Nuclear structure with Dirac phenomenology

PHYSICS REPORTS

--

ELSEVIER

Physics Reports 242 (1994) 233—251

Nuclear structure

____________________

With Dirac phenomenology

L. Zamicka,* D.C. Zhengb I

a Department of Physics and Astronomy, Rutgers University. Piscataway, NJ 08855, USA b Department of Physics and Astronomy, McMaster University, Hamilton, Onr., Canada L8S 4M1

Abstract With non-relativistic Bonn A G matrix elements, it appears that the spin—orbit interaction in a nucleus is too small. As a consequence the wave functions in the Op shell are too close to the LS limit. The introduction of a Dirac nucleon effective mass m* less than the free mass enhances the spin—orbit interaction and affects nuclear structure in a very 12C to the Jfl = ij, T= 1 state is increased by a factor of 2.5 when m*/m is significant way. For example, the B(Ml) in decreased from 1 to 0.67. In the above analyses, large-space shell-model calculations are essential to prevent collapse of 1 + states below the ground state. A superficial analysis suggests that the tensor interaction in the nucleus is too large despite the small percentage of the D-state admixture for Bonn A (about 4.4%). However, there are some complications in the analysis. It is emphasized that in order to see large effects of the Dirac phenomenology in nuclear structure, it is essential to calculate single-particle energies with the same interaction that is used for the particle—particle matrix elements in the open shell. This also holds for the core polarization corrections.

1. Personal note by L. Zamick I came to Princeton before Tom Kuo. I therefore had the privilege of being involved with two aspects of the problem of the effective interaction of nucleons in a nucleus. First there was the pre Kuo—Brown period. In this period Ben Bayman taught me about the nuclear shell model and particle—hole symmetries. I had the good fortune to join in a collaboration with him and John McCullen which lead to the paper “Spectroscopy in the Nuclear f 712 Shell”, also known as MBZ [1]. At about the same time, similar work was done at Rochester by Ginocchio and French [2]. The only difference was that they used an isospin formalism. Our wave functions were described explicitly in terms of neutron and proton quantum numbers. Our representation turns out to be very similar to that of IBM-2. Much of the data we tried to explain in 1963—64 came from Rubby Sherr’s group at Palmer Lab. Cyclotron in Princeton.

*

Corresponding author

0370-l573/94/~7.00© 1994 Elsevier Science B.V. All rights reserved. SSDI 03 70-1573 (94) 00011-Q

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Our philosophy was that of Talmi [3], to take the matrix elements from experiments. It seemed like a horrendous task to get the matrix elements from first principles. The spectra of 42Ca, 42Sc, and 42Ti are given in Table 1. The J~= 0~energy is obtained from binding energies, e.g. for 42Ca, 3.109 = 2EB(41Ca) — E 42Ca) — EB(40Ca). The rest of the spectra are given relative to the ~ = 0~state energy. The 8( two-body matrix elements (TBME) were identified with the data as follows: —

<(j2)JI VI(j2)~>=

(1)

(E~ — E°) + E°,

e.g., for fir = 3~<(12)3+1 VI(j2)31> = 1.490 + (—3.182) = — 2.322 MeV. With the assumption of a pure f 712 configuration, the secular matrix to be diagonalized can be written in terms of the eight numbers <(12) ~ I V 1(12) ‘> with J = 0—7. [Note that for odd J, T = 0 (singlet) and for even J, T = 1 (triplet). The near equality of the T = 1 excitation energies (fir = 2~,4~6~)is evidence of the goodness of charge independence of 42Ti the nucleon—nucleon (NN)42Sc interaction. the relative to 42Ca and is due to However, the repulsive substantially smaller “pairing energy” of Coulomb interaction.] When Kuo came to Princeton from Pittsburgh, he showed that with sufficient intelligence and fortitude one could get a reasonable handle on the effective interaction from first principles. Furthermore, he and Brown [4]and Bertsch [5] showed that in order to explain the above spectrum, it was woefully inadequate to use just the bare G-matrix. We list in Table 2 first the values of E~from 42Sc and then the correction term due to core G 3~_lh. 7~= 0~matrix element from experiment is — 3.182 MeV, the bare G-matrix Whereas the J element gives only 0.869 MeV. The core-polarization correction G3~lh is — 0.938 MeV, even larger than the bare value. The combination G + G 3~_ is — 1.807 MeV. There is a considerable —

lh

Table 1 E°(MeV) J~ 42Ca 42Sc 42Ti

0~ —3.109 —3.182 —2.648

E~—E°(MeV)

l~ 0.6111

2~ 1.5247 1.5863 1.5549

3~

4+ 2.7525 2.8153 2.6764

1.4904

5~

6’

7~

1.5101

3.1893 3.242 3.0430

0.6163

Table 2 0~

1~

experiment (42Sc)

—3.182

—2.571

—

—0.869

—0.230

1.596 —0.664

—

bare KB G-matrix (G)

1.692 —0.211

—0.367 —0.297

—0.938 — 1.807

—0.295 —0.525

—0.121 —0.785

0.003 —0.208

0.210 —0.087

J~

2~

3~

4+

5~

6~

7~

1.672 —0.604

—0.060 —0.120

—2.566 —2.185

0.102 —0.502

0.346 0.226

—0.014 —2.199

—

G 3~_1, G + G3~_jh

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235

contribution from G3~lhto the pairing energy. A good deal of the discrepancy between 1.807 and 3.182 MeV is removed by a matrix diagonalization, i.e., allowing the two nucleons to scatter from the configuration into other states like pL2~P~/2,ff12, and g~12.Note also that empirically the ~Jir= 6~matrix element is positive. The bare value is negative but G3~_lhgives a positive (repulsive) contribution. Thanks to the above monumental work by Kuo and Brown [4] and Bertsch [5], I was able to make a modest contribution [6] to explain the apparent repulsion between neutrons in a nucleus, which had previously been pointed out by Talmi [3].He had obtained a very interesting relationship between the binding energy of a nucleus and the number of particles of one kind, say neutrons, in a given shell. 1) EB(n = 0) = nC + ~n(n 1)cc + [n/2]/3. (2) E8(j’ We will focus on the quadratic term invoking This is defined as —

—

(f~12)J0

—

—

—

~.

