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Deformation, pairing and magnetic moments in rare-earth nuclei
[~ ELSEVIER
NUCLEAR PHYSICS A Nuclear Physics A 589 (1995) 222-238
Deformation, pairing and magnetic moments in rare-earth nuclei A n d r e w E. S t u c h b e r y Department of Nuclear Physics, Research School of Physical Sciences and Engineering, Australian National University, Canberra, ACT 0200, Australia Received 17 January 1995; revised 20 February 1995
Abstract The global behaviour of g-factors in the rare-earth region is examined. Magnetic moments of the 2+ states in even-even rare-earth nuclei with 60 ~< Z ~< 76 are calculated using the Migdal approximation for the moment of inertia. Attention is drawn to the implications of g-factors for the problem of identical bands, particularly to the application of the concept of F-spin multiplets. The magnetic properties of the heavy, transitional nuclei are discussed in terms of the possibility that the proton and neutron distributions in these soft nuclei might have different deformations.
1. Introduction The discovery of "identical bands" in nuclei at both superdeformation [1-3] and normal deformation [4-6] has renewed interest in the theory of moments of inertia. It has long been recognized that nuclear moments of inertia are reduced from those of a rigid body by pair correlations [7]. Recently, Zhang et al. [8] performed an empirical survey of ground-state rotational bands in even-even rare-earth nuclei, showing that changes in the moments of inertia are correlated with changes in the ratio of deformation to pairing gap ( e / d ) . Halbert and Nazarewicz [9] subsequently demonstrated that the global features of this empirical study can be understood theoretically using the Migdal estimate of the moment of inertia [ 10,11 ]. However, they concluded that while the main features of the correlation plots were accounted for, there was still no quantitative theory to explain the occurrence of identical bands in many cases (see also Ref. [ 12] ). In rotational nuclei the g-factors of the excited states reflect the ratio of the proton moment of inertia to the total moment of inertia and so provide a means of separating the behaviour of the proton and neutron fluids. Consequently, studies of magnetic moments 0375-9474/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSD1 0 3 7 5 - 9 4 7 4 ( 95 ) 0 0 0 5 4 - 2
A.E. Stuchbery/NuclearPhysicsA 589 (1995) 222-238
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may (i) provide an extension of the calculations of Halbert and Nazarewicz [9], and (ii) illuminate the identical bands problem. These considerations motivated the present study. After reviewing the relevant theory and exploring the properties of the Migdal formulation for g-factors in Section 2, calculations of g(2 +) are made in Section 3 for all even-even rare-earth nuclei between 146Nd and 192Os for which there are experimental data. Some implications of magnetic moments for the identical bands problem are discussed in Section 4, particularly in relation to the application of the concept of Fspin multiplets. As the g-factors of the heavy transitional nuclei, i.e. the isotopes of tungsten and osmium, stand out in comparisons of theory and experiment, attention is given to the magnetic moments of these nuclei in Section 5. A summary and the conclusions follow.
2. Review o f Migdal f o r m u l a e
In the late 1950's, the moment of inertia of a collective nucleus was calculated based on the BCS and cranking approximations. Migdal obtained approximate analytic solutions for the case where the density of single-particle states is high [ 10,11,7]. More recently, this approach has been used to estimate the dependence of both moments of inertia [9,13,14] and magnetic moments [15,16] on deformation and pairing. The relevant formulae appear in several places in the literature. For convenience, they are summarized in this section along with schematic calculations illustrating the parameter dependence of the Migdal formula for rotational g-factors.
2.1. Moments of inertia Migdal [ 10] writes the moment of inertia in the axially deformed harmonic-oscillator potential as the sum of two terms:
J=~
+ J2.
(1)
The J l term from AN = 0 Coriolis coupling dominates while J2 accounts for coupling and Y21 quadrupole pairing [7]. These quantities are given by [ I0]
ffl(A,~,A) = ffrig(A,B) (1
flx2 + f2x~) x21 + xZ9 '
AN = 2
(2)
and
/ ff2(A,t3, A)
= ffrig(A,t~) ~(X2
(fl + f2)2x2x~ ) + x2)(flx 2 + f2x 2) '
(3)
where
fi = f(xi) = I n ( x / + ~/1 + x 2)
(4)
A.E. Stuchbery/NuclearPhysicsA 589 (1995) 222-238
224
The quantities xi are dimensionless ratios involving the deformed harmonic-oscillator frequencies and the pair gap, Zi: x~ = Ito~ - o ~ . l / 2 a = 181ha,o/2zi,
(5)
x2 = la, z + to.j_l/2zi = (2 - 18)hwo/2zi,
(6)
and
where 8 is the harmonic-oscillator deformation parameter. The rigid-body moment of inertia, ffrig, is [7], ffrig(A,8) = 2MAR2(1 + ½8)
(7)
and the spherical oscillator frequency is hto0 = 41A- U3 (1 -4- - - N - ~ ) MeV,
(8)
where the plus sign applies to neutrons and the minus to protons. In Eq. (7) M is the nucleon mass and R = ro All3. As x2 "~ 10xl in typical nuclei, Eq.(2) becomes ffl = ffrig(A, 8) (1 - f l }
(9)
to a very good approximation. The further approximation J .~ ,]~ (A, t~, A)
(10)
is frequently made, i.e. the effects of AN = 2 coupling and quadrupole pairing are neglected. The above expressions apply to a one-fluid system. For the nucleus, which has both proton and neutron fluids, the total moment of inertia is ~o,= J. + Jp
= N f f ( A , Sn, An)+ AZ---J(A,Sp, Ap).
(11)
Clearly Zip 4= /in as protons and neutrons in heavy nuclei occupy different shells. While the deformations are usually assumed to be the same for protons and neutrons, the above expression retains the possibility that they may differ. Because of approximations made in its derivation, the Migdal formulation provides only a qualitative estimate of the moment of inertia. However, in many cases it gives a fairly good description of empirical moments of inertia and serves to identify global trends of the data; see Section 3.2 below, and Refs. [9,13]. For a recent application to superdeformed nuclei and the effects of quadrupole pairing, see Ref. [ 14].
A.E. Stuchbery/Nuclear Physics A 589 (1995) 222-238
225
2.2. Magnetic moments To calculate the rotational g-factor, the contributions of the nucleon spins are ignored (in the rare-earth region, calculations such as those of Prior et al. [ 17] show these are small and largely cancel), so that
g(A, t~p,t~n zip, zin) = ffp '
(12)
~ttot '
or
g(A, t~p,t~n, Ap, An) -- Z if(A, t~p, Ap) A
(13)
ffttot
This approach therefore provides a means of estimating the departure of the collective g-factor from Z/A. To illustrate the dependence of the g-factor on deformation and pairing, schematic calculations were made for the nominal case of Z = 68 and A = 168. Results are presented in Fig. 1. To reduce the (already small) dependence on the (arbitrarily) chosen values of Z and A, the diagram shows g~(Z/A) rather than the gfactor itself. Calculations were made using (i) the complete expressions, Eqs. (1)-(8), and (ii) the approximation of Eq. (10) which neglects the ,.72 term; for clarity, only results which neglect AN = 2 coupling and quadrupole pairing (the if2 term) are shown in Fig. 1. The upper panel of Fig. 1 shows schematically how the g-factor varies with changes in deformation for equal proton and neutron pair gaps. The calculated g-factor is less than Z/A even when zip = An because the harmonic-oscillator frequency hzo0 differs for protons and neutrons, Eq. (8). This is also responsible for the gradual reduction in g-factor as ,4 = zip = An increases, as well as the slow increase of the g-factor with increasing deformation. The middle and lower panels of Fig. 1 indicate, as expected, that the g-factor is most sensitive to differences in the behaviour of the proton and neutron fluids, i.e. to differences in both pairing and deformation. Including the ,.72 term would lead, in most cases, to an increase of the order of 10% in the calculated g-factor. Of course, the schematic calculations in Fig. 1 span an unrealistic range of parameters and hence exaggerate the departures of collective g-factors from Z/A compared with real nuclei. In previous applications to moments of inertia, it has generally been assumed that the proton and neutron deformations are the same [9,13]. Initially, this assumption will also be adopted here. It may be noted from the lower panel of Fig. 1, that the g-factor would be most sensitive to any differences in the proton and neutron deformations when the average deformation is small. This matter will be discussed further below in relation to the heavy transitional nuclei.
A.E. Stuchbery/Nuclear Physics A 589 (1995) 222-238
226
0.95~-------- J 800J I................................ i'I 1.00
400
________-
~
"~ 0.90 0.85
.....
,
12
. . . .
0.i0
I
. . . .
i
. . . .
i
. . . .
i
. . . .
t , , , , 1
0.15 0.20 0.25 0.30 0.35
i
i
i
i
'
I
i
1.5
4;o
-•.1.0
1200
v
0.5 0.0
Ap i
I
200
,
I
400
i
I
,
I
600 800 An (keY)
1000
1200 1400
2.0
1"5 I' ' ' ' I' ' r I I~J J ] ' '! I ' ' ' i' I r I ' r I ' ' ' ' I ' I 1 ~-" 1.0 0"5
o.o i,,
Oi~
ZZI,, i ,~ i p, ................... 1! 0.I0 0.15 0.20 0.25 0.30 0.35 6n Fig. 1. Dependence of rotational g-factor on deformation and pairing parameters in the Migdal approximation. Calculations were made for Z = 68 and A = 168. (A) Dependence on deformation, 8, for equal proton and neutron pair gaps with the values 400, 800 and 1200 keV. (B) Dependence on proton and neutron pair gaps, Ap and An, for deformation 6 = 0.2. (C) Dependence on proton and neutron deformations, •p and 8n, with the pair gaps set to Ap = An = 800 keV.
