Empirical strengths of spin operators in nuclei

Empirical Strengths of Spin Operators in Nuclei B. H. W I L D E N T H A L Michigan State University, Cyclotron Laboratory, East Laming, Michigan 48824, U.S.A.

ABSTRACT New shell-model wave functions, derived from a single parametrization of the effective Hamiltonian, have been calculated for all sd-shell nuclei. The energies and matrix elements for single-nucleon transfer and electric-multipole operators predicted from these calculations are briefly surveyed i n comparison with experimental values. The matrix elements predicted from these wave.functions for GamowTeller, magnetic-dipole and other related spin-type operators are then compared in detail with the magnitudes of the corresponding experimentally determined matrix elements from A = 17-39 nuclei. From these comparisons, the empirical corrections to the free-nucleon normalizations of these spin operators are extracted. These empirical values of the corrections are compared with theoretical estimates of the effects of many-hm-configuration mixing and of mesonic-exchange currents and isobar excitations. Finally, the predicted distributions of matrix-element magnitudes with excitation energy are studied and compared with experiment, the difference between Gamow-Teller and MI probes examined and the predicted momentumtransfer-dependence of the matrix elements compared with the results of electron scattering experiments. INTRODUCTION The theme of these lectures is the empirical study of the spin operator as i t acts on nuclear states. The ultimate goal of such studies is to understand nuclear structure in depth. With such an understanding, the interplay of quark, mesonic and isobar degrees of freedom with those of the neutron and proton would take place on an equal footing. Such a comprehensive theory is at present, however, far beyond our grasp. Hence, we ooncentrate on individual elements of a composite theory. Our goal in this study is to consolidate current efforts to link the details of experimentally measured matrix elements of spin-type operators, those of Gamow-Teller (GT) beta decay and magnetic dipole (M1) electromagnetic ~rocesses in particular, with theoretical estimates of the higher-order corrections to the standard nuclear-model estimates for these quantities. The spin operators should be particularly incisive probes of the importance of degrees of freedom omitted from conventional models of nuclear structure. Essentially all quantitative theoretical treatments of nuclear matrix elements factor the problem into a "lower-order", "nuclear model" component, which contains the

6

B.H. Wildenthal

description of the "local", (]tim motion of the nucleons, and a "higher-order" component, which contains the perturbative corrections to the lower-order "model" wave functions. Treatments of the higher-order corrections tend to separate into calculations of the "non-nucleonic" effects of mesonic-exchange currents and isobar excitations (quark degrees of freedom s t i l l are not yet considered at the same level of detail) and of the effects of "long-distance" configuration mixing into the nucleonic configurations of the "local" model space of small pieces of many, distant, highly excited configurations. The present study is organized in the context of this factored solution to the complete problem of nuclear structure. We compare as many experimental data as possible with the predictions of the best possible "local" wave functions. The f i r s t aim of these comparisons is to determine the degree to which confidence can be placed in the wave functions as good lower-order solutions to the motions of the neutrons and protons in the nuclear states. The second aim is to determine the degree to which differences between local model predictions and experiment can be concisely and precisely systematized. ( I t is not obvious that this can be done. The boundary between lower-order and higher-order effects could shift from one matrix element to another. I f there were frequent and large shifts between the two factors, however, the factorization approach, while not invalidated, would lose much of its practical appeal.) Finally, then, to the degree that such systematic differences can be established, the concluding aim is to evaluate how well recent calculations of higher-order corrections account for these differences. The essence of the problem we set ourselves is thus clear. Only the net results of combining lower-order and higher-order predictions can be tested against experiment. In order to evaluate the success of a theory for higher-order corrections, i t is necessary to have complete confidence in the lower-order solutions to which these corrections apply. We concentrate upon making the "local model" component as consistently accurate as possible and upon evaluating the degree to which this is achieved. In order to discuss the microscopic structure of nuclear states, some variety of the nuclear shell model (Mayer and Jensen, 1955) must be employed. In i t s elemental form, this model gives quantitative predictions only for a very small number of nuclear states. These are the ground states and, perhaps, a few excited states in nuclei which have either one nucleon more or one nucleon less than a "doubly magic" nucleus. The doubly-magic nuclei, ones whose ground states have both neutron and proton configurations of anomalous s t a b i l i t y , are generally regarded to be 4He, z60, ~°Ca, ~SCa, 1°°Sn, z32Sn and 2°sPb. Those which have different numbers of neutrons and protons, ( " j - j " closed shells) are subject to a particular variety of very strong higher-order (first-order in the conventional sense) correction, the so-called core-polarization correction of Arima and Horie (1954). Hence, only the A = 3, 15, 17, 39 and 41 systems provide nuclear states whose properties should be accurately represented with simple shell-model wave functions consisting of a single nucleon (or nucleon hole) orbiting closed shells of equal numbers (an "L-S" closed shell) of neutrons and protons. Conventionally, predictions of higher-order corrections to nuclear matrix elements have been evaluated by reference to experimental data from these few, l i g h t , "single-particle" nuclei. I t has been thought that only for these simple states is the lower-order description of nuclear structure well-enough understood to make consideration of the higher-order terms meaningful. The restriction of the data base to these few systems severely restricts the range and r e l i a b i l i t y of such evaluations because the small number of "well-understood" single-particle states limits the number of experimental matrix elements that can be obtained. Since the higher-order corrections are calculated for specific shell-model orbits, this lack of data limits the number of predictions which can be tested. In particular,

Empirical Strengths of Spin Operators in Nuclei there is a paucity of data on "off-diagonal" single-particle matrix elements. Moreover, since there is at most one experimental matrix element for a given combination of single-particle orbits, there is no possibility of making a s t a t i s t i cally meaningful estimate of the standard deviation between experiment and theory. Finally, as we shall note, the actual states in the "single-particle" nuclei are possibly subject to a variety of special "local" perturbations which could a l t e r their observed properties at the same level as do the general types of higherorder corrections about which we wish to learn. I t wouldhence be of the greatest interest to be able to expand the data base upon which predictions of higher-order corrections could be tested. We attempt to effect this expansion here by formulating wave functions for the states in nuclei two or more nucleons removed from the magic numbers which in all essential respects have a relationship between model and r e a l i t y which is equivalent to that which exists between simple, single-nucleon wave functions and the states of the nuclei which are one nucleon removed from L-S closed shells. There is one inescapable complication attendant to this progression into nuclei with more than a single nucleon active in lowest order, however. We must specify the model Hamiltonian which guides the mixing into the eigenfunctions of the various n-particle configurations which are degenerate in lowest order. This additional freedom in specifying wave functions does indeed provide opportunities to generate lower-order solutions whose imperfections relative to their own frame of reference are far more significant than the effects of higher-order corrections. Skepticism that such Hamiltonians are, or can be, formulated with the necessary accuracy is a legitimate reason to r e s t r i c t attention to the "Hamiltonian independent" single-particle systems when considering higher-order corrections. We w i l l attempt to allay this skepticism as we present the results we have obtained with a formulation of the Hamiltonian which gives a comprehensive accounting for the vast majority of low-lying states in the region A = 17-39. We will take advantage of the vast diversity of nuclear phenomena which characterize individual nuclear states in great detail and depth. We will argue that the comprehensive, internally consistent theoretical accounting we obtain for these diverse features is strong evidence that we now have a reliable lower-order description of the states at our disposal. DESCRIPTIONS OF NUCLEARSTATESWITH SHELL-MODELWAVEFUNCTIONS In this section we f i r s t sketch the details of how the shell-model calculations which provide us with our local-space nuclear model wave functions are carried out. We then give an overview of how well the particular calculations used in the present analyses reproduce observed features of level energies. We then review the formalism by which the shell-model wave functions are used to generate theoretical values for nuclear observables other than energies. Finally, we give a brief overview of the capability of this model approach to account for singlenucleon transfer data and the shape-collective type phenomena associated with electric quadrupole and hexadecupole operators. Specification of the Model Space and Determination of the Model Hamiltonian The wave functions we use as the models for the energy levels of nuclei in the A = 17-39 region are based in all instances on the assumption that 160 and ~°Ca are perfect closed shells. That is, we assume that the f i r s t 16 nucleons of an A-nucleon system are frozen into the Os and Op orbits and that no excitations are allowed into the Of, lp and higher orbits of the Mayer-Jensen model. Our wave functions are expansions over the complete space of 0d5/2- lSl/2 - Od3/2 configurations.

This is to say that the model wave function F for the #'th state

8

B.H. Wildenthal

of total angular momentum O and total isospin quantum number T in a nucleus of mass number A has the form

NTJ# ~NTJ FATJ# = Zi ai mi

(I)

of the space have the where, in the representation we use, the basis vectors .NTJ mi form BNTJI { { SS(Od5/2)n5'T5'J5×ss ( l S l / 2 ) n l ' T 1 ' J l IT51,J51 x SS(0d3/2)n3'T3'J3} N,T,Ji

(2)

where n5 + nl + n3 = N = A - 16. The statement that we use in all cases the complete space of sd-shell configurations is equivalent to saying that the summations in Eq. I are extended over all the basis vectors Bi of Eq. 2 which are allowed by the Pauli Principle. The amplitudes a~TJ# of the basis vectors Bi of the sd-shell space, which specify the "#"th eigenstate of the subspace NTJ of the complete sd shell space, are obtained from the diagonalization of the matrix of the model Hamiltonian for that subspace. The present results are obtained under the assumption, almost universal in shell-model calculations, that the effective Hamiltonian for the model space can be approximated with only one-body and two-body operators. I t follows from this assumption that such a Hamiltonian is completely specified by the values of i t s one-body and two-body matrix elements and, furthermore, that the matrix element HI, m of this Hamiltonian between the two multiparticle basis states BI and Bm can be expressed as a weighted sum of these one-body and two-body matrix elements Hl,m = EkI Glkl spek I + %k2 G2k2 2bmek2

(3)

A one-body matrix element, or s i n g l e - p a r t i c l e energy (spe) is to be understood as the binding energy of a single nucleon in the shell-model o r b i t k I to the shellmodel core. A two-body matrix element (2bme) is the expectation value of the twobody part of the model Hamiltonian evaluated between two-nucleon states of the model space: 2bmek2 = <(jl,J2)TJIH(2b)I(j3,J4)TJ>

(4)

where the Ji designate the single-nucleon orbits of the model space and T and J are the total isospin and total angular momentum coupling, respectively, of the two-body states formed from these o r b i t s . The coefficients GI and G2 of the spe and 2bme are obtained by carrying out the requisite fractional-parentage, a n t i symmetrization and angular-momentum-coupling algebra with a "shell-model" code. The results presented here were obtained with codes based on (Chung, 1976; Kruse, 1982) the conceptions and formulations of French and co-workers (1969). To summarize: with the a v a i l a b i l i t y of a shell-model code and the appropriate spe and 2bme, Hamiltonian matrices for the various NTJ subspaces of a model space can be constructed. These matrices can then be diagonalized by any of the standard techniques to obtain the eigen-energies and eigen-functions of as many states # of the NTJ set as desired. The results presented here were obtained with a diagonolization code based on a formulation by Whitehead and co-workers (1977) of the Lanczos algorithm. In the ideal, we would be able to carry out a shell-model calculation with a model Hamiltonian t h e o r e t i c a l l y extracted from the experimental features of the i n t e r action of two nucleons in free space. From these data we would obtain the

Empirical Strengths of Spin Operators in Nuclei

characteristics of the single-nucleon potential and of the two-body potential which governs the residual interactions between the "active" nucleons of the model space. From these potentials, the single-particle energies and the two-body matrix elements needed to construct the energy matrices and i n i t i a t e diagonalizations would be straightforward to obtain. For the conventional shell-model spaces for light nuclei, the p shell, the sd shell and the fp shell, nuclear theory has approached tantalizingly close to this ideal. The interactions of this type calculated by Kuo and co-workers (1966,1967,1983) for the sd-shell have perhaps been the most closely studied (Halbert and co-workers, 1971; Wildenthal and co-workers, 1971). I t was found that for nuclei less than five or six nucleons removed from the boundaries of the model space, e.g., A = 18-21 and A = 35-38 for the sd shell, these precalculated Hamiltonians produced energy level spectra and wave function structures which in some absolute sense could be evaluated as "good" and which in the relative sense certainly were as good as, or better than, the results obtained with alternative, more schematic approaches to formulating the Hamiltonian. In spite of the very impressive agreements with experiment obtained by using these precalculated Hamiltonians in shell-model calculations, however, there remain questions about the convergence of the perturbation series involved (Barrett and Kirson, 1974) and about the adequacy of the existing experimental data on the twobody system upon which such calculations must be based. The evolution of shellmodel calculations themselves, independently, have also called into question these precalculated interactions. In the sd shell, the A = 22 and 34 spectra obtained with the Kuo-type interactions show signs of divergence from experiment (Halbert and co-workers, 1971; Wildenthal and co-workers, 1971). The work of the Glasgow group (Cole, Watt and Whitehead, 1975) in sd-shell nuclei with s t i l l larger numbers of active nucleons, e.g. 26AI and 27AI, established that the simple application of the A = 18 Kuo-Brown interaction to nuclei of increasing particle number yielded theoretical energy level spectra which radically diverged from experiment. Problems of a similar nature with analogous interactions were encountered in the fp shell by McGrory, Wildenthal and Halbert (1970).

