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Magnetic moments of even rare-earth nuclei in the cranked HFB approach
NUCLEAR PHYSICS A
( 1993) 37-56
Nuclear Physics A552 North-Holland
Magnetic
moments of even rare-earth cranked HFB approach Saha Institute
nuclei in the
M. Saha and S. Sen ofNuclear Physics. Calcutta - 700 064, India Received 20 January (Revised 10 September
1992 1992)
Abstract: Magnetic moments of yrast levels of a large number of even-even rare-earth nuclei have been calculated using a simple version of the cranked Hartree-Fock-Bogoliubov (CHFB) model. In order to study the sensitivity of the calculated results to the input values of spherical single-particle energies, the calculation has been performed with two different sets of these energies, which are widely used in the literature. It is found that with proper choice of spherical single-particte energies, the CHFB formalism, in its simple version, is quite capable of reproducing the observed spin dependence of g-factors from Sm to OS nuclei.
1. Introduction The cranked Hartree-Fock-Bogoliubov (CHFB) formalism, in spite of its several sho~comings, has been extensively used ‘) to investigate the complexity of nuclear excitation spectra, arising from the interplay of single-particle and collective aspects of nuclear motion, as a function of rotational frequency. Since the cranking wave functions conserve total angular momentum only on “average”, they are clearly not suitable for calculations of those properties of nuclei which are sensitive to angularmomentum admixtures, such as transition probabilities or the static electric moments. However, calculation of the gyromagnetic ratio in this formalism is expected to be reliable since the magnetic moments involve only the expectation values of the total orbital and spin angular momenta. It is well known that the variation of g-factors along the yrast line in deformed nuclei can be used as a sensitive probe to understand the change in neutron and proton spin alignment with increasing rotational frequency 2”)_ For this reason, several workers have used the CHFB formalism to study this variation in a number of well-deformed nuclei 4-9). Magnetic-moment “-19) of the yrast-state g-factors have been done in several even measurements rare-earth nuclei over the last decade. Comparison of these experimental values with the results of the CHFB calculations offers a very good opportunity to test the reliability of this formalism in reproducing the g-factors of high-spin states.
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M. Saha, S. Sen / Magnetic
38
Earlier
experimental
moment.s
data f0312~‘3,‘7)in ‘70,t74Yb, ‘66Er, “‘Dy,
‘56*158Gdand ls4Sm
nuclei show a monotonic decrease of g-factors with increasing spin along the yrast line. Recent magnetic-moment studies 14*15)in 16*Er, ‘64Dy and ‘50.‘S2Sm, however, show almost
constant
g-factors
have been done and an initial
in 16*Er of all the levels for which measurements decrease followed by subsequent increasing trend in
the other nuclei. The decreasing trends of g-factors with increasing spin in ‘70*‘74Yb and 166Er have been well reproduced within the framework of CHFB formalism by Mantri et al. ‘) and Ansari et al. 6), respectively. Sugaware-Tanabe and Tanabe ‘) have calculated the g-factors in ‘66*168Erand obtained a good fit to the experimental data. Ansari “) has recently studied g-1 variation in “*Er in a variation-after-exact angular-momentum-projection approach. Although, all of them are unanimous in attributing the spin variations of the g-factors to rotational alignment effects, their calculational details differ in the choice of single-particle energies, form of residual interaction and their strengths etc. Moreover, none of these calculations have been simultaneously applied to all the rare-earth nuclei for which experimental data are available. However, based on their predictions, certain inferences about different aspects of the CHFB formalism and its reliability in predicting energy spectra and magnetic moments of the yrast bands have been drawn, which in our opinion need closer scrutiny. The rotational alignment effect in any nucleus depends sensitively upon the position of the high-spin intruder orbitals vis-a-vis the Fermi level. So it is our contention that one should clearly delineate the effect of the choice of single-particle energies on the predictions of g-factors in the CHFB formalism before making any study on the effect of inclusion of higher-order terms in the residual interaction, adjustment of its strength or projection of good particle number and of good angular momentum. The main objective of the present work is twofold: (i) to make a systematic study of g-I variations in the rare-earth region within the framework of the CHFB formalism to see how far this model, in its simplest version, is successful in reproducing the experimental trends; (ii) to study the sensitivity of the calculated results on the choice of input values of spherical single-particle energies. With these in view, we have calculated the g-factors for the yrast states in 152,154,156~~,
1%,i58,16oGd,
158,160,i62,164~y,
162,164,166,168.170~r,
168,170,172,174~,
184~
and
“‘0s nuclei within the framework of the CHFB formalism. The pairing-plusquadrupole hamiltonian of Baranger and Kumar *‘) is used. In certain nuclei, calculations have also been performed by including a hexadecapole-hexadecapole interaction term *2-24) in the ham’11 tonian because of experimental evidence 25) for appreciable hexadecapole deformation in their equilibrium shape. Since the model used in the present work is well known, only some of its relevant features are presented here. Results of the calculation are discussed in detail in subsequent sections keeping in view the twin objectives of the present study. Preliminary results of the study have been published elsewhere 26). Very recently, Cescato et al. *‘) have published the results of a systematic study of the yrast levels of a chain of Dy isotopes.
M. Saha, S. Sen / Magneiic
39
moments
2. CHFB formalism
2.1. MODEL
In the cranking perpendicular
model the hamiltonian
to the symmetry
axis (here,
H, is cranked z-axis)
about an axis (here, x-axis)
of the nucleus.
H,=H-wf,-Ai+,
(1)
where w is the Lagrange multiplier (given in energy units, interpreted as the cranking frequency for rotation about the x-axis) such that the total angular momentum of the system is given by the constraint
(24 along with the usual constraint on the particle number N, due to the presence of pairing in the hamiltonian H. Here A, interpreted as the Fermi level, is the Lagrange multiplier used to ensure the conservation of “average” particle number, ($CtWB (~)l~T,l~M3(~))
= NT,
(2b)
where NT denotes proton or neutron number outside the core. For compensating small numerical error in satisfying the above relation we have added a correction term in the energy E [refs. 6T23)]. In eq. (l), H is given by a more general form (to include the hexadecapole term) of the pairing plus quadrupole-model hamiltonian of Baranger and Kumar 2’): H =C E&C,
Here la), lb), . . . are the single-particle basis states to be specified later. Various symbols have their usual meanings; 7,~’ being neutron or proton.
In
general, Qhlr = r* yhl*(% cp)
(4a)
and pz and y are given by fi~#z
cos Y =
hw&,
D20,
sin y = ~6 D2z,
(4b)
where D2,
=x2
C
7
402,A
=
(-1)‘4-,
.
(4c)
4c
The radial
M. Saha, S. Sen / Magnetic
coordinate
moments
is given in units of oscillator
length,
6= (h/m,)“*, which is connected
with the oscillator
(54
energy,
hwO = 41 .2A-‘13 MeV .
(5b)
The parameters
(yT=
(22/A)“3
for protons
(2N/A)“’
for neutrons,
(SC)
have been introduced by Baranger and Kumar to ensure equal radii for protons and neutrons. The simplest approach to the hamiltonian (1) is to treat it in the Hartree-FockBogoliubov approximation, which is a self-consistent approximation in which pairing effects and quadrupole-field effects are treated on the same footing. The HFB approach determines at the same time the best shape of the self-consistent field and optimal pairing correlations for each angular momentum as a cranking term is added to the hamiltonian. In this calculation we have neglected the exchange terms. It is useful to introduce the symmetrized signature basis ‘*) given by
la>= la, a)=&[Ij,, m,)+a(-l)j~+“*lj,, where u = *l is the signature numbers necessary to specify
quantum a state,
number,
-??I,)],
and ljll, m,) denotes
(6) all quantum
The wave functions are those of an isotropic harmonic oscillator. of magnetic-moment The g-factors are calculated from j&., the x-component operator $, as
where
Fx=81Cjl,(i)+(g,-gr)C~~,(i), I
and free-particle
g-factors
,
are
g,=l,
g, = 5.586,
g/ = 0,
g, = -3.826,
for protons for neutrons.
(7b)
M. Saha, S. Sen / Magnetic moments
We have used attenuated
values
of g.? (spin g-factor)
41
as
g, = 0.6g, (free) . so g, = [(i:)+2.351((s^~)-O.98(s^~))]/~. 2.2. INCLUSION
OF HEXADECAPOLE
(7c)
TERM
We have treated all the components of the hexadecapole force self-consistently to minimise the energy. One straightforward way **), which we have adopted here, is to perform a self-consistent HFB calculation including the hexadecapole term. In this case also we have neglected the exchange matrix elements. The one-body hamiltonian corresponding to the hexadecapole term will look like h4=C
/_L= -4, -2,o,
Qu.+r
2,4,
@a)
.
(8b)
with D+ =x4c We may define
p4 through
7
oT(Q&
= (-l)@‘D,-,
the relation fi%P4 = 040.
It is obvious
that now we have five deformation
2.3. PARAMETER
PC) parameters
&, y, p4, Dd2, Dd4.
CHOICE
We have used the oscillator shells N = 4 and 5 for protons and N = 5 and 6 for neutrons as basis states. The inert core is taken as YtZr,,. This basis is identical to one used by Baranger and Kumar *I). In addition, the quadrupole matrix elements of the upper shells are scaled by factors # and g for protons and neutrons, respectively, to ensure the same r.m.s radii for both shells. The pairing and quadrupole force constants are given by G, = 271 A MeV, x2 = 70A-‘-4 MeV , When hexadecapole x~=x~=~O/A’.~
force is included, MeV,
G, = 221 A MeV , hw 0 = 41 .2A--“3 MeV . the interaction
G,=28/AMeV,
strengths
are changed
to 22*23)
G, = 231 A MeV .
