arXiv:nucl-ex/0605026v2 25 May 2006
Asymmetry dependence of proton correlations.
R. J. Charity1, L. G. Sobotka1,2, W. H. Dickhoff2
Departments of Chemistry1and Physics2, Washington University, St. Louis, Missouri 63130.
(Dated: February 8, 2008)
A dispersive optical model analysis of p+40Ca and p+48Ca interactions has been carried out. The
real and imaginary potentials have been constrained from fits to elastic scattering data, reaction
cross sections, and level properties of valence hole states deduced from (e,e′p) data. The surface
imaginary potential was found to be larger overall and the gap in this potential on either side of
the Fermi energy was found to be smaller for the neutron-rich p+48Ca system. These results imply
that protons with energies near the Fermi surface experience larger correlations with increasing
PACS numbers: 21.10.Pc,24.10.Ht,11.55.Fv
In the independent-particle model, nucleons in the nu-
cleus move in a mean-field potential generated by the
other nucleons.All nucleon levels up until the Fermi
energy (EF) are fully occupied, while those above are
empty. Although this model enables an understanding
of various aspects of nuclear structure, a full descrip-
tion of nuclei and nuclear matter requires consideration
of the correlations between the nucleons. These include
short-range, central and tensor interactions and longer
range correlations associated with low-lying collective ex-
citations . As a result, for closed-shell nuclei, single-
particle (sp) levels below EF have an occupancy of only
70-80% and the levels at higher energy have a nonzero
occupancy . The strength of the sp levels are spread
over energy, with narrow peaks or broad distributions
(depending on their separation from EF). In addition,
there is strength at very high momentum .
Although there are numerous studies of the effect of
correlations on the properties of sp levels for nuclei near
stability, there are only a few studies for very neutron
or proton-rich nuclei. From neutron knock-out reactions,
Gade et al.  infer the occupancy of the 0d5/2neutron
hole state in the proton-rich32Ar nucleus is considerably
reduced relative to those for stable nuclei.
An alternate method to study sp strength is through
the use of the dispersive optical model (DOM) developed
by Mahaux and Sartor . This description employs the
Kramers-Kronig dispersion relation that links the imag-
inary and real parts of the nucleon self-energy . This
procedure links optical-model (OM) analyses of reaction
data at positive energies to structural information at neg-
ative energies. In the present work, the properties of
proton levels in Ca nuclei as a function of asymmetry
are investigated with the DOM. Previously mea-
sured elastic-scattering and reaction-cross-section data
for protons on40Ca and48Ca as well as level proper-
ties of hole states in these nuclei, inferred from (e,e′p)
reactions, were simultaneously fit. The dependence on δ
is extracted and used to predict level properties of60Ca.
In the DOM, the complex energy-dependent potential
felt by the protons is comprised of a real V, volume Wv
and surface Ws imaginary components, plus spin-orbit
Vsoand Coulomb Vcpotentials,
U(r,E) = −V(r,E) + Vso(r) + Vc(r)
− iWv(E) f(r,rv,av) + 4iasWs(E)d
Wood-Saxon form factors f(r,ri,ai) = [1 + e
are used. The real part of the nuclear potential is as-
sumed to be given by two terms
∆V(r,E) = VHF(E) f(r,rHF,aHF) + ∆V(r,E) (2)
where VHF has a smooth energy dependence arising from
the nonlocality or momentum-dependence of the micro-
scopic self-energy. The dispersive correction ∆V has vol-
ume and surface parts,
∆V(r,E) = ∆Vv(E)f(r,rv,av)+4as∆Vs(E)d
and is related to the imaginary potential through the
dispersion relationship, i.e.,
where i = v,s and P stands for the principal value. The
dispersive corrections are a result of coupling to non-
elastic channels. The surface term accounts for the influ-
ence of low-lying collective states and giant resonances.
The form for the imaginary potential must take into
account the dominance of surface and volume absorp-
tion at low and high positive energies, respectively. In
addition around EF the imaginary potentials must be
zero. The potential should be approximately symmetric
around EF, but further away it must become asymmetric
as there are a finite number of hole states. In this work
Wv(E) = Av
(E − EF)4
(E − EF)4+ B4
+ ∆WNM.(E) (5)
0 50100 150
FIG. 1: (Color online) Calculated and experimental elastic-
scattering angular distributions of the differential cross sec-
tion dσ/dΩ and analyzing power. Left panels shows results for
p+40Ca, right for p+48Ca. Each curve is labeled by the pro-
ton energy in MeV. For display purposes, successively higher
energy curves and data for dσ/dΩ are scaled down by an addi-
tional factor of 4.5. For the lowest energies, compound-elastic
contributions were also included in the fits.
