Quenching of spin matrix elements in nuclei

QUENCHING OF SPIN MATRIX ELEMENTS IN NUCLEI

I.S. TOWNER

Atomic Energy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada KOJ 1JO

NORTH-HOLLAND

- AMSTERDAM

PHYSICS REPORTS (Review Section of Physics Letters) 155, No. 5 (1987) 263-377. North-Holland, Amsterdam

QUENCHING OF SPIN MATRIX ELEMENTS IN NUCLEI I.S. TOWNER Atomic Energy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada KOJ lJO Received April 1987

Contents: 1. Introduction 2. Effective interactions 2.1. Landau-Migdal interaction 2.2. Estimates of g' 2.3. Effective interactions in finite nuclei 2A. Isobar-hole interaction 2.5. g' for isobar-hole interactions 3. Rayleigh-Schr6dinger perturbation theory 3,1. First-order core polarisation 3,2. Extension to RPA 3,3. RPA in closed-shell nuclei 3.4. Core-polarisation blocking 3.5. Second-order core polarisation 4. Meson-exchange currents 4,1. S-matrix method 4.2. Non-Born terms of pion range

265 266 267 271 278 286 290 293 293 298 300 304 308 314 318 323

4.3. Pair and current diagrams of heavy-meson range 4.4. Exchange-current magnetic moment operator 4.5. Axial-vector meson-exchange currents 5. Isobar currents 5.1. Isobars as an MEC correction 5.2. Isobars as nuclear constituents 5.3. MI and GT giant resonances in Pb region 5.4. Beyond first order 6. Comparison with experiment 6.1. MEC and core polarisation 6.2. Relativistic corrections 6.3. Collation: closed LS shells 6.4. Comparison with previous works 6.5. Collation: closed jj shells in Pb region 6.6. Outlook References

326 336 340 347 348 352 355 357 359 360 361 362 367 370 371 373

Abstract: Matrix elements of spin operators evaluated in a nuclear medium are systematically quenched compared to their values in free space. There arc a number of contributing reasons for this. Foremost is the traditional nuclear structure difficulty of the inadequacy of the lowest-order shell-model wavefunctions. We use the Rayleigh-Schr6dinger perturbation theory to correct for this, arguing that calculations must be carried through at least to second order. This is a question of the appropriate effective interaction. We review the Landau-Migdal approach in which only RPA graphs are retained and discuss the strength of this interaction in the spin-isospin channel expressed in terms of the parameter g'. We also consider one-boson-exchange models and compare the two. The advantage of the OBEP models is that the two-nucleon meson-exchange current operators can be constructed to be consistent with the potential as required by the continuity equation for vector currents and the partial conservation (PCAC) equation for axial currents. We give a complete derivation of the MEC operators of heavy-meson range starting with the chiral Lagrangians used by Ivanov and Truhlik. Nonlocal terms are retained in the computations. We single out one class of MEC processes involving isobar excitation and demonstrate that in lowest order there is an equivalence between treating the isobar as an MEC correction and treating it as a nuclear constituent through the transition spin formalism. Differences occur in higher orders. There are a number of uncertainties in the isobar calculation involving the

Single orders for this issue PHYSICS REPORTS (Review Section of Physics Letters) 155, No. 5 (1987) 263-377. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must bc accompanied by check. Single issue price Dfl. 82.00, postage included.

0 370-1573/87/$40.25 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

I.S. Towner, Quenching of spin matrix elements in nuclei

265

neglect of the isobar's natural width, the relativistic propagator being off the mass shell and the coupling constants not being known with any precision. We present a comprehensive calculation of core-polarisation, meson-exchange current and isobar-current corrections to low-energy M1 and Gamow-Teller transitions in closed-shell-plus-one nuclei (at LS and jj closed shells) expressing the results in terms of equivalent effective one-body operators. We compare with the empirically-determined operators in the sd-shell of Brown and Wildenthal. While overall agreement is good, a closer inspection reveals two discrepancies which suggest two benchmark tests for newer and alternative models.

I. Introduction

This report, concerned with the response of a nucleus to spin-dependent electromagnetic and weak probes, concentrates on the traditional topics of low-energy nuclear physics: the study of nuclear magnetic moments, MI ~/-transitions and Gamow-Teller 13-decay. However recent advances in this field have been helped by the extension of the body of available data through the use of strongly interacting probes. For example, proton and pion probes in inelastic scattering and charge-exchange reactions have added significantly to our knowledge of spin-dependent matrix elements. The best known example is the charge-exchange (p, n) reaction, which at forward scattering angles and proton energies of the order of 200 MeV excites preferentially the giant Gamow-Teller resonance. The surprise here is that the strength of excitation of the resonance, being related to the square of a certain spin-dependent matrix element, is less than expected from sum rules based on nuclear structure models. The resonance is said to be quenched. Similar results are found in (p, p') and (e, e') inelastic scatterings to the M1 resonance. Indeed quenching of spin matrix elements is quite a general phenomenon in nuclear-structure physics. In this report, we will discuss some of the contributions to this quenching. Before that however, we have to face a difficulty that is common to all nuclear-structure calculations in finite nuclei, namely the construction of an effective interaction. This difficulty is extensively documented in review articles [1-7]. Briefly, there are two facets to the problem. Firstly the nucleon-nucleon interaction, in being strongly repulsive at short distances, is inappropriate in itself for perturbation expansions. But through the solution of the Bethe-Goldstone equation the bare interaction is converted to a well-behaved, modified one, known as a G-matrix. The price paid is that one must use correlated two-nucleon wavefunctions at short distances. Secondly, nuclear-structure calculations perforce are carried out in truncated model spaces and again lead to a requirement of an effective interaction. A formal solution to this problem exists and is given by the Bloch-Horowitz equation, but the effective interaction is energy dependent and non-Hermitian. Fortunately the energy dependence and non-Hermiticity turn out to be weak and a re-ordering of the perturbation series can successively, in an iterative scheme, push these undesirable features into higher and higher orders. This is the Rayleigh-Schr6dinger perturbation theory. Likewise transition operators, such as those describing weak or electromagnetic transitions in nuclei, are similarly modified to become effective operators in truncated model spaces. We will discuss this to first and second order in the Rayleigh-Schr6dinger theory and show that this step alone accounts for a large fraction of the observed quenching. The other major contribution to the quenching comes from the process in which the weak or electromagnetic probe, prompts a meson to be exchanged between two nucleons. These are the meson-exchange currents that produce two-nucleon spin-dependent operators, which likewise must be modified in the nuclear medium through the use of the Rayleigh-Schr6dinger equation. One particular meson-exchange process that has received a lot of attention is one in which, in the exchange of a pion between two nucleons, a nucleon is raised to its excited state, the isobar A, and then de-excited by the weak or electromagnetic current probe. This virtual isobar excitation is believed to be one of the

I.S. 1"owner, Quenching of spin matrix elements in nuclei

266

principal contributors to the quenching phenomenon. However, as we shall see, this is one of the difficult contributions to calculate reliably; there are several model-dependent assumptions that go into its calculation. The various chapters of this review, then, are the construction of the effective interaction, the use of Rayleigh-Schr6dinger perturbation theory to compensate for model space truncations, the role of meson-exchange currents and in particular isobar currents. We conclude with a comparison with experimental data in simple situations, such as closed-shell-plus-one configurations, where it is hoped nuclear-structure uncertainties will be minimised.

2. Effective interactions

The nucleon-nucleon interaction as deduced from NN phase-shift data has been analysed in detail in terms of meson exchanges [8, 9]. These treatments involve an approximate solution of the relativistic two-baryon problem and result in the parameterization of a one-boson-exchange potential arising from the exchange of different mesons. A very satisfactory description of the NN scattering data for laboratory energies less than 350MeV can be obtained with this approach. The most important contributions come from the exchange of v ( J ~ = 0 - , T = 1), 9(1 , 1), to(1 , 0) and ~r(0+, 0) mesons. The scalar ~-meson is a fictitious resonance simulating two-pion exchange which has coherence properties in the spin-zero, isospin-zero channel. Other mesons have been included in the analyses and are found to be of lesser importance. We will note later the use of the A~(1 +, 1) meson. The interaction is strongly replusive at short distances, due in the one-boson-exchange model to the exchange of vector o~-mesons. Intermediate-range attraction comes from ~-exchange and long-range attraction from v-exchange. The interaction, however, cannot be used as a starting point for a perturbation expansion because the short-range repulsion leads to very large matrix elements. To produce a well-behaved interaction, Brueckner [10] suggested that the calculation of the interaction between two nucleons in a nuclear medium should proceed in a similar way to the calculation of the NN scattering matrix. One allows the two nucleons to interact with each other any number of times, but since they are surrounded by other nucleons only intermediate states above the Fermi surface are allowed. The effective interaction so constructed is known as the Brueckner G-matrix. Mathematically, the G-matrix is obtained in the iterative solution of the Bethe-Goldstone [11] equation: Q Q Q G(o~) = V + V - V+ V- V- V + .... ~o - Ho ~o - H~, ~o - H~

Q V+ V- G(~o). ~o - H~,

(2.1)

Here Q is the Pauli projection operator which restricts intermediate states to just two-particle states above the Fermi surface. As it stands, the equations are of not much practical use because there is an infinite sum over the unoccupied two-particle states. For practical calculations Q is often approximated by an operator in the relative momentum of the interacting pair. The centre-of-mass momentum can be factored out and enters only parametrically. An appropriate average value is selected. The equations also depend parametrically on w, the energy of the two interacting nucleons. Finally, a choice has to be made for the single-particle Hamiltonian, H 0. The eigenstates of H 0 determine which are the occupied and which the unoccupied single-particle states. Thus there is a Hartree-Fock type self consistency problem here too. Common ways to proceed are to assume that H 0 is just the kinetic energy operator, or that H 0 has a simple modelistic form such as the harmonic oscillator Hamiltonian. For an oscillator

I.S. Towner, Quenching of spin matrix elements in nuclei

267

basis, Barrett, Hewitt and McCarthy [12] have given a fast and accurate algorithm for calculating the G-matrix. For the Paris meson-exchange potential [13], the G-matrix has been constructed by Shurpin, Kuo and Strottman [14] and by Hosaka, Kubo and Toki [15]. The G-matrix, then, is the first step in the renormalization procedure since it leads to an effective interaction that pays attention to the short-range NN correlations. For some nuclear structure calculations, especially those performed in large model spaces, the G-matrix approximation might be sufficient. However in finite shell-model spaces, a further renormalization is required that reflects the influence of the truncation of the model space. We will be concerned here with nuclear responses to various probes that excite the nucleus by a few tens of MeV. They are low-energy excitations when compared to the total nuclear binding energy of hundreds of MeV. The description of these excitations is principally in terms of single particle-hole excitations. Collective states, being a coherent superposition of particle-hole states, are a characteristic feature of nuclearstructure physics. Particle-hole calculations in their simplest form, the Tamm-Dancoff approximation (TDA), and extended in the random phase approximation (RPA) have been largely successful in this regard. The key ingredient, however, is the particle-hole interaction. In principle, this is given by the G-matrix. However in restricting the calculation to just one-particle one-hole excitations, the interaction is further modified by the effects of truncation. Because of the complexities of this, it has become commonplace to introduce a phenomenological particle-hole interaction and relate its parameters to the reaction G-matrix.

2.1. Landau-Midgal interaction Nucleons in nuclei, continually interacting with each other, become modified in the presence of others. A simple description of this has been developed by Landau [16] in the context of Fermi-liquid theory and extended for the study of finite nuclei by Migdal [17]. The basic assumption is that when the interaction is gradually turned on, the system develops adiabatically into a system of quasiparticles in such a way that its properties are not drastically changed. In other words the system retains certain essential properties of an ideal gas with, for example, a well-defined Fermi surface and the same classification of levels. The energy of the system, however, becomes a complicated function of the quasiparticle occupation numbers, n(k). Here k is the quasiparticle momentum. Landau showed the change in the ground state energy caused by a small number of elementary excitations is given by 1

BE = ~ e°(k) Bn(k) + -~ ~ ,~°(kl, k2) Bn(kl) Bn(k2) k

k1,k2

(2.2)

where Bn(k) represents the deviation from the sharp distribution function of n(k)= 1 for occupied states, k < kF, and n(k) = 0 for unoccupied states, k > k v. Here k F is the Fermi momentum, e°(k) the quasiparticle energy in the absence of other quasiparticles, and ~(kl, k2) the energy of interaction between two isolated quasiparticles. We can think of quasiparticles as nucleons in nuclei as modified by the presence of particle-hole excitations, and ~(kl, k2) as the particle-hole interaction. By definition, is the second functional derivative of the total energy with respect to the occupation numbers. For isotropic nuclear matter, the dependence of the interaction on spin and isospin can be summarised in the form ~-~(kl, k2) = f ( k l , k2) ~ - f t ( k l , k2) "/'1 "~2 "q- g(kl, k2)°'l ° 0"2 + g'(kl, k2) 0"1° 0-2 "l'l °I"2

(2.3)

268

I.S. Towner, Quenching of spin matrix elements in nuclei

where o. and ~" are the Pauli spin and isospin matrices respectively. We are only interested in small excitations, such that to a good approximation Ikl[ ~- k~ and Ikzl = k v and the momentum dependence reduces to that of the angle between k 1 and k 2. Thus an expansion is made in Legendre polynomials f ( k 1,

k2) = 2 f/P/(COS 8 ) l

(2.4)

where O is the angle between k I and k 2. The coefficients ft are functions of k v and hence of the density. Similar expressions hold for the other coefficients. It is convenient to introduce dimensionless quantities Fl = Nor ~

(2.5)

where N o is the number of states at the Fermi surface per unit energy and unit volume N o = 2m*kF/TrZh 2 .

(2.6)

The effective mass, m*, is defined as de(k)/dklk_kF = h 2 k v / m * .

(2.7)

For a short-range interaction only the first few terms in the Legendre expansion will be significant. Indeed for a very short-range force only the l = 0 term will be important. In this limit

o~(kl, k2) : No 1 [F¢~+ F[j ~-"~'2 + Go o.1" °'2 + G,; o.," o.2 ¢, "72l

(2.8)

and F 0, F 0, G 0, G o are called the Landau-Migdal parameters. We have left out tensor components. A Fourier transform to coordinate space leads to a zero-range interaction F(rl, r2) = N o 1 6(r, - r2) [F o + F[~ 71 • ~'2 + Go o.1" o'2 + Go o.1" o.2 ~'," 721.

(2.9)

This is the form, that is frequently used to generate RPA wavefunctions and transition amplitudes for collective states in nuclear-structure calculations. It is understood that the interaction is only used between one-particle one-hole states; the influence of more complicated configurations being assumed to be incorporated in the parameterization of the Landau-Migdal parameters. Furthermore it is assumed that only direct (rather than antisymmetrised) particle-hole matrix elements are calculated with this interaction. For a zero-range force the exchange matrix element is proportional to the direct one anyway so no physics is lost here; it is just that the exchange matrix elements are considered part of the Landau phenomenology. In the literature there are other normalizations of the particle-hole interaction. Here we have used No 1= 146.3MeVfm 3, k v = 1.4fm 1 and m * = m, the bare nucleon mass. Others, notably the Julich group [18], use essentially twice this figure CO= 302 MeV fm 3 for a slightly different value of k v = 1.36fm -1. Another choice [19] expresses the strength of the interaction in terms of the ~r-nucleon coupling constant F(rl,r2)-

4 ¢rf ~NN 6(r,-r2)[f m~

+f' ~','~'z+go.l'o.2+g'

o.l.o.2zt.T2]

(2.10)

I.S. Towner, Quenching of spin matrix elements in nuclei

269

where f~NN 2 =0.08 and 4 ~'f 2~NN/m 2~ = 392 MeW fm3: We use lower case letters to represent LandauMigdal parameters in these "pionic units". The various types of particle-hole collective states are sensitive to the different force parameters F(rl, r2). This is indicated [20] in table 1, where fitted values of the parameters f, f', g and g' in RPA calculations in the Pb region [21] are listed. For spin-dependent probes, we shall be mainly interested in the parameters g and g'. Calculations of the Landau parameters starting from the G-matrix have been given by the Tiibingen [22, 23] and Jiilich groups [24, 25, 18]. The leading contribution to F comes from the antisymmetrized G-matrix evaluated in nuclear matter with particle momenta restricted to the Landau limit, Iklt = kF and Ik21 : The Landau parameters are then functions of k F, the matter density. The results also depend on the relative momentum q = k 1 - k 2 and weakly on the centre-of-mass momentum. We quote some results in table 2 at normal densities in the limit q--->0 . The results also depend on the choice of effective mass, m*. We have scaled the Tiibingen results [22, 23] to correspond to a normalization No 1 being chosen with m*/m = 1. This is not completely correct since m* also influences the single-particle energies e(k)= h2k2/2m * used in evaluating the Bethe-Goldstone equation to find the G-matrix. There are some differences in the results depending on the choice of the input NN interaction, as discussed by Nakayama et al. [18], who investigated this point using several different versions of the Bonn potential [9]. However there is one common feature. At normal densities one obtains very 2 2 attractive values for the Landau parameter F 0 = N Ox (47rf~NN/m~)f=2.7 f that characterizes the strength of the particle-hole interaction for scalar-isoscalar excitations. In all cases one obtains for F 0 values that violate the stability condition F 0 < - 1 [26]. This means that in the G-matrix approximation nuclear matter at normal densities is unstable against small deformations, That is, the collective E0, AT = 0 mode has an excitation energy lower than the ground state. An RPA particle-hole calculation with this interaction would diverge. Therefore one has to go beyond a simple Brueckner G-matrix approach. It has been shown by Sjoberg [26] and Babu and Brown [27] that the inclusion of screening effects in the so-called "crossed channel" reduces strongly the attraction of the G-matrix. Schematically the equation to be solved is written as Fph = Gph + F~nduced(Fph)

(2.11)

where the induced i n t e r a c t i o n Finduced(Fph) sums all particle-hole bubbles in the crossed channel. A graphical representation is given in fig. 1. Intermediate isobar-hole states are also included in the screening. The equations are to be solved self-consistently. The major effect of the induced interaction Table 1 Class of collective state and component of zero-range Landau-Migdal interaction that influences it, from [20] Force parameter

Type of collective state

Examples

Parameter values ~

f f' g g'

Isoscalar electric Isovector electric Isoscalar spin-flip Isovector spin-flip

E0, E2 (A T = 0) El, E2 (AT = 1) M1 (AT = 0) M1 (AT= 1), Gamow-Teller

0.07 0.33 0.58 b 0.73

a From fits with RPA calculations to data in the Pb region by Speth, Werner and Wild [21], (Set II, table 12.2) in "pionic units". b Smaller values of g are now preferred [20].

270

I.S. Towner, Quenching of spin matrix elements in nuclei Table 2 Landau parameters at normal densities (k v = 1.4fm l) in 'pionic units'

f'

g

g'

T/ibingen"

I II

-0.78 -0.38

f

0.17 0.16

0.16 0.13

0.44 0.49

Jiilich b

1 II III

-0.90 -0.22 0.18

0.25 0.05 0.12

0.15 -0.05 0.12

0.49 0.58 0.56

Brooklyn ~

I IV

0.76 -0.23

0.19 0.23

0.13 0.14

0.44 0.38

a Dickhoff, Faessler, Muther and Wu [23]: I = bare G-matrix from the Bonn potential, HEA [9]; II = bare G-matrix plus induced interaction. b Nakayama, Krewald and Speth [25]: I = bare G-matrix from the Bonn potential, HEA [9]; II = bare G-matrix plus induced interaction; III = bare G-matrix plus induced interaction plus relativistic correction. CCelenza, Pong and Shakin [28]: I = bare G-matrix from the Bonn potential, HEA [9]; I V = b a r e G-matrix plus relativistic correction: There are small differences between the three results due to different numerical approximations and calculational strategies. Celenza et al. [28] point out there is considerable cancellation when the contribution from each individual meson exchanged in the NN interaction is separately itemized.

is to stabilize the parameter f since all induced contributions add coherently [24] to compensate the attractive value obtained from the bare G-matrix. In other channels, there are mutual cancellations among the induced contributions and the parameter changes are not very large. For example, the spin-isospin strength g' is enhanced by about 20%. A further improvement [25] comes from introducing relativistic effects, a nucleon in the nuclear medium being described by Dirac spinors, rather than nonrelativistic Pauli spinors. The justification for this is that the mean field felt by a nucleon of some 50 MeV (and hence small compared to the nucleon

(o)

(b)

(c)

(d)

(e)

(f)

Fig. 1. Graphical representation of the coupled equations to be solved for the effective particle-hole interaction, refs. [22-25]. Graphs (a) and (b) are the direct and exchange contributions from the Brueckner G-matrix while the remaining graphs together are the induced interaction.

I.S. Towner, Quenching of spin matrix elements in nuclei

271

mass) is the resultant of two large fields: a strong attractive scalar field Vs = -400 MeV and a repulsive vector field V0 = 350 MeV. The single-particle spinors obey the Dirac equation: [i y . p - 3'4 (E - V0) + m + Vs] gt = 0 which in nuclear matter reduces to the free Dirac equation with modified energy and mass: E* = E - V0 and m* = m + VS. Typically E* and m* attain only 60% of their value in free space. This has a significant effect on the lower component of the Dirac spinor qt = (G, F) where now F = (E* + m*) -1 o" "p G. Thus in the nuclear medium: Fmedium = {(E + m)/(E* + m*)}Ffree ~ 1.7 Ffree. Nakayama et al. [25] incorporate this difference in the ratio of upper and lower components of the Dirac spinor in a nuclear medium compared to free space in deducing the nucleon-nucleon interaction from a model of meson exchanges. Then with a modified NN interaction, they recompute the G-matrix and the Landau parameters. As was found by Celenza et al. [28], this relativistic modification also helps in the stabilizing of the Landau parameter F 0. Nakayama et al.'s [25] results are shown in table 2. As can be seen, quite reasonable values are obtained when compared to the empirically determined values [21] quoted in table 1 providing both the screening and relativistic effects are included. For the spin-dependent part of the particle-hole interaction, results indicate that g is small, and that g' is in the range 0.5 to 0.6.

2.2. Estimates of g' In a discussion of the nuclear response to spin-dependent probes, the Landau-Migdal parameter g' plays a crucial role. Its value, as calculated from the G-matrix in nuclear matter is not far off the empirical value determined in fits to data. So its understanding is basically at hand. Nevertheless it is of interest to give some simple model estimates of g' so as to get a better understanding of what ingredients in the nucleon-nucleon interaction influence its value. The model is the one-boson-exchange potential evaluated in nuclear matter in the Landau limit with some prescription for the short-range correlation function. The procedure is outlined in Brown, Backman, Oset and Weise [29]. We consider first ~- and p-meson exchange, anticipating that these will be the principal ingredients to g'. In momentum space the one-pion-exchange potential is written 2

47rf~Nrq (0"1" q)(0"2" q) V~(q)

--

mE~

q2 + m2

2 4'trf~NN 1 --

2

m~

3

q2

q2 2 O'1 " 0"2 71 " "/'2 "q- t e n s o r

+ m=

41rf,,NN 1 --

2

m,,

3

1

I"1 " 'T2

2

m,, --- 2 0"1"0"2 ~'1" ~'2 + tensor

q+m~,

(2.12)

where we have not written down the tensor term. Here f,,NS is the pion-nucleon coupling constant, f 2 = 0.08, and m,, the pion mass. The first term in the braces, upon Fourier transforming, leads to a 6-function interaction in coordinate space. The short-range correlation function essentially removes this term, because the short-range repulsion in the nucleon-nucleon interaction restricts two nucleons from coming close together. The removal of this term is a minimal requirement of short-range correlations. It amounts to assuming a correlated wavefunction that vanishes for zero separation of the nucleons and equals the uncorrelated wavefunction everywhere else. We will denote the potential with correlations included as I)(q). Thus

I.S. Towner, Quenching of spin matrix elements in nuclei

272

4"/rf2~rNN 1

l')'(q)-

m2

me

~ 0"1" 0"~ rl "r~ .

3 m; + q"

(2.13)

"

The direct matrix element in nuclear matter, graph (a) in fig. 2, corresponds to the transferred momentum q--0. The Landau parameter is defined as ? 9(q

= 0) -

"~f ;NN ' 2 g d i r 0"1 " 0"2 'l'l " 7"2 m

(2.14)

and for the one-pion-exchange potential with a minimal correlation function leads to the standard result ! z 1 g~,dir 3"

For the p-exchange potential, we proceed analogously

vo() --'q-

47rf2pNN 2

--

(0"1X q)'(0"2 X q ) 2

mp

2 {

47rfpyN 2 ~ m° 3

-

2

q +mp

71 " 'T~

m;}

1

(2.15)

2 -- - 2 0"1 " 0"2 r, • r 2 + tensor q + mp

where f0NN is the p-nucleon coupling constant, whose value is given roughly by (f2pNN/m2o)= 2(f2NN/m2). A spin-independent term is not included here but will be considered shortly. Again a minimal correlation function removes the a-function term and produces a Landau parameter of 2

2

, foNN/mp gp.dir = 2 2

f ,,NN/m~,

2

(2.16)

"3

which is of the order 1.3 and large. There is one other meson that generates a potential proportional to 0"1 " 0"2 I"1 72 in the one-bosonexchange model that is sometimes included and this is the Al-meson. With a mass of around 1100 MeV it leads to a very short-range interaction and so can probably be considered part of the short-range phenomenology. However, the A~-meson is the chiral partner of the p-meson in a chiral model of vector mesons. While chiral invariance cannot be said to be an established symmetry for vector mesons, it is perhaps worthwhile investigating its consequences for the nucleon-nucleon interaction [30, 31]. °

k2

k~

kz

k2

kl

Ca}

(b)

Fig. 2. The one-boson-exchange potential evaluated in nuclear matter in the limit (a) q : ~ I), where q is the transferred momentum for the direct graph, and (b) qZ~2k~(l cos 8 ) , where q is ]k~ - k21 for the exchange graph, to obtain an estimate of the Landau-Migdal parameters.

I.S. Towner, Quenching of spin matrix elements in nuclei

273

Furthermore, we will be interested in axial-vector currents in nuclei and, in vector meson dominance theories, the Al-meson plays the role of mediator of the current in much the same way that the p-meson plays the role of mediator of the vector current. Using pseudovector coupling for the AINN vertex, gAINN'~/.'~5 , the Al-meson potential takes the form 2

VAI(q)

= gAlSN2 --O'1"O'2--

q2 + mA

2

~'1'~'2

mA

(2.17)

where the second term comes from the gauge term in the Al-propagator [30]. In the work of Durso et al. [30] the A1NN coupling constant in the effective Lagrangian of Wess and Zumino [32] is related to the rrNN coupling constant: gA1NN/mA 2 2 = 4 7 r f ~2N N / m ~ ,2. In the hard pion model of Ivanov and Truhlik [33] which is based on the effective Lagrangian of Ogievetsky and Zupnik [34] the relation is expressed as gA~NN= gAgp,,,," Using SU(6) symmetry to relate gp~,. to gONN, gp.~, = 2gpr~N, the GoldbergerTrieman [35] relation to express the axial-vector coupling constant in terms of pion coupling constants, gA=f,g,,NN/M, the KFSR [36] relation to express the pion decay constant in terms of gp,,,: 2 f 2 2 g,,,~ = m~, and finally the Weinberg relation [37] to connect the masses of the p- and Al-mesons, m A = 2m2p, the same result is obtained 2

2

2

2

2

g A1NN/ m A = 4 7rf ,,yy/ m ~ = g=NN/ 4 M

2

(2.18)

.

Each of the relations just cited is satisfied by experimental observations to within 15%. Here M is the nucleon mass, g~NN the pion coupling constant for pseudoscalar coupling, which is related to the pion coupling constant for pseudovector coupling, f,,NN, by the expression shown. t Returning, then, to the calculation of gdir for the Al-meson exchange potential and introducing a minimal correlation function to remove the 6-function term yields 2

gA~NN ~'Al(q) -- q2+ m-----~2[--O.1" O'2 + ½0"1" O'2] 71 "72 + tensor

(2.19)

and hence a Landau parameter P

gA~,dir =

2 -- 3

(2.20)

"

The Al-meson cancels some of the large contributions from the p-meson. Next we consider the exchange contribution, graph (b) of fig. 2, obtained from the direct one through the operator: -P,,PTP(k 1 <--->k2). Here P~, and P, are the spin and isospin exchange operators respectively. The transferred momentum in the exchange graph is q2 = Ikl - k212. In the Landau limit, Ikll = kF and Ik21 = G , q2 =2kZF(1--COS O) where O is the angle between k I and k 2. As already discussed the Landau parameters are expanded in a Legendre series in Pt(cos O). We are only interested here in the lowest, l = 0, term. Thus the exchange contribution to g' is defined as +1

1 f

-I

dx

42

q: = 2kF(1 -- X))) -- 7rf"N~ ' • O.2"rl • ~'2 mZ gexchO'l

(2.21)

I.S. Towner, Quenching of spin matrix elements in nuclei

274

where x is cos O. Note that with the operators P and P, the contribution to g'exch in the one-bosonexchange model is not confined to just ~r-, P- and A~-exchanges but ¢, o~ and other mesons can all make a contribution. The properties of the spin exchange operators are 1 = - 4 ' ( 1 + % ' % + 00," % + 00, • 00~ r I " "r2)

-P,P,

- P P~ %" r 2 = - ¼(3 - r, • r 2 + 3 00, • 002 - 00, " 002 rl " "d'2) (2.22)

-P,~P,

00, " 002 = - 41( 3 + 3rl

-PP,

o'~ • 002 r~ • r 2 = - ¼(9 - 3~'~ • r 2 - 3 0 " 1 • 0" 2 + o ' , • 002 'rt " 'r2)

" re -

00, " 002 -

001

"

002 ~'1 " re)

thus it is a simple matter to pick out the coefficient of 00~ • 002 rj • r 2. The one-boson-exchange potential (omitting tensor, spin-orbit, and nonlocal terms) for mesons ~r, p, m, ¢ and A~ is 4

V~(q)

2 7J'f ,rrN N

m2

--

1

q

2

3 q2 + m2 00]

V o(q)

" "/'2

4 77fpNN 2 qm;~ 3 q2 + m 2p

1

~ ( q ) = gpNN q2 + m2p I"1 " 72 2

~']

2

2

=

00~

"

-

-

4

1

2

7 7 f toN N

m2

goNNq2+m2 ~

V ( q ) = - gZ,,NN

l 2 q +m

2

q

3

=

2

q2+m2

001" 00~

(2.23)

2 1

VA,(q)

001 " 002 Tl " 7"2

~ gA,NN q2 +1 mA 2

- - 001 " 0°2

3

q2 j 2 001 ° 00~, 'rl " gF/A

Again removing the g-function terms by the introduction of a minimal correlation function and projecting out the lowest, l--0, term in the Legendre expansion for the Landau parameter gives , g ..... h -

2m2~,__ ( m ~ ) 1 2k v Qo 1 + 2

,

mo

go .... h = ( ¼ O p - ~ C o ) ~

2k v

( Qo 1+

2

mp ) 2k{

?

ao~+~

~)~k~ eo 1+

2F

<(m:) lm ( mA) gAl,exch g ..... ,=~D~T77,2 Qo 1 + ~ 2k v 2kv 2

2

6 775,2 Qo 1 + 2k v 2kv

-y

(2.24)

l.S. Towner, Quenching of spin matrix elements in nuclei

275

where Qo(z) is the Legendre function of the second kind. Here C~ = ( f i 2Y N / m i2) / ( f ~ N2 N / m , )2 and Di = (giNN/mi22) /(47rf~,NN/m~) . 2 2 Numerical values are given in table 3 in the column whose short-range correlation function is labelled minimal. We see that the contribution to g' from the exchange graph is quite small in total, although contributions from individual mesons are sizeable. For example the large negative value for go,,exch is more or less cancelled by the positive contribution from g,~,exch" Furthermore it makes little difference whether the Al-meson is included or not. It gives a large contribution to t t both gait and gexch but with opposite signs, the summed effect being small. The resultant value of g ' = 1.5 is very large, much larger than the values calculated with realistic G-matrices (see lines I in table 2), and larger than the empirically deduced value. What is wrong? The large result comes principally from g0,air which being a short-range ingredient suggests that inadequate short-range correlations have been built into the calculation. Brown, Backman Oset and Weise [29] note that the ~r-exchange interaction has been constructed from zero-range ~rNN vertices. It is clear, however, that vertex corrections lead to a finite extension in space of these vertices. Describing this in terms of a form factor that depends only on the momentum transferred by the exchanged meson, the ~r-exchange potential can be written V~(q) -

;q0"2"q

2 N O"1 4'n'f,~N 2 2 rn q + rn

I r ( q 2 ) 1 2 'r, ' ' r 2

2 2

m~

2

3

1

2---

m~+q

2

I C ( q 2 ) l 2 0"1" °'2 ¢'1 "'r2 +

(2.25)

tensor

where a convenient way of parameterizing the form factor is F(q2) = (A2 - m~)/(A~ 2 2 + q2).

(2.26)

The cut-off mass, A~, is of order 1 GeV. Note that the first term in V~(q) is no longer a &function in coordinate space. I.nstead the effect of a finite-range vertex is to distribute the &function over a finite region and produce a strong interaction at short distances. Removal of this short-range piece by the repulsive core of the nucleon-nucleon force will not be complete and requires a proper treatment of Table 3 The Landau parameter, g', deduced from the one-boson-exchange potential evaluated in nuclear matter in the small-momentum limit for various choices of short-range correlation (SRC) function and vertex form factors (FF) Vertex FF SRC function

No No t

gd,~ Ir p to (r Aa Sum

0.00 0.00

No Minimal t

gexch

-1.00

0.07 0.12 -0.41 0.40 0.93

- 1.00

1.10

t

ga~, 0.33 1.48

No 1 - lo( qcr) t

g~x~h

1

t

gd~

-0.67

-0.01 -0.25 -0.37 0.40 0.59

0.32 0.75

-0.22

1.15

0.36

0.84

Yes

1 - Jo( qcr) I

gexch -0.01 -0.10 -0.14 0.22 0.15

g,x~h

0.12 0.23

0.02 -0.01 -0.10 0.19 0.06

-0.10

0.12 2

t

gdir

0.25 2

0.16 2

Parameter values: g,r~N= 13.41, gpr~N=2"63, g~NN= 7.89, g~r~,= 5.6, gA1NN= 7.79, f~N~ = 0.079, fpN, = 5.36, f~NN = 1.35, m~ = 139.6MeV, mp =770.3MeV, m . =782.6MeV, m =500MeV, MAl=X/2mo=lO89MeV, k F = 2 m = l . 4 1 f m -1, qc=3.93fm -1, A~ = IGeV, Ai= 1.44 GeV for i = p, to, a, A 1.

276

I.S. Towner, Quenching of spin matrix elements in nuclei

short-range correlations. Let us denote by g(r) the two-body correlation function that multiplies the one-pion-exchange potential in coordinate space, viz. I?(r) = V(r) g(r). In momentum space I? (q) :

f

- ~d37k- ~ V ( k ) g ( q - k )

(2.27)

where g(q) is the Fourier transform of g(r): g(q) = f d3r eiqrg(r).

(2.28)

Now we anticipate that g(r) mainly arises from the repulsion due to o~-exchanges in the nucleon-nucleon interaction. Thus the range of g(r) is expected to be comparable to the Compton wavelength of the ~o-meson. A parameterized form for g(r) that has been adjusted to reproduce the dominant Fourier components of a realistic two-body correlation function calculated with the Reid soft-core potential is [291:

g(r) = 1 - Jo( qcr)

(2.29)

with qc = 3.93 fm ~= mo. The Fourier transform is g(q) = (2~r)3 (~ ( 3 ) ( q ) _

2~r2~_6( q - qc) q~

(2.30)

and hence

(/(q) = V(q)

(q

-qc):_ V((q qc

qc):)= V(q) - V(q~).

(2.31)

Using this latter form as the definition of the nucleon-nucleon interaction corrected for short-range correlations it is easy to calculate the contributions to the Landau parameter g' following the method discussed above. Let us define 3

r b - q~ + m7 I g(q~)[ '

r b - q~ + m~ IV,(q?,)l"

+1

1 sCg(kr) = 5

f

2

2kv(1 - x) , IC(2k~(1 - x))l 2 dx 2k2( 1 _ x) + m 7

-I +I

,

1f

sCg(kF) = ~

m~

dx 2k2v(1 _ x) + m~

I

with i = rr, p, ~o, (r or A 1, then

1~(2k~.(1x))l 2 "

(2.32)

I.S. Towner, Quenching of spin matrix elements in nuclei

g~,dir

= ½~7~ ,

gAl.dir

' = 2Cot/o gp,dir I

I

-IFAi(0)I 2 + r/A, + 3~A1,

=

277

t

g~,~xch= -- ~(r/~ -- ~:~(k~)) (2.33)

g.,~x.h = }D.(~:;(kF)- r / ; ) - } Co(r/.- ~: (kF)) gco, exch

=

-

-

1D

4

,

co(~co(kF)

1

-- T~:) "1-

1

--

t

g.,¢xch = ~D.(~:.(kF) - 7/') gAl,exch

=

I ( ~ A I, ( k F )

, _ _ ~A1)

~(r/A ' -- ~:g,(kF) ) . 1

In the limit F ~ 1 (no vertex form factors) and qc ~4 ~ (only minimal correlations) then r / ~ 1, r / ' ~ 0, ~:(kF)~ 1 - ~:'(kF) and ~:'(kF)~ (m2/2k 2) Q0(1 + m2~2k2F) and these expressions reduce to those given previously, eq. (2.24). Numerical results are given in table 3. Firstly with the correlation function g(r) = 1 - Jo(qc r) but without the vertex form factors the contribution from the ~r-meson exchange to g' is little changed. This is because the light meson mass generates a long-range interaction that is little affected by the details of the correlation function. For the heavy mesons, however, their contributions to g' are cut down significantly by factors of two to three. Introducing vertex form factors further cuts down the contributions from all mesons by another factor of two or three. Thus the total g' has gone from 1.5 with minimal correlations to 1.0 with a more extensive correlation function to 0.4 with the further introduction of vertex form factors. Note the extensive G-matrix calculations of the Tfibingen [23] and Jfilich [25] groups find (table 2, line I) g' of 0.44 and 0.48 respectively. It is very clear that short-range correlations and vertex form factors are necessary ingredients in the simple one-bosonexchange model if a reasonable value of g' is to be obtained. However, the model calculations are very sensitive to the details of the correlation function and vertex form factors. We have used for the cut-off parameter A~ = 1 GeV, which may be on the small side (a larger A T would lead to a larger g'), and A i = 1.44 GeV for all other mesons. These values are taken from a fit to the electromagnetic nucleon form factor by Iachello, Jackson and Lande [38]. Further in the one-boson model, the contributions from to- and tr-mesons more or less cancel each other (this depends naturally on the choice of coupling

Table 4 The Landau parameter, g, deduced from the one-boson-exchange potential evaluated in nuclear matter in the small-momentum limit for various choices of short-range correlation (SRC) function and vertex form factors (FF) Vertex FF SRC function

No No

gdir ~r p to a A1 Sum

No Minimal gexch

0.00

-0.21 -0.36 -0.41 0.40 -0.69

0.00

-1.27

Parameters as recorded in table 3.

gd~,

No 1 - J0( qcr) g~x~h

0.18

0.04 0.76 -0.37 0.40 -0.44

0.18

0.39

gd,

Yes

1 - ]o( qcr) gexch

gd~,

gexch

0.09

0.04 0.29 -0.14 0.22 -0.11

0.03

-0.06 0.04 -0.10 0.19 -0.05

0.09

0.29

0.03

0.03

l.S. Towner, Quenching of spin matrix elements in nuclei

278

constants), while for the Al-meson, the direct and exchange contributions to g' likewise cancel. Thus it is quite a reasonable assumption to consider only the ~r- and p-mesons in the spin-isospin channel. Many of the simple effective interactions to be discussed shortly follow this ansatz. For completeness, we give in table 4 a similar analysis of the isospin-independent, spin-dependent Landau parameter, g. With the inclusion of short-range correlations and vertex form factors, a small resultant value of g -- 0.06 is obtained, somewhat smaller than the exact G-matrix result (table 2, line I) of g = 0.15. Again there are a lot of cancellations amongst the various contributions. These model estimates are only intended as a guide to the understanding of the Landau parameters. To make contact with values deduced from experimental data two further considerations are necessary. Firstly the calculation has to be taken at least to second order and, in particular, consideration given to the graph in which a particle-hole phonon is exchanged between the upgoing and downgoing line in graph (b) of fig. 2. This is the induced interaction discussed in the last section, which in the calculations of Nakayama et al. [25] give roughly a 20% increase in the value of g'. Estimates of these effects can be obtained by modifying the meson propagator 1/(q2 + m 2) to include the meson's self energy [29, 39]. We will not pursue this topic. Secondly, in finite nuclei as opposed to nuclear matter, some care is needed in discussing the finite range of the interaction. In particular, the tensor interaction which has been consistently left out here has a very important role to play. In the next section we discuss some extensions of the zero-range effective particle-hole interaction that include tensor interactions.