We have not made any adjustment of the strengths of the residual interactions (G,, G,, xz) from Sm to Yb (except, of course their normal mass dependence), although our study spans a mass region from A = 152 to 174 and 2 = 62 to 70. In order to understand the sensitivity of the calculated g-factors on the input values of spherical single-particle energies, we have performed the calculation with two
M. &ha,
S. Sen / Magnetic
Calculated
and experimental
42
moments
TABLE 1 g-factors
&I& Isotope
“‘Sm
I” exp “)
BK
NIL b,
4+ 6+ 8+
1.00 (13)
0.53
0.83 [0.93]
0.95 (15)
0.14
0.57 [0.83]
10+
0.88 (41)
0.83 (17)
-0.06
0.18 [OSS] 0.19 [OS81
12+ “‘Srn
‘%jrn
4+
0.99 (10)
0.91
6+
0.96 (11)
0.77
0.91 [0.98]
8+
0.92 (13)
0.62
0.83 [0.98]
10+
0.87 (18)
0.47
0.75 [0.97]
0.96 [0.99]
4+
0.99
0.97
6+
0.98
0.92
8+
0.98
0.86
10+
0.67
0.8 1 0.79
12+ ‘56Gd
0.32 [0.69]
4+
0.88
0.96
0.68
0.89
0.93 (7)
0.49
0.80
0.89 (12)
0.34
0.70
0.22
0.59
0.97
0.98
0.89
0.94
0.99 (2) 0.99 (20)
6+
0.97 (4) 0.92 (18)
8+ 10+ 12+ “‘Gd
4+
0.98 (1) 0.92 (23)
6+
0.95 (4) 0.91 (25)
8+
0.90 (7)
0.61
0.90
10+
0.84(11)
0.36
0.86
12+ 16’Gd
0.85
4+
0.99 (1)
0.99
0.97
6+
0.98 (4)
0.97
0.93
8+
0.96 (7)
0.96
0.89
10+
0.93 (12)
0.93
0.88
0.92
0.90
0.86
0.95
0.57
0.86 0.74
12+ “‘Dy
0.21
4+
0.98 (2) 0.95 (12)
6+
0.95 (4) 1.08 (32)
8+
0.91 (8)
0.33
10+
0.86 (13)
0.15
0.62
12+
0.79 (18)
0.03
0.51
14+
0.71 (25)
0.07
0.42
0.03
0.38
0.17
0.46
0.11 (30) ‘) 16+ 18+
0.62 (33)
M. Saha, S. Sen / Magnetic moments
43
TABLE l-continued
&I&?, Isotope
I” exp “)
BK
NIL ‘) 0.63
20+ 4+
0.98 (2)
0.92
0.97
6+ 8+
0.95 (5) 0.91 (10)
0.71 0.45
0.88
10+
0.86 (16)
0.26
0.83
12+
0.79 (23)
0.15
0.80
4+
0.97
0.96
6+
0.93
0.92
8+
0.89
0.87
0.93
10+
0.85
12+
0.88
4+
1.06 (34)
0.93
6+
0.80 (23)
0.82
0.96
8+
0.78 (26)
0.71
0.94
10+
1.01 (37)
0.64
0.92
0.62
12+
0.99
0.93
14+
0.98
16+
1.09
18+
1.18
4+
0.86
0.95
6+
0.58
0.88
s+
0.33
0.77
10+
0.17
12+
0.07
4+
0.95
0.94
6+
0.86
0.84
8+
0.73
10+
0.65
12+
0.60
4+
0.97 (5)
0.90
0.94
6+
0.83 (9)
0.76
0.84
8+
0.74 (13)
0.62
0.72
10+
0.62 (25)
0.53
0.60
12+
0.47
0.52
14+
0.44
0.48
16+
0.48
0.46
18+
0.54
0.53
4+ 6+
0.94 (12)
0.95
0.99
1.02 (14)
0.86
0.97
8+
1.03 (20)
0.75
0.93
lo+
1.OO (25)
0.64
0.87
0.54
0.81
12+
0.95
0.96
0.87
0.91
44
M. Saha, S. Sen / Magnetic
moments
TABLE l-continued
g,/g, Isotope
I” BK
exp “)
NIL”)
8+
0.78
0.85
10+
0.7 1
0.79
12+
0.77
0.77
4+
0.89
0.91
6+
0.76
0.76
8+
0.63
0.60
10+
0.53
0.47
12+
0.48
0.40
4+
0.97 (2)
0.95
0.98
6+
0.92 (5)
0.88
0.93 0.87
8+
0.86 (9)
0.77
10+
0.77 (15)
0.67
0.79
12+
0.66 (21)
0.58
0.71
14+
0.54 (29)
0.53
0.65
16+
0.39 (39)
0.61
0.64
4+
1.01 (2)
0.96
0.96
6+
1.03 (5)
0.90
0.91
8+
1.06 (9)
0.82
0.84
10+
1.10 (14)
0.75
0.79
0.83
0.76
12+ 4+
0.98 (1)
0.98
0.96
6+
0.96 (3)
0.96
0.90
8+
0.92 (6)
0.95
0.84
10+
0.87 (10)
0.96
0.80
12+
0.82 (14)
1.Ol
0.80
14+
0.75 (19)
1.06
0.82
16+
0.67 (26)
1.12
0.90
4+
1.01 (8)
0.91
0.93 [0.92]
6+
1.07 (11)
0.77
0.81 [0.82]
8+
1.25 (22)
0.63
0.71 [0.71]
0.53
0.70 [0.64]
10+
-
12+
[0.62]
4+
1.33 (13)
0.93
1.22 [l.lO]
6+
1.41 (22)
0.63
1.54[1.10]
0.20
8+
1.30 [0.94]
10+
1.05 [0.79]
12+
0.87 [0.67]
“) Taken from refs. ‘3.1’S”).
E x p erimental
errors are quoted
within
parentheses. b, Results
obtained
with
hexadecapole
within parentheses. ‘)
This value is for Iaverage= 14.1 [ref. “)I.
deformation
are shown
M. Saha, S. Sen / Magnetic
sets of spherical
single-particle
energies
(s.p.e.)
45
moments
which
are listed
in table
1 of our
earlier publication 26). One set, BK, is taken from Baranger and Kumar21), the other, NIL, from the prescription of Nilsson et al. 29). The first set, BK, has mass dependence
only through
the oscillator
parameter,
NIL, varies slowly over the mass region dence
of Z_Land k, prescribed
mass dependence
by Nilsson
under
fiw, = 41.2/A”3.
study according
The second
set,
to the mass depen-
et al. 29) and of course,
through
normal
of hwo. 3. Results
The calculated g-factors for two different sets of single-particle energies, BK and NIL, are listed along with the experimental data in table 1. The majority of the experimental data are taken from the recent compilation of Raghavan I’). The experimental data for 164Dy and 16’Er are taken from Doran et al. 14). Before going into the details of the agreement achieved in each individual isotope, a few comments can be made which are pertinent to the entire region of study. The calculated results in all the isotopes show an interesting feature. Two sets of s.p.e. produce a considerable difference in the predicted trends of variation of g-factors with spin. The magnitude of the difference in the calculated values is found in many cases to be larger than those achieved earlier 6,7) through higher-order modification in the residual interaction of the CHFB hamiltonian. Although BK produces in some isotopes very good agreement with experimental data (as in 16’Gd) but, in general, NIL gives better agreement with experimental trends over the entire region under study. In the present work, the pairing strengths G, and G, are taken from the work of Baranger and Kumar. In their work the strengths of the pairing interactions were fixed by fitting the experimental odd-even mass differences for the isotopes under study. Similarly, the qudrupole interaction strengths were deduced by comparing the calculated ground-state deformations with those deduced from experimental B(E2) values. Since their calculations were done with BK, one should be careful in using another s.p.e. set, e.g., NIL as model parameters while keeping the strengths of the residual interactions unchanged. However, it is found that so far as the agreement with the experimental deformation and odd-even mass difference are concerned, both the sets produce comparable results.
3.1.
15z,1s‘&.156~m
Experimental data ‘*-15) available in ‘52,‘54Sm show that the g-factor changes rather slowly with increasing spin. The calculated g-factors in ‘j2Sm using BK and NIL on the other hand show a much faster fall, indicating strong neutron alignment. In fact, BK shows such a strong neutron alignment that the g-factor becomes negative even for I” = 8+. However, several theoretical 22723*29,30) and experimental 25) studies indicate the presence of appreciable hexadecapole deformation not only in OS-Pt
46
M. Saha, S. Sen / ~~gneijc
moments
but also in Sm-Gd regions. Since it is also known that the hexadecapole deformation retards or accelerates particle alignments depending upon the sign of deformation, we have also calculated the g-factors in ‘52*‘54Smincluding hexadecapole interaction in the CHFB hamiltonian. The results obtained with the inclusion of p4 will be discussed in a subsequent section (sect. 4). In ‘54Sm, the g-factors obtained with NIL show much better agreement with experimental values than those obtained with BK. However, even NIL gives a neutron alignment which is stronger than what is suggested by the experimental trend of the g-1 curve. In “‘Sm, the calculated results using BK and NIL, show just the opposite feature. The g-factors calculated with BK show a very high degree of stability, whereas, those calculated with NIL show a gradual decrease of g-factor values with increasing spin. Both the sets predict better stability of the g-1 curve in ‘%m, than in ‘54Sm. However, the change observed with BK as one goes from A = 154 to 156 is much more abrupt.
Experimental data 17-19)are available in all these isotopes. The calculated results using NIL are in close agreement with the experimental results in 158,‘60Gd,whereas, in *56Gd, they show a somewhat faster fall than that observed experimentally. The set BK, on the other hand, predicts a rapid decrease of the g-factors with increasing spin in ‘56~i58Gdand a sudden reversal of trend in 16’Gd, showing high degree of stability in close agreement with the observed data. The g-factors in ‘583’60Gd, calculated with NIL, initially decreases with increasing spin, but then show an upward trend beyond I” = lO+ indicating early onset of h,,,2 proton alignment for these nuclei. The close correspondence between relative proton and neutron alignments (I!JIZ) and g-factors is shown in fig. 1. Variation of &, y, A, and A, with spin is also shown. It is found that the & remains more or less constant even upto I”= 12”-14+. This is true for most of the nuclei studied in the present work. The constancy of pz along the yrast line is not surprising in view of the fact that except for W and OS nuclei, all other nuclei considered here belong to the well-deformed category. So far as the variation of A, and A, with spin is concerned, it is found that in general, NIL causes a faster (slower) fall of A, (A,,) with increasing I than those produced by BK. This is partly responsible for relatively good stability of g-1 variation obtained with NIL. 3 3 . .
15S,160,162,164~Y
Early measurement 13) in Experimental data 13,‘4*‘7)are available in 1583160*164Dy. 15’Dy shows a gradual decrease of g-factors with increasing spin, whereas, later measurement *‘) shows more or less stable g-factors in the low-spin region and then
M. Saha, S. Sen / Magnetic
0.0
I
0
,
4
,
,
8
I
,
12
I
0
,
I
,
4
I
moments
I
1
12
0
,
4
I
,
8
I
,
12
IB
Fig. 1. Spin dependence of deformation parameters (pz, y), pairing gaps (Ap, A,), relative g-factors and Ie/I: values in ‘56~‘58~‘60Gd. Relative g-factors and It/I: values obtained with BK (NIL) are shown by circles (triangles) connected with solid (dashed) lines. flz and A,, values obtained with BK (NIL) are shown by circles (triangles) connected with solid lines. y and A, values obtained with BK (NIL) are shown by circles (triangles) connected with long-dashed lines. Pairing gaps and & are plotted relative to their ground-state values.
a very rapid fall with increasing
spin. In the high-spin
region,
the calculated
values
using BK also show a very rapid decrease, in agreement with the later measurement. On the other hand, NIL predicts a gradual decrease of the g-factors with increasing spin, a situation intermediate between the trends observed in earlier and later measurements. It can also be noted that in this isotope, the present work, using NIL, produces better stability of the g-factor along the yrast line than that achieved in an earlier work by Ansari et al. 637) with BK and quadrupole pairing in the hamiltonian. In 16’Dy, the theoretical values obtained with NIL are in very good agreement with experimental data, whereas, those obtained with BK show a sharper decrease by Doran et of the g-factors with increasing spin. In ‘64Dy, a recent measurement al. 14), indicates a decreasing trend of g-factors with increasing spin upto I” = 8+ but the effect is not as sharp as that observed in the neighbouring isotone 166Er. Theoretical values obtained in these isotones, using NIL, nicely reproduce this
48
M. Saha, S. Sen / Magnetic moments
experimental decrease
feature,
of g-factors
whereas
those obtained
with increasing
with BK predict
spin than that observed
a somewhat
sharper
experimentally.
