 0 [01 0.266 0.047 0 [01 0.259 0.057 0 [01 0.254 0.046 0 [01 0.247 0.057 0 [01 0.241 0.067 0 [01 0.230 0.072

173Lu 590 0.269 0.036 175Lu 626 0.265 0.044 177Lu 682 0.257 0.054

23 0.260 0.016 208.1 30 0.267 0.028 425.0 32 0.271 0.038 626.6 a) 46 0.267 0.047 569.6 73 0.260 0.057

177Ta

179Ta

XS~Ta

334 0.249 0.041 520.4 380 0.241 0.052 615.0 438 0.235 0.062

1aaTa

18iRe

18aRe

421 0.224 0.067 826.1 698 0.225 0.047 1102.0 775 0.217 0.057

851.1

9 - [5141

5+ [402]

½- [5411

~ - [505]

341.9 a) 1073 0.286 0.022 912.9

306 0.259 0.020 469.5 253 0.267 0.031 450 ") 184 0.272 0.042 396.3 149 0.269 0.051 150.4 115 0.261 0.060 73.6 133 0.252 0.046 30.7 133 0.246 0.056 6.3 142 0.240 0.065 73.1 146 0.229 0.070 262.2 295 0.230 0.051 496.2 350 0.223 0.060

199 0.254 0.013 295.8 253 0.261 0.024 356.8 284 0.266 0.036 343.4 292 0.264 0.046 457.9 301 0.256 0.056 70.5 42 0.250 0.044 238.7 44 0.244 0.054 482.0 55 0.237 0.064 459.1 44 0.227 0.070 0 [01 0.232 0.054 0 [01 0.225 0.0~54

1203 0.291 0.032 29.0 707 0.275 0.015 71.1 717 0.281 0.026 128.2 727 0.285 0.036 358.2 872 0.280 0.044 ~ 800 a) 1094 0.274 0.054 216.6 788 0.266 0.042 750.3 996 0.259 0.052 1213 0.254 0.064 1423 0.245 0.070 432.5 830 0.243 0.051 702 1035 0.237 0.060

988 0.220 0.047 1309 1039 0.215 0.057

387

EQUILIBRIUM DEFORMATIONS TABLE 2 (continued) {-[523]

{+[411]

lSSRe

~-+[404]

880.3 690 0.205 0.060 625.5 580 0.190 0.060

1s ~Re

9-[514]

{+[402]

{-[541]

~-[505]

387 352 0.212 0.064 206.2 340 0.197 0.064

0 [0] 0.213 0.067 0 [0] 0.200 0.067

1045 1169 0.227 0.064

1303 914 0.202 0.057 (1208) 762 0.187 0.057

1357 0.207 0.064

The first entry is the experimental band-head energy (keV), taken from ref. 4) unless otherwise noted, the second is the calculated energy (keV), the third is the calculated value o f e2 and the fourth is the calculated value of e4 . a) Values for 155.15VTb are from refs. 24, 25); 159,161Tb ' ref. 24); 159H0 ' ref. 26); 161Ho ' ref. 2T); 163H0 ' ref. 2s); 165H0 ' ref. 29); t65Tm ' ref. 30); 167Tm ' ref. 31); 169Tm ' ref. 32); 173Lu ' ref. 33); 17SLu ' refs. 33.34); 177Lu, ref. 35).

I

i

Ground-state deformations Q08

1811~1~7~ Lu 0.04

\

~"~" 0.02

1691

0

I

"m

165/ 165 I J I

I

/H ,//l

-0.02

1'171

0

'e

I~ . ~ T b -

155/ Eu

-0.04 I

015

I

Q20

I

025

I

Q30

E2

Fig. 7. Calculated ground-state deformations (e2, e4) for odd-Z nuclei. The dashed line represents the average trend o f the values.

