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Towards a unified microscopic description of nuclear deformation
Volume 69B, number 4
TOWARDS
A UNIFIED
PHYSICS LETTERS
MICROSCOPIC
DESCRIPTION
29 August 1917
OF NUCLEAR
DEFORMATION
P. FEDERMAN IFUNAM, Ap. Postal 20-364, M&co
20, D.F.
and S. PITTEL Bartol Research Foundation
of The Franklin Institute, Swarthmore,
Pennsylvania 19081,
USA
Received 24 May 1977 Nuclear deformation, as it occurs in both light and heavy nuclei, is discussed in a unified microscopic shell-model framework. The short-range 3S1 neutron-proton interaction plays an important role in this discussion.
The concept of deformation has been used extensively for over 25 years to describe “collective” properties of heavy nuclei [ I]. And yet very little is known about the microscopic structure of deformation in these nuclei. Since a short-range pairing force favors sphericity whereas a long-range quadrupole force favors deformation, it is generally accepted that deformation in neutron-rich heavy nuclei arises from the long-range (field-producing) part of the effective nuclear force [2]. But such a picture is not really microscopic, in the sense of relating the properties of deformed nuclei to the properties of the underlying nuclear force. Whereas the short-range pairing force reflects the ‘S, component of the nuclear force, the microscopic basis of the long-range quadrupole force is not nearly as clear. Unfortunately, any further microscopic insight into the structure of deformation in heavy nuclei (e.g., the insight that would accompany a shell-model study) is severely hindered by the complexity of the problem. It is therefore very convenient that deformation was observed in a region of light nuclei, namely in the (2sld) shell, where microscopic shell-model investigation is feasible. But here a seemingly different picture emerged. In light nuclei, deformation seems to arise from the short-range neutron-proton interaction [3]. In this Letter we show that it is not necessary to invoke different pictures to discuss deformation in light and heavy nuclei. More specifically, we show that the same microscopic shell-model picture that
has been used to discuss deformation in light nuclei can also be used in the heavier deformed regions. The relatively recent observation by Cheifetz et al. [4] of a new region of deformed nuclei is crucial to this work. By studying the fission fragments of 252Cf, they observed rotational spectra in several neutronrich isotopes of Zr and MO [see fig. I] as well as Ru and Pd. The observed B(E2) values were as enhanced as in the rare-earth and actinide regions. Like the actinide and rare-earth deformed nuclei these nuclei have large neutron excesses. On the other hand, like the deformed nuclei in the (2sId) shell, they can be conveniently discussed in a microscopic shell-model framework. As we shall see, this new region of deformed nuclei provides a kind of missing link between the light and heavy regions of deformation. We first review briefly what has been learned about deformation from the (2s-ld)-shell nuclei. The nucleus 2oNe is a classic example of a rotational nucleus in the (2sId) shell. It consists of two neutrons and two protons outside the doubly magic 160 core. It is instructive to compare the spectrum of 20Ne with the one of 2o0, which has four neutrons outside 160. Both spectra can be found in ref. [5]. Whereas the low-lying spec trum of 2oNe looks rotational, the one of 2o0 shows no analogous collective trend. The conclusion seems clear. Deformation in light nuclei is due to the T= 0 neutron-proton (n-p) interaction. This possibility was already pointed out in 1962 by Talmi [6], based on work of Unna [7]. 385
4’U
4*998
2’-ezI o*-
o*-
90
O’-
92
fJ*_
94
o*-
96
100
bl
4.1.14
2*L!a
2*9;T9
2+m ()*-
o*-
99
4+-221
2+=
42
102
Mo
c.+-1;QB ,*J& 4+33 4+JJ35
2*=
,+&&?
2+-a;iP o*_
o*_ 92
94
O’-
96
o+-
99
ry-
100
o*-
102
2*=
2+9Jl
O’-
o+-
104
106
Fig. 1. Experimental spectra of a) the even Zr isotopes and b) the even MO isotopes, from ref. [4]. Energies associated with each level are in MeV.
