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MICROCOPY RESOLUTION TEST CHART
MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS - 1963
6
: MASTER ORNL.P-2468 p
ORNL-P-2468
Conf.660906-25
SEP 26 1966
Dependence of Brueckner-Hartree-Fock Calculations for Light Nuclei
on Details of the Nucleon-Nucleon Interaction:
CFSTI PRICES
Comparison of the Yale, HJ, and Reid Potentials*
Richard L. Becker, A. D. MacKellar, and B. M. MorrisH.C. & Los
Oak Ridge National Laboratory, Oak Ridge, Tennessee
ABSTRACT
Calculations of matrix elements of Brueckner's reaction matrix
and the Bethe-Goldstone wave functions for pairs of nucleons in light
nuclei have been carried out for the three currently most prominent
phenomenological nucleon-nucleon lateractions. Significant differences
in binding energies, single particle energies, spin-orbit splittings
and effective interactions are found. The proton-proton correlation
function has been derived and evaluated.
RELEASED FOR ANNOUNCEMENT
LEGAL NOTICE
This report was prepared as an account of Government sponsored work. Neither the United
States, por the Commission, nor any person acting on behalf of the Commission:
A. Makes may varranty or representation, expressed or implied, with respect to the accu-
racy, completeness, or usefulness of the information contained in the report, or that to use
of any information, apparatus, method, or process disclosed in this report may not infringe
privately owned rights; or
B. Assumos any Uxbilities with respect to the whe of, or for damages rosuldas from the
use of any information, apparatus, method, or process disclosed in this report.
As used in the above, "person acting on behalf of the Commissiou" lacludes way on-
ployne or contractor of the Commission, or employee of such cool actor, to the extent that
such employee or contractor of the Commission, or omployee of such contractor preparos,
disseminates, or provides access to, any information pursuant to be omployment or contrast
with the Commission, or his employment wit such contractor.

IN NUCLEAR SCIENCE ABSTRACTS
Research jointly sponsored by the U. S. Atomic Energy Commission under
contract with the Union Carbide Corporation, and by the Massachusetts
Institute of Technology.
**
Permanent address: Massachusetts Institute of Technology, Cambridge,
Massachusetts.
Dependence of Brueckner-Hartree-Fock Calculations for Light Nuclei
on Details of the Nucleor.-Nucleon Interactiun:..
Comparison of the Yale, HJ, and Reid Potentials
Richard L. Becker, A. D. MacKellar, and B. M. Morris
Oak Ridge National Laboratory, Oak Ridge, Tennessee
The phenomenological nucleon-nucleon potentiels which account best
for the two nucleon data below the threshold for pion production seem to
be, at present, the Yale, the Hamada-Johnston, and the more recent Reid
potential, all of which involve hard repulsive cores at about haif a
fermi. The great advantage of Brueckner theory calcuiations over traditional
shell model calculations is that potentials containing strongly attractive
and strongly repulsive parts can be used. Recent calculations of
+
-
Brueckner's reaction matrix in a harmonic oscillator basis by the present
authors' and others have for the first time derived physically reasonable
values for absolute binding energies, the spin-orbit term in the shell
model and the "effective interaction" between two nucleons in light nuclei.
Structure calculations for finite nuclei employing reaction (t)
matrices are at the moment subject to several kinds of uncertainties in-
cluding (1) approximations introduced in the calculation of the Brueckner
reaction matrix, (ii) failure to calculate any but the lowest order terms
in the Goldstone series, and (111) inadequacies of the assumed phenomenological
nucleon-nucleon force. The current populer ity of Brueckner theory calcu-
lations in an oscillator basis is undoubtedly the result of the simplicity
achieved by the Moszkowski-Scott approximation to the reaction matrix in
which part of the potential 18 treated only to the second Born approximation.
=-
=
-
=
-
:
'
.
.
'
I
The present authors have developed an alternative nethod, which also has
several advantages: the potential is not separated into two parts so
there is no need for a perturbative treatment of any aspect of the po-
tential, the off-diagonal part of the tenser force is treated exactly
by solving coupled Bethe-Goldstone equations, and the bulk of the Pauli
exclusion effect is included in the first approximation to t by employing
the Eden-Emery approximation for the Pauli operator.
