- Email: [email protected]
Thomas-Fermi approach to nuclear mass formula
Nuclear Physics ASZg (1991) l-47 Noah-Holland
THOMAS-FERMI
APPROACH TO NUCLEAR MASS FORMULA
(III). Force fitting and construction of mass table* J.M. PEARSON, Y. ABOUSSIR, A.K. DU’ITA’, R.C. NAYAK’ and M. FARINE3 Luboratoire de Physique Nucl&aire, Dkpartement de Physique, Unioersite’de Mm&al, MontrL;af,Quc’, H3C 337 Canada F. TONDEUR
fnstitut d’Ast~nom~e et Astrophysiq#e, Universite’Libre de Bruxelles, and Znstitut Sup&w
Zndustriel de Bruxelles, Belgium
Received 3 September 1990 (Revised 29 November 1990) Abstract: The ETFSI method, developed
in two earlier papers, is here used to construct a complete mass table. Since the method allows for interpolation both in the (N, Z) plane and with respect to deformations, without losing the characteristic shell-model fluctuations, it is some 2000 times faster than the HF-BCS method for a given force. The present table is calculated using a preliminary Skyrme-type force with &function pairing, fitted to a restricted data set of 491 spherical nuclei. The resulting rms error for all 1492 measured nuclei, spherical and deformed, with Aa36 is erms= 0.868 MeV, achieved with just 9 parameters. The main experimental trends in ground-state deformations are well followed. The symmetry coefficient of nuclear matter corresponding to our force is 27.5 MeV. Ways of rapidly improving the frt are indicated.
1. Introduction In two earlier papers I,‘), hereafter referred to as (I) and (II), we have developed an approach to the nuclear mass formula that is based on the extended ThomasFermi (ETF) approximation 3*4)to the Hartree-Fock (HF) method for a Skyrme-type force, with shell corrections and BCS pairing added. In the present paper we (i) describe a preliminary fit of such a force to a restricted set of mass data, (ii) show how our method lends itself to interpolation both in the (N, 2) plane and with respect to defo~ation, an essential feature without which the computation time would be impra~ically high, (iii) use this interpolation procedure to construct a mass table for our preliminary force, (iv) indicate how this preliminary mass table can be improved. Much of the motivation for our work lies in the fact that the late stages of stellar nucleosynthesis, particularly the so-called r-process, depend crucially on the binding * Supported by NSERC (Canada) and FCAR (Quebec). ’ Present address: School of Physics, Devi Ahilya University, Indore, 452001 India. ’ On leave from: Dept. of Physics, Khallikote College, Berhampur, 760001 India. 3 Present address: Laboratoire de Spectroscopic Nuc%aire, Universite de Nantes, 44072 Nantes, France. 0375-9474/91/$03.50 @J 1991 - Elsevier Science Publishers B.V. (Noah-HoIland) June 1991
J.M. Pearson et al. / Nuclear mass formula (III)
2
energies
and fission barriers
of heavy nuclei
line that there is no possibility thus becomes of masses stability direction:
of the greatest
and fission barriers line, out towards extension
of being importance
lying so close to the neutron
able to measure
to be able to make reliable
away from the known
the n-drip
of the stability
(n)-drip
them in the laboratory.
region,
It
extrapolations
relatively
line. Also of interest
is extrapolation
line to investigate
the possible
close to the in another existence
of
superheavy nuclei. Furthermore, since our method involves a complete exploration of potential-energy surfaces we are also motivated by the possibility of extrapolating to highly deformed shapes, particularly beyond the saddle point. Any mass formula will fit all the available data if it has enough free parameters, but, if one is to have any confidence
in its ability to extrapolate
reliably,
the formula
must not only give a good fit to the data but also have a sound theoretical base. Clearly, if two different mass formulas give comparable fits to the data but extrapolate differently, the one with the better theoretical foundation would be preferred. Generally speaking, the theoretically superior of the two would be characterized by a smaller number of parameters. Now while the drop(let)-model (DM) mass formulas can give acceptable fits to the data, they do so only with a relatively large number of parameters. For example, the most recent and most sophisticated version, the so-called finite-range droplet model of Moller et al. ‘) (MMST), has rms errors of 0.769 MeV for 1593 masses, but requires some 25 parameters, as compared to 9 in the present work. [Both this version and the earlier finite-range drop model of Moller and Nix “) (MN) go beyond the usual leptodermous expansion ‘) in that some attempt is made to represent the diffusivity of the nuclear surface.] At the same time, it is open to theoretical criticism on two different counts: (a) The expansion in powers
of I = (N-Z)/A
appears
to be truncated
pre-
maturely with the result that when this model is fitted to the data, errors of as large as 15 MeV could be expected out at the n-drip line “). (b) It is difficult to relate the macroscopic and microscopic parts, i.e., DM and shell-model parts, coherently. To be specific, corrections involves the use of a single-particle be generated by the distribution of nucleons
the calculation (s.p.) potential, in the nucleus
of the microscopic and since this must it constitutes a link
between the microscopic and macroscopic parts of the mass formula. Now the actual way in which the s.p. potential is generated from the nucleon distribution is by folding some two-body force over the latter, but in no form of the drop(let) model is there an unambiguous prescription for choosing this force. (Actually, in both the MMST ‘) and MN “) versions the well depth is parametrized independently of the two-body force; which serves only to determine the form of the s.p. potential.) Furthermore, the density distribution itself is determined only in a very crude way in the drop(let) model. Both these classes of difficulty are avoided when the binding energy is calculated by the HF method (or rather by the HF+ BCS method, since pairing always has to
J. M. Pearson et al. / Nuclear mass formula (Ill)
be taken
into
account).
In the first place
power-series
expansion.
macroscopic
and microscopic
automatically therefore, method
guaranteed.
Secondly,
is no approximation
is no separation
so complete
A mass formula
be more theoretically represents,
there
parts,
there
based
3
on a
of the total energy
consistency
between
on the HF+BCS
secure than one based
based
on a drop(let)
in fact, the most fundamental
approach
that has any chance of succeeding, even though approach based on the “real” nuclear forces.
it is much
into
the two is method
will,
model.
This
to the mass formula less rigorous
than an
The ideal procedure to be followed with this method would consist in taking some suitable form of effective interaction and fitting its parameters, along with those of the pairing force, to all the data on masses, fission barriers, radii, etc., as in the published
DM fits. Unfortunately,
the method
suffers
from the defect
of
requiring a very large amount of computer time, especially for deformed nuclei, with the result that its systematic application has been somewhat limited. In particular, the available HF effective forces [see ref. ‘) for a convenient summary] have been fitted to relatively few of the available data, and even then not always very well, thereby detracting from the reliability of the method as a means of extrapolating. The present method (ETFSI) has been shown in papers (I) and (II) to be essentially equivalent to HF in the sense that when the underlying Skyrme-type force is fitted to the data the extrapolations out to the n-drip line are very close to those given by the HF method. Nevertheless, the method is computationally so much more rapid than HF that it offers a practical approach to the ultimate problem of constructing a mass table. We give a brief resume of the method in sect. 2. In sect. 3 we present a preliminary force determined by a fit to spherical nuclei alone, and then show that it performs acceptably well for a few selected deformed nuclei. In sect. 4 we discuss the properties of the a-function pairing force that we adopt in place of the more usual “constant-G” model. Sect. 5 describes the interpolation procedure, which is then used in sect. 6 to calculate with our preliminary force the masses of all known nuclei, spherical and deformed, with A 2 36. While the agreement of this preliminary mass table with experiment will be seen in general to be good, we show in sect. 7 that improvement will be relatively easy. 2. The ETFSI method We summarize here the main points of our method, referring the reader to papers (I) and (II) for full details. The basis of the method is the generalized Skyrme force Uij=to(l+XoP,)S(rij)+t,(l+x,P,)[p:S(rij)+h.a.]/2h*) +
t2(1
+x2pm)Pij
’ 6(rij)Pijlh2
(1)
4
J. M. Pearson et al. / Nuclear mass formula (Ill)
[see paper (I) for a discussion of the form we have adopted for the density dependence; the index q denotes n or p, according to whether the term in question relates to neutrons or protons, respectively]. To this we add the constraints x2 = -(4+5x,)/(5+4x,)
)
t,= +,(5+4x,),
(2)
in order for the effective nucleon mass to be equal to the real mass, M;=M,
(3)
a choice which allows the density of s.p. energies near the Fermi surface to be well reproduced lo) without having to take particle-vibration coupling into account I’). Fitting the density of s.p. energies near the Fermi surface helps to obtain correct masses 12), and has also been shown to be required for a good description of fission barriers I”) For the ETF semi-classical approximation to the kinetic-energy and spin-current densities, r4 and Jq, respectively, we use the full fourth-order (in powers of A) expansions of Grammaticos and Voros 3*4).The energy density 8(r), which gives the total energy as E ETF
=
J
8(r) d3r,
(4)
becomes a function of the nucleon densities, pq, and their gradients, the full expressions being given in papers (I) and (II). (Although those papers work in terms of t2 and x2, rather than t, and x,, in the present paper we discuss our fits in terms of the latter pair, since they are physically more significant. To simplify matters we show both pairs in table 2.) Note that we do not make the usual approximation of neglecting the terms in Jz and J2. Although the error made in EET, on neglecting these terms would be absorbed to some extent by the renormalization of the force on fitting to the data, retaining these terms might help to reproduce some data better “), in addition to the possibility of a more reliable extrapolation from the data, because of the greater consistency ‘). For spherical nuclei the density distributions are parametrized according to the Fermi form,
Pq(d=
PO4
l+ew[(r-Cq)f~qf’
(5)
the total ETF energy being minimized with respect to the four parameters C,, C,, a,, and up (the poq are fixed by normalization). Although more general forms lead to greater binding ‘), it is known that the ETF method tends to overbind anyway ‘), as compared to HF. Thus the choice (5) is preferred not only for the computer time that it saves, but also because it will actually lead to better agreement with HF energies, for a given force. In any case, it has been noted *‘) that the simple Fermi form (5) also reproduces HF and experimental density distributions better.
.I.M. Pearson et al. / Nuclear mass formula (Ill)
5
To save further computer time we parametrize the Q~ according to a,=a,ia,I+a,1~fa~r~+a,14,
(6)
where the upper sign refers to neutrons, and the lower to protons. The coefficients are determined once and for all (for each force) in a separate calculation. In this way we only have to minimize with respect to two parameters during production runs; we have checked that the error introduced in this way is quite negligible over the range of A considered here. Actually, using the density distributions (5) directly in the ETF functionals implies that they refer to point nucleons. Thus the Coulomb terms in the expressions given in papers (I) and (II) for the energy density a(r) are calculated for point protons. Nevertheless, protons are of finite size, and this will affect the energy. This effect could be taken into account simply in the fitting of the Skyrme force (1) to the data, but the resulting modification of the force would be isospin-independent, while the effect itself is not, of course. Since error could thus arise on extrapolating far away from the data, we attempt to take the finite proton size into account explicitly. We do this by noting that its effect is simply to add a short-range term to the Coulomb force between protons, which we approximate for simplicity by a S-function force, i.e., an extra Skyrme-type to term, for protons. The strength tof,,of this force is fitted to the results of a HF calculation in which the finite size of the proton has been taken into account “1: we find fOfP= - 4.8 MeV, corresponding to an rms charge radius of 0.8 fm for the proton. Turning to deformed nuclei, we adopt the parametrization of Brack et al. (BGH) 9), as described fully in paper (II). This means that only axially symmetric deformations can be considered, but allowing triaxial deformations may not be of much consequence for total binding energies. A basic feature of the BGH description is the definition of a reference surface, parametrized by the (c, h) variables r4), and enclosing the same volume for all deformations of a given nucleus. [In appendix A we present a detailed analysis of the relation between the (c, h) parameters and the more familiar (&, p4) parameters of a multipole expansion.] Deformed density distributions are then generated by a uniquely defined transfo~ation, characterized completely by the (c, h) parameters, from the spherical dist~bution (5). In principle, the C,, a, parameters of the latter could be chosen anew for each deformation (c, h) by minimizing the corresponding ETF energy, but in paper (II) we found that excellent agreement with HF was obtained by treating these parameters as deformation independent, determining them for a given nucleus in the spherical configuration. This suggests that the more general procedure might well have led to overbinding in the case of large deformations, although we did not test this point. We note now a problem that can arise for both spherical and deformed nuclei with the density distribution (5). Since the gradient of this distribution has an unphysical discontinuity at the origin, and at the centres of curvature in the deformed case, there is a danger of anomalousIy large currents being computed in the vicinity
6
J. M. Pearson et al. / Nuclear mass formula
{III)
of these points, whereas in reality it is easy to see on symmetry grounds that the currents should vanish there. Correcting for this problem in the computer programme is easy, and corresponds simply to flattening the distribution (5) at the origin. Shelf corrections. It is inevitable with the ETF method that the energy will vary smoothly from one nucleus to another: the shell effects are effectively lost in truncating the semi-classical expansions. Thus we are once more forced back to a macroscopic-microscopic approach, and are obliged to add shell corrections. However, compared to the DM mass formulas there is an important difference, since we can now determine quite unambiguously the s.p. fields appearing in the s.p. Schrodinger equation [(-&‘/2M)V*+
U,(r)+
W,(r) * {-iVxa)]+(r)
= &4(r),
(7)
the solutions to which are required for the calculation of the shell (and pairing) corrections: one simply folds the same Skyrme-type force involved in the ETF functional over the density distribution emerging from the macroscopic part of the calculation (see papers (I) and (II) for the expressions). There is thus a high degree of coherence between the macroscopic and microscopic parts, the unifying factor being the Skyrme-type force that underlies both; this presumably accounts for the excellent agreement with the HF method (see below). Neglecting pairing for the moment, we note that the shell corrections, which determine the total energy according to E = EETFi-6E,
01
can be written, according to the Strutinsky theorem (see, for example, ref. 15)), as SE=
r, 4=hP
(
C stz,q-F a/+?)
(9)
p
where the first term on the right-hand side represents the sum over all occupied states (neutrons and protons) of the s.p. energies corresponding to eq. (7), while the second term is a smoothed form of the first. To calculate this smoothed term we could have used the conventional Strutinsky energy-averaging method, as in refs. 5*6),for example. However, it is well-known that this method contains some ambiguities, mainly because of the continuum s.p. states, which means that they may be expected to become particularly troublesome towards the drip lines. Actually, some progress has been made towards resolving these problems 16*i7),but in the present case, where the macroscopic term is generated by the ETF method, we can avoid them altogether by a direct application of the Strutinsky theorem, as follows. Writing the HF energy in density-matrix notation as E HF= tr (tp) +5 tr tr (pop),
(IO)
(we are still neglecting pairing) we separate the density matrix into smooth and fluctuating parts, p=p”+sp.
(11)
J.M. Pearson et al, / Nuclear mass formula (Ill)
A straightforward
calculation
[see ref. 15), for example]
I
then leads to the Strutinsky
theorem,
E HF=~++E+O(Sp)*, where
6E is the shell correction
c
Ew7 = tr&= PL.4
(12)
(9) with the smoothed
terms given by
&G-~+~.U~+.I~.
C
W,
.
(13)
>
4=n,p
Here rq, pq and Jq are the smoothed
densities
obtained
in the macroscopic
part of
the calculation, while U, and W, are the corresponding fields, appearing in eq. (7). The error term O(Q)* in eq. (12) is known “) to be around l-2 MeV. This prescription for shell corrections is very simple to apply in our case and is quite unambiguous, even at the drip lines. Note, however, that it cannot be used with drop(let) models, mainly because the distribution of the DM energy between potential and kinetic terms is not specified, but also because the distributions pq and Jq giving rise to the s.p. fields are not known. We call this prescription the “Strutinsky-integral (SI)” method, to distinguish it from the familiar energyaveraging method, although in reality it does not deserve a special name, since it is nothing more than the Strutinsky theorem rendered explicit. Eq. (13) was apparently first written down by Chu et al. 18), but it does not seem to have been used very much in the past. A further feature of this method is the exact cancellation that takes place in eq. (8) between the kinetic-energy term of eq. (13) and the corresponding term in the ETF energy (4). Thus errors in the ETF expressions for kinetic energy will be of consequence only insofar as they affect the density distributions given by the minimization of E,,F. Executing this method requires solving the Schrodinger equation (7), in order to get the s.p. energies
appearing
in eq. (9). Since the s.p. field in eq. (7) results
the macroscopic
part of the calculation,
shell corrections
may be regarded
it follows
as doing
that this method
one iteration
from
of calculating
of HF on top of ETF.
Since this is by far the most time-consuming part of the whole calculation in the case of spherical nuclei we see that the ETFSI method is an order of magnitude faster than HF for these nuclei. As for deformed nuclei, where the ETF part of the calculation takes a significantly greater fraction of the total time, the ETFSI method is some five times faster than HF. Calculating the shell corrections in this way introduces a spurious c.m. (centre of mass) motion, exactly as in the HF method. We correct for this using the method of Butler ef al. 19); we have found that this method is essentially exact, being much more accurate than the (1 - l/A) method, as used, for example, in ref. 20). Pairing. We handle this by doing BCS with a S-function force 21) Dij
=
V,S(rij) .
(14)
8
J.M. Pearson et al. / Nuclear mass formula
Although
this increases
the computer
we shall see in the next section parameter,
the constant
and proton
pairing.
time as compared
that it allows
strength
(III)
to the “constant-G”
choice,
a good fit to the data with a single
V,, the same for all nuclei
We can thus expect a more reliable
and for both neutron
extrapolation
from the data
out to the exotic regions. Recalling
the main
points
of the BCS method,
we introduce
the occupation
probabilities
where eP.4 In these equations BCS equations
the chemical
=
{(%,q-A,)‘+
potential
A;,q}"2
A, and the gap A,,, are given by the usual
ML,, = N or 2, P A,,,
(16)
(17a)
=$C Gpud""
e “A
Y
(17b)
in which -GPy is the appropriate matrix element of the pairing force, whatever it may be (note that GPy is positive with an attractive pairing force). Solving the BCS equations for both neutrons and protons, eq. (8) for the total energy is replaced by E = E,,,+
6’E + Epair,
(18)
where
S’E= C
q=n,p
with the second
i\PC v:.,~,.,
(19)
c1
term still given by (13), and &iir=
C
&air,q
4’“.P
3
(20)
in which Epair, q =
-ac 2A2 P %,s .
(21)
When N or 2, or both, are odd, we apply blocking in the usual way, i.e., in the BCS equations (17a) and (17b) the “blocked” state is excluded from the summations, and n = N or 2 is replaced by n - 1, so that the right-hand side of (17a) is an even number (see, for example, ref. “)). (It is because of blocking that eq. (18) contains no quasi-particle terms.) In all the summations over s.p. states we include all quasi-bound continuum states up to 1 hw, it having been shown *‘) that higher cut-off energies do not affect the value of V,.
9
J.M. Pearson et al. / Nuclear mass formula (Ill)
Our interpolation procedures (sect. 5) approximate the basic b-function pairing force by defining an “equivalent constant G”, Geq, as follows. Let us rewrite eq. (18) for the total energy calculated with the S-function pairing force by the complete ETFSI method as
(22) where
(23)
E,“c’(S)= C u:.qeF,q + Epair,q P
(24)
with Epair,q defined by (21). Then for the nucleus in question, at the given deformation, we perform a constantG BCS calculation with the value of G chosen such that
EFS(G)= EtCS(G) .
(25)
This defines Geq,q; note particularly that it is defined separately for protons and neutrons, even though the strength V,.,of the b-function pairing force is the same for both. This definition of Geq,q ensures the correct reproduction of the net effects on the total energy (18) produced separately by the neutron and proton pairing with the b-function force. It follows, of course, that the total energy (18) obtained with the b-function pairing force also will be correctly reproduced. Correction for rotational energy. For deformed nuclei and fission barriers we subtract from the total computed energy the spurious rotational energy 23)
&t = f-j (J2> , where ,$ is the moment of inertia. We first calculated this by the cranking model [Inglis ‘“)I with pairing correlations included [Belyaev 24)], but found that this value, 9 crank, was consistently smaller than the values extracted from the measured spectra in well-deformed even-even nuclei, by about 30% for the rare earths and 50% for the actinides. Rather than adopt some more sophisticated method for calculating 8, we simply took a linear combination of B‘_rank with the rigid-body value of $,2rig, ~P(l-a)Bcrank+drig,
(27)
where the constant a was taken to be 0.2 for all even-even nuclei, this giving the best overall fit to the experimental values of 9 for a number of nuclei over different regions of the nuclear chart, as shown in table 1. The deformation parameters shown in this table correspond to equilibrium, and all results are found to be the same for all three of the forces developed in this paper (see sect. 3). We postpone until sect. 5
J.M. Pearson et al. / Nuclear mass formula (III) TABLE
1
Deformation parameters at equilibrium for all three forces (see appendix A for the relation between the two parametrizations). Moments of inertia (units of h* MeV-‘): &t.z&Ecalculated from eq. (27), db,,P experimental
100 106 150 154 156 162 162 168 172 174 184 194 224 232 236 238 240 246 246 250 250 252 256 256 260
38 44 58 64 62 66 72 76 68 70 74 76 90 90 92 96 94 94 100 96 100 98 100 104 106
1.25 1.16 1.24 1.22 1.27 1.20 1.15 1.13 1.16 1.16 1.10 0.89 1.25 1.24 1.26 1.30 1.26 1.20 1.24 1.20 1.20 1.17 1.17 1.17 1.14
0.00 0.00 -0.16 -0.11 -0.13 -0.02 -0.04 -0.07 0.12 0.12 0.17 0.13 -0.30 -0.21 -0.20 -0.25 -0.20 -0.07 -0.14 -0.06 -0.07 0.00 -0.01 0.01 0.09
0.38 0.26 0.24 0.28 0.33 0.32 0.23 0.17 0.33 0.33 0.27 -0.16 0.14 0.22 0.21 0.23 0.21 0.29 0.27 0.29 0.29 0.27 0.26 0.27 0.27
0.02 0.00 0.06 0.05 0.06 0.02 0.01 0.02 -0.03 -0.03 -0.05 -0.01 0.09 0.08 0.07 0.09 0.07 0.03 0.06 0.03 0.03 0.00 0.01 0.00 -0.03
23.1 14.6 31.8 31.2 38.1 35.0 25.0 23.2 40.5 42.8 33.2 24.4 51.9 56.5 63.6 61.0 64.5 60.3 65.6 62.3 64.7 62.1 62.2 65.1 65.3
11.1 30.6 24.3 39.5 37.2 39.5 39.2 26.3 13.8 32.2 60.7 66.0 70.0 65.2 62.5 65.6 62.3 -
the question of the moments of inertia of odd-A and odd-odd nuclei, since these nuclei are not involved in the determination of the force, and are always calculated by interpolation. While
this
procedure
is quite
adequate
for known
nuclei,
we realize
that
it
introduces an element of uncertainty into the extrapolation to unknown nuclei, since there is no guarantee that the parameter a would take the same value for nuclei far from stability. However, it should be realized that the same uncertainty surrounds the force parameters; indeed, one could argue that the parameter a has a status in our model quite similar to that of the force parameters, so that we should perhaps admit to 10 rather than 9 parameters in all. There is, however, the fundamental difference that the parameter a is determined from level spectra, and is quite independent of mass data. Di~en~i~na~i~y. We solve eq. (7) by expanding the eigensolutions in an oscillator basis. In the programme for spherical nuclei we take a base of dimensionality 10 in each subspace. The resulting truncation error on the total energy can amount to
J.&l. Pearson et al. / Nuclear mass formula (Ill}
11
2 MeV for heavy nuclei, but this is smaller than the error arising from the ETF approximation for a given force. In any case, both of these errors, which are of opposite sign, are for the most part absorbed by the force in fitting to the data, although they could in principle re-emerge as we extrapolate away from the data towards the drip lines; we return to this point below. As for deformed nuclei, we have two programmes, one for nuclei with left-right (lr) symmetry, and the other for nuclei with lr asymmetry. These codes are used only to calculate deformation energies: the former calculates the energy with respect to the spherical configuration, while the latter calculates the energy with respect to lr symmetric configurations. Since deformation energies converge much faster than absolute energies as a function of dimensionality this procedure permits a considerable saving of computer time for a given precision. Our lr symmetric code expands all states in a basis limited to 17fio for ground-state calculations of nuclei with As 204, and to 19%~ for A> 204, this guaranteeing an effective convergence. For the corresponding limit in the case of the code which calculates the lr asymmetry corrections, 15hu was found to be adequate. Comparison with HF. For a given set of force parameters the ETFSI method overbinds nuclei by between 3 and 7 MeV, as compared to HF ‘) (this represents the difference between the characteristic overbinding of the ETF method and the underbinding associated with the restricted form (5) of the density profile). Now while errors this large would be inacceptable in modern mass formulas it is not inevitable that this will pose a problem in practice, since whatever method is being used the force parameters are always fitted to the data. Rather, the crucial question is: when the ETFSI method is extrapolated out to the n-drip line, will it agree with the HF extrapolation? In papers (I) and (II) we showed that the discrepancy between the two methods at the n-drip line is less than 1 MeV for the total energy and fission barriers, and less than 0.5 MeV for neutron separation energies (S,) and beta-decay energies (QP). We thus conclude that while the ETFSI method is an order of magnitude faster than HF it gives essentially the same extrapolations from known nuclei out to the n-drip line, for a given form of force. Nevertheless, it should be realized that this comparison of the ETFSI and I-IF methods supposes that both methods use bases of the same dimensionality. Thus there is still the possibility that some of the truncation error of our spherical ETFSI calculations (~2 MeV), which is absorbed into the force by fitting the data, will re-emerge on extrapolating towards the drip lines. We conclude, then, that if our spherical programme had a larger basis the extrapolations out to the drip lines could shift slightly. However, the situation in this respect is no different than with the HF method. Interpolation. Once an effective interaction has been fitted to the data a further advantage of the ETFSI method becomes apparent. All the quantities in terms of which the method expresses the total energy, including the s.p. energies (but not their sum, of course), vary smoothly with N, 2, and deformation, even though the
J.M. Pearson et al. / Nuclear mass formula (III)
12
fluctuations
that are characteristic
it will be possible
to construct
of the shell model are correctly
the mass table by interpolation.
in more detail in sect. 5, where we show that the total computer by more than two further which interpolation
orders
of magnitude
as compared
reproduced.
