††thanks: Present Address: Denso Corporation, 1-1, Showa-cho, Kariya, Aichi
448-8661, Japan††thanks: Present Address: Nagoya University, Furocho, Chikusa,
Nagoya 464-8602, Japan††thanks: Present Address: New Japan Radio Corporation,
3-10, Nihonbashi Yokoyama-cho,Chuo-ku, Tokyo 103-8456, Japan

# Angular Distribution of $\gamma$-rays from Neutron-Induced Compound States
of 140La

T. Okudaira  Nagoya University, Furocho, Chikusa, Nagoya 464-8602, Japan  S.
Takada  Kyushu University, 744 Motooka, Nishi, Fukuoka 819-0395, Japan  K.
Hirota  Nagoya University, Furocho, Chikusa, Nagoya 464-8602, Japan  A. Kimura
Japan Atomic Energy Agency, 2-1 Shirane, Tokai 319-1195, Japan  M. Kitaguchi
Nagoya University, Furocho, Chikusa, Nagoya 464-8602, Japan  J. Koga  Kyushu
University, 744 Motooka, Nishi, Fukuoka 819-0395, Japan  K. Nagamoto  Nagoya
University, Furocho, Chikusa, Nagoya 464-8602, Japan  T. Nakao  Japan Atomic
Energy Agency, 2-1 Shirane, Tokai 319-1195, Japan  A. Okada  Nagoya
University, Furocho, Chikusa, Nagoya 464-8602, Japan  K. Sakai  Japan Atomic
Energy Agency, 2-1 Shirane, Tokai 319-1195, Japan  H. M. Shimizu  Nagoya
University, Furocho, Chikusa, Nagoya 464-8602, Japan  T. Yamamoto  Nagoya
University, Furocho, Chikusa, Nagoya 464-8602, Japan  T. Yoshioka  Kyushu
University, 744 Motooka, Nishi, Fukuoka 819-0395, Japan

###### Abstract

Angular distribution of individual $\gamma$-rays, emitted from a neutron-
induced compound nuclear state via radiative capture reaction of
139La(n,$\gamma$) has been studied as a function of incident neutron energy in
the epithermal region by using germanium detectors.

An asymmetry $A_{\mathrm{LH}}$ was defined as
$(N_{\mathrm{L}}-N_{\mathrm{H}})/(N_{\mathrm{L}}+N_{\mathrm{H}})$, where
$N_{\mathrm{L}}$ and $N_{\mathrm{H}}$ are integrals of low and high energy
region of a neutron resonance respectively, and we found that
$A_{\mathrm{LH}}$ has the angular dependence of $(A\cos\theta_{\gamma}+B)$,
where $\theta_{\gamma}$ is emitted angle of $\gamma$-rays, with $A=-0.3881\pm
0.0236$ and $B=-0.0747\pm 0.0105$ in 0.74 eV p-wave resonance.

This angular distribution was analyzed within the framework of interference
between s- and p-wave amplitudes in the entrance channel to the compound
nuclear state, and it is interpreted as the value of the partial p-wave
neutron width corresponding to the total angular momentum of the incident
neutron combined with the weak matrix element, in the context of the mechanism
of enhanced parity-violating effects. Additionally we used the result to
quantify the possible enhancement of the breaking of the time-reversal
invariance in the vicinity of the p-wave resonance.

compound nuclei, partial wave interference, neutron radiative capture
reaction, discrete symmetry

###### pacs:

13.75.Cs, 21.10.Hw, 21.10.Jx, 21.10.Re, 23.20.En, 24.30.Gd, 24.80.+y,
25.40.Fq, 25.70.Gh, 27.60.+j, 29.30.Kv

##  I Introduction

The magnitude of parity violating effects in effective nucleon–nucleon
interactions is $10^{-7}$, as observed in the helicity dependence of the total
cross section between nucleons Potter  _et al._ (1974); Yuan  _et al._ (1986);
Adelberger and Haxton (1985). Extremely large parity violation(P-violation)
was found in the helicity dependence of the neutron absorption cross section
in the vicinity of p-wave resonance of 139La+n Alfimenkov  _et al._ (1983).
The helicity dependence was measured as the ratio of the helicity-dependent
cross section to the p-wave resonance cross section, referred to as
longitudinal asymmetry, which amounts to (9.56 $\pm$ 0.35)%. The large
P-violation was explained as the interference between the amplitudes of the
p-wave resonance and the neighboring s-wave resonance Sushkov and Flambaum
(1982a, b). Longitudinal asymmetry was intensively studied in neutron
transmission and in (n,$\gamma)$ measurements Bowman  _et al._ (1989); Yuan
_et al._ (1991); Masuda  _et al._ (1989); Shimizu  _et al._ (1993). The
$\gamma$-ray energy dependence of the asymmetry was not found, which implies
that the interference occurs in the entrance channel to the compound state and
not in the exit channel Shimizu (1992). Under this assumption, the
longitudinal asymmetry $A_{\rm L}$ is given by

$A_{\rm L}\simeq-\frac{2xW}{E_{\rm p}-E_{\rm s}}\sqrt{\frac{\Gamma_{\rm
s}^{\rm n}}{\Gamma_{\rm p}^{\rm n}}}$ (1)

assuming s-wave and p-wave resonances, where $E_{\rm s}$ and $E_{\rm p}$ are
their respective energies, $\Gamma_{\rm s}^{\rm n}$ and $\Gamma_{\rm p}^{\rm
n}$ are the corresponding neutron widths and $W$ is the weak matrix element.
The value of $x$ is defined as $x^{2}=\Gamma_{{\mathrm{p}},j=\frac{1}{2}}^{\rm
n}/\Gamma^{\rm n}_{\rm p}$, where $\Gamma_{{\rm p},j=\frac{1}{2}}^{\rm n}$ is
the partial neutron width for the total angular momentum of the incident
neutron $j=1/2$. The detail definition of $x$ is described in Appendix E. A
theoretical model explaining the large enhancement of P-violation in compound
states was studied and summarized in Ref. Mitchell  _et al._ (2001). The
enhancement mechanism is expected to be applicable to P- and T-violating
interactions and to enable highly sensitive explorations of CP-violating
interactions beyond the Standard Model of elementary particles. The
sensitivity of T-violation can be quantified in relation to the magnitude of
P-violation as a function of $x$ Gudkov (1992); Gudkov and Shimizu (2017).

However, the values of $x$ and $W$ have not yet been measured individually.
The value of $x$ can be extracted from the energy dependence of the angular
distribution of $\gamma$-rays from a p-wave resonance in a neutron induced
compound nucleus, which has not yet been measured. Theoretically, it can be
deduced assuming interference between partial waves in the entrance channel
Flambaum and Sushkov (1985).

In this paper we report measurement results of the angular distribution of
individual $\gamma$-rays emitted from 0.74 eV p-wave resonance of 139La+n as a
function of incident neutron energy.

##  II Experiment

###  II.1 Experimental Setup

The angular distribution of individual $\gamma$-rays through the radiative
capture reactions induced by epithermal neutrons was measured by introducing a
pulsed neutron beam into the Accurate Neutron–Nucleus Reaction Measurement
Instrument (ANNRI) installed at the beamline BL04 of the Material and Life
science experimental Facility (MLF) of the Japan Proton Accelerator Research
Complex (J-PARC), as shown in Fig. 1 Igashira  _et al._ (2009). The primary
proton beam pulses were injected to the neutron production target in a single-
bunch mode with a repetition rate of 25 Hz and an average beam power of 150 kW
during the measurement. The disk chopper was operated synchronously with the
proton injection for the suppression of low energy neutrons, to avoid frame
overlap. The beam collimation was adjusted to define the neutron beam in a
$22$ mm diameter circle on the target, placed at $21.5$ m from the moderator
surface Kino  _et al._ (2011). A lead plate (thickness: $37.5$ mm) was placed
in the upstream optics to suppress the $\gamma$-ray background.

Figure 1: Schematic of the ANNRI installed at the beamline 04 of MLF at
J-PARC. (A) Collimator, (B) T0-chopper, (C) Neutron filter, (D) Disk chopper,
(E) Collimator, (F) Germanium detector assembly, (G) Collimator, (H) Boron
resin, and (I) Beam stopper (Iron).

The $z$-axis is defined in the beam direction, the $y$-axis is the vertical
upward axis, and $x$-axis is perpendicular to them, thus $xyz$ forms a right-
handed frame.

An assembled set of high-purity germanium spectrometers were used to detect
$\gamma$-rays emitted from the target Kimura  _et al._ (2012). The
configuration of the germanium spectrometer assembly is shown in Fig. 2.

Figure 2: Configuration of the germanium spectrometer assembly.

The assembly consisted of two types of detector units: Type-A (Fig. 3) and
Type-B (Fig. 4).

Two combined seven Type-A detectors were placed above and below the target.
The shape of the Type-A detector was hexagonal to enable clustering as shown
in Fig. 3. The polar angles between the center of the target and the center of
the crystal surface facing the target were $\theta=71^{\circ}$, $90^{\circ}$,
and $109^{\circ}$.

Figure 3:  Schematics of the upper seven Type-A detectors. (A) Electrode, (B)
Germanium crystal, (C) Aluminum case, (D) BGO crystal, (E) $\gamma$-ray shield
(Pb collimator), (F) Neutron shield-1 (LiH 22.3 mm thick), (G) Neutron
shield-2 (LiF 5 mm thick), (H) Neutron shield-3 (LiH 17.3 mm thick), and (I)
Photomultiplier tube for BGO crystal.

Eight Type-B detectors were placed on the $xz$-plane at $\theta=36^{\circ}$,
$72^{\circ}$, $108^{\circ}$, and $144^{\circ}$, as shown in Fig. 4.

Figure 4: Schematics of Type-B detectors. (A) Pb collimator, (B) Carbon board,
and (C) LiH powder.