The more s t r i k i n g disagreements between the experimental energy level spectra of many-active-nucleon nuclei and the spectra obtained from precalculated effective interactions can be removed by shifting the relative centers of gravity of the various single-nucleon o r b i t s , either by shifts of the single-particle energies themselves or by alterations of the average strengths of the o r b i t - o r b i t residual interactions. The s h i f t s in single-particle energies which are required to effect these remedies amount to many MeV relative to the values experimentally established to within several hundred keV in the experimental spectrum of 170. The centroid energies of the diagonal two-body matrix elements connecting d i f f e r e n t orbits need only be shifted by a few hundred keV to effect the same changes in the calculated spectra of the many-body systems. This is a consequence of the quadratic-like dependence of the eigenvalues upon the 2bme, r e l a t i v e to the linear dependence upon the spe. Chung (1976) and Wildenthal (1977), following the path of Preedom and Wildenthal (1972), studied the consequences of empirically altering the 2bme to correct the most obvious discrepancies between the results obtained with the Kuotype interactions and experiment. Chung and Wildenthal were unable to obtain a single set of spe and 2bme with which the energies of all sd-shell systems could be explained satisfactorily. Effective Hamiltonians which yielded acceptable agreement with the features of A = 17-22 nuclei predicted spectra for A = 34-39 nuclei which qualitatively disagreed with experiment. Similar results were obtained by starting with a Hamiltonian f i t t e d to A = 34-39 and extrapolating i t to A = 17-22. These attempts implicitly assumed A-independent Hamiltonians, as is the convention in shell-model calculations, even though the effects of introducing linear and quadratic dependencies of the spe upon A were studied (to no avail). The efforts of Chung and Wildenthal terminated

B.H. Wildenthal

i0

Table 1.

Values of the two-body matrix elements and single-particle energies o f the A = 18 version of the new empirical Hamiltonian for sd-shell model calculations, Units are MeV. The two-body matrix elements (2bme) <(jl,J2)jTIH2bI(j3,J4)JT> are labeled by the indices of the Ji according to the convention of "1" for Od5/2, "2" for lSl/2 and "3" for Od3/2 and by the values of twice the total angular momentum and total is,spin of the two-body states, 2J and 2T. The two-body matrix elements for other values of A are obtained by multiplying these values by (18/A) °'3 The values of the single-particle energies (spe) are independent of A.

Jl

J2

J3

J4

2J

2T

2bme

Jl

J2

J3

J4

2J

2T

2bme

1 1 1 1 I 1 1 I 1 I 1 I 1 1 I 1 I 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 3 3

1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 1 1 1 1 1 I 1 1 2 2 3 3 1 1

1 1 1 1 1 1 2 2 3 3 3 3 2 2 3 3 3 3 3 3 2 2 2 2 3 3 3 3 3 3 3 3 3 3

0 2 4 6 8 10 4 6 2 4 6 8 0 2 2 4 0 2 4 6 4 4 6 6 4 4 6 6 4 4 4 6 2 2

2 0 2 0 2 0 2 0 0 2 0 2 2 0 0 2 2 0 2 0 0 2 0 2 0 2 0 2 0 2 2 0 0 2

-2.8197 -1.6321 -1.0020 -1.5012 -0.1641 -4.2256 -0.8616 -1.2420 2. 5435 -0.2828 2.2216 -1.2363 -1.3247 -1.1756 -1.1026 -0.6198 -3.1856 0.7221 -1.6221 1.8949 -1.4474 -0.8183 -3.8598 0.7626 -0.0968 -0.4770 1.2032 -0.6741 -2.0664 -1.9410 -0.4041 0.1887 -6.5058 1.0334

1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3

3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3

1 1 1 1 1 1 2 2 2 2 2 3 3 3 2 2 2 3 3 2 2 2 2 3 3 3 3 3 3

3 3 3 3 3 3 2 3 3 3 3 3 3 3 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3

4 4 6 6 8 8 2 2 2 4 4 2 4 7 0 2 2 0 2 2 2 4 4 2 4 0 2 4 6

0 2 0 2 0 2 0 0 2 0 2 0 2 0 2 0 0 2 0 0 2 0 2 0 2 2 0 2 0

-3.8253 -0.3248 -0.5377 0.5894 -4.5062 -1.4497 2.1042 -1.7080 O. 1874 0.2832 -0.5247 0.5647 -0.6149 2.0337 -2.1246 -3.2628 1.2501 -1.0835 0.0275 -4.2930 0.6066 -1.8194 -0.4064 0.3983 -0.5154 -2.1845 -1.4151 -0.0665 -2.8842

spe(Od5/2) spe~lSl/2/r~ spe(Od3/2 )_.

-3.94780 -3.16354 1.64658

Empirical Strengths of Spin Operators in Nuclei

11

with two A-independent Hamiltonians, one for the A = 17-28 region and one for the A = 28-39 region. These Hamiltonians yielded good agreement with experiment for systems up through, respectively, nine particles or nine holes in the sd-shell. For the nuclei from A = 26 through A = 30, agreement of the predicted spectra with experiment was qualitatively not too bad, but definitely inferior to that obtained for the systems of three to nine particles or holes. The results of Chung and Wildenthal can be interpreted either as indicating an intrinsic difference between the structures of the lower and upper sd-shell regions or as indicating the necessity of introducing some A-dependence into the specification of the Hamiltonian. Several aspects of their results as well as general theoretical considerations suggested that the l a t t e r course be explored. Inspection of Kuo's results (Kuo, 1966; Wildenthal and co-workers, 1971) shows that the precalculated two-body matrix elements for the "°Ca region are typically smaller in absolute magnitude than those of the 160 region. Roughly, there is a simple diminution in the absolute magnitudes of the two-body matrix elements in going from i~0 to "°Ca of about 25%, although individual terms may d i f f e r s i g n i f i cantly from this norm, sometimes even increasing in magnitude. This result served as the rationale for concentrating upon the A-dependence of the 2bme. I t was assumed as a test that the 2bme appropriate for the Hamiltonian of a given value of the nucleon number A were related to the 2bme of A = 18 by the formula 2bme(A) = 2bme(18) x (18/A)P, where p ranged from 0.2 to 0.4. This assumption can be viewed as the minimal attempt to insert the effect of increasing nuclear size upon the matrix elements of a constant-strength interaction. Trials with the Chung-Wildenthal "Particle" interaction and the Particle and Hole wave functions immediately established that this sort of A-dependence in the 2bme instantly cured the major aspects of the disagreements previously found between experiment and the results obtained with A-independent 2bme. With this encouragement, we then embarked upon determining a new iterated set of 2bme with the technique of Chung (1976). The final values obtained yielded a stable best f i t to a set of 440 data, which sampled every sd-shell system with experimentally know energies, under the constraint that 2bme magnitudes decreased with A as (18/A) O.J (Wildenthal, 1982). The results are presented in Table 1.

Critique of Shell-Model Ei~envalues It is not our aim here to exhaustively review the eigenvalues obtained from the diagonalizations of the new A-dependent effective sd-shell Hamiltonian described in the preceding Section. Rather, we only wish to b r i e f l y glance at these results from several different perspectives so as to i l l u s t r a t e the efficacy of the present approach to formulating the Hamiltonian. One perspective we wish to emphasize is the "global" one, in which we examine the capability of the "universal sd-shell" Hamiltonian to account for the variations in observed energy-level phenomena as a function of the numbers of neutrons and protons in the nucleus as these range over all the values which f a l l within the confines of the shell. Another perspective emphasizes the consistency and accuracy with which the calculations reproduce the lowest several energy levels of the various angular momenta J and isospins T in all of the types of nuclei found in the shell. Lastly, we wish to i l l u s t r a t e the degree of comprehensiveness with which the degrees of freedom available within the sd-shell configurations are sufficient to reproduce the observed positiveparity level densities in the middle of the shell. Shell-wide systematics of binding energies. An overview of the agreement between the calculated and measured binding energies of the ground states of sd-shell nuclei is provided by the comparison of theoretical and experimental two-neutron separation energies shown in Fig. 1. The general run of the results shown in Fig.

12

B.H. Wildenthal

~I N

I

30

I

I

Mg

e Na t

AI

I

I

I

I

Si

P

S

CI

I

I

I

Ar K

t~ \

\ 25

2 o ~F

~E 15

v

cOJ

oo

"

0

I0

I0

II

12

13

14

15

16

17

18

19

20

N--* Fig. 1.

Calculated and measured values of the two-neutron separation energies for each sd-shell isotope chain. The calculated values are indicated by the lines, which connect the value at one neutron number with the value at the next. Those isotopes for which experimental values are available are indicated by the solid or open circles. The diameters of the circles and their placements relative to the lines indicate the magnitudes and directions of the differences between experiment and theory.

Empirical Strengths of Spin Operators in Nuclei

13

is consistent with the 150 keY rms deviation obtained between experiment and theory for the 440 level-energy data used in the f i t which determined the effective Hami!tonian. The only discrepancies between experiment and theory which are significant relative to this standard are those evident for the most neutron-rich isotopes of Mg and Na. The masses measured (Detraz and co-workers, 1979 and 1983; Thibault and co-workers, 1980) for these nuclei correspond to binding energies which are anomalously large with respect to most systematic estimates. Interpretation (Campi and co-workers, 1975; Watt and co-workers, 1981) of these anomalies has involved invoking an inversion of the ordering of the d3/2 and f7/2 orbits. The qualitatively striking differences between the experimental values for these isotopes and our version of sd-shell systematic energies is certainly suggestive that some foreign degree of freedom becomes operative in these ground-state wave functions. #

On the conservative side of the question of whether the sd-shell boundaries are broken for at least these few very-neutron-rich systems, we should note that the experimental masses are determined in extremely d i f f i c u l t experiments and have large quoted uncertainties. However, their masses are not the only evidence available for these systems which indicates they have structures inconsistent with those predicted by extrapolating (via our universal Hamiltonian) the features of the vast majority of sd-shell energy levels. We will note the nature of these additional anomalies as we proceed. I t would obviously be of great interest to experimentally delineate the boundaries of this possible "island of inversion" by establishing the masses of the neighboring N = 18, 19 and 20 isotopes. In addition, of course, more precise determination of these Mg and Na masses themselves would be invaluable. Embedded within the comparison of nuclear binding energies are very simple assumptions about Coulomb energies. The shell-model eigenvalues correspond to the nuclear part of the total binding energies of the last A - 16 nucleons to the 160 core. The experimental values used in the comparison in Fig. I and in the f i t are obtained from the mass tables of Wapstra and Bos (1981) and analog-state energies. No allowance is made for size effects upon the Coulomb energies as we move along a chain of isotopes to extract the nuclear parts of the binding energies. The effects of this simplification could well be noticeable in the comparisons for the very-neutron-rich systems. In addition, a cumulative error of several MeV in the putative "experimental" energies of the nuclei near "°Ca is possible due to our treatment of the Coulomb energy problem. The l a t t e r aspect should have l i t t l e effect on wave functions, however, since the compensating alterations to the effective 2bme are of a monopole character. Away from the Z = 11,12, N = 19,20 island, the comparison of theory with experiment shown in Fig. 1 suggests that the nuclear masses over the entire range of 8 ~ N, Z ~ 20 are consistent with sd-shell degrees of freedom. The essential point we wish to make is that the detailed variations of binding energy with respect to changes in the numbers of neutrons and protons which are experimentally observed are accurately reproduced with a fixed set of shell-model orbitals and a fixed one-body and two-body interaction. The independent variables in shell-model theory, their fundamental importance stemming from the dominant role of the Pauli Principle, are the numbers of neutrons and protons. Our present theoretical "function" of these variables follows the experimental "curves" through the N-Z space quite well. Shell-wide systematics of excitation energies. The usual "mass formulae" treatments of nuclear binding energies do not have the capability of simultaneously dealing with excitation energies. The f u l l shell-model approach on the other hand can treat ground-state masses and excitation energies on anequal footing. In Fig. 2 we i l l u s t r a t e the dependence upon N and Z of the measured and the calculated excitation energies of the 2+ and 4+ states in doubly-even nuclei. The spectra are

B.H. Wildenthal

14

I I

i.

I | I

iI I I

I

I

I

l

• •

".1 ~. •

"; I .i

i

l

I

: I

v

)-

i i

"~

~ ~

F.3

|

i

i

w

I

i

I .I t" I_I

6

I-

4" II

T=I 18

26

>

T:O 34

20

28

T:2 36

24

3 :>

T:3 T=4 22

30

A Fig. 2.