Thus
We will discuss
this
time can be reduced to the HF method,
for
is not possible. 3. Fitting procedure
Despite
the comparative
calculations parameters are obliged
rapidity
of the ETFSI
method,
the deformed-nucleus
are still so slow that in the automatic least-squares fit of the nine of our Skyrme-type force (1) and the pairing force (14) to the data we to assume spherical configurations, since each nucleus has to be calcu-
lated many times during the actual fit. Thus in the present paper we confine our automatic least-squares fit to the masses of 491 nuclei that are believed to be spherical, the data being taken from the 1988 compilation of Wapstra et al. “). This sample consists of all nuclei with Z 3 20, and N or Z not more than two units away from a magic number, excluding the few such nuclei known or suspected to be deformed. We first included nuclei with Z < 20 in the fits, but the fits were always poor, and since our primary interest lies in heavier nuclei
we decided
not to allow these light nuclei
to influence
the determination
of
the force parameters. Because of the Wigner effect we exclude likewise nuclei with N = Z (see sect. 6). The preliminary fit to the 491 nuclei assumed to be spherical led to three different parameter sets, labelled SkSCl, SkSC2, and SkSC3 in table 2 (SkSC = Skyrme semi-classical). The fits were made as follows. Of the eight Skyrme parameters, x-, and
f3 were expressed
momentum,
in terms
of J and kF, the symmetry coefficient and Fermi of nuclear matter. Then kF, J, x,, and y were held fixed
respectively,
at different values, while least-squares routine.
to, ti, x0, and
W,, were fitted to the data points
TABLE Force
parameters SkSCl
t, (MeV . fm3) t, ( MeV fm’) tz (MeV . fm’) t, ( MeV . fm4) X0
2
(y = 0.333333
-1788.59 282.623 -282.623 12775.3 0.72
for all forces) SkSC2
-1791.64 290.601 -290.601 12800.6 0.38
XI
-0.5
-0.5
X2
-0.5
-0.5
X3
W, ( MeV V, ( MeV
1.04564
. fm5) . fm3)
126.997 -220.0
0.592760 128.781 -220.0
SkSC3 -1788.11 289.901 -96.6337 12771.4 0.65 -1.0 1.0 0.960976 143.452 -220.0
with the
J.M. Pearson et al. / Nuclear mass formula (III)
13
In principle, we should have optimized the fit to the masses with respect to y, but to do this correctly would have consumed too much computer time. However, a rough mass fit led us to set y = f in all three forces, the corresponding values of the incompressibility K then always being around 235 MeV; we found this to be definitely better than y = 0.5, i.e., K = 270 MeV. This result is to be contrasted with the mass fit of MMST5), which favours a value of K closer to 300 MeV. The difference between the two fits appears to originate in the microscopic corrections, ours being much more sensitive to the value of K than are those of MMST ‘), essentially because when we change y in order to change K we induce changes in the diffusivity of the density distribution, and hence of the s.p. potential 26). Thus, while the value y = f is clearly indicated, it might be unwise to draw any conclusions from our work concerning K, since what we are really testing is the surface diffusivity, the coupling of which to K through the force might well be spurious. On the other hand, a DM-based fit such as MMST ‘) is more likely to be measuring K directly. Nevertheless, we note that our choice is in good agreement with the value K = 228 MeV given by an analysis of the differences between the charge-density distributions in various lead isotopes “). As for recent breathingmode estimates of K, these vary from 217 MeV (Gleissl et al. ‘“)) to 300 MeV (Sharma et al. ‘“)).
As for kF, requiring that the rms charge radii I, of heavy nuclei be fitted (table 3) leads to the value of 1.335 fm-‘, in good agreement with ref. 30). [We fold in the finite proton size, according to
where rr, is the rms radius for point protons, corresponding to the Fermi distribution (S), and sP = 0.8 fm is the rms charge radius of the proton.] The pairing-force strength V,, which was adjusted manually, was found to give a best fit to the masses with a value of -220 MeV for all three parametrizations (see below for further comments). An optimal fit to the 491 nuclei assumed to be spherical was then obtained with J = 27.5 MeV and x, = -0.5; this defines our first force SkSCl (see tables 4 and 6 for the errors). Support for this value of J comes from relativistic Brueckner-HF calculations of asymmetric nuclear matter by Miither et al. 3’), and ter Haar and TABLE
3
RMS charge radii (fm)
4sCa “‘Sn *08Pb
SkSCl
SkSC2
SkSC3
Exp “)
3.53 4.63 5.50
3.53 4.63 5.50
3.53 4.63 5.50
3.41 4.64 5.50
“) Data from De Vries et al. 56).
J.M. Pearson et al. / Nuclear mass formula (III)
14
Malfliet 32), which
give J = 27 and 26 MeV, respectively.
tivistic calculation
for the Reid soft-core
as reported
have added
but also the neutron
compare
Also Siemens’
non-rela-
33) gives 27 MeV after correction,
in ref. 34).
Our force would masses
potential
the SkSCl
Pandharipande
force
astrophysical
gas at nuclear in this respect
35) for the a,, potential
interest
if it fitted not only nuclear
and sub-nuclear
densities.
with the calculations et al. 36) obtain
[Cugnon
In fig. 1 we
of Friedman similar
and
results with
the Paris potential]. The poor agreement motivated us to define a second force, SkSC2: with J = 30 MeV and x, still equal to -0.5, it will be seen that there is an excellent fit to the neutron gas. However, the fit to the masses is not quite as good as before (tables 4 and 6), but it is still acceptable, and it remains to check the possibility that it will out-perform SkSCl for fission barriers. Finally, fit fission somewhat with x, =
in order to have a second degree of freedom which might be useful to barriers we take advantage of the fact that the fits to ground states are insensitive to the value of x, . Accordingly, we define a third force, SkSC3, -1.0 and J = 27.5 MeV. This too has an acceptable fit to the mass data,
but overall it is not quite as good as SkSCl (tables 4 and 6). We have checked that all three forces satisfy the conditions of Giai and Sagawa 37) for spin and spin-isospin stability. [The relevant expressions given in ref. 37) remain valid for the different kind of density dependences used here.] Errors in Jits to masses. Referring to table 4 for the fits to the 491 nuclei assumed to be spherical, we see that force SkSCl has the lowest rms error: .srms= 0.693 MeV. Ideally, one would also want the mean deviation, E, to be zero, in order for the fit to be free of systematic error. However, once the force parameters have been chosen
I
0
0.05
I
0.10
I
0.15
p(fme3) Fig. 1. Energy e per neutron in neutron matter of density p.
J.M. Pearson et al. / Nuclear mass formula (III) TABLE
15
4
Fitting errors for 491 spherical nuclei (MeV)
E,ltlS
E (nr) E,Ill, &‘,Ej
SkSCl
SkSC2
SkSC3
0.693 0.0610 0.587 0.965
0.729 0.0861 0.676 1.042
0.713 0.0899 0.636 0.960
by a least squares fit one has no remaining degree of freedom to control this (if one of these parameters is a simple additive constant, which is not the case here, then .? = 0 follows automatically). Thus it is gratifying to find that the best force from this point of view also has the lowest rms error: SkSCl. Table 4 also shows the rms error ~2: with which each force fits a sample of 68 of the most neutron-rich of the 491 nuclei to which the forces have been fitted. It will be remarked that in this respect SkSCl is still the best of the three forces. In the last line of table 4 ET: represents the rms error in a sample of 34 of the most proton-rich of our 491 spherical nuclei (fewer of these are proton-rich than neutronrich). It should be noted that SkSC3 now scores marginally better than SkSCl, but the difference is probably not statistically significant. More important is the fact that SkSC2 is distinctly worse than SkSCl, which means that J = 27.5 MeV works better than J = 30 MeV on both sides of the stability line. Deformed nuclei. Before using any of these forces for calculating a complete mass table, we must form some idea of how they perform for deformed nuclei. We make this test on the 25 nuclei of table 1, which are widely distributed over the nuclear chart. Table 5 shows the calculated energies, obtained in all cases with the code for lr symmetry; the equilibrium deformations, shown in table 1, are found to be the same for all three forces. From table 6 we see that the fit has now deteriorated rather badly, as compared to the spherical nuclei. However, we also see in table 6 that this deterioration is associated entirely with nuclei of A 3 250, and that if we exclude these nuclei the fit is as good as for the 491 assumed to be spherical. It will also be seen from this table that SkSC3 now performs somewhat better than SkSCl, but we do not regard
this as being
statistically
significant,
given the smallness
of the
sample. The problem with the nuclei having A2250 is clearly seen to lie in a tendency to underbind. We have checked that this is not a problem of the dimensionality of the codes. The dimensionality of the deformed codes is quite adequate, while as for the spherical code we made a trial calculation in which the dimensionality was increased from 10 to 15; we found that the binding energies of *08Pb and nucleus (Z = 106, A = 260) were increased by 1.03 and 1.09 MeV, respectively. This suggests that if the force were renormalized to refit the data with the greater dimensionality the final results for the heaviest nuclei would be essentially the same as before. Nor
16
J.M. Pearson et al. / Nuclear mass formula TABLE Energies
(MeV)
for 25 deformed
Z
A
even-even
SkSCl
(III)
5 nuclei
(axial
and left-right
symmetry)
SkSC2
SkSC3
Exp. -837.5
100
38
-837.3
-838.2
-837.6
106
44
-906.8
-906.9
-906.9
-907.3
150
58
-1230.1
-1230.4
-1230.2
-1230.1
154
64
-1267.5
-1267.7
-1267.7
- 1266.3
156
62
-1280.0
-1280.1
-1280.2
-1279.7
162
66
-1323.9
-1324.0
-1324.1
-1323.8
162
72
-1300.1
-1300.6
-1300.3
-1300.0
168
76
-1326.1
-1326.7
-1326.2
-1326.2
172
68
-1391.1
-1391.1
-1391.2
-1391.2
174
70
-1406.5
-1406.4
-1406.5
-1406.2
184
74
-1471.4
-1471.3
-1471.5
-1472.5
194
76
-1536.9
-1536.7
-1537.0
-1538.4
224
90
-1716.6
-1716.5
-1716.6
-1716.9
232
90
-1765.7
-1765.4
-1766.0
-1766.0
236
92
-1789.3
-1789.0
-1789.6
-1789.7
238
96
-1795.7
-1795.8
-1796.0
-1795.7
240
94
-1812.1
-1811.9
-1812.5
-1812.7
246
94
-1845.1
-1844.7
-1845.6
-1845.9
246
100
-1835.0
-1835.1
-1835.3
-1836.3
250
96
-1868.2
-1867.7
-1868.6
-1869.0
250
100
-1863.0
-1863.0
-1863.3
-1864.7
252
98
-1879.0
-1878.3
- 1878.9
-1880.5
256
100
-1900.2
-1899.9
-1900.6
-1901.7
256
104
-1888.1
-1888.0
-1888.3
-1889.7
260
106
- 1906.6
-1906.6
-1906.8
-1908.1
has the problem anything to do with left-right asymmetry, for we have confirmed that in the case of the heaviest nucleus of all, A = 260, removing the constraint of left-right symmetry does not in any way increase the binding. We therefore believe that this underbinding of the heaviest nuclei is not related specifically to our treatment of deformation, but rather reflects a defect in our three forces, since the heaviest nuclei in the sample to which the forces were fitted correspond to A = 218. Refitting the force to all nuclei, spherical and deformed,
TABLE Fitting errors (MeV)
6
for the 25 deformed SkSCl
nuclei of table 5
SkSC2
SkSC3
e,,, (all 25)
0.923
1.100
0.812
E,,,
0.665
0.809
0.604
(A < 250)
E (all 25)
-0.552
-0.564
-0.344
I (A < 250)
-0.274
-0.205
-0.074
J&f. Pearson et al. j Nuclear ~ass~o~ula
(IZI)
17
might thus be expected to lead to an improvement. We will return to this question in sects. 6 and 7. But more to the point is the fact that in the present work we are able to represent deformed nuclei almost as well as spherical nuclei, even though both the parametrization of the deformation and the subtraction of the rotational energy, as given by eqs. (26) and (27), could have given rise to further errors. In fact, with the SkSCl force we have achieved a very high-quality fit to both spherical and deformed nuclei, the rms error for 516 nuclei being 0.706 MeV. However, before choosing SkSCl as the force with which to calculate a mass table, we note that the other two forces have, so far, also given quite respectable fits, and that SkSC3 might even be better. We thus postpone our final choice of force until we have seen how all three perform in connection with fission barriers. Macroscopic parameters. Before considering the results for fission barriers, and making the final choice of force, we present (table 7) the so-called macroscopic coefficients of our forces, which appear in the droplet model ‘) but which are defined independently of it. Of the coefficients defined for infinite nuclear matter we give the volume coefficient av, the incompressibility K, and the symmetry-density coefficient L, as well as J and k, (already discussed). We also make a full ETF calculation of semi-infinite nuclear matter, assuming a Fermi profile, as in eq. (5), and extract the surface coefficient asf, the surface-symmetry stiffness Q, and the curvature coefficient aCv [for further details see, for example, ref. ‘)I. For reasons of simplicity our calculation of a_. omits the terms in J’, and J2 from the energy density appearing in (4). It can be seen from table 7 that L and Q are correlated with J in the usual manner 38,39).However, the values of Q we find for a given J are somewhat larger than would be expected on the basis of ref. ‘“): fig. 1 of that paper indicates that 60 MeV would be appropriate for SkSCl and SkSC3 (J = 27.5 MeV), and 45 MeV for SkSC2 (J = 30 MeV). Nevertheless, for two reasons this situation is far less Macroscopic parameters (MeV); e”’ = p~(d3e/dp3),=, (this is Ev of ref.9)). k,= 1.335fm-’ (p~=0.1607fm-3) for all forces SkSCl
SkSC2
SkSC3
-15.85 - 14.09 234.8 27.5 -2.32 17.4 94.2 -18.4 11.1
-15.90 -14.14 235.3 30.0 32.0 17.6 52.8 -38.1 11.3
-15.84 -14.09 234.7 27.5 2.74 17.4 88.3 -19.2 11.4
18
J.M. Pearson et al. / Nuclear mass formula
anomalous considered stability
than
appear
in ref. 39) were fitted to data of nuclear
masses
it should
concentrated
be noted much
work. Secondly,
on Q is given
more
that
the forces
closely
we should
by a first approximation
to the
note that to the
model ‘) as e = a, + a,fA-“3
where
to be. Firstly,
line than is the case with the present
the dependence droplet
might
(III)
e is the energy
+ (J + u~~A-“~) I2 + ~oulZ2A-4’3
per nucleon,
+ acvA-2’3 ,
(29)
and
ass= (2L/K)a,,-9J2/4Q
(30)
the so-called surface-symmetry coefficient, also shown in table 7. It is easy to show now that the shifts in e implied by the above shifts in Q for the given values of J are much smaller than the shifts implied by going, for example, from any of our forces to the MMST fit ‘) (J = 32.5 MeV, Q = 29.4 MeV, L = 0, whence ass = -80.8 MeV). The point is that when J*/Q is small, as with our fits (especially SkSCl and SkSC3), the fits to the masses become quite insensitive to Q. The values of acv that we obtain are quite normal for effective forces (see, for example, table 8 of ref. “). However, DM fits, and in particular the MMST fit of ref. 5), find acv much closer to zero. This anomaly has been discussed by Stocker et al. 40), and the simplest explanation is an insufficient convergence of the droplet model’s leptodermous expansion. There could be significant implications for the calculation of fission barriers, fission lifetimes, and particularly the dynamics of scission, since this is strongly influenced by the rapidly changing post-saddle curvature. Fission barriers. Table 8 shows the results we have calculated with our three forces for the experimentally known outer barriers of four nuclei, with lr asymmetry taken into account. We do not consider the inner barriers, since they are much less sensitive to the choice of force (this can be seen in table 9), and also are more likely to be afflicted by triaxiality, which we cannot yet handle. Forces SkSCl and 3 are seen to be virtually equivalent, while SkSC2 gives appreciably different, and worse, results for the two heaviest nuclei. This shows that TABLE 8 Fission
barriers
Nucleus
lS60s 2’0Po 240pu “‘Cm
(MeV)
for 4 known barriers
nuclei; only outer (last two nuclei)
SkSCl
SkSC2
SkSC3
24.6 22.0 3.1 3.0
24.2 21.7 2.1 2.3
25.0 21.7 3.2 3.2
“) Data from Moretto et al. 57) b, Data from Khodai-Joopari ‘*). “) Data from Back et ai. 59).
barriers
in case of double
Expt. 23.4kO.5 20.4kO.5 5.4kO.2 3.9*0.3
“) b, “) “)
J.M. Pearson et al. / Nuclear mass formula (III) TABLE
19
9
Calculated fission barriers and isomers of 262U(MeV)
1st barrier 1st isomer 2nd barrier 2nd isomer 3rd barrier
SkSCl
SkSCZ
SkSC3
2.7 1.3 5.8 2.9 13.0
2.5 0.9 4.6 1.3 9.1
2.9 1.4 5.9 3.0 13.0
fission barriers, like masses, favour the lower value of J, 27.5 MeV; however, the barriers are quite insensitive to x, . The general level of agreement with experiment given by forces 1 and 3 is satisfactory, especially when we bear in mind that no fission-barrier data were included in the parameter fitting. However, a slight tendency to underestimate barrier heights for heavy nuclei, and overestimate them for lighter nuclei is noted. Looking at the results for SkSC2, we see that reducing J increases the barriers, and that this effect is greater for heavier nuclei. This can be understood quite easily in terms of the fissility parameter
a,oul z2
X=za,Xr)A’
(31)
&f(I) = a$+ ff,,12.
(32)
where
We see that we might have obtained better barriers with a still smaller value of J; however, the fit to the masses will deteriorate significantly if J is reduced below 26.5 MeV. A smaller value of kF would have the same effect, since it would lead to a smaller value of c+Oul.(In any case, reducing the symmet~ coefficient f requires that kF be reduced also, if the stability line is to be unchanged.) The results of restricted calculations (h = 0, and lr symmetry) on the fission barriers of 262U, a nucleus lying on the r-process path, are instructive: see table 9. They confirm the sensitivity of the extrapolated barriers to the value of J, and also show that no sensitivity to x1 emerges beyond the region of known nuclei. Choice offorce. The results that we have obtained for masses and fission barriers, as described in this section, lead us, on balance, to SkSCl as our choice of force with which to calculate our first mass table. The associated value of J, 27.5 MeV, is significantly lower (and the value of Q correspondingly higher) than for any recent mass formula, although it is very close to the value of 27.63 MeV given by Myers and Swiatecki more than twenty years ago 41). Our low value could simply be a result of the large amount of new data we fitted [the MMST formula of ref. ‘) uses values of J and Q determined from earlier fits to more restricted data], but it could
20
J.M. Pearson et al. / Nuclear mass formula
also be a consequence that
for a given
of our model. We are referring
surface
stiffness
excesses,
to a softer neutron
because
the latter
omit
specifically
Q the ETF method
skin than do the droplet
some
higher-order
significant.
If then “malacodermous”
they would
have to be compensated
(Ill)
to the observation
“)
gives rise, at large neutron
models used to date, essentially
symmetry
terms
that
are apparently
effects of this sort are showing
up in the data
in our model by a larger value of Q, and hence
a lower J. In any case, our low value of J (along with its associated high value of Q) and the “malacodermous” effects that we implicitly include mean that we should expect our extrapolated masses and fission barriers of nuclei far from stability to be different from the MMST predictions ‘). (Note that these two sources of discrepancy work in opposite directions, but the latter will dominate close to the n-drip line.) We return
to this question
in sect. 6. 4. Comments on pairing properties.
In this section we examine in more detail some of the properties of the b-function pairing force (14), with particular emphasis on its advantages over the more usual “constant-G” model. All calculations are performed with the SkSCl parameter set. We begin by looking at the fourth-order even-odd mass differences, Ac4), as defined by Madland and Nix 42). The comparison with experiment is limited to spherical nuclei, since it is obscured in the case of deformed nuclei by the uncertainty in E,,, for odd-A and odd-odd nuclei. Fig. 2 shows for nearly all possible sequences of our 491 spherical nuclei the quantity x(A) = A?)-A:‘, in which
Ar’
corresponding
is the value
calculated
experimental
quantity
with
(33)
,
our
&function
force,
and
(we exclude
sequences
that include
Agb
the
a magic
number). It is apparent that with our value of V, (-220 MeV), determined like the Skyrme parameters by fitting to the total masses, there is a tendency for the even-odd mass differences to be over-estimated for A >80, in which range the majority of nuclei
(especially
those
relevant
to the r-process)
lie; the opposite
is the case for
the relatively few nuclei with A<80. On the other hand, when V, was optimized by fitting to the even-odd mass differences, leading to a value of around -200 MeV, the fit to total masses was found to deteriorate considerably, there being a tendency to underbind. Thus, even though we fit to the total masses, this problem with the even-odd mass differences sets a limit on the possible precision. The reason why we cannot optimize simultaneously the fit to the total masses and the even-odd mass differences is that the effect of the pairing force is not confined to producing even-odd fluctuations in the total energy E. Rather, the mean value of the term EtCS(G) in eq. (22) about which E fluctuates is significantly different
from zero. Moreover,
this mean value is strongly
shell-dependent
and hence
J. M. Pearson et al. / Nuclear mass formula (iii)
21
Cd
E
A: PROTONS
2 Y
*: NEUTRONS
0.8
A -0.8
***
*
*
40
80
120
160
200
24C
MASSNUMBERA Fig. 2. Errors with which force SkSCl reproduces experimental fourth-order even-odd mass differences: see eq. (33).
cannot be compensated by any adjustment of the Skyrme parameters, (at least, as long as we maintain the constraint of M* = M). A possible way out of this difficulty now suggests itself. Lipkin-Noganii method. Since the wave function given by the standard HF-BCS method is not exact, particle number not being conserved, the method will underestimate the total binding energy for a pairing force of given strength. Some improvement in this respect might be expected if states with a well-defined number of particles were projected out of the BCS ground state, and we see the possibility of getting a better fit to the total masses with V,= -200MeV, the value which optimizes the . even-odd mass differences. In an attempt to do this, we replaced the standard BCS method with the method of Lipkin and Nogami: see ref. 43) and papers cited therein. This simple and well tested method approximates the projected wave function by requiring that (fi’) = (fi)*, where 2 is the number operator. We generalize the results of ref. 43) to the case of an arbitrary pairing force, and find that our eq. (18) for the total energy is replaced by E’=E-2hJv:u2,, (34) n
J&f. Pearsonef af. / Nuclear mass formula (III)
22
where (35) u’,=l-v”,. This indeed
improves
(36)
the fit for V, = -200
MeV, increasing
the binding
energy
for any given pairing force over what is given by the standard method, but we found that we could never improve the fit over the original one given by the standard method with V, = -220 MeV, as already quoted. The general tendency for these fits with the Lipkin-Nogami
method
is that nuclei
fit become more so, and likewise for nuclei we vary all the parameters of the force.
that were underbound that were overbound,
in the original no matter
how
It would seem that in the present context of fitting the pairing strength along with the Skyrme parameters to the total energy, the standard BCS treatment of the pairing is superior to the Lipkin-Nogami method. We suggest that the averaging over adjacent nuclei characteristic of the BCS method, usually regarded as a defect, is actually an advantage in the present context: even with a S-function force for the pairing, fluctuating errors are inevitable, and by smoothing these over several adjacent nuclei it should be possible to reduce x2. Comparison with “constant-G” approximation. We now consider to what extent it is advantageous to use the a-function pairing force rather than the “constant-G” approximation. To investigate this point, we took our values of Aji4’, calculated as described above for the different sequences, and fitted to each of them an equivalent constant G Gceff’ in the “constant-G” approxima0 , b Y repeating the BCS calculations tion. (Thisis not exactly the same quantity as the G eq,q defined for each nucleus in sect. 2 for interpolation purposes; it would not have been appropriate to compare this latter quantity with GyPfq, determined as described below.) We found GET’ to show a strong variation, which could be represented, as in ref. 42), by the functional form G::’
= g,( 1 ‘F g, I + g,12)/‘A,
(37)
where the upper sign refers to neutrons and the lower to protons. A best fit to our values of GE:) was obtained with the following parameter values: g,= 18.95 MeV, g, = 0.79, and g, = 3.4 (all obtained with the same cut-off of the continuum spectrum in the BCS calculation as for the S-function force). This variation is represented by the curves in fig. 3, where we show G,fe*)/( 1 F g,I + g,12) versus l/A, and in figs, 4 and 5 where we show G’,““A vs I. The points in these same figures correspond to the eiperimental values of the equivalent constant G, GE:), obtained in the same way as G’,“” by fitting to the values of A&. It will be seen that despite the large scatter, our s-function force with its single parameter V, has well represented the trends in the actual variation of Gy:. To get the same quality of fit with the “constant-G” model of the pairing would have required two more parameters, i.e., 11 rather than 9, so we are led to believe that
J.M. Pearson et al. / Nuclear mass formula (Ill)
23
0.7
- NEUTRONS
0.6
. . .
and PROTONS .