The central crystals of the upper and lower Type-A detectors are denoted as d1
and d8, respectively, and the other surrounding six detectors are denoted
d2-d7 (d9-d14). The names of each Type-B detectors are shown in Fig. 4. In our
measurement, d16 and d17 were not used.

All germanium crystals were operated at a temperature of $77$ K. The typical
energy resolution for the $1.332$ MeV $\gamma$-rays was $4.2$ keV.

The output signal from each crystal was processed independently. The block
diagram of the signal processing is shown in Fig. 5. The output signals from
the preamplifier were fed into the signal processing module CAEN V1724 CAEN ,
which stored the combination of the pulse height digitized using the peak-
sensitive ADC and the timing of the zero-cross point measured from the timing
pulse of the injection of the primary proton beam bunch $t^{\rm m}$. The CAEN
V1724 module transferred the stored data to the computer when 1024 event data
are accumulated in the local buffer. Two pulses temporarily closer than
$0.4~{}\mu$s were processed as a single event, while their pulse heights were
recorded as a zero when their time difference was in the range of $0.4~{}\mu$s
to $3.2~{}\mu$s.

Figure 5:  Block diagram of the signal processing. A signal from the germanium
detector is divided into two branches: one for the timing and triggering and
the other for the pulse height. In the branch of timing and triggering, the
signal is converted to a bipolar signal. Signals over a threshold are
triggered and the time of zero crossing determines the timing of the signal.
In the branch of the pulse height, the signal is converted to a trapezoidal
signal and the height of the trapezoidal from a baseline determines the pulse
height of the signal.

Their response functions were simulated using GEANT4.9.6. Definitions of the
symbols to describe the detector characteristics and results of the simulation
are discussed in detail in Appendix A.

The relation between the pulse height of photo peaks and the deposit
$\gamma$-ray energy was determined by observing $\gamma$-rays emitted in
neutron capture reactions by aluminum. The effective photo-peak efficiency
including the solid angle coverage of each detector unit was determined
relatively based on the assumption that prompt $\gamma$-rays from
14N($\rm{n},\gamma)$ of a melamine target were emitted isotropically. The
relative photo-peak efficiencies are shown in Table 3.

###  II.2 Measurement

The target was a natural-abundance lanthanum plate at room temperature, with
the dimensions of $40\,{\rm mm}\times 40\,{\rm mm}\times 1\,{\rm mm}$, and
with a purity of $99.9$%. The total number of $\gamma$-ray events detected in
the experiment are denoted $I_{\gamma}$. Here, the corresponding neutron
energy $E^{\rm m}_{\rm n}$ is defined as

$\displaystyle E^{\rm m}_{\rm n}=\frac{m_{\rm{n}}}{2}\left(\frac{L}{t^{\rm
m}}\right)^{2},$ (2)

where $m_{\rm{n}}$ is the neutron mass and $L$ is the distance between the
target and the moderator surface. The deposited $\gamma$-ray energy $E^{\rm
m}_{\gamma}$ obtained from the calibration of the pulse height is defined as
well. The obtained results are shown as a 2-dimensional histogram
corresponding to $\partial^{2}I_{\gamma}/\partial t^{\rm m}\partial E^{\rm
m}_{\gamma}$ in Fig. 6.

Figure 6:  $\partial^{2}I_{\gamma}/\partial t^{\rm m}\partial E^{\rm
m}_{\gamma}$ 2-dimensional histogram of $\gamma$-rays with the lanthanum
target as a function of timing $t^{\rm m}$ and deposit $\gamma$-ray energy
$E^{\rm m}_{\gamma}$. The corresponding neutron energy $E^{\rm m}_{\rm n}$ is
also shown.

The histograms projected on $t^{\rm m}$ and $E^{\rm m}_{\gamma}$ are shown in
Fig. 7 and Fig. 8, respectively. In Fig. 7, $\gamma$-ray events with $E^{\rm
m}_{\gamma}$ are integrated and relatively corrected by the incident beam
spectrum for $t^{\rm m}$. The incident beam spectrum was obtained by measuring
the 477.6 keV $\gamma$-rays in 10B(n,$\alpha\gamma)^{7}$Li reactions, with a
boron target placed at the detector center.

Figure 7:  $\gamma$-ray counts relatively corrected by the incident beam
intensity as a function of $t^{\rm m}$ for $E^{\rm m}_{\gamma}\geq 2\,{\rm
MeV}$, which is referred as $\partial I_{\gamma}/\partial t^{\rm m}$.  Figure
8:  Pulse height spectrum of $\gamma$-rays $\partial I_{\gamma}/\partial
E^{\rm m}_{\gamma}$ from the (n,$\gamma$) reaction with lanthanum target as a
function of $E^{\rm m}_{\gamma}$.

The small peak at $t^{\rm m}\sim 1800~{}\mu\rm{s}$ is a p-wave resonance, and
the $1/v$ component is the tail of an s-wave resonance in the negative energy
region as listed in Table 1.

The neutron energy in the center-of-mass system $E_{\rm n}$ is given as

$\displaystyle E_{\rm n}=\frac{m_{\rm n}m_{\rm A}}{m_{\rm n}+m_{\rm
A}}\left(\frac{\bm{p}_{\rm n}}{m_{\rm n}}-\frac{\bm{p}_{\rm A}}{m_{\rm
A}}\right)^{2},$ (3)

where $\bm{p}_{\rm n}$ is the momentum of the neutron in the laboratory
system, $\bm{p}_{\rm A}$ is nuclear momentum of the target, and $m_{\rm A}$ is
the mass of the nucleus of the target.

The beam divergence is sufficiently small and the following can be assumed:

$\displaystyle\bm{p}_{\rm n}=\sqrt{2m_{\rm n}E_{\rm n}}\,\bm{e}_{z},$ (4)

where $\bm{e}_{z}$ is the unit vector parallel to the beam axis. The resonance
energy $E_{r}$ and the total width $\Gamma_{r}$ of the $r$-th resonance, which
are obtained by fitting $\partial I_{\gamma}/\partial t^{\rm m}$ with Eq. 16
are shown in Table 1, together with the published values. The formalism of the
neutron absorption cross section is described in Appendix B. The pulse shape
of the neutron beam and the Doppler effect of the target nucleus are
considered, as shown in Appendix C and Appendix D, respectively. As the
neutron width of the p-wave resonance is negligibly smaller than $\gamma$-ray
width of the p-wave resonance, the total width of p-wave resonance was used as
the $\gamma$-ray width of p-wave resonance. The fitted result is shown in Fig.
9.

$r$ | published values | this work
---|---|---
$E_{r}\,[{\rm eV}]$ | $J_{r}$ | $l_{r}$ | $\Gamma^{\gamma}_{r}\,[{\rm meV}]$ | $g_{r}\Gamma^{n}_{r}\,[{\rm meV}]$ | $g_{r}\Gamma_{r}^{{\rm n}l_{r}}\,[{\rm meV}]$ | $E_{r}\,[{\rm eV}]$ | $\Gamma^{\gamma}_{r}\,[{\rm meV}]$
$1$ |  $-48.63$(a) |  $4$(a) | $0$ | $62.2^{\rm(a)}$ |  | $82^{\rm(a)}$ |  |
$2$ |  $0.758\pm 0.001$(b) |  | $1$ |  $40.11\pm 1.94$(c) |  $(5.6\pm 0.5)\times 10^{-5}$(c) |  | $0.740\pm 0.002$ | $40.41\pm 0.76$
$3$ |  $72.30\pm 0.05$(b) |  | $0$ |  $75.64\pm 2.21$(c) |  $11.76\pm 0.53$(c) |  |  |
Table 1:  Resonance parameters of 139La. (a) taken from Ref. Mughabghab (2006)
and Ref. Shibata  _et al._ (2011). (b) taken from Ref. Terlizzi and others (n
TOF Collaboration). (c) calculated from Refs. Hacken  _et al._ (1976) and
Terlizzi and others (n TOF Collaboration). $\Gamma_{r}^{{\rm n}l_{r}}$ is a
reduced neutron width.  Figure 9:  Fitted result of the p-wave resonance. The
curve shows the best fit.

The level scheme related to 139La(n,$\gamma$)140La reaction is schematically
shown in Fig. 10 Nica (2007).

Figure 10:  Transitions from 139La+n to 140La. Dashed line shows separation
energy of 139La+n.

The $\gamma$-ray transitions to the ground state and low excited states of
140La were observed as shown in the expanded $\partial I_{\gamma}/\partial
E^{\rm m}_{\gamma}$ (Fig. 11). The highest peak at $E^{\rm
m}_{\gamma}$$=$$5161\,{\rm keV}$ corresponds to the $\gamma$-ray direct
transition to the ground state of 140La (spin of the final state:$F$=3), the
middle peak corresponds to the overlap of two transitions at $E^{\rm
m}_{\gamma}$$=$$5131\,{\rm keV}$ and $5126\,{\rm keV}$ to the first and second
excited states at excited energy $30\,{\rm keV}~{}(F\rm{=5}),35\,{\rm
keV}~{}($$F$=2), and the lower peak at $E^{\rm m}_{\gamma}$$=$$5098\,{\rm
keV}$ corresponds to the excited state at $63\,{\rm keV}~{}(F$=4).

Figure 11:  Expanded $\partial I_{\gamma}/\partial E_{\gamma}^{\rm m}$. The
dotted line shows the background determined by the simulation.

