Nuclear Physics Al34 (1969) 265-276;
@
North-Holland
Publishing
Co., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from tbe publisher
HARTREEFOCK
CALCULATIONS
WITH WOOD-SAXON WILLIAM
F. FORD
Lewis Research
BASIS FUNCTIONS
and RICHARD
Center,
Received
Cleveland,
C. BRALEY Ohio, USA
13 June 1969
Abstract: The Hartree-Fock method is used to study the doubly even N = 2 nuclei in the 2s-Id shell. Basis functions with correct asymptotic behavior are generated by use of a Wood-Saxon well. The results are compared with calculations using the conventional harmonic oscillator basis. It is found that most of the Wood-Saxon results may be reproduced with a harmonic oscillator basis if the oscillator parameter is carefully chosen.
1. Introduction In recent years the Hartree-Fock (HF) method has received considerable attention in studies of the energy levels of deformed nuclei ‘). This method is essentially a variational procedure, based on minimization of the energy of a many-particle system. While such a procedure may provide a sensitive test of the wave function in the nuclear interior, it is expected that the shape of the wave function beyond the nuclear surface will not affect nuclear energies significantly. For this reason, HF calculations up until now have made use of single-particle harmonic oscillator basis functions, in spite of their obvious incorrect asymptotic behaviour. There are, however, many quantities of interest which are expected to be much more sensitive than the total energy to the asymptotic part of the wave function. Electromagnetic transition rates, multipole moments, and form factors for elastic and inelastic scattering are examples which come readily to mind. It does not seem likely that very accurate calculations of such quantities can be performed using HF wave functions with a harmonic oscillator basis. Thus, a HF calculation using the basis functions of the more realistic Wood-Saxon potential is of interest both for the energy spectrum it yields and for the effect it shows of the long-range part of the wave function on other nuclear properties. In this article we present the results of a considerably restricted version of such a program. The HF variational calculation is carried out for five nuclei in the 2s-ld shell by using both Wood-Saxon (WS) and harmonic oscillator (HO) basis functions. A comparison is made of such properties as orbital energies, energy gap, and total (HF) energy. In addition, nuclear radii and intrinsic quadrupole moments are calculated and compared. 265
266
W.
F. FORD
AND
R. C. BRALEY
2. Theory
In the HF method, the nuclear wave function is approximated by the Slater determinant @ = (A!)-* det IpA(i
(I)
whose orbital functions cpA(i)are determined by minimizing the expectation value of the nuclear Hamiltonian. This variational problem leads to the eigenvalue equation hqA = (&J+P~,
= safes,
(2)
where
The operator ho is usually taken to be the Hamiltonian for a single nucleon moving in the average nuclear field, and Y represents the residual two-body force. The presence of the exchange operator P, is a natural consequence of the antisymmetry of the determinant. The summation in eq. (3) is taken over all the orbits which are to be varied. Since eqs. (2) and (3) are too difficult for a direct solution, approximation methods are required. UsualIy one begins by selecting a suitable set of basis functions and then making the expansion
The problem then reduces to diago~alizi~g h in the space spanned by the basis functions 9,. In practice, this is accomplished in an approximate way by first truncating the space, then guessing at an initial set Ct, and finahy iterating until two successive diagonalizations yield the same set Cz. (Often “a” will be written for the set j,,mrr, the setj,, or even the set n,l,j, when no confusion will result.) In order to carry out this procedure, one needs matrix elements of h. It generally assumed that h, is diagonal with eigenvalues deduced from experimental spectra. The matrix elements of u may be expressed “) in terms of the coupled two-body matrix elements of Y:
x &,
9
CZ>JT f (5)
f%l TM,Xf~, >~%@-~T)<4 VI
(In lez> the tilde indicates that the product function cpc(i)qpd(j) has been antisymmetrized.) The two-body matrix elements (abl Vl c&,, are of course independent of the orbitai functions, and they are therefore ideally suited for use as input in a general program designed to solve eq. (2). Calculation of the two-body matrix elements presents some problems which stem from the fact that V(ij) is a function of the separation between nucleons i and j. If
HARTREE-FOCK
CALCULATIONS
267
HO functions are used, one can overcome the dBiculty by transforming to c.m. and relative coordinates, and the resulting integrals can be expressed in terms of Moshinsky brackets ‘). In the general case, however, such a transformation is not possible, and the most promising approach seems to be an expansion of V(ij). For the moment the isotopicspindependence is ignored. The coupled matrix element may then be written <4 vrcz,
= G&f(I2)1 W2)l&(12)-
&f(2I)),
(6)
where
Y&12) =
c <.Lm,, jbmblJM)cp,(l)cpb(2)
= (- l)j”+jb-J&M(21).
