Reaction mechanism and characteristics of T_{20} in d + ^3He backward elastic scattering at intermediate energies

Abstract

For backward elastic scattering of deuterons by ^3He, cross sections \sigma and tensor analyzing power T_{20} are measured at E_d=140-270 MeV. The data are analyzed by the PWIA and by the general formula which includes virtual excitations of other channels, with the assumption of the proton transfer from ^3He to the deuteron. Using ^3He wave functions calculated by the Faddeev equation, the PWIA describes global features of the experimental data, while the virtual excitation effects are important for quantitative fits to the T_{20} data. Theoretical predictions on T_{20}, K_y^y (polarization transfer coefficient) and C_{yy} (spin correlation coefficient) are provided up to GeV energies. Comment: REVTEX+epsfig, 17 pages including 6 eps figs, to be published in Phys. Rev. C

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arXiv:nucl-th/9911044v1 15 Nov 1999
ISSN 1344-3879, RIKEN-AF-NP-334
Reaction mechanism and characteristics of T20in d +3He
backward elastic scattering at intermediate energies
M. Tanifujia, S. Ishikawaa, Y. Iserib, T. Uesakac, N. Sakamotoc, Y. Satouc, K. Itohc,
H. Sakaid,c, A. Tamiid, T. Ohnishid, K. Sekiguchid, K. Yakod, S. Sakodad, H. Okamurae,
K. Sudaeand T. Wakasaf
aDepartment of Physics, Hosei University, Fujimi, Chiyoda, Tokyo 102-8160, Japan
bDepartment of Physics, Chiba-Keizai College, Todoroki, Inage, Chiba 263-0021, Japan
cThe Institute of Physical and Chemical Research (RIKEN), Saitama 351-0198, Japan
dDepartment of Physics, University of Tokyo, Hongo, Bunkyo, Tokyo 113-0033, Japan
eDepartment of Physics, Saitama University, Saitama 338-8570, Japan
fResearch Center for Nuclear Physics, Osaka University, Osaka 567-0047, Japan
Abstract
For backward elastic scattering of deuterons by3He, cross sections σ and
tensor analyzing power T20are measured at Ed= 140 − 270 MeV. The data
are analyzed by the PWIA and by the general formula which includes virtual
excitations of other channels, with the assumption of the proton transfer from
3He to the deuteron. Using3He wave functions calculated by the Faddeev
equation, the PWIA describes global features of the experimental data, while
the virtual excitation effects are important for quantitative fits to the T20
data. Theoretical predictions on T20, Ky
and Cyy(spin correlation coefficient) are provided up to GeV energies.
y(polarization transfer coefficient)
25.10.+s, 24.70.+s, 25.45.Hi
Typeset using REVTEX
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I. INTRODUCTION
For the last few decades, elastic scattering of deuterons by protons at the backward angle
at intermediate energies has intensively been studied as an important source of information
of nuclear interactions and reaction dynamics, by including non-nucleonic degrees of freedom
in the consideration [1,2,3,4,5,6,7,8,9,10]. Since3He has the same spin as the proton’s, the
spin structure of the scattering amplitude of the d+3He (d3He) system is similar to that
of the d+p (dp) one when3He is considered as a single body [11]. Then the backward
elastic scattering of the deuteron by3He attracts our attention, for investigations of probable
differences as well as similarities of the information when compared to the dp scattering.
In the dp backward scattering, the assumption of mechanism by neutron transfer from
the deuteron to the proton (Fig. 1(a)) [2] has been fundamentally successful in explaining
energy dependence of observables. For example, the neutron-transfer mechanism produces
the overall agreement to the experimental cross section σ of the scattering up to GeV-
energy region when the empirical momentum distribution of the deuteron is employed in
the relativistic framework [7] and the simple PWIA calculation by the mechanism describes
qualitative features of the measured tensor analyzing power T20 and polarization transfer
coefficient Ky
scattering have broad resonance-like structures around Ed = 1 GeV in the plot against
the incident energy [5,9], which have been interpreted as the signal of excitations of the
transferred neutron to ∆-states (Fig. 1(b)) [5,6]. The PWIA calculation by the transfer
mechanism has reproduced the observed structure of T20 when effects of other reaction
channels like the ∆ excitation have been phenomenologically included [10]. For the transfer
mechanism, the spin observables are described by the w(k) to u(k) ratio, where u(k) and w(k)
are the deuteron S- and D- wave functions in the momentum space, and thus the information
of the nuclear interaction obtained by the spin observables is generally of different nature
from the one by the cross section [8].
