Extraction of Neutron Density Distributions from
Proton Elastic Scattering at Intermediate
Energies
H. Takeda , H. Sakaguchi , S. Terashima , T. Taki , M. Yosoi , M. Itoh ,
T. Kawabata, T. Ishikawa , M. Uchida , N. Tsukahara , Y. Yasuda ,
T. Noro† , M. Yoshimura† , H. Fujimura† , H.P. Yoshida† , E. Obayashi† ,
A. Tamii and H. Akimune‡
†
Department of Physics, Kyoto University, Kyoto 606-8502, Japan
Research Center for Nuclear Physics, Osaka University,Osaka 567-0047, Japan
Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
‡
Department of Physics, Konan University, Kobe 658-8501, Japan
Abstract. Cross sections, analyzing powers and spin rotation parameters of proton elastic scattering
from 58 Ni and 120 Sn have been measured at intermediate energies. Obtained data have been analyzed
in the framework of relativistic impulse approximations. In order to explain the 58 Ni data, it was
necessary to modify NN interactions in the nuclear medium by changing coupling constants and
masses of σ and ω mesons. For 120 Sn, by assuming the same modification of NN interactions
and by using proton densities deduced from charge densities, the neutron density distribution was
searched so as to reproduce 120 Sn data at 300 MeV.
INTRODUCTION
Research fields in nuclear study are remarkably spreading due to the recent developments of radioactive isotope beam facilities all over the world. Nuclei far from β stability line are expected to be different not only quantitatively but also qualitatively. For
instance neutron rich unstable nuclei are expected to have anomalous structures such as
neutron skin and halo. Neutron distributions in nuclei will provide fundamental information for nuclear structure study. Thus it is indispensable to establish procedures to
extract neutron density distributions from experimental information.
Protons at intermediate energies are considered to be suitable to extract information
inside the nucleus because of the large mean free path in the nuclear medium. Ambiguities due to the target nuclear structure are relatively small in the elastic scattering
since the ground state wave functions used for elastic scattering are restricted by charge
distributions measured by electron scattering. Although it is hard to obtain neutron distributions from charge distributions, they can be assumed to have the same shapes as
with protons for N Z nuclei. Thus the proton elastic scattering at intermediate energies has been used to discern various microscopic approaches for nuclear interactions.
Applying these models to N Z nuclei, neutron density distributions can be extracted
from proton elastic scattering data.
CP675, Spin 2002: 15th Int'l. Spin Physics Symposium and Workshop on Polarized Electron
Sources and Polarimeters, edited by Y. I. Makdisi, A. U. Luccio, and W. W. MacKay
© 2003 American Institute of Physics 0-7354-0136-5/03/$20.00
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medium effect
& realistic density
original Horowitz
Spin Rotation Parameters
Wallace
200MeV
(×100)
250MeV
(×10)
Analyzing Power
dσ/dΩ [mb/sr]
200MeV
250MeV
300MeV
300MeV
200MeV
300MeV
400MeV
(× 0.1)
58
Ni(p,p)58Ni
θCM [degree]
400MeV
400MeV
θCM [degree]
θCM [degree]
FIGURE 1. Cross sections, analyzing powers and spin rotation parameters of p - 58 Ni elastic scattering
at 200 – 400 MeV. See text for details.
EXPERIMENT
We measured cross sections, analyzing powers and spin rotation parameters of proton
elastic scattering from 58 Ni and 120 Sn at E p 200 – 400 MeV. Scattering angles were
up to 60Æ for measurements of cross sections and analyzing powers, and up to 45 Æ for
spin rotation parameters. We can determine the scattering amplitudes completely from
these measurements since there are only three independent observables in proton elastic
scattering from a spin 0 nucleus.
The experiment was performed at the Research Center for Nuclear Physics (RCNP).
Polarized protons were injected into the pre-accelerator AVF cyclotron, transported to
the six sector ring cyclotron, accelerated to a final energy of 200 – 400 MeV and directed
against the target. Scattered protons were momentum analyzed with a high momentum
resolution magnetic spectrometer, ‘Grand Raiden’, and detected by counters at the focal
plane. Momenta of scattered protons were measured by detecting their position with
vertical drift chambers. The polarization of scattered protons was determined using
the focal plane polarimeter (FPP), which measured scattering asymmetries in a carbon
analyzer block with multi-wire proportional chambers. Scintillators and hodoscopes
were used to trigger the data acquisition and for particle identifications.
NN INTERACTIONS IN MEDIUM
Obtained data were analyzed in the framework of the relativistic impulse approximation
(RIA). It has been pointed out[1] that the RIA model with density dependent coupling
constants and masses of exchanged σ and ω mesons has been able to explain cross
sections and analyzing powers of proton elastic scattering from 58 Ni at intermediate
energies. In that analysis the neutron distribution has been assumed to be same as protons
deduced from the charge distribution except for the normalization factor N Z. Figure 1
shows the experimental results and some model calculations of cross sections, analyzing
powers and spin rotation parameters of 58 Ni. Solid circles are our data. Dotted and
dashed curves are original RIA calculations by Horowitz et al.[2] and Wallace et al.[3],
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χ2 map (300MeV)
set3
bω
set1
set2
best
set4
bσ
FIGURE 2.
χ 2 -map in (b σ bω ) parameter space for 300 MeV.
respectively. Solid curves indicate the medium modified RIA model described above.
