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Folded diagrams and 1s-Od effective interactions derived from Reid and Paris nucleon-nucleon potentials
Nuclear Physics A408 (1983) 310-358 @ North-Holland Publishing Company
FOLDED DIAGRAMS AND Is-Od EFFECTIVE INTERACTIONS DERIVED FROM REID AND PARIS NUCLEON-NUCLEON POTENTIALS J. SHURPIN’
and T. T. S. KU0
Physics Dept., State university of New York at Stony Brook, Stony Brook, NY 11794, USA
D. STROTTMAN Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87544, USA Received 18 January 1983 (Revised 20 April 1983) Abstract: The sd-shell effective-interaction matrix elements are derived from the Paris and Reid potentials using a microscopic folded-diagram effective-interaction theory. A comparison of these matrix elements is carried out by calculating spectra and energy centroids for nuclei of mass 18 to 24. The folded diagrams were included by both solving for the energy-dependent effective interaction self-consistently and by including the folded diagrams explicitly. In the latter case the folded diagrams were grouped either according to the number of folds or as prescribed by the Lee and Suzuki iteration technique; the Lee-Suzuki method was found to converge better and yield the more reliable results. Special attention was given to the proper treatment of one-body connected diagrams in the calculation of the two-body effective interaction. We first calculate the (energy-dependent) G-matrix appropriate for theasd-shell for both potentials using a momentum-space matrix-inversion method which treats the Pauli exclusion operator essentially exactly. This G-matrix interaction is then used to calculate the irreducible and nonfolded diagrams contained in the &-box. The effective-interaction matrix elements are obtained by evaluating a [S-box folded diagram series. We considered four approximations for the basic @box. These were (Cl) the inclusion of diagrams up to 2nd order in G, (C2) 2nd order plus hole-hole phonons, (C3) 2nd order plus (bare TDA) particle-hole phonons, and (C4) 2nd order plus both hole-hole and particle-hole phonons. The contribution of the folded diagrams was found to be quite large, typically about 30x, and to weaken the interaction. Also, due to the greater energy dependence of higher-order diagrams, the effect of folded diagrams was much greater in higher orders. That_ is, the cont~bution from higher-order diagrams for most cases was greatly reduced by the folded diagrams. The convergence of the folded-diagram series deteriorates with the inclusion of higher-order &-box processes in the method which groups diagrams by the number of folds, but remains excellent in the Lee-Suzuki method. Whereas the inclusion of the particle-hole phonon was essential to obtain agreement with experiment in earlier work, when the folded diagrams are included the effect of the particle-hole phonon is to reduce the amount of binding. All four approximations to both potentials produce interactions which badly underbind nuclei. The excitation spectra given by these interactions are, however, all rather similar to each other. The Paris interaction produces more binding than does the Reid, but differences between results obtained with the two interactions were often less than
+ Present address: Telas Computer Systems, 71 West 35th St., NY, NY 10001. 310
J. Shurpin et al. / Folded diagrams
311
differences obtained in the four approximations. Essentially no difference was found between the effective noncentral interactions from the Reid ‘and Paris potentials after including the folded diagrams, although these two potentials themselves are quite different, especially in the strength of the tensor force. Comparisons between.calculated spectra and experiment were done for ‘*O, r8F, 19F, “‘0, “‘Ne, 22Ne, 22Na and *4Mg.
1. Introduction
An outstanding problem in nuclear physics is to describe the properties of finite nuclei in terms of the interactions between their constituent nucleons. To describe low-energy phenomena, usually a non-relativistic approximation of the nucleonnucleon (NN) interaction is carried out to construct a non-relativistic NN potential, V,,. One must then solve the non-relativistic many-body Schrodinger equation. This problem has been the subject of numerous articles and several reviews l, 2). As is well known, one cannot proceed directly with this ambitious prescription because of several major difficulties. To begin with, the NN potential is itself not completely understood. Until recently, to circumvent this problem one constructed phenomenological potentials which were required to reproduce as much of the experimental data, such as lowenergy phase shifts and deuteron properties, as possible. Examples of potentials of this type are the Yale potential 3), the Hamada-Johnston potential 4), and the Reid potential ‘), the last being by far the most widely used in recent years. This procedure, however, is not quite satisfactory because several different potentials can (and indeed do) yield the same phase shifts. Also, phenomenological potentials do not necessarily give a good theoretical basis for the interaction. A potential which is largely free of this latter shortcoming is the recently proposed Paris potentia16) which is largely theoretically derived. The long- and medium-range parts (r 5 0.8 fm) reflect n, 2rr,and 37r(w) exchange. Only the short-range part, which should be important for high-energy phenomena, is constructed so as to fit the phase shifts. So far, the Paris potential has been tested mainly in nuclear matter where it gives reasonable results ‘). A major aim of the present work is to study the results the Paris potential gives for finite nuclei. To solve the nuclear many-body problem, perturbation techniques adapted from quantum field theory and statistical mechanics were developed. A difficulty encountered immediately was that the matrix elements of V,, are either infinite (hard-core potentials) or at least very large (soft-core potentials) when the relative wave functions are non-vanishing near the origin (e.g. in a harmonic oscillator basis). Brueckner and his collaborators *) summed the interactions between pairs of nucleons to all orders to obtain the reaction matrix, or G-matrix interaction, whose matrix elements are finite. The perturbation theories were then expressed in terms of G rather than V,,. The earliest studies were carried out in infinite nuclear matter
312
J. Shurpin et al. / Folded diagrams
by Brueckner, Bethe and Goldstone ‘* 9). The Goldstone linked-cluster nondegenerate perturbation expansion lo) can also be used for closed-shell nuclei such as I60 and leads to the so-called Brueckner-Hartree-Fock (BHF) theory rr). For open-shell nuclei, i.e. nuclei with valence nucleons such as 180, several approaches have been developed. There are the energy-dependent theories of Feshbach 12) and of Bloch and Horowitz 13). In these theories, the effecttive interaction secular equation must be solved at the self-consistent energies. The energy-independent formulation, which leads to folded diagrams, has been developed via time-dependent perturbation theory by Morita 14), Oberlechner et al. 15), Johnson and Baranger 16), and by Kuo, Lee and Ratcliff “), while Brandow 18) used time-independent perturbation methods. The various energyindependent effective interactions are consistent with each other ‘). In addition, their form is similar to that of the empirical effective interaction, which is also energy independent. In this work, we employ effective interactions of the type obtained by Kuo, Lee and Ratcliff. In this formulation the non-folded diagrams are first grouped together to form the @box interaction, and then the folded diagrams are included in a very systematic way. Recent developments in this area have been made by Kuo and Krenciglowa 19). These will be examined and applied here. In addition, the energy-dependent effective interaction has been shown to be equivalent to the folded-diagram effective interaction 20). We will look at this point in actual calculations. Another difficulty which had plagued calculations in finite nuclei is the presence of the Pauli exclusion operator in the equation for the G-matrix. This has been largely overcome in the method of Tsai and Kuo 21), which is used in conjunction with momentum-space matrix inversion (to find G,, the G-matrix in free space, with no Pauli operator) as described by Krenciglowa, Kung, Kuo and Osnes”). This method is particularly suitable for the Paris potential (which is momentum dependent) and is the one we have used to construct our G-matrix. Thus the general procedure is as follows: We cannot solve the nuclear manybody problem directly. Instead, we must solye a model-space problem of linite (low) dimension whose solutions are in principle, the same as the low-lying states of the complete (infinite dimensional) problem. For example, in our prototype nucleus “0 we consider the shell-model problem of two sd-shell nucleons outside and 160 closed core. The model space in which we seek to solve this problem consists of all states with two sd-shell particles. Clearly, there are many other possible configurations for “0, such as 2 + n sd-shell (or higher-shell) particles and n holes in the 160 core. What we must do is to find within the model space an effective interaction between the two particles which takes into account those configurations we do not include explicitly. This is done by the $-box formulation l’) of the folded-diagram theory. There are still various remaining problems and open questions in the procedure referred to above, and we will address some of them here. The $-box contains
J. Shurpin et al. / Folded diagrams
313
diagrams to all orders in the interaction (I&+ or equivalently G). Using existing methods it is quite difficult to evaluate even the third-order diagrams. This difficulty is twofold. First, the formulae for the diagrams require tedious and lengthy derivations. Second, the computation time required to numerically evaluate the diagrams using the formulae so obtained makes the calculations practically impossible. A new method was recently proposed by Kuo, Shurpin, Tam, Osnes and Ellis23) which gives a simple and systematic way of writing down the diagram formulae almost by inspection. The formulae so obtained are often easier to interpret physically than previous ones. The method has the added advantage that it yields diagram formulae which are usually much more efficient for numerical computation. This method will be used in the present work. Perhaps the most important remaining question regarding the effective interaction is whether, and under what conditions, it converges. This question has received a great deal of attention in recent years ‘p2). In the early calculations of Kuo and Brown24) only two terms in the series were retained, the bare G-matrix and the 3plh (second-order) core-polarization diagrams. Yet their results agreed fairly well with experiment, and subsequent, far more elaborate calculations, yielded no dramatic improvement. This seemed to suggest that the effectiveinteraction series converged very rapidly, or at least that the higher-order terms cancelled among themselves. Not long afterward, the validity of this was questioned by the work of Barrett and Kirson 25). They showed that many of the third-order diagrams were of comparable size with the second-order corepolarization diagram. Moreover, the third order as a whole was generally of opposite sign to the second order. Thus truncating the series after the second order seemed not to be justified. In the folded-diagram theory the question of convergence can be divided into two parts ; first, whether the o-box converges and, second, whether the folded-diagram series converges. In view of the Kuo and Brown results, which included no folded diagrams, there is also the question of why, or indeed whether, we need to include the folded diagrams at all. This work will attempt to address these and related matters. We now briefly outline the structure and content of the sections which follow. We begin in sect. 2 with a brief review of the basic formalism of microscopic theories and model-space problems leading to the diagramatic expansion for the effective interaction. The notions of active and passive lines and linked and irreducible diagrams lead to a definition of the &-box, the basic entity in our expression for the effective interaction. The grouping of terms in the foldeddiagram series is then examined, especially as it.relates to one-body and two-body connected graphs. We briefly review also two techniques to obtain effective interactions which are also expressed in terms of the &-box. The first is the one derived by Kuo and Krenciglowa 19) starting from the equation of motion. We show how this method allows us to include the “hole-hole phonon” which considers the interaction between two hole lines to all orders. The second is the iteration formula for the effective interaction of Lee and Suzukiz6).
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Sect. 3 contains a brief discussion of the NN interaction. Special attention is given to its momentum-dependent form. We carry out various comparisons between the Paris and Reid potentials. We also do a series of phase shift calculations using the momentum-space matrix-inversion method 27). This is a necessary test of our computer code for our calculation of the Paris potential Gmatrix which we will use in all subsequent nuclear structure calculations. Sects. 4 and 5 describe an extensive series of calculations for sd-shell nuclei, based upon the ideas developed in the previous sections. Most of these are done in parallel for the Paris and Reid potentials. The hope is that we can perhaps trace the nuclear structure results back to the features of the free NN interaction, especially for the Paris potential. We study the effects of various refinements of the &-box, such the inclusion of hole-hole and TDA particle-hole phonons. Our calculation is done with an energy-dependent G-matrix and this enables us to include the off-energy-shell effects in the calculation of effective-interaction diagrams. Other components of the calculation such as our choice for the single-particle energies are also investigated. Our main effort is directed towards understanding the structure and function of the folded-diagram series and its convergence properties. The numerical results for the sd-shell matrix elements are tabulated. The spectra of several sd-shell nuclei (‘*O, ‘sF 19F 22Na, 22Ne) are presented and discussed. We will compare the various spectra both among themselves and with experiment. In sect. 6 we discuss the conclusions to be drawn from our various results and suggest several possibilities which may improve our results.
2. Folded-diagram effective interactions 2.1. INTRODUCTION
As outlined in sect. 1, our aim is to perform a microscopic nuclear-structure calculation based on a realistic NN potential V = I&,. That is, for a nucleus with A nucleons we try to solve the non-relativistic Schriidinger equation H’&1,2 ,... A)=E$(1,2
,... A),
(2.1)
where H=T+V,
(2.2)
T being the kinetic energy operator. It is clear that even for light nuclei we are unable to solve eq. (2.1), the A-body problem, exactly. The way to proceed is to project the wave function $ of the complete Hilbert space onto a model space of low dimension in which we are able to solve the problem in a more physical way.
J. Shurpin et al. / Folded diagrams
We first rewrite the hamiltonian
31.5
as
H=H,+H,,
(2.3a)
Ho= T+U,
(2.3b)
H,= V-U,
(2.3~)
where V is the NN interaction and U is some single-particle potential. U is chosen so as to define a convenient set of unperturbed basis functions 4, given by
&A = VP,.
(2.4)
Frequent choices for U are a harmonic oscillator potential, which we will use in the present calculation, or a Hartree-Fock single-particle potential. We then divide the complete Hilbert space into two regions P (the model space) and Q (its complement) such that P+Q
= 1,
(2.5a) (25b) (2.5~)
(ila)
= 0,
(2.5d)
where both Ii) and’ ICC)are eigenfunctions (i.e. hi and 4,) of the unperturbed hamiltonian H,.The division of the Hilbert space (i.e. what to choose as the model space) depends on the particular problem, and one is guided by physical intuition. For 180 a reasonable choice would be a closed 160 core with 2 particles in the sd shell. Thus in eqs. (2.5). Ii) is any state with a closed I60 core and 2 particles in thgsd shell while la) represents the respective complement (in which all states have holes in the I60 core and/or particles in shells above the sd shell). We seek to reduce eq. (2.1) to a more tractable problem:
PH,,,W = EW,
(2.6a)
H err
(2.6b)
where =
H, + V,,,.
Note that in eq. (2.6) E and tj are the same as in eq. (2.1). For a model space (P-
316
J. Shurpin et al. / Folded diagrams
space) of dimension d, Heff will reproduce d of the eigenstates of H. One must obtain the form of V,,, which must account for all states belonging to Q that are now being left out of the problem. There are several forms for V,,, to be found in the liteature. There are energydependent forms, I/,,,(E), which have been obtained by Feshbach 12) and by Bloch and Horowitz i3). In this work we will use the folded-diagram theory of I&, in the form derived by Kuo, Lee and Ratcliff l’) (henceforth ‘referred to as KLR). This form of V,,, is independent of E. This is a desirable feature, since the empirical matrix elements (which are obtained by constructing a V,,, which can best reproduce the experimentally observed properties of various sd-shell nuclei) are also independent of E. 2.2. FOLDED-DIAGRAM
The folded-diagram
EXPANSION
OF
I&
expansion l’) for Hi” (or V,,,) is
(2.8) where Q is the Q-box and 1 the generalized folding operation. The first Q in each term which is folded upon is written with a prime (Q’) to indicate that the lowestorder diagram it contains is at least second order in the interaction. The Q-box is composed of irreducible linked diagrams ” ’ ‘). It can be shown ‘) that V,,, of eq. (2.7) can be written in terms of derivatives of the Q-box (the so-called derivative formulation) as
v,,,= F,+F,+F,+F,+...,
(2.8)
where F, is the term with n foldings. As will be discussed later, a main part of our calculations will be carried out using this F, expansion for I!,,,. The KLR-model-space eigenvalue problem, after cancellation of the unlinked diagrams, is (2.9) where E; is the true l6 0 ground-state energy and (ii@) is the projection of the true wave function $ onto the model space. Note that the eigenvalue here is (En-E:) the energy relative to the 160 ground state. Observe that I$,, contains l-body diagrams such as diagram (b) in fig. 1. The sum of the l-body contributions is actually what determines the “0 energies. It is customary in eq. (2.9) to replace H, by HYP, i.e. by the experimental sd-shell single-particle levels (which are taken as the low-lying “0 states). As this accounts for the l-body diagrams, we must not
317
J. Shurpin et al. / Folded diagrams
include them a second time in V,,,. That is, V,,, should contain only the 2-body connected diagrams. For the first term, F,, this is easily done. We simply do not include the l-body diagrams in the $-box. For the subsequent terms, this cannot be done in this way. Consider fig. 2, where in each case the dashed line represents a &-box. Diagrams (a) and (c) belong to F, and diagrams (b) and (d) to F,. A “blob” attached to a line represents any sequence of interactions attached to that line. Thus (c) and (d) should be excluded from I/em since their contribution is already accounted for by HrP. Diagram (b), however, should still be in I/err,since it is 2-body connected. Indeed, its contribution can be quite large. It is only diagram (d) which we must remove. To this end, the g-box is introduced2a). The s-box is defined as the l-body part of the &-box (i.e. the set of all linked and irreducible diagrams in which all the interactions are connected to one line and the other line is merely a “spectator”). What must be removed from V,,, are all the terms in which the $-box folds upon itself only. Thus we have the following:
----_
_--.
_-_---
_D 10 -_-
---_iI
0
_-_- /’ _-_--___ t
-__ --__--_ 1$I ‘$ (0)
(c)
(b)
--0
_-_-
(d)
___
(e)
Fig. 1. Examples of linked and unlinked, reducible and irreducible diagrams.
318
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___ Ii 1+ “/
:
( ‘/3 ; 2
(a)
,’
‘8
\-
-__ !
(cl
Fig. 2. One-body and two-body connected diagrams in the folded-diagram series for qrr The dashed line represents the Q-box interaction between the two lines. A “blob” represents a series of V (or U) interactions attached to the line.
and so forth, where (2.11) with o0 being the degenerate model-space energy (PH, = Pw,). We now define
F”= MB-F,(9),
(2.12)
where s^ is the l-body box which we just introduced. Clearly the difference P, contains only the proper 2-body connected diagrams of V,,,. This gives us the equation we will actually use in our calculations,
v,,, = F,+F,+F,+F,+F,....
(2.13)
We include terms up to F, only. The number of terms in F, grows very rapidly with n, as can be seen from eq. (2.10). In sect. 4 we will explore the convergence properties of the folded-diagram series for V,,, when we group the terms according to the number of folds as in eq. (2.13).
J. Shurpin et al. 1 Folded diagrams 2.3. THE EQUATION-OF-MOTION
319
METHOD FOR v,,
Recently, Kuo and Krenciglowa 19) have proposed a somewhat different and more general method to obtain a folded-diagram expansion for the effective interaction. This method enables us to sum up a larger class of diagrams contained in V,,, as compared with the KLR method outlined in the previous section. The starting point of the KLR method is a model space whose states consist of two (sdshell) particles outside the unperturbed 160 core, [a:aT]lc). Thus the eigenfunctions we obtain in solving eq. (2.9) are the projections of the true wave functions onto this “uncorrelated” model space. At the heart of the newly proposed method is to use as a starting point lt&), the true (correlated) ground state of 160 in place of 1~). The model-space basis states are thus of the form [a+af]t&). An advantage of this is that when we obtain a secular equation of the form eq. (2.9) its eigenfunctions will be
which can be related to experimentally observable spectroscopic factors. This cannot be done if we use 1~). We do not give any details of the method here, except to point out some of its specific implications for our calculations described in sect. 4. In this method we write V,,, as (2.14) where the bar over the first Q-box in each term indicates the last interaction (attached to the outgoing valence lines) must take place at t = 0. All the other interactions are now constrained by cc > t > -co. As in eq. (2.9), the first 0 in each folded term is at least second order in the interaction. The value of any diagram (i.e. the diagram rules) will in general be different in the new and old schemes. We do not pursue this point except to evaluate in detail one diagram which will be important in our calculation. Consider the diagram in fig. 3. We suppress factors (such as I/ matrix elements) which do not depend upon which method we use and concentrate on the time integrations only. In the old scheme the time constraints are to = 0,
o>t,
> -co,
o>t,>
-co,
t,>t,>
-00,
whereas in the new scheme they are tz = 0,
o>t,>
-co,
co > t, > t,,
t,>t,>
-co.
J. Shurpin et al. 1 Folded diagrams
320
a
b
Fig. 3. The 4p2h diagram
I
which is evaluated
differently
in the old and new schemes.
Thus in the old scheme we have for this diagram the energy factor 1
1
wo-(&,+&A
1
0O-(&,+&b+&,+&d-&h,-Eh,)
(E~,+E&(E,+EB)’
(2.15)
while in the new scheme we have 1 oo-(sy+sA
1
1 -%-(-%,-+h,)
(2.16)
(sh*+E&(E,+EJ
where we have defined the starting energy o. as
Note the on-energy shell expression for the middle energy denominator obtained in the new scheme. This allows us to sum up-the 4p2h diagrams to infinite order as we will do in sect. 4. In the old scheme we could not do this. Note also that in fig. 3 the vertex a time t2 is the “last” interaction attached to the (outgoing) valence line. 2.4. THE
LEE-SUZUKI
ITERATION
METHOD
FOR
v,,,
In our calculation of l&, the first step is to calculate the @box. Then we calculate the folded-diagram series of eq. (2.7) or (2.14) using either the F, method of eq. (2.13) or the Lee-Suzuki iteration method 26) which is here outlined. The Lee-Suzuki method calculates the effective interaction V’,, in an iterative scheme. Denoting the nth iteration for the effective interaction as R,, we have R,
J. Shurpin et al. / Folded diagrams
321
related to R, (k< n)by II-1
R,=
n-1
(l-t-%- 1 &m
(2.17)
In=2
where Q is the Q-box interaction defined in subsect. 2.2 and &. represents the energy derivative of the Q-box as defined in eq. (2.14). An advantage of this method is that we only need to know Q and Q, at a given starting energy wO. As we will discuss later in sect. 4, the calculated V,,, is essentially independent of oO. (Theoretically, V,,, is independent of oO.) The Q-box here is the same as in the KLR theory or the method described in the preceeding section, and it enters in a perhaps more natural and convenient way. It contains all the linked and irreducible diagrams but no folded diagrams. The folded diagrams enter by way of eq. (2.17), which provides a way of regrouping the folded-diagram series. We will examine how this grouping affects the convergence of V’, in sect. 4, where V,,, will be calculated by first evaluating the Q-box and its derivatives and then calculating R, according to eq. (2.17). 3. Method of calculation In the last section, we obtained an expansion for the effective interaction in terms of matrix elements of the NN interaction I/. It has been common practice to use for I/ phenomenological NN potentials such as the Reid potential 5). In the present work we will use both this potential and the Paris potential ‘) as V. We will first compare some essential features of these two potentials, and then discuss some details ‘of the momentum-space matrix-inversion method on which our calculation will be based. 3.1. COMPARISON
BETWEEN
THE PARIS AND REID NN POTENTIAL
A main purpose of the present work is to compare the spectra of several sd-shell nuclei calculated from the Paris NN potential ‘) with those from the Reid NN potential. It is important to note that these two potentials are in fact quite different, although they give quite similar NN phase shifts. The Reid NN potential is energy independent, and for each partial wave channel is given as a sum of a few Yukawa-type terms, i.e. &e-~1~/p~r, where ci and CLiare-parameters and I is the internucleon distance. On the other hand, the Paris potential has a linear energy dependence which is, for convenience, transformed into a p2 dependence. Then, the Paris potential, in its parametrized form ‘), is given in r-space for each isospin T as
(3.1)
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where
(3.2)
(3.3) all in appropriate units. The labels s, t, LS, T, and SO, refer to the standard singlet, triplet, spin-orbit, tensor, and quadratic spin-orbit non-relativistic invariants, respectively. The V(r)‘s are expressed as a sum of Yukawa-type terms. For example, for the central singlet component of the potential one has f& = a(1 -br. igr)=
a,),
igjz
j=
1
(3.4) (3.5)
I
in eqs. (3.1) and (3.2) respectively. The potential is regularized at the origin (Y= 0) by requiring n-1
gn=-rn,C
%. j=l
(3.6)
mj
The form of the other components as well as the numerical values of the parameters gj and mj are found in ref. ‘). Although it is possible to work with this potential directly in configuration space, it is somewhat inconvenient to do so. Fortunately, the situation is much simpler if a plane-wave basis is used, as in the present work. Then the plane-wave matrix elements of the Paris potential are given simply as
($ +~)
(3.7)
for any partial wave *‘+lL. 3’ Here k and k’ are the magnitudes of the relative momenta of the outgoing and incoming nucleon pairs respectively. For convenience, numerical integration was used. This was checked against the analytic expression for the IS,, partial wave and found to be quite accurate. In fig. 4 we plot V(k, k) for the Paris potential ’ (solid line) and for the Reid potential (dashed line), for partial waves IS, to 3D,. One is immediately struck by the fact that, although both potentials were made to fit the same scattering data, the two potentials are distinctly different. Similar differences have also been found for other partial waves.