In the case of the3He target, the possible reaction mechanism of the backward scattering
of the deuteron will be proton transfer from3He to the deuteron (Fig. 1(c)) as follows. The
backward cross section of direct scattering, for example by potentials, is small compared to
the forward one by several order of magnitude at intermediate energies. On the other hand,
the cross section of the proton transfer reaction is large for the forward emission of3He, i.e.
the backward for the deuteron, as in usual pick-up reactions. When the transfer mechanism
is adopted, the analogy to the dp scattering suggests that the spin observables of the d3He
scattering in the PWIA are described by the w(k) to u(k) ratio, where u(k) and w(k) are
now the wave functions of the p-d relative motion in3He. The wave functions, which are
supplied by the Faddeev calculation, have ambiguities due to the choice of interactions as
the input [12]. The information of the w(k) to u(k) ratio obtained from the spin observables
will be useful in solving such ambiguities.
Further, the spin structure of the scattering amplitude for the d3He system is similar to
that for the dp one even in the nucleon transfer mechanism because in both scattering the
spin of the projectile is transformed in the same way, from one to one half, and similarly that
of the target from one half to one. Thus, the spin observables can be described in similar
forms in these scattering and therefore some of their characteristics in the dp scattering will
also be observed in the d3He one. These stimulate us to investigate the spin observables
y(d→p) at few hundreds MeV [5,9]. Further, the measured σ and T20of the
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of the d3He scattering at the backward angle by theories as well as by experiments, which
have not been attempted so far. In particular, T20will be investigated in detail because of
its sensitivity to the wave functions. The theoretical prediction is performed up to GeV-
energy region similarly to the dp case although the present measurements are limited to a
few hundreds MeV. As the first step of investigations, we will try to clarify global features
of T20in the scattering theoretically and to provide experimental evidence for the reaction
mechanism. The interests of the investigation will be focused on (i) the validity of the
assumption of the proton-transfer mechanism, (ii) effects due to the difference between the
3He wave function and the deuteron one and (iii) the overview of effects of coupling to other
reaction channels. In the following, experimental details are described in Sec. II, the PWIA
analyses are given in Sec. III, where in addition to the cross section and the analyzing power
the polarization transfer coefficient and the spin correlation one are briefly discussed, and
in Sec. IV the effects of other reaction channels are investigated. Section V is devoted to a
summary, discussion and perspective.
II. EXPERIMENTAL DETAILS AND RESULTS
The d3He experiment was carried out at RIKEN Accelerator Research Facility for σ and
T20. Polarized deuteron beam was provided by the high-intensity polarized ion source [13].
Three polarization modes (unpolarized and two tensor-polarized modes) were cycled every 5
seconds. The beam polarization was measured with a polarimeter based on the d+p elastic
scattering after acceleration to 140, 200 and 270 MeV. It was monitored continuously during
a run and obtained to be typically 60–80% of the ideal value. At Ed=200 and 270 MeV,
detectors were placed at the angle where T20vanishes so that the polarimeter could work as
a beam intensity monitor independent of beam polarization.
A cryogenic3He gas target was bombarded by the polarized deuteron beams. The size
of the target was 13 mm (20 mm) wide, 15 mm (20 mm) high and 20 mm (10 mm) thick
in the case of 270 MeV (140, 200 MeV) measurement. Entrance and exit windows were
6 µm thick Havar foils. The gas pressure (∼ 1 atm) was measured with a Baratron gauge.
Temperature of the target was monitored by using a diode thermo-sensor and found to be
about 11 K throughout the experiment. The density of the3He gas was 6.6 × 1020cm−3.
Scattered3He particles were momentum-analyzed in the magnetic spectrograph SMART
[14] and detected by a multi-wire drift chamber and plastic scintillators placed at the focal
plane. The total beam charge was measured with a Faraday cup placed in the first dipole
magnet and the beam intensity was monitored by the polarimeter described above.