(7)
After switching from j-j to L-S coupling “) by means of 44% =~~SW,
3SMSIJMMLM,XSM,
we obtain
Q&Q% x
where ML + MS = M and
v(12)l~~d,,(ij)xsMs(ij)),
(8)
(9) In this derivation V is assumed invariant under space and spin rotations separately. Usually a spin dependence of the form V(l2) =S~OK.(12)Q(l)
(10)
- 42)
is also assumed, where q, is the unit operator in spin space and cl is the ordinary Pauli spin operator. Then, since ~~(12) = (- 1)‘+sxs(21), <4
J”l~>, = c Q&Q:“,. c &4&&2)1~,.(12)I&(~2) LS
+(- lh’%21)>7
(11)
S’
where Ass, =
= 2(2s’+1)(-1)S-‘W(33-)3~s’s)
= l-4&&1.
(12)
Next the spherical harmonics in q~& and q~& are decoupled and then recoupled according to
&b*,,@)xdmd(2)
=
c L”M”
YL”lllb>(lbmb,
L”M”J1dmd)YL~,~,,(2).
(13b)
268
W.
F. FORD
AND
R. C. BRALEY
NOW, after a little more Racah algebra, the coupled matrix element may be written
= c Q&Q;;,
c A,,[B~$bF~t!b+(
LS
- l)L+S+‘c+‘dB~~~bF~?~Fb],
(14)
L’S’
where B 2:”
=
F 3 db =
&Xl)
(19
<(P~(l)IT/s(12)lcp~~(2)),
(16)
= ~,h>dh>YLd~l).
(171
In the derivation of eq. (14) V, is assumed central so that the sums over L” and M” reduce to a single term. The final step is the inclusion of isotopic spin dependence. It is assumed that Vs(12) = i: V&.(12)2,,(1) - z&2),
(19
T’=O
where r. and zr are defined like co and trl but in isotopic spin space. The treatment is evidently exactly like that given for the spin dependence. With the customary assumption that VYT’(~~)= PSY Wr, - 4,
(19)
where PS,r$ measures the relative strengths of VST,, the final result is obtained: (ablT/lcZ)~~ = C Q$JQ& LS
C [Ass, ATT,j?rT,B$?bFgsdb L’S’T’ _(
_
1)L+S+T+l,+lag~~dipb~~~,eb],
(20)
where F rTdb =
<(P&uwl~;~(2>>.
(20
The method used to evaluate integral (21) is based on a technique due to Sawaguri and Tobocman4), who developed a general formalism for evaluation of six-dimensional integrals. A simplified version adequate for the work herein is obtained as. follows. Let &Jr)
= ~Xr)K,(fi)
(22)
be a complete set of functions in terms of which the expansion
is made. The desired integral (21) may then be written ac, db
(24)
FL
where F”,: =
s
omrp:(r)9f(r)qc(r)r2dr.
(29
HARTREE-FOCK
269
CALCULATIONS
The usefulness of the method depends on whether the series in equation (24) is rapidly convergent and whether values of ‘/n,“, , are easy to obtain. Sawaguri and Tobocman showed that for the choice s:(r)
= .Fi(a, pr)
s
(~r)ze-*rrS2rZ1F1(- n; l+3;
f
P2r2)
one gets (27)
where
3r(rn++)m! (-cq
s O”
V(r).Fz(2,
+&?rJ2<)r2dr,
0
(28)
In practice, for appropriate choices of CIand /3, the series in eq. (24) can be made to converge sufficiently fast to make the entire scheme an excellent computational device. 3. Results The theory just described has been applied to the five s-d shell nuclei 2oNe, 24Mg, “Si, 32S, and 36Ar. The HF wave functions are assumed to have axial symmetry 5), and they are formed by putting particles into deformed orbits about a spherical, TABLE I70
Orbit
1% 1p*
1% 1% 2% 1%
single-particle
1
energies
Energy (MeV) harmonic oscillator Wood-Saxon Y = 0.27 fmv2 Y = 0.35 fmv2 -26.2 -15.0 -15.0 - 3.7 - 3.1 - 3.7
-32.6 -18.2 -18.2 - 3.1 - 3.7 - 3.1
- 34.68 -20.21 -16.53 - 6.02 - 4.54 - 0.53
exp.
