# Measurements of 160Dy($p,\gamma$) at energies relevant for astrophysical
$\gamma$ process

Hao Cheng  School of Physics, and International Research Center for Big-Bang
Cosmology and Element Genesis, Beihang University, Beijing 100191, P. R. China
China Institute of Atomic Energy, Beijing 102413, P. R. China
Bao-Hua Sun  School of Physics, and International Research Center for Big-Bang
Cosmology and Element Genesis, Beihang University, Beijing 100191, P. R. China
Li-Hua Zhu  School of Physics, and International Research Center for Big-Bang
Cosmology and Element Genesis, Beihang University, Beijing 100191, P. R. China
Motohiko Kusakabe  School of Physics, and International Research Center for
Big-Bang Cosmology and Element Genesis, Beihang University, Beijing 100191, P.
R. China  Yun Zheng  China Institute of Atomic Energy, Beijing 102413, P. R.
China
Liu-Chun He  School of Physics, and International Research Center for Big-Bang
Cosmology and Element Genesis, Beihang University, Beijing 100191, P. R. China
Toshitaka Kajino  School of Physics, and International Research Center for
Big-Bang Cosmology and Element Genesis, Beihang University, Beijing 100191, P.
R. China  National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka,
Tokyo 181-8588, Japan  Graduate School of Science, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Zhong-Ming Niu  School of
Physics and Material Science, Anhui University, Hefei 230039, P. R. China
Tian-Xiao Li  China Institute of Atomic Energy, Beijing 102413, P. R. China
Cong-Bo Li  China Institute of Atomic Energy, Beijing 102413, P. R. China
Dong-Xi Wang  China Institute of Atomic Energy, Beijing 102413, P. R. China
Meng Wang  School of Physics, and International Research Center for Big-Bang
Cosmology and Element Genesis, Beihang University, Beijing 100191, P. R. China
Guang-Shuai Li  School of Physics, and International Research Center for Big-
Bang Cosmology and Element Genesis, Beihang University, Beijing 100191, P. R.
China  Kang Wang  School of Physics, and International Research Center for
Big-Bang Cosmology and Element Genesis, Beihang University, Beijing 100191, P.
R. China  Lin Song  School of Physics, and International Research Center for
Big-Bang Cosmology and Element Genesis, Beihang University, Beijing 100191, P.
R. China  Ge Guo  School of Physics, and International Research Center for
Big-Bang Cosmology and Element Genesis, Beihang University, Beijing 100191, P.
R. China  Zhi-Yong Huang  China Institute of Atomic Energy, Beijing 102413, P.
R. China
Xiu-Lin Wei  School of Physics, and International Research Center for Big-Bang
Cosmology and Element Genesis, Beihang University, Beijing 100191, P. R. China
Fu-WeI Zhao  School of Physics, and International Research Center for Big-Bang
Cosmology and Element Genesis, Beihang University, Beijing 100191, P. R. China
Xiao-Guang Wu  China Institute of Atomic Energy, Beijing 102413, P. R. China
Yimuran Abulikemu  School of Physics, and International Research Center for
Big-Bang Cosmology and Element Genesis, Beihang University, Beijing 100191, P.
R. China  Jian-Cheng Liu  China Institute of Atomic Energy, Beijing 102413, P.
R. China
Ping Fan  China Institute of Atomic Energy, Beijing 102413, P. R. China

(Received Month date, 2020; Revised Month date, 2020; Accepted Month date,
2020)

###### Abstract

Rare information on photodisintegration reactions of nuclei with mass numbers
$A\approx 160$ at astrophysical conditions impedes our understanding of the
origin of $p$ nuclei. Experimental determination of the key ($p,\gamma$) cross
sections has been playing an important role in verifying nuclear reaction
models and providing rates of relevant ($\gamma,n$) reactions in $\gamma$
process. In this paper we report the first cross-section measurements of
160Dy($p,\gamma$)161Ho and 161Dy($p,n$)161Ho in the beam energy range of 3.4 -
7.0 MeV, partially covering the Gamow window. Such determinations are possible
by using two targets with various isotopic fractions. The cross section data
can put a strong constraint on the nuclear level densities and gamma strength
functions for $A\approx$ 160 in the Hauser-Feshbach statistical model.
Furthermore, we find the best parameters for TALYS that reproduce the
available A $\thicksim$ 160 data, 160Dy($p,\gamma$)161Ho and
162Er($p,\gamma$)163Tm, and recommend the constrained 161Ho($\gamma,n$)160Dy
reaction rates over a wide temperature range for $\gamma$ process network
calculations. Although the determined 161Ho($\gamma$, p) stellar reaction
rates at the temperature of 1 to 2 GK can differ by up to one order of
magnitude from the NON-SMOKER predictions, it has a minor effect on the yields
of 160Dy and accordingly the $p$ nuclei, 156,158Dy. A sensitivity study
confirms that the cross section of 160Dy($p$, $\gamma$)161Ho is measured
precisely enough to predict yields of $p$ nuclei in the $\gamma$ process.