388

B.S. NIELSEN AND M. E. BUNKER

odd-Z nuclei in the rare-earth region are given in table 2, together with the experimental excitation energies. It is noted that states within the same nucleus have predicted quadrupole deformations that ciiffer by as much as de 2 = 0.025. Fig. 7 displays the calculated ground-state deformations. A mean-value curve for the (/32, 84) coordinates is also indicated on this figure. The fact that the theoretical deformations calculated for even-A nuclei with our code are in reasonably good agreement with the empirical (fiE, f14) values [see, e.g., ref. 16)], gives one confidence in the calculational method. Figs. 8 and 9 show the single-particle level schemes calculated for average values of and # and for the mean values of (/~2,/34) indicated in fig. 7. In fig. 8 is also shown the ground-state orbital for each element considered and the range in (/~2, /34) spanned by the isotopes of the element. Except for the lighter Ta isotopes, the ground-state orbitals are those predicted by the single-particle diagram. However, in the light Ta isotopes the ~+ [404] and ,~-[514] states are experimentally and theoretically almost degenerate, and when the zero-point rotational energy for these two band heads is added, ~+ [404] becomes the ground state and not ~- [514], as observed. In figs. 10a-10c the experimental excitation energies of the non-collective states are compared with the calculated energies. In most cases, the energy trends are reproduced rather well by the theoretical calculations. The most obvious exceptions are the I + [402] state in the Ta isotopes and the ½+ [411] state in the Ho isotopes. It is noted that in most cases there is a lower level density near the ground state for the higher mass numbers of a given element. This is due in part to the tendency of Ap, for each series of isotopes, to decrease as A increases. However, most of the prominent trends - e.g., the increasing separation of 7+ [404] and 7-[523] as A increases in the Tb, Ho, and Ta isotopes - are attributable mainly to systematic changes in e2 and e4. For the specific case of the H o isotopes, ~ 45 % of the increasing divergence between ~+ [404] and ~-[523] is calculated to result from an increase in e2, ~ 40 % from the change in e4, and ~ 15 % from the systematic decrease in Ap. As another example, in the case of the Tm isotopes the change in excitation energy of the ½- [541] state relative to the ground state, ½+ [411], mainly reflects the strong change in e4. The quantitative agreement between the calculated and experimental excitation energies is quite good in many cases, but is relatively poor in others. The ½- [541] state, which is a "descending" state from the next major shell, lies consistently lower than we calculate, although we obtain a fair fit in the Re isotopes and the heaviest isotopes of Lu and T a ' . It is worth noting that we obtain a closer fit to the energy of the ½- [541] state than Gareev et al. 6), whose calculations are based on a Woods-Saxon single-particle model and include quasiparticle-phonon interactions.

t The relation between the deformation parameters (e2, e4) and (f12, f14)is specified in ref. 17) and is graphically displayedin fig. 9 of ref. 7). However,the sign of e4 in this figure must be reversed,as noted in ref. 18). qt It has recently been shown that part ( l ~ ~) of the ½-[541] energy discrepancycan be removed through the use of a "scaled" modified oscillator potential 37.as).

'

I

I Proton

389

i

levels

Kp = 0.0635

/.Lp= 0 . 6 0

•

7Z.7 .........

I ~

Tm

~

1

@

A

.

.

.

.

.

-

~ 3/2 404

-

-

3/2 422

I / 2 431 " - - ' - - - - - - "

_

-- ----

~

-

------

----

----

112301

I ......,:-........--~-~,-~:-z:--:z:--z-_-..... 4.5

I

~

I

J

. . . .

0.30

f 0.2~

0.2o -OD4

I -002

I 0.0

I

I

I

0.02

0.04

0.06

0.08

E4 Fig. 8. Single-proton Nilsson energy-level diagram, plotted as a function of the (e2, e,) deformation values shown at the bottom of the graph. The (e2, e4) curve corresponds to the mean-value curve of fig. 7. The states are labeled with the quantum numbers K, N, n=, A.

B. S. NIELSEN A N D M. E. B U N K E R

390 v

I

I

Neutron

levels

K n :0.0637

75

~91=2

734

i

/,/,.=0.42

7./2 624

1/2 501

ZO

~ ~ F

1/2 651

/

6.