3 z
.
0.5 ,-
N CI 2 + H
0.4
IT .-I \ G % -u
0.3
0.2
0. I
0.0 0.0
0.02
0.01
0.03
l/A Fig. 3. Variation
the &function
of Gf”/(
pairing
1 T g, I + g,I*) as function
of l/A
force has greater underlying
for force SkSCl,
physical
confidence in the reliability of its extrapolation is enhanced Deformed nuclei. We considered using the “constant-G”
with experimental
reality,
points.
and our a priori
accordingly. approximation
just for
the deformed-nucleus calculations, taking for G the value G,, that is equivalent the sense of sect. 2) to the S-function force for the spherical configuration of nucleus in question. However, G,, is deformation-dependent, and neglecting fact can lead to errors of at least 0.7 MeV for masses, and probably much more fission barriers. 5. Construction
(in the this for
of mass table by interpolation
Even though the total energy suffers from shell-model fluctuations, the ETFSI method (and indeed any method based on the Strutinsky theorem) expresses it in terms of quantities that vary smoothly with respect to N, 2, and the deformation parameters c and h. This enables us to make extensive use of interpolation in constructing the mass table, with the complete deformed ETFSI calculation, described in sect. 2, being performed only for a restricted number of key nuclei (all taken to be even-even) and key deformations, as explained in more detail below.
24
J.M. Pearson et al. / Nuclear mass formula (IIIj
3o.c
NEUTRONS
a =’ 5 -#
20.0
15.0
ICI0 - 0.1
0.0
0. I
0.2
0.3
I Fig. 4. Variation of GsW) A as function of I for force SkSCl, with experimental points: neutrons.
Without this possibility the amount of computer time required to calculate the some 6000 nuclei lying between the drip lines, only a small fraction of which can be supposed a priori to be spherical, would have been prohibitive. The quantities that we have to interpolate as functions of PJ, 2, c, and h are the ETF macroscopic energy EETF, as given by eq. (4), the s.p. energies Ed, the smoothed sum of s.p. energies %elr, given by eq. (13), the rotational correction E,,,, given by eq. (26), and the pairing matrix elements G,,, as used in eq. (17b). Only the last two of these quantities require any special mention. Rotational correction. The prescription (27) for the moment of inertia has been checked only for even-even nuclei (see table 1). All our key nuclei are of this type, so that the interpolated values of E,,, will always correspond to even-even nuclei also. Now in table 10 we show results obtained for complete ETFSI calculations on a number of strongly deformed odd-A and odd-odd nuclei; $,, denotes the moment of inertia that we calculated by neglecting the last unpaired nucleons, at the determined equilibrium deformation. We found that we could follow the main experimental trends (as determined by the first excited state) by multiplying $,, by 1.2 for odd-A nuclei, and by 1.4 for odd-odd nuclei: this corresponds to dp,,,, in
J.M. Pearson et al. / Nuclear mass formula (III)
25
PROTONS 25.0
>
3 a .
1o.o /
-0.1
0.0
0.1
0.2
0.3
I Fig. 5. Variation of Gg’) A as function of I for force SkSCl, with experimental points: protons.
table 10. These results agree with the experimental systematics shown in fig. 4.12 of ref. “). It follows that the interpolated E,,, has to be divided by these same factors in the appropriate cases. In principle, one should apply a further correction factor to the interpolated values of Erotin order to take account of any characteristic even-odd effect in the spurious angular momentum that should appear in the numerator of eq. (26). However, there seems to be no way in which one could improve our estimate of E,,, in this respect without making detailed calculations for each individual nucleus. We face here a basic limitation of the ETFSI approach, but it should be realized that hitherto all mass formulas based on the drop(let) model have totally neglected the spurious rotational effects that exist in their shell corrections, absorbing them into the fit of the parameters to the data. One might expect that some idea of any systematic error introduced by our treatment of E,,, for odd nuclei could be obtained from table 10 by comparing the calculated and experimental energies shown there (the calculated values EGa,Chave E,,,treated in exactly the way described above, i.e., as in inte~olation). The agreement with experiment is decidedly worse than in the case of the even-even deformed nuclei (table 5), there being a net tendency to underbind more beyond
J.M. Pearson et al. / Nuclear
26
mass formuta (III)
TABLE 10 Sample of 17 odd-A and odd-odd deformed nuclei (axial and left-right symmetry). Moments of inertia _P in units of h* MeV-‘; energies E in MeV A 153 157 161 162 165 166 174 183 225 231 237 239 239 241 250 251 255
Z
9a,,
2 CXP
9 c0rr
64 63 65 67 66 67 71 74 90 90 93 93 94 95 97 100 100
27.9 36.8 35.7 34.4 37.1 37.1 40.0 33.1 52.4 55.5 64.2 64.7 64.5 66.0 63.5 65.2 63.0
60.2 45.5 44.4 52.2 47.4
33.5 44.2 42.8 48.2 44.5 51.9 56.0 39.7 62.9 66.6 89.9 77.6 77.4 79.2 88.9 78.2 75.6
44.7 32.2 86.2 83.0 105.0 112.0 85.0 87.2 110.0 55.0
E talc -1258.6 -1287.2 -1315.7 -1321.0 -1342.9 -1349.4 -1403.5 -1463.8 -1721.7 -1758.7 - 1793.6 -1805.3 -1804.7 -1816.2 -1866.2 -1868.4 -1893.2
E exP -1257.4 -1287.1 -1315.8 -1320.9 -1343.4 -1350.2 -1404.1 - 1465.1 -1722.7 -1759.6 -1794.6 -1807.0 -1806.2 -1817.2 -1868.2 -1870.8 -1895.3
A = 200. At least a large part of this effect must be due to the excessive pairing already noted in sect. 4, but there could also be a contribution coming from a systematic underestimate of E,, : it is very hard to disentangle the two effects. Pairing matrix elements. The pairing matrix elements G,, are so numerous that if we had attempted to interpolate them directly the storage problem, already difficult, would have become impossible. We therefore computed for each key nucleus at each key deformation an “equivalent constant-G”, Geq,q, defined separately for neutrons and protons as in sect. 2. It is these G,, that are interpolated, rather than nucleus at a given deformation is then the G,,; the energy of each interpolated determined by performing a constant-G BCS calculation with the interpolated s.p. energies and Geg. Since the constant-G
BCS calculations
are much more sensitive
than the b-force
BCS calculations to the number of s.p. states taken into account, this number, which we denote by V, must vary slowly with respect to the number of nucfeons, n (= iV or 2). The actual prescription that we adopted is u = n + 2.5r1~/~ , which corresponds roughly to all states lying up to lhw above the Fermi level. When a key nucleus is magic, for the deformation in question, the &pairing force has no or very little effect on the total energy, so that our prescription for determining G,, breaks down. We handle this problem quite simply by applying our prescription to the two neighbouring nuclei, which will certainly be non-magic (for the given deformation), and then interpolating the two values of G,, linearly.
J.M. Pearson et al. / Nuclear mass formula (Ill)
Interpolation procedure. We interpolate (see below complete points
for the actual ETFSI
methods
calculations
in the (iV, Z) plane,
over the whole prohibitive,
nuclear
as shown
in the (N, 2) and (c, h) planes
of interpolation).
are performed
chart
and the overall
separately
The key nuclei
are chosen
at once,
but the memory excessively
on which
to lie at regularly
in fig. 6. In principle,
organization
27
one could requirements
complex.
lated over separate square blocks of 16 ( = 4 x 4) key nuclei; in fig. 6. The entire nuclear chart is covered by such blocks,
spaced
interpolate
We therefore
would
be
interpo-
a typical block is seen which are all inclined
at 45” and allowed to overlap, as shown in fig. 7 (the key nuclei common to two overlapping blocks need only be calculated once, of course). In most of the square blocks the spacing of the points representing key nuclei is IANI = IAZl = 6, so that there will be a total of 685 ( = 19*+ 18’) nuclei altogether in each of these blocks. Thus interpolating over the (N, Z) plane leads to a time-saving factor of 685/ 16 = 42.8 (actually, slightly more because of the key nuclei lying on the common boundary between adjacent blocks). For the single square covering nuclei with A < 60 we take IANI = lAZ[ = 4, simply because of the convergence of the drip lines (stability problems in the ETF part of the calculation arise for nuclei too far beyond the drip lines). In this smaller square there will be 313 ( = 13*+ 12’) nuclei. As for interpolation in the (c, h) plane, we find that all ground-state deformations fall in the parallelogram shown in fig. 8. All of the key nuclei are calculated at each of 56 key deformations inside the parallelogram. Actually, if we define x = c, y = c + h, our 56 points correspond to 8 values of x and 7 values of y.
.
.
.
.
.
.
.
.
Fig. 6. Typical 4 x 4 interpolation block in (N, Z) plane.
J.M. Pearson et al. / Nuclear mass formula (III)
28
I20
100
80
Z 60
20
40
60
Fig. 7. Mapping
80
of nuclear
100
120
chart by 4 x 4 interpolation
h (0.75, 0.64)
(0.75,O.lO)
Fig. 8. Interpolation
block in plane
160
140
of deformations.
blocks.
180
J.M. Pearson et al. / Nuclear mass formula (III)
29
With all 16 key nuclei of a given block computed at the 56 key deformations, construction of the mass table proceeds as follows. For a given deformation we interpolate to each of the 685 (or 313) nuclei in the block. In this way we obtain complete results for every nucleus of the block at each of the 56 key deformations. Finally, to determine the mass and deformation of the ground state of any particular nucleus we interpolate that nucleus over the (c, h) plane to each of 625 (25 x 25) points inside the parallelogram of fig. 8, and pick out the energy minimum. Tests showed that this number of points in the (c, h) plane was appropriate to a determination of the energy minimum without significant loss of precision. Thus interpolation over the (c, h) plane leads to a further time-saving factor of 625/56 = 11.1. It follows that the joint interpolation over the (N, 2) and (c, h) planes leads to the computer time being reduced by a factor of about 400. Since the ETFSI method itself is some five times faster than HF for a given deformed nucleus (sect. 2), we see that the overall time-reduction factor compared to HF (for which interpolation is impossible) is about 2000. In searching for the energy minimum one has the option of including the rotational correction (26) before minimization or adding it afterwards. The former procedure is the more logical, and our mass table is indeed constructed in this way. Nevertheless, we also studied the second option; see sect. 6. Whichever way the minimum is defined, we also determine by interpolation the energy of the spherical configuration for each nucleus, thereby obtaining the deformation energy. We have seen (sect. 2) that the dimensionality of our deformed code is sufficient only for this latter quantity, and not the total energy. Thus our final mass values are obtained by adding the interpolated deformation energy to the total energy of the spherical configuration, as calculated by the spherical code. The actual method used for interpolation over the (c, h) plane is a simple two-dimensional application of the cubic spline (CERN routine). Suppose that we wish to interpolate to the point (X, Y). Then for each of the 7 key values of y, yi, we interpolate in the x-direction to the 7 points (X, yi). A final eighth interpolation in the y-direction is then made over these 7 points. The same procedure could have been adopted for the interpolation over the (iV, 2) plane also, but since each linear interpolation is over only 4 points a special feature emerges. Normally, the cubic-spline method imposes continuity of first and second derivatives of the cubic expressions at each interpolation point. But with only 4 points it is possible to require continuity of the third derivatives as well, which means simply that the four points are fitted with a single cubic expression. In other words, this generalized 4-point cubic spline reduces to Lagrange interpolation (see, for example, Scheid “)). The Lagrange method itself can easily be generalized to two dimensions, the interpolating polynomial being in general
p(x, Y)= f
k=l
i _f(xksfi)Lm,k(x)Ln,j(Y) j=l
3
J. M. Pearson et al. / Nuclear
30
where f(xk, vj) is the function one-dimensional
Lagrange &k(X)
to be interpolated,
mass formula
(III)
and we have introduced
the usual
polynomials =
l? i=l(#k)
(x-x,) ~ (Xk
k-l,2
-Xi)
,...,
m,
(36)
’
etc; in our case m = n = 4, of course. Some idea of the loss of accuracy incurred by our joint interpolation over the (c, h) and (N, Z) planes can be had by taking a number of nuclei, selected more or less at random, and recalculating the energy by a direct application of the ETFSI method at the equilibrium deformation, as given by interpolation. The results are shown in table 11, where Eint denotes the final interpolated energy, i.e., the energy TABLE Errors of interpolation;
A
Edi’
AE
11
see text for explanations
A
2
E’“’
Ed”
2
E’“’
36
15
-301.0
-301.2
-0.2
230
81
-1711.9
-1712.0
81
40
,678.7
-678.9
-0.2
230
82
-1722.4
-1722.3
100
38
-837.4
-837.4
231
90
-1758.6
-1758.7
0.0
AE -0.1 0.1 -0.1
106
44
-907.3
-906.8
0.5
232
90
-1765.5
-1765.6
-0.1
149
62
-1231.7
-1231.5
0.2
236
92
-1789.1
-1789.2
-0.1
150 150
58 62
-1230.1
-1230.2
237
87
-1778.9
-1779.1
-0.2
-1240.0
-1239.8
0.2
237
93
-1793.8
-1793.6
151
62
-1245.9
-1245.7
0.2
238
84
-1763.5
-1763.3
152
62
-1253.9
-1253.9
0.0
238
96
-1794.6
-1795.1
153
62
-1259.6
-1259.5
0.1
239
93
-1805.5
-1805.3
153
64
-1258.9
-1258.6
0.3
239
94
-1804.5
-1804.7
154
64
-1267.7
-1267.6
156
62
-1280.0
-1280.1
157
63
-1287.2
-1287.2
161
65
-1315.7
162
67
-1321.1
162
66
162
72
-0.1
0.2 0.2 -0.5 0.2 -0.2
240
82
-1750.0
-1750.0
240
94
-1812.0
-1812.0
0.0
240
95
-1808.7
-1808.8
-1315.7 -1321.0
0.0 0.1
240 240
98 99
- 1800.8
-1800.7
0.1
-1793.3
-1793.2
0.1
-1324.1
-1324.1
0.0
241
95
-1816.3
-1816.2
0.1
-1300.6
-1300.3
0.3
246
94
-1845.2
-1845.2
0.0
0.1 -0.1
0.0 0.0 -0.1
165
66
-1342.5
-1342.9
-0.4
246
97
-1843.2
-1842.7
166
67
-1349.2
-1349.4
-0.2
246
100
-1835.0
-1835.3
-0.3 -0.5
0.5
168
76
-1326.4
-1326.3
0.1
246
103
-1813.9
-1814.4
172
68
-1391.4
-1391.2
0.2
250
96
-1868.2
-1868.2
174
70
-1406.4
- 1406.2
0.2
250
97
- 1866.4
-1866.2
174
71
-1403.5 -1463.8
0.7 -0.1
250
-1862.9
-1863.0
251
100 100
-1868.4
-1868.4
0.1
252
98
-1879.3
-1879.1
252
99
-1876.1
-1876.2
-0.1 -0.1
183
74
-1404.2 -1463.7
184
74
-1471.7
-1471.6
194
76
-1536.7
-1537.1
-0.4
0.0 0.2 -0.1 0.0 0.2
204
81
-1606.8
-1606.8
0.0
255
100
-1893.1
-1893.2
204
87
-1575.4
-1575.4
0.0
256
100
-1900.1
-1900.2
-0.1
205
79
-1612.5
-1612.5
0.0
256
104
-1887.4
-1888.2
-0.8
214
74
-1601.5
-1601.1
0.4
258
103
-1903.8
- 1904.2
-0.4
214
82
-1663.8
-1663.8
0.0
260
106
-1906.4
-1906.6
-0.2
224
90
-1716.2
-1716.3
264
97
-1932.2
-1932.2
225
90
-1721.7
-1721.7
-0.1 0.0
0.0
31
J.M. Pearson et al. / Nuclear mass formula (III) TABLE
12
Analysis of interpolation errors; see text for explanations
106 165 174 194 238 246 246 256
44 66 71 76 96 97 103 104
corresponding
0.82 1.21 1.18 0.89 1.25 1.21 1.21 1.16
0.3 1 -0.01 0.08 0.11 -0.19 -0.10 -0.10 0.02
-907.3 -1342.5 -1404.2 -1536.7 - 1794.6 -1843.2 -1813.9 -1887.4
to the mass tables,
-906.8 -1342.9 -1403.5 -1537.1 -1795.1 -1842.7 -1814.4 -1888.2
0.22 -0.03 0.15 -0.04 -0.04 -0.10 -0.08 -0.06
and Edir the energy
-0.10 -0.04 -0.20 0.14 0.01 -0.21 -0.20 0.04
given directly
-0.20 0.36 -0.33 0.25 0.43 -0.40 0.57 0.70
0.5 -0.4 0.7 -0.4 -0.5 0.5 -0.5 -0.8
by the ETFSI
calculation. The total interpolation errors A, which show no systematic trend, and are equally likely to have either sign, are generally of the order of ho.2 to 0.3 MeV, which is quite acceptable. Occasionally, the error can amount to 0.5 MeV, and in the worst case in our sample it is 0.8 MeV, which could not be tolerated if it were not an isolated instance. In table 12 we consider those cases of table 11 where the total error of interpolation lAE[ z 0.5 MeV, and show how it breaks down into the following three contributions: (i) A&,, , the error in the interpolated equilibrium energy, uncorrected for the spurious rotational energy (26), (ii) A&,, the error in the interpolated rotational correction (26), (iii) A&,,, the error in the interpolated energy for the spherical configuration. It will be seen that the bulk of the interpolation error arises in the last term, i.e., for the spherical configuration, even though this configuration corresponds to a key point in the (c, h) plane, so that only the (N, 2) plane has to be interpolated over. By contrast, the first two contributions to the interpolation errors are always acceptably small, even though they are associated with arbitrary deformations,
and thus involve
interpolations
over both the (c, h) and (N, 2) planes.
The larger interpolation errors found occasionally for the spherical configuration might be due to the large degeneracy of several s.p. levels for this configuration, resulting in a more systematic accumulation of errors in the sum of s.p. energies than in the case of deformed configurations. This problem could be handled, without an appreciable increase in computation time, by taking a finer interpolation grid in the (N, 2) plane for the spherical configuration than for the 55 deformed configurations. 6. Mass table for the SkSCl force Using the interpolation method described in sect. 5, we calculate with the SkSCl force, derived as described in sect. 3, the masses and ground-state deformations of all the 1492 nuclei with A 3 36 whose masses are known. A graphical summary of
32
J.M. Pearson et al. / Nuclear mass formula
the discrepancy values mean -0.014
is given value
E = Me_, - Mcalcbetween our calculated in fig. 9. The rms value of this error
d = 0.049 MeV.
This
with N and 2 2 8 of known parameters,
as compared
masses and the experimental is a,,,,, = 0.868 MeV, and the
is to be compared
MeV in the case of the MMST
(III)
with
E,,,=
mass. We recall,
however,
to 9 in our work. Moreover,
fit to the 491 nuclei
assumed
E=
all 1593 nuclei
that MMST
use some 25
we show in this section
the next how it should be possible to improve our own results. It will be seen that the final results obtained with our force good as for the original
0.769 MeV,
mass table 5), which includes
SkSCl
to be spherical,
and
are not as
for which we
had E,,, = 0.693 MeV (sect. 3). However, the checks that we made in sect. 5 (see table 11) show that interpolation can account only for a very small part of this deterioration. A part of the problem must lie in the systematic underbinding that occurs with our force SkSCl for nuclei beyond A = 240, none of which were included in the original fit. This tendency has already been noted in sect. 3, and is quite apparent in fig. 9. If we exclude these nuclei a,,, falls to 0.810 MeV. We saw in sect. 3 that this underbinding of heavy nuclei must be related to the force and not to the dimensionality of our codes. Now that we have completed the mass table it is possible to make a more systematic study of this question. In fig. 10 we display the underbinding of these heavy nuclei in terms of both 2 and A, limiting ourselves, because of pairing effects, to even values of 2 and odd values of A. It will be seen that the effect we are dealing with is Z-related rather than A-related; 4
t
L
I
,,,,,,l,,,lll\ 20
40
60
I 80
I
I,,
I
I,,
100
11 120
I,
I
‘,
i40
”
I
J
160
N
1 II
III
I!
ll,l,,,,l,,,,lllL1l~
20
40
60
80
I
100
Z
Fig. 9. Error of fit (E = Me._ -MC,,,)
of force SkSCl to mass data as functions of N and Z.
J.M. Pearson et al. / Nuclear mass formula (Ill) 250
240
A
3 260
I
I
O 2=96 98
;
100
X
102
+
104
Fig. 10. Error of fit (E = MC_,- MC,,,) as function
of A for different
heavy elements.
in fact, the binding energy increases slightly for A increasing at constant 2. Thi suggests that the problem lies with the Coulomb energy, this varying roughly a Z2/A-1’3. We can now see that if we had chosen a value of k, slightly smaller than thl 1.335 fm-’ that we actually took, then this problem of underbinding in heavy nucle might have been avoided. In fact, with the constraint of a smaller value of k, i should be possible in a new fit to adjust the Skyrme parameters in such a way tha we not only achieve the required increase in the binding of heavy nuclei but at thl same time correct the trend towards overbinding of light nuclei that is clearl: apparent in fig. 9. It remains to be seen what this would do to the calculated nuclea radii, but in view of the approximate nature of the Fermi distribution (5) one shoulc not insist too strongly on this point. Reducing kF, and with it the Coulomb energy might also help to eliminate the anomalous trends in the fission barriers noted ii the preliminary analysis of sect. 3 [see also ref. 46) for further details]. Another major factor contributing to the deterioration of the fit in going fron the original group of 491 spherical nuclei to the complete mass table lies in thl excessive even-odd mass differences for heavy nuclei, noted in sect. 4; these nucle carry much less weight in the restricted group than in the complete table. It is also apparent from fig. 9 that shell effects are somewhat exaggerated. Taking M* to be a little larger than M might improve this situation. Another degree o freedom which has not been completely exploited and which might help to modulat shell effects is y (this acts through the surface diffusivity, as discussed in sect. 3) Nuclei with N = 2 (even) are often underbound by more than 2 MeV, even thougl they lie in a region of the nuclear chart where the tendency is for overbinding. Thi is a consequence of the Wigner effect, in the sense of quartet correlations that canno be incorporated within the framework of any independent-particle description, am
34
J.M. Pearson et al. / Nuclear mass formula (III)
which
our method
nuclei
that their
does not take into account. effect on E,,, is negligible,
However,
there are so few of these
but our mass formula
should
not be
used for these nuclei. It will be noticed
in fig. 9 that there is a group
W that are anomalously axial symmetry of these
nuclei,
underbound.
