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g Factors in heavy nuclei and the proton-neutron interaction
Volume 190, number 1,2
PHYSICS LETTERS B
21 May 1987
g F A C T O R S I N HEAVY N U C L E I A N D T H E P R O T O N - N E U T R O N I N T E R A C T I O N A. W O L F Nuclear Research Centre-Negev, POB 9001, 84120 Beer Sheva, Israel and Brookhaven National Laboratory, Upton, NY 11973, USA R . F . CASTEN and D.D. W A R N E R t Brookhaven National Laboratory, Upton, NYl1973, USA
Received 12 January 1987; revised manuscript received 2 March 1987
It is shown that the experimental values of g(2 i~) for nuclei in the mass range A = 70-200 are well described within IBA-2using a simple linear relationship. Values ofg~, gv, the effective proton, neutron boson g factors, were extracted for six groups of nuclei in this mass range. The results indicate that g~ tends to increase, and gv to decrease with A. No obvious distinction is observed between spherical and deformed nuclei. A large deviation in the systematics is observed for Er, Yb isotopes. This anomaly is explained in terms of a change in the effective number of valence particles, due to the K dependence of the p-n interaction in the deformed field.
Magnetic properties o f nuclear states are effective probes of nuclear wave functions. In nuclei near closed shells, g factors are mainly determined by single-particle motion and configuration mixing. In deformed nuclei, the collective motion of the protons and neutrons is responsible for the g-factor values. A systematic analysis o f g factors of excited nuclear states is therefore expected to provide valuable information regarding the structure o f the respective states. Although spherical and deformed nuclei are quite different in m a n y aspects, the g factors o f low-lying states in these nuclei can be expressed in a simple way using the n e u t r o n - p r o t o n version of the interacting boson approximation, IBA-2. This is particularly true for 2~- states in even-even (spherical or deformed) nuclei. Assuming that these states are completely symmetrical in the proton and neutron degrees o f freedom (i.e., F-spin is a good q u a n t u m number), it has been shown [ 1 ] that g ( 2 ~ ) is given by g(2 + ) =g~N~/Nt + g = N J N t ,
(1)
Permanent address: Daresbury Laboratory, Daresbury, Warrington WA4 4AD, UK. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
where g~ (gv) are proton (neutron) boson g factors, N~ (Nv) are proton (neutron) boson numbers and N t = N ~ +Nv. Eq. (1) can be rewritten: ( N~/N~)g( 2 ~-) =g~+ g,~N,~/N~ .
(2)
Thus a plot o f the left side o f this equation against N J N v , using empirical g(2~-) values, should be a straight line with intercept gv and slope g~. Deviations from a straight line can be ascribed to non-constancy of gv and g= which would be expected if the underlying microscopic structure o f the correlated s and d pairs is changing dramatically in a given region due to, for example, subshell effects. Alternatively, it has recently been shown [ 2 ] that such effects can also be treated in terms o f effective N~, Nv values which differ from those normally assumed by counting valence particles (or holes) from the nearest major shell closures. In this approach, therefore, subshell effects are accounted for by an assumed change in the number o f nucleons taking part in the collective motion. The specific example used previously involved g ( 2 ~ ) values in the A = 150 region where it was demonstrated that it is possible to use constant values ofg~, g~ for a large number o f deformed nuclei around A = 150 and to extract effective values of N~, 19
Volume 190, number 1,2
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N~ for nuclei in which the dissipation o f the Z = 64 subshell causes a significant change in the n u m b e r of effective valence protons for different isotopes o f the same element. The purpose o f the present work is to extend this type o f treatment to study the behavior ofg~, gv and N~, N~ in other regions o f the periodic table, to compare the results for spherical and deformed nuclei, and to discuss an interpretation o f apparent variations in g~, gv for the latter in terms o f a reduced effective proton-neutron interaction. Included in our study are 65 nuclei in several mass regions from A = 70-200, for which experimental g(2 3 ) values are available. Most data values are weighted averages from the Table of Isotopes [ 3] and, in some cases, are adopted values from the most recent Nuclear Data Sheets. Some recent g-factor measurements [ 4] were also included. Recently, a fit o f g(2 ~ ) values using eq. (1) was reported by Van Isacker et al. [ 5 ] for all isotopes between Ba and Pt. Their results are consistent with the values o f g~, g~ found in the A = 150 region [2] and with some o f the results which will be presented here. In order not to obscure possible gross variations o f g~, gv with mass n u m b e r or effects due to changing from particles to holes, the nuclei studied were divided in the present work into several groups, following the major proton and neutron shells. A given group only contains nuclei with proton and neutron numbers in specific half major shells (i.e., above or below the midshell point). Nuclei in the immediate vicinity of closed shells (like the Te and Hg isotopes) were not included since their g factors might be dominated by non-collective effects. The various groups