$\gamma$ process, Cross section, Activation $\gamma$-ray measurement, Hauser-
Feshbach statistical model, nucleosynthesis

##  1 Introduction

The $\gamma$ process (Woosley & Howard, 1978), also referred to as
$p$-process, was proposed as one of the most promising candidates for
producing more than 30 stable neutron-deficient isotopes in the region between
74Se and 196Hg. It is essentially made of photodisintegrations of the
($\gamma,n$), ($\gamma,n$), or ($\gamma,\alpha$) types by burning the
preexisting seed $s$-isotopes in stellar environments of high enough
temperature ($T_{9}\equiv T/(10^{9}~{}{\rm K})=[2,3]$) and subsequent
$\beta^{+}$-decay and electron capture, possibly complemented with captures of
neutrons, protons or $\alpha$ particles at center-of-mass energies typically
far below 1 MeV or the Coulomb barrier in the case of charged particles
(Arnould et al., 2003; Gyurky et al., 2003). Stable nuclei and nearby unstable
nuclei are located on pathways on nuclear chart during the $\gamma$ process.
The required temperature conditions are fulfilled in the oxygen-/neon-rich
layers of type II supernovae (Woosley & Howard, 1978; Rayet et al., 1990;
Prantzos et al., 1990; Rayet et al., 1995; Costa et al., 2000; Rauscher, 2001;
Hayakawa et al., 2004; Iwamoto et al., 2005; Hayakawa et al., 2008) of massive
stars or type Ia supernovae (Howard et al., 1991; Howard & Meyer, 1992;
Goriely et al., 2002, 2005; Arnould & Goriely, 2006; Kusakabe et al., 2011;
Travaglio et al., 2011) of intermediate mass stars.

A reliable modeling of $\gamma$ process flows involves typically an extended
network of about 2000 nuclei and 10,000 reactions for nuclei with masses up to
around 210 (Arnould et al., 2003). The nucleosynthesis develops in the
neutron-deficient region of the chart of nuclides (Howard et al., 1991; Rayet
et al., 1995; Kusakabe et al., 2011), whose masses and decay rates are mostly
well known experimentally. The reaction rates of such critical reactions, in
particular for the photodissociations, then represent the main uncertainty
from the nuclear physics point view in predicting the elemental abundances
produced in $\gamma$ process (Rauscher et al., 2016; Nishimura et al., 2018).

Despite decades of considerable experimental effort, the scarcity of the
experimental reaction rates for $\gamma$ process makes it mandatory to rely
heavily on the Hauser-Feshbach (HF) statistical model (Hauser & Feshbach,
1952; Rauscher et al., 1997) and the various nuclear ingredients in such a
framework. In reality, it remains a challenge to validate the reliability of
the HF model and its inputs, and accordingly to put the $\gamma$ process
calculations on a more reliable base. Therefore, providing crucial new
experimental cross sections relevant to $\gamma$ process will play a key role
(see, e. g., Utsunomiya et al. (2003, 2006); Dillmann et al. (2006, 2008,
2010); Kiss et al. (2007); Mohr et al. (2007); Netterdon et al. (2014); Güray
et al. (2015); Larsen et al. (2016); Harissopulos et al. (2016); Wu et al.
(2020).) Experimentally, most of the photodissociations in the $\gamma$
process nucleosynthesis studies have been so far derived from the reverse
radiative capture on the basis of the reciprocity theorem.

Experiments in the $A\sim$ 160 mass region are particularly rare due to the
extremely low cross section at the energies of Gamow window and the difficulty
in target material purifications. The $\gamma$ nuclides in this mass region
are most efficiently produced in the stellar region which experiences peak
temperature of $T_{9}=$ 2.6–2.85 (Kusakabe et al., 2011). 162Er($p,\gamma$),
the only data available in this range was studied recently using the
activation method (Özkan et al., 2017).

Strong constraints for reaction models are helpful in improving and
guaranteeing reliable results of the $\gamma$ process calculations. The
present paper describes the first measurements of 161Dy($p,n$)161Ho and
160Dy($p,\gamma$)161Ho, partially covering the Gamow window in the $\gamma$
process. They are used to determine the ($\gamma$, $p$) rates to confirm
predictability of existing models for reaction cross section and obtain
realistic theoretical results of $\gamma$ process. The measurements and the
data analysis are described in Section 2. In Section 3, we compare the data
with TALYS (Koning, A. J. & Rochman, D., 2012) calculations obtained with
different level density, gamma strength function, optical model, compound
nucleus and pre-equilibrium. Moreover, we find the best calculation that can
reproduce the experimental data available in the $A=160$ range, and derive the
stellar photodisintegration rate for 160Dy of relevance for nucleosynthesis
calculations, and a sensitivity calculation is made. Final conclusions are
drawn in Section 4.

##  2  Experiment and data analysis

The experiment was performed with the activation method. This method is
suitable for radioactive reaction products with half-life between several
years and tens of minutes. One counts the number of radioactive products via
the activity after the irradiation took place. This method is free of prompt
beam-induced background at the irradiation, and avoids the details of decay
branching or angular correlation effects of $\gamma$ transitions in the prompt
$\gamma$ measurement.

