A simple approach to nuclear transition regions

Volume 152B, number

PHYSICS

3,4

7 March 1985

LETTERS

A SIMPLE APPROACH TO NUCLEAR TRANSITION

REGIONS

R.F. CASTEN Brookhaven National Laboratory, Upton, NY, USA and Institut fiir Kernphysik, Universitri’t K61n. Cologne, Fed. Rep. Germany Received 28 November 1984 Revised manuscript received 26 December

1984

A scheme for the interpretation of extended nuclear transition regions is introduced which is based on the importance of the neutron-proton interaction among valence nucleons. It substantially simplifies both the empirical parametrization and theoretical calculation of such regions. Data from near A - 100, -130, and -150 are discussed. In the first and last cases, information on the valence proton shell structure is extracted. In the A - 130 region an IBA calculation is presented which exploits the proposed method.

mass region on the left in figs. 1 and 2. The Eq:/E2; ratio has the limiting value of 2 for a quadrupole vibrator, 2.5 for a non-axial or y-soft rotor and 3.33 for an ideal symmetric rotor. For this same sequence, E2; is expected to drop continuously from near closed shell values of >l MeV toward the very small rotational values of the deformed rotor (loo-200

Nuclear transition regions have long been considered the most complex and challenging of all nuclear regions and have provided the favorite, albeit most stringent, testing ground for nuclear models. As an example of the data characterizing such a region two measures of the empirical structure, the energy ratio E4;/E2; and E2; are plotted for the A - 130

3.0 _

E+ -i

2.8-

E*+

1

2.6 -

1.8

I

I

I

I

I

I

I

I

I

I

I

80

78

76

7L

72

70

68

66

6L

0

L

N Fig. 1. Systematics of the energy ratio E4: /E2; product of proton and neutron boson numbers K = 0.025, K’ = -0.005 MeV and CY= 0.045, p = The solid lines on the right here and in figs. 3,4

I 8

I

I

I

I

I

I

12

16

20

2L

28

32

N;

Nk

for the A - 130 region plotted against neutron number (left) and against the (right). The dashed line is an IBA calculation described in the text with eo = 0.98, 0.0135, No = 13. The data are from refs. [l-3] and references contained therein. are drawn just to guide the eye.

0370-2693/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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PHYSICS LETTERS

800

_

600 R

01.

.

.

I

I

,

I

&

II.

.

80 78 76 7L 72 70 68 66

Ce

Nd

8

12

16

I

I

I.

20

2L

28

32 36

NIT’ NV

region.

keV). While the data plotted in figs. 1 and 2 clearly reflect this general trend, it is likewise evident that there is both a strong N and a strong Z dependence: that is, each element passes through the transition region differently. This is true, indeed to an even greater extent, for the A - 100 and A - 150 spherical-deformed transition regions as shown on the left

Ba

L

‘IBA-1

I1

I.

0

N Fig. 2. Systematics of E2+ for the A refs.

\x

Sm Gd

Dy

Er

in are

points dispersed throughout the It is the purpose of this to propose an approach proton neutron force,

Yb

I

I 96_

E,* I

2.8-

E2’

1

56

58

60

62

6L

66

68

70

2 Ed+/Ez+ for the A - 1.50 region. See caption to fig. 1. The fermion right. For simplicity, nuclei below mid-proton right. Those for 2 > 66 lie on a separate, nearly curve.

146

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LETTERS

Pd

1/

E

L*

1

E2:

b

.

0

+

,r +o

A

H

A

0

l

0

\

N=60

0

+

0

0

1.8_ 1.6_

I 38

LO

L2

I

I

LL

L6

Z

0

L

8

Fig. 4. Systematics of Ed; /Ez+l for the A - 100 region.

much simpler theoretical calculations. The importance of the proton-neutron interaction in nuclear physics has long been emphazised, in particular by Talmi [4] and also by Federman and Pittel [5,6] who explained the sudden onset of deformation near A = 100 as a consequence of such interactions. Subsequently, the ideas of refs. [5,6] were used [7] to re-interpret the,4 = 150 region in light of Z = 64 proton shell closure. The present suggestion builds on these and related [8] ideas. If the evolution of nuclear structure with mass is primarily a consequence of the proton-neutron interaction, then, largely independent of the detailed nature of the forces involved [9], it is clear that their average strength must be related to the number of neutrons and protons actively involved. It is reasonable to suppose that these are the respective numbers of valence (extra closed shell) protons and neutrons (or holes past midshell). It seems likely then that the product Np *Nn of these valence nucleon numbers will be a qualitative measure of the proton-neutron interaction strength. This would be the case, for example, for a proton neutron interaction of quadrupole type. In the language of the IBA [IO], the product Np *N,, is identical (except for a constant factor of 4) to the product N, *N,, of the proton and neutron boson numbers.

12

16

20

%r*

N,

See caption

2L

26

32

to fig. 1.