Experimental data ‘“*14,17)are available only in 166*‘68Er.The observed g-factors in 166Er show appreciable decrease with increasing spin, whereas in ‘68Er they remain almost constant for all the states (I” 6 lo+) for which measurements have been done. These experimental features are better reproduced in the theoretical calculation using NIL, rather than BK. If we compare the present result with those obtained by Ansari “) and Sugawara-Tanabe and Tanabe ‘), the following interesting features come out. Although the present work includes only the monopole part of the pairing and standard values of the interaction strengths, the predicted difference, using NIL, in the behaviour of the g-factor variations in ‘66Er and 16’Er is almost identical [fig. 2 of our earlier publication ‘“)I to that obtained by Sugawara-Tanabe and Tanabe. Inclusion of the quadrupole pairing term in the CHFB hamiltonian hinders rapid rotational alignment of single-particle orbitals. So the g-factors in both ‘66*‘68Er are seen to vary less rapidly with increasing spin in ref. “) than in the present work. The major part of the difference in the predictions of Ansari (i.e. the results obtained in the present work using BK) and those of ref. ‘) seems to arise primarily not because of different choices of the two-body interaction strengths, as mentioned in ref. *) but of spherical single-particle energies, BK and NIL, respectively.
Experimental
data 13*i7) are available
in ‘70~172~174Yb.In ‘70~‘74Yb, the results
obtained with NIL are in good agreement with the experimental values. In “‘Yb, the theoretical g-factors obtained in the present work are almost identical to those obtained by Mantri et al. ‘) upto spin value I” = 12+ [fig. 1 of our earlier publication ‘“)I. In ‘74Yb also, the quality of the fit obtained in the present and previous work is of the same order. The same single-particle parameter values p and K have been used in the present as well as in the earlier work. However, they performed the calculation for fixed values of &, p4 and y. The g-factors calculated with BK show good agreement with experimental data in “‘Yb, whereas in ‘74Yb they show strong proton alignment, a feature not an upward bend from I” = 8+ indicating supported by available experimental data in ‘74Yb or in “‘Yb. The experimental data “) in ‘72Yb, however, show an upward trend. Remeasurement of the g-factors in this isotope would be of considerable interest. 3.6. ls4W AND “‘0s
The tungsten region, belong
and osmium to a different
isotopes, category
situated at the extreme end of the rare-earth 22V23325330). They are classified as transitional
M. Saha, S. Sen / Magnetic
moments
49
nuclei and are known to exhibit shape transitions with increasing mass number 22330). Moreover, their shapes are also more complex in nature than those of other rare-earth nuclei studied in the present work. Some of them show large triaxiality**) and appreciable hexadecapole deformation in their equilibrium shape *“). Therefore, the proton/neutron
rotational
alignment
and
the g-Z
variation
in these
nuclei
are
expected to be somewhat different in nature. Results obtained in a detailed study of g-Z variations in W, OS and Pt nuclei will be reported in a separate communication. We have included the ls4W and “‘0s nuclei in the present discussion to show the effect of hexadecapole deformation on the rotational alignment and the g-Z variation.
4. Hexadecapole
deformation
and the g-I
variation
The g-factors in 1527’54Sm,ls4W and ‘880s have been calculated with and without the hexadecapole deformation. Although the triaxiality parameter y is small in all these isotopes, all the relevant terms Y40, Y4+*, Y4r4 have been included in the hamiltonian. These calculations have been performed with slightly different values *‘) of G, and G,. The results are shown in tables 1 and 2. It can be seen that the inclusion of hexadecapole deformation resulted in much better agreement with the determined observed g-Z variation in “*Sm. In ‘54Sm, the self-consistently hexadecapole parameters produce almost constant g-factors even upto spin value I” = 12+. In fact the observed g-Z variation is found to lie intermediate between those predicted by using NIL and with and without hexadecapole deformation. In the other extreme of the rare-earth region ( 1880s), inclusion of hexadecapole term makes the agreement with experimental data much worse than what can be achieved with quadrupole deformation only. The Z,P/Z: ratio shows that for low-spin values, the h,,,* proton alignment dominates but with increasing rotational 113/2 neutron tends to be aligned more and more along the rotational
Excitation
energies,
AE(I,
y-deformation
TABLE 2 parameters, pairing gaps and g-factors and hexadecapole deformation in “‘OS
I-2)
Y
calculated
frequency, the axis, resulting
with BK and NIL
k3
An
A*
I BK
NIL
0.197 0.403 0.533 0.606 0.63 1 0.553
0.198 0.389 0.501 0.587 0.624 0.632
0
2 4 6 8 10 12
BK
NIL
BK
NIL
BK
NIL
0.0
0.0
1.4 3.6 5.5 8.0 14.1 39.9
1.1 2.5 4.2 6.5 9.3 12.7
0.728 0.692 0.625 0.550 0.485 0.443 0.791
0.566 0.489 0.301 0.000 0.000 0.000 0.000
0.969 0.936 0.877 0.816 0.765 0.745 0.779
1.011 0.98 1 0.938 0.891 0.828 0.768 0.713
BK
0.37 0.37 0.36 0.33 0.30 -0.03
NIL
0.46 0.51 0.5 1 0.43 0.36 0.3 1
M. Saha, S. Sen / Magnetic
50
in considerable seems to enhance to changes
reduction
of the g-factors.
Inclusion
this effect to a considerable
in the Nilsson
tion. The yrast spectrum
energy spectrum in “‘0s
moments
extent.
produced
shows backbending
of the hexadecapole This happens
by the hexadecapole features
mode
primarily
due
deforma-
at I” = 12+ mainly
due to i1),2 neutron alignment. However, there is one significant difference between the results obtained with NIL and BK. The proton pairing gap with NIL shows (table 2) pronounced Coriolis antipairing effect (CAP) resulting in larger g-factors than those obtained with BK. Measurement of g-factors for higher spin states in *“OS would be very useful in estimating the relative importance of proton CAP and neutron rotational alignment (RAL) effects in this nucleus. In ls4W, the calculated g-factors, both with and without hexadecapole deformation, show a decreasing trend which is opposite to what has been observed experimentally. Of all the isotopes studied in the present work only in ls4W, the theoretical calculations using both NIL and BK have failed to make even qualitative prediction of the experimental trend with standard interaction strengths.
5. Dependence 5.1. STRENGTHS
of the calculated
OF RESIDUAL
g-factors
on various parameters
INTERACTIONS
The g-1 variations in almost all the isotopes for which experimental data are available, as well as the ground-state deformations have been reasonably reproduced with the s.p.e. set NIL and the standard values of the interaction strengths (G,, G,, x2, x4). But the same is not true for the neutron and proton pairing gaps. Moreover, the measured g-factors for the 2+ states are also not reproduced correctly in all the isotopes. These observations are true for both BK and NIL. Naturally, a question may arise whether it is possible to achieve the same quality of agreement with observed g-1 variations in these isotopes using strength parameters (especially G,, G,) which will produce a better fit to the experimental A,,, A, and the g, values. Since the deformation parameters have been more or less correctly reproduced in all the isotopes, we have repeated the calculations in two nuclei ls4Sm, 16’Er (where g-1 variations predicted by BK and NIL differ considerably) by adjusting the pairing strengths G, and G, to reproduce the experimental A, and A, and the g, as closely as possible for each individual set BK and NIL. We have also calculated the g-I variations in “‘Er for those values of G, and G, which produce almost identical values of A, and A, in both the cases. The results obtained in “‘Er are summarised in table 3. The results show that adjustment of the pairing strengths in each individual isotope may lead to a better agreement with the experimental A, value and g-factor data (but in general, not with A,,) than has been achieved in the present work. Moreover, it is found that in r6’Er, if one uses BK, it is very difficult to achieve good agreement with experimental pairing-strength parameters.
data for any reasonable
choice of the
M. &ha,
S. Sew / Magnetic
51
moments
TABLETS g-factors
in 16sEr calculated
with BK and p4 = 0.0 for different
values of G, and G,
AG,(MeV):
27.0
29.0
26.3
AG,(MeV):
22.0
23.2
22.5
A,(MeV)
“):
0.849
1.063
0.781
A,(MeV)
h):
0.680
0.834
0.741
I
g,/g,
g,
g,lg,
g/
izl
2
0.277
1.000
0.261
1.000
0.316
1.ooo
4
0.263
0.949
0.244
0.934
0.304
0.961
6
0.239
0.862
0.213
0.816
0.281
0.890
8
0.207
0.750
0.174
0.668
0.249
0.789
10
0.176
0.636
0.138
0.530
0.215
0.68 2
12
0.149
0.537
0.109
0.419
0.185
0.585
“) A,(exp.)
= 1.193 MeV.
h, A,(exp.)
= 0.884 MeV
TABLE g-factors
in 16sEr calculated
3b
with NIL and p4 = 0.0 for different values of G, and G,
AG,(MeV):
27.0
29.5
AG,(MeV):
22.0
22.5
A& MeV) “):
0.787
1.060
A,(MeV)
0.749
0.822
‘):
I
g,
2 4 6
0.305
0.965
0.223
0.888
8
0.293
0.927
0.197
0.783
10
0.276
0.872
0.166
0.66 1
“) A,(exp.) 5.2. RELATIVE FERMI
POSITION
OF
g,/gz
g,
0.316
1.ooo
0.251
1.000
0.312
0.988
0.241
0.959
= 1.193 MeV. THE
vi,3,2
h, A,(exp.) AND
vh,,,,
g,/g,
= 0.884 MeV. ORBITALS
VIS-A-VIS
SURFACE
The strength of the Coriolis force and interaction between the ground and s-bands have important consequences on the g-Z variation in any nucleus. They, in turn, depend sensitively on the position of Fermi surface (A) vis-a-vis the high-j neutron and proton intruder orbitals. It is well-known that the backbending behaviour of the yrast spectra arising from strong rotational alignment is expected in those nuclei for which the Fermi surface lies near low-L& high-j orbitals. On the other hand, the sharpness of the backbending behaviour depends upon the interaction matrix elements between the ground and s-bands (V,_,). It has been shown by several workers 31) that V,_, for the i13,z neutrons in the rare-earth region has maxima at N = 92, 98, 104 corresponding to the position of A between R = $2, I, p levels. The
M. Saha, S. Sen / Magnetic
52
moments
interaction is found to be zero where A is nearly equal to the energy of one of the high-spin intruder levels. In fig. 2 the g-factor ratios g,/g, and g,,/g, are shown as a function of neutron number N. The experimental ratios show a clear dip for N=98,
‘66Er nucleus.