0.~

02¢

-o.o4

-o!o2

I 00

0.02

I 004

O~

G08

44 Fig. 9. Single-neutron Nilsson energy-level diagram, plotted as described in fig. 8.

EQUILIBRIUM DEFORMATIONS 0.6

I

391

!

Eu 0.4

:;0.2

c w

g

o~J~° I 3/2+[41~:] Z~

0

0

"~ 0.2 W

O-

-0

A_

,,~ 7

5/2 + 1"413 "1 5/2- [5323

0.4 0

~0 Exp

Z~---Z~ Theory 0.6

I 153

I 155

A Fig. 10a. Comparison between calculated and experimental excitation energies of one-quasiparticle states in deformed Eu nuclei. For each nucleus, the plotted points correspond to band-head energies, with hole states plotted below the ground state (filled circle) and particle states plotted above.

The set of isotopes in which the overall energy fit is poorest is Ho (Z = 67). Here, all states lie at least 0.2 MeV farther from the ground state than the theory predicts; in fact, the ground-state assignment is incorrectly predicted as ~+ [411] for the three lightest isotopes ~. The overall agreement would obviously be much better if the ~-[523] ground state had a significantly lower total energy. One way to accomplish this is to attenuate the assumed energy contribution from the zero-point rotational energy, inasmuch as the calculated value for EzpR is much higher for ~- [523] than for the other states. [-Using h2/2j = 12keV, eq. (16)yields EzaR(~- [523]) = 258 keV, EzpR(~+ [411]) = 88 keV, Ezpe(½÷ [411]) = 65 keV.] In fig. 11, comparison of columns A and B shows the considerable influence of including unattenuated values of EzpR in the case of 165Ho. One must, of course, consider whether there is any justification for reducing our calculated Ezp R values. O n e a r g u m e n t that can be presented is that the semi-empirical fitting procedure used in choosing o p t i m u m values v) of the single-particle parameters x and/~ possibly takes into account, at least to some extent, the influence of zero-point rotation. Also, as pointed out by Ogle et al. 3), there m a y be correlation effects associated with residual interactions, not included in the present model, that would tend to reduce the expectation value o f ( j 2 ) (cf. eq. (8)), which is usually the m a j o r term in Ezp R. t This fact is not evident in fig. 10b since there is no means of plotting negative excitation energies (cf. calculated values in table 2).

o

.~

--

155

;

157

--

~Lr~ ~

I

Tb

159

:

I

161

--

~'~

I

I

]

5/2-1532]

5/2 ÷ ~4131

3/2 + [4111

7/2-1523]

712* [404]

5/2 + 14021

~

159

I

~1

: ~ ' -- ' ~

"'~.

~

~-

6- .....

I

.....

Ho

163

,,~

_a-"

I

A

~

-A

I

-

15231

5/2-15321

5/241413J

3/Z '" [4t IJ

?/2 -

V2+HIll

I/2-154oJ

~)

165

~

-"

I

3

167

--

.,..n - ~

I

--'"

?

I

169

Tm

c

: Exp

7,'2I~-z31

I/2 ÷ 1411]

7/2 + [404]

I/2-15411

171

t~.- - - . ~ Theory

-"

I

Fig. 10b. Comparison between calculated and experimental excitation energies ofone-quasiparticle states in Tb, Ho, and Tm nuclei. For further details see fig. 10a.

1,2

kO

O,8

O.E

0.4

uJ 0.2

0.4

o

O,G

O,8

t,.O

1.2

I

Z t~

Z > Z E;

z rn r"

W

169

I

171

I

l l

.

.

.

F

.

175

.

-

I

.

.

I_u

I

I

~

--

--

--

--

177

I

L

r~2--[5251

~

7,2"[*o4]

]..1 s,'~" [,4o~1

,"'"m 1 1/2-[541]

175

I

I

177

I

~

I

/

I

A

181

.......