In 2o0, the four valence neutrons interact via a predominantly short-range attractive force. Such a force leads to a ground state in which alike nucleons are coupled pairwise to angular momentum zero. Such “ airing correlations” yield a spherical nucleus. The 2g Ne system can experience T = 0 n-p correlations, in addition to n-n and p-p pairing correlations. As is already clear from the properties of the two-nucleon system (e.g., the spin of the deuteron), the 3SI n-p interaction is effectively the strongest component of the nuclear force. In 2oNe, the valence neutrons and protons are filling the same shell-model orbitals and therefore experience strongly the 3S1 T= 0 n-p force. The resulting n-p correlations dominate over the n-n and p-p correlations, and a deformed system results. For neutrons and protons to strongly experience the 3S, part of the nuclear force, the orbitals they occupy must overlap strongly [8]. In the (2sld) shell, 386
29 August 1977
PHYSICS LETTERS
Volume 69B. number 4
strong overlap occurs if either a) the neutron and proton occupy the same orbital or b) they occupy the Id,,, and ld312 spin-orbit-partner orbitals. Examination of effective interaction matrix elements appropriate to the (2sld) shell suggests that the T = 0 attraction of nucleons in spin-orbit-partner orbitals may be even stronger than the attraction of two nucleons in the same orbitals. To illustrate this point, we show in table 1 some representative diagonal effective interaction matrix elements calculated by Kuo [9]. Note that the interaction matrix element between a Id,,, nucleon and a ld3,2 nucleon in a J = 1, T = 0 state is significantly more attractive than either the ld,122(J = 5, T = 0)stretch-configuration matrix element or the ld,,22(J= 0, T = 1) pairin matrix element. The importance of the ! S, attraction between nucleans in the Id,,, and Id,,2 orbitals can be seen experimentally in the spectrum of “F, which has only a single neutron and proton outside 160. The 18F spectrum [5] shows four P = l+ levels below 4.5 MeV, only three of which can be understood without invoking a low-lying ld5i2 - Id,,, configuration [6,7]. Next we discuss the neutron-rich Zr and MO isotopes to see whether similar ideas may be used to discuss deformation in this region. The possibility that the 3S1 interaction is responsible for collective effects in heavier nuclei (sic. lowering of the first 2+ state) was already contained in work of de Shalit and Goldhaber [8] of 1953. Single-particle levels appropriate to a discussion of this region [lo] are shown in fig. 2, assuming an inert 88Sr core. While protons and neutrons added to 88Sr do not fill the same orbitals, they can simultaneously fill the lg 912 proton and 18712 neutron orbitals. The strong overlap of these spin-orbit-partner orbitals can lead to important n-p correlations in this region and thus to deformation. To see how the strong attraction between lgr,,2 protons and lg7,2 neutrons can lead to deformation in this region, we shall focus on the Zr isotopes. In a purely independent-particle picture of these isotopes, Table 1 Some (2s-Id)-shell effective interaction matrix elements, in MeV [9] in
i2
j3
i4
JT
tili2(JT)I~effIj3j4(JT))
dsiz
dsn
h/2
G/2
01
ds/2 ds/2
ds,z d,,,
dsi2 d,/,
ds/2 d3/2
50 10
-2.44 -3.66 -5.83
PHYSICS LETTERS
Volume 69B, number 4
protons
neutrons
Fig. 2. Single-particle levels atgropriate to a description of nuclei in the Zr-Mo region. A Sr core is assumed.