where the energy-asymmetric correction ∆WNM.(E) is
derived from nuclear-matter considerations . The sur-
face potential was taken as the difference of two functions
that cancel at large energies, i.e.,
∆Ws(E) = ω(E,A1
s,c,0) − ω(E,A2
ω(E,A,B,c,Q) = A Θ(X)
X4+ B4e−cX, (7)
where Θ(X) is Heavyside’s step function, X =|E − EF|−
The Hartree-Fock potential is often assumed to de-
crease linearly or exponentially with energy. We took
se−cQ, and Q = B1
FIG. 2: (Color online) Total reaction cross sections are dis-
played as a function of proton energy.
1 + exp(BHF(E − EF)/AHF)
which is approximately linear around EF and becomes
more exponential at larger energies. This form provided
a reasonable location for the 0s1/2level in40Ca.
The parameters of the DOM were fit for both40Ca
and48Ca from a large set of data covering both positive
and negative proton energies. For40Ca, 14 experimental
elastic-scattering angular distributions for energies from
18 to 135 MeV [8, 9, 10, 11, 12, 13] and seven data sets
for the analyzing power measured at energies from 21
to 80 MeV [12, 14, 15, 16, 17, 18] were included. For
48Ca the fitted data included 14 angular distributions
and seven sets of analyzing power at energies from 8 to
65 MeV [12, 13, 19, 20]. Reaction cross sections for both
targets were taken from the tabulations of Bauhoff .
For the 0d5/2, 1s1/2, and 0d3/2proton holes states, the
mean level energies, r.m.s. radii, spectroscopic factors,
and widths inferred from measured (e,e′p) cross sections
[22, 23] were also included. Lastly, the fits considered
the mean energies of the 0f7/2and 0f5/2(for48Ca only)
particle levels .
factors were varied, but kept identical for both targets.
Similarly the fit parameters B2
the decay of the Ws(E) at large energies, as well the
magnitude of the spin-orbit term, were also set identical
for both targets. Only AHF, BHF, A1
were allowed to differ for each isotope.
The final fit to the experimental data is shown in
Figs. 1-4 and the fitted potentials are displayed in Fig. 5a.
It was found possible to obtain similar quality fits with
different potentials, however a robust feature of all good
fits was that the magnitude of the surface imaginary po-
tential ( A1
around EF) was narrower for the neutron-rich48Ca nu-
cleus. There was however an ambiguity in determining
the rate at which this potential diminished at large en-
ergies. This ambiguity is coupled with an ambiguity in
s, ∆B and c defining
s, Av, and Bv
s) was larger and its width B1
s(of the minimum
FIG. 3: (Color online) Fitted level properties for the 0d5/2,
1s1/2, and 0d3/2proton hole states. The left (a,c,e) and right
(b,d,f) panels display the fits for40Ca and48Ca, respectively.
The fitted quantities include a,b) the root-mean-squared ra-
dius Rr.m.s, c,d) the spectroscopic factor S expressed as a
percent of the independent-particle-model value, and e,f) the
widths Γ of these levels.
determining the magnitude and the rate of increase of
Tornow et al.  fitted similar data for40Ca with the
DOM using a Ws(E) which decreased slowly and was
still substantial at the highest energy considered in the
work. With such a slow diminishing of Ws(E), one can
also obtain good fits to the48Ca data, however the fitted
Wv(E) potentials are substantially different for40Ca and
48Ca. On the other hand, if Ws(E) is made to diminish
faster, these differences can be reduced to zero.
Ambiguities in determining potentials in standard OM
fits are well known, however volume integrals of the po-
tentials have been shown to be better defined . Com-
parisons of the imaginary volume integral, JW(E) =
?W(r,E)dr, obtained from the OM fits in the referenced
experimental studies, indicate that JW is larger below
E ∼50 MeV in48Ca. However for higher energies, there
is no discernible difference for the two isotopes. Thus
for48Ca, if its larger surface potential is still significant
in this higher energy region, Wv(E) must be smaller to
produce similar values of JW. We believe this result is ar-
tificial, and therefore a solution where Ws(E) diminishes
faster is preferable. In any case, it is not possible to con-
strain any difference in Wv(E) for the two nuclei. Thus,
for the final fit we have taken Wv(E) to be independent
of asymmetry. This is consistent with global OM fits
[26, 27] which have a significant asymmetry dependence
for Ws(E), but none for Wv(E).
asymmetry dependence of Wv(E) would be expected and
exp DOM exp DOMDOM
FIG. 4: (Color online) Comparison of experimental proton
single-particle levels [22, 23, 24, 28, 29, 30, 31] to DOM cal-
culations. For the levels indicated with the solid dots, their
energies were included in the fits. The dashed lines indicate
the Fermi energy.
higher-energy48Ca data would provide sensitivity to this.