In the off-line analysis, data for θlab≤ 1.4◦were used. The background spectrum from
the (d,3He) reactions on Havar foils was subtracted to obtain the net yield for the d3He
events. Signal-to-noise ratio at the peak region was 4–20% depending on the beam energy.
The tensor analyzing power T20was deduced from the ratios of yields for three polarization
modes.
The measured cross sections and analyzing powers with statistical errors are given in
Table I. The systematic errors of σ and T20are ±10% and ±2%, respectively. The former is
mainly due to the uncertainties of both the beam intensity and the target thickness, whereas
the latter is due to the uncertainty of the beam polarization.
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III. PWIA ANALYSES
The theoretical analyses are performed first by the PWIA. The approximation is rather
crude but is still useful to obtain the qualitative feature of the reaction at the intermediate
energies [2,8]. Analyses by extended formulas which include the effect of virtual excitations
of other reaction channels will be presented in the next section.
The3He wave function in the pd cluster configuration is given by
ΨHe=
?
+
νd
?1
21νpνd|1
?1
2νHe
?
χνpχνd
u(k)
√4π
?
νd,m
21νpνd|3
2ν
? ?3
22νm|1
2νHe
?
χνpχνdY2m(ˆk)w(k), (1)
where k is the proton-deuteron relative momentum in3He, χνpand χνdare the wave functions
of the proton and the deuteron, and ν,s are the z-components of their spins. In the PWIA
for the d3He scattering, the proton transfer is induced by proton-deuteron interactions as
the neutron transfer by neutron-proton interactions in the dp scattering [8]. We denote
the proton-deuteron scattering amplitude at the momentum k by t(k), neglecting the spin
dependence similarly to the dp case. Referring to the calculation of the dp scattering, we get
non-vanishing independent T matrix elements ?ν′
the d3He backward elastic scattering as
3u(k)2+2√2
He,ν′
d|M|νHe,νd? for the proton transfer in
?1
2,1|M|1
2,1? =t(k)
4π
?2
3
u(k)w(k) +1
3w(k)2
?
,(2)
?1
2,0|M| −1
2,1? = ?−1
2,1|M|1
2,0? =t(k)
4π
?√2
3u(k)2−1
3u(k)w(k) −
√2
3w(k)2
?
, (3)
?−1
2,0|M| −1
2,0? =t(k)
4π
?1
3u(k)2−2√2
3
u(k)w(k) +2
3w(k)2
?
,(4)
where k is related to the incident deuteron momentum kd as k =
elements (2)-(4) are equivalent to those in the dp backward scattering except for the factor
2
3, when νHe is replaced by νpand t(k) by the n-p scattering amplitude. Then the cross
section σ and the tensor analyzing power T20are given as
1
3kd [8]. The matrix
σ = (µ
2π)21
3(|t(k)|
4π
)2(u(k)2+ w(k)2)2
(5)
and
T20=2√2r − r2
√2(1 + r2)
withr =w(k)
u(k). (6)
Here µ is the reduced mass of the d3He system. Using Eqs. (2)-(4), one can also calculate
other polarization observables, which are equivalent to those in the dp scattering.For
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example, the polarization transfer coefficient κ0 =
coefficient Cyyare given as
3
2Ky
y(d→3He) and the spin correlation
κ0=
1
1 + r2
?
1 −
1
√2r − r2
?
(7)
and
Cyy=
2
9(1 + r2)2
?
1 −
5
√2r + 3r2+
√2r3− 2r4
?
. (8)
The quantities T20and κ0satisfy the equation of a circle as in dp scattering,
(T20+
1
2√2)2+ κ02=9
8.(9)
At the low-energy limit where r = 0, T20= 0 and κ0= 1. With the increase of the incident
energy, the magnitude of r is increased and the point defined by a set of κ0and T20in the
κ0− T20plane moves along the above circle clockwise for r < 0 and counterclockwise for
r > 0. As will be seen later, the former is the case of the dp scattering and the latter that
of the d3He one. More details are referred to Refs. [8,9,10].
For the Faddeev calculation of3He, we will specify the nucleon-nucleon (NN) force to the
AV14 potential [15] and the three-nucleon (3N) force to the 2π-exchange Brazil model [16].
The Coulomb interaction is discarded and the3He nucleus is approximated by the triton.