Excitation energies of the f i r s t 2+ (solid lines), second 2+ (dashed lines) and f i r s t 4+ (dotted lines) states in the doubly-even nuclei of the sd-shell, labeled by mass number A and total isospin T. The lines connect the calculated values for each system. The circles indicate the existence of experimentally measured values corresponding to the calculated values. The diameters of the circles give the magnitudes of the deviations between experiment and theory and their placements the directions of these deviations, The ovals indicate situations in which the theoretical state corresponds to a m u l t i p l i c i t y of several experimental states.

Empirical Strengths of Spin Operators in Nuclei

15

grouped in Fig. 2 according to the differences N - Z = 2T and arranged in order of increasing value of A. We see that within each T group the observed variation with A is theoretically reproduced with about the 150 keV standard deviation characteri s t i c of the f i t . Likewise, the differences in the patterns of the different T groups are also reproduced. We call attention to the different behaviors of the f i r s t 2+ , the second 2+ and the f i r s t 4+ states as N - Z and A change. As is the case with the binding energies, the variations of the 2+ and 4+ excitation energies reflect the interplay of the model Hamiltonian and the Pauli Principle. The same "function", the model Hamiltonian, that yielded general agreement with the experimental finding energies also yields agreement of an equivalent quality with these experimental excitation energies. The most striking discrepancy between the predicted excitation energy for a f i r s t 2+ state and the corresponding experimental value (Detraz and co-workers, 1983) occurs for 32Mg. This nucleus was part of the island of very-neutron-rich Na and Mg isotopes whose experimental binding energy values were anomalous. The very low measured excitation energy of this state is further evidence that the lowest energy configurations of this system may be dominated by non-sd-shell configurations. There are no clear-cut discrepancies between theoretical and experimental " f i r s t 4+'' energies. For the second 2+ states the correspondences between experiment and theory are ambiguous near the shell boundaries. The experimental spectra have more 2+ states at these excitation energies than do the theoretical spectra, and the character of the model state would seem to be fragmented over several physical states. The "extra" states are referred to as "intruder states". Accumulated evidence suggests that near the shell boundaries certain preferred many particle-many hole excitations can bridge the shell-closure barrier with surprising ease and create states with parentages qualitatively different from the nominally "normal" configurations at low excitation energies. In the sd-shell, these "intruder" configurations would come from the Op shell near N,Z = 8 and the Of,lp + shell near N,Z = 20. Both the A = 18 and A = 38 spectra apparently have intruder2 states. (A similar situation obtains for the second 0+ states, but these are not part of the display of Fig. 2.) The island of inversion we discussed as an explanation of the binding energy, and now excitation energy, anomalies in the veryneutron-rich Na and Mg isotopes would constitute a case of the intruder configuration becoming the ground state. We emphasize that our results, here and later, strongly suggest that intruder states compete most effectively with "orthodox" states near the shell boundaries for the neutron and proton numbers. Near the "middle" of the shell, intruder states seem to be pushed up to significantly higher excitation energies. That this should be so is not self evident. I t could be imagined that large numbersof active nucleons serve to polarize the nucleus and soften the core, so as to destroy the shell boundaries more completely in between the magic numbers than near them. Indeed, this is more the conventional view. However, here in the sd-shell, empirical evidence, counting our f i t t e d Hamiltonian as part of this evidence, is to the contrary. The A and N dependencesof some simple excitations in odd-mass nuclei are shown in Figs. 3 and 4. The relative energies of the lowest I/2 + , 3/2 + and 5/2 + states are shown, grouped according to the number of the last odd nucleon and ordered with increasing A. This arrangement is in the s p i r i t of the Nilsson (1956) model, which emphasizes the role of the last unpaired nucleon as i t moves in a deformed well generated by the other nucleons. I t is seen from Figs. 3 and 4 that the model Hamiltonian automatically generates, from the interactions among all the active nucleons in concert with Pauli Principle effects, the observed changes in the spacings between these simple excitations as A and N change.

16

B.H. Wildenthal

5/2 --- 3/2 ..... I / 2 > u

=E )4

!

I I

.=

I

Q

UJ

!

z3

0

I I ! !

0

I"

6 9

iI"

-% Q

I

•

jO

LIJ

°

)

•

0

Q,

°

"•

I

e:

\

. Q

•

~ .°

o

•

Ii

•

it

O

".

~ ,.kI.Q "'o. 6M "'.: • o

e oO

0

D O

o 0.,.I 0

I?0 21F 2sF 29F 190 23 N 27 No 31No210 2SMg 29AI 33AI 19F 23 F 27F 9 - EVEN Fig. 3.

21Ne 25No 29No II-EVEN

23Ne 27AI 31AI 13-EVEN

Energies of the lowest 1/2+ and 3/2+ states in the odd-even nuclei with nine, eleven and thirteen odd nucleons, relative to those of the lowest 5/2+ states. The theoretical and experimental energies of the 5/2+ states are set to zero to form the base lines. The theoretical energies are indicated by the lines (dotted for 1/2+ , dashed for 3/2+ ) which connect the values at one A value with the next. The circles indicate experimental measurements, with the diameters of the circles giving the sizes of the deviations between experiment and theory and their orientations the directions of the deviations.

Empirical Strengths of Spin Operators in Nuclei

17

5/2 D

0

0

I

0

3/2 I/2

>5 v

~4 n,. W

z

0 F.

x

i,~2

o

". '

Xx /E~.,o,

.."0

X, O ,+

O".0" +

•

,.,.,...,.... 230 271Vlg31p 35p 25029Mg33S 37CI 270 311Vlq35 S 39K 25Ne29Si 33F, 27Ne31Si 35CI 29Ne 33Si 3~r 15- EVEN Fig. 4.

17-EVEN

19-EVEN

Relative energies of the lowest 5/2 + (solid lines), 1/2 + (dotted lines) and 3/2 + (dashed lines) states in the odd-even nuclei with f i f t e e n , seventeen and nineteen odd nucleons. For the " f i f t e e n " group, the energies of the I/2 + states are set to zero, for the "seventeen" and "nineteen" groups, the energies of the 3/2 + states are set to zero. The conventions of the presentation are as described in the caption to Fig. 3.

18

B.H. Wildenthal

The most significant discrepancy shown in Figs. 3 and 4 involves 31Na. The ground state spin is measured to be 3/2 and the calculated ground state is 5/2 +. Hence, the binding energy discrepancy for this member of the group of anomalous masses is even more anomalous than is indicated by Fig. 1. Again, this is further evidence that the actual ground-state structure in this region is qualitatively different from that of the rest of the sd-shell systems. The large circles for the 3/2 + in A = 17 and the 5/2+ state in A = 39 reflect the same sort of fragmentation of the model state over several physical states that we discussed in the case of the second 2+ states in the doubly-even nuclei. Accuracy of individual spectral details. How well can the present formulation of the sd-shell Hamiltonian reproduce the detailed orderings of, and spacings between, the many-body eigenstates characterizedby the various values of total angular momentum J found at low excitation energy? As examples, we examine the sequence of nuclei 2~Al, 28Si and 29Si. Comparisons between calculation and experiment for these odd-even and doubly-even nuclei are shown in Fig. 5. Equivalent results for the example of a doubly-odd system will be included in later paragraphs in the context of a discussion of level densities. These nuclei were l i g h t l y represented in the set of energy-level data used to f i x the parameters of the model Hamiltonian. This was merely a practical consequence of the fact that the dimensionalities of their wave functions, and hence the times required to construct and diagonalize their energy matrices, are so large. Only the J = 1/2 states of A = 27 and 29 and the J = 0 and 2 states of A = 28 were part of the set of f i t t i n g data. Hence, general agreement between the calculated and observed spectra can be taken as a significant validation of the predictive power of the new Hamiltonian. There would, in our opinion, be minimal excuses for poor agreement for these nuclei, because of their position in the middle of the shell. As we have argued (although the present results are part of the argument) the lowenergy spectra of these nuclei should be relatively free of positive-parity intruder states. That is, we think that the complete Od5/2 - lSl/2 - Od3/2 basis space should, at the least, provide as good a model for these nuclei as for any others in the A = 17-39 region. Inspection of Fig. 5 shows that all of the observed features of the low-lying energy-level spectra of these nuclei appear in the spectra of the shell-model eigenvalues. Over the f i r s t 6 or so MeV of excitation energy in each nucleus, the total number of observed positive-parity levels is correctly predicted, the number of levels of each J value are correctly predicted, and, within the standard deviation for the f i t t e d data themselves, the details of spin sequences and inter-level spacings are correctly predicted. The only level within this range for which we think the results indicate a significantly anomalous character is the third J = 0 level of 28Si. Thus, the observed spectra around the middle of the shell, within which all three sd-shell orbits play important roles, are predicted from a Hamiltonian which simultaneously reproduces the single-particle and single-hole spectra of A = 17 and 39, respectively, the essential features of the two-particle and two-hole spectra of A = 18 and 38, respectively, and with increasing detail, the features of all other sd-shell systems in between. As a final example of the power of the shell-model to give a detailed accounting of nuclear spectra we present in Table 2 (T = 0 states) and Table 3 (T = 1, 2 and 3 states) calculated and measured level energies for A = 26, This mass should be, like the cases shown in Fig. 5, the optimum environment for the model to match reality over an extended range of excitation energies. As for the A = 27, 28 and 29 nuclei, only a few of the levels of A = 26 were included in the f i t of the Hamiltonian. (These are indicated in the Tables.)

Empirical Strengths of Spin Operators in Nuclei

19

.

I ,,,

3

6 >

~

7 9 ~-__.___~~ II

>- 5 (.9 r~ i,I Z

5 95 ~

7 I I - - ~ - - - - ~j .-. . . - . -

~

4

I

5,..

9

7~

5 /

ua 4 z 0

I

~3

,,

X W

O

,

5

5

7-----.-.___.__

2

2

~ -

~

5--------___ i _

0

5

0 27AI

Fig. 5.

I 28Si

29Si

Excitation spectra of low-lying states of 27AI, 28Si and 29Si. The calculated and measured groundstate energies are set equal to each other in these plots. In each of the three columns, the left-hand terminus of each line has the value of the calculated excitation energy of the state of the labeled J and the right-hand terminus has the corresponding experimental value. Knownnegative-parity states in the experimental spectra are not shown. All other experimentally known states are entered up to the energy limits plotted.

B.H. Wildenthal

20

TABLE 2.

Calculated, E(th), and measured, E(exp), energies of T = 0 states in 26AI. The angular momenta of the states are denoted by the values of 2J in the columns of that heading and the ordering nuB~er within an ATJ group is given in the "#" columns. The "nuclear" binding energies relative to 160 (Wapstra and Bos, 1981) are given for the lowest state in each spectrum. Excitation energies relative to these ground state values are l i s t e d for the other states. All experimental levels below 6 MeV (before the line "xxxxx") which are known to exist and not to have T = 1 are l i s t e d . Only experimental levels thought to correspond to theoretical states are included for higher excitation energies. States of known negative parity are listed in the E(exp) columns, with identification given in the last, "c", columns, but they, of course, do not have correspondences in the E(th) columns. The experimental entries not otherwise annotated are given unambiguous assignments of spin, isospin and parity in the compilation of Endt and van der Leun (1978). The symbol "a" in the "c" columns indicates some ambiguity in the assignment of J, the symbol "p" uncertain parity and "?" denotes only that no contrary evidence to the assumed correspondence is known. The i d e n t i f i e r s "E" and "F" in the columns "c" indicate unpublished assignments by Endt (1983) and Fox (1982).

E(th)

2J

#

2T

E(exp)

c

-105.617 0.712 0.818 1.326 1. 737 2.004 2.121 2.303 2.325 2.588 2.749 2.899 3.069 3.309 3.334 3.357 3.422 3.655 3,685 3.749 3.923 4.052 4.103 4.380

I0 6 2 4 2 2 6 8 6 4 4 2 6 8 12 6 10 6 2 14 4 8 6 6

1 1 1 1 2 3 2 1 3 2 3 4 4 2 I 5 2 6 5 i 4 3 7 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

-105. 756 0.417 1.058 1.759 1.850 2.072 2.365 2.069 2. 545 2.661 2.913 2.740 3.074 3.675 3.507 3.596 3.403 3.681 3.724 3.922 3.751 4.025 3.963 4.349 4.430 4.480 4.622 4.941

*

4.481 4.698 4.721 4.938 5.014 5.154 5.211 5.288

10 0 8 2 6 2 8 4

3 1 4 6 9 7 5 5

0 0 0 0 0 0 0 0

4.773 5.010 4.952 5.585 5.245 5.494

E(th)

2J

#

2T

* * *

*

* * E E 4O4F E E a,E E E a,E

5.348 5.453 5.533 5.579 5.668

8 4 10 10 2

6 6 4 5 8

0 0 0 0 0

5.944 6.003 6.031 6.069 6.149 6.176 6.241

2 4 8 10 4 12 6

9 7 7 6 8 2 10

0 0 0 0 0 0 0

6.243 6.321 6.448 6.520 6.658 6.667 6.688 6.941 7.017 7.114 7.167 7.265

14 10 10 12 2 8 4 2 10 4 10 8

2 7 8 3 10 8 9 11 9 10 10 9

0 0 0 0 0 0 0 0 0 0 0 0

E(exp)

c

5. 396 5.431 5.456 5.513 5.849 5.488 5.568 5.671 5. 598 6.203 5.949 5.882 6.084 5.949

826E E F F E 6E p,E ? a,E ?