Calculated and experimental sp level schemes for40Ca
and48Ca are displayed in Fig. 4. Apart from the levels
included in the fit (indicated with the solid dots), the
other known levels are well reproduced. The 0s1/2level
of40Ca is very wide and even though the DOM prediction
for its energy is low, it lies within the experimentally
determined width .
Present DOM calculations have been extrapolated to
60Ca assuming the parameters AHF, BHF, A1
s vary linearly with δ.
dependence of Ws(E) is shown in Fig. 5a as the thin-
lined curve and the predicted sp level scheme is displayed
in Fig. 4. The surface dispersive correction is large for
this nucleus and its effects are quite apparent. The lev-
els in the immediate vicinity of EF are focused closer to
EF, increasing the density of sp levels. A reduced gap
between the particle and hole valence levels implies that
the closed-shell nature of this nucleus has diminished and
proton pairing may be important. The levels further from
EF have been pushed away and as a result there are big
gaps between the 0p1/2and the 0d5/2and also between
the 1p1/2and 0g9/2levels.
The occupation probabilities as defined in Ref.  are
plotted in Fig. 5c. The results for48Ca agree with the
theoretical work of Ref. . For proton hole states just
below EF, the occupation probabilities have decreased
for the more neutron-rich48Ca while the opposite is true
for the particle states. This trend is further accentuated
in our extrapolation to60Ca. On the other hand, the
The extrapolated energy-
FIG. 5: (Color online) Quantities derived from DOM fits. a)
The energy dependence of the Hartree-Fock potential VHF,
and the surface Ws and volume Wv imaginary potentials. b)
The radial dependence of the effective mass at EF. c) The
occupation probabilities (points indicate sp levels). For all
plots, the thick (back), medium (red) and thin (green) curves
give the results for40Ca,48Ca, and60Ca, respectively.
more deeply-bound 0s1/2, 0p3/2, and 0p1/2levels show
very little sensitivity. Their occupancies are more sensi-
tive to the volume imaginary component whose asymme-
try dependence was not constrained.
Our observation that the occupancies of valence pro-
ton hole states are reduced in neutron-rich48Ca can be
compared to the reduced occupancy of the valence neu-
tron hole state in the proton-rich32Ar inferred by Gade
et al. . Thus a preponderance of one type of parti-
cle, reduces the occupancies of valence hole states for the
other type. This indicates that correlations are stronger
for these valence nucleons. Recent calculations of asym-
metric nuclear matter by Frick et al.  also predict
such a result as a purely volume effect.
probably similar in both cases reflecting that p-n inter-
actions are stronger than n-n or p-p, partly because of
the tensor force. Thus, protons in neutron-rich systems
are more strongly correlated as illustrated from the in-
ferred spectroscopic factors of 65%, 56% and 50% for the
0d3/2proton level in40Ca,48Ca and60Ca, respectively.
Conversely for neutrons, the opposite is expected.
The nucleon effective mass m∗(r,E)/m
dV (r,E)/dE (m is the nucleon mass) at E=EFinferred
from this work is displayed in Fig. 5b. Only the surface
contribution around r=4-5 fm has been constrained and
it increases significantly with asymmetry. This suggests
an asymmetry dependence of the surface component of
the level-density parameter.
In conclusion, the properties of proton single-particle
Its origin is
states in the vicinity of EF for40Ca and48Ca have been
studied with a comprehensive dispersive-optical-model
analysis of elastic-scattering and bound-level data. The
analysis indicates that the imaginary surface potential is
∼50% larger and the minimum around the Fermi energy
is narrower for the neutron-rich48Ca nucleus. This im-
plies that, with increasing asymmetry, the occupancies of
proton levels vary more smoothly across the Fermi sur-
face, a consequence of increased correlations. The present
observations and those of Gade et al.  can be under-
stood from the larger strength of p-n relative to the p-p
and n-n interactions. Hence, protons (neutrons) experi-
ence larger (weaker) correlations in neutron-rich matter.
The reversed is true for proton-rich matter.
This work was supported by the U.S. Department of
Energy, Division of Nuclear Physics under grant DE-
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