3. Gyromagnetic ratio systematics in rare-earth nuclei
3.1. Details of calculations The formulae in Section 2 were used to evaluate the rotational g-factors and moments of inertia of even-even rare-earth nuclei between 146Nd and m2Os for which experimental gyromagnetic ratios are available. Calculations were also made for some further rare-earth nuclei which have "identical" ground-state bands [6]. The calculations use the same approach that Halbert and Nazarewicz [9] employed to estimate moments of inertia. As such, the results presented here are an extension of their work. Deformations for protons and neutrons were assumed identical and, where available, the theoretical values from the Woods-Saxon-Strutinsky calculations of Ref. [ 18] were adopted. These reproduce well experimental quadrupole deformations. For the Nd isotopes (not calculated in Ref. [ 18] ), the experimental quadrupole deformations deduced from experimental B(E2; 0 + ~ 2 +) data were used [ 19] with f14 from Ref. [20]. As the sign
A.E. Stuchbery/Nuclear PhysicsA 589 (1995) 222-238
227
of the deformation of 192Os obtained in Ref. [ 18] is the opposite of experiment, the present calculations for this nucleus were made for both prolate and oblate shapes. The schematic calculations in Section 2 indicate that the g-factor is not directly sensitive to the deformation; however it may be indirectly sensitive to deformation, should the relative pair gaps be sensitive to deformation. The pair gaps for protons and neutrons were calculated microscopically using the BCS approximation with the same Woods-Saxon potential and pairing force strengths as in the calculations of [9,18]. (Details of the potential parameters and the pairing force strengths given in Ref. [ 18] are not repeated here. The present BCS calculations were effected by modifying routines described in Ref. [21 ] to evaluate the pair gaps for even-even nuclei.) The adopted deformation parameters and calculated pair gaps are listed in Tables I and 2. In calculations with the ,.7"2term, i.e. including the contributions of AN = 2 coupling and quadrupole pairing to the moment of inertia, the pair gaps should be evaluated with both monopole and quadrupole pairing forces. This has not been done here. Where the ,72 term is retained in the following calculations, the intention is to show its influence schematically. In Ref. [ 16] it was shown that the main features of the mass-dependent trends in the g-factors of the transitional isotopes of Nd, Sm and Gd can be described by the Migdal approach with empirical pair gaps estimated from odd-even mass differences. As the empirical pair gaps for these nuclei include deformation and other effects, it is more appropriate to calculate them microscopically as described here. 3.2. Comparison of theory and experiment 3.2.1. Moments of inertia The experimental and theoretical moments of inertia are compared in Fig. 2, where E(2~-) = 6h2/2fftot is plotted versus mass number. Calculations are shown for the cases where the ,.72 term is included (lower panel) or disregarded (upper panel). Clearly, the main trends of the mass dependence of the moment of inertia are reproduced in both cases. Inclusion of the ,72 term increases the moment of inertia (i.e. decreases the excitation energy of the 2 + state), improving the agreement between theory and experiment for the more deformed nuclei (where the Migdal approach is most applicable), but making the agreement worse for less deformed nuclei near the ends of the shell. As detailed quantitative agreement with experiment cannot be expected, the level of agreement between theory and experiment is satisfactory. 3.2.2. Magnetic moments The calculated g-factors are compared with experiment in Tables 1 and 2, and Fig. 3. The experimental g-factors were compiled from Refs. [22,23,16,24-30]. Given that no parameters have been adjusted to fit the data, the agreement between the experimental and theoretical g(2 +) values for nuclei between 146Nd and 192Os is good on the whole, and particularly good for 62 ~< Z ~< 72. For the heavier nuclei, Z > 72, theory and experiment begin to diverge. Consistent with the schematic calculations in Section 2.2,
A.E. Stuchbery/Nuclear Physics A 589 (1995) 222-238
228
Table 1 Deformations, pair gaps and gyromagnefic ratios in even--even rare-earth nuclei Nucleus
f12a
,04a
8b
Ap
An
gR
(keV)
(keV) theoryc experimenta
I~Nd86 l~Nd88 l~Ndgo
0.131 0.173 0.227
0.040 0.052
0.120 1358 0.156 1264 0.210 1053
1028 1120 1050
0.261 0.315 0.357
0.292 (08) 0.355 (12) 0.409 (22)
1~Sm86 16~Sm88 152o-62,am90 l~Sm92
0.128 0.198 0.236 0.257
0.021 0.044 0.057 0.066
0.117 0.177 0.209 0.226
1418 1212 1081 1020
973 1070 1023 948
0.238 0.329 0.352 0.344
0.270 0.394 0.416 0.392
l~Gdss I~Gdgo l~Gd92
0.205 0.234 0.255
0.033 0.041 0.049
0.183 0.207 0.225
1177 1103 1068
1201 1039 975
0.390 0.360 0.348
0.444 (40) 0.430 (30) 0.387 (04)
l~Gd94 1~Gd96
0.269 0.277
0.049 0.042
0.236 1055 0.242 1056
945 913
0.339 0.325
0.381 (04) 0.364 (17)
l~Dy9o 1~Dy92 1~Dy94 1~Dy96 l~Dy9s
0.229 0.252 0.264 0.273 0.280
0.028 0.035 0.033 0.025 0.019
0.203 1093 0.222 1060 0.232 1051 0.239 1041 0.245 1040
1047 998 975 934 892
0.375 0.367 0.357 0.343 0.327
0.390 0.355 0.362 0.343 0.342
(40) (42) (09) (12) (12)
l~Er96 l~68Er9s ~Erloo 17ot~_ 68 P--,1102
0.270 0.278 0.284 0.285
0.013 0.007 -0.001 -0.013
0.237 0.243 0.248 0.249
1034 1027 1017 1001
947 897 836 810
0.357 0.340 0.321 0.314
0.353 0.320 0.329 0.321
(10) (03) (06) (09)
w°Ybloo 17o2Yblo2 174yblo4 17~yblo6
0.281 0.283 0.280 0.269
-0.010 -0.022 -0.032 -0.044
0.246 0.247 0.245 0.236
1023 986 960 941
839 808 758 712
0.329 0.325 0.313 0.297
0.337 0.335 0.338 0.320
(04) (08) (08) ( 11 )
1762Hflo4 172SHf~o6 l~Hflo 8
0.263 0.258 0.252
-0.036 -0.045 -0.056
0.231 0.227 0.222
1063 1029 1031
779 690 659
0.294 0.272 0.256
0.270 (20) 0.240 (14) 0.267 (15)
1so,,, 74 w 106 1782W1o8 t~Wll0 ~W~I2
0.238 0.232 0.223 0.206
-0.042 -0.052 --0.060 --0.064
0.211 1100 0.206 1077 0.198 1055 0.184 1040
698 678 686 707
0.257 0.250 0.249 0.248
0.260 0.265 0.289 0.312
(20) (10) (07) (11)
739 764 742 665 549
0.280 0.290 0.271 0.178 0.161
0.281 0.298 0.340 0.398 0.398
(08) (11) (12) (13) (13)
l~OSll0 ~8~OsH2 179~OSll4 192Os116
17~OSll6
0.201 --0.054 0.185 --0.057 0.164 --0.055 -0.149 -0.022 0.149 --0.022
0.180 0.166 0.148 --0.147 0.135
998 962 960 1195 1088
(15) (21) (25) (18)
A.E. Stuchbery/Nuclear Physics A 589 (1995) 222-238
229
Table l--continued a Calculated deformations from Ref. [ 18], except for u~-lS°Nd where ~2 values are from measured B(F.2)s [19] using the prescription in Ref. [18] and f14 values are from Ref. [20]. b 8 = 0.946/32 - 0.256fl22. e Evaluated neglecting the AN = 2 coupling and quadrupole pairing term if2, Eq. (1). d Experimental g-factors evaluated from Refs. [22,23,16,24-30]. adding the contributions from the ,.72 term increases all o f the calculated g-factors by about 10%. The overall agreement with experiment is only slightly improved. Generally, the quantitative agreement between theory and experiment appears to be better for the magnetic m o m e n t s than for the m o m e n t s o f inertia. This is particularly so for the lighter, less-deformed isotopes of N d and Sm. It appears that even when the total m o m e n t o f inertia is not well reproduced by the Migdal formula, there can still be a rather good a p p o r t i o n i n g o f the angular m o m e n t u m between the proton and neutron fluids. The level o f agreement between theory and experiment shown in Fig. 3 suggests the a s s u m p t i o n s u n d e r l y i n g the calculations are essentially correct. Specifically, the spin c o n t r i b u t i o n s to the g-factors o f the l o w - l y i n g rotational states are relatively small; the g-factors are sensitive m a i n l y to differences in pairing for protons and neutrons; and the Table 2 Deformations, pair gaps and gyromagnetic ratios in rare-earth nuclei with identical ground-state rotational bands Nucleus
IB-le: l~Dyg0 l~sEr92 l~Yb94 l~Hf96 172~*, 74,98 l~Osl04
~2a
f14a
8b
0.229 0.028 0.203 0.242 0.020 0.214 0.250 0.008 0.221 0.246 0.006 0.217 0.228 --0.002 0.202 0.224 --0.028 0.199
Ap
An
(keV)
(keY) theory¢ experimentA
1093 1067 1022 1114 1098 1085
1047 1032 1072 956 961 843
0.375 0.382 0.411 0.351 0.356 0.312
0.390 (40) -
934 897 808
0.343 0.340 0.325
0.343 (12) 0.320 (03) 0.335 (08)
IB-2r: 1¢~Dy96 0.273 0.025 0.239 1041 1~68F-,r98 0.278 0.007 0.243 1027 172v~ 0.283 --0.022 0.247 986 70 1 o102
gR
a Calculated deformations from Ref. [ 18]. b See footnote b of Table 1. c See footnote c of Table I. d See footnote d of Table 1. e The nuclei in IB-l, 156Dy,16°lr, 164yb, t68Hf, 172Wand 18°Os, have "identical" ground-state bands with E(2 +) ~ 125 keV. f The nuclei in IB-2, 162 Dy, 166 Er and 172 Yb, have "identical" ground-state bands with E(2 +) _ 80 keV.