A number of corrections to our first approximation have now been
investigated: The "Inside the core" contributions are of the order of
a per cent or less. The two nucleon wave function renormalization
effect* for finite systems is also only ! one per cent correction for
occupied states. The correction for the error introduced by employing
the Eden-Emery approximation for the Pauli operator Qout, can be obtained
from the expression
<$/ +(B) - 6(7145)/6) - <40/6(1)(8)+ - +(2)(8) 14.3 +
+5 ege re) - emale) <.1+(2)(5) 43\* <&.lt(s)4.)
by replacing t by tl) in the last term. Here S is the "starting energy,"
which equals e, if . is occupied in the chosen configuration. Pre-
liminary results indicate that this correction will not be large, mainly
because of the small "phase space" (in the angular momentum representation)
permitted by Qom - Qout. In order to obtain sufficient numerical accuracy,
the calculations have been performed in double precision on the IBM 360
computer at ORNL.
It is of considerable interest to examine whether or not the
currently best available potentials give essentially the same predictions
of nuclear structure. In order to minimize the errors arising from
neglect, of higher order terms in the many-body perturbation series, it is
advantageous to consider at first doubly closed shell nuclei. In Table I
are presented results of calculations for the ground state of the doubly
closed shell nucleus -0. Approximate Brueckner-Hartree-Fock (BHF) self-
consistency has been achieved within the context of a single oscillator
configuration by adjusting the value of the oscillator range parameter
a = (mW/26) ? and adding state dependent constants to the single particle
potential, as described in references 1 and 2.
Table I shows calculated
single particle BHF energies (including a large splitting between the 1p, 12
and 1pz/sta.tes); the binding energy per nucleon in the BHF approximation
(i.e., to first order in t) incluüing the Coulomb energy and the Bethe- Rose
correction for the kinetic energy of the center of mass; and the second
order contribution A E, to the binding energy per nucleon from single
particle-single hole excitations of energy anw. AE, would vanish 1f
exact BHF self-consistency had been achieved; its magnitude 18 minimized
by the choice of a. The best value of a for each potential le also given
in the table.
It is seen that the Reid and HJ potentials give generally similar
results for all quantities, whereas the results obtained with the Yale
potential are significantly different. The Yale potential gives single
particle energies closer to those measured experimentally.
In the present
approximation the HJ und Reid potentials also give a little too much
binding. Unfortunately, the theoretical framework will have to be improved
somewhat before such calculations can be used to favor one of these
potentials over another.
The "effective interaction" in the lp shell nuclei has been cal-
culated to first order in t as a function of a. Table II shows a few
important relative matrix elements for the case in which the center of mass
of the pair 16 in its lowest state. The state dependent constants con-
tained in the single particle potential are identified by the parameter
E. as explained in reference 1. The Pauli operator Qost appropriate
for tºo has been used. The potentials are first compared for the same a
and E, and then are compared at the values of a and E giving approximate
self-consistency in 16.
The fact that three potentials which give fits of similar quality
to the on-the-energy-shell two nucleon data give somewhat different results
for structure calculations, which involve also off-energy-shell matrix
elements, indicates that further improvement of the phenomenological
interaction would be of great value to nuclear theory. Here is great
concern even over the important question as to whether or not there is a
strong repulsion at short distances. One means of investigating this is
through experiments measuring the pair correlation function.' Eder, Emery,
and Sanpanther (EES) calculated an approximation to the first cluster order
nucleon-nucleon correlation function, which includes nn, pp, and np cor-
relations, using the BGT potential. For the Coulomb contribution to the
inelastic fixed momentum transfer electron scattering sum rule' one needs
instead the proton-proton correlation function. We have derived an ex-
pression for this quantity including the parts of the Bethe-Goldstone wave
functions brought in by the coupling of different orbital angular momenta
by the tensor force. Omitting these terms for brevity and to simplify
comparison with the nucleon-nucleon correlation function of EES we find
para peg(P) = x® San PopCar) = (1/56) {3-2013 + (512) vees +
2u6602 + 34 kel2 + (9/2) veel + (2/3) vēsos + (2/3) "c01*.