the lgg/2 proton orbital is empty and the lg7i2 neutron orbital does not start to fti until after N = 62. The residual interaction between valence nucleons can substantially modify the lggi2 proton and 1g7i2 neutron occupation probabilities. The n-n and p-p interactions distribute nucleons over all the active orbitals through configuration mixing, as illustrated by the fact that the ground state of “Zr has a roughly 30% lgg/2 proton occupation probability [lO,ll]. These pairing correlations, by themselves, stabilize the spherical shape of the system. The n-p interaction, if sufficiently strong, can break these correlations through a kind-of polarization effect. For example, as neutrons are added to the 1g712 orbital, some protons in the 2p1,2 orbital may be promoted into the lgg12 orbital, so as to take better advantage of the n-p force. Similarly, the addition of 1ggj2 protons may polarize the neutron part of the wave function, by promoting additional neutrons into the 1g7j2 orbital. Such mutual polarization effects can only occur if the resulting gain in n-p interaction energy exceeds the loss in single-particle energy plus n-n and p-p pairing energy, in other words if the n-p interaction is sufficiently strong. It is through this n-p polarization effect that the heavy Zr isotopes develop the spatial correlations necessary for deformation. There is experimental evidence supporting such polarization effects in this region. Neutron stripping data indicates that the “lg7 2 level” is at 2.67 MeV in 8gSr (ref. [12]), at 2.19 Me G. m ‘lZr (ref. [13]) and at 1.37
29 August 1977
MeV in g3Mo (ref. [14]), supporting the idea that the addition of 1ggi2 protons leads to a gradual energetic favoring of 18712 neutrons. A similar conclusion can be drawn from the spectrum of g8Nb. Experiment [15] suggests that the ground state of g8Nb is dominated by the configuration 1ggi2(p) - 1g7j2(n), even though single-particle considerations strongly favor a lgg,2(p) - 3s1j2(n) configuration. Although polarization effects are important for a detailed discussion of deformation in this region, reasonable qualitative predictions as to which nuclei are deformed can already be made on the basis of singleparticle considerations. For example, just filling the single-particle orbitals of fig. 2 leads to the prediction that deformation in the MO isotopes first occurs for lo6Mo, only two neutrons from where it is found experimentally. The idea that the n-p force favors large lggi2 proton and lg7/2 neutron occupation probabilities and that this is closely related to the onset of deformation is consistent with the Nilsson model. By examining the relevant deformed proton orbitals [ 161, we find that for sufficiently large deformation-either oblate or prolate* - the two lowest Nilsson orbitals beyond 2 = 38 correspond to 1gg,2 and not 2pI,2 protons. Similar remarks apply to the deformed neutron orbitals as well. Thus, it would appear that the same mechanism which produces deformation in light (2s1 d)-shell nuclei, namely that deformation results from strong n-p correlations induced by the 3S, part of the nuclear force, may also be responsible for the onset of deformation in the neutron-rich Zr-Mo region. As we shall see, the same mechanism can also be used to discuss deformation in the heavier rare-earth and actinide regions. At this point it is useful to generalize our earlier remarks as to when strong n-p correlations should occur. As noted earlier, the crucial criterion is that the neutrons and protons occupy orbitals with good overlap. It was pointed out long ago [8] that the overlap between two orbitals (nN IN &) and (np 1, ip) is maximum if nN = np and IN sz:I,. So far, we have focussed on cases in which nN = nP and 1, = lp, although we have emphasized that & need not be the same as ip. * Preliminary HFB calculations now in progress seem to indicate that the deformed nuclei in the Zr-Mo region are oblate rather than prolate, in agreement with the results of earlier Nilsson calculations [ 171.
387
Volume 69B, number 4
PHYSICS LETTERS
It is also true that good overlap occurs for nN = np and I, = 1, f 1, particularly if the orbital angular momenta I, and 1, are large. In our qualitative discussions of the (2sld) shell and the neutron-rich Zr and MO isotopes, it has not been necessary to consider the latter possibility. A discussion of the rare-earth region requires a relevant ordering of single-particle levels. Because of the lack of a nearby doubly-magic core, we prefer not to extract the single-particle levels from experiment, but rather to use the levels obtained from Hartree-FockBogolyubov calculations [ 181. These calculated levels already include some important polarization effects. For N - 90, the relevant ordering of proton orbitals (above Z = 50) is lg7i2, 2d,j2, lh, Ij2, with the lh,,,2 and 2d,/2 orbitals nearly degenerate. Similarly for 2% 60, the neutron orbitals (above N= 82) are 2f7i2, lh,,,, li13j2. Thus, for 2 - 60 and N- 90, the IhI,,, proton and lh9j2 neutron orbitals are starting to fill and deformation is predicted. This is precisely where the rare earth region begins. Further into the rare earth region, strong correlations between Ih,,,, protons and li,,,, neutrons can occur. On this basis, we would expect important n-p correlations to persist until the lh,I,2 proton and li,,,, neutron orbitals become closed, which occurs for Z - 76 and N N 114. This is very near the observed end of the deformed rare-earth region. The actinide nuclei involve a 208Pb core [ 191 with valence protons filling the 1h9j2, 2f7,, , li 13j2, and 2f512 orbitals and valence neutrons the 2g712, li11,2, lj ,2 and 3d,,, orbitals. As we move away from zbd Pb, this ordering of levels can be modified by polarization effects. In the absence of HFB calculations for this region, we conjecture that as the li, 1,2 neutron orbital fills, an inversion of the 2f7,2 and li13/2 proton orbitals should occur. On this basis, we would predict that the li13i2 proton orbital and the li11,2 neutron orbital start filling for Z - 92 and N- 136, just about where the deformed actinide region begins. Further into the actinide region, correlations between lj IS,2 neutrons and li13,2 protons should become important. In summary, we have shown that it is possible to discuss nuclear deformation as it occurs in heavy nuclei in the same microscopic shell-model framework that has historically been reserved for light nuclei. We have seen that all known regions of deformed nuclei 388
29 August 1977
are characterized by neutrons and protons simultaneously filling orbitals with good overlap. Under such circumstances, important spatial correlations between neutrons and protons can develop through an n-p polarization mechanism*. Finally, whereas our picture places the primary responsibility for nuclear deformation on the strong n-p interaction between nucleons in selected shell-model orbitals, it does not rule out the likely possibility that all shell-model orbitals play a role in dictating whether the deformation is oblate or prolate [21]. Helpful discussions with G. Dussel, H.T. Fortune B.R. Mottelson are gratefully acknowledged.