A.E. Stuchbery/NuclearPhysics A 589 (1995) 222-238
230
600
r
,
,
,
, , , , 1 1 , , , 1 1 , , , [ r , , , 1 1 , , ,
(a) 500 400 300 +,-~
200 i00 0 6OO 500 4O0 30O cq v
200
i
,
I I ,
experiment: filled points theory: open points
o
,
J
,
I
,
J
,
,
t
,
,
t
,
I
,
,
t
,
I
,
t
,
,
i
i
,
(B) experiment: filled points theory: open points
~=~l +~2
v
i00 0 140
,
i
o ~2
i
150
i
,
0
i
]
160
i
i
i
i
I
i
s
,
r
I
170 180 mass number
,
,
,
Q
,
I
190
,
200
Fig. 2. Comparison of experimental and theoretical 2+-state excitation energies in the rare-earth region. The nuclei included are those in Table 1 for which there are measured magnetic moments. The contribution of the ,72 term in Eq. ( 1 ) is omitted (included) in the lower (upper) panel.
microscopic calculations of the relative proton and neutron pair gaps are substantially correct.
4. Magnetic moments and identical bands Casten et al. [6] have drawn attention to two groups of nuclei in the rare-earth region having "identical" rotational bands. One group consists of the nuclei 156Dy, 16°Er, 164yb, 168Hf, 172W and 18°Os (IB-1); the other group is 162Dy, 166Er and 172yb (IB-2). The second group (IB-2) is an example of identical bands in a region of saturated collectivity and deformation near mid-shell having E(2 +) ~- 80 keV, while those in the first group (IB-1) are less deformed having E(2 +) ~ 125 keV. Calculations of g-factors and moments of inertia were made. In calculations with the ,72 term included, the average value and standard deviation of E(2+)±eory/E(2+)expfor IB-1 (IB-2) was 0.77 -4- 0.07
A.E. Stuchbery/Nuclear Physics A 589 (1995) 222-238
0.5
I
I
I
I
I
I
I
231
I
0.4 o-
o
0.3 0.2
|
0.0
I
I
I
I
I
144 146 148 150 0.5
I
I
I
I
Gd (Z=64)
Sm (Z=62)
Nd (Z=60)
0.i
I
i
I
I
148 150 152 154 I
I
I
I
I
I
i
I
152 154 156 158 160 I
I
I
I
I
0.4 o
0.3
£) u~
&
0.2
0.0
:-
Dy (Z:66)
0.I I
I
I
I
I
I
156 158 160 162 164 0.5
I
I
I
Yb (Z=70)
Er (Z=68) I
I
I
164 166 168 170 I
I
I
]
i
i
I
I
170 172 174 176 I
I
I
0.4
I |
|
o
0.3
t) r~
0.2 0.i 0.0
Hf (Z=72) I
I
I
176 178 180
Os (Z=76)
W (Z=74) I
I
I
I
180 182 184 186 Mass ntarber
I
I
I
I
186 188 190 192
Fig. 3. Experimental and theoretical g-factors of 2+ states in rare-earth nuclei. The solid (dotted) lines are calculations with the ,7"2term in Eq. (1) included (neglected). (0.95 -t- 0.08). Thus the calculated moments of inertia are nearly constant for both groups, in qualitative agreement with experiment. As expected, the agreement is better for the more collective group, IB-2. The g-factors are also predicted to be nearly constant across each multiplet and, where data are available, are in satisfactory agreement with experiment (see Table 2 and Figs. 4 and 5). While the g-factors of rotational states cannot be measured with the same precision as moments of inertia (energies), they can nevertheless give insight into the phenomenon of "identical bands". Discussion here will focus on the suggestion [6] that the classification of these nuclei with identical bands using the proton-neutron interacting boson model (IBM-2) concept of F-spin [31] multiplets may be relevant for seeking a microscopic interpretation of their identical moments of inertia. As g-factors and M1 transition strengths are sensitive to F-spin breaking, they can serve to assess the concept of F-spin in relation to these bands. In a nucleus with boson numbers N~ and N~, the z component of F-spin is F0 = ½(N~r - N~), while the allowed values of the F-spin quantum number, F, range from
A.E. Stuchbery/NuclearPhysics A 589 (1995)222-238
232
0.8 IB-I
0.6
/A"
~ o
0.4
0.2
0.0
180Os i
I
172W 168Hf 164yb 160Er 156Dy I
,
I
,
-0.4
-0.6
I
,
-0.2
I
,
-0.0
F0 /
0.2
,
I
,
0.4
Fma x
Fig. 4. Gyromagnetic ratios of rotational states in the designated (IB-I) nuclei which have "identical" ground-state rotational bands. Squares are the present calculations using the Migdal formulae, while triangles show the interacting boson model prediction in the limit of F-spin symmetry. The only available experimental result, for 156Dy, is shown.
IF0{ to Fmax = 1 (N~r + N~ ) (in unit steps). In the limit of good F-spin, the low excitation states have F = Fmax. Nuclei in an F-spin multiplet have constant Fmax (i.e. the same number of valence nucleons) and F0 values that range from - F to +F. Thus in IB-1 all but 18°Os belong to an F-spin multiplet with F = 6; in IB-2 162Dy and 166Er have F = 7.5, while 172yb has F = 8. A s M 1 transitions between the low-lying states of a collective even-even nucleus are small, a n d M 1 transitions are forbidden between states with F = Fmax, F-spin b r e a k i n g
is generally considered to be small in these nuclei. Recently it has been considered
appropriate [ 32-37] to use the measured magnetic moments of the states, in conjunction with the M1 strengths, to determine the degree of F-spin mixing. In the limit of F-spin purity, the g-factors of the states with F = Fmax have the values
N,~ N~ +gv g=g=(N~r+N~) (N,r+N~)'
(14)
where g~(~) is the proton (neutron) boson g-factor. It is helpful to rewrite Eq. (14) in the form 0.6
i
i
IB-2 0.4
0.2
0.0
i
.A-'"
•
~.-
172yb 166Er 16213,] I
-0.3
,
I
i
I
-0.i 0.i F 0 / Fma x
Fig. 5. As for Fig. 4, but for the designated nuclei in IB-2. Experimentaldata are designated by the circles.
A.E. Stuchbery/Nuclear Physics A 589 (1995) 222-238
Fo
g = g+ W g- Fm---'-~'
233
(15)
where g+ = ½(g~r 4-g~). Because the boson g-factors are not expected [35] to depart far from their bare values, g,~ = 1, g~ = 0, g_~
(N,r + Np)
(16)
and g+ _~ 0.5. According to Eq. (15), in an F-spin multiplet the rotational g-factors scale linearly with the F-spin projection, F0. For this reason, the g-factors of the nuclei in IB-1 and IB-2 are plotted as a function of Fo/Fmax in Figs. 4 and 5, wherein the contrast between the predicted g-factors in the limit of F-spin symmetry and experiment is clear. In IB-2, for which the g-factor data are complete, the IBM-2 prediction in the limit of good F-spin is not in agreement with experiment or with the present calculations based on the Migdal approximation. Similar comments apply for the case of 156Dy in IB-1. Obviously, F-spin mixing is required in the IBM calculations to reproduce the gfactors. Previously published IBM-2 parameters for 162Dy and 166Er which reproduce the M1 transition strengths [38] imply F-spin admixtures in the 2 + states of ~ 0.1%, and g-factor values close to those given by Eq. (16). In the transitional nuclei it has been found [36,37] that the F-spin admixtures required to reproduce g-factors are of the order of a few percent and, although the influence on energies is relatively small, it is not so small as to be dismissed in a discussion of "identical" bands. It is beyond the scope of the present paper to determine the degree of F-spin breaking through appropriate IBM-2 fits to the spectra, E2 and M1 properties of the nuclei in IB-1 and IB-2. In fact, this might not be possible in the absence of experimental magnetic moments for most the members of the multiplet IB-1. (Some work along these lines has been reported in Ref. [33].) Once there are IBM-2 calculations which reproduce the g-factors, and the extent of F-spin symmetry breaking through these multiplets is reliably characterized, the import of the F-spin multiplet concept for the occurrence of identical bands can be further assessed. In consideration of both the magnetic moments and the moments of inertia in identical bands, the present work favours a microscopic approach in which there is a play off between deformation and pairing effects. The concept of an F-spin multiplet has appeal if energies alone are considered, but the magnetic properties, which imply F-spin mixing, mitigate against a simple connection between the concept of F-spin and the phenomenon of identical bands. Any microscopic interpretation in terms of F-spin will have to take into account that the F-spin symmetry is broken.