Modifications of coupling constants and masses are parameterized as
g2j ḡ2j
m j m̄ j
g2j
ḡ2j
1 a j ρ rρ0 1 ā j ρ rρ0
(1)
m j 1 b j ρ rρ0 m̄ j 1 b̄ j ρ rρ0 (2)
where j refers to the σ or ω mesons and ρ0 stands for the normal density. Newly
measured spin rotation parameters are also well explained by the medium modified
RIA model. Figure 2 shows χ 2 -map in (bσ bω ) parameter space for 300 MeV. Other
six parameters were optimized at each grid point. A very strong correlation between b σ
and bω is indicated by a narrow valley. The same correlation can be found between a σ
and aω and in other energies also.
NEUTRON DISTRIBUTION SEARCH
For N Z nuclei such as 120 Sn it can not be expected that the neutron distribution has
the same shape as with protons. However, since the elastic scattering is sensitive to both
NN interactions in nuclear medium and density distributions of the target nucleus, the
neutron density distribution can be extracted from elastic scattering, assuming the same
medium modifications fixed by the 58 Ni data.
In order to search the neutron distribution we used a sum of Gaussians (SOG) type
distribution
ρn r Qi
rRi 2 γ 2
rRi 2 γ 2
e
e
∑
2π 32 γ 3 i 1 2R2i γ 2
N
Ê
(3)
The normalization condition ρn r d 3r N results in the constraint for Qi (∑ Qi 1).
Qi are searched so as to reproduce 120 Sn data at 300 MeV, whereas width γ and position
Ri of each Gaussian are fixed with the values listed in a reference[4]. All resulting SOG
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-3
Density (fm )
0.15
0.1
0.05
0
0
5
10
Radius (fm)
10
6
10
5
10
4
10
3
10
2
120 Sn.
1
med.eff. & srch.den.
original Horowitz
0
200MeV
-1
1
200MeV
(×100)
10
1
-1
250MeV
10
(×10)
Analyzing Power
dσ/dΩ [mb/sr]
FIGURE 3. Obtained neutron distribution in
0
250MeV
-1
1
0
-2
10
300MeV
400MeV
-4
120
-5
0
0
(× 0.1)
10
10
300MeV
-1
1
-3
10
Sn(p,p)
120
20
Sn
40
θCM [degree]
60
-1
400MeV
0
20
40
θCM [degree]
FIGURE 4. Cross sections and analyzing powers of proton elastic scattering from
400 MeV.
60
120 Sn
at E p 200 –
distributions with good reduced χ ν2 χ 2 ν χν2 χν2min 1
(4)
where ν is the number of degrees of freedom and χν2min is the minimum value of χ ν2 , are
possible densities and their superposition determines an error band, which is displayed
in Fig. 3 as shaded area. The obtained neutron distribution has a bump structure at the
nuclear center. This result is consistent with the wave function of neutrons in 3s 12 orbit
that is expected to be occupied in 120 Sn nuclei. Solid curves in Fig. 4 are the calculations
using the best fit neutron density. Original unmodified RIA calculations with relativistic
Hartree densities are also displayed in Fig. 4 by dotted curves. It is notable that our data
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0.15
0.15
Density (fm )
0.1
-3
-3
Density (fm )
best
set1
set2
0.05
0
best
set1
set2
set3
set4
0.1
0.05
0
5
10
0
0
5
10
Radius (fm)
Radius (fm)
FIGURE 5. The left part shows the distributions deduced with ‘set1’ and ‘set2’ parameters. Deduced
distributions with all sets in Fig. 2 are superposed in the right part.
indicated by solid circles are well explained by the deduced density not only at 300 MeV
but also at other energies although the density search was performed using 300 MeV data
only. The difference of the neutron and proton root mean square radii can be evaluated
as ∆rnp 0116 0015 fm, which agrees with a result deduced from the SDR sum rule
in Ref. [5].
In order to estimate the uncertainty in the deduced distribution due to the ambiguities
in our medium modification parameters, we also searched the distributions using various
parameter sets indicated in Fig. 2 as ‘set1’ to ‘set4’. Reduced χν2 of the ‘set1’ and ‘set2’
parameters are about χν2min 1, while it is χν2 χν2min 2 for the ‘set3’ and ‘set4’
parameters. The left part of Fig. 5 shows the distribution deduced with the ‘set1’ and
‘set2’ parameters. Changes of the distributions are relatively small compared to the
error band. However changes become larger if we use the ‘set3’ and ‘set4’ parameters as
displayed in the right part of Fig. 5. In other words, we can obtain the neutron distribution
with small errors if the NN interactions in medium are well determined.
ACKNOWLEDGMENTS
We would like to thank Prof. Hatanaka and the staff members of the RCNP for their
support and tuning to obtain a clean and high intensity beam during the experiment.
REFERENCES
1.
2.
3.
4.
5.
Sakaguchi, H., et al., Phys. Rev. C57, 1749 (1998) and references therein.
Murdock, D.P., and Horowitz, C.J., Phys. Rev. C35, 1442 (1987); Horowitz, C.J.,et al., Computational Nuclear Physics 1, Springer-Verlag, Berlin, 1991, Chap. 7.
Tjon, J.A., and Wallace, S.J., Phys. Rev. C32, 1667 (1985); Phys. Rev. C36, 1085
(1987).
de Vries, H., et al., Atomic Data and Nuclear Data Tables 36, 495 (1987).
Krasznahorkay, A., et al., Phys. Rev. Lett. 82, 3216 (1999).
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