###  2.1 Target preparation

Natural dysprosium is composed of seven stable isotopes with mass numbers 156,
158, 160, 161, 162, 163 and 164. 156Dy and 158Dy have only tiny natural
abundances and reactions on them are out of reach in the current experiment
conditions. Of all the ($p,\gamma$) products, 165Ho is stable and 163Ho has a
half-life of 4570(20) years thus is too long to be studied. At 1.65 MeV proton
bombarding energy the 161Dy($p,n$)161Ho reaction channel opens, which results
in the same final nucleus as 160Dy($p,\gamma$)161Ho. The threshold energies
for the 162Dy($p,n$)162Ho, 163Dy($p,n$)163Ho, and 164Dy($p,n$)164Ho are 2.94,
0.79, and 1.78 MeV, respectively. Hence using natural or enriched but not 100%
abundance isotope Dy targets above the ($p,n$) reaction thresholds, ($p,n$)
and ($p,\gamma$) reaction channels cannot be distinguished, and the resulting
cross section is the weighted sum of the two cross sections.

Table 1: Isotopic abundances in the natural Dy (${}^{\textrm{nat}}$Dy) and the enriched 160Dy targets (160Dy2O3). Type | Material | Isotopic Composition |
---|---|---|---
1 |  ${}^{\textrm{nat}}$Dy |  160Dy(2.34%), 161Dy(18.91%), 162Dy(25.51%), 163Dy(24.90%), 164Dy(28.18%) |
2 |  160Dy2O3 |  160Dy(51.82%), 161Dy(13.87%), 162Dy(5.79%), 163Dy(3.05%), 164Dy(1.68%) |

One way to extract the ($p,\gamma$) cross sections is to employ two kinds of
targets with different isotopic abundances. By coupling the ($p,\gamma$) and
($p,n$) channels to the same radioactive nuclide, the individual cross section
can be deduced in the case of good statistics. The information of two targets
used is summarized in Table 1, one with natural Dy and the other with 160Dy-
enriched abundance. With the abundance distributions in two targets, we can
determine the cross sections for the two reactions 160Dy($p,\gamma$)161Ho and
161Dy($p,n$)161Ho.

The thicknesses of natural targets are around 1-2 mg cm-2 and those of the
enriched targets are 2 mg/cm-2 as displayed in Table 2. These targets are
ellipses with an axis length of 9 mm $\times$ 12 mm, and the sizes of target
frames are 25 mm $\times$ 12.5 mm. The self-supported natural dysprosium
targets were made by rolling method. Enriched 160Dy (51.82%) targets were
sputtered onto a 197Au backing. The mass abundances of contamination elements
are determined typically to be less than 0.1% and 1% for the natural targets
and enriched targets by the energy-dispersive X-ray spectroscopy,
respectively.

###  2.2 Experiment Setup

The investigated energies in this work ranging from 3.4 to 7 MeV are above the
relevant ($p,n$) reaction thresholds. These energies partially cover the Gamow
window for the $\gamma$ process in this mass region for $T_{9}$ = [2, 2.9].

Schematic drawing of the target chamber. Refer to the text for details.

The proton beams were delivered by 2$\times$1.7 MeV and H1-13 tandem
accelerator at China Institute of Atomic Energy, Beijing, respectively. The
maximum energy provided by the 2$\times$1.7 MeV tandem accelerator and the
minimum energy that can be steadily and intensely provided by the H1-13 tandem
accelerator are 3.4 MeV and 5 MeV, respectively. The proton current throughout
the irradiations was between 150 nA and 700 nA for different beam energies.

A schematic drawing of the target chamber is shown in Figure 2.2. The target
chamber also servers as a Faraday cup with a $\Phi$5 mm-hole collimator and
insulated from the beam tube. The distance from the hole collimator to the
target surface is 20 cm. A suppression voltage of -300 volts was applied to
suppress secondary electrons emitted from the target. A water-cooling system
for the target chamber was used to maintain the temperature and minimize
surface deformation of the target. The beam current was digitized with a
frequency of 1 Hz by the Keithley 6517B electrometer.

Figure 2.2 shows the beam current during the beam time. The variance in beam
currents $I(t)$ can bring an up to 35% difference in the cross section when
considering the maximum and minimum beam intensity. This indicates the
necessity of recording the precision current with time.

Variation of the beam current during the beam time.

The effective center-of-mass energy ($E_{\text{c.m.}}$) for each run is
calculated by considering the proton energy loss $\Delta E$ in the target
material and follows

$E_{\text{c.m.}}=\frac{M_{A}}{M_{p}+M_{A}}(E_{p}-\frac{\Delta E}{2})\;,$ (1)

where $M_{A}$ and $M_{p}$ are the mass of the target nuclide and the proton,
$E_{p}$ is the proton energy. Energy losses are calculated using ATIMA, a
built-in Physical Calculator in LISE++ (Tarasov & Bazin, 2008). The energies
used in the experiments are summarized in Table 2. The relevant uncertainties
include the energy straggling and the difference caused by using different
thickness of targets.