Since
we might have here an indication although
previously
of nuclei
in the region
our deformed
performed
code
of triaxiality triaxial
always
of OS and assumes
in the ground
HF calculations
state do not
suggest an effect on the binding energy as large as the one we find 47). S, and Qp. We show in fig. 11 the errors with which our force SkSCl reproduces the neutron-separation energies S, for the Sn, Pb, and Cm isotopes, and compare with the results of MMST. We see that for the Sn isotopes we do as well as, or slightly better than MMST, especially near the magic number N = 82. Actually, while force SkSCl gives a little too much pairing for all Sn isotopes, MMST has this problem only for N < 70, there actually being too little pairing for N > 70. In any case, the error for our force never exceeds 0.5 MeV, which is astrophysically acceptable. As for the Pb isotopes, we see that once again SkSCl has too much pairing for all isotopes, although the error still never exceeds 0.5 MeV. MMST would perform somewhat better, were it not for the serious error that arises close to the magic number
N = 126. However,
our results
for the Cm isotopes
are much worse
Fig. 1 I. Deviation MMST fit (dotted
from experimental neutron-separation energies S, for force SkSCl lines) [E(&) = S,(exp) - SJcalc)]. (a) Sn isotopes (b) Pb isotopes,
(solid lines) and (c) Cm isotopes.
J.M. Pearson et al. / Nuclear mass formufa (III)
35
than those of MMST, and in fact they would be of doubtful reliability in r-process calculations. In fig. 12 we show the errors with which force SkSCl reproduces the experimental beta-decay energies QB (always defined for /3- decay) for A = 154, 155, 234 and 235, and compare with the MMST fit. We do not do as well here as do MMST, particularly for the heavier nuclei, and our errors can be as large as 1 MeV. Again, our problem is that the pairing force is too strong. Actually, our results for the A = 235 chain suggest that it is the neutron pairing in particular that is too strong, and there is some indication of a similar difference between neutron and proton pairing in the A= 155 results as well (in the A = 235 case, at least, the effect is probably too large to be attributable to the moments of inertia). Both the S, and QP results show clearly that we have a growing problem of excess pairing as A increases. Even if this has little effect on the overall rms error of the mass table, it is undesirable in any tabie that is to be used for astrophysical
1
a’;
A=154
1
(MeV) -1
A= 155
A=XE
Fig. 12. Deviatim frOM experimentai beta-decay energies Q@(always defined for @- decay) far force SkSCt (solid lines) and MMST fit (dotted lines) [e( Qs) = Q@fexp) - Qrs(calc)]. (a) A = 154, (b) A = 155, (c) A = 234, (d) A = 235.
36
J.M. Pearson et al. / Nuclear mass formula
(III)
applications, for example. While the need for a smaller value of V, is clearly indicated, we recall that this will lead to a deterioration in the overall fit to the masses (sect. 4), even if the Lipkin-Nogami method is applied. It was always clear that some compromise was necessary, and it now seems that the one we have made here may not be the most favourable, although this only becomes apparent after constructing the entire mass table. One way in which the conflict might be resolved without any such compromise would be to take M” to be slightly bigger than M. But even if this helped us to eliminate the excess pairing in heavy nuclei without reducing the overall precision of the fit to the masses, it would make still worse the problem of too little pairing in nuclei with A < 60 (see fig. 2). Reducing the pairing in heavy nuclei and simultaneously increasing it in light nuclei might be achieved by allowing it to act preferentially in the surface through an appropriate density dependence. Defirmntion configuration. In calculating the masses, the equilibrium deformation of each nucleus is automatically determined. However, this is not a quantity of primary interest in the present work, and we therefore limit our discussion of this aspect to the question of whether or not we predict the correct position in the (N, 2) plane for the transitions from spherical to deformed shapes. In figs. 13-17 we compare for a few isotopic series, selected with this end in view, our calculated ground-state deformation parameters & [see appendix A for the relation to the (c, h) parameters] with the experimental values, as determined from the B(E2) data 48). Note that these data determine only /p 2,/ so that when the calculated value of & is negative we show i/321 as well. In order to facilitate the comparison with experiment we connect the points corresponding to the calculated values of l&j by a solid line; negative calculated values of & are connected by a dotted line. (We
PP
Fig. 13. Expe~mental
and calculated
(SkSCt)
values of & for Ra isotopes.
37
J. M. Pearson et al. / Nuclear mass formula (III)
, I I 1
I I
180
t I
190
?OO :
A
-0.5 Fig. 14. Experimental and calculated (SkSCl) values of & for pt isotopes (dotted Lines correspond to negative calculated values: see text).
-130
140
150
160
Fig. 15. Experimental and calculated (SkSCl) values of & for Sm isotopes.
J.M. Pearson et al. / Nuclear mass formula (III)
38 0.5 r
54X=
P2
0 110
120
Fig. 16. Experimental
and calculated
130
(SkSCl)
values
140
of & for Xe isotopes.
r
r
Fig. 17. Experimental
SkSCl
and calculated (SkSCl) values of & for Zr isotopes negative calculated values: see text).
(dotted
lines correspond
to
J.M. Pearson et al. / Nuclear mass formula (III)
do not consider
here the often tenuous
since it is irrelevant The overall when
we bear
dynamical tions.
to the question
agreement in mind
deformations,
While
between
experimental
of demarcating
while our calculated
both quantities
dynamical shape fluctuations a more significant dynamical
indications the regions
theory and experiment
that the experimental should
39
values values
on the sign of p2 of spherical
is rather
nuclei.)
good, especially
of p2 include
the effect of
relate only to static deforma-
agree for well deformed
nuclei,
in which the
remain quite small compared to the static deformation, contribution to p2 occurs in nuclei that are spherical
or only weakly deformed, with the result that the measured values of p2 in these nuclei are expected to be larger than the values calculated in static theories such as the present. A particularly interesting case is the shape transition in the heavy 4,,Zr isotopes. The B(E2) data show that while the N = 56 isotope is close to spherical, a strong deformation has set in at A = 100. (The experimental excitation spectra indicate that the transition is even sharper: while isotope N = 58 is close to spherical, isotopes with N 2 59 are strongly deformed.) We show our results for force SkSCl in fig. 17, while figs. 18 and 19, respectively, show the results for HF calculations with the S3 parametrization of the Skyrme force 49), and for the MMST calculation “). It will be seen that we reproduce this situation better than do either of the latter two calculations, which both predict a rather smooth transition beginning already at N = 54. We believe that the problem with the S3 force is a consequence of a wrong level ordering “) arising from an excessively high value of y, on which the surface diffusivity, and hence the s.p. potential, depend, as explained in sect. 3. As for the MMST calculation, we speculate that here too the s.p. potential is wrong, for the simple reason that the adopted parametrization (folded Yukawa) does not permit the diffuseness to vary off the stability line; such limitations are automatically avoided in the HF method, and also in the ETFSI approximation Of course, our results of fig. 17 have been obtained by including correction (26) before minimization, as throughout our calculations. interesting to note that we obtain a still better description of the from spherical to deformed shapes in the heavy 4oZr isotopes if correction after minimization: see fig. 20 (we denote this case
thereto. the rotational However, it is sharp transition we include this by SkSCl*). In
particular, while the N = 96-98 isotopes are oblate in the former calculation they are spherical in the latter. Actually, we must conclude that we are unable to obtain an unambiguous description of this shape transition because spherical-oblate energy differences smaller than 0.5 MeV, as in 96-98Zr, cannot be considered as significant in any model (droplet, ETFSI, or HF) that does not include a treatment of zero-point motions. At the same time, it will be seen by comparing figs. 17 and 20 that the shape transition in the light Zr isotopes is much better described by the usual prescription of including the rotational correction (26) before minimization. ExtrupoZution. Despite the success with which our force follows the experimental deformation trends, we have shown that the fit to the masses can be improved. We
40
J. M. Pearson et al. / Nuclear mass formula (III)
-0.5
L
Fig. 18. Experimental and caicuiated (force S3) values of & for Zr isotopes (dotted Iines correspond to negative calculated values: see text).
shall, therefore, not extrapolate the present mass table out to the drip lines, but rather await the improvements that will be relatively easy to make, and which are, in fact, already underway. Nevertheless, before embarking on still more computation, it is worthwhile to compare some of the predictions of our mass formula with those of the best existing microscopic-macroscopic formula, that of MMST ‘), in order to see whether any differences can be expected. The extension of the latter table out to the drip lines has not yet been published, but Dr. Peter MSller has kindly sent us some of his preliminary results. We consider here just the case of the Pb isotopes, plotting in fig. 21 the difference between the MMST results and the ETFSI results for force SkSCl. As we go beyond the heaviest known isotope we see that we begin to underbind significantly compared to MMST. This can easily be understood from eqs. (29) and (30) in terms of our value of ass = -18.4 MeV, as compared to -80.8 MeV in the case of MMST: the coefficient of I2 in eq. (29) is larger in our case, even though f is smaller. But beyond N = 170 the MMST energy climbs much more rapidly than ours, until at the magic number iV = 184 we have some 6.0 MeV more binding than MMST, a clear indication of the expected malacodermous effects 8), as discussed in sect. 3. One may expect,
J.M. Pearson et al. / Nuclear mass formula (III) 0.5
P*
41
r
O
-0.5
Fig. 19. Experimental and calculated (MMST) values of pz for Zr isotopes (dotted lines correspond to negative calculated values: see text).
may expect, therefore, that ETFSI will predict that the n-drip line lies further out than will MMST. Nevertheless, although these differences between the predictions of MMST and SkSCl are significant, we note that there are still wider differences between MMST and earlier versions of the droplet model 51,52). We have also compared the extrapolation of force SkSCl with that of SkSC2. Since the models are identical here what is being tested is the importance of the quality of fit to the data, which is slightly worse for the latter force on account of the different symmetry properties. Limiting ourselves to the Pb isotopes, calculated in the spherical approximation, we show the difference by the dotted line in fig. 21, and see that it never exceeds 2 MeV. This shows that the choice of the underlying model can be more important than the precision of the fit. Comparison with other Skyrmeforces. Although the force SkSCl presented here is not to be considered as our final force, it is interesting to compare it with other Skyrme-type forces currently available. Perhaps the best known of these are S3 [ref. ‘“)I and SkM* [refs. 9,53);in this latter paper the force is called “SkM modified”], and it is interesting to compare these forces with our SkSCl. For doubly-magic nuclei close to stability both these forces give fits to the masses
J.M. Pearson et al. / Nuclear mass formula (III)
42
-0.5
1
Fig. 20. Experimental and calculated {SkSCI*) values of & for Zr isotopes (dotted lines correspond to negative calculated values: see text).
comparable to our own, but the published results show that S3 works much less well than our force for open-shell nuclei *‘). This is probably because for S3 the effective mass (M* = 0.76M at nuclear-matter densities) has been chosen to fit the deep-lying s.p. levels, while our value of M* = M fits rather the levels close to the Fermi surface. (53 would give improved masses if particle-vibration coupling were taken into account “).) For similar reasons SkM* (M* = 0.79M at nuclear-matter densities) is not expected to give very good masses for open-shell nuclei, although nothing has been published on this point. In any case, SkM* deteriorates badly away from the stability line, even for closed-shell nuclei. Another respect in which our force is found to perform better than 53 concerns the incompressibility K, which at 350 MeV is excessively high in the case of S3. As discussed in sect. 3, this is linked with the surface diffusivity, and hence will have an impact on the shell structure. We have seen above that this might explain why we have a better description of the onset of deformation in the heavy Zr isotopes. As for fission barriers, 53 is known to perform very badly 54), while semi-classical calculations show that SkM* is much more promising in this respect, at least close
J.M. Pearson et al. / Nuclear mass formula (III) -
MWMSl’I-M6kSCl)
------
M 6kSC2) - M (SkSCI)
43
Rangeoi MassData
&
(MN
-
-5 -
Fig. 21. Comparison of extrapolations for Pb isotopes.
to the stability line 9*53).H owever, the necessary shell-model corrections to these semi-classical calculations (or complete HF calculations) for the SkM* fission barriers have not yet been performed, and the value of M* leads one to expect that the final results might not be very satisfactory 13). In any case, we have already noted that SkM* is not reliable for nuclei far from stability. We conclude that for the purpose of calculating masses, fission barriers, and, in general, potential-energy surfaces, especially in the astrophysically crucial regions far from stability, our force SkSCl is superior to any other Skyrme force presently available. However, we have in no sense produced a universal Skyrme-type force: its effective mass of M* = M makes it quite unsuitable for calculating giant resonances, for example. In comparing our force with other Skyrme-type forces it should, of course, be realized that because of the slight differences between the HF and ETFSI approaches ‘,*) the force SkSCl should only be used with an ETFSI code, not a HF code. It would, however, be quite easy to modify SkSCl in such a way as to obtain comparable results in HF calculations.
7. Concluding remarks In the present work we have presented a preliminary version of a complete mass table based on the ETFSI method. Although strictly speaking a microscopicmacroscopic method, the underlying Skyrme-type force establishes a much greater
44
J.M. Pearson et al. / Nuclear mass formula
(III)
level of unity between the two parts than is usual. In fact, we have shown lv2)that
the method is essentially equivalent to the HF method, in the sense that for a given from the data out to the drip lines, and to strongly deformed shapes. Nevertheless, we have seen how with a given force a mass table can be constructed some 2000 times more rapidly with the ETFSI method than with the HF method, The present mass table is based on a set of force parameters, SkSCl, that were fitted to a restricted sample of just 491 spherical nuclei. With this force we calculated the masses of all 1492 nuclei with A236 for which mass data are available. The rms discrepancy between our results and experiment is E,,,,~= 0.868 MeV, which is to be compared with E,,~= 0.769 MeV in the case of the MMST mass table 5), the most refined droplet-model formula available. Since we use only 9 parameters, as compared to some 25 in the case of the MMST table, one would in principle have greater confidence in extrapolations based on our mass formula. However, we note the slightly worse value of E,,, in our case, and recall also the problem with the neutron separation energies, S,, and beta-decay energies, Qp. Although our errors in these latter quantities may have relatively little impact on the overall mass fit, they are significant in the astrophysical context. Nevertheless, we have shown in sect. 6 that even though our force SkSCl has only 9 parameters we did not exhaust all possible degrees of freedom in constructing it, and that it should thus be possible to improve the force substantially. It is for this reason that we regard the present mass table as preliminary, and refrain from extending our mass table out to the drip lines until this improvement has been achieved. The degrees of freedom likely to permit an improved fit to the data correspond to kr, y, M”, and the pairing. Only one other parameter is still available, x,, but our experience with the SkSC3 force (sect. 3) shows that this is almost totally ineffective. We point out now that much improvement of the force should be possible without repeating all the vast amount of computation that has gone into the present work. In particular, refitting the Skyrme parameters with the small changes in kF that we contemplate is unlikely to change appreciably either the deformation parameters at equilibrium or the associated deformation energy. Thus, provided M*, y, and the pairing remain unchanged, we can use the deformation energies calculated here with force SkSCl to renormalize all measured masses to their “equivalent sphericalconfiguration” values. Subsequent fits of the force can then be made to all nuclei, spherical and deformed, while always assuming a spherical configuration, thereby keeping the computer time within the limits of feasibility*. As for reducing V,,, which will have to be done if acceptable S, and QP are to form of Skyrme force the two methods give very similar extrapolations
* Note added in proofi This refit of the force to ail 1492 masses, with unchanged M*, y, and pairing, has now been performed: E,_, s_ falls to 0.729 MeV, with the optimal value of k, remaining (somewhat unexpectedly) unchanged at 1.335 fm-‘, while I becomes 27.0 MeV.
J.M. Pearson et al. / Nuclear mass formula (Ill)
45
be obtained, it is unlikely that much additional work will be necessary. New deformation energies may be obtained with the already interpolated s.p. energies, using new values of Geq,qadjusted according to the modification of V,, but no other new calculations on the key nuclei will be necessary. Refitting the Skyrme parameters will then proceed as described above. To conclude, we have shown in the present paper that the ETFSI method provides a feasible approach to a mass table having an accuracy comparable to, if not better than, the best currently available mass formulas based on the droplet model. The fact that this can be done with far fewer parameters means that ETFSI mass formulas will provide more reliable extrapolation, both out to the drip lines, and to extremely deformed shapes. Even though our present best force, SkSCl, is not fully acceptable, we have shown how it might be improved. We stress that while there were clear indications at the outset that there might be defects in this force, improvement could not have been made without first calculating the present mass table with it. We wish to thank P. MGller for several helpful communications, and P. Haustein for sending us the 1988 mass compilation prior to publication. The continuing interest of M. Brack is acknowledged. We are indebted to B. Jennings for first drawing our attention to the potentialities of the SI method. The extremely generous allocation of computer time by the Centre de Calcul at the Universite de Montreal is gratefully acknowledged, and we would like to express our deep appreciation for the invaluable service rendered by Jean-Pierre Ponnau and Daniel Raymond. Appendix RELATION
BETWEEN
(c, h) AND
(p2, p4) PARAMETRIZATION
The reference surface used in our parametrization of the deformed density distributions with axial and left-right symmetry (see sect. 2 and refs. ‘“)) is defined in terms of the parameters (c, h) as follows. Working in cylindrical coordinates (17,z), we have
n’~r’-z2=(c’R’-z’)(A+B~), B>O
(A.la)
~2=r2-z2=A(c2R2-z2)exp
(A.lb)
or B
(note the error in the second line of eq. (3.2) of ref. ‘)). Here A and B are defined in terms of c and h according to B=2h+;(c-1),
A=c-~-+B
(A-2)
while R is just the radius of the reference surface in the spherical limit, defined in eq. (3.6) of ref. ‘).
J.M. Pearson et al. / Nuclear mass formula (III)
46
With z = r cos 8, we can rewrite which can be expanded
eq. (37) for the surface
in spherical
r=R{l+&Y,O(cos
harmonics
according
e)+p,Y,“(cos
!I)+.
in polar
form,
r = r(e),
to * *},
64.3)
where p,+
7r I( 19)Y,O(cos 0) sin I0
8 de.
(A-4)
Computation of the multipole coefficients &, is now straightforward (for the case B < 0 we must make the usual assumption that Ip) < l), and we have constructed a table for p2 and p4 in terms of c and h over a sufficiently fine mesh. Graphical representations of this transformation have already been given in ref. 14) (Fig. VS), and ref. 55) but neither is easy to use. Moreover, ref. 14) has a limited range of application (c > l), and we find occasional small disagreements with our own table, no doubt because of the different method used in that work (least-squares fit of the multipole coefficients). On the other hand, we agree completely with ref. 55). References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28)
A.K. Dutta, J.-P. Arcoragi, J.M. Pearson, R. Behrman and F. Tondeur, Nucl. Phvs. A458 (1986) 77 F. Tondeur, A.K. Dutta,J.M. Pearson and R. Behrman, Nucl. Phys. A470 (1987), 93 B. Grammaticos and A. Voros, Ann of Phys. 123 (1979) 359 B. Grammaticos and A. Voros, Ann of Phys. 129 (1980) 153 P. Moller, W.D. Myers, W.J. Swiatecki and J. Treiner, At. Data Nucl. Data Tables 39 (1988), 225 P. Miiller and J.R. Nix, Nucl. Phys. A361 (1981) 117 W.D. Myers and W.J. Swiatecki, Ann. of Phys. 55 (1969), 395 A.K. Dutta, J.-P. Arcoragi, J.M. Pearson, R. Behrman, and M. Farine, Nucl. Phys. A454 (1986), 374 M. Brack, C. Guet and H.-B. Hakansson, Phys. Reports 123 (1985) 275 M. Barranco and J. Treiner, Nucl. Phys. A351 (1981) 269 V. Bernard and Nguyen Van Giai, Nucl. Phys. A348 (1980) 75 F. Tondeur, M. Brack, M. Farine and J.M. Pearson, Nucl. Phys. A420 (1984) 297 F. Tondeur, Nucl. Phys. A442 (1985) 460 M. Brack, J. Damgaard, AS. Jensen, H.C. Pauli, V.M. Strutinsky and C.Y. Wong, Rev. Mod. Phys. 44 (1972) 320 M. Brack and P. Quentin, Nucl. Phys. A361 (1981) 35 F. Ivanjuk and V.M. Strutinsky, Z. Phys. A286 (1978) 291 F. Tondeur, J.-P. Arcoragi and J.M. Pearson, J. de Phys. 45 (1984) C6-125 Y.H. Chu, B.K. Jennings and M. Brack, Phys. Lett. B68 (1977) 407 M.N. Butler, D.W.L. Sprung and J. Martorell, Nucl. Phys. A422 (1984) 157 M. Beiner, H. Flocard, Nguyen Van Giai and P. Quentin, Nucl. Phys. A238 (1975) 29 F. Tondeur, Nucl. Phys. A315 (1979) 353 J.M. Eisenberg and W. Greiner, Microscopic theory of the nucleus (North-Holland, 1972) pp. 332-336 C. Ko, H.C. Pauli, M. Brack and G.E. Brown, Nucl. Phys. A236 (1974) 269 D.R. Inglis, Phys. Rev. 96 (1954) 1059; ST. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31 (1959) No. 11 A.H. Wapstra, G. Audi and R. Hoekstra, At. Data Nucl. Data Tables 39 (1988) 281 F. Tondeur, Nucl. Phys. A303 (1978) 185 J.M. Cavedon, B. Frois, D. Goutte, M. Huet, P. Leconte, X.H. Phan, S.K. Platchkov, C.N. Papanicolas, S.E. Williamson, W. Boeglin, I. Sick and J. Heisenberg, Phys. Rev. Lett. 58 (1987) 195 P. Gleissl, M. Brack, J. Meyer and P. Quentin, Ann. Phys. 197 (1990) 205
J. M. Pearson et al. / Nuclear mass formula (Ill) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48) 49) 50) 51) 52) 53) 54) 55) 56) 57) 58) 59)
l
47
M.M. Sharma, W. Stocker, P. Gleissl and M. Brack, Nucl. Phys. A504 (1989) 337 F. Tondeur, D. Berdichevsky and M. Farine, Z. Phys. A325 (1986) 405 H. Miither, M. Prakash and T.L. Ainsworth, Phys. Lett. B199 (1987) 469 B. ter Haar and R. Malfliet, Phys. Rev. Lett. 59 (1987) 1652 P.J. Siemens, Nucl. Phys. A141 (1970) 225 H.A. Bethe, G.E. Brown, J. Applegate and J.M. Lattimer, Nucl. Phys. A324 (1979) 487 B. Friedman and V.R. Pandharipande, Nucl. Phys. A361 (1981) 502 J. Cugnon, P. Deneye and A. Lejeune, Z. Phys. A328 (1987) 409 Nguyen Van Giai and H. Sagawa, Phys. Lett. B106 (1981) 379 M. Farine, J.M. Pearson and B. Rouben, Nucl. Phys. A384 (1978) 317 M. Farine, J. C&e and J.M. Pearson, Phys. Rev. C24 (1981) 303 W. Stocker, J. Bartel, J.R. Nix and A.J. Sierk, Nucl. Phys. A489 (1988) 252 W.D. Myers and W.J. Swiatecki, Ark. Fys. 36 (1967), 343 D.G. Madland and J.R. Nix, Nucl. Phys. A476 (1988) 1 H.C. Pradhan, Y. Nogami and J. Law, Nucl. Phys. A201 (1973) 357 A. Bohr and B.R. Mottelson, Nuclear Structure, vol. 2 (Benjamin, New York, 1975) F. Scheid, Schaum’s Outline of Theory and Problems of Numerical Analysis (McGraw-Hill, New York, 1968) p. 53 F. Tondeur, Y. Aboussir, A.K. Dutta, and J.M. Pearson, Proc. of AMCO 8 Conf. (1990, Jerusalem)* N. Redon, J. Meyer, M. Meyer, P. Quentin, M.S. Weiss, P. Bonche, H. Flocard and P.-H. Heenan, Phys. Lett. B181 (1986) 223 S. Raman, C.H. Malarkey, W.T. Milner, C.W. Nestor Jr. and P.H. Stelson, At. Data Nucl. Data Tables 36 (1987) 1 P. Bonche, H. Flocard, P.H. Heenan, S.J. Krieger and M.S. Weiss, Nucl. Phys. A443 (1985) 39 F. Tondeur, Nucl. Phys. A359 (1981) 278 L. Spanier and S.A.E. Johansson, At. Data Nucl. Data Tables 39 (1988) 259 W.D. Myers, At. Data Nucl. Data Tables 17 (1976) 411 J. Bartel, P. Quentin, M. Brack, C. Guet and H.-B. Hakansson, Nucl. Phys. A386 (1982), 79 H. Flocard, P. Quentin, D. Vautherin, M. VtnCroni and A.K. Kerman, Nucl. Phys. A231 (1974) 176 U. G&z, H.C. Pauli, K. Alder and K. Junker, Nucl. Phys. Al92 (1972) 1 H. De Vries, C.W. De Jager and C. De Vries, At. Data Nucl. Data Tables 36 (1987), 495 L.G. Moretto, S.G. Thompson, J. Routi and R.C. Gatti, Phys. Lett. B38 (1972) 471 A. Khodai-Joopari, Rep. UCRL-16489 (1966), Univ. of California Radiation Laboratory, Berkeley (unpublished) B.B. Back, 0. Hansen, H.C. Britt and J.D. Garrett, Phys. Rev. C9 (1974) 1924
Note added in proof: Conference
cancelled.
