Experimental Studies on Three-Nucleon Systems
at RCNP
K. Hatanaka∗, K. Sagara†, Y. Shimizu∗ , T. Yagita† , Y. Sakemi∗ , T. Wakasa ∗,
H.P. Yoshida∗, J. Kamiya∗, M. Yoshimura∗ , H. Sakai∗∗, A. Tamii∗∗ , K.
Yako∗∗ , Y. Maeda ∗∗ , T. Saito∗∗ , T. Ishida† , S. Minami† , K. Tsuruta†, T.
Noro† , K. Sekiguchi‡ , H. Akiyoshi‡ and V.P. Ladygin§
†
∗
RCNP, Osaka University, Mihogaoka, Ibaraki, Osaka 560-0047, Japan
Department of Physics, Kyushu University, Hakozaki, Fukuoka 812-8581, Japan
∗∗
Department of Physics, University of Tokyo, Bunkyo, Tokyo 113-0033, Japan
‡
RIKEN, Hirosawa, Wako, Saitama 351-0198, Japan
§
Joint Institute for Nuclear Research, 141980 Dubna, Russia
Abstract. The results of experimental studies of three-nucleon systems will be presented, i.e. the
pd elastic scattering at E p = 250 MeV and the d p →3 He+γ capture reaction at Ed = 200 MeV.
The angular distributions of the cross section and all the proton spin observables were measured
for elastic scattering, and the cross section and analyzing powers, Ay , Axx and Ayy for the capture
reaction. The results are compared with theoretical predictions based on exact solutions of the threenucleon Faddeev equations and modern realistic nucleon-nucleon potentials combined with threenucleon forces.
INTRODUCTION
One of the fundamental interests in nuclear physics is to establish the nature of nuclear forces and understand nuclear phenomena based on the fundamental Hamiltonian.
Studies of few-nucleon systems offer a good opportunity to investigate these forces. Realistic two-nucleon forces (2NF) [1] fail to reproduce experimental binding energies for
light nuclei, clearly showing underbinding. Correct three-nucleon (3N) and four-nucleon
(4N) binding energies can be achieved by including the Tucson-Melbourne (TM) [2]
or Urbana IX [3] three-nucleon forces (3NF) which are refined versions of the FujitaMiyazawa force [4], a 2π -exchange between three nucleons with an intermediate ∆ excitation. In recent years, it became possible to perform rigorous numerical Faddeev-type
calculations for the 3N scattering processes by the tremendous advances in computational capabilities. In addition to the first signal on 3NF effects resulting from discrete
states, strong 3NF effects were observed in a study of the minima of the Nd elastic
scattering cross section at incoming nucleon energies above 60 MeV. The discrepancy
between the data and predictions based exclusively on NN forces could be largely removed by including the TM 3NF [5]. For spin observables, however, a recent study at
RIKEN [6] showed that the inclusion of the 3NF does not always improve the description
of precise data taken at intermediate deuteron energies. Proton vector analyzing power
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
705
data at 70–200 MeV have revealed the deficiency of the 3NF [7], which produces large
but wrong effects. These results may be caused by a wrong spin structure of present-day
3NF. Clearly the present situation is only the very beginning of the investigation of the
spin structure of the 3NF. Precise data at intermediate energies including higher-rank
spin observables are needed to provide constraints on theoretical 3NF models.
In the present paper, we present two experimental studies on 3N systems recently
performed at the Research Center for Nuclear Physics (RCNP), Osaka University; the
pd elastic scattering at 250 MeV [8] and the d p →3He+γ reaction at Ed = 200 MeV. The
experimental results are compared with the theoretical predictions.
EXPERIMENTAL RESULTS AND DISCUSSION
Measurements were performed at the RCNP cyclotron facility. Polarized protons or
deuterons were produced in an atomic beam polarized ion source [9], injected into and
accelerated by the K = 120 MeV AVF (azimuthally varying field) cyclotron. Subsequently the beam was injected into the K = 400 MeV Ring cyclotron and accelerated to
the final energy.
pd elastic scattering at 250 MeV
Differential cross sections, analyzing powers, and a complete set of polarization transfer (PT) coefficients were measured for pd elastic scattering using self-supporting 99%
isotopically enriched deuterated polyethylene foils (CD 2) [10] with total thicknesses of
21 and 44 mg/cm2. In a later measurement, a gaseous target was used to normalize cross
sections taken with the solid CD2 target. Scattered protons or recoil deuterons in the pd
scattering were momentum analyzed by the Grand Raiden spectrometer [11]. The horizontal and vertical acceptance of the Grand Raiden was limited by a slit system to ±20
and ±30 msr, respectively. The polarization of elastically scattered protons from CD 2
targets was measured at center of mass scattering angles from 10◦ to 95◦ by the focal
plane polarimeter (FPP) [12]. The experimental results for the differential cross section,
the vector analyzing power and the PT coefficients are shown in Figs. 1 and 2. Only
statistical errors, mostly smaller than the size of the data point are shown. The overall
uncertainty in the absolute normalization calibrated by the gaseous target measurements
is estimated to be 3%. There is also the relative uncertainty of 2.5% attributed to the
inhomogeneity of the CD 2 foils. The analyzing power has an uncertainty of only 1% in
the absolute normalization owing to the precise calibration of the beamline polarimeter
[8]. The the normalization of the PT coefficients have an uncertainty of 2.5% [12].