and
* The same mechanism has also been shown to be important for nuclear deformation in the framework of the Interacting Boson Model [ 201.
References [l] A. Bohr and B.R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27 No. 16 (1953). [2] B.R. Mottelson, in: Proc. Intern. School of Physics Enrico Fermi XV, Varenna, Italy, 1960, ed. G. Racah (Academic Press, New York, 1962) p. 44. [3] M. Danos and V. Gillet, Phys. Rev. 161 (1967) 1034. [4] E. Cheifetz, R.C. Jared, S.G. Thompson and J.B. Wilhelmy, Phys. Rev. Letters 25 (1970) 38. [5] F. Ajzenberg-Selove, Nucl. Phys. A190 (1972) 1. [6] I. Talmi, Rev. Mod. Phys. 34 (1962) 704. [7] I. Unna, Phys. Rev. 132 (1963) 2225. [8] A. de Shalit and M. Goldhaber, Phys. Rev. 92 (1953) 1211. [9] T.T.S. Kuo, Nucl. Phys. A103 (1967) 71. [lo] N. Auerbach and I. Talmi, Nucl. Phys. 64 (1965) 458. [ 111 B.M. Preedom, E. Newman and J.C. Hiebert, Phys. Rev. 166 (1968) 1156. [12] D.C. Slater, E.R. Cosman and D.J. Pullen, Nucl. Phys. A206 (1973) 433. [13] C.R. Bingham and M.L. Halbert, Phys. Rev. C2 (1970) 2297. [ 141 J.B. Moorehead and R.A. Moyer, Phys. Rev. 184 (1969) 1205. [15] L.R. Medsker and H.T. Fortune, Phys. Letters 58B (1975) 297. [ 161 A. de Shalit and H. Feshbach, Theoretical nuclear physics, Vol. 1 (John Wiley & Sons, New York, 1974) p. 448. [ 171 D.A. Arseniev, A. Sobiczewski and V.G. Soloviev. Nucl. Phys. Al39 (1969) 269. 1181 A.L. Goodman, J.P. Vary and R.A. Sorenson, Phys. Rev. Cl3 (1976) 1674. [I91 P. Ring and J. Speth, Nucl. Phys. A235 (1974) 315. A. Arima, T. Ohtsuka, F. Iachello and I. Talmi, Phys. WI Letters 66B (1977) 205. 1211 B.R. Mottelson, private communication.
TOWARDS
A UNIFIED
PHYSICS LETTERS
MICROSCOPIC
DESCRIPTION
29 August 1917
OF NUCLEAR
DEFORMATION
P. FEDERMAN IFUNAM, Ap. Postal 20-364, M&co
20, D.F.
and S. PITTEL Bartol Research Foundation
of The Franklin Institute, Swarthmore,
Pennsylvania 19081,
USA
Received 24 May 1977 Nuclear deformation, as it occurs in both light and heavy nuclei, is discussed in a unified microscopic shell-model framework. The short-range 3S1 neutron-proton interaction plays an important role in this discussion.