5. Magnetic moments in the transitional nuclei
There is a clear deviation of theory from experiment for the heavy transitional nuclei, Z > 72. Whereas the magnetic moments of the W and Os nuclei increase with mass
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A.E. Stuchbery/Nuclear Physics A 589 (1995) 222-238
number, the calculations always show a decrease. This decrease in the g-factors with increasing mass seems unavoidable in the theory. It comes about because neutron pairing tends to collapse as the neutron Fermi surface approaches the N = 126 shell closure, thus making the neutron fluid more rigid and the g-factors smaller. There is some dependence on deformation, but this trend occurs in the pair-gap calculations for the Os isotopes whether one assumes prolate, oblate or triaxial shapes. It is tempting to dismiss the divergence of theory and experiment for these nuclei by noting that the assumptions used to derive the Migdal formula may begin to fail. For example, compared with the mid-shell nuclei, the transitional nuclei are soft, weakly deformed and the density of single-particle levels is low. However one would still need to explain why the present calculations track the g-factor trends so well in the light transitional nuclei (Nd, Sm, Gd), but fail spectacularly for 190'192Os, There have been several previous theoretical studies of the osmium nuclei. The cranking model calculations of Prior, Boehm and Nilsson [ 17] agree reasonably well with the present, simpler approach. However, those workers were not able to explain the increasing g-factors in the Os isotopes using microscopically calculated pair gaps. With a constant pair gap for protons and a neutron pair gap that increases with mass number, g-factors that increase with mass could be obtained. Likewise, the calculations of Kumar and Baranger [ 39] also fail to reproduce the increase in g-factors observed for 186-192Os. More recently, Ansari [40] has examined the g-factors of the even Os isotopes using the pairing-plus-quadrupole hamiltonian, finding it necessary to vary the pairing force constants from nucleus to nucleus to obtain the increase in g-factors with mass. In contrast with these calculations based on the cranking approximation, the protonneutron interacting boson model (IBM-2) in its simplest conception [Eq. (16) ] gives an excellent description of the g(2 +) values in 186-192Os [29], although the g-factors of the 23 states and the M1 transitions have presented a challenge to theory for some time [41 ]. Recently, it has been demonstrated that a consistent description of the g-factors of the low-lying states and of the M1 transitions among them can be obtained using the F-spin breaking formalism in the proton-neutron interacting boson model [36,37]. It can be shown [42] that F-spin breaking implies different deformations for the proton and neutron boson condensates, i.e. F-spin breaking means (flI~M/fllUBM) ~ 1. In view of the success of the IBM for these transitional nuclei, and the failure of the present geometrical model-based calculations, it is worth considering the possibility that the proton and neutron deformations may differ in these soft, transitional nuclei. It is therefore of interest to relate the deformation of the boson condensate, flmM, to the deformation in the geometrical model flCM. These quantities cannot be related precisely because of the basic differences in the two models; however, with the caution that actual numerical values should not be taken too seriously, Ginocchio and Kirson [43] write for IBM-I 2NB
/~GM < 1.18 ~
•IBM,
(17)
where NB is the number of bosons (i.e. half the number of valence nucleons). With
A.E. Stuchbery/Nuclear Physics A 589 (1995) 222-238
2.0
i
I
I
235
I
~-'-' 1.0 t.OIz~ 0.5 0.0
i
186
Os i (Z = 76) 188
190
i
192
A
Fig. 6. Qualitative trend for proton to neutron deformation ratio in the 186-19205 isotopes implied by the magnetic moment data. Circles indicate the results of an interacting boson model analysis, whereas squares are from the present geometrical model approach based on the Migdal formulae. As stressed in the text, the numerical values are not to be interpreted quantitatively.
similar caveats, for IBM-2, one may use as a crude estimate ~
As
N
(fl~BM/fl~BM) is
(18)
almost constant for 186-192Os [37], the IBM analysis of M1 properties predicts that the ratio of proton to neutron deformation may change between 186Os and 192Os with the qualitative trend shown by the circles in Fig. 6. It must be stressed that the numerical values cannot be taken literally; however the trend - which is certainly exaggerated - has the qualitative mass-dependence that one would expect intuitively: namely, the neutron deformation decreases faster than the proton deformation as the neutron number approaches N = 126. To investigate the possibility of different deformations for protons and neutrons in terms of the Migdal formulation, the proton deformations were fixed to the experimental values consistent with the quadrupole moment and B(E2) data, and t~n was varied (including reevaluation of the neutron pair gap) from isotope to isotope to reproduce the experimental g-factors. The resulting deformation ratios are shown by the squares in Fig. 6. In this analysis, the ratio of proton to neutron deformation will be exaggerated (as it is in the IBM analysis), because the disparity between theory and experiment has been attributed entirely to it. (For example, triaxiality, which may be of considerable importance, has been ignored.) However, the level of agreement between the IBM and the geometrical (Migdal) model shown in Fig. 6 is remarkable. The fact that these two very different approaches to nuclear structure both imply the same (qualitative) massdependent trend for differences in proton and neutron deformation is suggestive. It would be interesting to evaluate this mechanism along with others in a more comprehensive theory for 186-192Os. In the first instance, the influence of triaxiality, which has been neglected here, should be evaluated. The triaxial description of 188Os reported by Sahu et al. [44] goes some way towards describing the experimental observation [41] that g(2~-)/g(2+) = 1.45 4- 0.18 in that nucleus.
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6. Summary and condusions The Migdal approximation, which has been used recently to examine trends in moments of inertia in the rare-earth region [9], has been employed to study g-factor systematics as well. In this model the average behaviour of the nucleon superftuid is treated approximately, while account is taken of the underlying single-particle motion through microscopic calculations of the deformations and pair gaps. The theory, with no free parameters, reproduces well the global g(2 +) data for 62 ~ Z ~< 72, but begins to fail for the transitional nuclei, particularly 19°,192Os. The global success recommends this approach as a simple method for estimating g-factors in the rare-earth region, particularly for more deformed nuclei. In groups of nuclei with "identical" ground-state bands, the present geometrical-model based approach leads to very different g-factor predictions than the proton-neutron interacting boson model in the limit of F-spin symmetry. Where available, the measured g-factors agree with the present calculations and imply that the F-spin symmetry must be broken in the interacting boson model description. This mitigates against a simple relationship between the concept of F-spin and the phenomenon of identical bands. A full assessment of the degree of F-spin mixing, and hence its importance, requires further g-factor measurements in the (unstable) nuclei 16°Er, 164yb, 168Hf, 172W and 18°Os. While less precise data would be sufficient to investigate the degree of F-spin mixing, measurements with a precision of about 10% might expose any g-factor variations within this group of nuclei of the order predicted by the Migdal theory. An attempt has been made to understand the failure of many calculations, including the present ones, to account for the g-factors of the heavy transitional nuclei. A possible contributing factor may be that the proton and neutron fluids in these soft, weakly deformed nuclei have different deformations. It has been shown that simplified analyses of the magnetic moment data in 186-192Os, in terms of both the interacting boson model and the geometrical model, are in qualitative agreement, both suggesting that the neutron deformation may decrease faster than the proton deformation between 186Os and 192Os as the N = 126 shell closure is approached. This possibility, which is contrary to the expectation that proton and neutron deformations in nuclei are similar, requires further investigation and evaluation. In particular, the effects of triaxial deformations on the magnetic properties should be fully investigated. A more challenging theoretical task is to interpret quantitatively the phenomenological IBM-2 descriptions of the transitional nuclei, in terms of geometrical model parameters and, eventually, microscopics.
Acknowledgements The author wishes to thank Dr. W. Nazarewicz for discussions, and particularly for the suggestions which led to the present work. Discussions with Dr. S. Kuyucak are gratefully acknowledged. This work is supported in part by the Australian Research Council.
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References [ 1 ] T. Byrski, EA. Beck, D. Curien, C. Schuck, E Fallon, A. Alderson, I. Ali, M.A. Bentley, A.M. Bruce, P.D. Forsyth, D. Howe, J.W. Roberts, J.E Sharpey-Schafer, G. Smith and EJ. Twin, Phys. Rev. Lett. 64 (1990) 1650. [2] ES. Stephens, M.A. Deleplanque, J.E. Draper, R.M. Diamond, C.W. Beausang, W. Korten, W.H. Kelly, E Azaiez, J.A. Becker, E.A. Henry, N. Roy, M.J. Bdnkman, J.A. Cizewski, S.W. Yates and A. Kuhnert, Phys. Rev. Lett. 64 (1990) 2623. [3] ES. Stephens, M.A. Deleplanque, J.E. Draper, R.M. Diamond, A.O. Macchiavelli, C.W. Beausang, W. Korten, W.H. Kelly, E Azaiez, J.A. Becker, E.A. Henry, S.W. Yates, M.J. Brinkman, A. Kuhnert and J.A. Cizewski, Phys. Rev. Lett. 65 (1990) 301. [4] 1. Ahmad, M.P. Carpenter, R.R. Chasman, R.V.E Janssens and T.L. Khoo, Phys. Rev. C 44 (1991) 1204. [5] C. Baktash, J.D. Garrett, D.E Winchell and A. Smith, Phys. Rev. Lett. 69 (1992) 1500. [6] R.E Casten, N.V. Zamfir, E yon Brentano and W.-T. Chou, Phys. Rev. C 45 (1992) R1413. [7] A. Bohr and B.R. Mottelson, Nuclear structure, vol. 2 (Benjamin, New York, 1975). l 8] J.-Y. Zhang, R.E Casten, W.-T. Chou, D.S. Brenner, N.V. Zamfir and E von Brentano, Phys. Rev. Lett. 69 (1992) 1160. 19] E.C. Halbert and W. Nazarewicz, Phys. Rex,. C 48 (1993)R2158. [10] A.B. Migdal, Nucl. Phys. 13 (1959) 655. J i l l S.T. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31 (11) (1959). [12] C. Baktash, W. Nazarewicz and R. Wyss, Nucl. Phys. A 555 (1993) 375. [13] L. Goettig and W. Nazarewicz, J. Phys. G 16 (1990) L51. [ 14] I. Hamamoto and W. Nazarewicz, Phys. Rev. C 49 (1994) 2489. [15] R. Bengtsson and S. Abet,g, Phys. Lett. B 172 (1986) 277. [ 16] A.E. Stuchbery, G.J. Lampard and H.H. Bolotin, Nucl. Phys. A 516 (1990) 119. [17] O. Prior, E Boehm and S.G. Nilsson, Nucl. Phys. A ! 10 (1968) 257. [ 18] W. Nazarewicz, M.A. Riley and J.D. Garrett, Nucl. Phys. A 512 (1990) 61. [ 19] S. Raman, C.H. Malarkey, W.T. Milner, C.W. Nestor Jr. and P.H. Stelson, At. Data Nucl. Data Tables 36 (1987) 1. [20] I. Ragnarsson, A. Sobiczewski, R.K. Sheline, S.E. Larsson and B. Nerlo-Pomorska, Nucl. Phys. A 233 (1974) 329. [211 P.B. Semmes and I. Ragnarsson, The particle + triaxial rotor model. A user's guide, unpublished, distributed at the Hands-on nuclear theory workshop, Oak Ridge (5-16 August 1991), and references therein. [22] Table of isotopes, 7th ed., eds. C.M. Lederer and V.S. Shirley (Wiley, New York, 1978) appendix VII. [23] P. Raghavan, At. Data Nucl. Data Tables 42 (1989) 189. [24] A.P. Byrne, A.E. Stuchbery, H.H. Bolotin, C.E. Doran and G.J. Lampard, Nucl. Phys. A 466 (1990) 419. [25] N. Benczer-Koller, D.J. Ballon and A. Pakou, Hyperfne Interactions 33 (1987) 37. [26] D. Bazzacco, E Brandolini, K. L6wenich, P. Pavan, C. Rossi Alvarez, M. De Poli and A.M.I. Haque, Hyperfine Interactions 59 (1990) 133. [ 27 ] A. Alzner, E. Bodenstedt, B. GemUnden, J. van den Hoff, S. Piel, R. Sajok, H. Koch, Th. Sch~ifer and R. Vianden, Z. Phys. A 331 (1988) 277. [28] A.E. Stuchbery, A.G. White, G.D. Dracoulis, K.J. Schiffer and B. Fabricius, Z. Phys. A 338 (1991) 135. [29] A.E. Stuchbery, G.J. Lampard and H.H. Bolotin, Nucl. Phys. A 528 (1991) 447. [30[ A.E. Stuchbery, S.S. Anderssen, H.H. Bolotin, A.P. Byrne, G.D. Dracoulis, B. Fabficius and T. Kib6di, Z. Phys. A 338 (1991) 135. 131] A. Arima, T. Otsuka, E Iachello and I. Talmi, Phys. Lett. B 66 (1977) 205. 132] J.N. Ginocchio, W. Frank and E von Brentano, Nucl. Phys. A 541 (1992) 211. [33] A. Wolf, O. Scholten and R.E Casten, Phys. Lett. B 312 (1993) 372. [34] S. Kuyucak, in Proc. 8th Int. Syrup. on Capture y-ray spectroscopy, Fribourg, Switzerland, ed. J. Kern (World Scientifc, Singapore, 1994) p. 251. [ 35 ] E.D. Davis and P. Navr,'ttil, Phys. Rev. C, to be published.