However,
if we consider
the 164Dy (2 = 66, N = 98) nucleus,
the &,/g, ratio fits the systematics but the glO/g, becomes almost equal to 1. Earlier study of Bengtsson and Frauendorf 3’) has shown that the interaction strength for the hill2 protons shows a maximum near Z = 66. Therefore this sudden rise of the g,,/gz ratio in 164Dy may arise from significant admixture of a h,,,2 proton-aligned band in the high-spin yrast states. In the present work, it is found that two different s.p.e. sets produce considerable difference in predicted g-1 variations. Replacement of a particular set of s.p.e. by another s.p.e. set would naturally lead to a change in the position of the Fermi surface vis-a-vis the high-j orbitals. The observed g-1 variations in ‘64Dy, ‘66Er and 16’Er show marked dependence on change in neutron and proton numbers. Corresponding theoretical trends in these isotopes also show strong dependence on the input values of the spherical single-particle energies. The occupation probabilities V’, (where (Y is the single-particle basis which diagonalises the density matrix) for several states in these isotopes near the proton and neutron Fermi surfaces at w = 0 are listed in table 4. The dominant shell-model components (in signature basis) for these states are also shown. In ‘663168Ernuclei, A, predicted by BK lies above the (h ,,,*,3) orbital, whereas A, lies between the (i13,2,$) and (i13,2,$ orbitals. As the neutron number changes from N = 98 (‘66Er) to N = 100 (16*Er) A, moves closer to
0.4
I 90
I
, 92
1
, 94
I
, 96
I
, 98
N
1
, I , I , 102 104 100
I
Fig. 2. Experimental g,/gz (asterisks) and g,,/g, (triangles) are shown as a function of neutron number. Proton numbers are given near corresponding data points. For N = 98 and 100 g-factor ratios are shown for two neighbouring isotones. Errors in measurements (given in table 1) are not plotted.
M. Saha, S. Sen / Magnetic TABLE 4 The occupation probabilities Vi for proton/neutron in ‘64Dy and ‘66,‘68Er for w = 0. The single-particle density matrix p. Single-particle states
Neutrons
12jp lh 1,/Z,<
166Er “‘Er ‘64Dy
pairs near the Fermi surface basis la) diagonalises the
Protons
Isotopes
Occupation BK NIL BK NIL NIL
0.9256 0.9360 0.9319 0.9382 0.8764
53
moments
11)” s li 13/Z,?
probabilities
0.6370 0.6569 0.6443 0.6593 0.3330
12)” l113,2>Z
(V’,) 0.8235 0.8717 0.9298 0.9343 0.8865
0.1743 0.2759 0.3972 0.5292 0.2943
0 =z which results in better stability in the calculated g-factors in ‘68Er compared to that in ‘66Er. On the other hand, A, in ‘68Er predicted by NIL almost overlaps with the fi =I orbital. It will result not only in relatively weak neutron alignment (compared to that obtained with BK) but also vanishingly small g-s band interaction. Consequently NIL produces much better stability of g-factors against increasing spin than that produced by BK. No appreciable change in A, (relative to the h,,,z proton orbitals) is observed when BK is replaced by NIL. So the nature of proton alignment will be the same for both BK and NIL. An interesting situation arises when the proton number changes from 66 to 68 for the N = 98 isotones, ‘64Dy and ‘66Er. The A, predicted by NIL lies between the (h ll,~,?) and (h ,,,zr g) orbitals in 164Dy and above the 0 =g orbital in ‘66Er. Consequently the proton alignment as well as the interaction between the ground and proton-aligned bands are stronger in ‘64Dy, thus offsetting to a considerable extent the effect of neutron alignment on the g-1 variation in this isotope. The observed stability of the g-factors in 164Dy, therefore, is not due to absence of neutron alignment, rather it signifies a keen competition between both proton and neutron alignments.
5.3. BACKBENDING
FEATURE
AND
g-I
VARIATION
In our earlier publication 26), it was pointed out that although the g-Z trends calculated in 16*Er using BK and NIL came out to be quite different, corresponding excitation energies of the yrast states showed identical behaviour. The same is found to be true in 156Gd also. On the other hand, in several nuclei it is found that both the E-I and g-1 trends predicted by BK and NIL differ considerably. It shows that the correct prediction of g-factor variation along the yrast line is not entirely
54
M. Saha, S. Sen / ~agnetjc mmnenfs
dependent
upon exact reproduction
This is not unexpected
because
of the experimental
BB feature
backbending
is a sensitive
(BB) feature.
probe of the behaviour
of
total (neutron and proton) alignment with increasing rotational frequency, whereas the g-factor variation depends upon the detailed nature of relative alignment of neutrons
and protons.
5.4. g-1 VARIATIONS
WITH
EMPIRICAL
SINGLE-PARTICLE
ENERGIES
On the basis of their experimental investigations in l”,14’Gd nuclei (with Z = 64 shell closure), Kleinheinz et al. ‘*) have suggested a set of spherical single-panicle energies for the valence neutron and proton orbitals. It is interesting to note that these empirical energies show a h,l,z-d 5/z 0.436) spacing for protons, (Ej/
Am0
=
which is much larger than suggested by BK (&j/fiw, = 0.1 IS) or NIL ( ej/ hwO = 0.121). A calculation of the g-factor variation with spin in ‘66Y168Erusing these empirical values shows an increasing trend due to early proton alignment (e.g. g,/g, = 1.350 in 16*Er). It has been pointed out by Sorensen 33) that in any calculation using schematic interactions like the quadrupole-plus-pairing forces, choice of an appropriate set of single-particle energies becomes very important. A lot of experimental information on the energy levels and B(E2) values have been accumulated in the ‘46Gd region 34). Based on these data, a set of spherical single-panicle energies can be deduced as has been done earlier by Reehal and Sorensen 35)_ The same can then be used in a CHFB
calculation
and the results
may be compared
with the present
one.
6. Conclusions On the basis of a comparative study of the results of the present work with those of earlier theoretical and experimental investigations, the following conclusions may be drawn: (1) Choice of spherical single-particle energies is probably the most important factor in correct prediction of the observed spin dependence of g-factors in welldeformed nuclei within the framework of CHFB formalism. (2) A reasonable fit to the observed g-i variations in the rare-earth region can be obtained in a simple CHFB calculation using standard values of the monopole pairing and the quadrupole-quadrupole interaction strength parameters along with the spherical single-particle energies prescribed by Nilsson et al. (3) The g-factor trend is very much sensitive to the relative alignment of the proton and neutron spins, whereas the backbending feature depends on total alignment. Therefore, a correct reproduction of the observed backbending feature in a nucleus does not automatically lead to simultaneous reproduction of g-1 variation.
55
M. Saha, S. Sen / Magnetic moments
(4) If one is interested
in studying
the finer aspects
of the CHFB
formalism,
e.g.
the effect of inclusion of quadrupole pairing or other modifications of the residual interaction, number/angular-momentum projection etc., the sensitivity of the calculated results on the input values of the single-particle energies should be taken into account. The present work also suggests that the following theoretical and experimental investigations would lead to a better understanding of the related problems: (i) Measurement of g-factors of high-spin states in ‘*‘OS, remeasurement of g-factors in r’%b, IE4W etc.; (ii) Number-projection CHFB calculation in cases where the observed spin dependence of the g-factors changes significantly in the neighbouring isotopes and isotones e.g. in ‘64Dy, 166V’6REr; (iii) Angular-momentum projection CHFB calculation in some cases where the observed g-factors change abruptly, (not monotonically~ with increasing spin; (iv) CHFB calculation with spherical single-particle energies based on empirical data available in ‘46Gd region. The authors
are indebted
to Professor
A. Ansari
for many helpful
discussions.
