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Nuclear masses from a unified macroscopic-microscopic model
ATOMIC
DATA AND NUCLEAR
NUCLEAR
DATA TABLES
39,2 13-223 ( 1988)
MASSES FROM A UNIFIED P. MILLER Theoretical
MACROSCOPIC-MICROSCOPIC
MODEL
and J. R. NIX
Division, Los Alamos National Laboratory Los Alamos, New Mexico 87545
We calculate ground-state masses for 4678 nuclei ranging from I60 to 3’8122 by means of an improved version of the macroscopic-microscopic model employed in our 198 1 mass calculation, which uses a Yukawa-plus-exponential model for the macroscopic term and a folded-Yukawa singleparticle potential as starting point for the microscopic term. Some of the new features now incorporated are a new model for the uveruge pairing strength, the solution of the microscopic pairing equations by use of the Lipkin-Nogami method with approximate particle-number conservation, the use of experimental mass uncertainties in determining the model parameters, and an estimation of the theoretical error of the model by application of the maximum-likelihood method. Only five parameters are determined from least-squares fitting to the nuclear masses; the other constants in the model are taken from previous work with no adjustment. The resulting theoretical error in the calculated ground-state masses of 1593 nuclei ranging from I60 to *‘j3106 is 0.832 MeV. We also extend the calculation to some additional nuclei in the heavy and superheavy region that were not considered in 198 1. The present calculation represents an interim report on a project in which we plan to make further improvements and extend the region of nuclei considered to the neutron and proton drip lines. 0 1988 Academic P, Inc.
0092-640X/88 $3.00 Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.
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Data and Nuclear
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P. MtSLLER AND J. R. NIX
Unified Macroscopic-Microscopic
Model
CONTENTS 1. INTRODUCTION
.....................................
2. MACROSCOPIC-MICROSCOPIC MODEL 2.1. Macroscopic Model ................................ 2.2. Microscopic Model .................................
214 ................
2 14 2 15 217
3. ESTIMATION OF PARAMETERS AND ERROR 3.1. Maximum-LikelihoodMethod :::::::::: ............. 3.2. Resulting Formulas ................................. 3.3. Previous Models for Estimating flth ....................
217 218 2 19 220
4. CALCULATED
221
MASSES
...............................
1. INTRODUCTION
state masses’ for 4023 nuclei ranging from I60 to 279112 and fission barriers7 for 28 nuclei throughout the periodic system were calculated. The root-mean-square deviation between experimental and calculated groundstate masses was 0.835 MeV for a set of 1323 masses and 1.33 1 MeV for the 28 fission barriers. Many other prop erties such as ground-state deformations were also well described by that calculation, as is extensively discussed in Ref. 9. The model represents a unified approach to the study of many features of nuclear structure, fission, and heavy-ion reactions. Here we study the effect of some new features in the treatment of pairing and extend the calculation to additional nuclei in the heavy and superheavy regions. These heavy nuclei were previously studied within the framework of the original model in Ref. 10. Below we present some details of the model that are of particular relevance to the present study, but refer the reader to Refs. 5-9 for a more complete presentation of the original model. In the macroscopic-microscopic model the nuclear energy, which is calculated as a function of shape,
The advent of the Strutinsky shell-correction method1p2 about 20 years ago made it possible to culculute the shell correction in nuclear mass models of the macroscopic-microscopic type. With this method the potential energy of a nucleus can be calculated for arbitrary shapes, within given shape parameterizations. Coupled with a wealth of new experimental results this has led to an enormous increase in our understanding of nuclear ground-state and fission properties and of the stability of elements at the end of the periodic system. For an extensive review of some of these developments see Refs. 3 and 4. Here we apply the model to the study of nuclear masses.
2. MACROSCOPIC-MICROSCOPIC
MODEL
Our original model has been discussed extensively in Refs. 5-9. The present calculation is similar to the investigation described in Refs. 7 and 8, where ground214
Atomic
Data and Nuclear
Data Tables.
Vol. 39. NO. 2. July 1933
P. MiiLLER
Unified Macroscopic-Microscopic
AND J. R. NIX
proton number Z, and neutron number N, is the sum of a macroscopic term and a microscopic term. Thus the total nuclear potential energy can be written as
exponential model6 for the macroscopic term and a folded-Yukawa single-particle potential” as a starting point for calculating the microscopic term. They are briefly discussed below.
E(Z, N shape) = Emacr(Z, N, shape) + &AZ
N shape).
2.1. Macroscopic
( 1)
Model
In the present version of our model the Yukawaplus-exponential macroscopic energy is given by
There exist several different models for both the macroscopic and microscopic terms. We use a Yukawa-plus-
L&,
Model
N shape) = i&Z
mass excesses of Z hydrogen atoms and N neutrons
+ M,N
-
a,(
1 -
Kv1’)A
volume and volume-asymmetry
+
a,(
1 -
KJ2)&A2’3
generalized surface and surface-asymmetry
+
clplo
direct Coulomb
Z 413 c4 A’/3
+ f(kfrp)
%
- c&N - Z) 1/A, 0,
+
I
energies
A0 energy
+ Cl -$3 -
energies
Z and N odd and equal otherwise
+A,+ a, - drip,
Z and N odd
+&,
Z odd and N even
+a,,
Z even and N odd .
+o,
energy
exchange Coulomb
energy
proton-form-factor
correction
charge-asymmetry
energy
Wigner energy
average pairing energy
Z and N even
- a,1Z2,39
energy of the bound electrons
(2)
where A = N + Z is the mass number and I = (N - Z)/A is the relative neutron excess. This expression differs from the corresponding one used in our earlier calculations’** only in the form of the average pairing energy appearing in the next-to-last term. For the average neutron pairing gap &, and average proton pairing gap ii, we now use1 ‘,I* rB, --s&II2 &=-e
These results, which were derived”“* by use of the BCS approximation applied to a uniform distribution of levels, take into account the dependencies of &, and z\, upon both the relative neutron excess I and the relative surface energy B,. For the average neutron-proton interaction energy 6,, , we now use the new form’ ‘J*
and
The zero reference point for the pairing energy now corresponds to even-even nuclei rather than to halfway between even-even and odd-odd nuclei.
h
(5)
N’/3
i&=-e z’f3r&
-+sI-II1
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P. MtiLLER
AND J. R. NIX
Unified Macroscopic-Microscopic
one obtains
In the above expressions the quantities cl and c4 are defined in terms of the elementary charge e and the nuclear-radius constant ro by 3 e2 “‘=3-K
The proton form factorfis
B,=l-;
+(I
+,,(2+$f-$)e-a
(6)
11
e-2m . (14)
5 3 =I3 . 4 ( 2* 1
c4=q--
Model
(71
The expression B3 for the relative Coulomb energy yields the energy for an arbitrarily shaped, homogeneously charged, diffuse-surface nucleus to all orders in the diffuseness parameter aden. The constants in front of B1 and B3 have been chosen so that BI and B3 are 1 for a sphere in the limit in which the range a and diffuseness aden of the Yukawa-plus-exponential term and of the charge density, respectively, in the macroscopic energy go to zero, in analogy with the definition of the quantities B, and Bc in the standard liquid-drop model.
given by
where r, is the proton root-mean-square Fermi wave number is given by
radius and the
Values of Constants (9)
The constants appearing in the expression for the Yukawa-plus-exponential macroscopic model fall into three categories. ‘*’ The first category, which represents constants that were taken from previous work with no adjustment whatsoever, includes7~8~‘2
The relative surface energy B, , which is the ratio of the surface area of the nucleus at the actual shape to the surface area of the nucleus at the spherical shape, is given by ~-213
A& = 7.289034 MeV
n
Ad,, = 8.071431 MeV e2 = 1.4399764 MeV fm The quantity BI represents the relative generalized surface or nuclear energy in a model that accounts for the effect of the finite range of the nuclear force. It is given by
B, = &
j-j-
(2
- !d)
V
0
a
,““I;‘;, r
a de0 = 0.99/2’j2 fm
ael = 1.433 X 10v5 MeV
&&.
r = 5.72 MeV
(11)
s = 0.118 The relative Coulomb
energy B3 is given by
B3 =
d3rd3r’
t = 8.12
Ir-fl
h = 6.82 MeV
15 A-‘13 577 ss
v
r, = 0.80 fm It includes diffuseness corrections to all orders. For spherical shapes one can calculate the quantities BI and B3 analytically. With roA’J3 x0 = and a
ii-J1f3 y. = -
hydrogen-atom mass excess neutron mass excess elementary charge squared range of Yukawa function used to generate nuclear charge distribution electroniobinding constant preexponential pairing constant linear exponential pairing constant quadratic exponential pairing constant neutron-proton interaction constant proton root-meansquare radius
The values of some of the above constants have been revised by more recent work. However, the changes are so small that the use of the more current values would usually change the calculated masses by less than 0.0 1 MeV.
(13)
aden
216
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Data and Nuclear
Data Tsbles.
Vol. 39. NO. 2. July 1988
P. MijLLER
AND J. R. NIX
Unified Macroscopic-Microscopic
The second category, representing those constants whose values were determined from considerations other than nuclear ground-state masses, includes’** r. = 1.16 fm a = 0.68 fm
where I = (N - Z)/A + O.O112Z2/A5’3 1 + 3.15//Q/3
nuclear-radius constant range of Yukawa-plus-exponential potential surface-energy constant surface-asymmetry constant
a, = 21.13 MeV K, = 2.3
R den= l.16,4”3(l
+!q
The third category represents five constants whose values are determined from a least-squares adjustment to nuclear ground-state masses. These are the constants a,, K,, co, c,, and W, and their values are given later.