21 May 1987
are listed in table 1. This division is to some extent arbitrary, and is by no means unique. However, the quality o f the fits for the different groups and the relatively large n u m b e r o f nuclei included in each fit suggest that the conclusions o f this analysis are not strongly affected by the particular division used. For the discussion below it is also useful to classify the groups according to the degree of deformation in the nuclei concerned. As indicated in the table we used a rough criterion that groups in which most nuclei have E ( 4 ~-)/E(2 ? ) >i 2.7 are labelled deformed, and other groups as spherical. For each group we performed a linear least squares fit o f g(2 ~-)Nt/Nv versus NJNv, and extracted g~, gv. As an example, the fits for the Ge-Se, Xe-Ba, E r - Y b and H f - P t regions are presented graphically in fig. 1. The resulting g~, gv values are given in table 1 together with the respective X 2 / ( n - 2 ) values, where n is the number o f nuclei in each group. In cases where the normalized X2 value is > 1, the errors ofg~, g~ have been multiplied by x / X 2 / ( n - 2 ) , to account for the quality o f the fit. The poor accuracy o f the extracted g~, gv for group 1 is due to the low accuracy of the experimental data. Nevertheless, from table 1 and fig. 1, it is clear that the linear relation o f eq. (2) describes reasonably well the experimental data for each group. The rather large X2/( n - 2) for the H f - P t group is due to the small experimental errors, and it may signify that there is some "fine structure" in the A or Z dependence o f g~, gv. More experimental data is required to clarify this point. In fig. 2 we plotted the values orgy, g~ against the average mass number A in each group. The results in fig. 2 and table 1 suggest the general feature that g~
Table 1 The effective boson g factors for various groups of nuclei. Group no.
Isotopes
Number of nuclei n
Type of nuclei a)
g~
gv
z2/n-- 2
g~ + gv
1
0.30(16) 0.48(12)
0.55(14) 0.33(5)
sph.
0.50(5)
0.16(4)
6
176Hf-19spt
8 14 9 11 8 15
sph. sph.
3 4 5
7°Ge-S2Se 98Mo-I14Cd t24:Xe-lS6Ba 14~Ba-164Dy 164Er-176yb
def. def. def. b)
0.63 ( 5 ) 0.30(4) 0.68(7)
0.05 ( 5 ) 0.35(3) 0.12(3)
1.0 0.5 1.4 1.0 3.0 5.0
0.85(21 ) 0.81 (13) 0.66(6) 0.68 (7) 0.65(5) 0.80(8)
2
") Sph.: spherical, E( 4 ~-)/E( 2 ~-) < 2.7; def.: deformed, E( 4 3 )/E( 2 ~ ) > 2.7. b) The four Pt isotopes inlcuded in this grouP have E(4 3 )/E(2 3 ) = 25. 20
Volume 190, n u m b e r 1,2
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0.8 1.5 0.6 1.0-
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(d)
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& o!4 o; o; 11o
1'o 1; 21o 2; 31o N=/N~
Fi:;. 1. g ( 2 ~ )~=p N,IN~ versus N=IN~for: (a) Xe-Ba, (b) Ge-Se, (c) Er-Yb, and (d) Hf-Pt. The straight lines are least squares fits. Where not plotted, errors are o f the size of or smaller than the points.
tends to increase with A, while g~ is decreasing. The major exception to this trend is the Er-Yb group which will be discussed in detail below. In the simplest description of boson g factors, in which only the orbital contributions from the fermions are considered, one expects g~= 1.0 and gv=0.0. The present analysis clearly shows that this is not the case. While it is possible that part of the discrepancy is due to spin contributions, Dieperink [ 6] has pointed out that there is a contribution from the core which effectively increases the size of the valence space. If this argument is correct, our use of 0.8 0.70.6
& 0.5
the normal N~, Nv in eq. (2) would indeed lead to a g~ff smaller than 1. However, one might expect [6] that for spherical nuclei this effect will be smaller and g~fr closer to 1.0. This is contrary to the trends in table 1 where, even allowing for the experimental uncertainties, the g~ values for spherical nuclei (groups 1-3 ) are certainly not larger than those for deformed ones (groups 4-6) and may, in fact, be smaller. The reason for the anomalous values of g~, gv remains unclear. It is interesting to recall, however, that these values give an improved description [2] of the recently observed [ 7 ] B(M 1 ) strength of the isovector J~ = 1 ÷ states in certain rare earth nuclei. Table 1 and fig. 2 also show that g~ + & stays remarkably constant for all groups, at about 0.7. In fact, the weighted average ofg~ +gv is (& +&)a~e=0.68_+0.03,
0.4 08 0.2 0.1 I