References 1) P. Ring and P. Schuck, The nuclear many-body problem (Springer, Berlin, 1980); A. Faessler, M. Ploszajczak and K.W. Schmid, in Progress in particle and nuclear physics, vol. 5, ed. D. Wilkinson (Pergamon, Oxford, 1981) p. 79; Z. Szymanski, Fast nuclear rotation (Clarendon, Oxford, 1983); M.J.A. de Voigt, J. Dudek and 2. Szymanski, Rev. Mod. Phys. 55 (1983) 949; R. Bengtsson and J.D. Garrett, in Collective phenomena in atomic nuclei, International review of nuclear physics vol. 2, ed. T. Engeland, J. Rekstad and J.S. Vaagen, (World Scientific, Singapore, 1984) p. 193; I. Hamamoto, in Nuclear structure 1985, ed. R. Broglia, G.B. Hagemann and B. Herskind (NorthHolland, Amsterdam 1985) p. 129; K. Tanabe and K. Sugawara-Tanabe, Phys. Lett. B259 (1991) 12 2) R. Bengtsson and S. Aberg, Phys. Lett. B172 (1986) 277; S. Frauendorf, Phys. Lett. BlOO (1981) 219 3) Y.S. Chen and S. Frauendorf, Nucl. Phys. A393 (1983) 135 4) M. Diebel, A.N. Mantri and U. Mosel, Nucl. Phys. A345 (1980) 72 5) A.N. Mantri, M. Diebel and U. Mosel, Phys. Rev. Lett. 47 (1981) 308 6) A. Ansari, E. Wiist and K. Miihthans, Nucl. Phys. A415 (1984) 215 7) E. Wiist, A. Ansari and l-l. Mosel, Nucl. Phys. A435 (1985) 477 8) K. Sugawara-Tanabe and K. Tanabe, Phys. Lett. 8207 (1988) 243 9) M. Diebel, A.N. Mantri and U. Mosel, Phys. Ser. 24 (1981) 331 10) C.E. Doran, H.H. Bolotin, A.E. Stuchbery and A.P. Byrne, Z. Phys. A325 (1986) 28.5 11) A. Alzner, E. Bodenstedt, J. Gemiinden, J. van den Hoff and H. Rief, Z. Phys. A322 (1985) 467 12) H.R. Andrews, 0. Hiusser, D. Ward, P. Taras, R. Nicole, J. Keinonen, P. Skensved and B. Haas, Nucl. Phys. A383 (1982) $09 13) H.R. Andrews, 0. Hausser, D. Ward, P. Taras, N. Rud, B. Haas, R.M. Diamond, D. Fossan, H. Kluge, M. Niemann, C. Roulet and F.S. Stephens, Phys. Rev. Lett. 45 (1980) 1835 14) C.E. Doran, A.E. Stuchbery, H.H. Bolotin, A.P. Byrne and G.J. Lampard, Phys. Rev. C40 (1989) 2035
56
M. Saha, S. Sen / Magnetic
moments
15) A.P. Byrne, A.E. Stuchbery, H.H. Bolotin, C.E. Doran and G.J. Lampard, Nucl. Phys. A466 (1987) 419 16) D. Ward, 0. Hiusser, H.R. Andrews, P. Taras, P. Skensved, N. Rud and C. Broude, Nucl. Phys. A330 (1979) 225 17) P. Raghavan, At. Data Nucl. Data Tables 42 (1989) 189 18) A.E. Stuchbery, A.G. White, G.D. Dracoulis, K.J. Schiffer and B. Fabricius, Z. Phys. A338 (1991) 135 19) Th. Schafer, R. Vianden and E. Bodenstedt, Z. Phys. A335 (1990) 387 20) A. Ansari, Phys. Rev. C41 (1990) 782 21) M. Baranger and K. Kumar, Nucl. Phys. All0 (1968) 490, 529 22) A. Ansari, Phys. Rev. C33 (1986) 321 23) A. Ansari, Phys. Rev. C38 (1988) 323 24) M. Ploszajczak, A. Faessler, G. Leander and S.G. Nilsson, Nucl. Phys. A301 (1978) 477; A. Ansari, 0. Civitarese and A. Faessler, Nucl. Phys. A334 (1980) 93; S.G. Rohozinski and A. Sobiczewski, Acta. Phys. Pol. B12 (1981) 1001 25) W. Nazarewicz and P. Rozmej, Nucl. Phys. A369 (1981) 396 26) M. Saha and S. Sen, Phys. Lett. 8252 (1990) 181 27) M.L. Cescato, Y. Sun and P. Ring, Nucl. Phys. AS33 (1991) 455 28) A.L. Goodman, Nucl. Phys. A230 (1974) 466; A. Faessler, K.R. Sandhya Devi, F. Griimmer, K.W. Schmid and R.R. Hilton, Nucl. Phys. A256 (1976) 106; A.L. Goodman, Nucl. Phys. A265 (1976) 113; A. Ansari and S.C.K. Nair, Nucl. Phys. A283 (1977) 326 29) S.G. Nilsson, CF. Tsang, A. Sobiczewski, Z. Szymanski, S. Wycech, C. Gustafson, I.L. Lamm, P. Moller and B. Nilsson, Nucl. Phys. A131 (1969) 1 30) A. Faessler, M. Ploszajczak and K.R. Sandhya Devi, Nucl. Phys. A301 (1978) 382; A. Ansari, Phys. Rev. C38 (1988) 953; A.E. Stuchbery, G.J. Lampard and H.H. Bolotin, Nucl. Phys. A528 (1991) 447; F.M. Bernthal, B.D. Jeltema, J.S. Boyno, T.L. Khoo and R.A. Warner, Phys. Rev. Lett. 33 (1974) 915; U. GBtz, H.C. Pauli, K. Alder and K. Junker, Nucl. Phys. Al92 (1972) 1 31) R. Bengtsson, 1. Hamamoto and B. Mottelson, Phys. Lett. B73 (1978) 259; R. Bengtsson and S. Frauendorf, Nucl. Phys. A314 (1979) 27; A327 (1979) 139 32) P. Kleinheinz, R. Broda, P.J. Daly, S. Lunardi, M. Ogawa and J. Blomqvist, Z. Phys. A290 (1979) 279 33) R.A. Sorensen, Nucl. Phys. A420 (1984) 221 34) H. Walters et al., Z. Phys. A333 (1989) 413; R.M. Lieder, Nucl. Phys. A520 (1990) 59~; K. Zuber et al., Nucl. Phys. AS20 (1990) 195~ 35) B. Reehal and R. Sorensen, Phys. Rev. C2 (1970) 819
( 1993) 37-56
Nuclear Physics A552 North-Holland
Magnetic
moments of even rare-earth cranked HFB approach Saha Institute
nuclei in the
M. Saha and S. Sen ofNuclear Physics. Calcutta - 700 064, India Received 20 January (Revised 10 September
1992 1992)
Abstract: Magnetic moments of yrast levels of a large number of even-even rare-earth nuclei have been calculated using a simple version of the cranked Hartree-Fock-Bogoliubov (CHFB) model. In order to study the sensitivity of the calculated results to the input values of spherical single-particle energies, the calculation has been performed with two different sets of these energies, which are widely used in the literature. It is found that with proper choice of spherical single-particte energies, the CHFB formalism, in its simple version, is quite capable of reproducing the observed spin dependence of g-factors from Sm to OS nuclei.
1. Introduction The cranked Hartree-Fock-Bogoliubov (CHFB) formalism, in spite of its several sho~comings, has been extensively used ‘) to investigate the complexity of nuclear excitation spectra, arising from the interplay of single-particle and collective aspects of nuclear motion, as a function of rotational frequency. Since the cranking wave functions conserve total angular momentum only on “average”, they are clearly not suitable for calculations of those properties of nuclei which are sensitive to angularmomentum admixtures, such as transition probabilities or the static electric moments. However, calculation of the gyromagnetic ratio in this formalism is expected to be reliable since the magnetic moments involve only the expectation values of the total orbital and spin angular momenta. It is well known that the variation of g-factors along the yrast line in deformed nuclei can be used as a sensitive probe to understand the change in neutron and proton spin alignment with increasing rotational frequency 2”)_ For this reason, several workers have used the CHFB formalism to study this variation in a number of well-deformed nuclei 4-9). Magnetic-moment “-19) of the yrast-state g-factors have been done in several even measurements rare-earth nuclei over the last decade. Comparison of these experimental values with the results of the CHFB calculations offers a very good opportunity to test the reliability of this formalism in reproducing the g-factors of high-spin states.
0375-9474/93/$06.00
@ 1993 - Elsevier
Science
Publishers
B.V. All rights reserved
M. Saha, S. Sen / Magnetic
38
Earlier
experimental
moment.s
data f0312~‘3,‘7)in ‘70,t74Yb, ‘66Er, “‘Dy,
‘56*158Gdand ls4Sm
nuclei show a monotonic decrease of g-factors with increasing spin along the yrast line. Recent magnetic-moment studies 14*15)in 16*Er, ‘64Dy and ‘50.‘S2Sm, however, show almost
constant
g-factors
have been done and an initial
in 16*Er of all the levels for which measurements decrease followed by subsequent increasing trend in
the other nuclei. The decreasing trends of g-factors with increasing spin in ‘70*‘74Yb and 166Er have been well reproduced within the framework of CHFB formalism by Mantri et al. ‘) and Ansari et al. 6), respectively. Sugaware-Tanabe and Tanabe ‘) have calculated the g-factors in ‘66*168Erand obtained a good fit to the experimental data. Ansari “) has recently studied g-1 variation in “*Er in a variation-after-exact angular-momentum-projection approach. Although, all of them are unanimous in attributing the spin variations of the g-factors to rotational alignment effects, their calculational details differ in the choice of single-particle energies, form of residual interaction and their strengths etc. Moreover, none of these calculations have been simultaneously applied to all the rare-earth nuclei for which experimental data are available. However, based on their predictions, certain inferences about different aspects of the CHFB formalism and its reliability in predicting energy spectra and magnetic moments of the yrast bands have been drawn, which in our opinion need closer scrutiny. The rotational alignment effect in any nucleus depends sensitively upon the position of the high-spin intruder orbitals vis-a-vis the Fermi level. So it is our contention that one should clearly delineate the effect of the choice of single-particle energies on the predictions of g-factors in the CHFB formalism before making any study on the effect of inclusion of higher-order terms in the residual interaction, adjustment of its strength or projection of good particle number and of good angular momentum. The main objective of the present work is twofold: (i) to make a systematic study of g-I variations in the rare-earth region within the framework of the CHFB formalism to see how far this model, in its simplest version, is successful in reproducing the experimental trends; (ii) to study the sensitivity of the calculated results on the choice of input values of spherical single-particle energies. With these in view, we have calculated the g-factors for the yrast states in 152,154,156~~,
1%,i58,16oGd,
158,160,i62,164~y,
162,164,166,168.170~r,
168,170,172,174~,
184~
and
“‘0s nuclei within the framework of the CHFB formalism. The pairing-plusquadrupole hamiltonian of Baranger and Kumar *‘) is used. In certain nuclei, calculations have also been performed by including a hexadecapole-hexadecapole interaction term *2-24) in the ham’11 tonian because of experimental evidence 25) for appreciable hexadecapole deformation in their equilibrium shape. Since the model used in the present work is well known, only some of its relevant features are presented here. Results of the calculation are discussed in detail in subsequent sections keeping in view the twin objectives of the present study. Preliminary results of the study have been published elsewhere 26). Very recently, Cescato et al. *‘) have published the results of a systematic study of the yrast levels of a chain of Dy isotopes.
M. Saha, S. Sen / Magneiic
39
moments
2. CHFB formalism
2.1. MODEL
In the cranking perpendicular
model the hamiltonian
to the symmetry
axis (here,
H, is cranked z-axis)
about an axis (here, x-axis)
of the nucleus.
H,=H-wf,-Ai+,
(1)
where w is the Lagrange multiplier (given in energy units, interpreted as the cranking frequency for rotation about the x-axis) such that the total angular momentum of the system is given by the constraint
(24 along with the usual constraint on the particle number N, due to the presence of pairing in the hamiltonian H. Here A, interpreted as the Fermi level, is the Lagrange multiplier used to ensure the conservation of “average” particle number, ($CtWB (~)l~T,l~M3(~))
= NT,
(2b)
where NT denotes proton or neutron number outside the core. For compensating small numerical error in satisfying the above relation we have added a correction term in the energy E [refs. 6T23)]. In eq. (l), H is given by a more general form (to include the hexadecapole term) of the pairing plus quadrupole-model hamiltonian of Baranger and Kumar 2’): H =C E&C,
Here la), lb), . . . are the single-particle basis states to be specified later. Various symbols have their usual meanings; 7,~’ being neutron or proton.
In
general, Qhlr = r* yhl*(% cp)
(4a)
and pz and y are given by fi~#z
cos Y =
hw&,
D20,
sin y = ~6 D2z,
(4b)
where D2,
=x2
C
7
402,A
=
(-1)‘4-,
.
(4c)
4c
The radial
M. Saha, S. Sen / Magnetic
coordinate
moments
is given in units of oscillator
length,
6= (h/m,)“*, which is connected
with the oscillator
(54
energy,
hwO = 41 .2A-‘13 MeV .
(5b)
The parameters
(yT=
(22/A)“3
for protons
(2N/A)“’
for neutrons,
(SC)
have been introduced by Baranger and Kumar to ensure equal radii for protons and neutrons. The simplest approach to the hamiltonian (1) is to treat it in the Hartree-FockBogoliubov approximation, which is a self-consistent approximation in which pairing effects and quadrupole-field effects are treated on the same footing. The HFB approach determines at the same time the best shape of the self-consistent field and optimal pairing correlations for each angular momentum as a cranking term is added to the hamiltonian. In this calculation we have neglected the exchange terms. It is useful to introduce the symmetrized signature basis ‘*) given by
la>= la, a)=&[Ij,, m,)+a(-l)j~+“*lj,, where u = *l is the signature numbers necessary to specify
quantum a state,
number,
-??I,)],
and ljll, m,) denotes
(6) all quantum
The wave functions are those of an isotropic harmonic oscillator. of magnetic-moment The g-factors are calculated from j&., the x-component operator $, as
where
Fx=81Cjl,(i)+(g,-gr)C~~,(i), I
and free-particle
g-factors
,
are
g,=l,
g, = 5.586,
g/ = 0,
g, = -3.826,
for protons for neutrons.