THOMAS-FERMI
APPROACH TO NUCLEAR MASS FORMULA
(III). Force fitting and construction of mass table* J.M. PEARSON, Y. ABOUSSIR, A.K. DU’ITA’, R.C. NAYAK’ and M. FARINE3 Luboratoire de Physique Nucl&aire, Dkpartement de Physique, Unioersite’de Mm&al, MontrL;af,Quc’, H3C 337 Canada F. TONDEUR
fnstitut d’Ast~nom~e et Astrophysiq#e, Universite’Libre de Bruxelles, and Znstitut Sup&w
Zndustriel de Bruxelles, Belgium
Received 3 September 1990 (Revised 29 November 1990) Abstract: The ETFSI method, developed
in two earlier papers, is here used to construct a complete mass table. Since the method allows for interpolation both in the (N, Z) plane and with respect to deformations, without losing the characteristic shell-model fluctuations, it is some 2000 times faster than the HF-BCS method for a given force. The present table is calculated using a preliminary Skyrme-type force with &function pairing, fitted to a restricted data set of 491 spherical nuclei. The resulting rms error for all 1492 measured nuclei, spherical and deformed, with Aa36 is erms= 0.868 MeV, achieved with just 9 parameters. The main experimental trends in ground-state deformations are well followed. The symmetry coefficient of nuclear matter corresponding to our force is 27.5 MeV. Ways of rapidly improving the frt are indicated.
1. Introduction In two earlier papers I,‘), hereafter referred to as (I) and (II), we have developed an approach to the nuclear mass formula that is based on the extended ThomasFermi (ETF) approximation 3*4)to the Hartree-Fock (HF) method for a Skyrme-type force, with shell corrections and BCS pairing added. In the present paper we (i) describe a preliminary fit of such a force to a restricted set of mass data, (ii) show how our method lends itself to interpolation both in the (N, 2) plane and with respect to defo~ation, an essential feature without which the computation time would be impra~ically high, (iii) use this interpolation procedure to construct a mass table for our preliminary force, (iv) indicate how this preliminary mass table can be improved. Much of the motivation for our work lies in the fact that the late stages of stellar nucleosynthesis, particularly the so-called r-process, depend crucially on the binding * Supported by NSERC (Canada) and FCAR (Quebec). ’ Present address: School of Physics, Devi Ahilya University, Indore, 452001 India. ’ On leave from: Dept. of Physics, Khallikote College, Berhampur, 760001 India. 3 Present address: Laboratoire de Spectroscopic Nuc%aire, Universite de Nantes, 44072 Nantes, France. 0375-9474/91/$03.50 @J 1991 - Elsevier Science Publishers B.V. (Noah-HoIland) June 1991
J.M. Pearson et al. / Nuclear mass formula (III)
2
energies
and fission barriers
of heavy nuclei
line that there is no possibility thus becomes of masses stability direction:
of the greatest
and fission barriers line, out towards extension
of being importance
lying so close to the neutron
able to measure
to be able to make reliable
away from the known
the n-drip
of the stability
(n)-drip
them in the laboratory.
region,
It
extrapolations
relatively
line. Also of interest
is extrapolation
line to investigate
the possible
close to the in another existence
of
superheavy nuclei. Furthermore, since our method involves a complete exploration of potential-energy surfaces we are also motivated by the possibility of extrapolating to highly deformed shapes, particularly beyond the saddle point. Any mass formula will fit all the available data if it has enough free parameters, but, if one is to have any confidence
in its ability to extrapolate
reliably,
the formula
must not only give a good fit to the data but also have a sound theoretical base. Clearly, if two different mass formulas give comparable fits to the data but extrapolate differently, the one with the better theoretical foundation would be preferred. Generally speaking, the theoretically superior of the two would be characterized by a smaller number of parameters. Now while the drop(let)-model (DM) mass formulas can give acceptable fits to the data, they do so only with a relatively large number of parameters. For example, the most recent and most sophisticated version, the so-called finite-range droplet model of Moller et al. ‘) (MMST), has rms errors of 0.769 MeV for 1593 masses, but requires some 25 parameters, as compared to 9 in the present work. [Both this version and the earlier finite-range drop model of Moller and Nix “) (MN) go beyond the usual leptodermous expansion ‘) in that some attempt is made to represent the diffusivity of the nuclear surface.] At the same time, it is open to theoretical criticism on two different counts: (a) The expansion in powers
of I = (N-Z)/A
appears
to be truncated
pre-
maturely with the result that when this model is fitted to the data, errors of as large as 15 MeV could be expected out at the n-drip line “). (b) It is difficult to relate the macroscopic and microscopic parts, i.e., DM and shell-model parts, coherently. To be specific, corrections involves the use of a single-particle be generated by the distribution of nucleons
the calculation (s.p.) potential, in the nucleus
of the microscopic and since this must it constitutes a link
between the microscopic and macroscopic parts of the mass formula. Now the actual way in which the s.p. potential is generated from the nucleon distribution is by folding some two-body force over the latter, but in no form of the drop(let) model is there an unambiguous prescription for choosing this force. (Actually, in both the MMST ‘) and MN “) versions the well depth is parametrized independently of the two-body force; which serves only to determine the form of the s.p. potential.) Furthermore, the density distribution itself is determined only in a very crude way in the drop(let) model. Both these classes of difficulty are avoided when the binding energy is calculated by the HF method (or rather by the HF+ BCS method, since pairing always has to
J. M. Pearson et al. / Nuclear mass formula (Ill)
be taken
into
account).
In the first place
power-series
expansion.
macroscopic
and microscopic
automatically therefore, method
guaranteed.
Secondly,
is no approximation
is no separation
so complete
A mass formula
be more theoretically represents,
there
parts,
there
based
3
on a
of the total energy
consistency
between
on the HF+BCS
secure than one based
based
on a drop(let)
in fact, the most fundamental
approach
that has any chance of succeeding, even though approach based on the “real” nuclear forces.
it is much
into
the two is method
will,
model.
This
to the mass formula less rigorous
than an
The ideal procedure to be followed with this method would consist in taking some suitable form of effective interaction and fitting its parameters, along with those of the pairing force, to all the data on masses, fission barriers, radii, etc., as in the published
DM fits. Unfortunately,
the method
suffers
from the defect
of
requiring a very large amount of computer time, especially for deformed nuclei, with the result that its systematic application has been somewhat limited. In particular, the available HF effective forces [see ref. ‘) for a convenient summary] have been fitted to relatively few of the available data, and even then not always very well, thereby detracting from the reliability of the method as a means of extrapolating. The present method (ETFSI) has been shown in papers (I) and (II) to be essentially equivalent to HF in the sense that when the underlying Skyrme-type force is fitted to the data the extrapolations out to the n-drip line are very close to those given by the HF method. Nevertheless, the method is computationally so much more rapid than HF that it offers a practical approach to the ultimate problem of constructing a mass table. We give a brief resume of the method in sect. 2. In sect. 3 we present a preliminary force determined by a fit to spherical nuclei alone, and then show that it performs acceptably well for a few selected deformed nuclei. In sect. 4 we discuss the properties of the a-function pairing force that we adopt in place of the more usual “constant-G” model. Sect. 5 describes the interpolation procedure, which is then used in sect. 6 to calculate with our preliminary force the masses of all known nuclei, spherical and deformed, with A 2 36. While the agreement of this preliminary mass table with experiment will be seen in general to be good, we show in sect. 7 that improvement will be relatively easy. 2. The ETFSI method We summarize here the main points of our method, referring the reader to papers (I) and (II) for full details. The basis of the method is the generalized Skyrme force Uij=to(l+XoP,)S(rij)+t,(l+x,P,)[p:S(rij)+h.a.]/2h*) +
t2(1
+x2pm)Pij
’ 6(rij)Pijlh2
(1)
4
J. M. Pearson et al. / Nuclear mass formula (Ill)
[see paper (I) for a discussion of the form we have adopted for the density dependence; the index q denotes n or p, according to whether the term in question relates to neutrons or protons, respectively]. To this we add the constraints x2 = -(4+5x,)/(5+4x,)
)
t,= +,(5+4x,),
(2)
in order for the effective nucleon mass to be equal to the real mass, M;=M,
(3)
a choice which allows the density of s.p. energies near the Fermi surface to be well reproduced lo) without having to take particle-vibration coupling into account I’). Fitting the density of s.p. energies near the Fermi surface helps to obtain correct masses 12), and has also been shown to be required for a good description of fission barriers I”) For the ETF semi-classical approximation to the kinetic-energy and spin-current densities, r4 and Jq, respectively, we use the full fourth-order (in powers of A) expansions of Grammaticos and Voros 3*4).The energy density 8(r), which gives the total energy as E ETF
=
J
8(r) d3r,
(4)
becomes a function of the nucleon densities, pq, and their gradients, the full expressions being given in papers (I) and (II). (Although those papers work in terms of t2 and x2, rather than t, and x,, in the present paper we discuss our fits in terms of the latter pair, since they are physically more significant. To simplify matters we show both pairs in table 2.) Note that we do not make the usual approximation of neglecting the terms in Jz and J2. Although the error made in EET, on neglecting these terms would be absorbed to some extent by the renormalization of the force on fitting to the data, retaining these terms might help to reproduce some data better “), in addition to the possibility of a more reliable extrapolation from the data, because of the greater consistency ‘). For spherical nuclei the density distributions are parametrized according to the Fermi form,
Pq(d=
PO4
l+ew[(r-Cq)f~qf’
(5)
the total ETF energy being minimized with respect to the four parameters C,, C,, a,, and up (the poq are fixed by normalization). Although more general forms lead to greater binding ‘), it is known that the ETF method tends to overbind anyway ‘), as compared to HF. Thus the choice (5) is preferred not only for the computer time that it saves, but also because it will actually lead to better agreement with HF energies, for a given force. In any case, it has been noted *‘) that the simple Fermi form (5) also reproduces HF and experimental density distributions better.
.I.M. Pearson et al. / Nuclear mass formula (Ill)
5
To save further computer time we parametrize the Q~ according to a,=a,ia,I+a,1~fa~r~+a,14,
(6)
where the upper sign refers to neutrons, and the lower to protons. The coefficients are determined once and for all (for each force) in a separate calculation. In this way we only have to minimize with respect to two parameters during production runs; we have checked that the error introduced in this way is quite negligible over the range of A considered here. Actually, using the density distributions (5) directly in the ETF functionals implies that they refer to point nucleons. Thus the Coulomb terms in the expressions given in papers (I) and (II) for the energy density a(r) are calculated for point protons. Nevertheless, protons are of finite size, and this will affect the energy. This effect could be taken into account simply in the fitting of the Skyrme force (1) to the data, but the resulting modification of the force would be isospin-independent, while the effect itself is not, of course. Since error could thus arise on extrapolating far away from the data, we attempt to take the finite proton size into account explicitly. We do this by noting that its effect is simply to add a short-range term to the Coulomb force between protons, which we approximate for simplicity by a S-function force, i.e., an extra Skyrme-type to term, for protons. The strength tof,,of this force is fitted to the results of a HF calculation in which the finite size of the proton has been taken into account “1: we find fOfP= - 4.8 MeV, corresponding to an rms charge radius of 0.8 fm for the proton. Turning to deformed nuclei, we adopt the parametrization of Brack et al. (BGH) 9), as described fully in paper (II). This means that only axially symmetric deformations can be considered, but allowing triaxial deformations may not be of much consequence for total binding energies. A basic feature of the BGH description is the definition of a reference surface, parametrized by the (c, h) variables r4), and enclosing the same volume for all deformations of a given nucleus. [In appendix A we present a detailed analysis of the relation between the (c, h) parameters and the more familiar (&, p4) parameters of a multipole expansion.] Deformed density distributions are then generated by a uniquely defined transfo~ation, characterized completely by the (c, h) parameters, from the spherical dist~bution (5). In principle, the C,, a, parameters of the latter could be chosen anew for each deformation (c, h) by minimizing the corresponding ETF energy, but in paper (II) we found that excellent agreement with HF was obtained by treating these parameters as deformation independent, determining them for a given nucleus in the spherical configuration. This suggests that the more general procedure might well have led to overbinding in the case of large deformations, although we did not test this point. We note now a problem that can arise for both spherical and deformed nuclei with the density distribution (5). Since the gradient of this distribution has an unphysical discontinuity at the origin, and at the centres of curvature in the deformed case, there is a danger of anomalousIy large currents being computed in the vicinity
6
J. M. Pearson et al. / Nuclear mass formula
{III)
of these points, whereas in reality it is easy to see on symmetry grounds that the currents should vanish there. Correcting for this problem in the computer programme is easy, and corresponds simply to flattening the distribution (5) at the origin. Shelf corrections. It is inevitable with the ETF method that the energy will vary smoothly from one nucleus to another: the shell effects are effectively lost in truncating the semi-classical expansions. Thus we are once more forced back to a macroscopic-microscopic approach, and are obliged to add shell corrections. However, compared to the DM mass formulas there is an important difference, since we can now determine quite unambiguously the s.p. fields appearing in the s.p. Schrodinger equation [(-&‘/2M)V*+
U,(r)+
W,(r) * {-iVxa)]+(r)
= &4(r),
(7)
the solutions to which are required for the calculation of the shell (and pairing) corrections: one simply folds the same Skyrme-type force involved in the ETF functional over the density distribution emerging from the macroscopic part of the calculation (see papers (I) and (II) for the expressions). There is thus a high degree of coherence between the macroscopic and microscopic parts, the unifying factor being the Skyrme-type force that underlies both; this presumably accounts for the excellent agreement with the HF method (see below). Neglecting pairing for the moment, we note that the shell corrections, which determine the total energy according to E = EETFi-6E,
01
can be written, according to the Strutinsky theorem (see, for example, ref. 15)), as SE=
r, 4=hP
(
C stz,q-F a/+?)
(9)
p
where the first term on the right-hand side represents the sum over all occupied states (neutrons and protons) of the s.p. energies corresponding to eq. (7), while the second term is a smoothed form of the first. To calculate this smoothed term we could have used the conventional Strutinsky energy-averaging method, as in refs. 5*6),for example. However, it is well-known that this method contains some ambiguities, mainly because of the continuum s.p. states, which means that they may be expected to become particularly troublesome towards the drip lines. Actually, some progress has been made towards resolving these problems 16*i7),but in the present case, where the macroscopic term is generated by the ETF method, we can avoid them altogether by a direct application of the Strutinsky theorem, as follows. Writing the HF energy in density-matrix notation as E HF= tr (tp) +5 tr tr (pop),
(IO)
(we are still neglecting pairing) we separate the density matrix into smooth and fluctuating parts, p=p”+sp.
(11)
J.M. Pearson et al, / Nuclear mass formula (Ill)
A straightforward
calculation
[see ref. 15), for example]
I
then leads to the Strutinsky
theorem,
E HF=~++E+O(Sp)*, where
6E is the shell correction
c
Ew7 = tr&= PL.4
(12)
(9) with the smoothed
terms given by
&G-~+~.U~+.I~.
C
W,
.
(13)
>
4=n,p
Here rq, pq and Jq are the smoothed
densities
obtained
in the macroscopic
part of
the calculation, while U, and W, are the corresponding fields, appearing in eq. (7). The error term O(Q)* in eq. (12) is known “) to be around l-2 MeV. This prescription for shell corrections is very simple to apply in our case and is quite unambiguous, even at the drip lines. Note, however, that it cannot be used with drop(let) models, mainly because the distribution of the DM energy between potential and kinetic terms is not specified, but also because the distributions pq and Jq giving rise to the s.p. fields are not known. We call this prescription the “Strutinsky-integral (SI)” method, to distinguish it from the familiar energyaveraging method, although in reality it does not deserve a special name, since it is nothing more than the Strutinsky theorem rendered explicit. Eq. (13) was apparently first written down by Chu et al. 18), but it does not seem to have been used very much in the past. A further feature of this method is the exact cancellation that takes place in eq. (8) between the kinetic-energy term of eq. (13) and the corresponding term in the ETF energy (4). Thus errors in the ETF expressions for kinetic energy will be of consequence only insofar as they affect the density distributions given by the minimization of E,,F. Executing this method requires solving the Schrodinger equation (7), in order to get the s.p. energies
appearing
in eq. (9). Since the s.p. field in eq. (7) results
the macroscopic
part of the calculation,
shell corrections
may be regarded
it follows
as doing
that this method
one iteration
from
of calculating
of HF on top of ETF.
Since this is by far the most time-consuming part of the whole calculation in the case of spherical nuclei we see that the ETFSI method is an order of magnitude faster than HF for these nuclei. As for deformed nuclei, where the ETF part of the calculation takes a significantly greater fraction of the total time, the ETFSI method is some five times faster than HF. Calculating the shell corrections in this way introduces a spurious c.m. (centre of mass) motion, exactly as in the HF method. We correct for this using the method of Butler ef al. 19); we have found that this method is essentially exact, being much more accurate than the (1 - l/A) method, as used, for example, in ref. 20). Pairing. We handle this by doing BCS with a S-function force 21) Dij
=
V,S(rij) .
(14)
8
J.M. Pearson et al. / Nuclear mass formula
Although
this increases
the computer
we shall see in the next section parameter,
the constant
and proton
pairing.
time as compared
that it allows
strength
(III)
to the “constant-G”
choice,
a good fit to the data with a single
V,, the same for all nuclei
We can thus expect a more reliable
and for both neutron
extrapolation
from the data
out to the exotic regions. Recalling
the main
points
of the BCS method,
we introduce
the occupation
probabilities
where eP.4 In these equations BCS equations
the chemical
=
{(%,q-A,)‘+
potential
A;,q}"2
A, and the gap A,,, are given by the usual
ML,, = N or 2, P A,,,
(16)
(17a)
=$C Gpud""
e “A
Y
(17b)
in which -GPy is the appropriate matrix element of the pairing force, whatever it may be (note that GPy is positive with an attractive pairing force). Solving the BCS equations for both neutrons and protons, eq. (8) for the total energy is replaced by E = E,,,+
6’E + Epair,
(18)
where
S’E= C
q=n,p
with the second
i\PC v:.,~,.,
(19)
c1
term still given by (13), and &iir=
C
&air,q
4’“.P
3
(20)
in which Epair, q =
-ac 2A2 P %,s .
(21)
When N or 2, or both, are odd, we apply blocking in the usual way, i.e., in the BCS equations (17a) and (17b) the “blocked” state is excluded from the summations, and n = N or 2 is replaced by n - 1, so that the right-hand side of (17a) is an even number (see, for example, ref. “)). (It is because of blocking that eq. (18) contains no quasi-particle terms.) In all the summations over s.p. states we include all quasi-bound continuum states up to 1 hw, it having been shown *‘) that higher cut-off energies do not affect the value of V,.
9
J.M. Pearson et al. / Nuclear mass formula (Ill)
Our interpolation procedures (sect. 5) approximate the basic b-function pairing force by defining an “equivalent constant G”, Geq, as follows. Let us rewrite eq. (18) for the total energy calculated with the S-function pairing force by the complete ETFSI method as
(22) where
(23)
E,“c’(S)= C u:.qeF,q + Epair,q P
(24)
with Epair,q defined by (21). Then for the nucleus in question, at the given deformation, we perform a constantG BCS calculation with the value of G chosen such that
EFS(G)= EtCS(G) .
(25)
This defines Geq,q; note particularly that it is defined separately for protons and neutrons, even though the strength V,.,of the b-function pairing force is the same for both. This definition of Geq,q ensures the correct reproduction of the net effects on the total energy (18) produced separately by the neutron and proton pairing with the b-function force. It follows, of course, that the total energy (18) obtained with the b-function pairing force also will be correctly reproduced. Correction for rotational energy. For deformed nuclei and fission barriers we subtract from the total computed energy the spurious rotational energy 23)
&t = f-j (J2> , where ,$ is the moment of inertia. We first calculated this by the cranking model [Inglis ‘“)I with pairing correlations included [Belyaev 24)], but found that this value, 9 crank, was consistently smaller than the values extracted from the measured spectra in well-deformed even-even nuclei, by about 30% for the rare earths and 50% for the actinides. Rather than adopt some more sophisticated method for calculating 8, we simply took a linear combination of B‘_rank with the rigid-body value of $,2rig, ~P(l-a)Bcrank+drig,
(27)
where the constant a was taken to be 0.2 for all even-even nuclei, this giving the best overall fit to the experimental values of 9 for a number of nuclei over different regions of the nuclear chart, as shown in table 1. The deformation parameters shown in this table correspond to equilibrium, and all results are found to be the same for all three of the forces developed in this paper (see sect. 3). We postpone until sect. 5
J.M. Pearson et al. / Nuclear mass formula (III) TABLE
1
Deformation parameters at equilibrium for all three forces (see appendix A for the relation between the two parametrizations). Moments of inertia (units of h* MeV-‘): &t.z&Ecalculated from eq. (27), db,,P experimental
100 106 150 154 156 162 162 168 172 174 184 194 224 232 236 238 240 246 246 250 250 252 256 256 260
38 44 58 64 62 66 72 76 68 70 74 76 90 90 92 96 94 94 100 96 100 98 100 104 106
1.25 1.16 1.24 1.22 1.27 1.20 1.15 1.13 1.16 1.16 1.10 0.89 1.25 1.24 1.26 1.30 1.26 1.20 1.24 1.20 1.20 1.17 1.17 1.17 1.14
0.00 0.00 -0.16 -0.11 -0.13 -0.02 -0.04 -0.07 0.12 0.12 0.17 0.13 -0.30 -0.21 -0.20 -0.25 -0.20 -0.07 -0.14 -0.06 -0.07 0.00 -0.01 0.01 0.09
0.38 0.26 0.24 0.28 0.33 0.32 0.23 0.17 0.33 0.33 0.27 -0.16 0.14 0.22 0.21 0.23 0.21 0.29 0.27 0.29 0.29 0.27 0.26 0.27 0.27
0.02 0.00 0.06 0.05 0.06 0.02 0.01 0.02 -0.03 -0.03 -0.05 -0.01 0.09 0.08 0.07 0.09 0.07 0.03 0.06 0.03 0.03 0.00 0.01 0.00 -0.03
23.1 14.6 31.8 31.2 38.1 35.0 25.0 23.2 40.5 42.8 33.2 24.4 51.9 56.5 63.6 61.0 64.5 60.3 65.6 62.3 64.7 62.1 62.2 65.1 65.3
11.1 30.6 24.3 39.5 37.2 39.5 39.2 26.3 13.8 32.2 60.7 66.0 70.0 65.2 62.5 65.6 62.3 -
the question of the moments of inertia of odd-A and odd-odd nuclei, since these nuclei are not involved in the determination of the force, and are always calculated by interpolation. While
this
procedure
is quite
adequate
for known
nuclei,
we realize
that
it
introduces an element of uncertainty into the extrapolation to unknown nuclei, since there is no guarantee that the parameter a would take the same value for nuclei far from stability. However, it should be realized that the same uncertainty surrounds the force parameters; indeed, one could argue that the parameter a has a status in our model quite similar to that of the force parameters, so that we should perhaps admit to 10 rather than 9 parameters in all. There is, however, the fundamental difference that the parameter a is determined from level spectra, and is quite independent of mass data. Di~en~i~na~i~y. We solve eq. (7) by expanding the eigensolutions in an oscillator basis. In the programme for spherical nuclei we take a base of dimensionality 10 in each subspace. The resulting truncation error on the total energy can amount to
J.&l. Pearson et al. / Nuclear mass formula (Ill}
11
2 MeV for heavy nuclei, but this is smaller than the error arising from the ETF approximation for a given force. In any case, both of these errors, which are of opposite sign, are for the most part absorbed by the force in fitting to the data, although they could in principle re-emerge as we extrapolate away from the data towards the drip lines; we return to this point below. As for deformed nuclei, we have two programmes, one for nuclei with left-right (lr) symmetry, and the other for nuclei with lr asymmetry. These codes are used only to calculate deformation energies: the former calculates the energy with respect to the spherical configuration, while the latter calculates the energy with respect to lr symmetric configurations. Since deformation energies converge much faster than absolute energies as a function of dimensionality this procedure permits a considerable saving of computer time for a given precision. Our lr symmetric code expands all states in a basis limited to 17fio for ground-state calculations of nuclei with As 204, and to 19%~ for A> 204, this guaranteeing an effective convergence. For the corresponding limit in the case of the code which calculates the lr asymmetry corrections, 15hu was found to be adequate. Comparison with HF. For a given set of force parameters the ETFSI method overbinds nuclei by between 3 and 7 MeV, as compared to HF ‘) (this represents the difference between the characteristic overbinding of the ETF method and the underbinding associated with the restricted form (5) of the density profile). Now while errors this large would be inacceptable in modern mass formulas it is not inevitable that this will pose a problem in practice, since whatever method is being used the force parameters are always fitted to the data. Rather, the crucial question is: when the ETFSI method is extrapolated out to the n-drip line, will it agree with the HF extrapolation? In papers (I) and (II) we showed that the discrepancy between the two methods at the n-drip line is less than 1 MeV for the total energy and fission barriers, and less than 0.5 MeV for neutron separation energies (S,) and beta-decay energies (QP). We thus conclude that while the ETFSI method is an order of magnitude faster than HF it gives essentially the same extrapolations from known nuclei out to the n-drip line, for a given form of force. Nevertheless, it should be realized that this comparison of the ETFSI and I-IF methods supposes that both methods use bases of the same dimensionality. Thus there is still the possibility that some of the truncation error of our spherical ETFSI calculations (~2 MeV), which is absorbed into the force by fitting the data, will re-emerge on extrapolating towards the drip lines. We conclude, then, that if our spherical programme had a larger basis the extrapolations out to the drip lines could shift slightly. However, the situation in this respect is no different than with the HF method. Interpolation. Once an effective interaction has been fitted to the data a further advantage of the ETFSI method becomes apparent. All the quantities in terms of which the method expresses the total energy, including the s.p. energies (but not their sum, of course), vary smoothly with N, 2, and deformation, even though the
J.M. Pearson et al. / Nuclear mass formula (III)
12
fluctuations
that are characteristic
it will be possible
to construct
of the shell model are correctly
the mass table by interpolation.
in more detail in sect. 5, where we show that the total computer by more than two further which interpolation
orders
of magnitude
as compared
reproduced.