N,
shape)
=
&hell(~~
shape)
+
&miXZ,
N
shape).
vs.,.= --A - h 2a*vv, ( 2mc 1
(15)
(19)
V, = (52.5 - 48.78) MeV,
(20)
(24)
xp h
A 240
’
(1
x,=31.5+4.5 Finally, given by
the Coulomb V&) = ep,
(25)
potential s ~lr-fl'
.
for protons
d3r’
(27) is (28)
where the charge density pc is given by (29)
3. ESTIMATION OF PARAMETERS AND ERROR
(18)
VP = (52.5 + 48.78) MeV
(23)
A, = 28.0 + 6.0 &( 1
and
where the integration is over the volume of the generating shape, whose volume is held fixed at $TR,.,,,~~as the shape is deformed. The potential radius R,, potential depth V. = VP for protons, and potential depth V. = V,, for neutrons are given by + 0.82 fm - 0.56 frn2jhen
.
where X is the spin-orbit interaction strength, m is the mass of either a neutron or a proton, u is the Pauli spin matrix, and p is the nucleon momentum. The spin-orbit strength has been determined from adjustments to experimental levels in the rare-earth and actinide regions. It has been shown14y7,9that many nuclear properties throughout the periodic system are well reproduced with X given by a function linear in A through the values determined in these two regions. This gives for X
The first term is the spin-independent nuclear part of the potential, which is calculated in terms of the foldedYukawa potential e-lr-r’lln,t d3rr V,(r) = s (17) pt3 s v lr-+llawt ’
= &,,
+ 0.330z2 + o*“y;z2
of the Yukawa function in Eq. ( 17) has been determined from an adjustment of calculated single-particle levels to experimental data in the rare-earth and actinide regions. It is kept constant for nuclei throughout the periodic system. The spin-orbit potential is given by the expression
Both terms are evaluated from a set of calculated singleparticle levels. As before, the shell correction is calculated by use of Strutinsky’s method.‘s2 For the pairing correction we now use the Lipkin-Nogami’3 version of the BCS method with approximate particle-number conservation. Since our model is extensively discussed in Ref. 5 and references quoted therein, we present here only the major features of the model and some changes relative to that reference. The single-particle potential felt by a nucleon is given by v= v, + v,,. + v,. (16)
R,
(22)
apt = 0.8 fm
Model
N,
+ ?) fm
The range
The microscopic-energy term arises because of the nonuniform distribution of single-particle levels in the nucleus. It is the sum of a shell-correction term and a pairing term: K&Z,
(21)
and A
2.2. Microscopic
Model
It is possible to use several approaches to characterize the error of a model. Here we investigate a model that calculates nuclear masses for a set of Z and N values. For a subset of these Z and N values experimen-
and
217
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Data and Nuclear
Data Tables.
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P. MijLLER
AND J. R. NIX
Unified Macroscopic-Microscopic
tal data values exist. Thus, a very detailed definition of the error is to give the deviation between the calculated values and the available experimental data points. This type of information is available in the Table. However, both the experimental and the theoretical error contribute to this deviation. It is useful to develop an error concept that allows us to characterize the error of a model by a single number that does not contain contributions from the experimental errors. We now introduce such an error concept. In this approach one can compare the error of two models and also compare the error of a model for the region to which its parameters were adjusted to the error for new regions of nuclei. We show in the beginning of Section 4 that our model, with the error definition introduced below, shows no systematic increases in the error for new regions of nuclei far from stability. In most earlier studies, for example those in Ref. 7, the error of a theoretical mass model was taken to be the root-mean-square (rms) deviation
Model
tude. Since we assume that the error is due to a large number of missing terms that are small and fluctuating, we treat the problem with statistical methods. We assume here that there are no systematic effects on the error, but give some comments on this problem at the end of Section 3.2. 3.1. Maximum-Likelihood
Method
A general method for determining unknown parameters of a known statistical distribution if one has observed events distributed according to the distribution is the maximum-likelihood (ML) method. For an indepth discussion of this method, notation, and fundamental concepts of statistics we refer to the books by Cramer” and von Mises. I6 According to the ML method one estimates the parameters so that the events that have been observed have the maximum likelihood of being observed. Suppose that 5’ is distributed according to Ai(x, e), with B unknown. If we have observed t1 = x’, t2 = x2, . . . ) E” = x”, we can determine 6 so that our observations have maximum likelihood. This is done by maximizing the likelihoodfunction L (8), where
L(B) =hl(xI, e)fi(x2,e) . . . A~x”,e).
and. .the. .parameters of the model were determined by mmlmtzmg a,, in Eq. (30). Here ML,, is the measured mass for a particular value of the proton number 2 and neutron number N, and Mfh is the corresponding calculated quantity. There are II such measurements for different pairs of 2 and N. The theoretical model for Mfh depends on 2 and N and also on I)I adjustable parametersn,p2,. -. , pm. The use of the rms deviation (30) is reasonable when all the errors u&, associated with the measurements are small compared to u,, . However, for the large experimental errors a&, that are associated with many recent mass measurements far from stability, the above definition is unsatisfactory, since both the theoretical and experimental errors contribute to u,,,. Consequently the rms deviation (30) is always an overestimate of the true theoretical error. We must therefore use an approach that decouples the theoretical and experimental errors from one another. To proceed systematically we first define precisely what we mean by the error of a theory. We then derive expressions from which this error can be estimated. Our model Mth for nuclear masses is a sum of 20 or so terms. There are a few large terms but many more terms that give fairly small contributions to Mth . This structure of the model has arisen over the years through the addition of new terms to describe previously neglected effects. It is reasonable to assume that there are an even larger number of terms that account for even smaller effects that have been left out of the model. Thus the model has an error associated with these missing terms. We now want to characterize this error and estimate its magni-
Maximizing maximizing
function L(8) is equivalent to In L(B), or to solving
the likelihood the logarithm Wn
(31)
w)i
=
de
o
*
We denote the solution of Eq. (32) by e*(x*, x2, x”). By solving Eq. (32) for several sets of observa...) tions xi’, XT, . . . , x; of the set of random variables 5 ‘, t2, - - * 9F, we obtain several estimates et of the parameter 8 of the distribution functions&(x, 0). Thus 8*(x’, x2, . . . , x”) is an observation of a random variable e*([ ‘, t2, * * - 7 t”) with a distribution function ge=. Since the distribution function of the multidimensional random variable ([l, t2, . . . , t”) is known if theS,i are known (it is actually the product in the right member of Eq. (3 I)), the distribution function of e*([‘, E2, . . . , 5”) can be determined, at least in principle. This is of fundamental importance. It is desirable that its distribution function ge* have certain properties. This is an important field of mathematical statistics, and we refer to the books by Cramer” and von Mises16 for a discussion of these questions. Here we just mention that the ML estimate is consistent, that is, lim E(8*) = 8, lim D2(0*) = 0, n-m n-m
(33)
where E(d*) denotes the mean and D2(O*) the variance of the distribution function of 8*. One can also show that if an eficient estimate B* of 8 exists, then the likelihood equation will have a unique solution equal to 8*. An 218
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P. M6LLER
AND J. R. NIX
Unified Macroscopic-Microscopic
efficient estimate is an estimate that is unbiased and has minimum variance. The estimate is said to be unbiased
unknown terms that are missing from the theory and Cr,h is by definition the error of the theory. We note that in practice there are correlations in the error term, in the sense that the error for calculated masses is of similar magnitude for nearby nuclei. However, this is of no consequence for the considerations below, since we consider a large number of masses throughout the periodic system. Were one to consider the differences between theoretical masses for nearby nuclei, correlations would be important and would result in errors in the differences that are much smaller than \IZQth, which would be expected in the absence of correlations. The experimental quantities are not observations of precisely ufr , but are instead observations of the random variable z)&, = ufr + e& = M:‘h + eL + e&, because there are random errors associated with the measuring process. We assume that e&, E N(0, UC%, i *). The uiXp2 are different for different i because the measuring device is different for each i, but they are all known. From the above conditions we find that u&,
if E(B*) = B
(34)
for all n. However, many ML estimates have bias. For the case in which the ML estimate is biased, one may ask if an unbiased estimate is a better one. However, even if the ML estimate is biased, it is consistent and therefore the parameter estimates are arbitrarily close to the true values for sufficiently large n, provided that the number of parameters does not increase as n increases. One may therefore agree with von Mises (Ref. 16, p. 560) that if for the actual purpose of an investigation the difference between an ML estimate and an unbiased estimate is of importance, a larger sample should have been chosen. For the specific case of nuclear mass formulas, it is not possible to substantially increase the number of data points. Thus one should consider only models in which relatively few parameters are adjusted to the data. If one persists in studying models with a large number of adjustable parameters compared to the available data one should study the effect of bias, since for a large number m of adjustable parameters we expect that (Ref. 16, p.
E N(ii&, uLxp2 + uth2). Thus, for the distribution tion f,i we have
555)
et&,
E(O*) N y
3.2. Resulting
0.