~=

where

E2

(2J+2)E2—E~° 2j + 1

(3)

is the mean energy of the states with = ~Jo:Jeven(2~1 ~.jo:jeven(2j

J ~

0: (4)

+ 1)E~ +

1)

The quantity ~ can be interpreted as the slope of the two-neutron separation energy as a function of the number of neutrons. Since it is easier to remove neutrons from a nucleus with neutron excess, ~ is usually negative, e.g., for the calcium isotopes, Talmi [3] obtained the following parameters: C = 8.38 MeV,

~ = —0.23 MeV,

/3

=

3.33MeV.

The problem is that with the bare G-matrix, one gets ~ to be positive because the free neutron— neutron interaction is attractive. Where does the repulsion come from? The answer is from core polarization: from the exchange of a phonon between the two nucleons (i.e., the Bertsch—Kuo— Brown diagram, also denoted by BKB in this work). The phonon can be thought of as a particle—hole excitation [(p’h T”] where J” and T” are the angular momentum and the isospin of the phonon. Now, I showed that the phonons with J” = 0 (monopole phonons) act in the wrong way: They cause x to become more positive (more attractive). However, other phonons, especially the quadrupole phonons, act in the right way causing ~ to be negative (repulsive). Furthermore, they dominate over the contribution of the monopole phonons. So I was fortunate at Princeton to be involved in the two waves of activity involving the effective interaction: The purely phenomenological approach of taking the matrix elements from experiment; and the more fundamental approach of calculating this interaction from first principles. Today, almost 30 years later, both approaches are still being used. This means that although considerable progress has been made in the effective-interaction theory and a good deal of it by Tom Kuo and his collaborators, we still have quite a way to go before we make the phenomenological approach obsolete. 1).!”.

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In the next sections we will discuss some new ideas concerning the application of the Dirac phenomenology to the effective-interaction problems.

2. Kuo vadis? We wish to test the Bonn A interaction [7, 8] in a nucleus. To do this, we perform shell-model calculations with G matrix elements obtained from this interaction which has a “weaker tensor force”, e.g., 4.4% D-state admixture in the deuteron. We feel that to truly test this (or any other) interaction, one should not only calculate the two-body matrix elements (TBME) for the valence space but also calculate the single-particle energies (SPE) with the same interaction. For practical reasons, we have also constructed a schematic interaction of the form: V = V~+ xJ~0+ y~,where c, s.o. and t stand for central, spin—orbit and tensor, respectively [9, 10]. For x = 1 and y = 1, we get a reasonable fit to the Bonn A G matrix elements. By varying x and y, we can see the first-order effects of changing the spin—orbit and tensor interactions. For example, by setting y = 0, we turn off the tensor interaction. Thus we can get a feeling for what various parts of the Bonn A interaction are doing. Note, however, the second-order effects are not obtained by such a procedure. For example, it is well known that a substantial part of the central G matrix elements of a realistic interaction comes from the second-order ladder diagram with the tensor interaction. When we vary y, for example, we do not change This difference should be kept in mind. ~

3. Particle—hole states in a closed

LS

shell

Does one get pion condensation with Bonn A interaction or at least a precursor to pion condensation? To find this out, we have performed a one-particle, one-hole (1 p—i h) calculation for negative-parity states in 160. Let us focus on fir = states. The wave function is of the form =

a[1s112 Opj~]° + b[0d312 Op~]°

,

~_

(5)

with a b for the lowest state. Experimentally, the lowest f’~= 0, T = 0 state is at 10.952 MeV and the 0, T = 1 state is at 12.797 MeV. The T = 1—T = 0 splitting is thus 1.845 MeV. We first give in Table 3 the results for the schematic interaction with the tensor interaction off and on. >~.

Table 3 Tensor force

off(x

=

on (x

=

1, y I, y

E(0, T = 0)

E(O, T

=

1)

AE (MeV)

=

0)

16.9

16.9

0.0

=

1)

14.1

17.1

3.0

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237

There are several comments to be made. First, nothing very dramatic happens. The pionic mode (0, T = 1) does not come down low in energy, as would be required for pion condensation. In fact, when the tensor force is turned on, the energy goes slightly up. Note that for a central plus spin—orbit interaction, the splitting of the 0, T = 1 and 0, T = 0 states is negligibly small. It would appear then that most of the splitting comes from the tensor interaction. This point has previously been made by Blomqvist and Molinari [ii] and by Millener and Kurath [12]. From the above analysis, one could naively infer that the tensor interaction of Bonn A, which is considered to be very weak, is nevertheless too strong in the nuclear medium. With the schematic interaction which is similar to the Bonn A, the T = I T = 0 splitting of the 0 states is 3 MeV but it is only 1.845 MeV experimentally. The situation is however a bit more complicated. First of all, there are central contributions to the splitting of the 0, T= I and 0, T= 0 states which are shown by Millener and Kurath [12]. In their calculation, and indeed also in our calculation with the schematic interaction, this difference is negligible. However, with other central interactions, this might not be the case. To show that the situation is indeed more complicated, we give results for the splitting for the real Bonn A and the real Bonn C. The numbers are (for a relativistic version with a Dirac effective nucleon mass equal to the free mass) 3.08 and 2.72 MeV, respectively. These results are very puzzling because whereas the Bonn A gives 4.4% D state in the deuteron, the Bonn C gives 5.6%. One would a priori expect the splitting to be larger with Bonn C which has a stronger tensor component. It has been suggested to us by Machleidt [13] that we are finding an effect that is sensitive to the shape of the tensor interaction. In order to get the same deuteron quadrupole moment with Bonn A and Bonn C, the Bonn A tensor interaction, which is weaker at short distances, must be stronger at large distances. The 160 0_ state calculation may be giving more weight to the large-distance behavior. —