J. Shurpin et al.
323
/ Folded diagrams jr
-\ \ \
lL.A
4
\
\
\
3
jSI
\
\
‘1
‘\
2
I
2
3
4
5
k
I
2
2
3
4
5
3
4
5
k
3 k
I
4
5
I
2 k
Fig. 4. Plot of V(k, k) for the Paris (solid line) and Reid (dashed line) potentials for selected partial waves. The k are in fm- ’ and V is in units of 41.47 MeV . fm3.
In the Paris potential the components other than the central ones are not energy dependent and are functions of I alone. Thus they can be compared with their counterparts in the Reid potential directly in configuration space. We do this in fig. 5 for the important T = 0 S = 1 tensor force +. As shown, the tensor forces for these two potentials are quite different. It is of interest to see how this difference manifests itself in nuclear-structure calculations. Before proceeding to the actual nuclear structure calculation we study the Paris potential NN phase shifts and compare with those of the Reid potential. A more compelling reason for looking at the phase shifts is that the method used to calculate them (momentum-space matrix inversion) is also used in the calculation of the G-matrix and thus provides a check of our computer programs.
+ In the early stages of this work, the Paris potential parameters were taken from a preprint. In the published version’) the T = 0 parameters are somewhat different. Thus, in fig. 4 the preprint parameters were used, as they were in the phase-shift calculations of the following section. Fig. 5, however, used the published parameters. (The preprint had an even weaker tensor force.) Of course, for the G-matrix elements the published parameters were used.
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!
I I II 0.5
0
I 1.0
I 2.0
I 1.5
2.5
r (fm)
Fig. 5. Comparison
of the T = 0, S = 1 tensor component between the Paris (solid line) and Reid (dashed line) potential.
3.2. PHASE-SHIFT
CALCULATIONS
To calculate NN scattering phase shifts, we use the momentum-space matrixinversion method as outlined in ref. “). Briefly, for an uncoupled partial wave, the phase shift is given by (3.12)
6,(k) = -tan-‘[kR,(k,k)], where R,(k, k’) is given by the principal value integral equation R,(k, k’) = I/;(k, k’)-
a
m I/;@,q)R,(q, U q2dq
P s0
q2
-
k”
(3.13)
This integral can be rewritten to avoid the principal-value singularity”). A similar expression exists for the coupled-channels case. To solve eq. (3.13) we use gausian integration. To determine the validity of this procedure, we chose several quite different sets of Gauss points and looked at the resulting phase shifts to see if and how they were affected. The integral (3.13) from 0 to cc is broken up into several intervals. The intervals
325
J. Shurpin et al. 1 Folded diagr~s
are specified by S,, the start of the nth interval and by N,, the number of Gauss points in the interval between S,_ 1 and S,,. For finite intervals, the Gauss points qi (and weights) are mapped from the region - 1 I xi 5 -+ 1 onto the interval. If the interval extends from S, to infinity, we use a tangent mapping to obtain the Gauss points qi = S, + C tan&( 1+ xi).
(3.14)
N, is the number of points for which the tangent mapping is used. The constant C is for adjusting the distribution of the tangent Gauss points. The total number of points is N,,. The procedure is essentially the same as that of ref. “). The choices of Gauss points for several calculations are specified in table 1, all with C = 1. Tables 2a and 2b show the phase shifts for the Paris potential calculated with Gauss point sets 1 and 2 of table 1. Note that they agree remarkably well with each other despite the large difference between the Gauss points used. We have found that the distribution of Gauss points is far more important than the number of points used. In table 2c we quote the phase shifts obtained in ref. ‘). Note that the phase shifts we calculate are the eigenphases whereas the ones from ref. 7, are the nuclear bar phase shifts. The two can be related by equating the expressions for the S-matrix obtained from the two methods. [See e.g. ref. ““).] We have done this and found the agreement with the publish~ values to be very good for table 2b. We found it to be important to map the Gauss points onto several intervals rather than one interval covering the entire range from 0 to cx). The phase shifts (and hence the R-matrix) were found to be quite stable with respect to reasonable variations of the Gauss points. For the G-matrix calculation we have used the Gauss points set (G) of table 1. For comparison, the Reid-potential phase shifts are given in table 2d. [These are the ones we calculated using Gauss points set 3 of table 1, i.e. eigenphases, but when converted into the nuclear bar phases they agree closely with those given by Reid in his paper ‘).I The significant difference between the two potentials is in the
TABLE1 Specification of th,e Gauss points Set
So
s,
S?.
S,
N,
N,
1
0 0 0 0
3.0 2.75 2.6 2.5
10.0 8.0 7.0 7.0
25.0 18.0 15.0 15.0
6 11 11 15
5 7 7 10
2 3 G
N3
N,
4 4 4 6
0 3 3 4
N,,
15 2.5 25 35
The boundaries of the integration intervals are the S, (in fm-‘) and N, and N, are the number of points in the interval. The last row (G) refers to the Gauss points used in the G-matrix calculation of subsect. 3.4.
326
J. Shurpin et al. 1 Folded diagrams TABLE 2a
Results of phase-shift calculations (in degrees) for the Paris potential calculated with the momentumspace matrix inversion method using Gauss points set (1) of table 1 Et,, (MeV) =
‘So ‘D* 3P0 -1 -2 E2 3F2
‘P, % “1 3D, -2
25
50
95
142
210
330
48.68 0.75 9.20 - 5.24 2.91 - 17.59 -0.19
38.42 1.84 11.93 -8.60 6.53 - 16.53 -0.18
25.25 3.79 10.26 - 12.91 11.63 -14.11 0.10
14.92 5.65 6.12 - 16.55 14.97 - 12.02 0.44
3.18 7.76 -0.45 -21.21 17.21 -9.74 0.62
- 12.46 9.47 - 11.01 - 28.50 16.85 - 6.94 -0.12
- 6.99 80.51 1.84 - 2.94 3.86
- 10.66 62.31 2.20 -6.79 9.59
- 14.75 43.43 2.79 - 12.22 18.10
- 17.65 30.16 3.80 - 16.44 23.84
-20.81 15.97 6.06 - 20.81 28.00
-25.00 - 1.46 13.86 - 25.49 28.75
‘P, partial wave. Otherwise, the phase shifts are very similar. Of course it must be remembered that both potentials were constructed to reproduce the phase shifts. Still, the potentials themselves are markedly different and one cannot predict in advance their results in a nuclear structure calculation. 3.3. THE G-MATRIX
In sect. 2 the effective interaction is given as a series of diagrams, each containing matrix elements of the NN interaction V. In general, matrix elements of TABLE2b Same as table 2a, but using Gauss points set 2 of table 1 E,,, (MeV) =
‘SO ‘D2 3P0 3P, jp2 82 3F2
‘Pl % &2 -* ‘D2
25
50
95
142
210
330
48.58 0.75 9.19 - 5.25 2.91 - 17.60 -0.19
38.31 1.84 11.93 -8.60 6.51 - 16.53 -0.18
25.18 3.79 10.25 - 12.94 11.59 - 14.12 0.10
14.90 5.64 6.10 - 16.61 14.90 - 12.03 0.44
3.21 7.75 - 0.45 -21.32 17.11 -9.75 0.62
- 12.48 9.44 -11.04 - 28.75 16.74 -6.95 -0.12
- 7.01 so.25 1.80 -2.90 3.87
- 10.67 62.04 2.18 - 6.77 9.59
- 14.79 43.16 2.78 - 12.30 18.03
- 17.74 29.95 3.81 - 16.54 23.73
- 20.93 15.91 6.09 - 20.87 27.85
- 25.21 - 1.44 13.83 - 25.70 28.48
J. Shurpin et al. / Folded diagrams
321
TABLE2c Same as table 2a, but for the Paris potential quoted from ref. ‘); see text for explanation E,,, (MeV) =
‘So
25
‘D, ‘Pll 3P, 3P, s2 3P*
48.51 0.75 8.87 -5.04 2.44 -0.85 0.11
38.74 1.81 11.82 - 8.41 5.13 - 1.78 0.35
25.85 3.12 10.35 - 12.76 10.66 -2.6-l 0.77
15.68 5.56 6.31 - 16.47 14.06 -2.91 1.04
4.03 1.66 -0.16 -21.27 16.49 -2.71 1.08
- 11.80 9.31 - 10.79 - 28.90 16.44 -2.00 0.14
‘PI js, 82 3D, 3D,
- 7.03 19.81 1.72 -2.95 3.97
- 10.75 61.56 1.96 -6.16 9.62
- 14.87 42.12 2.24 - 12.29 17.96
- 17.87 29.61 2.69 - 16.52 23.58
-21.13 15.57 3.59 - 20.73 21.51
- 25.52 -2.61 5.54 -29.15 28.13
I/ are infinite or very large even for soft-core potentials. This is especially true in an oscillator basis with non-vanishing relative wave function at the origin. To overcome this, the reaction matrix, or G-matrix was introduced *). In operator language we have in our case 21) G(w,) = I’+ VQ wg
_‘eTQ Q’%Ad
(3.15)
where Q is the Pauli exclusion operator and oO is the starting energy. Diagrammatically, eq. (3.15) is expanded in fig. 6. Note that intermediate lines TABLE2d Same as table 2a, but for the Reid potential using Gauss points 3 of table 1 E,,, (MeV) =
25
50
95
142
210
330
“P,
49.09 0.68 8.60 -4.60 3.02 - 15.86 -0.15
38.19 1.66 11.55 - 8.09 7.10 - 14.40 -0.12
25.64 3.39 10.22 - 13.16 13.10 - 12.09 0.18
15.52 5.11 5.98 - 17.39 16.97 - 10.54 0.55
4.32 7.18 - 1.18 - 22.23 19.51 -9.11 0.90
- 10.14 9.31 - 13.29 - 28.46 19.78 - 7.26 0.79
‘PI 3S, s1 3D, “D,
- 1.94 81.17 1.90 -2.91 4.13
-4.26 62.80 2.58 -6.91 10.10
- 10.67 43.85 3.91 - 12.28 17.80
- 17.47 30.90 5.98 - 16.15 22.13
- 26.07 17.80 9.12 -20.18 24.98
- 31.94 2.51 18.05 - 26.04 26.08
‘SO ‘D* jPll jP, 3P* 61
J. Shurpin et al. / Folded diagrams
328
=
+
--+. +II_----
Fig. 6. Definition of the G-matrix between two particle lines.
must be passive lines - the railed lines which represent the single-particle states higher than the pf shell of fig. 7. Otherwise the diagram is not allowed in the Q-box (recall subsect. 2.2). Thus all the diagrams in the Q-box and hence all the diagrams contributing to V’,, have their V-vertices “replaced” by G-vertices in analogy with fig. 6. Since the matrix elements of G are finite we can proceed to calculate the matrix elements of V,,,. We do not go into the details of the G-matrix calculation, as we closely follow the procedure of ref. “). We only make several notes in passing. As pointed out, eq. (3.15) is similar in structure to eq. (3.13) and we use the same momentum space method to solve it. We use 35 k-space Gauss points distributed as indicated by the last line of table 1. This choice should be adequate, judging by the phase-shift calculations described in the previous section. Also, in eq. (3.15) we use a twoparticle Pauli operator Q specified by (n,, n,,, nrrr) = (3, 10, 21). That is, the intermediate states included in G must lie outside the shaded region in fig. 7. The choice of Q is very important in determining which G-matrix diagrams are allowed in the Q-box as well as in determining the allowed orbits for intermediate states of
pfh -
nI
nr
ht
Fig. 7. Specification of the Pauli operator Q used in the calculation of the G-matrix. It is defined by (n,, n,,, n,,,) = (3, 10, 21) with the orbit numbering as explained by the circled numbers.
J. Shurpm
et al. / Folded diagrams
329
$-box diagrams. [See e.g. ref. ““).] Our procedure was as follows. We started from the Reid or Paris Potential, obtained V(k, k’) using gaussian integration, and then G,(k, k’), the “free” reaction matrix by gaussian integration and matrix inversion. Gaussian integration is again used to obtain (~~~ZlG~(~)l~~~‘~) in a rel-GM oscillator basis. The Pauli operator was treated by the Tsai-Kuo method *I), which expresses eq. (3.15) in terms of P, the complement of Q, and G,. Finally, everything is gathered together and transformed to the laboratory frame oscillator basis by means of a Moshinsky transformation. We obtain JT G(ab, cd; JT)
JT (3.16)
n l-l = (ab(Glcd),
where a, b, c, d refer to the oscillator orbits specified by (n871rl,j 0)* etc. J and T are total angular momentum and isospin, respectively, and the coupling is as indicated by the arrows in eq. (3.16). The matrix elements (3.16) are precisely what we need for nuclear-structure calculations. Extensive computer programs have been constructed for carrying out the above G-matrix calculation. Tables of G-matrix elemer,ts will be given later in sect. 4. 3.4. CALCULATION OF &-BOX DIAGRAMS
In the previous sections we have sketched the development of the effective interaction V,,, in terms of the $-box formulation of the folded-diagram theory. The &-box is the sum of all the linked and irreducible diagrams. Thus we need a set of rules to enable us to evaluate the diagrams in a simple systematic way. The diagram rules in the uncoupled representation or m-scheme are well known 2, 30).
3
J’M’
Fig. 8. The 3plh
,J’M’
core-polarizationdiagram(a) is more easily evaluated when recoupled as in (b).
330
J. Shurpin et al. / Folded diagrams
For actual nuclear-structure calculations, we will need to evaluate diagrams in the angular-momentum-coupled representation, such as the core-polarization diagram of fig. 8. Recently, a set of simple diagram rules has been developed 23) to evaluate these angular-momentum-coupled diagrams, and we will employ it to calculate all the @box diagrams in the present work. To illustrate the above method, we give in the following the formula for calculating diagram (a) of fig. 8. This diagram is first recoupled to become diagram (b), using the usual X or 9-j transformation coefficients. Then diagram (a) is readily given 23) as
(a) =
c
d
j
[‘J,
b J’
j o
r
1 &J+1)(2T+1)
C+ J’T’
; &2J’+1)(2T’+l)
I
J’T
(3.17)
where the last matrix elements are of the cross-coupled type 23) and the energy variables are or = E, + E,, and o2 = E, +.sd - E, +E~, the E’Sbeing the single-particle energies.
4. Results for two-body effective interactions 4.1. THE &-BOX
DIAGRAMS
The next step in calculating an effective interaction is to evaluate the &-box (and its derivatives). The Q-box is composed of all the linked and irreducible (G-matrix) diagrams. We take as a first approximation to the Q-box all the diagrams up to second order in the G-matrix. The diagrams we include are Dl through D7 and U (the one-body potential) as shown in fig. 9. In diagram D5 the intermediate particle states are restricted to the pf-shell because the G-matrix was calculated with a Puli operator defined by (n,, n,,, n,,,) = (3, 10,21) which takes into account passive particle states of orbit 11 and above only. Orbits 7-10 (the pf-shell) must be included explicitly. For the same reason, particles in the pf-shell are also included in the intermediate states of diagram D2. Our @box, expressed in terms of the G-matrix, takes into account the particleparticle interaction (outside the model space) to all orders. If we use the equation of motion method of Kuo and Krenciglowa (subsect. 2.3) we can extend our Q-box
J. Shurpin et al. 1 Folded diagrams
331
0
P 2 C
P2
P3
Fig, 9a. The one-body diagrams included in various approximations to the Q-box. The shaded circle represents a (bare) TDA particle-hole phonon in diagrams P2 and P3.
P6
P7
Fig. 9b. The two-body diagrams included in various approximations to the Q-box. The shaded circle represents a (bare) TDA particle-hole phonon in diagram P7 and a hole-hole phonon in P6.
332
J. Shurpin et al. 1 Folded diagrams
to include the hole-hole interaction to all orders as well. That is, the two hole lines of diagram D6 are allowed to interact any number of times so that D6 is replaced by diagram P6. (See fig. 9.) In P6, the blob represents a hole-hole phonon. This refinement of the Q-box approximation can have a significant effect on the matrix elements of V,,, and the resulting spectra. Another modification is in the treatment of the core-polarization processes. In place of the particle-hole pair in D2, D3 and D7 we can include a TDA particle-hole phonon. That is, D2, D3 and D7 are replaced in the &box by P2, P3 and P7, respectively. In obtaining the TDA eigenvalues and eigenvectors, we include only the bare (first order) G-matrix particle-hole interaction vertex as shown by diagram (a) of fig. 10. Because we include the TDA phonons only as a “correction” to the particle-hole intermediate states, inclusion of only this bare vertex should be sufficient. Also, it has been demonstrated that the inclusion of all the second order particle-hole interaction vertices, such as diagram (b) of fig. 10, in RPA-phonon calculations gives results very similar to the ones obtained in bare TDA only22 32). One result of including the TDA states rather than simple particle-hole states is that we can now isolate the spurious l- state in diagrams P2 and P3. This cannot be done if we include only the particle-hole intermediate state by second-order perturbation theory because then all the l- states (there are 7 of them) are mixed. The TDA phonon intermediate state allows us to isolate the spurious l- state and remove it in lowest order. This is done by explicitly removing the spurious lcontributions in the calculation of diagrams P2 and P3. The removal of the spurious state changes P2 and P3 by about 0.6 MeV. The results we will quote here are from calculations in which the spurious state is removed. However, to gauge the effect of including TDA phonons in place of simple particle-hole states, we really should leave the spurious state in for the comparison to be valid, since we do not remove it in the simple particle-hole case. The calculation was done in a harmonic oscillator basis with hQ = 14 MeV. The unperturbed starting energy oO, the point at which we calculated the effective interaction, was taken to be - 10 MeV (for two sd-shell particles). The different
(a)
(b)
Fig. 10. The bare TDA vertex (a), and one of the second-order TDA vertex (b).
J. Shurpin et al. / Folded diagrams
333
approximations to the &box are referred to as Cl, C2, C3 and C4 for the &-box with 2nd order diagrams only, 2nd order plus hole-hole phonon, and 2nd order plus particle-hole phonon, and 2nd order plus both hole-hole and particle-hole phonons, respectively. Thus, the Cl interaction includes diagrams D4, D5, D6 and D7 of fig. 9b ; for the C2 interaction, D6 is replaced by P6, etc. Effective interactions obtained with these four methods will be compared in sect. 5. 4.2. INCLUSION OF THE FOLDED DIAGRAMS
In the early microscopic calculations of this type23 24) the folded diagrams were generally left out and the effective interaction was taken to be the &-box alone (in some low order approximation such as the bare G plus 3plh core-polarization diagrams). Barrett and Kirson 25) have investigated several once-folded diagrams and found their effects important. In the present work, we shall investigate the effect of folded diagrams in a more extensive and systematic way. When more processes are included in the &-box, it will become apparent that the $-box alone gives substantial overbinding as well as poor level spacing. (See e.g. fig, 11 for n = 1, which corresponds to V,,, = 6.) Thus we must include the folded diagrams to renormalize the effective interaction. This we can do in one of the three ways described below. It has been shown2’) that the Q-box evaluated at a self-consistent starting energy is equivalent to the effect of V,,, with all the folded diagrams included. That is,
[Ho+ K,tll’&> = [Ho-f &(E,)]I ‘y,> = E,l ‘y,),
(4.1)
where E, is the total energy of the nuclear system relative to the exact ground-state energy of the core system. For example, E, is E,(190)-Eo(‘60) when the above equation is used to calculate the energy spectrum of r90. As discussed earlier, V,,, is related to the @boxes by, for example, eqs. (2.13~(2.16) and in calculating I$,, we need to evaluate the folded diagrams involving the &-boxes. Eq. (4.1) is in some aspects more convenient for-calculation because we just evaluate the &box at the self-consistent energy without the need for calculating any &-box folded diagrams. For a given o-box, this self-consistent energy can usually be obtained by a graphical method. There is, however, a rather important disadvantage associated with the above self-consistent equation. Namely, &(E,) is dependent on the total energy of the nuclear system and for different nuclei and different nuclear states the @boxes will need to be recalculated. In other words, this formulation does not lead to a set of matrix elements of V,,, which can be used to treat the low-lying states of all the sd-shell nuclei, unlike the case of the empirical shell model. For example, we would like to have a set of two-body matrix elements of V,,, which can be used to calculate both the ground state of ‘*O and the excited states of
334
J. Shurpin et al. 1 Folded diagrams
20Ne. The matrix elements of $(E,) of eq. (4.1) do not meet this requirement. Another inconvenience of &!Z,,) is that it contains unlinked valence diagrams, whereas V,,, contains only linked valence diagrams. Thus in the present work we will not use the self-consistent formulation of eq. (4.1) for major calculations except for checking the convergence of the folded-diagram series for simple cases I80 and i8F. A second method is to sum the folded-diagram series explicitly as described in sect. 2. Using the method of grouping terms according to the number of folds we can proceed in one of two ways. We can use eqs. (2.12) and (2.13) to obtain what are usually called the sd-shell effective-interaction matrix elements. In this case, the energy levels obtained are relative to l’0 since the one-body terms are removed from V,,, and are replaced by the experimental I70 low-lying states. Alternatively, we can retain the one-body diagrams in V,,, (i.e. use F, = F,(Q)) in which case the energy levels obtained are relative to the 160 ground state. In either case, all the & (see eqs. (2.10~(2.13)) are evaluated at the unperturbed starting energy oo, at which point all the matrix multiplication is carried out. A third way to include the folded diagrams is to group them as prescribed by the Lee-Suzuki iteration method. Here one also uses the 0, evaluated at w. but the terms are grouped differently. One solves for the eigenvalues of the Ho + R,, where R, is defined by eq. (2.17), increasing n until the eigenvalues converge. At that point one has essentially included all of the folded diagrams and
Kff = &
(4.2)
is obtained. As written, eq. (2.17) contains the one-body graphs in the Q-boxes so that the eigenvalues obtained are the energy levels relative to the I60 ground state. However, one may obtain the usual sd-shell matrix elements (i.e. two-bodyconnected only) by removing the one-body graphs from V,,,. This is done by doing two parallel calculations R,e = (I--&
- ‘.x- Q, m=2
R;r = (l-9,
-
c $m
RF)-‘0,
(4.3a)
k=n-m+l
(4.3b)
m=2
and then, after convergence is reached, taking the eigenvalues of (Ho +Rf -Rz). In this case, of course, the one-body part of V,,, is taken from experiment. We thus have three ways to take into account the folded diagrams: (i) the selfconsistent graphical method (SC), with an energy-dependent effective interaction, and (ii) the method of grouping according to the number of folds (F,), and (iii) the Lee-Suzuki iteration method (LS), where V,,, from either method (ii) or (iii) is
J. Shurpin et al. / Folded diagrams
335
energy-independent. Before going on to compare the results of the three methods in the following sections, we make several observations. For the energy-dependent effective interaction (SC method) we need to know (only) the &-box, but at all energies. -For the energy-independent interaction (F, or LS method) we need to know the &box and all its derivatives, but at a single energy only. As mentioned earlier, the F, and LS method directly lead to a set of energy-independent matrix elements of V,,, which can be used for any sd-shell nuclei, but not for the SC method.