Model
M:‘h,
‘&p*
+
‘-%h2)
=-
(35)
func-
-(X-M~)‘/2(d,,2+o,hz). (36)
The values A4iXp constitute observations of utxp. We can now estimate both the parameters pv of &, and the quantity (Tthby maximizing the likelihood function L. In our case
Formulas
The unknown parameters of our theory can be estimated by use of the ML method only if they are parameters of the distributions involved. In particular, they cannot be parameters of the prescription used to generate the variables of the distribution function. Thus, we must formulate the problem so that A4& becomes a parameter of the distribution. This can be done in the following way. We write ufr = A&, + efh, where ef,, E N(0, ati’) is an error term and A!{,, is our model for the nuclear mass. We are now in a position to give a precise definition of the error of the theory, which we designate by the quantity ati. The variable u& is the true mass of the nucleus i, where i stands for a particular pair of values of the proton number Z and neutron number A? However, uf is also a random variable, by construction. The following somewhat drastic example clarifies this duality. Suppose that when God originally created the Universe and fixed each nuclear mass i, He used our expression for Mfh, with some parameters He fixed and which it is now up to us to estimate. To the number He obtained by calculating M:‘,, He added a random number efh which He obtained by throwing dice such that ei,, E N(0, at,.,*). The random variable ef,, represents the
UPl,
***,
P2,
Pm,
uth*)
= fvb, * ’ ‘fv.(x,
Maximizing tions
dxp* +
M:h,
uth*)
&.p* +
M;h,
flth2).
(37)
L is equivalent to solving the m + 1 equa-
and 5 (M&p
-
Mfh)* (a;,,2
i=l
+
(&;
+
‘?h2*)
=
o
ut,,**)*
’
(3%
Here the notation f&h** means that by solving Eq. (39) we obtain the estimate 6,h2* of the true u,h*. The above equations are equivalent to minimizing S with respect to D,, where _ (40) and solving i i=l
219
(M&p
-
Mfh)* (b;,;
+
b&,2
+
%**)
uth2*)*
Atomic Data and Nudear
=
o ’
Data Tables,
(41)
Vol. 39, NO. 2, July 1988
P. MtjLLER
AND J. R. NIX
Unified Macroscopic-Microscopic
Thus we are led to one additional equation relative to the usual least-squares equations that arise when model parameters are estimated by adjustments to experimental data under the assumption of a perfect theory, that is, uti2 = 0. Equations (38) and (39), or alternatively Eqs. (40) and (4 1), constitute a system of m + 1 equations that are to be solved together. It is instructive to rewrite Eq. (39) as
distributed evenly in the interval 0 to 10. The simulation shows that although the experimental errors are most often much larger than the error of the theoretical model, it was possible to estimate the theoretical error E(B*). In an actual application of the model discussed here only one set of theoretical values exists. In such a case the uncertainty in the estimate of the error E(B*) would be approximately D(e*) m 0.15. From a simulation with a;,, distributed approximately as those for the 200 most recently discovered masses, we obtain D(e*) es 0.05. The case of systematic effects in the error can also be treated by use of the above ideas. If there are systematic effects in Z and Non the error it means that one can write uth as a function of Z, N, and a set of parameters. The equations for estimating the parameters of this error expression can be derived by use of the maximum-likelihood method discussed above. To use the example from above this type of error means that God throws a different set of dice for each mass. We have not carried through such an analysis here, but we conclude from other studies below that there are no systematic trends in the error of the present model over large ranges of Z and N, except that the amplitude of the fluctuations in the error is slightly larger in the light region than in the heavier region. This will be discussed when the results are presented below.
(42)
uth
where 1
Wi =
+ tTth2*p
b&D
and k= 2.
(44)
Written in this way, the unknown uth2* enters on both sides of the equal sign in Eq. (42), but the equation is suitable for an iterative determination of uth2*. We have found that the convergence is extremely rapid. Another benefit of rewriting Eq. (39) in the form given by Eq. (42) is that one may ask if k = 2 is the only possibility, or if other values of k are possible. We shall not discuss this question in depth here but instead will comment that if all a&,2 are equal, then all values of k yield the same equation. However, the more interesting case is how k should be chosen if the u&2 have different values. According to our discussion above, one can prove that the ML estimate has several desirable properties. Therefore, we choose k = 2, which is the ML estimate. To gain some additional insight into the properties of the above method we compare the ML estimate corresponding to k = 2 with estimates obtained with k = 1, 3, and 4 by simulating the random variables efh and e&,. We do several simulations, each with n = 200. In each simulation we generate one set of 200 values for e&. In each case we also generate 2000 sets of 200 values for efh, with uth = 1.O in all cases. Equation (42) is solved for each of the 2000 sets. In this way we estimate the mean and variance of bth2*, which is distributed according to go,. Table A shows the results obtained with the a&,
3.3. Previous Models for Estimating
s’
EC@*)
me*)
0.9884 0.025 11
We*)
0.15848
3
under the auxiliary
1.0177
1.0167
1.0194
0.02583 0.16071
0.02825 0.16807
5
(M&J ‘-&J~
-
M:‘h)’
+
‘-Theu2
(45)
condition
Thus, with this heuristic method the error d&u is determined by requiring that x2 per degree of freedom be equal to 1. In practice, this suggestion leads to results close to those obtained with the formulas (38)-(44) that we have derived here. However, since Eqs. (45) and (46) were obtained heuristically, it is not possible to discuss their significance by accepted statistical methods in a quantitative manner, in contrast to the results in the previous section. Since Eqs. (45) and (46) have been used in some investigations, we interpret them in the following qualitative way. When there are no adjustable parameters, corresponding to ~yt= 0, Eq. (46) is identical
4
0.02307 0.15187
=
i=l
k 2
cth
It was realized earlier that the rms deviation (30) for the error of a mass formula is inadequate when the experimental errors are large. By use of heuristic arguments it was suggested I’,‘* that the parameters and error uh,” be estimated by minimizing S’ with respect to the model parameters, where S’ is given by
TABLE A Results for 0 G uf, Q 10
1
Model
220
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Data and Nuclear
Data Tables,
Vol. 39, NO. 2. July 1968
P. MOLLER
AND J. R. NIX
Unified Macroscopic-Microscopic
to our Eqs. (42) and (43) but with k = 1. However, k = 1 does not correspond to an ML estimate. According to our discussion above, the ML estimate is under certain conditions the estimate with minimum variance. Therefore, the heuristic equations (45) and (46) yield an estimate with a larger variance than do our rigorous equations (38)-(44). The fact that the number of degrees of freedom for x2 is n - m can, in a heuristic way, be expected to reduce the bias of the estimate, and is comparable to our approximate suggestion expressed by Eq. (35). In our previous work’,* we used the rms deviation (30) for the error of our mass formula. For comparison we shall also sometimes give this result here, denoting it by Q,,. For some of our earlier results obtained in this investigation, we use the heuristic method of Eqs. (45) and (46) and characterize the error by (The”.For our final results presented here, we use the more rigorously derived equations (38)-(44) and characterize the error by the quantity (Tth. 4. CALCULATED
Model
A. Although this interpolation accounts for the effect of a changing radius in the single-particle potential, it takes no account of the change in potential as a function of distance from the line of j3 stability. One improvement over the 198 1 calculation is the calculation of the shellplus-pairing correction at the ground-state deformation from a single-particle well appropriate to each particular nucleus. This improvement in the model actually leads to an increase of ah,,, to 0.885 MeV. As the next step we changed our experimental data set from Refs. 19 and 20 to Ref. 22. For the nuclei with 2 >, 8 and N > 8 that we consider, the first set contains 1323 masses with an experimental error less than 1.0 MeV. We now consider all measured masses regardless of the magnitude of the error in the experimental point, provided that it is known. For nuclei with Z 2 8 and N 2 8, the new data set contains 1593 masses. After a readjustment of the five parameters we find that dheUis reduced by about 2%. In our next step we changed the model for the average neutron and proton pairing gaps from that used in our 198 1 calculation to one that takes into account their dependencies on relative neutron excess and nuclear shape.’ ‘*I2 In this model the average neutron pairing gap is different from the average proton pairing gap. Therefore, one cannot, as was done in Ref. 7, use a point halfway between even-even and odd-odd nuclei as the reference point for the odd-even pairing effects. We therefore change the reference point to the even-even system, as indicated in Eq. (2). With the old expressions &, = &, = 12 MeV/A’12 and a,, = 20 MeVIA for the average pairing gaps and residual neutron-proton interaction energy, the effect of changing the reference point to the even-even system increases dheu to 0.882 MeV. When Eqs. (3)-(5) are used for the average pairing strengths, the more rigorous measure of error given by Eqs. (38)-(44) becomes uth = 0.886 MeV. When studying the deviations between calculated and experimental masses, for example those in Fig. 8b of Ref. 7, one observes that the largest fluctuations in the error occur near magic numbers. The BCS method for solving the pairing equations collapses close to magic numbers, or more precisely when the pairing strength G is smaller than some critical value in relation to the gap in the single-particle level spectrum at the Fermi surface.13 In addition, close to the point of collapse the BCS approximation yields inaccurate solutions. Such collapses occur frequently in our calculations, especially for light nuclei, but also for deformed nuclei at N = 142 and N = 152, where there are well-developed gaps. As the last step in the set of improvements we study here, we go beyond the usual BCS approximation and solve the pairing equations by use of the Lipkin-Nogami method, with approximate particle-number conservation.13 A
MASSES
The present calculation represents a study of the effect of various improvements to the model used in a 198 1 calculation of 4023 nuclear masses.‘.* First, we now use Eqs. (40) and (41) for adjusting the model parameters and for determining the error of the theoretical model. For the 198 1 results we find that, with no readjustment of parameters, Uheu= 0.833 MeV, compared to g,, = 0.835 MeV. The difference is small because most experimental errors are small in the experimental data set 19*20available in 198 1, where, in addition, all data with an error larger than 1 MeV were excluded. We also compared our earlier model to a set of 22 1 new masses” that were not included in our 198 1 study. For this set of data we find that c,,,,, = 1.006 MeV and Uheu= 0.8 10 MeV. In this case there is a significant difference between the two results because the new data include many with large uncertainties. When the theoretical error is calculated properly, it demonstrates that the original model does not diverge outside the region in which the parameters were adjusted. We now proceed to investigate improvements to the calculation of the potential energy itself. At each step we readjust the five parameters a,, K,, W, c,, and co to nuclear masses. For the improvements studied here we neglect any change in the ground-state shape of the nucleus, which is a good approximation.’ In our 198 1 calculation, shell-plus-pairing corrections were calculated for several hundred nuclei from the same set of calculated single-particle levels even though the parameters of the single-particle potential depend on 2 and A. An improved shell-plus-pairing correction was then determined by a linear interpolation in 221
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P. MijLLER
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Unified Macroscopic-Microscopic
computer code for this method has been kindly provided to us by Leander and Nazarewicz. The pairing correction is calculated as the difference between the pairing correlation energy for the actual set of levels minus the pairing correlation energy for a uniform set of levels, as discussed in Ref. 5. The uniform distribution of levels also gives a relation between the average pairing gap and the pairing strength G.’ By use of this relation and expressions (3) and (4) we determine the pairing strength G for both the usual BCS approximation and the Lipkin-Nogami method. In both cases the average pairing energy is determined by the method described in Ref. 5. Because of its improved particle-number conservation, the Lipkin-Nogami method yields a pairing correlation energy that is lower on the average than that obtained with the usual BCS approximation. In the future one should therefore investigate how the average pairing correlation energy corresponding to a uniform distribution of energy levels can be extracted by use of the Lipkin-Nogami method. Since the average pairing correlation energy has a very smooth behavior with A, our present inconsistency in the treatment of the average pairing correlation energy is absorbed as slightly different values of the coefficients of the macroscopic model, especially as a slightly larger value of the coefficient of the A0 term. On the other hand, the entire average pairing correlation energy should possibly be considered part of the macroscopic energy. The effect of treating the pairing correlation energy by use of the Lipkin-Nogami method is to decrease the value of uti from 0.886 to 0.832 MeV. The masses calculated in this way are listed in the Table for 4678 nuclei ranging from I60 to 318122. The values obtained for the five parameters adjusted to nuclear masses are a, = K, = W= co = c, =
16.000 MeV 1.911 35 MeV 5.8 MeV 0.145 MeV
s
Model
10 k Experimental
40
60
80
100
120
140
160
NeutronNumberN Figure 1. Comparison of experimental and calculated ground-state microscopic energies for 1593 nuclei. Isotopes are connected by lines, even though some mass measurements are missing within certain isotopic chains.