In the left panel of Fig. 1, the measured differential cross sections are compared
with theoretical predictions [13]. The various 2NF predictions are very similar and are
depicted by a narrow band (light shaded). The inclusion of the TM 3NF (dark shaded
band) leads to a much better description at angles larger than 70 ◦. This supports the
claim of the clear evidence [5, 6, 14, 15] of the 3NF from the systematic analysis of the
energy dependence of the cross section data. The inclusion of the TM (dashed curve)
and the Urbana IX (solid curve) 3NF also leads to a good agreement with the data.
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FIGURE 1. The differential cross section d σ /dΩ (left) and proton analyzing powers (right) of elastic
pd scattering at E p = 250 MeV. The light shaded bands contain several NN force predictions (AV18, CDBonn, Nijm I, II, and 93), the dark shaded bands contain the NN + TM 3NF predictions. The solid and
dashed lines are the AV18 + Urbana IX and CD-Bonn + TM predictions, respectively.
However, discrepancies remain at angles larger than 120◦ . In the right panel of Fig. 1, we
compare the experimental analyzing power A y with different nuclear-force predictions.
The differences (narrow light shaded band) between the 2NF predictions are rather small
at forward angles and become larger at backward angles. These predictions are in good
agreement with the experimental data at forward angles, but deviate dramatically at
backward angles larger than 60◦. By including the TM 3NF (dark shaded band) the
agreement with the data becomes better in the minimum around θ c.m. = 60◦ –100◦ but
the discrepancies at more backward angles remain. The discrepancy between data and
theoretical predictions, which increases with increasing energy [7, 16], may be due to
relativistic effects not accounted for in the present nonrelativistic calculations.
The measured PT data are shown in Fig. 2 together with theoretical predictions.
The PT coefficients in the horizontal plane (K xx , K zx , K xz , and K zz ) are reasonably well
described by calculations with 2NF only (light shaded bands). The inclusion of the TM
3NF (dark shaded bands) rather deteriorates the agreement with the experimental data.
FIGURE 2. Polarization transfer coefficients ( K xx , K zx , K xz , K zz , and K yy ) of elastic pd scattering at E p
= 250 MeV. For the description of bands and lines see legend of Fig. 1
707
The TM (dashed curves) and the Urbana IX (solid curves) 3NF do not have a large effect
on these PT coefficients and give a reasonably good agreement with the data. In the case
of the PT coefficient in the vertical plane (K yy ), the inclusion of the TM 3NF (dark shaded
band) and especially the Urbana IX 3NF (solid curve) give results in better agreement
with the measurements. This is similar to the case of the analyzing power which is also
a polarization observable in the vertical plane. These results clearly indicate that the
spin-dependent parts of 3NF are not well described in present-day models.
pd radiative capture at Ed = 200 MeV
A liquid hydrogen target was used instead of a solid CH 2 foil to cope with the small
cross section of the pd capture reaction and large (d,3 He) cross sections on carbon
and other materials. The target thickness was about 1.5 mm ( 11 mg/cm 2). The liquid
hydrogen was obtained by cooling hydrogen gas with a cryogenic refrigerator. Recoil
3
He particles from the capture reaction were emitted into a come out at forward angle
cone of ±5◦ in the laboratory frame with energies in the range from 105 to 145 MeV.
They were detected by the Large Acceptance Spectrometer (LAS) which has an angular
acceptance of ±60 mr and ±100 mr in the horizontal and vertical plane, respectively,
and a momentum acceptance of ±15 %. The angular distributions of the cross section
and analyzing powers, Ay , Axx and Ayy , were measured from 20◦ to 160◦ in the center
of the mass frame. The absolute value of the cross section was calibrated by a separate
measurement with CH2 foil target around θ cm =90◦.
0.5
0.2
MEC(AV18)
MEC+3NF(URIX)
Present DATA
MEC(AV18)
MEC+3NF(URIX)
Present DATA
0.1
0.05
0.3
Ay
dσ/dΩ [µb/sr]
0.4
0.15
0.2
0
-0.05
-0.1
0.1
-0.15
0
-0.2
0
20
40
60
80
100
120
θCM
140
160
180
0
0.2
20
40
60
80
100 120 140 160 180
θCM
0.2
MEC(AV18)
MEC(BonnB)
Present DATA
0.1
MEC(AV18)
MEC+3NF(URIX)
Present DATA
0.1
0
0
Axx
Ayy
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
-0.4
-0.4
-0.5
-0.5
-0.6
0
20
40
60
80
100
θCM
120
140
160
180
0
20
40
60
80
100
θCM
120
140
160
180
FIGURE 3. Cross section and analyzing powers, Ay , Ayy , and Axx , of the pd radiative capture at Ed =
200 MeV. Curves represent MEC calculations[13] based on the AV18 NN potential with (solid curves) or
without (dashed ones) Urbana IX 3NF.