The targets were irradiated for 30-60 minutes (irradiation time) and then
followed by 30-1000 minute measurements (measurement time), while in between
there were 10 to 20 minutes (waiting time) to release the vacuum, dismount the
target and place the target in the position for off-line measurement. The
details of measurement are displayed in Table 2.

Table 2: Details of the Measurements. Target and it thickness (in units of mg cm-2), and the center-of-mass energy $E_{\textrm{c.m.}}$ (in units of MeV), the relevant irradiation time (in units of seconds), waiting time (in units of seconds), measurement time (in units of seconds) are listed.  Target | Thickness | $E_{\textrm{c.m.}}$ | Irradiation Time | Waiting Time | Measurement Time
---|---|---|---|---|---
| (mg cm-2) | (MeV) | (s) | (s) | (s)
${}^{\textrm{nat}}$Dy | 1.92(10) | 3.34(8) | 2484 | 620 | 3311
160Dy2O3 | 1.85(19) | 3.33(9) | 2157 | 796 | 55297
${}^{\textrm{nat}}$Dy | 1.92(10) | 4.94(6) | 3524 | 745 | 4742
160Dy2O3 | 2.13(21) | 4.92(8) | 3767 | 814 | 9280
${}^{\textrm{nat}}$Dy | 1.72(9) | 5.44(5) | 3551 | 1046 | 4725
160Dy2O3 | 1.92(19) | 5.42(7) | 3597 | 1172 | 9555
${}^{\textrm{nat}}$Dy | 0.96(5) | 5.95(3) | 2108 | 1084 | 2776
160Dy2O3 | 2.13(21) | 5.92(7) | 2059 | 1011 | 7973
${}^{\textrm{nat}}$Dy | 1.07(5) | 6.45(3) | 1757 | 1175 | 3307
160Dy2O3 | 1.60(16) | 6.43(7) | 2059 | 934 | 7714
${}^{\textrm{nat}}$Dy | 0.92(5) | 6.95(2) | 1182 | 840 | 2608
160Dy2O3 | 1.85(19) | 6.92(5) | 1528 | 832 | 2165

The $\gamma$ radiations following the electron-capture decays of the produced
Ho isotopes were measured with a CLOVER detector in a low background shielding
system as shown in Figure 2.2. The CLOVER consists of four coaxial $N$-type
high-purity Germanium detectors, each with a diameter of 60 mm and a length of
60 mm. The energy resolution for the CLOVER is 2.2 keV (FHWM) for the 1.332
MeV $\gamma$ rays of 60Co. The relative efficiency is 38% for each Germanium
crystal. The shielding system is a cylinder with a radius of 64 cm and a
height of 66.1 cm. It consists of layers of iron, lead, copper, and plexiglass
from the outside to the inside. The lead layer is used to shield most of the
low energy environment background, and the copper layer aims to absorb the
characteristic X-rays of lead. This system has been used to investigate the
intrinsic radiation background of a B380 LaBr3(Ce) detector (Cheng et al.,
2020).

Schematic drawing of the $\gamma$ activity setup. It contains a CLOVER crystal
and a low-background shielding room. The shielding room is composed of
plexiglass, copper, lead, and iron from the inside to the outside. Their
relevant layer thicknesses are 5 mm, 2 mm, 84 mm and 20 mm, respectively. The
irradiated targets were placed 3 cm away from the front surface of the
detector.

The irradiated targets were placed 3 cm away from the front surface of the
detector. The absolute efficiency was calibrated with one 152Eu and six
monoenergetic sources (137Cs, 241Am, 54Mn, 88Y, 104Cd, and 65Zn). The sources
were placed at the same position as the targets. The absolute full-energy peak
efficiency curve was described using the EFFIT program in the Radware
(Radford, 1995) package. The efficiency curve was further corrected for the
summing coincidence effect by a dedicated GEANT4 simulation (He et al., 2018).
The impact of geometric acceptance due to the size of irradiation target, is
estimated to be less than 1%. This effect is much smaller than the statistic
error.

Signals of the CLOVER detector were recorded by the VME data acquisition
system (DAQ). The timestamp of each event was digitized in a precision of 10
ms. This information is useful to determine the decay curve of a specified
$\gamma$ activity and can be used as an independent check of the radioactive
product. Dead time correction was also added in DAQ for the absolute counting.

###  2.3 Cross-section Determination

There is a low-lying 1/2+ isomeric state with a half-life of 6.67 (7) s in
161Ho. Nevertheless, this isomer does not affect the determination of cross
sections because its half-life is much shorter than the irradiation time of
typically more than half an hour. All the isomeric states would have decayed
to the ground state by isomeric transition (IT) in the off-line activity
measurement.

Figure 1: Activation $\gamma$-ray spectra below 1500 keV taken from the
measurements using the natural Dy target (a) and the enriched 160Dy target (b)
at the proton beam energy of 6.5 MeV. Labeled are the characteristic
$\gamma$-rays from 160Ho (asterisk), 161Ho (circles), 162Ho (triangles), and
164Ho (box) , and the energies marked by C are contaminants. The X-rays from
Ho, Dy, and Au at the low energy are also visible.