+ 15 vãpos + 154p23 + 3u poz + 3u1pl1 + (7/6) 44403 + (23/6) viac.}
where the first two subscripts indicate the unprturbed orbital nl values,
the third is the total (energy) quantum number of the center of mass state,
and the fourth specifies a singlet or triplet spin state. All of the
correlated radial wave functions ulr) share the features of vanishing at
the core radius and "healing" quickly to the corresponding oscillator
function. Consequently it is not surprising that Pop(r) looks very much
the same for all three potentials and, furthermore, differs little from
the nucleon-nucleon pair correlation function.
-
=
REFERENCES
* Research jointly sponsored by the U. S. Atomic Energy Commission under
contract with the Union Carbide Corporation and by the Massachusetts
Institute of Technology.
Permanent address: Massachusetts Institute of Technology, Cambridge,
Massachusetts.
1. A. D. MacKellar and R. L. Becker, Physics Letters 18, 308 (1965);
A. D. MacKellar, ORNL-IM-1374, Feb. 1966: and Thesis, Texas A&M
University, Feb. 1966 (unpublished); R. L. Becker and A. D. MacKellar,
Physics Letters 21, 201 (1966).
2. R. J. Eden and V. J. Emery, Proc. Roy. Soc. (London) A 248, 266 (1958);
R. J. Eden, V. J. Emery, and S. Sampanthar, loc. cit. A 253, 177, 186
(1959).
3. H. Be the and J. Goldstone, Proc. Roy. Soc. (London) A 238, 551 (1956).
4. Richard L. Becker, Phys. Rev. 127, 1328 (1962).
5. K. W. McVoy and L. Van Hove, Phys. Rev. 125, 1034 (1962); y.n.
Srivastava, Phys. Rev. 135, B612 (1964).
TABLE I
Comparison of ground state properties of bºo
as calculated with three nucleon-nucleon interactions
Yale
Kind of Exp.
HJ
-63.4
Reid
-64.0
Experiment
5-3423.52
1 - 4472
8/2(0)
-52.1
-21.7.
-29.6
-25.4
P3/2(n)
P3/2(p)
P1/2(n)
21/2(p)
-37.5
-33.2
-26.5
-22.2
11.0
(0,2p)
150*
(p,2p)
15.-160
-37.7
-33.5
-27.9
-23.7
9.8
-21.7
-1981
-15.6
-17.5
-12.451
(p,2p)
P1/2-P3/2
-
7.9
56.6t2
16.1
BE/A
- 6.7
- 9.3
- 8.0
48/A
- 0.6
- 8.9
- 0.75
0.512
- 0.7
0.474
0.507
TABLE II
Some diagonal relative t matrix elements, < nel t(gs) / ne>
Yale HJ Reia
(a - 0.5, 2, 3 1.0)
Yale
a s 0.474
4 = 1.27
HJ
g = 0.512
6 = 0.73
Reid
g = 0.507
4 = 0.13
n
:
3
S
0 0 0 0 8.98 - 9.30
0 0 1 1 -9.59 -11.56
• 9.23
-12.30
*8.30
49.49
9.60)
-1.13
- 9.16
-li.80
+2.73
+ 3.64
0
()
0
0
1
1
1
1
1
0
1
2
0
1
1
1
43.38 + 3.3
-2.7k • 2.60
0.19 4 3.2
-2.29 - 2.18
3.88
- 3.28
+ 3.94
- 2.20
+ 3.88
- 3.28
+ 3.94
- 2.18
+3.50
-1.88
44.03
- 2.37
มาที่ ๗ dณ์ ที่
0
0
0
1
1
2 2
2 1
2 2
0 0
0 1
0
1
1
0
1
-1.38 * 1.23
+2.03 + 2.
-4.34 - 4.60
3.65 , 4.8)
1.63 9.89
* 1.19
+ 2.75
- 5.23
- 4.64
- 9.82
-1.il
+l.67
4.12
* 1.35
+ 2.97
- 4.96
- 4.70
- 1.19
+ 2.80
- 5.23
3.62
" " " + +
- 4.59
. 9.45
-8, 16
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