~

17g

I

I

I

I

183

To. ~".." .n

A"..-

I

,i2-[4.1

~- [5,,1 ~/2"[..o*]

5/z +[4°z]

-1,,~-15,~]

~

,i

181

I

•

I

183

I

"%•

--

I

Re

185

I

--

I

5/2* [402]

i,,2-15o.,]

I

187

iX--- --~ Theory

~

I

Fig. 10c. Comparison between calculated and experimental excitation energies of one-quasiparticle states in Lu, Ta, and Re nuclei. For further details see fig. 10a.

1.2

1.0

0.8

0.6

0.4

0.2

I

~

o -

0.2

OA

.~,

v

J

0.6

0.8

1.0

1.2p

I

:z

~7 "rl © ~o > -]

~0

t'rl ,0

394

B.S. NIELSEN AND M. E. BUNKER

165Ho

1.4 1.2

/"

I / 2 - [541]

/"

/

f

1.0

/

0.8

7/2 +

> ~0.4

/ r

/

I/2 +

I~

7/2÷[404]

j II

t~ 0.2 ._~0.0

~

I/2"[411] 7/2-

,/

0.2

3/2 +

w

,,'

-

/

-

7/z-

[s23]

3/2+ [411 ] /

0.4

--5/2+

[413]

0.6

/ 0.8

1.0 1.2

5/2 + Exp.

/ A

B

C

Colculoted

D

Fig. 11. Energies of non-collective states in 165Ho. Experimental vslues are shown at the left. In calculation A, Ap = 0.854 MeV, EzpR = 0 and off-diagonal RPC terms have been neglected. Calculation B is the same as A except that values of Ezpafrom eq. (16) are included. Calculation C is the same as B except that off-diagonal RPC terms have been included. Calculation D is the same as C except that Ap = 0.546 MeV. To our knowledge, a rigorous theoretical analysis o f this problem has not yet been undertaken. The present semi-empirical evidence is that a significant reduction in our estimate o f 4,/2) (and hence o f EzpR) would improve the overall energy fit for the H o isotopes, but would in general yield a poorer fit for the other (non-Ho) odd-Znuclides. We have also explored the influence on the H o states of including the off-diagonal Coriolis matrix elements. The results for 165Ho are shown in column C of fig. 11. Comparison of columns B and C in that figure shows that ~ - [523 ] is indeed lowered in energy more than the other states by the Coriolis coupling, but the net effect is almost negligible. Another way to force the excited H o states farther from the Fermi level is to reduce Ap. As indicated previously, we have used Ap = Pp. However, the value of Pp is subject to some uncertainty. For example, the equation we have used for Pp, given in ref. 11), yields Ap( 165Ho) = 0.854 MeV, whereas the equation given in ref. 19) yields 0.546 MeV. Columns C and D in fig. 11 were calculated using these very different values of Ap. Although the use of Ap = 0.546 MeV improves the fit, we cannot offer further justification for selecting this value. Some additional spreading of the levels near the Fermi level would presumably result if blocking were taken into account in the BCS pairing calculations 3, 20). This would improve the fit to the H o levels, but might not improve the overall odd-Z energy fit significantly.

EQUILIBRIUM DEFORMATIONS

395

As mentioned above, the calculated values 0f/32, the quadrupole deformation parameter, for states within the same nucleus often exhibit significant differences [cf. ref. 21)], which reflects the simple fact that the deformation depends on which orbitals are occupied. The orbitals that have the largest influence on the deformation are obviously those with large slopes, dEs.p./d/32, on the usual Nilsson diagram 14). For example, in the Re isotopes, there are two such orbitals near the Fermi level, ½- [541] with a large negative slope and ~t- [505] with a large positive slope. This results in rather different deformations for the corresponding one-quasiparticle states; e.g. for 185Re: /32(½- [541]) = 0.227,

/34(½-[541]) = 0.064,

/32(L~t - [505]) = 0.202,

/34(~- [505]) = 0.057.