Figure 12 shows the magnified 2-dimensional histogram of
$\partial^{2}I_{\gamma}/\partial t^{\rm m}\partial E^{\rm m}_{\gamma}$ in the
vicinity of the p-wave resonance and the $\gamma$-ray transition to the ground
state of 140La. The p-wave resonance was selectively observed only for two
ridges corresponding to the transition at $E^{\rm m}_{\gamma}$$=$$5161\,{\rm
keV}$ and the sum of transitions at $E^{\rm m}_{\gamma}$$=$$5131\,{\rm
keV},5126\,{\rm keV}$, but not for the ridge at $E^{\rm
m}_{\gamma}$$=$$5098\,{\rm keV}$. According to the dependence of
$\partial^{2}I_{\gamma}/\partial t^{\rm m}\partial E^{\rm m}_{\gamma}$ on
$t^{\rm m}$, and therefore on the incident neutron energy, the s-wave
resonance in the negative region contributes to all three $\gamma$-ray
transitions, and the p-wave resonance contributes to the $5161\,{\rm keV}$
transition and $5131\,{\rm keV}$ and/or $5126\,{\rm keV}$ transitions.

Figure 12:  Magnified 2-dimensional histogram.

###  II.3 Angular Distribution

Hereafter, we concentrate on the $5161\,{\rm keV}$ transition to the ground
state of 140La, in order to study the interference between the s- and p-wave
amplitudes.

The photo-peak efficiency, including both the detection efficiency and the
solid angle coverage, was readjusted using the photo-peak counts of the
$\gamma$-rays at $E^{\rm m}_{\gamma}$$=$$5262\,{\rm keV}$ from the
14N(n,$\gamma)$ reaction measured using the melamine target. It can be
reasonably assumed that the $\gamma$-rays are emitted isotropically, as the
14N does not have any resonance below $400\,{\rm eV}$ and p-wave or higher
angular momentum components of the incident neutron is negligibly small in
this energy region.

The photo-peak counts of the $5161\,{\rm keV}$ transition were determined by
subtracting the background counts caused by Compton scattering of the more
energetic $\gamma$-rays from targets other than the lanthanum target. To
evaluate the background, two energy regions were used: (I) $5200\,{\rm
keV}$$\leq$$E^{\rm m}_{\gamma}$$\leq$$5290\,{\rm keV}$ and (II) $4900\,{\rm
keV}$$\leq$$E^{\rm m}_{\gamma}$$\leq$$4980\,{\rm keV}$. The contribution of
Compton scattering of $\gamma$-rays corresponding to the three photo-peaks are
contained in region (II). The amount of this contribution from Compton
scattering is estimated using the response function $\bar{\psi}$ given in Eq.
30 and obtained by simulation. The background in region (II) is estimated by
subtracting the Compton contribution from the $\gamma$-ray counts in region
(II). The background is estimated using a best-fit third-order polynomial of
$E^{\rm m}_{\gamma}$ in regions (I) and (II).

There still remains a possible contamination of prompt $\gamma$-rays from
impurities overlapping with the $5161\,{\rm keV}$ photo-peak. The possible
contamination was examined over the entire pulse height spectrum, and was
determined to be less than $0.08\%$ of the photo-peak. The possibility of
contamination was neglected as the determined upper limit of $0.08\%$ is
smaller than the statistical error of the photo-peak.

According to the data acquisition system, two pulses detected within
$3.2~{}\mu$s did not have amplitude information, which amounted to $2\%$ of
the total $\gamma$-ray counts in the vicinity of the p-wave resonance. The
$2\%$ loss was corrected in the following analysis.

Two pulses detected within $0.4~{}\mu{\rm s}$ were processed as a single
pulse. The corresponding loss of the events were estimated as $0.2\%$ of the
total $\gamma$-ray counts in the vicinity of the p-wave resonance, which is
negligibly small compared with the statistical error of the corresponding
$\gamma$-ray counts, and is ignored in the following analysis.

Equation 16 was extended to describe the angular distribution of $\gamma$-rays
as

$\displaystyle\frac{\partial^{2}I_{\gamma}}{\partial t^{\rm
m}\partial\Omega_{\gamma}}(t^{\rm m},\Omega_{\gamma})$
$\displaystyle\quad=I_{0}\int{\rm d}E^{\prime}{\rm d}^{3}p_{\rm A}\Phi(t^{\rm
m},E^{\prime},\bm{p}_{\rm A})\frac{{\mathrm{d}}{\sigma_{{\rm
n}\gamma}}}{{\mathrm{d}}{\Omega_{\gamma}}}(E,\Omega_{\gamma}),$
$\displaystyle\Phi(t^{\rm m},E^{\prime},\bm{p}_{\rm A})$
$\displaystyle\quad=\frac{\partial^{2}\phi}{\partial E_{\rm n}\partial
t}\left(E^{\prime},t^{\rm m}-L\sqrt{\frac{m_{\rm n}}{2E^{\prime}}}\right)$
$\displaystyle\quad\quad\times\frac{1}{(2\pi m_{\rm A}k_{\rm
B}T)^{3/2}}e^{-p_{\rm A}^{2}/2m_{\rm A}k_{\rm B}T}$
$\displaystyle\quad\quad\times\frac{1}{\sigma_{\rm
t}(E)}\left(1-e^{-n\sigma_{\rm t}(E)\,\Delta z}\right).$ (5)

The $\gamma$-ray counts to be measured by the $d$-th detector can be written
as

$\displaystyle\frac{\partial{N}}{\partial{t^{\rm m}}}(t^{\rm
m},\bar{\theta}_{d})$ $\displaystyle=$ $\displaystyle\int_{\Omega_{d}}{\rm
d}\Omega_{\gamma}\int_{(E^{\rm m}_{\gamma})_{d}^{{w^{-}}}}^{(E^{\rm
m}_{\gamma})_{d}^{{w^{+}}}}{\rm d}(E^{\rm m}_{\gamma})_{d}$
$\displaystyle\times\frac{\partial^{2}I_{\gamma}}{\partial t^{\rm
m}\partial\Omega_{\gamma}}(t^{\rm
m},\Omega_{\gamma})\psi_{d}(E_{\gamma},\Omega_{\gamma},(E^{\rm
m}_{\gamma})_{d}),$

where the photo-peak region is taken as the full-width at quarter-maximum,
which implies $w=1/4$. Figure 13 shows the $N(t^{\rm m},\bar{\theta}_{d})$ for
$5161\,{\rm keV}$ $\gamma$-rays.

Figure 13:  $\partial N/\partial t^{\rm m}$ in the vicinity of p-wave
resonance for each $\bar{\theta}_{d}$. The central figure shows degrees in the
direction of neutron momentum of the type-A detectors and the type-B
detectors. The hexagons and the circles in the center of the figure denote
each crystal of the type-A detector and the type-B detector, respectively.

The peak shape of the p-wave resonance varies according to $\bar{\theta}_{d}$.
Here, we define $N_{\rm L}$ and $N_{\rm H}$ as

$\displaystyle N_{\rm L}(\theta_{\gamma})$ $\displaystyle=$
$\displaystyle\int_{E_{\rm p}-2\Gamma_{\rm p}}^{E_{\rm
p}}\frac{\partial{N}}{\partial{t^{\rm
m}}}(t^{\prime},\bar{\theta}_{\gamma}){\rm d}t^{\rm m}\frac{dt^{\rm
m}}{dE_{\rm n}}dE_{\rm n},$ $\displaystyle N_{\rm H}(\theta_{\gamma})$
$\displaystyle=$ $\displaystyle\int_{E_{\rm p}}^{E_{\rm p}+2\Gamma_{\rm
p}}\frac{\partial{N}}{\partial{t^{\rm
m}}}(t^{\prime},\bar{\theta}_{\gamma}){\rm d}t^{\rm m}\frac{dt^{\rm
m}}{dE_{\rm n}}dE_{\rm n}.$ (7)

Figure 14:  Visualization of the definition of $N_{\rm L}$ and $N_{\rm H}$.

The angular dependences of $N_{\rm L}$ and $N_{\rm H}$ are shown in Fig. 15.

Figure 15:  Angular dependences of $N_{\rm L}$ and $N_{\rm H}$. The white
point and black point show $N_{\rm L}$ and $N_{\rm H}$, respectively.

As $N_{\rm L}$ and $N_{\rm H}$ have certain angular dependence in Fig. 15, we
define an asymmetry between $N_{\rm L}$ and $N_{\rm H}$ to determine the
angular dependence of the shape of p-wave resonance as

$\displaystyle A_{\rm LH}=\frac{{N_{\rm L}}-{N_{\rm H}}}{{N_{\rm L}}+{N_{\rm
H}}}.$ (8)

The angular dependence of $A_{\rm LH}$ is shown in Fig. 16.

Figure 16:  Angular dependence of $A_{\rm LH}$. The solid line shows the best
fit.

The asymmetry $A_{\rm LH}$ has a correlation with $\cos\bar{\theta}_{\gamma}$
as

$\displaystyle A_{\rm LH}=(A\cos\bar{\theta}_{\gamma}+B)$ (9)

where

$\displaystyle A=-0.3881\pm 0.0236,\ B=-0.0747\pm 0.0105,$ (10)

which are the best fit results of Fig. 16. This result implies that clear
energy dependence of the angular distribution of $\gamma$-rays was observed.

The $x$ value of the p-wave resonance can be obtained using this result. The
analysis and the interpretation of this result are discussed in section III
and IV.

##  III Analysis

Our experimental results are analyzed using the formulation of possible
angular correlations of individual $\gamma$-rays, emitted in (n,$\gamma$)
reactions induced by low energy neutrons according to s- and p-wave amplitudes
Flambaum and Sushkov (1985). The formalism of the differential cross section
of the (n,$\gamma$) reaction induced by unpolarized neutrons is described in
Appendix E. We use $\bm{I}$ as the spin of the target nuclei, $\bm{J}$ as the
spin of the compound nucleus, $\bm{F}$ as the spin of the final state of the
$\gamma$-ray transition, and $\bm{l}$ as the orbital angular momentum of the
incident neutron. The total neutron spin is defined as $\bm{j}=\bm{l}+\bm{s}$,
where $\bm{s}$ is the neutron spin. The value of $j$ is $1/2$ for s-wave
neutrons ($l$$=$$0$) and $j$$=$$1/2$, $3/2$ for p-wave neutrons ($l$$=$$1$).