5.676 5.692 5.916 xxxxx 6.694 6.496 6.550

E 64-

6.551 6.680

a,E E

7.015

E

7.292

E

F E E

21

Empirical Strengths of Spin Operators in Nuclei TABLE 3.

Calculated and measured energies of states of T = I , 2 and 3, in 2~g, 26Na and 26Ne, respectively. The conventions, definitions and sources are as given in the heading to Table 2. The i d e n t i f i e r "G" in the "c" columns indicate unpublished assignments by Glatz and co-workers (1983).

E(th)

2J

#

2T

-105.536 1.929 3.153 3.681 3.921 4.511 4.533 4.541 4.932 5.000 5.204 5.404 5.473 5.833 6.009 6.062 6.268 6.647 6.777

0 4 4 0 6 6 8 4 8 4 0 4 8 2 8 0 6 4 8

I I 2 2 I 2 1 3 2 4 3 5 3 I 4 4 3 6 5

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

6. 798

2

2

2

6.843

4

7

2

7.038 7.093 7. 282 7.411 7.465 7.473 7.602 7.721 7.941 8.005 8.131 8.194 8.391 8.404 8.413 8.428 8.443 8.520 8.645

10 4 6 8 10 4 6 2 8 6 0 12 4 6 8 12 2 10 0

i 8 4 6 2 9 5 3 7 6 5 1 10 7 8 2 4 3 6

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

E(exp) -105.528 1.809 2.938 3.588 3.940 4.350 4.319 4.332 4.900 4.834 4.972 5.291 5.474 5.690 5.716 6.256 6.125 6.744 6.621 6.887

c

E(th)

2J

#

2T

*

8.716 8.790 8.935 8.959 9.034 9.081 9.115 9.146 9.190 9.236 9.304 9.387 9.458 9.576 9.643 9.724 9.768 9.781 9.827 9.908

4 8 10 8 12 2 6 2 4 8 6 8 10 6 2 12 10 8 2 14

11 9 4 10 3 5 8 6 12 11 9 12 5 10 7 4 6 13 8 I

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

-92.227 0.182 0.187 0.413 1.429 1.531 1.600 1.652 1.829

2 6 4 4 6 2 0 8 8

-81.625 2.011 3.448 3.662 3.812 4.356 4.717

0 4 4 8 0 2 4

*

* a 6-

7.062 7.099 xxxxx 6.978

G

7.395

G

E(exp)

c

9.065

p,G

9.112

G

9.989

G

9.829

G

I I 1 2 2 2 1 I 2

4 0.088 4 -92.265 4 0.241 4 0.420 4 4 4 4 4

*,? * ? ?

1 1 2 I 2 I 3

6 -81.427 6 6 6 6 6 6

*

2"

8.202

G

8.472

G

8.670

?,G

We see from these tables that up to about 6 MeV excitation energy, the complete observed spectra of T = 0 and 1 positive-parity states, totaling several dozen states in a l l , are accurately predicted. At excitation energies higher than 6 MeV i t is increasingly d i f f i c u l t to completely characterize the experimental spectra. Intruder states must s t a r t to appear at energies not much higher. Nonetheless, the correspondences between model and physical states, p a r t i c u l a r l y for states of high spin, can be pursued up to 10 MeV at the present state of experimental knowledge.

22

B.H. Wildenthal

Concurrent with this very detailed and extensive correspondence between model and experiment for the T = 0 and I spectra, agreement of a comparable sort is obtained for the lowest states of T = 2 and 3, even though the ordering of the ground-state multiplet of 26Na is not predicted correctly. Calculations of Matrix Elements of One-Body Operators The v a l i d i t y of the Hamiltonian we have obtained by insisting on the best possible match between shell-model eigenvalues and experimental energies can be evaluated most objectively by testing the wave functions which are associated with the calculated energies. These tests involve evaluating the matrix elements of operators assumed to correspond to the various measurements typical of nuclear spectroscopy. We follow the convention of assuming that these measurements correspond to the action of one-body operators on the wave functions, the "impulse approximation". That is, we assume that the physical process changes the state of only one nucleon in a wave function FNTJ# (which is to say, in any of the basis vectors B~TJ~ at a time. Thus, the prediction of the shell-model wave functions F and F' f~r a process of rank L in angular momentum is contained in the elements of the one-body density matrix D(N,J'T'#', JT#;L,AT;j,j'): D(N,J'T'#',JT#;L,AT;j,j') = < FNT'J'#' Ill (a; x aj, )L'~TIIIFNTJ#> (2L + 1)½(2AT + 1) ½

(5)

The total matrix element M(Op;N,T'J'#',TJ#) predicted to correspond to an experimental measurement for some process "Op" is obtained by combining these algebraic reductions of the mixed-configuration wave functions with the elemental matrix elements corresponding to the action of the physical process on a single nucleon in one of the single-particle orbits of the shell-model space. These elemental, or "single-particle", matrix elements, S(Op,j,j';AT), are the condensation of the nature of the physical probe of the nuclear state. The total matrix element M is obtained by a sum over the pairs of orbits of the model space. We have M(Op;N,T'J'#',TJ#;AT) = Z D(N,T'J'#',TJ#;j,j';L,~T)S(Op,j,j';AT) j,j'

(6)

where the particular isospin projection of the nucleus under consideration is dealt with by combining the AT = 0 and 1 matrix elements of Eq. 6 with the appropriate isospin vector coupling coefficients to obtain the total matrix element M(Tz).

M(Tz) = AT ~ i-I)

Tf-Tz(T f \ AT TzTi)M(AT) t-T Z 0

(7)

The values of the single-particle matrix elements S are obtained in "lowest order" simply by taking the expectation values of the conventional forms of the spectroscopic operators with the wave functions of the individual shell-model singleparticle orbits. The coefficients of the terms in these operators would be given the values consistent with the measured properties of the free neutron and proton. We refer to the resulting S values and the total theoretical matrix elements obtained with them as the "free-nucleon" predictions of the model. In the context of the factored theory for nuclear structure which we outlined in the introduction, the higher-order effects of the orbits of the total shell-model space which were excluded in the construction of the e x p l i c i t model wave functions and of the

Empirical Strengths of Spin Operators in Nuclei

23

non-nucleonic components of the physical wave functions which were also neglected in the lower-order model representation can be incorporated into the calculation of the total theoretical matrix by modifying, or "renormalizing", the values of the single-particle matrix elements from S(free-nucleon O p ; j , j ' ) to S(effective Op;j , j ' ) . One could use the shell-model one-body density matrix elements D in combination with single-particle matrix elements S(effective) which incorporate predictions for higher-order corrections to produce theoretical values with which to compare experimental values. We will take an alternate approach. We will analyze measured matrix elements with the D elements and extract empirical values of the S(effective O p ; j , j ' ) . We will then compare these empirical values of the effective single-particle matrix elements with the corresponding values from calculations of higher-order corrections. The empirical values of S ( j , j ' ) are just the least-squares solutions to sets of linear equations of the form of Eq. 6, in which the D ( j , j ' ) ' s are the constants, the S ( j , j ' ) are the unknowns and measured values of the total matrix elements are inserted in place of the theoretical values of M. Thus an experimental measurement does not in general correspond to a single S ( j , j ' ) but to a linear combination. This is the complication resulting from going from single-particle representations of nuclear states to multi-particle representations corresponding to the necessity of introducing the two-body part of the Hamiltonian. As recompense for having to go through the complication of a least squares f i t rather than simply dividing experiment by lower-order theory, we can now use as many pieces of data as the complete sd-shell and experimental ingenuity combined can give us when we try to determine the empirical renormalization for a spectroscopic operator. In the displays and discussions of the predictions of the shell-model wave functions which will occupy us in the remainder of this text we w i l l , in the context of this section, refer to ',free-nucleon" predictions and " f i t t e d " , or "empirical", predictions. Critique of Predicted Single-Particle Structure In this section we very b r i e f l y review the predictions of our shell-model wave functions for single-nucleon-transfer processes. Experiments in which a single nucleon is added to or subtracted from a target state, i n a " s t r i p p i n g " or "pick-up" "direct reaction", yield very clearcut evidence about those states in the residual system which are strongly populated. These states are identified as having components in their wave functions whose structures are those of the target wave function coupled to a nucleon (or nucleon hole in the case of pick-up) in one of the single-particle orbits of the shell-model sequence. States weakly populated are shown not to have these components, of course. These data thus clearly ident i f y , in the language of the shell model, some of the microscopic structural elements of the states. Historically, they have provided some of the most compelling empirical support for the whole shell-model approach to understanding nuclear structure. We u t i l i z e the details of such experiments to evaluate whether our model wave functions are correct in their basic construction. We want to evaluate the predicted mix of particles in the ground-state wave functions and we want to evaluate the predicted response of these wave functions to the addition or subtraction of a nucleon in one of the active orbits. The data of interest in this context are the sums and the d i f f e r e n t i a l energy distributions of the "spectroscopic factors" (basically reduced cross sections) for stripping and pick-up of nucleons of the various orbits j of the model space. The theoretical values are the matrix elements of the creation operator a~ between the states of mass number A and A + 1. J

24

B.H. Wildenthal

o.sJ

I s I/2

0.6 0.4

,I

0"2f

I: 0.8 "

0 ds/2

~ 0.4

o.6 F

0.2"I I., 0.8

0 d 3/z

0.6 OA 0.2

I

I

Fig. 6.

I

!

II

!

2 ~3 4 5 6 7 E×CITATION ENERGY (MeV)

II

Comparison of predicted and measured spectroscopic factors for proton stripping onto 22Ne. The square roots of the spectroscopic factors S(j), as calculated (solid lines) and measured (dashed lines) are plotted at the calculated and observed energies at which they occur for each of the three orbits of the sd shell.

Empirical Strengths of Spin Operators in Nuclei

25

We show in Fig. 6 typical results from a comparison of predicted and measured (Endt, 1977) spectroscopic factors, the example being the stripping of protons onto 22Ne to form states in the spectrum of 23Na. We have separated the results for the three sd-shell orbits so that we show the responses of the ground state of 22Ne to the creation of a proton in the 0d3/2, the Od5/2 and the ISl/2 orbits individually. We see from the close correspondences between experiment and theory in all respects of this comparison that the single-particle structure of the shellmodel wave functions is completely consistent with experimental evidence. This quality of agreement is found throughout the shell for transfer on even-mass targets. The spectra obtained for odd-mass targets are much more complex than the one shown in Fig. 6 because more than one J orbit can participate in the transition to a single final state and because the level densities in the doubly-odd systems are so high. Even so, the agreement between theory and experiment is equivalently good for transfers onto odd-mass targets, with the proviso that in a few cases in which two levels of the same spin are almost degenerate in a double-odd final system, only the sum of spectroscopic factors to the doublet is reproduced, rather than the detailed s p l i t t i n g of the strength. Critique of PredictedShape-Collective Features In this section we b r i e f l y review aspects of the shell-model wave functions that are related to the size and shape of nuclear states. The spectroscopic operators of interest are those of the even electric multipoles, EO, E2 and E4. The angular momenta of the orbits of the sd shell and the restriction to one-body operators rule out E6 and higher multipolarity transitions. The electric operators (Brussaard and Glaudemans, 1977) involve combinations of spherical harmonics and powers of the radial coordinates of the model nucleons. The shell-model formulation we follow in this work does not directly address the description of the radial component of the single-particle wave functions since we specify the Hamiltonian s t r i c t l y in terms of matrix elements. We follow the conventional course of assuming, or imposing the constraint, that the radial extensions of the single-particle wave functions are such that the multi-particle wave functions constructed from them have radii consistent with the measured radii of the nuclei of concern (Brown and co-workers, 1977; 1980; 1982). Thus, the basic empirical scale of nuclear sizes is inserted into our wave functions independently (or almost so) of the details of the predicted configuration mixing, by empirically fixing the sizes of the single particle wave functions. All of the results presented here are obtained with the assumption that the radial wave functions have harmonic o s c i l l a t o r form, with the size parameters set so that the measured rms charge radius of the stable ground state of the A value treated is, after the usual corrections for nucleon f i n i t e size and center of mass motion are made, reproduced by the rms radius of the shell model wave function. This simple approach of coupling a single additional radial scale parameter to all the various mixed-configuration wave functions of a given A value works well enough to make further improvements d i f f i c u l t to establish against the background of imperfections from all other sources in the model. This is true even in the l i m i t of choosing the simplest, i . e . , harmonic o s c i l l a t o r , form for the generating potential. As an example of how this approach reproduces elastic electron scattering, we show in Fig. 7 a comparison of measured (Ryan and co-workers, 1983) and calculated (Radhi, Brown and Wildenthal, 1983) elastic electron scattering on 27AI. In this non-zero spin nucleus, the EO, E2 and E4 moments of the ground state all contribute to the form factor.