(7b)
M. Saha, S. Sen / Magnetic moments
We have used attenuated
values
of g.? (spin g-factor)
41
as
g, = 0.6g, (free) . so g, = [(i:)+2.351((s^~)-O.98(s^~))]/~. 2.2. INCLUSION
OF HEXADECAPOLE
(7c)
TERM
We have treated all the components of the hexadecapole force self-consistently to minimise the energy. One straightforward way **), which we have adopted here, is to perform a self-consistent HFB calculation including the hexadecapole term. In this case also we have neglected the exchange matrix elements. The one-body hamiltonian corresponding to the hexadecapole term will look like h4=C
/_L= -4, -2,o,
Qu.+r
2,4,
@a)
.
(8b)
with D+ =x4c We may define
p4 through
7
oT(Q&
= (-l)@‘D,-,
the relation fi%P4 = 040.
It is obvious
that now we have five deformation
2.3. PARAMETER
PC) parameters
&, y, p4, Dd2, Dd4.
CHOICE
We have used the oscillator shells N = 4 and 5 for protons and N = 5 and 6 for neutrons as basis states. The inert core is taken as YtZr,,. This basis is identical to one used by Baranger and Kumar *I). In addition, the quadrupole matrix elements of the upper shells are scaled by factors # and g for protons and neutrons, respectively, to ensure the same r.m.s radii for both shells. The pairing and quadrupole force constants are given by G, = 271 A MeV, x2 = 70A-‘-4 MeV , When hexadecapole x~=x~=~O/A’.~
force is included, MeV,
G, = 221 A MeV , hw 0 = 41 .2A--“3 MeV . the interaction
G,=28/AMeV,
strengths
are changed
to 22*23)
G, = 231 A MeV .
We have not made any adjustment of the strengths of the residual interactions (G,, G,, xz) from Sm to Yb (except, of course their normal mass dependence), although our study spans a mass region from A = 152 to 174 and 2 = 62 to 70. In order to understand the sensitivity of the calculated g-factors on the input values of spherical single-particle energies, we have performed the calculation with two
M. &ha,
S. Sen / Magnetic
Calculated
and experimental
42
moments
TABLE 1 g-factors
&I& Isotope
“‘Sm
I” exp “)
BK
NIL b,
4+ 6+ 8+
1.00 (13)
0.53
0.83 [0.93]
0.95 (15)
0.14
0.57 [0.83]
10+
0.88 (41)
0.83 (17)
-0.06
0.18 [OSS] 0.19 [OS81
12+ “‘Srn
‘%jrn
4+
0.99 (10)
0.91
6+
0.96 (11)
0.77
0.91 [0.98]
8+
0.92 (13)
0.62
0.83 [0.98]
10+
0.87 (18)
0.47
0.75 [0.97]
0.96 [0.99]
4+
0.99
0.97
6+
0.98
0.92
8+
0.98
0.86
10+
0.67
0.8 1 0.79
12+ ‘56Gd
0.32 [0.69]
4+
0.88
0.96
0.68
0.89
0.93 (7)
0.49
0.80
0.89 (12)
0.34
0.70
0.22
0.59
0.97
0.98
0.89
0.94
0.99 (2) 0.99 (20)
6+
0.97 (4) 0.92 (18)
8+ 10+ 12+ “‘Gd
4+
0.98 (1) 0.92 (23)
6+
0.95 (4) 0.91 (25)
8+
0.90 (7)
0.61
0.90
10+
0.84(11)
0.36
0.86
12+ 16’Gd
0.85
4+
0.99 (1)
0.99
0.97
6+
0.98 (4)
0.97
0.93
8+
0.96 (7)
0.96
0.89
10+
0.93 (12)
0.93
0.88
0.92
0.90
0.86
0.95
0.57
0.86 0.74
12+ “‘Dy
0.21
4+
0.98 (2) 0.95 (12)
6+
0.95 (4) 1.08 (32)
8+
0.91 (8)
0.33
10+
0.86 (13)
0.15
0.62
12+
0.79 (18)
0.03
0.51
14+
0.71 (25)
0.07
0.42
0.03
0.38
0.17
0.46
0.11 (30) ‘) 16+ 18+
0.62 (33)
M. Saha, S. Sen / Magnetic moments
43
TABLE l-continued
&I&?, Isotope
I” exp “)
BK
NIL ‘) 0.63
20+ 4+
0.98 (2)
0.92
0.97
6+ 8+
0.95 (5) 0.91 (10)
0.71 0.45
0.88
10+
0.86 (16)
0.26
0.83
12+
0.79 (23)
0.15
0.80
4+
0.97
0.96
6+
0.93
0.92
8+
0.89
0.87
0.93
10+
0.85
12+
0.88
4+
1.06 (34)
0.93
6+
0.80 (23)
0.82
0.96
8+
0.78 (26)
0.71
0.94
10+
1.01 (37)
0.64
0.92
0.62
12+
0.99
0.93
14+
0.98
16+
1.09
18+
1.18
4+
0.86
0.95
6+
0.58
0.88
s+
0.33
0.77
10+
0.17
12+
0.07
4+
0.95
0.94
6+
0.86
0.84
8+
0.73
10+
0.65
12+
0.60
4+
0.97 (5)
0.90
0.94
6+
0.83 (9)
0.76
0.84
8+
0.74 (13)
0.62
0.72
10+
0.62 (25)
0.53
0.60
12+
0.47
0.52
14+
0.44
0.48
16+
0.48
0.46
18+
0.54
0.53
4+ 6+
0.94 (12)
0.95
0.99
1.02 (14)
0.86
0.97
8+
1.03 (20)
0.75
0.93
lo+
1.OO (25)
0.64
0.87
0.54
0.81
12+
0.95
0.96
0.87
0.91
44
M. Saha, S. Sen / Magnetic
moments
TABLE l-continued
g,/g, Isotope
I” BK
exp “)
NIL”)
8+
0.78
0.85
10+
0.7 1
0.79
12+
0.77
0.77
4+
0.89
0.91
6+
0.76
0.76
8+
0.63
0.60
10+
0.53
0.47
12+
0.48
0.40
4+
0.97 (2)
0.95
0.98
6+
0.92 (5)
0.88
0.93 0.87
8+
0.86 (9)
0.77
10+
0.77 (15)
0.67
0.79
12+
0.66 (21)
0.58
0.71
14+
0.54 (29)
0.53
0.65
16+
0.39 (39)
0.61
0.64
4+
1.01 (2)
0.96
0.96
6+
1.03 (5)
0.90
0.91
8+
1.06 (9)
0.82
0.84
10+
1.10 (14)
0.75
0.79
0.83
0.76
12+ 4+
0.98 (1)
0.98
0.96
6+
0.96 (3)
0.96
0.90
8+
0.92 (6)
0.95
0.84
10+
0.87 (10)
0.96
0.80
12+
0.82 (14)
1.Ol
0.80
14+
0.75 (19)
1.06
0.82
16+
0.67 (26)
1.12
0.90
4+
1.01 (8)
0.91
0.93 [0.92]
6+
1.07 (11)
0.77
0.81 [0.82]
8+
1.25 (22)
0.63
0.71 [0.71]
0.53
0.70 [0.64]
10+
-
12+
[0.62]
4+
1.33 (13)
0.93
1.22 [l.lO]
6+
1.41 (22)
0.63
1.54[1.10]
0.20
8+
1.30 [0.94]
10+
1.05 [0.79]
12+
0.87 [0.67]
“) Taken from refs. ‘3.1’S”).
E x p erimental
errors are quoted
within
parentheses. b, Results
obtained
with
hexadecapole
within parentheses. ‘)
This value is for Iaverage= 14.1 [ref. “)I.
deformation
are shown
M. Saha, S. Sen / Magnetic
sets of spherical
single-particle
energies
(s.p.e.)
45
moments
which
are listed
in table
1 of our
earlier publication 26). One set, BK, is taken from Baranger and Kumar21), the other, NIL, from the prescription of Nilsson et al. 29). The first set, BK, has mass dependence
only through
the oscillator
parameter,
NIL, varies slowly over the mass region dence
of Z_Land k, prescribed
mass dependence
by Nilsson
under
fiw, = 41.2/A”3.
study according
The second
set,
to the mass depen-
et al. 29) and of course,
through
normal
of hwo. 3. Results
The calculated g-factors for two different sets of single-particle energies, BK and NIL, are listed along with the experimental data in table 1. The majority of the experimental data are taken from the recent compilation of Raghavan I’). The experimental data for 164Dy and 16’Er are taken from Doran et al. 14). Before going into the details of the agreement achieved in each individual isotope, a few comments can be made which are pertinent to the entire region of study. The calculated results in all the isotopes show an interesting feature. Two sets of s.p.e. produce a considerable difference in the predicted trends of variation of g-factors with spin. The magnitude of the difference in the calculated values is found in many cases to be larger than those achieved earlier 6,7) through higher-order modification in the residual interaction of the CHFB hamiltonian. Although BK produces in some isotopes very good agreement with experimental data (as in 16’Gd) but, in general, NIL gives better agreement with experimental trends over the entire region under study. In the present work, the pairing strengths G, and G, are taken from the work of Baranger and Kumar. In their work the strengths of the pairing interactions were fixed by fitting the experimental odd-even mass differences for the isotopes under study. Similarly, the qudrupole interaction strengths were deduced by comparing the calculated ground-state deformations with those deduced from experimental B(E2) values. Since their calculations were done with BK, one should be careful in using another s.p.e. set, e.g., NIL as model parameters while keeping the strengths of the residual interactions unchanged. However, it is found that so far as the agreement with the experimental deformation and odd-even mass difference are concerned, both the sets produce comparable results.
3.1.
15z,1s‘&.156~m
Experimental data ‘*-15) available in ‘52,‘54Sm show that the g-factor changes rather slowly with increasing spin. The calculated g-factors in ‘j2Sm using BK and NIL on the other hand show a much faster fall, indicating strong neutron alignment. In fact, BK shows such a strong neutron alignment that the g-factor becomes negative even for I” = 8+. However, several theoretical 22723*29,30) and experimental 25) studies indicate the presence of appreciable hexadecapole deformation not only in OS-Pt
46
M. Saha, S. Sen / ~~gneijc
moments
but also in Sm-Gd regions. Since it is also known that the hexadecapole deformation retards or accelerates particle alignments depending upon the sign of deformation, we have also calculated the g-factors in ‘52*‘54Smincluding hexadecapole interaction in the CHFB hamiltonian. The results obtained with the inclusion of p4 will be discussed in a subsequent section (sect. 4). In ‘54Sm, the g-factors obtained with NIL show much better agreement with experimental values than those obtained with BK. However, even NIL gives a neutron alignment which is stronger than what is suggested by the experimental trend of the g-1 curve. In “‘Sm, the calculated results using BK and NIL, show just the opposite feature. The g-factors calculated with BK show a very high degree of stability, whereas, those calculated with NIL show a gradual decrease of g-factor values with increasing spin. Both the sets predict better stability of the g-1 curve in ‘%m, than in ‘54Sm. However, the change observed with BK as one goes from A = 154 to 156 is much more abrupt.
