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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

asymmetry.

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 [2]. 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 [3]. 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 [4].

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. [5] 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 [6]. This description employs the

Kramers-Kronig dispersion relation that links the imag-

inary and real parts of the nucleon self-energy [7]. 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

δ=N−Z

A

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

drf(r,rs,as). (1)

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

r−riA1/3

ai

]−1

∆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

drf(r,rs,as),

(3)

and is related to the imaginary potential through the

dispersion relationship, i.e.,

∆Vi(E) =P

π

∞

?

−∞

Wi(E′)

?

1

E′− E−

1

E′− EF

?

dE′

(4)

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

we assume

Wv(E) = Av

(E − EF)4

(E − EF)4+ B4

v

+ ∆WNM.(E) (5)

Page 2

2

0 50100150

θ

-11

-8

10

-5

10

-2

10

10

4

10

18

20

21

25

26

30

35

40

45

48

61

65

80 135

Ca

40

Ca (p,p)

40

0 50100 150

8

10

12

14

15

16

21

25

30

35

40

45

48 65

Ca

48

Ca (p,p)

48

[mb/sr]

Ω

/d

σ

d

-0.5

0

0.5

21

-0.5

0

0.5

26.3

-0.5

0

0.5

30.3

-0.5

0

0.5

40

-0.5

0

0.5

49

-0.5

0

0.5

65

-0.5

0

0.5

80.2

8

10

12

14

15

15.6

c.m.

θ

65

[deg]

c.m.

Analysing Power

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 [6]. 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,B1

s,c,0) − ω(E,A2

s,B2

s,c,Q), (6)

ω(E,A,B,c,Q) = A Θ(X)

X4

X4+ B4e−cX, (7)

where Θ(X) is Heavyside’s step function, X =|E − EF|−

Q, A2

The Hartree-Fock potential is often assumed to de-

crease linearly or exponentially with energy. We took

s= A1

se−cQ, and Q = B1

s+ ∆B.

0 1020

E [MeV]

30 4050

400

600

800

1000

1200

Ca

40

p+

Ca

48

p+

[mb]

react

σ

FIG. 2: (Color online) Total reaction cross sections are dis-

played as a function of proton energy.

the form

VHF(E) =

2AHF

1 + exp(BHF(E − EF)/AHF)

(8)

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 [21].

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 [24].

Thesevengeometric

(rHF,aHF=av,rv,rs,as,rso,aso)

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

parameters

thedefiningform

s, ∆B and c defining

s,B1

s, Av, and Bv

s) was larger and its width B1

s(of the minimum

Page 3

3

2

3

4

5

Ca

40

a)

40

60

80

c)

0

2

4

6e)

Ca

48

b)

d)

f)

[MeV]

Γ

S [%]

[fm]

r.m.s

R

Levels

5/2

0d

1/2

1s

3/2

0d

5/2

0d

1/2

1s

3/2

0d

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

Wv(E).

Tornow et al. [25] 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 [6]. 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

Theoretically some

E [MeV]

-60

-40

-20

0

2

1

0s

2

3

0p

2

1

0p

2

5

0d

2

1

1s

2

3

0d

2

7

0f

2

3

1p

2

1

1p

2

1

0s

2

3

0p

2

1

0p

2

5

0d

2

1

1s

2

3

0d

2

7

0f

1p3/2

1p1/2

2

5

0f

2

1

0s

2

3

0p

2

1

0p

2

5

0d

2

1

1s

2

3

0d

2

7

0f

2

3

1p

1f5/2

1p1/2

2

9

0g

2

1

2s

2

5

1d

Ca

40

Ca

48

Ca

60

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 [28].

Present DOM calculations have been extrapolated to

60Ca assuming the parameters AHF, BHF, A1

B1

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. [6] are

plotted in Fig. 5c. The results for48Ca agree with the

theoretical work of Ref. [32]. 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

s, and

The extrapolated energy-

Page 4

4

-1000100200

Potential [MeV]

0

5

10

15

6)

÷

(

HF

V

v

W

s

W

a)

02468 10

m*/m

0.5

1

1.5

2

F

E=E

b)

Ca

40

Ca

48

Ca

60

-40-20

[MeV]

F

E-E

0

Occupation

0

0.5

1

c)

[MeV]

F

E-E

r [fm]

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. [5]. 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. [33] 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

=1 −

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. [5] 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-

FG02-87ER-40316.

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