4.3. CONVERGENCE
Before stating the results of the calculations we have outlined above, we first investigate the reliability of the methods used to obtain them. The most important questions at this point are whether the diagramatic series for V,,, converges and what are some of the factors which influence convergence. In the previous section, we described three ways in which we can take into account the contribution from the folded diagrams. In the SC method, there is no problem of convergence (unless perhaps at points where Q(w) is singluar. [See ref. ‘).I We now examine the structure of the folded-diagram series in the F, method as it is embodied in eqs. (2.10) through (2.13). The energy (0) dependence of the &-box is of central importance. The @box is defined as a sum of diagrams, so that its energy dependence is the sum of the energy dependences of the individual diagrams. The energy dependence of the first-order diagrams (Dl and D4) is practically linear. The second-order diagrams have a more pronounced energy dependence. When phonons are included, the energy dependence is even stronger because most of the energy dependence comes from the energy denominators associated with low-lying intermediate states whereas the G-matrix itself has a relatively weak energy dependence. Thus, the higher order the process, the stronger the energy dependence. The result of this is that as one goes from Cl to C4 (i.e. includes higher-order processes as discussed in subsect. 4.1) the size of &(“), the nth derivative of 0, increases. The higher derivatives show an especially large increase. The effect of this on the folded-diagram series terms F, is easily predicted by eqs. (2.10~(2.13). As seen in table 3, as one goes from Cl to C4 the size of F, increases. Also, the relative increase is larger for higher n. The above discussion of how well V,,, converges when the terms are grouped according to the number of foldings is clearly reflected in the resulting eigenvalues (spectra). In fig. 11 we show the spectra (relative to l’0) of V,,, where
K,, = i ?I=0
F,,
(4.4)
336
J. Shurpin et al. / Folded diagrams TABLE3
Representative V,, matrix elements in the F, method using the Paris potential with starting energy w0 = - 10 MeV T
Jabcd
FO
104444 104455 104466 105544 105555 105566 106644 106655 106666 1 0 104455 104466 105544 105555 105566 106644 106655 106666
4
4
4
4
- 2.9467 -4.2159 - 1.1982 -4.2159 - 0.9605 -0.9140 - 1.1982 -0.9140 -2.3319
0.9698 0.8920 0.3679 1.3098 - 0.0552 0.2817 0.4570 0.2052 0.8047
-0.3173 0.0841 -0.0913 0.0232 0.0206 - 0.0054 -0.1436 0.0055 -0.0935
0.0611 0.0032 0.0109 - 0.0408 0.0535 - 0.0084 0.0285 0.0050 - 0.0436
-0.0060 - 0.0362 0.0009 - 0.0490 0.0124 - 0.0077 -0.0031 - 0.0098 0.0184
-2.2391 - 3.2728 - 0.9098 - 2.9726 - 0.9293 - 0.6539 - 0.8594 - 0.7080 - 1.6485
- 3.0753 -4.3978 - 1.3937 -4.3978 - 1.0489 - 0.9222 - 1.3937 - 0.9222 - 2.9024
1.1547 0.8670 0.4764 1.4623 -0.1873 0.3616 0.8599 0.5364 1.3358
- 0.3888 0.4197 0.1102 0.4520 -0.1352 0.0153 - 0.491 0.1605 -0.0541
- 0.0648 0.3979 -0.0219 0.3467 - 0.0785 0.0222 0.0921 0.1410 -0.2569
0.0287 - 0.2491 0.0170 0.1094 -0.0324 0.0174 0.0125 -0.0518 0.0378
- 2.3455 - 2.9623 - 1.0324 - 2.0275 - 1.4823 - 0.5057 -0.9533 -0.1361 - 1.8399
The upper half is for calculation Cl and the lower half for C4. The orbits (4, 5, 6) represent respectively (Od,,,, Od,,,, lsliz). All entries are in MeV.
for N = 0 to 4 using the Paris interaction. For the Cl (i.e. second-order diagrams only) calculation the convergence is quite satisfactory and is not shown. For the C4 calculation, however, convergence has markedly deteriorated. To study the convergence of V,,, when the folded diagrams are grouped according to the LS scheme we look at fig. 12 where we give the spectra of V,,, = R, (actually, of Rf - Ri) for n = 1 to 6. Fig. 12 shows the “worst case” calculation C4 for T = 1 and T = 0. It is clear from the figure that even for this case the LS series converges very rapidly for all the states. It appears, on the basis of its relatively rapid and smooth convergence, that the LS method is preferred over the F, method. The argument in favor of LS over F, is strengthened by the following. Consider table 4 in which we compare the eigenvalues of all three methods, SC, LS, and F, for all four &-box approximations. (All are for the Paris potential.) Since we include the SC method in the comparison, the eigenvalues in table 4 are relative to r60. The agreement between the SC and LS eigenvalues is striking. (Note that our numbers are really only valid in the interval -20 to -6 MeV because we have calculated the &box only in this energy interval.) Actually, the F, numbers are also in good agreement generally, except for the TJ = 01 ground state for which the F, method gives a rather poor result. Thus we &n say that the LS iteration formula reaches the “exact result” (i.e.
.I. Shurpin et al. / Folded diagrams
331
‘80
4_
/------
32-
;:
__I
-1;
/-2-z-
o-4 -5 -‘;
r I 0
I I
I 2
1 3 n
1 4
LS
Fig. 11. The A = 18 spectra as a function of the number of terms n in the F, method. The LS results (after six iterations) are included for comparison. These results are for the Paris potential in approximationC4 which converged the most poorly of those considered.
the self-consistent solution) after only 6 iterations. [In the SC calculation, (i) selfconsistency was considered to have been reached with agreement to within 0.00005 MeV, and (ii) we know the &-box at 8 points only so that the higher derivatives are less reliable. In view of these two limitations, the agreement attained is indeed remarkable.] We therefore use the LS spectra as the standard against which we measure the convergence of the F, series in fig. 11. If we know the @box and its derivatives, we can find V,,, with only a small number of iterations when using the LS method for calculating the folded diagrams. The other half of the problem, what to include in the basic &-box, still remains. 4.4. THE EFFECTIVE
INTERACTION
We now present results of the various calculations we have been describing. We first examine the importance of the spurious particle-hole TJ” = Ol- state which enters into diagrams D2(P2) and D3(P3). As pointed out earlier we can isolate its effect when we include diagrams P2 and P3 in place of D2 and D3. The spurious state’s contribution to the diagrams is very large, often more than 10%. The reason, of course, is that because it has a particle-hole eigenvalue o_+,,,close to
338
J. Shurpin et al. 1 Folded diagrams
4 I
-4
-2
-2 2 /
-3
of------6
-
-7
-
.
-0
-4, -5;
-0
I
I
1
I
I
I
I
2
3
4
5
6
”
-
I
Exp
I
I 123456
I
1
I
I
n
Fig. 12. The A = 18 spectra after n iterations in the LS method for calculation C4 using the Paris potential. The experimental spectra are included for comparison. The ground-state energies, relative to “0, are from ref. 33) and the excitation energies are from ref. 34).
zero the energy denominator (wO-mph) is relatively small. However, P2 and P3 enter with. opposite sign so that the spurious state contribution to the effective interaction tends to cancel out. When the one-body diagrams are included (i.e. if we do not use the experimental l’0 levels so that the spectra are relative to r60) the effect can be as large as 0.1 MeV or more. When the one-body part of V’,, is taken from experiment (i.e. diagrams P2 and P3 contribute only in the folded terms which are two-body connected) the effect is quite small, 20 keV or less. Thus we can proceed to make valid comparisons between the different @box approximations (calculations Cl, C2, C3 and C4) even though the spurious state is not treated consistently (Cl and C2 include it while C3 and C4 remove it in lowest order). Unperturbed single-particle energies E were used in both the hole-hole and particle-hole phonon calculations. Our results are not to be compared with some of those to be found in the literature [see e.g. ref. “‘)I which use the experimental single-particle energies for the E. The Paris and Reid results are very similar with no surprises. The important thing is that both the hole-hole and particle-hole energies are significantly renormalized from their unperturbed values when we allow the two hole lines or particle and hole lines to interact. The resulting smaller denominators (i) make the &box “larger” (i.e. IP21 > ID21 and IP31 > ID3), and (ii)
J. Shurpin et al. / Folded diagrams
339
TABLE4 Low-lying A = 18 levels TJ
Cl
c2
c3
c4
10
- 10.59920 - 10.59920 - 10.59920
- 10.71672 - 10.71670 - 10.727
- 10.35508 - 10.35502 - 10.371
- 10.46468 - 10.46462 - 10.490
2
- 8.88340 - 8.88341 - 8.882
- 8.89500 - 8.89499 - 8.894
- 8.75340 - 8.75343 - 8.756
- 8.76300 - 8.76306 - 8.766
4
- 7.44860 - 7.44858 - 7.450
- 7.44860 - 7.44858 - 7.450
- 7.08640 - 7.08653 - 7.083
- 7.08640 - 7.08653 - 7.083
01
- 12.10408 - 12.10405 - 12.197
- 12.73150 - 12.73148 - 13.223
- 11.86850 - 11.86847 - 12.287
- 12.46800 - 12.46801 - 13.786
3
- 10.49140 - 10.49140 - 10.496
- 10.52120 - 10.52123 - 10.526
- 10.45800 - 10.45797 - 10.478
- 10.48412 - 10.48408 - 10.506
5
- 10.11280 - 10.11280 - 10.113
- 10.11280 - 10.11280 - 10.113
-9.63784 -9.63786 -9.637
- 9.63784 -9.63786 -9.637
2
-9.16022 -9.16024 -9.158
- 9.25420 -9.25423 - 9.252
- 9.03048 -9.03052 - 9.029
-9.10274 -9.10277 -9.101
All entries are in MeV and are relative to the I60 ground state. In each case the three rows correspond to the SC, LS and F,, respectively. The Paris potential were used with a starting energy of - 10 MeV.
make the &-box more energy dependent, so that the folded diagrams become much more important. As we shall see, these two effects tend to cancel each other. The sd-shell matrix elements of V,,, for I = 1, .I = 0 are given in table 5 for calculations Cl through C4 for both the Paris and Reid potentials. As more higher-order processes are introduced into the &-box, V,,, becomes more nonhermitian. This is entirely due to the contribution of the folded terms. Although the effective interaction obtained by this manner is not hermitian, the eigenvalues are all real. The non-hermitian matrix V,,, may be transformed to an hermitian form by use of a similarity transformation, although the columns of the transformation matrix will not be mutually orthogonal. Alternatively, the two-body interaction may be forced to be hermitian by the simple expedient of using for the matrix element V&the average of 6, and Vki[ref. ‘“)I. This method was used for the calculations described in the next section. The error introduced by this
340
J. Shurpin er al. / Folded diagrams TABLE 5
The sd-shell
effective-interaction
matrix
elements
for 7’ = 1, J = 0
TJ
abed
Cl
c2
c3
c4
10
4444
- 2.2360 - 2.0899
- 2.2925 -2.1472
-2.1822 - 1.9961
- 2.2464 - 2.0583
10
4455
- 3.2670 -3.1363
- 3.3960 - 3.2610
-3.1239 - 3.2353
- 3.2296 - 3.3460
10
4466
- 0.9096 -0.8976
- 0.9304 -0.9193
- 1.0031 - 0.9940
- 1.0227 - 1.0144
10
5544
- 2.9536 -2.8348
- 3.0050 - 2.8866
- 2.5074 - 2.5420
- 2.5609 - 2.5960
10
5555
-0.9409 -0.8450
- 1.0253 -0.9273
- 1.1018 -0.9555
- 1.1576 - 1.0171
10
5566
- 0.6505 - 0.6428
- 0.6669 - 0.6598
-0.5517 -0.5591
-0.5663 - 0.5746
10
6644
-0.8596 -0.8327
-0.8801 -0.8533
- 0.8594 -0.8523
- 0.8809 -0.8733
10
6655
-0.7021 -0.6893
- 0.7357 -0.7216
-0.3381 - 0.4075
- 0.3584 - 0.4305
10
6666
- 1.6478 - 1.4433
- 1.6575 - 1.4532
- 1.7779 - 1.5039
- 1.7863 - 1.5126
In each case the upper row is for the Paris potential and the lower for the Reid potential. The full non-hermitian V,,, is given. The orbits (4, 5, 6) represent respectively (Od,,,, Od,,*, Is,,,). All entries are in MeV. In all cases, the Lee-Suzuki (LS) iteration method was used.
approximation is easily checked for the A = 18 case. The largest array occurs for JT = 10; in this instance the difference between the lowest eigenvalue of the hermitian and non-hermitian array is 0.4 keV when using the Paris potential. The greatest error for this JT is only 2.4 keV.
5, Results for several-particle
systems
In this section we shall discuss results obtained using the newly derived matrix elements as shown in table 5 for several-nucleon systems. We feel this to be important for several reasons. First of all the matrix elements were derived to predict nuclear spectra, and their success or failure to do so can only be found by actual comparison with experiment. In cases where agreement is poor (and there shall be many such cases) one may perhaps obtain clues to where improvements may be
341
J. Shurpin et al. / Folded diagrams
made. A comparison between results obtained using the different approximations will be possible. Another point is that the matrix elements of table 5 are only the two-body parts of the model-space effective interaction. Their results for the spectra of several-nucleon systems may serve to indicate the potential importance of the many-body effective interactions. Finally, from a knowledge of the nature of the nuclear wave functions, one may establish the relative importance of the differing spin-orbit and tensor components of the two interactions. Before discussing the results obtained with the Paris and Reid interactions, we briefly recall the four approximations used in obtaining the effective interaction: Cl: @box with 2nd order diagrams; c2:
2nd order plus the hole-hole phonon ;
c3: 2nd order plus the particle-hole phonon ; c4: 2nd order plus hole-hole and particle-hole phonon. For convenience we shall refer to the interaction interaction with approximation Cl as Pl, etc.
obtained
using the Paris
5.1. CENTROIDS
A global assessment of an interaction may be obtained centroids of an interaction defined in jj-coupling by
E
ob
by calulating energy
rCTCJICTI(J’,J’,JTlVlj,j,JT)
1.
-
=
(5.1)
and in LS-coupling by
+)
= c CJlETl((~~)~STJI~I(~CL)~STJ) g(A/l)(2S+ 1)(2T + 1)
*
(5.2)
In eq. (5,2) (Q) labels the possible SU(3) representations and g(Lp) is the dimension of (APL).In the sd-shell the three possible SU(3) representations for two particles are (40) and (02) which are spatially symmetric and (21) which is spatially antisymmetric. The quantity [k] E 2k + 1. If the two-body interaction V were written as I/ = i (R(k)k=O
S”‘(U,U,)),
(5.3)
342
J. Shurpin et al. / Folded diagrams
then a tensor decomposition (LaII
of two-body matrix elements is obtained from
R’k’lJlh’)(ST~II Sk’l(S’T’P’) = (-)“‘“j-(-)”
;,; ;1
(LSTaPJII/I~S’Ta’P’J).
J
i
In eq. (5.4) c( and /I are any labels necessary to completely specify the matrix element; we shall take them to be irreducible representations of SU(3) and SU(4), respectively. We shall refer to the k = 0, 1 and 2 interactions as central, spin-orbit and tensor interactions, although their actual origin may be from other components. By applying eq. (5.4) and then transforming the two-body matrix elements to jjcoupling, one may determine the amount of effective central, spin-orbit and tensor interactions contributing to each two-body matrix element. It is trivial to prove that non-central components of the interaction will not contribute to an SU(3) centroid; hence, an SU(3) energy centroid is a measure of the effective central interaction. An analogous statement is not true for a jj-coupling centroid. In fig. 13 are shown contributions to the energy centroids s1 = .stt and .ss = E++($ for d; and $ for s,) for several interactions. Only a central or rank-zero interaction can contribute to sr. The twelve interactions shown in fig. 13 include the eight interactions obtained in this work and four interactions which have been widely used in shell-model calculations. The interaction “KB” is the Kuo-Brown interaction 24) and “K” is the Kuo interaction 3s), both derived from the Hamada-Johnston potential “). The interactions “K12” and “R” are interactions obtained in ref. 36), namely “Kuo plus 12 free parameters” or K12fp and “RIP”. The former interaction is essentially the Kuo interaction, but with those diagonal matrix elements involving only the d+ or s+ particles being varied to produce a better tit to spectra. The RIP interaction is a purely central interaction obtained by varying the values of radial integrals. From fig. 13 one finds the spin-orbit (6”) and tensor (.sT)contributions to ss are small and of opposite sign with the obvious exception of RIP. Further, the magnitude is essentially the same for the eight interactions obtained in the present work. The contribution to the .sg of the central (8) effective interaction does vary slightly from interaction to interaction and is appreciably different for the eight new interactions from the Kuo and K12fp interactions. Since K12fp was tit to inter alia binding energies, this suggests the new interactions may underbind nuclei. The st centroid, sr, does depend upon the original two-body interaction with the Paris potential producing more attraction as expected. However, again the three Kuo interactions are more attractive. It is of interest to compare these results with those obtained for the Q-box or unfolded interaction. The sb centroid is -3.71 MeV and -3.16 MeV for the Paris
343
J. Shurpin et al. / Folded diagrams I
. 0
xx
. -I
.
-2
‘I
2 I
-3
g L
I RI
I R2
I
I
I
I
I
I
R3
R4
PI
P2
P3
P4
I KB
I K
I Kl2
T--
I
I
R
Fig. 13. The centroids of the Od,,, matrix elements, ss, and Is,,, matrix elements, E; for 12 interactions. The quantities ES, ET and s: are centroids of the rank-O, rank-l and rank-2 parts of the interaction, respectively. The four interactions Rl, R2, R3 and R4 are derived from the Reid potential with the four approximations Cl, C2, C3 and C4, respectively. The interactions derived from the Paris interaction are similarly labelled, Pl, P2, P3 and P4. The remaining interactions are KB: Kuo-Brown 24), K: Kuo 35), K12 : “Kuo + 12 free parameters” and R: RIP 36).
and Reid interaction, respectively. (These numbers are for the &-box with phonons, i.e. diagrams D4, D5, P6 and P7 and assuming a starting energy of q, = - 10 MeV.) Hence, the inclusion of folded diagrams results in less attraction of 1.5 MeV! The contribution to Ed of the central, spin-orbit and tensor components of the Paris interaction is - 1.41, -0.12, and 0.25 MeV and for the Reid interaction - 1.42, -0.17 and 0.26 MeV. The difference between the two interactions is the effective two-body spin-orbit interaction. The main effect of including folded diagrams is on the central components: there is less attraction of approximately 0.5 MeV. The noncentral contributions are almost unaffected. In the beginning of the sd-shell, the SU(3) model provides a much more viable coupling scheme than does jj-coupling as well as providing some physical insight into the nuclear structure calculations. In fig. 14 are plotted energy centroids of the three SU(3) representations of two sd particles. As remarked above, the (40) and (02) are spatially symmetric, or belong to the SU(6) representation [2], and (21) belongs to [ll]. The (21) centroid is positive (repulsive) as expected from phenomenological potentials. Although no great differences were discernable in jj-coupling amongst the four methods of calculating an interaction, substantial differences do appear in SU(3).
344
J. Shurpin et al. / Folded diagrams
G
-3
0 0
-4
i
RI
R2
R3
R4
PI
P2
P3
P4
KB
K
Kl2
R
Fig. 14. Energy centroids of the three W(3) representations occuring in the two-particle system in the sd shell. The labelling of the 12 interactions is explained in the caption of fig. 13.
Calculations C3 and C4 are for both two-body potentials less attractive than for Cl and C2. From the centroids one may establish that for the Paris interaction the inclusion of the hole-hole phonon produces an added attraction of 0.038,0.049 and 0.042 MeV for (40), (02) and [2], respectively, and the particle-hole phonon produces less attraction of 0.355,0.148 and 0.296 MeV. The hole-hole phonon does not affect [ll] and the particle-hole phonon moves the [ll] centroid up by 0.053 MeV. Similar, albeit smaller, numbers result from the Reid potential. The effect of the 3plh diagram in the original work of Kuo and Brown 24) was to lower the 0’ state and raise the 4+ state of ‘*O, thereby producing better agreement between theory and experiment. An examination of fig. 14 as well as the calculated spectra of “0 in fig. 16 gives rise to an apparent paradox in that by adding the particle-hole phonon to the Q-box produces less attraction for both the spatially symmetric centroids and the ground state of **O. The behavior of the (40) energy centroid may be easily explained since the J = 4 matrix element enters with a weight of nine greater than for J = 0 and the J = 4 state is raised by the particle-hole renormalization. However, the puzzle of less binding for the O+ of I80 remains. In table 6 are listed diagonaL matrix elements of the (40) and (02) representations ; these two
J. Shurpin
et al. /
345
Folded diagrams
TABLE 6
(@I L
w
0 2
(02) “0 2
ww C(O2)
D7
P7
Pl-P3
- 0.983 -0.210 0.273 - 0.432 0.168
- 1.335 -0.267 0.339 -0.525 0.155
0.058 0.420 0.313 - 0.079 0.075
0.038 0.031
-0.017 0.028
0.319 0.099
The contributions to the diagonal two-body T = 1 matrix elements and SU(3) centroids arising from diagrams D7 and P7 of fig. 9b using the Paris potential and o0 = - 10 MeV (columns headed by D7 and P7) and the difference of the matrix elements obtained from interaction Pl and P3 (last column).
representations account for approximation 90% of the wave function of the lowlying states. The results are for the Paris potential. As in the earlier work the contributions from the simple 3plh graphs are to lower the O+ by almost one MeV and to raise the 4+ by 0.3 MeV. The results for the particle-hole phonon are even larger. The net effect on the centroids are virtually nil. The diagonal (40) L = 0 matrix elements for the full &box interaction (D4 +D5 +D6 +D7) is - 5.67 MeV and -6.02 MeV when D7 is replaced by P7. Renormalizing the interaction by including the folded diagrams produces an appreciably less attractive interaction : the matrix element becomes - 4.14 and -4.08 MeV for Pl and P4, respectively. Further, the energy separation between the L = 0 and L = 4 states of the (40) representation is 2.36 MeV (Pl) and 2.61 MeV (P4) rather than 3.20 MeV or 3.61 MeV for the &-box interaction containing D7 or P7, respectively. Thus, an effect of including folded diagrams is to obtain less binding energy and a more compact spectrum; much of the effect of the phonon on the @box interaction disappears. A similar, though less dramatic, effect occurs for hole-hole phonon. The above discussion may be summarized as follows. As one goes from Cl to C4 the strength of the Q-box increases. When one also includes the folded terms in V,,, it is found that the matrix elements are typically weaker for C3 than Cl, for example. For C4 the spectra obtained in lowest order (i.e. from V,,, = 6) are noticeably lower than the Cl spectra but when the folded terms are included in V,,, the C4 spectra are quite similar to those of Cl, while the C3 spectra actually are less bound than the Cl. Thus, when we include higher-order processes in the &box the folded diagrams become more important. The core renormalization due to 4p2h states was characterized in ref. 38) as similar to a pairing interaction. The hole-hole phonon also exhibits some characteristics of pairing interaction in that [11] states are essentially unaffected
J. Shurpin et al. / Folded diagrams
346
‘8F -__3-
-
-
-
-9
-’
-
-2
-
I
-3
-3
-
-I
-
-
-I
-
-
-
-2
-3
-2
_, 4
-
___3
-10 -
-22 z
___2 -II -
-_5-3
-
-_5
_
-3
-
-
-3 -3
-5
-I
-3
-
-
-I -I
-13 RI
Fig.
15. Energy
spectra
-__I
-
-12 I
-
R2
R3
R4
of 18F as calculated
with
PI
P2
the eight
-I
P3
P4
interactions
EXP
derived
in this work. The
labelling of the eight interactions is explained in the caption of fig. 13.
and low angular momentum states of the two particle system are affected the most (see figs. 15 and 16). However, a pairing interaction would affect (40) more than (02) in the ratio of 5 : 1, unlike the hole-hole phonon. The inclusion of the particle-hole phonon makes the relative separation of (40) and (02) centroids smaller - 0.207 MeV less for the Paris potential - which has unfortunate consequences in several-nucleon nuclei. Bands built from W(3) representations other than the ground state or leading SU(3) representation will now lie at a lower excitation energy. The energy of an SU(3) centroid in a nparticle system is 3g) E(n(&)) = (2n - n2)E(20) +&-15n+7nZ +&(21n+n2
- 3( Cc49 + 2( Cc3’))E(40) -3
+$(n2-5n+(C’4’))E(21),
(5.5)
J. Shurpin et al. 1 Folded diagrams
I
__o_
-8
-
__4-
-9
2 z
-II
-
-
-
-
-_2-
=
__4-
-
=s -4
-2
=-P i
I -_2-
-10
-
-0
-2
341
-
-
-
-2
_,j-“- - RI
R2
R3
R4
-
__o_
PI
-o_j
-
P2
P3
P4
EXP
Fig. 16. Energy spectra of I80 as calculated with the eight interactions derived in this work. The labelling of the eight interactions is explained in the caption of fig. 13.
with the expectation values of the Casimir operators (03’)
= A2+$+&J+3@+~),
@“‘>
=
Cck)of SU(k) given by 40)
.