In such cases one can expect to find corresponding differences in the intrinsic (or Coriolis-unpetturbed) rotational constants, (h2/2J)intr, in contrast to the claim by variousauthors that the puristic approach in Coriolis calculations is to use the same (h2/2J)intrvalues for all bands. Neglecting the e4 dependence 22), one can estimate 23) the influence on(h2/2J)intr of a change in/32 by using (h2/2j) ~ e2 2,which predicts 3 keV difference between the rotational constants for the two Re states mentioned above. 4. Conclusions

The present calculations give a reasonable and consistent explanation for many of the systematic energy trends of low-lying one-quasiparticle states in the odd-Z deformed nuclei of the rare-earth region. It appears that the majority of these trends are a consequence of systematic changes in the nuclear deformation, and that changes in/34 are sometimes more important than changes in/32. The absolute energy agreement is reasonably good in most cases, but there is still much room for improvement. There is no indication that inclusion of particle-phonon interactions and of off-diagonal Coriolis matrix elements would significantly improve the overall energy fit. As a possible extension of the present approach, one could attempt to optimize further the Nilsson single-particle parameters x and # as well as the pairing strength, although it is not obvious how great an improvement in the energy fit could be derived in this way. As a preliminary step in any such optimization of parameters, it would seem advisable to first adjust the empirial data for the energy contributions associated with nonsingle-particle effects, such as the zero-point nuclear rotational motion. We are indebted to Prof. S. G. Nilsson and Dr. P. M¢ller for many helpful discussions. We also wish to thank Mr. J. W. Starner for assistance with the computer calculations. One of the authors (B.S.N.) gratefully acknowledges a grant from the Danish National Science Research Council and also expresses his appreciation for the hospitality of the Los Alamos Scientific Laboratory.