PPNP-B

26

B.H. Wildenthal

l

I

!

I

ZTAI(e, e) 27AI

_

I0 -I

!

elastic

io-2. ÷÷÷÷÷÷e,eeeeee

10-3 _

4"

4'

4'

O" 10-4 04 LL

÷

e÷

÷

• ÷ 4' 4'

'4, %•

I 0 "5 .

÷

÷

Y Y Y Y Y

io - s _

10 -7

Y Y

I0-8

I

0

I

I

I

q(fm -I ) Fig. 7.

I

2

I

5

Comparison of measured and calculated (DWBA) form factors for elastic electron scattering on 27AI. The solid line is the sum of the EO (dotted line), E2 ("++++" signs) and E4 ("YYYY" signs) contributions.

The E2 and E4 operators connect single-nucleon states within the active model space to orbits both below and above the shell-closure boundaries. Thus, in A = 17-39 nuclei, the ground states of closed-shell-core plus sd-shell configurations are coupled with the 2~ E2 and E4 "giant resonances" as well as to states of sd-shell parentage. The 2~ giant resonances of E2 and E4 character hence mix into the sd-shell states of the appropriate multipolarities. This mixing is a particular type of "higher-order" correction and one which has very strong effects on the matrix elements of the local model space. The results of the higher-order corrections can be inserted very concisely into the model-space calculations by the "effective charge" model, in which the neutrons and protons of the model space are given additional charges beyond their free values of 0 and le. At a more detailed level, the orbit dependence of the effects can be considered (Brown, Arima and McGrory, 1977), as well as the model for the radial dependence of the additional charge (Brown, Radhi and Wildenthal, 1983 )~ In the results for E2 and E4 phenomena we show here we have assumed that the nucleons have added charges of 0,35e for E2 matrix elements and O.50e for E4 matrix elements.

Empirical Strengths of Spin Operators in Nuclei

27

The clearest manifestations of nuclear shape are the electric quadrupole moments. These can be determined for ground states and, with difficulty, for a few excited states. We compare the values of the electric quadrupole moments which have been measured (Spear, 1981) for sd-shell states with the predictions (Wildenthal and Schwalm, 1983) of our shell-model wave functions in Fig. 8. The format of the

t F- + 2 8 z

~E + 2 4 ' rr +20' bJ a. x +16' I.=J

,L +12, +8' +4'

-28 --24 -20 -16 -I 2 -8 - 4

+4 +8 +12 +16+20+24 +28 ;

+-THEORY - *

-12'

-16' -20'

-28

Fig. 8.

Comparison of measured and predicted electric quadrupole moments of sd-shell states. The units are e fm2. Each entry corresponds to plotting the experimental value of the moment on the vertical scale and the theoretical moment on the horizontal scale. The dots indicate measurements with classical atomic physics techniques, the triangles indicate muonic-atom measurements, the crosses indicate reorientation-effect measurements and the "x's" show the values for the single-particle nuclei z70 (negative) and 39K (positive).

comparison of theory with experiment used in Fig. 8 is one which we will use again for other observables. Each point shows the theoretical value (along the horizontal axis) and the experimental value (along the vertical axis) for a particular state. Thus, i f there were uniformly perfect agreement of theory with experiment,

28

B.H. Wildenthal

all points would fall along the 135°-45° diagonal. The trend of theresults shown in Fig. 8 obviously is oriented along this line. Since the quadrupole moment experiments measure the moments, not the transition rates (matrix elements squared), the signs as well as the magnitudes of the charge • deformations of the nuclear states are determined. Negative signs correspond to oblate intrinsic shapes and positive signs to prolate shapes. The only incorrect predictions for shapes occur for 36CI and 3°Si. In both instances, the measured and predicted numbers are close to or consistent with zero. The moment data as a whole are evidence for strong collective effects in the nuclear deformation. The magnitudes for single-nucleon wave functions are shown by the "x" symbolsin Fig. 8, 170 being negative and 39K being positive. Most values have magnitudes much larger than these single-particle values, indicative of a coherent amplification of the matrix elements via interference in the sums over the orbits of Eq. 6. In conclusion, the clustering of points in Fig. 8 along the 135°-45° diagonal out to magnitudes three or four times larger than the single-particle values shows that the observed collective effects in deformations are reproduced both in sign and relative size by the model-space matrix elements. The simple, constanteffective-charge correction for higher-order configuration mixing suffices to put theory and experiment into absolute agreement. Hence, the fixed model space, fixedmodel Hamiltonian approach we have adopted encompasses the degrees of freedom.necessary to deal with these aspects of nuclear structure, and the specific formulation of the Hamiltonian we have determined gives a rather comprehensive agreement with the experimental details. Agreement between theory and experiment for E2 matrix elements similar to that seen in Fig. 8 also is obtained in comparisons of B(E2) values for transitions between low-lying states. As for the moments, the measured transition rates show evidence for strong collective enhancements and, with the same effective-charge renormalization, the model matrix elements reproduce these enhancements with quant i t a t i v e acuracy (Wildenthal, 1981; Brown and co-workers, 1982). As individual examples of how well the present calculations account for the details of electric-multipole strength, we consider some of the states of 27AI whose energies were inspected in Fig. 5. The availability of inelastic electron scattering form factors (Ryan and co-workers, 1983) allows us to contrast E2 features with E4 features and, looking ahead, provides us with examples of how the theoretical wave functions encompass not just these two matrix elements but also the M1 matrix elements andthe momentum-transfer, "q", dependence of these and s t i l l other matrix elements. We show the measured and (PWBA) calculated (Radhi, Brown and Wildenthal, 1983 ) inelastic scattering form factors for the f i r s t 7/2+ and second 5/2+ states of 2~Al in Fig. 9. In Figs. 10 and 11 we show these same data and predictions in representations (Radhi, Brown and Wildenthal,1983a) that remove the exponential q dependence and permit the matrix elements at q = O, the square roots of (2J i + 1) times the B(E2) and B(M1) values, to be displayed simultaneously with the scattering data. In Fig. 12 we show the form factors for the second 7/2+ state. At this point in our discussion we note just a few simple features of the three excited states considered in Figs. 9-12. These states have spins such that they can be populated from the 5/2+ ground state of 2~Al via several different multipoles, in particular, E2 and E4. We see from Figs. 9 and 12 that in the overview provided by the electron scattering form factors, the transitions to the second 5/2+ and 7/2+ states are experimentally determined to have dominant E4 components, while that to the f i r s t 7/2 has a dominant E2 component. These experimental features are exactly reproduced in the structure of the model wave functions. Independently, from Figs. 10 and 11, we note that the transitions to the f i r s t 7/2+

Empirical Strengths of Spin Operators in Nuclei

29

L', 2,2iMeV, 7/2 + no-3

I0-3 ICr 4

,-~ i 0 - 5

vyY

u_ I0" 6 10-7

+;~YYyyy

yyY

'

~.l

Yy

+

$

10-6

YYyy

10"7

I

10. 9

... . ...i'.. + .... y\~yb.

Y

Y~,

yY •. .

++

..',~

"..

\

\

yv ..

.~.~

10-8 I

I

I

I

103

I0-3 10-4

0.10-5 ","

4, ~i ="~elb.e.o,,~e

N

i"~;. * ~ . ~ - . - ~----- -

10.6

11

~- ,o"e

10.7

,

*

i0-8 -

•

X x

,';'

3X : :÷ ~

"

++

~,' ~f

;o°

'

'

."

"

"".I~I

'

1

10-4

u. 10.6

,~10 "5 ~J 10-6

10"7

10.7

10.8

i0-8

~ q(fm "l)

,

3

+x i +x

i. x

".

y~A

. Y&"

-%.~,.. Y ~ +

***

V

"..

;,

",

"~

-%

x x

++

yA

"'%

~l

90 °

10"4

0

l",: ÷,x t ~

~ x

f0-9 10.3

10"5

/

"

10. e

10-3

Fig. 9.

T

J°'4I

,-~K)-5 I~.

t

10.9

0

I

2 q ( f m -I)

Form factors of inelastic electron scattering to the f i r s t 7/2 + and second 5/2 + states of 27AI. The top panels show the separated longitudinal form factors, the middle panels the separated transverse form factors and the bottom panels the total form factors at 90 ° . The theoretical curves in the longitudinal panels follow the conventions of dots, "+'s", and "Y's" for the EO, E2 and E4 components, respectively. The theoretical curves in the transverse panels follow the conventions of dots, "+'s", "x's", "Y's", and triangles for the M1, E2, M3, E4 and M5 components. The dot-dashed lines in the bottom panels show the transverse components multiplied by 1.5.

30

B.H. Wildenthal

q

,

,

,

22.5

,

,

,

D

,

,

,

,

/

T

'

'

'

'

'

L, 2.21MeV,

2G

17.~ O"

~12.-j I0 7.~

2.~ i

i

I

|

,

i

4...'

.

,

2 i

,

,

3 i

,

,

4

i

5

q2(fm'2) i

i

,

6

,

,

7

,

i

8

9

i

T, 2.21MeV, 712 +

4.(: 3.5 ~2.~ 2c

1 . 5 ~ I.O

1"5I °o Fig. 10.

'o.'~

'

i:o ' ,'.5' q2(fm-2)

Lo

Longitudinal and transverse f i r s t 7/2+ state in 2~Al in which removes most of t h e i r allows t h e i r display on the q = 0 matrix elements.

Ls'

3.o

form factors for the a representation q-dependence and same scale with the

Empirical Strengths of Spin Operators in Nuclei

,

,

,

,

,

,

,

|

,

,

,

,

,

,

,

,

,

L, 2.735MeV,512+ ZO

I0 ~ ( 2 J i + l ) B(E2)

~ ' J s ~ 0

I

2 i

4.5

3 i

i

!. .... 4 5 q2(fm-2) i

6

7

8

9

i

T, 2.735MeV, 5/2 +

4.0 3.5 3.0

.....~z.5

2.0

'-°I ,o

Fig. 11.

qZ(fm'2)

,o

,5

30

Longitudinal and transverse form factors for the second 5/2 + state in 27AI in the same representation used in Fig. 10.

31

32

B.H. Wildenthal

i

i

i

T, 4 . 5 8

i

MeV,

712 +

10-3 10-4 10"5 oJ ,,

io-6 10-7

•

.i.

10-s _+

X

.--

+

+

X

x

x

~ ,,

* ~Y"

vY

~Y"

y'

-+,: i0 -9

~,,.

".

. ii.....

~

",,"-

Jr

x.

.--

1"~" ~,

',ry,, . _

'Yy'.

"--Zi,~...'Yv',,

:~~:

~.

z i

-

•

,. ,"+ |

i

i

w

90 °

i0 -3

10-4

~ 1o-5 u_ IO.6 io-T

,, to-8

. ÷÷

' Y

io-9 0

%.,

I

I

I

I

I

2

I

3

q(fm -I) Fig. 12.

Transverse and t o t a l form f a c t o r s f o r the second 7/2 + s t a t e of 27A1. The l i n e conventions are the same as those used in Fig. 9.

Empirical Strengths of Spin Operators in Nuclei

33

and second 5/2 + states have predicted B(E2) values in close agreement with the experimental measurements. The quality of the agreement between theory and experiment for E2 and E4 phenomena shown in Figs. 9-12 is atypical only in that the quality of the data now available for 27AI is considerably better than the norm. All available evidence indicates that the matrix elements of the (effective-charge-corrected) E2 and E4 operators evaluated with our shell-model wave functions for the lowest few states of each angular momentum in a nucleus agree with the corresponding experimental values both in shell-wide trends and state-by-state details. MATRIX ELEMENTS OF L = 0 SPIN OPERATORSAT q = 0 We now turn to the study of nuclear observables which incorporate the nucleon spin operator. The essential feature of this operator, which gives i t i t s special role in the study of nuclear structure, is that i t does not connect s i n g l e - p a r t i c l e states of d i f f e r e n t orbital or principal quantum numbers. Hence, the action of the spin operator is confined to within a major shell and, for L-S shell closures, the higher-order corrections begin at second-order (in the s t r i c t sense of the term) in perturbation theory. Hence, a "good" model for the lower-order aspects of many-body structure should comprise a major part of the explanation of observed matrix elements, thus leaving the higher-order corrections cleanly v i s i b l e , even i f small. The primary data we consider are the measured values of Gamow-Teller (GT) beta decay strengths and magnetic dipole (MI) moments and decay rates. We w i l l compare the matrix element magnitudes extracted from these data with theoretical matrix elements obtained from our shell-model wave functions and the GT and MI operators. The agreement between experimental data and the predictions of the shell-model calculations for energies, single-nucleon structure and shape-collective features provides the rationale for assuming that these same wave functions constitute a r e l i a b l e guide to the effects of the lower-order, "local" multi-nucleon nuclear structure upon the magnitudes of the GT and MI matrix elements. The antecedents of the present studies (Brown and Wildenthal, 1983 b-e) are the pioneering papers of Wilkinson (1973, 1974) and t h e i r subsequent elaborations by Brown, Chung and Wildenthal (1978) and Wildenthal and Chung (1979).