~fi2+3(fl-f,)+f2-f3~
for representations (APL)and [j], respectively. The SU(4) representation [f must be used in evaluating eq. (5.5) which is congruent to the W(6) representation [f]. From eq. (5.5) the energy difference between (80), the ground-state band of 2oNe, and (42), the first excited band, is E(80) - E(42) = @(40) - E(02)]. Hence, including the particle-hole phonon moves the two centroids 0.48 MeV closer together for the Paris potential. Though this may seem small, the two centroids are already too near each other as compared to experiment. Further, the changes will be larger in nuclei with more particles such as 28Si.
348
J. Shurpin et al. 1 Folded diagrams
A comparison with the centroids of the K12fp interaction is useful to predict possible consequences in heavier nuclei and, since the twelve parameters were varied to produce a better lit with experiment, will indicate some problems which will arise. The K12fp interaction has a separation of the (40) and (02) energy centroids 0.22 MeV larger than the C2 approximation to the Paris interaction (P2). There is also a larger separation between [2] and [ll]. Hence, the interactions obtained in this work will have too many states of low energy and both SU(3) and SU(4) will be more badly violated than with K12fp. Further, the energy centroid of K12fp for [2] is lower representing more binding. Hence, we may anticipate the current interactions will badly underbind nuclei throughout the sd-shell. The SU(3) centroids obtained for the o-box interaction suggests that results obtained using the &-box or unfolded interaction would be very similar to that obtained using the earlier Kuo or Kuo-Brown i@eractions. The centroids for the (40), (02) and (21) representations are -4.03, - 1.71 and 0.46 MeV for the Reid interaction and -4.17, - 1.73 and 0.54 MeV for the Paris interaction. (These results are for a o-box which includes phonons, i.e. diagrams D4, D5, P6 and P7 with o0 = - 10 MeV.) These centroids are not appreciably different from those of the Kuo interaction. The inclusion of the folded diagrams results in less attraction for (40) of 1.35 MeV and 0.55 MeV for (02) whereas the (21) centroid is essentially unaffected. One may tentatively conclude that the folded diagrams do not affect the spatially antisymmetric states and the more correlated a state, the greater the effect of folded diagrams. 5.2. SPECTRA
We now turn to a consideration of the predictions of the matrix elements in nuclei. Although all nuclei with A equal to 18 to 22 and 24Mg were calculated, we shall for brevity present results only for the more illuminating cases. In all cases the single-particle energies were taken from experiment. The calculations were performed using a modified version of the Glasgow shell-model code4i). In fig. 15 are shown results for ‘*F and experiment. Although many of the features anticipated from an examination of the centroids appear, the spectra are obviously more detailed. The inclusion of the phonons affect the low-spin states (l+ and 2+) more than the 3+ and 5+; in particular, the second 3+ level is essentially unaffected. Although the P2 interaction reproduces the ground-state binding energy, the other interactions do rather worse. All interactions fail to produce the proper energy for 3: and 5:, although P3 reproduces the excitation spectrum reasonably well. Since these energies enter the evaluation of the centroid and position of levels in heavier nuclei - with more weight than the l+, the expectations based on the centroids appear valid. In fig. 16 are shown results for “0. Again, the limits provided by the centroids
J. Shurpin et al. / Folded diagrams
349
are validated by the detailed calculation of structure. The Paris and Reid interactions do not provide sufficient binding although the Paris interaction is somewhat stronger than the Reid. Again, the higher-spin states appear to be relatively unaffected by the inclusion of phonons. Although the 4: level is correctly predicted, the 0: and 2: are too high, resulting in a compressed spectrum. This will also occur in heavier nuclei. In fig. 17 are results for 19F. The wave functions of the ground-state band are dominated by the SU(3) representation (60) and have S equal to one-half. Hence, as in ‘a0 9 the energies depend primarily on the central interactions with small contributions from the spin-orbit interaction. The error in the predicted binding energy ranges from approximately 1 MeV for the P2 interaction to 2.7 MeV for the R3 interaction. Inclusion of the hole-hole phonon not only shifts the centroids down, but also expands the spectrum; the position of the y’ is essentially unaffected by inclusion of phonons. However, the particle-hole phonon tends to contract the spectrum. The final result with both hole-hole and particle-hole phonons included produces a spectrum which has the correct level ordering but with insufficient binding.
-2 -17
-
-7
-IS
-7 -,3
_I3
__7
--7
-_5,,
-18
-3
-13
--5,7=:
=:
=:’
=;,,-7
=y
-7
-I3
-I3
-13
-_:
--:
-5 -7
=:” 7
-
-19 -
--9 -9
-_9
13
-9
-2o-
-9 -9
-3
B H
-9
-9
-3
=5 -_7 ---3
-3
-3 -3
-21
-
=T -5
-22
_ -1 -
-3 =:
-9--
_-3 =:
-5 -1
--3 -5 -I
=: -3
-5 -1
-23 -5-, -24
RI
R2
R3
R4
PI
P2
P3
w
EXP
Fig. 17. Energy spectra of “F as calculated with the eight interactions derived in this work. The labelling of the eight interactions is explained in the caption of fig. 13. The dashed line represents a 3’ state dominantly 7p4h.
350
J. Shurpin et al. / Folded diagrams
The calculation of levels in 2o0 is of interest because only the T = 1 interaction may contribute. The results are not shown. The second O+ and 2+ states both lie about one MeV lower than the experimental position. Again the spectrum is too compressed and the amount of binding is about one MeV too little. The calculated spectra of 20Ne may be found elsewhere42 ). Briefly, deficiencies found in other nuclei also appear in 20Ne: there is too little binding energy, 3.2 MeV for Pl and 4.8 MeV for Rl ; the spectrum is too compressed with the 8+ appearing 0.5 MeV for Pl and 1.4 MeV for the Rl interaction too low relative to the ground state; the (42) excited band lies at too low an energy relative to the ground state. The tensor interaction of the Paris potential is appreciably weaker than that of the Reid interaction, but the contribution from the tensor components to the Qbox two-body matrix elements are only 2% larger for the Reid potential than for the Paris potential. However, the process of including the folded diagrams may alter this situation: the many angular momentum recouplings and matrix products may cause the effective non-central components to change or conceivably even disappear. Since it is in general difficult to choose an appropriate weighting of the many two-body matrix elements (recall the non-central components do not contribute to SU(3) energy centroids), the expectation value was evaluated between several wave functions generated for mass 20 nuclei using the Cl interactions. The non-central components were obtained using eq. (5.4). The results are given in table 7. The lowest T = 0 wave functions are dominated by S = 0 components for which the non-central forces cannot contribute. However, the lowest T = 1 wave functions have S = 1 and both the two-body spin-orbit and tensor force may contribute. An examination of table 7 demonstrates several points. First, the effective tensor - and also spin-orbit - force of the interaction obtained from the Reid potential is not very different from that of the Paris potential. This is somewhat surprising. Second, the contribution from the two-body effective spinorbit interaction is appreciably larger than that from the effective tensor interaction. Third, in most instances the effective tensor interaction is repulsive. Fourth, essentially all the differences between results obtained using the two interactions arise from the effective central interaction. Finally, the contribution from the non-central components to the energy of the T = 0, S = 0 states is approximately the same as that to the T = 1 states. As remarked in subsect. 5.1 the inclusion of the particle-hole phonon and then folding results in less binding, even for the low-spin states, contrary to expectation based on previous interactions which included only the &-box with no folded diagrams. In table 7 are also listed the contributions to the energy of mass-20 levels of three diagrams D4, P6 and P7. The quantities listed are the contribution to the interaction from only the ,@box and does not include any effects such diagrams have on the interaction after including folded diagrams.
TABLE7 Contributions
to the energy of 7’ = 0 and T = 1 states in mass 20 from specified parts of the interaction
C
v
T
D4
P6
P7
- 24.68 - 23.01 - 18.21 - 16.82 - 22.97 -21.52 - 20.60 - 19.27 - 15.91 - 14.99
-0.24 -0.19 -0.16 -0.12 -0.33 -0.27 -0.41 -0.37 -0.65 -0.61
- 0.07 - 0.05 0.15 0.20 0.05 0.07 - 0.03 - 0.02 0.16 0.18
- 26.76 - 25.55 -20.18 - 19.00 - 25.45 - 24.39 - 23.53 - 22.59 - 19.76 - 18.99
-2.71 - 2.88 - 2.01 - 1.91 - 1.90 - 2.23 - 1.58 - 1.94 -0.13 -0.62
- 1.37 - 2.29 -2.13 - 1.41 -0.80 - 2.03 -0.39 - 1.27 0.59 -0.48
-
-0.88 -0.86 -0.89 -0.82 -0.83 -0.80 -0.77 -0.72
-
- 1.36 0.40 - 1.07 0.08 -0.91 0.45 - 1.20 0.06
0.23 -0.69 - 0.07 -0.91 0.21 -0.80 - 0.02 -0.55
J
T=O 0, 0, 2 4 6
T=l 1 2 3 4
13.52 12.60 13.97 13.04 13.33 12.39 12.84 11.97
0.06 0.07 0.09 0.11 0.26 0.29 0.36 0.38
16.10 15.48 16.64 15.93 16.10 15.34 15.31 14.60
The columns labelled “c’, “V” and “T’ contain the expectation values of the effective centroid, spinorbit and tensor interactions; the columns headed ‘TM”, “P6’ and “P7” contain the expectation values of diagram D4, P6 and P4 of fig. 9. The wave functions used were generated using the Cl interaction with “0 single-particle energies. For each J the first line contains results for the Paris interactions and the second for Reid interactions.
From the table it is seen the contribution from the particle-hole phonon diagram (P7) is much larger for the Reid interaction than for the Paris interaction. This should be related to the relative strength of the tensor force of these two interactions. For both interactions the effect is to lower the energy of the low-spin states and increase the energy of the high-spin state. This reinforces our earlier conclusion that the apparent repulsion resulting from the inclusion of the particlehole phonon was a manifestation of the folded diagrams. One may also investigate the extent the hole-hole phonon contribution simulates a pairing interaction. One expects the pairing interaction to be attractive and, in *‘Ne it can contribute to states of J I 4. From table 7 one observes the contribution from P6 obeys these expectations. Finally, we note that if one were to take as the interaction just the contributions from D4 and P7 using a starting energy of o. = - 10 MeV [similar to the original work of Kuo and Brown24)], agreement with experiment would be quite good. Using the Pl interaction, the calculated binding energy would be in error by only
352
J.ShUrpin etal./ Folded&QgrQmS
70 keV and the excitation energy of the 8+ is only 0.5 MeV too small. The Rl interaction underbinds “Ne by 170 keV but produces an 8+ within 100 keV of experiment. In fig. 18 are results for 22Na . Since 22Na is odd-odd, the low-lying states have S = 1 and all parts of the interaction may contribute. The level structure of 22Na has routinely defied successful calculation with both phenomenological interactions 43) and realistic interactions 36). The first l+ (and sometimes also the second 1’) states tend to lie lower than the experimental 3+ ground state. The present calculations have the same failings. Interactions which are lit to the structure of mass-22 nuclei can produce the correct ordering by either using a repulsive triplet-even tensor interaction 44) or by varying a number of two-body matrix elements3’). Because of the number of matrix elements in the latter approach, it is difficult to establish the physical origin of the discrepancy. From fig. 18 it is apparent the spectra is too compressed. Experimentally there are seven levels below 3 MeV excitation; the Rl interaction produces 12 levels and P4 produces 11 levels below this energy. The 2+ level appears 1.5 to 2 MeV too low.
-,
-------I
-32.21
-3t.s3
RI
R4
Fig. 18. Ei4ergy spectra of “Na
-/
- 34.01
PI
-’
- 52.73
P4
-?a.52
EXP
calculated with four of the interactions derived in this work.
353
22Ne -. 5-
-4
-I
-2,1 -2
-1 -2
------I
-3
-2 -3
-4
-___
4
1
-I
-4
-2 -----I -------2
-7
-3
-2
-2
-2
-4
-4
3
-----2
-2
4-
-2 -2
3-4
2 z
I
-4
-4
-
2-
------2
-2
-2
-2
-----0 -5149
-0 - 54.03
_
-2
-0 -52 21 RI
R4
PI
-0 -52 00 P4
-----0 -5775 EXP
Fig. 19. Energy spectra of “Ne calculated with four of the interactions derived in this paper.
In fig. 19 are shown results for “Ne. The low-lying states are dominated by (82) S = 0. Again the spectra is too compact with the 2: and 4: having too small an excitation energy and the 6: which is not shown, even worse, although the excitation spectrum for the 2: and 4: states is acceptable. The most obvious failure is the position of the 2: which is a member of the (82) K = 2 band. Its incorrect position is now almost traditional 36*43). Although its energy may be corrected by explicit inclusion of pf excitations45), interactions may be found which do predict its energy correctly 37*42) although the physical origin of the modifications is not known. Modifying the matrix elements which violate S which are substantial in the present interactions - do not substantially modify the results 46). Finally, we present results for ‘*Mg. The amount of underbinding is substantial, ranging from 5.5 MeV for the PI interaction to over 10 MeV for R4. The K = 2
354
J. Shurpin et al. / Folded diagrams
band again appears much too low. As predicted by the centroids, the excitation energy of the 0: is too low when the particle-hole phonon is included.
6. Discussion and conclusion The aim of this paper was to perform within the framework of the shell model a microscopic nuclear-structure calculation, the starting point of which was the free NN interaction. The first step in this procedure, the calculation of the G-matrix, is probably quite sound. The methods used, momentum space matrix inversion to obtain G, and the Tsai-Kuo method to treat that part of G which depends on the Pauli operator Q, are very accurate. The second step is the calculation of the &box, and it is in this step that we have to make some approximations. Four approximation schemes, denoted as Cl, C2, C3 and C4 have been used. We will return to discuss them later. In the final step, we obtain V’,, by “folding” the Qboxes together, i.e. evaluating the folded diagram Q-box series of V,,,. For a given Q-box, this step can be quite accurately carried out by using the Lee-Suzuki iteration method. This method was shown to yield essentially exact results after only a small number of iterations. We have found that the effect of higher-order &box diagrams is suppressed significantly by the inclusion of the corresponding folded diagrams. Generally speaking, the effect of the folded diagrams is quite significant in the calculation of V,,,. Another aim of these calculations was to see how the Paris potential would fare in finite nuclei. To our knowledge, this is the first microscopic calculation of finite open-shell nuclei for which the Paris potential was used. Our results indicate that the Paris potential is as good as, if not better than, the Reid potential. Indeed, in all nuclei we considered, the Paris potential seemed to give better, albeit inadequate, ground-state binding energy. However, in most cases differences in spectra produced with or without the inclusion of phonons were as great as those between the Paris and Reid potential. We demonstrated that in mass-20 nuclei the contributions to the energy from the effective non-central components of the Reid and Paris interaction were essentially the same. Differences in spectra and binding energies arose mainly from the effective central interaction, although there were appreciable differences in the contribution to the energies from diagrams P6 and P7. However, the Reid interaction tended to produce wave functions with a slightly larger occupation of the dt than did the Paris interaction. The third aim of these calculations was to do a systematic study of the contribution of the folded diagrams to V’,,. It is quite clear that the inclusion of the folded diagrams have a much greater effect on the A = 18-24 spectra than do various infinite-order partial summations (hole-hole and particle-hole phonons) that this calculation and others include. Until now, it has been practically taken for
J. Shurpin et al.
/ Folded diagrams
355
granted that the inclusion of TDA (or RPA, for that matter) phonons would give increased binding. Our calculations have shown that when the folded diagrams are included, this may not be the case. The C3 spectra (for which the TDA particlehole phonons were included) show appreciably less binding than do the Cl spectra (for which we included no phonons). By explicitly calculating the expectation value of the particle-hole diagram for mass-20 states, we demonstrated this apparent anomaly was a result of the folded diagrams. Why did the early calculations, starting with those of Kuo and Brown, reproduce the experimental spectra as well as they did, even though all diagrams above second order were ignored ? The answer suggested by the current work seems to be that although the &-box they used contain only some low-order diagrams, the effect of the higher-order Q-box diagrams would have been cancelled in large part if the folded diagrams were included in V,,, as well. This is because the higher-order diagrams have a relatively greater energy dependence so that they significantly renormalize the &box interaction. Thus if higher-order diagrams are included (in the basic Q-box), the folded diagrams must be included as well (in the expansion for V,,,). Indeed, using an effective interaction defined by the sum of contributions from diagram D4, the bare G-matrix, and P7, the particle-hole phonon, of fig. 9, with no folded diagrams included, much better agreement with experiment was obtained. When compared with experiments the excitation spectra of several nuclei are reproduced reasonably well but our calculated binding energies are generally too small by several MeV, especially for nuclei in the middle of the sd-shell. It is not clear how to remedy this, and will be an interesting problem for further study. Several directions may be pursued. In the present work we have included only the two-body components of I&,. I$,, clearly has many-body components. For example, I90 will have 3-body and 22Ne should have 6-body effective interactions. Calculation of such many-body components is rather complicated. In fig. 20 we give some representative diagrams which contribute to 3-body effective interactions. Diagram (i) belongs to the non-folded term (j of eq. (2.9); it originates from our model-space restriction that nucleons are confined to certain low-lying shell-model states. (Note that a railed line represents a particle outside the model space.) Diagram (ii) is a 3-body effective interaction diagram due to non-nucleonic degrees of freedom. Here nucleons 1 and 2 interact by exchanging a pion, exciting nucleon 2 into an isobar. Then it is deexcited back to a nucleon. Three-body forces can also arise from folding two 2-body irreducible boxes together, as shown by diagram (iii). Polls et ~1.~‘) have recently investigated the contribution of these types of diagrams to the three-body effective interaction for the sd-shell nuclei using the Reid “) and Jtilich-Bonn 48) NN potentials. They found that the explicit inclusion of A degrees of freedom (diagram (ii) of fig. 20) has rather a negligible contribution to the three-body effective interaction. But the three-body effective interaction due to the truncation of the shell-model space
356
J. Shurpin et al. / Folded diagrams
I
2 (1)
3
I
2 (Ii)
3
I
2
3
(iii)
Fig. 20. Some of the diagrams which contribute to the three-body effective interaction.
(diagrams (i) and (iii) is significant, although not overwhelmingly large. They give reduced binding energy of approximately 0.5 MeV to the A = 19 nuclei. [A similar result has been obtained by Huang et al. 49) who performed a folded-diagram calculation for the low-lying states of 190 using mainly the M3Y “) potential as the underlying two-body effective interaction without considering the effect due to the d-degrees of freedom.] Another possibility to improve the present calculation is to include still more irreducible diagrams in the @box than were included in the present work. In fact, one would like at this point to have the exact o-box. One possible step toward this goal would be the inclusion of the 3rd order diagrams in the Q-box. Such a calculation is facilitated by the new techniques 23) developed to evaluate angularmomentum-coupled diagrams. However, the net result of a systematic 3rd order calculation is not clear. Without the folded diagrams the spectra would presumably show less binding than for 2nd order. [See e.g. ref. ‘“).I However, the derivative (slope) of the 3rd order contribution would also be positive, and moreover, large compared to the 2nd order contribution so that the net result upon inclusion of the folded diagrams is not at all clear. By solving a set of self-consistent integral equations, Chakravarty et al. ‘l) have calculated several large classes of &-box diagrams to all orders. It is interesting that their results of the &box interaction for the 1sOd shell are generally not very different from ours with the C4 approximation. They have not calculated the folded o-box series for V’,,. It would be of interest to do so to compare the V,,, so obtained with ours. Contributions from highly excited states to the $-box due to the tensor interaction, the so-called Vary-Sauer-Wong effect ‘I), were not evaluated in the present work. However, recent work s2) using the Bonn potential 48) showed that two-body potentials with a weak tensor interaction give rise to a negligible VarySauer-Wong effect. The Paris potential has a weak tensor component.
..f. Shurpin et al. / Fofakd diagrams
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We should like to thank I. M. Richard for providing the routines used to evaluate the Paris potential. This work was supported in part by the DOE under contract DE-AC0276ER13001. Note added in proof: The coupled-cluster method of Kiimmel and his collaborators has been recently extended and applied to open-shell nuclei 54-46). It will be of interest to compare their method and results with those of the present work. We are very grateful to Prof. Kiimmel for discussions in this regard.