The concept of deformation has been used extensively for over 25 years to describe “collective” properties of heavy nuclei [ I]. And yet very little is known about the microscopic structure of deformation in these nuclei. Since a short-range pairing force favors sphericity whereas a long-range quadrupole force favors deformation, it is generally accepted that deformation in neutron-rich heavy nuclei arises from the long-range (field-producing) part of the effective nuclear force [2]. But such a picture is not really microscopic, in the sense of relating the properties of deformed nuclei to the properties of the underlying nuclear force. Whereas the short-range pairing force reflects the ‘S, component of the nuclear force, the microscopic basis of the long-range quadrupole force is not nearly as clear. Unfortunately, any further microscopic insight into the structure of deformation in heavy nuclei (e.g., the insight that would accompany a shell-model study) is severely hindered by the complexity of the problem. It is therefore very convenient that deformation was observed in a region of light nuclei, namely in the (2sld) shell, where microscopic shell-model investigation is feasible. But here a seemingly different picture emerged. In light nuclei, deformation seems to arise from the short-range neutron-proton interaction [3]. In this Letter we show that it is not necessary to invoke different pictures to discuss deformation in light and heavy nuclei. More specifically, we show that the same microscopic shell-model picture that
has been used to discuss deformation in light nuclei can also be used in the heavier deformed regions. The relatively recent observation by Cheifetz et al. [4] of a new region of deformed nuclei is crucial to this work. By studying the fission fragments of 252Cf, they observed rotational spectra in several neutronrich isotopes of Zr and MO [see fig. I] as well as Ru and Pd. The observed B(E2) values were as enhanced as in the rare-earth and actinide regions. Like the actinide and rare-earth deformed nuclei these nuclei have large neutron excesses. On the other hand, like the deformed nuclei in the (2sId) shell, they can be conveniently discussed in a microscopic shell-model framework. As we shall see, this new region of deformed nuclei provides a kind of missing link between the light and heavy regions of deformation. We first review briefly what has been learned about deformation from the (2s-ld)-shell nuclei. The nucleus 2oNe is a classic example of a rotational nucleus in the (2sId) shell. It consists of two neutrons and two protons outside the doubly magic 160 core. It is instructive to compare the spectrum of 20Ne with the one of 2o0, which has four neutrons outside 160. Both spectra can be found in ref. [5]. Whereas the low-lying spec trum of 2oNe looks rotational, the one of 2o0 shows no analogous collective trend. The conclusion seems clear. Deformation in light nuclei is due to the T= 0 neutron-proton (n-p) interaction. This possibility was already pointed out in 1962 by Talmi [6], based on work of Unna [7]. 385
4’U
4*998
2’-ezI o*-
o*-
90
O’-
92
fJ*_
94
o*-
96
100
bl
4.1.14
2*L!a
2*9;T9
2+m ()*-
o*-
99
4+-221
2+=
42
102
Mo
c.+-1;QB ,*J& 4+33 4+JJ35
2*=
,+&&?
2+-a;iP o*_
o*_ 92
94
O’-
96
o+-
99
ry-
100
o*-
102
2*=
2+9Jl
O’-
o+-
104
106
Fig. 1. Experimental spectra of a) the even Zr isotopes and b) the even MO isotopes, from ref. [4]. Energies associated with each level are in MeV.
In 2o0, the four valence neutrons interact via a predominantly short-range attractive force. Such a force leads to a ground state in which alike nucleons are coupled pairwise to angular momentum zero. Such “ airing correlations” yield a spherical nucleus. The 2g Ne system can experience T = 0 n-p correlations, in addition to n-n and p-p pairing correlations. As is already clear from the properties of the two-nucleon system (e.g., the spin of the deuteron), the 3SI n-p interaction is effectively the strongest component of the nuclear force. In 2oNe, the valence neutrons and protons are filling the same shell-model orbitals and therefore experience strongly the 3S1 T= 0 n-p force. The resulting n-p correlations dominate over the n-n and p-p correlations, and a deformed system results. For neutrons and protons to strongly experience the 3S, part of the nuclear force, the orbitals they occupy must overlap strongly [8]. In the (2sld) shell, 386
29 August 1977
PHYSICS LETTERS
Volume 69B. number 4