2.3. Effective interactions in finite nuclei The response of a nucleus to a spin-dependent probe depends critically on the strength of the residual particle-hole interaction in the o-~ • % r~ • r 2 channel as characterized by the Landau parameter, g'. Estimates from nuclear matter in the last section suggest that the ~r- and p-exchange components of the one-boson-exchange force are the principal ingredients. Thus most models adopt the ~r- and p-exchange potentials to describe the finite-range character of the interaction in this spin-isospin channel, including now their tensor components, and augment the potential with a phenomenological zero-range piece whose strength is characterized by ~g': 2

F

h(r,,r2)=

4rrf ~NN 2 m

",

~ 6(r,

-

r2).

(2.34)

This interaction has a rather speci/]c interpretation. It is only to be used in particle-hole calculations of the TDA and RPA type. All other graphs are presumed to lead to short-range ingredients and are effectively included in the parameterization 8g'. Furthermore only direct matrix elements (as opposed to antisymmetrized matrix elements) are calculated for the zero-range piece. The exchange term, anyway, is proportional to the direct one and so can be considered included in the definition of 8g'. However, for the finite-range ~r- and p-exchange potentials both direct and exchange pieces are calculated. The argument for this is as follows [39]. Although V is of long range due to the small mass of the pion, its exchange term is not. Denoting by k 1 and k 2 the momenta of the interacting nucleons, then the transferred momentum in the exchange graph, fig. 2(b), is q = ]k; - k2] which in the Landau limit of Ik, I ~ [kz] = kv has an average value of ( q 2)av = 2k~. , The meson propagator in the Landau limit, instead of being 1/m2 as it is in the direct graph, becomes 1/(m2~ + 2k2). This indicates that the exchange graph is also of short range in nuclear matter and therefore could be considered included in g'. However the situation is very different for finite nuclei. The average taken over momenta k I and k 2 depends strongly on the particular geometry of a given particle-hole configuration. Meyer-ter-Vehn [39]

I.S. Towner, Quenching of spin matrix elements in nuclei

279

finds for the J " = 0- configuration in particular the exchange matrix element is very large. Thus it is argued that for calculations in finite nuclei, the exchange contributions of V,, and, for consistency, Vp should be explicitly calculated and separated from ~g'. Some of the model interactions used in the literature in the last few years are: a) Meyer-ter-Vehn [39]. This work is based on the one-pion-exchange potential 2

2

Fph(q) = "V',,(q) + ag' 4~'f ,~NN/m~ O"1 "°'2 71 " TZ

(2.35)

with just the minimal correlation function in 17"(q) that removes the delta-function piece from the potential. Vertex form factors are included of monopole type F~(q2) = (A2 _ m=)/(A= 2 2 + q2)

(2.36)

with the,cut-off parameter A,, = 1 GeV. In this limit there is no contribution to g' in nuclear matter from 17' (q) in the direct graph, fig. 2(a), and only a small contribution from the exchange graph, fig. 2(b). Thus g' comes almost entirely from the additional phenomenological term in ~g'. Meyer-ter-Vehn [39] determines ~g' in finite nuclei by calculating the excitation energy of the low-lying isovector unnatural parity states in the closed-shell nuclei 4He, 160 and 4°Ca in a large-space RPA calculation. Harmonic oscillator wavefunctions are used as the basis functions. The unperturbed position of these particle-hole excitations are obtained from the observed single-particle and single-hole energies in the neighbouring odd-A nuclei. The shift in the excitation energy calculated in RPA depends on ~g' and its value is determined in a fit to the experimental shifts. There is one further point, and this concerns the introduction of isobar-hole states in the RPA calculation. Meyer-ter-Vehn does not explicitly include them, but following Brown and Weise [40], modifies the pion propagator by including a self-energy term. The model interaction is then written

2

(

41rf=NN Fph(q)= --~ m~,

O.1° q o"2 • q +~g' °'1" o.2}/N(q) q m,

(2.37)

with the renormalization N(q) = 1 + Ua

t

_

q

2

q2 + m2

+ ~g'

)

.

(2.38)

Here ~g' in N(q) actually comes from the isobar-hole coupling and it is assumed that (~g')AN = (~g')NN" More on this in the next section. The Lindhard function [40] is approximately given by 2

(2.39) where f=Na is the pion coupling to the isobar, f,,ya = 2f,,NN, t°n the isobar-nucleon mass difference, p the nuclear density and P0 the normal value for the density of nuclear matter P0- 0.46 m3. With this renormalized interaction, Meyer-ter-Vehn obtains ~g' = 0.7 +- 0.1

(2.40)

I.S. Towner, Quenching of spin matrix elements in nuclei

280

in fits to shifts in excitation energies of the unnatural parity states. It should be noted that ~ ( q ) leads to an attractive particle-hole interaction that lowers the excitation energy ^while the 8g' term is repulsive. Thus the calculated shift is a balance between the attraction from V~(q) and the repulsion from 8g'. Indeed without the repulsive term, one could get low-lying collective states of unnatural parity that would be the signature of a pion condensation phenomenon. The fact that such states are not observed provides a lower bound on the value of 8g'. The principal emphasis of Meyer-ter-Vehn's work is the demonstration that at normal nuclear densities the spectra of unnatural parity states in light nuclei show no evidence for precursor phenomena of pion condensates. b) Anastasio and Brown [41]. The model interaction here is quite simple being

Fph(q) = l~(q) + ~,(q)

(2.41)

with a minimal correlation function that removes the delta-function piece in the "rr- and p-potentials. As discussed in the last section this correlation function alone is not in itself sufficient. Rather V ( q ) and Vp(q) should be multiplied by an extended correlation function whose principal effect is to cut down the contribution to g' from heavy mesons. A very rough estimate [41] is given by multiplying the p-meson 2 '~ contribution by 0.4, so that f 2 ~ f ~ = O . 4 f2. With f~/m2o = 2 f / m ~ the contribution from the direct graph, fig. 2(a), to g' is then t

1

2

gd~r = ~ + ~ X 2 X 0.4 = 0.87

(2.42)

the first term being the contribution from v-exchange and the second from p-exchange. The derivation of the factor 0.4 involves making a zero-range approximation for the p-exchange potential ~

=

2 2 ~fpm~ r, "r e o', • o%, (A/m~) 6(r12 )

(2.43)

where [41]

mpr -

m ~3'

f exp(-mr) m;r [g(r) - g(0)] d3r.

(2.44)

Here g(r) is the two-body correlation function from ~0- and c-exchange. The factor 0.4 is not applied to the tensor interaction, because short-range correlations are of little influence here since the tensor force chiefly connects relative-S states with relative-D states and the centifugal barrier effectively is keeping the nucleons apart in the D-state. The exchange contribution, fig. 2(b), is not calculated in this simplified interaction because as Brown, Osnes and Rho [42] argue its contribution to g' of -0.12 is more or less cancelled by the contribution from higher-order graphs principally the induced interaction. Results in table 2 from the Tiibingen [23] and Jfilich groups [25] indicate contributions to g' from the induced interaction ranging from +0.05 to +0.10 in nuclear matter. Thus the g' for this simplified interaction is 0.87 and somewhat larger than the empirical value determined for example in a particle-hole calculation [43] of the energy of the giant Gamow-Teller

I.S. Towner, Quenching of spin matrix elements in nuclei

281

resonance in Pb. As noted by Brown and Rho [43], if one includes only nucleon particle-hole states then there is a further renormalization of the interaction because of the neglect of isobar-hole states in the RPA calculation. They argue that isobar-hole states screen the nucleon-nucleon interaction reducing the Landau parameter to an effective one to be used in the nucleon subspace of !

geff =

Yg'.

(2.45)

Brown and Rho [43] calculate 3' to be ~, = 0.72 and hence g'e, = 0.62 consistent with values obtained in zero-range particle-hole calculations for the energy of the giant Gamow-Teller resonance by Bertsch, Cha and Toki [44], who find g' = 0.56 - 0.05 for a range of medium and heavy nuclei, and Gaarde et al. [45], who find g ' = 0.62 for Pb. c) Cha and Speth [46]. In this work the spin-isospin dependent effective particle-hole interaction is written 2

Fph(q)=9(q)+Sg'

47rf~m~ 2 0"1 " 0.2 "rl "'T2 • mr

(2.46)

Note we are using 'pionic units' for the normalization of the zero-range piece and have scaled the values of ~g' from [46] accordingly. Here the finite-range 97"(q) is constructed from the bare "tr + p exchange and the two-body correlation function g: C

d3k g(q - k)IVy(k)

9(q) = J g(q) =

+ Vp(k)]

(2.47)

2,rr 2

(2"n') 3 6(q) - ----5- 6(Iql- qc) q~

(2.48)

which is the Fourier transform of g(r)=l-jo(qcr) with q c = 3 . 9 3 f m -~ roughly the Compton wavelength of the o-meson responsible for the short-range repulsion. The finite-range potentials are 2

V(q) -

4'rrf,,,NN 2

m,~

IC (q )l

(0.," q) (0" 2 • q) m~ + q

(2.49) Vp(q)-

47rf:pNN

m;

I (q )l

( O ' 1 X q ) ' ! % X q)

m; +

where the spin-independent piece of the p-potential has not been included. Parameter values are f2~,sr~ = 0.081, fpN~ 2 = 4.86, m r = 0.699 fm -I, mp = 3.9fm -1 and in the monopole vertex form factors A,~ = 6 fm -1 and Ap = 10 fm -1. The g' evaluated in nuclear matter for this finite-range interaction is for the direct graph, fig. 2(a):

g'dir= 0.15 + 0.35 = 0.50

(2.50)

where the first term comes from "rr-exchange and the second from p-exchange and for the exchange graph, fig. 2(b):

I.S. Towner, Quenching of spin matrix elements in nuclei

282 t

gexch ----0.02 - 0.05 = -0.03.

(2.51)

The sum is g' = 0.47. To find the strength of the zero-range piece, 8g', Cha and Speth [46] calculated the energy of the Gamow-Teller giant resonance in 4SCa, 9°Zr and 2°spb in an RPA calculation that included both particle-hole and isobar-hole states. Note that both direct and exchange graphs for the finite-range part of the interaction V + V0 are calculated but only the direct graph for the zero-range piece is retained. To fit the experimental location of the giant resonance, Cha and Speth [46] require an additional contribution of 8g' = 0.37. Thus the sum g' + 8g' is 0.47 + 0.37 = 0.84, which is very close to the value of 0.87 obtained in the simplified interaction of Anastasio and Brown [41] just discussed. If the interaction is to be used in the smaller subspace of just nucleon particle-hole states in an RPA calculation, then the interaction would be further renormalized by a factor y = 0.72 as discussed by Brown and Rho [43]. d) Towner and Khanna [47]. The last two interactions are both based on a v + p-exchange model with some short-range phenomenology whose strength is adjusted in fits to energies of giant resonance states in an RPA calculation. By construction these interactions are only intended to be used in particle-hole calculations. More complicated configurations such as 2p-2h states are not explicitly included, rather their contribution is presumed to be contained in the phenomenology. Towner and Khanna [47] chose not to include any such phenomenology and based their interaction on the one-boson-exchange potential of the Bonn type [9], but reduced to the four or five more important meson exchanges. For use in finite nuclei, this interaction must be converted to a G-matrix, which is only crudely done by the introduction of a short-range correlation function. All graphs at least through to second order (not just the RPA graphs) need to be calculated. This has only been accomplished in light nuclei. The estimate of g' in nuclear matter from lowest order graphs, fig. 2, for such an interaction is small, typically g ' = 0.4 as seen in table 2, but the result is quite sensitive to parameter choices particularly in the vertex form factors. Second-order graphs still need to be evaluated. In coordinate space the potential is a sum of rr-, p-, to-, and ~r-meson exchanges and in some applications the Al-meson as well:

V(r) = V(r) + Vf,(r) + V(r) + V(r) + VA,(r ) V

(r) = 1

2

[(<

Vo2(x0+

.

_

o

2

2

gpNN

Vf,(r) = l m f p N s [2(~ 1 " O'2) Vo2(xo) - SI2V=(xp) ] (r 1 • r2) + m r 47r 2

(2.52)

2

mp

mo

x Voo(Xp)+ (2M) 2 (1 + 2Kp) Vo2(xo) - ~ 5

(3 + 4Ko) L" S Vl2(Xp) ] (~'1' 1"2)

m; Vo2(x ) - ~m: L" S V,2(x,, ) ]

V~(r) = m ~g'~NN[_Voo(x, ) + ~ - / gA~NN

VAl(r) = ma, - 47r -

[-(tr, • ~r2) ~,o(XA) + ~(~r, • O'2) Vo2(XA) -~-

1SI2 V22(XA)] (r," r2).

The form of the to-exchange potential, V ( r ) , is the same as Vo(r ) but without the isospin r~ • r 2 factor.

I.S. Towner, Quenching o[ spin matrix elements in nuclei

283

Here the tensor operator is S12 = 3(o"1 • ~) (o.2" r) - (o.1" o'2). Monopole form factors [(A 2 - m z)/(A 2 + q2)] are introduced at the boson-nucleon-nucleon vertices in momentum space which lead to the following radial dependences in the coordinate space representation: A2

Voo(X) = Yo(x ) _ A Y0(xa) m

2

- m XAYo(XA) 2Am

A3 A(AZ - m2) (2 - XA) Yo(XA) V02(x) = Yo(x) - --~ Y0(xa) + 2m 3 (2.53) Vn(x ) = 1 Yl(X) X

A 3 1 y1(xa) m3 XA

A3 V2z(x) = Y2(x) - --3 Y2(XA) m

A(A 2 - m 2) Yo(XA) ' 2m 3

A(A z - m 2) 2m 3

(1 '~ XA) Y0(Xa)

'

where Yo(x) = e-X/x, Yl(X) = (1 + 1/x) Yo(x), Y2(x) = (1 + 3/x + 3/x 2) Yo(x), x = mr and x a = Ar. The range of the form factors have been set at A~ = 1 GeV (consistent with NN interaction results) and Ai = 1.44 GeV for i = p, co, cr and A 1 (from the monopole fit [38] to the nucleon electromagnetic form factor). Other coupling constants are discussed shortly. Because of the strong short-range repulsion from o~-meson exchange, Towner and Khanna [47] in their use multiply this interaction by a short-range correlation function:

(/(r) = V(r) g(r) .

(2.54)

Their choice of g(r) is O(r - d) with d = 0.5 h/m,,c = 0.7 fm. Here O(x) = 0 if x < 0 and O(x) = 1 if x ~ 0. The same correlation function was used by Shimizu et al. [48] in a second-order perturbation calculation for magnetic moments in light nuclei. The claim was that with d = 0.5 h/m~c essentially the same result was obtained as with a more complicated correlation function calculated by Hadjimichael et al. [49] from the Hamada-Johnston potential. There are other choices. For example Brown et al. [29] chose a form for the correlation function of g(r) = 1 - ]o(qc r) with qc = 3.93 fm-1, where qc has been adjusted to reproduce the dominant Fourier components of a realistic two-body correlation function calculated with the Reid soft-core potential. While Adelberger and Haxton [50] in their work on parity violation in nuclei chose a form g(r)= 1 - e x p ( - a r 2) (1 - br 2) with a = 1.1 fm -2 and b = 0.68 fm -2 obtained from the work of Miller and Spencer [51]. No doubt other forms are available in the literature. For ease of calculation a simple functional form is required, but at the same time a certain amount of model dependence is being introduced at this step. In calculating matrix elements of D'(r) it is convenient to transform from single-particle coordinates to relative and centre-of-mass coordinates. With harmonic oscillator functions this is a fairly trivial step using the Moshinsky transformation [52]. This correlation function is contained in the radial integral in the relative coordinate, the so-called Talmi integral:

Ip = 47r J dr r 2p+ 2 e-r 2 V ( V ~ r/a) g(X/2 r/a)

(2.55)

o

where r is a dimensionless variable, r = a(r 1 - r E ) / V ~ , and a = 1/b is the inverse of the oscillator length parameter. The introduction of a factor X/2 comes from the Moshinsky transformation where a

I.S. Towner, Quenching of spin matrix elements in nuclei

284

symmetric definition of relative and centre-of-mass coordinates r = (r] - r 2 ) / V ~ and R = (rl + r2)/V-2 is used in the evaluation of the transformation coefficients. In table 5, we give the lowest few Talmi integrals for the central part of the ~r- and p-meson exchange potentials for various choices of the correlation function. Note that the lowest-order Talmi integrals (smallest values of p) are sensitive to the small r behaviour and hence are very sensitive to the choice of correlation function. While for the higher-order Talmi integrals (larger values of p) the converse is true. Thus the tensor force, for which the minimum value of p is 1, is little affected by the choice of correlation function, but the central force, as displayed in table 5, is strongly influenced. For ~r-exchange, we see the lowest Talmi integral, Io, is smaller than Ip (p >-2) and quite a bit smaller than I 0 for p-exchange. This is as expected since the light pion mass generates a long-range interaction, while the heavier p-meson gives a short-range interaction. Thus I 0 is dominated by p-exchange (and other heavy meson contributions) while Ip (p->2) is principally given by w-exchange. The integral I 0 is also the one most influenced by the choice of correlation function. Looking at the p-meson contribution to I0, the Towner-Khanna choice of g(r) = O(r-d) with d =0.5h/m~c gives a value similar to that of Brown et al. [29] with g(r)= 1 - jo(qcr) and somewhat larger than that given by the Adelberger-Haxton correlation function [50]. Finally the choice of coupling constants is itemized in table 6. The ~rNN coupling is well established at g~NN = 13.41 for pseudoscalar coupling from dispersion relations for wN scattering and the long-range part of the nucleon-nucleon interaction. For vector mesons, two constants are used for the vector and tensor couplings: &NN[3', -- Kp~,q~/2M]r in the usual notation. In vector dominance models, Kp is constrained to the empirical value from photon-nucleon coupling Kp -- K v = 3.7, which comes from the known isovector anomalous magnetic moment. The p-exchange potential strength is proportional to (1 + Kp) 2 showing the importance of the tensor coupling term. There is growing evidence that Ko, in fact, should be larger than the vector dominance value. Hohler and Pietarinen [53] examining the isovector J = 1 - channel in rrN scattering suggest Kp = 6.6 with g~NN = 2.63. Large values for K o are also found in the Bonn one-boson-exchange potential [9] in fits to nucleon-nucleon scattering data. By contrast the tensor coupling for the o~-meson is known to be small [54], thus it is sufficient to use the vector dominance value of K~, = -0.12. In SU(3), the ~o-meson coupling constant is related to that of

Table 5 Talmi integrals, lp, in MeV for the central part of "rr- and p-exchange meson exchange forces, V = (1/3)f2NN m ~Vo2(X~) and Vp = (2/3)fpNNm 2 p o2(Xo), with vertex form factors included at a characteristic oscillator energy of h~o = 14 MeV for various choices of the short-range correlation function

O(r- d) 1 jo(q~r)

exp(-ar2)(1-br)

d=0.3

d=0.4

d=0.5

d=0.6

Pion p =0 1 2 3

1.341 1.311 2.005 3.926

-0.396 1.403 2.027 3.934

-1.627 1.299 2.011 3.927

-0.271 1.354 2.013 3.927

0.726 1.421 2.018 3.927

1.347 1.483 2.024 3.928

Rho p=0 1 2 3

4.880 1.I18 0.310 0.126

3.948 0.953 0.287 0.125

5.459 1.050 0.294 0.124

5.671 1.055 0.294 0.124

4.792 0.994 0.289 0.123

3.583 0.872 0.277 0.122

f~NN2

f0NN~

Parameter values: =0.079, = 5.358, qc=3.93fm 1, a = l . l f m z, b=0.68fm 2 and d is expressed in pion Compton wavelength units.

I.S. Towner, Quenching of spin matrix elements in nuclei

285

Table 6 Coupling constants in the one-boson-exchange potential

a

g~N

2 g~Ns/41r

xr p to

13.41 2.63 7.89* 6.2l ++ 7.79

14.3 0.55 4.95 2.84 4.82

A~

K~

f~N

m (MeV)

6.6 -0.12 -

0.079 5.36 1.35 0.20 1.63

139.57 770.34 782.58 500.00 1089.43"*

+ g, NN= 3 goNN, the SU(3) relation. Larger values are often preferred. Some results will be given with g,NN = 11.21. Remember g~Ns must be adjusted accordingly. ++This value must be adjusted from nucleus to nucleus, see table 7. Result quoted is for 160. *it r e A l = V'2mp; fiNN = (1/47r)(g~NNma/2M)2(1 + K~):.

the p-meson: go,NN = 3gpNN = 7.89. Durso et al. [31], however, show that in second order in NNscattering, the 'par-box' graph with intermediate isobars also leads to a short-range repulsion that can effectively be included in the one-boson-exchange model by enhancing the toNN coupling constant. 2 4 rr in the range of 5 to 8, i.e. the effective toNN They suggest an additional contribution of 8g~,r~N/ coupling constant g~,NN is roughly in the range of 11 to 13. We will quote some results with g~,SN = 11.21, however we must be careful that in doing specific second-order calculations with intermediate isobar states, there is no double counting with the 'par-box' diagram. The Bonn potential [9] contains a large toNN coupling constant in fits to NN scattering data, g,oNN----"17, while the Paris potential [13] has goNr~--~15. The coupling constant for the Al-meson is, in chiral Lagrangians, related 2 2 g2NN/4M2 where M is the nucleon mass. to that for the pion, as discussed in section 2.2: gA~NN/mA We adopt this value, gA1NN = 7.79. It is not clear whether the Al-meson should be included in the one-boson-exchange potential or not. The heavy At-meson mass generates a short-range interaction that perhaps should be part of the short-range phenomenology. Certainly its range is comparable to that of the vertex form factors. In the Bonn potential [9], for example, the Al-meson is not explicitly included while the Paris potential [13] includes it with a coupling constant gA1NN--~g*NN" Finally we come to the (r-exchange potential that provides the intermediate-range attraction in the NN interaction. The (r-meson is unphysical in that it is simulating 2ar-exchange so its mass is a parameter, unlike the other mesons in the force whose masses are taken at their physical values. We will fix the mass at r n = 500 MeV. It is the attraction from ~r-meson exchange counterbalanced by the repulsion from to-meson exchange that are the key ingredients to a nuclear binding calculation. Our procedure, then, has been to adjust g(rNN SO that a Hartree-Fock calculation (in a large spherical harmonic oscillator basis) using the one-boson-exchange potential and multiplied by a short-range correlation function gives the correct binding energy for the closed-shell, finite-size nucleus. Some results are given in table 7 for nuclei 4He, 160 and 4°Ca. Notice that g~NN decreases with increasing nuclear size. This phenomenon was noted with the Bonn potential when two-nucleon data were compared to nuclear matter calculations. Furthermore as g,oNN is increased, so does g~NN" This is expected since more repulsion has to be balanced by more attraction. In Hartree-Fock the relation seems to be linear in the square of the coupling constants. For example, an empirical relation for 160 2 2 that fits the data in table 7 (for Kp = 6.6 and no Al-meson ) is g,~NN/47r = 0.21 gosN/4"n" + 1.79. Lastly we note the A~-meson has little impact on nuclear binding. The values of g~NN are only slightly changed when the A~-meson is included in the one-boson-exchange force. =

I.S. Towner, Quenching of spin matrix elements in nuclei

286

Table 7 Values of g.Nr~ from Hartree-Fock calculations with the one-boson-exchange potential multiplied by a short-range correlation function g(r) = O(r - d) with d = 0.5 h / m c . Vertex form factors are included with parameters A T = 1 GeV and ,1, = 1.44 GeV with i = p, to, ~r and A~. Harmonic oscillator energy is hw = 20.4 MeV for 4He, 13.3 MeV for 160 and 11.2 MeV ~C~Ca NO A~-meson g~Ni = 7.89

4He ~60 4°Ca

With A~-meson

g~NN = 11.21

g,NN = 7.89

g~NY = 11.21

K 0 = 6.6

Kp = 3.7

K o = 6.6

K0 = 3.7

K = 6.6

K, = 3.7

K 0 = 6.6

K = 3.7

7.23 5.98 5.70

8.18 6.66 6.34

8.17 7.01 6.74

9.02 7.6(I 7.30

7.55 6.21 5.92

8.46 6.86 6.54

8.45 7.21 6.92

9.27 7.79 7.46

All other parameter values as in table 6.

2.4. Isobar-hole interaction

In the last section, we have made reference to the inclusion of isobar-hole states in RPA calculations in finite nuclei and the impact they have on 'screening' the particle-hole interaction. To be more quantitative about this we need to specify the isobar-hole interaction and introduce the concept of transition spin. Consider the well-studied process of pion scattering from a single nucleon as shown by the graphs in fig. 3. The scattering proceeds principally through the formation of an intermediate isobar state. There are both direct and exchange graphs to be considered. (In symmetric nuclear matter, the exchange graph cancels the direct graph for intermediate nucleon states, which is why the isobar excitation process dominates in ~N scattering.) The isobar production vertex is given by a Lagrangian

~N~ = i

(47r) 1/2 m

~ (Q ) k. ~r(k) u(p)

(2.56)

where u(p) is a nucleon spinor of momentum p, 7r(k) pion wavefunction of momentum k and N~,(Q) the isobar Rarita-Schwinger spin-3/2 spinor (a spin-1/2 spinor coupled to a four-vector of index/z) of momentum Q = p + k. Here f~Na is the ~rNA coupling constant normalized in analogy to the -rrNN coupling constant. Isospin labels have been suppressed. The matrix element corresponding to the direct graph in fig. 3 is

/

/

//

P

q /

Q=p-q /

Q=p+k

/

/ / /

k..-

Ca)

Cb)

Fig. 3. ~rN scattering through an intermediate isobar state: (a) the direct graph, and (b) the exchange (or crossed pion) graph,

I,S. Towner, Quenching of spin matrix elements in nuclei

M = i (4rr)f2~'Na a 2 6(p')rt+(q) q~ S~,.(Q) k~ 7r(k) u(p)

287

(2.57)

m~r A

where S~,~(Q) is the isobar propagator

Sa~(Q)=N~,(Q)N~(Q) iQ + m a

-~

(~Y"Q~ + Q~%~)

"

(2.58)

We will be interested in the low-energy limit in which k ~ 0 and k 4 ~ im,~ ~ 0, p ~ 0 and P4 ~ iM and hence Q ~ 0 and Q 4 ~ i M , where M is the nucleon mass. Note the isobar is off its mass shell in this limit. Then the propagator becomes

Sa(Q-->O, Q4--* iM) =

- i ( m a + My4)

-

-

+

2 _ M 2 m A

(2.59)

and the matrix element in the nonrelativistic limit (for/z and v both space-like) is M-

4

2 7Tf'trNA

2

m~

+

ma + M

"gl'+(q)x1/2mf --f--;-.2 { q . k - l o " q o " k } x 1 / 2 m i T r ( k ) ma - M

2

-

4rrf~Na 1 + 2 7r+(q) - - X 1 / 2 z f { 2 q ' k - l i o " ( q x k ) } x 1 / 2 m i ¢ r ( k ) m~ ma - M

(2.60)

where X1/2m is the nonrelativistic nucleon Pauli spinor. An alternative but equivalent nonrelativistic description introduces the concept of transition spin [55]. The isobar can be thought of as a composite state of a pion and a nucleon whose wavefunction is constructed by combining the angular momentum l = 1 of the pion and the spin s = ½ of a nucleon. Denote by e r a unit vector that transforms as l = 1, then a spin s = -~object is formed by vector coupling

N~s= E (lmslrl3Ms) e~X1/2zs.

(2.61)

r,m s

Note that the isobar N~ s contains a spatial index i as well as spinor properties associated with The Lagrangian, then, for the rrNA vertex takes the following nonrelativistic form [40]

5E,~A = i (a ~_,1/2 Tr, f ~ a N + "k 7r(k) X

Xl/2Zs. (2.62)

m= or as a matrix element ~ N a = i (47r)l/z f~Na ( 3MsIS " kl 1ms ) It(k) m~ where S is the so-called transition spin defined by its matrix element

(2.63)

I.S. Towner, Quenching of spin matrix elements in nuclei

288

(3MslS.kl½m,)=~,(-)r (3MslSrl½m ~) k ~ = ~ (-) ~ (½m, lrl3Ms) k_ r • r

(2.64)

r

If we use the Wigner-Eckhart theorem in the conventions of Brink and Satchler [56]

(2.65)

( 3MslSrl ½m,) = ( l mslrl 3 M,) ( 31lSll ½)

then the reduced matrix element of the transition operator is identified as < llsl[ > = 1. Thus the transition spin 'connects' a two-component Pauli spinor on the right with a four-component isobar spinor on the left. Note also the Hermitian conjugate of these matrix elements for isobar annihilation: 1 (~m, lSrl+ 7Ms) 3 1 = {3Mslrl~ms) <½11S+II~_>

(2.66) <111s÷11 > =

< llsll

:

Returning now to the direct graph in fig. 3 and writing down a nonrelativistic version for its matrix element, we get

m a -

M

_ 4 " ~ f 2vNA - - 2 ¢r+(q) mA 1_ M ~'M,(½mflS+ "ql3M~)

(3mslS'kl½m~) ~r(k) .

(2.67)

m~

This is seen to be identical to the expression obtained before, once it has been shown that the transition spins have the property

S + . q S . k = 27 q . k - ~icr.(q x k).

(2.68)

This result is easily proven [57] by taking matrix elements of both sides between nucleon spinors. Thus by construction the nonrelativistic transition spin formalism is identical in leading terms to relativistic Rarita-Schwinger formalism for spin-3/2 fields. Now let us look at the isospin factor. The isobar is an isospin-3/2 object constructed by combining the isospin angular momentum of the pion of one with the isospin t = i of a nucleon. Thus in complete analogy to the spin, the isospin isobar function is

NY T= E (½m,ljl~MT) e{ X,¢2,,,,

(2.69)

j,m t

where e[ is a unit vector characterising the isospin of the pion and X l / 2 m t the two-component lsospin spinor of a nucleon. The nonrelativistic Lagrangian for the arNA vertex is then generalized to be I/'2

~ N a = i (47r)

LNA N j+ " k 7rJ(k) X

or as a matrix element

(2.70)

l.S. Towner, Quenching of spin matrix elements in nuclei

289

1/2

LP,N a = i (art)

f=Na

m~

(2_3M s, ~MrlS'k Tq2-lm,, ~m,) 7rJ(k)

(2.71)

where j represents the isospin index and the implied sum over j represents the (cartesian) scalar product of the transition spin T and the pion wave-function, 7r. All the properties of the isospin transition spin T are identical to those of the transition spin S. In particular

(3MrlTqlm,) = (lmtlj]3Mr) (~llrll½)

(2.72)

with (~IITII½ > = 1. The similarity with the nonrelativistic Lagrangian for the ~rNN vertex is obvious 1/2

(47r) ~NN = 1 • m~

lmt) zd(k)

(21M s, X M r l o. . k rJl ~ms, '

(2.73)

and so we arrive at a useful prescription for writing down the nonrelativistic vertices involving isobar production: replace the coupling constant f~NN by f~,Na and the Pauli spin and isospin operators o. and ¢ by the corresponding transition spins S and T. For example the one xr-exchange potential connecting an NN-state with an AN-state, see fig. 4, is vNN--,aN, , ( q) =

4¢rf*NNf~NaS1 "q °'2" q T 1 " z2 . m2 q2 + m2

(2.74)

This is known as the transition potential. Similarly for the one p-exchange potential the same replacements are made vNN--*AN., x o tq) --

47rfpmJ, Na (S 1 x q ) . ( a 2 × q) 2 2 2 mo q + mp

TI°

¢2 •

(2.75)

Finally we need to write down the coupling constant for isobar production. There is some uncertainty here. For example, one can derive an expression [19] for the decay width at resonance for the process A--->~rN in the static limit that involves the coupling constant and known masses. Hence, from the experimentally known width, the coupling constant is found to be f~Na/f~NN = 2.15. This is close to the Chew-Low theory [58] value of f~NA/f,,NN = 2. Nonrelativistic quark models lead to a somewhat smaller value of f~Na/f~NN = 6V~/5 = 1.7. Relativistic corrections in, for example, the MIT bag model [59] do not alter this quark-model value substantially. Thus in discussing the response of a nucleus to some spin-dependent probe and in particular in discussing the process in which an isobar is first excited and

+ 7/"

ill LI

Fig. 4. The N N ~ AN transition potential in the one-boson-exchange model.

I.S. Towner, Quenching of spin matrix elements in nuclei

290

then de-excited there will be an uncertainty in the calculated amplitude of some 40% due simply to the range of values available for the square of the -rrNA coupling constant. For the pNA coupling there is no experimental information that can be brought to bear. Thus it is commonplace to use the quark-model prediction of LNJLNN = LN /LN .

2.5. g' for isobar-hole interactions To get an appreciation of the strength of the NN-AN transition potential in the particle-hole channel it is useful to evaluate the potential in nuclear matter in the Landau limit. Recall that this limit corresponds to the situation when all particles in the initial and final states are on the Fermi surface and the momentum transfer is q ~ 0 in the direct graph, fig. 5(a), and q2_+ 2k~(1 - c o s O) in the exchange graph, fig. 5(b). In this way a Landau parameter (g')AN is determined for the isobar-hole channel from the equations vNN-AN(q

= 0 ) --

47rf~N~f~Na_ ( g d i r ) A N r

S1 ° °'v,

T 1 "r~.

(2.76)

mr +I

1 (

-~ o dx<-P~P,

9NN-AN(q2

21

=2kv(

-x)))-

4rrf=NNf~Na ' m2 (gexcOANS,'mr,'r2

(2.77)

7T

--1

in complete analogy to the method outlined for (g')NN in section 2.2. Here 1? is the one-bosontransition potential multiplied by a short-range correlation function, S and T are the transition spins introduced in the last section and x = cos O. The value of (g')aN has been the subject of intensive discussions in recent years. This is because the response of a nucleus to an isovector spin probe, such as the nuclear response in a (p, n) reaction, shows a characteristic quenching in relation to the lowest-order shell-model prediction. One mechanism for the quenching comes from the inclusion of isobar-hole states in the RPA calculation. Other mechanisms involve 2p-2h states. The efficiency of isobar-hole states in the quenching phenomenon depends on the strength of the transition potential and hence on the value of (g')aN. With a fairly large value such as (g')aN = 0.6 most of the observed quenching can be attributed to isobar currents as seen in the calculations in refs. [60-64]. With a smaller value such as ( g ' ) a N - 0 . 4 [65], significant contributions to the quenching must come from alternative sources. Since (g')NN in the nucleonnucleon channel is of the order of g ' - 0 . 6 , arguments for a large value of (g')aN often invoke

N

N/5 (a]

l..... (b)

(c)

Fig. 5. The (a) direct, and (b) exchange graph for the N N ~ A N transition potential in particle-hole coupling evaluated in nuclear matter in the Landau limit to determine the parameter (g')AN. Graph (c) is the higher order 'induced' graph. The shaded area represents a phonon, an RPA sum of particle-hole interactions.

I.S. Towner, Quenching of spin matrix elements in nuclei

291

'universality' under which (g')Nr~ = (g')aN" Quark model arguments would support this. However in the framework of the one-boson-exchange model it is quite clear that (g')Ny ~ (g')aN as emphasized by Arima et al. [65, 66], although the model does not rule out the possibility that the two g' values could be similar in numerical evaluations. Indeed it will be hard to settle the matter, because the estimates of (g')aN are very sensitive to the short-range correlation function and the short-range phenomenology, as we will show, and these ingredients are not under control. The evaluation of (g')aN follows analogously the steps for the evaluation of (g')NN given in section 2.2. However because the excitation of the isobar involves a spin and an isospin transfer of one only the °'1 " o'2 ~'1"~'2 components of the one-boson-exchange force contribute, namely ~r-exchange, p-exchange and possibly At-exchange. For the direct graph, fig. 5(a), the same expressions are obtained and 'universality' is certainly true for these contributions: t

f

(g=,dir)aN = (g~,dir)NN = ½r/~ t

t

(2.78)

(gp,dir)aN = (gp,dir)NN = 2Cpr/p p

t

(gA,,a,r)aN = (gA,,d,,INN = -Ir

,(o)l 2 +

t

1 + 3"~A 1

where the expressions on the right are defined in eq. (2.32) and have been obtained for the correlation function g(r)= 1-jo(qcr) and with vertex form factors included. The exchange contributions on the other hand are four times larger in the AN-channel than in the NN-channel because the exchange operators satisfy (P~,P, Sl " 0"2 T l " 7 2 ) =

(2.79)

( S l • O"2 T1 • ~r2)

without the factor 1/4 evident in eq. (2.22). Thus the exchange contributions are

(g'-~,e~ch)aN= 4( g'~,~x~h)NN =

-- ½(~I~ -

St-.(kF)) (2.80)

(g'o,ex~h)aN=4(g'o,ex~h)NN= -~C,07p- ~:,(kF)) t

I

t

r

(gAl,exch)aN = 4(gAl,eXch)NN = ~AI(kF) -- ~A 1 -- 1 ( ~ A 1 -

~A1(kF))

and clearly do not satisfy 'universality'. Note also that in the limit kF---->0 , then so--->0 and ~:'--->I/(0)12 and the exchange contribution to (g')aN exactly cancels the direct contribution. Clearly (g')aN depends quite sensitively on the density, a point noted by Cheon et al. [66]. In table 8 we give some numerical results for two choices of the short-range correlation function: g(r) = O(r- d) used by Arima et al. [65] and g(r)= 1 - ]o(qcr) used by Brown et al. [29]. Without vertex form factors, there is quite a difference between the estimates for g' from the two correlation functions as has been noted by Toki [67]. The contribution from the p-meson is about a factor of two larger with g(r)= 1-jo(qcr). In both cases, however, (g')aN is between 30 and 40% smaller than (g')Nr~ because the negative contribution from the exchange graph (principally from the p-meson) is four times larger in the AN-channel than in the NN-channel. Including, now, vertex form factors the results from the two different correlation functions are much closer together. Indeed for the p-meson contribution they are almost identical which is what we have noted before in calculating Talmi integrals

I.S. Towner, Quenching of spin matrix elements in nuclei

292

Table 8 Estimates of (g')aN evaluated from the transition potential in the Landau limit for two choices of the short-range correlation (SRC) function with and without vertex form factors (FF), compared to estimates of (g')yN Vertex FF SRC function t

No

No

O(r- d)

l - j,~(q r) !

t

Yes

O(r !

t

d) t

Yes

Yes

1 - j~( qcr)

1 j,~(q~r)

t

t

t

t

~__~Y "~ p A~

0.30 0.36

0.03 -0.18

0.32 0.75

0.05 -(11.45

0.20 0.22

0.02 (I.10

0.12 0.23

0.07 -(I.10

0.12 0.23 -0.10

0.07 -0.10 /).25

Sum

0.66

0.21

1.07

0.50

0.42

-0.08

0,35

-0.03

0.25

0.._

NN av p to, (r, A~

0.30 0.36

0.01 -0.04

0.32 0.75

0.01 -0, I 1

0.20 0.22

0.00 -0.02

0.12 0.23

0.02 0.03

(I.12 0.23 0.10

0.02 0.01 0.13

Sum

0.66

(I.05

1.07

I).13

0.42

-0.02

0.35

-I).01

0.25

11.13

Parameters as recorded in table 3 with d = 0.7 fm.

in finite nuclei (table 5). Vertex form factors reduces the calculated g' quite considerably. We obtain (g')~N -0.33 and (g')NN = 0.37 as the average values from the two choices of correlation function with (g')~N being about 10% smaller than (g')NN" A different calculation by Sagawa, Lee and Ohta [68] in reaction matrix theory using a model Hamiltonian for 'rr, N and A obtain (g')aN = 0.36 in agreement with these estimates from boson-exchange models augmented by short-range correlations. Finally in the last column in table 8 we give results on including other mesons, especially the A~-meson in the calculation. The Al-meson gives a sizeable contribution in both the direct and exchange graphs in the NN-channel but because the contributions are of opposite sign the net effect is rather small (see table 3). However for the AN-channel, the exchange graph contribution is four times bigger and the Ai-meson gives a considerable boost to (g')~N. The results in table 8 show the sum of direct and exchange contributions to be (g')aN = 0.47 and (g')NN = 0.38. It is clear that one is never going to resolve the g' 'universality' argument. The estimates given here depend on the short-range ingredients in the calculation (choice of short-range correlation function, inclusion or not of vertex form factors, and inclusion or not of the At-meson ) and these ingredients are not under firm control. Furthermore in finite nuclei the density, especially near the Fermi surface, will be somewhat less than in nuclear matter. This affects the calculated exchange contributions to (g')aN, which depend on the choice of k~, and smaller values for (g')~N can be anticipated in finite nuclei than in nuclear matter [66]. Indeed in the limit kv~ 0 the exchange contribution will exactly cancel the direct contribution. Finally we should not stop the calculation in lowest order but consider the induced graph in which a particle-hole phonon is exchanged between the up-going and down-going line in the exchange graph, fig. 5(c). The importance of this term has been stressed by Nakayama et al. [24, 25]. Three comments are in order. First, this induced graph gives a rather small contribution to (g')NN because all components of the interaction, f, f', g, g' contribute to the phonon and there are cancellations amongst them. Typically (g')yN is enhanced by 20%. For the AN-channel, however, only the g' component of the force can contribute so this mutual cancellation is not present. Second, being exchange graphs, there is a factor of 1/4 coming from the spin and isospin exchange operators in the NN-channel as discussed in connection with fig. 5(b). For AN-channels this factor of 1/4 becomes a factor of 1; the graphs give contributions four times larger. Thirdly the graph on the right in fig. 5(c) is quite small

I.S. Towner, Quenching of spin matrix elements in nuclei

293

Table 9 Landau parameters (g')NN and (g')ar~ in pionic units (g')NN

(g')aN

Jiilich a

I II III

0.49 0.58 0.56

0.35 0.56 0.68

Tokyo b

I II

0.52 0.61

0.35 0.45

a Nakayama, Krewald and Speth [25]: II = bare G-matrix from the Bonn potential, HEA [9]; II = bare G-matrix plus induced interaction; III = bare G-matrix plus induced interaction plus relativistic correction. b Cheon, Shimizu and Arima [66]: I = bare G-matrix from Dickhoff et al. [22]; II = bare G-matrix plus induced interaction.

compared to the graph on the left because the m a - M N mass difference contributes to the energy denominator of the intermediate states in the former. Thus in summing RPA particle-hole interactions to form the photon, only half of the RPA graphs should be kept, those of the structure of the graph on the left in fig. 5(c). In table 9 we give some results obtained by the Jiilich group [25] and the Tokyo group [66]. In each case a series of nonlinear coupled equations is solved self-consistently for the particle-hole and isobar-hole interactions. Both groups find (g')aN enhanced significantly by the inclusion of the induced interaction, but they differ quantitatively by roughly a factor of two. The Jiilich group [25] also considers relativistic corrections that give a further boost to (g')aN" For the time being the discussion of (g')aN remains in a rather unsatisfactory state. Its value is clearly model dependent and certainly strongly density dependent.