to the difference between the experimental and calculated ground-state masses. One should note that the experimental microscopic energy is not entirely experimental, since it depends on the macroscopic model. Figure 1 shows that there are no dramatic increases in the deviations far from stability. However, the remaining errors still show strong correlations, which leads us to believe that one can still understand some of the deviations in terms of neglected effects. For example, it has been shown that in the region beyond lead the remaining discrepancies decrease dramatically if c3- and 66shape degrees of freedom are included in the minimization search for the ground state.7*23 The fluctuations in the deviations grow toward the lighter region and appear to arise from deficiencies in the single-particle model. Whether this represents a natural collapse of the macroscopic-microscopic method for few-particle systems or whether the deviations can be decreased by improving the single-particle model or by including other previously neglected effects in this region remains an outstanding problem for the future. One of the assumptions that entered into the derivations of Eqs. (40) and (4 1) was that the true mass ub of In&US i Can be W&en Z& = i#h + &, where efh E N(0, 0th’). In the general case the difference Mt,, - M:‘h contains contributions from the experimental error, but for the cases where the experimental error is small and its contribution can be neglected we would according to the above assumptions have ML,, - Mfh E N(0, @th’). Figure 2, in which we have plotted this difference for 1443 nuclei for which the experimental error is less than 0.1 MeV, illustrates that this assumption is approximately fulfilled. We have also derived a theoretical error for these 1443 nuclei by use of Eq. (41) and plotted the corresponding Gaussian in Fig. 2. For
volume-energy constant volume-asymmetry constant Wigner constant A0 constant charge-asymmetry constant
The rounding off of the parameter values leads to an increase of ath that is less than 0.0003 MeV above its minimum value. We compare in Fig. 1 the experimental and calculated ground-state microscopic energies for the 1593 nuclei ranging from 160 to 263106 that are considered here. For this purpose the ground-state microscopic energy is defined as the difference between the groundstate mass and the spherical macroscopic energy obtained from Eq. (2), with the shape-dependent functions B, and Bj obtained from Eq. (14) and with B, equal to unity. The difference, labeled discrepancy, is equivalent 222
Atomic
Data and hudear
Data TaMa.
Vol. 39. No. 2. July 1988
P. MGLLER
AND J. R. NIX
Unified Macroscopic-Microscopic
Model
3. J. R. Nix, Annu. Rev. Nucl. Sci. 22, 8 1 (1972) Yukawa-plus-exponential model
4. S. Bjomholm 725 (1980)
* 6 g ‘5 c= z 2‘ gj s t”
80 70 60 50 40 30 20 10 0 -5
and J. E. Lynn, Rev. Mod. Phys. 52,
5. M. Bolsterli, E. 0. Fiset, J. R. Nix, and J. L. Norton, Phys. Rev. C 5, 1050 (1972) 6. H. J. Krappe, J. R. Nix, and A. J. Sierk, Phys. Rev. c 20,992 (1979)
-4
-3
-2
-1
0
1
2
3
4
5
7. P. Miiller (1981)
and J. R. Nix, Nucl. Phys. A 361, 117
8. P. Mijller
and J. R. Nix, ATOMIC DATA AND NuTABLES 26, 165 (198 1)
CLEAR DATA
Error(MeV)
9. R. Bengtsson, P. Moller, J. R. Nix, and Jing-ye Zhang, Phys. Ser. 29,402 (1984)
Figure 2. Plot of error frequency per interval of 0.1 MeV for the quantity M& - M{,, . Only data points for which the experimental error is less than 0.1 MeV have heen included. Since we have chosen only data points for which the experimental error is small the plotted quantity is approximately the experimental value of the theoretical error term e,h. The solid line shows the Gaussian that is obtained from the maximum-likelihood estimate with the assumption that e,,, E N(0, CJ,,,~).
10. P. Moller, G. A. Leander, and J. R. Nix, Z. Phys. A 323,41 (1986) 11. D. G. Madland and J. R. Nix, Bull. Am. Phys. Sot. 31,799 (1986) 12. D. G. Madland (1988)
and J. R. Nix, Nucl. Phys. A 476, 1
13. H. C. Pradhan, Y. Nogami, A 201,357 (1973)
this set of nuclei we find that Q, = 0.842 MeV, which is also the standard deviation corresponding to the Gaussian curve in Fig. 2. This value is slightly larger than the value 0.832 MeV that was obtained for the entire set of 1593 masses considered above. Usually masses with large errors in the experimental values are masses far from stability. Thus we conclude that in the present model there is no increase in the error as we go far from stability.
and J. Law, Nucl. Phys.
14. P. Moller, S. G. Nilsson, and J. R. Nix, Nucl. Phys. A 229,292 (1974) 15. H. Cramer, Mathematical Methods of Statistics (Princeton Univ. Press, Princeton, NJ, 1946) 16. R. von Mises, Mathematical Theory of Probability and Statistics (Academic Press, New York, 1964) 17. A. H. Wapstra, 1982)
Acknowledgments We acknowledge valuable discussions with G. A. Leander, D. G. Madland, W. D. Myers, P. Myklund, W. J. Swiatecki, and the UCB statistics consulting service. We are grateful to G. A. Leander and W. Nazarewicz for. providing us with a pairing code based on the Lipkin-Nogami method. P. Mijller thanks the Los Alamos National Laboratory, the Lawrence Livermore National Laboratory, and the Lawrence Berkeley Laboratory for their hospitality during the past 16 months and for the support that made this investigation possible. This work was supported by the U.S. Department of Energy.
private
communication
(April
18. P. Miiller, J. R. Nix, W. D. Myers, and W. J. Swiatecki, in “Proceedings, Workshop on Nuclear Dynamics IV, Copper Mountain, Colorado, 1986,” National Technical Information Service Report No. CONF-860270 (1986), p. 26 19. A. H. Wapstra and K. Bos, ATOMIC DATA NUCLEAR DATA TABLES 19, 175 (1977)
AND
20. A. H. Wapstra and K. Bos, ATOMIC DATA NUCLEAR DATA TABLES 20, 126 (1977)
AND
21. A. H. Wapstra, private communication
(1983)
22. A. H. Wapstra, G. Audi, and R. Hoekstra, ATOMIC DATA AND NUCLEAR DATA TABLES 39,28 1 ( 1988)
References 1. V. M. Strutinsky,
Nucl. Phys. A 95,420 (1967)
2. V. M. Strutinsky,
Nucl. Phys. A 122, 1 (1968)
23. W. Nazarewicz, P. Olanders, I. Ragnarsson, J. Dudek, G. A. Leander, P. Moller, and E. Ruchowska, Nucl. Phys. A 429,269 (1984) 223
Atomic
Data and Nuclear
Data TaMes.