I
I
I
I
I
'
8'0 i00 120 140 160 180 200 A Fig. 2. The effective boson g factors g~, gv versus mass numberA. 6'0
(3)
and again emphasizes the difference from the simplest expectation that g~ +gv= 1.0. An important feature of the results in table 1 and fig. 2 centers on the Er and Yb nuclei. The sharp drop in g~ and the near equality ofg~ and & for these nuclei is certainly surprising: there is no obvious reason why 21
Volume 190, number 1,2
PHYSICS LETTERS B
these deformed nuclei should behave differently than the neighboring B a - D y and H f - P t groups. In this connection it is interesting that Wesselborg et al. [ 8 ] and Richter [ 8 ] have recently discussed a sharp drop in B(M1 ) values to isovector 1 + states just in the Er isotopes. These B(M1 ) values are proportional to
(g~ -gO 2. As noted earlier, an alternative interpretation for the g(2 ~-) data can be constructed in terms of effective N~ and Nv values. It is interesting to discuss the Er-Yb anomaly in this latter context by assuming "normal" g~, g~ values, constant for the entire A = 150-200 region, and extracting effective N~, N~ values. In principle, one can deduce the effective number of either protons or neutrons from experimental g factors of 2 + states. This was done previously [2] for transitional Ce, Nd, and Sm isotopes. However, since this uses only one experimental quantity for each nucleus [i.e., g(2 ~-)], one can extract only N~fr or N~eft , assuming certain values for the other. For the Er-Yb nuclei we cannot say a priori whether the change in the number of valence particles is mainly a proton effect, a neutron effect, or both. Thus, a second experimental quantity is needed in order to extract both N~ ff and N f . B(E2) data are widely available and provide an appropriate observable for this purpose. In order to use this data, we need a relation similar to eq. (1), giving B(E2) as a function of N~, N~. However, no universal relation of this kind exists for the E2 operator. Analytical formulas exist [ 9 ] only for the three limiting symmetries [ i.e., SU ( 3 ), O (6), and U (5)]. Recently, an approximate analytical formula for calculating B(E2) values in deformed and transitional nuclei was proposed [ 10]: B(E2: 2~- --,0~+) = 0 . 2 5 ( 1 - 0 . 1 X ) 2 × [(Art + 1)/Nt]2(e~N~+e~N~) 2 ,
(4)
where e~ (e~) are the proton (neutron) effective charges and Z is the parameter of the quadrupole interaction. In the 0 ( 6 ) limit, X=0, while in the SU(3) limit X= - 1.32. Eq. (4) was shown [ 10] to provide B(E2) values accurate to ~ _+ 12% or better for - I < x < 0 and thus covers practically all deformed and transitional nuclei. In analogy to eq. (2), we now rewrite eq. (4): 22
2
Art
21 May 1987
x/B(E2:2 + ~ 0 + )
(N,+ 1)N~
(1-0.1Z)
=ev+e~NJNv.
(5)
We extracted e~, ev from experimental B(E2) values [ 11 ] in deformed Dy, Hf, W isotopes by fitting the left side of eq. (5) versus N J N v , in a fashion analogous to the procedure used for determining g~, gv- We used X = - 0 . 5 for all nuclei included in the fit. A similar procedure was used previously by other authors [ 9,12 ] to determine boson effective charges in the U ( 5 ) and 0 ( 6 ) limits. Here, we obtained the results e~ =0.20(2),
e~=0.07(2).
(6)
These values are very close to those deduced by Ginocchio and Van Isacker [9] for the 0 ( 6 ) limit, using data for Pd isotopes. The next step is to use eqs. (1) or (2) and ( 5 ), the experimental values of g ( 2 i~) and B ( E2:2 + --, 0 + ), and the deduced e~, ev, g~, gv constants, to extract N~ff and Nveff. The procedure will be described in more detail in a further publication [ 13 ]. For g~, g~ we use the average values of the B a - D y and H f - P t regions, namely g~ = 0.65, g~-- 0.08. The resulting N~fr, N f for the Er and Yb isotopes studied are plotted against neutron number in fig. 3. The error bars include only the experimental errors of B (E2) and g( 2 + ). The results suggest a reduction in the total number of effective valence particles, relative to that obtained in a normal counting scheme, especially in the number of neutrons for 17°Er and 170-176yboThis reduction stems from both the empirical g factors and the observed saturation of B(E2 : 2 + - , 0 + ) values in the middle of the rare-earth region. As discussed previously [ 14,15 ], the empirical B(E2) values do not scale with N 2 (or N~ or N~) but reach a maximum saturation value near Z = 6 8 and N = 94-96. There is a possible interpretation for this reduction, if it is recalled that the principal non-pairing residual interaction is the p - n interaction and if the reduced valence particle numbers are viewed as an effective reduction in the product N~Nv, that is, in the integrated proton-neutron interaction which determines the overall collectivity. The p - n interaction is largest for proton and neutron orbits of large
Volume 190, number 1,2
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16 N t ett 14 12 i