3. Rayleigh-Schr6dingerperturbation theory Besides the effective interaction discussed in the last section we are also interested in the matrix elements of transition operators. Since our calculations are approximate and carried out in truncated model spaces we will be dealing with effective operators. Indeed we are seeking an effective transition operator whose matrix elements between the eigenstates of the effective Hamiltonian will yield the same results as the matrix elements of the true transition operator evaluated between the eigenstates of the true Hamiltonian. A formal solution to this problem exists as reviewed, for example, by Brandow [1], and Ellis and Osnes [7], but there are many practical difficulties. However in simple situations, such as closed-shell nuclei with one or two nucleons in a few valence orbitals, a practical calculation can be mounted. The solution involves a perturbation expansion in the effective interaction evaluated in first and second order. 3.1. First-order core polarisation The simplest situation to discuss involves a single nucleon outside a closed-shell core. The Hamiltonian is divided into a one-body Hamiltonian and a residual interaction: H = H o + ~ where H o = T + U, the sum of kinetic energy and one-body potential energy operators, and ~ = V - U where

I.S. Towner, Quenching of spin matrix elements in nuclei

294

V is the two-body potential energy operator. The eigenfunctions of H 0 form the basis of the calculation. The closed shell is defined as the full occupancy of the lowest-energy eigenstates of the one-body Hamiltonian H o. The valence orbitals are the next few unoccupied eigenstates. The single nucleon occupies one of these valence orbitals. For example, the ground state of 2°9Bi in the lowest approximation would be considered as a closed shell 2°*Pb with a single proton in the hg/2 orbital. Likewise its magnetic moment in the lowest-order approximation is simply the bare magnetic moment operator evaluated in the h9/2 single-particle state. Corrections to this come from a perturbative expansion in the residual interaction. In general, then, to first order in a closed-shell-plus-one nucleus the matrix element of a one-body operator F is given by

=+E + E <~lFla> ~,~a,b

~.~

Ei - E

(3.1)

Ef - E

Here a and b are single-particle valence states and c~ an infinite set of single-particle or two-particle one-hole (2p-lh) states constructed from the eigenfunctions of H 0. Likewise the energy denominators are constructed from the single-particle eigenenergies of H 0. Note that energy denominators are negative quantities. A graphical representation of all the possible terms is given in fig. 6. Two comments are in order. (a) Consider graph (b) whose matrix element is given by the expression (b[fig. 6(b)la) = E

(bIFIp){phlV- Ulah)

pCb.a

E a --

(3.2)

Ep

h

where s are the single-particle eigenenergies of H 0, viz. E~ = E . . . . q ' - s a and, in this graph, E = Ecore + 8p. This graph contains what is called a Hartree-Fock insertion. If the unperturbed Hamiltonian were to be chosen as the Hartree-Fock Hamiltonian that minimized the energy of a single Slater determinant characterizing the closed-shell core, then the potential U would be defined in the Hartree-Fock procedure by its matrix elements as

(blUla) = E (bhlVlah)

(3.3)

h

where the sum over h is a sum over all the occupied states in the closed-shell core. Thus with H o being the Hartree-Fock Hamiltonian, graphs (b), (d), (e) and (g) are all identically zero. In practice, it is rare that calculations are done in a Hartree-Fock basis. It is more usual to use harmonic oscillator functions or in some cases Saxon-Woods functions. Then there is no reason to drop the Hartree-Fock insertion graphs except that one might argue that oscillator or Saxon-Woods functions are close to Hartree-Fock

x. O

oblx O

(a)

xo? oh • Q b oh, x oh> hx • x

-

(b)

(c)

(d)

(e)

(f)

(g)

Fig. 6. Zeroth-order, diagram (a), and first-order, diagrams (b) to (g), perturbation corrections to the matrix element of a one-body operator in a closed-shell-plus-one nucleus.

I.S. Towner, Quenching of spin matrix elements in nuclei

295

functions so the insertion graphs would be small. Explicit calculations by Ellis and Mavromatis [69] in light nuclei with oscillator functions fail to support this idea. Hartree-Fock insertion graphs give a non-negligible contribution. Fortunately the magnetic moment operator is a special case. This operator being simply a combination of L and S, the orbital and spin angular momentum operators, it is di~/gonal in orbital space. Thus in the single-particle matrix element, graph (a), the spatial part of the single-particle wavefunctions, la) and Ib), must be the same. In the Hartree-Fock insertion graph (b), the matrix element (phlV- Ulah) requires that the particle state ]p) has the same total angular momentum as la), while the matrix element requires the spatial part of IP) to be the same as that of Ib). In short, these restrictions limit Ip) to being equal to la), and this is expressly forbidden. Thus graph (b) is identically zero for the magnetic moment operator. This argument holds for all Hartree-Fock insertion graphs in all orders for the magnetic moment operator. Of course, if one looks at higher multipoles, or in magnetic electron scattering at finite momentum transfers then these arguments break down and Hartree-Fock insertion graphs have to be considered. (b) The second comment concerns the infinite sum over intermediate states a. In principle these summations are unlimited, but in the first-order graphs shown in fig. 6, the selection rules on the one-body operators restrict the number of possible intermediate states. (This limitation is not present in second order.) Again the magnetic moment operator provides a special case. Consider graph (c) given by the expression

(b[fig. 6(c)[a)= E (hlFlp)(bp[Vlah) p,h

(3.4)

'lEa + Eh -- Ep -- Eb

The restrictions require the spatial parts of Ip) and [h) to be the same. This will limit orbitals p and h to being just spin-orbit partners and the number of terms in the summation a to just one or two. Indeed in light nuclei where the shell closures occur at LS closed shells, spin-orbit partners are either both particles or both holes and no terms survive at all. Thus for the magnetic moment operator at closed LS-shell-plus-one nuclei, such as 170 and 41Ca, all first-order corrections are identically zero. In heavy nuclei where the shell closures divide spin-orbit partners, such as 2°9Bi, there are just two contributions to graph (c) coming from proton h l l / 2 and h9/2 and neutron i13/2 and i11/2 orbitals. Thus we see that magnetic moments (and the Gamow-Teller operator of B-decay and the M1 operator of ~-decay in the long wavelength approximation) play a rather important role in the tests of effective interaction and effective operator theory. As an example we give, in table 10, calculated first-order corrections to magnetic moments and M1 transition rates in single-particle (hole) states in closed-shell-plus (minus)-one nuclei in the vicinity of 2°spb. We consider three of the effective interactions discussed in the last section, viz. a ~-function interaction (direct matrix elements only) whose strength is given by the Landau parameter g' = 0.6; the interaction ~r + p + 8 of Cha and Speth [46] being a sum of finite-range one-pion and one-rho exchange potentials and a zero-range piece of strength ~g' = 0.37; and the one-boson-exchange potential with a short-range correlation function [47], including the Al-meson and with the o~NN-couplingconstant taken at its SU(3) value: go,NN= 3gpNN= 7.89. Since no isobars are included in the calculation, the Cha and Speth interaction [46] has been further renormalized by a screening factor of 3' = 0.72, as discussed earlier. The first-order correction to the reduced matrix element from diagram (c) in fig. 6 in an angularmomentum coupled representation is given by

1.S. Towner, Quenching of spin matrix elements in nuclei

296

I I I

I I I I

I I I I I

I I I I

I

I

I

I

I

I

0

~

I

I

c~ ~

I

II

I

I

I

I

I

/

I

I

I

I

I

I

c~ ~

~

I

c:; c~ c~ c~

~

I

c~ c~ c~ ~ I

~

I

I

I

"g

0

8

I

I

I

I

I

I

I

I

]

I

I

I

I

I

F

~

g. I

1

I

"7'.~

<

8 ? b:

~

~

~

~= ~ . ~

~

.~ a= "~

0

II

I

I

I

~ 1 1

o.2

G~o ..=~G E . . E , , . . ~ . ~ ~

,~ ~ . , , . ~ . z . . . - ~ . ~

~.

o=

I.S. Towner, Quenching of spin matrix elements in nuclei

297

(bllfig. 6(c)[[a> = E (-)z~b-1 (0llF(*)ll(h-lp)Z) <(h-lp)AlVl(b-aa)A) h.p

(e~ + e h -- ep -- eb)

(3.5)

where A is the tensorial rank of the one-body operator F (~ (A = 1 for magnetic moment operator), and f = (2j + 1) 1/2. Note the use of particle-hole coupled matrix elements. The Hermitian diagram (f) in fig. 6 is

(bllfig. 6(f)lla)= E (__)a+b-A/~/~-I ((a-lb)AlVl(h-lp)A)((h-lp)AIIF¢*)[]0) h,p

= (_)o-b-k a

(E b 31- Eh -- Ea -- ep)

(allfig. 6(c)llb>

(3.6)

where k is determined from the Hermitian property of the operator F(~): (F~)) * = ( - ) k - ' ~ F ( ) ) m . For the magnetic moment operator k = 0. The magnetic moment and M1 transition operator F (~) is defined through its single-particle matrix elements as a

t 1/2

~z=k~--~]-/

(allgLL +gsSlla)

3 n(Ml; b--,a) = ~ I(bllgL L + gsSIla)l 2

(3.7)

where gL = 1 and gs = 5.586 for a proton, and gL = 0 and gs = -3.826 for a neutron. We are using the definitions of Brink and Satchler [56] for the reduced matrix element with the initial state for transitions written on the left. Matrix elements were calculated using harmonic oscillator wavefunctions (hto = 7MeV), while energy denominators used experimental energies: e p - e h = 5.6MeV for the proton h9/2-h11/2 spin-orbit splitting, ep - e h = 5.85 MeV for the neutron i11/2-i13/2 splitting, and e b - e a = 0.0 for magnetic moments and e b - e a Ev, the photon energy for an M1 transition. There is an inconsistency here since we are using oscillator functions for the single-particle wavefunctions and experimental energies for the single-particle energies. Ideally both should come as the eigenfunctions and eigenenergies of the single-particle Hamiltonian H 0 used in establishing the perturbing interaction. For the illustrative purpose of the first-order calculations discussed here, this inconsistenc~, is not serious. Some groups, e.g. [46], have used Saxon-Woods potentials whose parameters are adjusted so that the potential's eigenenergies match the experimental values. The inconsistency, however, is a much more serious problem for magnetic transition operators of multipolarity higher than A = 1, or for magnetic electron scattering where the momentum dependence of the form factors is being examined. It is more than thirty years since the first core-polarisation calculations were performed by Arima and Horie [70] and by Blin-Stoyle [71] for magnetic moments. Many calculations have been mounted since. The core-polarisation phenomenon has been called the Arima-Horie effect and was the subject of a commemorative conference held in 1983 [72] which was subtitled "thirty years of configuration mixing". We will not attempt to review all these past endeavours but will discuss the results in table 10 as being typical of many that have gon e before them. One very useful device introduced by Arima and Lin [73] characterises the results of the first-order calculation in terms of an effective Ml-operator: =

Fef f = gL,effL "[- gs,effS "[- gp, eff[Y2 X S] (1)

(3.8)

298

I.S. Towner, Quenching of spin matrix elements in nuclei

where gc,e, = gc + ~gc and so on. Note the presence of a tensor one-body operator [Y2 x S] (~j which is absent in the bare one-body operator. The corrections ~gL, ~gs, ~gp are obtained by performing the core-polarisation calculation three times for (l a + 111fllla >,
-

-

,

3.2. Extension to RPA

The first-order results of the last section can easily be extended to higher orders in the random phase approximation (RPA). The pertinent second-order graphs are shown in fig. 7. It must be stressed that these are not the only second-order graphs; there are many more which we will discuss in section 3.5. However these graphs are the simplest to evaluate. The selection rules that limited the number of intermediate states in the first-order graph still apply to this particular set of second-order graphs. Furthermore in the Landau phenomenology these are the only second-order graphs that should be

I.S. Towner,Quenchingof spin matrixelementsin nuclei

299

___~-x

(o)

[b)

Cc)

Fig. 7. First-order, graph (a), and second-order, graphs (b) and (c), perturbation corrections to the matrix element of a one-body operator representing the start of the RPA series in a closed-shell-plus-one nucleus. The Hermitian adjoint graphs, which must also be included, have not been drawn.

considered. All other second-order graphs are presumed to have been summed in determining the Landau parameters. The correction to the reduced matrix element from diagrams (a), (b) and (c) in fig. 7 in an angular-momentum coupled representation is given by (bllfig. 711a)=E L -La - + E L - - - - +Aas Ls a

Ea

aS

~a

~S

(_)~+k E T a - -Bas - - LS aS

Ea

(3.9)

Es

where we are using Greek letters a, /3,... to represent particle-hole coupled states, viz. la ) = l( h--1 ~ p~)A) of multipolarity A and the following notation is being introduced:

L = (011F~)[l(h-~ ~ Pa) A) La = ()2a -

/~-1 (( h-ta Pa)AlVl(b-la) A)

Aa s = ((h~'pa)AlVl(h~'ps)A)

(3.10)

Bas = (OlVl(h~lpa)A(h-~lps)h) = (_)h.-p.+A ((p~lha)A[Vl(h~,ps)A) Ea = Eha-- Epa.

In second and higher orders it is assumed that in the energy denominators the photon energy, E~ = eb - ea, is negligibly small. This assumption enables the two graphs in fig. 7(c), which differ only in their energy denominators, to be conveniently summed together to give an expression of similar form to that from fig. 7(b). The matrix elements A s and Bas are exactly the matrix elements of RPA theory [75]. The sign (_)k is determined from the Hermitian property of the operator F ~x). For the M1 operator (_)~+k = --1. There is also a Hermitian set of graphs to be added to those displayed in fig. 7. The correction from this set can be incorporated in eq. (3.9) by replacing L~ by a generalized form:

L~ ~ L~ + (_)a-b-k -~ ~ L~(a~-b).

(3.11)

I.S. Towner, Quenching of spin matrix elements in nuclei

300

In the special case of diagonal matrix elements, viz. a = b, the Hermitian set equals that displayed in fig. 7, and the result from eq. (3.9) is simply multiplied by a factor of two. These results, from first and second order, can be algebraically summed to all orders. In a matrix notation the all-orders expression is

=

T[I-(A +

(3.12)

where the matrix [ I - (A - B)/el is first constructed and then inverted. Here I is the unit matrix. Again Lt3 is to be interpreted in its more general form, eq. (3.11). In table 11, we give some results using this expression for the RPA correction to magnetic moments and B(M1) transitions in closed-shell-plus-one nuclei in the Pb region for the same three choices of residual interaction as used in table 10 for the first-order corrections. Again oscillator wavefunctions (hw = 7 MeV) and experimental single-particle energies are used. The results show the corrections are between 30 and 40% smaller than the first-order calculations in table 10. The core-polarisation calculation for spin operators leads, in perturbation theory, to an alternating sign series; quenching in first order, enhancement in second order and so on. We see, for example, the calculated RPA correction for an h9/z-proton is of order 8/x = 0.45 - 0.50/aN. The magnetic moment of 2°9Bi differs from the single-particle estimate by 1.49/x N and the RPA correction only gives one third of this. There are other corrections, principally from meson-exchange currents which in this case lead to a large correction. Thus we postpone for now any comparison with experiment until we have discussed meson-exchange currents.

3.3. RPA in closed-shell nuclei Another, in principle, simple calculation involves magnetic dipole excitations in closed-shell nuclei. These excitations are described in terms of one particle-one hole states and from the M1 selection rules there are relatively few of them. For example in 2°8pb there are two possible configurations for the 1 * state, viz. 'rrhl-ll/2h9/2 and 1)113/2111/2.' • -I The RPA calculation, then, is to construct the matrices A and B, eq. (3.10), and solve the RPA eigenvalue problem (B

AB)(f)=E(%)'

(3.13)

To the diagonal elements of A, the particle-hole energies are added. With the eigenvectors, the B(M1) from the ground state to the 1 + state is evaluated using \--z

5)

a

a

(3.14) B(M1; 0+-+ 1+)=

~3 I<0+11F¢ 111+) i

where T = <01lF~ll(hS~p.)X) and the sum a is over the particle-hole basis states. For the M1 operator, the sign (_)~+k is - 1 . In the limit that the matrices B are set to zero, and hence Y~ is zero, the calculation reduces to the Tamm-Dancoff approximation (TDA). As a benchmark for the calculation, it is useful to consider the summed strength S = Ef B(M1; 0+---~ 1 + ), to all final states in the

I.S. Towner, Quenching of spin matrix elements in nuclei

I I I

I l l l

I I I I I

I I I I

II

o

II

t~

I

II

I

I I I

I I I I I I I I

I I I I I

I

I

I

I

I

I

o II

8 .

.

I

.

II

I I I

~ I

I

I

I

I

I

I

1

"

.

~

<

8

TTTT

II

. .

I

1"

I

$7~

I I I

I

-~--"-. $ ~.~

1" t t

. ~ . ~ 't~--~ 't~

°°i

I.S. Towner, Quenching of spin matrix elements in nuclei

302

TDA calculation. This value of S is independent of the choice of residual interaction [75], but does depend on the number of basis states in the calculation and the coupling constants in the M1 operator, F ~*). In 2°spb with just two particle-hole basis states and free-nucleon coupling constants this summed strength is S -- 50/x 2. There have been many TDA and RPA calculations [76, 77, 78] of the B(M1) strength in 2°spb. Typically in TDA there is a 1 + state very close in energy to the unperturbed particle-hole energy that is -1 • -1 • 2 the 'isoscalar' combination of 'trh11/2h9/2 and 1)113/2111/2states and with a B(M1) - 1 ~N, and a second state shifted by about 2 MeV, which is the 'isovector' combination with a B(M1) - 49/z 2. Ground state correlations, as included in the RPA, influence the energies and transition strengths: the principal effect is to reduce the isovector strength by about 20% to a value typically B(M1) = 40/z 2. In table 12 we give some RPA results using the three residual interactions discussed in the last section. The experimental situation regarding B(M1) strength in 2°spb has had a chequered history, as documented in earlier [79] and recent [80] reviews. The current consensus is that the sum of the M1 strength currently known in 2°8pb is on the order of about 9/z 2, only 20% of the TDA sum rule. There is a significant theoretical problem of missing strength. As just mentioned RPA correlations reduce the sum from 50/z 2 to 40 #2. The large amount of fragmentation suggested by the experimental work indicates that the prediction with just two states that carry all the transition strength is too simplified. A coupling of l p - l h states with a background of 2p-2h states gives a mechanism for the spreading. Calculations [77, 78] still yield a relative localization of M1 strength below 10 MeV, but in the work of Cha et al. [78] there is a long tail distributing a small amount of strength as far away as 60 MeV of excitation. In the main peak this spreading corresponds to another 20% reduction in the summed strength. A further reduction of 20% comes from meson-exchange currents, and in particular isobar currents that effectively renormalize the coupling constants of the one-body operator. We will discuss this in some detail later. In summary, the best that might reasonably be expected from theoretical predictions is a summed strength of around 20/~2N which is still a factor of two larger than the experimentally known situation. Laszewski and Wambach [80] comment that, from a theoretical point of view, the most interesting observation is the cumulative importance of a series of 20% effects including RPA correlations, 2p-2h mixing, isobar couplings and meson-exchange currents that all reduce the summed M1 strength. It is a similar story for the Gamow-Teller operator. In this case there are about a dozen proton-particle, neutron-hole 1 + states that would be evident in the 13 decay of 2°*pb (if the energy systematics were right), but which are accessible in the equivalent (p, n) reaction on 2°spb at forward angles. The coherent linear superposition of these l p - l h states form the Gamow-Teller giant Table 12 RPA calculations in 2°spb for M1 and GT operators with harmonic oscillator wavefunctions and experimental single-particle energies, for three residual interactions discussed in the text M1

GT

Interaction

Ex

B(MI; ~' )

%Sum

EX

B(GT; T )

%Sum

Zero-range, g' = 0.6

5.8 7.5

0.6 37.5

l 75

10.0 15. I

55 134

26 64

7r + p + 8g'

5.8 7.3

0.7 39.3

1 79

11.1 18.0

28 176

13 84

OBEP

5.8 7.0

0.9 43.9

2 88

10.2 16.2

41 156

20 75

1.S. Towner, Quenching of spin matrix elements in nuclei

303

resonance. The (p, n) reaction [45, 81] clearly identifies the resonance but the transition strength is found to be less than that anticipated. The one-body operator for Gamow-Teller transitions is F± = WgA ~

1

O'r±l

(3.15)

where .r+l w(1/X/-2)(rx +- irr) are the components of a spherical tensor and zx, re, r z are the usual Cartesian Pauli matrices for isospin. The uppe~ sign is for 13÷ decays and the lower sign for 13- decays. Here gA is the axial-vector coupling constant, Which from free neutron decay is gA = 1.26. Again it is useful to define sum rules =

B(GT;

f)= I(illFIIf>l2 (3.16)

S=~

B(GT; i--->f) f

and from the commutator algebra of isospin [rx, Zy] = 2izz obtain the Ikeda result [82] S(p,n ) -- S(n,p ) = 3 ( N -

Z)g2A .

(3.17)

The difference in the sum rule for the (p, n) reaction and the (n, p) reaction on a particular target nucleus such as 2°8pb depends only on the neutron excess in the target and is quite independent of any nuclear structure model dependence. In fact, in a heavy nucleus the S(n.p~ is very weak so for practical purposes 3 ( N - Z ) g 2 can be viewed as a close lower bound on the summed strength S(p,n~. Typically (p, n) reactions [81] find between 50 and 60% of this lower bound strength in the Gamow-TeUer giant resonance, after using Osterfeld's RPA results [83] for background subtraction. In table 12 we give some sample RPA calculations for the Gamow-Teller resonance in 2°8Bi, the excitation energy, E x, being expressed relative to the ground state of the target nucleus of the (p, n) reaction. Harmonic oscillator wavefunctions (hoJ = 7 MeV) and experimental single-particle energies are used, so the calculations are not truly consistent. It is preferable to be using Hartree-Fock, or at least Saxon-Woods, functions and energies in RPA calculations. Since we are using a sharply defined occupancy of the orbitals, all B-matrices are in fact zero and the calculations reduce to the TDA approximation. The summed B(GT; 0+---> 1 ÷) strength for the dozen 1 ÷ states exactly saturates the 3 ( N - Z ) g 2 sum rule. The calculations show this strength is concentrated in one or two states. In table 12, we list the two strongest states, the upper being associated with the Gamow-Teller giant resonance. These two states contain between 80 and 90% of the sum rule. These results are typical of the many RPA calculations irt the literature [84, 85]. Experimentally [45, 81] the observed strength is a single continuous peak at around 19.2 MeV (in 2°spb) but spreading to lower energies. Simple RPA theory with l p - l h states is not able to describe the spreading of the strength. The observed resonance [81] contains only about 50-60% of the sum rule bound. As with the M1 resonance, background mixing with 2p-2h states [86], and inclusion of meson-exchange currents, in particular isobar currents [46, 87], lead to a diminution of strength. We will return to this in section 5. Note that with the use of experimental single-particle energies the calculated energy of the Gamow-Teller resonance is too low compared to its experimental location of 19.2 MeV. This is a well-known phenomenon [43, 88]. As already mentioned we should be using, in RPA theory, singleparticle energies evaluated in Hartree-Fock (HF) theory in the closed-shell nucleus. Typically the HF

304

I.S. Towner, Quenching of spin matrix elements in nuclei

level spacings are wider apart than the experimental observed spacings. This is because the coupling to vibrational states compresses the levels near the Fermi surface which leads to the observed singleparticle energies. Many of the RPA calculations in the literature are based on HF calculations using the Skyrme III interaction [85]. The interaction is characterized by an effective mass m*/m = 0.75. Very crudely, we can correct the experimental single-particle energies by multiplying them by m/m*. In RPA this raises the location of the Gamow-Teller resonance by about 4 MeV and reduces the transition strength in the resonance by about 10%.

3.4. Core-polarisation blocking We discussed in section 3.1 first-order core-polarisation calculations in closed-shell-plus-one nuclei. We now want to move further away from closed shells and start with the next simplest case namely that of closed-shell-plus-two nuclei. Our purpose is to demonstrate that additional two-body core-polarisation graphs occur as soon as extra valence nucleons are added. Consider the graphs in fig. 8. Diagram (a) is the zeroth-order graph with the one-body operator interacting with one of the two valence nucleons. (Naturally there should be a second graph with the one-body operator interacting with the other valence nucleon. This has not been drawn, but its presence is implied. Hermitian and topologically equivalent graphs have been omitted in fig. 8.) Diagram (b) is a first-order core-polarisation graph but only one of the two valence nucleons is actively involved. We could proceed in this case by first calculating the graph in closed-shell-plus-one and finding, in the example of an M1 operator, an equivalent effective operator: Fef f = gL,effL + gs,effS + gp, eff[Y2, S]. With the effective coupling constants determined, it is then only necessary to evaluate diagram (a) with Feff rather than F and core-polarisation would be included in the calculation. This, then, is the method of attacking open-shell nuclei in the nuclear shell model: use effective one-body operators in the shell-model computation whose effective coupling constants are determined from the closed-shell-plus-one situation. However there is a flaw in this argument. Consider again graph (b). It is possible that the upgoing particle line in the bubble could be in the same quantum state as the noninteracting particle line, i.e. p = a t. If this were so then the Pauli Exclusion Principle would require this contribution to be zero. But graph (b) would not give zero because nucleon a I is not antisymmetrized with respect to nucleon p in forming the 3p-lh intermediate state. This error, however, is corrected in perturbation theory by graphs (c) and (d). Thus it is essential if one wants to maintain full antisymmetry in the theory that graphs (b), (c) and (d) be taken together. In first order in the residual interaction there are no further core-polarisation graphs. Therefore in open-shell nuclei the correct way to proceed is to allow for effective one-body and two-body operators to be evaluated between the many-body wavefunctions where the effective coupling

o,f-xt (o)

°tt °I

a I

OI - _ -ta 2

(b)

(c)

al

.

.

.

o la2

.

(d)

Fig. 8. Zeroth-order, diagram (a), and first-order, diagrams (b) to (d), perturbation corrections to the matrix element of a one-body operator in a closed-shell-plus-two nucleus. Hermitian and topologically equivalent graphs have been omitted.

I.S. Towner, Quenching of spin matrix elements in nuclei

305

constants have been obtained from closed-shell-plus-one and closed-shell-plus-two situations. In all but the simplest cases, this is rarely done. Let us consider the particular case where all the valence nucleons occupy a single j-shell. We will use the seniority scheme to classify the many-body states, [j"vaJM), where J is the total angular momentum, M its magnetic projection, v the seniority quantum number (essentially the number of unpaired nucleons) and a any other quantum number necessary to complete the specification of the state. Then the matrix elements of the one-body magnetic multipole operator connecting states [j"vaJM) and Ij"v'a'J'M): (i) vanish unless v = v', and (ii) are independent of n. Proofs can be found in standard shell-model textbooks [89, 90]. In particular for the one-body magnetic moment operator, the many-body matrix element can be related to the single-nucleon matrix element

(j"vJllF(l llj%J)

(JlIF('IIJ)

(J"vJllJIIJ nvJ)

(JllJIIJ>

(3.18)

This result holds for any one-body operator, a many-body state is (

j

F (1), of tensorial rank h = 1. Thus the magnetic moment of

~1/2

g(j"uJ)=-\)-_i_--f/

(j"vJIIF"

IIJ"uJ) = j g(j)/j

(3.19)

and we arrive at the well-known result that the g-factor (being the magnetic moment divided by the spin, g = tz/J) for the state [j"vaJM) is identical to the g-factor of the single-particle state [jm). This result has been tested in many cases and found to be true to within 10%. However there is a systematic departure that grows with n. This departure cannot be due to any renormalization of the one-body operator, such as given by the one-body core-polarisation graph of fig. 8(b), but is indeed a signal of the presence of two-body graphs, such as those in fig. 8(c) and fig. 8(d). There are two cases to be considered. In case (i) valence nucleons occupy a j = l + ½orbital, while the spin-orbit partner orbital j = l - ½ remains empty. Examples [91] would be the calcium isotopes where neutrons are filling the f7/2 orbital or the N = 28 isotones where protons are filling the f7/2 orbital; in either case the f5/2 orbital is unoccupied. In this case graph, fig. 8(c), contributes to the two-body core polarisation with the intermediate particle line being the spin-orbit partner orbital, j = l - ½. The graph, fig. 8(d), gives no contribution. In case (ii) valence nucleons occupy a j = l - ½orbital, while the spin-orbit partner orbital j = l + ½is fully occupied. Examples [92, 93] would be the N -- 126 isotones where protons are filling the hg/2 orbital. In this case it is fig. 8(d) that contributes to the core polarisation, while fig. 8(c) gives no contribution. For two particles in the valence orbit j, [j2vjM), the expression for the normalized and antisymmetrized reduced matrix element corresponding to fig. 8(c) is

(J%Jllfig. 8(c)ll j%J)

= v"2 (1

+ (_)k) ~. U( Aj'Jj; jJ)(JlIF( i'

)IIJ')(jj'; JIVIj=; (3.20)

where k relates to the Hermitian property of the one-body operator F (~). For the particular case of the M1 operator, k = 0 and j' is restricted to being just the spin-orbit partner orbital to j. Likewise for fig. 8(d), we have

306

l.S. Towner, Quenching of spin matrix elements in nuclei

(j2vjllfig.

8(d)lij2vj) = -~/2 (1 + ( _ ) k ) ~ U(Aj'Jj;

jJ)(jIIF¢~)IIj')(jj'; JlVlj2j)/(ej,- ej).

J'

(3.21)

We will write these expressions, eqs. (3.20) and (3.21), as (j:vJIIG¢~llj~vJ) where G represents a two-body operator. What we are interested in is the number dependence when this operator is evaluated between many-body states of configuration, jnvJM. Nomura [94] has given the general result for two-body operators of odd tensorial rank, G ~) with A -- odd, which simplifies for the diagonal case with A = 1 to be:

( f wJiiG~l~llJ.wj)

_ 2j + 1 - 2n 2j + 1 2v

(F°JIIG ' llf oJ) +

rl - o

2j+ 1 -2v

[ J ( J -1-- 1)] 1/2

m0

(3.22)

where _

M°

1

[K(K+1)]":(j2; KIIG,I II j ;K).

2K+l

.2

[j(j + 1)11'2 K.even ~ 2j + 1 L ~7-+-1)

(3.23)

Note the very simple linear n-dependence. Contrast this with one-body magnetic operators, F ~) with h = odd, which are independent of n. For the particular case of low seniority, v = 1 or 2, the expression simplifies even further and the contribution to the g-factor (being the magnetic moment divided by spin) from two-body operators reduces to [94]:

g(fvJ)=

[j(j+

7 - _ 1 M 0 + ½ ( l + ( - ) n)

~--~

Z(J)

(3.24)

where

Z(J) =

(jglIG~I)IlJg) [ j ( j + 1)]1/2

1

2j- 1 M°"

(3.25)

Note that from its construction Z(J) satisfies ~ J(J + 1)(2J + 1 ) Z ( J ) = 0, the sum being over even values of J. Nomura [94] gives arguments why Z(J) is expected to be small. Thus on examining the magnetic moments of odd mass nuclei, with assumed seniority v = 1 ground states, and of excited states in even-even nuclei with assumed seniority v = 2 a linear dependence on the number of nucleons in the single valence shell is to be expected in first-order core-polarisation theory. We examine examples of the two cases cited above. (i) The N = 28 isotones. Magnetic moments of 5~V, 53Mn and 55Co are known [95] together with the magnetic moments of the 2 + and 6 + states in 54Fe. We assume the configurations for these states are 'rrf7/2n with seniority, v = 1 for the odd mass cases, and seniority v = 2 in 54Fe. The contribution to the g-factors from one-body operators will be written as gsp and is independent of n. Thus for one-body and two-body operators taken together, we expect

g(7/2nvJ) = gsp + ~(n - 1)M 0 + J(1 + (--)n)(4

-

-

n) Z(J).

(3.26)

In principle, gsp should be determined from the magnetic moment of 49Sc, but since this moment is not

I.S. Towner, Quenching of spin matrix elements in nuclei

307

Table 13 Experimental and calculated g-factors for many-body states whose configuration is presumed to be j"vJ with lowest seniority, o N = 28 isotones

N = 126 isotones

State

Expt [95]

Theo

State

Expt [96]

Then

5'V(7/2-) 53Mn(7/2-) 54Fe(2+) 54Fe(6÷) 55Co(7/2-)

1.472 1.435(2) 1.43(28) 1.35(3) 1.378(1)

1.472' 1.354 1.340 1.285 1.236

2°9Bi(9/2-) 21°Po(8+) 2'°Po(6+) 212Rn(8÷) 2~4Ra(S+)

0.915(2) 0.919(5) 0.913(6) 0.896(15) 0.881(12)

0.915' 0.906 0.899 0.876 0.846

* Fitted

known, we will treat gsp as a parameter to be fitted in the analysis to 51V. We have calculated M 0 and Z(J) using eqs. (3.22) and (3.25) with the one-boson-exchange potential for V, the free-nucleon M1 operator for F (~), and the spin-orbit splitting between the proton f7/2 and f5/2 orbitals taken as 5 MeV. Harmonic oscillator wavefunctions (hto = 11 MeV) are used. The results are given in table 13. The theoretical results show a reduction in the g-factor with n, as expected, although the calculated fall-off is too fast, viz. A = g(55Co) - g(51V) has the value A = -0.094 experimentally and A = -0.236 theoretically. This is a characteristic feature of first-order calculations that we will discuss shortly. ~ (ii) The N = 126 isotones. Magnetic moments of the ground state of 2°9Bi and the 8 + states in 2 1 U rto, 2~2Rn and 2~4Ra [96] are known. We assume the configurations for these states a r e ,11"h9/2n with lowest seniority. Again the contribution from one-body operators, g~p, is fixed to the known experimental g-factor in 2°9Bi, but with one additional comment. Consider the one-body graph from first-order core polarisation given in fig. 8(b). There is a contribution to this graph with the particle line in the bubble being the valence orbital, p -= h9/2, and the hole line being the spin-orbit partner orbital, h = h11/2. We will single out this contribution and evaluate it to get

(Jllfig. 8(b)l[ j) =

(2K + 1) [(2j + 1)(2j' + 1)]1/2 U(AjjK; j'j)

(1 + (_)k)

× (JllF< )llJ')(

jj';

KIVIy2K)/(ej,- ej)

= - [ ] ( ] + 1)]1/2M0

(3.27)

where j = h9/2 and j' = hit/2, and where M 0 is defined in eq. (3.23) and obtained using fig. 8(d) for the two-body operator. Thus this particular contribution to the one-body core polarisation is intimately connected with the two-body contribution. Indeed the g-factor from this one-body contribution is just g(fig. 8(b)) = - M 0. We will separate this piece out from gsp; that is, we define gsp such that for 2°9Bi: g(2°gBi) = g~p - M 0. With this definition, the g-factor for other N = 126 isotones is g(9/2"vJ) = gsp - 1( 9 - n)Mo + ~(1 + (-)")(5 - n) Z ( J ) .

(3.28)

Note that at the closed-shell-minus-one configuration, n = 9, the contribution from the term proportional to M 0 is zero; that is, the contribution from fig. 8(b) is exactly cancelled from the contribution fig. 8(d). This is what is required to satisfy the Pauli Exclusion Principle. This phenomenon is called core-polarisation blocking. Again we have calculated M 0 and Z(J) using eqs. (3.22), (3.23) and (3.25)

308

I.S. Towner, Quenching of spin matrix elements in nuclei

with the one-boson-exchange potential for V and the spin-orbit splitting between the proton h9/2 and hu/2 orbitals taken as 5.6 MeV. Harmonic oscillator wavefunctions (hw --7 MeV) are used. The results are given in table 13. Again the theoretical g-factors show a characteristic fall-off with n that is about a factor of two larger than experiment. Namely a - g ( 2 1 4 R a ; 8 + ) - g(2X°po; 8 + ) has the value A = -0.038 experimentally and A = - 0 . 0 6 0 theoretically. Furthermore the difference A , = g ( Z l ° P o ; 8 + ) g(2°9Bi; 9/2 ) as calculated has the wrong sign as has been noted before [92]. As is seen here, and in previous works [91, 92], first-order core-polarisation calculations universally overestimate the core-polarisation blocking. This deficiency is largely removed when the calculation is taken beyond first order. Towner et al. [93] in an all-orders calculation involving l p - l h configurations: n -1 n • --I n + Z h - 2 / 2 ) , h1-9/2 n + l hHI,/2 -1 [ h 9 / 2 h l l / 2 ) , 1h9/2(1)111/2, 113/2)), a n d 2 p - 2 h configurations: h.-9/2-( v i , , / 2 , i 13/2))'-1 ]h9/2 (pi211/2113/2)) obtains an acceptable fit to the data on the N-- 126 isotones. Their result is shown in fig. 9. The conclusion is that two-body core-polarisation graphs have an important role to play in understanding the state-dependence of magnetic moments of many-body systems, but as was the case discussed in section 3.1, first-order calculations tend to overestimate the effect. There is also the possibility that two-body operators coming from meson-exchange currents will also give a contribution to
3.5. Second-order core polarisation In eq. (3.1) in section 3.1, we give the expression for the first-order core-polarisation correction to the expectation value of a one-body operator in a closed-shell-plus-one nucleus. Extending this expression, now, to second order the following additional terms are to be added

<,~lvl/3) _ E

(~-~ -~--~~(-~. _-~-~) a,/3,~a,b

(E a

c~,~a,b

(blV[/3)

+ ~.~a,bE(Eb---E;) 1 •

2 ~,'a.b

a#a,b

ot,~,~a,b

(ylVla) (/3IF[')')(E-a---E-~)

(E a - Eta) 2

-E,)-

-

E 2



1 Z _ )2 2 ~a.b (E b E~


where, as before, a and b are single-particle valence states, a an infinite set of 2p-lh states and/3, 3' an

I.S. Towner, Quenching of spin matrix elements in nuclei I

I

I

I

309

I

2 I/20.92 9/2 0.91

o\

0.90

0.89

u0.88

,,q LA.