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[36] A.E. Stuchbery, in Proc. Int. Conf. on Perspectives for the interacting boson model on the occasion of its 20th Anniversary, ed. R.E Casten (World Scientific, Singapore, to be published). [37] S. Kuyucak and A.E. Stuchbery, to be published. [38] P. Van Isacker, P.O. Lipas, K. Helim~i, I. Koivistoinen and D.D. Warner, Nucl. Phys. A 476 (1988) 301. [ 39 ] K. Kumar and M. Baranger, Nucl. Phys. A 110 (1968) 529. [40] A. Ansari, Phys. Lett. B 207 (1988) 361. [41 ] A.E. Stuchbery, I. Morrison, L.D. Wood, R.A. Bark, H. Yamada and H.H. Bolotin, Nucl. Phys. A 435 (1985) 635. [421 A. Leviatan, J.N. Ginocchio and M.W. Kirson, Phys. Rev. Lett. 65 (1990) 2853. ]43[ J.N. Ginocchio and M.W. Kirson, Nucl. Phys. A 350 (1980) 31. 144] R. Sahu, M. Satpathy, A. Ansari and L. Satpathy, Phys. Rev. C 19 (1979) 511.
NUCLEAR PHYSICS A Nuclear Physics A 589 (1995) 222-238
Deformation, pairing and magnetic moments in rare-earth nuclei A n d r e w E. S t u c h b e r y Department of Nuclear Physics, Research School of Physical Sciences and Engineering, Australian National University, Canberra, ACT 0200, Australia Received 17 January 1995; revised 20 February 1995
Abstract The global behaviour of g-factors in the rare-earth region is examined. Magnetic moments of the 2+ states in even-even rare-earth nuclei with 60 ~< Z ~< 76 are calculated using the Migdal approximation for the moment of inertia. Attention is drawn to the implications of g-factors for the problem of identical bands, particularly to the application of the concept of F-spin multiplets. The magnetic properties of the heavy, transitional nuclei are discussed in terms of the possibility that the proton and neutron distributions in these soft nuclei might have different deformations.
1. Introduction The discovery of "identical bands" in nuclei at both superdeformation [1-3] and normal deformation [4-6] has renewed interest in the theory of moments of inertia. It has long been recognized that nuclear moments of inertia are reduced from those of a rigid body by pair correlations [7]. Recently, Zhang et al. [8] performed an empirical survey of ground-state rotational bands in even-even rare-earth nuclei, showing that changes in the moments of inertia are correlated with changes in the ratio of deformation to pairing gap ( e / d ) . Halbert and Nazarewicz [9] subsequently demonstrated that the global features of this empirical study can be understood theoretically using the Migdal estimate of the moment of inertia [ 10,11 ]. However, they concluded that while the main features of the correlation plots were accounted for, there was still no quantitative theory to explain the occurrence of identical bands in many cases (see also Ref. [ 12] ). In rotational nuclei the g-factors of the excited states reflect the ratio of the proton moment of inertia to the total moment of inertia and so provide a means of separating the behaviour of the proton and neutron fluids. Consequently, studies of magnetic moments 0375-9474/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSD1 0 3 7 5 - 9 4 7 4 ( 95 ) 0 0 0 5 4 - 2
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may (i) provide an extension of the calculations of Halbert and Nazarewicz [9], and (ii) illuminate the identical bands problem. These considerations motivated the present study. After reviewing the relevant theory and exploring the properties of the Migdal formulation for g-factors in Section 2, calculations of g(2 +) are made in Section 3 for all even-even rare-earth nuclei between 146Nd and 192Os for which there are experimental data. Some implications of magnetic moments for the identical bands problem are discussed in Section 4, particularly in relation to the application of the concept of Fspin multiplets. As the g-factors of the heavy transitional nuclei, i.e. the isotopes of tungsten and osmium, stand out in comparisons of theory and experiment, attention is given to the magnetic moments of these nuclei in Section 5. A summary and the conclusions follow.
2. Review o f Migdal f o r m u l a e
In the late 1950's, the moment of inertia of a collective nucleus was calculated based on the BCS and cranking approximations. Migdal obtained approximate analytic solutions for the case where the density of single-particle states is high [ 10,11,7]. More recently, this approach has been used to estimate the dependence of both moments of inertia [9,13,14] and magnetic moments [15,16] on deformation and pairing. The relevant formulae appear in several places in the literature. For convenience, they are summarized in this section along with schematic calculations illustrating the parameter dependence of the Migdal formula for rotational g-factors.
2.1. Moments of inertia Migdal [ 10] writes the moment of inertia in the axially deformed harmonic-oscillator potential as the sum of two terms:
J=~
+ J2.
(1)
The J l term from AN = 0 Coriolis coupling dominates while J2 accounts for coupling and Y21 quadrupole pairing [7]. These quantities are given by [ I0]
ffl(A,~,A) = ffrig(A,B) (1
flx2 + f2x~) x21 + xZ9 '
AN = 2
(2)
and
/ ff2(A,t3, A)
= ffrig(A,t~) ~(X2
(fl + f2)2x2x~ ) + x2)(flx 2 + f2x 2) '
(3)
where
fi = f(xi) = I n ( x / + ~/1 + x 2)
(4)
A.E. Stuchbery/NuclearPhysicsA 589 (1995) 222-238
224
The quantities xi are dimensionless ratios involving the deformed harmonic-oscillator frequencies and the pair gap, Zi: x~ = Ito~ - o ~ . l / 2 a = 181ha,o/2zi,
(5)
x2 = la, z + to.j_l/2zi = (2 - 18)hwo/2zi,
(6)
and
where 8 is the harmonic-oscillator deformation parameter. The rigid-body moment of inertia, ffrig, is [7], ffrig(A,8) = 2MAR2(1 + ½8)
(7)
and the spherical oscillator frequency is hto0 = 41A- U3 (1 -4- - - N - ~ ) MeV,
(8)
where the plus sign applies to neutrons and the minus to protons. In Eq. (7) M is the nucleon mass and R = ro All3. As x2 "~ 10xl in typical nuclei, Eq.(2) becomes ffl = ffrig(A, 8) (1 - f l }
(9)
to a very good approximation. The further approximation J .~ ,]~ (A, t~, A)
(10)
is frequently made, i.e. the effects of AN = 2 coupling and quadrupole pairing are neglected. The above expressions apply to a one-fluid system. For the nucleus, which has both proton and neutron fluids, the total moment of inertia is ~o,= J. + Jp
= N f f ( A , Sn, An)+ AZ---J(A,Sp, Ap).
(11)
Clearly Zip 4= /in as protons and neutrons in heavy nuclei occupy different shells. While the deformations are usually assumed to be the same for protons and neutrons, the above expression retains the possibility that they may differ. Because of approximations made in its derivation, the Migdal formulation provides only a qualitative estimate of the moment of inertia. However, in many cases it gives a fairly good description of empirical moments of inertia and serves to identify global trends of the data; see Section 3.2 below, and Refs. [9,13]. For a recent application to superdeformed nuclei and the effects of quadrupole pairing, see Ref. [ 14].