These potentials give the binding energy of the three-nucleon system as 8.34 (7.68) MeV
with (without) the 3N force [12]. In Fig. 2(a), the calculated u(k) and w(k) are shown as the
functions of k and are compared to those for the deuteron by the same NN force. The wave
functions of3He have similar k-dependence to those of the deuteron in a global view but
with the opposite sign of w(k) for most k. Due to the different sign of w(k), the calculated
r of3He has the opposite sign to that of the deuteron except at large k as is shown in Fig.
2(b). This characteristic of r is consistent with the result of Ref. [17] that the asymptotic
D-state to S-state ratio in3He has the opposite sign to that in the deuteron. The quantity
r is infinite at the zero point of u(k), k = k0u, which is located at almost the same k for3He
and the deuteron, k0u= 0.40 GeV/c. The zero point of w(k) for3He, k = k0w, is located at
about k0w=0.79 GeV/c (0.96 GeV/c) with (without) the 3N force.
The analysis of the measured cross section will not fully been performed because the
cross section depends on the proton-deuteron scattering amplitude t(k) as is seen in Eq. (5),
and reliable information of t(k) is not available at the present. In Fig. 3, however, a crude
estimation which assumes the k-dependence of t(k) to be negligible for the relevant energies
is presented. The calculation describes the global feature of the energy dependence of the
measured cross section. This will encourage further investigations of the scattering by the
assumption of the transfer mechanism.
The characteristics of r in Fig. 2 are reflected to T20of the scattering as shown in Fig.
4(a), where comparisons will be made in two ways; one is the comparison of T20between
the d3He scattering and the dp one and the other is that of T20between the calculated and
the measured for the same scattering. Due to the opposite sign of r, in a small-k region,
the calculated T20 for the d3He scattering has the opposite sign to that for the dp one.
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In the figure, the measured T20for the d3He scattering by the present experiment is shown
together with that for the dp scattering in Ref. [9] and their signs are opposite to each other.
Since the dp and d3He scattering in the transfer model are essentially (d,p) stripping and
(d,3He) pickup reactions, although the incident energies are higher than the conventional
stripping and pickup reactions, the above feature is compared to that of measured T20for
(d,p) and (d,t) reactions at low energies, where T20of the former reactions has the opposite
sign to that of the latters in a wide angular range [17,18]. This nature of the sign of the
measured T20indicates that the proton-pickup mechanism is a reasonable model for the d3He
backward scattering. In both of the d3He and dp scattering, the calculated T20at the small
k reproduces the qualitative feature of the k dependence of the measured one, although
the magnitudes of the calculated are larger than those of the measured. This qualitative
agreement of the calculation to the experiment supports the assumption of the proton-
transfer mechanism for the d3He scattering. In Fig. 4(b), as an example of calculations by
other potentials, T20by the recently proposed NijmII potential [19] is compared to that by
the AV14 one. The calculated T20are very similar to each other up to k ≈ 0.6 GeV/c and
the difference due to the potential employed is seen at larger k. In the figure, the effect of
the 3N force, which arises through high-momentum components of the3He wave function,
is shown for the AV14 potential. In large k region, the contribution of the 3N force to T20
becomes large and the second zero point of T20 is shifted from k=0.96 GeV/c to k=0.79
GeV/c due to the 3N force.
In Figs. 5(a) and 5(b), the calculated κ0and Cyyare shown for the AV14 and NijmII
potentials. The dependence of κ0and Cyyon the choice of the potential is weak, up to about
k ≈ 0.6 GeV/c. The 3N force effect is shown in the figure but is less remarkable compared
to that in T20.