Figure 1 shows the typical activation $\gamma$-ray spectra below 1500 keV
taken from the natural Dy and enriched Dy targets irradiated by 6.5 MeV
protons. We identified the characteristic $\gamma$-rays at 103 keV from 161Ho,
the $\gamma$-rays at 197, 728, 879, etc., from 160Ho, the $\gamma$-rays at
185, 283, 937, 1220, 1320, and 1373 keV from 162Ho, and the $\gamma$-ray at 91
keV from 164Ho. The 103-keV $\gamma$ transition with a relative intensity of
3.9% per decay of 161Ho was used for data analysis. The purity of the $\gamma$
ray is verified by examining its time evolution. The deduced half-life of
148.5(24) minutes is in a good agreement with 148.8(30) minutes reported in
Ref. (Reich et al., 2011)

The count of the 103 keV $\gamma$ peak was analyzed using the Radware package
(Radford, 1995) and corrected with the dead time of DAQ. Decay parameters of
Ho isotopes, the decay constant ($\lambda$) and the relative intensity
($\eta_{\gamma}$) of the characteristic $\gamma$ ray, are from the Nuclear
Data Sheets (NDS; (Singh & Chen, 2014)). By decoupling the total counts of
characteristic $\gamma$ rays measured in both types of targets at the same
proton energies, we can deduce the cross sections for 160Dy($p,\gamma$)161Ho
and 161Dy($p,n$)161Ho. The derivation of cross section from the activation
$\gamma$ rays is described in detail in Section appendix .

##  3 Results and discussion

Table 3: Determined cross sections and astrophysical $S$ factors at the relevant Center-of-mass energy $E_{\text{c.m.}}$.  $E_{c.m.}$ | $\sigma_{{}^{160}\textrm{Dy}(p,\gamma)^{161}\textrm{Ho}}$ |  $S$ Factor | $\sigma_{{}^{161}\textrm{Dy}(p,n)^{161}\textrm{Ho}}$
---|---|---|---
(MeV) | (mb) | (MeV b) | (mb)
3.34(6) |  3.25(58)$\times 10^{-3}$ |  3.22$\times 10^{10}$ |  3.72(70)$\times 10^{-3}$
4.93(5) | 0.106(18) |  2.84$\times 10^{9}$ | 0.60(11)
5.43(4) | 0.152(25) |  1.09$\times 10^{9}$ | 1.60(30)
5.94(4) | 0.243(40) |  5.72$\times 10^{8}$ | 4.47(84)
6.44(3) | 0.310(51) |  2.79$\times 10^{8}$ | 9.15(17)
6.93(3) | 0.507(84) |  1.92$\times 10^{8}$ | 19.4(36)

The cross sections of 160Dy($p,\gamma$)161Ho and 161Dy($p,n$)161Ho are
summarized in Table 3. The uncertainties in cross sections come from the
propagation of errors in $\gamma$-ray counts (0.6%$\thicksim$6.4%), gamma
intensity from NDS (15%), half-life of characteristic $\gamma$-rays (2%),
target thicknesses (5% for the natural target and 10% for the enriched
target), and $\gamma$ detection efficiencies (3%). The total uncertainties are
dominated by the relative poor precision in $\gamma$ intensity. The cross
sections caused by the difference in the thicknesses of the natural and
enriched targets is less than 5%. This effect is taken into account by
introducing the uncertainties of proton energies. Moreover, as an independent
examination of our experimental setup and data analysis, we performed a cross-
section measurement of 63Cu($p,n$) at a proton energy of 9.2 MeV. The target
consisted of a 1.1 mg/cm-2 foil of natural Cu with a backing of 6.14 mg/cm-2
thick 197Au. The deduced cross section of 0.38 (3) mb is consistent with 0.367
(27) mb at 9.2 MeV reported by Colle et al. (1974).

The experimental cross sections are compared to theoretical calculations using
the codes NON-SMOKER (Rauscher et al., 2001), TALYS (Koning, A. J. & Rochman,
D., 2012) and EMPIRE (Herman et al., 2007) in Figure 3. The NON-SMOKER
calculations are widely adapted in current network calculations and have been
included in the JINA REACLIB database (Cyburt et al., 2010). The TALYS and
EMPIRE calculations were performed using the default parameters. As for the
($p,\gamma$) reaction, all three calculations can reproduce the 3.34 MeV data
and agree with each other nicely at the energy up to around 4 MeV, the neutron
threshold, which corresponds to the excited state near the neutron separation
energy of 161Ho. However, deviations present above the neutron threshold,
particularly for the NON-SMOKER results, which overpredict the cross sections
at 5 $\thicksim$ 7 MeV by a factor of 3-6. Moreover, NON-SMOKER predicts a
kink point at around 5.5 MeV, which seems to be different from the
experimental data. The EMPIRE calculations reproduce nicely the ($p,n$) data
and the 3.34 MeV ($p,\gamma$) data, but underestimate the experimental
centroid values by a factor of 2 at 5 $\thicksim$ 7 MeV although they are
generally consistent within the experimental uncertainties. Overall, a good
agreement between the data and the TALYS calculations with default input
parameters is observed for both reactions at the energies investigated.
However, we note that the calculated cross section of
162Er$(p,\gamma)^{163}$Tm using the same default parameters is two times
smaller than the experimental data at 6.5 to 9 MeV (Özkan et al., 2017).