Experimental data 17-19)are available in all these isotopes. The calculated results using NIL are in close agreement with the experimental results in 158,‘60Gd,whereas, in *56Gd, they show a somewhat faster fall than that observed experimentally. The set BK, on the other hand, predicts a rapid decrease of the g-factors with increasing spin in ‘56~i58Gdand a sudden reversal of trend in 16’Gd, showing high degree of stability in close agreement with the observed data. The g-factors in ‘583’60Gd, calculated with NIL, initially decreases with increasing spin, but then show an upward trend beyond I” = lO+ indicating early onset of h,,,2 proton alignment for these nuclei. The close correspondence between relative proton and neutron alignments (I!JIZ) and g-factors is shown in fig. 1. Variation of &, y, A, and A, with spin is also shown. It is found that the & remains more or less constant even upto I”= 12”-14+. This is true for most of the nuclei studied in the present work. The constancy of pz along the yrast line is not surprising in view of the fact that except for W and OS nuclei, all other nuclei considered here belong to the well-deformed category. So far as the variation of A, and A, with spin is concerned, it is found that in general, NIL causes a faster (slower) fall of A, (A,,) with increasing I than those produced by BK. This is partly responsible for relatively good stability of g-1 variation obtained with NIL. 3 3 . .
15S,160,162,164~Y
Early measurement 13) in Experimental data 13,‘4*‘7)are available in 1583160*164Dy. 15’Dy shows a gradual decrease of g-factors with increasing spin, whereas, later measurement *‘) shows more or less stable g-factors in the low-spin region and then
M. Saha, S. Sen / Magnetic
0.0
I
0
,
4
,
,
8
I
,
12
I
0
,
I
,
4
I
moments
I
1
12
0
,
4
I
,
8
I
,
12
IB
Fig. 1. Spin dependence of deformation parameters (pz, y), pairing gaps (Ap, A,), relative g-factors and Ie/I: values in ‘56~‘58~‘60Gd. Relative g-factors and It/I: values obtained with BK (NIL) are shown by circles (triangles) connected with solid (dashed) lines. flz and A,, values obtained with BK (NIL) are shown by circles (triangles) connected with solid lines. y and A, values obtained with BK (NIL) are shown by circles (triangles) connected with long-dashed lines. Pairing gaps and & are plotted relative to their ground-state values.
a very rapid fall with increasing
spin. In the high-spin
region,
the calculated
values
using BK also show a very rapid decrease, in agreement with the later measurement. On the other hand, NIL predicts a gradual decrease of the g-factors with increasing spin, a situation intermediate between the trends observed in earlier and later measurements. It can also be noted that in this isotope, the present work, using NIL, produces better stability of the g-factor along the yrast line than that achieved in an earlier work by Ansari et al. 637) with BK and quadrupole pairing in the hamiltonian. In 16’Dy, the theoretical values obtained with NIL are in very good agreement with experimental data, whereas, those obtained with BK show a sharper decrease by Doran et of the g-factors with increasing spin. In ‘64Dy, a recent measurement al. 14), indicates a decreasing trend of g-factors with increasing spin upto I” = 8+ but the effect is not as sharp as that observed in the neighbouring isotone 166Er. Theoretical values obtained in these isotones, using NIL, nicely reproduce this
48
M. Saha, S. Sen / Magnetic moments
experimental decrease
feature,
of g-factors
whereas
those obtained
with increasing
with BK predict
spin than that observed
a somewhat
sharper
experimentally.
Experimental data ‘“*14,17)are available only in 166*‘68Er.The observed g-factors in 166Er show appreciable decrease with increasing spin, whereas in ‘68Er they remain almost constant for all the states (I” 6 lo+) for which measurements have been done. These experimental features are better reproduced in the theoretical calculation using NIL, rather than BK. If we compare the present result with those obtained by Ansari “) and Sugawara-Tanabe and Tanabe ‘), the following interesting features come out. Although the present work includes only the monopole part of the pairing and standard values of the interaction strengths, the predicted difference, using NIL, in the behaviour of the g-factor variations in ‘66Er and 16’Er is almost identical [fig. 2 of our earlier publication ‘“)I to that obtained by Sugawara-Tanabe and Tanabe. Inclusion of the quadrupole pairing term in the CHFB hamiltonian hinders rapid rotational alignment of single-particle orbitals. So the g-factors in both ‘66*‘68Er are seen to vary less rapidly with increasing spin in ref. “) than in the present work. The major part of the difference in the predictions of Ansari (i.e. the results obtained in the present work using BK) and those of ref. ‘) seems to arise primarily not because of different choices of the two-body interaction strengths, as mentioned in ref. *) but of spherical single-particle energies, BK and NIL, respectively.
Experimental
data 13*i7) are available
in ‘70~172~174Yb.In ‘70~‘74Yb, the results
obtained with NIL are in good agreement with the experimental values. In “‘Yb, the theoretical g-factors obtained in the present work are almost identical to those obtained by Mantri et al. ‘) upto spin value I” = 12+ [fig. 1 of our earlier publication ‘“)I. In ‘74Yb also, the quality of the fit obtained in the present and previous work is of the same order. The same single-particle parameter values p and K have been used in the present as well as in the earlier work. However, they performed the calculation for fixed values of &, p4 and y. The g-factors calculated with BK show good agreement with experimental data in “‘Yb, whereas in ‘74Yb they show strong proton alignment, a feature not an upward bend from I” = 8+ indicating supported by available experimental data in ‘74Yb or in “‘Yb. The experimental data “) in ‘72Yb, however, show an upward trend. Remeasurement of the g-factors in this isotope would be of considerable interest. 3.6. ls4W AND “‘0s
The tungsten region, belong
and osmium to a different
isotopes, category
situated at the extreme end of the rare-earth 22V23325330). They are classified as transitional
M. Saha, S. Sen / Magnetic
moments
49
nuclei and are known to exhibit shape transitions with increasing mass number 22330). Moreover, their shapes are also more complex in nature than those of other rare-earth nuclei studied in the present work. Some of them show large triaxiality**) and appreciable hexadecapole deformation in their equilibrium shape *“). Therefore, the proton/neutron
rotational
alignment
and
the g-Z
variation
in these
nuclei
are
expected to be somewhat different in nature. Results obtained in a detailed study of g-Z variations in W, OS and Pt nuclei will be reported in a separate communication. We have included the ls4W and “‘0s nuclei in the present discussion to show the effect of hexadecapole deformation on the rotational alignment and the g-Z variation.
4. Hexadecapole
deformation
and the g-I
variation
The g-factors in 1527’54Sm,ls4W and ‘880s have been calculated with and without the hexadecapole deformation. Although the triaxiality parameter y is small in all these isotopes, all the relevant terms Y40, Y4+*, Y4r4 have been included in the hamiltonian. These calculations have been performed with slightly different values *‘) of G, and G,. The results are shown in tables 1 and 2. It can be seen that the inclusion of hexadecapole deformation resulted in much better agreement with the determined observed g-Z variation in “*Sm. In ‘54Sm, the self-consistently hexadecapole parameters produce almost constant g-factors even upto spin value I” = 12+. In fact the observed g-Z variation is found to lie intermediate between those predicted by using NIL and with and without hexadecapole deformation. In the other extreme of the rare-earth region ( 1880s), inclusion of hexadecapole term makes the agreement with experimental data much worse than what can be achieved with quadrupole deformation only. The Z,P/Z: ratio shows that for low-spin values, the h,,,* proton alignment dominates but with increasing rotational 113/2 neutron tends to be aligned more and more along the rotational
Excitation
energies,
AE(I,
y-deformation
TABLE 2 parameters, pairing gaps and g-factors and hexadecapole deformation in “‘OS
I-2)
Y
calculated
frequency, the axis, resulting
with BK and NIL
k3
An
A*
I BK
NIL
0.197 0.403 0.533 0.606 0.63 1 0.553
0.198 0.389 0.501 0.587 0.624 0.632
0
2 4 6 8 10 12
BK
NIL
BK
NIL
BK
NIL
0.0
0.0
1.4 3.6 5.5 8.0 14.1 39.9
1.1 2.5 4.2 6.5 9.3 12.7
0.728 0.692 0.625 0.550 0.485 0.443 0.791
0.566 0.489 0.301 0.000 0.000 0.000 0.000
0.969 0.936 0.877 0.816 0.765 0.745 0.779
1.011 0.98 1 0.938 0.891 0.828 0.768 0.713
BK
0.37 0.37 0.36 0.33 0.30 -0.03
NIL
0.46 0.51 0.5 1 0.43 0.36 0.3 1
M. Saha, S. Sen / Magnetic
50
in considerable seems to enhance to changes
reduction
of the g-factors.
Inclusion
this effect to a considerable
in the Nilsson
tion. The yrast spectrum
energy spectrum in “‘0s
moments
extent.
produced
shows backbending
of the hexadecapole This happens
by the hexadecapole features
mode
primarily
due
deforma-
at I” = 12+ mainly
due to i1),2 neutron alignment. However, there is one significant difference between the results obtained with NIL and BK. The proton pairing gap with NIL shows (table 2) pronounced Coriolis antipairing effect (CAP) resulting in larger g-factors than those obtained with BK. Measurement of g-factors for higher spin states in *“OS would be very useful in estimating the relative importance of proton CAP and neutron rotational alignment (RAL) effects in this nucleus. In ls4W, the calculated g-factors, both with and without hexadecapole deformation, show a decreasing trend which is opposite to what has been observed experimentally. Of all the isotopes studied in the present work only in ls4W, the theoretical calculations using both NIL and BK have failed to make even qualitative prediction of the experimental trend with standard interaction strengths.