4. Open shell nuclei

—

that’s where the action is

The first-order contribution of the tensor interaction to the SPE of a closed

LS

shell plus or

minus one nucleon vanishes. However, this is not true for a closed minor shell. This suggests that

we should examine open shell nuclei to get a better idea of what the tensor interaction does in nuclei. Consider, for example, a naive picture of ‘2C as a closed p3/2 shell (p~/2).The famous 1~,T = 0 and T = 1 states at 12.71 and 15.11 MeV, respectively would be lp—lh states with the configuration [p112 p~1]~.T. The SPE splitting of p1/2 and p3/2 orbits, which is obtained by their interaction with the core (0si~2and Op~/2)would get contributions from the tensor interaction. In fact, it was shown many years ago by Wong [14] and Scheerbaum [15] that the tensor interaction is strong enough to give an inversion of the SPE. Indeed in a calculation with the Bonn A interaction, we find that the splitting 8pz~~p3~~is equal to 2.75MeV for 4He, 4.16MeV for 160 but is —3.01 MeV for 12C [9]. Does this minus sign of the single-particle splitting in 12C cause any difficulties in nuclear structure calculations? The answer is a bit complicated. If we perform a minimal calculation in which the ground state is and the 1 + states are of the form p1/2p~, the answer is a resounding p~/2

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yes. In fact we get a collapse! The 1~,T = 0 state comes below the ground state and the 1~,T = state just barely above. This collapse is however easily explained. If indeed the P1/2 single-particle level came below the P3/2 level, it would make no sense to assume that the ground state be PL2. The next step is to perform a complete p shell-model diagonalization as was done many years ago by Cohen and Kurath [16]. We reemphasize that in this calculation as in the simpler one, we calculate the SPE with the same interaction as is used to calculate the valence-space TBME. We feel that to truly test an interaction, it makes no sense to do a hybrid calculation in which the SPE are obtained empirically. Using the OXBASH code [17], we find, in a complete p shell calculation with Bonn A, the energies of the 1~,T = 0 and T = 1 states emerge very close to the experimental values of 12.71 and 15.11 MeV. This would seem to indicate that we could live with Wong’s single-particle inversion as long as we do a large shell-model calculation instead of a smaller one. But is this all there is to the story? The answer is no. We should also look at transition rates in particular the magnetic dipole (Ml) excitation rate: B(M 1)0 = + = o,~.Let us focus on the isovector transition since B(M 1) is much larger for this transition than for the isoscalar one. We find that the calculated value of B(Mi) to the 1~,T = 1 state is 0.64~.This is a factor of more than four too small as compared with the experimental value of 2.85~.So there seems to be some remnants of the collapse. —

1

,

5. Dirac phenomenology to the rescue In this section we present results in which the Dirac phenomenology of Serot and Walecka [18] has been used to address the problems mentioned in the previous sections. This relativistic approach was originally used to explain striking discrepancies in spin-independent properties of proton—nucleus scattering in the energy range from about 200—800 MeV [19,20]. The same phenomenology has been used in nuclear structure but up to now there have not been striking differences between this approach and the non-relativistic approach merely gratification that the two approaches agree. However, in work done in collaboration with Müther, we do find striking differences in nuclear structure [21, 22]. Consider the previously mentioned problem that in a non-relativistic approach, the calculated value of B(M1) in ‘2C to the 15.11MeV 1~, T= 1 state is over a factor of four too small as compared with the experimental value. In the Dirac phenomenology, there is a parameter called the Dirac effective nucleon mass m*. If the single-particle potential in infinite matter is written as U = S + Vy°, where S is the scalar potential due to the sigma meson exchange and V the vector potential due to the omega meson exchange, then (in units where c = 1) m* = m + S. Now S is very large and negative (attractive) and V is large and positive (repulsive), e.g., S 400 MeV and V ~ 350 MeV. The effective single particle central potential is S + V ~ 50 MeV. In infinite matter, the Dirac spinor which is a solution of the equation m U)ü(k,s) = 0 is

—

—

(~ —

~(k, s) =

~J(E*

+

m*)/2E*

[~.

k/(E* +

m*)] X~

—

—

(6)

L. Zamick, D.C. Zheng/Physics Reports 242 (1994) 233—251

239

where E* = .~/k2 + m*2. The difference from a true spinor is the replacement of the free mass m by the Dirac effective mass m* = m + S. For more details, consult the book by Celenza and Shakin [20]. We will present results for m* = m = 938.9 MeV/c2 (this is a value typical for the density at the surface of a light nucleus), m* = 630.0 MeV/c2 (a value more typical for the global density of a light nucleus), and an intermediate value m* 729.1 MeV/c2. The corresponding ratios m*/m are 1,0.67 and 0.78. We now give the results in Table 4, for the reduced Ml rate B(M1) in ‘2C from the 0~,T = 0 ground state to the 1 T = 1 state at 15.11 MeV. We see from the above that changing from a nonrelativistic approach to a relativistic one with no medium correction (m*/m = 1) does not lead to any significant change of the B(M1) value. However, as m* is made smaller, the value of B(M1) becomes larger and is in better agreement with experiment. The best value is 1.57~u~ when m*/m = 0.67. This is still far away from the experimental value of 2.85p~but it is about two and half times the non-relativistic result. Relativity with medium correction makes a big difference. It should be mentioned that there is no problem in fitting this and other data in the p shell with empirical TBME and empirical SPE. This was demonstrated many years ago by Cohen and Kurath [16]. Why are we getting such a large effect from relativity? We believe the answer in part is that lowering the Dirac effective mass increases the spin—orbit interaction. We will show this later on. This has been discussed in the context of proton—nucleus scattering by Shepard et al. [23]. This increase in the spin—orbit interaction will partially offset Wong—Scheerbaum [14, 15] effect of reversing the order of the SPE (~p1,2 and 8p 12C core due to the tensor 32) with respect to interaction. Increasing the spin—orbit splitting will take us away from the LS limit and towards the f/limit. There is learned theorem that in the LS limit, the spin part of the Ml transition amplitude vanishes, leaving only the orbital part. Since gl (isovector) is much smaller than g~(isovector), the value of B(M1) will be very small. We will find in general that with the non-relativistic Bonn A interaction, we get results too close to the LS limit and that increasing the spin—orbit splitting will make things better. Let us digress and discuss briefly why B(Ml)~~ 1~ vanishes in the LS limit. First let us consider an axially symmetric Nilsson model. In the asymptotic limit, the wave functions of the occupied states