References 1) T. T. S. Kuo, Ann. Rev. Nucl. Sci. 24 (1974) 101, and references therein 2) P. J. Ellis and E. Osnes; Rev. Mod. Phys. 49 (1977) 777, and references therein 3) G. Breit, Rev. Mod. Phys. 34 (1962) 776; K. E. Lassila et al., Phys. Rev. 126 (1962) 881 4) T. Hamada and 1. D. Johnston, Nucl. Phys. 34 (1962) 382 5) R. V. Reid, Ann. of Phys. 50 (1968) 411 6) R. Vinh Mau, in Mesons in nuclei, ed. M. Rho and D. Wilkinson (North-Holland, Amsterdam, 1979) p. 151 7) M. Lacombe et al., Phys. Rev. C21 (1980) 861 8) K. A. Brueckner, C. A. Levinson and H. M. Mahmoud, Phys. Rev. 95 (1954) 217 9) H. A. Bethe, Phys. Rev. 103 (1956) 1353; Ann. Rev. Nucl. Sci. 21 (1971) 93; H. A. Bethe and J. Goldstone, Proc. Roy. Sot. A238 (1957) 551 10) J. Goldstone, Proc. Roy. Sot. A239 (1957) 267 11) K. T. R. Davies, M. Baranger, R. M. Tarbutton and T. T. S. Kuo, Phys. Rev. 177 (1969) 1519 12) H. Feshbach, Ann. of Phys. 19 (1962) 287 13) C. Bloch and J. Horowitz, Nucl. Phys. 8 (1958) 51 14) T. Morita, Prog. Theor. Phys. 29 (1963) 351 15) G. Oberlechner et al., Nuovo Cim. B6ii (1970) 23 16) M. B. Johnson and M. Baranger, Ann. of Phys. 62 (1971) 172 17) T. T. S. Kuo, S. Y. Lee and K. F. Ratcliff, Nucl. Phys. Al76 (1971) 65 18) B. H. Brandow, Rev. Mod. Phys. 39 (1967) 771; Int. J. Quantum Chem. 15 (1979) 207 19) T. T. S. Kuo and E. M. Krenciglowa, Nucl. Phys. A342 (1980) 454 20) E. M. Krenciglowa and T. T. S. Kuo, Nucl. Phys. A235 (1974) 171 21) S. F. Tsai and T. T. S. Kuo, Phys. Lett. 3!?B (1972) 427 22) E. M. Krenciglowa, C. L. Kung, T. T. S. Kuo and E. Osnes, Ann. of Phys. 101 (1976) 154 23) T. T. S. Kuo, J. Shurpin, K. C. Tam, E. Osnes and P. J. Ellis, Ann. of Phys. 132 (1981) 237 24) T. T. S. Kuo and G. E. Brown, Nucl. Phys. A85 (1966) 40 25) B. R. Barrett and M. W. Kirson, Nucl. Phys. A148 (1970) 145 26) S. Y. Lee and K. Suzuki, Phys. Lett. 91B (1980) 173; Prog. Theor. Phys. 64 (1980) 2091 .27) G. E. Brown, A. D. Jackson and T. T. S. Kuo, Nucl. Phys. Al33 (1969) 481 28) J. Shut-pin, D. Strottman, T. T. S. Kuo, M. Conze and P. Manakos, Phys. Lett. 69B (1977) 395; J. Shut-pin, H. Muther, T. T. S. Kuo and A. Faessler, Nucl. Phys. A293 (1977) 61 29) H. P. Stapp, T. J. Ypsilantis and N. Metropolis, Phys. Rev. 105 (1957) 302 30) T. T. S. Kuo and E. Osnes, Many-body theory for nuclear structure, monograph in preparation 31) E. Osnes, T. T. S. Kuo and C S. Warke, Nucl. Phys. A168 (1971) 190 32) T. T. S. Kuo and E. Osnes, Nuci. Phys. A226 (1974) 204 33) A. H. Wapstra and K. Bos, Atom. Nucl. Dam Tables 19 (1977) 175 34) F. Ajzenberg-Selove, Nucl. Phys. A300 (1978) 1 35) T. T. S. Kuo, Nucl. Phys. A103 (1967) 71
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36) E. C. Halbert, J. B. McGrory, B. H. Wildenthal and S. P. Pandya in Advances in nuclear physics,
37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48) 49) 50) 51) 52) 53) 54) 55) 56)
vol. 4, ed. M. Baranger and E. Vogt (Plenum, New York, 1970) B. M. Preedom and B. H. Wildenthal, Phys. Rev. C6 (1972) 1633 T. T. S. Kuo and G. E. Brown, Nucl. Phys. A92 (1967) 481 J. C. Parikh, Phys. Lett. 41B (1972) 468 S. I. So and D. Strottman, J. Math. Phys. 20 (1970) 153 R. R. Whitehead, A. Watt, D. J. Cole and 1. Morrison, in Advances in nuclear physics, vol. 10, ed. J. Negele and E. Voyt (Plenum, New York, 1938) J. Shurpin, Ph. D. thesis, State University of New York at Stony Brook (J980), unpublished Y. Akiyama, A. Arima and T. Sebe, Nucl. Phys. 59 (1964) 1; 85 (1966) 184 N. Anyas-Weiss and D. Strottman, Nucl. Phys. A306 (1978) 201 A. Arima and D. Strottman, Nucl. Phys. Al62 (1971) 605 M. Conze, H. Feldmeier and P. Manakos, Phys. Lett. 43B (1973) 101; private communication A. Polls, H. Mtither, A. Faessler, T. T. S. Kuo and E. Osnes, preprint (University of Tubingen, 1982) K. Holinde, R. Machleidt, M. R. Anastasio, A. Faessler and H. Mtither, Phys. Rev. Cl8 (1978) 870 W. Z. Huang, H. Q. Song, Z. X. Wang and T. T. S. Kuo, Chinese J. Nucl. Phys., in press (1982) G. Bertsch, J. Borysowich, H. McManus and W. G. Love, Nucl. Phys. A284 (1977) 399 S. Chakravarty, P. J. Ellis, T. T. S. Kuo and E. Osnes, Phys. Lett. 109B (1982) 141 J. P. Vary, P. U. Sauer and C. W. Wong, Phys. Rev. C7 (1973) 1776 H. M. Sommermann, H. Mtither, K. C. Tam, T. T. S. Kuo and A. Faessler, Phys. Rev. C23 (1981) 1765 R. Offerman, W. Ey and H. Ktimmel, Nucl. Phys. A273 ‘(1976) 349 W. Ey, Nucl. Phys. A296 (1978) 189 J. G. Zabolitzkv and W. Ey, Nucl. Phys. A328 (1979) 507
FOLDED DIAGRAMS AND Is-Od EFFECTIVE INTERACTIONS DERIVED FROM REID AND PARIS NUCLEON-NUCLEON POTENTIALS J. SHURPIN’
and T. T. S. KU0
Physics Dept., State university of New York at Stony Brook, Stony Brook, NY 11794, USA
D. STROTTMAN Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87544, USA Received 18 January 1983 (Revised 20 April 1983) Abstract: The sd-shell effective-interaction matrix elements are derived from the Paris and Reid potentials using a microscopic folded-diagram effective-interaction theory. A comparison of these matrix elements is carried out by calculating spectra and energy centroids for nuclei of mass 18 to 24. The folded diagrams were included by both solving for the energy-dependent effective interaction self-consistently and by including the folded diagrams explicitly. In the latter case the folded diagrams were grouped either according to the number of folds or as prescribed by the Lee and Suzuki iteration technique; the Lee-Suzuki method was found to converge better and yield the more reliable results. Special attention was given to the proper treatment of one-body connected diagrams in the calculation of the two-body effective interaction. We first calculate the (energy-dependent) G-matrix appropriate for theasd-shell for both potentials using a momentum-space matrix-inversion method which treats the Pauli exclusion operator essentially exactly. This G-matrix interaction is then used to calculate the irreducible and nonfolded diagrams contained in the &-box. The effective-interaction matrix elements are obtained by evaluating a [S-box folded diagram series. We considered four approximations for the basic @box. These were (Cl) the inclusion of diagrams up to 2nd order in G, (C2) 2nd order plus hole-hole phonons, (C3) 2nd order plus (bare TDA) particle-hole phonons, and (C4) 2nd order plus both hole-hole and particle-hole phonons. The contribution of the folded diagrams was found to be quite large, typically about 30x, and to weaken the interaction. Also, due to the greater energy dependence of higher-order diagrams, the effect of folded diagrams was much greater in higher orders. That_ is, the cont~bution from higher-order diagrams for most cases was greatly reduced by the folded diagrams. The convergence of the folded-diagram series deteriorates with the inclusion of higher-order &-box processes in the method which groups diagrams by the number of folds, but remains excellent in the Lee-Suzuki method. Whereas the inclusion of the particle-hole phonon was essential to obtain agreement with experiment in earlier work, when the folded diagrams are included the effect of the particle-hole phonon is to reduce the amount of binding. All four approximations to both potentials produce interactions which badly underbind nuclei. The excitation spectra given by these interactions are, however, all rather similar to each other. The Paris interaction produces more binding than does the Reid, but differences between results obtained with the two interactions were often less than
+ Present address: Telas Computer Systems, 71 West 35th St., NY, NY 10001. 310
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differences obtained in the four approximations. Essentially no difference was found between the effective noncentral interactions from the Reid ‘and Paris potentials after including the folded diagrams, although these two potentials themselves are quite different, especially in the strength of the tensor force. Comparisons between.calculated spectra and experiment were done for ‘*O, r8F, 19F, “‘0, “‘Ne, 22Ne, 22Na and *4Mg.
1. Introduction
An outstanding problem in nuclear physics is to describe the properties of finite nuclei in terms of the interactions between their constituent nucleons. To describe low-energy phenomena, usually a non-relativistic approximation of the nucleonnucleon (NN) interaction is carried out to construct a non-relativistic NN potential, V,,. One must then solve the non-relativistic many-body Schrodinger equation. This problem has been the subject of numerous articles and several reviews l, 2). As is well known, one cannot proceed directly with this ambitious prescription because of several major difficulties. To begin with, the NN potential is itself not completely understood. Until recently, to circumvent this problem one constructed phenomenological potentials which were required to reproduce as much of the experimental data, such as lowenergy phase shifts and deuteron properties, as possible. Examples of potentials of this type are the Yale potential 3), the Hamada-Johnston potential 4), and the Reid potential ‘), the last being by far the most widely used in recent years. This procedure, however, is not quite satisfactory because several different potentials can (and indeed do) yield the same phase shifts. Also, phenomenological potentials do not necessarily give a good theoretical basis for the interaction. A potential which is largely free of this latter shortcoming is the recently proposed Paris potentia16) which is largely theoretically derived. The long- and medium-range parts (r 5 0.8 fm) reflect n, 2rr,and 37r(w) exchange. Only the short-range part, which should be important for high-energy phenomena, is constructed so as to fit the phase shifts. So far, the Paris potential has been tested mainly in nuclear matter where it gives reasonable results ‘). A major aim of the present work is to study the results the Paris potential gives for finite nuclei. To solve the nuclear many-body problem, perturbation techniques adapted from quantum field theory and statistical mechanics were developed. A difficulty encountered immediately was that the matrix elements of V,, are either infinite (hard-core potentials) or at least very large (soft-core potentials) when the relative wave functions are non-vanishing near the origin (e.g. in a harmonic oscillator basis). Brueckner and his collaborators *) summed the interactions between pairs of nucleons to all orders to obtain the reaction matrix, or G-matrix interaction, whose matrix elements are finite. The perturbation theories were then expressed in terms of G rather than V,,. The earliest studies were carried out in infinite nuclear matter
312
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by Brueckner, Bethe and Goldstone ‘* 9). The Goldstone linked-cluster nondegenerate perturbation expansion lo) can also be used for closed-shell nuclei such as I60 and leads to the so-called Brueckner-Hartree-Fock (BHF) theory rr). For open-shell nuclei, i.e. nuclei with valence nucleons such as 180, several approaches have been developed. There are the energy-dependent theories of Feshbach 12) and of Bloch and Horowitz 13). In these theories, the effecttive interaction secular equation must be solved at the self-consistent energies. The energy-independent formulation, which leads to folded diagrams, has been developed via time-dependent perturbation theory by Morita 14), Oberlechner et al. 15), Johnson and Baranger 16), and by Kuo, Lee and Ratcliff “), while Brandow 18) used time-independent perturbation methods. The various energyindependent effective interactions are consistent with each other ‘). In addition, their form is similar to that of the empirical effective interaction, which is also energy independent. In this work, we employ effective interactions of the type obtained by Kuo, Lee and Ratcliff. In this formulation the non-folded diagrams are first grouped together to form the @box interaction, and then the folded diagrams are included in a very systematic way. Recent developments in this area have been made by Kuo and Krenciglowa 19). These will be examined and applied here. In addition, the energy-dependent effective interaction has been shown to be equivalent to the folded-diagram effective interaction 20). We will look at this point in actual calculations. Another difficulty which had plagued calculations in finite nuclei is the presence of the Pauli exclusion operator in the equation for the G-matrix. This has been largely overcome in the method of Tsai and Kuo 21), which is used in conjunction with momentum-space matrix inversion (to find G,, the G-matrix in free space, with no Pauli operator) as described by Krenciglowa, Kung, Kuo and Osnes”). This method is particularly suitable for the Paris potential (which is momentum dependent) and is the one we have used to construct our G-matrix. Thus the general procedure is as follows: We cannot solve the nuclear manybody problem directly. Instead, we must solye a model-space problem of linite (low) dimension whose solutions are in principle, the same as the low-lying states of the complete (infinite dimensional) problem. For example, in our prototype nucleus “0 we consider the shell-model problem of two sd-shell nucleons outside and 160 closed core. The model space in which we seek to solve this problem consists of all states with two sd-shell particles. Clearly, there are many other possible configurations for “0, such as 2 + n sd-shell (or higher-shell) particles and n holes in the 160 core. What we must do is to find within the model space an effective interaction between the two particles which takes into account those configurations we do not include explicitly. This is done by the $-box formulation l’) of the folded-diagram theory. There are still various remaining problems and open questions in the procedure referred to above, and we will address some of them here. The $-box contains
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diagrams to all orders in the interaction (I&+ or equivalently G). Using existing methods it is quite difficult to evaluate even the third-order diagrams. This difficulty is twofold. First, the formulae for the diagrams require tedious and lengthy derivations. Second, the computation time required to numerically evaluate the diagrams using the formulae so obtained makes the calculations practically impossible. A new method was recently proposed by Kuo, Shurpin, Tam, Osnes and Ellis23) which gives a simple and systematic way of writing down the diagram formulae almost by inspection. The formulae so obtained are often easier to interpret physically than previous ones. The method has the added advantage that it yields diagram formulae which are usually much more efficient for numerical computation. This method will be used in the present work. Perhaps the most important remaining question regarding the effective interaction is whether, and under what conditions, it converges. This question has received a great deal of attention in recent years ‘p2). In the early calculations of Kuo and Brown24) only two terms in the series were retained, the bare G-matrix and the 3plh (second-order) core-polarization diagrams. Yet their results agreed fairly well with experiment, and subsequent, far more elaborate calculations, yielded no dramatic improvement. This seemed to suggest that the effectiveinteraction series converged very rapidly, or at least that the higher-order terms cancelled among themselves. Not long afterward, the validity of this was questioned by the work of Barrett and Kirson 25). They showed that many of the third-order diagrams were of comparable size with the second-order corepolarization diagram. Moreover, the third order as a whole was generally of opposite sign to the second order. Thus truncating the series after the second order seemed not to be justified. In the folded-diagram theory the question of convergence can be divided into two parts ; first, whether the o-box converges and, second, whether the folded-diagram series converges. In view of the Kuo and Brown results, which included no folded diagrams, there is also the question of why, or indeed whether, we need to include the folded diagrams at all. This work will attempt to address these and related matters. We now briefly outline the structure and content of the sections which follow. We begin in sect. 2 with a brief review of the basic formalism of microscopic theories and model-space problems leading to the diagramatic expansion for the effective interaction. The notions of active and passive lines and linked and irreducible diagrams lead to a definition of the &-box, the basic entity in our expression for the effective interaction. The grouping of terms in the foldeddiagram series is then examined, especially as it.relates to one-body and two-body connected graphs. We briefly review also two techniques to obtain effective interactions which are also expressed in terms of the &-box. The first is the one derived by Kuo and Krenciglowa 19) starting from the equation of motion. We show how this method allows us to include the “hole-hole phonon” which considers the interaction between two hole lines to all orders. The second is the iteration formula for the effective interaction of Lee and Suzukiz6).
314
J. Shurpin et al. / Folded diagrams
Sect. 3 contains a brief discussion of the NN interaction. Special attention is given to its momentum-dependent form. We carry out various comparisons between the Paris and Reid potentials. We also do a series of phase shift calculations using the momentum-space matrix-inversion method 27). This is a necessary test of our computer code for our calculation of the Paris potential Gmatrix which we will use in all subsequent nuclear structure calculations. Sects. 4 and 5 describe an extensive series of calculations for sd-shell nuclei, based upon the ideas developed in the previous sections. Most of these are done in parallel for the Paris and Reid potentials. The hope is that we can perhaps trace the nuclear structure results back to the features of the free NN interaction, especially for the Paris potential. We study the effects of various refinements of the &-box, such the inclusion of hole-hole and TDA particle-hole phonons. Our calculation is done with an energy-dependent G-matrix and this enables us to include the off-energy-shell effects in the calculation of effective-interaction diagrams. Other components of the calculation such as our choice for the single-particle energies are also investigated. Our main effort is directed towards understanding the structure and function of the folded-diagram series and its convergence properties. The numerical results for the sd-shell matrix elements are tabulated. The spectra of several sd-shell nuclei (‘*O, ‘sF 19F 22Na, 22Ne) are presented and discussed. We will compare the various spectra both among themselves and with experiment. In sect. 6 we discuss the conclusions to be drawn from our various results and suggest several possibilities which may improve our results.
2. Folded-diagram effective interactions 2.1. INTRODUCTION
As outlined in sect. 1, our aim is to perform a microscopic nuclear-structure calculation based on a realistic NN potential V = I&,. That is, for a nucleus with A nucleons we try to solve the non-relativistic Schriidinger equation H’&1,2 ,... A)=E$(1,2
,... A),
(2.1)
where H=T+V,
(2.2)
T being the kinetic energy operator. It is clear that even for light nuclei we are unable to solve eq. (2.1), the A-body problem, exactly. The way to proceed is to project the wave function $ of the complete Hilbert space onto a model space of low dimension in which we are able to solve the problem in a more physical way.
J. Shurpin et al. / Folded diagrams
We first rewrite the hamiltonian
31.5
as
H=H,+H,,
(2.3a)
Ho= T+U,
(2.3b)
H,= V-U,
(2.3~)
where V is the NN interaction and U is some single-particle potential. U is chosen so as to define a convenient set of unperturbed basis functions 4, given by
&A = VP,.
(2.4)
Frequent choices for U are a harmonic oscillator potential, which we will use in the present calculation, or a Hartree-Fock single-particle potential. We then divide the complete Hilbert space into two regions P (the model space) and Q (its complement) such that P+Q
= 1,
(2.5a) (25b) (2.5~)
(ila)
= 0,
(2.5d)
where both Ii) and’ ICC)are eigenfunctions (i.e. hi and 4,) of the unperturbed hamiltonian H,.The division of the Hilbert space (i.e. what to choose as the model space) depends on the particular problem, and one is guided by physical intuition. For 180 a reasonable choice would be a closed 160 core with 2 particles in the sd shell. Thus in eqs. (2.5). Ii) is any state with a closed I60 core and 2 particles in thgsd shell while la) represents the respective complement (in which all states have holes in the I60 core and/or particles in shells above the sd shell). We seek to reduce eq. (2.1) to a more tractable problem:
PH,,,W = EW,
(2.6a)
H err
(2.6b)
where =
H, + V,,,.
Note that in eq. (2.6) E and tj are the same as in eq. (2.1). For a model space (P-
316
J. Shurpin et al. / Folded diagrams
space) of dimension d, Heff will reproduce d of the eigenstates of H. One must obtain the form of V,,, which must account for all states belonging to Q that are now being left out of the problem. There are several forms for V,,, to be found in the liteature. There are energydependent forms, I/,,,(E), which have been obtained by Feshbach 12) and by Bloch and Horowitz i3). In this work we will use the folded-diagram theory of I&, in the form derived by Kuo, Lee and Ratcliff l’) (henceforth ‘referred to as KLR). This form of V,,, is independent of E. This is a desirable feature, since the empirical matrix elements (which are obtained by constructing a V,,, which can best reproduce the experimentally observed properties of various sd-shell nuclei) are also independent of E. 2.2. FOLDED-DIAGRAM
The folded-diagram
EXPANSION
OF
I&
expansion l’) for Hi” (or V,,,) is
(2.8) where Q is the Q-box and 1 the generalized folding operation. The first Q in each term which is folded upon is written with a prime (Q’) to indicate that the lowestorder diagram it contains is at least second order in the interaction. The Q-box is composed of irreducible linked diagrams ” ’ ‘). It can be shown ‘) that V,,, of eq. (2.7) can be written in terms of derivatives of the Q-box (the so-called derivative formulation) as
v,,,= F,+F,+F,+F,+...,
(2.8)
where F, is the term with n foldings. As will be discussed later, a main part of our calculations will be carried out using this F, expansion for I!,,,. The KLR-model-space eigenvalue problem, after cancellation of the unlinked diagrams, is (2.9) where E; is the true l6 0 ground-state energy and (ii@) is the projection of the true wave function $ onto the model space. Note that the eigenvalue here is (En-E:) the energy relative to the 160 ground state. Observe that I$,, contains l-body diagrams such as diagram (b) in fig. 1. The sum of the l-body contributions is actually what determines the “0 energies. It is customary in eq. (2.9) to replace H, by HYP, i.e. by the experimental sd-shell single-particle levels (which are taken as the low-lying “0 states). As this accounts for the l-body diagrams, we must not
317
J. Shurpin et al. / Folded diagrams
include them a second time in V,,,. That is, V,,, should contain only the 2-body connected diagrams. For the first term, F,, this is easily done. We simply do not include the l-body diagrams in the $-box. For the subsequent terms, this cannot be done in this way. Consider fig. 2, where in each case the dashed line represents a &-box. Diagrams (a) and (c) belong to F, and diagrams (b) and (d) to F,. A “blob” attached to a line represents any sequence of interactions attached to that line. Thus (c) and (d) should be excluded from I/em since their contribution is already accounted for by HrP. Diagram (b), however, should still be in I/err,since it is 2-body connected. Indeed, its contribution can be quite large. It is only diagram (d) which we must remove. To this end, the g-box is introduced2a). The s-box is defined as the l-body part of the &-box (i.e. the set of all linked and irreducible diagrams in which all the interactions are connected to one line and the other line is merely a “spectator”). What must be removed from V,,, are all the terms in which the $-box folds upon itself only. Thus we have the following:
----_
_--.
_-_---
_D 10 -_-
---_iI
0
_-_- /’ _-_--___ t
-__ --__--_ 1$I ‘$ (0)
(c)
(b)
--0
_-_-
(d)
___
(e)
Fig. 1. Examples of linked and unlinked, reducible and irreducible diagrams.
318
J. Shurpin et al. / Folded diagrams
___ Ii 1+ “/
:
( ‘/3 ; 2
(a)
,’
‘8
\-
-__ !
(cl
Fig. 2. One-body and two-body connected diagrams in the folded-diagram series for qrr The dashed line represents the Q-box interaction between the two lines. A “blob” represents a series of V (or U) interactions attached to the line.
and so forth, where (2.11) with o0 being the degenerate model-space energy (PH, = Pw,). We now define
F”= MB-F,(9),
(2.12)
where s^ is the l-body box which we just introduced. Clearly the difference P, contains only the proper 2-body connected diagrams of V,,,. This gives us the equation we will actually use in our calculations,
v,,, = F,+F,+F,+F,+F,....
(2.13)
We include terms up to F, only. The number of terms in F, grows very rapidly with n, as can be seen from eq. (2.10). In sect. 4 we will explore the convergence properties of the folded-diagram series for V,,, when we group the terms according to the number of folds as in eq. (2.13).
J. Shurpin et al. 1 Folded diagrams 2.3. THE EQUATION-OF-MOTION
319
METHOD FOR v,,
Recently, Kuo and Krenciglowa 19) have proposed a somewhat different and more general method to obtain a folded-diagram expansion for the effective interaction. This method enables us to sum up a larger class of diagrams contained in V,,, as compared with the KLR method outlined in the previous section. The starting point of the KLR method is a model space whose states consist of two (sdshell) particles outside the unperturbed 160 core, [a:aT]lc). Thus the eigenfunctions we obtain in solving eq. (2.9) are the projections of the true wave functions onto this “uncorrelated” model space. At the heart of the newly proposed method is to use as a starting point lt&), the true (correlated) ground state of 160 in place of 1~). The model-space basis states are thus of the form [a+af]t&). An advantage of this is that when we obtain a secular equation of the form eq. (2.9) its eigenfunctions will be
which can be related to experimentally observable spectroscopic factors. This cannot be done if we use 1~). We do not give any details of the method here, except to point out some of its specific implications for our calculations described in sect. 4. In this method we write V,,, as (2.14) where the bar over the first Q-box in each term indicates the last interaction (attached to the outgoing valence lines) must take place at t = 0. All the other interactions are now constrained by cc > t > -co. As in eq. (2.9), the first 0 in each folded term is at least second order in the interaction. The value of any diagram (i.e. the diagram rules) will in general be different in the new and old schemes. We do not pursue this point except to evaluate in detail one diagram which will be important in our calculation. Consider the diagram in fig. 3. We suppress factors (such as I/ matrix elements) which do not depend upon which method we use and concentrate on the time integrations only. In the old scheme the time constraints are to = 0,
o>t,
> -co,
o>t,>
-co,
t,>t,>
-00,
whereas in the new scheme they are tz = 0,
o>t,>
-co,
co > t, > t,,
t,>t,>
-co.
J. Shurpin et al. 1 Folded diagrams
320
a
b
Fig. 3. The 4p2h diagram
I
which is evaluated
differently
in the old and new schemes.