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2.2. Magnetic moments To calculate the rotational g-factor, the contributions of the nucleon spins are ignored (in the rare-earth region, calculations such as those of Prior et al. [ 17] show these are small and largely cancel), so that
g(A, t~p,t~n zip, zin) = ffp '
(12)
~ttot '
or
g(A, t~p,t~n, Ap, An) -- Z if(A, t~p, Ap) A
(13)
ffttot
This approach therefore provides a means of estimating the departure of the collective g-factor from Z/A. To illustrate the dependence of the g-factor on deformation and pairing, schematic calculations were made for the nominal case of Z = 68 and A = 168. Results are presented in Fig. 1. To reduce the (already small) dependence on the (arbitrarily) chosen values of Z and A, the diagram shows g~(Z/A) rather than the gfactor itself. Calculations were made using (i) the complete expressions, Eqs. (1)-(8), and (ii) the approximation of Eq. (10) which neglects the ,.72 term; for clarity, only results which neglect AN = 2 coupling and quadrupole pairing (the if2 term) are shown in Fig. 1. The upper panel of Fig. 1 shows schematically how the g-factor varies with changes in deformation for equal proton and neutron pair gaps. The calculated g-factor is less than Z/A even when zip = An because the harmonic-oscillator frequency hzo0 differs for protons and neutrons, Eq. (8). This is also responsible for the gradual reduction in g-factor as ,4 = zip = An increases, as well as the slow increase of the g-factor with increasing deformation. The middle and lower panels of Fig. 1 indicate, as expected, that the g-factor is most sensitive to differences in the behaviour of the proton and neutron fluids, i.e. to differences in both pairing and deformation. Including the ,.72 term would lead, in most cases, to an increase of the order of 10% in the calculated g-factor. Of course, the schematic calculations in Fig. 1 span an unrealistic range of parameters and hence exaggerate the departures of collective g-factors from Z/A compared with real nuclei. In previous applications to moments of inertia, it has generally been assumed that the proton and neutron deformations are the same [9,13]. Initially, this assumption will also be adopted here. It may be noted from the lower panel of Fig. 1, that the g-factor would be most sensitive to any differences in the proton and neutron deformations when the average deformation is small. This matter will be discussed further below in relation to the heavy transitional nuclei.
A.E. Stuchbery/Nuclear Physics A 589 (1995) 222-238
226
0.95~-------- J 800J I................................ i'I 1.00
400
________-
~
"~ 0.90 0.85
.....
,
12
. . . .
0.i0
I
. . . .
i
. . . .
i
. . . .
i
. . . .
t , , , , 1
0.15 0.20 0.25 0.30 0.35
i
i
i
i
'
I
i
1.5
4;o
-•.1.0
1200
v
0.5 0.0
Ap i
I
200
,
I
400
i
I
,
I
600 800 An (keY)
1000
1200 1400
2.0
1"5 I' ' ' ' I' ' r I I~J J ] ' '! I ' ' ' i' I r I ' r I ' ' ' ' I ' I 1 ~-" 1.0 0"5
o.o i,,
Oi~
ZZI,, i ,~ i p, ................... 1! 0.I0 0.15 0.20 0.25 0.30 0.35 6n Fig. 1. Dependence of rotational g-factor on deformation and pairing parameters in the Migdal approximation. Calculations were made for Z = 68 and A = 168. (A) Dependence on deformation, 8, for equal proton and neutron pair gaps with the values 400, 800 and 1200 keV. (B) Dependence on proton and neutron pair gaps, Ap and An, for deformation 6 = 0.2. (C) Dependence on proton and neutron deformations, •p and 8n, with the pair gaps set to Ap = An = 800 keV.
3. Gyromagnetic ratio systematics in rare-earth nuclei
3.1. Details of calculations The formulae in Section 2 were used to evaluate the rotational g-factors and moments of inertia of even-even rare-earth nuclei between 146Nd and m2Os for which experimental gyromagnetic ratios are available. Calculations were also made for some further rare-earth nuclei which have "identical" ground-state bands [6]. The calculations use the same approach that Halbert and Nazarewicz [9] employed to estimate moments of inertia. As such, the results presented here are an extension of their work. Deformations for protons and neutrons were assumed identical and, where available, the theoretical values from the Woods-Saxon-Strutinsky calculations of Ref. [ 18] were adopted. These reproduce well experimental quadrupole deformations. For the Nd isotopes (not calculated in Ref. [ 18] ), the experimental quadrupole deformations deduced from experimental B(E2; 0 + ~ 2 +) data were used [ 19] with f14 from Ref. [20]. As the sign
A.E. Stuchbery/Nuclear PhysicsA 589 (1995) 222-238
227
of the deformation of 192Os obtained in Ref. [ 18] is the opposite of experiment, the present calculations for this nucleus were made for both prolate and oblate shapes. The schematic calculations in Section 2 indicate that the g-factor is not directly sensitive to the deformation; however it may be indirectly sensitive to deformation, should the relative pair gaps be sensitive to deformation. The pair gaps for protons and neutrons were calculated microscopically using the BCS approximation with the same Woods-Saxon potential and pairing force strengths as in the calculations of [9,18]. (Details of the potential parameters and the pairing force strengths given in Ref. [ 18] are not repeated here. The present BCS calculations were effected by modifying routines described in Ref. [21 ] to evaluate the pair gaps for even-even nuclei.) The adopted deformation parameters and calculated pair gaps are listed in Tables I and 2. In calculations with the ,.7"2term, i.e. including the contributions of AN = 2 coupling and quadrupole pairing to the moment of inertia, the pair gaps should be evaluated with both monopole and quadrupole pairing forces. This has not been done here. Where the ,72 term is retained in the following calculations, the intention is to show its influence schematically. In Ref. [ 16] it was shown that the main features of the mass-dependent trends in the g-factors of the transitional isotopes of Nd, Sm and Gd can be described by the Migdal approach with empirical pair gaps estimated from odd-even mass differences. As the empirical pair gaps for these nuclei include deformation and other effects, it is more appropriate to calculate them microscopically as described here. 3.2. Comparison of theory and experiment 3.2.1. Moments of inertia The experimental and theoretical moments of inertia are compared in Fig. 2, where E(2~-) = 6h2/2fftot is plotted versus mass number. Calculations are shown for the cases where the ,.72 term is included (lower panel) or disregarded (upper panel). Clearly, the main trends of the mass dependence of the moment of inertia are reproduced in both cases. Inclusion of the ,72 term increases the moment of inertia (i.e. decreases the excitation energy of the 2 + state), improving the agreement between theory and experiment for the more deformed nuclei (where the Migdal approach is most applicable), but making the agreement worse for less deformed nuclei near the ends of the shell. As detailed quantitative agreement with experiment cannot be expected, the level of agreement between theory and experiment is satisfactory. 3.2.2. Magnetic moments The calculated g-factors are compared with experiment in Tables 1 and 2, and Fig. 3. The experimental g-factors were compiled from Refs. [22,23,16,24-30]. Given that no parameters have been adjusted to fit the data, the agreement between the experimental and theoretical g(2 +) values for nuclei between 146Nd and 192Os is good on the whole, and particularly good for 62 ~< Z ~< 72. For the heavier nuclei, Z > 72, theory and experiment begin to diverge. Consistent with the schematic calculations in Section 2.2,
A.E. Stuchbery/Nuclear Physics A 589 (1995) 222-238
228
Table 1 Deformations, pair gaps and gyromagnefic ratios in even--even rare-earth nuclei Nucleus
f12a
,04a
8b
Ap
An
gR
(keV)
(keV) theoryc experimenta
I~Nd86 l~Nd88 l~Ndgo
0.131 0.173 0.227
0.040 0.052
0.120 1358 0.156 1264 0.210 1053
1028 1120 1050
0.261 0.315 0.357
0.292 (08) 0.355 (12) 0.409 (22)
1~Sm86 16~Sm88 152o-62,am90 l~Sm92
0.128 0.198 0.236 0.257
0.021 0.044 0.057 0.066
0.117 0.177 0.209 0.226
1418 1212 1081 1020
973 1070 1023 948
0.238 0.329 0.352 0.344
0.270 0.394 0.416 0.392
l~Gdss I~Gdgo l~Gd92
0.205 0.234 0.255
0.033 0.041 0.049
0.183 0.207 0.225
1177 1103 1068
1201 1039 975
0.390 0.360 0.348
0.444 (40) 0.430 (30) 0.387 (04)
l~Gd94 1~Gd96
0.269 0.277
0.049 0.042
0.236 1055 0.242 1056
945 913
0.339 0.325
0.381 (04) 0.364 (17)
l~Dy9o 1~Dy92 1~Dy94 1~Dy96 l~Dy9s
0.229 0.252 0.264 0.273 0.280
0.028 0.035 0.033 0.025 0.019
0.203 1093 0.222 1060 0.232 1051 0.239 1041 0.245 1040
1047 998 975 934 892
0.375 0.367 0.357 0.343 0.327
0.390 0.355 0.362 0.343 0.342
(40) (42) (09) (12) (12)
l~Er96 l~68Er9s ~Erloo 17ot~_ 68 P--,1102