The p-wave resonance and two neighboring s-wave resonances are considered in
the negative and positive energy region, listed in Table 2, in the following
analysis.

$r$ | $E_{r}\,[{\rm eV}]$ | $J_{r}$ | $l_{r}$ | $\Gamma^{\gamma}_{r}\,[{\rm meV}]$ | $g_{r}\Gamma^{\rm{n}}_{r}\,[{\rm meV}]$ |
---|---|---|---|---|---|---
$1$ | $-48.63^{\rm(a)}$ | $4^{\rm(a)}$ | $0$ | $62.2^{\rm(a)}$ | $(571.8)^{\rm(a)\ast}$ | ${\rm s}_{1}$
$2$ | $0.740\pm 0.002$ | $4$ | $1$ | $40.41\pm 0.76$ |  $(5.6\pm 0.5)\times 10^{-5}$(c) | ${\rm p}$
$3$ | $72.30\pm 0.05^{\rm(b)}$ | $3$ | $0$ | $75.64\pm 2.21^{\rm(c)}$ | $11.76\pm 0.53^{\rm(c)}$ | ${\rm s}_{2}$
Table 2:  Resonance parameters of 139La used in the analysis. (a) taken from
Ref. Mughabghab (2006) and Ref. Shibata  _et al._ (2011). (b) taken from Ref.
Terlizzi and others (n TOF Collaboration). (c) calculated from Refs. Hacken
_et al._ (1976) and Terlizzi and others (n TOF Collaboration). ∗The neutron
width for the negative resonance was calculated using $\left|{E_{1}}\right|$
instead of $E_{1}$.

The resonance energy and resonance width measured in this work is adopted to
the p-wave resonance ($r$$=$$2$) and the values in Ref. Mughabghab (2006) for
the negative resonance and positive s-wave resonance($r$$=$$1$, $r$$=$$3$).
The compound nuclear spin of the negative s-wave resonance is $J_{1}$$=$$4$
Mughabghab (2006). The compound nuclear spin of the p-wave resonance is
assumed to be the same as that of the negative s-wave resonance, as both the
negative s-wave and the p-wave components were observed in the $5161\,{\rm
keV}$ $\gamma$-ray transition, which implies that $J_{2}$$=$$4$. Nevertheless,
the compound nuclear spin of the positive s-wave resonance is taken as
$J_{3}$$=$$3$, as the $5161\,{\rm keV}$ transition was not adequately observed
in the resonance, as shown in Fig. 17.

Figure 17:  Comparison of the expanded $\partial I_{\gamma}/\partial E^{\rm
m}_{\gamma}$ gated in the vicinities of s1-wave resonance ($E^{\rm m}_{\rm
n}$=0.2-0.4 eV: black line), the p-wave resonance ($E^{\rm m}_{\rm n}$=0.6-0.9
eV: shaded area with diagonal line), and the s2-wave resonance ($E^{\rm
m}_{\rm n}$=70-75 eV : solid shaded area).

The contributions of far s-wave resonances are assumed to be negligibly small,
that is, $\alpha_{1}=0$ in Eq. 55.

The ratios of the $\gamma$ width from each resonance to the ground state can
be determined by a comparison of the peak height ratio of the neutron
resonance gated in the 5161 keV photo-peak between s1-wave, p-wave, and
s2-wave as

$\displaystyle\frac{\Gamma_{\rm{s_{1},gnd}}^{\rm\gamma}}{\Gamma_{\rm{s_{1}}}^{\rm\gamma}}:\frac{\Gamma_{\rm{p,gnd}}^{\rm\gamma}}{\Gamma_{\rm{p}}^{\rm\gamma}}:\frac{\Gamma_{\rm{s_{2},gnd}}^{\rm\gamma}}{\Gamma_{\rm{s_{2},gnd}}^{\rm\gamma}}=1:0.796\pm
0.020:0.009\pm 0.006.$ (11)

As shown in Fig. 17 and in Eq. 11, the branching ratio from the s2-wave
resonance to the ground state is very small. We define
$(\overline{a_{0}})_{\rm L}$, $(\overline{a_{1}})_{\rm L}$,
$(\overline{a_{3}})_{\rm L}$, $(\overline{a_{0}})_{\rm H}$,
$(\overline{a_{1}})_{\rm H}$, and $(\overline{a_{3}})_{\rm H}$ as

$\displaystyle(\overline{a}_{0,1,3})_{\rm L}$ $\displaystyle=$
$\displaystyle\int_{E_{\rm p}-2\Gamma_{\rm p}}^{E_{\rm p}}{\rm
d}E^{\prime}\int{\rm d}^{3}p_{\rm A}\,a_{0,1,3}\Phi(t^{\rm
m},E^{\prime},\bm{p}_{A}),$ $\displaystyle(\overline{a}_{0,1,3})_{\rm H}$
$\displaystyle=$ $\displaystyle\int_{E_{\rm p}}^{E_{\rm p}+2\Gamma_{\rm
p}}{\rm d}E^{\prime}\int{\rm d}^{3}p_{\rm A}\,a_{0,1,3}\Phi(t^{\rm
m},E^{\prime},\bm{p}_{A}).$

Here, the $a_{3}$ term is ignored as it is proportional to
${\lambda_{2f}}^{2}$ and it is suppressed relative to the s-wave neutron
width, according to the centrifugal potential by the factor of $(kR)^{2}$.
Under this approximation, Eq. LABEL:eq:Flambaum1 is reduced to

$\displaystyle\frac{{\mathrm{d}}{\sigma_{{\rm
n}\gamma_{f}}}}{{\mathrm{d}}{\Omega_{\gamma}}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(a_{0}+a_{1}\cos\theta_{\gamma}\right),$

Substituting Eq. LABEL:eq:Flambaum2 into Eq. 5, the angular dependence of the
$\gamma$-ray counts in the neutron energy regions $E_{\rm p}-2\Gamma_{\rm
p}\leq E_{\rm{n}}\leq E_{\rm p}$ and $E_{\rm p}\leq E_{\rm{n}}\leq E_{\rm
p}+2\Gamma_{\rm p}$ can be written as

$\displaystyle\left(\frac{\partial^{2}I_{\gamma}}{\partial t^{\rm
m}\partial\Omega_{\gamma}}(t^{\rm m},\Omega_{\gamma})\right)_{\rm L}$
$\displaystyle=$ $\displaystyle\frac{I_{0}}{2}\left((\overline{a_{0}})_{\rm
L}+(\overline{a_{1}})_{\rm L}P_{1}(\cos\theta_{\gamma})\right),$
$\displaystyle\left(\frac{\partial^{2}I_{\gamma}}{\partial t^{\rm
m}\partial\Omega_{\gamma}}(t^{\rm m},\Omega_{\gamma})\right)_{\rm H}$
$\displaystyle=$ $\displaystyle\frac{I_{0}}{2}\left((\overline{a_{0}})_{\rm
H}+(\overline{a_{1}})_{\rm H}P_{1}(\cos\theta_{\gamma})\right).$

By convoluting with Eq. LABEL:eq:Ndgamma, the $\gamma$-ray counts
$\left(I_{\gamma,d}\right)_{\rm L}$ and $\left(I_{\gamma,d}\right)_{\rm H}$ to
be measured by the $d$-th detector can be written as

$\displaystyle\left(I_{\gamma,d}\right)_{\rm L,H}$ $\displaystyle=$
$\displaystyle\frac{I_{0}}{2}\left((\overline{a_{0}})_{\rm
L,H}\overline{P_{d,0}}+(\overline{a_{1}})_{\rm L,H}\overline{P_{d,1}}\right).$
(15)

As the energy dependence of $x_{2}$ and $y_{2}$ is negligibly small in the
vicinity of the p-wave resonance ($r_{\rm p}=2$), $(\overline{a_{1}})_{\rm L}$
and $(\overline{a_{1}})_{\rm H}$ are linear functions of $x_{2}$ and $y_{2}$,
thus a function of $\phi_{2}$. The value of $\phi_{2}$ is determined by
comparing $\left(\left(I_{\gamma,d}\right)_{\rm
L}-\left(I_{\gamma,d}\right)_{\rm
H}\right)/\left(\left(I_{\gamma,d}\right)_{\rm
L}+\left(I_{\gamma,d}\right)_{\rm H}\right)$ with the measured values $A$ in
Eq. 10.

$\displaystyle A$ $\displaystyle=$ $\displaystyle\frac{(\overline{a_{1}})_{\rm
L}-(\overline{a_{1}})_{\rm H}}{(\overline{a_{0}})_{\rm
L}+(\overline{a_{0}})_{\rm H}}$ (16) $\displaystyle=$ $\displaystyle
0.295\cos\phi_{2}-0.345\sin\phi_{2}.$

Two solutions can be obtained as

$\displaystyle\phi_{2}=(99.2_{-5.3}^{+6.3})^{\circ},\quad(161.9_{-6.3}^{+5.3})^{\circ}.$
(17)

The visualization of $\phi_{2}$ is shown in Fig. 18.

Figure 18:  Visualization of the value of $\phi_{2}$ on the $xy$-plane. The
solid line and shaded area show the central values of $\phi_{2}$ and 1$\sigma$
area, respectively. $\phi_{2}$ denotes an angle of a point on a unit circle.

##  IV Discussion

As the value of $\phi_{2}$ was obtained in the previous section, the
T-violation sensitivity is discussed in this section. We obtain $x_{2}$ from
Eq. 17 and Eq. 58 with resonance number $r_{\mathrm{p}}=2$ as

$\displaystyle x_{2}=-0.16_{-0.11}^{+0.09},\quad-0.95_{-0.03}^{+0.04}.$ (18)

This leads to the value of $W$ which is given by Eq. 1 as

$\displaystyle W=(13.2_{-5.3}^{+18.1})\
\mathrm{meV},\quad(2.21_{-0.06}^{+0.10})\ \mathrm{meV}.\ $ (19)

The published value of $A_{\rm L}=(9.56\pm 0.35)\times 10^{-2}$ in Ref. Bowman
_et al._ (1989) and resonance parameters in Table. 1 are used in the
calculation. Note that the neutron width of the negative s-wave resonance
$\Gamma^{\mathrm{n}}_{\mathrm{1}}$ at the resonance energy of the p-wave
resonance is adopted.