-35 -21.83 -15.67 - 4.14 - 3.27 + 0.94
closed-shell, inert ’ 6O core. Each deformed orbit is described as a mixture of Id,, 2s+ and Id+ basis vectors, each with the appropriate spin projection m,. The coefficients for the negative m, orbits are related to those for the positive m, orbits by time reversal symmetry: c;< - m,) = Cf(mJ( - l)+Y
(30)
270
W.
P. FORD
AND
R. C. BRALEY
The eigenvalues of h, used in the calculation are taken from the experimental spectrum of l’0, and they are listed in the fourth column of table 1. The residual twobody force V(ij> is a Rosenfeld mixture with Gaussian shape, much like that used by Ripka ‘): V(g) = $V,z, - ~~~0.3+0.7~~(~) *ut(j)]e-puz (31) with Y,, = -50 MeV (instead of Ripka’s Ya = -55 MeV) and ~1= 0.29 fm-‘. For the basis vectors themselves two sets of harmonic oscillator basis functions were used and one set of Wood-Saxon basis functions “). The latter were chosen by varying the parameters V, and 0: in the Wood-Saxon potential (32)
p(r) =
“0
l+exp
(33)
[(r-R)ial’
so as to reproduce as closely as possible the experimental eigenvalues of he as given in table 1. The eigenvalues thus obtained are given in the third column of table 1.
**r .l
0
12345678012345678 r tfm)
r ffm)
012345618 r ifm)
Fig. 1.
Choosing the harmonic oscillator basis functions poses something of a problem namely, what criteria to use. At length it was decided that one set of functions (HO - 1)
HARTREE-POCK
271
CALCULATIONS
should be chosen on the basis of similarity to the Wood-Saxon basis functions within the nuclear interior, as measured by comparison of eigenvalues and location of peaks and nodes in the functions. For this set an optimum choice was v = 0.35 fm-‘. This
(d)2s1/~
-. 6-
I
c
s
I
I
I r
L
I
I
I
I
(fm)
f .7-
.6--
r Itm) 1. Single-particlefunctions used in HF calculations: ful1 curves - Wood-Saxon, dashed CUIWS harmonic oscillator with Y = 0.35 fm- f , dot-and-dash curves - harmonic oscillator with Y = 0.27 fms2.
Fig.
-132.45 -116.80
-179.64 -157.89
-238.56 -209.66
28Si
32s
=Ar
72.28 68.26
40.21 35.27
-
-
-
(MeV)
J%F
Hartree-Fock energy
24Mg
2oNe
Nucleus
Energy gap
8.1 6.9
5.9 5.0
8.2 7.1
4.9 5.0
10.6 8.5
&eV)
TABLE 2
0.7965 -0.4162
: :
-18.55 -15.14 -16.22 -14.35
0.6691 0.7411 0.5498
t # ; : * 4 :
-25.07 -21.41 -23.05 -20.38 -22.27 -19.47 -18.94 -16.16 -17.18 -14.57
0.3607
0.8167
!i
0.4083
0.6066
-20.24 -18.11
t
-15.86 - 13.98 :
:
-19.95 -17.16
0.4798
0.8414 0.8552
0.6781 0.7782
%,3
3
0.9969
0.9968
0.7030 0.7889
0.6440 0.6959
0.9559 0.9550
%
1.000
l.ooo
1.000
1.000
1.000
1.000
1% a
of the Hartree-Fock
-0.0781 -0.0801
Projection
-22.66 -18.35
t
:
-13.83 -12.78 -21.93 -18.37
f
:
Projection of angular momentum on z-axis ml
-15.88 - 14.78
-17.01 -14.46
(MeV)
Single-particle Hartree-Fock energy
*
-0.2172 -0.5332
0.7419
0.9268
-0.4685 0.8972
0.4494
0.8830
0.7345
0.8320
-0.1080 -0.0997
-0.6342 -0.4943
%
0.6352 0.6429
0.0431
-0.1040
-0.3823 0.1481
-0.3730
-0.2315
-0.3040
-0.2787
0.5294 0.5086
-0.3715 -0.3874
%,3
0.9969 0.9968
0.0781
0.0801
0.7112 0.6145
0.7650 0.7128
-0.2938 -0.2835
l%+
orbits on spherical basis, C”,
Energies and wave functions for axially symmetric doubly even, 2s-1 d shell nuclei (a) Harmonic oscillator parameter Y = 0.35 fmW2
P
f: h,
-211.89 -209.66
-159.06 -157.89
Results using a Wood-Saxon
36Ar
ZTCJ
--I 16.41 -116.80
YSi
65.99 68.26
-
=%lg
34.82 35.27
-
=oNe
:
z
-l_l_l
type; light type corresponds
-15.42 -14.57
-16.87 -16.16
:
:
-20.40 -20.38 -19.81 - 19.47
$
-20.68 -21.41
x
%”
-0.0895 -0.0801
0.9960
0.9968
0.7333 0.7889
0.6682 0.6959
0.9562 0.%50
1.000 1.000
1.005
1.000
1.000
Loo@
-0.3386 -0.5332
0.7419
_~ 0.8716
-0.6495 0.8972
0
0.7598
0*X345
0.7961
- 0.08W -o.o9!n
-0.5729 - 0.4943
,~~
0.6475 0.6429
0.0431
-0.0576
~--
-0.3133 0.1481
-0.3730
-0.3020
-0.3040
-0.2922
OS469 0.5086
-0.3780 -0.3874
~-__
0*9960 0.9968
0.0895
O.0801