396

B.S. NIELSEN AND M. E. BUNKER

References 1) B. R. Mottelson and S. G. Nilsson, Mat. Fys. Skr. Dan. Vid. Selsk. 1 (1959) no. 8 2) O. Nathan and S. G. Nilsson, in Alpha-, beta- and gamma-ray spectroscopy, vol. 2, ed. K. Siegbahn (North-Holland, Amsterdam, 1965) p. 863 3) W. Ogle, S. Wahlborn, R. Piepenbring and S. Fredriksson, Rev. Mod. Phys. 43 (1971) 424 4) M. E. Bunker and C. W. Reich, Rev. Mod. Phys. 43 (1971) 348 5) V. G. Soloviev and P. Vogel, Nucl. Phys. A92 (1967) 449; V. G. Soloviev, P. Vogel and G. Jungklaussen, Izv. Akad. Nauk SSSR (ser. fiz.) 31 (1967) 518 I-Bull. Acad. Sci. USSR (phys. ser.) 31 (1967) 515]; L. A. Malov, V. G. Soloviev and U. M. Fainer, Izv. Akad. Nauk SSSR (ser. fiz.) 33 (1969) 1244 [Bull. Acad. Sci. USSR (phys. ser.) 33 (1969) 1155] 6) F. A. Gareev, S. P. Ivanova, V. G. Soloviev and S. I. Fedotov, Physics of elementary particles and the atomic nucleus, vol. 4 (1973) no. 2 7) S. G. Nilsson, C. F. Tsang, A. Sobiczewski, Z. Szymanski, S. Wycech, C. Gustafson, I.-L. Lamm, P. MOller and B. Nilsson, Nucl. Phys. A131 (1969) 1 8) P. MOiler, Nucl. Phys. A142 (1970) 1 9) W. D. Myers and W. J. Swiatecki, Nucl. Phys. 81 (1966) 1 10) V. M. Strutinsky, Nucl. Phys. A95 (1967) 420 11) S. G. Nilsson and O. Prior, Mat. Fys. Medd. Dan. Vid. Selsk. 32 (1961) no. 16 12) A. Bohr and B. R. Mottelson, Nuclear structure, vol. I (Benjamin, New York, 1969) p. 169 13) A. H. Wapstra and N. B. Gove, Nucl. Data Tables 9A (1971) no. 4-5 14) I.-L. Lamm, Nucl. Phys. A125 (1969) 504 15) M. A. Preston, Physics of the nucleus (Addison-Wesley, 1962) ch. 10 16) D. L. Hendrie, N. K. Glendenning, B. G. Harvey, O. N. Jarvis, H. H. Duhm, J. Saudinos and J. Mahoney, Phys. Lett. 26B (1968) 127 17) B. Nilsson, Nucl. Phys. A129 (1969) 445 18) P. Kleinheinz, R. F. Casten and B. Nilsson, Nucl. Phys. A203 (1973) 539 19) L. S. Kisslinger and R. A. Sorensen, Rev. Mod. Phys. 35 (1963) 853 20) S. Wahlborn, Nucl. Phys. 37 (1962) 554 21) D. A. Arseniev, S. I. Fedotov, V. V. Pashkevich and V. G. Soloviev, Phys. Lett. 40B (1972) 305 22) K. Pomorski, B. Nerlo-Pomorska, I. Ragnarsson, R. K. Sheline and A. Sobiczewski, Nucl. Phys. A205 (1973) 433 23) F. S. Stephens, R. M. Diamond, J. R. Leigh, T. Kammuri and K. Nakai, Phys. Rev. Lett. 29 (1972) 438 24) J. C. Tippett and D. G. Burke, Can. J. Phys. 50 (1972) 3152; J. S. Boyno and J. R. Huizenga, Phys. Rev. C6 (1972) 1411 25) G. Winter, L. Funke, K.-H. Kaun, P. Kemnitz and H. Sodan, Nucl. Phys. A176 (1971) 609 26) J. S. Geiger, R. L. Graham and M. W. Johns, Bull. Am. Phys. Soc. 14 (1969) 1225; P. E. Haustein, Bull. Am. Phys. Soc. 18 (1973) 37, and private communication 27) J. L. Wood, Nucl. Phys. A185 (1972) 58; K.-H. Kaun, L. Funke, P. Kemnitz, H. Sodan, E. Will, G. Winter, K. Ya. Gromov, W. G. Kalinnikov, S. M. Kamalchodjaev and H. Strusny, Nucl. Phys. A194 (1972) 177; L. Funke, K.-H. Kaun, P. Kemnitz, H. Sodan and G. Winter, Nucl. Phys. A170 (1971) 593 28) L. Funke, K.-H. Kaun, P. Kemnitz, H. Sodan and G. Winter, Nucl. Phys. A190 (1972) 576 29) J. W. Starner, B. S. Nielsen and M. E. Bunker, Bull. Am. Phys. Soc. 19 (1974) 645 ; D. G. Burke, private communication (1973) 30) J. Gizon, A. Gizon, S. A. Hjorth, D. Barneoud, S. Andr6 and J. Treherne, Nucl. Phys. A193 (1972) 193 31) L. Funke, K.-H. Kaun, P. Kerr/nitz, H. Sodan, G. Winter, R. Arlt, K. Ya. Gromov, S. M. Kamalchodjaev, A. F. Novgorodov, D. De Frenne and E. Jacobs, Nucl. Phys. A175 (1971) 101 32) L. Funke, P. Kemnitz, H. Sodan, E. Will and G. Winter, Zentralinstitut f'tir Kernforschung annual report Zfk-262 (1973) p. 83; Proc. Int. Conf. Munich (1973) p. 186 33) R. A. O'Neil, D. G. Burke and W. P. Alford, Nucl. Phys. A167 (1971) 481 34) C. Foin, S. Andr6 and S. A. Hjorth, Nucl. Phys. A219 (1974) 347 35) P. Manfrass and W. Andrejtscheff, Nucl. Phys. A194 (1972) 561 36) F. S. Stephens, Lawrence Berkeley Laboratory Report LBL-1251 (1972) 37) S. G. Nilsson, private communication 38) R. Bengtsson and S. G. ~berg, unpublished