Accordingly, the data on the left in figs. 1 and 2 are now replotted on the right, against N,*NV instead of N or Z. Although this parametrization is extremely simple its effect is dramatic. Instead of a number of separate curves there is now a simple functional dependence on the single quantity N, *NV. It is worth noting that, while the proton-neutron interaction may well scale approximately linearly with N, -NV, empirical observables may exhibit a different functional dependence. For example, both E4+, /E2’, and E2+, may change rapidly for low N,, . N, but are expected to flatten out as they approach the asymptotic values of the rotor. (Incidentally, similar plots against total boson number NB = N,, + N,, reflect some of the same simplification but not to the same degree as the N;NV plots. Presumably, and not surprisingly, the product N, *NV(or some function thereof) better simulates the average behaviour of the proton neutron force than the total boson number N, .) Thus, a sequence of nuclei, of different A, N, or 2, but the same product N, -NV (or, to a lesser approximation, the same NB) display very similar structure and spectra. In fig. 1, the solid curve on the right is drawn to guide the eye through a simple evolution from vibrational-like [SU(5)] nuclei near N = 80, through a y-soft [O(6)] region [ 1 ,11,12] near A - 130, towards 147

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a deformed region [3] in the lightest nuclei (largest N,, *NV). Note that the curve is almost linear in N, -NV until the asymptotic limit of 3.33 is approached. Figs. 3,4 show similar plots for the A - 100 and -150 regions. It is important to understand how the right-hand sides of these figures have been constructed. The proton-neutron interaction is strongest [.5,6,8] when the shell model orbits involved are socalled spin orbit partners, that is nl(j = 1 - l/2) and nl(‘j = 1 + l/2). Thus, when neutrons begin to fill the g7j2 orbit near N = 60 in the Sr-Zr isotopes, the attractive proton-neutron interaction effectively lowers the gg/2 proton single particle energy, making it energetically favourable [5,6] to elevate protons from beneath the closed shell at Z = 38 into this orbit and thus eliminating the Z = 38 gap at N = 60. Similarly, in the rare earth region, the Z = 64 gap disappears when neutrons begin filling the hg12 orbit near N = 90 [7]. Thus, in fig. 3, the valence proton shell is taken as Z = 38-50 for N < 60, while, for N > 60, Z = 28-50 is used instead. For example for Zr , with Z = 40, there are two valence protons (one boson) for N < 60, but 10 (5 bosons) for N > 60. Relative to a prescription that maintained the Z = 38 closure, this shifts the N > 60 data points of nuclei with Z < 42 far to the right. Similarly, in the Z = 64 region, a Z = 50-64 proton shell is used for N < 90, while the full Z = 50-82 shell applies for N > 90. Thus, Sm, with Z = 62, has one proton (hole) boson forN < 90,but 6 proton (particle) bosons for N > 90. These criteria are essential to producing the smooth data systematics of figs. 3,4. With these considerations one notices in figs. 3,4 that, for the most part, as in the A - 130 region, the data now lie on well defined universal curves spanning the region from vibrator toward rotor. These curves rise sharply for low N, *NVas the nuclear structure changes rapidly and then flatten out to approach asymptotically the limit rotor value of 3.33. The only significant exceptions occur at N = 60 and N = 90 where the points deviate systematically from the curve. The explanation for the latter behaviour seems clear: the assumption of a radically sharp change in shell structure precisely at these two neutron numbers is a, not unexpected, oversimplification. Analogous to the procedure used in ref. [ 131, one can exploit the otherwise smooth character of the curves to reverse the technique applied so far, by demanding 148

7 March 1985

LETTERS

that the N = 60 and N = 90 data fall on the now well established systematics (i.e., by shifting these data points leftward) and deducing the effective proton boson numbers required to do this. The results are shown in fig. 5 where the earlier definitions of proton boson numbers for N # 60 and N # 90 are connected to each other through the empirically determined values at N = 60 and N = 90. They indicate that the Z = 38 and N = 64 shell gaps remain partially intact at N = 60 and N = 90. It is interesting that this qualitative result, based solely on energy ratios, gives a picture of the breakup of the z = 64 shell very similar to that obtained from the g-factors used in [ 131. The main difference is that, here, the proton shell breakup seems to occur for slightly larger neutron numbers. Recent microscopic calculations of Scholten [ 141 also predict a neutron number dependence of N,, albeit more gradual than suggested by the data here and in ref. [13]. The results of figs. l-4 also suggest the possibility of a simplified approach to the calculation of nuclear transition regions in the context of the IBA. This model is characterized by a group structure based on

5mil_

NJ 2

1

0

56

56

60

62

61

66

90

92

9L

N

0 7

6

NTri 2 1

0 04

66

08

N Fig. 5. Effective proton-boson numbers in the transition regions near N = 60 and N = 90 extracted from the data of figs. 3,4.