Table 4 Approach non-relativistic Dirac spinor (m*/m Dirac spinor (m*/m Dirac spinor (m*/m

Cohen—Kurath experiment

B(M1)o~.T,o...l ,T=

= = =

I) 0.78) 0.67)

0.64 0.69 1.14 1.57

2.50 2.85

iCON)

240

L. Zamick, D.C. Zheng/Physics Reports 242 (1994) 233—251

I

are exp( F2)z and exp( 12)(x + iy) where ~2 = x2/2b~+ y2/2b~+ z2/2b~where b~,b~,and b 2/mb~,etc.]. We can 2 areget thea oscillator lengthinparameters in the ax,nucleon y, and zfrom directions [hwy now spin excitation 12C by exciting the orbit (x =+ hiy) ~ to (x + iy) However, the LS limit does not correspond to an axial Nilsson model but rather a triaxial one. In the asymptotic triaxial limit, the wave functions are exp( F2)x, exp( F2)y, and exp( 12)z. It is not possible for the spin part of the Ml operator to change x into y, y into z, and z into x, etc. Hence, B(M1),~ 1~ vanishes. 12C and the amplitude We next comment_upon how the energy of the 1~, T = 1 state in \/B(Mi)0;,T0~l+,Tl change as we change the spin—orbit strength. We do this with our schematic interaction V= i7~+ xV~ 0.+ yV1. As shown in Fig. 1, the amplitude for B(M1) for a large range of x varies almost linearly in x. Of course for very large x, the result will have to approach to thejj limit: \/‘~iT) = \/1l.25~. However, the fact is that in the region of interest, there is a nice simple behavior with increasing x. Hence, the transition is a good indication of where we are between jj and LS coupling limits. The energy of the 1 ~ T = 1 state varies in a more complicated manner. We show in Fig. 2 the behavior of the energy of the 1 ~ T = 1 state in both a full p space calculation with the shell-model code (OXBASH) [17] and a simpler ip—lh calculation on a pL2 closed shell. In the latter calculation, the configuration of the 1 + state is P1/2P3~ so it is not surprising that the energy of this spin—orbit state varies linearly with x. For very small x, the 1 + state comes alarmingly close to the ground state. In the full p shell calculation, the response for very large x is almost the same as in the lp—lh case. This is not surprising because we are close to the jj limit in that extreme, where there is very little configuration mixing. However, as we decrease x from very large values to the realistic one (x = 1), the energy does not keep decreasing. Rather it turns around and starts to increase slowly. In fact, there is a rather flat plateau from about x = 0 to x = 2 (no spin—orbit interaction to twice the Bonn A spin—orbit strength) where the energy of the 1~,T = I state hardly changes. Thus, the energy of the state is not a very good indication of what the spin—orbit strength should be. —

—

.~.

—

3C

~

—

I

—

I

I

I

I

____ Spin-orbit strength x Fig. I.

Spin-orbit strength x

Fig. 2.

L. Zamick. D.C. Zheng/Physics Reports 242 (1994) 233—251

241

Table 5 2N

Approach

ft(’

non-relativistic Dirac spinor (m*/m = 1) Dirac spinor (m*/m = 0.78) Dirac spinor (m*/m = 0.67) experiment

91573

—~

~C)

73114 35368 22753 13178

Of course, even though the (excitation) energy is not changing much, the wave function is changing continuously. This manifests itself in the changing transition rate; the amplitudes of which is nearly linear in x from x = 0 to x = 2. The moral of this story is that when we seek evidence for the change of the nuclear interaction in a nucleus (i.e., the enhancement of the spin—orbit interaction due to a decreased Dirac effective mass or the reduction of the tensor interaction due to the universal scaling), we should look not only at the excitation energies, but also and even more importantly, at the transition rates. To reemphasize that the Dirac phenomenology can have a large effect on nuclear structure, we now give results for a process closely related to Ml excitation: The Gamow—Teller (GT) transition from the ground state of 12N (fir = 1~,T = 1) to the ground state of 12C. We can think of this as isospin rotation of the Ml process, except that we set the orbital part to zero in the GT case (we thank Denys Wilkinson for suggesting that we look at this). We present the results in the same way as they were given by experimentalists Alburger and Nathan [24] in terms of ft. The relation of ft to B(GT) for the transition 12C ‘2N is 1og 10(ft) 3.790 12B 1og10(B(GT)/3). results are in Table 5. (The experimental for the is ft = The 11669.) Ironically, whereas everyone talks of value the need for mirror transition a quenching of the GT operator, we find that in this particular case (‘2N ‘2C) we need an enhancement to explain the data. Again we must remember that we are using the calculated SPE in our large shell model calculation. —~

—

—~ ‘2C

—+

6. Bare single—particle energies Having claimed in the previous section that underlying the “Ml problem” was in part a singleparticle spin—orbit splitting problem, it behooves us to take a more careful look at the spin—orbit splitting for a hole relative to 160 and also for 160 plus a valence particle. In the former case we consider the P1/2—P3/2 splitting, in the latter the d 312—d5~2splitting. The results for various Dirac effective masses are in Table 6. In the above we identify the spin—orbit splitting with the lowest excitation energy for 3/2 (hole) state in 150 and for 3/2k (particle) state in ‘TO. For the “hole” system, the splitting ~2_e~32 is 3.95 MeV when there are no medium effects (i.e., m* = m). The results vary remarkably linearly in (m/m*) and for m*/m = 0.67, the spin—orbit

242

L. Zamick, D.C. Zheng/Phvsics Reports 242 (1994) 233—251

Table 6 m*/m

.