Vol. 39. No. 2. July 1988
DATA AND NUCLEAR
NUCLEAR
DATA TABLES
39,2 13-223 ( 1988)
MASSES FROM A UNIFIED P. MILLER Theoretical
MACROSCOPIC-MICROSCOPIC
MODEL
and J. R. NIX
Division, Los Alamos National Laboratory Los Alamos, New Mexico 87545
We calculate ground-state masses for 4678 nuclei ranging from I60 to 3’8122 by means of an improved version of the macroscopic-microscopic model employed in our 198 1 mass calculation, which uses a Yukawa-plus-exponential model for the macroscopic term and a folded-Yukawa singleparticle potential as starting point for the microscopic term. Some of the new features now incorporated are a new model for the uveruge pairing strength, the solution of the microscopic pairing equations by use of the Lipkin-Nogami method with approximate particle-number conservation, the use of experimental mass uncertainties in determining the model parameters, and an estimation of the theoretical error of the model by application of the maximum-likelihood method. Only five parameters are determined from least-squares fitting to the nuclear masses; the other constants in the model are taken from previous work with no adjustment. The resulting theoretical error in the calculated ground-state masses of 1593 nuclei ranging from I60 to *‘j3106 is 0.832 MeV. We also extend the calculation to some additional nuclei in the heavy and superheavy region that were not considered in 198 1. The present calculation represents an interim report on a project in which we plan to make further improvements and extend the region of nuclei considered to the neutron and proton drip lines. 0 1988 Academic P, Inc.
0092-640X/88 $3.00 Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.
213
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Data and Nuclear
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P. MtSLLER AND J. R. NIX
Unified Macroscopic-Microscopic
Model
CONTENTS 1. INTRODUCTION
.....................................
2. MACROSCOPIC-MICROSCOPIC MODEL 2.1. Macroscopic Model ................................ 2.2. Microscopic Model .................................
214 ................
2 14 2 15 217
3. ESTIMATION OF PARAMETERS AND ERROR 3.1. Maximum-LikelihoodMethod :::::::::: ............. 3.2. Resulting Formulas ................................. 3.3. Previous Models for Estimating flth ....................
217 218 2 19 220
4. CALCULATED
221
MASSES
...............................
1. INTRODUCTION
state masses’ for 4023 nuclei ranging from I60 to 279112 and fission barriers7 for 28 nuclei throughout the periodic system were calculated. The root-mean-square deviation between experimental and calculated groundstate masses was 0.835 MeV for a set of 1323 masses and 1.33 1 MeV for the 28 fission barriers. Many other prop erties such as ground-state deformations were also well described by that calculation, as is extensively discussed in Ref. 9. The model represents a unified approach to the study of many features of nuclear structure, fission, and heavy-ion reactions. Here we study the effect of some new features in the treatment of pairing and extend the calculation to additional nuclei in the heavy and superheavy regions. These heavy nuclei were previously studied within the framework of the original model in Ref. 10. Below we present some details of the model that are of particular relevance to the present study, but refer the reader to Refs. 5-9 for a more complete presentation of the original model. In the macroscopic-microscopic model the nuclear energy, which is calculated as a function of shape,
The advent of the Strutinsky shell-correction method1p2 about 20 years ago made it possible to culculute the shell correction in nuclear mass models of the macroscopic-microscopic type. With this method the potential energy of a nucleus can be calculated for arbitrary shapes, within given shape parameterizations. Coupled with a wealth of new experimental results this has led to an enormous increase in our understanding of nuclear ground-state and fission properties and of the stability of elements at the end of the periodic system. For an extensive review of some of these developments see Refs. 3 and 4. Here we apply the model to the study of nuclear masses.
2. MACROSCOPIC-MICROSCOPIC
MODEL
Our original model has been discussed extensively in Refs. 5-9. The present calculation is similar to the investigation described in Refs. 7 and 8, where ground214
Atomic
Data and Nuclear
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Vol. 39. NO. 2. July 1933
P. MiiLLER
Unified Macroscopic-Microscopic
AND J. R. NIX
proton number Z, and neutron number N, is the sum of a macroscopic term and a microscopic term. Thus the total nuclear potential energy can be written as
exponential model6 for the macroscopic term and a folded-Yukawa single-particle potential” as a starting point for calculating the microscopic term. They are briefly discussed below.
E(Z, N shape) = Emacr(Z, N, shape) + &AZ
N shape).
2.1. Macroscopic
( 1)
Model
In the present version of our model the Yukawaplus-exponential macroscopic energy is given by
There exist several different models for both the macroscopic and microscopic terms. We use a Yukawa-plus-
L&,
Model
N shape) = i&Z
mass excesses of Z hydrogen atoms and N neutrons
+ M,N
-
a,(
1 -
Kv1’)A
volume and volume-asymmetry
+
a,(
1 -
KJ2)&A2’3
generalized surface and surface-asymmetry
+
clplo
direct Coulomb
Z 413 c4 A’/3
+ f(kfrp)
%
- c&N - Z) 1/A, 0,
+
I
energies
A0 energy
+ Cl -$3 -
energies
Z and N odd and equal otherwise
+A,+ a, - drip,
Z and N odd
+&,
Z odd and N even
+a,,
Z even and N odd .
+o,
energy
exchange Coulomb
energy
proton-form-factor
correction
charge-asymmetry
energy
Wigner energy
average pairing energy
Z and N even
- a,1Z2,39
energy of the bound electrons
(2)
where A = N + Z is the mass number and I = (N - Z)/A is the relative neutron excess. This expression differs from the corresponding one used in our earlier calculations’** only in the form of the average pairing energy appearing in the next-to-last term. For the average neutron pairing gap &, and average proton pairing gap ii, we now use1 ‘,I* rB, --s&II2 &=-e
These results, which were derived”“* by use of the BCS approximation applied to a uniform distribution of levels, take into account the dependencies of &, and z\, upon both the relative neutron excess I and the relative surface energy B,. For the average neutron-proton interaction energy 6,, , we now use the new form’ ‘J*
and
The zero reference point for the pairing energy now corresponds to even-even nuclei rather than to halfway between even-even and odd-odd nuclei.
h
(5)
N’/3
i&=-e z’f3r&
-+sI-II1
215
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Vol. 39, No. 2, July 1998
P. MtiLLER
AND J. R. NIX
Unified Macroscopic-Microscopic
one obtains
In the above expressions the quantities cl and c4 are defined in terms of the elementary charge e and the nuclear-radius constant ro by 3 e2 “‘=3-K
The proton form factorfis
B,=l-;
+(I
+,,(2+$f-$)e-a
(6)
11
e-2m . (14)
5 3 =I3 . 4 ( 2* 1
c4=q--
Model
(71
The expression B3 for the relative Coulomb energy yields the energy for an arbitrarily shaped, homogeneously charged, diffuse-surface nucleus to all orders in the diffuseness parameter aden. The constants in front of B1 and B3 have been chosen so that BI and B3 are 1 for a sphere in the limit in which the range a and diffuseness aden of the Yukawa-plus-exponential term and of the charge density, respectively, in the macroscopic energy go to zero, in analogy with the definition of the quantities B, and Bc in the standard liquid-drop model.
given by
where r, is the proton root-mean-square Fermi wave number is given by
radius and the
Values of Constants (9)
The constants appearing in the expression for the Yukawa-plus-exponential macroscopic model fall into three categories. ‘*’ The first category, which represents constants that were taken from previous work with no adjustment whatsoever, includes7~8~‘2
The relative surface energy B, , which is the ratio of the surface area of the nucleus at the actual shape to the surface area of the nucleus at the spherical shape, is given by ~-213
A& = 7.289034 MeV
n
Ad,, = 8.071431 MeV e2 = 1.4399764 MeV fm The quantity BI represents the relative generalized surface or nuclear energy in a model that accounts for the effect of the finite range of the nuclear force. It is given by
B, = &
j-j-
(2
- !d)
V
0
a
,““I;‘;, r
a de0 = 0.99/2’j2 fm
ael = 1.433 X 10v5 MeV
&&.
r = 5.72 MeV
(11)
s = 0.118 The relative Coulomb
energy B3 is given by
B3 =
d3rd3r’
t = 8.12
Ir-fl
h = 6.82 MeV
15 A-‘13 577 ss
v
r, = 0.80 fm It includes diffuseness corrections to all orders. For spherical shapes one can calculate the quantities BI and B3 analytically. With roA’J3 x0 = and a
ii-J1f3 y. = -
hydrogen-atom mass excess neutron mass excess elementary charge squared range of Yukawa function used to generate nuclear charge distribution electroniobinding constant preexponential pairing constant linear exponential pairing constant quadratic exponential pairing constant neutron-proton interaction constant proton root-meansquare radius
The values of some of the above constants have been revised by more recent work. However, the changes are so small that the use of the more current values would usually change the calculated masses by less than 0.0 1 MeV.
(13)
aden
216
Atwric
Data and Nuclear
Data Tsbles.
Vol. 39. NO. 2. July 1988
P. MijLLER
AND J. R. NIX
Unified Macroscopic-Microscopic
The second category, representing those constants whose values were determined from considerations other than nuclear ground-state masses, includes’** r. = 1.16 fm a = 0.68 fm
where I = (N - Z)/A + O.O112Z2/A5’3 1 + 3.15//Q/3
nuclear-radius constant range of Yukawa-plus-exponential potential surface-energy constant surface-asymmetry constant
a, = 21.13 MeV K, = 2.3
R den= l.16,4”3(l
+!q
The third category represents five constants whose values are determined from a least-squares adjustment to nuclear ground-state masses. These are the constants a,, K,, co, c,, and W, and their values are given later.