the parent group, U(6), which leads to three dynamical symmetries commonly denoted U(S), SU(3), and O(6). These can be schematically represented in terms of the symmetry triangle of fig. 6, where each vertex denotes one of the symmetries, and the legs represent transition regions between symmetries. A convenient IBA- 1 hamiltonian [ 151 describing this symmetry structure is H= ennd -

KQ*Q

-

K’L*L

where Q = (s+z + d+s) + (x/fi)(d+d”)(*)

,

and where e, K, K ’ and x are parameters. The U(5) limit is obtained when E dominates (E/K + -). The SU(3) and O(6) symmetries correspond to x = a/2 = -2.958 and 0, respectively (with K finite and E = 0). Complex transition regions are calculated very easily in this scheme. Changes in nuclear structure along one of the legs of the symmetry triangle just involve the variation of a single parameter. Thus, in an O(6) + SU(3) transition region (e.g., Pt-0s) x varies from 0 toward -2.958 and, in a U(5) + SU(3) region (e.g., Sm, Cd), E/K decreases from some finite value

towards 0. (Of course, to fit excitation energies, one needs also to specify each parameter separately.) Such nuclear transition regions have been calculated before (e.g., refs. [ 16,17]), but, invariably, with separate parameter variations for each element. The present results suggest a simpler approach in which the parameters are varied globally for a region and as a function only of N, ‘N,. To test this in an even further simplified way, a highly schematic calculation for the A - 130 region has been carried out in which K and K’ are actually held constant and x and E are forced to vary linearly, according to the expressions c = eO - odv, *N, and x = -p(N;N, - No) (with E, -x > 0). Since the A - 130 nuclei in fact pass through a double transition region from U(5) + O(6) + SU(3), this should be a particularly stringent test. The parameter scheme is indicated in fig. 6, where, for low N;N,, the sharply dropping values of E/K induce the first leg of this transition while, for large N;N,, the increasing x values (coupled with a now small e) generate the second leg. Some of the results are shown in figs. 1 and 2 where the agreement with the data (also for the 2; levels which are not shown) is excellent, particularly in view of the constraints in the calculation. The systematics and even the absolute excitation energies for a double transition region comprising - 32 nuclei is reproduced by a calculation specified by Only six constants (co, K , K ‘, a,

SU(3)

U(5)

E’xxx 20

10

P,

and

NoI.

To summarize, the N;N, construction, though trivial in concept and application, may be a very useful aid in the unified treatment of the changes in nuclear structure over extended regions, by providing a more economical classification based directly on a simple and plausible physical mechanism.

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Nr;N, Fig. 6. Schematic description of the double transition region nearA - 130 in terms of the IBA symmetry triangle (top) and of the parameter determination in the schematic calculations described in the text (bottom).

I am grateful for numerous fruitful discussions with S. Pittel, I. Talmi, D.D. Warner, K. Heyde, A. Aprahamian, P. von Brentano, and A. Gelberg. Work supported by US Department of Energy, Contract NO. DE-AC 02-76CH00016, by the Von-HumboldtFoundation and by the BMFT. References

[ 1I RF. Casten and P. von Brentano, to be published. [*I L.M. Lederer and V. Shirley, eds., Table of Isotopes, 7thEd. (Wiley, New York, 1978). 149

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[3] B.J. Varley et al., Contrib. 5th Intern. Symp. on Capture gamma-ray spectroscopy and related topics (Knoxville, TN, September 1984). [4] 1. Talmi, in: Interacting bosons in nuclei, ed. F. Iachello (Plenum, New York, 1979) p. 79. [S] P. Federman and S. Pittel, Phys. Lett. 69B (1977) 385; 77B (1978) 29; Phys. Rev. C20 (1979) 820. [6] P. Federman, S. Pittel and R. Campos, Phys. Lett. 82B (1979) 9. [7] R.F. Casten, D.D. Warner, D.S. Brenner, and R.L. Gill, Phys. Rev. Lett. 20 (1981) 1433. [B] G. Scharff-Goldhaber, C.B. Dover, and A.L. Goodman, Ann. Rev. Nucl. Sci 26 (1976) 239. [9] A. de Shalit and M. Goldhaber, Phys. Rev. 92 (1953) 1211. [lo] A. Arima and F. Iachello, Ann. Phys. (NY) 99 (1976) 253; 111 (1978) 201; 123 (1979) 468.

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LETTERS [ 1 l] R.A. Meyer et al., Phys. Rev. Cl4 (1976) B.P. Pathak et al., Phys. Rev. Cl4 (1976) [ 121 G.

2024; 1573.

[ 131 A. Wolf and D.D. Warner, to be published; see also, A. Wolf et al., Phys. Lett.

123B (1983)

165.

[ 151 D.D. Warner and R.F. Casten, Phys. Rev. Lett. 48 (1982)

1385.