Thus in the old scheme we have for this diagram the energy factor 1
1
wo-(&,+&A
1
0O-(&,+&b+&,+&d-&h,-Eh,)
(E~,+E&(E,+EB)’
(2.15)
while in the new scheme we have 1 oo-(sy+sA
1
1 -%-(-%,-+h,)
(2.16)
(sh*+E&(E,+EJ
where we have defined the starting energy o. as
Note the on-energy shell expression for the middle energy denominator obtained in the new scheme. This allows us to sum up-the 4p2h diagrams to infinite order as we will do in sect. 4. In the old scheme we could not do this. Note also that in fig. 3 the vertex a time t2 is the “last” interaction attached to the (outgoing) valence line. 2.4. THE
LEE-SUZUKI
ITERATION
METHOD
FOR
v,,,
In our calculation of l&, the first step is to calculate the @box. Then we calculate the folded-diagram series of eq. (2.7) or (2.14) using either the F, method of eq. (2.13) or the Lee-Suzuki iteration method 26) which is here outlined. The Lee-Suzuki method calculates the effective interaction V’,, in an iterative scheme. Denoting the nth iteration for the effective interaction as R,, we have R,
J. Shurpin et al. / Folded diagrams
321
related to R, (k< n)by II-1
R,=
n-1
(l-t-%- 1 &m
(2.17)
In=2
where Q is the Q-box interaction defined in subsect. 2.2 and &. represents the energy derivative of the Q-box as defined in eq. (2.14). An advantage of this method is that we only need to know Q and Q, at a given starting energy wO. As we will discuss later in sect. 4, the calculated V,,, is essentially independent of oO. (Theoretically, V,,, is independent of oO.) The Q-box here is the same as in the KLR theory or the method described in the preceeding section, and it enters in a perhaps more natural and convenient way. It contains all the linked and irreducible diagrams but no folded diagrams. The folded diagrams enter by way of eq. (2.17), which provides a way of regrouping the folded-diagram series. We will examine how this grouping affects the convergence of V’, in sect. 4, where V,,, will be calculated by first evaluating the Q-box and its derivatives and then calculating R, according to eq. (2.17). 3. Method of calculation In the last section, we obtained an expansion for the effective interaction in terms of matrix elements of the NN interaction I/. It has been common practice to use for I/ phenomenological NN potentials such as the Reid potential 5). In the present work we will use both this potential and the Paris potential ‘) as V. We will first compare some essential features of these two potentials, and then discuss some details ‘of the momentum-space matrix-inversion method on which our calculation will be based. 3.1. COMPARISON
BETWEEN
THE PARIS AND REID NN POTENTIAL
A main purpose of the present work is to compare the spectra of several sd-shell nuclei calculated from the Paris NN potential ‘) with those from the Reid NN potential. It is important to note that these two potentials are in fact quite different, although they give quite similar NN phase shifts. The Reid NN potential is energy independent, and for each partial wave channel is given as a sum of a few Yukawa-type terms, i.e. &e-~1~/p~r, where ci and CLiare-parameters and I is the internucleon distance. On the other hand, the Paris potential has a linear energy dependence which is, for convenience, transformed into a p2 dependence. Then, the Paris potential, in its parametrized form ‘), is given in r-space for each isospin T as
(3.1)
322
J. Shurpin et al. 1 Folded diagrams
where
(3.2)
(3.3) all in appropriate units. The labels s, t, LS, T, and SO, refer to the standard singlet, triplet, spin-orbit, tensor, and quadratic spin-orbit non-relativistic invariants, respectively. The V(r)‘s are expressed as a sum of Yukawa-type terms. For example, for the central singlet component of the potential one has f& = a(1 -br. igr)=
a,),
igjz
j=
1
(3.4) (3.5)
I
in eqs. (3.1) and (3.2) respectively. The potential is regularized at the origin (Y= 0) by requiring n-1
gn=-rn,C
%. j=l
(3.6)
mj
The form of the other components as well as the numerical values of the parameters gj and mj are found in ref. ‘). Although it is possible to work with this potential directly in configuration space, it is somewhat inconvenient to do so. Fortunately, the situation is much simpler if a plane-wave basis is used, as in the present work. Then the plane-wave matrix elements of the Paris potential are given simply as
($ +~)
(3.7)
for any partial wave *‘+lL. 3’ Here k and k’ are the magnitudes of the relative momenta of the outgoing and incoming nucleon pairs respectively. For convenience, numerical integration was used. This was checked against the analytic expression for the IS,, partial wave and found to be quite accurate. In fig. 4 we plot V(k, k) for the Paris potential ’ (solid line) and for the Reid potential (dashed line), for partial waves IS, to 3D,. One is immediately struck by the fact that, although both potentials were made to fit the same scattering data, the two potentials are distinctly different. Similar differences have also been found for other partial waves.
J. Shurpin et al.
323
/ Folded diagrams jr
-\ \ \
lL.A
4
\
\
\
3
jSI
\
\
‘1
‘\
2
I
2
3
4
5
k
I
2
2
3
4
5
3
4
5
k
3 k
I
4
5
I
2 k
Fig. 4. Plot of V(k, k) for the Paris (solid line) and Reid (dashed line) potentials for selected partial waves. The k are in fm- ’ and V is in units of 41.47 MeV . fm3.
In the Paris potential the components other than the central ones are not energy dependent and are functions of I alone. Thus they can be compared with their counterparts in the Reid potential directly in configuration space. We do this in fig. 5 for the important T = 0 S = 1 tensor force +. As shown, the tensor forces for these two potentials are quite different. It is of interest to see how this difference manifests itself in nuclear-structure calculations. Before proceeding to the actual nuclear structure calculation we study the Paris potential NN phase shifts and compare with those of the Reid potential. A more compelling reason for looking at the phase shifts is that the method used to calculate them (momentum-space matrix inversion) is also used in the calculation of the G-matrix and thus provides a check of our computer programs.
+ In the early stages of this work, the Paris potential parameters were taken from a preprint. In the published version’) the T = 0 parameters are somewhat different. Thus, in fig. 4 the preprint parameters were used, as they were in the phase-shift calculations of the following section. Fig. 5, however, used the published parameters. (The preprint had an even weaker tensor force.) Of course, for the G-matrix elements the published parameters were used.
324
J. Shurpin et al. / Folded diagrams
!
I I II 0.5
0
I 1.0
I 2.0
I 1.5
2.5
r (fm)
Fig. 5. Comparison
of the T = 0, S = 1 tensor component between the Paris (solid line) and Reid (dashed line) potential.
3.2. PHASE-SHIFT
CALCULATIONS
To calculate NN scattering phase shifts, we use the momentum-space matrixinversion method as outlined in ref. “). Briefly, for an uncoupled partial wave, the phase shift is given by (3.12)
6,(k) = -tan-‘[kR,(k,k)], where R,(k, k’) is given by the principal value integral equation R,(k, k’) = I/;(k, k’)-
a
m I/;@,q)R,(q, U q2dq
P s0
q2
-
k”
(3.13)
This integral can be rewritten to avoid the principal-value singularity”). A similar expression exists for the coupled-channels case. To solve eq. (3.13) we use gausian integration. To determine the validity of this procedure, we chose several quite different sets of Gauss points and looked at the resulting phase shifts to see if and how they were affected. The integral (3.13) from 0 to cc is broken up into several intervals. The intervals
325
J. Shurpin et al. 1 Folded diagr~s
are specified by S,, the start of the nth interval and by N,, the number of Gauss points in the interval between S,_ 1 and S,,. For finite intervals, the Gauss points qi (and weights) are mapped from the region - 1 I xi 5 -+ 1 onto the interval. If the interval extends from S, to infinity, we use a tangent mapping to obtain the Gauss points qi = S, + C tan&( 1+ xi).
(3.14)
N, is the number of points for which the tangent mapping is used. The constant C is for adjusting the distribution of the tangent Gauss points. The total number of points is N,,. The procedure is essentially the same as that of ref. “). The choices of Gauss points for several calculations are specified in table 1, all with C = 1. Tables 2a and 2b show the phase shifts for the Paris potential calculated with Gauss point sets 1 and 2 of table 1. Note that they agree remarkably well with each other despite the large difference between the Gauss points used. We have found that the distribution of Gauss points is far more important than the number of points used. In table 2c we quote the phase shifts obtained in ref. ‘). Note that the phase shifts we calculate are the eigenphases whereas the ones from ref. 7, are the nuclear bar phase shifts. The two can be related by equating the expressions for the S-matrix obtained from the two methods. [See e.g. ref. ““).] We have done this and found the agreement with the publish~ values to be very good for table 2b. We found it to be important to map the Gauss points onto several intervals rather than one interval covering the entire range from 0 to cx). The phase shifts (and hence the R-matrix) were found to be quite stable with respect to reasonable variations of the Gauss points. For the G-matrix calculation we have used the Gauss points set (G) of table 1. For comparison, the Reid-potential phase shifts are given in table 2d. [These are the ones we calculated using Gauss points set 3 of table 1, i.e. eigenphases, but when converted into the nuclear bar phases they agree closely with those given by Reid in his paper ‘).I The significant difference between the two potentials is in the
TABLE1 Specification of th,e Gauss points Set
So
s,
S?.
S,
N,
N,
1
0 0 0 0
3.0 2.75 2.6 2.5
10.0 8.0 7.0 7.0
25.0 18.0 15.0 15.0
6 11 11 15
5 7 7 10
2 3 G
N3
N,
4 4 4 6
0 3 3 4
N,,
15 2.5 25 35
The boundaries of the integration intervals are the S, (in fm-‘) and N, and N, are the number of points in the interval. The last row (G) refers to the Gauss points used in the G-matrix calculation of subsect. 3.4.
326
J. Shurpin et al. 1 Folded diagrams TABLE 2a
Results of phase-shift calculations (in degrees) for the Paris potential calculated with the momentumspace matrix inversion method using Gauss points set (1) of table 1 Et,, (MeV) =
‘So ‘D* 3P0 -1 -2 E2 3F2
‘P, % “1 3D, -2
25
50
95
142
210
330
48.68 0.75 9.20 - 5.24 2.91 - 17.59 -0.19
38.42 1.84 11.93 -8.60 6.53 - 16.53 -0.18
25.25 3.79 10.26 - 12.91 11.63 -14.11 0.10
14.92 5.65 6.12 - 16.55 14.97 - 12.02 0.44
3.18 7.76 -0.45 -21.21 17.21 -9.74 0.62
- 12.46 9.47 - 11.01 - 28.50 16.85 - 6.94 -0.12
- 6.99 80.51 1.84 - 2.94 3.86
- 10.66 62.31 2.20 -6.79 9.59
- 14.75 43.43 2.79 - 12.22 18.10
- 17.65 30.16 3.80 - 16.44 23.84
-20.81 15.97 6.06 - 20.81 28.00
-25.00 - 1.46 13.86 - 25.49 28.75
‘P, partial wave. Otherwise, the phase shifts are very similar. Of course it must be remembered that both potentials were constructed to reproduce the phase shifts. Still, the potentials themselves are markedly different and one cannot predict in advance their results in a nuclear structure calculation. 3.3. THE G-MATRIX
In sect. 2 the effective interaction is given as a series of diagrams, each containing matrix elements of the NN interaction V. In general, matrix elements of TABLE2b Same as table 2a, but using Gauss points set 2 of table 1 E,,, (MeV) =
‘SO ‘D2 3P0 3P, jp2 82 3F2
‘Pl % &2 -* ‘D2
25
50
95
142
210
330
48.58 0.75 9.19 - 5.25 2.91 - 17.60 -0.19
38.31 1.84 11.93 -8.60 6.51 - 16.53 -0.18
25.18 3.79 10.25 - 12.94 11.59 - 14.12 0.10
14.90 5.64 6.10 - 16.61 14.90 - 12.03 0.44
3.21 7.75 - 0.45 -21.32 17.11 -9.75 0.62
- 12.48 9.44 -11.04 - 28.75 16.74 -6.95 -0.12
- 7.01 so.25 1.80 -2.90 3.87
- 10.67 62.04 2.18 - 6.77 9.59
- 14.79 43.16 2.78 - 12.30 18.03
- 17.74 29.95 3.81 - 16.54 23.73
- 20.93 15.91 6.09 - 20.87 27.85
- 25.21 - 1.44 13.83 - 25.70 28.48
J. Shurpin et al. / Folded diagrams
321
TABLE2c Same as table 2a, but for the Paris potential quoted from ref. ‘); see text for explanation E,,, (MeV) =
‘So
25
‘D, ‘Pll 3P, 3P, s2 3P*
48.51 0.75 8.87 -5.04 2.44 -0.85 0.11
38.74 1.81 11.82 - 8.41 5.13 - 1.78 0.35
25.85 3.12 10.35 - 12.76 10.66 -2.6-l 0.77
15.68 5.56 6.31 - 16.47 14.06 -2.91 1.04
4.03 1.66 -0.16 -21.27 16.49 -2.71 1.08
- 11.80 9.31 - 10.79 - 28.90 16.44 -2.00 0.14
‘PI js, 82 3D, 3D,
- 7.03 19.81 1.72 -2.95 3.97
- 10.75 61.56 1.96 -6.16 9.62
- 14.87 42.12 2.24 - 12.29 17.96
- 17.87 29.61 2.69 - 16.52 23.58
-21.13 15.57 3.59 - 20.73 21.51
- 25.52 -2.61 5.54 -29.15 28.13
I/ are infinite or very large even for soft-core potentials. This is especially true in an oscillator basis with non-vanishing relative wave function at the origin. To overcome this, the reaction matrix, or G-matrix was introduced *). In operator language we have in our case 21) G(w,) = I’+ VQ wg
_‘eTQ Q’%Ad
(3.15)
where Q is the Pauli exclusion operator and oO is the starting energy. Diagrammatically, eq. (3.15) is expanded in fig. 6. Note that intermediate lines TABLE2d Same as table 2a, but for the Reid potential using Gauss points 3 of table 1 E,,, (MeV) =
25
50
95
142
210
330
“P,
49.09 0.68 8.60 -4.60 3.02 - 15.86 -0.15
38.19 1.66 11.55 - 8.09 7.10 - 14.40 -0.12
25.64 3.39 10.22 - 13.16 13.10 - 12.09 0.18
15.52 5.11 5.98 - 17.39 16.97 - 10.54 0.55
4.32 7.18 - 1.18 - 22.23 19.51 -9.11 0.90
- 10.14 9.31 - 13.29 - 28.46 19.78 - 7.26 0.79
‘PI 3S, s1 3D, “D,
- 1.94 81.17 1.90 -2.91 4.13
-4.26 62.80 2.58 -6.91 10.10
- 10.67 43.85 3.91 - 12.28 17.80
- 17.47 30.90 5.98 - 16.15 22.13
- 26.07 17.80 9.12 -20.18 24.98
- 31.94 2.51 18.05 - 26.04 26.08
‘SO ‘D* jPll jP, 3P* 61
J. Shurpin et al. / Folded diagrams
328
=
+
--+. +II_----
Fig. 6. Definition of the G-matrix between two particle lines.
must be passive lines - the railed lines which represent the single-particle states higher than the pf shell of fig. 7. Otherwise the diagram is not allowed in the Q-box (recall subsect. 2.2). Thus all the diagrams in the Q-box and hence all the diagrams contributing to V’,, have their V-vertices “replaced” by G-vertices in analogy with fig. 6. Since the matrix elements of G are finite we can proceed to calculate the matrix elements of V,,,. We do not go into the details of the G-matrix calculation, as we closely follow the procedure of ref. “). We only make several notes in passing. As pointed out, eq. (3.15) is similar in structure to eq. (3.13) and we use the same momentum space method to solve it. We use 35 k-space Gauss points distributed as indicated by the last line of table 1. This choice should be adequate, judging by the phase-shift calculations described in the previous section. Also, in eq. (3.15) we use a twoparticle Pauli operator Q specified by (n,, n,,, nrrr) = (3, 10, 21). That is, the intermediate states included in G must lie outside the shaded region in fig. 7. The choice of Q is very important in determining which G-matrix diagrams are allowed in the Q-box as well as in determining the allowed orbits for intermediate states of
pfh -
nI
nr
ht
Fig. 7. Specification of the Pauli operator Q used in the calculation of the G-matrix. It is defined by (n,, n,,, n,,,) = (3, 10, 21) with the orbit numbering as explained by the circled numbers.
J. Shurpm
et al. / Folded diagrams
329
$-box diagrams. [See e.g. ref. ““).] Our procedure was as follows. We started from the Reid or Paris Potential, obtained V(k, k’) using gaussian integration, and then G,(k, k’), the “free” reaction matrix by gaussian integration and matrix inversion. Gaussian integration is again used to obtain (~~~ZlG~(~)l~~~‘~) in a rel-GM oscillator basis. The Pauli operator was treated by the Tsai-Kuo method *I), which expresses eq. (3.15) in terms of P, the complement of Q, and G,. Finally, everything is gathered together and transformed to the laboratory frame oscillator basis by means of a Moshinsky transformation. We obtain JT G(ab, cd; JT)
JT (3.16)
n l-l = (ab(Glcd),
where a, b, c, d refer to the oscillator orbits specified by (n871rl,j 0)* etc. J and T are total angular momentum and isospin, respectively, and the coupling is as indicated by the arrows in eq. (3.16). The matrix elements (3.16) are precisely what we need for nuclear-structure calculations. Extensive computer programs have been constructed for carrying out the above G-matrix calculation. Tables of G-matrix elemer,ts will be given later in sect. 4. 3.4. CALCULATION OF &-BOX DIAGRAMS
In the previous sections we have sketched the development of the effective interaction V,,, in terms of the $-box formulation of the folded-diagram theory. The &-box is the sum of all the linked and irreducible diagrams. Thus we need a set of rules to enable us to evaluate the diagrams in a simple systematic way. The diagram rules in the uncoupled representation or m-scheme are well known 2, 30).
3
J’M’
Fig. 8. The 3plh
,J’M’
core-polarizationdiagram(a) is more easily evaluated when recoupled as in (b).
330
J. Shurpin et al. / Folded diagrams
For actual nuclear-structure calculations, we will need to evaluate diagrams in the angular-momentum-coupled representation, such as the core-polarization diagram of fig. 8. Recently, a set of simple diagram rules has been developed 23) to evaluate these angular-momentum-coupled diagrams, and we will employ it to calculate all the @box diagrams in the present work. To illustrate the above method, we give in the following the formula for calculating diagram (a) of fig. 8. This diagram is first recoupled to become diagram (b), using the usual X or 9-j transformation coefficients. Then diagram (a) is readily given 23) as
(a) =
c
d
j
[‘J,
b J’
j o
r
1 &J+1)(2T+1)
C+ J’T’
; &2J’+1)(2T’+l)
I
J’T
(3.17)
where the last matrix elements are of the cross-coupled type 23) and the energy variables are or = E, + E,, and o2 = E, +.sd - E, +E~, the E’Sbeing the single-particle energies.
4. Results for two-body effective interactions 4.1. THE &-BOX
DIAGRAMS
The next step in calculating an effective interaction is to evaluate the &-box (and its derivatives). The Q-box is composed of all the linked and irreducible (G-matrix) diagrams. We take as a first approximation to the Q-box all the diagrams up to second order in the G-matrix. The diagrams we include are Dl through D7 and U (the one-body potential) as shown in fig. 9. In diagram D5 the intermediate particle states are restricted to the pf-shell because the G-matrix was calculated with a Puli operator defined by (n,, n,,, n,,,) = (3, 10,21) which takes into account passive particle states of orbit 11 and above only. Orbits 7-10 (the pf-shell) must be included explicitly. For the same reason, particles in the pf-shell are also included in the intermediate states of diagram D2. Our @box, expressed in terms of the G-matrix, takes into account the particleparticle interaction (outside the model space) to all orders. If we use the equation of motion method of Kuo and Krenciglowa (subsect. 2.3) we can extend our Q-box
J. Shurpin et al. 1 Folded diagrams
331
0
P 2 C
P2
P3
Fig, 9a. The one-body diagrams included in various approximations to the Q-box. The shaded circle represents a (bare) TDA particle-hole phonon in diagrams P2 and P3.
P6
P7
Fig. 9b. The two-body diagrams included in various approximations to the Q-box. The shaded circle represents a (bare) TDA particle-hole phonon in diagram P7 and a hole-hole phonon in P6.
332
J. Shurpin et al. 1 Folded diagrams
to include the hole-hole interaction to all orders as well. That is, the two hole lines of diagram D6 are allowed to interact any number of times so that D6 is replaced by diagram P6. (See fig. 9.) In P6, the blob represents a hole-hole phonon. This refinement of the Q-box approximation can have a significant effect on the matrix elements of V,,, and the resulting spectra. Another modification is in the treatment of the core-polarization processes. In place of the particle-hole pair in D2, D3 and D7 we can include a TDA particle-hole phonon. That is, D2, D3 and D7 are replaced in the &box by P2, P3 and P7, respectively. In obtaining the TDA eigenvalues and eigenvectors, we include only the bare (first order) G-matrix particle-hole interaction vertex as shown by diagram (a) of fig. 10. Because we include the TDA phonons only as a “correction” to the particle-hole intermediate states, inclusion of only this bare vertex should be sufficient. Also, it has been demonstrated that the inclusion of all the second order particle-hole interaction vertices, such as diagram (b) of fig. 10, in RPA-phonon calculations gives results very similar to the ones obtained in bare TDA only22 32). One result of including the TDA states rather than simple particle-hole states is that we can now isolate the spurious l- state in diagrams P2 and P3. This cannot be done if we include only the particle-hole intermediate state by second-order perturbation theory because then all the l- states (there are 7 of them) are mixed. The TDA phonon intermediate state allows us to isolate the spurious l- state and remove it in lowest order. This is done by explicitly removing the spurious lcontributions in the calculation of diagrams P2 and P3. The removal of the spurious state changes P2 and P3 by about 0.6 MeV. The results we will quote here are from calculations in which the spurious state is removed. However, to gauge the effect of including TDA phonons in place of simple particle-hole states, we really should leave the spurious state in for the comparison to be valid, since we do not remove it in the simple particle-hole case. The calculation was done in a harmonic oscillator basis with hQ = 14 MeV. The unperturbed starting energy oO, the point at which we calculated the effective interaction, was taken to be - 10 MeV (for two sd-shell particles). The different
(a)
(b)
Fig. 10. The bare TDA vertex (a), and one of the second-order TDA vertex (b).