IV. VIRTUAL EXCITATION OF OTHER REACTION CHANNELS
Now we will extend the theoretical framework to a more general one so as to include
effects of other reaction channels [10]. Let us first transform the original T-matrix elements
?ν′
9
2√2?1
He,ν′
d|M|νHe,νd? into U, T and T′as
2,1|M|1
U =
2,1? + 3?1
2,0|M| −1
2,1? +
3
√2?−1
2,0|M| −1
2,0?, (10)
T = 2?1
2,1|M|1
2,1? −
√2?1
2,0|M| −1
2,1? − 2?−1
2,0|M| −1
2,0?, (11)
T′=1
4?1
2,1|M|1
2,1? −
1
√2?1
2,0|M| −1
2,1? +1
2?−1
2,0|M| −1
2,0?. (12)
Here, U, T and T′are free from the PWIA in principle. It is shown by the invariant-
amplitude method [10] that U is the scalar amplitude in the spin space and describes the
scattering by central interactions and T and T′, the second-rank tensor ones, describe the
scattering by tensor interactions. In the PWIA limit, as will be seen later, contribution of the
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S-state of3He (u(k)2) is included in U, that of the interference between the S-state and the D-
state (u(k)w(k)) in T and that of the D-state (w(k)2) is divided into two amplitudes, U and
T′, according to their tensorial character. Secondly, we introduce the relative magnitudes
and phases between U, T and T′as
|T|
|U|≡ R,
|T′|
|U|≡ R′,θT− θU≡ Θ,andθT′ − θU≡ Θ′.(13)
Then we get the general form of T20 in terms of R,R′,Θ and Θ′as
T20= {2√2RcosΘ − R2− 32R′2+ 12RR′cos(Θ′− Θ)}/NR,(14)
NR=
√2 + 2√2R2+ 34√2R′2− 4R′cosΘ′,(15)
which is exact except for the proton-transfer assumption. Since these formulas are the same
as those in the dp case [10], we will follow their analyses.
We will calculate R and R′by the PWIA and treat Θ and Θ′as the adjustable parameters.
This treatment of Θ and Θ′takes into account effects of virtual excitations of other reaction
channels phenomenologically since the virtual excitations of the channel induces imaginary
parts in the transition amplitudes to vary Θ and/or Θ′. These effects can be neglected
in R and R′since the neglect has not induced significant errors in the calculation of T20
of the dp scattering except at very large k [10]. The range of Θ and Θ′will generally be
−π ≤ Θ(Θ′) ≤ π. In the following, Θ and Θ′are treated to be k-independent for simplicity,
although Θ and Θ′vary with k in principle. The amplitudes U, T and T′in the PWIA limit
are calculated as
U = 3√2(u(k)2+1
4w(k)2)t(k), (16)
T = 3√2u(k)w(k)t(k)(17)
and
T′=3
4w(k)2t(k). (18)
Then R and R′are obtained as
R =
4|r|
4 + r2
andR′=
r2
√2(4 + r2). (19)
In addition, when we choose the PWIA limit for Θ and Θ′, i.e. Θ = 0 for k < k0uand
k > k0wand π for k0u< k < k0w, and Θ′= 0, Eqs. (14), (15) and (19) give the previous
result, Eq. (6). Since Θ changes at k = k0uand k = k0w, in the following we will identify Θ
by the value for k < k0u, for simplicity.
By the numerical calculation, we will study the effects on T20by varying Θ and/or Θ′,
specifying the potential to the AV14 one. First we will examine the calculation without the
3N force. In Fig. 6(a), effects of variations of Θ are described for typical Θ by fixing Θ′to
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the PWIA limit, where the result is shown for positive Θ since T20in this case is independent
of the sign of Θ as seen in Eqs. (14) and (15). The k dependence of T20varies remarkably
with Θ. At k = k0u(r = ∞) and k = k0w(r = 0), T20are independent of Θ and are −1
and zero, respectively. The calculation by Θ = 60◦describes the present data very well
and indicates the importance of the virtual excitation of other channels for the quantitative
description of the data. In Fig. 6(b), we show the Θ′dependence of T20with typical Θ′by
fixing Θ = 0, where the result is shown for positive Θ′because of the independence on the
sign of Θ′(see Eqs. (14) and (15)). None of the calculations with variations of Θ′reproduces
our data of T20 in the small-k region. However, the calculations for Θ′= 120◦and 180◦
produce resonance-like structures around k = k0u, which are similar to the structure found
in the dp scattering observed at k = 0.3 − 0.45 GeV/c in Fig. 4(a). There the calculation
by Eqs. (14) and (15) with Θ = 180◦(PWIA limit) and Θ′= 120◦for the dp scattering is
shown to exhibit that the virtual-excitation effect reproduces the structure.
As examples of combined effects of Θ and Θ′, we vary Θ′fixing Θ = 60◦in Fig. 6(c).