N,
shape)
=
&hell(~~
shape)
+
&miXZ,
N
shape).
vs.,.= --A - h 2a*vv, ( 2mc 1
(15)
(19)
V, = (52.5 - 48.78) MeV,
(20)
(24)
xp h
A 240
’
(1
x,=31.5+4.5 Finally, given by
the Coulomb V&) = ep,
(25)
potential s ~lr-fl'
.
for protons
d3r’
(27) is (28)
where the charge density pc is given by (29)
3. ESTIMATION OF PARAMETERS AND ERROR
(18)
VP = (52.5 + 48.78) MeV
(23)
A, = 28.0 + 6.0 &( 1
and
where the integration is over the volume of the generating shape, whose volume is held fixed at $TR,.,,,~~as the shape is deformed. The potential radius R,, potential depth V. = VP for protons, and potential depth V. = V,, for neutrons are given by + 0.82 fm - 0.56 frn2jhen
.
where X is the spin-orbit interaction strength, m is the mass of either a neutron or a proton, u is the Pauli spin matrix, and p is the nucleon momentum. The spin-orbit strength has been determined from adjustments to experimental levels in the rare-earth and actinide regions. It has been shown14y7,9that many nuclear properties throughout the periodic system are well reproduced with X given by a function linear in A through the values determined in these two regions. This gives for X
The first term is the spin-independent nuclear part of the potential, which is calculated in terms of the foldedYukawa potential e-lr-r’lln,t d3rr V,(r) = s (17) pt3 s v lr-+llawt ’
= &,,
+ 0.330z2 + o*“y;z2
of the Yukawa function in Eq. ( 17) has been determined from an adjustment of calculated single-particle levels to experimental data in the rare-earth and actinide regions. It is kept constant for nuclei throughout the periodic system. The spin-orbit potential is given by the expression
Both terms are evaluated from a set of calculated singleparticle levels. As before, the shell correction is calculated by use of Strutinsky’s method.‘s2 For the pairing correction we now use the Lipkin-Nogami’3 version of the BCS method with approximate particle-number conservation. Since our model is extensively discussed in Ref. 5 and references quoted therein, we present here only the major features of the model and some changes relative to that reference. The single-particle potential felt by a nucleon is given by v= v, + v,,. + v,. (16)
R,
(22)
apt = 0.8 fm
Model
N,
+ ?) fm
The range
The microscopic-energy term arises because of the nonuniform distribution of single-particle levels in the nucleus. It is the sum of a shell-correction term and a pairing term: K&Z,
(21)
and A
2.2. Microscopic
Model
It is possible to use several approaches to characterize the error of a model. Here we investigate a model that calculates nuclear masses for a set of Z and N values. For a subset of these Z and N values experimen-
and
217
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P. MijLLER
AND J. R. NIX
Unified Macroscopic-Microscopic
tal data values exist. Thus, a very detailed definition of the error is to give the deviation between the calculated values and the available experimental data points. This type of information is available in the Table. However, both the experimental and the theoretical error contribute to this deviation. It is useful to develop an error concept that allows us to characterize the error of a model by a single number that does not contain contributions from the experimental errors. We now introduce such an error concept. In this approach one can compare the error of two models and also compare the error of a model for the region to which its parameters were adjusted to the error for new regions of nuclei. We show in the beginning of Section 4 that our model, with the error definition introduced below, shows no systematic increases in the error for new regions of nuclei far from stability. In most earlier studies, for example those in Ref. 7, the error of a theoretical mass model was taken to be the root-mean-square (rms) deviation
Model
tude. Since we assume that the error is due to a large number of missing terms that are small and fluctuating, we treat the problem with statistical methods. We assume here that there are no systematic effects on the error, but give some comments on this problem at the end of Section 3.2. 3.1. Maximum-Likelihood
Method
A general method for determining unknown parameters of a known statistical distribution if one has observed events distributed according to the distribution is the maximum-likelihood (ML) method. For an indepth discussion of this method, notation, and fundamental concepts of statistics we refer to the books by Cramer” and von Mises. I6 According to the ML method one estimates the parameters so that the events that have been observed have the maximum likelihood of being observed. Suppose that 5’ is distributed according to Ai(x, e), with B unknown. If we have observed t1 = x’, t2 = x2, . . . ) E” = x”, we can determine 6 so that our observations have maximum likelihood. This is done by maximizing the likelihoodfunction L (8), where
L(B) =hl(xI, e)fi(x2,e) . . . A~x”,e).
and. .the. .parameters of the model were determined by mmlmtzmg a,, in Eq. (30). Here ML,, is the measured mass for a particular value of the proton number 2 and neutron number N, and Mfh is the corresponding calculated quantity. There are II such measurements for different pairs of 2 and N. The theoretical model for Mfh depends on 2 and N and also on I)I adjustable parametersn,p2,. -. , pm. The use of the rms deviation (30) is reasonable when all the errors u&, associated with the measurements are small compared to u,, . However, for the large experimental errors a&, that are associated with many recent mass measurements far from stability, the above definition is unsatisfactory, since both the theoretical and experimental errors contribute to u,,,. Consequently the rms deviation (30) is always an overestimate of the true theoretical error. We must therefore use an approach that decouples the theoretical and experimental errors from one another. To proceed systematically we first define precisely what we mean by the error of a theory. We then derive expressions from which this error can be estimated. Our model Mth for nuclear masses is a sum of 20 or so terms. There are a few large terms but many more terms that give fairly small contributions to Mth . This structure of the model has arisen over the years through the addition of new terms to describe previously neglected effects. It is reasonable to assume that there are an even larger number of terms that account for even smaller effects that have been left out of the model. Thus the model has an error associated with these missing terms. We now want to characterize this error and estimate its magni-
Maximizing maximizing
function L(8) is equivalent to In L(B), or to solving
the likelihood the logarithm Wn
(31)
w)i
=
de
o
*
We denote the solution of Eq. (32) by e*(x*, x2, x”). By solving Eq. (32) for several sets of observa...) tions xi’, XT, . . . , x; of the set of random variables 5 ‘, t2, - - * 9F, we obtain several estimates et of the parameter 8 of the distribution functions&(x, 0). Thus 8*(x’, x2, . . . , x”) is an observation of a random variable e*([ ‘, t2, * * - 7 t”) with a distribution function ge=. Since the distribution function of the multidimensional random variable ([l, t2, . . . , t”) is known if theS,i are known (it is actually the product in the right member of Eq. (3 I)), the distribution function of e*([‘, E2, . . . , 5”) can be determined, at least in principle. This is of fundamental importance. It is desirable that its distribution function ge* have certain properties. This is an important field of mathematical statistics, and we refer to the books by Cramer” and von Mises16 for a discussion of these questions. Here we just mention that the ML estimate is consistent, that is, lim E(8*) = 8, lim D2(0*) = 0, n-m n-m
(33)
where E(d*) denotes the mean and D2(O*) the variance of the distribution function of 8*. One can also show that if an eficient estimate B* of 8 exists, then the likelihood equation will have a unique solution equal to 8*. An 218
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Data and Nuclear
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P. M6LLER
AND J. R. NIX
Unified Macroscopic-Microscopic
efficient estimate is an estimate that is unbiased and has minimum variance. The estimate is said to be unbiased
unknown terms that are missing from the theory and Cr,h is by definition the error of the theory. We note that in practice there are correlations in the error term, in the sense that the error for calculated masses is of similar magnitude for nearby nuclei. However, this is of no consequence for the considerations below, since we consider a large number of masses throughout the periodic system. Were one to consider the differences between theoretical masses for nearby nuclei, correlations would be important and would result in errors in the differences that are much smaller than \IZQth, which would be expected in the absence of correlations. The experimental quantities are not observations of precisely ufr , but are instead observations of the random variable z)&, = ufr + e& = M:‘h + eL + e&, because there are random errors associated with the measuring process. We assume that e&, E N(0, UC%, i *). The uiXp2 are different for different i because the measuring device is different for each i, but they are all known. From the above conditions we find that u&,
if E(B*) = B
(34)
for all n. However, many ML estimates have bias. For the case in which the ML estimate is biased, one may ask if an unbiased estimate is a better one. However, even if the ML estimate is biased, it is consistent and therefore the parameter estimates are arbitrarily close to the true values for sufficiently large n, provided that the number of parameters does not increase as n increases. One may therefore agree with von Mises (Ref. 16, p. 560) that if for the actual purpose of an investigation the difference between an ML estimate and an unbiased estimate is of importance, a larger sample should have been chosen. For the specific case of nuclear mass formulas, it is not possible to substantially increase the number of data points. Thus one should consider only models in which relatively few parameters are adjusted to the data. If one persists in studying models with a large number of adjustable parameters compared to the available data one should study the effect of bias, since for a large number m of adjustable parameters we expect that (Ref. 16, p.
E N(ii&, uLxp2 + uth2). Thus, for the distribution tion f,i we have
555)
et&,
E(O*) N y
3.2. Resulting
0.