I IM

0.87

0.86

0.85

0.Sq

n

I I

209Bi

2 I

210po

3 I

2llAt

4 I

212Rn

5 I

213Fr

6 I

2t4Ra

Fig. 9. The g-factors for ['rth9/2; J] states. The solid lines illustrate core-polarisation calculations taken (a) to first order in perturbation theory; (b) to all orders for lp-lh intermediate states; (c) to all orders for lp-lh and 2p-2h intermediate states, and (d) to include in addition configuration mixing with proton f7/2 and i13/2 orbitals. All curves were separately normalized to the 2°9Bi datum. Experimental data are taken from ref. [93].

infinite set of 2 p - l h and 3p-2h states. We are not considering any Hartree-Fock insertions. When F is the M1 magnetic moment operator, the selection rules on (blFla) severely limit the number of intermediate states of type c~. However there is no such restriction on the intermediate states of type and 3'. Second-order calculations therefore are computationally quite time consuming even for the M1 operator. If we limit the discussion to light nuclei and closed LS shells, then there is no first-order correction to the magnetic moment operator as already discussed in section 3.1. The reason is that the one-body M1 operator cannot create (or annihilate) a particle-hole state at LS closed shells because the M1 selection rules require the particle and the hole state to have the same orbital structure. The same reasoning eliminates the first four terms of eq. (3.29) for the second-order core-polarisation correction. This leaves the fifth, sixth and seventh terms, each of which carries intermediate state summations over 2 p - l h and 3p-2h states. Note the last two terms represent simply a normalization correction to the single-particle matrix element (blFla). In fig. 10, we give a graphical representation of these three

310

I.S. Towner, Quenching of spin matrix elements in nuclei

--x

--x

(o)

(d)

2

(b)

(e)

--x

+

b.c.

(c)

(f)

(g)

Fig. 10. Second-order core-polarisation graphs that give a correction to the magnetic moment of a closed-LS-shell-plus-one nucleus. Graphs (a), (b), (c) involve 2p-lh intermediate states, graphs (d), (e), (f) and (g) involve 3p-2h intermediate states. Graphs (c) and (g) are 'folded' graphs that correct (to second order) the normalization of the single-particle state.

terms for a closed-shell-plus-one nucleus. The normalization corrections are represented by folded graphs [1, 7, 75]. These particular groups of graphs are sometimes called the number conserving set [98]. This is because if the one-body operator were to be replaced by the number operator the sum from these graphs would be zero. This is quickly seen if we rewrite the last three terms in eq. (3.29) for the diagonal matrix element with a = b in the following equivalent form

Here the operator Q/e represents g¢]¢!){ ~ l / ( E a - g ~ ) . Thus if the one-body operator, F, commutes with both the energy denominator, e, and the residual interaction, V, then the second-order correction to the diagonal matrix element will vanish. The number operator does just that. In second-order calculations the unperturbed Hamiltonian is usually the harmonic oscillator and the unperturbed energies simple multiples of the characteristic oscillator energy, h~o. Thus the one-body operator will always commute with e. The residual interaction, V, can be decomposed into tensors in orbital and spin space: V= ~ ~(k). 5g
(3.31)

k

where k = 0, 1 or 2 corresponds to central, spin-orbit and tensor components respectively. Consider, first, the isoscalar magnetic moment operator comprising operators L and S. This operator clearly commutes with the central component of the residual interaction. Thus only noncentral and in particular tensor interactions can generate a second-order correction to the single-particle isoscalar moment. A similar argument holds for the spin component of the isovector magnetic moment operator, St, which commutes with any spin-independent central interaction such as the Wigner and Majorana exchange forces. If the spin dependence of the residual interaction is weak, then one can expect the second-order configuration mixing due to the central interaction to produce only a small correction to the matrix element of S~-. The same argument does not apply to the orbital operator, L~-, which does not commute with any central interaction. From these considerations it is clear that the tensor force will play an important role in the second-order core-polarisation corrections to magnetic moments.

I.S. Towner, Quenching of spin matrix elements in nuclei

311

There have been several calculations of second-order core polarisation reported in the literature. Ichimura and Yazaki [99] using only central interactions obtained no correction to the isoscalar magnetic moments and only a small correction to S~'. Mavromatis and Zamick [100] used the G-matrix of Kuo and Brown [101], which contains noncentral components, but limited the intermediate-state summation to states whose oscillator energy were just 2hto above that of the valence nucleon. Shimizu, Ichimura and Arima [48] were the first to point out the importance of extending the intermediate-state summation beyond 2hw since the tensor force, in particular, has a strong coupling to highly excited intermediate states. In that work the Hamada-Johnston potential was used (with a short-range correlation function). This potential has a 'strong' tensor force in that the short-range phenomenological additions to the one-pion-exchange potential in the Hamada-Johnston parameterization enhances the tensor force over that of pion exchange. Towner and Khanna [47] have repeated these earlier calculations using the one-boson-exchange potential. In this case, the potential is said to have a 'weak' tensor force because the short-range additions, coming from p-exchange, have the opposite sign and weaken the pion tensor force. Despite this there is still a need for extensive intermediate-state summations. In the following, we will discuss some numerical results from Towner and Khanna [47]. (The values here differ slightly from the published ones in that there have been some small changes in meson coupling constants and the Al-meson has been explicitly included here. The coupling constants are listed in table 6.) Again it is convenient to express the results of calculations such as these in terms of an equivalent effective one-body operator. For M1, we write Feet = gL,effL + gs,effS + ge, eft[Y2, S]

(3.32)

where as before gL,eff = gL + ~gL, etc. Here gL is the free-nucleon single-particle coupling constant and ~gL is the calculated correction. We will also introduce a superscript, e.g. ~g~L°) and 8g~L1), to distinguish isoscalar and isovector coupling constants. The free-nucleon values are gel°)= 0.5, g~0)= 0.88 and g~p0)= 0.0, and g~L1) = 0.5, g~l) = 4.706 and g~p1)= 0.0. For the Gamow-Teller [3-decay operator, we write Fef f = gLA,effL + gA,efftlr + gpA,eff[Y2, or].

(3.33)

Note the traditional use of the spin operator tr = 2S. The flee-nucleon values are gLA = 0 . 0 , ~.A = 1.26 and gPA = 0.0. The computed second-order core-polarisation corrections were obtained using harmonic oscillator wavefunctions with hto = 20.4 MeV for closed shell core A = 4, hto = 13.3 MeV for A = 16, and hto = 10.8 MeV for A = 40. These values are chosen such that the mean square radius of the nuclear charge density of the closed-shell nucleus calculated with oscillator functions reproduces the experimental value. We note the following points from the second-order calculations. a) Intermediate-state summations. In fig. 11 we show the impact the short-range tensor force has on the truncation of the intermediate-state summation. The upper panel shows the contribution to the total that comes from intermediate states of specific energy, Nfito, above the energy of the single valence nucleon, in this case a 0d nucleon outside an 160 core. The lower panel gives the accumulated sum when the intermediate-state summation is truncated at energy Nhto. With just central forces, 70% of the total comes from 2hw excitations while summing through to 4hto gives 90% of the total correction. However with tensor forces included, only 40% of the total comes from 2hto excitations and the summation has to be extended to 10 hw before 90% of the total correction is obtained. It is clear that even with a 'weak' tensor force employed here intermediate-state summations must be taken to high

I.S. Towner, Quenchingof spin matrix elements in nuclei

312

ISOVECTOR HI

70%

~I

NTRAL ONLY

t~

so%

~

30%

t...I ¢Y

o_

10%

2

Orl PARTICLE

TZI?

4

6

8

I0

*

TENSOR

12

N 100%

cl ua i..-

80%

s

s

/

ONLY

. ~

.- * *~'~ CENTRAL t..j LJ

60%

11" r

[" 2

J

I

I

J

I

I

4

6

8

I0

12

N Fig. 1l. The correction from second-order core polarisation to the isovector magnetic moment in mass A = 17 as a function of the energy of the intermediate states, Nhw, expressed as a percentage of the sum. In the lower panel the accumulated correction with intermediate states summed to NhoJ is expressed as a percentage of the total sum.

energy. This is reminiscent of the work of Vary, Sauer and Wong [102], who in computing the corrections from 3 p - l h intermediate states to the two-nucleon effective interaction in 180, found the tensor force required the intermediate-state summation be taken at least to 10 h~o. This phenomenon is sometimes called tensor correlations. To date, these second-order calculations are only practicable in light nuclei. However Shimizu [103] has given an approximate summation that works quite well in heavy nuclei for LS closed shells, but the role of excess neutrons still have to be evaluated. b) Residual interaction. Our choice of residual interaction for these second-order core-polarisation calculations is the one-boson-exchange potential. This choice is not guided by its pedigree for nuclear-structure calculations in light nuclei but rather by the requirement that there be a consistency between the nuclear Hamiltonian and the meson-exchange current processes to be discussed in the next section. The one-boson-exchange potential enables this consistency to be established. It is of interest to see what ingredients in the potential influence the calculated corrections to magnetic moments and Gamow-Teller matrix elements. This can be assessed in table 14, where the results of the successive addition of another meson to the one-boson-exchange potential is tabulated. (Remember these calculations are quadratic in the residual interaction.) Of the nine effective coupling constants listed all but the isovector Bg[ 1) are very little influenced by the short-range to-, ~- and Al-mesons. For example, for the isoscalar magnetic moments and Gamow-Teller matrix elements we could use just the simple ar + p meson exchange force and get essentially the same results. This is very good news because there is little ambiguity and uncertainty in these ingredients of the nuclear force. Here we are using 'strong' p coupling and are including vertex form factors. The results are also influenced by the choice of short-range correlation function. Despite that the calculated corrections to isoscalar magnetic moments

1.S. Towner, Quenching of spin matrix elements in nuclei

I

II

I

I

I

I

313

I I I I I

I

I I I I I

I I I I

I

I

I

I

I

I

I I I I I

I I I I

I I I I

~TTTT

II

li

I

I I I I I

o

I I I I I

l

i

l

l

i l l

l

~ i l

8 .=_ i

l

l

l

l

e~

bb

bb

÷÷

÷÷

~

~b

0

I;

314

I.S. Towner, Quenching of spin matrix elements in nuclei

and Gamow-Teller matrix elements remain under very good control. For the isovector magnetic moment, the contribution from the spin operator Sz likewise is governed principally by ~r + p exchange as can be seen by the results for ~g~/and ~g(p~). It is only the orbital operator Lz that is sensitive to the to- and ~r-mesons. In SU(2), the to-meson coupling constant is related to that of the p-meson, goNN = 3gpNN = 7.89, whereas potentials fitted to NN scattering such as the Bonn or Paris potentials tend to support larger values of g,oNN" This is explained by Durso et al. [31] as an effective coupling constant that is effectively including in go,NN the short-range repulsion from the 'p'rr-box' graph with intermediate isobars. Since later on we will be discussing such second-order graphs as 'Vrr-box' explicitly, we have used g~NN = 7.89 in table 14. Fortunately the choice is not critical. This is because what happens to ~g(~l) mirrors what happens in binding energy calculations: there is a cancellation between contributions from to- and (1) G-mesons. For example, in table 14 for a 0p-valence nucleon, the calculated og c goes from -0.131 for v + p, to -0.099 for v + p + to, to -0.196 for v + p + to + ~ residual interactions. That is to-mesons give a positive contribution to ~g~) and G-mesons a negative contribution. If instead we use go~NN= 11.21 instead of 7.89 and correspondingly increase g,~NNfrom 6.21 to 7.21 (table 7) in order to maintain the same binding energy in a Hartree-Fock calculation for 160, the calculated ~g~) goes from -0.131 for + p, to -0.080 for v + p + to, to -0.208 for v + p + to + or. Thus, to within 5%, we get the same result as before. The Al-meson contribution is rather small, so it makes little difference whether this meson is included in the residual interaction or not. Quoted results in the next section include the A~-meson. All this is related to the use of a short-range correlation function that cuts down the contribution from heavy mesons. c) Results. In table 15, we summarise the results of second-order core-polarisation calculations in light nuclei for valence orbitals at the closed LS-shells A =4, 16 and 40. Quoted there are the corrections to the magnetic moment expressed as a percentage of the single-particle matrix element and the corrections to the coupling constants, ~g, in the equivalent effective one-body operator. These latter values enable the correction to magnetic moments in the single-particle states p~(j= l + ½) and ~(j = l - ½ ) and the off-diagonal matrix element (j = I+ ½[l llJ: l - ½ ) t o be obtained since, by construction, these ~g values only depend on the orbital quantum numbers of the state, nl, and not the total angular momentum, j. We also give the ~ge values for/-forbidden M1 transitions. Note the results show only a weak state dependence for ~gc and ~gs with a tendency for the hole orbits to have a larger correction than the particle orbits. The ~gp values are more variable, but the matrix elements ~gp(Y2, S) are generally small when compared to those of ~gs(S). For isoscalar M1 and GamowTeller operators the calculated ~gi~ and ~gp are negligible. For the other coupling constants we find ~g(L1) = --0.13 -- 0.06, 8g~1) = --0.06 ----0.2, 8g~°) ------0.12 --+0.02 and ~gA = --0.16 --+0.04 span the range of results tabulated in table 15. Expressed as a fraction of the free-nucleon coupling constant, the three spin operators are all in the range ~gs/gs = -0.13--0.3; that is, second-order core polarisation gives a quenching to spin operators of around 13 -- 3%. We postpone a comparison with experiment until after a discussion of meson-exchange currents and isobar contributions.

4. Meson-exchange currents Standard nuclear physics is based on the nonrelativistic Schr6dinger equation in which wavefunctions for a many-body state are computed to some approximation as a first step and expectation values of

I.S. Towner, Quenching of spin matrix elements in nuclei

¢-L

I l l l ~

il

II

E

I

I

E I I I I I

II

II

El

II

II

I

~1

E

l i l

II

eS~

•

tl

I

~

I

"e5

II

I

II

% I

.a=

.=_

I I I I I

I I I

II

II

I

II

]

I

I

[I

H

i

8 ? II

li

II

II

H

II

II

II

II

II

315

l.S. Towner, Quenching of spin matrix elements in nuclei

316

observables calculated as a second step. For processes such as gamma or beta decay the observable is described by one-body operators, whose particular form can easily be deduced once the relativistic interaction between the currents and the fields is written down. For example, the interaction of a nucleon charge current with an electromagnetic field is given by a Hamiltonian Y( = - J , A where A , is the vector potential describing the field and J is the charge current of a single nucleon: ' p) = i ti(p') [Fly~, -

L( p ,

F2

(4.1)

Here ti(p') and u(p) are plane-wave Dirac spinors, whose energy and mass are assumed here to be on 1 s the mass shell, E Z = p Z + m 2, with F 1 = ½(El + F ~ -3) and F2= 5(F2+ F ; r 3) being the Dirac and anomalous coupling constants. In general these coupling constants are functions of k 2, where k = p ' - p . We will be mainly interested in their values at k2---~0, which from the free-nucleon magnetic moments are F~(0) = 1.0, FSl(0) = 1.0, F~(0) = 3.7 and F2(0 ) = -0.12. A nonrelativistic form of the charge current Jr = (P, J) is obtained by multiplying out the Dirac spinors with the Dirac matrices and keeping only terms to leading order in 1 / M - this produces a one-body operator sandwiched between Pauli spinors: P ( P ' , P ) = F 1 + O(1/M z)

(4.2) 1( p', p) =

F1

F 1 + F2 .

( p ' + p) + - 2M

lO" × k + O(1/M 3)

where for economy of notation we have left out the Pauli spinors associated with g ( p ' ) and u(p). A Fourier transform to coordinate space and a multipole decomposition would lead to the familiar one-body electromagnetic operators, whose matrix elements are then evaluated in the many-body system using for example shell-model wavefunctions. When the coupling constants at the currentnucleon vertex as used in the many-body system are taken from the free-nucleon system then the scheme is called the impulse approximation. It has been widely successful, but at some point it is known to break down. This is because the nucleons in the nucleus are interacting through the exchange of mesons and the perturbing electromagnetic or weak current can disturb this exchange and even interact with the exchanged meson itself. This is most easily demonstrated by recalling that the charge current has to be a conserved current, viz.

k~,J. = k ' J -

koP = 0 (4.3)

k. 1 = [14, p]

where in coordinate space k 0 becomes O/Ot and is replaced by the commutator with the full nuclear many-body Hamiltonian. We will consider this equation of continuity in leading order in the nonrelativistic reduction of the one-body current given in eq. (4.2): k.l=

F1 ( p, _ p ) . ( p , + p ) =

F~

= T(p', p') p(p', p) - p(p', p) T(p, p) = [T, p]

(4.4)

1.S. Towner, Quenching of spin matrix elements in nuclei

317

where T(p, p) = p2/2M, the kinetic energy part of the Hamiltonian. Note that the anomalous part of the current, the term proportional to F2, is entirely transverse, viz. k. J = 0. Thus, if the equation of continuity is to be satisfied, a second term must be added to the current such that

k. jox = [v, p]

= V(p;, P2; Pl + k, P2) P(Pl + k, p,) - p(p;, p',- k) V(p~- k, p~; Pl, P2) + ( 1 ~ 2 ) . (4.5) This extra current, jex, the meson-exchange current, is two-body in character since V is a two-body potential. Furthermore on identifying V with the exchange of an isovector meson, such as the pion, with an isospin dependence V= v~2 ~-1• Cz, then the commutator with the isovector part of the charge density does not vanish: l~,v 3, = i(¢1 X ¢2)3 F v1/)12" [V, p] = [v12 ¢1" "rE, ~rlrll

(4.6)

Thus current continuity and the isospin dependence of the nuclear force guarantee the existence of exchange currents. The magnetic moment due to the exchange current is

/,rex ___=1 f r X jex d3r

(4.7)

which can be broken into two pieces [104, 74, 105] in the following way:

e x = 1Rx fjex d3r+ 1 f i r

l(rl + r2)] x jex dar

(4.8)

where g = (r I + r2)/2. Since jex depends only on the relative coordinates r - rl, r - r 2 the second term in eq. (4.8) is translationally invariant. By contrast, the first term depends on the choice of origin and is not translationally invariant. Using Green's theorem and the fact that currents J°" vanish exponentially at large distances, the integral in the first term of eq. (4.8) becomes f jex d3r = _ f r(17. jex)d3r

(4.9)

and hence, through the equation of continuity, is known unambiguously. Therefore the magnetic moment terms, which are translationaUy noninvariant, are determined uniquely: #.x = - ½(R x r,2 ) ('r, x ,r2) 3 F [ v,2

(4.10)

where rl2 = r I -- r 2. This is known as the Sachs moment and is completely model independent inasmuch as it is determined by the nucleon-nucleon potential, v12. The Sachs moment is the dominant contribution to the meson-exchange correction to the magnetic moment. Note, in particular, that this correction is isovector in character. This approach, however, tells us nothing about the translationally-invariant and hence modeldependent second term in eq. (4.8). For this reason the method of constructing the exchange-current operator directly from the continuity equation has largely been superceded by an ab initio approach

318

I.S. Towner, Quenching of spin matrix elements in nuclei

rather similar to the way the one-boson-exchange model is constructed for the nuclear force: all Feynman diagrams in which one meson is exchanged between two nucleons are drawn and the external photon line linked in all possible places. Graphs involving two meson exchanges are generally too complicated to consider, but the essence of the one-boson-exchange model is that the influence of such graphs can be simulated by the exchange of a boson, for example two pions in a singlet state are described by a fictitious scalar ~r-meson. When carried out consistently, the exchange current from this approach will automatically satisfy the continuity equation, with the potential V in the commutator with the charge density being the one-boson-exchange potential. This Feynman graph approach was pioneered by Chemtob and Rho [106] and is called the S-matrix method. 4.1. S-matrix method

We start by considering graphs involving pion exchange. These are the most important graphs because the pion, being the lightest mass meson, generates the longest-range exchange current operators. Heavy-meson exchange will lead to short-range operators, which in nuclear physics are less important since the nuclear wavefunction at short distances heals rapidly to zero. This is a consequence of the short-range repulsion in the nucleon-nucleon interaction. The basic graph, then, is as illustrated in fig. 12 and comprises three pieces: a pion-production amplitude by some external-probe current, a pion propagator and a pion-absorption amplitude. The propagator and the ~NN-vertex are well known from the one-boson-exchange model of the nuclear force. The interest lies in the pion-production amplitude. At low energies in which the current-probe momentum tends to zero, k ~ 0, the photo- or weak-pion production amplitude can be expressed in terms of the nonradiative amplitude to leading order in 1/k. The next order is also available if the divergence of the current is known, Low theorem [107]. There is a second form of low-energy theorem applicable in the limit that the pion momentum tends to zero, q~ ~ 0, the soft-pion limit which is discussed in detail in the papers of Adler and Dothan [108]. Chemtob and Rho [106] have harnessed these theorems to obtain a pion-production amplitude that is essentially model independent. We will by-pass the details. The end result [109] is that the evaluation of the amplitude comes down to an evaluation of a commutator, -(1/f~) [Qs, J~], involving the probe current and the axial-charge operator, Qs, the integral over all space of the time component of the axial current. The coefficient to the commutator depends on the pion-decay constant, f . Were we dealing with zero-mass quarks with point couplings then the evaluation of this commutator would be trivial: the charges would be generators of closed SU(2) x SU(2) algebra (we are dealing with only upand down-quarks) and the commutator would be determined from the group algebra. These same commutators are nonetheless adopted in this context, following the premise of current algebra [110,111] that the currents given by the quark model exhibit algebraic properties that are of more I

Pl

I

P2

Pz Fig. 12. Basic meson-exchange graph for a two-nucleon system interacting with a vector-current or axial-vector-current probe of momentum, k.

319

I.S. Towner, Quenching of spin matrix elements in nuclei

general validity. (Alternatively, imposing chiral symmetry on nucleons and massless pions would lead to the same commutation relations [112].) Thus for a vector current probe, V~, or an axial-vector current probe, A~,, (the superscript is an isospin index), the commutator from SU(2) x SU(2) algebra is [Qs, " V~,] ="len/ta~,'

,

[ Q ,,5, a~,] ='le,#tV~,'

(4.11)

where e,j~ is the completely antisymmetric tensor in the isospin indices. What this says is that the leading term in a photo-(weak-) pion production amplitude is a current proportional to an axial-vector (vector) current, 3'~,Y5(3',). It is the presence of the pion that brings about this parity change. Making a nonrelativistic reduction of the pion-production amplitude shows it to be of order O(1) if the probe current is the space part of a vector current or the time part of an axial-vector current and of order O(1/M) if it is the time part of a vector current or the space part of an axial-vector current. Recall that for the photo- (or weak-) vertex with a nucleon without the associated pion production (the normal impulse-approximation graphs leading to one-body operators) the converse is true. Hence we get the repeatedly emphasized conclusion [113] that pion-exchange currents should be important for the magnetic part of the vector current (space part of Vj,) and the axial charge (time part of A~,). While conversely for the electric charge (time part of V~,) and Gamow-TeUer beta transitions (space part of A~,) the pion exchange currents are suppressed and low-energy theorems powerless. For reviews of low-energy theorems for meson-exchange currents, see Ivanov and Truhlik [114], and Chemtob [115]. An alternative method that in leading order gives the identical result to that from low-energy theorems, is the ab initio approach in which the Feynman diagram in fig. 12 is interpreted as representing a sum of many elementary contributions. The principal ones are shown in fig. 13. Those involving just pions and nucleons are the Born terms, graphs (a), (b) and (c). The remaining graphs reflect the contribution to the amplitude from composite states, such as the isobar, A, or heavy mesons, p, to, A1,... and are known as the non-Born terms. The statement is that the Born graphs alone, whether evaluated with pseudoscalar or pseudovector ~rNN coupling, will reproduce the low-energy result. We will demonstrate this for pseudoscalar coupling for the isovector electromagnetic current. Consider graphs (a) and (b) of fig. 13 (and topologically similar graphs in which the photon is coupled to the other external nucleon lines) and write down an expression for the two-body current using standard Feynman rules: v ~-~-~t%,~k~ F2 ] ~r l J1 SF(Q1) J~=it~(p~) {[ FIT

x

g~rNN]/SqIl+ g~rNNT57.11SF(Q2)

,y,~-~--MO'~k~ ~rl u(p~)Zl~(q2)~i(p;)[g~NNYSrt2]u(p2)+(l~2)

(o)

(b)

(c)

(d]

(4.12)

(e)

Fig. 13. Meson-exchange graphs of pion range: (a) nucleon intermediate states, (b) antinucleon intermediate states, pair graph, (c) pionic-current graph, (d) isobar intermediate states, and (e) heavy-meson current graph.

I.S. Towner, Quenching of spin matrix elements in nuclei

320

where Q1 = P l - k, Q2=Pl + k and q = P 2 - P 2 are respectively the momenta on the intermediate nucleon states and the pion. The nucleon propagator, SF(Q ), is divided into its positive- and negative-frequency parts

Sv(Q) = S~v+)(Q) + SCv-)(Q) 1

1

(4.13)

i S~v+)(Q) = (M - Qo) 2M [-i Q . T + My4 + M] 1 [_iQ.y_My4+M] i S~-)(Q) - (2M)2

"

The pion propagator is i A (q 2 ) = l / ( q 2+m2) and is taken in the static limit. We consider first the positive-frequency part of the nucleon propagator (fig. 13(a)) and make a nonrelativistic reduction by multiplying out the Dirac spinors and matrices and keeping terms to leading order in 1/M. For the case when the index ~ is space-like, we obtain

111~_M{VF~(pl+Q,)+(F~+FV2)io JJ=-~o . , , 1

×k}

~T 11 J

o. 1 . q TI'

1 ]] g'~NN O"2"q

g=NN2M0.1 " q TI'

l

v x -~--h~{Fl(Q2 + p~)+(F~ + FV2)i 0.1x k } 7"c~J 2M q2 + m2 7 2 + (1 ~ 2 )

(4.14)

which can be rewritten in the form t. JJ=ff- [V ~(Pl,t P2,i. PI + k, p2)JJ(pl + k, pl)-JJ(p~, Plt _ k)V~(pl t _ k, P2,P,, "

P2)] + ( 1 ~ 2 ) (4.15)

where V is the pion-exchange potential 2

V(p~, P'2; Pt, P 2 ) -

g=NN (0.1" q)(0.z'q) (2M)2 q2 + mZ

(4.16)

and JJ(Pl, Pl) the one-body current, eq. (4.2). This shows that the two-body current is singular in the limit that the photon energy tends to zero, k 0~ 0. More importantly, the positive-frequency part of the propagator has led to a current that is not a true two-body exchange current, but is a product of a one-body current and a potential. This term is already included in the impulse approximation, when the matrix elements of the one-body current are evaluated with shell-model wavefunctions obtained from the solution of the Schr6dinger equation that contains the one-pion exchange potential. Therefore to avoid double counting with the impulse approximation fig. 13(a) should not be included in the exchange-current operator. For the negative-frequency part of the nucleon propagator, the two-body exchange current in the

I.S. Towner, Quenching of spin matrix elements in nuclei

321

nonrelativistic limit corresponding to fig. 13(b) is 2

v g~,NN °'~(tr2"q) i(~', × 1"z)j + ( 1 ~ 2 ) . jJ= _F1 (2M) 2 q2 + m2

(4.17)

We have said that the same result is obtained in both pseudoscalar and pseudovector pion coupling. What happens is that the pair diagram is negligible in pseudovector coupling. But because of the derivative ~rNN-coupling there is an additional Born graph, not drawn in fig. 13, known as a contact or seagull graph which nonetheless leads to the identical exchange current to order 1/M 2. Let us return to the alternative derivation based on fig. 12, which uses the low-energy theorem and writes the photo-pion production vertex as a commutator with the axial charge. This leads to the following expression for the exchange current

J~ = -i ~,d(p,) [Q"5,V~] U(pa) a ( qZ) g(P2) g=NNTSrz u(P2) + (1 ~ 2). t

(4.18)

n • l l On replacing [Qs, V~] by le,jtA~, and writing the axial current as A,,/ = i gAy~,ySrl/2 where ga is the axial-vector coupling constant, a nonrelativistic reduction leads to

jj=

gA g~NN O'~(O'2"q)

2f~ 2M q2+m2 i ( z 1 × z 2 ) ' + ( l ~ 2 )

(4.19)

which is identical to eq. (4.17) on using the Goldberger-Trieman relation f~ = MgA/g,,NN and taking the low-energy limit with Fl(k2)---> F~(0)= 1. Next, we consider fig. 13(c), the pion-current diagram. To describe the interaction of vector currents (or axial-vector currents) with mesons, we will appeal to the vector-dominance model in which isovector currents are mediated by p-mesons, isoscalar currents by o~-mesons and weak axial currents by A~-mesons. Then for the case of an isovector current, its coupling to a p-meson is given by 2

j~_

mp j 2g0Nr~ P"

(4.20)

and the p--->2"rr vertex is described by the Lagrangian

•p•r

~r =

gp ejlmP j~ Tr 1 tPl~ Tr m

(4.21)

where p~ and 7r" are meson fields (superscripts are isospin indices) and 2gpNN = gp, the pxr'tr coupling constant. Note that G-parity arguments forbid to'trot and A1rr~r vertices, so the pion-current diagram only contributes for isovector currents. In fig. 14, we have redrawn fig. 13(c) displaying the role of the p-meson. The exchange current generated by this diagram, with p = P'I- Pl, q = P~- Pz and k = p + q, is J~ = &,NNt~(P'l) 75 u(p,) Ztx A=(p 2) goejtm(p - q)~,a,(k 2) mZo/g, x d•(q 2) ti(p;) 3'5 u(p2) r 2 .

(4.22)

322

I.S. Towner, Quenching of spin matrix elements in nuclei !

!

Pl

132

Ii

P Pl

q

P2

×

Fig. 14. The pion-current MEC diagram in the vector-dominancemodel where the isovector current is mediated by the p-meson.

Pseudoscalar ~rNN coupling has been used, but pseudovector coupling leads to the same result in the nonrelativistic limit, namely jj_

2

g;NN (-~2

mo

0-, "P

m2p + k 2 i(r~ x r2) j p2 + m2

( p _ q) _ ~_' q 2 o q +m~

(4.23)

,

To satisfy the equation of continuity, we will need to identify F~(k 2) = mp/(m'p 2 9 + k 2) which in fact is the vector-dominance result for the coupling of an isovector photon (mediated by the p-meson) to a nucleon. For the Born graphs the equation of continuity for the pion exchange currents reads k . J ex = k. (Jpair +'/current) 2 g=rNN v " ( 2 M ) 2 F 1 i(r, x r 2 ) J (

/k-ylF -q

k.0- 0"l.p

q2 + m2

p2 + m2

(0"--7---5-C,,~'P)k" (p - q) 0-2 " q } (p + m )(q +m2=)

2

g~NN " 0"1 " q 0"2" q (2M) 2 F ~ i ( r 1 x r2)' --Sq----Y+m~, + ( 1 ~ 2 ) .

(4.24)

i j Using the identity i(r 1 x r2) j• = [r I • r2, ~rl] and identifying

2

V~ - -

g~rNN O"1"q 0"2 " q (2M)2 q2 + m2 rl . r2,

/ v j p =½ F 1 T 1

we obtain the required result k . J ex = I V , pJ].

(4.25)

Thus with the use of vector dominance in the evaluation of pion-current graph, we see the equation of continuity is strictly satisfied by the Born terms alone. This places a requirement on the non-Born terms, that they must lead to transverse currents satisfying k. jex = 0. We will return to this point in the next section. Finally, we comment on the meson-nucleon couplings. In this section these coupling constants have been written as simple constants appropriate for point nucleons when in principle they are functions of q2, the meson momentum squared. It is quite common, in fact, to simulate the compositeness of a

I.S. Towner, Quenching of spin matrix elements in nuclei

323

nucleon by multiplying the coupling constant by a vertex form factor g~NN~ g~NN/, (q2) where one common parameterization is the monopole form: F(q2) = (A2 _ mZ)/(A 2 + q2)

(4.26)

with the parameter A,, being of order A , , - 1 GeV. The pion pair current, then, is still given by eq. (4.17) with the first term multiplied by Ir (q2)l 2 and the second term with 1 ~ 2 multiplied by Ir3p2)l 2. However, it would be wrong to take the pionic current from eq. (4.23), and multiply it by F ( p 2) F(q2) as that would lead to a current which no longer satisfies the equation of continuity. This point has been stressed by Riska [116] and by Mathiot [117]. Let us denote by F(A,~) the function that is to modify the pionic current, eq. (4.23), and use the equation of continuity to solve for F(A,,). The solution is

p2 + m2 + q2+ m2 F(A,~) = F~(p 2) F~(q 2) {1 + - q2 + A2 p2 + A2 j •

(4.27)

This result is easily interpreted if one imagines the vertex function as being simulated by a "heavy pion" of mass A,, that is emitted by the nucleon and in turn "converted" to a pion. Then in the pionic-current graph we should allow the isovector photon to couple not just to the pion but to the "heavy pion" as well. Thus there are two extra graphs to consider, whose algebra is the same, except for changes in the propagators. When the photon couples to the 'heavy pion' on the left side of the diagram a propagator 2 -1 (p2 + m2)-1 is lost to be replaced by (q2 + A,,) , while when it couples to the 'heavy pion' on the right 2 -1 a propagator (q2 + m2)-1 is lost to be replaced by (p2 + A,,) . Note that these pion-exchange current operators even when correctly modified by vertex form factors remain model independent in the sense that they involve no parameter not already present in the pion-exchange potential.

4.2. Non-Born terms of pion range We turn now to the non-Born graphs, diagrams (d) and (e) of fig. 13 involving either isobar intermediate states or heavy mesons. Our purpose is to obtain an expression for the exchange current corresponding to these graphs and show that this current is indeed transverse, viz. k . J = O. We postpone to section 5, the isobar graphs and consider here the heavy-meson graphs, p-at, ~0-'rr and Al-'rr. From G-parity arguments, the p-rr vertex couples to an isoscalar vector current (or isovector axial-vector current to be discussed in section 4.5) while the to-~ and AI-aT vertices both couple to isovector vector currents. Again we will use vector meson dominance, so the graphs are to be interpreted as redrawn in fig. 15. We consider each in turn.

:--] II;" x

(a)

,6,I

ll;-\

x

X

(b}

(c)

Fig. 15. Heavy-meson current graphs of pion range: (a) p-~r graph, (b) co - ~r graph, and (c) A~-Ir graph.

I.S. Towner, Quenching of spin matrix elements m nuclei

324

p--rr graph For the p-rr graph, we assume co-meson dominance and write 2

j~ - mo, u 2go,NN %

(4.28)

and the Lagrangian for the prow-vertex as [111] ~

= igo~G~p ~ 0 % Oppi~Tr i

(4.29)

where % , f,~ and 7r~ are meson fields (superscripts are isospin indices) and go,o~ the coupling constant. Using Feynman rules to evaluate fig. 15(a), we obtain for the current 2

s m~ - N J ~ = i r 1 • r 2 goNNg~ssgo~ -2g~N

a(pl) Ix,

Kv

x Ap(p2) i G~p,, k pp A~o(k:) A~(q 2) g(p;)[ys] u(p2) + ( 1 ~ 2 )

(4.30)

where Ap(p2), Ao~(k2) and A ( q 2) are propagators for the respective mesons and p = p ' 1 - P l , q = P 2 - P2 and k = p + q. Finally multiplying out the Dirac spinors with Dirac matrices and keeping terms to leading order in 1/M, we obtain a nonrelativistic form for the current (with index/x being space-like):

(k × q) (o-2. q)

J~=-iF~(k2)

rl"rZ &NN 2M

2go,NN ( p 2 + m Z p ) ( q 2 + m 2 ) + ( 1 ~ 2 )

(4.31)

where F~(k 2) is identified from the vector-dominance result, F~(k 2) = m 2 / ( m 2 + k2). Note this current is transverse as required. It remains to evaluate the coupling constant. Again through the vectordominance model, we can model the radiative decay of p, viz. p ~ ~r-/, as being mediated through the co-meson, then the radiative width is given by [118] P(p--+'rry)=

m p ( 1 - m~/mo)2" 2,3

a

\ 2KoNN /

(4.32)

where a = e2/hc, the fine structure constant. A recent determination [119] gives F = 71 _+7 keV and hence go~p~rm p

g,=~ = 2g,oNN

0.578 --+0.028.

(4.33)

It is amusing to note that de Alfaro et al. [111] gives a relation goomo = 2 g , ~ , which together with SU(3) relations gp~ = 2gpNN and goNN = 3gONN, leads to g0~v _- 2-~' This result is within 15% the value deduced from the radiative width. to-~ graph The evaluation of the o~-Tr graph, fig. 15(b), follows analogously from the p-'rr result on essentially

I.S. Towner, Quenchingof spin matrix elements in nuclei

325

exchanging the role played by the O- and to-mesons. The result for the nonrelativistic form of the current is •

g,'NN

go, e,,

(k×q)(°'2"q)

j i = _iF~(kZ) r~ g~,Nrq 2M 2geNrq (p2 + mZ)(q2 + m2)

+(1~-~-2)

(4.34)

where again vector dominance is used in identifying F~(k 2) = mp/(mp 2 2 + k2). The current is transverse as required. We can use the same model assumptions to deduce the top'rr-coupling constant from the radiative decay of to, viz. to~w~. The radiative width is given by [118]

= z4,

~2 m~(1

-

2, m 2,,3 m,'/ ~) .

(4.35)

The Particle Data Group's fit [120] to the partial decay widths of the three principal decay modes of the ~0-meson yields for the radiative width F = 8 5 2 - 52 keV and hence

g'°e~m°' - 1.98 - 0.06.

(4.36)

g ~ = 2geNN Again we can use the relation go,e,'me = 2ge,',., the SU(3) relations ge~," = 2geNN and g~,NN= 3geNN, and the approximate equality m e = m,~ to deduce g~,~v= 2. This is in excellent agreement with the value deduced from the radiative width. Indeed with SU(3) we have g,,,'v = 3ge~v which is approximately satisfied by the experimental widths. At-It graph Finally for the Al-'rr graph, fig. 15(c), we will need a Lagrangian to describe the A lp~r vertex. We will adopt the phenomenological Lagrangian of Ogievetsky and Zupnik [34] as used in the work of Ivanov and Truhlik [33] and Adam and Truhlik [121]. The Lagrangian is minimal, chiral and approximately gauge-invariant under SU(2) x SU(2) transformation. The gauge invariance is assumed to be broken only by the nonzero heavy meson (O, A1) masses. Likewise the usual chiral invariance is broken by the pion mass. PCAC is incorporated and the Lagrangian reproduces all the standard PCAC results in the soft-pion limit. Being minimal (no more than two field derivatives in each term), this Lagrangian contains only four independent parameters gp(=ge~), gAx( = gA~NN/gA), me and m A and are related to f,', the pion decay constant, and amongst themselves by the Weinberg sum rules and the KSFR relation [36] f2

2

2.

,'go = rnp,

2

2.

gA1 = go,

2

2p

m A = 2m .

(4.37)

With this Lagrangian the Alpcr coupling is described by t~A 1Plr -

1

f'r¢

1

m

elran(a~plv--OvP~)[AJz(Ov'n"

n

)-

1

n

m

~7r (O,,A~

-OVA,,)] m

(4.38)

where pt,,, A~ and 7r" are meson fields with superscripts being isospin indices. Using o-meson dominance, we obtain for the isovector current corresponding to fig. 15(c)

I.S. Towner, Quenching of spin matrix elements in nuclei

326 •

mp

1

JJ~, = - i (~', x 72)' ~ gAINNg~rNN2gpNN /~(Pl)[Y,~Ys]u(Pl)

XAA(P2)[k,(q,+ lp~,)-(k'q+ ½k'p) 6,~,]Ap(k2)A~(qZ)~(p~)[Zs]u(p2)

(4.39)

where AA(p2), Ap(k2) and A(q2) are propagators for the respective mesons. Multiplying out the Dirac spinors with Dirac matrices and keeping terms to leading order in 1/M, we obtain for the nonrelativistic form of the current j j = F~(k2) i (~.1 × 72)j gnNNgA,NN

2MmA [(d,'k) q - ( k ' q ) 6 " ~ + ( d , ' k ) k - k

2~,1

~r2" q

x (p2+m2A)(q2+m2)+ (1~2)

(4.40)

where again F]'(k 2) = mp2/.(mp2 + k 2) and the Goldberger-Trieman relation f~g~NN = MgA is used. Furthermore we define

6"l = ~q + (o-, .p) p/m A

(4.41)

where the second term came from the second term in the A~-meson propagator: i AA(p2) = [ 6 +p,p/m~]/(p2+ m~). Note that the current is transverse. This result has been obtained by Adam and Truhlik [121]. Note that these heavy-meson graphs of pion range have all been found to be transverse and as such give no contribution to the equation of continuity. In this sense there is no constraint on these terms. They are model dependent. They exploit the vector dominance ideas and use for the three-meson vertices Lagrangians which are phenomenological satisfying certain symmetry requirements and being minimal in the number of field derivatives retained. The Lagrangians, however, are not unique. The main advantage of this method is that it leads to a scheme that is convenient for practical calculation. It enables one, applying the standard Feynman rules, to write down expressions for the exchange currents for any process for which the component Lagrangians can be specified.