Thus
We will discuss
this
time can be reduced to the HF method,
for
is not possible. 3. Fitting procedure
Despite
the comparative
calculations parameters are obliged
rapidity
of the ETFSI
method,
the deformed-nucleus
are still so slow that in the automatic least-squares fit of the nine of our Skyrme-type force (1) and the pairing force (14) to the data we to assume spherical configurations, since each nucleus has to be calcu-
lated many times during the actual fit. Thus in the present paper we confine our automatic least-squares fit to the masses of 491 nuclei that are believed to be spherical, the data being taken from the 1988 compilation of Wapstra et al. “). This sample consists of all nuclei with Z 3 20, and N or Z not more than two units away from a magic number, excluding the few such nuclei known or suspected to be deformed. We first included nuclei with Z < 20 in the fits, but the fits were always poor, and since our primary interest lies in heavier nuclei
we decided
not to allow these light nuclei
to influence
the determination
of
the force parameters. Because of the Wigner effect we exclude likewise nuclei with N = Z (see sect. 6). The preliminary fit to the 491 nuclei assumed to be spherical led to three different parameter sets, labelled SkSCl, SkSC2, and SkSC3 in table 2 (SkSC = Skyrme semi-classical). The fits were made as follows. Of the eight Skyrme parameters, x-, and
f3 were expressed
momentum,
in terms
of J and kF, the symmetry coefficient and Fermi of nuclear matter. Then kF, J, x,, and y were held fixed
respectively,
at different values, while least-squares routine.
to, ti, x0, and
W,, were fitted to the data points
TABLE Force
parameters SkSCl
t, (MeV . fm3) t, ( MeV fm’) tz (MeV . fm’) t, ( MeV . fm4) X0
2
(y = 0.333333
-1788.59 282.623 -282.623 12775.3 0.72
for all forces) SkSC2
-1791.64 290.601 -290.601 12800.6 0.38
XI
-0.5
-0.5
X2
-0.5
-0.5
X3
W, ( MeV V, ( MeV
1.04564
. fm5) . fm3)
126.997 -220.0
0.592760 128.781 -220.0
SkSC3 -1788.11 289.901 -96.6337 12771.4 0.65 -1.0 1.0 0.960976 143.452 -220.0
with the
J.M. Pearson et al. / Nuclear mass formula (III)
13
In principle, we should have optimized the fit to the masses with respect to y, but to do this correctly would have consumed too much computer time. However, a rough mass fit led us to set y = f in all three forces, the corresponding values of the incompressibility K then always being around 235 MeV; we found this to be definitely better than y = 0.5, i.e., K = 270 MeV. This result is to be contrasted with the mass fit of MMST5), which favours a value of K closer to 300 MeV. The difference between the two fits appears to originate in the microscopic corrections, ours being much more sensitive to the value of K than are those of MMST ‘), essentially because when we change y in order to change K we induce changes in the diffusivity of the density distribution, and hence of the s.p. potential 26). Thus, while the value y = f is clearly indicated, it might be unwise to draw any conclusions from our work concerning K, since what we are really testing is the surface diffusivity, the coupling of which to K through the force might well be spurious. On the other hand, a DM-based fit such as MMST ‘) is more likely to be measuring K directly. Nevertheless, we note that our choice is in good agreement with the value K = 228 MeV given by an analysis of the differences between the charge-density distributions in various lead isotopes “). As for recent breathingmode estimates of K, these vary from 217 MeV (Gleissl et al. ‘“)) to 300 MeV (Sharma et al. ‘“)).
As for kF, requiring that the rms charge radii I, of heavy nuclei be fitted (table 3) leads to the value of 1.335 fm-‘, in good agreement with ref. 30). [We fold in the finite proton size, according to
where rr, is the rms radius for point protons, corresponding to the Fermi distribution (S), and sP = 0.8 fm is the rms charge radius of the proton.] The pairing-force strength V,, which was adjusted manually, was found to give a best fit to the masses with a value of -220 MeV for all three parametrizations (see below for further comments). An optimal fit to the 491 nuclei assumed to be spherical was then obtained with J = 27.5 MeV and x, = -0.5; this defines our first force SkSCl (see tables 4 and 6 for the errors). Support for this value of J comes from relativistic Brueckner-HF calculations of asymmetric nuclear matter by Miither et al. 3’), and ter Haar and TABLE
3
RMS charge radii (fm)
4sCa “‘Sn *08Pb
SkSCl
SkSC2
SkSC3
Exp “)
3.53 4.63 5.50
3.53 4.63 5.50
3.53 4.63 5.50
3.41 4.64 5.50
“) Data from De Vries et al. 56).
J.M. Pearson et al. / Nuclear mass formula (III)
14
Malfliet 32), which
give J = 27 and 26 MeV, respectively.
tivistic calculation
for the Reid soft-core
as reported
have added
but also the neutron
compare
Also Siemens’
non-rela-
33) gives 27 MeV after correction,
in ref. 34).
Our force would masses
potential
the SkSCl
Pandharipande
force
astrophysical
gas at nuclear in this respect
35) for the a,, potential
interest
if it fitted not only nuclear
and sub-nuclear
densities.
with the calculations et al. 36) obtain
[Cugnon
In fig. 1 we
of Friedman similar
and
results with
the Paris potential]. The poor agreement motivated us to define a second force, SkSC2: with J = 30 MeV and x, still equal to -0.5, it will be seen that there is an excellent fit to the neutron gas. However, the fit to the masses is not quite as good as before (tables 4 and 6), but it is still acceptable, and it remains to check the possibility that it will out-perform SkSCl for fission barriers. Finally, fit fission somewhat with x, =
in order to have a second degree of freedom which might be useful to barriers we take advantage of the fact that the fits to ground states are insensitive to the value of x, . Accordingly, we define a third force, SkSC3, -1.0 and J = 27.5 MeV. This too has an acceptable fit to the mass data,
but overall it is not quite as good as SkSCl (tables 4 and 6). We have checked that all three forces satisfy the conditions of Giai and Sagawa 37) for spin and spin-isospin stability. [The relevant expressions given in ref. 37) remain valid for the different kind of density dependences used here.] Errors in Jits to masses. Referring to table 4 for the fits to the 491 nuclei assumed to be spherical, we see that force SkSCl has the lowest rms error: .srms= 0.693 MeV. Ideally, one would also want the mean deviation, E, to be zero, in order for the fit to be free of systematic error. However, once the force parameters have been chosen
I
0
0.05
I
0.10
I
0.15
p(fme3) Fig. 1. Energy e per neutron in neutron matter of density p.
J.M. Pearson et al. / Nuclear mass formula (III) TABLE
15
4
Fitting errors for 491 spherical nuclei (MeV)
E,ltlS
E (nr) E,Ill, &‘,Ej
SkSCl
SkSC2
SkSC3
0.693 0.0610 0.587 0.965
0.729 0.0861 0.676 1.042
0.713 0.0899 0.636 0.960
by a least squares fit one has no remaining degree of freedom to control this (if one of these parameters is a simple additive constant, which is not the case here, then .? = 0 follows automatically). Thus it is gratifying to find that the best force from this point of view also has the lowest rms error: SkSCl. Table 4 also shows the rms error ~2: with which each force fits a sample of 68 of the most neutron-rich of the 491 nuclei to which the forces have been fitted. It will be remarked that in this respect SkSCl is still the best of the three forces. In the last line of table 4 ET: represents the rms error in a sample of 34 of the most proton-rich of our 491 spherical nuclei (fewer of these are proton-rich than neutronrich). It should be noted that SkSC3 now scores marginally better than SkSCl, but the difference is probably not statistically significant. More important is the fact that SkSC2 is distinctly worse than SkSCl, which means that J = 27.5 MeV works better than J = 30 MeV on both sides of the stability line. Deformed nuclei. Before using any of these forces for calculating a complete mass table, we must form some idea of how they perform for deformed nuclei. We make this test on the 25 nuclei of table 1, which are widely distributed over the nuclear chart. Table 5 shows the calculated energies, obtained in all cases with the code for lr symmetry; the equilibrium deformations, shown in table 1, are found to be the same for all three forces. From table 6 we see that the fit has now deteriorated rather badly, as compared to the spherical nuclei. However, we also see in table 6 that this deterioration is associated entirely with nuclei of A 3 250, and that if we exclude these nuclei the fit is as good as for the 491 assumed to be spherical. It will also be seen from this table that SkSC3 now performs somewhat better than SkSCl, but we do not regard
this as being
statistically
significant,
given the smallness
of the
sample. The problem with the nuclei having A2250 is clearly seen to lie in a tendency to underbind. We have checked that this is not a problem of the dimensionality of the codes. The dimensionality of the deformed codes is quite adequate, while as for the spherical code we made a trial calculation in which the dimensionality was increased from 10 to 15; we found that the binding energies of *08Pb and nucleus (Z = 106, A = 260) were increased by 1.03 and 1.09 MeV, respectively. This suggests that if the force were renormalized to refit the data with the greater dimensionality the final results for the heaviest nuclei would be essentially the same as before. Nor
16
J.M. Pearson et al. / Nuclear mass formula TABLE Energies
(MeV)
for 25 deformed
Z
A
even-even
SkSCl
(III)
5 nuclei
(axial
and left-right
symmetry)
SkSC2
SkSC3
Exp. -837.5
100
38
-837.3
-838.2
-837.6
106
44
-906.8
-906.9
-906.9
-907.3
150
58
-1230.1
-1230.4
-1230.2
-1230.1
154
64
-1267.5
-1267.7
-1267.7
- 1266.3
156
62
-1280.0
-1280.1
-1280.2
-1279.7
162
66
-1323.9
-1324.0
-1324.1
-1323.8
162
72
-1300.1
-1300.6
-1300.3
-1300.0
168
76
-1326.1
-1326.7
-1326.2
-1326.2
172
68
-1391.1
-1391.1
-1391.2
-1391.2
174
70
-1406.5
-1406.4
-1406.5
-1406.2
184
74
-1471.4
-1471.3
-1471.5
-1472.5
194
76
-1536.9
-1536.7
-1537.0
-1538.4
224
90
-1716.6
-1716.5
-1716.6
-1716.9
232
90
-1765.7
-1765.4
-1766.0
-1766.0
236
92
-1789.3
-1789.0
-1789.6
-1789.7
238
96
-1795.7
-1795.8
-1796.0
-1795.7
240
94
-1812.1
-1811.9
-1812.5
-1812.7
246
94
-1845.1
-1844.7
-1845.6
-1845.9
246
100
-1835.0
-1835.1
-1835.3
-1836.3
250
96
-1868.2
-1867.7
-1868.6
-1869.0
250
100
-1863.0
-1863.0
-1863.3
-1864.7
252
98
-1879.0
-1878.3
- 1878.9
-1880.5
256
100
-1900.2
-1899.9
-1900.6
-1901.7
256
104
-1888.1
-1888.0
-1888.3
-1889.7
260
106
- 1906.6
-1906.6
-1906.8
-1908.1
has the problem anything to do with left-right asymmetry, for we have confirmed that in the case of the heaviest nucleus of all, A = 260, removing the constraint of left-right symmetry does not in any way increase the binding. We therefore believe that this underbinding of the heaviest nuclei is not related specifically to our treatment of deformation, but rather reflects a defect in our three forces, since the heaviest nuclei in the sample to which the forces were fitted correspond to A = 218. Refitting the force to all nuclei, spherical and deformed,
TABLE Fitting errors (MeV)
6
for the 25 deformed SkSCl
nuclei of table 5
SkSC2
SkSC3
e,,, (all 25)
0.923
1.100
0.812
E,,,
0.665
0.809
0.604
(A < 250)
E (all 25)
-0.552
-0.564
-0.344
I (A < 250)
-0.274
-0.205
-0.074
J&f. Pearson et al. j Nuclear ~ass~o~ula
(IZI)
17
might thus be expected to lead to an improvement. We will return to this question in sects. 6 and 7. But more to the point is the fact that in the present work we are able to represent deformed nuclei almost as well as spherical nuclei, even though both the parametrization of the deformation and the subtraction of the rotational energy, as given by eqs. (26) and (27), could have given rise to further errors. In fact, with the SkSCl force we have achieved a very high-quality fit to both spherical and deformed nuclei, the rms error for 516 nuclei being 0.706 MeV. However, before choosing SkSCl as the force with which to calculate a mass table, we note that the other two forces have, so far, also given quite respectable fits, and that SkSC3 might even be better. We thus postpone our final choice of force until we have seen how all three perform in connection with fission barriers. Macroscopic parameters. Before considering the results for fission barriers, and making the final choice of force, we present (table 7) the so-called macroscopic coefficients of our forces, which appear in the droplet model ‘) but which are defined independently of it. Of the coefficients defined for infinite nuclear matter we give the volume coefficient av, the incompressibility K, and the symmetry-density coefficient L, as well as J and k, (already discussed). We also make a full ETF calculation of semi-infinite nuclear matter, assuming a Fermi profile, as in eq. (5), and extract the surface coefficient asf, the surface-symmetry stiffness Q, and the curvature coefficient aCv [for further details see, for example, ref. ‘)I. For reasons of simplicity our calculation of a_. omits the terms in J’, and J2 from the energy density appearing in (4). It can be seen from table 7 that L and Q are correlated with J in the usual manner 38,39).However, the values of Q we find for a given J are somewhat larger than would be expected on the basis of ref. ‘“): fig. 1 of that paper indicates that 60 MeV would be appropriate for SkSCl and SkSC3 (J = 27.5 MeV), and 45 MeV for SkSC2 (J = 30 MeV). Nevertheless, for two reasons this situation is far less Macroscopic parameters (MeV); e”’ = p~(d3e/dp3),=, (this is Ev of ref.9)). k,= 1.335fm-’ (p~=0.1607fm-3) for all forces SkSCl
SkSC2
SkSC3
-15.85 - 14.09 234.8 27.5 -2.32 17.4 94.2 -18.4 11.1
-15.90 -14.14 235.3 30.0 32.0 17.6 52.8 -38.1 11.3
-15.84 -14.09 234.7 27.5 2.74 17.4 88.3 -19.2 11.4
18
J.M. Pearson et al. / Nuclear mass formula
anomalous considered stability
than
appear
in ref. 39) were fitted to data of nuclear
masses
it should
concentrated
be noted much
work. Secondly,
on Q is given
more
that
the forces
closely
we should
by a first approximation
to the
note that to the
model ‘) as e = a, + a,fA-“3
where
to be. Firstly,
line than is the case with the present
the dependence droplet
might
(III)
e is the energy
+ (J + u~~A-“~) I2 + ~oulZ2A-4’3
per nucleon,
+ acvA-2’3 ,
(29)
and
ass= (2L/K)a,,-9J2/4Q
(30)
the so-called surface-symmetry coefficient, also shown in table 7. It is easy to show now that the shifts in e implied by the above shifts in Q for the given values of J are much smaller than the shifts implied by going, for example, from any of our forces to the MMST fit ‘) (J = 32.5 MeV, Q = 29.4 MeV, L = 0, whence ass = -80.8 MeV). The point is that when J*/Q is small, as with our fits (especially SkSCl and SkSC3), the fits to the masses become quite insensitive to Q. The values of acv that we obtain are quite normal for effective forces (see, for example, table 8 of ref. “). However, DM fits, and in particular the MMST fit of ref. 5), find acv much closer to zero. This anomaly has been discussed by Stocker et al. 40), and the simplest explanation is an insufficient convergence of the droplet model’s leptodermous expansion. There could be significant implications for the calculation of fission barriers, fission lifetimes, and particularly the dynamics of scission, since this is strongly influenced by the rapidly changing post-saddle curvature. Fission barriers. Table 8 shows the results we have calculated with our three forces for the experimentally known outer barriers of four nuclei, with lr asymmetry taken into account. We do not consider the inner barriers, since they are much less sensitive to the choice of force (this can be seen in table 9), and also are more likely to be afflicted by triaxiality, which we cannot yet handle. Forces SkSCl and 3 are seen to be virtually equivalent, while SkSC2 gives appreciably different, and worse, results for the two heaviest nuclei. This shows that TABLE 8 Fission
barriers
Nucleus
lS60s 2’0Po 240pu “‘Cm
(MeV)
for 4 known barriers
nuclei; only outer (last two nuclei)
SkSCl
SkSC2
SkSC3
24.6 22.0 3.1 3.0
24.2 21.7 2.1 2.3
25.0 21.7 3.2 3.2
“) Data from Moretto et al. 57) b, Data from Khodai-Joopari ‘*). “) Data from Back et ai. 59).
barriers
in case of double
Expt. 23.4kO.5 20.4kO.5 5.4kO.2 3.9*0.3
“) b, “) “)
J.M. Pearson et al. / Nuclear mass formula (III) TABLE
19
9
Calculated fission barriers and isomers of 262U(MeV)
1st barrier 1st isomer 2nd barrier 2nd isomer 3rd barrier
SkSCl
SkSCZ
SkSC3
2.7 1.3 5.8 2.9 13.0
2.5 0.9 4.6 1.3 9.1
2.9 1.4 5.9 3.0 13.0
fission barriers, like masses, favour the lower value of J, 27.5 MeV; however, the barriers are quite insensitive to x, . The general level of agreement with experiment given by forces 1 and 3 is satisfactory, especially when we bear in mind that no fission-barrier data were included in the parameter fitting. However, a slight tendency to underestimate barrier heights for heavy nuclei, and overestimate them for lighter nuclei is noted. Looking at the results for SkSC2, we see that reducing J increases the barriers, and that this effect is greater for heavier nuclei. This can be understood quite easily in terms of the fissility parameter
a,oul z2
X=za,Xr)A’
(31)
&f(I) = a$+ ff,,12.
(32)
where
We see that we might have obtained better barriers with a still smaller value of J; however, the fit to the masses will deteriorate significantly if J is reduced below 26.5 MeV. A smaller value of kF would have the same effect, since it would lead to a smaller value of c+Oul.(In any case, reducing the symmet~ coefficient f requires that kF be reduced also, if the stability line is to be unchanged.) The results of restricted calculations (h = 0, and lr symmetry) on the fission barriers of 262U, a nucleus lying on the r-process path, are instructive: see table 9. They confirm the sensitivity of the extrapolated barriers to the value of J, and also show that no sensitivity to x1 emerges beyond the region of known nuclei. Choice offorce. The results that we have obtained for masses and fission barriers, as described in this section, lead us, on balance, to SkSCl as our choice of force with which to calculate our first mass table. The associated value of J, 27.5 MeV, is significantly lower (and the value of Q correspondingly higher) than for any recent mass formula, although it is very close to the value of 27.63 MeV given by Myers and Swiatecki more than twenty years ago 41). Our low value could simply be a result of the large amount of new data we fitted [the MMST formula of ref. ‘) uses values of J and Q determined from earlier fits to more restricted data], but it could
20
J.M. Pearson et al. / Nuclear mass formula
also be a consequence that
for a given
of our model. We are referring
surface
stiffness
excesses,
to a softer neutron
because
the latter
omit
specifically
Q the ETF method
skin than do the droplet
some
higher-order
significant.
If then “malacodermous”
they would
have to be compensated
(Ill)
to the observation
“)
gives rise, at large neutron
models used to date, essentially
symmetry
terms
that
are apparently
effects of this sort are showing
up in the data
in our model by a larger value of Q, and hence
a lower J. In any case, our low value of J (along with its associated high value of Q) and the “malacodermous” effects that we implicitly include mean that we should expect our extrapolated masses and fission barriers of nuclei far from stability to be different from the MMST predictions ‘). (Note that these two sources of discrepancy work in opposite directions, but the latter will dominate close to the n-drip line.) We return
to this question
in sect. 6. 4. Comments on pairing properties.
In this section we examine in more detail some of the properties of the b-function pairing force (14), with particular emphasis on its advantages over the more usual “constant-G” model. All calculations are performed with the SkSCl parameter set. We begin by looking at the fourth-order even-odd mass differences, Ac4), as defined by Madland and Nix 42). The comparison with experiment is limited to spherical nuclei, since it is obscured in the case of deformed nuclei by the uncertainty in E,,, for odd-A and odd-odd nuclei. Fig. 2 shows for nearly all possible sequences of our 491 spherical nuclei the quantity x(A) = A?)-A:‘, in which
Ar’
corresponding
is the value
calculated
experimental
quantity
with
(33)
,
our
&function
force,
and
(we exclude
sequences
that include
Agb
the
a magic
number). It is apparent that with our value of V, (-220 MeV), determined like the Skyrme parameters by fitting to the total masses, there is a tendency for the even-odd mass differences to be over-estimated for A >80, in which range the majority of nuclei
(especially
those
relevant
to the r-process)
lie; the opposite
is the case for
the relatively few nuclei with A<80. On the other hand, when V, was optimized by fitting to the even-odd mass differences, leading to a value of around -200 MeV, the fit to total masses was found to deteriorate considerably, there being a tendency to underbind. Thus, even though we fit to the total masses, this problem with the even-odd mass differences sets a limit on the possible precision. The reason why we cannot optimize simultaneously the fit to the total masses and the even-odd mass differences is that the effect of the pairing force is not confined to producing even-odd fluctuations in the total energy E. Rather, the mean value of the term EtCS(G) in eq. (22) about which E fluctuates is significantly different
from zero. Moreover,
this mean value is strongly
shell-dependent
and hence
J. M. Pearson et al. / Nuclear mass formula (iii)
21
Cd
E
A: PROTONS
2 Y
*: NEUTRONS
0.8
A -0.8
***
*
*
40
80
120
160
200
24C
MASSNUMBERA Fig. 2. Errors with which force SkSCl reproduces experimental fourth-order even-odd mass differences: see eq. (33).
cannot be compensated by any adjustment of the Skyrme parameters, (at least, as long as we maintain the constraint of M* = M). A possible way out of this difficulty now suggests itself. Lipkin-Noganii method. Since the wave function given by the standard HF-BCS method is not exact, particle number not being conserved, the method will underestimate the total binding energy for a pairing force of given strength. Some improvement in this respect might be expected if states with a well-defined number of particles were projected out of the BCS ground state, and we see the possibility of getting a better fit to the total masses with V,= -200MeV, the value which optimizes the . even-odd mass differences. In an attempt to do this, we replaced the standard BCS method with the method of Lipkin and Nogami: see ref. 43) and papers cited therein. This simple and well tested method approximates the projected wave function by requiring that (fi’) = (fi)*, where 2 is the number operator. We generalize the results of ref. 43) to the case of an arbitrary pairing force, and find that our eq. (18) for the total energy is replaced by E’=E-2hJv:u2,, (34) n
J&f. Pearsonef af. / Nuclear mass formula (III)
22
where (35) u’,=l-v”,. This indeed
improves
(36)
the fit for V, = -200
MeV, increasing
the binding
energy
for any given pairing force over what is given by the standard method, but we found that we could never improve the fit over the original one given by the standard method with V, = -220 MeV, as already quoted. The general tendency for these fits with the Lipkin-Nogami
method
is that nuclei
fit become more so, and likewise for nuclei we vary all the parameters of the force.
that were underbound that were overbound,
in the original no matter
how
It would seem that in the present context of fitting the pairing strength along with the Skyrme parameters to the total energy, the standard BCS treatment of the pairing is superior to the Lipkin-Nogami method. We suggest that the averaging over adjacent nuclei characteristic of the BCS method, usually regarded as a defect, is actually an advantage in the present context: even with a S-function force for the pairing, fluctuating errors are inevitable, and by smoothing these over several adjacent nuclei it should be possible to reduce x2. Comparison with “constant-G” approximation. We now consider to what extent it is advantageous to use the a-function pairing force rather than the “constant-G” approximation. To investigate this point, we took our values of Aji4’, calculated as described above for the different sequences, and fitted to each of them an equivalent constant G Gceff’ in the “constant-G” approxima0 , b Y repeating the BCS calculations tion. (Thisis not exactly the same quantity as the G eq,q defined for each nucleus in sect. 2 for interpolation purposes; it would not have been appropriate to compare this latter quantity with GyPfq, determined as described below.) We found GET’ to show a strong variation, which could be represented, as in ref. 42), by the functional form G::’
= g,( 1 ‘F g, I + g,12)/‘A,
(37)
where the upper sign refers to neutrons and the lower to protons. A best fit to our values of GE:) was obtained with the following parameter values: g,= 18.95 MeV, g, = 0.79, and g, = 3.4 (all obtained with the same cut-off of the continuum spectrum in the BCS calculation as for the S-function force). This variation is represented by the curves in fig. 3, where we show G,fe*)/( 1 F g,I + g,12) versus l/A, and in figs, 4 and 5 where we show G’,““A vs I. The points in these same figures correspond to the eiperimental values of the equivalent constant G, GE:), obtained in the same way as G’,“” by fitting to the values of A&. It will be seen that despite the large scatter, our s-function force with its single parameter V, has well represented the trends in the actual variation of Gy:. To get the same quality of fit with the “constant-G” model of the pairing would have required two more parameters, i.e., 11 rather than 9, so we are led to believe that
J.M. Pearson et al. / Nuclear mass formula (Ill)
23
0.7
- NEUTRONS
0.6
. . .
and PROTONS .