0.270 0.278 0.284 0.285
0.013 0.007 -0.001 -0.013
0.237 0.243 0.248 0.249
1034 1027 1017 1001
947 897 836 810
0.357 0.340 0.321 0.314
0.353 0.320 0.329 0.321
(10) (03) (06) (09)
w°Ybloo 17o2Yblo2 174yblo4 17~yblo6
0.281 0.283 0.280 0.269
-0.010 -0.022 -0.032 -0.044
0.246 0.247 0.245 0.236
1023 986 960 941
839 808 758 712
0.329 0.325 0.313 0.297
0.337 0.335 0.338 0.320
(04) (08) (08) ( 11 )
1762Hflo4 172SHf~o6 l~Hflo 8
0.263 0.258 0.252
-0.036 -0.045 -0.056
0.231 0.227 0.222
1063 1029 1031
779 690 659
0.294 0.272 0.256
0.270 (20) 0.240 (14) 0.267 (15)
1so,,, 74 w 106 1782W1o8 t~Wll0 ~W~I2
0.238 0.232 0.223 0.206
-0.042 -0.052 --0.060 --0.064
0.211 1100 0.206 1077 0.198 1055 0.184 1040
698 678 686 707
0.257 0.250 0.249 0.248
0.260 0.265 0.289 0.312
(20) (10) (07) (11)
739 764 742 665 549
0.280 0.290 0.271 0.178 0.161
0.281 0.298 0.340 0.398 0.398
(08) (11) (12) (13) (13)
l~OSll0 ~8~OsH2 179~OSll4 192Os116
17~OSll6
0.201 --0.054 0.185 --0.057 0.164 --0.055 -0.149 -0.022 0.149 --0.022
0.180 0.166 0.148 --0.147 0.135
998 962 960 1195 1088
(15) (21) (25) (18)
A.E. Stuchbery/Nuclear Physics A 589 (1995) 222-238
229
Table l--continued a Calculated deformations from Ref. [ 18], except for u~-lS°Nd where ~2 values are from measured B(F.2)s [19] using the prescription in Ref. [18] and f14 values are from Ref. [20]. b 8 = 0.946/32 - 0.256fl22. e Evaluated neglecting the AN = 2 coupling and quadrupole pairing term if2, Eq. (1). d Experimental g-factors evaluated from Refs. [22,23,16,24-30]. adding the contributions from the ,.72 term increases all o f the calculated g-factors by about 10%. The overall agreement with experiment is only slightly improved. Generally, the quantitative agreement between theory and experiment appears to be better for the magnetic m o m e n t s than for the m o m e n t s o f inertia. This is particularly so for the lighter, less-deformed isotopes of N d and Sm. It appears that even when the total m o m e n t o f inertia is not well reproduced by the Migdal formula, there can still be a rather good a p p o r t i o n i n g o f the angular m o m e n t u m between the proton and neutron fluids. The level o f agreement between theory and experiment shown in Fig. 3 suggests the a s s u m p t i o n s u n d e r l y i n g the calculations are essentially correct. Specifically, the spin c o n t r i b u t i o n s to the g-factors o f the l o w - l y i n g rotational states are relatively small; the g-factors are sensitive m a i n l y to differences in pairing for protons and neutrons; and the Table 2 Deformations, pair gaps and gyromagnetic ratios in rare-earth nuclei with identical ground-state rotational bands Nucleus
IB-le: l~Dyg0 l~sEr92 l~Yb94 l~Hf96 172~*, 74,98 l~Osl04
~2a
f14a
8b
0.229 0.028 0.203 0.242 0.020 0.214 0.250 0.008 0.221 0.246 0.006 0.217 0.228 --0.002 0.202 0.224 --0.028 0.199
Ap
An
(keV)
(keY) theory¢ experimentA
1093 1067 1022 1114 1098 1085
1047 1032 1072 956 961 843
0.375 0.382 0.411 0.351 0.356 0.312
0.390 (40) -
934 897 808
0.343 0.340 0.325
0.343 (12) 0.320 (03) 0.335 (08)
IB-2r: 1¢~Dy96 0.273 0.025 0.239 1041 1~68F-,r98 0.278 0.007 0.243 1027 172v~ 0.283 --0.022 0.247 986 70 1 o102
gR
a Calculated deformations from Ref. [ 18]. b See footnote b of Table 1. c See footnote c of Table I. d See footnote d of Table 1. e The nuclei in IB-l, 156Dy,16°lr, 164yb, t68Hf, 172Wand 18°Os, have "identical" ground-state bands with E(2 +) ~ 125 keV. f The nuclei in IB-2, 162 Dy, 166 Er and 172 Yb, have "identical" ground-state bands with E(2 +) _ 80 keV.
A.E. Stuchbery/NuclearPhysics A 589 (1995) 222-238
230
600
r
,
,
,
, , , , 1 1 , , , 1 1 , , , [ r , , , 1 1 , , ,
(a) 500 400 300 +,-~
200 i00 0 6OO 500 4O0 30O cq v
200
i
,
I I ,
experiment: filled points theory: open points
o
,
J
,
I
,
J
,
,
t
,
,
t
,
I
,
,
t
,
I
,
t
,
,
i
i
,
(B) experiment: filled points theory: open points
~=~l +~2
v
i00 0 140
,
i
o ~2
i
150
i
,
0
i
]
160
i
i
i
i
I
i
s
,
r
I
170 180 mass number
,
,
,
Q
,
I
190
,
200
Fig. 2. Comparison of experimental and theoretical 2+-state excitation energies in the rare-earth region. The nuclei included are those in Table 1 for which there are measured magnetic moments. The contribution of the ,72 term in Eq. ( 1 ) is omitted (included) in the lower (upper) panel.
microscopic calculations of the relative proton and neutron pair gaps are substantially correct.
4. Magnetic moments and identical bands Casten et al. [6] have drawn attention to two groups of nuclei in the rare-earth region having "identical" rotational bands. One group consists of the nuclei 156Dy, 16°Er, 164yb, 168Hf, 172W and 18°Os (IB-1); the other group is 162Dy, 166Er and 172yb (IB-2). The second group (IB-2) is an example of identical bands in a region of saturated collectivity and deformation near mid-shell having E(2 +) ~- 80 keV, while those in the first group (IB-1) are less deformed having E(2 +) ~ 125 keV. Calculations of g-factors and moments of inertia were made. In calculations with the ,72 term included, the average value and standard deviation of E(2+)±eory/E(2+)expfor IB-1 (IB-2) was 0.77 -4- 0.07
A.E. Stuchbery/Nuclear Physics A 589 (1995) 222-238
0.5
I
I
I
I
I
I
I
231
I
0.4 o-
o
0.3 0.2
|
0.0
I
I
I
I
I
144 146 148 150 0.5
I
I
I
I
Gd (Z=64)
Sm (Z=62)
Nd (Z=60)
0.i
I
i
I
I
148 150 152 154 I
I
I
I
I
I
i
I
152 154 156 158 160 I
I
I
I
I
0.4 o
0.3
£) u~
&
0.2
0.0
:-
Dy (Z:66)
0.I I
I
I
I
I
I
156 158 160 162 164 0.5
I
I
I
Yb (Z=70)
Er (Z=68) I
I
I
164 166 168 170 I
I
I
]
i
i
I
I
170 172 174 176 I
I
I
0.4
I |
|
o
0.3
t) r~
0.2 0.i 0.0
Hf (Z=72) I
I
I
176 178 180
Os (Z=76)
W (Z=74) I
I
I
I
180 182 184 186 Mass ntarber
I
I
I
I
186 188 190 192
Fig. 3. Experimental and theoretical g-factors of 2+ states in rare-earth nuclei. The solid (dotted) lines are calculations with the ,7"2term in Eq. (1) included (neglected). (0.95 -t- 0.08). Thus the calculated moments of inertia are nearly constant for both groups, in qualitative agreement with experiment. As expected, the agreement is better for the more collective group, IB-2. The g-factors are also predicted to be nearly constant across each multiplet and, where data are available, are in satisfactory agreement with experiment (see Table 2 and Figs. 4 and 5). While the g-factors of rotational states cannot be measured with the same precision as moments of inertia (energies), they can nevertheless give insight into the phenomenon of "identical bands". Discussion here will focus on the suggestion [6] that the classification of these nuclei with identical bands using the proton-neutron interacting boson model (IBM-2) concept of F-spin [31] multiplets may be relevant for seeking a microscopic interpretation of their identical moments of inertia. As g-factors and M1 transition strengths are sensitive to F-spin breaking, they can serve to assess the concept of F-spin in relation to these bands. In a nucleus with boson numbers N~ and N~, the z component of F-spin is F0 = ½(N~r - N~), while the allowed values of the F-spin quantum number, F, range from
A.E. Stuchbery/NuclearPhysics A 589 (1995)222-238
232
0.8 IB-I
0.6
/A"
~ o
0.4
0.2
0.0
180Os i
I
172W 168Hf 164yb 160Er 156Dy I
,
I
,
-0.4
-0.6
I
,
-0.2
I
,
-0.0
F0 /
0.2
,
I
,
0.4
Fma x
Fig. 4. Gyromagnetic ratios of rotational states in the designated (IB-I) nuclei which have "identical" ground-state rotational bands. Squares are the present calculations using the Migdal formulae, while triangles show the interacting boson model prediction in the limit of F-spin symmetry. The only available experimental result, for 156Dy, is shown.
IF0{ to Fmax = 1 (N~r + N~ ) (in unit steps). In the limit of good F-spin, the low excitation states have F = Fmax. Nuclei in an F-spin multiplet have constant Fmax (i.e. the same number of valence nucleons) and F0 values that range from - F to +F. Thus in IB-1 all but 18°Os belong to an F-spin multiplet with F = 6; in IB-2 162Dy and 166Er have F = 7.5, while 172yb has F = 8. A s M 1 transitions between the low-lying states of a collective even-even nucleus are small, a n d M 1 transitions are forbidden between states with F = Fmax, F-spin b r e a k i n g
is generally considered to be small in these nuclei. Recently it has been considered
appropriate [ 32-37] to use the measured magnetic moments of the states, in conjunction with the M1 strengths, to determine the degree of F-spin mixing. In the limit of F-spin purity, the g-factors of the states with F = Fmax have the values
N,~ N~ +gv g=g=(N~r+N~) (N,r+N~)'
(14)
where g~(~) is the proton (neutron) boson g-factor. It is helpful to rewrite Eq. (14) in the form 0.6
i
i
IB-2 0.4
0.2
0.0
i
.A-'"
•
~.-
172yb 166Er 16213,] I
-0.3
,
I
i
I
-0.i 0.i F 0 / Fma x
Fig. 5. As for Fig. 4, but for the designated nuclei in IB-2. Experimentaldata are designated by the circles.