Measured cross sections of the 160Dy($p,\gamma$)161Ho (solid square) and
161Dy($p,n$)161Ho (open square) reactions compared with the predictions by the
NON-SMOKER (solid line), EMPIRE (dotted line) code, and the TALYS (dashed
line) codes using their default parameters. The relevant Gamow window for
($p,\gamma$) is indicated by the shadowed area. The threshold for
161Dy($p,n$)161Ho is 1.65 MeV.

###  3.1 Constraining the Hauser-Feshbach model

The key ingredients in the Hauser-Feshbach calculations (Rauscher et al.,
2013) include $\gamma$ strength functions (GSF), optical model potential
(OMP), the nuclear level densities (NLD) and mass models. TALYS offers a
variety of options for the description of these inputs and is used hereafter
to make the sensitivity studies (Mei et al., 2015; Rapp et al., 2006).

In Figure 3.1(a), the ($p,\gamma$) experimental results are compared with the
predictions by the TALYS-1.9 code utilizing different NLD models, while all
the other inputs are kept to be the same as the default ones. Using different
NLDs gives similar trends, so we present the selective calculations including
the one with the default parameters as well as those of the lowest and highest
cross sections, as limits of these calculations. The default NLD in TALYS is
the Constant temperature $+$ Fermi gas (CTFG) level density (Gilbert et al.,
1965). It sets the lower limit of cross sections, while the upper limit is
obtained by using the microscopic NLD from Hilaire’s table (HT) (Hilaire et
al., 2008), As shown in Figure 3.1(a), the ($p,\gamma$) cross section is
sensitive to the NLDs from about 4 MeV on, and has a less significant impact
in the Gamow window, i.e., the low energy data. Variation of the level density
modes could result in a factor of 4.5 difference in cross section at 7 MeV.
Our data are close to the predicted lowest limit, and thus provide an direct
constraint for the NLD and rule out many NLDs in this mass range.

(a) Comparison of the measured 160Dy($p,\gamma$)161Ho cross sections (squares)
with the predictions of TALYS using different level density models while
keeping the other inputs as the default ones. The solid line represents the
optimized calculations for $A\thicksim$ 160\. (b) Same as (a) but using
different $\gamma$ strength functions. Refer to the text for details.

Similarly, we present TALYS predictions using three GSFs in Figure 3.1(b). The
default GSF in the TALYS is the traditional standard Lorentzian model of
Kopecky-Uhl generalized Lorentzian (KUGL) (Kopecky et al., 1990). The lowest
and highest cross sections are obtained by using the KUGL and the Gogny D1M
HFB+QRPA models (Goriely et al., 2018), respectively. Generally, the
($p$,$\gamma$) cross section depends sensitively on both GSF and NLD modeling.

Table 4: Deduced Stellar Rate (s-1) of the 161Ho($\gamma$, $p$)160Dy in Temperatures $T_{9}$ = [0.1,10].  $T_{9}$ | Rate | $T_{9}$ | Rate
---|---|---|---
0.10 |  6.89$\times 10^{-279}$ | 1.50 |  3.41$\times 10^{-12}$
0.15 |  8.16$\times 10^{-279}$ | 2.00 |  7.45$\times 10^{-06}$
0.2 |  3.17$\times 10^{-279}$ | 2.50 |  6.78$\times 10^{-02}$
0.3 |  1.09$\times 10^{-95}$ | 3.00 |  3.22$\times 10^{+01}$
0.4 |  2.09$\times 10^{-71}$ | 4.00 |  6.46$\times 10^{+04}$
0.5 |  2.59$\times 10^{-56}$ | 5.00 |  5.18$\times 10^{+06}$
0.6 |  6.05$\times 10^{-46}$ | 6.00 |  9.81$\times 10^{+07}$
0.7 |  2.43$\times 10^{-38}$ | 7.00 |  9.71$\times 10^{+08}$
0.8 |  1.68$\times 10^{-32}$ | 8.00 |  6.07$\times 10^{+09}$
0.9 |  7.38$\times 10^{-28}$ | 9.00 |  2.41$\times 10^{+10}$
1.00 |  4.54$\times 10^{-24}$ | 10.00 |  6.39$\times 10^{+10}$

We also examined the ($p,\gamma$) cross sections by using various compound
models, pre-equilibrium models and optical models in TALYS. They have only a
minor effect on the computed cross sections. The optical potential is expected
to be important at the astrophysically relevant low energies for charged
particle captures and photodisintegrations. The one and only data point at
3.34 MeV can already help to exclude several models.