4.3. Pair and current diagrams of heavy-meson range We next consider meson-exchange currents of heavy-meson range. These will lead to short-range operators whose expectation values are strongly damped by the short-range correlations in the nucleus. Thus these terms in any practical calculation are sensitive to the details of any short-range phenomenology and are subject to considerable uncertainty. However for the construction of the exchange current, we will follow the techniques of the last two sections and use phenomenological Lagrangians, vector-dominance theories and insist that the current satisfy the equation of continuity. Consider, first, diagrams (a) and (b) of fig. 13 where the meson exchanged may be v, p, ~0, ~ or A~. In general the current corresponding to these diagrams is written J,=-i

ff(Pl){[F~%, - 2-~ o - k ]

SF(Q1 )/" ½(a¢+)+a~ ))+ F SF(Q2 ) [Fty,

x l(a(+~ - a (-)) u(pl) ~¢(q ) li(p;) F~ u(p2) + ( 1 ~ 2 )

1:2 o- k,] (4.42)

I.S. Towner, Quenching of spin matrix elements in nuclei

327

where SF(Q) and zl, t3(q2) a r e the nucleon and meson propagators respectively, a ~÷) and a ~-) are isospin anticommutators and commutators as detailed in table 16, and F~ describes the meson-nucleon coupling, viz. F = i g~NN~5 =

- i gpNN('Ya --

for xr

Kpo'~,p,/2M)

for p

= - i g,oNN(Y~ -- Ko,°'~,P,/2M)

for

= g~r~N 1

for or

= --i gAlSNY ~/5

for A 1 .

(4.43)

Taking the divergence of this current gives a continuity equation

k~, J~ = -iF~ ~(p~) { k~. y~, SF( Q1)F~I (a ( +) - a ~-)) + F~ SF(Q2) k , 7 , 1(a(+) - a (-)} u(pl ) a~t3(q 2) u(p2) Ft3 u(P2).

(4.44)

Writing k = p ~ - Q1 in the first term, k = Qz -P~ in the second term and using the Dirac equation p~, y, u(p) = i M u(p) and the identity Q, 7, SF(Q) = - 1 + i M SF(Q), the continuity equation reduces to

k~.J~. = -iF] a (-) ff(p~) F~ u(pl) a~t3(q 2) ff(P2) rt3 u(p2) •

(4.45)

Thus in cases where the isospin commutator, a (-), is zero the current corresponding to diagrams (a) and (b) of fig. 13 is in itself conserved, From table 16, this occurs for isoscalar currents and for isoscalar mesons. Only for isovector currents with isovector meson exchange must additional graphs be found in order to construct a conserved current. These additional graphs are the contact and current graphs, which we will discuss explicitly below. Furthermore it is convenient to separate the contributions of fig. 13(a) from fig. 13(b) corresponding to the division of the nucleon propagator into a positive frequency part and a negative frequency part: SF(Q) = S~+)(Q) + S~-)(Q). As shown explicitly in eq. (4.15) for the pion current, the positive-frequency part contains the one-body impulse-approximation current multiplied by the potential and hence is not truly part of the exchange current. Defining the exchange current as the pair graph, fig. 13(b), together with the appropriate contact and current graphs in cases where a ~-) is not zero, the continuity equation becomes [121] eX

k~,J~,

-k~,J~÷)=[V,p]

(4.46)

Table 16 Isospin anticommutator, a t+) = ½{¢~, ¢i} + ¢i, and commutator a ~-) = ½[~'~, ¢~]_r~ arising in meson-exchange graphs, fig. 13(a), for the different isospin structures of the current and the exchanged meson Current

Meson

a~ ÷)

a ~-)

Isovector Isoscalar Isovector Isoscalar

Isovector Isoveetor Isoscalar Isoscalar

r~ rl. r 2 r~ 1

- i ( r I x ~2)j 0 0 0

l.S. Towner, Quenching of spin matrix elements in nuclei

328

where JT~ is the current from the positive-frequency part of the nucleon propagator in fig. 13(a) and can be expressed as a commutator of the heavy meson potential, V, with the one-body charge density, p. Furthermore on separating k~,J~~ = k . J . .- k. o .p and writing kop ~ as the commutator [T, pex], where T is the one-body kinetic energy operator and pex the two-body charge density of heavy-meson range, the continuity equation takes its final form k.J

ex =

[V(1/M°), p(1/M2)] + [V(1/M2), p(1/M°)] + [T(1/M), peX(1/M)] .

(4.47)

Here the arguments for these quantities show the order in 1/M to which they will be evaluated in the nonrelativistic reduction. In the following we construct the exchange current to order 1/M 2 for each heavy meson in turn and demonstrate it satisfies the continuity equation. Note that in the continuity equation only the longitudinal part of the charge density, the term proportional to coupling constant F1, is involved as stressed by Adam and Truhlik [121]. Then P(Pl, P l ) to order 1/M ° and 1/M 2 are p(1/M °) = F 1 (4.48) 1

r2

2

,

p(1/M 2) = -F~ -(2M)2 [½Pl + ½ P l - P , ' P , - i

t

o"1 "Pl xp,]

and in particular

p(p, +

p , ) - o(p; p ; - I,) =

F 1

i < × ( p l - p , ) .k

(4.49)

t Z IVI )

where F 1 = ~(F 1 s1 -1-F v1%), 3 the standard decomposition into isoscalar and isovector components. We begin by considering the isoscalar mesons, cr and to, for which only pair diagrams, fig. 13(b), contribute to the exchange current. or=meson The expression for the~-pair current follow analogously from eq. (4.12) except that the ~rNNLagrangian, ~7~NN = g,,NNNN05, replaces the pseudoscalar "rrNN Lagrangian, 5fN N = i g~NNN75rt2NTrt, where N, rr and 05 are nucleon, pion and scalar meson fields. Inserting the negative frequency part of the nucleon propagator, we obtain for the nonrelativistic ~r-pair current to order l/M2: 2

F, g~NN {2P + i o', X k} + ( 1 ~ 2 ) J = 2 (2M)---5 q2 + m2

(4.50)

where p = Pl' - Pl, q = P2' - P2, 2P -- Pl' + Pl , 2 Q = P2' + P2 and k = p + q. In particular, we have 2

Fx k . J = 4 ( 2 ~M )

g,NN P ' k + ( 1 ~ 2 ) . q2+rn2

For the right-hand side of the equation of continuity, we have V ( 1 / M 0) =

2

q2 + m o)

(4.51)

1.S. Towner, Quenching of spin matrix elements in nuclei

V~(1/MZ)

-

1 g2~N~ 1 2 , , (2M)2 q2 + m,2 [~Px + ½p~2 +Pl "Pl + i o"I "Pl

XPl

329

"Jr-( 1 ~ 2 ) ]

(4.52)

and specifically 2

1 q2g~NN V.(p~, P2; Pl ÷ k, P2) - V ~ ( p ,l - k, p~; Pl, P2) - (2M)2 + m2 [4P" k + i 0"1 x p. k].

(4.53)

Then

[V(1/M°),p(1/M2)] -

F1 g2'rNN i t r 1 x p . k + ( l ~ 2 ) (2M)2 q2 + m 2 (4.54)

[V(1/ME),p(1/MO)]-

FI g2,rNN [ 4 P . k + i o . l x p . k ] + ( l ~ 2 ) (2m)Z qZ + m 2

and the sum of these two terms equals eq. (4.51) and the continuity equation is satisfied. to4-meson The oNN-Lagrangian is written 5~,r~N = -i g~,NN1V(y~ -- K~,tr~t3qt3/2M)Nto. We will treat separately the cases when the index a is time-like (o4) and when it is space-like (to). First, for the time-like case, the pair current is 2

F1 g,~Nr~ (l+K~,)[icrlxp_io.lxk]+(1,,_~__2) J = 2 (2M)--~ qZ + mZ,

(4.55)

and the potential

V~,(1/M o) = g~NN/(q z z + mZ,o) Vo,(1/M2 ) =

1

g2°NN

(2M)2 q2 + mZ

(1 +2K,~) [ ~1( P l , - Pl )2 - iorl "Plt × Pl + (1 ~ 2 ) ]

(4.56)

2

1 q2g~,NN V~,4(P~, P~; Pl + k, P2) - Vo,4(P~ - k, P2; Pl, P2) - (2M)2 + m2 ( l + 2 K ~ , ) i t r l x p . k

"

Thus the commutators for the right-hand side of the equation of continuity are

[V(1/MO),p(1/M2)]-

FI g2,om~ i o . l × p . k + ( l ~ 2 (2M)2 q2 + mZ,o

) (4.57)

[V(1/M2),p(1/M°)] =

F1 g2NN z ( l + 2 K o , ) i ° ' l X p ' k + ( l ~ - - - 2 ) (2M)2 q2 + m~

and, by inspection, the continuity equation is seen to be satisfied.

I.S. Towner, Quenching of spin matrix elements in nuclei

330

o~-meson For the space-like part of the coNN-Lagrangian, the pair current is FI J = - 2 -(2M) - 2

2 gtoNN

q2+m2[2~

(4.58)

+ ( 1 + K,o)i0"2 x q] + (l *--~---~-2)

o)

and the potential 2

gmNN [_i(p;+pt)+(l+K)0",X(p;_p~)]. V ( 1 / M 2) - _ _ 1 (2M)2 q2 + m2 [-i(P'2 + p2 ) + (1 + Ko,)

0" 2 X

(4.59)

(P2-P2)] •

After some rearranging, we obtain 2

g~NN [20 • k + ( l + K , o ) i 0 " z × q . k ] + ( l ~ 2 ) [V~(1/M2), p(1/M°)] = - 2 -F, (2M)2 q2 + m2o,

(4.60)

showing the continuity equation is satisfied. These results for the ~- and co-pair graphs differ from those given by Riska [122], in that here we have kept the full momentum dependence of the antinucleon propagator rather than take it just in its static limit: iS(F-)(Q)= ( 1 - y4)/4M. p4-meson (isoscalar current) For the isovector mesons, p and A 1, it is convenient to treat the isoscalar and isovector exchange currents separately. Similarly we will separate the cases when the index ~ in the pNN-Lagrangian, £PpNN=--ig,NNN(Y~- Ko%t3q¢/2M)Npi~ ri is first time-like (P4) and second space-like (p). For the isoscalar 94 case only the pair diagram contributes to the exchange current, namely

jr=_

FSl g2pNN 2 (l+Kp) r,'r, i 0 " l × q + ( l ~ 2 ) . (2M)2 q2 + mp

(4.61)

Likewise the potential is V4(1/MO)_

gpNN q 2 + m 2p 'rl

"']'2

(4.62)

2

V,4(1/M2)_

1

g,NY (1 + 2Kp) r, • 'r2 [ l ( p ; - p 1 ) 2 - i

(2M)2 q2 + m~

0"I " p l( ×1 P~ l- +- - - - 2 ) 1 '

and the required commutators are

[V(1/M°), p(1/M2)]-

1

s

F1

2

gpNN

2 (2M) 2 q2 + m2p ~'1

"¢2i o"1 >( q ' k + ( 1 ~ 2 )

1.S. Towner, Quenching of spin matrix elements in nuclei

331

2

1 FSl gpNN 2 ( l + 2 K e ) ~ ' , ' ~ ' 2 i o " , x q ' k + ( l ~ 2 ) 2 (2M) 2 q2+me

[V(1/M2), p(1/M°)]-

(4.63)

showing once again the equation of continuity to be satisfied. p-meson (isoscalar current) For the space-like part of the pNN-Lagrangian, the isoscalar pair current is s

j

S

u

F1

2

g¢~N

(2M)2 q2

2 ~'l"~'2[2Q+(l+Kp) i o " z × q ] + ( l ~ 2 )

+ mp

(4.64)

and the potential 2

1 geNN Ve(1/M z) - _ _ (2M)2 q2 + me2

'1"1"q'2

[-i(p~ +Pl) + (1 + Ke)

o"1 X (ff~ - - p l ) ] °

[ - i ( p ~ + P2) + (1 + Ke) o"2 x (p~ - P2)].

(4.65)

The commutator with the charge density is [Vp(1/MZ),p(1/MO)]_

FSl g2pN~2~..7212Q.k+(1+Ke)io.zXq.k]+(l~_~_2) (2M)2 q2 + me

(4.66)

and the continuity equation is satisfied. p4-meson (isovector current) For the isovector currents, we have to consider not just pair graphs but contact and p-current graphs as well. First the pair graph for the time-like part of the p-Lagrangian we obtain J~air-

F~ g2pNN ( l + K p ) [ i o . 1 X p ~ . ~ _ i o . l X k . r ~ _ q i ( ~ r l X 7 2 ) y ] + ( l ~ 2 ) . (2M)2 q2 + mZo

(4.67)

The contract graph is shown in fig. 16(a), where again vector-dominance theories are used in which the isospin current is mediated by the p-meson. We need, therefore, a contact ppNN Lagrangian which we take from the chiral Lagrangian of Adam and Truhlik [121]: 2 Ko ~-r i . r k l '•oeNN = gpr~r~2M lvtr~, r lvp`'p~ Ekli

(4.68)

where N and pk are nucleon and p-meson fields and Roman letters are isospin indices. From Feynman rules, the exchange current corresponding to fig. 16(a) is J~ = i F l ( k 2) q: + mp2 2M 0"i x

~2) ]

ti(pl)[or] U(px)ff(p2) 3',, - - ~ o'`'t3q~ u(P2) + (1~-2) (4.69)

I.S. Towner, Quenchingof spin matrix elements in nuclei

332

//

P

P

x

×

(a)

(b)

Fig. 16. Two additional graphs, other than pair graphs, that contribute in p-meson range: (a) contact graph, and (b) p-current graph. Both are evaluated assuming vector-meson dominance and chiral Lagrangians.

where from vector-dominance theories we identify Fl(k 2) = m2p/(m2p+ k2). A nonrelativistic reduction to order 1/M 2 for the case when the index p, is space-like and the index a time-like leads to the required exchange current J~ontact-

F~ g2pNN K p i ( 7 1 X 7 2 ) J [ p + 2 i o ' ~ x e ] + ( l ~ 2 ) . (2M)2 q2 + m2o

(4.70)

Likewise the current graph is shown in fig. 16(b) with p-meson dominance again assumed. We need, therefore, the Lagrangian LPoop which can be obtained from the free p-meson Lagrangian, 5~ = -41-Pixy, 2 by picking out the term trilinear in p-meson fields. Here p~,, is the covariant curl of the p-meson field which in the chiral Lagrangian is coupled with A 1- and v-meson fields: k i j ~1 e~jk[Ai. O.TrJ- A.i O. Tr' + 7/ Or Z j. - 7riaZJ] p~k =(O.p.k _ _ o .p.)-goe,jkp.p.+

(4.71)

i and Ti.i are p-, A 1- and ~-meson fields and superscripts are isospin indices. Thus we where p~, A~, obtain [121]

i j

k

(4.72)

~POO~= --go eijkP~ pv O~p~

where go = gp~ = 2gpNN" From Feynman rules, the exchange current corresponding to fig. 16(b) is [

Y~ = F~(k2) (p2

x[~(p

goNN z 2 z i('rlX'rz)'t~(P;) )'~ + mp)(q + mo)

- q) +6t~,(k+q) - ~

K~

]

-~--~%vP, u(p,)

(k+p)~]a(p;)[y ~ - ~K° ~rtjsq~] u(p2)

(4.73)

with as before F~(k 2) = mp/(mp 2 2 + k2). A nonrelativistic reduction to order 1 / M 2 for the case when the index # is space-like and indices a and/3 time-like leads to the required exchange current 2

gpNN + mZo) i(71 X ~'2)j (p -- q) J{urrent = F1v (pC + m2p)(q2 x 1

(1+ 2Kp) p2 i z (2M) 2 [1 +~q _ i o . . p × P _ i ~ r 2 . q x Q ]

}

"

(4.74)

333

I.S. Towner, Quenching of spin matrix elements in nuclei

Note that for the equation of continuity, we will need to evaluate 2

gpNN { (1 + 2Kp) k Jcurrent = F1 -q2 -+ mp2i('rlx~'21' 1 - (2M) z •

i

v

x

[½p2 + ~q2 _ i o"1 .p

X

P - i 0"2"q

x

Q]} + (1 ~ 2 ) .

(4.75)

Note also there is a term of order O(1/M °) as well as a term of order O(1/M 2) and the continuity equation is expected to be satisfied at both orders. For isovector currents, the commutators of the potential with the charge density contain isospin factors that do not commute. It in convenient to write V--> V 71 • ~'2 and p---> trr~ and explicitly evaluate the isospin commutators (4.76)

[V, p]----)[V, p] "r~ + (V, p} i('r 1 × "rz)'.

We will consider the continuity equation first for terms in z~ involving commutators and second for terms in (71 x ~-2)j involving anticommutators. First, then, for commutators 2

F~ 2 q2g,NN [Vp4(1/M°)' P(1/ME)]rJ2-• 21 (2M) + m2p i0"lxp.kr~+(l<_~.__2 ) v

(4.77)

2

• 21 (2M) F 1 2 q2g,NN [Vp4(1/MZ)'P(1/M°)]rJ2+ mZp (l+2Kp) i0"lxp.kz~+(l(__.__2) and the sum of these equals k. Jpair for the term in eq. (4.67) involving r~. Second for anticommutators

{Vp4(1/M°), P(1/M°)}iO'l

2 • F1v q2gpNN × '72)1= + m~ i ( , r 1 × 7 2 ) J + ( l ~ _ . 2

)

(4.78)

and equals the term of order O(1/M °) in eq. (4.74) for k "Jc.rrent J as required. To order O(1/M 2) we have 2

(Vp.(1/M2), p(1/M°)}

F~ gPYY 2 i(71 x ,r2)j (1 + 2Kp) (2M)2 q2 + mo

i(71 × 72) j -

X

[q2 + i O"1 "q X P - i 0"2"q x Q] + (1 ~-~--2) v

{Vp4(1/M°), p(1/M2)} i0" 1 x ~r2)j -

(4.79)

2

F1 gpNr~ (2M) 2 q2 + rno2 i 0"1 x ~'2)' [½k2 + i 0"1 x P. k] + (1 ~ 2 )

which after some manipulation can be shown to equal the remaining terms in k'(Jpair Jr Jcontact qJc.rrent). The exchange-current therefore satisfies the equation of continuity to order O(1/M2).

334

I.S. Towner, Quenching of spin matrix elements in nuclei

p-meson (isovector currents) For the space-like part of the pNN Lagrangian, the pair current is 2

F~

Jpair -

goNN 2.{[2Q + (1 + Kp) i (r2 × q] r~

(2M)2 q2 + mp

+[2i (r, x Q + (1 + Ko) (((rl ° 0"2) q -- ((r," q) (r2)l i (r, × r2) j} + (1 ~ 2 ) .

(4.80)

The contact diagram, fig. 16(a), from eq. (4.69) yields 2 J{ontact -

F~ gpNN K , [ 2 i ( r ~ × Q + ( l + K o ) ( ( ( r . ( r 2 ) q _ ( ( r . q ) ( r 2 ) ] i ( r , × r z ) J + ( l ~ 2 ) (2M)2 q2 + m2 (4.811

and the current diagram, fig. 16(b), from eq. (4.73) J~ . . . . . t = J { u r r

1 °l-J{urr-2 v

2

F~

gpNN

(2M)2

i (¢~ × r2) j (p -- q)[-4P" Q - 2i(1 + Kp) (r~ × q. P

(p2+m2)(q2+m2p)

-2i (1 + Kp) (rx × p" Q + (1 + K p ) 2 (or, × p). (O"2 X q)] v

F1 J~urr-2 = 2 - -

(2M12

(4.82)

2

g.NN 2

2

2

(p2 + mo)( q + m, )

i ( r 1 x r,) j

x {[2P + (1 + K ) i O"I X p] [2Q" k + (1 + Kp)i (r2 x q "P - 2Mk,,] -[2Q + (1 + Ko) i (re × q] [ 2 P . k + (1 + Kp)i (r, × p . q - 2 M k o ] ) where Jcurr-1 c o m e s from the term in (p - q) at the ppp-vertex and Jcurr-2 from the remaining terms. This latter term is purely transverse in that k~,J~ ..... -2 = 0, or more explicitly k . J c u r r _ 2 = kop. . . . 2 = [T, &urr-Z]" Here Pours-2 is the two-body charge density. In evaluating J~ur~ 2, we will approximate 2Mk o = 2P. p + 2Q. q. Adam and Truhlik [121] point out that the current, J~ur~-2, as discussed here, is a factor of 2 larger than that obtained in earlier derivations [105,123]. This arises from the use of a chiral Lagrangian and vector-dominance theories. In the earlier derivations a pp'~,-Lagrangian is constructed through a minimal substitution, 0~ ~ 0~, -ieA~, where A~, is the vector potential of the electromagnetic field, in the free p-meson Lagrangian. Both derivations are valid. The factor of 2 difference is just indicative of the model dependence in these types of calculations. Furthermore the equation of continuity cannot be used as a constraint since Jcurr-2 is transverse. In principle, an experimental test could distinguish between the two results [124]. In practice this is unlikely to be feasible since this term is numerically small when compared to other terms in the p-meson exchange current. To conclude, we show the remaining terms in the exchange current satisfy the equation of continuity.

I.S. Towner, Quenching of spin matrix elements in nuclei

335

Note that for Jcu~-~ we can simplify the divergence of the current to read k

J

"J~u~-a

_

F~

g2pNN

(2M)2 (q2 + mZp)

iO.lX~.2)J[_4p.Q_2i(l+Kp)tr2Xq. P

-2i (1 + Kp)o-1 x p. Q + (1 + Kp) 2 (o-1 x p). (o-2 x q)] + (1 ~-2).

(4.83)

Recall that with the presence of isospin commutators, eq. (4.76), we need to consider the continuity equation first for terms in r~ involving the commutator [V, p] and second for terms in 0"1 x ~-2)j involving the anticommutator {V, p}. First, then, for terms in r~ we have •

2

F~

g¢~r~

(2M) z q2 + mp2 [2Q'k+(l+Kp) i°r2xq'k]~'~+(l~-2)

[Vp(1/M2), p(1/M°)] r~ -

(4.84)

which equals k. J~air from eq. (4.80). Second for the terms in 0"1 x ~-2)j we have 2

F~ 2 qZgpNN {Vp(1/MZ),p(1/M°)}i(~" 1 x ,r2)j• _ (2M) + mZp i0"1X~'2) j [ - 4 P . Q - 2 ( I + K . ) i o +2(1 + Kp)i tr I xq'Q-(a+Kp)2(o'~

2xq'P

x q)'(o'2 x q)] + ( 1 ~ 2 ) (4.85)

which after some manipulation can be shown to be equal to k.(Jpair +.]contact + Jcurr-1) from eqs. (4.80), (4.81) and (4.83). Thus all terms in the p-meson exchange current are now identified and all are consistent with the continuity equation. A 1-meson Because the Al-meson mass is large, m a = 1100 MeV, the corresponding exchange current will be very short ranged and its contribution in finite nuclei to the types of calculation to be discussed in the next section is small. Thus it is sufficiently accurate to calculate the exchange current to order O(1/M°), rather than to order O(1/M 2) used for the other heavy mesons. Note also the Al-meson potential is only evaluated to order O(1/M°). For this reason the pair diagram is not considered; it is of order O(1/M2). The only remaining graph is the Al-current, analogous to that in fig. 16(b). Again we appeal to the chiral Lagrangian of Adam and Truhlik [121] for the A1Alp-vertex, viz. j

,~AIA1 p = gpejkt(P~A,,

k

--

j

k

l

p~A~,) O~,A~

(4.86)

where p~ and A k are p- and Al-meson fields (superscripts are isospin indices) and gp = gp,~ = 2gpN.. The exchange current is then

j~(k)=_F~(k 2) gA1NN 2 10" • 1 X ~r2)j t/(pl) y~ 75 u(p,) AA(p2 ) x [(p - q)~, 6,~ + q, 8,~, - p , 8~,~1 AA(qz ) if(P2) Y, 75 u(p2)

(4.87)

336

I.S. Towner, Quenching of spin matrix elements in nuclei

where AA(q 2) is the propagator for the A~-meson and from vector dominance F~(k 2) is identified as

mZo/(mZp+ k2). A nonrelativistic reduction to order O(1/M °) for the index > being space-like yields 2 gAINN

JJ(k) = - F ~ ( k 2) (p2 + m2)(q2 + m 2) i(rl × r2) j [(P - q) 0"~"02 + (81" q) ~2 - al (82 .p)] (4.88) where

6.1 = trl + (0-1"P) P/m2 ,

dr2 = % + (0"2"q) q/m2.

This result has been given by Adam and Truhlik [121]. Finally, we check that this current is consistent with the continuity equation evaluated to order O(1/M°). First the divergence of the current can be rearranged to read

k'JJ(k) = - F l ( k 2 )

2 gA1NN

5-----2 i(rl x r2) j tr,. 6"2 + (1~-2) q +mA

(4.89)

which after consideration of the isospin commutators, eq. (4.76), is easily seen to equal the anticommutator {VA(1/M°), p(1/M°)} i (rl x r2) j as required. In summary, then, the heavy-meson exchange currents have been derived from a chiral invariant Lagrangian with the assumption of vector-meson dominance. Each current has been explicitly shown to satisfy the continuity equation. This is a desirable property necessary to preserve the gauge invariance character of our calculations.

4.4. Exchange-current magnetic moment operator The exchange currents deduced in the previous three sections all have the structure: J ( k ) J(k; p, q, P, Q) (2"n')3 a(3)(k - p - q), where p, q, P and Q are relative and center-of-mass momenta in the two-nucleon system P =Pl-Pl,

2P =p~ +Pl (4.90)

q =P; - P 2 ,

2Q = p ; +P2

and the delta function expresses the conservation of momentum. Here k is the momentum associated with the electromagnetic current. Our requirement is to calculate the expectation value of this two-nucleon current in a finite-nucleus many-body system. It is convenient, therefore, to rewrite the current in a configuration space representation following a multiple Fourier transformation j(k;xl, x~,x,,x2)

-

1 f (27r) 12

d3pdaqd3pdBQj(k; p,q,p,Q)exp{i[p.½(x1+ x,t)+ q.½(x2 + x2)

+ P ' ( x ; - x,) + Q . ( x 2 - x2)]} (2~r) 3 6 ( k - p -

q).

(4.91)

I.S. Towner, Quenching of spin matrix elements in nuclei

337

Note that if the current is independent of P, Q then it is said to be local, since the integrations over P and Q lead to the delta functions 6<3)(x'1 - Xl) and 6<3)(x~ - x2). To handle nonlocal terms in P and Q we use the replacement P exp(iP, x) = - i Vx exp(iP • x) and interchange the order of differentiation and integration in the Fourier transform. In this way the dependence on P and Q can be transformed away to become derivatives on the delta functions. They produce in configuration space derivative operators and are simply obtained through the replacement rule: P--->-iV~ and Q--->-iV2. It is essential for the usefulness of this scheme that the dependence on P and Q be sufficiently weak; a quadratic dependence, for example, is barely tolerable. Here we will limit ourselves to nonlocal terms that are no higher than linear in derivative operators. The construction of the one-boson-exchange potential makes the same approximation. The spin-orbit potential is an example of a linear nonlocal term. Thus the current in configuration space is written

J(k; Xl,X2): ~-~

d3p d3qJ(k; p, q , - i V 1, - i ~ ) e x p ( i p . x, + iq. x2} (2~r) 3 6(k

- p - q). (4.92)

Introducing relative and center-of-mass coordinates r = x~ - x 2, 2R = x I + x 2 and choosing to eliminate the integral in p gives

J(k;r,R)-

1 f d3qj(k;k_q,q,_iV1,_iV2)exp(ik.xl_iq.r}.

(4.93)

(2rr) 3

One integral is left, which in all cases can be handled analytically [115]. There remains the practical step of projecting out the required multipole operator. For transverse magnetic multipole moments of interest here the projection is

_Tmag(b'" j~ ,,~, r, R )

:

- ~--~ 1

i_jf

d2k Y~l([C) "J(k; r, R)

(4.94)

M

where Ym(k) is the vector spherical harmonic. The multipole operators are as defined in ref. [125]. The usual magnetic moment operator,/~, is the special case of the J = 1 multipole evaluated in the long wavelength approximation, k--->0. An equivalent definition is /~(r'R)=-I

i ( - ~1) 3 f d3qe -iq'r Vk xJ(k;k-q'q'-iV~'-iV2)

eik.xl

k--,o

(4.95)

We will not list here the expressions for the magnetic moment operator corresponding to the two-body exchange currents obtained in the previous three sections. Their derivation is straightforward but tedious. For the one-body current, eq. (4.2), we proceed analogously and first Fourier transform to configuration space:

J(k; x 1,' Xl) = ~ - ~ 1 f d3pd3pj(k; p,P)exp{i[p, l(xl + x,1)+ e.(x~_x1)}(27r)3 8(k_p ) (4.96)

I.S. Towner, Quenching of spin matrix elements in nuclei

338

and use the same trick to replace the momentum operator, P, by a derivative J(k; x 1) = J(k, - i ~ ) exp(ik • x, ).

(4.97)

Then we project out the magnetic moment operator to obtain ~tg(Xl)= - ½ i V k × { - ~F1 i V~ +

F~+F2io.×k)exp(ik. Xl) k--,o 2----M--

1 (1) 3 = 2M [g(L°)L + gL L~'l +

(1),, 3, ar~l

g~°)S + gs

(4.98)

where L = --ix I X V l , S = ½Orand ge _(0) = ~FI(0), ~ s _(1) = ~FI(0), 1 v gL g~0)= F~(0) + F2(0 ) andg~l) = F~(0) + F2(0 ). The coefficient, 1/2M, is the nuclear magneton, eh/2Mc, expressed here in natural units. We will calculate the expectation value of these one-body and two-body magnetic moment operators in closed-shell-plus-one configurations. For the one-body operator the calculation reduces to that of a single-particle matrix element in the valence orbital outside the closed shell. For the two-body operator we will be computing two-body matrix elements between the valence nucleon and one of the nucleons in the closed-shell core and summing over all nucleons in the core. It will be useful to express the result of this computation in terms of an effective equivalent one-body operator, eq. (3.32): Jt~eff = gL,eff L + g s , e f f S + gP,eff []12, S ] .

(4.99)

We give some sample results in table 17 for a 0p-hole and 0d-particle at an ~60 closed-shell core. Harmonic oscillator wavefunctions, hw = 13.3 MeV, are used. With two-body matrix elements some Table 17 Meson-exchange current corrections to magnetic moments in closed-shell-plus (or minus)-one configurations at mass, A = 16, expressed as a percentage of the single-particle Schmidt value

OP1/12 lsovector ~r-pair ~r-current

bgL

~gs

bgP

%

~gL

~gs

~gP

%

0.292 -0.215

-0.300 -0.300 0.224 0.019

- 34.9 14.4 6.1 0.6 - 1.4 - 1.1 7.7 - 3.0 1.2

O. 165 -O. 125

0.011 0.001 0.037 0.018 -0.006

O.183 -0.134 -0.016 0.001 0.018 -0.006 0.026 0.020 -0.005

-0.369 -0.369 0.255 0.020

0.015 0.001 0.060 0.031 -0.010

0.465 -0.232 -0.014 0.001 0.024 -0.003 0.047 0.046 -0.008

12.0 - t0.1 (1.2 0.1 0.9 - 0.1 2.6 1.5 - 0.4

0.175

0.326

-0.403

-25.8

0.102

0.089

-0.498

6.5

-0.002 0.024 -0.010 0.019

0.025 -0.003 -0.005

1.6 3.3 - 0.3 - 1.2

0.011 -0.003 0.000

0.002 0.018 -0.007 0.011

0.026

0.015 -0.004 0.001

-0.003 -0.005

0.0 2.2 - 0.6 0.4

0.012

0.032

0.017

0.2

0.009

0.020

0.018

2.0

to-~r A,-'n

~r-pair to-pair p-pair + contact p-current A~-current Total

Od~, 2

-0.025 -0.017 -0.004

-0.025 -0.015 0.006

lsoscalar p-~r

~-pair o~-pair p-pair + contact Total

I.S. Towner, Quenching of spin matrix elements in nuclei

339

consideration must be given to the role of short-range correlations. Our ansatz is that the same procedure must be used as in the construction of the effective one-boson exchange potential, eq. (2.54), in order to maintain the consistency requirement between the potential and the exchange current. Accordingly, we take for the two-body magnetic moment operator (4.100)

fz(r, R) = It(r, R) g(r)

where the correlation function is taken as g(r) = O(r - d) with d = 0.5 h/m~c = 0.7 fm. We also include vertex form factors at all meson-nucleon vertices but we take care to ensure that the equation of continuity remains satisfied. In table 17, the results are given for a typical light nucleus, 0Px/2-hole and 0ds/2-particle relative to an 160 closed-shell core, while in table 18 results are given for a proton 0h9/2-particle and a neutron 0i~3/2-particle relative to a 2°spb closed shell. It is more efficient to perform these calculations in LS coupling. Then, for LS closed shells, the tensorial rank of the two-body exchange magnetic moment operator matches uniquely the tensorial rank of the equivalent one-body operator. For example, the terms in the exchange magnetic moment operator with tensorial rank L = 1, S = 0 are the only ones that will contribute to ~gL" This is because the closed shells have L = 0, S = 0. However, for closed shells with neutron excess orbits, such as those occurring at 2°spb, it is necessary to use jj-coupling for the sums over the excess neutron orbits and the unique identification between terms in the two-body exchange operator and the equivalent effective one-body operator is lost. For example in the a-pair graph there are no terms in the two-body exchange operator of tensorial rank L = 2, S = 1 and hence no contribution to ~gp from LS closed shells as evident in table 17. However the neutron excess orbits generate an effective contribution to 8gp as seen in table 18. Let us look first at the isovector ~SgL value, which comes principally from the Sachs moment, eq. (4.10). In the S-matrix approach, represented as a sum of low-order diagrams, the pion Born terms give a large contribution but there is a significant cancellation between the pair and current diagrams. For example, for the h9/2 proton at 2°9Bi we compute ~igL = 0.302- 0.246 = 0.056 with vertex form factors and ~gL = 0.325- 0.230 = 0.095 without. This latter is very close to the estimate first obtained by Miyazawa [126] of ~gL=0.1 from pion-exchange currents. The non-Born terms of pion range are negligible, whereas the heavy mesons make a significant contribution. Our result for a n h 9 / 2 proton is Table 18 Meson-exchange current corrections to magnetic moments in closed-shell-plus-one configurations at 2°8Pb expressed as a percentage of the single-particle Schmidt value Proton

~r-pair ~r-current p-~r to-lr A~-~r ~-pair oJ-pair p-pair + contact p-current Al-current Total

0h9/2

Neutron

0i13/2

~gL

~gs

8gp

%

BgL

~gs

~gP

%

0.302 -0.246 -0.000 -0.000 0.000 0.034 -0.003 0.077 0.036 -0.013

0.725 -0.382 -0.002 -0.052 0.003 0.057 -0.022 0.067 0.049 -0.008

0.301 -1.044 0.029 0.158 0.024 -0.015 -0.042 -0.075 0.031 0.005

44.0 -35.8 - 0.1 0.I - 0.1 5.6 - 0.1 13.7 5.8 - 2.3

-0.168 0.136 -0.000 0.000 0.000

-0.335 0.214 -0.003 0.049 -0.002

0.003 1.160 0.026 -0.195 -0.019

61.5 -53.1 0.0 - 0.6 0.2

-0.005 -0.041 -0.020 0.007

0.002 -0.005 -0.016 0.004

0.038 0.048 -0.032 -0.004

1.3 12.8 6.7 - 2.2

0.187

0.435

-0.627

30.7

-0.090

-0.092

1.025

26.4

I.S. Towner, Quenching of spin matrix elements in nuclei

340

~gL = 0.056 + 0.131 = 0.187 and for an i13/2 neutron it is ~gL = -0.032 - 0.058 --- -0.090, where the first figure represents the contributions from pions, the second from heavy mesons both with vertex form factors included. Arima and Hyuga [74, 105] use the phenomenological Hamada-Johnston potential and the Sachs moment prescription to estimate the heavy-meson contribution. They obtain for an h9/2 proton ~gL = 0.093 + 0.062 = 0.155 and for an i13/2 neutron ~gL = - 0 . 0 5 2 - 0.034 = -0.086. There are no vertex form factors in the pion contributions here. Note that the sum is quite similar irrespective of whether the Hamada-Johnston or the one-boson-exchange potential is used. This, of course, relates to the model independence of the ~gL calculation inasmuch as it is constrained to the nucleon-nucleon potential through the Sachs moment, eq. (4.10). The first experimental indication that the proton gL value is enhanced in nuclei by about 10% over its free-nucleon value came in the measurement of Yamazaki et al. [127] of the magnetic moment of the 11- isomer in 21°po. Nagamiya and Yamazaki [128] later showed the enhancement to be a general phenomenon present throughout the whole mass region. A systematic analysis [129] of all the magnetic moment data in the Pb region produces as best-fit values proton

ggL = 0.15 -+ 0.02

neutron

~gL = --0.03 --+0.02.

Our results in table 18 are in reasonable accord with this expectation. Note that first-order corepolarisation, section 3.1, and isobar currents, section 5, give negligible contribution to ~gL, so meson-exchange currents are the only significant Source of enhancement to ~gL. Arima and Hyuga [74, 105] have estimated the contributions to ~gL of a number of different second-order processes in the Pb region and although any one process might be significant they find interestingly that the resultant sum of them all is rather small. We will not pursue this. Turning to the isovector spin operator, we again find cancellation among the two pion Born graphs, a small contribution from the non-Born graphs of pion range and a significant contribution from heavy mesons. These three ingredients give for a proton in 0h9/2 orbit ~gs = 0.34 - 0.05 + 0.15 = 0.44 and for a neutron in 0i13/2 orbit ~ g s - - - 0 . 1 2 + 0 . 0 4 - 0.01 = - 0 . 0 9 respectively. Compared to the free-nucleon values of gs = 5.586 and -3.826 for protons and neutrons, we see that meson-exchange currents enhance gs by ~gs/gs = 5-10%. This is in contrast to core-polarisation, which as we have seen, gives a quenching to gs. Finally, in table 17 we see the contribution from meson-exchange currents to isoscalar magnetic moments is quite small, a few percent at most. This is because there is no contribution from pion Born graphs. The only pion-range component comes from the non-Born p-Tr graph, which is small, and heavy-meson pair contributions are cut down by short-range correlations. We postpone a detailed comparison with experiment to section 6.