T~
I
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I
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i
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~
I 100
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~J
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916
9
N Fig. 3. The number of effective bosons N~fr, N~~, and NTfr= N~fr+N~a versus neutron number N, for the Er and Yb isotopes considered in this study. The solid lines show the numbers of particles or holes obtained by "normal" counting. spatial overlap. In spherical nuclei, this overlap depends only on the nlj shell model q u a n t u m numbers. In a deformed field, an important part of the p - n interaction is the quadrupole component. For this component, the p - n interaction for a specific pair o f orbits will be given by the product of the quadrupole moments o f the respeciive orbits. It is well known that the quadrupole moment depends strongly on the orientation of an orbit, that is, on the K value and, indeed, changes sign for prolate (low K) and oblate (high K) orbits. Thus, the integrated p - n interaction will clearly depend on the distribution o f K values in the occupied orbits. It will increase most rapidly with N~ and Nv when protons and neutrons are filling orbits of similar K values such as in the B a - D y region and will increase more slowly or even decrease when very different K values are included, as in the Er, Yb nuclei. In the latter case, a smaller interaction, within the framework o f the approach discussed earlier, would indeed manifest itself as a need for lower values of N~ and/or Nv. Although it is possible to construct a schematic model for the dependence o f the
21 May 1987
n e u t r o n - p r o t o n interaction on A K = K ~ - K v that yields a 30-40% reduction in the strength o f the interaction, a more realistic calculation of spatial overlaps as a function o f K~, Kv and o f the resulting changes in effective integrated p - n interaction strength compared with the simple estimate N~Nv, would be very useful. Such calculations are in progress. In conclusion, it has been shown that the g factors of 2+ states in a large number o f nuclei can be described in a simple way by eq. (1) and that g~, g~ values can be extracted that are constant within the empirical errors for certain mass groups. Despite the relatively large experimental errors, especially in the lighter nuclei studied, there seems to be an indication that the boson g factors have a specific mass dependence, namely, a tendency for g. to increase and g~ to decrease with A. There is no obvious distinction between spherial and deformed nuclei. Large deviations from this behavior occur only in the immediate vicinity of closed shells and in the E r - Y b region. This latter anomaly may be interpreted in terms o f a change in the effective number o f valence particles by virtue o f the K dependence of the p - n interaction in the deformed field. More accurate experimental results, especially in the 70-120 mass region, are needed to further study the g~, gv systematics in spherical nuclei and to reveal possible fine structure in the mass dependence ofg~, gv. Research was performed under contract DE-AC0276CH00016 with the US Department o f Energy. The authors thank S. R a m a n for making available the compilation o f B ( E 2 ) data prior to publication. Discussions with A.E.L. Dieperink, B. Barrett, and P. von Brentano are gratefully acknowledged.
References [ 1] M. Sambataro and A.E.L. Dieperink, Phys. Lett. B 107 (1981) 249. [2] A. Wolf, D.D. Warner and N. Benczer-Koller, Phys. Lett. B 158 (1985) 7. [ 3 ] C.M. Lederer and V.S. Shirley, eds., Table of isotopes, 7th Ed. [4] A.E. Stuchbery et al., Nucl. Phys. A 435 (1985) 635. [5] P. Van Isacker et al., Ann. Phys. (NY) 171 (1986) 253. [6] A.E.L. Dieperink, in: Proc. Intern. Conf. on Nuclear structure, reactions and symmetries (Dubrovnik, June 1986), 23
Volume 190, number 1,2
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eds. R.A. Meyer and V. Paar, to be published; and private communication. [7] D. Bohle et al., Phys. Lett. B 148 (1984) 260. [ 8 ] K. Wesselborg et al., Contribution to the Intern. Conf. on Nuclear physics (Harrogate, August 1986 ); A. Richter, Nuclear Structure, 1985, Niels Bohr Centennial Conf., eds. R. Broglia, G. Hagemann and B. Herskind (North-Holland, Amsterdam, 1985) p. 469. [9] J.N. Ginoechio and P. Van Isacker, Phys. Rev. C 33 (1986) 365.
24
21 May 1987
[ 10] R.F. Casten and A. Wolf, to be published. [ 11 ] S. Raman et al., At. Data Nucl. Data Tables, to be published. [ 12] I. Morrison, Phys. Rev. C 23 (1981) 1831; W.D. Hamilton, A. Irback and J.P. Elliott, Phys. Rev. Lett. 53 (1984) 2469. [ 13] A. Wolf and R.F. Casten, to be published. [ 14 ] D.D. Warner and R.F. Casten, Phys. Rev. C 28 ( 1983 ) 1798. [ 15 ] R.F. Casten, W. Frank and P. von Brentano, Nucl. Phys. A444 (1985) 133.