J. Shurpin et al. / Folded diagrams
333
approximations to the &box are referred to as Cl, C2, C3 and C4 for the &-box with 2nd order diagrams only, 2nd order plus hole-hole phonon, and 2nd order plus particle-hole phonon, and 2nd order plus both hole-hole and particle-hole phonons, respectively. Thus, the Cl interaction includes diagrams D4, D5, D6 and D7 of fig. 9b ; for the C2 interaction, D6 is replaced by P6, etc. Effective interactions obtained with these four methods will be compared in sect. 5. 4.2. INCLUSION OF THE FOLDED DIAGRAMS
In the early microscopic calculations of this type23 24) the folded diagrams were generally left out and the effective interaction was taken to be the &-box alone (in some low order approximation such as the bare G plus 3plh core-polarization diagrams). Barrett and Kirson 25) have investigated several once-folded diagrams and found their effects important. In the present work, we shall investigate the effect of folded diagrams in a more extensive and systematic way. When more processes are included in the &-box, it will become apparent that the $-box alone gives substantial overbinding as well as poor level spacing. (See e.g. fig, 11 for n = 1, which corresponds to V,,, = 6.) Thus we must include the folded diagrams to renormalize the effective interaction. This we can do in one of the three ways described below. It has been shown2’) that the Q-box evaluated at a self-consistent starting energy is equivalent to the effect of V,,, with all the folded diagrams included. That is,
[Ho+ K,tll’&> = [Ho-f &(E,)]I ‘y,> = E,l ‘y,),
(4.1)
where E, is the total energy of the nuclear system relative to the exact ground-state energy of the core system. For example, E, is E,(190)-Eo(‘60) when the above equation is used to calculate the energy spectrum of r90. As discussed earlier, V,,, is related to the @boxes by, for example, eqs. (2.13~(2.16) and in calculating I$,, we need to evaluate the folded diagrams involving the &-boxes. Eq. (4.1) is in some aspects more convenient for-calculation because we just evaluate the &box at the self-consistent energy without the need for calculating any &-box folded diagrams. For a given o-box, this self-consistent energy can usually be obtained by a graphical method. There is, however, a rather important disadvantage associated with the above self-consistent equation. Namely, &(E,) is dependent on the total energy of the nuclear system and for different nuclei and different nuclear states the @boxes will need to be recalculated. In other words, this formulation does not lead to a set of matrix elements of V,,, which can be used to treat the low-lying states of all the sd-shell nuclei, unlike the case of the empirical shell model. For example, we would like to have a set of two-body matrix elements of V,,, which can be used to calculate both the ground state of ‘*O and the excited states of
334
J. Shurpin et al. 1 Folded diagrams
20Ne. The matrix elements of $(E,) of eq. (4.1) do not meet this requirement. Another inconvenience of &!Z,,) is that it contains unlinked valence diagrams, whereas V,,, contains only linked valence diagrams. Thus in the present work we will not use the self-consistent formulation of eq. (4.1) for major calculations except for checking the convergence of the folded-diagram series for simple cases I80 and i8F. A second method is to sum the folded-diagram series explicitly as described in sect. 2. Using the method of grouping terms according to the number of folds we can proceed in one of two ways. We can use eqs. (2.12) and (2.13) to obtain what are usually called the sd-shell effective-interaction matrix elements. In this case, the energy levels obtained are relative to l’0 since the one-body terms are removed from V,,, and are replaced by the experimental I70 low-lying states. Alternatively, we can retain the one-body diagrams in V,,, (i.e. use F, = F,(Q)) in which case the energy levels obtained are relative to the 160 ground state. In either case, all the & (see eqs. (2.10~(2.13)) are evaluated at the unperturbed starting energy oo, at which point all the matrix multiplication is carried out. A third way to include the folded diagrams is to group them as prescribed by the Lee-Suzuki iteration method. Here one also uses the 0, evaluated at w. but the terms are grouped differently. One solves for the eigenvalues of the Ho + R,, where R, is defined by eq. (2.17), increasing n until the eigenvalues converge. At that point one has essentially included all of the folded diagrams and
Kff = &
(4.2)
is obtained. As written, eq. (2.17) contains the one-body graphs in the Q-boxes so that the eigenvalues obtained are the energy levels relative to the I60 ground state. However, one may obtain the usual sd-shell matrix elements (i.e. two-bodyconnected only) by removing the one-body graphs from V,,,. This is done by doing two parallel calculations R,e = (I--&
- ‘.x- Q, m=2
R;r = (l-9,
-
c $m
RF)-‘0,
(4.3a)
k=n-m+l
(4.3b)
m=2
and then, after convergence is reached, taking the eigenvalues of (Ho +Rf -Rz). In this case, of course, the one-body part of V,,, is taken from experiment. We thus have three ways to take into account the folded diagrams: (i) the selfconsistent graphical method (SC), with an energy-dependent effective interaction, and (ii) the method of grouping according to the number of folds (F,), and (iii) the Lee-Suzuki iteration method (LS), where V,,, from either method (ii) or (iii) is
J. Shurpin et al. / Folded diagrams
335
energy-independent. Before going on to compare the results of the three methods in the following sections, we make several observations. For the energy-dependent effective interaction (SC method) we need to know (only) the &-box, but at all energies. -For the energy-independent interaction (F, or LS method) we need to know the &box and all its derivatives, but at a single energy only. As mentioned earlier, the F, and LS method directly lead to a set of energy-independent matrix elements of V,,, which can be used for any sd-shell nuclei, but not for the SC method.
4.3. CONVERGENCE
Before stating the results of the calculations we have outlined above, we first investigate the reliability of the methods used to obtain them. The most important questions at this point are whether the diagramatic series for V,,, converges and what are some of the factors which influence convergence. In the previous section, we described three ways in which we can take into account the contribution from the folded diagrams. In the SC method, there is no problem of convergence (unless perhaps at points where Q(w) is singluar. [See ref. ‘).I We now examine the structure of the folded-diagram series in the F, method as it is embodied in eqs. (2.10) through (2.13). The energy (0) dependence of the &-box is of central importance. The @box is defined as a sum of diagrams, so that its energy dependence is the sum of the energy dependences of the individual diagrams. The energy dependence of the first-order diagrams (Dl and D4) is practically linear. The second-order diagrams have a more pronounced energy dependence. When phonons are included, the energy dependence is even stronger because most of the energy dependence comes from the energy denominators associated with low-lying intermediate states whereas the G-matrix itself has a relatively weak energy dependence. Thus, the higher order the process, the stronger the energy dependence. The result of this is that as one goes from Cl to C4 (i.e. includes higher-order processes as discussed in subsect. 4.1) the size of &(“), the nth derivative of 0, increases. The higher derivatives show an especially large increase. The effect of this on the folded-diagram series terms F, is easily predicted by eqs. (2.10~(2.13). As seen in table 3, as one goes from Cl to C4 the size of F, increases. Also, the relative increase is larger for higher n. The above discussion of how well V,,, converges when the terms are grouped according to the number of foldings is clearly reflected in the resulting eigenvalues (spectra). In fig. 11 we show the spectra (relative to l’0) of V,,, where
K,, = i ?I=0
F,,
(4.4)
336
J. Shurpin et al. / Folded diagrams TABLE3
Representative V,, matrix elements in the F, method using the Paris potential with starting energy w0 = - 10 MeV T
Jabcd
FO
104444 104455 104466 105544 105555 105566 106644 106655 106666 1 0 104455 104466 105544 105555 105566 106644 106655 106666
4
4
4
4
- 2.9467 -4.2159 - 1.1982 -4.2159 - 0.9605 -0.9140 - 1.1982 -0.9140 -2.3319
0.9698 0.8920 0.3679 1.3098 - 0.0552 0.2817 0.4570 0.2052 0.8047
-0.3173 0.0841 -0.0913 0.0232 0.0206 - 0.0054 -0.1436 0.0055 -0.0935
0.0611 0.0032 0.0109 - 0.0408 0.0535 - 0.0084 0.0285 0.0050 - 0.0436
-0.0060 - 0.0362 0.0009 - 0.0490 0.0124 - 0.0077 -0.0031 - 0.0098 0.0184
-2.2391 - 3.2728 - 0.9098 - 2.9726 - 0.9293 - 0.6539 - 0.8594 - 0.7080 - 1.6485
- 3.0753 -4.3978 - 1.3937 -4.3978 - 1.0489 - 0.9222 - 1.3937 - 0.9222 - 2.9024
1.1547 0.8670 0.4764 1.4623 -0.1873 0.3616 0.8599 0.5364 1.3358
- 0.3888 0.4197 0.1102 0.4520 -0.1352 0.0153 - 0.491 0.1605 -0.0541
- 0.0648 0.3979 -0.0219 0.3467 - 0.0785 0.0222 0.0921 0.1410 -0.2569
0.0287 - 0.2491 0.0170 0.1094 -0.0324 0.0174 0.0125 -0.0518 0.0378
- 2.3455 - 2.9623 - 1.0324 - 2.0275 - 1.4823 - 0.5057 -0.9533 -0.1361 - 1.8399
The upper half is for calculation Cl and the lower half for C4. The orbits (4, 5, 6) represent respectively (Od,,,, Od,,,, lsliz). All entries are in MeV.
for N = 0 to 4 using the Paris interaction. For the Cl (i.e. second-order diagrams only) calculation the convergence is quite satisfactory and is not shown. For the C4 calculation, however, convergence has markedly deteriorated. To study the convergence of V,,, when the folded diagrams are grouped according to the LS scheme we look at fig. 12 where we give the spectra of V,,, = R, (actually, of Rf - Ri) for n = 1 to 6. Fig. 12 shows the “worst case” calculation C4 for T = 1 and T = 0. It is clear from the figure that even for this case the LS series converges very rapidly for all the states. It appears, on the basis of its relatively rapid and smooth convergence, that the LS method is preferred over the F, method. The argument in favor of LS over F, is strengthened by the following. Consider table 4 in which we compare the eigenvalues of all three methods, SC, LS, and F, for all four &-box approximations. (All are for the Paris potential.) Since we include the SC method in the comparison, the eigenvalues in table 4 are relative to r60. The agreement between the SC and LS eigenvalues is striking. (Note that our numbers are really only valid in the interval -20 to -6 MeV because we have calculated the &box only in this energy interval.) Actually, the F, numbers are also in good agreement generally, except for the TJ = 01 ground state for which the F, method gives a rather poor result. Thus we &n say that the LS iteration formula reaches the “exact result” (i.e.
.I. Shurpin et al. / Folded diagrams
331
‘80
4_
/------
32-
;:
__I
-1;
/-2-z-
o-4 -5 -‘;
r I 0
I I
I 2
1 3 n
1 4
LS
Fig. 11. The A = 18 spectra as a function of the number of terms n in the F, method. The LS results (after six iterations) are included for comparison. These results are for the Paris potential in approximationC4 which converged the most poorly of those considered.
the self-consistent solution) after only 6 iterations. [In the SC calculation, (i) selfconsistency was considered to have been reached with agreement to within 0.00005 MeV, and (ii) we know the &-box at 8 points only so that the higher derivatives are less reliable. In view of these two limitations, the agreement attained is indeed remarkable.] We therefore use the LS spectra as the standard against which we measure the convergence of the F, series in fig. 11. If we know the @box and its derivatives, we can find V,,, with only a small number of iterations when using the LS method for calculating the folded diagrams. The other half of the problem, what to include in the basic &-box, still remains. 4.4. THE EFFECTIVE
INTERACTION
We now present results of the various calculations we have been describing. We first examine the importance of the spurious particle-hole TJ” = Ol- state which enters into diagrams D2(P2) and D3(P3). As pointed out earlier we can isolate its effect when we include diagrams P2 and P3 in place of D2 and D3. The spurious state’s contribution to the diagrams is very large, often more than 10%. The reason, of course, is that because it has a particle-hole eigenvalue o_+,,,close to
338
J. Shurpin et al. 1 Folded diagrams
4 I
-4
-2
-2 2 /
-3
of------6
-
-7
-
.
-0
-4, -5;
-0
I
I
1
I
I
I
I
2
3
4
5
6
”
-
I
Exp
I
I 123456
I
1
I
I
n
Fig. 12. The A = 18 spectra after n iterations in the LS method for calculation C4 using the Paris potential. The experimental spectra are included for comparison. The ground-state energies, relative to “0, are from ref. 33) and the excitation energies are from ref. 34).
zero the energy denominator (wO-mph) is relatively small. However, P2 and P3 enter with. opposite sign so that the spurious state contribution to the effective interaction tends to cancel out. When the one-body diagrams are included (i.e. if we do not use the experimental l’0 levels so that the spectra are relative to r60) the effect can be as large as 0.1 MeV or more. When the one-body part of V’,, is taken from experiment (i.e. diagrams P2 and P3 contribute only in the folded terms which are two-body connected) the effect is quite small, 20 keV or less. Thus we can proceed to make valid comparisons between the different @box approximations (calculations Cl, C2, C3 and C4) even though the spurious state is not treated consistently (Cl and C2 include it while C3 and C4 remove it in lowest order). Unperturbed single-particle energies E were used in both the hole-hole and particle-hole phonon calculations. Our results are not to be compared with some of those to be found in the literature [see e.g. ref. “‘)I which use the experimental single-particle energies for the E. The Paris and Reid results are very similar with no surprises. The important thing is that both the hole-hole and particle-hole energies are significantly renormalized from their unperturbed values when we allow the two hole lines or particle and hole lines to interact. The resulting smaller denominators (i) make the &box “larger” (i.e. IP21 > ID21 and IP31 > ID3), and (ii)
J. Shurpin et al. / Folded diagrams
339
TABLE4 Low-lying A = 18 levels TJ
Cl
c2
c3
c4
10
- 10.59920 - 10.59920 - 10.59920
- 10.71672 - 10.71670 - 10.727
- 10.35508 - 10.35502 - 10.371
- 10.46468 - 10.46462 - 10.490
2
- 8.88340 - 8.88341 - 8.882
- 8.89500 - 8.89499 - 8.894
- 8.75340 - 8.75343 - 8.756
- 8.76300 - 8.76306 - 8.766
4
- 7.44860 - 7.44858 - 7.450
- 7.44860 - 7.44858 - 7.450
- 7.08640 - 7.08653 - 7.083
- 7.08640 - 7.08653 - 7.083
01
- 12.10408 - 12.10405 - 12.197
- 12.73150 - 12.73148 - 13.223
- 11.86850 - 11.86847 - 12.287
- 12.46800 - 12.46801 - 13.786
3
- 10.49140 - 10.49140 - 10.496
- 10.52120 - 10.52123 - 10.526
- 10.45800 - 10.45797 - 10.478
- 10.48412 - 10.48408 - 10.506
5
- 10.11280 - 10.11280 - 10.113
- 10.11280 - 10.11280 - 10.113
-9.63784 -9.63786 -9.637
- 9.63784 -9.63786 -9.637
2
-9.16022 -9.16024 -9.158
- 9.25420 -9.25423 - 9.252
- 9.03048 -9.03052 - 9.029
-9.10274 -9.10277 -9.101
All entries are in MeV and are relative to the I60 ground state. In each case the three rows correspond to the SC, LS and F,, respectively. The Paris potential were used with a starting energy of - 10 MeV.
make the &-box more energy dependent, so that the folded diagrams become much more important. As we shall see, these two effects tend to cancel each other. The sd-shell matrix elements of V,,, for I = 1, .I = 0 are given in table 5 for calculations Cl through C4 for both the Paris and Reid potentials. As more higher-order processes are introduced into the &-box, V,,, becomes more nonhermitian. This is entirely due to the contribution of the folded terms. Although the effective interaction obtained by this manner is not hermitian, the eigenvalues are all real. The non-hermitian matrix V,,, may be transformed to an hermitian form by use of a similarity transformation, although the columns of the transformation matrix will not be mutually orthogonal. Alternatively, the two-body interaction may be forced to be hermitian by the simple expedient of using for the matrix element V&the average of 6, and Vki[ref. ‘“)I. This method was used for the calculations described in the next section. The error introduced by this
340
J. Shurpin er al. / Folded diagrams TABLE 5
The sd-shell
effective-interaction
matrix
elements
for 7’ = 1, J = 0
TJ
abed
Cl
c2
c3
c4
10
4444
- 2.2360 - 2.0899
- 2.2925 -2.1472
-2.1822 - 1.9961
- 2.2464 - 2.0583
10
4455
- 3.2670 -3.1363
- 3.3960 - 3.2610
-3.1239 - 3.2353
- 3.2296 - 3.3460
10
4466
- 0.9096 -0.8976
- 0.9304 -0.9193
- 1.0031 - 0.9940
- 1.0227 - 1.0144
10
5544
- 2.9536 -2.8348
- 3.0050 - 2.8866
- 2.5074 - 2.5420
- 2.5609 - 2.5960
10
5555
-0.9409 -0.8450
- 1.0253 -0.9273
- 1.1018 -0.9555
- 1.1576 - 1.0171
10
5566
- 0.6505 - 0.6428
- 0.6669 - 0.6598
-0.5517 -0.5591
-0.5663 - 0.5746
10
6644
-0.8596 -0.8327
-0.8801 -0.8533
- 0.8594 -0.8523
- 0.8809 -0.8733
10
6655
-0.7021 -0.6893
- 0.7357 -0.7216
-0.3381 - 0.4075
- 0.3584 - 0.4305
10
6666
- 1.6478 - 1.4433
- 1.6575 - 1.4532
- 1.7779 - 1.5039
- 1.7863 - 1.5126
In each case the upper row is for the Paris potential and the lower for the Reid potential. The full non-hermitian V,,, is given. The orbits (4, 5, 6) represent respectively (Od,,,, Od,,*, Is,,,). All entries are in MeV. In all cases, the Lee-Suzuki (LS) iteration method was used.
approximation is easily checked for the A = 18 case. The largest array occurs for JT = 10; in this instance the difference between the lowest eigenvalue of the hermitian and non-hermitian array is 0.4 keV when using the Paris potential. The greatest error for this JT is only 2.4 keV.
5, Results for several-particle
systems
In this section we shall discuss results obtained using the newly derived matrix elements as shown in table 5 for several-nucleon systems. We feel this to be important for several reasons. First of all the matrix elements were derived to predict nuclear spectra, and their success or failure to do so can only be found by actual comparison with experiment. In cases where agreement is poor (and there shall be many such cases) one may perhaps obtain clues to where improvements may be
341
J. Shurpin et al. / Folded diagrams
made. A comparison between results obtained using the different approximations will be possible. Another point is that the matrix elements of table 5 are only the two-body parts of the model-space effective interaction. Their results for the spectra of several-nucleon systems may serve to indicate the potential importance of the many-body effective interactions. Finally, from a knowledge of the nature of the nuclear wave functions, one may establish the relative importance of the differing spin-orbit and tensor components of the two interactions. Before discussing the results obtained with the Paris and Reid interactions, we briefly recall the four approximations used in obtaining the effective interaction: Cl: @box with 2nd order diagrams; c2:
2nd order plus the hole-hole phonon ;
c3: 2nd order plus the particle-hole phonon ; c4: 2nd order plus hole-hole and particle-hole phonon. For convenience we shall refer to the interaction interaction with approximation Cl as Pl, etc.
obtained
using the Paris
5.1. CENTROIDS
A global assessment of an interaction may be obtained centroids of an interaction defined in jj-coupling by
E
ob
by calulating energy
rCTCJICTI(J’,J’,JTlVlj,j,JT)
1.
-
=
(5.1)
and in LS-coupling by
+)
= c CJlETl((~~)~STJI~I(~CL)~STJ) g(A/l)(2S+ 1)(2T + 1)
*
(5.2)
In eq. (5,2) (Q) labels the possible SU(3) representations and g(Lp) is the dimension of (APL).In the sd-shell the three possible SU(3) representations for two particles are (40) and (02) which are spatially symmetric and (21) which is spatially antisymmetric. The quantity [k] E 2k + 1. If the two-body interaction V were written as I/ = i (R(k)k=O
S”‘(U,U,)),
(5.3)
342
J. Shurpin et al. / Folded diagrams
then a tensor decomposition (LaII
of two-body matrix elements is obtained from
R’k’lJlh’)(ST~II Sk’l(S’T’P’) = (-)“‘“j-(-)”
;,; ;1
(LSTaPJII/I~S’Ta’P’J).
J
i
In eq. (5.4) c( and /I are any labels necessary to completely specify the matrix element; we shall take them to be irreducible representations of SU(3) and SU(4), respectively. We shall refer to the k = 0, 1 and 2 interactions as central, spin-orbit and tensor interactions, although their actual origin may be from other components. By applying eq. (5.4) and then transforming the two-body matrix elements to jjcoupling, one may determine the amount of effective central, spin-orbit and tensor interactions contributing to each two-body matrix element. It is trivial to prove that non-central components of the interaction will not contribute to an SU(3) centroid; hence, an SU(3) energy centroid is a measure of the effective central interaction. An analogous statement is not true for a jj-coupling centroid. In fig. 13 are shown contributions to the energy centroids s1 = .stt and .ss = E++($ for d; and $ for s,) for several interactions. Only a central or rank-zero interaction can contribute to sr. The twelve interactions shown in fig. 13 include the eight interactions obtained in this work and four interactions which have been widely used in shell-model calculations. The interaction “KB” is the Kuo-Brown interaction 24) and “K” is the Kuo interaction 3s), both derived from the Hamada-Johnston potential “). The interactions “K12” and “R” are interactions obtained in ref. 36), namely “Kuo plus 12 free parameters” or K12fp and “RIP”. The former interaction is essentially the Kuo interaction, but with those diagonal matrix elements involving only the d+ or s+ particles being varied to produce a better tit to spectra. The RIP interaction is a purely central interaction obtained by varying the values of radial integrals. From fig. 13 one finds the spin-orbit (6”) and tensor (.sT)contributions to ss are small and of opposite sign with the obvious exception of RIP. Further, the magnitude is essentially the same for the eight interactions obtained in the present work. The contribution to the .sg of the central (8) effective interaction does vary slightly from interaction to interaction and is appreciably different for the eight new interactions from the Kuo and K12fp interactions. Since K12fp was tit to inter alia binding energies, this suggests the new interactions may underbind nuclei. The st centroid, sr, does depend upon the original two-body interaction with the Paris potential producing more attraction as expected. However, again the three Kuo interactions are more attractive. It is of interest to compare these results with those obtained for the Q-box or unfolded interaction. The sb centroid is -3.71 MeV and -3.16 MeV for the Paris
343
J. Shurpin et al. / Folded diagrams I
. 0
xx
. -I
.
-2
‘I
2 I
-3
g L
I RI
I R2
I
I
I
I
I
I
R3
R4
PI
P2
P3
P4
I KB
I K
I Kl2
T--
I
I
R
Fig. 13. The centroids of the Od,,, matrix elements, ss, and Is,,, matrix elements, E; for 12 interactions. The quantities ES, ET and s: are centroids of the rank-O, rank-l and rank-2 parts of the interaction, respectively. The four interactions Rl, R2, R3 and R4 are derived from the Reid potential with the four approximations Cl, C2, C3 and C4, respectively. The interactions derived from the Paris interaction are similarly labelled, Pl, P2, P3 and P4. The remaining interactions are KB: Kuo-Brown 24), K: Kuo 35), K12 : “Kuo + 12 free parameters” and R: RIP 36).
and Reid interaction, respectively. (These numbers are for the &-box with phonons, i.e. diagrams D4, D5, P6 and P7 and assuming a starting energy of q, = - 10 MeV.) Hence, the inclusion of folded diagrams results in less attraction of 1.5 MeV! The contribution to Ed of the central, spin-orbit and tensor components of the Paris interaction is - 1.41, -0.12, and 0.25 MeV and for the Reid interaction - 1.42, -0.17 and 0.26 MeV. The difference between the two interactions is the effective two-body spin-orbit interaction. The main effect of including folded diagrams is on the central components: there is less attraction of approximately 0.5 MeV. The noncentral contributions are almost unaffected. In the beginning of the sd-shell, the SU(3) model provides a much more viable coupling scheme than does jj-coupling as well as providing some physical insight into the nuclear structure calculations. In fig. 14 are plotted energy centroids of the three SU(3) representations of two sd particles. As remarked above, the (40) and (02) are spatially symmetric, or belong to the SU(6) representation [2], and (21) belongs to [ll]. The (21) centroid is positive (repulsive) as expected from phenomenological potentials. Although no great differences were discernable in jj-coupling amongst the four methods of calculating an interaction, substantial differences do appear in SU(3).
344
J. Shurpin et al. / Folded diagrams
G
-3
0 0
-4
i
RI
R2
R3
R4
PI
P2
P3
P4
KB
K
Kl2
R
Fig. 14. Energy centroids of the three W(3) representations occuring in the two-particle system in the sd shell. The labelling of the 12 interactions is explained in the caption of fig. 13.