We choose to work here with the particular formulations of the M1 and GT singleparticle operators defined by the following general expression: Op = gs[S(d-d) + S(s-s)] + g~L(d-d) + gs60P

(8)

where 60p= 6s(d-d)S(d-d) + ~s(S-s)S(s-s) + 6~L(d-d) + 6p(S-d)P(s-d) + 6p(d-d)P(d-d)

(9)

S = E si , L = Z. hi , P = Z. Pi

(10)

and I

where

1

1

p = (8~)½[y(~) x S] (1)

The values of the coefficients g and 6 depend upon the observable. The g coefficients are obtained from the free-nucleon manifestations of the operator and the 8 coefficients characterize the renormalizations which are needed when working within the sd-shell model space. The parenthesis (d-d), (s-d) and (s-s) identify the respective pairs of orbits, ~=2 (Od5/2 and 0d3/2) and 4=0 (Isi/2) which are PPNP-B*

34

B.H. Wildenthal

acted on by the operators. The reduced single-particle matrix elements for the individual operator components s, ~ and p are given in Table 4. TABLE 4.

Reduced single-particle matrix elements of the operators s, ~ and p.

The

values of for x = s, ~ and p, where p - (8~)½[Y(2) × s] (I) The matrix elements are obtained as a product of C times . J'

C

<~>

112

~I(2~ + 3)

+ 1/2

+ I/2

[2(~+ 1)(2~ + 3)/(2~ + I]

0

~L- 1/2

- 1/2

[(2~)(2~ - 1)I(2C + I ) ]

I

+ 1/2

1/2

[2(~ + 1)(~)/(2~ + 1)]

I

[2(~ + 2)(~ + I)/(2~ + 3)]

0

+ I/2

(~ + 2) - 1/2

a)R is the radial overlap integral.

- 1 1 2 -(~ + 1)/(2~ - 1) -i 0

1/2 -3R/2a)

For the Os-ld shell R = -(2/5) ½.

The term 60p has been multiplied by g@ so that I00 6s can be regarded as the perc,entage renormalization of the S matrix element or, alternatively, as the percentage renormalization of the spin g factor gs; these are equivalent ways of expressing the same result. For the M1 operator, (gs/g~) 100 6~ can be regarded as the percentage deviation in the L matrix element or in g~. In Eq. 8 the ISM1 and IVM1 values of gs and ge are the free-nucleon "g-factors" in the isoscalar and isovector combinations, respectively, and the GT gs value is consistent with the h a l f - l i f e of the free neutron and the 0+ to 0+ pure Fermi decays (Wilkinson, 1973) gs(ISM1) = 0.880, g~(ISM1) = 0.550, gs(IVM1) = 4.706, g~(IVM1) = 0.500,

(11)

gs(GT) = IgA/gVl = 1.251 ± 0.009 g~(GT) = 0 In the textbook d e f i n i t i o n s of these three operators, the a c o e f f i c i e n t s vanish. We use the adjective "free-nucleon" or "free" to refer to these l i m i t s of the operators and to the values of s i n g l e - p a r t i c l e matrix elements and total multip a r t i c l e shell-model matrix elements which are calculated from them. For the isoscalar MI moments one can make use of the relation J =

+

(12)

to obtain the well known r e s u l t [~(ISMI) - ~]/0.380 = + (0.880/0.380~iI~0pli>

(13)

where the numbers in brackets are gs(ISM1)/[gs(ISM1) - g~(ISM1)]. We take the left-hand side of Eq. 13 as the observable for the total (isoscalar) spin contribution with an operator of the form of Eq. 8 with

Empirical Strengths of Spin Operators in Nuclei

gs(IS) = 1 and g~(IS) = 0 .

35

(14)

Because of this t r i v i a l J dependence in the isoscalar moments, defects in the shell model calculations show up much more clearly in the matrix elements than in the isoscalar moments themselves. The very good percentage agreement which appears in the usual comparisons of isoscalar moments with theory stems from this large contribution of a model independent term. For this reason, the experimental error in the isoscalar moment must be rather small (usually at most a few percent) for a meaningful comparison to nuclear-structure theory. Analysis of Mirror States We f i r s t consider (Brown and Wildenthal, 1983b) the subset of data on M1 and GT matrix elements in which only the ground-state mirror wave functions of the sd shell, together with a few T = 0 levels, participate. This allows us to extract corrections to the isoscalar MI (ISMI), the isovector MI (IVMI) and the GT singleparticle matrix elements from the same experimental and theoretical components. We express the M1 and GT observables in terms of u(ISM1/IVMI) =

[~(Tz = +T) ± ~(TZ = -T)] 2

M(GT) = [(2J i + 1)B(GT)]½

(15) (16)

The u(Tz) are the magnetic dipole moments in units of nuclear magnetons, where our convention is that T7 = +1/2 for the proton. The B(GT) are related to the partial half-lives for beta ~ecay, t½, by -+4

t½ = fvB~O+ fAB(GT)

(17)

The B(F) are the Fermi beta-decay nuclear matrix elements and the terms fv and fA are the vector iV) and axial-vector CA) beta-decay phase space factors. The M1 moments are proportional to Eq. 6, with the proportionality constant

For the Tz = 1/2 to Tz = -I/2 GT decay, C(GT) : (2/3) ½ The results of our comparison of the experimental values of the ISMI, IVM1 and GT matrix elements for the mirror sd-shell states with the theoretical values are shown in Fig. 13. We show theoretical values which incorporate alternatively the free-nucleon values of the single-particle matrix elements and values of these terms which include the empirical values of the delta terms determined in the least-squares f i t s of equations of the form of Eq. 6 to the respective data sets. We omit the A = 17 and 39 data from these f i t s to c l a r i f y any differences between the multi-particle data and the single-particle data. We have redefined the IVM1 and GTmatrix element, so that the expectation value of the total spin has a unit coefficient.

36

B.H. Wildenthal

0.7 0.6 0.5 0.4

_~-~:FREE NUCLEONSPME •",.T----EMPIRICAL SPME

/--

0.3 0.2

°-'-L-"

0.1 I

!

!

!

!

!

|

0.7 0.6 0.~ 0,4

.- ~

-4

0.~ 0,2 0.I

1.4 1.2 1.0

0.8 0.6 ,

,

17

,

21

,

,

25

,

,

2g

,

,

33

,

,

ILV' ~I ' ~S' ~9' ~ '

37

:~7'

A Fig. 13.

Comparisons of normalized experimental values of the isoscalar and isovector magnetic dipole moments (ISM1 and IVM1, respectively) and GamowTeller beta-decay matrix elements (GT) for the mirror ground states of A = 17-39 nuclei with predictions of mixed-configuration sd-shell model wave functions. The solid lines are obtained with single-particle matrix elements set to the free-nucleon values, while the dashed lines are obtained with single-particle matrix elements set to the "empirically corrected" values. The top three panels show the normalized experimental and both free-nucleon and "final f i t " theoretical values. The middle three panels show only the spin components of these matrix elements. The "experimental" values are obtained by subtracting the "final f i t " L and P theoretical values from the experimental values shown in the top panels. The bottom three panels show the ratios of experimental to theoretical values for the spin components of these matrix elements as they are defined for the middle three panels.

Empirical Strengths of Spin Operators in Nuclei

37

We see from Fig. 13 that, with the t r i v i a l coefficients and additive terms removed, the experimental isoscalar matrix elements are less well reproduced by theory than are either the IVM1 or the GT matrix elements. The significant influence of the contributions of the orbital angular momentum terms to the M1 matrix elements is evident in the differences between the top panels and the middle panels, which show only the spin parts of the theoretical matrix elements. The experimental values shown in comparison with theory in the middle panels have been manipulated by subtracting the values of the predicted (with empirical values of the singleparticle matrix elements) orbital parts. Necessarily, the predicted free-nucleon values in the middle IVM1 and GT panels are the same. The key results of the analysis can be seen easily in either the middle or the bottom panels. These results are that, empirically, there is need for a significant correction to the ISM1 lower-order, free-nucleon theory, l i t t l e evidence of the need for a s i g n i f i cant net correction to the IVM1 theory and a very unan~biguous need for a correction to the lower-order, free-nucleon GT theory. At the simplest level, the corrections which are needed to bring the ISM1 and GT shell-model predictions, which are based on the free-nucleon normalizations, into better agreement with experiment can be approximated in terms of quenching factors. The empirical corrections we actually use in Fig. 13 and in later discussions are more complex than this in that each term in the operator is allowed in the f i t to have i t s own correction. We could, however, have adopted a simpler ansatz and made one-parameter rather than four or five parameter corrections. The results which would be obtained in the one-parameter adjustments obviously would be that, averaged over the shell, the ISMI and GT operators need to be quenched below their free-nucleon normalizations in order for their matrix elements with the O~m shellmodel wave functions of the sd-shell to agree with experiment. As can be seen from Fig. 13, the A = 17 and 39 data on their own also require a comparable degree of quenching, as has been long known. One important consequence of the present results for the nuclei between A = 17 and 39 lies in their independent confirmation of the conclusions based on the single-particle systems. The consistency shown to obtain over the entire shell is strong evidence that the simple single-particle structure assumed for A = 17 and 39 is, in nature, not significantly contaminated by abnormal "intruder state" admixtures. Moreover, the over-determination of the corrections by the multiple-entry data set allows us to estimate the "nuclear model" uncertainties in the extracted values of the correction terms. Finally, while the A = 17 and 39 data together can yield only two of the five correction terms which in principle can affect the M1 and GT matrix elements, the data set displayed in Fig. 13 determines four of the five with s t a t i s t i c a l l y meaningful accuracy, the two 6p terms not being determined separately with meaningful accuracy. Analysis of Expanded Data Sets We can pursue the analysis of M1 and GT data further by relaxing the requirement that the values come from mirror states. We have carried out analyses similar to those just described in which the data bases comprise, in turn, all of the magnetic moment data of the sd shell, a collection of the experimentally welldetermined B(MI) values of the sd shell and all of the Gamow-Teller beta decay data of the sd shell (Brown and Wildenthal, 1983c, 1983d, 1983e). All of these data bases are larger than our mirror-state data set. The transition-strength data tend to more heavily weight the d5/2 - d3/2 "spin-flip" single-particle matrix element than do the moment data. In Fig. 14 we compare all the experiment a l l y measured sd-shell magnetic moments with the shell-model predictions as they

38

B.H. Wildenthal

incorporate the four higher-order correction terms. As could be inferred from the comparisons in Fig. 13 of the free nuclear values with the corrected values of the isovector components of the M1 moments, the agreement with experiment of these corrected shell-model predictions is only a slight improvement over what is achieved with the free-nucleon predictions.

l

l

I

I

I

I

I

I

I

I

I

|

I

I

I

I

I

I

4'-

-

E: 2 -

E x

I-

0

-I

-2 -2

-I

0

I

2

3

4

fitted theory Fig. 14.

Comparison of experimentally measured magnetic dipole moments of sd-shell nuclear states with shell-model predictions which incorporate empirical higher-order corrections in the single-particle matrix elements. Units are nuclear magnetons. The format of the presentation is similar to that used in Fig. 8.

The same M1 operator which acts on the wave functions to yield the magnetic moments must also generate the M1 component of electromagnetic transitions between d i f f e r ent states in the spectra of the various nuclei. We have selected roughly 200 MI transitions whose strengths are experimentally determined with good precision (Endt and van der Leun, 1978) and extracted the empirical higher-order corrections to the M1 operator from least-squares solutions of equations of the form of Eq. 6.

39

Empirical Strengths of Spin Operators in Nuclei

We have carried out an analysis of all Gamow-Teller beta decay data from sd-shell nuclei in the same context as the analyses of the M1 data just described. The experimental data sets for the GT process tend to be more complete and more clearly circumscribed than for MI decays. Accordingly, in displaying the results of the comparison of experiment with theory, we condense the individual transitions by summing the magnitudes of all the observed matrix elements (and the magnitudes of the corresponding theoretical matrix elements) and plotting them as the fractions they represent of the total possible strength from the decaying state. We display in Fig. 15 the comparison of experiment with the theoretical matrix elements which incorporate the empirical higher-order corrections. In the case of the w

0.9

w

i

l

t

•

l

!