Further, it is worth noting that nuclear reaction calculations involve several
(sub)models to compute the key input quantities like level density, $\gamma$
strength function, optical potential, nuclear masses, pre-equilibrium, and the
compound process. Their interplay makes the model evaluation almost
impossible. Instead of developing a consistent reaction model, we attempt to
find the best model parameters of TALYS, which can well describe the known
experimental data. We are aware that the parameters might be mass range
dependent or only hold to a certain precision in applying to nuclei nearby;
therefore we concentrate on the mass range of 160. This strategy is
practically useful to reduce the nuclear physics inputs in $\gamma$ process
simulations (Rauscher et al., 2016; Nishimura et al., 2018). We thus made a
Monte Carlo calculation by randomly varying the key physical input parameters
(Koning et al., 2019), and employed the least-square method to find the
calculation that matches best both the 160Dy($p,\gamma$)161Ho and
162Er($p,\gamma$)163Tm data. The results are shown in Figure 3.1.

###  3.2 Evaluation of the Stellar Rate

The TALYS results constrained by the present experimental data, as presented
in the previous section, represent a solid basis for the reaction rate
calculation. We thus employ the optimized parameters to compute the
astrophysical rate of 161Ho($\gamma,n$). It is obtained by integrating the
cross section over a Maxwell-Boltzmann distribution of energies $E$ at the
given temperature $T$. In addition, the thermal excitation effect in hot
astrophysical plasma is taken into account.

The calculated stellar reaction rates sharply depend on the temperature and
span over 250 order of magnitudes in the temperature range of $T_{9}$ =
[0.1,1.0]. The rates are summarized in Table 4. Figure 3.2 shows and the
ratios to the NON-SMOKER predictions in the temperature of $T_{9}$ = [0.1,10].
The theoretical uncertainty (Koning et al., 2019) is shown by the shadowed
band. The ratio is up to one order of magnitude, and NON-SMOKER underproduced
the rates at $T_{9}$ = [1, 2]. The difference would affect the yield of 160Dy
in the $\gamma$ process and will be discussed in the next subsection.

The ratio of the deduced reaction rates (red solid line) to the NON-SMOKER
predictions in the temperature of $T_{9}=[0.2,10]$ . The shadowed area
represents the theoretical uncertainty.

###  3.3 Nucleosynthesis Calculation for $A\sim$ 160 $p$ nuclei

As a possibly important region for the $\gamma$ process of $A\sim 160$ region,
we tested trajectories with peak temperatures $T_{9p}=[2,2.9]$ and peak
density $\rho_{\rm p}=10^{6}$ g cm-3 assuming the $\gamma$ process layer in
Supernovae (SNe) Ia. They are assumed to decrease exponentially as
$T_{9}=T_{9p}\exp[-t/(3\tau)]$ and $\rho=\rho_{\rm p}\exp(-t/\tau)$ with
$\tau=1$ s. The initial nuclear composition is adopted from Case A1 of
Kusakabe et al. (2011) which is based on an efficient $s$-process during
thermal pulses in the presupernova stage. The nucleosynthesis calculation is
then performed with a variable-order network code (Kusakabe et al., 2006)
which is updated with new theoretical estimates on the rates of
16O($n$,$\gamma$)17O (He et al., 2020) and 17O($n$,$\gamma$)18O (Zhang et al.,
2021).

In the SNe $\gamma$ process environment, a sudden heating followed by a nearly
adiabatic cooling occurs. The heating is triggered either by a deflagration or
detonation wave in thermonuclear explosion corresponding to SNe Ia, or a shock
in gravitational collapse type explosion corresponding to SNe II. Then, early
in the heating epoch, first neutron source nuclei are burned to proceed with
the neutron-capture process. As a result, neutron-rich nuclei are produced
from seed $s$ nuclei. As the temperature increases further, the ($\gamma$,
$n$) reactions start operating, and abundant nuclei shift from neutron-rich
nuclei to neutron-deficient nuclei. If the peak temperature is high enough,
($\gamma$, $p$) and ($\gamma$, $\alpha$) reactions also work and heavy nuclei
are disintegrated to lighter nuclei (e.g., Kusakabe et al., 2011). We estimate
a sensitivity of nuclear yields to the reaction rate of
161Ho($\gamma,n$)160Dy. Any effect of changing the 161Ho($\gamma,n$)160Dy rate
always occurs via changes of 161Ho and/or 160Dy abundances in the nuclear
reaction network in the $\gamma$ process. We confirm that the current rate
determination indicates that the possible uncertainty in the reaction cross
section is small enough that it does not remarkably affect evolution of 161Ho
or 160Dy abundances in the $\gamma$ process. Accordingly, the change of the
reaction rate does not affect yields of nuclei with $A<160$ produced through
the pathway of 161Ho($\gamma,n$)160Dy.