The calculated T20in the small-k region is little affected by the variation of Θ′, reproducing
our data. On the other hand, the calculations by Θ′= −60◦, and −120◦produce structures
similar to those in Fig. 6(b). Therefore Θ and Θ′play different roles in the calculation of
T20; i.e. T20 in the small-k region is mainly governed by the magnitude of Θ, while the
resonance-like structure in the medium k is produced by proper choices of Θ′. In Fig. 6(c),
the calculated T20at k = k0uis concentrated in a narrow range of the magnitude of T20. This
is due to the weak dependence of T20on Θ′at k = k0uas seen in Eqs. (14) and (15), and
thus the measurement at k = k0uwill provide less ambiguous examinations of the reaction
mechanism. These features of T20are similar to those in the dp scattering [10].
In Figs. 6(a) and 6(b), we show the effect of the 3N force, as examples, for Θ = 120◦
in Fig. 6(a), and Θ′= 120◦in Fig. 6(b). The qualitative nature of the effect is similar to
that in the PWIA limit in Fig. 4(b); T20for large k is remarkably affected by the 3N force,
reflecting that the 3N force dominantly affects the p-d wave function at small distance. The
second zero point of T20, which is independent of Θ and Θ′, is moved to k = 0.79 GeV/c
from 0.96 GeV/c by the force.
√2
V. SUMMARY, DISCUSSION AND PERSPECTIVE
In the present paper, we have investigated the cross section and the spin observables in
the d3He backward elastic scattering by the theory and the experiment. The theory which
assumes the proton-transfer mechanism has predicted T20, κ0and Cyyin a wide range of the
intermediate energies. In the low-energy region of the range, σ and T20have been measured.
The global feature of the energy dependence of σ is reproduced by the PWIA calculation
with the energy-independent p-d scattering amplitude and the measured T20 is described
qualitatively by the PWIA prediction. The3He wave function is obtained by solving the
Faddeev equation. At small k, the sign of w(k)/u(k) of3He is different from that of the
deuteron and this explains the sign of the measured T20in the d3He scattering to be opposite
to that in the dp scattering. These give strong support to the assumption that the dominant
reaction mechanism is the proton transfer from3He to the deuteron. The quantitative differ-
ence between the measured T20and the PWIA one has been explained by including the effect
of the coupling to other channels. More details will be discussed later. The formula of T20in
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the d3He scattering looks similar to the one in the dp scattering. However, the difference of
the internal wave functions between3He and the deuteron produces the different features of
T20for most k. The calculated T20depends weakly on the NN interaction employed, up to
k ≈ 0.6 GeV/c. The 3N force contributes to T20through the high-momentum components
of the p-d wave functions in3He. The 3N forces, which have a long history since the pioneer
works in Ref. [20], have recently been investigated in detail by analyzing cross sections, an-
alyzing powers and some spin-correlation effects in the n-d scattering [21,22] and the cross
sections in the n-t one [23]. The measurement of T20at large k in the d3He backward elastic
scattering will provide additional information of the 3N force.
The effect of the virtual excitation of other reaction channels is considered through the
imaginary part of the scattering amplitudes, which are parameterized by Θ and Θ′. The
calculations with Θ = 60◦are successful in reproducing the small-k data. This effect will
be interpreted as the virtual-breakup effects, mainly of the deuteron, by the analogy to the
dp case. In the case of dp scattering the calculations by Θ = 120◦∼ 135◦which differ from
the PWIA limit (Θ = 180◦) by 60◦∼ 45◦have described the measured T20at the small k
[10] and this Θ effect has been interpreted as the deuteron-breakup contribution, because
below the pion threshold the breakup is the only one open channel strongly coupled to the
dp one. In the present scattering, the large magnitude of Θ′produces the resonance-like
structure, which is similar to that observed in the dp scattering. If the virtual excitation of
the transferred nucleon to the ∆-state (Fig. 1(b)) is responsible for producing the structure
in the dp scattering, similar effects will also be expected in the d3He scattering as shown
in Fig. 1(d). Thus it will be interesting to examine by experiments if such structures are
observed. It should be noted that the effects of Θ and Θ′on T20have been studied by using
the AV14 NN potential and the Brazil 3N force. However, the essential features of the effects
will be unchanged by other choices of the nuclear interactions except in the large-k region.