The ratio of P-odd T-odd cross sections to P-odd cross sections is given as

$\displaystyle\frac{\Delta\sigma_{\rm PT}}{\Delta\sigma_{\rm
P}}=\kappa(J)\frac{W_{\rm T}}{W},$ (20)

where $\Delta\sigma_{\rm PT}$ is the P-odd T-odd cross section,
$\Delta\sigma_{\rm P}$ the P-odd cross section, $W_{\rm T}$ the P-odd T-odd
matrix element and $W$ the P-odd matrix element Gudkov and Shimizu (2017). The
calculations of these matrix elements were performed in Ref Flambaum and Vorov
(1993) and Flambaum and Vorov (1995). The spin factor $\kappa(J)$ is defined
as

$\displaystyle\kappa(J)=\left\\{\begin{array}[]{cc}(-1)^{2I}\left(1+\frac{1}{2}\sqrt{\frac{2I-1}{I+1}}\frac{y}{x}\right)&\quad(J=I-\frac{1}{2})\\\
(-1)^{2I+1}\frac{I}{I+1}\left(1-\frac{1}{2}\sqrt{\frac{2I+3}{I}}\frac{y}{x}\right)&\quad(J=I+\frac{1}{2})\end{array}\right..$
(23)

The magnitude of $\kappa(J)$ indicates the sensitivity to the P-odd T-odd
interaction. The $J=I+\frac{1}{2}$ case corresponds to the p-wave resonance of
the 139La+n at $E_{\rm n}=E_{2}$. The value of $\kappa(J)$ corresponding to
the $\phi_{2}$ obtained is

$\displaystyle\kappa(J)=4.84_{-1.69}^{+5.58},\quad 0.99_{-0.07}^{+0.08}$ (25)

and $\left|{\kappa(J)}\right|$ is shown in Fig. 19.

Figure 19:  Value of $\left|{\kappa(J)}\right|$ as a function of $\phi_{2}$.
The solid line and shaded area show the central values of $\phi_{2}$ and the
1$\sigma$ area from central value, respectively.

In the previous section, the $a_{3}$ term was ignored, as the centrifugal
potential of the p-wave resonance is small. Hereafter we discuss the case when
the $a_{3}$ term in Eq. LABEL:eq:Flambaum1 is activated. We analyze the
angular dependences of $N_{\rm L}-N_{\rm H}$ and $N_{\rm L}+N_{\rm H}$ fitted
by the functions of
$f(\bar{P}_{d,1}/\bar{P}_{d,0})=A^{\prime}\bar{P}_{d,1}/\bar{P}_{d,0}+B^{\prime}$
and
$g(\bar{P}_{d,2}/\bar{P}_{d,0})=C^{\prime}\bar{P}_{d,2}/\bar{P}_{d,0}+D^{\prime}$,
respectively, with fitting parameters $A^{\prime}$, $B^{\prime}$,
$C^{\prime}$, and $D^{\prime}$. The equations of $a_{3}$ can be written as

Figure 20:  Angular dependences of $N_{\rm L}-N_{\rm H}$ and $N_{\rm L}+N_{\rm
H}$. The solid line indicates the best fit.

$\displaystyle\frac{A^{\prime}}{D^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{(\overline{a_{1}})_{\rm L}-(\overline{a_{1}})_{\rm
H}}{(\overline{a_{0}})_{\rm L}+(\overline{a_{0}})_{\rm H}}$ (26)
$\displaystyle=$ $\displaystyle 0.295\cos\phi_{2}-0.345\sin\phi_{2},$
$\displaystyle\frac{C^{\prime}}{D^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{(\overline{a_{3}})_{\rm L}+(\overline{a_{3}})_{\rm
H}}{(\overline{a_{0}})_{\rm L}+(\overline{a_{0}})_{\rm H}}$ (27)
$\displaystyle=$
$\displaystyle-0.295\cos\phi_{2}\sin\phi_{2}+0.050\sin^{2}\phi_{2}.$

The fitted results of $C^{\prime}/D^{\prime}$ and $A^{\prime}/D^{\prime}$ are

$\displaystyle\frac{C^{\prime}}{D^{\prime}}=0.191\pm 0.028,\ \
\frac{A^{\prime}}{D^{\prime}}=-0.409\pm 0.024.$ (28)

The value of $\phi_{2}$ is determined by combining the equation of $a_{1}$(Eq.
26) and $a_{3}$(Eq. 27) on the $xy$-plane. The result is shown in Fig. 21.

Figure 21:  $a_{1}$(straight lines) and $a_{3}$(curved lines) on the
$xy$-plane for the cases of $J_{1}=J_{2}=4,J_{3}=3$. The solid line, shaded
area, dashed line, and dotted line show the central values of $\phi_{2}$,
1$\sigma$ areas, 2$\sigma$ contours and 3$\sigma$ contours, respectively.

The restriction from the $a_{3}$ term is not consistent with that of the
$a_{1}$ term. The $a_{3}$ term deviats from the requirement of
$x_{2}^{2}+y_{2}^{2}=1$ by more than 2$\sigma$.

In this analysis, $J_{1}=J_{2}=4,J_{3}=3$ are assumed. However, there is a
possibility of the case of $J_{1}=J_{2}=J_{3}=3$. As the effect of the s2-wave
is negligibly small in this discussion, we discuss combinations of $J_{1}$ and
$J_{2}$ only. The result of the case of $J_{1}=J_{2}=J_{3}=3$ is shown in Fig.
22. Both $a_{1}$ and $a_{3}$ in the case of $J_{1}=J_{2}=J_{3}=3$ have no
solution. As both $a_{1}$ and $a_{3}$ in the case of $J_{1}=J_{2}=4,J_{3}=3$
have solution in 3$\sigma$, we support $J_{1}=J_{2}=4$.

Figure 22:  $a_{1}$(straight lines) and $a_{3}$(curved lines) on the
$xy$-plane in the case of $J_{1}=J_{2}=J_{3}=3$. The solid line, shaded area,
dashed line, and dotted line show the central values of $\phi_{2}$, 1$\sigma$
areas, 2$\sigma$ contours, and 3$\sigma$ contours, respectively.

The origin of the inconsistency has not been identified in the present study.
The inconsistency may be due to possible incompleteness of the reaction
mechanism based on the interference between s- and p-wave amplitudes with the
Breit–Wigner approximation.

##  V Conclusion

We observed clear angular distribution of emitted $\gamma$-rays in the
transition from the p-wave resonance of 139La+n to the ground state of 140La
as a function of incident neutron energy. The angular distribution was
analyzed by assuming interference between s- and p-wave amplitudes, and the
partial neutron width of the p-wave resonance was obtained. This result
suggests that the T-violating effect can be enhanced on the same order of the
P-violating effect for 0.74 eV p-wave resonance of 139La+n. Therefore an
experiment to explore T-violation in compound nuclear states is feasible. In
addition, the analysis under this assumption leads to results that are
consistent with theoretical expectation, and we therefore believe the
assumption of s-p mixing is correct.

###### Acknowledgements.

The authors would like to thank the staff of ANNRI for the maintenance of the
germanium detectors, and MLF and J-PARC for operating the accelerators and the
neutron production target. We would like to thank Dr. K. Kino for the
calculation of the pulse shape of neutron beam. We also appreciate the
continuous help by Prof. V. P. Gudkov for the interpretation of the measured
results. The neutron scattering experiment was approved by the Neutron
Scattering Program Advisory Committee of IMSS and KEK (Proposal No. 2014S03,
2015S12). The neutron experiment at the Materials and Life Science
Experimental Facility of the J-PARC was performed under a user program
(Proposal No. 2016B0200, 2016B0202, 2017A0158, 2017A0170, 2017A0203). This
work was supported by MEXT KAKENHI Grant number JP19GS0210 and JSPS KAKENHI
Grant Number JP17H02889.

##  Appendix A Definitions of symbols describing detector characteristics and
the results of the simulation

In this section, we describe the definition of characteristics of the
germanium detectors and the simulation results. Herein, we use
$\psi_{d}(E_{\gamma},\Omega_{\gamma},(E^{\rm m}_{\gamma})_{d})$ to denote the
probability of the case where the energy of $(E^{\rm m}_{\gamma})_{d}$ is
deposited in the $d$-th detector, when a $\gamma$-ray with an energy of
$E_{\gamma}$ is emitted in the direction of
$\Omega_{\gamma}$$=$$(\theta_{\gamma},\varphi_{\gamma})$. The polar angle and
the azimuthal angle of the direction of the emitted $\gamma$-ray are denoted
by $\theta_{\gamma}$ and $\varphi_{\gamma}$, respectively. The
$\psi_{d}(E_{\gamma},\Omega_{\gamma},(E^{\rm m}_{\gamma})_{d})$ satisfies

$\displaystyle\int_{0}^{E_{\gamma}}\psi_{d}(E_{\gamma},\Omega_{\gamma},(E^{\rm
m}_{\gamma})_{d}){\rm d}(E^{\rm m}_{\gamma})_{d}=1.$ (29)

We define the distribution of the energy deposit as

$\displaystyle\bar{\psi}_{d}(E_{\gamma},(E^{\rm
m}_{\gamma})_{d})=\int_{\Omega_{d}}\psi_{d}(E_{\gamma},\Omega_{\gamma},(E^{\rm
m}_{\gamma})_{d}){\rm d}\Omega_{\gamma},$ (30)

where $\Omega_{d}$ is the geometric solid angle of the $d$-th detector. The
photo-peak efficiency of $d$-th detector for $\gamma$-rays with the energy of
$E_{\gamma}$ is defined as

$\displaystyle\epsilon_{d}^{{\rm pk},w}(E_{\gamma})=\int_{(E^{\rm
m}_{\gamma})_{d}^{w^{-}}}^{(E^{\rm
m}_{\gamma})_{d}^{w^{+}}}\bar{\psi}_{d}(E_{\gamma},(E^{\rm
m}_{\gamma})_{d}){\rm d}(E^{\rm m}_{\gamma})_{d},$ (31)

where $(E^{\rm m}_{\gamma})_{d}^{w^{+}}$ and $(E^{\rm
m}_{\gamma})_{d}^{w^{-}}$ are the upper and lower limits of the region of the
energy deposit for defining the photo-peak region, as schematically shown in
Fig. 23. For the definition of the photo-peak efficiency $w=1/4$ was used.