Model
M:‘h,
‘&p*
+
‘-%h2)
=-
(35)
func-
-(X-M~)‘/2(d,,2+o,hz). (36)
The values A4iXp constitute observations of utxp. We can now estimate both the parameters pv of &, and the quantity (Tthby maximizing the likelihood function L. In our case
Formulas
The unknown parameters of our theory can be estimated by use of the ML method only if they are parameters of the distributions involved. In particular, they cannot be parameters of the prescription used to generate the variables of the distribution function. Thus, we must formulate the problem so that A4& becomes a parameter of the distribution. This can be done in the following way. We write ufr = A&, + efh, where ef,, E N(0, ati’) is an error term and A!{,, is our model for the nuclear mass. We are now in a position to give a precise definition of the error of the theory, which we designate by the quantity ati. The variable u& is the true mass of the nucleus i, where i stands for a particular pair of values of the proton number Z and neutron number A? However, uf is also a random variable, by construction. The following somewhat drastic example clarifies this duality. Suppose that when God originally created the Universe and fixed each nuclear mass i, He used our expression for Mfh, with some parameters He fixed and which it is now up to us to estimate. To the number He obtained by calculating M:‘,, He added a random number efh which He obtained by throwing dice such that ei,, E N(0, at,.,*). The random variable ef,, represents the
UPl,
***,
P2,
Pm,
uth*)
= fvb, * ’ ‘fv.(x,
Maximizing tions
dxp* +
M:h,
uth*)
&.p* +
M;h,
flth2).
(37)
L is equivalent to solving the m + 1 equa-
and 5 (M&p
-
Mfh)* (a;,,2
i=l
+
(&;
+
‘?h2*)
=
o
ut,,**)*
’
(3%
Here the notation f&h** means that by solving Eq. (39) we obtain the estimate 6,h2* of the true u,h*. The above equations are equivalent to minimizing S with respect to D,, where _ (40) and solving i i=l
219
(M&p
-
Mfh)* (b;,;
+
b&,2
+
%**)
uth2*)*
Atomic Data and Nudear
=
o ’
Data Tables,
(41)
Vol. 39, NO. 2, July 1988
P. MtjLLER
AND J. R. NIX
Unified Macroscopic-Microscopic
Thus we are led to one additional equation relative to the usual least-squares equations that arise when model parameters are estimated by adjustments to experimental data under the assumption of a perfect theory, that is, uti2 = 0. Equations (38) and (39), or alternatively Eqs. (40) and (4 1), constitute a system of m + 1 equations that are to be solved together. It is instructive to rewrite Eq. (39) as
distributed evenly in the interval 0 to 10. The simulation shows that although the experimental errors are most often much larger than the error of the theoretical model, it was possible to estimate the theoretical error E(B*). In an actual application of the model discussed here only one set of theoretical values exists. In such a case the uncertainty in the estimate of the error E(B*) would be approximately D(e*) m 0.15. From a simulation with a;,, distributed approximately as those for the 200 most recently discovered masses, we obtain D(e*) es 0.05. The case of systematic effects in the error can also be treated by use of the above ideas. If there are systematic effects in Z and Non the error it means that one can write uth as a function of Z, N, and a set of parameters. The equations for estimating the parameters of this error expression can be derived by use of the maximum-likelihood method discussed above. To use the example from above this type of error means that God throws a different set of dice for each mass. We have not carried through such an analysis here, but we conclude from other studies below that there are no systematic trends in the error of the present model over large ranges of Z and N, except that the amplitude of the fluctuations in the error is slightly larger in the light region than in the heavier region. This will be discussed when the results are presented below.
(42)
uth
where 1
Wi =
+ tTth2*p
b&D
and k= 2.
(44)
Written in this way, the unknown uth2* enters on both sides of the equal sign in Eq. (42), but the equation is suitable for an iterative determination of uth2*. We have found that the convergence is extremely rapid. Another benefit of rewriting Eq. (39) in the form given by Eq. (42) is that one may ask if k = 2 is the only possibility, or if other values of k are possible. We shall not discuss this question in depth here but instead will comment that if all a&,2 are equal, then all values of k yield the same equation. However, the more interesting case is how k should be chosen if the u&2 have different values. According to our discussion above, one can prove that the ML estimate has several desirable properties. Therefore, we choose k = 2, which is the ML estimate. To gain some additional insight into the properties of the above method we compare the ML estimate corresponding to k = 2 with estimates obtained with k = 1, 3, and 4 by simulating the random variables efh and e&,. We do several simulations, each with n = 200. In each simulation we generate one set of 200 values for e&. In each case we also generate 2000 sets of 200 values for efh, with uth = 1.O in all cases. Equation (42) is solved for each of the 2000 sets. In this way we estimate the mean and variance of bth2*, which is distributed according to go,. Table A shows the results obtained with the a&,
3.3. Previous Models for Estimating
s’
EC@*)
me*)
0.9884 0.025 11
We*)
0.15848
3
under the auxiliary
1.0177
1.0167
1.0194
0.02583 0.16071
0.02825 0.16807
5
(M&J ‘-&J~
-
M:‘h)’
+
‘-Theu2
(45)
condition
Thus, with this heuristic method the error d&u is determined by requiring that x2 per degree of freedom be equal to 1. In practice, this suggestion leads to results close to those obtained with the formulas (38)-(44) that we have derived here. However, since Eqs. (45) and (46) were obtained heuristically, it is not possible to discuss their significance by accepted statistical methods in a quantitative manner, in contrast to the results in the previous section. Since Eqs. (45) and (46) have been used in some investigations, we interpret them in the following qualitative way. When there are no adjustable parameters, corresponding to ~yt= 0, Eq. (46) is identical
4
0.02307 0.15187
=
i=l
k 2
cth
It was realized earlier that the rms deviation (30) for the error of a mass formula is inadequate when the experimental errors are large. By use of heuristic arguments it was suggested I’,‘* that the parameters and error uh,” be estimated by minimizing S’ with respect to the model parameters, where S’ is given by
TABLE A Results for 0 G uf, Q 10
1
Model
220
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Data and Nuclear
Data Tables,
Vol. 39, NO. 2. July 1968
P. MOLLER
AND J. R. NIX
Unified Macroscopic-Microscopic
to our Eqs. (42) and (43) but with k = 1. However, k = 1 does not correspond to an ML estimate. According to our discussion above, the ML estimate is under certain conditions the estimate with minimum variance. Therefore, the heuristic equations (45) and (46) yield an estimate with a larger variance than do our rigorous equations (38)-(44). The fact that the number of degrees of freedom for x2 is n - m can, in a heuristic way, be expected to reduce the bias of the estimate, and is comparable to our approximate suggestion expressed by Eq. (35). In our previous work’,* we used the rms deviation (30) for the error of our mass formula. For comparison we shall also sometimes give this result here, denoting it by Q,,. For some of our earlier results obtained in this investigation, we use the heuristic method of Eqs. (45) and (46) and characterize the error by (The”.For our final results presented here, we use the more rigorously derived equations (38)-(44) and characterize the error by the quantity (Tth. 4. CALCULATED
Model
A. Although this interpolation accounts for the effect of a changing radius in the single-particle potential, it takes no account of the change in potential as a function of distance from the line of j3 stability. One improvement over the 198 1 calculation is the calculation of the shellplus-pairing correction at the ground-state deformation from a single-particle well appropriate to each particular nucleus. This improvement in the model actually leads to an increase of ah,,, to 0.885 MeV. As the next step we changed our experimental data set from Refs. 19 and 20 to Ref. 22. For the nuclei with 2 >, 8 and N > 8 that we consider, the first set contains 1323 masses with an experimental error less than 1.0 MeV. We now consider all measured masses regardless of the magnitude of the error in the experimental point, provided that it is known. For nuclei with Z 2 8 and N 2 8, the new data set contains 1593 masses. After a readjustment of the five parameters we find that dheUis reduced by about 2%. In our next step we changed the model for the average neutron and proton pairing gaps from that used in our 198 1 calculation to one that takes into account their dependencies on relative neutron excess and nuclear shape.’ ‘*I2 In this model the average neutron pairing gap is different from the average proton pairing gap. Therefore, one cannot, as was done in Ref. 7, use a point halfway between even-even and odd-odd nuclei as the reference point for the odd-even pairing effects. We therefore change the reference point to the even-even system, as indicated in Eq. (2). With the old expressions &, = &, = 12 MeV/A’12 and a,, = 20 MeVIA for the average pairing gaps and residual neutron-proton interaction energy, the effect of changing the reference point to the even-even system increases dheu to 0.882 MeV. When Eqs. (3)-(5) are used for the average pairing strengths, the more rigorous measure of error given by Eqs. (38)-(44) becomes uth = 0.886 MeV. When studying the deviations between calculated and experimental masses, for example those in Fig. 8b of Ref. 7, one observes that the largest fluctuations in the error occur near magic numbers. The BCS method for solving the pairing equations collapses close to magic numbers, or more precisely when the pairing strength G is smaller than some critical value in relation to the gap in the single-particle level spectrum at the Fermi surface.13 In addition, close to the point of collapse the BCS approximation yields inaccurate solutions. Such collapses occur frequently in our calculations, especially for light nuclei, but also for deformed nuclei at N = 142 and N = 152, where there are well-developed gaps. As the last step in the set of improvements we study here, we go beyond the usual BCS approximation and solve the pairing equations by use of the Lipkin-Nogami method, with approximate particle-number conservation.13 A
MASSES
The present calculation represents a study of the effect of various improvements to the model used in a 198 1 calculation of 4023 nuclear masses.‘.* First, we now use Eqs. (40) and (41) for adjusting the model parameters and for determining the error of the theoretical model. For the 198 1 results we find that, with no readjustment of parameters, Uheu= 0.833 MeV, compared to g,, = 0.835 MeV. The difference is small because most experimental errors are small in the experimental data set 19*20available in 198 1, where, in addition, all data with an error larger than 1 MeV were excluded. We also compared our earlier model to a set of 22 1 new masses” that were not included in our 198 1 study. For this set of data we find that c,,,,, = 1.006 MeV and Uheu= 0.8 10 MeV. In this case there is a significant difference between the two results because the new data include many with large uncertainties. When the theoretical error is calculated properly, it demonstrates that the original model does not diverge outside the region in which the parameters were adjusted. We now proceed to investigate improvements to the calculation of the potential energy itself. At each step we readjust the five parameters a,, K,, W, c,, and co to nuclear masses. For the improvements studied here we neglect any change in the ground-state shape of the nucleus, which is a good approximation.’ In our 198 1 calculation, shell-plus-pairing corrections were calculated for several hundred nuclei from the same set of calculated single-particle levels even though the parameters of the single-particle potential depend on 2 and A. An improved shell-plus-pairing correction was then determined by a linear interpolation in 221
Atomic
Data and Nuclear
Data Tabs.