4.5. Axial-vector meson-exchange currents 5

We turn to the construction of the axial-vector meson-exchange current, J r , and in particular to the case when the index/x is space-like as this is the current relevant for allowed Gamow-Teller [3-decay in nuclei. To start with, low-energy theorems are not going to be of any use. These theorems give a model-independent construction of the exchange current of ~ion range for the space part of vector currents, J,, and the time part of axial-vector currents, J , , but can give no model-independent

I.S. Towner, Quenching of spin matrix elements in nuclei

341

statement in other cases [113]. Indeed we will have to resort to the ab initio approach of considering the lowest-order Feynman diagrams of single-meson exchange and, in deriving the nonrelativistic form of the axial-vector current, keep terms to order O(1/M3). Our model-dependent assumptions will be: (a) the use of vector dominance ideas in which the axial-vector current is mediated by the Al-meson in its interaction with mesons and nucleons, and (b) the use of a chiral Lagrangian for the isovector mesons "rr, 9 and A~ taken from Ogievetsky and Zupnik [34] as discussed by Ivanov and Truhlik [33] and more recently by Adam and Truhlik [121]. In this chiral Lagrangian model, the axial-vector current is [33, 34] 2

mo

j

jsj~" = __ gp A~, - f~ O. 7rj + f=gpektjP~ Trt

(4.101)

where A~,, p~, ~rj are the A~-, p- and "rr-meson fields respectively and superscripts are isospin indices. Here gp is the p'mr coupling constant (gp - gp,~,, = 2gom~) and f~, the pion decay constant. The first term represents the mediation of the axial current by the Al-meson, the second term the mediation by the ~-meson and the third is a construct of the Lagrangian involving the two-meson p'rr fields. First we consider the single-nucleon response to the axial-vector current mediated by the A~-meson:

J~, (Pl,' Pl," A t ) = --gA1Nr~ a(p;) Y~, 3'5 u ( p , )

JaA(k 2) mp/g, 2

(4.102)

where aa(k 2) is the A~-meson propagator i AA(k2) = (6~, + k~k~,/m~) k 2 + m zA On using the relations galen = g, ga, where ga is the axial-vector coupling constant, and m a2 = 2m 20 we obtain 5j (Pl,, Pl; A1) l~,

=

i gA(k2) ti(p~) [3'~,3'5 + 3'ak ~k~,3'5/mA] u(p~) -½r

=i gA(k2) a(p~)[3',,3'5 +2iMk~,3'5/m~] u(Pl) lrJ

(4.103)

where gA(k2) = gAmz/(k2+ m2). The second term here, which arises from the second term in the Al-propagator, has been simplified using the Dirac equation, 3"~p~ u ( p ) = i M u ( p ) , assuming the spinors represent on-mass-shell nucleons. Here k = P'I- Pl and M is the nucleon mass. Note that the range of the form factor associated with the axial-vector coupling constant is given in this model by the mass of the Al-meson. Likewise for the pion term in the axial-vector current, the single-nucleon matrix element is

J~,

, Pa; ~r) = g,,N~ a(p~) 3'5 u(p,) "d A ( k 2) i f~k,

(4.104)

where A~,(k2) is the xr-meson propagator: i A,,(k2) = 1/(k2 + m2). We are using pseudoscalar wNN coupling. Rearranging this expression slightly we get 5j

-- p

J~, (/-'1, Pl; "rr)= i gp(k 2) a(p~)[-ik~, ys] u(Pl) 1C

(4.105)

342

I.S. Towner, Quenching of spin matrix elements in nuclei

where gp(k 2) is the pseudoscalar coupling constant: gp(k 2) = 2f.~g~NN/(k 2 + m2). Equations (4.103) and (4.105) together give the single-nucleon axial-vector current. This current satisfies the partially conserved axial current (PCAC) hypothesis. Consider the divergence of the current in momentum space, then mA

k, JSui(Pl, P l ; A , ) = i M g A k2+mA 1 +

_

,

i u(p,)z, u(p,)r j

= i f~ g~NN i ti(p;) TS u(p,) ~'J

(4.106)

using the Goldberger-Trieman relation: m g A =frrg~rNN" Similarly k2

k, J~(PI' Pl; rr) -

k2 + m 2 i f~ g~NN i t~(p;) Ys u(p,) ~'J

(4.107)

and on adding these two pieces together we get the required result [130] 2

k, J~(A I + 7r)=i f~ k2 m~ + m 2 M J(Pl,' Pl ; "rr)

(4.108)

where M j is the single-nucleon pion absorption amplitude in pseudoscalar coupling: MJ(pl, p l ; r r ) = i g~YN u(Pl)3'5 u(pl)r( Thus the divergence of the axial-vector current is proportional to the pion 2 0, the current is exactly absorption amplitude, the PCAC result. Note that in the chiral limit, m~,~ conserved. We will be mainly interested in the low-energy limit, k Z ~ 0, in which case the second term in eq. (4.103) from the A~-propagator and the pionic term will not contribute and the continuity equation becomes simply k , J ~ ( A 1 ) = i f~, MJ(~r). This will be the strategy for constructing the two-nucleon axial-vector current. We consider just the A 1- and pTr-dominated currents and show in the k 2 ~ 0 limit that they satisfy this simpler PCAC equation. This low-energy limit is perfectly adequate for nuclear beta decay, but would not be a good approximation for muon capture. The nonrelativistic reduction of the single-nucleon axial current leads to the familiar one-body operators for the axial charge and current:

jsq,_, ~r.P rj + O(1/M3) 0 ~/)l, Pl,.k2___~0) = _½ gA ---M---(4.109)

JSJ(pl, p,; k2-'*O): --½gA~rZJ + O(1/M 2) where 2P = Pl' + 1°1• For nuclear-structure calculations it is necessary to make a Fourier transformation to coordinate space and project out the required multipoles. For the two-nucleon axial-vector current we again follow the S-matrix method and consider all Feynman diagrams of one-pion range. These are drawn in fig. 17. The first row of the figure illustrates the Born graphs involving just pions and nucleons in intermediate states (apart from the Acmeson mediating the axial current). The second row shows the non-Born graphs involving composite intermediate states such as heavy mesons, e.g. p, or the isobars, A. Note that graphs (b), (d) and (g) in fig. 17 show the axial current being mediated by the pion. As mentioned earlier these graphs in the low-energy limit, k 2 ~ O, do not contribute to the axial current. However it is these same graphs (with

343

I.S. Towner, Quenching of spin matrix elements in nuclei

x

(b)

Ca)

(e)

(c]

(f)

(d)

(g)

(h)

Fig. 17. Two-nucleon meson-exchange graphs of pion range for the axial-vector current: (a)~r-pair graph for Al-dominated part of axia! current, (b) ~r-pair graph for ~-dominated part of axial current, (c) Al~r-contact graph, (d) ~r~r-contact graph, (e) p~-current graph, (f) plrA~-current graph, (g) p~r~r-current graph, and (h) isobar-current graph.

the wiggly line representing the axial current removed) that represent the two-nucleon pion absorption amplitudes. Indeed the pion-mediated axial current and pion absorption amplitudes are trivially related: k. Mj(~) . J~(Tr)---i f~ k2 + m2~

(4.110)

In what follows we will write down from Feynman rules expressions for the axial current and pion absorption amplitudes corresponding to the graphs in fig. 17 and show they satisfy the PCAC relation. This is straightforward once the Lagrangian for each vertex is specified. We start with the Born graphs in pseudoscalar coupling 2

J~(17a;A1)

=

g~NN 2M q2O"2°q + m2 gg ff(P'l)[(Y. 3'5 SF(QI) 3'5 + 3'5 SF(Q2) 3'. 3'5) ~T 1 2 --(3'.

3'5

SF(Q1) 3'5 -- 3'5 SF(Q2) 3'. 3'5) ½(rl x ,r2)j] u(p~) + (1 ~ 2 ) (4.111)

2

i f~ M/(17b; ~r) : i g~rNl~ q gA a(P'l)[(3'5 SF(Q1) 3'5 + 3'5 SF(Q2) 3'5) ~'T 2M q20"2° + m2 1 2J -- (3'5 S F ( Q 1 ) 3'5 - 3'5

Sv(Qz) 3'5) ½ ('1"1 X

~'2) j] U ( p l ) "[- (1 ~ 2 )

where we have written the pion-nucleon interaction at nucleon 2 in its nonrelativistic form: g~Nr~°'2" q~ (2M). Here i SF(Q) = 1/(iQ. 3'. + M) is the nucleon propagator and Q1 = P~ - k, Q2 = P l + k. For fig. 17(c) the AtcrNN-contact Lagrangian is specified in the phenomenological Lagrangian [33, 34, 121] as g0

i j k

LegionN = 1 i ~ (1 -- 2g 2) ~l 7. N eijk z ~r'A. 1

gp Kp

4 L 2M

i

N

j k

j

+ 3.

k

j

-

(4.112)

344

I.S. Towner, Quenching of spin matrix elements in nuclei

where A k~ = (O,Ak - OAk,) and N', 7rj," A k~, are nucleon, pion and Al-meson fields and Kp the tensor coupling constant in the pNN Lagrangian. The term, -2g~, is Adler's PCAC constraint term and arises naturally in the phenomenological Lagrangian on making a chiral rotation from pseudovector to pseudoscalar ~rNN coupling as discussed by Ivanov and Truhlik [33]. We will consider just the PCAC constraint term, deferring the remainder to when we discuss the non-Born terms. Thus for diagram 17(c) we have g'#NN 0"2 " q J~J(17c, PCAC term) = - i (2M) 2 ~-- -2 gA u(Pl) Yu u(p~)i (r~ × r~) j + (1 ~ 2 ) . q +m~

(4.113)

Lastly for the Born terms we need the contact av~rNN Lagrangian 2

~/~-~NN -- g-~NN 2M 1~

,3

(2M)2 N Yu N eijk ./7r j 0 7rk

N 7r i 7r j ~ij + i g=NN

(4.114)

which comprises two terms. The corresponding two-nucleon pion absorption amplitudes are i f~ MJ(17d, first t e r m ) -

g~,YN q gA t~(Pl) u(p,) Z~ + ( 1 2 2 ) 2M q2~r2" + m2 (4.115)

i f~ MJ(17d, second term) -

g,NN q2(72" q gA ti(p~) i TA kA u ( P i ) i ('r 1 × '1"2)j + (1 ~--2). (2M)2 + m2

These Born terms satisfy the PCAC relation. Consider eq. (4.111) for fig. 17(a). On multiplying by k # and using the identities t~(p;) k, y, Z~ SF(Q1)= ff(Pl) 7s [-1 + 2Mi SF(Q,) l (4.116)

SF(Q2 ) k 7~ 7~ u(pl)= [-1 +2Mi SF(Q2)]Zsu(p,) which follow from the use of the free-nucleon Dirac equation, then

k. J~5j (17a; A~) = i f~, MJ(17b; ~r) + i f~ MJ(17d, first

term).

(4.117)

Similarly for the PCAC constraint term k~, J~(17c, PCAC term) = i f, MJ(17d, second term)

(4.118)

thus proving that the Born-term two-nucleon axial-vector current is related to the Born-term twonucleon pion absorption amplitude k J~(17a + 17c, PCAC term) = i f~ MJ(17b + 17d). Next for the non-Born terms we have

(4.119)

I.S. Towner, Quenching of spin matrix elements in nuclei

2

345

[

I<,

]

J~(17c, remaining terms) = g,,NN O'2" q 1 i (~'~ X ~'2)j t/(p~) i -i u(p 0 (2M)2 q ~ +m,~ - " - 2 2g A y~, ~-~ o - p~ jsi(17e) _ g~NN O"2 " q 1 mo Kp (2M) z 2 q2 + m2 2g A i(~'~ x ~2)Jp2 + 2m____~t/(p~) p [ i y. -- ~ i tr.. p. ] U(pl) 2

J~(17f) -

g~rl~S "q 1 • 1 O"2 i(~'~ x ~'2)J p2 2 (2M)2 q2 + m.2 2g g + mp

xti(p~)[-ip~(y~-~--~Gt~KP

PtJ)(G . lk~)+(p . . .q

½p . k)(iy~

K0

~'-Mitr p~)]u(Pl)

2

i f~MJ(17g) - g,,NN OZ" q 1 (2M)2 q2 + m2 2g---~i0"1 x ~'2)' p- 2-+mp2 ti(p~) ikuy~ - ~

i G, k~, p~ u(Pl) (4.120)

where for fig. 17(f) the Alp~r-Lagrangian is given in eq. (4.38) and for fig. 17(g) the p~'rr-Lagrangian is given in eq. (4.21). After some rearranging it is seen that k~,jsJ(17c; remaining terms + 17e + 17f) = i f. M~(17g)

(4.121)

showing that this group of non-Born terms satisfy the required PCAC relation. We defer to the next section, a discussion of the isobar graph, fig. 17(h), but it is clear that it too must satisfy its own PCAC relation. Thus in this phenomenological Lagrangian model, the two-nucleon axial-vector current is given by graphs (a), (c), (e) and (f) in fig. 17. The next step is to write down a nonrelativistic approximation to these currents evaluated to order 1/M 3. As was the case of the two-nucleon vector current, the pair graph, fig. 17(a), needs special consideration. The nucleon propagator SF(Q) is written as a sum of positive frequency and negative frequency pieces, eq, (4.13), and the positive frequency piece in the nonrelativistic reduction of graph 17(a) becomes (1 / k 0) IV,,, js~(1)]. That is, the contribution is singular in the limit k 0 ~ 0 and is simply a product of the pion-exchange potential, V,~, and the one-nucleon axial-vector current, JSJ(1). This term is already part of the impulse approximation and should not be included in the exchange current. For the negative frequency part of the propagator we get 2

jsj(17a, pair)_

(2M) 3g~NN q2O"2"q+ mE gA [z~(k+2i °'1 × P ) + i 0 " l X ~'2)' i o"1 X q]+ ( 1 ~ 2 )

(4.122)

for the axial-vector exchange current. Likewise 2

jsJ(17c ) = g,,NN (2M13 2

O"2 "q 1 - f - -2 2g A i(~'l x ~rz)' [(1 - 2g 2 ) 2e + (1 - 2g~ + K.)i O"1x p] + (1 ~.~--21 q +m~ 2

jsj(17e ) _ g,,NN Oz "q 1 i (~'1 X ~'2)'• 2"--"-'--5 m o [2P+(l+K,)io.lxp]+(1,~__2) ' (2M)3 q2 + m2 2g A p + m.

I.S. Towner, Quenching of spin matrix elements in nuclei

346 2

jsj(17f ) _ g-~NN 0"2 • q -- 1 (2M)3 q 2"---2 2gA + m~,

i('r,

x ,r2) j - 2-

1

2 [-½p(2P'k+ p +mp

+ ( 2 P + (1 + K ) i o'~ x p ) ( ½ p . k - p 2 ) ]

+ (1~2)

(1 + K0)i ~r~ x p . k - 2 M k o ) (4.123)

where p = Pl - Pl, q = P£ - 1o2 and k = p + q. Again we want to calculate the expectation value of this two-nucleon axial-vector current in a finite-nucleus many-body system. As in eq. (4.91), we Fourier transform to coordinate space and use the replacement rule: P ~ - i ~ and Q ~ - i ~ to handle the nonlocal operators. For nuclear [3-decay in the allowed approximation only the lowest multipole (L =0, J = 1) is required and the multipole projection becomes a trivial step. The two-body Gamow-Teller operator is just the two-nucleon axial-vector current evaluated in the low-energy limit: AJ(2-body) =JSJ(k2---~0)

(4.124)

where jsj is the sum of the four graphs 17(a), 17(c), 17(e) and 17(f). Note that in this limit, the local operator proportional to i o"1 x q in the pion pair graph is exactly cancelled by the PCAC constraint term from fig. 17(c). Furthermore the remaining local terms in fig. 17(c) when added to terms from figs. 17(e) and 17(f) produce a resultant operator that in the low-energy limit is just twice the contribution of the p - v graph, 17(e), alone. Indeed this derivation from a phenomenological Lagrangian model leads to a two-body Gamow-Teller operator that agrees with the one given earlier by Chemtob and Rho [106] from just the p-~r coupling taken from the KSFR relation [36]. This intriguing result has been noted and commented on by Ivanov and Truhlik [33]. To estimate the size of MEC corrections we will calculate the expectation value of the one-body and two-body Gamow-Teller operator in closed-shell-plus-one configurations. For the two-body operator we will be calculating matrix elements between the valence nucleon and one of the nucleons in the closed-shell core and summing over all nucleons in the core. It is useful to express the result of this computation in terms of an effective one-body operator, eq. (3.33) A~ff = - ½{gLa.effL + gA,effO"+ gr,Axff[ Y2, O']}~"j

(4.125)

where gA,e, = gA + 8gA etc. Recall that the corresponding operator from impulse approximation is A J(1) = - ½ga ¢~'j

(4. 126)

with gA = 1.26. We give some sample results in table 19 for a 0p-hole and 0d-particle at an 160 closed-shell core representing the ground state to ground state mirror Gamow-Teller transition in A = 15 and A = 17 respectively. Harmonic oscillator wavefunctions, hw = 13.3MeV, are used, a short-range correlation function, g(r), introduced as in eq. (4.100) and vertex form factors included at all meson-nucleon vertices. The results show that MEC corrections are typically at a few per cent level for the A = 16 closed shells, much less than the 12% discrepancy between the experimentally determined Gamow-Teller matrix element and the single-particle estimate. We postpone a detailed comparison with experiment until section 6. So far we have only considered MEC corrections to the axial-vector current that are of pion range. It is quite straightforward to construct terms of heavy-meson range coming mainly from pair graphs. We

I.S. Towner, Quenching of spin matrix elements in nuclei

347

Table 19 Meson-exchange current correction to Gamow-Teller matrix element between mirror ground states of closed-shell-plus (or minus)-one configuration at mass, A = 16, expressed as a percentage of the single-particle value 0p~

0dsj 2

~gLA

~gA

~gv^

%

Al~r-contact p~r-current plrAl-current It-pair

0.000 0.000 0.000 -0.001

0.001 -0.005 -0.005 0.002

-0.049 -0.023 0.012 -0.003

-3.0 - 1.9 0.4 0.2

0.000 0.000 0.000 -0.001

~gLA

0.001 -0.005 -0.002 0.002

~gA

-0.047 -0.025 0.013 -0.005

~gPA

-0.4 -0.6 -0.1 -0.1

%

Sum

-0.001

-0.008

-0.063

-4.3

-0.002

-0.004

-0.065

- 1.2

have computed the leading terms from these graphs and found their contribution to the Gamow-Teller matrix element at mass A = 16 to be very small, typically a few tenths of a percent of the single-particle matrix element. Thus we will not consider heavy-meson pair graphs any further.

5. Isobar currents

In considering various ways in which the presence of isobars in nuclei might be experimentally verified, Lipkin and Lee [133] begin their discussion with the following cautionary remarks: 'There has been considerable debate of the possibility that isobars may be present in nuclei. However, it is not clear exactly what this means. Ths isobar is a highly unstable particle with a natural width much larger than the spacing between nuclear levels. It is, therefore, unclear whether an isobar present in a nucleus can be considered as an elementary fermion (or skyrmion) in some approximation, as a three-quark composite, or as a pion-nucleon resonance. Furthermore, an isobar present in a normal nuclear ground state is very far off shell, and it is not clear how to extrapolate the properties of such a complicated object far off shell. All this confuses the issue and makes it difficult to test experimentally for the presence of such isobars.' These remarks emphasize a little appreciated point: there is considerable uncertainty attached to isobar calculations. In this section we will treat the isobar as an elementary spin-3/2 fermion and consider two approaches. In the first we use the Rarita-Schwinger [132] equations to describe the isobar and write down the lowest order Feynman diagrams in which an isobar is excited in the exchange of a meson between two nucleons and de-excited by the vector or axial-vector current probe as pictured in fig. 17(h). The calculation then parallels that of the meson-exchange currents described in section 4. The required input are the relativistic Lagrangians for the meson-isobar-nucleon vertices and a knowledge of the coupling constants. The leading terms for this method match the alternative approach in which the isobar in the nucleus is considered as a nonrelativistic object and shell-model states are constructed involving the isobar in much the same way as shell-model states are constructed with protons and neutrons. The uncertainties revolve around the neglect of the isobar's natural width, the relativistic propagator of the isobar being far off its mass shell, and the coupling constants not being known with any precision. There has been a great surge of interest in isobars in the past decade as the large matrix element between a nucleon and an isobar in the spin-isospin channel is believed to be one of the principal reasons why cross-sections for M1 and Gamow-Teller transitions are systematically quenched relative to the lowest-order shell-model expectations. Some of the first discussions of isobars in nuclei are to be

348

I.S. Towner, Quenching of spin matrix elements in nuclei

found in the work of Ericson [133], Rho [134] and Brown and Weise [40], and pursued by many authors [43-46, 60-68, 87, 88] since. For a recent survey of the status of this work we refer to the contributions of Gaarde, Marty, Madey, Krewald and Osterfeld in the 1986 Heidelberg conference [135]. 5.1. Isobars as an M E C correction Since the isobar has isospin 3/2 it has to be excited from a nucleon by an isovector meson. This limits the possible meson-exchanges to v-, p- and Al-exchange. We start by writing down the relativistic Lagrangian for the meson-isobar-nucleon from the hard-pion model of Ivanov and Truhlik [33]: ~NA-

g~Na 2M ~

Tj N 0~ 7r j + h.c.

~A1NA = gA1NA / ~ T J N A ~ +h.c.

T j N ( O p~

GI XpNa=--go~YsT, •

(5.1) / Gz -- O~P+,)+½go~IV Ts(~-b)T/u(O p~-O p~)+h.c.

j

where N~, N, rr j, A~and p~ are respectively isobar, nucleon, pion, Al-meson and p-meson field operators with superscripts being isospin indices. Here T / is a transition isospin operator [55] connecting an isospin-1/2 nucleon with an isospin-3/2 isobar, as defined in eq. (2.72). We introduce coupling constants g~NA, gA1Na = (g,,Na/g~NN) gAINN, G~ and G 2 for the isobar coupling and recall that gA1NN is defined as gpgA with gA the axial-vector couplings constant and g o - g o ~ = 2goNN" The unknowns are the ratio, g~Na/g, NN, G 1 and G 2. In the simple quark model, g~NA/g~NN = 6V~/5 and Gz/G 1 = - M / m a where M is the nucleon mass and mA the isobar mass. Finally we fix G~ by requiring that the isobar current as calculated from these relativistic Lagrangians agree in the nonrelativistic limit with that obtained from the alternative method of constructing isobar-nucleon shell-model states as we will show. This fixes G 1 to be 4G 1 = --(g~NA/g~rNy)(1 + Kv), where K, is either Kp, the ratio of tensor to vector coupling in the p-meson Lagrangian, or F2(O)/F~(O ), the ratio of tensor to vector coupling in the photon Lagrangian. With strict vector dominance these two values would be the same. However with strong tensor coupling in the pNN Lagrangian, such as Kp = 6.6 in the analysis of Hohler and Pietarinen [53], there is a dilemma in deciding which is the appropriate value to assign to GI. Rather than use the quark model, or strict SU(2), one can appeal to experiment. For example, the Chew-Low theory [58] predicts ~-nucleon scattering will proceed principally through the P33-channel. Using, then, an isobar model to describe "rr-nucleon scattering, a value of g~rNA/gTrNN = 2 will yield the same scattering phase shifts as the static Chew-Low theory. Alternatively fitting the resonance width in the isobar model [19] determines g~NA/g~N N ~--2.15, which is some 25% larger than the quark-model value of 6X/2/5. A more recent suggestion [136, 137] is to examine the electric quadrupole to magnetic dipole transition amplitudes (EMR) in the process A---~N + ~/. With the assumption of vector dominance that the photon couples to the N and A through the p-meson, the EMR becomes sensitive to the ratio G2/G ~ in the pNA-Lagrangian. Indeed with the quark model of G z / G 1 = - M / m a , the EMR is predicted to be zero. Davidson et al, [136] determine G~ and G 2 from fitting the available M1 and E1 multipole data on the photoproduction of pions from nucleons in the regime of the P33 resonance. They obtain + G I = -2.4 -+ 0.05, G 2 = 1.43 -+ 0.05 and hence derive EMR = (-1.5 -+ 0.2)%. ~In the notation of Davidson et al. [136], G 1 = -(1/2)g~,

G 2 =

(1/4)g 2.

349

I.S. Towner, Quenching of spin matrix elements in nuclei

Starting with the hard-pion Lagrangians, eq. (5.1), and the assumption that the vector current couples to a nucleon through the p-meson and the axial-vector current through the Al-meson it is a straightforward application of Feynman rules to write down the expression for the two-nucleon MEC current corresponding to fig. 17(h). First for the vector current of pion range we obtain

J~, =

-i

Fl(k 2) G1 g~NA SA +j k M 2M t~(p;)[7,(k.y~, - k,y,6.~,) 75 ]/4 F,~(Q]) qt3 T1 T, + qt3 SF,a.(Q2)

X 3,5(k~7~,--k,y,6.~) r~k r{]u(p,)Zl~(q2) g=NNa(P2)nu(p2)rk + ( l ~ 2 )

(5.2)

where the second term in the pNA-Lagrangian has been dropped, since in the nonrelativistic reduction it gives a contribution 1/M 2 smaller than the leading term. In eq. (5.2), A~(q2) is the Non propagator and S aF,~ (Q) the isobar propagator:

i[

,

]

S~'~t~(Q)- iQ'+ ma 6~ - 1y~Tt3 3Q2 (~Y~Qo + Q~3't~) •

(5.3)

This particular form of the propagator is advocated by Williams [138] as being valid both off and on the mass shell. In our application the isobar is far off the mass shell. We have Q1 = P~ - k and Q2 = Pl + k, such that in the static limit, Q--->0, the fourth component becomes Qa--->iM, the nucleon mass rather than the isobar mass. We will use for the propagator SaF..~(Q) =

2 -- i- 2 ( - i T ' Q + M y 4 + m a ) ( 6 , , - ½ Y , 7 , + ~ .~. M 21 ( ~ 7 , Q t 3 + Q . 7 , ~ ) ma - M

) •

(5.4)

Similar expressions to eq. (5.2) can be written down for the vector current of p-meson and Al-meson range. In the nonrelativistic reduction, we keep terms of order 1/M 3 for "rr- and p-exchange and order 1/M for A,-exchange to obtain for the vector currents

JJ(~r)-

1 0"2°q 4 J g~Nag~r~NF~(k2) (2M) 2 2M 4 G 1 -m- a - M q2 + m2 [ u r 2 i q x k + }i (7"1 X ,r2) j (k" q 0"1

-- 0"," k q ) ]

+(1~2)

g2pNN F~(k 2) 1 1 JJ(p) = ~ (1 + K.) ~ (4G1)2 m - a- _ M qE +m2o [ 4 r ~ ( i 0 " 2 × q k ' q - i 0 " 2 x q ' k q ) +1i (rx × 72) j (20"1 x q.k0" 2 x q - 2 0 " 1 × q 0"2 x q . k - 0",'0"2 x q q x k

+q'k0-1"q0-2-q0-1"q0-2"k-2q z 0-1 "k 0-2 + 2q 2 0-, 0-2"k-2k'q0-1 0-2"q "~'2q O"1" k o"2" q)] + (1 ~ 2 )

F~(k 2) 4G 1 1 1 2 [ 4 r2i j 2 × k JJ(A1) = --gA1Na gA1NN 2M m a - M q2 + mA + 9'- i

x

j (0-i

k - 0-,. k

+ (1=2)

(5.5)

350

I.S.

Towner, Quenching o f

spin

matrix elements in nuclei

where 6"2 = ~2 + (or:. q)q/m 2. Note that in each case these currents are transverse, viz. k. JJ -- 0. Thus these isobar meson-exchange currents satisfy the requirements of the equation of continuity. In deriving JJ(1) we have neglected the contribution from the time part of INN- and iNA-Lagrangians and dropped the terms involving the nonlocal operators P and Q. Following the discussion in section 4.4, we will project out from these currents the two-nucleon magnetic moment operator and calculate its expectation value in closed-shell-plus-one configurations. Again it will be useful to express the results in terms of an equivalent one-body operator eq. (3.32):

(5.6)

~[Leff = gL,effL + g s , e f f S + g p , e f f [ Y 2 , S ] .

We give some sample results in table 20 for a 0p-hole and 0d-particle at an 160 closed-shell core (h~o = 13.3MeV) and for a proton 0hp/2-particle and a neutron 0il3/2-particle relative to a 2°spb closed-shell core (hoJ = 7 MeV). We use harmonic oscillator wavefunctions, introduce a short-range correlation function, eq. (4.100), and include vertex form factors at all meson-nucleon vertices. We choose the Chew-Low value of 2 for scaling all the meson-nucleon-isobar vertex coupling constants, e.g. g~Na/g~NN,and adopt the following ansatz for choosing K v in the expression for GI: 4G 1 = --(g~Na/g~NN) (1 + Kv). If the i-meson is coupled to a photon then K V= 3.7, but if it is coupled to another nucleon, as in the P potential, then K v = 6.6 is selected. This ansatz leads to the identical results as those to be obtained in the alternative approach, section 5.2. The results in table 20 indicate the isobar corrections to magnetic moments are rather small and not necessarily quenching. This can be understood in terms of the effective coupling constants, 5g s and 5gp. Consider the case of a nucleon in a j = l + i orbit. In the diagonal matrix element, such as the magnetic moment, these two components of the effective one-body operator come together in the combination: ~gs = 5gs + (2l/(21 + 3)) (87r)-l/2~gp. Since for isobar currents 5g s is found to be of opposite sign to Bgp Table 20 Correction to the magnetic moment, 8p., and the M1 spin-flip j = l + 1 / 2 - - > j = l - 1/2 matrix element, 8(M1), of a closed-shell-plus (or minus)-one configuration as a percentage of the single-particle Schmidt value coming from isobar currents for a 0p-hole and 0d-particle relative to t60 closed shell and for a 0h-proton and 0J-neutron relative to 2°spb closed shell

Op~,J2

~-isobar p-isobar Al-isobar Sum

Od~2 5(M1) {M1)

og s

ogp

-

-0.110 -0.072 0.011

0.879 -0.080 0.006

21.9 5.0 0.6

-4.7 - 1.5 0.3

-

-0.172

0.805

17.4

-6.0

9.

8g s

Bgp

5# /~

-

-0.118 0.076 0.011

1.031 -0.105 0.008

0.0 - 1.3 0.2

-5.3 1.5 0.2

-

-0.184

0.934

-1.2

-6.6

5gL

Proton 0hp/2

v-isobar p-isobar A¢isobar Sum

5(M1) (M1)

Neutron 0i13/2

Bgc

8g s

8gp

~9. %

B(M1) (M1) %

-0.003 0.001 -0.000

-0.323 -0.151 0.022

0.513 -0.032 0.002

2.3 2.6 -0.4

-0.003

-0.452

0.484

4.5

89. -#- %

-8 ( -M I%) (MI)

~gc

~gs

8gp

- 8.1 - 3.2 0.5

0.003 -0.001 0.000

0.286 0.135 -0.020

-0.752 0.077 -0.006

-5.2 -3.7 0.5

- 9.4 - 3.3 0.5

- 10.8

0.002

0.402

-0.680

-8.4

- 12.2

351

I.S. Towner, Quenching of spin matrix elementsin nuclei

there is a cancellation between the central and tensor pieces with the result that the corrections to magnetic moments are rather small. Indeed depending on the interplay between the two terms one can even have enhancements to the magnetic moment from isobar currents as seen in table 20. By contrast, M1 spin-flip matrix elements between j = l + i and j = l - 1 states bring the two components together in the combination: 8gs = 8gs - 1 (87r)-l/2~gp. Thus for off-diagonal matrix elements isobar currents are clearly inducing quenching. Since most of the experimental evidence for quenching in M1 transitions involves spin-flip matrix elements, there is no inconsistency with large quenching in M1 transitions and small quenching in magnetic moments. It simply signals the important role played by the induced tensor term ~gp. Thus in the Pb region the spin-flip M1 matrix element is quenching = 11% by isobar currents (the B(M1) transition rate quenched b y - 2 2 % ) . Here we are using Chew-Low values for the isobar couplings. If instead we were to use the quark model value of g~,r~a/g,,r~ = 1.7 instead of 2, all the entries in table 20 would be scaled by 0.72. Then the B(M1) transition rate in the Pb region would be quenched by - 16%. Lastly in this section we consider the axial-vector isobar current. Again we use vector dominance and assume the axial-vector current couples to a nucleon through the Al-meson. The two-nucleon MEC current is then given by an equation analogous to eq. (5.2) except the A~NA-Lagrangian replaces the pNA-Lagrangian. This current also satisfies the two-nucleon PCAC relation which requires the divergence of the current to be proportional to the two-nucleon pion-absorption amplitude. This follows because on multiplying the A1NA-Lagrangian by k~, in the expression for the axial-vector current J~J the Lagrangian essentially converts into the form of the rrNA-Lagrangian. Thus it is trivial to show that the axial-vector isobar current satisfies

5j.

k~ J . (Isobar) = i f~ M~(isobar)

(5.7)

where M~, is the two-nucleon pion absorption amplitude with isobar intermediate states. We will just write down the form of the axial-vector isobar current following a nonrelativistic reduction keeping terms to order 1 / M 2 for ~r- and p-exchange and to order 1 / M ° for Ax-exchange:

\g---~NN/ (2M) 2 g A ( k z ) m - a-

1

0"2"q [4~.~q+ li (~.1 x ~.2)j i o-1 x q] + (1 = 2 ) M q2 + m2

2

1 jsj(p) : _(g.~Na 'g~,NN ]/ 4G 1 ~gpNN (1 + Kp) gA(k 2) -m a- -

1

M q2 + m2o

(5.8) 4 j x[~z2(o2.

JSJ(A1)

=

q q - q 2 o - 2 ) + 1 i ( % x 72)'(io- 1 • qo-2 x q - i q o - 1 .o- 2 X q ) ] + ( l ~ 2 ) ,

g~Na 2 2 1 1 --\g---~NN/ gAINN gA(k2) rn a - M q2 + mA2

[4~'l'20"2 j"{-

~i(,rl X 'T2) /

i O" 1

x6.2]+(1~2)

.

Again in deriving the expression for the p-meson isobar current we have neglected the contribution from the time part of the pNN- or pNA-Lagrangian and dropped the terms involving the non-local operators P and Q. To see how important isobar currents are in nuclear beta decay, we will calculate the expectation value of this two-nucleon axial-vector isobar current in a finite-nucleus many-body system. We will take the long-wavelength approximation, eq. (4.124), and express the results of a computation in a

I.S. Towner, Quenching of spin matrix elements in nuclei

352

closed-shell-plus-one configuration in terms of an effective one-body operator, eq. (3.33) A~ff = 1

(5.9)

{ g L A , e e f L + g A , e f f O r + g P A , e f f [ ] 1 2 ' Or]} ,r j .

We give some sample results in table 21 for a 0p-hole and 0d-particle at an 160 closed-shell core representing the ground state to ground state mirror transition in A - - 1 5 and A = 17 respectively. Harmonic oscillator wavefunctions, hw = 13.3 MeV, are used, a short-range correlation function introduced as in eq. (4.100) and vertex form factors included at all meson-nucleon vertices. The results show a similar story to that for magnetic moments. The ~gA and ~gPA are found to be of opposite sign such that they give interfering contributions in diagonal matrix elements and constructive contributions in spin-flip matrix elements. This similarity is not accidental. Indeed forming the magnetic moment operator: M j = -½iV~ × JJ from the expressions eq. (5.5) in the low-energy limit k ~ 0 yield, to within a constant, the same expressions as the axial-vector isobar current in eq. (5.8). Thus the results for the effective one-body operator for isobar currents, eq. (5.9), can be immediately obtained from that for magnetic moments by the replacement:

(5.1o)

~gA/Sg~1) =~gpA/~g~1)= gA/g~'> where gA = 1.26 and g~l) = 4.706, the free-nucleon coupling constants.

5.2. Isobars as nuclear constituents An alternative nonrelativistic description of the isobar treats it as a nuclear constituent in just the same way as protons and neutrons are elementary nuclear constitutents interacting through two-body forces. The isobar is a spin 3/2, isospin 3/2 baryon described as a bound state in a harmonic oscillator potential of the same characteristic frequency as that used for nucleons. Shell-model states are constructed but there is no antisymmetry requirement between isobars and nucleons. Two-body interactions among the constituents are described by the one-boson-exchange model in which the potential between nucleons and isobars is obtained from that between nucleons by substituting g-~NN~ g~,Na, O"~ S, ~"~ T at the appropriate vertices [40], where S and T are generalizations of the Pauli spin and isospin matrices acting as transition operators between spin 1/2 and 3/2 spinors, as discussed in section 2.4. We will estimate the degree of isobar admixtures in nuclear ground-state wavefunctions using Table 21 Correction to the ground state Gamow-Teller matrix element of a closed-shell-plus (or minus)-one configuration as a percentage of the single-particle value coming from isobar currents for a 0p-hole and 0d-particle relative to 160 closed shell

Op~,'~ ~gLA ~r-isobar p-isobar A 1-isobar Sum

Od~ ~(GT) (GT) %

~gLA

0.235 -0.021 0.002

12.6 - 2.9 0.3

0.215

10.0

~gA

~gr,a

-0.029 -0.019 0.003 -0.046

~{GT) (GT/ %

~gA

~gPA

-

-0.032 -0.020 0.003

0.276 -0.028 0.002

-0.0 - 1.9 (1.3

-

-0.049

0.250

- 1.6

l.S. Towner, Quenching of spin matrix elements in nuclei

353

perturbation theory. For example consider a closed-shell-plus-one nucleus, where the dominant piece of the ground-state wavefunction is the single-particle configuration. There will be a small piece of nucleon-isobar-hole configuration, but the admixture will be small. Nonetheless it can have a significant impact on the expectation value of a spin matrix element, such as the magnetic moment, because the admixture amplitude comes in linearly (rather than quadratically as in the ground state eigenenergy) and the coupling matrix element between a nucleon and an isobar is large. The calculation parallels that given in section 3 where the impact of two-particle one-hole states on ground state properties is estimated in perturbation theory. In fig. 18 we show the two first-order graphs in which isobars are introduced as intermediate states. They are analogous to figs. 6(c) and 6(f). The first-order correction to the reduced matrix element from fig. 18(a) in an angular-momentum coupled representation (cf. eq. (3.5)) is (bllfig. 18(a)lla) = ~ (_)2,/~-1 h,A

(h-lA)A[Vl(b-la)A) (Ea + eh -- e a -- eu) + ( M - ma)

(5.11)

where h is the tensorial rank of the one-body operator F (A) (h = 1 for the magnetic moment operator), and f = (2i + 1) 1/2 Note the presence of the additional factor m a - M , the isobar-nucleon mass difference, in the energy denominator. The Hermitian diagram, fig. 18(b), is given by a similar expression, cf. eq. (3.6). Here in the one-body operator, F (~), connecting nucleons and isobars the coupling constant is taken from the nucleons-only operator but multiplied by g~N,,/g~NNand the spin and isospin matrices replaced by transition spins, viz. or ~ S, ~ ' ~ T - the identical replacements used in constructing the transition potential. From eq. (5.11) and its Hermitian equation it is straightforward to calculate the correction to magnetic moments and M1 matrix elements in closed-shell-plus-one nuclei coming from isobar currents. The results, however, are identical to those given in table 20. This is not surprising. The transition spin formalism was developed, as discussed in section 2.4, so that pion scattering from a nucleon in the isobar model would be the same as that calculated with the relativistic Rarita-Schwinger formalism. This correspondence carries through to the meson-exchange current calculations. In the MEC approach a two-body operator, F2, is deduced whose matrix element in a closed-shell-plus-one configuration amounts to a sum of two-body matrix elements between the valence nucleon and all the nucleons in a closed shell core. Schematically this MEC operator can be factored, F 2 = Fy, where F 1 is a one-body spin-isospin operator and V the meson-exchange potential. Thus the calculation based on eq. (5.11) will give the same result as the MEC approach when F 1 corresponds to F (A) and V in eq. (5.11) is the meson-exchange potential. b' A

__•-x

(a)

h

o

(b)

Fig. 18. First-order perturbation corrections from isobar-hole states to the matrix element of an isovector one-body operator in a dosed-s6ell-plusone nucleus.

I.S. Towner, Quenching of spin matrix elements in nuclei

354

The advantage of the transition spin approach is that it gives a natural framework within which the calculation can be extended. For example, the potential V does not necessarily have to be the meson exchange potential. It may be more appropriate to use an effective interaction in finite nuclei, as discussed in section 2.3, and many of the calculations in the literature adopt this approach. In table 22 we give some sample first-order calculations for isobar-hole states using three different choices of the effective interactions: a zero-range interaction with (g')aN ----0.6, the interaction of Cha and Speth [46] and the one-boson exchange potential. Results similar to OBEP have also been given by Lawson [139] and by Hyuga et al. [105]. The results for the two effective interactions are quite different from that obtained with the one-boson-exchange potential. We will concentrate our discussion on the spin-flip M1 matrix element and define quenching, 8 ( M 1 ) / ( M 1 ) , to be the reduction in the matrix element as a fraction of the single-particle matrix element. We note the following points: (a) The quenching due to isobar currents from a zero-range interaction is between 1.5 and 4 times greater than that calculated with OBEP varying from case to case. Since these are first-order calculations the results for the zero-range interaction scale with the value of (g')aN' Thus the OBEP potential can be said to have an effective (g')AN that varies from 0.15 to 0.40. This variation from nucleus to nucleus is not unexpected. We noted in section 2.5 that (g')AN depended sensitively on the density having a value at normal nuclear matter densities of (g')aN =0.47 (table 8) and anticipated being smaller than this in finite nuclei. This is discussed at some length in the work of Cheon et al. [66] and Arima et al. [67]. (b) Since these are first-order calculations the derived 8g s value listed in table 22 is directly proportional to the strength of the central part of the isobar-hole effective interaction and the 8gp value proportional to the strength of the tensor part. Thus for the zero-range interaction the 8gp value is zero, while for the Cha-Speth interaction [46] and OBEP very similar values are obtained. This reflects the use of similar tensor interactions coming in the first case from "rr + p exchange and in our use of OBEP from ~r + p + A t exchange. With a short-range correlation function included, the role of the Al-meson is very minor. (c) The 8gL values in table 22 come from the spin-orbit potential, which in our OBEP calculations, is carried through to the transition potential. The contribution from this term is negligible.