PHYSICS LETTERS B
21 May 1987
g F A C T O R S I N HEAVY N U C L E I A N D T H E P R O T O N - N E U T R O N I N T E R A C T I O N A. W O L F Nuclear Research Centre-Negev, POB 9001, 84120 Beer Sheva, Israel and Brookhaven National Laboratory, Upton, NY 11973, USA R . F . CASTEN and D.D. W A R N E R t Brookhaven National Laboratory, Upton, NYl1973, USA
Received 12 January 1987; revised manuscript received 2 March 1987
It is shown that the experimental values of g(2 i~) for nuclei in the mass range A = 70-200 are well described within IBA-2using a simple linear relationship. Values ofg~, gv, the effective proton, neutron boson g factors, were extracted for six groups of nuclei in this mass range. The results indicate that g~ tends to increase, and gv to decrease with A. No obvious distinction is observed between spherical and deformed nuclei. A large deviation in the systematics is observed for Er, Yb isotopes. This anomaly is explained in terms of a change in the effective number of valence particles, due to the K dependence of the p-n interaction in the deformed field.
Magnetic properties o f nuclear states are effective probes of nuclear wave functions. In nuclei near closed shells, g factors are mainly determined by single-particle motion and configuration mixing. In deformed nuclei, the collective motion of the protons and neutrons is responsible for the g-factor values. A systematic analysis o f g factors of excited nuclear states is therefore expected to provide valuable information regarding the structure o f the respective states. Although spherical and deformed nuclei are quite different in m a n y aspects, the g factors o f low-lying states in these nuclei can be expressed in a simple way using the n e u t r o n - p r o t o n version of the interacting boson approximation, IBA-2. This is particularly true for 2~- states in even-even (spherical or deformed) nuclei. Assuming that these states are completely symmetrical in the proton and neutron degrees o f freedom (i.e., F-spin is a good q u a n t u m number), it has been shown [ 1 ] that g ( 2 ~ ) is given by g(2 + ) =g~N~/Nt + g = N J N t ,
(1)
Permanent address: Daresbury Laboratory, Daresbury, Warrington WA4 4AD, UK. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
where g~ (gv) are proton (neutron) boson g factors, N~ (Nv) are proton (neutron) boson numbers and N t = N ~ +Nv. Eq. (1) can be rewritten: ( N~/N~)g( 2 ~-) =g~+ g,~N,~/N~ .
(2)
Thus a plot o f the left side o f this equation against N J N v , using empirical g(2~-) values, should be a straight line with intercept gv and slope g~. Deviations from a straight line can be ascribed to non-constancy of gv and g= which would be expected if the underlying microscopic structure o f the correlated s and d pairs is changing dramatically in a given region due to, for example, subshell effects. Alternatively, it has recently been shown [ 2 ] that such effects can also be treated in terms o f effective N~, Nv values which differ from those normally assumed by counting valence particles (or holes) from the nearest major shell closures. In this approach, therefore, subshell effects are accounted for by an assumed change in the number o f nucleons taking part in the collective motion. The specific example used previously involved g ( 2 ~ ) values in the A = 150 region where it was demonstrated that it is possible to use constant values ofg~, g~ for a large number o f deformed nuclei around A = 150 and to extract effective values of N~, 19
Volume 190, number 1,2
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N~ for nuclei in which the dissipation o f the Z = 64 subshell causes a significant change in the n u m b e r of effective valence protons for different isotopes o f the same element. The purpose o f the present work is to extend this type o f treatment to study the behavior ofg~, gv and N~, N~ in other regions o f the periodic table, to compare the results for spherical and deformed nuclei, and to discuss an interpretation o f apparent variations in g~, gv for the latter in terms o f a reduced effective proton-neutron interaction. Included in our study are 65 nuclei in several mass regions from A = 70-200, for which experimental g(2 3 ) values are available. Most data values are weighted averages from the Table of Isotopes [ 3] and, in some cases, are adopted values from the most recent Nuclear Data Sheets. Some recent g-factor measurements [ 4] were also included. Recently, a fit o f g(2 ~ ) values using eq. (1) was reported by Van Isacker et al. [ 5 ] for all isotopes between Ba and Pt. Their results are consistent with the values o f g~, g~ found in the A = 150 region [2] and with some o f the results which will be presented here. In order not to obscure possible gross variations o f g~, gv with mass n u m b e r or effects due to changing from particles to holes, the nuclei studied were divided in the present work into several groups, following the major proton and neutron shells. A given group only contains nuclei with proton and neutron numbers in specific half major shells (i.e., above or below the midshell point). Nuclei in the immediate vicinity of closed shells (like the Te and Hg isotopes) were not included since their g factors might be dominated by non-collective effects. The various groups