strong overlap occurs if either a) the neutron and proton occupy the same orbital or b) they occupy the Id,,, and ld312 spin-orbit-partner orbitals. Examination of effective interaction matrix elements appropriate to the (2sld) shell suggests that the T = 0 attraction of nucleons in spin-orbit-partner orbitals may be even stronger than the attraction of two nucleons in the same orbitals. To illustrate this point, we show in table 1 some representative diagonal effective interaction matrix elements calculated by Kuo [9]. Note that the interaction matrix element between a Id,,, nucleon and a ld3,2 nucleon in a J = 1, T = 0 state is significantly more attractive than either the ld,122(J = 5, T = 0)stretch-configuration matrix element or the ld,,22(J= 0, T = 1) pairin matrix element. The importance of the ! S, attraction between nucleans in the Id,,, and Id,,2 orbitals can be seen experimentally in the spectrum of “F, which has only a single neutron and proton outside 160. The 18F spectrum [5] shows four P = l+ levels below 4.5 MeV, only three of which can be understood without invoking a low-lying ld5i2 - Id,,, configuration [6,7]. Next we discuss the neutron-rich Zr and MO isotopes to see whether similar ideas may be used to discuss deformation in this region. The possibility that the 3S1 interaction is responsible for collective effects in heavier nuclei (sic. lowering of the first 2+ state) was already contained in work of de Shalit and Goldhaber [8] of 1953. Single-particle levels appropriate to a discussion of this region [lo] are shown in fig. 2, assuming an inert 88Sr core. While protons and neutrons added to 88Sr do not fill the same orbitals, they can simultaneously fill the lg 912 proton and 18712 neutron orbitals. The strong overlap of these spin-orbit-partner orbitals can lead to important n-p correlations in this region and thus to deformation. To see how the strong attraction between lgr,,2 protons and lg7,2 neutrons can lead to deformation in this region, we shall focus on the Zr isotopes. In a purely independent-particle picture of these isotopes, Table 1 Some (2s-Id)-shell effective interaction matrix elements, in MeV [9] in
i2
j3
i4
JT
tili2(JT)I~effIj3j4(JT))
dsiz
dsn
h/2
G/2
01
ds/2 ds/2
ds,z d,,,
dsi2 d,/,
ds/2 d3/2
50 10
-2.44 -3.66 -5.83
PHYSICS LETTERS
Volume 69B, number 4
protons
neutrons
Fig. 2. Single-particle levels atgropriate to a description of nuclei in the Zr-Mo region. A Sr core is assumed.
the lgg/2 proton orbital is empty and the lg7i2 neutron orbital does not start to fti until after N = 62. The residual interaction between valence nucleons can substantially modify the lggi2 proton and 1g7i2 neutron occupation probabilities. The n-n and p-p interactions distribute nucleons over all the active orbitals through configuration mixing, as illustrated by the fact that the ground state of “Zr has a roughly 30% lgg/2 proton occupation probability [lO,ll]. These pairing correlations, by themselves, stabilize the spherical shape of the system. The n-p interaction, if sufficiently strong, can break these correlations through a kind-of polarization effect. For example, as neutrons are added to the 1g712 orbital, some protons in the 2p1,2 orbital may be promoted into the lgg12 orbital, so as to take better advantage of the n-p force. Similarly, the addition of 1ggj2 protons may polarize the neutron part of the wave function, by promoting additional neutrons into the 1g7j2 orbital. Such mutual polarization effects can only occur if the resulting gain in n-p interaction energy exceeds the loss in single-particle energy plus n-n and p-p pairing energy, in other words if the n-p interaction is sufficiently strong. It is through this n-p polarization effect that the heavy Zr isotopes develop the spatial correlations necessary for deformation. There is experimental evidence supporting such polarization effects in this region. Neutron stripping data indicates that the “lg7 2 level” is at 2.67 MeV in 8gSr (ref. [12]), at 2.19 Me G. m ‘lZr (ref. [13]) and at 1.37
29 August 1977
MeV in g3Mo (ref. [14]), supporting the idea that the addition of 1ggi2 protons leads to a gradual energetic favoring of 18712 neutrons. A similar conclusion can be drawn from the spectrum of g8Nb. Experiment [15] suggests that the ground state of g8Nb is dominated by the configuration 1ggi2(p) - 1g7j2(n), even though single-particle considerations strongly favor a lgg,2(p) - 3s1j2(n) configuration. Although polarization effects are important for a detailed discussion of deformation in this region, reasonable qualitative predictions as to which nuclei are deformed can already be made on the basis of singleparticle considerations. For example, just filling the single-particle orbitals of fig. 2 leads to the prediction that deformation in the MO isotopes first occurs for lo6Mo, only two neutrons from where it is found experimentally. The idea that the n-p force favors large lggi2 proton and lg7/2 neutron occupation probabilities and that this is closely related to the onset of deformation