Vol. 39. No. 2, July 1998
P. MijLLER
AND J. R. NIX
Unified Macroscopic-Microscopic
computer code for this method has been kindly provided to us by Leander and Nazarewicz. The pairing correction is calculated as the difference between the pairing correlation energy for the actual set of levels minus the pairing correlation energy for a uniform set of levels, as discussed in Ref. 5. The uniform distribution of levels also gives a relation between the average pairing gap and the pairing strength G.’ By use of this relation and expressions (3) and (4) we determine the pairing strength G for both the usual BCS approximation and the Lipkin-Nogami method. In both cases the average pairing energy is determined by the method described in Ref. 5. Because of its improved particle-number conservation, the Lipkin-Nogami method yields a pairing correlation energy that is lower on the average than that obtained with the usual BCS approximation. In the future one should therefore investigate how the average pairing correlation energy corresponding to a uniform distribution of energy levels can be extracted by use of the Lipkin-Nogami method. Since the average pairing correlation energy has a very smooth behavior with A, our present inconsistency in the treatment of the average pairing correlation energy is absorbed as slightly different values of the coefficients of the macroscopic model, especially as a slightly larger value of the coefficient of the A0 term. On the other hand, the entire average pairing correlation energy should possibly be considered part of the macroscopic energy. The effect of treating the pairing correlation energy by use of the Lipkin-Nogami method is to decrease the value of uti from 0.886 to 0.832 MeV. The masses calculated in this way are listed in the Table for 4678 nuclei ranging from I60 to 318122. The values obtained for the five parameters adjusted to nuclear masses are a, = K, = W= co = c, =
16.000 MeV 1.911 35 MeV 5.8 MeV 0.145 MeV
s
Model
10 k Experimental
40
60
80
100
120
140
160
NeutronNumberN Figure 1. Comparison of experimental and calculated ground-state microscopic energies for 1593 nuclei. Isotopes are connected by lines, even though some mass measurements are missing within certain isotopic chains.
to the difference between the experimental and calculated ground-state masses. One should note that the experimental microscopic energy is not entirely experimental, since it depends on the macroscopic model. Figure 1 shows that there are no dramatic increases in the deviations far from stability. However, the remaining errors still show strong correlations, which leads us to believe that one can still understand some of the deviations in terms of neglected effects. For example, it has been shown that in the region beyond lead the remaining discrepancies decrease dramatically if c3- and 66shape degrees of freedom are included in the minimization search for the ground state.7*23 The fluctuations in the deviations grow toward the lighter region and appear to arise from deficiencies in the single-particle model. Whether this represents a natural collapse of the macroscopic-microscopic method for few-particle systems or whether the deviations can be decreased by improving the single-particle model or by including other previously neglected effects in this region remains an outstanding problem for the future. One of the assumptions that entered into the derivations of Eqs. (40) and (4 1) was that the true mass ub of In&US i Can be W&en Z& = i#h + &, where efh E N(0, 0th’). In the general case the difference Mt,, - M:‘h contains contributions from the experimental error, but for the cases where the experimental error is small and its contribution can be neglected we would according to the above assumptions have ML,, - Mfh E N(0, @th’). Figure 2, in which we have plotted this difference for 1443 nuclei for which the experimental error is less than 0.1 MeV, illustrates that this assumption is approximately fulfilled. We have also derived a theoretical error for these 1443 nuclei by use of Eq. (41) and plotted the corresponding Gaussian in Fig. 2. For
volume-energy constant volume-asymmetry constant Wigner constant A0 constant charge-asymmetry constant
The rounding off of the parameter values leads to an increase of ath that is less than 0.0003 MeV above its minimum value. We compare in Fig. 1 the experimental and calculated ground-state microscopic energies for the 1593 nuclei ranging from 160 to 263106 that are considered here. For this purpose the ground-state microscopic energy is defined as the difference between the groundstate mass and the spherical macroscopic energy obtained from Eq. (2), with the shape-dependent functions B, and Bj obtained from Eq. (14) and with B, equal to unity. The difference, labeled discrepancy, is equivalent 222
Atomic
Data and hudear
Data TaMa.
Vol. 39. No. 2. July 1988
P. MGLLER
AND J. R. NIX
Unified Macroscopic-Microscopic
Model
3. J. R. Nix, Annu. Rev. Nucl. Sci. 22, 8 1 (1972) Yukawa-plus-exponential model
4. S. Bjomholm 725 (1980)
* 6 g ‘5 c= z 2‘ gj s t”
80 70 60 50 40 30 20 10 0 -5
and J. E. Lynn, Rev. Mod. Phys. 52,
5. M. Bolsterli, E. 0. Fiset, J. R. Nix, and J. L. Norton, Phys. Rev. C 5, 1050 (1972) 6. H. J. Krappe, J. R. Nix, and A. J. Sierk, Phys. Rev. c 20,992 (1979)
-4
-3
-2
-1
0
1
2
3
4
5
7. P. Miiller (1981)
and J. R. Nix, Nucl. Phys. A 361, 117
8. P. Mijller
and J. R. Nix, ATOMIC DATA AND NuTABLES 26, 165 (198 1)
CLEAR DATA
Error(MeV)
9. R. Bengtsson, P. Moller, J. R. Nix, and Jing-ye Zhang, Phys. Ser. 29,402 (1984)
Figure 2. Plot of error frequency per interval of 0.1 MeV for the quantity M& - M{,, . Only data points for which the experimental error is less than 0.1 MeV have heen included. Since we have chosen only data points for which the experimental error is small the plotted quantity is approximately the experimental value of the theoretical error term e,h. The solid line shows the Gaussian that is obtained from the maximum-likelihood estimate with the assumption that e,,, E N(0, CJ,,,~).
10. P. Moller, G. A. Leander, and J. R. Nix, Z. Phys. A 323,41 (1986) 11. D. G. Madland and J. R. Nix, Bull. Am. Phys. Sot. 31,799 (1986) 12. D. G. Madland (1988)
and J. R. Nix, Nucl. Phys. A 476, 1
13. H. C. Pradhan, Y. Nogami, A 201,357 (1973)
this set of nuclei we find that Q, = 0.842 MeV, which is also the standard deviation corresponding to the Gaussian curve in Fig. 2. This value is slightly larger than the value 0.832 MeV that was obtained for the entire set of 1593 masses considered above. Usually masses with large errors in the experimental values are masses far from stability. Thus we conclude that in the present model there is no increase in the error as we go far from stability.
and J. Law, Nucl. Phys.
14. P. Moller, S. G. Nilsson, and J. R. Nix, Nucl. Phys. A 229,292 (1974) 15. H. Cramer, Mathematical Methods of Statistics (Princeton Univ. Press, Princeton, NJ, 1946) 16. R. von Mises, Mathematical Theory of Probability and Statistics (Academic Press, New York, 1964) 17. A. H. Wapstra, 1982)
Acknowledgments We acknowledge valuable discussions with G. A. Leander, D. G. Madland, W. D. Myers, P. Myklund, W. J. Swiatecki, and the UCB statistics consulting service. We are grateful to G. A. Leander and W. Nazarewicz for. providing us with a pairing code based on the Lipkin-Nogami method. P. Mijller thanks the Los Alamos National Laboratory, the Lawrence Livermore National Laboratory, and the Lawrence Berkeley Laboratory for their hospitality during the past 16 months and for the support that made this investigation possible. This work was supported by the U.S. Department of Energy.
private
communication
(April
18. P. Miiller, J. R. Nix, W. D. Myers, and W. J. Swiatecki, in “Proceedings, Workshop on Nuclear Dynamics IV, Copper Mountain, Colorado, 1986,” National Technical Information Service Report No. CONF-860270 (1986), p. 26 19. A. H. Wapstra and K. Bos, ATOMIC DATA NUCLEAR DATA TABLES 19, 175 (1977)
AND
20. A. H. Wapstra and K. Bos, ATOMIC DATA NUCLEAR DATA TABLES 20, 126 (1977)
AND
21. A. H. Wapstra, private communication
(1983)
22. A. H. Wapstra, G. Audi, and R. Hoekstra, ATOMIC DATA AND NUCLEAR DATA TABLES 39,28 1 ( 1988)
References 1. V. M. Strutinsky,
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2. V. M. Strutinsky,
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23. W. Nazarewicz, P. Olanders, I. Ragnarsson, J. Dudek, G. A. Leander, P. Moller, and E. Ruchowska, Nucl. Phys. A 429,269 (1984) 223
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Data and Nuclear
Data TaMes.
Vol. 39. No. 2. July 1988