Calculations C3 and C4 are for both two-body potentials less attractive than for Cl and C2. From the centroids one may establish that for the Paris interaction the inclusion of the hole-hole phonon produces an added attraction of 0.038,0.049 and 0.042 MeV for (40), (02) and [2], respectively, and the particle-hole phonon produces less attraction of 0.355,0.148 and 0.296 MeV. The hole-hole phonon does not affect [ll] and the particle-hole phonon moves the [ll] centroid up by 0.053 MeV. Similar, albeit smaller, numbers result from the Reid potential. The effect of the 3plh diagram in the original work of Kuo and Brown 24) was to lower the 0’ state and raise the 4+ state of ‘*O, thereby producing better agreement between theory and experiment. An examination of fig. 14 as well as the calculated spectra of “0 in fig. 16 gives rise to an apparent paradox in that by adding the particle-hole phonon to the Q-box produces less attraction for both the spatially symmetric centroids and the ground state of **O. The behavior of the (40) energy centroid may be easily explained since the J = 4 matrix element enters with a weight of nine greater than for J = 0 and the J = 4 state is raised by the particle-hole renormalization. However, the puzzle of less binding for the O+ of I80 remains. In table 6 are listed diagonaL matrix elements of the (40) and (02) representations ; these two
J. Shurpin
et al. /
345
Folded diagrams
TABLE 6
(@I L
w
0 2
(02) “0 2
ww C(O2)
D7
P7
Pl-P3
- 0.983 -0.210 0.273 - 0.432 0.168
- 1.335 -0.267 0.339 -0.525 0.155
0.058 0.420 0.313 - 0.079 0.075
0.038 0.031
-0.017 0.028
0.319 0.099
The contributions to the diagonal two-body T = 1 matrix elements and SU(3) centroids arising from diagrams D7 and P7 of fig. 9b using the Paris potential and o0 = - 10 MeV (columns headed by D7 and P7) and the difference of the matrix elements obtained from interaction Pl and P3 (last column).
representations account for approximation 90% of the wave function of the lowlying states. The results are for the Paris potential. As in the earlier work the contributions from the simple 3plh graphs are to lower the O+ by almost one MeV and to raise the 4+ by 0.3 MeV. The results for the particle-hole phonon are even larger. The net effect on the centroids are virtually nil. The diagonal (40) L = 0 matrix elements for the full &box interaction (D4 +D5 +D6 +D7) is - 5.67 MeV and -6.02 MeV when D7 is replaced by P7. Renormalizing the interaction by including the folded diagrams produces an appreciably less attractive interaction : the matrix element becomes - 4.14 and -4.08 MeV for Pl and P4, respectively. Further, the energy separation between the L = 0 and L = 4 states of the (40) representation is 2.36 MeV (Pl) and 2.61 MeV (P4) rather than 3.20 MeV or 3.61 MeV for the &-box interaction containing D7 or P7, respectively. Thus, an effect of including folded diagrams is to obtain less binding energy and a more compact spectrum; much of the effect of the phonon on the @box interaction disappears. A similar, though less dramatic, effect occurs for hole-hole phonon. The above discussion may be summarized as follows. As one goes from Cl to C4 the strength of the Q-box increases. When one also includes the folded terms in V,,, it is found that the matrix elements are typically weaker for C3 than Cl, for example. For C4 the spectra obtained in lowest order (i.e. from V,,, = 6) are noticeably lower than the Cl spectra but when the folded terms are included in V,,, the C4 spectra are quite similar to those of Cl, while the C3 spectra actually are less bound than the Cl. Thus, when we include higher-order processes in the &box the folded diagrams become more important. The core renormalization due to 4p2h states was characterized in ref. 38) as similar to a pairing interaction. The hole-hole phonon also exhibits some characteristics of pairing interaction in that [11] states are essentially unaffected
J. Shurpin et al. / Folded diagrams
346
‘8F -__3-
-
-
-
-9
-’
-
-2
-
I
-3
-3
-
-I
-
-
-I
-
-
-
-2
-3
-2
_, 4
-
___3
-10 -
-22 z
___2 -II -
-_5-3
-
-_5
_
-3
-
-
-3 -3
-5
-I
-3
-
-
-I -I
-13 RI
Fig.
15. Energy
spectra
-__I
-
-12 I
-
R2
R3
R4
of 18F as calculated
with
PI
P2
the eight
-I
P3
P4
interactions
EXP
derived
in this work. The
labelling of the eight interactions is explained in the caption of fig. 13.
and low angular momentum states of the two particle system are affected the most (see figs. 15 and 16). However, a pairing interaction would affect (40) more than (02) in the ratio of 5 : 1, unlike the hole-hole phonon. The inclusion of the particle-hole phonon makes the relative separation of (40) and (02) centroids smaller - 0.207 MeV less for the Paris potential - which has unfortunate consequences in several-nucleon nuclei. Bands built from W(3) representations other than the ground state or leading SU(3) representation will now lie at a lower excitation energy. The energy of an SU(3) centroid in a nparticle system is 3g) E(n(&)) = (2n - n2)E(20) +&-15n+7nZ +&(21n+n2
- 3( Cc49 + 2( Cc3’))E(40) -3
+$(n2-5n+(C’4’))E(21),
(5.5)
J. Shurpin et al. 1 Folded diagrams
I
__o_
-8
-
__4-
-9
2 z
-II
-
-
-
-
-_2-
=
__4-
-
=s -4
-2
=-P i
I -_2-
-10
-
-0
-2
341
-
-
-
-2
_,j-“- - RI
R2
R3
R4
-
__o_
PI
-o_j
-
P2
P3
P4
EXP
Fig. 16. Energy spectra of I80 as calculated with the eight interactions derived in this work. The labelling of the eight interactions is explained in the caption of fig. 13.
with the expectation values of the Casimir operators (03’)
= A2+$+&J+3@+~),
@“‘>
=
Cck)of SU(k) given by 40)
.
~fi2+3(fl-f,)+f2-f3~
for representations (APL)and [j], respectively. The SU(4) representation [f must be used in evaluating eq. (5.5) which is congruent to the W(6) representation [f]. From eq. (5.5) the energy difference between (80), the ground-state band of 2oNe, and (42), the first excited band, is E(80) - E(42) = @(40) - E(02)]. Hence, including the particle-hole phonon moves the two centroids 0.48 MeV closer together for the Paris potential. Though this may seem small, the two centroids are already too near each other as compared to experiment. Further, the changes will be larger in nuclei with more particles such as 28Si.
348
J. Shurpin et al. 1 Folded diagrams
A comparison with the centroids of the K12fp interaction is useful to predict possible consequences in heavier nuclei and, since the twelve parameters were varied to produce a better lit with experiment, will indicate some problems which will arise. The K12fp interaction has a separation of the (40) and (02) energy centroids 0.22 MeV larger than the C2 approximation to the Paris interaction (P2). There is also a larger separation between [2] and [ll]. Hence, the interactions obtained in this work will have too many states of low energy and both SU(3) and SU(4) will be more badly violated than with K12fp. Further, the energy centroid of K12fp for [2] is lower representing more binding. Hence, we may anticipate the current interactions will badly underbind nuclei throughout the sd-shell. The SU(3) centroids obtained for the o-box interaction suggests that results obtained using the &-box or unfolded interaction would be very similar to that obtained using the earlier Kuo or Kuo-Brown i@eractions. The centroids for the (40), (02) and (21) representations are -4.03, - 1.71 and 0.46 MeV for the Reid interaction and -4.17, - 1.73 and 0.54 MeV for the Paris interaction. (These results are for a o-box which includes phonons, i.e. diagrams D4, D5, P6 and P7 with o0 = - 10 MeV.) These centroids are not appreciably different from those of the Kuo interaction. The inclusion of the folded diagrams results in less attraction for (40) of 1.35 MeV and 0.55 MeV for (02) whereas the (21) centroid is essentially unaffected. One may tentatively conclude that the folded diagrams do not affect the spatially antisymmetric states and the more correlated a state, the greater the effect of folded diagrams. 5.2. SPECTRA
We now turn to a consideration of the predictions of the matrix elements in nuclei. Although all nuclei with A equal to 18 to 22 and 24Mg were calculated, we shall for brevity present results only for the more illuminating cases. In all cases the single-particle energies were taken from experiment. The calculations were performed using a modified version of the Glasgow shell-model code4i). In fig. 15 are shown results for ‘*F and experiment. Although many of the features anticipated from an examination of the centroids appear, the spectra are obviously more detailed. The inclusion of the phonons affect the low-spin states (l+ and 2+) more than the 3+ and 5+; in particular, the second 3+ level is essentially unaffected. Although the P2 interaction reproduces the ground-state binding energy, the other interactions do rather worse. All interactions fail to produce the proper energy for 3: and 5:, although P3 reproduces the excitation spectrum reasonably well. Since these energies enter the evaluation of the centroid and position of levels in heavier nuclei - with more weight than the l+, the expectations based on the centroids appear valid. In fig. 16 are shown results for “0. Again, the limits provided by the centroids
J. Shurpin et al. / Folded diagrams
349
are validated by the detailed calculation of structure. The Paris and Reid interactions do not provide sufficient binding although the Paris interaction is somewhat stronger than the Reid. Again, the higher-spin states appear to be relatively unaffected by the inclusion of phonons. Although the 4: level is correctly predicted, the 0: and 2: are too high, resulting in a compressed spectrum. This will also occur in heavier nuclei. In fig. 17 are results for 19F. The wave functions of the ground-state band are dominated by the SU(3) representation (60) and have S equal to one-half. Hence, as in ‘a0 9 the energies depend primarily on the central interactions with small contributions from the spin-orbit interaction. The error in the predicted binding energy ranges from approximately 1 MeV for the P2 interaction to 2.7 MeV for the R3 interaction. Inclusion of the hole-hole phonon not only shifts the centroids down, but also expands the spectrum; the position of the y’ is essentially unaffected by inclusion of phonons. However, the particle-hole phonon tends to contract the spectrum. The final result with both hole-hole and particle-hole phonons included produces a spectrum which has the correct level ordering but with insufficient binding.
-2 -17
-
-7
-IS
-7 -,3
_I3
__7
--7
-_5,,
-18
-3
-13
--5,7=:
=:
=:’
=;,,-7
=y
-7
-I3
-I3
-13
-_:
--:
-5 -7
=:” 7
-
-19 -
--9 -9
-_9
13
-9
-2o-
-9 -9
-3
B H
-9
-9
-3
=5 -_7 ---3
-3
-3 -3
-21
-
=T -5
-22
_ -1 -
-3 =:
-9--
_-3 =:
-5 -1
--3 -5 -I
=: -3
-5 -1
-23 -5-, -24
RI
R2
R3
R4
PI
P2
P3
w
EXP
Fig. 17. Energy spectra of “F as calculated with the eight interactions derived in this work. The labelling of the eight interactions is explained in the caption of fig. 13. The dashed line represents a 3’ state dominantly 7p4h.
350
J. Shurpin et al. / Folded diagrams
The calculation of levels in 2o0 is of interest because only the T = 1 interaction may contribute. The results are not shown. The second O+ and 2+ states both lie about one MeV lower than the experimental position. Again the spectrum is too compressed and the amount of binding is about one MeV too little. The calculated spectra of 20Ne may be found elsewhere42 ). Briefly, deficiencies found in other nuclei also appear in 20Ne: there is too little binding energy, 3.2 MeV for Pl and 4.8 MeV for Rl ; the spectrum is too compressed with the 8+ appearing 0.5 MeV for Pl and 1.4 MeV for the Rl interaction too low relative to the ground state; the (42) excited band lies at too low an energy relative to the ground state. The tensor interaction of the Paris potential is appreciably weaker than that of the Reid interaction, but the contribution from the tensor components to the Qbox two-body matrix elements are only 2% larger for the Reid potential than for the Paris potential. However, the process of including the folded diagrams may alter this situation: the many angular momentum recouplings and matrix products may cause the effective non-central components to change or conceivably even disappear. Since it is in general difficult to choose an appropriate weighting of the many two-body matrix elements (recall the non-central components do not contribute to SU(3) energy centroids), the expectation value was evaluated between several wave functions generated for mass 20 nuclei using the Cl interactions. The non-central components were obtained using eq. (5.4). The results are given in table 7. The lowest T = 0 wave functions are dominated by S = 0 components for which the non-central forces cannot contribute. However, the lowest T = 1 wave functions have S = 1 and both the two-body spin-orbit and tensor force may contribute. An examination of table 7 demonstrates several points. First, the effective tensor - and also spin-orbit - force of the interaction obtained from the Reid potential is not very different from that of the Paris potential. This is somewhat surprising. Second, the contribution from the two-body effective spinorbit interaction is appreciably larger than that from the effective tensor interaction. Third, in most instances the effective tensor interaction is repulsive. Fourth, essentially all the differences between results obtained using the two interactions arise from the effective central interaction. Finally, the contribution from the non-central components to the energy of the T = 0, S = 0 states is approximately the same as that to the T = 1 states. As remarked in subsect. 5.1 the inclusion of the particle-hole phonon and then folding results in less binding, even for the low-spin states, contrary to expectation based on previous interactions which included only the &-box with no folded diagrams. In table 7 are also listed the contributions to the energy of mass-20 levels of three diagrams D4, P6 and P7. The quantities listed are the contribution to the interaction from only the ,@box and does not include any effects such diagrams have on the interaction after including folded diagrams.
TABLE7 Contributions
to the energy of 7’ = 0 and T = 1 states in mass 20 from specified parts of the interaction
C
v
T
D4
P6
P7
- 24.68 - 23.01 - 18.21 - 16.82 - 22.97 -21.52 - 20.60 - 19.27 - 15.91 - 14.99
-0.24 -0.19 -0.16 -0.12 -0.33 -0.27 -0.41 -0.37 -0.65 -0.61
- 0.07 - 0.05 0.15 0.20 0.05 0.07 - 0.03 - 0.02 0.16 0.18
- 26.76 - 25.55 -20.18 - 19.00 - 25.45 - 24.39 - 23.53 - 22.59 - 19.76 - 18.99
-2.71 - 2.88 - 2.01 - 1.91 - 1.90 - 2.23 - 1.58 - 1.94 -0.13 -0.62
- 1.37 - 2.29 -2.13 - 1.41 -0.80 - 2.03 -0.39 - 1.27 0.59 -0.48
-
-0.88 -0.86 -0.89 -0.82 -0.83 -0.80 -0.77 -0.72
-
- 1.36 0.40 - 1.07 0.08 -0.91 0.45 - 1.20 0.06
0.23 -0.69 - 0.07 -0.91 0.21 -0.80 - 0.02 -0.55
J
T=O 0, 0, 2 4 6
T=l 1 2 3 4
13.52 12.60 13.97 13.04 13.33 12.39 12.84 11.97
0.06 0.07 0.09 0.11 0.26 0.29 0.36 0.38
16.10 15.48 16.64 15.93 16.10 15.34 15.31 14.60
The columns labelled “c’, “V” and “T’ contain the expectation values of the effective centroid, spinorbit and tensor interactions; the columns headed ‘TM”, “P6’ and “P7” contain the expectation values of diagram D4, P6 and P4 of fig. 9. The wave functions used were generated using the Cl interaction with “0 single-particle energies. For each J the first line contains results for the Paris interactions and the second for Reid interactions.
From the table it is seen the contribution from the particle-hole phonon diagram (P7) is much larger for the Reid interaction than for the Paris interaction. This should be related to the relative strength of the tensor force of these two interactions. For both interactions the effect is to lower the energy of the low-spin states and increase the energy of the high-spin state. This reinforces our earlier conclusion that the apparent repulsion resulting from the inclusion of the particlehole phonon was a manifestation of the folded diagrams. One may also investigate the extent the hole-hole phonon contribution simulates a pairing interaction. One expects the pairing interaction to be attractive and, in *‘Ne it can contribute to states of J I 4. From table 7 one observes the contribution from P6 obeys these expectations. Finally, we note that if one were to take as the interaction just the contributions from D4 and P7 using a starting energy of o. = - 10 MeV [similar to the original work of Kuo and Brown24)], agreement with experiment would be quite good. Using the Pl interaction, the calculated binding energy would be in error by only
352
J.ShUrpin etal./ Folded&QgrQmS
70 keV and the excitation energy of the 8+ is only 0.5 MeV too small. The Rl interaction underbinds “Ne by 170 keV but produces an 8+ within 100 keV of experiment. In fig. 18 are results for 22Na . Since 22Na is odd-odd, the low-lying states have S = 1 and all parts of the interaction may contribute. The level structure of 22Na has routinely defied successful calculation with both phenomenological interactions 43) and realistic interactions 36). The first l+ (and sometimes also the second 1’) states tend to lie lower than the experimental 3+ ground state. The present calculations have the same failings. Interactions which are lit to the structure of mass-22 nuclei can produce the correct ordering by either using a repulsive triplet-even tensor interaction 44) or by varying a number of two-body matrix elements3’). Because of the number of matrix elements in the latter approach, it is difficult to establish the physical origin of the discrepancy. From fig. 18 it is apparent the spectra is too compressed. Experimentally there are seven levels below 3 MeV excitation; the Rl interaction produces 12 levels and P4 produces 11 levels below this energy. The 2+ level appears 1.5 to 2 MeV too low.
-,
-------I
-32.21
-3t.s3
RI
R4
Fig. 18. Ei4ergy spectra of “Na
-/
- 34.01
PI
-’
- 52.73
P4
-?a.52
EXP
calculated with four of the interactions derived in this work.
353
22Ne -. 5-
-4
-I
-2,1 -2
-1 -2
------I
-3
-2 -3
-4
-___
4
1
-I
-4
-2 -----I -------2
-7
-3
-2
-2
-2
-4
-4
3
-----2
-2
4-
-2 -2
3-4
2 z
I
-4
-4
-
2-
------2
-2
-2
-2
-----0 -5149
-0 - 54.03
_
-2
-0 -52 21 RI
R4
PI
-0 -52 00 P4
-----0 -5775 EXP
Fig. 19. Energy spectra of “Ne calculated with four of the interactions derived in this paper.
In fig. 19 are shown results for “Ne. The low-lying states are dominated by (82) S = 0. Again the spectra is too compact with the 2: and 4: having too small an excitation energy and the 6: which is not shown, even worse, although the excitation spectrum for the 2: and 4: states is acceptable. The most obvious failure is the position of the 2: which is a member of the (82) K = 2 band. Its incorrect position is now almost traditional 36*43). Although its energy may be corrected by explicit inclusion of pf excitations45), interactions may be found which do predict its energy correctly 37*42) although the physical origin of the modifications is not known. Modifying the matrix elements which violate S which are substantial in the present interactions - do not substantially modify the results 46). Finally, we present results for ‘*Mg. The amount of underbinding is substantial, ranging from 5.5 MeV for the PI interaction to over 10 MeV for R4. The K = 2
354
J. Shurpin et al. / Folded diagrams
band again appears much too low. As predicted by the centroids, the excitation energy of the 0: is too low when the particle-hole phonon is included.
6. Discussion and conclusion The aim of this paper was to perform within the framework of the shell model a microscopic nuclear-structure calculation, the starting point of which was the free NN interaction. The first step in this procedure, the calculation of the G-matrix, is probably quite sound. The methods used, momentum space matrix inversion to obtain G, and the Tsai-Kuo method to treat that part of G which depends on the Pauli operator Q, are very accurate. The second step is the calculation of the &box, and it is in this step that we have to make some approximations. Four approximation schemes, denoted as Cl, C2, C3 and C4 have been used. We will return to discuss them later. In the final step, we obtain V’,, by “folding” the Qboxes together, i.e. evaluating the folded diagram Q-box series of V,,,. For a given Q-box, this step can be quite accurately carried out by using the Lee-Suzuki iteration method. This method was shown to yield essentially exact results after only a small number of iterations. We have found that the effect of higher-order &box diagrams is suppressed significantly by the inclusion of the corresponding folded diagrams. Generally speaking, the effect of the folded diagrams is quite significant in the calculation of V,,,. Another aim of these calculations was to see how the Paris potential would fare in finite nuclei. To our knowledge, this is the first microscopic calculation of finite open-shell nuclei for which the Paris potential was used. Our results indicate that the Paris potential is as good as, if not better than, the Reid potential. Indeed, in all nuclei we considered, the Paris potential seemed to give better, albeit inadequate, ground-state binding energy. However, in most cases differences in spectra produced with or without the inclusion of phonons were as great as those between the Paris and Reid potential. We demonstrated that in mass-20 nuclei the contributions to the energy from the effective non-central components of the Reid and Paris interaction were essentially the same. Differences in spectra and binding energies arose mainly from the effective central interaction, although there were appreciable differences in the contribution to the energies from diagrams P6 and P7. However, the Reid interaction tended to produce wave functions with a slightly larger occupation of the dt than did the Paris interaction. The third aim of these calculations was to do a systematic study of the contribution of the folded diagrams to V’,,. It is quite clear that the inclusion of the folded diagrams have a much greater effect on the A = 18-24 spectra than do various infinite-order partial summations (hole-hole and particle-hole phonons) that this calculation and others include. Until now, it has been practically taken for
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/ Folded diagrams
355
granted that the inclusion of TDA (or RPA, for that matter) phonons would give increased binding. Our calculations have shown that when the folded diagrams are included, this may not be the case. The C3 spectra (for which the TDA particlehole phonons were included) show appreciably less binding than do the Cl spectra (for which we included no phonons). By explicitly calculating the expectation value of the particle-hole diagram for mass-20 states, we demonstrated this apparent anomaly was a result of the folded diagrams. Why did the early calculations, starting with those of Kuo and Brown, reproduce the experimental spectra as well as they did, even though all diagrams above second order were ignored ? The answer suggested by the current work seems to be that although the &-box they used contain only some low-order diagrams, the effect of the higher-order Q-box diagrams would have been cancelled in large part if the folded diagrams were included in V,,, as well. This is because the higher-order diagrams have a relatively greater energy dependence so that they significantly renormalize the &box interaction. Thus if higher-order diagrams are included (in the basic Q-box), the folded diagrams must be included as well (in the expansion for V,,,). Indeed, using an effective interaction defined by the sum of contributions from diagram D4, the bare G-matrix, and P7, the particle-hole phonon, of fig. 9, with no folded diagrams included, much better agreement with experiment was obtained. When compared with experiments the excitation spectra of several nuclei are reproduced reasonably well but our calculated binding energies are generally too small by several MeV, especially for nuclei in the middle of the sd-shell. It is not clear how to remedy this, and will be an interesting problem for further study. Several directions may be pursued. In the present work we have included only the two-body components of I&,. I$,, clearly has many-body components. For example, I90 will have 3-body and 22Ne should have 6-body effective interactions. Calculation of such many-body components is rather complicated. In fig. 20 we give some representative diagrams which contribute to 3-body effective interactions. Diagram (i) belongs to the non-folded term (j of eq. (2.9); it originates from our model-space restriction that nucleons are confined to certain low-lying shell-model states. (Note that a railed line represents a particle outside the model space.) Diagram (ii) is a 3-body effective interaction diagram due to non-nucleonic degrees of freedom. Here nucleons 1 and 2 interact by exchanging a pion, exciting nucleon 2 into an isobar. Then it is deexcited back to a nucleon. Three-body forces can also arise from folding two 2-body irreducible boxes together, as shown by diagram (iii). Polls et ~1.~‘) have recently investigated the contribution of these types of diagrams to the three-body effective interaction for the sd-shell nuclei using the Reid “) and Jtilich-Bonn 48) NN potentials. They found that the explicit inclusion of A degrees of freedom (diagram (ii) of fig. 20) has rather a negligible contribution to the three-body effective interaction. But the three-body effective interaction due to the truncation of the shell-model space
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I
2 (1)
3
I
2 (Ii)
3
I
2
3
(iii)
Fig. 20. Some of the diagrams which contribute to the three-body effective interaction.
(diagrams (i) and (iii) is significant, although not overwhelmingly large. They give reduced binding energy of approximately 0.5 MeV to the A = 19 nuclei. [A similar result has been obtained by Huang et al. 49) who performed a folded-diagram calculation for the low-lying states of 190 using mainly the M3Y “) potential as the underlying two-body effective interaction without considering the effect due to the d-degrees of freedom.] Another possibility to improve the present calculation is to include still more irreducible diagrams in the @box than were included in the present work. In fact, one would like at this point to have the exact o-box. One possible step toward this goal would be the inclusion of the 3rd order diagrams in the Q-box. Such a calculation is facilitated by the new techniques 23) developed to evaluate angularmomentum-coupled diagrams. However, the net result of a systematic 3rd order calculation is not clear. Without the folded diagrams the spectra would presumably show less binding than for 2nd order. [See e.g. ref. ‘“).I However, the derivative (slope) of the 3rd order contribution would also be positive, and moreover, large compared to the 2nd order contribution so that the net result upon inclusion of the folded diagrams is not at all clear. By solving a set of self-consistent integral equations, Chakravarty et al. ‘l) have calculated several large classes of &-box diagrams to all orders. It is interesting that their results of the &box interaction for the 1sOd shell are generally not very different from ours with the C4 approximation. They have not calculated the folded o-box series for V’,,. It would be of interest to do so to compare the V,,, so obtained with ours. Contributions from highly excited states to the $-box due to the tensor interaction, the so-called Vary-Sauer-Wong effect ‘I), were not evaluated in the present work. However, recent work s2) using the Bonn potential 48) showed that two-body potentials with a weak tensor interaction give rise to a negligible VarySauer-Wong effect. The Paris potential has a weak tensor component.
..f. Shurpin et al. / Fofakd diagrams
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We should like to thank I. M. Richard for providing the routines used to evaluate the Paris potential. This work was supported in part by the DOE under contract DE-AC0276ER13001. Note added in proof: The coupled-cluster method of Kiimmel and his collaborators has been recently extended and applied to open-shell nuclei 54-46). It will be of interest to compare their method and results with those of the present work. We are very grateful to Prof. Kiimmel for discussions in this regard.
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