1 0.78 0.67 experiment

t~,,

3.95 5.24 6.16 6.2

1d,

~

5.80 7.52 8.72 5.1

splitting increases to a value of 6.16 MeV which is remarkably close to the “experimental” value of

6.2 MeV. Had we stopped there everything would be fine. However, for the “particle system”, we find that for m*/m = 1 we already have too large a spin—orbit splitting of 5.80 MeV as compared with the empirical value of 5.1 MeV. For m*/m = 0.67, the spin—orbit splitting is much too large: 8.72 M~..’T We are running into the long standing “(21 + 1)” problem. We would a priori expect the d 3/2—d5/2 splitting to be larger than the P1/2—P3/2 splitting. This is because the spin—orbit splitting for a simple interaction s (~ = constant) varies as ~(21+ I). The d3/2—dS/2 splitting should be times the P1/2—P3/2 splitting. One gets basically the same answer for more realistic one-body spin—orbit interactions for which ~ does vary as a function of r. Indeed our calculations give something like ~. A way out of this dilemma, which we mention in our published work [22], is to argue that for the hole system one is justified in using the small effective mass m*/m = 0.67 because p orbit is in a dense medium, mainly of other nucleons in the p shell. However, for the particle system the valence d particle has a larger mean square radius than the core particles. It is therefore in a less dense medium for which it is appropriate to use an effective mass close to the free value. Indeed, with the extreme choices of m*/m = 0.67 for the hole system and m*/m = 1 for the particle system, the spin—orbit splittings for the single particle states in 170 and for the single-hole states in ‘~Oare, respectively 6.16 and 5.80 MeV. It should be remarked here that 12C) one reason thatdo wenot getisstriking from relativistic medium and others that weeffects use calculated SPE. More often corrections (e.g., for the Ml rate in than not, other workers, even when they used realistic TBME, performed a hybrid calculation in which the single-particle matrix elements were taken from experiment. If we had followed this latter procedure, we would have thrown away the entire physics associated with the fact that lowering the Dirac effective mass leads to an increase in the single-particle splitting. At any rate we feel that one is not really testing a realistic interaction unless one calculates everything with it. —

~-

7. Core polarization for SPE including Hartree—Fock insertions We next consider core-polarization corrections for the single hole and SPE relative to We will include not only the Kuo—Bertsch [25, 26] second-order corrections but also Hartree—Fock (HF) insertions [22, 27]. The results are shown in Table 7. 160.

L. Zamick, D.C. Zheng/Phvsics Reports 242 (1994) 233—251

243

Table 7 (a) The single-hole splitting e~,~ in 150 and (b) The single-particle splitting Sd~.—Ed,, in ‘TO. In the table, “Bare”, “Kuo—Bertsch” and “HF insertions” refer to diagrams in Fig. 1 of Ref. [22]: Bare = Fig. 1(a), Kuo—Bertsch = Fig. 1(b) + Fig. 1(c), HF insertions = Fig. 1(d) + Fig. 1(e) m*/m

Bare

(a)

1.00 0.78 0.67

3.95 5.24

4.11 5.33

6.16

6.19

6.06 6.92 7.22

1.00 0.78 0.67

5.80 7.52 8.72

5.50 7.14 8.26

6.95 7.80 7.99

(b)

+ Kuo—Bertsch

+ HF insertions

Several comments are in order. The Kuo—Bertsch diagrams do not change the single-particle spin—orbit splitting very much. However they do have the beneficial effect of making the splitting for the hole system (150) slightly larger and for the particle system (‘TO) somewhat smaller. The Hartree—Fock insertions give a large contribution to the single-particle splitting than do the Kuo—Bertsch diagrams. This is undoubtedly a manifestation of a more basic problem. In a selfconsistent calculation, the Bonn A matrix elements give too small a radius for The TBME that we are using are evaluated with the harmonic oscillator wave functions with a size parameter fitted to the empirical radius of 160 rather than to the self-consistent radius approximate for the Bonn A interaction. There is therefore some doubt about whether it is meaningful to include the HF diagrams. In a self-consistent calculation with Bonn A, the radius of 160 comes out much too small [28]. Note that the HF diagrams give a larger splitting for the hole system than for the particle system. This also will help to explain the “(21 + 1)” anomaly. In fact, whereas in general the HF diagrams lead to an increase in the “spin—orbit” splitting for most of the cases, there is an exception. For with m*/m = 0.67, the HF diagrams decrease the spin—orbit splitting ~ from 8.26 to 7.99 MeV. One may still worry though that one gets almost the same answer for the splitting 2 if one has m*/m = 0.67 and no core polarization (6.16 MeV) as one does if one has m*/m = 1 and all core-polarization effects (Kuo—Bertsch diagram plus HF insertions) (6.06 MeV). 16Q~

1

~,

2~P3

8. The value of B(M1 )o and Dirac phenomenology

~ -..i

We study the effects on

~

,

=1

B(Ml)

in ‘2C due to combined effect of core polarization for the transition

0j~

--i

1jF, T = 1 of

the Dirac phenomenology.