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The experimental results are shown in Fig. 3 where only statistical errors are show.
The uncertainties in the absolute normalization are estimated to be about 5 %, 2 % and
3 % for the cross section, the tensor and vector analyzing powers, respectively. These
data are compared with Faddeev calculations [13] in which the meson (π and ρ ) exchange current (MEC) is explicitly taken into accounts [17]. In the calculations, AV18
NN interactions up to j = 3 are used with or without Urbana IX 3NF which is adjusted to
reproduce 3N binding energies. The cross section and Ay disagree with predictions without 3NF, and are well reproduced by including 3NF, as seen in Fig. 3. This fact confirms
the existence and necessity of 3NF. Since Ay of the pd elastic scattering is reproduced
by calculations with 3NF at some energies and disagrees with the calculations at other
energies, reproduction of Ay of the pd capture reaction, shown in this paper at Ed = 200
MeV, has to be examined in a wide energy range. Contrary to the good discription of the
cross section and Ay , calculations cannot reproduce tensor analyzing powers, Ayy and
Axx , with or without 3NF. The disagreement of Ayy is moderate and similar to the disagreement of tensor analyzing powers of the elastic scattering in the same energy range
[6]. However, Axx differs completely from the calculations. The experimental values of
Axx and Ayy are nearly the same, while the calculated values are quite different.
The energy dependence of Ayy and Axx at θcm = 90◦ is shown in Figure 4. There are
several measurements of Ayy below Ed = 200 MeV. However, Axx have been measured
only at 17.5 MeV and 200 MeV. Measured Ayy are fairly well reproduced by calculations
in whole the energy range below 200 MeV. Measured Axx agrees with calculations at
17.5 MeV, however, remarkably disagrees at 200 MeV. Measured Axx have nearly the
same values as measured Ayy at 17.5 and 200 MeV. Therefore, Axx and Ayy are expected
to have nearly the same values below 200 MeV. Calculated Axx and Ayy agree to each
other at low energy and disagree above about 50 MeV. The difference increases with the
energy up to 200 MeV. It is expected, therefore, that discrepancies in Axx between the
experiment and calculation begin at about 50 MeV and increase with energy up to 200
MeV.
0.1
Axx(90o), Ayy(90o)
0
Axx MEC
-0.1
Axx MEC+3NF
-0.2
Ayy MEC+3NF
Axx Akiyoshi et al. (2001)
Ayy Akiyoshi et al. (2001)
Ayy Jourdan et al. (1986)
Ayy Anklin et al. (1998)
Ayy Pitts et al. (1988)
-0.3
-0.4
Ayy MEC
Ayy
Present
Axx DATA
-0.5
0
50
100
150
Ed [MeV]
200
250
300
FIGURE 4. Energy dependence of Ayy and Axx at θcm = 90◦ . Experimental data at 17.5 MeV[18],
29.2MeV[19], 45 MeV[20] and 200 MeV are compared with MEC calculations[13] based on AV18 NN
potential with or without Urbana IX 3NF.
709
SUMMARY
Recent results of experimental studies on three-nucleon systems were presented, i.e. the
pd elastic scattering at E p = 250 MeV and the d p →3He+γ capture reaction at Ed = 200
MeV. The experimental results are compared with theoretical predictions based on exact
solutions of the three-nucleon Faddeev equations and modern realistic nucleon-nucleon
potentials combined with three-nucleon forces. For the elastic scattering, the differential cross sections and the vector analyzing powers are reasonably well explained by
calculations including 3NF around the cross section minima, but the discrepancies at
more backward angles remain. PT data are not always better described by calculations
with 3NF. For the capture reaction, calculations with 3NF improve the description of the
cross section and the vector analyzing power Ay . There are large discrepancies between
tensor analyzing power data and theoretical predictions. These results clearly indicate
that the spin-dependent parts of 3NF’s are not well described in present-day models.
More theoretical studies are needed including relativistic treatments and chiral parturbation theories. From the experimental point of view, a rich spectrum of spin observables
will be measured not only for elastic scattering but also for the Nd breakup and capture
processes over wide energy range in order to offer further valuable information.
ACKNOWLEDGMENTS
We thank the RCNP staff for their support during the experiment. We also wish to thank
Professor H. Toki for his encouragements throughout the work. We are grateful to Dr.
G.P.A. Berg for his critical reading of the manuscript.
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