21 May 1987
are listed in table 1. This division is to some extent arbitrary, and is by no means unique. However, the quality o f the fits for the different groups and the relatively large n u m b e r o f nuclei included in each fit suggest that the conclusions o f this analysis are not strongly affected by the particular division used. For the discussion below it is also useful to classify the groups according to the degree of deformation in the nuclei concerned. As indicated in the table we used a rough criterion that groups in which most nuclei have E ( 4 ~-)/E(2 ? ) >i 2.7 are labelled deformed, and other groups as spherical. For each group we performed a linear least squares fit o f g(2 ~-)Nt/Nv versus NJNv, and extracted g~, gv. As an example, the fits for the Ge-Se, Xe-Ba, E r - Y b and H f - P t regions are presented graphically in fig. 1. The resulting g~, gv values are given in table 1 together with the respective X 2 / ( n - 2 ) values, where n is the number o f nuclei in each group. In cases where the normalized X2 value is > 1, the errors ofg~, g~ have been multiplied by x / X 2 / ( n - 2 ) , to account for the quality o f the fit. The poor accuracy o f the extracted g~, gv for group 1 is due to the low accuracy of the experimental data. Nevertheless, from table 1 and fig. 1, it is clear that the linear relation o f eq. (2) describes reasonably well the experimental data for each group. The rather large X2/( n - 2) for the H f - P t group is due to the small experimental errors, and it may signify that there is some "fine structure" in the A or Z dependence o f g~, gv. More experimental data is required to clarify this point. In fig. 2 we plotted the values orgy, g~ against the average mass number A in each group. The results in fig. 2 and table 1 suggest the general feature that g~
Table 1 The effective boson g factors for various groups of nuclei. Group no.
Isotopes
Number of nuclei n
Type of nuclei a)
g~
gv
z2/n-- 2
g~ + gv
1
0.30(16) 0.48(12)
0.55(14) 0.33(5)
sph.
0.50(5)
0.16(4)
6
176Hf-19spt
8 14 9 11 8 15
sph. sph.
3 4 5
7°Ge-S2Se 98Mo-I14Cd t24:Xe-lS6Ba 14~Ba-164Dy 164Er-176yb
def. def. def. b)
0.63 ( 5 ) 0.30(4) 0.68(7)
0.05 ( 5 ) 0.35(3) 0.12(3)
1.0 0.5 1.4 1.0 3.0 5.0
0.85(21 ) 0.81 (13) 0.66(6) 0.68 (7) 0.65(5) 0.80(8)
2
") Sph.: spherical, E( 4 ~-)/E( 2 ~-) < 2.7; def.: deformed, E( 4 3 )/E( 2 ~ ) > 2.7. b) The four Pt isotopes inlcuded in this grouP have E(4 3 )/E(2 3 ) = 25. 20
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1'o 1; 21o 2; 31o N=/N~
Fi:;. 1. g ( 2 ~ )~=p N,IN~ versus N=IN~for: (a) Xe-Ba, (b) Ge-Se, (c) Er-Yb, and (d) Hf-Pt. The straight lines are least squares fits. Where not plotted, errors are o f the size of or smaller than the points.
tends to increase with A, while g~ is decreasing. The major exception to this trend is the Er-Yb group which will be discussed in detail below. In the simplest description of boson g factors, in which only the orbital contributions from the fermions are considered, one expects g~= 1.0 and gv=0.0. The present analysis clearly shows that this is not the case. While it is possible that part of the discrepancy is due to spin contributions, Dieperink [ 6] has pointed out that there is a contribution from the core which effectively increases the size of the valence space. If this argument is correct, our use of 0.8 0.70.6
& 0.5
the normal N~, Nv in eq. (2) would indeed lead to a g~ff smaller than 1. However, one might expect [6] that for spherical nuclei this effect will be smaller and g~fr closer to 1.0. This is contrary to the trends in table 1 where, even allowing for the experimental uncertainties, the g~ values for spherical nuclei (groups 1-3 ) are certainly not larger than those for deformed ones (groups 4-6) and may, in fact, be smaller. The reason for the anomalous values of g~, gv remains unclear. It is interesting to recall, however, that these values give an improved description [2] of the recently observed [ 7 ] B(M 1 ) strength of the isovector J~ = 1 ÷ states in certain rare earth nuclei. Table 1 and fig. 2 also show that g~ + & stays remarkably constant for all groups, at about 0.7. In fact, the weighted average ofg~ +gv is (& +&)a~e=0.68_+0.03,
0.4 08 0.2 0.1 I
I
I
I
I
I
'
8'0 i00 120 140 160 180 200 A Fig. 2. The effective boson g factors g~, gv versus mass numberA. 6'0