both core polarization and

244

L. Zamick, D.C. Zheng/Physics Reports 242 (1994) 233—251

Table 8

Graph m*/m m*/m m*/m

= = =

1 0.78 0.67

Bare

Bubble

Hole—hole

Particle—particle

0.69 1.14 1.56

1.31 1.89 2.36

0.91 1.40 1.80

0.60 0.95 1.25

First we calculate core renormalized TBME relative to a 4He core, although we use the G-matrix elements appropriate for The results for B(M1) are presented in Table 8 up to and including a certain graph (“Bubble” = Bertsch—Kuo—Brown (also denoted by BKB) bubble, Fig. 1(b) in Ref. [21]; “hole—hole” = hole—hole diagram, Fig. 1(d) in Ref. [21]; “particle—particle” = particle—particle ladder diagram, Fig. 1(c) in Ref. [21]). Looking down the first column of Table 8, we have a repeat of what we said before: lowering the Dirac effective mass causes the Ml rate to increase and get closer to experiment. In the second column we have the combined effects of the bare diagram and the Bertsch—Kuo--Brown bubble [4, 25]. At this point the results look even more encouraging. The value of B(M 1) for m*/m = 0.67 is now 2.36j~,rather close to the experimental value of 2.85~.It would appear that the medium effects due to sigma exchange and the phonon exchange between two nucleons (i.e., the BKB diagram) act in the same way. Although this is indeed encouraging, we may worry that one can confuse the two effects in a phenomenological analysis. At any rate, we continue to the third column which has the combined effects of the bare, BKB bubble and hole—hole diagrams. We see that the results are now smaller again. In the last column of Table 8 we have the three previous diagrams and the particle—particle ladders. The value of B( Ml) gets even smaller. It appears that the three diagrams (phonon exchange, hole—hole and particle—particle) wipe each other out. It is still true though that as the Dirac effective mass decreases, the value of B(M 1) increases. We then decided it would make more sense to do the calculations relative to an core since the G-matrix elements were determined for From here on we will give results for ‘2C pictured as four holes relative to 160. We will furthermore give the results with and without HF insertions (Table 9). In Table 9, for the first four rows there are no HF insertions for the SPE. The last three rows do have these insertions. One point of interest is that when the HF insertions are put only into the SPE, the results for B(M1) get larger. This is consistent with the fact that the P1/2—P3/2 splitting is larger due to these insertions and we move away from the LS limit towards the jf limit. However, when we put the HF insertions also into the TBME, the value of B(M1) decreases. This can be seen by comparing rows 3, 6 and 7. In row 3, there are no HF diagrams anywhere and the value of B(M1) is 0.92R~.In row 6, when we put HF insertions into the SPE alone, the value goes up to 1.31. In row 7, we also add HF insertions into the hole—hole matrix elements and the value of B(M1) goes down to 1.08. Clearly, if one is going to put in HF insertions, one must put them in everywhere. 160

160

160

L. Zamick, D.C. Zheng/Phvsics Reports 242 (1994) 233—251

245

Table 9 2C for m*/m = 0.67 with various degrees of core polarization. The calculations are Values of B(Ml) ‘ In the table, for the SPE, “G”, “i” and “HF” refer to bare, 2p—lh plus 3p—2h, and HF performed relative0~..1.1.1 to an 160 in core. insertion diagrams, respectively. (In terms of Fig. 1 in Ref. [22]: G = Fig. 1(a), ~ = Fig. 1(b) + Fig. 1(c), HF = Fig. 1(d) + Fig. 1(e).) For the TBME, “G”, “BKB”, “L” and “HF” refer to bare, Bertsch—Kuo—Brown bubble, particle—particle plus hole—hole, and HF insertion diagrams, respectively. (In terms of Fig. I in Ref. [21]: G = Fig. 1(a), BKB = Fig. 1(b), L = Fig. 1(c) + Fig. 1(d), and HF = Fig. 1(e).) Single-particle energy

Two-body matrix element

B(M1)

(1) G (2)G+~

G G+BKB

1.56 1.26

(3)G+~ (4)G+Z

G+BKB+L G+BKB+L+HF

0.92 0.74

(5) G + E + HF (6)G+~+HF (7)G+~+HF

G + BKB G+BKB+L G+BKB+L+HF

1.83 1.31 1.08

(~)

Without HF insertions the most complete answer is in row 3. The value of B(Mi) is 0.92p~~, which unfortunately is smaller than the value using bare matrix elements which was i.56p~.

9.

The

A =

14 system

Another problem of interest is the A = 14 system. In particular, the Gamow—Teller (GT) transition from the ground state in ‘4C (fir = 0~,T = 1) to the ground state in (fir = 1~, T = 0). All the quantum numbers are right for this to be an allowed GT transition. However, the transition is strongly inhibited. The value of log ft is 9.04 (for the 13 + transition ‘~O—p ‘4N, the value is 7.27). For all intents and purposes, B(GT) is essentially zero. Note that in the LS limit, B(GT) would be 6(CA/Cv)2 = 6 X 1.25 12 = 9.39. This transition is especially interesting because it was shown by Inglis [29]in the fifties that if one assumed a configuration of the two holes in the p shell, the B(GT) could not vanish unless there ‘4N

was a tensor interaction present.

We here present results for

B(GT)

B(GT) = i.2512~
for the ~3 decay from 14C to 14N. The definition is

0lI~a(i)t(i)I~0~, T=

i>~2,

(7)

i.e., we include the factor (CA/Cy)2. We first present in Table 10 results with a bare interaction for the various values of theDirac effective mass. We note that with m* = m, the value of B(GT) is large (3.97). When the Dirac

246

L. Zamick, D.C. Zheng/Physics Reports 242 (1994) 233—251

Table 10 Approach

B(GT)o~.Tl.i~,To

non-relativistic Dirac spinor m*/m = 1 Dirac spinor m*/m = 0.78 Dirac spinor m*/m = 0.67

4.80 3.967 0.035 0.060

effective mass is decreased to 0.78m, the value decreases dramatically to 0.035. We are starting to get close to the experimental value which is essentially zero. When we go down further to m*/m = 0.67, there is an apparent increase to 0.067. What is undoubtedly happening though, is that we are getting a sign change in the amplitude as we decrease m*/m from 0.78 to 0.67. In that sense the above result is not surprising. Why are we getting this dramatic decrease in B(GT) as we decrease m*? Here the Inglis analysis [29] is important. The point is not that we do not have a tensor interaction, we do, but rather that for m*/m = 1, the ratio of the tensor interaction strength to the spin—orbit interaction strength is too large. What happens when we decrease m*? To an excellent approximation, the tensor interaction does not change at all. Rather the spin—orbit interaction gets stronger (as we demonstrated previously) so the ratio of tensor to spin—orbit goes down. In more detail, we find with our schematic interaction V~+ x V,.0. + y V1 that the focus of points for which

vanishes seems to form, to an excellent approximation, a straight line:

B(GT)

y = 1.41x 0.87 (for x > 0.85), no solution (for x < 0.85). —

That is to say, if the spin—orbit interaction is too small less than 85% of the value we get to fit the Bonn A matrix elements then there is no solution to the equation B(GT) = 0. On the other hand, for x = 1, the value of the tensor strength parameter y needed to get B(GT) = 0 is 0.54, i.e., about ~ of the Bonn A strength. But there are other solutions: We can get B(GT) to vanish for the Bonn A tensor strength y = 1 provided we make the spin—orbit strength larger: x = 1.33. In fact, the latter is what lowering the Dirac effective mass seems to be doing. By the way, our contention that lowering the Dirac effective mass does not change the tensor interaction significantly can be seen by examining the TBME for the Bonn A interaction: <(Os112 Os1~2)~~T01 VI(0sii20d3i2)J~~T~o>. Note that the left-hand configuration must have L = 0, S = 1 and the right-hand configuration must have L = 2, S = 1. Only a tensor interaction will connect these two states. The values of the above matrix element for m*/m = 1, 0.78, and 0.67 are, respectively 3.756, 3.722, and 3.696MeV. We next show in Table 11 the core-polarization 2C. effects on B(GT) for the A = 14 system using the same as was used fortable. the B(M1) ‘ across we see indeed at all levels of core polarization, We format now comment on that As we in read the value of B(GT) goes towards zero as the Dirac effective mass decreases. —

—

L. Zamick, D.C. Zheng/Phvsics Reports 242 (1994) 233—251

247

Table 11 4C:Ot —a ‘4N: 1 ~, T = 1 for various degrees of core polarization. The calculations are Values of B(GT) for the transition ‘ performed relative to an 160 core. See Table 9 for notations Single particle energies

Two-body matrix elements

B(GT)

N.R.

m*/m

=

1

m*/m = 0.78

,n*/m

(1) G

G

4.80

3.97

0.04

0.06

(2) G + ~ (3) G + ~ (4) G + Z

G + BKB G + BKB + L G + BKB + L + HF

2.91 5.85 6.86

1.84 5.25 6.04

0.26 2.99 3.61

0.02 1.31 1.54

(5) G + ~ + HF (6) G + ~ + HF (7) G + ~ + HF

G + BKB G + BKB + L G + BKB + L + HF

1.04 2.78 4.27

0.025 2.063 2.808

0.007 0.708 0.9 17

0.02 0.36 0.40

=

0.67

As we read down we get a scatter of results. For the calculation with m*/m = 1, the addition of the Bertsch—Kuo—Brown bubble diagram causes a reduction in B(GT), but the inclusion of ladder diagrams brings the results up to a value of 5.85. The answer with HF diagrams is 4.27. For m*/m = 0.67, the results and range of results are much smaller. The smallest value of B(GT) is 0.02 and the largest is 1.54. Clearly, since the experimental result is very close to zero, the calculations will be sensitive to small effects. It will therefore be difficult to get too fine an understanding about what is happening. The best we can do is noting that the results do get smaller and therefore in better agreement with experiment when the Dirac effective mass is O.67m instead of m.

10. The magnetic moment of the ground state of 13C As a contrast to the results previously presented for the transition rates, we will now consider a static property: The magnetic moment of the ground state of ‘3C. This state has quantum numbers fir = 1/2. In the Schmidt model, the magnetic moment is taken to be that of a neutron. For odd N nuclei with j = 1 1/2, the Schmidt magnetic moment is given by p= [j/j + 1)]p(free) = [j/(j + 1)]( 1.913). In the largerj limit, we get the classical result p= p(free) because the spin is pointing in the opposite direction to the angular momentum. However for j = 1/2, because of quantum effects, we get p = ~p(free) = °.64PN. This is the J/ coupling extreme. Configuration mixing will tend to make the magnetic moment larger. The measured value is O.7O2/iN. We present results for the calculated magnetic moment in Table 12 in the same format as in the previous section. We see that with the bare Bonn A G-matrix, we get a result p = °~976PNfor Dirac spinor with m*/m = 1, much bigger than experiment. This is an indication of too much configuration mixing. As we decrease the Dirac effective mass, the results become better: °.887PN for m*/m 0.78 and —

P1/2

—

—

—

—

—

248

L. Zamick, D.C. Zheng/Phvsics Reports 242 (1994) 233—251

Table 12 3C for various approaches. The experimental value is O.702/IN. The The magnetic moment of the ground state in ‘ calculations are performed relative to an 160 core. See Table 9, for notations Single-particle energies

Two-body matrix elements

~i(in

units of tIN)

N.R.

m*/in

=

I

m*/m =0.78

m*m

(1) G

G

0.994

0.976

0.887

0.814

(2) G + ~ (3) G + ~ (4) G + E

G + BKB G + BKB + L G + BKB + L + HF

0.952 1.028 1.035

0.935 1.015 1.015

0.880 0.963 0.973

0.831 0.910 0.923

(5) G + ~ + HF (6) G + ~ + HF (7) G + E + HF

G + BKB G + BKB + L G + BKB + L + HF

0.823 0.945 0.986

0.812 0.929 0.958

0.771 0.867 0.902

0.759 0.840 0.864

°.814PN for

m*/m

=

=

0.67

0.67. The result is still larger than experiment (O.7O2PN). The results are not

improved by including core-polarization corrections.