-
0.6799 0.6%45
0.7440 0.7182
-
-0.2928 -0.2835
to a harmonic oscillator basis with stated harmonic oscillator parameter Y.
0.6828 0.5498 -
0.6691
0.4868
0.6928 -0.4162
-16.11 -15.14 - 14.48 -14.35
0.8167
0.5757
:
__l_l_
0.6066
-17.76 -18.11
t
-14.07 -13.98 ?k t
:
-17.55 -17.16
0.5290
0.8333 0.8552
0.7273 0.7782
-18.76 -18.35
t
r t
-12.52 - 12.78 -17.94 -l&37
t
t
-14.14 - 14.78
-14.27 - 14.46
basis are in boldface
7.1 6.9
5.0 5.0
7.0 7.1
4.8 5.5
8.2 8.5
(6) Harmonic oscillator parameter v = 0.27 fmv2
274
W.
P. FORD
AND
R. C. BRALEY
choice of Yis based on an approximate method for calculating the “correct” nuclear radius ‘). A second set (HO-2) was chosen eventually on the basis of similarity to the WoodSaxon basis functions as measured by comparison of the resulting nuclear binding energies, orbital energies, energy gaps, and intrinsic quadrupole moments. For this set, v = 0.27 fm-‘. In table 1 the energy eigenvalues of the various basis functions are compared, while in figs. l(a)-(e) their shapes are compared. Little can be determined from the energy eigenvalues, but figs. 1(a) to (e) reveal clearly for the Is-1 p shell functions that HO-1 is closer to WS than HO-2 is. In the Zs-Id shell the situation is somewhat ambiguous: HO-I has the same peaks and nodes as WS, as before, whereas HO-2 has different peaks and nodes but has perhaps an overall better match in magnitude. The results of HF calculations using the three sets of basis functions just described are compared in tables 2(a) and (b). Comparison of the binding energies, energy gaps, and single particle energies reveals that HO-2 yields a much better match to WS than does HO-l. While the binding energies in 2(b) differ very little for the cases compared there, a substantial difference in binding energies is seen to exist between HO-1 and WS. The level ordering in *‘Ne and 24Mg is found to be the same for the three cases. However, although the level ordering for 28Si and 32S is the same in both oscillator bases, the level ordering in the WS basis is different. For the case of 36Ar, it is found that the level ordering of the first three occupied orbits is not the same for any two of the three cases being compared. This is not surprising for the orbitals in 2(b) since the energies of the orbits differ only slightly. 3.1, ROOT-MEAN-SQUARE
RADII
The rms radii are obtained from evaluation of the expectation value of the operator R2
=,$
between HF states #. The results, shown in fig. 2, differ considerably for the three sets of basis functions, and only the WS results bear a reasonable resemblance to experiment and then only in an average sense. As the mass number increases toward 40, the WS results begin to deviate even more from experiment. The HO results are quite poor in both cases. 3.2. INTRINSIC
QUADRUPOLE
MOMENT
The intrinsic quadrupole moment is defined as the expectation value of the operator Q. = e(qx)*
; i=l
[
F@j rf Y,“(Q,,)
(35)
with respect to HF states #. The operator t3 is an isospin operator with eigenvalues +1 and - I for protons and neutrons, respectively; e is the proton charge. In table 3
HARTREE-FOCKCALCULATIONS
275
the results are given for the intrinsic quadrupole moments. The set HO-2 yields a much better match to the WS results than does HO-I. The sign of (Qe) for any particular nucleus does not depend on the choice of basis, and the signs are in agreement with the accepted type of deformation (i.e., prolate or ablate) for these nuclei. Perhaps the most surprising finding of this study is the sensitivity of the energy to the shape of the basis functions. This is illustrated in table 2, which reveals that choos3.5
3.4F
u A
Wood-Saxon Harmonic oscillator,
0
Expermlent
0
Harmontc
OIcillator,
Y -0.35
v .0.21
/
3. 3 -
3.2 E 2 62 e5
3. I -
.s
3.0-
h .+ .F
2.9--
2 za-
2.1-
7.6 i-
Mass numwr.