3 z
.
0.5 ,-
N CI 2 + H
0.4
IT .-I \ G % -u
0.3
0.2
0. I
0.0 0.0
0.02
0.01
0.03
l/A Fig. 3. Variation
the &function
of Gf”/(
pairing
1 T g, I + g,I*) as function
of l/A
force has greater underlying
for force SkSCl,
physical
confidence in the reliability of its extrapolation is enhanced Deformed nuclei. We considered using the “constant-G”
with experimental
reality,
points.
and our a priori
accordingly. approximation
just for
the deformed-nucleus calculations, taking for G the value G,, that is equivalent the sense of sect. 2) to the S-function force for the spherical configuration of nucleus in question. However, G,, is deformation-dependent, and neglecting fact can lead to errors of at least 0.7 MeV for masses, and probably much more fission barriers. 5. Construction
(in the this for
of mass table by interpolation
Even though the total energy suffers from shell-model fluctuations, the ETFSI method (and indeed any method based on the Strutinsky theorem) expresses it in terms of quantities that vary smoothly with respect to N, 2, and the deformation parameters c and h. This enables us to make extensive use of interpolation in constructing the mass table, with the complete deformed ETFSI calculation, described in sect. 2, being performed only for a restricted number of key nuclei (all taken to be even-even) and key deformations, as explained in more detail below.
24
J.M. Pearson et al. / Nuclear mass formula (IIIj
3o.c
NEUTRONS
a =’ 5 -#
20.0
15.0
ICI0 - 0.1
0.0
0. I
0.2
0.3
I Fig. 4. Variation of GsW) A as function of I for force SkSCl, with experimental points: neutrons.
Without this possibility the amount of computer time required to calculate the some 6000 nuclei lying between the drip lines, only a small fraction of which can be supposed a priori to be spherical, would have been prohibitive. The quantities that we have to interpolate as functions of PJ, 2, c, and h are the ETF macroscopic energy EETF, as given by eq. (4), the s.p. energies Ed, the smoothed sum of s.p. energies %elr, given by eq. (13), the rotational correction E,,,, given by eq. (26), and the pairing matrix elements G,,, as used in eq. (17b). Only the last two of these quantities require any special mention. Rotational correction. The prescription (27) for the moment of inertia has been checked only for even-even nuclei (see table 1). All our key nuclei are of this type, so that the interpolated values of E,,, will always correspond to even-even nuclei also. Now in table 10 we show results obtained for complete ETFSI calculations on a number of strongly deformed odd-A and odd-odd nuclei; $,, denotes the moment of inertia that we calculated by neglecting the last unpaired nucleons, at the determined equilibrium deformation. We found that we could follow the main experimental trends (as determined by the first excited state) by multiplying $,, by 1.2 for odd-A nuclei, and by 1.4 for odd-odd nuclei: this corresponds to dp,,,, in
J.M. Pearson et al. / Nuclear mass formula (III)
25
PROTONS 25.0
>
3 a .
1o.o /
-0.1
0.0
0.1
0.2
0.3
I Fig. 5. Variation of Gg’) A as function of I for force SkSCl, with experimental points: protons.
table 10. These results agree with the experimental systematics shown in fig. 4.12 of ref. “). It follows that the interpolated E,,, has to be divided by these same factors in the appropriate cases. In principle, one should apply a further correction factor to the interpolated values of Erotin order to take account of any characteristic even-odd effect in the spurious angular momentum that should appear in the numerator of eq. (26). However, there seems to be no way in which one could improve our estimate of E,,, in this respect without making detailed calculations for each individual nucleus. We face here a basic limitation of the ETFSI approach, but it should be realized that hitherto all mass formulas based on the drop(let) model have totally neglected the spurious rotational effects that exist in their shell corrections, absorbing them into the fit of the parameters to the data. One might expect that some idea of any systematic error introduced by our treatment of E,,, for odd nuclei could be obtained from table 10 by comparing the calculated and experimental energies shown there (the calculated values EGa,Chave E,,,treated in exactly the way described above, i.e., as in inte~olation). The agreement with experiment is decidedly worse than in the case of the even-even deformed nuclei (table 5), there being a net tendency to underbind more beyond
J.M. Pearson et al. / Nuclear
26
mass formuta (III)
TABLE 10 Sample of 17 odd-A and odd-odd deformed nuclei (axial and left-right symmetry). Moments of inertia _P in units of h* MeV-‘; energies E in MeV A 153 157 161 162 165 166 174 183 225 231 237 239 239 241 250 251 255
Z
9a,,
2 CXP
9 c0rr
64 63 65 67 66 67 71 74 90 90 93 93 94 95 97 100 100
27.9 36.8 35.7 34.4 37.1 37.1 40.0 33.1 52.4 55.5 64.2 64.7 64.5 66.0 63.5 65.2 63.0
60.2 45.5 44.4 52.2 47.4
33.5 44.2 42.8 48.2 44.5 51.9 56.0 39.7 62.9 66.6 89.9 77.6 77.4 79.2 88.9 78.2 75.6
44.7 32.2 86.2 83.0 105.0 112.0 85.0 87.2 110.0 55.0
E talc -1258.6 -1287.2 -1315.7 -1321.0 -1342.9 -1349.4 -1403.5 -1463.8 -1721.7 -1758.7 - 1793.6 -1805.3 -1804.7 -1816.2 -1866.2 -1868.4 -1893.2
E exP -1257.4 -1287.1 -1315.8 -1320.9 -1343.4 -1350.2 -1404.1 - 1465.1 -1722.7 -1759.6 -1794.6 -1807.0 -1806.2 -1817.2 -1868.2 -1870.8 -1895.3
A = 200. At least a large part of this effect must be due to the excessive pairing already noted in sect. 4, but there could also be a contribution coming from a systematic underestimate of E,, : it is very hard to disentangle the two effects. Pairing matrix elements. The pairing matrix elements G,, are so numerous that if we had attempted to interpolate them directly the storage problem, already difficult, would have become impossible. We therefore computed for each key nucleus at each key deformation an “equivalent constant-G”, Geq,q, defined separately for neutrons and protons as in sect. 2. It is these G,, that are interpolated, rather than nucleus at a given deformation is then the G,,; the energy of each interpolated determined by performing a constant-G BCS calculation with the interpolated s.p. energies and Geg. Since the constant-G
BCS calculations
are much more sensitive
than the b-force
BCS calculations to the number of s.p. states taken into account, this number, which we denote by V, must vary slowly with respect to the number of nucfeons, n (= iV or 2). The actual prescription that we adopted is u = n + 2.5r1~/~ , which corresponds roughly to all states lying up to lhw above the Fermi level. When a key nucleus is magic, for the deformation in question, the &pairing force has no or very little effect on the total energy, so that our prescription for determining G,, breaks down. We handle this problem quite simply by applying our prescription to the two neighbouring nuclei, which will certainly be non-magic (for the given deformation), and then interpolating the two values of G,, linearly.
J.M. Pearson et al. / Nuclear mass formula (Ill)
Interpolation procedure. We interpolate (see below complete points
for the actual ETFSI
methods
calculations
in the (iV, Z) plane,
over the whole prohibitive,
nuclear
as shown
in the (N, 2) and (c, h) planes
of interpolation).
are performed
chart
and the overall
separately
The key nuclei
are chosen
at once,
but the memory excessively
on which
to lie at regularly
in fig. 6. In principle,
organization
27
one could requirements
complex.
lated over separate square blocks of 16 ( = 4 x 4) key nuclei; in fig. 6. The entire nuclear chart is covered by such blocks,
spaced
interpolate
We therefore
would
be
interpo-
a typical block is seen which are all inclined
at 45” and allowed to overlap, as shown in fig. 7 (the key nuclei common to two overlapping blocks need only be calculated once, of course). In most of the square blocks the spacing of the points representing key nuclei is IANI = IAZl = 6, so that there will be a total of 685 ( = 19*+ 18’) nuclei altogether in each of these blocks. Thus interpolating over the (N, Z) plane leads to a time-saving factor of 685/ 16 = 42.8 (actually, slightly more because of the key nuclei lying on the common boundary between adjacent blocks). For the single square covering nuclei with A < 60 we take IANI = lAZ[ = 4, simply because of the convergence of the drip lines (stability problems in the ETF part of the calculation arise for nuclei too far beyond the drip lines). In this smaller square there will be 313 ( = 13*+ 12’) nuclei. As for interpolation in the (c, h) plane, we find that all ground-state deformations fall in the parallelogram shown in fig. 8. All of the key nuclei are calculated at each of 56 key deformations inside the parallelogram. Actually, if we define x = c, y = c + h, our 56 points correspond to 8 values of x and 7 values of y.
.
.
.
.
.
.
.
.
Fig. 6. Typical 4 x 4 interpolation block in (N, Z) plane.
J.M. Pearson et al. / Nuclear mass formula (III)
28
I20
100
80
Z 60
20
40
60
Fig. 7. Mapping
80
of nuclear
100
120
chart by 4 x 4 interpolation
h (0.75, 0.64)
(0.75,O.lO)
Fig. 8. Interpolation
block in plane
160
140
of deformations.
blocks.
180
J.M. Pearson et al. / Nuclear mass formula (III)
29
With all 16 key nuclei of a given block computed at the 56 key deformations, construction of the mass table proceeds as follows. For a given deformation we interpolate to each of the 685 (or 313) nuclei in the block. In this way we obtain complete results for every nucleus of the block at each of the 56 key deformations. Finally, to determine the mass and deformation of the ground state of any particular nucleus we interpolate that nucleus over the (c, h) plane to each of 625 (25 x 25) points inside the parallelogram of fig. 8, and pick out the energy minimum. Tests showed that this number of points in the (c, h) plane was appropriate to a determination of the energy minimum without significant loss of precision. Thus interpolation over the (c, h) plane leads to a further time-saving factor of 625/56 = 11.1. It follows that the joint interpolation over the (N, 2) and (c, h) planes leads to the computer time being reduced by a factor of about 400. Since the ETFSI method itself is some five times faster than HF for a given deformed nucleus (sect. 2), we see that the overall time-reduction factor compared to HF (for which interpolation is impossible) is about 2000. In searching for the energy minimum one has the option of including the rotational correction (26) before minimization or adding it afterwards. The former procedure is the more logical, and our mass table is indeed constructed in this way. Nevertheless, we also studied the second option; see sect. 6. Whichever way the minimum is defined, we also determine by interpolation the energy of the spherical configuration for each nucleus, thereby obtaining the deformation energy. We have seen (sect. 2) that the dimensionality of our deformed code is sufficient only for this latter quantity, and not the total energy. Thus our final mass values are obtained by adding the interpolated deformation energy to the total energy of the spherical configuration, as calculated by the spherical code. The actual method used for interpolation over the (c, h) plane is a simple two-dimensional application of the cubic spline (CERN routine). Suppose that we wish to interpolate to the point (X, Y). Then for each of the 7 key values of y, yi, we interpolate in the x-direction to the 7 points (X, yi). A final eighth interpolation in the y-direction is then made over these 7 points. The same procedure could have been adopted for the interpolation over the (iV, 2) plane also, but since each linear interpolation is over only 4 points a special feature emerges. Normally, the cubic-spline method imposes continuity of first and second derivatives of the cubic expressions at each interpolation point. But with only 4 points it is possible to require continuity of the third derivatives as well, which means simply that the four points are fitted with a single cubic expression. In other words, this generalized 4-point cubic spline reduces to Lagrange interpolation (see, for example, Scheid “)). The Lagrange method itself can easily be generalized to two dimensions, the interpolating polynomial being in general
p(x, Y)= f
k=l
i _f(xksfi)Lm,k(x)Ln,j(Y) j=l
3
J. M. Pearson et al. / Nuclear
30
where f(xk, vj) is the function one-dimensional
Lagrange &k(X)
to be interpolated,
mass formula
(III)
and we have introduced
the usual
polynomials =
l? i=l(#k)
(x-x,) ~ (Xk
k-l,2
-Xi)
,...,
m,
(36)
’
etc; in our case m = n = 4, of course. Some idea of the loss of accuracy incurred by our joint interpolation over the (c, h) and (N, Z) planes can be had by taking a number of nuclei, selected more or less at random, and recalculating the energy by a direct application of the ETFSI method at the equilibrium deformation, as given by interpolation. The results are shown in table 11, where Eint denotes the final interpolated energy, i.e., the energy TABLE Errors of interpolation;
A
Edi’
AE
11
see text for explanations
A
2
E’“’
Ed”
2
E’“’
36
15
-301.0
-301.2
-0.2
230
81
-1711.9
-1712.0
81
40
,678.7
-678.9
-0.2
230
82
-1722.4
-1722.3
100
38
-837.4
-837.4
231
90
-1758.6
-1758.7
0.0
AE -0.1 0.1 -0.1
106
44
-907.3
-906.8
0.5
232
90
-1765.5
-1765.6
-0.1
149
62
-1231.7
-1231.5
0.2
236
92
-1789.1
-1789.2
-0.1
150 150
58 62
-1230.1
-1230.2
237
87
-1778.9
-1779.1
-0.2
-1240.0
-1239.8
0.2
237
93
-1793.8
-1793.6
151
62
-1245.9
-1245.7
0.2
238
84
-1763.5
-1763.3
152
62
-1253.9
-1253.9
0.0
238
96
-1794.6
-1795.1
153
62
-1259.6
-1259.5
0.1
239
93
-1805.5
-1805.3
153
64
-1258.9
-1258.6
0.3
239
94
-1804.5
-1804.7
154
64
-1267.7
-1267.6
156
62
-1280.0
-1280.1
157
63
-1287.2
-1287.2
161
65
-1315.7
162
67
-1321.1
162
66
162
72
-0.1
0.2 0.2 -0.5 0.2 -0.2
240
82
-1750.0
-1750.0
240
94
-1812.0
-1812.0
0.0
240
95
-1808.7
-1808.8
-1315.7 -1321.0
0.0 0.1
240 240
98 99
- 1800.8
-1800.7
0.1
-1793.3
-1793.2
0.1
-1324.1
-1324.1
0.0
241
95
-1816.3
-1816.2
0.1
-1300.6
-1300.3
0.3
246
94
-1845.2
-1845.2
0.0
0.1 -0.1
0.0 0.0 -0.1
165
66
-1342.5
-1342.9
-0.4
246
97
-1843.2
-1842.7
166
67
-1349.2
-1349.4
-0.2
246
100
-1835.0
-1835.3
-0.3 -0.5
0.5
168
76
-1326.4
-1326.3
0.1
246
103
-1813.9
-1814.4
172
68
-1391.4
-1391.2
0.2
250
96
-1868.2
-1868.2
174
70
-1406.4
- 1406.2
0.2
250
97
- 1866.4
-1866.2
174
71
-1403.5 -1463.8
0.7 -0.1
250
-1862.9
-1863.0
251
100 100
-1868.4
-1868.4
0.1
252
98
-1879.3
-1879.1
252
99
-1876.1
-1876.2
-0.1 -0.1
183
74
-1404.2 -1463.7
184
74
-1471.7
-1471.6
194
76
-1536.7
-1537.1
-0.4
0.0 0.2 -0.1 0.0 0.2
204
81
-1606.8
-1606.8
0.0
255
100
-1893.1
-1893.2
204
87
-1575.4
-1575.4
0.0
256
100
-1900.1
-1900.2
-0.1
205
79
-1612.5
-1612.5
0.0
256
104
-1887.4
-1888.2
-0.8
214
74
-1601.5
-1601.1
0.4
258
103
-1903.8
- 1904.2
-0.4
214
82
-1663.8
-1663.8
0.0
260
106
-1906.4
-1906.6
-0.2
224
90
-1716.2
-1716.3
264
97
-1932.2
-1932.2
225
90
-1721.7
-1721.7
-0.1 0.0
0.0
31
J.M. Pearson et al. / Nuclear mass formula (III) TABLE
12
Analysis of interpolation errors; see text for explanations
106 165 174 194 238 246 246 256
44 66 71 76 96 97 103 104
corresponding
0.82 1.21 1.18 0.89 1.25 1.21 1.21 1.16
0.3 1 -0.01 0.08 0.11 -0.19 -0.10 -0.10 0.02
-907.3 -1342.5 -1404.2 -1536.7 - 1794.6 -1843.2 -1813.9 -1887.4
to the mass tables,
-906.8 -1342.9 -1403.5 -1537.1 -1795.1 -1842.7 -1814.4 -1888.2
0.22 -0.03 0.15 -0.04 -0.04 -0.10 -0.08 -0.06
and Edir the energy
-0.10 -0.04 -0.20 0.14 0.01 -0.21 -0.20 0.04
given directly
-0.20 0.36 -0.33 0.25 0.43 -0.40 0.57 0.70
0.5 -0.4 0.7 -0.4 -0.5 0.5 -0.5 -0.8
by the ETFSI
calculation. The total interpolation errors A, which show no systematic trend, and are equally likely to have either sign, are generally of the order of ho.2 to 0.3 MeV, which is quite acceptable. Occasionally, the error can amount to 0.5 MeV, and in the worst case in our sample it is 0.8 MeV, which could not be tolerated if it were not an isolated instance. In table 12 we consider those cases of table 11 where the total error of interpolation lAE[ z 0.5 MeV, and show how it breaks down into the following three contributions: (i) A&,, , the error in the interpolated equilibrium energy, uncorrected for the spurious rotational energy (26), (ii) A&,, the error in the interpolated rotational correction (26), (iii) A&,,, the error in the interpolated energy for the spherical configuration. It will be seen that the bulk of the interpolation error arises in the last term, i.e., for the spherical configuration, even though this configuration corresponds to a key point in the (c, h) plane, so that only the (N, 2) plane has to be interpolated over. By contrast, the first two contributions to the interpolation errors are always acceptably small, even though they are associated with arbitrary deformations,
and thus involve
interpolations
over both the (c, h) and (N, 2) planes.
The larger interpolation errors found occasionally for the spherical configuration might be due to the large degeneracy of several s.p. levels for this configuration, resulting in a more systematic accumulation of errors in the sum of s.p. energies than in the case of deformed configurations. This problem could be handled, without an appreciable increase in computation time, by taking a finer interpolation grid in the (N, 2) plane for the spherical configuration than for the 55 deformed configurations. 6. Mass table for the SkSCl force Using the interpolation method described in sect. 5, we calculate with the SkSCl force, derived as described in sect. 3, the masses and ground-state deformations of all the 1492 nuclei with A 3 36 whose masses are known. A graphical summary of
32
J.M. Pearson et al. / Nuclear mass formula
the discrepancy values mean -0.014
is given value
E = Me_, - Mcalcbetween our calculated in fig. 9. The rms value of this error
d = 0.049 MeV.
This
with N and 2 2 8 of known parameters,
as compared
masses and the experimental is a,,,,, = 0.868 MeV, and the
is to be compared
MeV in the case of the MMST
(III)
with
E,,,=
mass. We recall,
however,
to 9 in our work. Moreover,
fit to the 491 nuclei
assumed
E=
all 1593 nuclei
that MMST
use some 25
we show in this section
the next how it should be possible to improve our own results. It will be seen that the final results obtained with our force good as for the original
0.769 MeV,
mass table 5), which includes
SkSCl
to be spherical,
and
are not as
for which we
had E,,, = 0.693 MeV (sect. 3). However, the checks that we made in sect. 5 (see table 11) show that interpolation can account only for a very small part of this deterioration. A part of the problem must lie in the systematic underbinding that occurs with our force SkSCl for nuclei beyond A = 240, none of which were included in the original fit. This tendency has already been noted in sect. 3, and is quite apparent in fig. 9. If we exclude these nuclei a,,, falls to 0.810 MeV. We saw in sect. 3 that this underbinding of heavy nuclei must be related to the force and not to the dimensionality of our codes. Now that we have completed the mass table it is possible to make a more systematic study of this question. In fig. 10 we display the underbinding of these heavy nuclei in terms of both 2 and A, limiting ourselves, because of pairing effects, to even values of 2 and odd values of A. It will be seen that the effect we are dealing with is Z-related rather than A-related; 4
t
L
I
,,,,,,l,,,lll\ 20
40
60
I 80
I
I,,
I
I,,
100
11 120
I,
I
‘,
i40
”
I
J
160
N
1 II
III
I!
ll,l,,,,l,,,,lllL1l~
20
40
60
80
I
100
Z
Fig. 9. Error of fit (E = Me._ -MC,,,)
of force SkSCl to mass data as functions of N and Z.
J.M. Pearson et al. / Nuclear mass formula (Ill) 250
240
A
3 260
I
I
O 2=96 98
;
100
X
102
+
104
Fig. 10. Error of fit (E = MC_,- MC,,,) as function
of A for different
heavy elements.
in fact, the binding energy increases slightly for A increasing at constant 2. Thi suggests that the problem lies with the Coulomb energy, this varying roughly a Z2/A-1’3. We can now see that if we had chosen a value of k, slightly smaller than thl 1.335 fm-’ that we actually took, then this problem of underbinding in heavy nucle might have been avoided. In fact, with the constraint of a smaller value of k, i should be possible in a new fit to adjust the Skyrme parameters in such a way tha we not only achieve the required increase in the binding of heavy nuclei but at thl same time correct the trend towards overbinding of light nuclei that is clearl: apparent in fig. 9. It remains to be seen what this would do to the calculated nuclea radii, but in view of the approximate nature of the Fermi distribution (5) one shoulc not insist too strongly on this point. Reducing kF, and with it the Coulomb energy might also help to eliminate the anomalous trends in the fission barriers noted ii the preliminary analysis of sect. 3 [see also ref. 46) for further details]. Another major factor contributing to the deterioration of the fit in going fron the original group of 491 spherical nuclei to the complete mass table lies in thl excessive even-odd mass differences for heavy nuclei, noted in sect. 4; these nucle carry much less weight in the restricted group than in the complete table. It is also apparent from fig. 9 that shell effects are somewhat exaggerated. Taking M* to be a little larger than M might improve this situation. Another degree o freedom which has not been completely exploited and which might help to modulat shell effects is y (this acts through the surface diffusivity, as discussed in sect. 3) Nuclei with N = 2 (even) are often underbound by more than 2 MeV, even thougl they lie in a region of the nuclear chart where the tendency is for overbinding. Thi is a consequence of the Wigner effect, in the sense of quartet correlations that canno be incorporated within the framework of any independent-particle description, am
34
J.M. Pearson et al. / Nuclear mass formula (III)
which
our method
nuclei
that their
does not take into account. effect on E,,, is negligible,
However,
there are so few of these
but our mass formula
should
not be
used for these nuclei. It will be noticed
in fig. 9 that there is a group
W that are anomalously axial symmetry of these
nuclei,
underbound.