11. States which are not so sensitive to the Dirac effective mass It is popular to test out the nuclear interaction in a nucleus by looking at the low lying spectrum of a closed shell plus two nucleons, e.g., the spectrum of 180 We would argue that this spectrum will not be as sensitive to the variation in the Dirac effective mass as the spin—orbit state in ‘2C. The reason is that the two-particle states have wave functions which already are close to thejj limit, e.g., d~/ 2in 180 As we lower the Dirac effective mass, we increase the spin—orbit strength and thus make thej] limit better. This means that the main configuration will not change significantly, rather there will be a slight decrease in the configuration mixing. 18F. This state is known to be strongly mixed A may possible this is the 1 LS T =limit 0 state in in thejj limit. So the energy of this state might and evenexception be better to described in the than well be sensitive to the Dirac effective mass. Next we consider deformed states, e.g., the ground state band in 12C or 20Ne. The intrinsic state is dominantly an S = 0 state. In the limit of pure S = 0, both the expectation value of the spin—orbit and tensor interactions will vanish. Thus the leading component will not be sensitive to the changes in the spin—orbit or tensor interactions. Of course there will be S = 1 and S = admixtures due to the presence of these interactions. However, if they are small it will be hard to detect their presence in the “usual” experiments such as B(E2) 0~=2~’ and even harder to detect changes in these admixtures. 2C from the ground state to 4.1 MeV 2~state. some numbers, the consider B(E2) ‘ In Just a fullto pgive space calculation, valuesthe with ourinschematic interaction V~+ xV~. 0.+ yV1 are 2fm4, tensor off(x = 1, y = 0) B(E2) = 16.12e

~,

2

L. Zamick, D.C. Zheng/Phvsics Reports 242 (1994) 233—251

tensor on (x

=

1, y = 1)

B(E2)

=

249

15.62e2fm4

Note that there is not very much difference between the two cases: full tensor and no tensor, despite the fact that the tensor interaction has an enormous effect on the SPE, causing an inversion of P1/2 and P3/2. The reason, as mentioned above, is that in a full p space calculation, the 0~and states are mainly S = 0 states and the tensor interaction vanishes in an S = 0 state. 2j~

12. Closing remarks We are able to see large effects due to the Dirac phenomenology in nuclear structure of the open shell nuclei because we select the states which should be sensitive to the variation of the Dirac effective mass, and because we calculate SPJE rather than take them from experiment. This last point is very important. For example, as we have emphasized before, the Dirac effective mass affects the two-body spin—orbit interaction in the nucleus which in turn affects the one-body spin—orbit interaction. In our calculation over the range of m*/m from 0.67 to 1, the spin—orbit splitting is, to an excellent approximation, proportional to m/m*. We should emphasize that this spin—orbit effect is a finite nucleus effect since in infinite nuclear matter, the first order spin—orbit contribution is zero. The effects of changing the spin—orbit interaction are best seen in transitions which connect (mainly) S = 0 states with S = 1 states. On the other hand, many ground state deformed rotational bands are mainly S = 0. For pure S = 0, the expectation values of the spin—orbit and tensor interactions vanish so we are insensitive to the effects of the Dirac phenomenology. In the formulation presented here, which is basically the Serot—Walecka model [18], it is the spin—orbit and not the tensor interaction which changes. In analyzing data, one must be aware of the danger of attributing a given effect to one component of the interaction when in fact other components can be equally important. For example, in the beta decay ‘4C —+ ‘4N, the fact that B(GT) is essentially zero is due to the fact that for the bare Bonn A G-matrix, the tensor interaction is too strong relative to the spin—orbit interaction. One cannot from this experimental result conclude that the tensor interaction is too strong or that the spin—orbit interaction is too weak. One must look at a variety of experiments before making more detailed conclusions, e.g., the splitting between the 0, T= 1 and T= 0 states in 160 There are alternate formulations in which the tensor interaction does get weaker in the nucleus, e.g., the universal scaling model of Brown and Rho [30], and the application of this to finite nuclei by Hosaka and Toki [31]. In this model, the p-meson in a nuclear medium becomes less massive. This increases the range of the p-exchange interaction in a nucleus. Since the p-exchange gives a repulsive contribution to the tensor interaction, making the range larger will cause the increased cancellation with the attractive contribution with it-exchange. In this model, the spin—orbit interaction is also enhanced. We have no hard numbers, but there are indications that phenomenologically, at least, the incorporation of such effects will improve agreement with experiment in nuclei. Indeed, since in the model we have presented here, we go towards but do not actually reach experimental agreement, it may be that the extra kick given by weakening the tensor interaction will do the job.

250

L. Zamick, D.C. Zheng/Phvsics Reports 242 (1994) 233—251

Core polarization must be handled correctly before we can ascertain how important more exotic effects in the nucleus are. We have noted that for the problems we address, all the diagrams (Bertsch—Kuo—Brown bubble, particle—particle ladder and hole—hole diagrams) are important. We also find the HF insertions are important and the question of whether they should or should not be included has not been completely answered. There is the underlying problem that the Bonn A in a Brueckner Hartree—Fock calculations leads to too small a radius for 160 We close by reminding the reader of the large results with the bare Bonn A interaction that we obtained for the transition rates. In the Dirac spinor formulation the value of B(M1)o,.,11T. 1 increased from O.69p~ to 1.57p~ (exp = 2.85p~) decreased m*/m I to 0.67. 2N ‘2C changed fromft as= we 73114 to 22753 (expfrom = 13178) andThe the value value of of ft for the transition ‘ B(GT) for the transition ‘4C —* 14N decreased from 3.97 to 0.06 (exp 0) over the same range of m*/m. These large effects give us hope that exotic effects will show themselves up in nuclear structure calculations. —*

Acknowledgments We thank H. Müther for his advice and help. We also thank P. Ellis and D.J. Millener for useful comments. This work was supported by the Department of Energy under Grant DE-FGO5-. 86ER-40299 and by NSERC, Canada under operating grant A-3198.

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