A
Fig. 2. TABLE 3 Intrinsic quadrupole Nucleus
Quadrupole
moment
moment
<&>(e
harmonic oscillator Y = 0.27 frnm2
* fm’) Wood-Saxon
v = 0.35 fms2
ZONC
29.00
22.30
l*Mg
15.80
12.60
*%i 3% JdAr
-42.59 - 12.74 -25.92
-33.18 - IO.25 -19.17
29.24 14.90 -42.51 -11.74 -29.65
ing v to be 0.27 fme2 brings the values calculated using harmonic oscihator functions into agreement with those obtained using Wood-Saxon functions. An explanation of this can be found in the fact that the basis functions are normalized to unity. Thus
276
W. F. FORD
AND
R. C. BRALEY
a function which falls off more slowly at large distances than another must be correspondingly smaller at small distances to achieve the same normalization. The effect is more pronounced for states with smaller binding energies, as can be seen from figs. l(a)-(e). Since only the relatively weakly bound 2s and Id stated entered the present calculation, a noticeable effect was obtained. The short range of the interaction (matrix elements of which are used in computing the energy) makes it probable that it is the decrease in magnitude of the basis functions at short distances rather than any specific long range behavior which is directly responsible for the smaller nuclear binding energies obtained with the Wood-Saxon functions and the harmonic oscillator functions when v = 0.27 fm-‘. Because of the preli~na~ nature of this study, no definite conclusions can be drawn about the results to be expected when a Wood-Saxon basis is used to represent an active, polarized core. Based on the strong similarity between the WS and v = 0.35 fm-’ functions for Is and lp, however, one might conjecture that if oscillator parameters for core and valence states are chosen separately, their ultimate values may be quite different. In a sense one could introduce limited radial variation into the HF calculation in this way. This has important consequences for more extended calculations in which the 2p-lf shell is included, for with a Wood-Saxon well some if not all of these states will be unbound. If one can replace these functions by harmonic oscillator counterparts known to yield good results for observables sensitive to details of the nuclear surface, considerable numeri~l di%culties may be avoided. References 1) G. Ripka, Hartree-Fock theory and nuclear deformations, Lectures in theoretical physics, Vol. VIIIc (University of Colorado Press, 1965) pp. 237-298; M. Bouten, P. van Leuven and H. Depuydt, Nucl. Phys. A94 (1967) 687 2) M. Moshiiky, Nucl. Phys. 13 (1959) 104 3) D. M. Brink and G. R. Satchlcr, Angular momentum (Clarendon Press, Oxford, 1962) 4) T. Sawaguri and W. Tobocman, J. Math. Phys. 8 (1967) 2223 5) M. R. Gunye and C. S. Warke, Phys. Rev. 156 (1967) 1087 6) R. S. Caswell, Tech. Note. 410, National Bureau of Standards, Sept. 30,1966 7) S. A. Moszkowski, Handbuch der Physik, vol. 39; S. Fhigge (Springer-Verlag, Berlin, 1957) p. 411 8) J. Bar-Touv and I. Kelson, Phys. Rev. 138 (1965) 1035 9) H. A. Jahn, Proc. Roy, Sot. 205 (1951) 192