(3)
and again emphasizes the difference from the simplest expectation that g~ +gv= 1.0. An important feature of the results in table 1 and fig. 2 centers on the Er and Yb nuclei. The sharp drop in g~ and the near equality ofg~ and & for these nuclei is certainly surprising: there is no obvious reason why 21
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PHYSICS LETTERS B
these deformed nuclei should behave differently than the neighboring B a - D y and H f - P t groups. In this connection it is interesting that Wesselborg et al. [ 8 ] and Richter [ 8 ] have recently discussed a sharp drop in B(M1 ) values to isovector 1 + states just in the Er isotopes. These B(M1 ) values are proportional to
(g~ -gO 2. As noted earlier, an alternative interpretation for the g(2 ~-) data can be constructed in terms of effective N~ and Nv values. It is interesting to discuss the Er-Yb anomaly in this latter context by assuming "normal" g~, g~ values, constant for the entire A = 150-200 region, and extracting effective N~, N~ values. In principle, one can deduce the effective number of either protons or neutrons from experimental g factors of 2 + states. This was done previously [2] for transitional Ce, Nd, and Sm isotopes. However, since this uses only one experimental quantity for each nucleus [i.e., g(2 ~-)], one can extract only N~fr or N~eft , assuming certain values for the other. For the Er-Yb nuclei we cannot say a priori whether the change in the number of valence particles is mainly a proton effect, a neutron effect, or both. Thus, a second experimental quantity is needed in order to extract both N~ ff and N f . B(E2) data are widely available and provide an appropriate observable for this purpose. In order to use this data, we need a relation similar to eq. (1), giving B(E2) as a function of N~, N~. However, no universal relation of this kind exists for the E2 operator. Analytical formulas exist [ 9 ] only for the three limiting symmetries [ i.e., SU ( 3 ), O (6), and U (5)]. Recently, an approximate analytical formula for calculating B(E2) values in deformed and transitional nuclei was proposed [ 10]: B(E2: 2~- --,0~+) = 0 . 2 5 ( 1 - 0 . 1 X ) 2 × [(Art + 1)/Nt]2(e~N~+e~N~) 2 ,
(4)
where e~ (e~) are the proton (neutron) effective charges and Z is the parameter of the quadrupole interaction. In the 0 ( 6 ) limit, X=0, while in the SU(3) limit X= - 1.32. Eq. (4) was shown [ 10] to provide B(E2) values accurate to ~ _+ 12% or better for - I < x < 0 and thus covers practically all deformed and transitional nuclei. In analogy to eq. (2), we now rewrite eq. (4): 22
2
Art
21 May 1987
x/B(E2:2 + ~ 0 + )
(N,+ 1)N~
(1-0.1Z)
=ev+e~NJNv.
(5)
We extracted e~, ev from experimental B(E2) values [ 11 ] in deformed Dy, Hf, W isotopes by fitting the left side of eq. (5) versus N J N v , in a fashion analogous to the procedure used for determining g~, gv- We used X = - 0 . 5 for all nuclei included in the fit. A similar procedure was used previously by other authors [ 9,12 ] to determine boson effective charges in the U ( 5 ) and 0 ( 6 ) limits. Here, we obtained the results e~ =0.20(2),
e~=0.07(2).
(6)
These values are very close to those deduced by Ginocchio and Van Isacker [9] for the 0 ( 6 ) limit, using data for Pd isotopes. The next step is to use eqs. (1) or (2) and ( 5 ), the experimental values of g ( 2 i~) and B ( E2:2 + --, 0 + ), and the deduced e~, ev, g~, gv constants, to extract N~ff and Nveff. The procedure will be described in more detail in a further publication [ 13 ]. For g~, g~ we use the average values of the B a - D y and H f - P t regions, namely g~ = 0.65, g~-- 0.08. The resulting N~fr, N f for the Er and Yb isotopes studied are plotted against neutron number in fig. 3. The error bars include only the experimental errors of B (E2) and g( 2 + ). The results suggest a reduction in the total number of effective valence particles, relative to that obtained in a normal counting scheme, especially in the number of neutrons for 17°Er and 170-176yboThis reduction stems from both the empirical g factors and the observed saturation of B(E2 : 2 + - , 0 + ) values in the middle of the rare-earth region. As discussed previously [ 14,15 ], the empirical B(E2) values do not scale with N 2 (or N~ or N~) but reach a maximum saturation value near Z = 6 8 and N = 94-96. There is a possible interpretation for this reduction, if it is recalled that the principal non-pairing residual interaction is the p - n interaction and if the reduced valence particle numbers are viewed as an effective reduction in the product N~Nv, that is, in the integrated proton-neutron interaction which determines the overall collectivity. The p - n interaction is largest for proton and neutron orbits of large
Volume 190, number 1,2