In the present investigation, the k dependence of Θ and Θ′is not considered. Since they
depend on k in principle, a particular set of Θ and Θ′might be valid in a limited range of
k. At present, Θ = 60◦with any Θ′reproduces the data between k = 0.1 and 0.2 GeV/c,
indicating that such sets of Θ and Θ′are valid in this range of k. In the dp scattering, as
shown in Fig. 4(a), a set of Θ and Θ′reproduces the global feature of the data in a wide
range of k. This suggests a set of Θ and Θ′to be applicable in a wide range of k in the d3He
scattering as well.
Finally, although numerical results are not displayed, the effects of virtual excitation
of other channels on κ0and Cyy are small at small k. However, their contributions have
appreciable magnitudes at k ∼ 0.2 GeV/c. Therefore, the measurements of these quantities
in this k region will give information of Θ and Θ′, such as the validity of the phenomenological
value, Θ ≈ 60◦. Because the predicted T20has interesting features, experiments at higher
energies are desirable, which will provide valuable information of the nuclear interaction
and the reaction mechanism. In particular, the measurement of T20at k ∼ 0.4 GeV/c will
be one candidate to obtain the convincing evidence of the reaction mechanism because T20
there is almost independent of Θ and Θ′. Further refinements of the theory, which include
relativistic effects, will be made when higher-energy data become available.
9

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REFERENCES
[1] G. Barry, Ann. Phys. 73, 482 (1972).
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22 (1997).
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[13] H. Okamura et al., AIP Conf. 293, 84 (1993).
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[18] L. D. Knutson, Proc. of the Fourth Symposium on Polarization Phenomena in Nuclear
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10

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TABLES
TABLE I. Measured cross sections and tensor analyzing powers in d3He backward elastic scat-
tering with statistical errors.
ELab(MeV)
σ (mb/sr)
T20
140200270
16.3 ± 0.1
0.07 ± 0.02
8.56 ± 0.03
0.15 ± 0.01
0.97 ± 0.08
0.17 ± 0.03
11

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FIGURES
n
p
∆ ∆
∆ ∆
(d)
(c)
(b)
(a)
3He
3He
3He
3He
pp
p
p
d
d
d
d
d
d
d
d
FIG. 1. Neutron transfer in dp scattering and proton transfer in d3He scattering. (b) and (d)
show the excitations of the transferred nucleons to ∆-states by the exchange of mesons, for example
pions, and include varieties of the combination of the charges of the related nucleons and those
of the exchanged mesons. In (d), the four combinations of the spectator nucleons are described
representatively by one combination.
12

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0.00.2 0.40.6 0.81.0
0
5
(b)
r = w(k) / u(k)
k (GeV/c)
-0.2
-0.1
0.0
0.1
w(k)
u(k)
(a)
u(k), w(k)
FIG. 2. (a) Wave functions of3He and deuterons normalized properly. u(k) and w(k) are the
S and D components, respectively. (b) The D to S wave-function ratio r for3He and deuterons.
The solid lines are for3He and the thin solid lines for the deuteron. The dash-dotted lines are for
3He including the contribution of the Brazil 3N force.
13

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0.10.20.3
1
10
σ σ(k) (mb/sr)
k (GeV/c)
FIG. 3. Cross sections of d3He backward elastic scattering. The closed circles are the present
experimental data. The solid line is calculated by σ ∼ (u(k)2+w(k)2)2, which is suitably normalized
to the data.
14

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-1.5
-1.0
-0.5
0.0
0.5
1.0
(a)


T20
0.00.20.40.6 0.8 1.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
(b)
T20
k (GeV/c)
FIG. 4. T20as a function of k. In (a), the solid and dashed lines describe the PWIA calculations
by the AV14 potential for the d3He scattering and that for the dp scattering, respectively. The
closed circles are the present experimental data for the d3He scattering and the open circles are
the data for the dp scattering in Ref. [9]. The thin solid line is for dp scattering, which includes
effects of virtual excitations of other reaction channels. (b) describes the dependence of T20on the
nuclear interactions employed for the d3He scattering. The solid and dotted line are for the AV14
and NijmII potentials, respectively and the dashed line includes the Brazil 3N force in the case of
the AV14 potential.
15