Figure 23:  Schematic of the definition of the photo-peak in the pulse height
spectrum.

The relative photo-peak efficiency is also defined as

$\displaystyle\bar{\epsilon}_{d}^{{\ \rm
pk},w}(E_{\gamma})=\frac{\epsilon_{d}^{{\rm
pk},w}(E_{\gamma})}{\epsilon_{1}^{{\rm pk},w}(E_{\gamma})}.$ (32)

The value of $\bar{\psi}_{d}(E_{\gamma},(E^{\rm m}_{\gamma})_{d})$ was
obtained using the simulation to reproduce the pulse height spectra for
$\gamma$-rays from the radioactive source of 137Cs
($E_{\gamma}$$=$$0.662\,{\rm MeV}$), as shown in Fig. 24. Subsequently, the
reproducibility was checked by comparing the pulse height spectra for 60Co at
$E_{\gamma}$$=$$1.173\,{\rm MeV},1.332\,{\rm MeV}$ and 22Na at
$E_{\gamma}$=$1.275\,{\rm MeV}$. Finally, we confirmed that the simulation
program is applicable to higher energies by comparing the numerical simulation
with the pulse height spectrum for prompt $\gamma$-rays emitted by the
14N(n,$\gamma$) reaction at $E_{\gamma}$$=$$10.829\,{\rm MeV}$, as shown in
Fig. 25.

Figure 24:  Comparison of the pulse height spectrum for $\gamma$-rays from a
radioactive source 137Cs (gray line) and the numerical simulation (black line)
for a type-B detector unit (left) and a type-A detector unit (right). As the
simulation reproduces the experimental data faithfully, these almost overlap.
Figure 25:  Comparison of pulse height spectrum for prompt $\gamma$-rays
emitted in 14N(n,$\gamma$) reaction (gray line) and the numerical simulation
(black line) for a type-A detector unit (left) and a type-B detector unit
(right). As the simulation reproduces the experimental data faithfully, these
almost overlap.

The photo-peak efficiency of the detector assembly for 1.332 MeV $\gamma$-ray
is determined to be 3.64$\pm$0.11% Kin  _et al._ (2009). In the case of
angular distribution of $\gamma$-rays this is expanded using Legendre
polynomials as $\sum_{p=0}^{\infty}c_{p}P_{p}(\cos\theta_{\gamma})$, and the
photo-peak counts of the $d$-th detector can be written as

$\displaystyle N_{d}(E_{\gamma})$ $\displaystyle=$ $\displaystyle
N_{0}\sum_{p=0}^{\infty}c_{p}\bar{P}_{d,p},$ $\displaystyle\bar{P}_{d,p}$
$\displaystyle=$ $\displaystyle\frac{1}{4\pi}\int_{(E^{\rm
m}_{\gamma})_{d}^{w^{-}}}^{(E^{\rm m}_{\gamma})_{d}^{w^{+}}}{\rm d}(E^{\rm
m}_{\gamma})_{d}\int{\rm
d}\Omega_{\gamma}P_{p}(\cos\theta_{\gamma})\bar{\psi}_{d}(E_{\gamma},\Omega_{\gamma}).$

The determiend values of $\bar{P}_{d,p}$ are listed in Table 3 for $p=0,1,2$.
The quantity $\bar{P}_{0}$ corresponds to $\epsilon_{d}^{{\rm pk},w}$. The
table also contains the weighted average of the viewing angle of each detector
$\bar{\theta}_{d}$ determined as
$P_{p}(\cos\bar{\theta}_{d})=\bar{P}_{d,p}/\bar{P}_{d,0}$.

$d$ | $E_{\gamma}=0.662\,{\rm MeV}$ | $E_{\gamma}=5.262\,{\rm MeV}$
---|---|---
$\bar{P}_{d,1}/\bar{P}_{d,0}$ | $\bar{P}_{d,2}/\bar{P}_{d,0}$ | $\bar{\theta}_{d}$ | $\bar{\epsilon}_{d}^{{\rm pk},1/4}$ | $\bar{P}_{d,1}/\bar{P}_{d,0}$ | $\bar{P}_{d,2}/\bar{P}_{d,0}$ | $\bar{\theta}_{d}$
$1$ | $0.00001\pm 0.00002$ | $-0.488480\pm 0.000003$ | $90.000\pm 0.001$ | $1$ | $-0.00004\pm 0.00004$ | $-0.49036\pm 0.00001$ | $90.002\pm 0.002$
$2$ | $-0.00003\pm 0.00003$ | $-0.489485\pm 0.000003$ | $90.002\pm 0.001$ | $0.997\pm 0.027$ | $0.00006\pm 0.00004$ | $-0.49115\pm 0.00001$ | $89.997\pm 0.002$
$3$ | $0.30297\pm 0.00002$ | $-0.35283\pm 0.00002$ | $72.299\pm 0.001$ | $0.920\pm 0.026$ | $0.29887\pm 0.00004$ | $-0.35791\pm 0.00003$ | $72.556\pm 0.002$
$4$ | $0.30294\pm 0.00002$ | $-0.35285\pm 0.00002$ | $72.301\pm 0.001$ | $1.041\pm 0.028$ | $0.29889\pm 0.00004$ | $-0.35791\pm 0.00003$ | $72.555\pm 0.002$
$5$ | $0.00008\pm 0.00002$ | $-0.489493\pm 0.000003$ | $89.996\pm 0.001$ | $1.025\pm 0.028$ | $-0.00012\pm 0.00004$ | $-0.49116\pm 0.00001$ | $90.007\pm 0.002$
$6$ | $-0.30293\pm 0.00002$ | $-0.35288\pm 0.00002$ | $107.698\pm 0.001$ | $1.092\pm 0.029$ | $-0.29886\pm 0.00004$ | $-0.35792\pm 0.00003$ | $107.443\pm 0.002$
$7$ | $-0.30297\pm 0.00002$ | $-0.35284\pm 0.00002$ | $107.700\pm 0.001$ | $0.999\pm 0.027$ | $-0.29885\pm 0.00004$ | $-0.35794\pm 0.00003$ | $107.442\pm 0.002$
$8$ | $0.00004\pm 0.00002$ | $-0.488259\pm 0.000003$ | $89.998\pm 0.001$ | $0.923\pm 0.026$ | $0.00005\pm 0.00004$ | $-0.49030\pm 0.00001$ | $89.997\pm 0.002$
$9$ | $-0.00002\pm 0.00003$ | $-0.48927\pm 0.000003$ | $90.001\pm 0.001$ | $0.915\pm 0.026$ | $-0.00003\pm 0.00004$ | $-0.49107\pm 0.00001$ | $90.002\pm 0.002$
$10$ | $0.30292\pm 0.00002$ | $-0.35288\pm 0.00002$ | $72.302\pm 0.001$ | $0.998\pm 0.027$ | $0.29894\pm 0.00004$ | $-0.35785\pm 0.00003$ | $72.552\pm 0.002$
$11$ | $0.30297\pm 0.00002$ | $-0.35284\pm 0.00002$ | $72.300\pm 0.001$ | $0.945\pm 0.026$ | $0.29886\pm 0.00004$ | $-0.35793\pm 0.00003$ | $72.557\pm 0.002$
$12$ | $-0.00001\pm 0.00003$ | $-0.489256\pm 0.000003$ | $90.001\pm 0.001$ | $0.979\pm 0.027$ | $0.00004\pm 0.00004$ | $-0.49108\pm 0.00001$ | $89.997\pm 0.002$
$13$ | $-0.30293\pm 0.00002$ | $-0.35286\pm 0.00002$ | $107.698\pm 0.001$ | $1.046\pm 0.028$ | $-0.29892\pm 0.00004$ | $-0.357867\pm 0.00003$ | $107.447\pm 0.002$
$14$ | $-0.30297\pm 0.00002$ | $-0.35284\pm 0.00002$ | $107.700\pm 0.001$ | $1.142\pm 0.030$ | $-0.29887\pm 0.00004$ | $-0.35791\pm 0.00003$ | $107.444\pm 0.002$
$15$ | $-0.80431\pm 0.00001$ | $0.47410\pm 0.00004$ | $143.824\pm 0.001$ | $0.892\pm 0.025$ | $-0.80420\pm 0.00002$ | $0.47340\pm 0.00005$ | $143.780\pm 0.002$
$16$ | $-0.30772\pm 0.00003$ | $-0.35142\pm 0.00003$ | $107.967\pm 0.002$ | $0.356\pm 0.014$ | $-0.30775\pm 0.00004$ | $-0.35229\pm 0.00004$ | $107.963\pm 0.003$
$17$ | $0.30697\pm 0.00002$ | $-0.34904\pm 0.00002$ | $72.058\pm 0.001$ | $0.945\pm 0.026$ | $0.30715\pm 0.00004$ | $-0.34994\pm 0.00003$ | $72.054\pm 0.002$
$18$ | $0.80434\pm 0.00001$ | $0.47419\pm 0.00004$ | $36.173\pm 0.001$ | $1.170\pm 0.031$ | $0.80418\pm 0.00002$ | $0.47337\pm 0.00005$ | $36.221\pm 0.002$
$19$ | $0.80425\pm 0.00001$ | $0.47396\pm 0.00004$ | $36.182\pm 0.001$ | $0.917\pm 0.026$ | $0.80415\pm 0.00002$ | $0.47328\pm 0.00005$ | $36.225\pm 0.002$
$20$ | $0.30683\pm 0.00002$ | $-0.34917\pm 0.00002$ | $72.066\pm 0.001$ |  | $0.30705\pm 0.00004$ | $-0.35004\pm 0.00003$ | $72.060\pm 0.002$
$21$ | $-0.30689\pm 0.00002$ | $-0.34911\pm 0.00002$ | $107.938\pm 0.001$ |  | $-0.30709\pm 0.00004$ | $-0.34999\pm 0.00003$ | $107.943\pm 0.002$
$22$ | $-0.80427\pm 0.00001$ | $0.47401\pm 0.00004$ | $143.820\pm 0.001$ | $1.113\pm 0.030$ | $-0.80420\pm 0.00002$ | $0.47341\pm 0.00005$ | $143.781\pm 0.002$