is consistent with the Nilsson model. By examining the relevant deformed proton orbitals [ 161, we find that for sufficiently large deformation-either oblate or prolate* - the two lowest Nilsson orbitals beyond 2 = 38 correspond to 1gg,2 and not 2pI,2 protons. Similar remarks apply to the deformed neutron orbitals as well. Thus, it would appear that the same mechanism which produces deformation in light (2s1 d)-shell nuclei, namely that deformation results from strong n-p correlations induced by the 3S, part of the nuclear force, may also be responsible for the onset of deformation in the neutron-rich Zr-Mo region. As we shall see, the same mechanism can also be used to discuss deformation in the heavier rare-earth and actinide regions. At this point it is useful to generalize our earlier remarks as to when strong n-p correlations should occur. As noted earlier, the crucial criterion is that the neutrons and protons occupy orbitals with good overlap. It was pointed out long ago [8] that the overlap between two orbitals (nN IN &) and (np 1, ip) is maximum if nN = np and IN sz:I,. So far, we have focussed on cases in which nN = nP and 1, = lp, although we have emphasized that & need not be the same as ip. * Preliminary HFB calculations now in progress seem to indicate that the deformed nuclei in the Zr-Mo region are oblate rather than prolate, in agreement with the results of earlier Nilsson calculations [ 171.
387
Volume 69B, number 4
PHYSICS LETTERS
It is also true that good overlap occurs for nN = np and I, = 1, f 1, particularly if the orbital angular momenta I, and 1, are large. In our qualitative discussions of the (2sld) shell and the neutron-rich Zr and MO isotopes, it has not been necessary to consider the latter possibility. A discussion of the rare-earth region requires a relevant ordering of single-particle levels. Because of the lack of a nearby doubly-magic core, we prefer not to extract the single-particle levels from experiment, but rather to use the levels obtained from Hartree-FockBogolyubov calculations [ 181. These calculated levels already include some important polarization effects. For N - 90, the relevant ordering of proton orbitals (above Z = 50) is lg7i2, 2d,j2, lh, Ij2, with the lh,,,2 and 2d,/2 orbitals nearly degenerate. Similarly for 2% 60, the neutron orbitals (above N= 82) are 2f7i2, lh,,,, li13j2. Thus, for 2 - 60 and N- 90, the IhI,,, proton and lh9j2 neutron orbitals are starting to fill and deformation is predicted. This is precisely where the rare earth region begins. Further into the rare earth region, strong correlations between Ih,,,, protons and li,,,, neutrons can occur. On this basis, we would expect important n-p correlations to persist until the lh,I,2 proton and li,,,, neutron orbitals become closed, which occurs for Z - 76 and N N 114. This is very near the observed end of the deformed rare-earth region. The actinide nuclei involve a 208Pb core [ 191 with valence protons filling the 1h9j2, 2f7,, , li 13j2, and 2f512 orbitals and valence neutrons the 2g712, li11,2, lj ,2 and 3d,,, orbitals. As we move away from zbd Pb, this ordering of levels can be modified by polarization effects. In the absence of HFB calculations for this region, we conjecture that as the li, 1,2 neutron orbital fills, an inversion of the 2f7,2 and li13/2 proton orbitals should occur. On this basis, we would predict that the li13i2 proton orbital and the li11,2 neutron orbital start filling for Z - 92 and N- 136, just about where the deformed actinide region begins. Further into the actinide region, correlations between lj IS,2 neutrons and li13,2 protons should become important. In summary, we have shown that it is possible to discuss nuclear deformation as it occurs in heavy nuclei in the same microscopic shell-model framework that has historically been reserved for light nuclei. We have seen that all known regions of deformed nuclei 388
29 August 1977
are characterized by neutrons and protons simultaneously filling orbitals with good overlap. Under such circumstances, important spatial correlations between neutrons and protons can develop through an n-p polarization mechanism*. Finally, whereas our picture places the primary responsibility for nuclear deformation on the strong n-p interaction between nucleons in selected shell-model orbitals, it does not rule out the likely possibility that all shell-model orbitals play a role in dictating whether the deformation is oblate or prolate [21]. Helpful discussions with G. Dussel, H.T. Fortune B.R. Mottelson are gratefully acknowledged.
and
* The same mechanism has also been shown to be important for nuclear deformation in the framework of the Interacting Boson Model [ 201.
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