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16 N t ett 14 12 i
T~
I
I-
I
I
I
[
i
i
i
i
o
~
,
•
--Er iyb
~
I 100
I 102
r 104
I 106
lo-J N~ elf
8-
~J
6-
i
N . err 6-
916
9
N Fig. 3. The number of effective bosons N~fr, N~~, and NTfr= N~fr+N~a versus neutron number N, for the Er and Yb isotopes considered in this study. The solid lines show the numbers of particles or holes obtained by "normal" counting. spatial overlap. In spherical nuclei, this overlap depends only on the nlj shell model q u a n t u m numbers. In a deformed field, an important part of the p - n interaction is the quadrupole component. For this component, the p - n interaction for a specific pair o f orbits will be given by the product of the quadrupole moments o f the respeciive orbits. It is well known that the quadrupole moment depends strongly on the orientation of an orbit, that is, on the K value and, indeed, changes sign for prolate (low K) and oblate (high K) orbits. Thus, the integrated p - n interaction will clearly depend on the distribution o f K values in the occupied orbits. It will increase most rapidly with N~ and Nv when protons and neutrons are filling orbits of similar K values such as in the B a - D y region and will increase more slowly or even decrease when very different K values are included, as in the Er, Yb nuclei. In the latter case, a smaller interaction, within the framework o f the approach discussed earlier, would indeed manifest itself as a need for lower values of N~ and/or Nv. Although it is possible to construct a schematic model for the dependence o f the
21 May 1987
n e u t r o n - p r o t o n interaction on A K = K ~ - K v that yields a 30-40% reduction in the strength o f the interaction, a more realistic calculation of spatial overlaps as a function o f K~, Kv and o f the resulting changes in effective integrated p - n interaction strength compared with the simple estimate N~Nv, would be very useful. Such calculations are in progress. In conclusion, it has been shown that the g factors of 2+ states in a large number o f nuclei can be described in a simple way by eq. (1) and that g~, g~ values can be extracted that are constant within the empirical errors for certain mass groups. Despite the relatively large experimental errors, especially in the lighter nuclei studied, there seems to be an indication that the boson g factors have a specific mass dependence, namely, a tendency for g. to increase and g~ to decrease with A. There is no obvious distinction between spherial and deformed nuclei. Large deviations from this behavior occur only in the immediate vicinity of closed shells and in the E r - Y b region. This latter anomaly may be interpreted in terms o f a change in the effective number o f valence particles by virtue o f the K dependence of the p - n interaction in the deformed field. More accurate experimental results, especially in the 70-120 mass region, are needed to further study the g~, gv systematics in spherical nuclei and to reveal possible fine structure in the mass dependence ofg~, gv. Research was performed under contract DE-AC0276CH00016 with the US Department o f Energy. The authors thank S. R a m a n for making available the compilation o f B ( E 2 ) data prior to publication. Discussions with A.E.L. Dieperink, B. Barrett, and P. von Brentano are gratefully acknowledged.
References [ 1] M. Sambataro and A.E.L. Dieperink, Phys. Lett. B 107 (1981) 249. [2] A. Wolf, D.D. Warner and N. Benczer-Koller, Phys. Lett. B 158 (1985) 7. [ 3 ] C.M. Lederer and V.S. Shirley, eds., Table of isotopes, 7th Ed. [4] A.E. Stuchbery et al., Nucl. Phys. A 435 (1985) 635. [5] P. Van Isacker et al., Ann. Phys. (NY) 171 (1986) 253. [6] A.E.L. Dieperink, in: Proc. Intern. Conf. on Nuclear structure, reactions and symmetries (Dubrovnik, June 1986), 23
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eds. R.A. Meyer and V. Paar, to be published; and private communication. [7] D. Bohle et al., Phys. Lett. B 148 (1984) 260. [ 8 ] K. Wesselborg et al., Contribution to the Intern. Conf. on Nuclear physics (Harrogate, August 1986 ); A. Richter, Nuclear Structure, 1985, Niels Bohr Centennial Conf., eds. R. Broglia, G. Hagemann and B. Herskind (North-Holland, Amsterdam, 1985) p. 469. [9] J.N. Ginoechio and P. Van Isacker, Phys. Rev. C 33 (1986) 365.
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[ 10] R.F. Casten and A. Wolf, to be published. [ 11 ] S. Raman et al., At. Data Nucl. Data Tables, to be published. [ 12] I. Morrison, Phys. Rev. C 23 (1981) 1831; W.D. Hamilton, A. Irback and J.P. Elliott, Phys. Rev. Lett. 53 (1984) 2469. [ 13] A. Wolf and R.F. Casten, to be published. [ 14 ] D.D. Warner and R.F. Casten, Phys. Rev. C 28 ( 1983 ) 1798. [ 15 ] R.F. Casten, W. Frank and P. von Brentano, Nucl. Phys. A444 (1985) 133.