Table 3:  Values of $\bar{P}_{d,p}/\bar{P}_{d,0}$ for $p=1,2$ are shown for
all detectors together with the weighted average angle of each detector
$\bar{\theta}_{d}$ at $E_{\gamma}$=0.662 MeV and $E_{\gamma}$=5.262 MeV. The
relative peak efficiency : $\bar{\epsilon}_{d}^{{\rm pk},1/4}$ was measured
using 14N(n,$\gamma$) reactions are also shown.

##  Appendix B Neutron absorption cross section

The formula used to describe the neutron cross section $\sigma_{\rm t}$ is
given as a function of the neutron energy in the center-of-mass system:

$\displaystyle\sigma_{\rm t}(E)=\sigma_{\rm s}+\sigma_{\rm
n\gamma}(E_{\rm{n}}),$ (34) $\displaystyle\sigma_{\rm s}=4\pi a^{2},$ (35)
$\displaystyle\Gamma_{r}^{\gamma}$ $\displaystyle=$
$\displaystyle\sum_{f}\Gamma_{r,f}^{\gamma},$ (36) $\displaystyle\sigma_{\rm
n\gamma}(E_{\rm n})$ $\displaystyle=$ $\displaystyle\sum_{f}\sigma_{{\rm
n}\gamma_{f}}(E_{\rm n}),$ (37) $\displaystyle\sigma_{{\rm
n}\gamma_{f}}(E_{\rm n})$ $\displaystyle=$
$\displaystyle\sum_{r}\frac{\pi\hbar^{2}}{2m_{\rm n}E_{\rm
n}}\left[\frac{E_{\rm n}}{E_{r}}\right]^{l_{r}+1/2}$ (38)
$\displaystyle\quad\times\frac{\displaystyle g_{r}\Gamma_{r}^{\rm
n}\Gamma_{r,f}^{\gamma}}{\displaystyle(E_{\rm
n}-E_{r})^{2}+(\Gamma_{r}/2)^{2}},$

where $\sigma_{\rm s}$ is the scattering cross section, $\sigma_{\rm n\gamma}$
is the radiative capture cross section, $k_{\rm n}=\sqrt{2m_{\rm n}E_{\rm
n}}/\hbar$, $a$ are scattering length, $E_{r}$ is the resonance energy of the
$r$-th resonance, $g_{r}=(2J_{r}+1)/(2(2I+1))$ is the statistical factor with
the target nuclear spin $I$ and the spin of the $r$-th resonance $J_{r}$,
$\Gamma_{r}^{\rm n}$ is the neutron width of $r$-th resonance, and
$\Gamma_{r,f}^{\gamma}$ is the $\gamma$-width of the $\gamma$-ray transition
from the $r$-th resonance to the final state $f$. The quantity $l_{r}$
represents the orbital angular momentum of the incident neutrons contributing
to $r$-th resonance. The above formula is valid for $l_{r}=0,1$. Higher
angular momenta can be ignored in the incident neutron energy region of our
interest.

##  Appendix C Pulse shape of the neutron beam

The pulse shape of neutron beam depends on the neutron energy $E_{\rm n}$
according to the time delay during the moderation process. The double
differential of the flux of the pulsed neutron beam $I_{\rm{n}}$ as a function
of $E_{\rm n}$, and the time measured from the primary proton beam injection
$t$ is known to be well reproduced by the Ikeda–Carpenter function, defined as

$\displaystyle\frac{\partial^{2}I_{\rm{n}}}{\partial E_{\rm n}\partial
t}(E_{\rm n},t)=\frac{\alpha C}{2}\left\\{(1-R)(\alpha t)^{2}e^{-\alpha
t}+2R\frac{\alpha^{2}\beta}{(\alpha-\beta)^{3}}\right.$
$\displaystyle\quad\quad\quad\times\left.\left[e^{-\beta t}-e^{-\alpha
t}\left(1+(\alpha-\beta)t+\frac{(\alpha-\beta)^{2}}{2}t^{2}\right)\right]\right\\},$
(39)

where parameters $\alpha$, $\beta$, $C$, $R$ depend on $E_{\rm n}$. The
Ikeda–Carpenter function was originally proposed to explain the pulse shape of
the cold source of polyethylene moderator at the Intense Pulsed Neutron Source
of the Argonne National Laboratory Ikeda and Carpenter (1985). The double
differential of the neutron beam flux on the moderator surface of the J-PARC
spallation source was calculated using MCNPX Ikeda (2009), and was fitted with
the Ikeda–Carpenter function. The dependence of neutron energy on the fitting
parameters $t_{0},\alpha,\beta,R$, and C were obtained by fitting of the pulse
shape of neutron beam with a polynomial functionKino  _et al._ (2011).

The energy spectrum at a given time $t^{\rm m}$ at the distance of $L$ from
the moderator surface is given as

$\displaystyle\frac{\partial{I_{\rm{n}}}}{\partial{t^{\rm m}}}(t^{\rm m})$
$\displaystyle=$ $\displaystyle\int{\rm
d}E^{\prime}\frac{\partial^{2}I_{{\mathrm{n}}}}{\partial E_{\rm n}\partial
t}\left(E^{\prime},t^{\rm m}-L\sqrt{\frac{m_{\rm n}}{2E^{\prime}}}\right).$

##  Appendix D Thermal motion of the target nucleus

We adopted the free gas model for the thermal motion of the target nuclei,
which leads to the $\gamma$-ray yield in the form of

$\displaystyle\frac{\partial{I_{\gamma}}}{\partial{t^{\rm m}}}(t^{\rm m})$
$\displaystyle=$ $\displaystyle I_{0}\int{\rm d}E^{\prime}{\rm d}^{3}p_{\rm
A}\,\frac{\partial^{2}I_{\rm{n}}}{\partial E_{\rm n}\partial
t}\left(E^{\prime},t^{\rm m}-L\sqrt{\frac{m_{\rm n}}{2E^{\prime}}}\right)$
$\displaystyle\times\frac{1}{(2\pi m_{\rm A}k_{\rm B}T)^{3/2}}e^{-p_{\rm
A}^{2}/2m_{\rm A}k_{\rm B}T}$ $\displaystyle\times\frac{\sigma_{\rm
n\gamma}(E_{\rm n})}{\sigma_{\rm t}(E_{\rm n})}\left(1-e^{-n\sigma_{\rm
t}(E)\,\Delta z}\right),$

as long as the target is sufficiently thin, such that multiple scattering is
negligible. $I_{0}$ is the normalization constant, $\Delta z$ is the target
thickness, $n$ is the number density of target nuclei, $k_{\rm B}$ is the
Boltzmann constant, and $T$ is the effective temperature of the target, which
can be used as a fitting parameter.

##  Appendix E Formula describing the (n,$\gamma$) angular dependence

The differential cross section of the (n,$\gamma$) reaction induced by
unpolarized neutrons can be written as

$\displaystyle\frac{{\mathrm{d}}{\sigma_{{\rm
n}\gamma_{f}}}}{{\mathrm{d}}{\Omega_{\gamma}}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(a_{0}+a_{1}\cos\theta_{\gamma}+a_{3}\left(\cos^{2}\\!\\!\theta_{\gamma}-\frac{1}{3}\right)\right),$
$\displaystyle a_{0}$ $\displaystyle=$ $\displaystyle\sum_{r_{\rm
s}}\left|{V_{r_{\rm s}}}\right|^{2}+\sum_{r_{\rm p}}\left|{V_{r_{\rm
p}}}\right|^{2},$ $\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle
2{\rm{Re}}\sum_{r_{\rm s}i_{\rm p}j}V_{r_{\rm{s}}}V_{r_{\rm
p}}^{\ast}z_{r_{\rm p}j}P(J_{r_{\rm s}}J_{r_{\rm p}}\frac{1}{2}j1IF),$
$\displaystyle a_{3}$ $\displaystyle=$ $\displaystyle 3\sqrt{10}\
{\rm{Re}}\sum_{r_{\rm p}jr_{\rm p}^{\prime}j^{\prime}}V_{r_{\rm p}}V_{r_{\rm
p}^{\prime}}^{\ast}z_{r_{\rm p}j}z_{r_{\rm p}j^{\prime}}$ (45)
$\displaystyle\times P(J_{r_{\rm p}}J_{r_{\rm
p}^{\prime}}jj^{\prime}2IF)\left\\{\\!\begin{array}[]{ccc}{2}&{\\!1}&{\\!1}\\\
{0}&{\\!1/2}&{\\!1/2}\\\ {2}&{\\!r_{\rm
p}}&{\\!j^{\prime}}\end{array}\\!\right\\}$ $\displaystyle V_{r_{\rm
s}f}=-\frac{\sqrt{g_{r_{\rm s}}}}{2k_{\rm n}}\frac{\sqrt{\Gamma_{r_{\rm
s}}^{\rm n}\Gamma_{r_{\rm s}f}^{\gamma}}(1+\alpha_{r_{\rm s}})}{E_{\rm
n}-E_{r_{\rm s}}+i\Gamma_{r_{\rm s}}/2}$ $\displaystyle V_{r_{\rm
p}f}=-\frac{\sqrt{g_{r_{\rm p}}}}{2k_{\rm n}}\frac{\sqrt{\Gamma_{r_{\rm
p}}^{\rm n}\Gamma_{r_{\rm p}f}^{\gamma}}}{E_{\rm n}-E_{r_{\rm
p}}+i\Gamma_{r_{\rm p}}/2}$ $\displaystyle z_{r_{\rm
p}j}=\sqrt{\frac{\Gamma_{r_{\rm p}j}^{\rm n}}{\Gamma_{r_{\rm p}}^{\rm
n}}}=\left\\{\begin{array}[]{ll}x_{r_{\rm p}}&\quad(j=1/2)\\\ y_{r_{\rm
p}}&\quad(j=3/2)\end{array}\right.,$ (49) $\displaystyle
P(JJ^{\prime}jj^{\prime}kIF)=(-1)^{J+J^{\prime}+j^{\prime}+I+F}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\times\frac{3}{2}\sqrt{(2J+1)(2J^{\prime}+1)(2j+1)(2j^{\prime}+1)}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\times\left\\{\\!\begin{array}[]{ccc}{k}&{\\!j}&{\\!j^{\prime}}\\\
{I}&{\\!J^{\prime}}&{\\!J}\end{array}\\!\right\\}\left\\{\\!\begin{array}[]{ccc}{k}&{\\!1}&{\\!1}\\\
{F}&{\\!J}&{\\!J^{\prime}}\end{array}\\!\right\\},$ (54) (55)