Since
we might have here an indication although
previously
of nuclei
in the region
our deformed
performed
code
of triaxiality triaxial
always
of OS and assumes
in the ground
HF calculations
state do not
suggest an effect on the binding energy as large as the one we find 47). S, and Qp. We show in fig. 11 the errors with which our force SkSCl reproduces the neutron-separation energies S, for the Sn, Pb, and Cm isotopes, and compare with the results of MMST. We see that for the Sn isotopes we do as well as, or slightly better than MMST, especially near the magic number N = 82. Actually, while force SkSCl gives a little too much pairing for all Sn isotopes, MMST has this problem only for N < 70, there actually being too little pairing for N > 70. In any case, the error for our force never exceeds 0.5 MeV, which is astrophysically acceptable. As for the Pb isotopes, we see that once again SkSCl has too much pairing for all isotopes, although the error still never exceeds 0.5 MeV. MMST would perform somewhat better, were it not for the serious error that arises close to the magic number
N = 126. However,
our results
for the Cm isotopes
are much worse
Fig. 1 I. Deviation MMST fit (dotted
from experimental neutron-separation energies S, for force SkSCl lines) [E(&) = S,(exp) - SJcalc)]. (a) Sn isotopes (b) Pb isotopes,
(solid lines) and (c) Cm isotopes.
J.M. Pearson et al. / Nuclear mass formufa (III)
35
than those of MMST, and in fact they would be of doubtful reliability in r-process calculations. In fig. 12 we show the errors with which force SkSCl reproduces the experimental beta-decay energies QB (always defined for /3- decay) for A = 154, 155, 234 and 235, and compare with the MMST fit. We do not do as well here as do MMST, particularly for the heavier nuclei, and our errors can be as large as 1 MeV. Again, our problem is that the pairing force is too strong. Actually, our results for the A = 235 chain suggest that it is the neutron pairing in particular that is too strong, and there is some indication of a similar difference between neutron and proton pairing in the A= 155 results as well (in the A = 235 case, at least, the effect is probably too large to be attributable to the moments of inertia). Both the S, and QP results show clearly that we have a growing problem of excess pairing as A increases. Even if this has little effect on the overall rms error of the mass table, it is undesirable in any tabie that is to be used for astrophysical
1
a’;
A=154
1
(MeV) -1
A= 155
A=XE
Fig. 12. Deviatim frOM experimentai beta-decay energies Q@(always defined for @- decay) far force SkSCt (solid lines) and MMST fit (dotted lines) [e( Qs) = Q@fexp) - Qrs(calc)]. (a) A = 154, (b) A = 155, (c) A = 234, (d) A = 235.
36
J.M. Pearson et al. / Nuclear mass formula
(III)
applications, for example. While the need for a smaller value of V, is clearly indicated, we recall that this will lead to a deterioration in the overall fit to the masses (sect. 4), even if the Lipkin-Nogami method is applied. It was always clear that some compromise was necessary, and it now seems that the one we have made here may not be the most favourable, although this only becomes apparent after constructing the entire mass table. One way in which the conflict might be resolved without any such compromise would be to take M” to be slightly bigger than M. But even if this helped us to eliminate the excess pairing in heavy nuclei without reducing the overall precision of the fit to the masses, it would make still worse the problem of too little pairing in nuclei with A < 60 (see fig. 2). Reducing the pairing in heavy nuclei and simultaneously increasing it in light nuclei might be achieved by allowing it to act preferentially in the surface through an appropriate density dependence. Defirmntion configuration. In calculating the masses, the equilibrium deformation of each nucleus is automatically determined. However, this is not a quantity of primary interest in the present work, and we therefore limit our discussion of this aspect to the question of whether or not we predict the correct position in the (N, 2) plane for the transitions from spherical to deformed shapes. In figs. 13-17 we compare for a few isotopic series, selected with this end in view, our calculated ground-state deformation parameters & [see appendix A for the relation to the (c, h) parameters] with the experimental values, as determined from the B(E2) data 48). Note that these data determine only /p 2,/ so that when the calculated value of & is negative we show i/321 as well. In order to facilitate the comparison with experiment we connect the points corresponding to the calculated values of l&j by a solid line; negative calculated values of & are connected by a dotted line. (We
PP
Fig. 13. Expe~mental
and calculated
(SkSCt)
values of & for Ra isotopes.
37
J. M. Pearson et al. / Nuclear mass formula (III)
, I I 1
I I
180
t I
190
?OO :
A
-0.5 Fig. 14. Experimental and calculated (SkSCl) values of & for pt isotopes (dotted Lines correspond to negative calculated values: see text).
-130
140
150
160
Fig. 15. Experimental and calculated (SkSCl) values of & for Sm isotopes.
J.M. Pearson et al. / Nuclear mass formula (III)
38 0.5 r
54X=
P2
0 110
120
Fig. 16. Experimental
and calculated
130
(SkSCl)
values
140
of & for Xe isotopes.
r
r
Fig. 17. Experimental
SkSCl
and calculated (SkSCl) values of & for Zr isotopes negative calculated values: see text).
(dotted
lines correspond
to
J.M. Pearson et al. / Nuclear mass formula (III)
do not consider
here the often tenuous
since it is irrelevant The overall when
we bear
dynamical tions.
to the question
agreement in mind
deformations,
While
between
experimental
of demarcating
while our calculated
both quantities
dynamical shape fluctuations a more significant dynamical
indications the regions
theory and experiment
that the experimental should
39
values values
on the sign of p2 of spherical
is rather
nuclei.)
good, especially
of p2 include
the effect of
relate only to static deforma-
agree for well deformed
nuclei,
in which the
remain quite small compared to the static deformation, contribution to p2 occurs in nuclei that are spherical
or only weakly deformed, with the result that the measured values of p2 in these nuclei are expected to be larger than the values calculated in static theories such as the present. A particularly interesting case is the shape transition in the heavy 4,,Zr isotopes. The B(E2) data show that while the N = 56 isotope is close to spherical, a strong deformation has set in at A = 100. (The experimental excitation spectra indicate that the transition is even sharper: while isotope N = 58 is close to spherical, isotopes with N 2 59 are strongly deformed.) We show our results for force SkSCl in fig. 17, while figs. 18 and 19, respectively, show the results for HF calculations with the S3 parametrization of the Skyrme force 49), and for the MMST calculation “). It will be seen that we reproduce this situation better than do either of the latter two calculations, which both predict a rather smooth transition beginning already at N = 54. We believe that the problem with the S3 force is a consequence of a wrong level ordering “) arising from an excessively high value of y, on which the surface diffusivity, and hence the s.p. potential, depend, as explained in sect. 3. As for the MMST calculation, we speculate that here too the s.p. potential is wrong, for the simple reason that the adopted parametrization (folded Yukawa) does not permit the diffuseness to vary off the stability line; such limitations are automatically avoided in the HF method, and also in the ETFSI approximation Of course, our results of fig. 17 have been obtained by including correction (26) before minimization, as throughout our calculations. interesting to note that we obtain a still better description of the from spherical to deformed shapes in the heavy 4oZr isotopes if correction after minimization: see fig. 20 (we denote this case
thereto. the rotational However, it is sharp transition we include this by SkSCl*). In
particular, while the N = 96-98 isotopes are oblate in the former calculation they are spherical in the latter. Actually, we must conclude that we are unable to obtain an unambiguous description of this shape transition because spherical-oblate energy differences smaller than 0.5 MeV, as in 96-98Zr, cannot be considered as significant in any model (droplet, ETFSI, or HF) that does not include a treatment of zero-point motions. At the same time, it will be seen by comparing figs. 17 and 20 that the shape transition in the light Zr isotopes is much better described by the usual prescription of including the rotational correction (26) before minimization. ExtrupoZution. Despite the success with which our force follows the experimental deformation trends, we have shown that the fit to the masses can be improved. We
40
J. M. Pearson et al. / Nuclear mass formula (III)
-0.5
L
Fig. 18. Experimental and caicuiated (force S3) values of & for Zr isotopes (dotted Iines correspond to negative calculated values: see text).
shall, therefore, not extrapolate the present mass table out to the drip lines, but rather await the improvements that will be relatively easy to make, and which are, in fact, already underway. Nevertheless, before embarking on still more computation, it is worthwhile to compare some of the predictions of our mass formula with those of the best existing microscopic-macroscopic formula, that of MMST ‘), in order to see whether any differences can be expected. The extension of the latter table out to the drip lines has not yet been published, but Dr. Peter MSller has kindly sent us some of his preliminary results. We consider here just the case of the Pb isotopes, plotting in fig. 21 the difference between the MMST results and the ETFSI results for force SkSCl. As we go beyond the heaviest known isotope we see that we begin to underbind significantly compared to MMST. This can easily be understood from eqs. (29) and (30) in terms of our value of ass = -18.4 MeV, as compared to -80.8 MeV in the case of MMST: the coefficient of I2 in eq. (29) is larger in our case, even though f is smaller. But beyond N = 170 the MMST energy climbs much more rapidly than ours, until at the magic number iV = 184 we have some 6.0 MeV more binding than MMST, a clear indication of the expected malacodermous effects 8), as discussed in sect. 3. One may expect,
J.M. Pearson et al. / Nuclear mass formula (III) 0.5
P*
41
r
O
-0.5
Fig. 19. Experimental and calculated (MMST) values of pz for Zr isotopes (dotted lines correspond to negative calculated values: see text).
may expect, therefore, that ETFSI will predict that the n-drip line lies further out than will MMST. Nevertheless, although these differences between the predictions of MMST and SkSCl are significant, we note that there are still wider differences between MMST and earlier versions of the droplet model 51,52). We have also compared the extrapolation of force SkSCl with that of SkSC2. Since the models are identical here what is being tested is the importance of the quality of fit to the data, which is slightly worse for the latter force on account of the different symmetry properties. Limiting ourselves to the Pb isotopes, calculated in the spherical approximation, we show the difference by the dotted line in fig. 21, and see that it never exceeds 2 MeV. This shows that the choice of the underlying model can be more important than the precision of the fit. Comparison with other Skyrmeforces. Although the force SkSCl presented here is not to be considered as our final force, it is interesting to compare it with other Skyrme-type forces currently available. Perhaps the best known of these are S3 [ref. ‘“)I and SkM* [refs. 9,53);in this latter paper the force is called “SkM modified”], and it is interesting to compare these forces with our SkSCl. For doubly-magic nuclei close to stability both these forces give fits to the masses
J.M. Pearson et al. / Nuclear mass formula (III)
42
-0.5
1
Fig. 20. Experimental and calculated {SkSCI*) values of & for Zr isotopes (dotted lines correspond to negative calculated values: see text).
comparable to our own, but the published results show that S3 works much less well than our force for open-shell nuclei *‘). This is probably because for S3 the effective mass (M* = 0.76M at nuclear-matter densities) has been chosen to fit the deep-lying s.p. levels, while our value of M* = M fits rather the levels close to the Fermi surface. (53 would give improved masses if particle-vibration coupling were taken into account “).) For similar reasons SkM* (M* = 0.79M at nuclear-matter densities) is not expected to give very good masses for open-shell nuclei, although nothing has been published on this point. In any case, SkM* deteriorates badly away from the stability line, even for closed-shell nuclei. Another respect in which our force is found to perform better than 53 concerns the incompressibility K, which at 350 MeV is excessively high in the case of S3. As discussed in sect. 3, this is linked with the surface diffusivity, and hence will have an impact on the shell structure. We have seen above that this might explain why we have a better description of the onset of deformation in the heavy Zr isotopes. As for fission barriers, 53 is known to perform very badly 54), while semi-classical calculations show that SkM* is much more promising in this respect, at least close
J.M. Pearson et al. / Nuclear mass formula (III) -
MWMSl’I-M6kSCl)
------
M 6kSC2) - M (SkSCI)
43
Rangeoi MassData
&
(MN
-
-5 -
Fig. 21. Comparison of extrapolations for Pb isotopes.
to the stability line 9*53).H owever, the necessary shell-model corrections to these semi-classical calculations (or complete HF calculations) for the SkM* fission barriers have not yet been performed, and the value of M* leads one to expect that the final results might not be very satisfactory 13). In any case, we have already noted that SkM* is not reliable for nuclei far from stability. We conclude that for the purpose of calculating masses, fission barriers, and, in general, potential-energy surfaces, especially in the astrophysically crucial regions far from stability, our force SkSCl is superior to any other Skyrme force presently available. However, we have in no sense produced a universal Skyrme-type force: its effective mass of M* = M makes it quite unsuitable for calculating giant resonances, for example. In comparing our force with other Skyrme-type forces it should, of course, be realized that because of the slight differences between the HF and ETFSI approaches ‘,*) the force SkSCl should only be used with an ETFSI code, not a HF code. It would, however, be quite easy to modify SkSCl in such a way as to obtain comparable results in HF calculations.
7. Concluding remarks In the present work we have presented a preliminary version of a complete mass table based on the ETFSI method. Although strictly speaking a microscopicmacroscopic method, the underlying Skyrme-type force establishes a much greater
44
J.M. Pearson et al. / Nuclear mass formula
(III)
level of unity between the two parts than is usual. In fact, we have shown lv2)that
the method is essentially equivalent to the HF method, in the sense that for a given from the data out to the drip lines, and to strongly deformed shapes. Nevertheless, we have seen how with a given force a mass table can be constructed some 2000 times more rapidly with the ETFSI method than with the HF method, The present mass table is based on a set of force parameters, SkSCl, that were fitted to a restricted sample of just 491 spherical nuclei. With this force we calculated the masses of all 1492 nuclei with A236 for which mass data are available. The rms discrepancy between our results and experiment is E,,,,~= 0.868 MeV, which is to be compared with E,,~= 0.769 MeV in the case of the MMST mass table 5), the most refined droplet-model formula available. Since we use only 9 parameters, as compared to some 25 in the case of the MMST table, one would in principle have greater confidence in extrapolations based on our mass formula. However, we note the slightly worse value of E,,, in our case, and recall also the problem with the neutron separation energies, S,, and beta-decay energies, Qp. Although our errors in these latter quantities may have relatively little impact on the overall mass fit, they are significant in the astrophysical context. Nevertheless, we have shown in sect. 6 that even though our force SkSCl has only 9 parameters we did not exhaust all possible degrees of freedom in constructing it, and that it should thus be possible to improve the force substantially. It is for this reason that we regard the present mass table as preliminary, and refrain from extending our mass table out to the drip lines until this improvement has been achieved. The degrees of freedom likely to permit an improved fit to the data correspond to kr, y, M”, and the pairing. Only one other parameter is still available, x,, but our experience with the SkSC3 force (sect. 3) shows that this is almost totally ineffective. We point out now that much improvement of the force should be possible without repeating all the vast amount of computation that has gone into the present work. In particular, refitting the Skyrme parameters with the small changes in kF that we contemplate is unlikely to change appreciably either the deformation parameters at equilibrium or the associated deformation energy. Thus, provided M*, y, and the pairing remain unchanged, we can use the deformation energies calculated here with force SkSCl to renormalize all measured masses to their “equivalent sphericalconfiguration” values. Subsequent fits of the force can then be made to all nuclei, spherical and deformed, while always assuming a spherical configuration, thereby keeping the computer time within the limits of feasibility*. As for reducing V,,, which will have to be done if acceptable S, and QP are to form of Skyrme force the two methods give very similar extrapolations
* Note added in proofi This refit of the force to ail 1492 masses, with unchanged M*, y, and pairing, has now been performed: E,_, s_ falls to 0.729 MeV, with the optimal value of k, remaining (somewhat unexpectedly) unchanged at 1.335 fm-‘, while I becomes 27.0 MeV.
J.M. Pearson et al. / Nuclear mass formula (Ill)
45
be obtained, it is unlikely that much additional work will be necessary. New deformation energies may be obtained with the already interpolated s.p. energies, using new values of Geq,qadjusted according to the modification of V,, but no other new calculations on the key nuclei will be necessary. Refitting the Skyrme parameters will then proceed as described above. To conclude, we have shown in the present paper that the ETFSI method provides a feasible approach to a mass table having an accuracy comparable to, if not better than, the best currently available mass formulas based on the droplet model. The fact that this can be done with far fewer parameters means that ETFSI mass formulas will provide more reliable extrapolation, both out to the drip lines, and to extremely deformed shapes. Even though our present best force, SkSCl, is not fully acceptable, we have shown how it might be improved. We stress that while there were clear indications at the outset that there might be defects in this force, improvement could not have been made without first calculating the present mass table with it. We wish to thank P. MGller for several helpful communications, and P. Haustein for sending us the 1988 mass compilation prior to publication. The continuing interest of M. Brack is acknowledged. We are indebted to B. Jennings for first drawing our attention to the potentialities of the SI method. The extremely generous allocation of computer time by the Centre de Calcul at the Universite de Montreal is gratefully acknowledged, and we would like to express our deep appreciation for the invaluable service rendered by Jean-Pierre Ponnau and Daniel Raymond. Appendix RELATION
BETWEEN
(c, h) AND
(p2, p4) PARAMETRIZATION
The reference surface used in our parametrization of the deformed density distributions with axial and left-right symmetry (see sect. 2 and refs. ‘“)) is defined in terms of the parameters (c, h) as follows. Working in cylindrical coordinates (17,z), we have
n’~r’-z2=(c’R’-z’)(A+B~), B>O
(A.la)
~2=r2-z2=A(c2R2-z2)exp
(A.lb)
or B
(note the error in the second line of eq. (3.2) of ref. ‘)). Here A and B are defined in terms of c and h according to B=2h+;(c-1),
A=c-~-+B
(A-2)
while R is just the radius of the reference surface in the spherical limit, defined in eq. (3.6) of ref. ‘).
J.M. Pearson et al. / Nuclear mass formula (III)
46
With z = r cos 8, we can rewrite which can be expanded
eq. (37) for the surface
in spherical
r=R{l+&Y,O(cos
harmonics
according
e)+p,Y,“(cos
!I)+.
in polar
form,
r = r(e),
to * *},
64.3)
where p,+
7r I( 19)Y,O(cos 0) sin I0
8 de.
(A-4)
Computation of the multipole coefficients &, is now straightforward (for the case B < 0 we must make the usual assumption that Ip) < l), and we have constructed a table for p2 and p4 in terms of c and h over a sufficiently fine mesh. Graphical representations of this transformation have already been given in ref. 14) (Fig. VS), and ref. 55) but neither is easy to use. Moreover, ref. 14) has a limited range of application (c > l), and we find occasional small disagreements with our own table, no doubt because of the different method used in that work (least-squares fit of the multipole coefficients). On the other hand, we agree completely with ref. 55). References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28)
A.K. Dutta, J.-P. Arcoragi, J.M. Pearson, R. Behrman and F. Tondeur, Nucl. Phvs. A458 (1986) 77 F. Tondeur, A.K. Dutta,J.M. Pearson and R. Behrman, Nucl. Phys. A470 (1987), 93 B. Grammaticos and A. Voros, Ann of Phys. 123 (1979) 359 B. Grammaticos and A. Voros, Ann of Phys. 129 (1980) 153 P. Moller, W.D. Myers, W.J. Swiatecki and J. Treiner, At. Data Nucl. Data Tables 39 (1988), 225 P. Miiller and J.R. Nix, Nucl. Phys. A361 (1981) 117 W.D. Myers and W.J. Swiatecki, Ann. of Phys. 55 (1969), 395 A.K. Dutta, J.-P. Arcoragi, J.M. Pearson, R. Behrman, and M. Farine, Nucl. Phys. A454 (1986), 374 M. Brack, C. Guet and H.-B. Hakansson, Phys. Reports 123 (1985) 275 M. Barranco and J. Treiner, Nucl. Phys. A351 (1981) 269 V. Bernard and Nguyen Van Giai, Nucl. Phys. A348 (1980) 75 F. Tondeur, M. Brack, M. Farine and J.M. Pearson, Nucl. Phys. A420 (1984) 297 F. Tondeur, Nucl. Phys. A442 (1985) 460 M. Brack, J. Damgaard, AS. Jensen, H.C. Pauli, V.M. Strutinsky and C.Y. Wong, Rev. Mod. Phys. 44 (1972) 320 M. Brack and P. Quentin, Nucl. Phys. A361 (1981) 35 F. Ivanjuk and V.M. Strutinsky, Z. Phys. A286 (1978) 291 F. Tondeur, J.-P. Arcoragi and J.M. Pearson, J. de Phys. 45 (1984) C6-125 Y.H. Chu, B.K. Jennings and M. Brack, Phys. Lett. B68 (1977) 407 M.N. Butler, D.W.L. Sprung and J. Martorell, Nucl. Phys. A422 (1984) 157 M. Beiner, H. Flocard, Nguyen Van Giai and P. Quentin, Nucl. Phys. A238 (1975) 29 F. Tondeur, Nucl. Phys. A315 (1979) 353 J.M. Eisenberg and W. Greiner, Microscopic theory of the nucleus (North-Holland, 1972) pp. 332-336 C. Ko, H.C. Pauli, M. Brack and G.E. Brown, Nucl. Phys. A236 (1974) 269 D.R. Inglis, Phys. Rev. 96 (1954) 1059; ST. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31 (1959) No. 11 A.H. Wapstra, G. Audi and R. Hoekstra, At. Data Nucl. Data Tables 39 (1988) 281 F. Tondeur, Nucl. Phys. A303 (1978) 185 J.M. Cavedon, B. Frois, D. Goutte, M. Huet, P. Leconte, X.H. Phan, S.K. Platchkov, C.N. Papanicolas, S.E. Williamson, W. Boeglin, I. Sick and J. Heisenberg, Phys. Rev. Lett. 58 (1987) 195 P. Gleissl, M. Brack, J. Meyer and P. Quentin, Ann. Phys. 197 (1990) 205
J. M. Pearson et al. / Nuclear mass formula (Ill) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48) 49) 50) 51) 52) 53) 54) 55) 56) 57) 58) 59)
l
47
M.M. Sharma, W. Stocker, P. Gleissl and M. Brack, Nucl. Phys. A504 (1989) 337 F. Tondeur, D. Berdichevsky and M. Farine, Z. Phys. A325 (1986) 405 H. Miither, M. Prakash and T.L. Ainsworth, Phys. Lett. B199 (1987) 469 B. ter Haar and R. Malfliet, Phys. Rev. Lett. 59 (1987) 1652 P.J. Siemens, Nucl. Phys. A141 (1970) 225 H.A. Bethe, G.E. Brown, J. Applegate and J.M. Lattimer, Nucl. Phys. A324 (1979) 487 B. Friedman and V.R. Pandharipande, Nucl. Phys. A361 (1981) 502 J. Cugnon, P. Deneye and A. Lejeune, Z. Phys. A328 (1987) 409 Nguyen Van Giai and H. Sagawa, Phys. Lett. B106 (1981) 379 M. Farine, J.M. Pearson and B. Rouben, Nucl. Phys. A384 (1978) 317 M. Farine, J. C&e and J.M. Pearson, Phys. Rev. C24 (1981) 303 W. Stocker, J. Bartel, J.R. Nix and A.J. Sierk, Nucl. Phys. A489 (1988) 252 W.D. Myers and W.J. Swiatecki, Ark. Fys. 36 (1967), 343 D.G. Madland and J.R. Nix, Nucl. Phys. A476 (1988) 1 H.C. Pradhan, Y. Nogami and J. Law, Nucl. Phys. A201 (1973) 357 A. Bohr and B.R. Mottelson, Nuclear Structure, vol. 2 (Benjamin, New York, 1975) F. Scheid, Schaum’s Outline of Theory and Problems of Numerical Analysis (McGraw-Hill, New York, 1968) p. 53 F. Tondeur, Y. Aboussir, A.K. Dutta, and J.M. Pearson, Proc. of AMCO 8 Conf. (1990, Jerusalem)* N. Redon, J. Meyer, M. Meyer, P. Quentin, M.S. Weiss, P. Bonche, H. Flocard and P.-H. Heenan, Phys. Lett. B181 (1986) 223 S. Raman, C.H. Malarkey, W.T. Milner, C.W. Nestor Jr. and P.H. Stelson, At. Data Nucl. Data Tables 36 (1987) 1 P. Bonche, H. Flocard, P.H. Heenan, S.J. Krieger and M.S. Weiss, Nucl. Phys. A443 (1985) 39 F. Tondeur, Nucl. Phys. A359 (1981) 278 L. Spanier and S.A.E. Johansson, At. Data Nucl. Data Tables 39 (1988) 259 W.D. Myers, At. Data Nucl. Data Tables 17 (1976) 411 J. Bartel, P. Quentin, M. Brack, C. Guet and H.-B. Hakansson, Nucl. Phys. A386 (1982), 79 H. Flocard, P. Quentin, D. Vautherin, M. VtnCroni and A.K. Kerman, Nucl. Phys. A231 (1974) 176 U. G&z, H.C. Pauli, K. Alder and K. Junker, Nucl. Phys. Al92 (1972) 1 H. De Vries, C.W. De Jager and C. De Vries, At. Data Nucl. Data Tables 36 (1987), 495 L.G. Moretto, S.G. Thompson, J. Routi and R.C. Gatti, Phys. Lett. B38 (1972) 471 A. Khodai-Joopari, Rep. UCRL-16489 (1966), Univ. of California Radiation Laboratory, Berkeley (unpublished) B.B. Back, 0. Hansen, H.C. Britt and J.D. Garrett, Phys. Rev. C9 (1974) 1924
Note added in proof: Conference
cancelled.