Table 22 Correction to the magnetic moment, 89., and the M1 spin-flip j - l + 1/2--) j - l - 1/2 matrix element, 8(M1 ), of a closed-shell-plus (or minus)-one configuration as a percentage of the single-particle value from first-order calculations of isobar currents for three choices of effective interaction Op))2

Ods<2

9. Zero-range ~r+p+Sg' OBEP

0.004

1.077 -0.902 -0.172

0.835 0.805

-39.8 - 8.7 16.8

8(MI) (M1) -25.6 -23.4 - 6.1

~,, 8g,

~) 8g s

(I.003

-0.692 -0.669 -0.184

Proton 0hgj 2

Zero-range rr + P + 8g' OBEP

~gL

8g s

-0.003 -0.0(/2

- 1.261 -1.344 -0.452

(,)

0.969 0.935

8p. # - 10.3 - 8.3 - 1.0

8(M1) (M1) 16.5 18.2 6.7

Neutron 0it3,2

8gr

8,u. %

0.499 0.484

19.7 18.3 4.7

8(M1) - % (M1) -27.5 -30.3 -10.9

8gt.

~gs

0.002 0.001

0.744 1.139 0.402

8gv

-0.704 -0.680

89. -- % 9.

8(M1) - % (M1)

- 19.5 -27.6 - 8.0

- 19.5 31.6 - 12.3

I.S. Towner, Quenching of spin matrix elements in nuclei

355

5.3. M1 and GT giant resonances in Pb region

We return to the problem of quenching in the M1 and Gamow-Teller giant resonances in 2°8pb introduced in section 3.3. The effects of meson-exchange currents and isobar currents can now be introduced into the previous results by simply replacing the one-body operator by an effective one, eq. (5.6) or eq. (5.9), where the corrections ~ge, ~gs etc. have been computed as recorded in tables 18 and 22. Thus the wavefunctions used are still given by a nucleons-only RPA calculation but the transition operator connecting the closed-shell and l p - l h configurations has become modified. This procedure is correct to first order in perturbation theory. The results of this calculation are given in table 23. The first line reproduces the RPA results of table 12 and gives the percentage of the sum rule strength in the strongest state for the same three choices of effective interaction used before. The meson-exchange current corrections are seen to have a negligible influence on the GamowTeller resonance, while for the M1 resonance they actually enhance the strength in the strongest state by about 4/z~ in the B(M1; 0 + ~ 1 +). This is understood as coming principally from a modification to the ge value from the Sachs moment. The isobar currents, on the other hand, noticeably reduce the strength in the strongest state but the degree of quenching depends very much on the choice of the isobar-hole interaction. For the two effective interactions, zero-range with g ' = 0.6 and the Cha-Speth [46] interaction, isobar currents reduce the resonance strength by roughly 50%. Typically, a B(M1; 0 ÷ ~ 1+) = 20 tz2 is obtained, still a factor of two more than has been experimentally measured [80], but nevertheless a big improvement. Similarly for the GT resonance, the two effective interactions give roughly a 70% reduction in strength in the strongest state resulting in a B(GT; 0 + ~ 1 +) that is about 50% of 3 ( N - Z ) g 2 sum rule and comparable to what has been experimentally observed [81]. The parameter 8g' in the Cha-Speth interaction is a phenomenological one in that it was adjusted to bring about roughly the correct amount of quenching in a RPA calculation of the GT resonance. It was introduced because the boson-exchange potential, even with short-range correlations included, is not sufficiently strong to shift the GT and M1 giant resonances to the higher excitation energies observed experimentally as commented by Suzuki et al. [64]. The OBEP interaction also gives noticeably less quenching from isobar-hole states to the transition strength as can be seen in table 23. Unlike the two effective interactions, which are constructed for use in RPA calculations alone, the OBEP interaction being a more fundamental one does not have this privilege. It can not be singled out for use in just a certain subset of possible graphs such as the RPA but all second-order and possibly higher-order processes must be considered on an equal footing. For nuclei in the Pb region, a calculation of all second-order processes that contribute to the giant resonance excitation energy and transition strength would be a formidable task. It is likely, however, that the most important graphs beyond the RPA are the second-order core polarisation graphs discussed in section 3.5. Shimizu [103] has estimated the impact of these graphs on such properties as the magnetic moments of single-particle states using a closure approximation. His results are expressed in the form of an equivalent effective M1 operator, eq. (5.6), with the values of ~gL and 8gs listed as an addended note in ref. [74]. We incorporated these values into the effective transition operator used here and hence estimated the contribution from second-order core-polarisation to the quenching of giant resonances. The results are listed in line 4 of table 23. It is seen that the combination of isobar currents and second-order core polarisation reduces the calculated B(M1) strength by roughly a factor of two, much the same as the reduction from isobar currents alone with the other two effective interactions. Thus the interpretation of the parameterised zero-range interaction is clear: The additional ~g' term

356

I.S. Towner, Quenching of spin matrix elements in nuclei

E

.,-.

< d

& o

÷

=

++ ÷ + +

+ + + +

1.S. Towner, Quenching of spin matrix elements in nuclei

357

added to the ~r + p exchange potential in the Cha-Speth interaction is compensating for the other second-order processes not included in the RPA. Cha and Speth [46] give some quantitative estimates of this. We also include in line 5 an estimate of the core-polarisation correction to the meson-exchange current (see section 6.1) taken from the effective M1 operator of Hyuga et al. [105]. The contribution gives a small enhancement to the B(M1) strength in the giant resonance. We must stress that these estimates of second-order effects in the OBEP model in the Pb region are rather crude. The calculations themselves are approximate and there are still other second-order graphs not yet included. Thus the values listed in table 23 are only an indication of the sort of results that might be obtained in more complete calculations.

5.4. Beyond first order It is obvious from the results in tables 22 and 23 that the one-boson-exchange model of isobar currents predicts smaller corrections to the matrix elements of spin operators than does either of the two effective interactions listed there. Part of the reason for this is the very strong cancellation that occurs between the exchange and direct pieces of the two-body matrix elements in the effective interaction in the isobar-hole channel. Indeed with a zero-range interaction the exchange piece exactly cancels the direct piece. (This is why only direct matrix elements are calculated in the phenomenological models with a zero-range interaction of strength determined by (Sg')aN.) This is in contrast to particle-hole matrix elements in spin channels involving nucleons only where the exchange piece is typically four times smaller than the direct piece. This is also the reason why isobar calculations are sensitive to the nuclear density because the exchange matrix element, as discussed in section 2.5, is in nuclear matter a function of the Fermi momentum, k F, and hence the density. In light nuclei one expects values of k F much less than the nuclear matter value to be relevant. Indeed in the limit k F---->0 the exchange piece exactly cancels the direct piece even for finite-range meson-exchange potentials, see eq. (2.80). It is important in the one-boson-exchange model to take calculations beyond first order. It is doubly important in this case because Nakayama et al. [14, 25] have argued that a certain class of higher-order graphs strongly screen the exchange term in the isobar-hole interaction leaving the direct term largely unaffected. Their argument follows from a calculation of (g')aN in nuclear matter starting from a G-matrix (see section 2.1). It is pointed out that if the particle-hole interaction in the spin-isospin channel is collective then the interaction between the particle and hole line (graph (b) of fig. 1) should also allow the particle and hole line to exchange a collective phonon in the crossed channel (graphs (c) to (f) in fig. 1). This can lead to a strong renormalization of the residual interaction in the particle-hole channel. This phenomenon was first discussed by Babu and Brown [27] for liquid 3He and applied to nuclear matter by Sj6berg [26]. The coupled equations illustrated in fig. 1 have recently been solved by Dickhoff et al. [22, 23], Nakayama et al. [24, 25] and Cheon et al. [66]. There is some divergence of results between the groups but in general the (g')NN is increased by roughly 20% by the induced interaction whereas in the isobar-hole channel there is a significant boost to (g')aN even by as much as a factor of two. It is clear that the induced interaction enhances drastically the isobar-hole coupling by screening out the exchange term in the interaction. Returning, then, to the calculation of single-particle spin matrix elements in closed-shell-plus-one nuclei, it is quite clear that the analogues of the induced interaction must be calculated. Presumably they will screen some of the exchange contributions, which we have argued are responsible for the

I.S. Towner, Quenching of spin matrix elements in nuclei

358

+

+

,?

f,

.~.

-X

+

t:0 -X

+

_-

+

-

-X

+...

+

+

....

-

-X

4" "

•

Fig. 19. Sample second-order perturbation diagrams involving isobar excitations: First line contains diagrams contributing to the RPA series, second line contains graphs of TVV structure (T = one-body operator, V - residual interaction) that are called vertex corrections, third line contains graphs of VTV structure and are called box corrections.

smaller quenching obtained in one-boson-exchange models. In fig. 19 we show some sample secondorder diagrams involving isobar excitations. The first row shows diagrams which when taken with the first-order graphs in fig. 18 represent the start of the RPA series. Notice both nucleon-hole and isobar-hole intermediate states occur starting in second order. The second row shows graphs that are called vertex corrections. These are the analogous graphs to the induced interaction in the effective interaction. They are also the graphs that correct the antisymmetry (to second order) which is violated in the RPA series beginning in second order. The third row shows yet another group of graphs analogous to the second-order core-polarisation graphs (fig. 10) except that one of the particle lines has been replaced by an isobar. We call these the 'box' graphs. Not all relevant graphs are shown in fig. 19, but in our calculations all second-order diagrams involving one isobar excitation are included. We give some results for a 0p-hole and 0d-particle relative to an 160 closed shell in table 24. We use harmonic oscillator wavefunctions (hoJ -- 13.3 MeV) and a one-boson-exchange potential multiplied by a short-range correlation function with vertex form factors included. In the first two rows, we repeat the first-order result of table 22 except that the direct and exchange contributions to the matrix element are separately tabulated. Notice there is a cancellation in the ~gs coupling constant of the effective one-body operator but not in the induced tensor coupling constant ~gp. In light nuclei, the ~gp term has an important role to play (for the magnetic moment of a Pl/2 orbit it can even dominate), but in heavier nuclei its role tends to diminish. In line 4 we give the computed results for the vertex graphs which are anticipated to screen the first-order exchange graphs of line 2. This indeed is what is found for the 3gs value; between 40 and 60% of the exchange value of Bgs is cancelled by the vertex graph. However it does not happen for the induced tensor term ~gp where both contributions have the same sign. Furthermore the other second-order graphs, line 3 and line 5, give comparable contributions with signs such that there is considerable cancellation among all the second-order terms. Indeed just taking first-order alone is not an unreasonable approximation to the sum of first and second order taken together at least for the results shown in table 24.

1.S. Towner, Quenching of spin matrix elements in nuclei

359

Table 24 Correction to the magnetic moment, ~/z, and the M1 spin-flip j = l + 1/2---, j = l - 1/2 matrix element, ~(M1 ), of a closed-shell-plus (or minus)-one configuration as a percentage of the single-particle value from second-order isobar graphs in the OBEP model 0p~,12 gL

~g~l,

5g~1)

~/Z %

0.003 0.001 -0.002 0.002 -0.008

-0.396 0.224 0.060 -0.093 0.089

0.636 0.170 -0.216 0.208 0.079

3.7 13.1 -3.9 2.4 6.8

-11.0 4.9 2.0 - 2.7 2.1

0.002 0.001 -0.001 0.003 -0.005

-0.294 0.110 0.016 -0.069 0.059

-0.005

-0.116

0.876

22.2

- 4.7

0.000

-0.178

(1)

First-order, direct First-order, exchange Second-order RPA Vertex graphs Box graphs Sum

Od~,2 5(M1)

-

~/z %

~(M1)

0.758 0.176 -0.233 0.161 0.097

-3.0 2.0 -0.2 -0.6 0.8

-8.8 2.2 1.0 -2.1 1.3

0.960

-1.0

-6.5

-

- -

The result here computes just the second-order graphs, whereas the argument for screening from the induced interaction involves iterating a series of coupled equations and finding a self-consistent solution. In essence a subset of graphs have been evaluated to all orders. Our result, however, indicates there are other graphs not in the subset that are equally important in this application.

6. Comparison with experiment In the last three sections, we have discussed the principal corrections to the single-particle M1 and Gamow-Teller matrix elements for a closed-shell-plus-one configuration: core polarisation, mesonexchange currents and isobar excitations. We stress that none of these corrections occur for a free nucleon. They each rely on the presence of other nucleons being, in our applications, in closed-shell cores. The results depend upon the size of the core and as such they represent what is fashionably called a medium modification to the single-particle operator. There are alternative candidate theories, other than the explicit consideration of meson exchanges, which also lead to medium-modified single-particle operators. One such alternative is the idea that nucleons may swell in size in a nuclear environment. First suggested by Noble [140] in connection with understanding quasi-elastic electron scattering from nuclei, the idea was brought sharply into focus by the EMC collaboration [141] which showed that a nucleon's electromagnetic structure function changed in nuclei. Shakin [142] has demonstrated that several other nuclear phenomena over a broad range of momentum transfer are also explained by an increase in nucleon size in a medium. Rho [143] argues that the experimental determination of the effective axial-vector coupling in a nucleus of g A , e f f = 1 being quenched from the free-nucleon value of gA = 1.26 also implies an expansion in size. Various mechanisms for this expansion have been suggested ranging from an increase of the quark-confinement volume [144] or, in other versions, the radius of a skyrmion [143] in QCD-inspired models to a modification of the pion cloud [145] in a more traditional nuclear physics approach. A second alternative for medium modification comes from the Dirac-based relativistic theories of nuclear structure as exemplified in the tr-to model of Walecka et al. [146]. In this model the average force on a nucleon in a nuclear medium is assumed to arise from a strong attractive scalar field, Vs, and a repulsive vector field, V0, which is almost as strong. The effect of these fields is a modification of the mass and energy terms in the Dirac equation: M ~ M* = M + Vs, E ~ E* = E - V0. In a nonrelativistic

360

I.S. Towner, Quenching of spin matrix elements in nuclei

reduction of the Dirac to the Schr6dinger equation, the relatively weak central optical potential one observes experimentally is explained as being due to a strong cancellation between these fields. In the small components of the Dirac spinors, by contrast, the mass and energy occur in the combination M + E and the potentials come together constructively. The resulting effect is very large. Among other things, this leads to a strong increase in the spin-orbit potential, which was one of the main reasons for considering the model in the first place [147]. It also leads to a strong modification of the matrix elements of 'odd' Dirac operators, such as the non-anomalous part of the magnetic moment operator [147-149]. Both the swelling of nucleons idea [150] and the or-co model [148] have been used to explain medium modifications seen in quasi-free electron scattering [151] and in coincident quasi-free proton knockout, (e, e'p), reactions [152] and identified in the longitudinal-transverse separation of the electromagnetic structure functions. The data on the EMC effect, coming at high energies, and the quasi-free electron scattering at intermediate energies suggest the manifestation of medium modification should also be evident at even lower energies in, for example, nuclear resonance excitation (M1 transitions) and elastic magnetic moments. Indeed Karl et al. [153] observing that the magnetic moments of 3He and 3H are roughly 10% larger than that of a free proton and a free neutron interpret this as a change in length scale for a nucleon inside nuclei. This interpretation conflicts with that of Sick [154] whose observation of scaling in electron scattering from 3He limits the radius increase to 6%. It would also be hard to accommodate with the EMC effect where Close [144] has argued for a 4% increase in nucleon size in 3He. Any contributions to the nuclear moment, that arise from configuration mixing, meson-exchange and isobar effects will reduce the amount of radius increase, probably to within the above limits. Indeed Ericson and Richter [155] considering the mass A = 3 and 12 systems found the upper limit for the change of the nucleon magnetic moment inside nuclei due to a rescaling of the nucleon size to be about 2% once pion-exchange processes are explicitly included. Thus before conclusions can be drawn from the swollen nucleons idea or the relativistic ~r-to model it is necessary to have as complete an account as possible of the single-particle M1 and Gamow-Teller operators as given by the traditional nuclear physics model of nucleons interacting through the exchange of mesons. It has been our purpose here to provide this. Before confronting our calculations with experiment there are two further asides that we must attend to in sections 6.1 and 6.2. 6. I. M E C and core polarisation

The core-polarisation calculation, discussed in section 3, corrects the matrix element of a one-body operator evaluated in the closed-shell-plus-one configuration for the presence of 2 p - l h and 3p-2h admixtures in the single-particle wavefunction. The admixture amplitudes are estimated in perturbation theory. We found that the calculation must be taken at least to second order in the residual interaction (or to the fourth power in the meson-nucleon coupling constants). It is logical therefore that the matrix element of the two-body meson-exchange operators should likewise be corrected for 2p-lh and 3p-2h admixtures. Since the MEC operator itself involves the meson-nucleon coupling constants to the second power, it is sufficient to estimate this correction to first order in the residual interaction. The relevant Goldstone diagrams are shown in fig. 20, where the double horizontal line represents the two-body MEC operator. Note that in graph (a), there is a limitation on the number of contributing particle-hole intermediate states coming from the restrictions imposed by the angular momentum algebra. The particle-hole state must be coupled to the same angular momentum, A, as the multipolarity of the two-body MEC

I.S. Towner, Quenching of spin matrix elements in nuclei

(a)

361

(b)

Fig. 20. Core-polarization corrections to a two-body MEC operator in a closed-shell-plus-one nucleus.

operator. We noted a similar limitation in the first-order core-polarisation corrections for one-body operators discussed in section 3.1. For this reason the contribution from fig. 20(a) turns out to be small. By contrast, the contributions from fig. 20(b) are by no means negligible. Here there are no angular-momentum restrictions on the construction of intermediate states. As was the case of the second-order core polarisation of one-body operators, discussed in section 3.5, the tensor component of the residual interaction and the tensor component of the two-body MEC operator couple strongly to high-lying intermediate states. Thus the intermediate-state summation required in the evaluation of fig. 20(b) must not be prematurely terminated. In our computations we explicitly took these sums to 10 hw and geometrically extrapolated beyond that. The importance of this graph was first pointed out by Arima et al. [74, 105], where it is known as the 'crossing term'. Its importance is that the graph gives a contribution to the renormalization of spin operators with the opposite sign to that from the second-order core-polarisation graph involving one-body operators and cancels a large part of it. This has been demonstrated explicitly for triton 13-decayin the work of Green and Schucan [156], Ichimura, Hyuga and Brown [157] and Riska and Brown [158]. It is argued by Oset and Rho [60] that this cancellation should persist in heavier nuclei. In the summary tables, which we will come to in section 6.3, such a cancellation is clearly present but in detail it varies from case to case and the quoted numbers are model-dependent and parameter dependent. Nevertheless the indications are clear. It would be a mistake to omit this MEC contribution to the tensor correlations. 6.2. Relativistic corrections

The one-body magnetic moment and Gamow-Teller operators used in obtaining impulse-approximation matrix elements are derived from the one-body currents, eqs. (4.2) and (4.109), evaluated in the nonrelativistic limit to order 1 / M for the vector current and 1 / M ° for the axial-vector current. The relativistic correction estimates the next-order terms by expanding the normalization factor of the 1/2 1/2 1/2 2 2 free-nucleon spinor (multiplied by ( M / E ) ) to get ( M / E ) ((E + M ) / 2 M ) 1 = 1 - p / 8 M and the factor (E + M)- 1 in the lower component of the Dirac spinor to get (E + M)- - (2M)- 1(1 - p /24 M 2 ) using the on-mass-shell relation E 2 = p2 + M 2. The magnetic moment operator can then be written gc{L(1 --pZ/2M2) - ( p : / 2 M z ) ( s - (S "/~)/~)} + gs S(1 -pZ/2M2)

(6.1)

and the Gamow-Teller operator gA { tr - ( p2/ 2MZ)(cr - (~r .ff)/~)}.

(6.2)

362

I.S. Towner, Quenching of spin matrix elements in nuclei

In terms of the equivalent effective one-body operator, eqs. (5.6) and (5.9), we can pick out from these expressions the values of 8g L, 8 g s , . ' . etc., viz. 8gL=--½gL (P 2/M2) 8g s = (_ 3gL 1 _ i ~gS)(P 2/M2} 8gp = --~(87r) 1/2 gL

(P 2/M2)

(6.3)

8 g A = - - ~ g A(p2/M2} 8gPA = -- ~ (8~') '/a gA (p2/M2)

•

It remains to estimate the expectation value (p2/M2) for which we use single-particle harmonic oscillator wavefunctions in keeping with all other calculations presented in this report. However since it involves the second derivative this expectation value is very sensitive to the choice of single-particle radial wavefunction. Other estimates of this relativistic correction can be found in the literature [159]. The main result is that the impulse approximation values of gL, gs and gA are reduced by roughly 2 to 3%.

6.3. Collation: closed LS shells For the LS closed shells in light nuclei, A = 4, 16, 40, we summarise all the corrections to the single-particle matrix elements coming from core polarisation, meson-exchange currents and isobar excitations evaluated to second order. In table 25, we give a detailed breakdown for the two cases of a 0pl/2-hole and 0ds/2-particle relative to an 160 core. In tables 26 and 27 we give just the final summed results and compare with experimental data. In each case the computed result is expressed in terms of the equivalent effective one-body operator, 8g L, 8gs,.-- etc., as defined in eqs. (5.6) and (5.9). (a) Isoscalar M1. The isoscalar calculation is dominated by second-order core polarisation; isobar excitations and meson-exchange currents give rather small contributions. Note, in particular, the enhancement to 8g{L°) from MEC, coming mainly from {y-pair graphs (see table 17), is more or less offset by the relativistic correction. Our MEC calculation is about 50% smaller than that calculated by the Tokyo group [160,161] but depends sensitively on the choice of g~NN and go,NN coupling constants. Like that of the Tokyo group, our MEC result is in strong disagreement with the suggestion of Riska [122] that the isoscalar exchange current obtained from a minimal substitution in the spin-orbit force leads to a large quenching of 8g~°) in nuclear matter. This disagreement is analysed in detail in the work of the Tokyo group [160,161]. The computed corrections to the isoscalar magnetic moments are in reasonable agreement with experimental data. A large enhancement is predicted for j = l - ½hole states and a small reduction for the j = l + ½ particle states as required. Indeed the success of the calculation suggests that the tensor force in the one-boson-exchange model has about the right strength. Recall that central forces give no contribution to second-order core-polarisation corrections to isoscalar magnetic moments at LS closed shells. This calculation uses a 'weak' tensor force corresponding to a 'strong' coupling, Kp = 6.6, in the p-meson Lagrangian.

I.S. Towner, Quenching of spin matrix elements in nuclei

363

Table 25 Corrections from core polarisation, isobar currents, meson-exchange currents, core polarisation of MEC, and relativistic terms in closed-shell-plus (or minus)-one configurations to magnetic moments and Gamow-Teller matrix elements expressed as a percentage of the single-particle matrix element Op~/12

Isovector Core pol. Isobar MEC M E C - CP

Relativistic Sum

Isoscalar Core pol. Isobars MEC MEC + CP

Relativistic Sum Gamow-Teller* Core pol.

Isobars MEC MEC + CP

~gL

~gs

~gr

%

~gL

5gs

5g~

%

-0.183 -0.005 0.175 0.125 -0.009

-0.699 -0.116 0.326 0.275 -0.089

0.177 0.876 -0.403 0.302 -0.015

6.5 22.2 -25.8 0.6 - 2.4

-0.101 0.000 0.102 0.080 -0.012

-0.508 -0.178 0.089 0.181 -0.125

0.146 0.960 -0.498 0.517 -0.021

-13.3 - 1.0 6.5 8.3 - 2.6

0.104

-0.303

0.937

1.1

0.068

-0.541

1.104

- 2.1

0.013 0.001 0.012 0.005 -0.009

-0.138 -0.002 0.032 0.013 -0.022

-0.022 -0.003 0.017 -0.010 -0.015

18.5 0.7 0.2 1.2 - 0.2

0.010 0.001 0.009 0.003 -0.012

-0.097 -0.003 0.020 0.012 -0.030

-0.012 -0.003 0.018 -0.012 -0.021

- 2.1 - 0.0 2.0 0.8 - 2.9

0.022

-0.117

-0.032

20.5

0.010

-0.097

-0.030

- 2.1

0.013 0.001 -0.001 0.001

-0.181 -0.030 -0.008 0.049 -0.015

0.052 0.251 -0.063 0.031 -0.037

-13.1 13.4 - 4.3 5.6 - 3.5

0.011 0.002 -0.002 0.001

-0.136 -0.046 -0.004 0.032 -0.021

0.005 0.264 -0.065 0.026 -0.052

- 9.0 - 0.9 - 1.2 3.0 - 2.1

0.013

-0.185

0.234

- 1.9

0.013

-0.175

0.179

-10.2

Relativistic Sum

Od~,2

* quoted are 8gLA, 8gA, ~gPA and the correction to the ground state to ground state GT matrix element as a percentage of the single-particle value. Table 26 also lists the correction to the off-diagonal spin-flip matrix element j = l + ½--->j = l 21

which is obtained from ~ ( M 1 ) / ( M 1 ) = (Sg L - 5gs

+ 1(87/') -1/2

~gP)/( gL -- gs)'

(6.4)

Electron scattering is sensitive to both orbital and spin contributions as given by this formula, however, isoscalar transitions are very weak compared to isovector transitions and are not easily measurable. By contrast, isoscalar (p,p')1 ÷ transitions have been measured [162] and show strong quenching, Q = O'expt/O'cal¢= ] l + ~ ( M 1 ) / ( M 1 ) 1 2 - 4 0 % . The published calculation has been renormalized by a factor of 1.67 [163]. In this case (p, p') is only sensitive to the spin part of the M1 operator and the appropriate reduction factor in the matrix element is ( ~ g s - l(8'rr) -1/2 ~gv)/gs. For the middle of the sd-shell, a value intermediate between that of the two shell closures of A = 16 and 40 of ~g~O)... -0.13 for the 0d orbit corresponds to Q --- 70%, a substantial reduction but not as large as the (p, p') data are suggesting. However there are many uncertainties here; we merely note that quenching in isoscalar transitions implies configuration mixing beyond the lowest-order 0hto shell-model calculation. (b) Isovector M1. The isovector calculation is characterized by the conflicting interplay of many competing processes. For example, core polarisation and MEC both give large corrections but they

I.S. Towner, Quenching of spin matrix elements in nuclei

364

A

]

A

f

o~ iv

F

I

I

I

I

i

?~

I

I

I

|

I

I

I

I

I

I

i

I H

I f l l l

i

F

I

F eq +1

~A

t~r~

o~ jv

z

Q

I I I I

I

J

I

I

-~

E

.... ©

If

H

il

II

II

~[

II

II

II

If

II

~1

I.S. Towner, Quenching of spin matrix elements in nuclei

365

come with the opposite sign. In addition, for the j = l - ½states, the orbital and spin contributions both for core polarisation and MEC have the opposite sign. Thus the final result in a delicate balance between all these contributing pieces. Indeed it is something of an achievement to get the right sign for the correction to the Schmidt isovector magnetic moment. Our calculation underestimates the correction in the 0pl/~2 moment in A = 15 and 0f7/2 moment in A = 41, but is in reasonable agreement in other cases. Table 26 also lists the correction to the off-diagonal spin-flip matrix element as given by eq. (6.4). Again looking at data in 28Si, the summed cross-section over about seven T = 1 states shows an overall quenching of Q = 77% in electron scattering [164] and Q ---55% in (p, p') reactions [162,163]. We can get a crude estimate of this quenching by assuming it comes from the spin-flip ds/2-d3/2 matrix element and taking for a 0d orbital in the middle of the shell ~g(L1) = 0.07, ~g~X) _~ -0.56, ~g(X) --_-1.0. We obtain Q =68% for electron scattering (including orbital contributions) and Q =74% for (p,p') scattering coming from the spin part of the operator only. Again there are large uncertainties here so we can say little other than that an understanding of the quenching is qualitatively at hand if not in detail. There is one other example of a well-studied isovector spin-flip transition and this involves the 1 ÷ state at 10.23 MeV in 48Ca seen in (e, e') and (p, p') experiments. This excited state has a dominant f7--/12f5/2 neutron particle-hole structure. The independent particle shell model predicts a value B(M1) = 12 #2 for the pure f7/2---> f5/2 transition. Configuration mixing among the fp-shell orbitals reduces this 2 2 figure to 8.96/~N in the calculation of McGrory and Wildenthal [165] and to 8.56/z N in the work of Muto and Horie [166]. Experimentally the B(M1) to the strong 1 ÷ state at 10.23MeV has been measured [167] to be (3.9 -+ 0.3)/L 2 indicating significant quenching. In addition there are several more weakly excited 1 ÷ states [167] identified so the total B(M1) strength is around (5.3 +- 0.6)/z~. Comparing this with Muto and Horie's value we deduce an additional reduction of ~ ( M 1 ) / ( M 1 ) = -(18-+ 4)% is required in the neutron spin-flip matrix element over and above the configuration mixing in the fp-shell. We can estimate this from eq. (6.4) using the results for A = 41 given in table 26, by combining the isoscalar and isovector values to obtain ~g[V~= -0.06, ~g~V~= 0.50 and ~gCp~= -0.97, to get ~(M1}/(M1 } = -17%. This cannot be immediately compared with the mass A = 48 data since the neutron excess will introduce same subtle effects (e.g. Pauli blocking in the core-polarisation calculation, meson-exchange currents among the valence nucleons). Nonetheless, to the extent that these subtle effects can be ignored, our calculation shows that an understanding of the required additional reduction is at hand. In our calculation, -11% of the reduction comes from second-order core polarisation and - 6 % from isobar excitations. The meson-exchange currents and relativistic corrections more or less cancel each other. There would be less ambiguity in the discussion of spin-flip matrix elements if there were available data in the closed-shell-plus (or minus)-one nucleus itself involving the spin-orbit partners. One such datum occurs in 15N. The lifetime and E2/M1 mixing ratio are known [168] for the ~/-transition between the 6.32 MeV 3/2- and the ground state yielding an M1 width of F = 3.07 - 0.08 eV. Interpreting this as a pure proton p3/2--->pl/2 transition a reduction of ~(M1)/(M1) = -(20.8-+ 1.0)% is required in the single-particle M1 matrix element to fit the datum. From table 26 combining the isoscalar and isovector corrections we calculate a reduction of -13.9% coming from core polarisation (-14.9%), isobar excitations (-4.4%), meson-exchange currents (+7.4%) and relativistic corrections (-2.0%). Thus only about two-thirds of the experimentally observed reduction is explained in our calculation. One other datum available comes from the "O('y, no) experiments of Holt et al. [169], which when analysed in R-matrix theory yields a ground-state radiative width for the 5.08 MeV M1 excitation that is anomalously small for a pure d3/2---->d5/2 spin-flip transition. Their B(M1) value of 0.66 is one third the

I.S. Towner, Quenching o f spin matrix elements in nuclei

366

single-particle value and represents a reduction in the M1 matrix element of ~ ( M 1 ) / ( M 1 ) = - 4 4 % . From the values in table 26 we calculate a reduction of -16% in the neutron matrix element, much less than the required value. (c) Isovector Gamow-Teller. A number of ground states in closed-LS-shells-plus (or minus)-one nuclei are t3-unstable and the allowed Gamow-Teller matrix element between mirror states has been measured [170, 171] in A = 3, 15, 17, 39 and 41 nuclei. In each case there is a quenching phenomenon evident when the deduced matrix element is compared with the single-particle estimate. There is a similarity between the calculation of corrections to the Gamow-Teller matrix element and the isovector M1 matrix element. For example the second-order core-polarisation correction in [3-decay is identical to that from the spin part of the M1 operator. Likewise the isobar contributions are the same (with the exception of the second-order box graphs), see eq. (5.10). The calculations are compared to the experimental data in table 27 and generally speaking they give insufficient quenching. This is particularly noticeable in the A = 15 datum, where we also found the isovector magnetic moment calculation to be in poor agreement with experiment. Recently (p, n) measurements [172-174] have enabled the spin-flip matrix element to be determined and these results are also included in table 27. Again the indications from experiment are that the calculated reduction in the Gamow-Teller matrix element is about a factor of two too small. (d) l-forbidden transitions. In the mass A =39 system, the /-forbidden Od3/2----~lsl/2 M1 and Gamow-Teller transitions have been measured. These are interesting transitions because in the long wavelength approximation with the impulse-approximation one-body operators they are strictly forbidTable 27 Summary of all corrections to the ground-state diagonal Gamow-Teller matrix element, ~(GT)~, and the off-diagonal spin-flip matrix element, 8(GT)f, expressed as a percentage of the single-particle value for closed-shell-plus (or minus)-one configurations

A=4 A =4 A = 16 A = 16 A = 16 A = 16 A =40 A = 40 A=40 A =40 A =40 A =40

0s~)2 0p3. 2 0p~2 0ds/2 ls~ 2 (0d3/2-1sl~2) 0d3)2 ls~)2 (0d3,2-1s~2) 1 0fT/2 lp3, 2 (0fs~2-1p3/z)

~gLA

~gA

~gPA

0.015 0.013 0.013

-0.076 -0.124 -0.185 -0.175 -0.190

0.309 0.234 0.179

0.009

-0.225 -0.269

0.010 0.007

-0.221 -0.244

0.273 0.167 0.267 0.105 0.149 0.246

g(GT)d {GT)~ % calc

g(GT)d (GT)d % expt"

-- 6.0 -- 6.7 -- 1.9 10.2 t5.1

-- 4.0--+0.4 h

--17.1 --21.3 --14.2 -- 17.9

g(GT)f (GT}, % calc

8(GT)f (GT), % expt

--13.1 --+(1.5 13.8--+0.3

--12.8 17.0 --15.8

= --38 c'd =--33 c

--33.7 --+ 1.0

--21.9

~ --45 c~

f --26.2--+0.4

--18.8 --20.8

Deduced from experimental [3-decay data recorded in [170] with gA = 1.260 --+0.004. b From Simpson [171]. c From (p, n) measurements of Watson et al. [174], where we have renormalized the (p, n) cross-section so the deduced GT matrix element for the ground-state transition agrees with/3-decay measurements. Watson et al., however, prefer to normalize their results to distorted-wave impulse approximation calculations and find for masses A = 15, 39, notable discrepancies between their deduced values of the ground-state Gamow-Teller matrix element and those deduced from [3-decay. d From Goodman et al. [173]. e From Rapaport et al. [172]. f The/-forbidden GT matrix element has been measured by Adelberger et al. [177] yielding ~gPA = 0.105--+ 0.014.

I.S. Towner, Quenching of spin matrix elements in nuclei

367

den. A selection rule does not permit a change of two units in orbital angular momentum. However as soon as configuration admixtures are included in the single-particle wavefunctions (core polarisation) or meson-exchange currents and isobar excitations allowed for, then a nonzero matrix element can be obtained. In terms of the equivalent effective one-body operators, eqs. (5.6) and (5.9), these transitions depend only on the induced tensor terms gp and gva. In tables 26 and 27 we express the experimental results in terms of these effective coupling constants using the relations

[B(M1;d3/2--->s,/2)]

= -0.053 gm (6.5)

[B(GT; d3/2 ~ S1/2)11/2 = -0.219 gPA where the B(GT) value for 13-decay is related to the ft-value by the formula: ft = 6166/B(GT) secs. These transitions have been studied by Towner and Khanna [175] who found the dominant contribution to gp and geA came from isobar excitations. This also led to a dilemma. As already mentioned the isobar calculations (except for the second-order box graphs) for M1 and GT transitions are strictly related, eq. (5.10),

g~l)/gpA = g~l)/g a = 3.73

(6.6)

and the calculated values with all contributions included are, to within a factor of two, close to this ratio. Yet the experimental data give a ratio 17 + 3 and cannot be accommodated by the theory. This dilemma persists despite a remeasurement of one of the experimental quantities, the lifetime of the first excited state in 39Ca, and a recalculation of the corrections as reported recently by Alexander et al. [176]. This dilemma is also evident in the present calculation listed in tables 26 and 27, which differs from that reported in [176] only in that a stronger isobar coupling constant has been used here, viz. g,,Na/g~NN = 2 rather than the quark model 6V~/5. The only way the theoretical value of the ratio g~)/gPA can be altered significantly from 3.73 is for the core-polarisation correction from the orbital part of the M1 operator and the meson-exchange current corrections in either the M1 or GT operator to be quite different from the values computed here.

6.4. Comparison with previous works We will briefly comment on the differences between the present calculation and our previous one [47] and compare these results with the recent work of the Tokyo group [161] and the empirical values of the effective one-body operator deduced from data in the sd-shell by Brown and Wildenthal [178-180]. First, since the 1983 calculation of Towner and Khanna [47] a number of improvements have been introduced, the principal ones being: (a) The Al-meson of mass ll00MeV has been added to the one-boson-exchange force. Being a short-range ingredient its contribution is largely damped by the short-range correlation function so the numerical consequences of introducing the Al-meson are minimal. (b) The meson-exchange current (MEC) operators for heavy-meson exchange are taken from the hard-pion model of Ivanov and Truhlik [33] and include the Al-meson. We insist that the MEC operators are consistent with the one-boson-exchange potential as given by the continuity equation. This is why the Al-meson is added to the force. (c) To the MEC calculation, the pair diagram involving the cr-meson is added, again a requirement

I.S. Towner, Quenching of spin matrix elements in nuclei

368

from the equation of continuity. Furthermore nonlocal terms in all pair and current diagrams, previously omitted, are here retained in both vector and axial-vector MEC operators. (d) The isobar coupling constant is increased from the quark-model value, g~Ya/g~NN= 6V~/5, to the Chew-Low value, g~Na/g~NN= 2. This increases the isobar contribution by 40%. (e) A relativistic correction is introduced in the one-body M1 operator, see section 6.2, comparable to the one already in place in the Gamow-Teller operator. A comparison of the two calculations is given in table 28 where it is seen that the modifications have had rather little impact on the summed result. The most noticeable difference is in the isovector ~g~1) value which has decreased from -0.370 to -0.541. Part of this decrease, -0.04, results from the stronger isobar couplings and the remainder, -0.12, comes from the relativistic correction. In the isovector ~gAvalue, however, the increases from stronger isobar couplings have been offset by changes in the axial-vector MEC operator coming from the hard-pion model. We note also that the isoscalar ~g~0) value is the same in the two calculations. This is accidental. The introduction of or-pair graphs gives a boost of +0.012 by ~g~0) that is cancelled by the relativistic correction. We also list in table 28 the results from the recent work of the Tokyo group [161]. There are a number of differences in the way the two calculations were done stemming from different choices for the residual interaction, the use or not of vertex form factors, the choice of isobar coupling constants, and the inclusion or not of relativistic corrections. Arima et al. [161] argue that relativistic corrections have their origin in two-body pair currents leading to a possibility of double counting here. It is our belief that the relativistic correction comes not so much from the pair graph but from the singular term in the positive frequency nucleon propagator in the Born graph (fig. 13(a)) evaluated to higher order in I / M 2. With the positive frequency term in the propagator specifically excluded from the MEC current there should not be a problem. However with the ~r-meson simulating 2~r-exchange there is always the danger of double counting in the one-boson-exchange model so a more thorough assessment of this concern seems necessary. Arima et al. [161], in evaluating the core-polarisation correction to MEC, also allow for the possibility of an intermediate isobar state in the graphs given in fig. 20. We have not included these. However in treating the isobar as a nuclear constituent, rather than as a mesonexchange current, and including all graphs to second order in the residual interaction as discussed in section 5.4, we are in fact including a number of the graphs shown in fig. 6 of Hyuga et al. [105]. Despite these differences there remains nonetheless a good accord between the two calculations. This is because they agree on the major contributions, core polarisation and the lowest-order isobar and meson-exchange current corrections. These terms are dominated by 7r- and p-exchange about which there is little uncertainty once one has made a choice of the appropriate coupling constants. A quite different approach to the determination of the effective one-body operator is that of Brown and Wildenthal [178-180]. The coupling constants of the effective M1 and Gamow-Teller operators are Table 28 Comparison of the effective one-body operator for a 0d configuration in A = 17 from the present calculation with that from the 1983 Towner-Khanna (TK) calculation [47], the Arima et al. (ASBH) calculation [161] and the empirical values [178-180] deduced by Brown and Wildenthal (BW) Isoscalar

5g(i' Present work TK ASBH Empirical, BW

0.010 0.010 0.024 0.015(3)

Isovector

~g~)

5g~°)

-0.097 -0.087 -0.107 -0.10(2)

-0.030 -0.007 0.016 0.10(10)

~g~L~' 0.068 0.054 0.040 0.089(14)

Gamow-Teller

5gls~)

~ (~) Ogp

~gLA

-0.541 -0.370 0.362 -0.59(5)

1.104 1.073 1.096 1.6(2)

0.013 0.012 0.013 0.01(1)

~gA -0.175 -0,186 -0.180 -0.26(i)

~gPA 0.179 0.101 0.224 0.09(4)

I.S. Towner, Quenching of spin matrix elements in nuclei

369

determined in a fit to a large number of data in the sd-shell using shell-model wavefunctions calculated without truncation in the complete sd-sheU model space with the empirically determined effective interaction of Wildenthal [181]. The assumption is that the effective one-body operator is only a weakly varying function of nuclear mass; a proportionality of A °'35 is assumed [178]. There is a reasonable agreement between these empirically determined operators and the calculated ones as can be seen in table 28. However there are a number of differences that are worth highlighting. (a) The most glaring discrepancy is with the ~gn value in the Gamow-Teller operator, where the calculated value, -0.18, is only two-thirds the empirically deduced value, - 0 . 2 6 - 0.01. Since the ~gLA and ~gPA terms give only a small contribution to the Gamow-Teller matrix element this discrepancy immediately translates into a discrepancy in the total matrix element. Indeed it is evident in table 27 that theory is underestimating the quenching in Gamow-Teller transitions in closed-shell-plus-one nuclei. If we assume that the core-polarisation calculation is about right (because it is the principal correction to isoscalar magnetic moments which are well described), then we can use the empirically determined value to solve for the isobar correction. The MEC and relativistic corrections are small for Gamow-Teller transitions. Thus we write, in an obvious notation, ~gg(A) = 8gA(BW ) - ~gA(CP) - ~gA(MEC) - ~gg(Rel) = -0.13 + 0.01.