/

t

STRENGTHS OF GT MATRIX ELEMENTS / SUMMED OVER 0

0.8 0.7

~0.6 uJ ~0.5

wXO.4 0.3 0.2 0.1 00

Fig. 15.

I

l

0.2

I

I

0.4

|

I

0.6 THEORY

I

I

O.8

t

1.0

Comparison of the sums of Gamow-Teller matrix elements measured from beta-unstable nuclei of the sd-shell with predictions of the shell-model wave functions which incorporate empirical higher-order correction. The values plotted correspond to the fractions of the total possible strengths for the decays which appear in the visible spectrum. The higher-order corrections amount essentially to quenching the theoretical matrix elements by the factor (0.60)½.

GT operator, these corrections amount basically to simply quenching the singleparticle matrix elements by the factor (0.6)½. (This is essentially the result

40

B.H. Wildenthal

already found with the mirror-state subset of the GT data displayed in Fig. 13) This result says in other words that shell-model predictions for GT decay strengths of sd-shell nuclei which incorporate the free-nucleon normalization of the operator (see Eq. 11) are systematically larger than the experimontally measured strengths such that the ratio B(GT)exp/B(GT)sm,free_nucleon is 0.60 ± 0.02. Again, from Fig. 13 i t can be seen that this result from the t o t a l i t y of multiparticle sd-shell data is consistent with the average of the A = 17 and 39 data alone. Comparison of Empirical and Theoretical Higher-Order Corrections We now draw together all of the foregoing analyses of M1 and GT phenomena with shell-model wave functions, summarize their results and compare them to a recent, very complete set of predictions for higher,order corrections by Towner and Khanna (1983). The results for the isoscalar M1 (ISM1), isovector M1 (IVM1) and GamowTeller operators and data are displayed in Figs. 16~ 17 and 18, respectively. In each figure we show the percentage corrections to the terms of the operator as formulated in Eq. 10, with the two "p" terms lumped together. For each of the four correction terms we consider five different values. In order from l e f t to right, the f i r s t values plotted (solid lines) for the corrections are those based on analysis of the data from the "single-particle", A= 17 and 39, nuclei. These data provide information on the ~ and s parts of the "d" orbit only. The second values of the five plotted (dashed lines) are those obtained from analysis of the mirror-state data set under the constraints that the "p" term is zero and that the other correction terms have no A dependence. The third values plotted (dotted lines) also result from the mirror-state data set. They are obtained under the constraints that the "p" term is free to vary and that all the corrections have a mass-dependence (suggested by the calculations of Towner and Khanna (1983)) of (A/28)°'~5 The fourth values in the plots (dotdashed lines) of the IVM1 corrections are obtained from the analysis of the B(MI) data, under the constraints of no A dependence and isoscalar terms fixed to values obtained in the f i t to the moments. The fourth values in the plots (dot-dashed lines) of the GT corrections are obtained from the f i t to the complete set of sdshell GT data. The f i f t h values ("xxx" lines) in all of the plots are the results of the Towner-Khanna calculations. A key element in the results of the present empirical analyses is the estimate of the uncertainty with which the correction terms are determined by the data and the model. These uncertainties are indicated in the second, third and fourth values by the horizontal bars. The agreement of the Towner-Khanna predictions with our empirical values of the higher-order corrections for the sd-shell M1 and GT operators is in qualitative terms very impressive. The dominance of the spin component, the approximate overall size of the corrections and the smaller effects for the IVM1 operator relative to those for either the analogous GT operator or the ISM1 operator all emerge from these predictions. The empirical results allow a more detailed critique Of the theory, although at some level the uncertainties in the calculated results will be comparable to the empirically quoted uncertainties. The largest and most precisely determined corrections for each of the three operators are for the as (d-d) terms. The various empirical estimates are consistent among themselves and in excellent agreement with the theoretical predictions in each case. The largest deviation involves the estimate from the analysis of the B(M1) data for the IVM1 correction. The ~ (d-d) corrections for the three operators are much smaller in magnitude (and opposite in sign), but they also are determined with relatively l i t t l e uncertainty and the empirical values agree among themselves and with theory for each case. The ~p corrections are determined to be

Empirical Strengths of Spin Operators in Nuclei

%

41

ISM1

+15

m m

+I0

-

8s

~S

81

=

$-S

d-d

d-d

=-

+5 z

1

0

•

~

~"

"

7

•

-5

-

----..

i _z =~.

--X ------

-15

.

t

.-

X

I

----'Z

-I0

t

8p s-d, d-d

I X X

=

,~

=

~,

t

.

I I

-:~0

=

,,

,,"

3It M

-25

!: I

1 1 1

N

-50

N N

-55

Fig. 16.

Corrections to the isoscalar MI operator for the sd shell, as obtained empirically in the analyses of data with shell-model wave functions and as calculated by Towner and Khanna (1983). Details of the presentation are explained in the text.

consistently small for the IVMI and GT operators. The empirical values agree with theory. The only determination of 6p for the ISMI operator has a large uncertainty and is of the opposite sign from the very small theoretical prediction. This discrepancy does not appear to be very meaningful in the overall context.

42

B.H. Wildenthal

IVM1 % +15

8s s-s

8s d-d

81 d-d

+I0 +5 0

-5

-I0

,i

~ Z

i-

-=@'

m

=+!!

I i! !+

_= -15

8p s-d, d-d

=

=i

Z

-2o Fig. 17.

Corrections to the isovector M1 operator for the sd shell, as obtained empirically in the analyses of data with shell-model wave functions and as calculated by Towner and Khanna (1983). Details of the presentation are explained in the text.

The empirical corrections to the ~s(s-s) terms for the M1 operator exhibit the most variations among themselves and the most discrepancy with prediction. These values depend totally upon the configuration-mixing details of the shell-model calculations. Hence i t might be that here we are seeing evidence for model errors However, the results for the GT operator are internally very consistent with each other and with theory, which does not support blaming the shell model structures. We would conclude that the situation for this term of the IVM1 operator is unclear The moment data indicate a vanishing correction, the transition rate data a moderate value somewhat larger than theory. The situation for this term of the ISM1 operator seems to be the only clear quantitative discrepancy between our results and the predictions of Towner and Khanna (1983). None of the empirical values comes to one-half of the theoretical estimate. We conclude at this point that a very large number of experimental data on M1 and GT processes which have been measured for sd-shell nuclear states are, when

Empirical Strengths of Spin Operators in Nuclei

43

% +15

GT

+10 +5

3s

8s

S-S

d-d

0

-I0 -15

°

•

-20 -25

•

X

z!

_---:i- =i

N

= I 1N

i : l 1N

I : 1 =N

8p s-d, d - d

~:1:

"!:

-5~

81 d-d

|

!

•

N

• •

N N N

i'" . NN X

IN

|

•

°

=

•

°

i=-

-30 Fig. 18.

Corrections to the Gamow-Teller operator for the sd-shell, as obtained empirically in the analyses of data with shell-model wave functions and as calculated by Towner and Khanna (1983). Details of the presentation are explained in the text.

reduced of their many-body complexities by analysis with shell-model wave functions, consistent with each other and with the single-particle states of A = 17 and 39. The relationship of these data to the shell-model predictions which incorporate the properties of the free neutron and proton can be formulated in terms of simple, systematic corrections to the properties of the single-nucleon orbits of the model space. The values of these empirical corrections are remarkably similar to the most detailed current predictions of the effects of "long-distance" configuration mixing and non-nucleonic component admixtures to the shell-model wave functions.

44

B.H. Wildenthal

SPIN MATRIX ELEMENTSAT HIGH ENERGYAND NON-ZERO q The frontiers of the study of spin and nuclear structure involve many facets of experiment and theory. In these paragraphs we will touch on some extensions to higher excitation energies and larger momentum transfers of the analyses we have described for the matrix elements of the M1 and GT moments and decays of low-lying states. These added dimensions to our perspective of nuclear structure both amplify and secure our understanding of the more classical data and offer the poss i b i l i t y of more directly probing the boundaries between the local model description of nuclei and those higher-order processes which are the focus of much of our current concern. A v i t a l l y important and invigorating experimental advance in the study of spin phenomena, and of nuclear structure in general, has been led by Goodman and coworkers (1980) as they have demonstrated that at medium (100-200 MeV) energies, the (p,n) reaction at 0° simulates GT beta decay with remarkable accuracy. This discovery allows the Q-value constraints which classically limited our access to GT matrix elements to be completely surmounted. This is even more important than might be obvious at f i r s t thought, because the distribution of GT strength with excitation energy puts the major portion of this strength out of reach of ordinary beta decay studies. (We recall that in Fig. 15 the large majority of entries correspond to small fractions of the total possible strength.) l"l t I t I t

5.0 2.5

A' I

r-I

I

,

'l I "~t

m 1,5

,

I

'~% t

I

~ l

I

~ t ~ l

1.0

/%

I .! \ e .'7-A

t

t i

~, /.." ,'.'~ ~..',,,

0.5 0

i ~' •

l

~. 2.(

,

0 Fig. 19.

2

,."T ....... 4 6 8 I0 12 i4 16 18 EXCITATION ENERGY (MeV)

The experimental (solid line) and theoretical (dashed and dotted lines) strength distributions of the 26Mg (p,n) reaction. Both experimental and theoretical spectra have been a r t i f i c a l l y broadened in resolution to simplify the comparisons. The two theoretical curves differ by a factor of 0.60.

Empirical Strengths of Spin Operators in Nuclei

45

The (p,n) data provide a measure of GT strength up to very high excitation energy and can be thought to sample the total strength accessible within the nucleonic degrees of freedom. Analysis of such data suggests that the simple sum rule relating total GT strength to the neutron excess is violated systematically, with the conclusion that, in accord with analyses of beta decay, the normalization of the GT operator appropriate to the shell-model context must be quenched by a factor ranging between 0.4 and 0.7. We show in Fig. 19 the results of a recent medium-energy (p,n) experiment on 26Mg (Madey and co-workers, 1983) and its analysis with our shell-model wave functions. The actual experimental data, measured with high resolution in order to identify individual signatures, have in Fig. 19 been a r t i f i c a l l y broadened to a resolution of 1.5 MeV in order to highlight the central issues of the comparison with theory. The theoretical energies and matrix elements for t h i r t y 1+ states each of isospins T = 0, 1 and 2 were also individually calculated (a subset of these energies appears in Tables 2 and 3) and given an equivalent resolution broadening. The solid line in Fig. 19 shows the experimental results, the dashed line shows the theoretical predictions which use the free-nucleon normalization for the GT operator and the dotted line shows the theoretical results in which the GT operator is quenched by the factor (0.60) 3 deduced from the f i t to the data from low-lying beta decay.

I0

8 I "1-

6 Z

i,I

r~ |

F-4 (f)

2

2O Fig. 20.

24

28

:32

:36

Total strengths of the ot operator from the T = 0 ground states of the sd-shell, as evaluated with the full configuration-mixed shell-model wave functions (solid lines) and with one-component wave functions (dotted lines) consisting of the j - j coupling basis vector which has the largest amplitude in the full wave function.

46

B.H. Wildenthal

We see from Fig. 19 that the renormalized theoretical curve correctly predicts the overall strength observed in the experiment as well as its detailed (at the 1MeV scale) distribution in energy over a 20 MeV range. This result is a very important confirmation of the results we obtained in analyzing the ordinary data of beta decay. I t t e l l s us that the ground state and lowlying state configurations predicted by our Hamiltonian have the correct configuration admixtures as these are tested by the total strength of the GT operator and that the correct distribution of GT strength with excitation energy emerges simultaneously from this same formulation. The essential role of quantitatively accurate shell-model predictions for the analysis of such data is illustrated in Figs. 20 and 21, in which we show the d i f ferences between the predictions of the full configuration-mixed shell model wave functions and those of their leading single-j-j coupled basis vectors.

6 4

-1I(.9

2

Z

18

b.I n," I(/)

E] 22 I!:.. 26 I

30

34

J

38

A.-*

-2 -4 -6 8

Fig. 21.

Total strengths of the or operator from the T : 1 ground states of the sd-shell, as evaluated with the full configuration-mixed shell-model wave functions (solid lines) and with one-component wave functions (dotted lines) consisting of the j - j coupling basis vector which has the largest amplitude in the full wave function.