where $r_{\rm s}$ is the r${}^{\text{th}}$ s-wave resonance number ($l_{r_{\rm
s}}$$=$$0$) and $r_{\rm p}$ is the r${}^{\text{th}}$ p-wave resonance
number($l_{r_{\rm p}}$$=$$1$). Amplitudes $V_{r_{\rm s}}$ and $V_{r_{\rm p}}$
are the s- and p-wave amplitudes, respectively, and $\alpha_{r_{\rm s}}$
represents the contribution from far s-wave resonances. Width $\Gamma_{r_{\rm
p}j}^{\rm n}$ is the partial neutron width for the incident neutrons of total
angular momentum of $j$, and $x_{r_{\rm p}}$ and $y_{r_{\rm p}}$ are defined
as

$\displaystyle x_{r_{\rm p}}=\sqrt{\frac{\Gamma_{\mathrm{p},j=1/2}^{\rm
n}}{\Gamma^{\rm n}_{\rm p}}},\ y_{r_{\rm
s}}=\sqrt{\frac{\Gamma_{\mathrm{p},j=3/2}^{\rm n}}{\Gamma^{\rm n}_{\rm p}}}.$
(56)

$x_{r_{\rm p}}$ and $y_{r_{\rm p}}$ satisfy

$\displaystyle x_{r_{\rm p}}^{2}+y_{r_{\rm p}}^{2}=1$ (57)

due to the relation $\Gamma_{\rm p}^{\rm n}=\Gamma_{{\rm p},j=1/2}^{\rm
n}+\Gamma_{{\rm p},j=3/2}^{\rm n}$. The resonance energy and the resonance
width obtained from this experiment, the published values listed in Table 1,
and $I=7/2$ are used to determine the value of $\phi_{r_{\rm p}}$, defined as

$\displaystyle x_{r_{\rm p}}=\cos\phi_{r_{\rm p}},\quad y_{r_{\rm
p}}=\sin\phi_{r_{\rm p}}.$ (58)

In the case of 139La, negative s-wave amplitude $V_{1}$, p-wave amplitude
$V_{2}$, and positive s-wave amplitude $V_{3}$ can be written as,

$\displaystyle V_{1f}$ $\displaystyle=$
$\displaystyle-\lambda_{1f}\left(\frac{\left|{E_{1}}\right|}{E_{\rm
n}}\right)^{\frac{1}{4}}\frac{\Gamma_{1}/2}{E_{\rm n}-E_{1}+i\Gamma_{1}/2},$
$\displaystyle V_{2f}$ $\displaystyle=$
$\displaystyle-\lambda_{2f}\left(\frac{E_{\rm
n}}{E_{2}}\right)^{\frac{1}{4}}\frac{\Gamma_{2}/2}{E_{\rm
n}-E_{2}+i\Gamma_{2}/2},$ $\displaystyle V_{3f}$ $\displaystyle=$
$\displaystyle-\lambda_{3f}\left(\frac{E_{3}}{E_{\rm
n}}\right)^{\frac{1}{4}}\frac{\Gamma_{3}/2}{E_{\rm n}-E_{3}+i\Gamma_{3}/2},$

where the absolute value of $E_{1}$ is adopted simply to avoid the imaginary
neutron width. The terms $a_{0}$, $a_{1}$, and $a_{3}$ are given as

$\displaystyle a_{0}$ $\displaystyle=$
$\displaystyle\lambda_{1f}^{2}\sqrt{\frac{\left|{E_{1}}\right|}{E_{\rm
n}}}\frac{\Gamma^{\ 2}_{1}/4}{(E_{\rm n}-E_{1})^{2}+\Gamma_{1}^{\ 2}/4}$
$\displaystyle+\lambda_{2f}^{2}\sqrt{\frac{E_{\rm
n}}{E_{2}}}\frac{\Gamma_{2}^{\ 2}/4}{(E_{\rm n}-E_{2})^{2}+\Gamma_{2}^{\
2}/4}$ $\displaystyle+\lambda_{3f}^{2}\sqrt{\frac{E_{3}}{E_{\rm
n}}}\frac{\Gamma_{3}^{\ 2}/4}{(E_{\rm n}-E_{3})^{2}+\Gamma_{3}^{\ 2}/4}$
$\displaystyle a_{1}$ $\displaystyle=$
$\displaystyle\lambda_{1f}\lambda_{2f}\left(\frac{\left|{E_{1}}\right|}{E_{2}}\right)^{\frac{1}{4}}$
$\displaystyle\times\frac{\Gamma_{1}\Gamma_{2}(E_{\rm n}-E_{1})(E_{\rm
n}-E_{2})+\Gamma_{1}^{\ 2}\Gamma_{2}^{\ 2}/4}{2\left((E_{\rm
n}-E_{1})^{2}+\Gamma_{1}^{\ 2}/4\right)\left((E_{\rm
n}-E_{2})^{2}+\Gamma_{2}^{\ 2}/4\right)}$
$\displaystyle\times\frac{5}{8}\left(-x+\sqrt{\frac{7}{5}}y\right)$
$\displaystyle+\lambda_{3f}\lambda_{2f}\left(\frac{E_{3}}{E_{2}}\right)^{\frac{1}{4}}$
$\displaystyle\times\frac{\Gamma_{3}\Gamma_{2}(E_{\rm n}-E_{3})(E_{\rm
n}-E_{2})+\Gamma_{3}^{\ 2}\Gamma_{2}^{\ 2}/4}{2\left((E_{\rm
n}-E_{3})^{2}+\Gamma_{3}^{\ 2}/4\right)\left((E_{\rm
n}-E_{2})^{2}+\Gamma_{2}^{\ 2}/4\right)}$
$\displaystyle\times\frac{3\sqrt{3}}{8}\left(x+\sqrt{\frac{5}{7}}y\right)$
$\displaystyle a_{3}$ $\displaystyle=$
$\displaystyle\lambda_{2f}^{2}\sqrt{\frac{E_{\rm
n}}{E_{2}}}\frac{\Gamma_{2}^{\ 2}/4}{(E_{\rm n}-E_{2})^{2}+\Gamma_{2}^{\
2}/4}\frac{33}{280}\left(-\sqrt{35}xy+y^{2}\right).$

It can be assumed that the energy dependence of the neutron width of the
$r$-th resonance is given as

$\displaystyle\Gamma_{r}^{\rm n}=(k_{\rm n}R)^{2l_{r}}\sqrt{\frac{E_{\rm
n}}{1\,{\rm eV}}}\Gamma_{r}^{{\rm n}l_{r}}$ (61)

for $E_{\rm n}>0$ where $R$ is the radius of target nuclei and
$\Gamma_{r}^{{\rm n}l_{r}}$ is the reduced neutron width. This energy
dependence is implemented as

$\displaystyle\Gamma_{r}^{\rm n}=\left(\frac{E_{\rm
n}}{\left|{E_{r}}\right|}\right)^{l_{r}+\frac{1}{2}}\overline{\Gamma_{r}^{\rm
n}},$ (62)

where $\overline{\Gamma_{\rm r}^{\rm n}}$ is a constant independent of the
energy . As the phase shift due to the optical potential is negligibly small,
each amplitude can be written as

$\displaystyle V_{rf}$ $\displaystyle=$
$\displaystyle-\lambda_{rf}\left(\frac{E_{\rm
n}}{\left|{E_{r}}\right|}\right)^{\frac{l_{r}}{2}-\frac{1}{4}}\frac{\Gamma_{r}/2}{E_{\rm
n}-E_{r}+i\Gamma_{r}/2}$ . (63)

where $\lambda_{rf}$ is defined as

$\displaystyle\lambda_{rf}=\frac{\hbar}{2}\sqrt{\frac{2g_{r}\overline{\Gamma_{r}^{\rm
n}}\Gamma_{rf}^{\rm\gamma}}{m_{\rm n}\left|{E_{r}}\right|\Gamma_{r}^{2}}}.$
(64)

$\Gamma_{rf}^{\rm\gamma}$ is the $\gamma$-width from the $r$-th resonance to
the final state.

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