(6.7)

Our calculated value, from table 25, is ~gA(A) ------- 0. 05 nearly three times smaller. This strongly suggests the one-boson-exchange model is underestimating the contribution from isobar currents, an inference first pointed out by Brown and Wildenthal [178]. Now if the one-boson-exchange potential for isobars is augmented by a phenomenological zero-range term in the spin-isospin channel of strength (~g')aN, as in the Cha-Speth interaction [46], then much larger isobar corrections are easily obtained as discussed in section 5. For example, from table 22, the first-order isobar calculations from such effective interactions yields ~gA = ~g~1) x 1.26/4.706 = - 0 . 1 8 exceeding slightly the required value, eq. (6.7). This additional phenomenological term is presumably compensating for some missing ingredient in the one-boson-exchange model. In the work of Cha and Speth [46] only RPA graphs are computed and the phenomenological term represents the other uncalculated second-order graphs. Here all second-order graphs are computed so the phenomenological correction must be representing even higher-order corrections. Desplanques [182] has estimated a subset of third-order graphs and finds quenching from them too. However, if one is going to invoke a large increase in ~gA(A) from such sources then there is going to be a correspondingly large increase in ~g~l)(A), which could spoil the agreement with the empirical ~g~l) value. (b) Next we consider the difference in the isovector Bg~) value for the M1 operator and the isovector ~gA value (scaled by 4.706/1.26) for the Gamow-TeUer operator. The difference arises almost exclusively from the meson-exchange currents which are quite different for vector and axial-vector currents. (The orbital part of the M1 operator which in principle can also lead to a difference in core-polarisation calculations mainly influences the ~g(L~) value.) From the empirical results of Brown and Wildenthal [178-180] we have ~g(sl) 4.706 1.26 8gA = 0.39 -+ 0.07 compared to the calculated value of

370

I.S. Towner, Quenching of spin matrix elements in nuclei

~g~l)

4.706 1.26 ~gA ----0.11 .

Thus again we are faced with a significant discrepancy involving the spin part of the MEC calculation. (c) Lastly we consider the orbital ~gL values in the isovector and isoscalar M1 operator. For the isovector case there is a large cancellation between the core polarisation and the MEC contributions so the resultant value is somewhat dependent on subtle details. Our result can be considered to be in reasonable agreement with the empirical value. It is a similar story in the isoscalar case. The core polarisation, meson-exchange current (mainly from the nonlocal terms in the ~r-pair graph) and the relativistic corrections are of comparable magnitude with signs such that ~g~L°~ = 0.010 + 0.012 -- 0.012 = 0.010 (see table 25). Again this is probably close enough to the empirical value ~g~L°~ = 0.016 --+0.005 not to pose a serious concern.

6.5. Collation: closed ]j shells in Pb region In heavy nuclei with shell closures separating the single-particle spin-orbit partner states t h e ~ is a large correction to the M1 and Gamow-Teller matrix elements coming from first-order core polarisation as discussed in section 3.1. Second-order effects (other than the RPA series) are time-consuming to compute, but estimates have been given by Shimizu [103] using a closure approximation. In tables 29 and 30 we bring together all the computed corrections from core polarisation (CP), meson-exchange currents (MEC) in lowest order, isobar excitations (A) in lowest order, core polarisation of the two-body MEC operators (MEC-CP) and the relativistic correction (Rel) introduced in section 6.2. In addition we must remember that single-particle states in the Pb region, especially the high-spin states, are not of pure single-particle configuration but have some core admixtures. For example, the lowest 13/2 ÷ state in 2°9Bi is not simply a proton i13/2 state but has a sizeable admixture of h9/2 × 3-. Hamamoto [183] has estimated the impact of core excitations on the magnetic moment expectation values in a particle-vibration model, and we include her results in the tables. The correction is small except for the i13/e states. There is a small amount of double counting between this correction and the second-order core polarisation, but we will ignore this. In table 29 we express the results in terms of the equivalent effective one-hotly operator Ior tile ca~e of a proton in the h9/2 orbit and a neutron in the i13/2 orbit. Consider first the orbital ~gL value. As was the case in light nuclei, there is considerable cancellation between core polarisation and mesonexchange currents. Our results of ~g~L~) = 0.107 and ~g~L~) = --0.065 are not too far from the empirical values determined in a best-fit analysis of magnetic moment data in Pb region of Yamazaki [129] of 0.15---0.02 and - 0 . 0 3 - 0.02 respectively. The spin 8g s values are dominated by core polarisation, all ~(~r)/_(~) ~(~)/-(~) other effects more or less cancelling each other. Our results of ~/~s ~gs = - 4 2 % and ~gs 'gs = - 4 6 % show a large quenching in the gs value consistent with the empirical relation often used: gs,eff = 0.5 gs" Also in table 29 we give the correction to the spin-flip M1 matrix element. Again a large quenching is evident as is required if one hopes to understand the weak excitation of the giant M1 resonance in 2°8pb. Finally in table 30 we compare these corrections with all the known magnetic moment and Ml-transition data in closed-shell-plus (or minus)-one nuclei in the Pb region. The agreement between theory and experiment is within 0.2/h~ in all cases except one, the neutron i13/2 state in a°Tpb, where the strong cancellation between core polarisation and meson-exchange currents seems to be too severe.

I.S. Towner, Quenching of spin matrix elements in nuclei

371

Table 29 Contributions to the equivalent effective one-body operator from all sources in the OBEP model for a 0h-proton and 0i-neutron in the Pb region Neutron 0i~3/2

l ~ o t o n Oh9/2

CP(RPA) MEC A CP(2ndf MEC-CPb Vib Rel Sum Expt

~gL

8gs

8gr

~/J" # %

5(M1) (M1) %

0.005 0.187 -0.003 -0.150 0.122 -0.030 -0.024

-1.167 0.435 -0.452 -1.030 0.352 -0.338 -0.152

0.481 -0.627 0.484

17.1 30.8 4.5 -12.0 17.3 - 0.3 - 2.0

0.107

-2.352

55.5 56.8

-0.041 0.297

~gL

5gs

8gP

-26.6 6.8 -10.8 -19.2 5.0 - 6.7 - 2.7

-0.005 -0.090 0.002 0.080 -0.062 0.010 0.000

1.034 -0.092 0.402 0.350 -0.176 0.118 0.107

0.102 1.025 -0.680

54.3

-0.065

1.743

0.447

0.000

5/z -/~ - %

5(M1) (M1) %

-25.9 26.4 - 8.4 -34.2 24.1 - 6.3 - 2.8

-26.9 2.7 -12.2 - 7.1 3.0 - 2.8 - 2.8

-27.0 -47.0

-46.1

From ref. [74]. b From ref. [105]. Table 30 Sum of all corrections to magnetic moments and B(M1) in the Pb re#on in the OBEP model compared with experiment ~/~ or 8[B(M1; i---*f)],:2 CP(RPA)

MEC

A

CP(2nd) a

MEC-CPb

"/r ~r Ir "#

hg/2 i13~2 s1-:2 d3-/12 f5:2""~f7/2 lr f7/2~ h9/2 ~r d3-:2~ s~-:2

0.45 -0.44 -0.45 0.21 -0.37 0.08 -0.06

0.81 0.99 0.26 0.25 0.08 -0.01 0.01

0.12 -0.17 -0.29 0.10 -0.21 0.05 -0.04

-0.32 -1.15 -0.61 0.05 -0.32

0.46 0.74 0.22 0.13 0.09

v g9/2 v Pl/2 V f5-/12 V il-31/2 V i11:2~ g9/2 V p3112"-~p ~/12 -1 -1 V f7:2-'~fs/2 V p3-~12~f~:2

0.38 -0.09 --0.30 0.50 0.02 --0.23 -0.30 -0.02

-0.31 -0.10 -0.31 --0.51 --0.03 0.02 0.03 0.04

0.19 -0.01 --0.11 0.16 0.05 --0.17 -0.18 -0.07

0.55 -0.01 0.13 0.66

-0.37 -0.01 --0.15 --0.46

--0.10 -0.11

0.05 0.05

Vib'

Rel

Sum

Exptd

-0.01 -0.54 0.01 -0.01 0.10 0.03 0.02

-0.05 -0.26 -0.06 0.01 -0.05 -0.00 0.00

1.45 -0.83 -0.92 0.74 -0.67 0.14 -0.07

1.49 -+0.01 -0.78-+0.10 -0.92-+ 0.01 ° 0.64-+ 0.19 f -0.82-+0.09 -+0.05-+0.01 -+0.15-+0.02

-0.06 -0.01 0.02 0.12 --0.01 0.02 0.07 0.00

0.05 -0.02 --0.03 0.05 0!00 --0.03 -0.03 0.00

0.44 -0.24 --0.74 0.52 0.03 --0.44 -0.47 -0.05

0.58-+0.06 -0.05-+0.00 --0.58-+ 0.03 0.90-+ 0.03 -+0.10-+0.01 --0.44-+0.05 -0.52-+0.11 -+0.22 -+0.03

a Second-order core polarisation estimated from effective operator of Arima and Hyuga [74]. b Core-polarisation correction to the MEC operator (crossing term) estimated from effective operator of Hyuga, Arima and Shimizu [105]. c Case 1 of Hamamoto [183]. d From references cited in Towner, Kharma and Hausser [93]. ° From Neugart et al. [184]. fFrom Hausser et al. [185].

For the/-forbidden M1 transitions, we obtain small corrections as required, but the comparison with experiment is not very revealing. 6.6. Outlook Our goal has been to provide as complete a calculation as possible, in the Hilbert space of nucleons and mesons, of the spin properties of nuclei as evident in magnetic moments and low-energy M1 and

I.S. Towner, Quenching of spin matrix elements in nuclei

372

Gamow-Teller transitions. The model is based on the one-boson-exchange potential and we have striven for consistency in the sense that the weak and electromagnetic two-nucleon exchange currents are related to the potential through their respective continuity equations. The calculations here are for finite nuclei. Thus there will always be some uncertainty due to the impreciseness of nuclear-structure calculations. We try to minimize this by considering states of simple configuration mainly just closed-shell-plus-(or minus)-one nucleon states. Nevertheless, the choice of the appropriate effective residual interaction has to be made and for consistency with the exchange current we have stayed with the one-boson-exchange potential. Others might choose differently. We also have to grapple with short-range correlations (from the repulsion in the nucleon-nucleon interaction) and the long-range tensor correlations each of which is handled in a model-dependent way. The bottom line is the comparison with the experimental data which we have given in section 6. It should be stressed that although some choices had to be made, such as the selection of values for certain coupling constants, no parameter was adjusted to fit magnetic moment or Gamow-Teller data. In summary, the results are very encouraging and an understanding of the very general phenomenon of quenching in spin matrix elements seems to be at hand. The comparison with experiment has become much more illuminating following the systematic studies in the sd-shell of Brown and Wildenthal [178-180]. By determining an empirical one-body operator from fits to a large number of data idiosyncracies in individual pieces of data are ironed out. A comparison between their empirical operators and our calculated ones, while good overall, does hint at a couple of deficiencies in our work. There is a suggestion of insufficient isobar contributions in the quenching of the Gamow-Teller matrix element and a hint that the spin part of the meson-exchange current M1 operator is too small. It is very popular these days to discuss nuclear properties in the quark model looking for evidence of quark degrees of freedom in nuclei. There are a number of hybrid models, such as that of Ito and Kisslinger [186], where short-distance behaviour is governed by a quark model joined at some radius to the more traditional model of nucleons and mesons for the large-distance behaviour. These models have been used to discuss magnetic moments and M1 transitions. There are approaches mentioned at the start of this section involving nucleons swelling in size within a nuclear medium. Again all these approaches rely on there being in place a very accurate and precise treatment in the traditional framework against which the newer models can be tested. In this sense, the deficiencies that have been identified in the comparison between our calculations and the empirical operators of Brown and Wildenthal could become very important. If these deficiencies stand up to further scrutiny then they should become the testbench for future calculations. For any new model, we suggest the following be computed: for a 0d nucleon in the sd-shell, compute the diagonal ds/z-ds/2, d3/z-d3/2 and off-diagonal ds/2-d3/2 matrix element for the M1 and GamowTeller operators. Express the results in terms of the equivalent effective one-body operators eqs. (5.6) and (5.9). The test quantities are then: (a)

~g~0~ _ ~gA × 0.88/1.26

(b)

~g~l)- ~gA × 4.706/1.26

which should be compared to the empirical values of Brown and Wildenthal of 0.08 + 0.02 and 0.39 + 0.07 respectively in A = 17. Our present calculation fails to meet these values by more than a factor of two. The former, (a), tests the isobar content of the calculation, the latter, (b), the meson-exchange currents (spin parts).

L S. Towner, Quenching of spin matrix elements in nuclei

373

In closing we note that magnetic moments have been a fascination for nuclear-structure physicists for nearly four decades. Despite the urge to move on to other things, they continue to demand our attention.

References [1] B.H. Brandow, Rev. Mod. Phys. 39 (1967) 771. [2] M. Baranger, in: Proc. Int. School of Physics 'Enrico Fermi', Course XL, Varenna, 1967, ed. M. Jean (Academic, New York) p. 511. [3] M.H. Madarlane, in: Proc. Int. School of Physics 'Enrico Fermi', Course XL, Varenna, 1%7, ed. M. Jean (Academic, New York) p. 457. [4] G.E. Brown, Unified theory of nuclear models and forces (North-Holland, Amsterdam, 1971). [5] B.R. Barrett and M.W. Kirson, in: Advances in Nuclear Physics Vol. 6, eds. M. Baranger and E. Vogt (Plenum, New York, 1973) p. 219. [6] T.T.S. Kuo, Ann. Rev. Nud. Sci. 24 (1974) 101. [7] P,J. Ellis and E. Osnes, Rev. Mod. Phys. 49 (1977) 777. [8] K. Erkeleuz, Phys. Reports 13 (1974) 191. [9] K. Holinde, K. Erkelenz and R. Alzetta, Nucl. Phys. A194 (1972) 161; K. Holinde and R. Machleidt, Nucl. Phys. A280 (1977) 429; K. Hofinde, Phys. Reports 68 (1981) 121; R. Machleidt, K. Holinde and Ch. Elster, Phys. Reports 149 (1987) 1. [10] K.A. Brueckner and C.A. Levinson, Phys. Rev. 97 (1954) 1344. [11] H.A. Bethe and J. Goldstone, Proc. Roy. Soc. (London) A238 (1957) 551. [12] B.R. Barrett, R.G.L. Hewitt and R.J. McCarthy, Phys. Rev. C3 (1971) 1137. [13] M. Lacombe et al., Phys. Rev. C21 (1980) 861. [14] J. Shurpin, T.T.S. Kuo and D. Strottman, Nucl. Phys. A408 (1983) 310. [15] A. Hosaka, K.I. Kubo and H. Toki, Nucl. Phys. A444 (1985) 76. [16] L.D. Landau, Soy. Phys. JETP 3 (1956) 920; 5 (1957) 101; 8 (1959) 70. [17] A.B. Migdal, Theory of finite Fermi systems and applicationsto atomic nuclei (Interscience, London, 1967). [18] K. Nakayama, S. Krewald, J. Speth and W.G. Love, Nud. Phys. A431 (1984) 419. [19] E. Oset, H. Toki and W. Weise, Phys. Reports 83 (1982) 281. [20] W.G. Love and J. Speth, Comments Nucl. Part. Phys. 14 (1985) 185. [21] J. Speth, E. Werner and W. Wild, Phys. Reports 33 (1977) 127. [22] W.H. Dickhoff, A. Faessler, J. Meyer-ter-Vehnand H. Miither, Phys. Rev. C23 (1981) 1154; Nucl. Phys. A368 (1981) 445. [23] W.H. Dickhoff, A. Faessler, H. Miither and S.S. Wu, Nucl. Phys. A405 (1983) 534. [24] K. Nakayama, S. Krewald, J. Speth and G.E. Brown, Phys. Rev. Lett. 52 (1984) 500. [25] K. Nakayama, S. Krewald and J. Speth, Phys. Lett. B145 (1984) 310. [26] O. Sjoberg, Ann. Phys. 78 (1973) 39. [27] S. Babu and G.E. Brown, Ann. Phys. 78 (1973) 1. [28] L.S. Celenza, W.S. Pong and C.M. Shakin, Phys. Rev. C25 (1982) 3115; M.R. Anastasio, L.S. Celenza, W.S. Pong and C.M. Shakin, Phys. Reports 100 (1983) 327. [29] G.E. Brown, S.O. Backman, E. Oset and W. Weise, Nucl. Phys. A286 (1977) 191. [30] J.W. Durso, G.E. Brown and M. Saarela, Nucl. Phys. A430 (1984) 653. [31] J.W. Durso, M. Saarela, G.E. Brown and A.D. Jackson, Nucl. Phys. A278 (1977) 445. [32] J. Wess and B. Zumino, Phys. Rev. 163 (1967) 1727. [33] E. Ivanov and E. Truhlik, Nucl. Phys. A316 (1979) 437. [34] V.I. Ogievetskyand B.M. Zupnik, Nucl. Phys. B24 (1970) 612. [35] M.L. Goldberger and S.B. Treiman, Phys. Rev. !10 (1958) 1178. [36] K. Kawarabayshi and M. Suzuki, Phys. Rev. Lett. 16 (1%6) 255; Riazuddin and Fayazuddin, Phys. Rev. 147 (1966) 1071. [37] S. Weinberg, Phys. Rev. 18 (1%7) 507. [38] F. IacheUo, A. Jackson and A. Lande, Phys. Lett. B43 (1973) 191. [39] J. Meyer-ter-Vehn,Phys. Reports 74 (1981) 323. [40] G.E. Brown and W. Weise, Phys. Reports 22C (1975) 279; W. Weise, Nucl. Phys. A278 (1977) 402. [41] M.R. Anastasio and G.E. Brown, Nucl. Phys. A285 (1977) 516. [42] G.E. Brown, E. Osnes and M. Rho, Phys. Lett. B163 (1985) 41. [43] G.E. Brown and M. Rho, Nucl. Phys. A372 (1981) 397.

374

I.S. Towner, Quenching of spin matrix elements in nuclei

[44] G. Bertsch, D. Cha and H. Toki, Phys. Rev. C24 (1981) 533. [45] C. Gaarde et al., Nucl. Phys. A369 (1981) 258. [46] D. Cha and J. Speth, Phys. Lett. B143 (1984) 297; F. Osterfeld, D. Cha and J, Speth, Phys. Rev. C31 (1985) 372. [47] I.S. Towner and F.C. Khanna, Nucl. Phys. A399 (1983) 334; Phys. Rev. Lett. 42 (1979) 51. [48] K. Shimizu, M. Ichimura and A. Arima, Nucl. Phys. A226 (1974) 282. [49} E. Hadjimichael, S.N. Yang and G.E. Brown, Phys. Lett. A196 (1972) 17. [50] E.G. Adelberger and W.C. Haxton, Ann. Rev. Nucl. Part. Sci. 35 (1985) 501. [51] G.A. Miller and J.E. Spencer, Ann. Phys. i00 (1976) 562. [52] T.A. Brody and M. Moshinsky, Tables of transformation brackets, monografias del instituto de fisica, Mexico, 1960. [53] G. Hohler and E. Pietarinen, Nucl. Phys. B95 (1975) 210. [54] O. Dumbrajs et al., Nucl. Phys. B216 (1983) 277. [55] H. Sugawara and F. von Hippel, Phys. Rev. 172 (1968) 1764. [56] D.M. Brink and G.R. Satchler, Angular momentum (Clarendon, Oxford, 1968). [57] F. Scheck, Leptons, Hadrons and Nuclei (North-Holland, Amsterdam 1983). [58] G.F. Chew and F.E. Low, Phys. Rev. 101 (1956) 101. [59] J. Katz and S. Tatur, Nuo. Cimento 40A (1977) 135. [60] E. Oset and M. Rho, Phys. Rev. Lett. 42 (1979) 47. [61] H. q;oki and W. Weise, Phys. Lett. B97 (1980) 12. [62] A. Harting, W. Weise, H. Toki and A. Richter, Phys. Lett. B104 (1981) 261. [63] A. Bohr and B. Mottelson, Phys. Lett. BI00 (1981) I0. [64] T. Suzuki, S. Krewald and J. Speth, Phys. Lett. B107 (1981) 9. [65] A. Arima, T. Cheon, K. Shimizu, H. Hyuga and T. Suzuki, Phys. Lett. B122 (1983) 126. [66] T. Cheon, K. Shimizu and A. Arima, Phys. Lett. B138 (1984) 345. [67] H. Toki, Proc. Int. Symp. on Electromagnetic Properties of Atomic Nuclei, Tokyo, 1983, eds. H. Horie and H. Ohnuma (Tokyo Institute of Technology) p. 307. [68] H. Sagawa, T.S.H. Lee and K. Ohta, Phys. Rev. C33 (1986) 629. [69] P.J. Ellis and H.A. Mavromatis, Nucl. Phys. A175 (1971) 309. [70] A. Arima and H. Horie, Prog. Theor. Phys. 11 (1954) 504; 12 (1954) 623. [71] R.J. Blin-Stoyle, Proc. Phys. Soc. A66 (1953) 1158. [72] Proc. Int. Symp. on Electromagnetic Properties of Atomic Nuclei, Tokyo, 1983, eds. H. Horie and H. Ohnuma (Tokyo Institute of Technology). [73] A. Arima and L.J. Huang-Lin, Phys. Lett. B41 (1972) 429; B41 (1972) 435. [74] A. Arima and H. Hyuga, in: Mesons in Nuclei, eds. D.H. Wilkinson and M. Rho (North-Holland, Amsterdam, 1979) p. 685. [75] I.S. Towner, A Shell Model Description of Light Nuclei (Clarendon, Oxford, 1977). [76] R.A. Broglia, A. Molinari and B. Sorensen, Nucl. Phys. AI09 (1%8) 353; J.D. Vergados, Phys. Lett. B36 (1971) 12; P. Ring and J. Speth, Phys. Lett. B44 (1973) 447; M.R. Anastasio and G.E. Brown, Nucl. Phys. A285 (1977) 516; S. Cwiok and M. Wygonowska, Act. Phys. Pol. B4 (1973) 233; W. Knupfer et al., Phys. Lett. B77 (1978) 367; R. Frey et al., Phys. Lett. B74 (1978) 45; G.E. Brown, J.S. Dehesa and J. Speth, Nucl. Phys. A330 (1979) 290; W. Knupfer, M. Dillig and A. Richter, Phys. Lett. B95 (1980) 349; J. Speth, V. Klemt, J. Wambach and G.E. Brown, Nucl. Phys. A343 (1980) 382; E. Lipparini and A. Richter, Phys. Lett. B144 (1984) 13. [77] T.S.H. Lee and S. Pittel, Phys. Rev. Cll (1975) 607; J.S. Dehesa, J. Speth and A. Faessler, Phys. Rev. Lett. 38 (1977) 208. [78] D. Cha, B. Schwesinger, J. Wambach and J. Speth, Nucl. Phys. A430 (1984) 321. [79] G.E. Brown and S. Raman, Comments Nucl. Part. Phys. 9 (1980) 79; R.J. Holt, H.E. Jackson, R.M. Laszewski and J.R. Specht, Phys. Rev. C20 (1979) 93; S. Raman, in: Neutron Capture ~/-Ray Spectroscopy, eds. R.E. Chrien and W.R. Kane (Plenum, New York, 1979) p. 193. [80] R.M. Laszewski and J. Wambach, Comments on Nucl. and Part. Phys. 14 (1985) 321. [81] D.E. Bainum et al., Phys. Rev. Lett. 44 (1980) 1751; D.J. Horen et al., Phys. Lett. B95 (1980) 27; J. Rapaport, AlP Conf. Proc. No. 42, ed. M.O. Meyer (American Institute of Physics, New York, 1983) p. 365. [82] K. Ikeda, S. Fujii and J.I. Fujita, Phys. Lett. 3 (1963) 271. [83] F. Osteffeld, Phys. Rev. C26 (1982) 762;

LS. Towner, Quenching of spin matrix elements in nuclei

375

F. Osterfeld and A. Schulte, Phys. Lett. B138 (1984) 23. [84] K. Ikeda, Prog. Theor. Phys. 31 (1964) 434; G. Bertsch, D. Cha and H. Toki, Phys. Rev. C24 (1981) 533; N. Auerbach, L. Zamick and A. Klein, Phys. Lett. Bl18 (1982) 256; K. Grotz et al., Phys. Lett. B126 (1983) 417; G.F. Bertsch, P.F. Bortignon and R.A. Broglia, Rev. MOd. Phys. 55 (1983) 287; EF. Bortignon and R.A. Broglia, Nucl. Phys. A371 (1981) 405. [85] N. van Giai and H. Sagawa, Phys. Lett. B106 (1981) 379. [86] G.F. Bertsch and I. Hamamoto, Phys. Rev. C26 (1982) 1323; H.R. Fiebig and J. Wambach, Nud. Phys. A386 (1982) 381; K. Muto, T. Oda and H. Horie, Phys. Lett. Bl18 (1982) 261; B. Schwesingerand J. Wambach, Phys. Lett. B134 (1984) 29; Nucl. Phys. A426 (1984) 253; D. Cha, B. Schwesinger,J. Wambach and J. Speth, Nucl. Phys. A430 (1984) 321; K. Takayanagi, K. Shimizu and A. Arima, Nucl. Phys. A444 (1985) 436. [87] H. Sagawa and N. van Giai, Phys. Lett. Bl13 (1982) 119; K. Grotz et al., Phys. Lett. B132 (1983) 22. [88] S. Krewald, F. Osterfeld, J. Speth and G.E. Brown, Phys. Rev. Lett. 46 (1981) 103. [89] A. de Shalit and I. Talmi, Nuclear Shell Theory (Academic, New York, 1963). [90] R.D. Lawson, Theory of the Nuclear Shell Model (Clarendon, Oxford, 1980). [91] A. Arima, The Structure of lf7~2 Nuclei, ed. R.A. Ricci (Editorice Compositori, Bologna, 1971) p. 385; H. Horie, J. Phys. Soc. Jpn. Suppl. 34 (1973) 461; L. Zamick, J. Phys. Soc. Jpn. Suppl. 34 (1973) 470; K. Arita, J. Phys. Soc. Jpn. Suppl. 34 (1973) 516; E Goode and L. Zamick, Part. and Nucl. 3 (1972) 125. [92] A. Arima, J. Phys. Soc. Jpn. Suppl. 34 (1973) 205; I. Tonozuka, K. Sasaki and K. Harada, J. Phys. Soc. Jpn. Suppl. 34 (1973) 475; K. Arita, J. Phys. Soc. Jpn. Suppl. 34 (1973) 516; K. Arita, Proc. Int. Conf. on Nuclear Physics, Munich, 1973, vol. 1, eds. J. de Boer and H.J. Mang (North-Holland, Amsterdam, 1973) p. 264. [93] I.S. Towner, F.C. Khanna and O. Hausser, Nucl. Phys. A277 (1977) 285. [94] M. Nomura, Phys. Lett B40 (1972) 522. [95] C.M. Lederer and V.S. Shirley, Table of Isotopes (7th ed., New York, 1968). [96] O. Hausser et al., Hyperfine Interactions 4 (1978) 219. [97] I.S. Towner, Nucl. Phys. A444 (1985) 402. [98] EJ. Ellis and S. Siegel, Phys. Lett. B34 (1971) 177. [99] M. Ichimura and K. Yazaki, Nucl. Phys. 63 (1965) 401. [100] H.A. Mavromatis and L. Zamick, Phys. Lett. 20 (1966) 191; Nucl. Phys. A104 (1967) 19. [101] T.T.S. Kuo and G.E. Brown, Nucl. Phys. 85 (1966) 40. [102] J.E Vary, EU. Saner and C.W. Wong, Phys. Rev. C7 (1973) 1776. [103] K. Shimizu, Z. Physik A278 (1976) 201. [104] R.H. Dalitz, Phys. Rev. 95 (1954) 799. [105] H. Hyuga, A. Arima and K. Shimizu, Nucl. Phys. A336 (1980) 363. [106] M. Chemtob and M. Rho, Nucl. Phys. A163 (1971) 1; M. Chemtob and M. Rho, Phys. Lett. B29 (1969) 540. [107] F.E. Low, Phys. Rev. 110 (1958) 974. [108] S.L. Adler and Y. Dothan, Phys. Rev. t51 (1966) 1267; S.L. Adler, Ann. of Phys. 50 (1968) 189; S.L. Adler, Proc. 6th Hawaii Topical Conf. in Particle Physics, Honolulu, Univ. of Hawaii, 1975, pp. 1-191. [109] M. Rho, Prog. in Part. Nucl. Phys. 1 (1978) 105. [110] S.L. Adler and R.F. Dashen, Current Algebras (Benjamin, New York, 1968). [111] V. de Alfaro, S. Fubini, G. Furlan and C. Rossetti, Currents in Ha&on Physics (North-Holland, Amsterdam, 1973). [112] M. Rho and G.E. Brown, Comments Nucl. Part. Phys. 10 (1981) 201; M. Rho, Ann. Rev. Nucl. Part. Sci. 34 (1984) 531. [113] K. Kubodera, J. Delorme and M. Rho, Phys. Rev. Lett. 49 (1978) 755. [114] E.A. Ivanov and E. Truhlik, Soy. J. Part. Nucl. 12 (1981) 198. [115] M. Chemtob, in: Mesons in Nuclei, eds. D.H. Wilkinson and M. Rho (North-Holland, Amsterdam, 1979). [116] D.O. Riska, Prog. in Part. Nud. Phys. 11 (1984) 199. [117] J.F. Mathiot, Nucl. Phys. A412 (1984) 201.

376

I.S. Towner, Quenching of spin matrix elements in nuclei

[118] O. Dumbrajs et al., Nucl. Phys. B216 (1983) 277. [119] T. Jensen et al., Phys. Rev. D27 (1983) 26. [120] Particle Data Group, Phys. Lett. B170 (1986) 1. [121] J. Adam and E. Truhlik, Czech..1. Phys. B34 (1984) 1157. [122] D.O. Riska, Physica Scripta 31 (1985) 107. [123] D.O. Riska, Nucl. Phys. A350 (1980) 227. [124] E. Truhlik, private communication. [125] T. deforest and D. Walecka, Adv. Phys. 15 (1966) 1; H. Uberall, Electron Scattering from Complex Nuclei (Academic, New York, 197l); T.W. Donnelly and J.D. Walecka, Ann. Rev. Nucl. Sci. 25 (1975) 329: T.W. Donnelly and I. Sick, Rev. Mod. Phys. 56 (1984) 461. [126] H. Miyazawa, Prog. Theo. Phys. 6 (1951) 801. [127] T. Yamazaki, T. Nomura, S. Nagamiya and T. Katou, Phys. Rev. Lett. 25 (1970) 547. [128] S. Nagamiya and T. Yamazaki, Phys. Rev. C4 (1971) 1961. [129] T. Yamazaki, in: Mesons in Nuclei, eds. D.H. Wilkinson and M. Rho (North-Holland, Amsterdam, 1979) p. 651. ]130] J. Adam, E. Truhlik and P.U. Sauer, Czech. J. Phys. B36 (1986) 383. [131] H.J. Lipkin and T.S.H. Lee, Phys. Lett. B183 (1987) 22. [132] W. Rarita and J. Schwinger, Phys. Rev. 60 (1941) 61. [133] M. Ericson, A. Figureau and C. Thevenet, Phys. Lett. B45 (1973) 19; J. Delorme, M. Ericson, A. Figureau and C. Thevenet, Ann. Phys. 102 (1976) 273. [134] M. Rho, Nucl. Phys. A231 (1974) 493. [135] Proc. of Int. Syrup. on Weak and Electromagnetic Interactions in Nuclei, ed. H.V. Klapdor (Springer-Verlag, Heidelberg, 1986). [136] R. Davidson, N.C. Mukhopadhyay and R. Wittman, Phys. Rev. Lett. 56 (1986) 804. [137] P. Noelle, Prog. Theo. Phys. 60 (1978) 778; W.F. Grushin et al., JETP Lett. 39 (1984) 491; H. Tanabe and T. Ohta, Phys. Rev. C31 (1985) 1876: S.N. Yang, J. Phys. Gll (1985) 1205. [138] H.T. Williams, Phys. Rev. C31 (1985) 2297. [139] R.D. Lawson, Phys. Lett. B125 (1983) 255. [140] J.V. Noble, Phys. Rev. Lett. 46 (1981) 421. ]1411 J.J. Aubert et al., Phys. Lett. B123 (1983) 275: R.G. Arnold et al., Phys. Rev. Lett. 52 (1984) 727. [142] L.S. Celenza, A. Rosenthal and C.M. Shakin, Phys. Rev. Lett. 53 (1984) 892; L.S. Celenza, H. Harindranath, A. Rosenthal and C.M. Shakin, Phys. Rev. C31 (1985) 63; C31 (1985) 1944. [143] M. Rho, Phys. Rev. Lett. 54 (1985) 767. [144] F.E. Close, R.G. Roberts and G.G. Ross, Phys. Lett. B129 (1983) 346; R.L. Jaffe, F.E. Close, R.G. Roberts and G.G. Ross, Phys. Lett. B134 (1984) 449; F.E. Close, R.L. Jaffe, R.G. Roberts and G.G. Ross, Phys. Rev. D31 (1985) 1004. [145] C.H. Llewellyn Smith, Phys. Lett. B128 (1983) 107; M. Ericson and A.W. Thomas, Phys. Lett. B128 (1983) 112; B148 (1984) 191: G.V. Dunne and A.W. Thomas, Nucl. Phys. A455 (1986) 701. [146] J.D. Walecka, Ann. Phys. 83 (1974) 491; B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16 (1986) l. [147] L.D. Miller, Ann. Phys. 91 (1975) 40. [148] T. de Forest, Phys. Rev. Lett. 53 (1984) 855. [149] A. Bouyssy, S. Marcos and J.F. Mathiot, Nucl. Phys. A415 (1984) 497; H. Kurasawa and T. Suzuki, Phys. Lett. B165 (1985) 234; W. Bentz et al., Nucl. Phys. A346 (1985) 593; J.A. McNeil et al., Phys. Rev. C34 (1986) 746; P.G. Blunden, Nucl. Phys. A464 (1987) 525; G.E. Brown, W. Weise, G. Baym and J. Speth, Comments Nucl. Part. Phys. 17 (1987) 39; J. Delorme and I.S. Towner, to be published. [150] P.J. Mulders, Phys. Rev. Lett. 54 (1985) 2560. [151] R. Barreau et al., Nucl. Phys. A402 (1983) 515; C. Marchand et al., Phys. Lett. B153 (1985) 29; Z.E. Meziani et al., Phys. Rev. Lett. 52 (1982) 52; 54 (1985) 1233. [152] G. van der Steenhoven et al., Phys. Rev. Lett. 57 (1986) 182. [153] G. Karl, G.A. Miller and J. Rafelski, Phys. Lett. B143 (1984) 326.

I.S. Towner, Quenching of spin matrix elements in nuclei

377

[154] I. Sick, Nucl. Phys. A434 (1984) 677c; Phys. Lett. B157 (1985) 13. [155] T.E.O. Ericson and A. Richter, Phys. Lett. B183 (1987) 249. [156] A.M. Green and T.H. Schucan, Nucl. Phys. A188 (1972) 289. [157] M. Ichimura, H. Hyuga and G.E. Brown, Nucl. Phys. A196 (1972) 17. [158] D.O. Riska and G.E. Brown, Phys. Lett. B32 (1971) 662. [159] J.S. Bell and R.J. Blin-Stoyle, Nucl. Phys. 6 (1957) 87; H. Ohtsubo, M. Sano and M. Morita, Prog. Theo. Phys. 49 (1973) 877; N.C. Mukhopodhyay and L.D. Miller, Phys. Lett. B47 (1973) 415. [160] S. Ichii, W. Bentz and A. Arima, Nucl. Phys. A464 (1987) 575. [161] A. Arima, K. Shimizu, W. Bentz and H. Hyuga, Nuclear magnetic properties and Gamow-Teller transitions, Adv. in Nucl. Phys., to be published. [162] N. Anantaraman et al., Phys. Rev. Lett. 52 (1984) 1409, [163] B.A. Brown, unpublished, as quoted in Arima et al. [161]. [164] R. Schneider et al., Nucl. Phys. A323 (1979) 13. [165] J.B. McGrory and B.H. Wildenthal, Phys. Lett. B103 (1981) 173. [166] K. Muto and H. Horie, Nucl. Phys. A440 (1985) 254. [167] W. Steffen et al., Phys. Lett. B95 (1980) 23; W. Steffen et al., Nucl. Phys. A404 (1983) 413. [168] F. Ajzenberg-Selove, Nucl. Phys. A449 (1986) 1. [169] R.J. Holt, H.E. Jackson, R.M. Laszewski, J.E. Monahan and J.R. Specht, Phys, Rev. C18 (1978) 1962. [170] S. Raman, C.A. Houser, T.A. Walkiewicz and I.S. Towner, At. Data Nucl. Data Tables 21 (1978) 567. [171] J.J. Simpson, Phys. Rev. C35 (1987) 752. [172] J. Rapaport et al., Nucl. Phys. A431 (1984) 301. [173] C.D. Goodman et al., Phys. Rev. Lett. 54 (1985) 877. [174] J.W. Watson et al., Phys. Rev. Letr 55 (1985) 1369. [175] I.S. Towner and F.C. Khanna, Jour de Physique 45 (1984) C4-519. [176] T.K. Alexander et al., submitted to Nucl. Phys. [177] E.G. Adelberger, J.L. Osborne, H.E. Swanson and B.A. Brown, Nucl. Phys. A417 (1984) 269. [178] B.A. Brown and B.H. Wildenthal, Phys. Rev. C28 (1983) 2397. [179] B.A. Brown and B.H. Wildenthal, At. Data Nucl. Data Tables 33 (1985) 347. [180] B.A. Brown, Proc. Int. Conf. on Nuclear Physics, Harrogate 1986, eds. J.L. Durell, J.M. Irvine and G.C. Morrison (IOP Publishing) p. 119; B.A. Brown and B.H. Wildenthal, Empirically optimum M1 operator for sd-shell nuclei, Nucl. Phys., in press. [181] B.H. Wildenthal, Prog. in Part. and Nucl. Phys. 11 (1984) 5. [182] B. Desplanques, Phys. Lett. B141 (1984) 285. [183] I. Hamamoto, Phys. Lett. B61 (1976) 343. [184] R. Neugart et al., Phys. Rev. Lett. 55 (1985) 1559. [185] O. Hausser, Proc. Int. Symp. on Electromagnetic Properties of Atomic Nuclei, Tokyo, 1983, eds. H. Horie and H. Ohnuma (Tokyo Institute of Technology) p. 145. [186] H. Ito and L.S. Kisslinger, Ann. of Phys. 174 (1987) 169.