Empirical Strengths of Spin Operators in Nuclei

47

Results are shown for each doubly-even sd-shell ground state, the T = 0 systems in Fig. 20, the T = I systems in Fig. 21. For each T = 0 ground state, the total strength predicted for the ~ operator from the f u l l configuration-mixed wave function is shown by the solid line. The corresponding strength predicted from the one-component wave function consisting of the basis vector of that state which has the largest amplitude in the f u l l wave function is shown by the dotted line. In Fig. 21, the total strengths from the T = 1 ground states are shown to the three isospin systems, T = O, I and 2, which have Tz = 0 and the T = 2 system which has TZ = -2. The solid and dotted lines have the same meanings as in Fig. 20. For the T = I systems, the sum rule for the ~ strength is 6. This value is arrived at by adding the three TZ = 0 contributions and subtracting the TZ = -2 contribution. We see from Figs. 20 and 21 that the general effect of configuration mixing within the sd-shell orbits is to strongly reduce the strengths of the individual T channels from their pure-configuration values. The sum rule is maintained during this reduction by the fact that the T = 2 contribution is reduced more than the T = 0 and I contributions. Since this term enters into the sum with a large negative coefficient from the TZ = -2 channel, i t s reduction compensates for the relatively smaller reductions on the positive side. The A = 18 and 38 systems do not have a TZ = -2 channel open to them in the general shell-model context, so in these cases t~e strength lost in one channel must appear in the other, as observed. We conclude from Figs. 20 and 21 that careful consideration of the Ohm configuration mixing is an absolute requirement for the extraction of quantitative values of the quenching of spin strength from (p,n) or other "giant resonance" data. I t is also apparent from Figs. 20 and 21 that the higher isospin channels, being more strongly affected by the local configuration mixing, yield a more model-dependent measure of the effect of higher-order corrections upon the GT strength than does the T = 0 channel. Conversely, given that we understand the general properties of the spin probes, the T = 2 channel (and the analogous channels in other shell-model regions) offers the most sensitive arena in which to test theoretical predictions of configuration mixing within the local space. Inelastic electron scattering offers an alternate probe with which to measure spintype matrix elements over a wide energy range, one which includes the dominant part of the M1 strength, the so-called "M1 giant resonance". This probe, of course, can sample only the isospin channels of values equal to and one greater than the target value. Thus, i t is automatically oriented towards measurement of some combination of configuration mixing and operator renormalization. A principal interest in MI scattering stems from the contrasting view i t affords of nuclear structure when compared with the analogous GT excitation data. The major differences between the two probes arise, as was clear from the results shown in Fig. 13, from the presence of the orbital components of the M1 operator. In Fig. 22 we i l l u s trate the differences which are predicted for the MI and GT excitation of the giant-resonance region of 28Si-28P. High-quality data, which are now within experimental reach, can provide additional very discriminating tests of the nuclear structure of these important features. Obviously, Fig. 22 implies that a major portion of the MI strength in 26Si is associated with the orbital part of the operator. Since the higher-order corrections for the spin and orbital operators are quite different (see Fig. 17) a quantitative understanding of the meaning of such data will depend upon understanding the underlying shell-model structure of these excitations. The fundamental importance of electron scattering as a probe of nuclear structure resides in the fact that the electron is not absorbed in the nucleus. Thus, electron scattering experiments at the appropriate momentum trnasfer values q can sample the i n t e r i o r properties of the nuclear wave functions as well as their surface and integral properties. Other scattering probes and conventional transition strength measurements are limited to these l a t t e r aspects only. Thus, while in

48

B.H. Wildenthal

practical terms electron scattering can be extremely valuable as a sourceof experi mental estimates of B(M1) and BEE2) values and, more importantly, of higher multipolarity transition strength values, i t s unique role is to provide these measures as a function of momentum transfer, which is to say, as a function of radius.

3.5

3.0 2.5 2.0 1.5 i.O 0.5

1.0 0.5

-4

-2

0

2

4

6

8

I0 i2 14 16 18 20 22

EXCITATION ENERGY (MeV)

Fi g. 22.

Relative M1 (upper) and GT (lower) response functions of 28Si. The individual 1+ states are indicated by the vertical lines. The envelope curves result from summing these strengths with a 1MeV gaussian broadening function,

We have already presented examples of experimental inelastic electron scattering form factors in comparison with the predictions of our shell-model wave functions in Figs. 9-12. At that point in the discussion we touched briefly upon the facts

Empirical Strengths of Spin Operators in Nuclei

49

that the observed B(E2) values of the transitions and the relative strengths of the E2 and E4 contributions to the longitudinal scattering were theoretically reproduced. We emphasise here that the agreement between experiment and theory for the longitudinal scattering is very good out to 3 fm- I . The theoretical curves incorporate the shell-wide constant effective charges for E2 and E4 described e a r l i e r and the Tassie model for the shape of this higher-order contribution to the transitions (Brown, Radhi and Wildenthal, 1983a). We emphasize at this point more the results for the spin-type contributions to the data and predictions shown in Figs. 9-12. In particular, we note that Figs. 10 and 11 show that the shell-model wave functions correctly predict not only the B(MI) values of these two transitions but the momentum-transfer dependences (rather different in the two cases) of the associated matrix elements out to 3 inverse fermis. These data are embodied in the transverse components of the electron scattering cross sections. In the sd-shell, the M1, M3 and M5 magnetic excitations contribute to the transverse scattering along with those of the transverse E2 and E4 operators, these l a t t e r also having a spin-type nature. The predictions we show incorporate the free-nucleon parametrizations of these operators. The only clear indication (not evident from Figs. 9-12) that corrections to these normalizations are needed comes from data which emphasize the M3 operator. Such studies suggest that a quenching of the M3 operator by as much as a factor of 0.6 is appropriate. As is evident from Figs. 9 and 12, the transverse scattering from a non-zero-spin target samples a plethora of matrix elements. I t is encouraging that the envelopes of the predicted combined strengths follow the data as well as they do. However, progress in elucidating the individual aspects of these operators will come more easily by studying examples in which selection rules simplify the situation. CONCLUSIONS In one sense, the results we have shown might be considered discouraging. Too much of the experimental phenomena might seem to be merely the consequences of "uninteresting" shell-model configuration mixing. I t might seem that i t is too d i f f i c u l t to isolate and study "interesting and exciting" features. Certainly, I think that these results contain the message that d i s t i l l i n g the essence out of nuclear structure studies is not easy. However, i t seems to me that i t would be naive to think that such a complex construction as the nucleus could be dealt with casually. Moreover, I do not see that a r e a l i s t i c level of d i f f i c u l t y is i n t r i n sically an unhealthy feature of serious science. The important and positive feature we can see in these results is a substantive progress in theoretically understanding experimental results in nuclear physics. At the same time, progress in experimental capabilities is continuing to open more aspects of nuclear structure to our view. As long as this progress continues I do not think we could or should ask for more. ACKNOWLEDGMENTS

I wish principally to acknowledge the contributions to this work of B.A. Brown. Many of the ideas and results presented here are his, although he is not to be held responsible for my presentation of them. I am indebted to him for his intellectual guidance, his massive help with practical details and the continuing stimulation and pleasure of his collaboration. In the i d e a l , l w o u l d have comprehensively cited in these lectures the almost innumerable papers of the numerous investigators

50

B.H. Wildenthal

in the fields I have touched upon, work from which I have learned much, even i f far from enough. In the reality I have gone to the other extreme and I offer ~tY apologies. Shari Conroy, Mamar Blosser and Orilla McHarris have been both patient and efficient in helping me prepare the manuscript. The research discussed was supported in part by the U.S. National Science Foundation, Grant No. PHY 80-17605. REFERENCES Arima, A- and H. Horie (1954). Pro9. Theor..Phys., 11, 509 and 12, 623. Barrett, B,R., and M.W. Kirson (1973). In M. Baranger and E. Vogt (Ed.), Advances in Nuclear Physics, Vol. 6, New York, Plenum, p. 219. Brown, B.A., A. Arima and J.B. McGrory (1977). Nucl. Phys. A277, 77. Brown, B.A., W. Chung and B.H. Wildenthal (1978). Phys. Rev. Lett. 40, 1631. Brown, B.A., W. Chung and B.H. Wildenthal (1980) Phys. Rev., C22, 774. Brown, B.A., R. Radhi and B.H. Wildenthal (1983): Electric quadrupole and hexadecupole nuclear excitations from the perspectives of electron scatterin 9 and modern shell-model theory, to be published. Brown, B.A. and B.H. Wildenthal (1983b). Corrections to the free-nucleon values of the single-particle matrix elements of the M1 and Gamow-Teller operators from a comparison of shell-model predictions with sd-shell data, to be published. BFown, B.A. and B.H. Wildenthal (1983c). Analysis of sd-shell magpetic moments, to be published. Brown, B.A. and B.H. Wildenthal (1983d). Analysis of sd-shell magnetic dipole transitions, to be published. Brown, B.A. and B.H. Wildenthal (1983e). Analysis of sd-shell Gamow-Teller beta decay, to be published. Brussaard, P.J. and P.W.M. Glaudemans (1977). Shell Model Applications in Nuclear Spectroscopy. North Holland-Amsterdam. p. 452. Campi, X., H. Flocard, A.K. Kerman and S. Koonin (1975). Nucl. Phys., A251, 193. Chung, W. (1976). Ph.D. Thesis, Michigan State University, E. Lansing, MI. Cole, J., A. Watt, and R.R. Whitehead (1975). J. of Physics G2, 213. Detraz, C. D. Guillemaud, G. Huber, R. Klapisch, M. Langevin, F. Naulin, C. Thibault, L.C. Carraz and F. Touchard (1979). Phys. Rev., C19, 164. Detraz, C., M. Langevin, M.C. Goffri-Koussi and D. Guillemaud (1983). Nucl. Phys. A394, 378. Endt, P.M. (1977). Atom. Data Nucl. Data Tables, 19, 23. Endt, P.M. and C. van der Leun (1978). Nucl. Phys., A310, 1. Endt, P.M. (1983) private communication. Fox, J. (1982) private con~munication. French, J.B., E.C. Halbert, J.B. McGrory and S.S.M. Wong (1969). Advances in Nuclear Physics, Vol. 3, Plenum Press, New York. Chap. 3. Glatz, F. (1983) private communication. Goodman, C.D., C.A. Goulding, M.B. Greenfield, J. Rapaport, D.E. Bainum, C.C. Foster, W.G. Love and F. Petrovish (1980). Phys. Rev. Lett. 44, 1755. Halbert, E.C., J.B. McGrory, B.H. Wildenthal and S.P. Pandya (1971). Advances in Nuclear Physics, Vol. 4. Plenum Press, New York. Chap. 6. Kuo, T.T.S. (1967). Nucl. Phys., A103, 71. Kuo, T.T.S. and G.E. Brown (1966). Nucl. Phys., 85, 40. Kuo, T.T.S. (1983) private communication. Kruse, H. (1982) private communication. Madey, R., B.D. Anderson, J.W. Watson, A.R. Baldwin, C. Lebo, B.S. Flanders, C.C. Foster, S.M. Austin, A. Galonsky and B.H. Wildenthal (1983). Indiana University Cyclotron Facility 1982 Scientific and Technical Report, p. 41. Mayer, M.G. and J.H.D. Jensen (1955). Elementary Theory of Nuclear Shell Structure. Wiley, New York. McGrory, J.B., B.H. Wildenthal and E.C. Halbert (1970). Phys. Rev. C2, 186.

Empirical Strengths of Spin Operators in Nuclei

51

Nilsson, S.G. (1955). Mat. Fys. Medd. Dan. Vid. Selsk., 29, 16. Preedom, B.M. and B.H. Wildenthal (1972). Phys. Rev., C6, 1633. Radhi, R. B.A. Brown and B.H. Wildenthal (1983). Analysis of inelastic electron scatterin 9 on 2~AI, to be published. Ryan, P.J., R.S. Hicks, A. Hotta, J. Dubach, G.A. Peterson and D.V. Webb (1983). Phys. Rev., C27, 2515. Spear, R.H (1981). Physics Reports, 73, p. 369. Thibault, C. et al (1980). In J.A. Nolen and W. Benenson (Ed.), Atomic Masses and Fundamental Constants 6. Plenum Press, New York. p. 291. Towner, I.S. and F.C. Khanna (1983). Nucl. Physl, A399, 334. Wapstra, A.H. and K. Box (1981). October 1981 Atomic Mass Table for Nuclides, private communication. Watt, A., R.P. Singhal, M.H. Storm and R.R. Whitehead (1981). J. Phys., G7, 145. Whitehead, R.R., A. Watt, B.J. Cole and I. Morrison (1977). Advances in Nuclear Physics, Vol. 8. Plenum Press. Chap. 3. Wildenthal, B.H. (1977). In R. Broglia and A. Bohr (Ed.), Ele~ntary Modes of Nuclear Excitation, North-Holland, Amsterdam. P. 545. Wildenthal, B.H. (1982). Bull. Am. Phys. Soc., 27, 725. Wildenthal, B.H. and W. Chung (1979). In M. Rho and D.H. Wilkinson (Ed.), Mesons in Nuclei. North-Holland, Amsterdam. p. 723. Wildenthal, B.H., E.C. Halbert, J.B. McGrory and T.T.S. Kuo (1971). Phys. Rev., C4, 1266. Wildenthal, B.H. and D. Schwalm (1983). Static Quadrupole Moments of sd-shell Nuclei, to be published. Wilkinson, D.H. (1973). Nucl. Phys., A209, 470. Wilkinson, D.H. (1974). Nucl. Phys., A225, 365.