A.E. Stuchbery/Nuclear Physics A 589 (1995) 222-238
Fo
g = g+ W g- Fm---'-~'
233
(15)
where g+ = ½(g~r 4-g~). Because the boson g-factors are not expected [35] to depart far from their bare values, g,~ = 1, g~ = 0, g_~
(N,r + Np)
(16)
and g+ _~ 0.5. According to Eq. (15), in an F-spin multiplet the rotational g-factors scale linearly with the F-spin projection, F0. For this reason, the g-factors of the nuclei in IB-1 and IB-2 are plotted as a function of Fo/Fmax in Figs. 4 and 5, wherein the contrast between the predicted g-factors in the limit of F-spin symmetry and experiment is clear. In IB-2, for which the g-factor data are complete, the IBM-2 prediction in the limit of good F-spin is not in agreement with experiment or with the present calculations based on the Migdal approximation. Similar comments apply for the case of 156Dy in IB-1. Obviously, F-spin mixing is required in the IBM calculations to reproduce the gfactors. Previously published IBM-2 parameters for 162Dy and 166Er which reproduce the M1 transition strengths [38] imply F-spin admixtures in the 2 + states of ~ 0.1%, and g-factor values close to those given by Eq. (16). In the transitional nuclei it has been found [36,37] that the F-spin admixtures required to reproduce g-factors are of the order of a few percent and, although the influence on energies is relatively small, it is not so small as to be dismissed in a discussion of "identical" bands. It is beyond the scope of the present paper to determine the degree of F-spin breaking through appropriate IBM-2 fits to the spectra, E2 and M1 properties of the nuclei in IB-1 and IB-2. In fact, this might not be possible in the absence of experimental magnetic moments for most the members of the multiplet IB-1. (Some work along these lines has been reported in Ref. [33].) Once there are IBM-2 calculations which reproduce the g-factors, and the extent of F-spin symmetry breaking through these multiplets is reliably characterized, the import of the F-spin multiplet concept for the occurrence of identical bands can be further assessed. In consideration of both the magnetic moments and the moments of inertia in identical bands, the present work favours a microscopic approach in which there is a play off between deformation and pairing effects. The concept of an F-spin multiplet has appeal if energies alone are considered, but the magnetic properties, which imply F-spin mixing, mitigate against a simple connection between the concept of F-spin and the phenomenon of identical bands. Any microscopic interpretation in terms of F-spin will have to take into account that the F-spin symmetry is broken.
5. Magnetic moments in the transitional nuclei
There is a clear deviation of theory from experiment for the heavy transitional nuclei, Z > 72. Whereas the magnetic moments of the W and Os nuclei increase with mass
234
A.E. Stuchbery/Nuclear Physics A 589 (1995) 222-238
number, the calculations always show a decrease. This decrease in the g-factors with increasing mass seems unavoidable in the theory. It comes about because neutron pairing tends to collapse as the neutron Fermi surface approaches the N = 126 shell closure, thus making the neutron fluid more rigid and the g-factors smaller. There is some dependence on deformation, but this trend occurs in the pair-gap calculations for the Os isotopes whether one assumes prolate, oblate or triaxial shapes. It is tempting to dismiss the divergence of theory and experiment for these nuclei by noting that the assumptions used to derive the Migdal formula may begin to fail. For example, compared with the mid-shell nuclei, the transitional nuclei are soft, weakly deformed and the density of single-particle levels is low. However one would still need to explain why the present calculations track the g-factor trends so well in the light transitional nuclei (Nd, Sm, Gd), but fail spectacularly for 190'192Os, There have been several previous theoretical studies of the osmium nuclei. The cranking model calculations of Prior, Boehm and Nilsson [ 17] agree reasonably well with the present, simpler approach. However, those workers were not able to explain the increasing g-factors in the Os isotopes using microscopically calculated pair gaps. With a constant pair gap for protons and a neutron pair gap that increases with mass number, g-factors that increase with mass could be obtained. Likewise, the calculations of Kumar and Baranger [ 39] also fail to reproduce the increase in g-factors observed for 186-192Os. More recently, Ansari [40] has examined the g-factors of the even Os isotopes using the pairing-plus-quadrupole hamiltonian, finding it necessary to vary the pairing force constants from nucleus to nucleus to obtain the increase in g-factors with mass. In contrast with these calculations based on the cranking approximation, the protonneutron interacting boson model (IBM-2) in its simplest conception [Eq. (16) ] gives an excellent description of the g(2 +) values in 186-192Os [29], although the g-factors of the 23 states and the M1 transitions have presented a challenge to theory for some time [41 ]. Recently, it has been demonstrated that a consistent description of the g-factors of the low-lying states and of the M1 transitions among them can be obtained using the F-spin breaking formalism in the proton-neutron interacting boson model [36,37]. It can be shown [42] that F-spin breaking implies different deformations for the proton and neutron boson condensates, i.e. F-spin breaking means (flI~M/fllUBM) ~ 1. In view of the success of the IBM for these transitional nuclei, and the failure of the present geometrical model-based calculations, it is worth considering the possibility that the proton and neutron deformations may differ in these soft, transitional nuclei. It is therefore of interest to relate the deformation of the boson condensate, flmM, to the deformation in the geometrical model flCM. These quantities cannot be related precisely because of the basic differences in the two models; however, with the caution that actual numerical values should not be taken too seriously, Ginocchio and Kirson [43] write for IBM-I 2NB
/~GM < 1.18 ~
•IBM,
(17)
where NB is the number of bosons (i.e. half the number of valence nucleons). With
A.E. Stuchbery/Nuclear Physics A 589 (1995) 222-238
2.0
i
I
I
235
I
~-'-' 1.0 t.OIz~ 0.5 0.0
i
186
Os i (Z = 76) 188
190
i
192
A
Fig. 6. Qualitative trend for proton to neutron deformation ratio in the 186-19205 isotopes implied by the magnetic moment data. Circles indicate the results of an interacting boson model analysis, whereas squares are from the present geometrical model approach based on the Migdal formulae. As stressed in the text, the numerical values are not to be interpreted quantitatively.
similar caveats, for IBM-2, one may use as a crude estimate ~
As
N
(fl~BM/fl~BM) is
(18)
almost constant for 186-192Os [37], the IBM analysis of M1 properties predicts that the ratio of proton to neutron deformation may change between 186Os and 192Os with the qualitative trend shown by the circles in Fig. 6. It must be stressed that the numerical values cannot be taken literally; however the trend - which is certainly exaggerated - has the qualitative mass-dependence that one would expect intuitively: namely, the neutron deformation decreases faster than the proton deformation as the neutron number approaches N = 126. To investigate the possibility of different deformations for protons and neutrons in terms of the Migdal formulation, the proton deformations were fixed to the experimental values consistent with the quadrupole moment and B(E2) data, and t~n was varied (including reevaluation of the neutron pair gap) from isotope to isotope to reproduce the experimental g-factors. The resulting deformation ratios are shown by the squares in Fig. 6. In this analysis, the ratio of proton to neutron deformation will be exaggerated (as it is in the IBM analysis), because the disparity between theory and experiment has been attributed entirely to it. (For example, triaxiality, which may be of considerable importance, has been ignored.) However, the level of agreement between the IBM and the geometrical (Migdal) model shown in Fig. 6 is remarkable. The fact that these two very different approaches to nuclear structure both imply the same (qualitative) massdependent trend for differences in proton and neutron deformation is suggestive. It would be interesting to evaluate this mechanism along with others in a more comprehensive theory for 186-192Os. In the first instance, the influence of triaxiality, which has been neglected here, should be evaluated. The triaxial description of 188Os reported by Sahu et al. [44] goes some way towards describing the experimental observation [41] that g(2~-)/g(2+) = 1.45 4- 0.18 in that nucleus.
236
A.E. Stuchbery/Nuclear Physics A 589 (1995) 222-238
6. Summary and condusions The Migdal approximation, which has been used recently to examine trends in moments of inertia in the rare-earth region [9], has been employed to study g-factor systematics as well. In this model the average behaviour of the nucleon superftuid is treated approximately, while account is taken of the underlying single-particle motion through microscopic calculations of the deformations and pair gaps. The theory, with no free parameters, reproduces well the global g(2 +) data for 62 ~ Z ~< 72, but begins to fail for the transitional nuclei, particularly 19°,192Os. The global success recommends this approach as a simple method for estimating g-factors in the rare-earth region, particularly for more deformed nuclei. In groups of nuclei with "identical" ground-state bands, the present geometrical-model based approach leads to very different g-factor predictions than the proton-neutron interacting boson model in the limit of F-spin symmetry. Where available, the measured g-factors agree with the present calculations and imply that the F-spin symmetry must be broken in the interacting boson model description. This mitigates against a simple relationship between the concept of F-spin and the phenomenon of identical bands. A full assessment of the degree of F-spin mixing, and hence its importance, requires further g-factor measurements in the (unstable) nuclei 16°Er, 164yb, 168Hf, 172W and 18°Os. While less precise data would be sufficient to investigate the degree of F-spin mixing, measurements with a precision of about 10% might expose any g-factor variations within this group of nuclei of the order predicted by the Migdal theory. An attempt has been made to understand the failure of many calculations, including the present ones, to account for the g-factors of the heavy transitional nuclei. A possible contributing factor may be that the proton and neutron fluids in these soft, weakly deformed nuclei have different deformations. It has been shown that simplified analyses of the magnetic moment data in 186-192Os, in terms of both the interacting boson model and the geometrical model, are in qualitative agreement, both suggesting that the neutron deformation may decrease faster than the proton deformation between 186Os and 192Os as the N = 126 shell closure is approached. This possibility, which is contrary to the expectation that proton and neutron deformations in nuclei are similar, requires further investigation and evaluation. In particular, the effects of triaxial deformations on the magnetic properties should be fully investigated. A more challenging theoretical task is to interpret quantitatively the phenomenological IBM-2 descriptions of the transitional nuclei, in terms of geometrical model parameters and, eventually, microscopics.
Acknowledgements The author wishes to thank Dr. W. Nazarewicz for discussions, and particularly for the suggestions which led to the present work. Discussions with Dr. S. Kuyucak are gratefully acknowledged. This work is supported in part by the Australian Research Council.
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