5. Dependence 5.1. STRENGTHS
of the calculated
OF RESIDUAL
g-factors
on various parameters
INTERACTIONS
The g-1 variations in almost all the isotopes for which experimental data are available, as well as the ground-state deformations have been reasonably reproduced with the s.p.e. set NIL and the standard values of the interaction strengths (G,, G,, x2, x4). But the same is not true for the neutron and proton pairing gaps. Moreover, the measured g-factors for the 2+ states are also not reproduced correctly in all the isotopes. These observations are true for both BK and NIL. Naturally, a question may arise whether it is possible to achieve the same quality of agreement with observed g-1 variations in these isotopes using strength parameters (especially G,, G,) which will produce a better fit to the experimental A,,, A, and the g, values. Since the deformation parameters have been more or less correctly reproduced in all the isotopes, we have repeated the calculations in two nuclei ls4Sm, 16’Er (where g-1 variations predicted by BK and NIL differ considerably) by adjusting the pairing strengths G, and G, to reproduce the experimental A, and A, and the g, as closely as possible for each individual set BK and NIL. We have also calculated the g-I variations in “‘Er for those values of G, and G, which produce almost identical values of A, and A, in both the cases. The results obtained in “‘Er are summarised in table 3. The results show that adjustment of the pairing strengths in each individual isotope may lead to a better agreement with the experimental A, value and g-factor data (but in general, not with A,,) than has been achieved in the present work. Moreover, it is found that in r6’Er, if one uses BK, it is very difficult to achieve good agreement with experimental pairing-strength parameters.
data for any reasonable
choice of the
M. &ha,
S. Sew / Magnetic
51
moments
TABLETS g-factors
in 16sEr calculated
with BK and p4 = 0.0 for different
values of G, and G,
AG,(MeV):
27.0
29.0
26.3
AG,(MeV):
22.0
23.2
22.5
A,(MeV)
“):
0.849
1.063
0.781
A,(MeV)
h):
0.680
0.834
0.741
I
g,/g,
g,
g,lg,
g/
izl
2
0.277
1.000
0.261
1.000
0.316
1.ooo
4
0.263
0.949
0.244
0.934
0.304
0.961
6
0.239
0.862
0.213
0.816
0.281
0.890
8
0.207
0.750
0.174
0.668
0.249
0.789
10
0.176
0.636
0.138
0.530
0.215
0.68 2
12
0.149
0.537
0.109
0.419
0.185
0.585
“) A,(exp.)
= 1.193 MeV.
h, A,(exp.)
= 0.884 MeV
TABLE g-factors
in 16sEr calculated
3b
with NIL and p4 = 0.0 for different values of G, and G,
AG,(MeV):
27.0
29.5
AG,(MeV):
22.0
22.5
A& MeV) “):
0.787
1.060
A,(MeV)
0.749
0.822
‘):
I
g,
2 4 6
0.305
0.965
0.223
0.888
8
0.293
0.927
0.197
0.783
10
0.276
0.872
0.166
0.66 1
“) A,(exp.) 5.2. RELATIVE FERMI
POSITION
OF
g,/gz
g,
0.316
1.ooo
0.251
1.000
0.312
0.988
0.241
0.959
= 1.193 MeV. THE
vi,3,2
h, A,(exp.) AND
vh,,,,
g,/g,
= 0.884 MeV. ORBITALS
VIS-A-VIS
SURFACE
The strength of the Coriolis force and interaction between the ground and s-bands have important consequences on the g-Z variation in any nucleus. They, in turn, depend sensitively on the position of Fermi surface (A) vis-a-vis the high-j neutron and proton intruder orbitals. It is well-known that the backbending behaviour of the yrast spectra arising from strong rotational alignment is expected in those nuclei for which the Fermi surface lies near low-L& high-j orbitals. On the other hand, the sharpness of the backbending behaviour depends upon the interaction matrix elements between the ground and s-bands (V,_,). It has been shown by several workers 31) that V,_, for the i13,z neutrons in the rare-earth region has maxima at N = 92, 98, 104 corresponding to the position of A between R = $2, I, p levels. The
M. Saha, S. Sen / Magnetic
52
moments
interaction is found to be zero where A is nearly equal to the energy of one of the high-spin intruder levels. In fig. 2 the g-factor ratios g,/g, and g,,/g, are shown as a function of neutron number N. The experimental ratios show a clear dip for N=98,
‘66Er nucleus.
However,
if we consider
the 164Dy (2 = 66, N = 98) nucleus,
the &,/g, ratio fits the systematics but the glO/g, becomes almost equal to 1. Earlier study of Bengtsson and Frauendorf 3’) has shown that the interaction strength for the hill2 protons shows a maximum near Z = 66. Therefore this sudden rise of the g,,/gz ratio in 164Dy may arise from significant admixture of a h,,,2 proton-aligned band in the high-spin yrast states. In the present work, it is found that two different s.p.e. sets produce considerable difference in predicted g-1 variations. Replacement of a particular set of s.p.e. by another s.p.e. set would naturally lead to a change in the position of the Fermi surface vis-a-vis the high-j orbitals. The observed g-1 variations in ‘64Dy, ‘66Er and 16’Er show marked dependence on change in neutron and proton numbers. Corresponding theoretical trends in these isotopes also show strong dependence on the input values of the spherical single-particle energies. The occupation probabilities V’, (where (Y is the single-particle basis which diagonalises the density matrix) for several states in these isotopes near the proton and neutron Fermi surfaces at w = 0 are listed in table 4. The dominant shell-model components (in signature basis) for these states are also shown. In ‘663168Ernuclei, A, predicted by BK lies above the (h ,,,*,3) orbital, whereas A, lies between the (i13,2,$) and (i13,2,$ orbitals. As the neutron number changes from N = 98 (‘66Er) to N = 100 (16*Er) A, moves closer to
0.4
I 90
I
, 92
1
, 94
I
, 96
I
, 98
N
1
, I , I , 102 104 100
I
Fig. 2. Experimental g,/gz (asterisks) and g,,/g, (triangles) are shown as a function of neutron number. Proton numbers are given near corresponding data points. For N = 98 and 100 g-factor ratios are shown for two neighbouring isotones. Errors in measurements (given in table 1) are not plotted.
M. Saha, S. Sen / Magnetic TABLE 4 The occupation probabilities Vi for proton/neutron in ‘64Dy and ‘66,‘68Er for w = 0. The single-particle density matrix p. Single-particle states
Neutrons
12jp lh 1,/Z,<
166Er “‘Er ‘64Dy
pairs near the Fermi surface basis la) diagonalises the
Protons
Isotopes
Occupation BK NIL BK NIL NIL
0.9256 0.9360 0.9319 0.9382 0.8764
53
moments
11)” s li 13/Z,?
probabilities
0.6370 0.6569 0.6443 0.6593 0.3330
12)” l113,2>Z
(V’,) 0.8235 0.8717 0.9298 0.9343 0.8865
0.1743 0.2759 0.3972 0.5292 0.2943
0 =z which results in better stability in the calculated g-factors in ‘68Er compared to that in ‘66Er. On the other hand, A, in ‘68Er predicted by NIL almost overlaps with the fi =I orbital. It will result not only in relatively weak neutron alignment (compared to that obtained with BK) but also vanishingly small g-s band interaction. Consequently NIL produces much better stability of g-factors against increasing spin than that produced by BK. No appreciable change in A, (relative to the h,,,z proton orbitals) is observed when BK is replaced by NIL. So the nature of proton alignment will be the same for both BK and NIL. An interesting situation arises when the proton number changes from 66 to 68 for the N = 98 isotones, ‘64Dy and ‘66Er. The A, predicted by NIL lies between the (h ll,~,?) and (h ,,,zr g) orbitals in 164Dy and above the 0 =g orbital in ‘66Er. Consequently the proton alignment as well as the interaction between the ground and proton-aligned bands are stronger in ‘64Dy, thus offsetting to a considerable extent the effect of neutron alignment on the g-1 variation in this isotope. The observed stability of the g-factors in 164Dy, therefore, is not due to absence of neutron alignment, rather it signifies a keen competition between both proton and neutron alignments.
5.3. BACKBENDING
FEATURE
AND
g-I
VARIATION
In our earlier publication 26), it was pointed out that although the g-Z trends calculated in 16*Er using BK and NIL came out to be quite different, corresponding excitation energies of the yrast states showed identical behaviour. The same is found to be true in 156Gd also. On the other hand, in several nuclei it is found that both the E-I and g-1 trends predicted by BK and NIL differ considerably. It shows that the correct prediction of g-factor variation along the yrast line is not entirely
54
M. Saha, S. Sen / ~agnetjc mmnenfs
dependent
upon exact reproduction
This is not unexpected
because
of the experimental
BB feature
backbending
is a sensitive
(BB) feature.
probe of the behaviour
of
total (neutron and proton) alignment with increasing rotational frequency, whereas the g-factor variation depends upon the detailed nature of relative alignment of neutrons
and protons.
5.4. g-1 VARIATIONS
WITH
EMPIRICAL
SINGLE-PARTICLE
ENERGIES
On the basis of their experimental investigations in l”,14’Gd nuclei (with Z = 64 shell closure), Kleinheinz et al. ‘*) have suggested a set of spherical single-panicle energies for the valence neutron and proton orbitals. It is interesting to note that these empirical energies show a h,l,z-d 5/z 0.436) spacing for protons, (Ej/
Am0
=
which is much larger than suggested by BK (&j/fiw, = 0.1 IS) or NIL ( ej/ hwO = 0.121). A calculation of the g-factor variation with spin in ‘66Y168Erusing these empirical values shows an increasing trend due to early proton alignment (e.g. g,/g, = 1.350 in 16*Er). It has been pointed out by Sorensen 33) that in any calculation using schematic interactions like the quadrupole-plus-pairing forces, choice of an appropriate set of single-particle energies becomes very important. A lot of experimental information on the energy levels and B(E2) values have been accumulated in the ‘46Gd region 34). Based on these data, a set of spherical single-panicle energies can be deduced as has been done earlier by Reehal and Sorensen 35)_ The same can then be used in a CHFB
calculation
and the results
may be compared
with the present
one.
6. Conclusions On the basis of a comparative study of the results of the present work with those of earlier theoretical and experimental investigations, the following conclusions may be drawn: (1) Choice of spherical single-particle energies is probably the most important factor in correct prediction of the observed spin dependence of g-factors in welldeformed nuclei within the framework of CHFB formalism. (2) A reasonable fit to the observed g-i variations in the rare-earth region can be obtained in a simple CHFB calculation using standard values of the monopole pairing and the quadrupole-quadrupole interaction strength parameters along with the spherical single-particle energies prescribed by Nilsson et al. (3) The g-factor trend is very much sensitive to the relative alignment of the proton and neutron spins, whereas the backbending feature depends on total alignment. Therefore, a correct reproduction of the observed backbending feature in a nucleus does not automatically lead to simultaneous reproduction of g-1 variation.
55
M. Saha, S. Sen / Magnetic moments
(4) If one is interested
in studying
the finer aspects
of the CHFB
formalism,
e.g.
the effect of inclusion of quadrupole pairing or other modifications of the residual interaction, number/angular-momentum projection etc., the sensitivity of the calculated results on the input values of the single-particle energies should be taken into account. The present work also suggests that the following theoretical and experimental investigations would lead to a better understanding of the related problems: (i) Measurement of g-factors of high-spin states in ‘*‘OS, remeasurement of g-factors in r’%b, IE4W etc.; (ii) Number-projection CHFB calculation in cases where the observed spin dependence of the g-factors changes significantly in the neighbouring isotopes and isotones e.g. in ‘64Dy, 166V’6REr; (iii) Angular-momentum projection CHFB calculation in some cases where the observed g-factors change abruptly, (not monotonically~ with increasing spin; (iv) CHFB calculation with spherical single-particle energies based on empirical data available in ‘46Gd region. The authors
are indebted
to Professor
A. Ansari
for many helpful
discussions.
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