Figure 2 (a) shows time evolution of abundances on the test trajectory with
$T_{9p}=3$. The temperature is rather high, and first neutrons are released
shortly. Instantaneous neutron captures make a flow of nuclear abundances to
the neutron-rich region on the nuclear chart at $t\sim 10^{-7}$ s, and
abundances of 159,160Dy and 161Ho decrease. After the neutron abundance
decreases, successive ($\gamma,n$) reactions move back the flow toward the
proton-rich region. Then, abundances of 159,160Dy and 160,161Ho increase at
$t\sim 10^{-5}$–$10^{-4}$ s and eventually decrease due to the
photodisintegration into a more proton-rich and low $A$ region. Figure 2 (b)
shows fractional differences in the abundances between the standard case and
the case of the JINA REACLIB rate for 161Ho($\gamma,n$)160Dy. Each band
bounded by two lines delineates a variation caused by the uncertainty in the
evaluated rate. Variations of ${\mathcal{O}}(1)$ % are seen when the
photodisintegration is effectively destroying those nuclei. The smaller
161Ho($\gamma,n$)160Dy rate in our standard case at $T_{9}=3$ increases
abundances of 161Ho which survives that reaction and 160Ho produced via
161Ho($\gamma,n$)160Ho. Also, the 160Dy abundance slightly decreases due to
the hindered production via the 161Ho($\gamma,n$)160Dy. The uncertainty in the
current rate corresponds to at most ${\mathcal{O}}(0.1)$ % differences in
abundances.

Figure 2: (a) Time evolution of nuclear mole fractions on the test trajectory
with $T_{9p}=3$. (b) Fractional differences in the mole fractions between the
standard case and the case of JINA REACLIB rate for 161Ho($\gamma,n$)160Dy.
Each narrow band that is almost invisible corresponds to a variation caused by
the uncertainty in the evaluated rate.

Figure 3: (a) Mole fractions of stable nuclei with mass number $A\lesssim 160$
in the $\gamma$ process in SN Ia as a function of the peak temperature
$T_{9p}$. (b) Fractional differences in the final mole fractions in the case
with the 161Ho($\gamma,n$)160Dy rate enhanced by a factor of 100 compared to
the standard case.

The effect of the change in the 161Ho($\gamma,n$)160Dy rate is small even at
$t\sim 10^{-5}$–$10^{-4}$ s when the photodisintegration operates
predominantly. This is because the 161Ho($\gamma,n$) rate is much larger than
the ($\gamma,n$) rate for $T_{9}\geq 2$ relevant to the $\gamma$ process. At
such high temperatures of $T_{9}\lesssim 3$, all heavy seed nuclei are
disintegrated, and yields are zero (Figure 3). At lower temperatures, although
the changes in abundances could remain in the final abundances, those are
constrained to be at ${\mathcal{O}}(0.1)$ % level.

Figure 3 (a) shows final yields of stable nuclei with mass numbers $A\lesssim
160$ in the $\gamma$ process in SN Ia as a function of $T_{9p}$. Since those
nuclei are located immediately downstream of the pathway
161Ho($\gamma,n$)160Dy, changes are expected in abundances of those nuclei if
that reaction is important in determining the final abundances. As a standard
case, we adopt the rate for 161Ho($\gamma,n$)160Dy derived from the 160Dy Best
parameters. Figure 3 (b) shows fractional differences in the final yields
between the case with the 161Ho($\gamma,n$)160Dy rate 100 times as large as
the adopted rate and the standard case. Even when the rate is thus extremely
large, the yields vary by at most ${\mathcal{O}}(10)$ %. Since the uncertainty
in the 161Ho($\gamma,n$)160Dy rate is estimated to be much smaller (Figure
3.2), it is found that the current determination is sufficiently precise to
obtain accurate results of the $\gamma$ process.

##  4 Conclusion

In this work we determined for the first time the cross sections of
160Dy($p,\gamma$)161Ho and 161Dy($p,n$)161Ho in the energy range of 3.4 to 7.0
MeV using the activation techniques. This measurement extends the scarce
experimental database for charged-particle-induced reactions on neutron-
deficient nuclei.

The measured cross sections were compared to statistical model calculations
obtained from the widely used Hauser-Feshbach codes: NON-SMOKER, EMPIRE, and
TALYS. Using the default input parameters it was found that cross sections for
both reaction channels agreed with TALYS theoretical prediction. The results
allowed us to constrain strongly the $\gamma$ strength function and nuclear
level density used in the HF statistical model.

With the TALYS code, we find the optimized TALYS input for the $A\sim$ 160
mass range, and further recommend the stellar rates for 161Ho($\gamma,n$)160Dy
over a large temperature range for $\gamma$ process network calculations. This
approach is practically useful to compute the reaction rates in short-range
extension to the experimentally known region. We would like to point out that
the large uncertainties in the weak $\gamma$ intensity of interest in Dy
isotopes, impedes the improvement of accuracy in cross-section determinations.
A dedicated experiment to further improve the gamma intensity will help to
reduce the uncertainty.

Adopting the derived rate for the photodisintegration 161Ho($\gamma,n$)160Dy,
evolutions of nuclear abundances around $A=160$ in the $\gamma$ process were
analyzed. We conclude that employing the determined 161Ho($\gamma,n$)160Dy
rate does not affect the yields of nuclei with $A\sim 160$. Moreover, a
sensitivity study shows that the present experimental cross section is precise
enough for the $\gamma$ process calculations.

The low yields make the proton or $\alpha$ irradiation measurements relevant
for $\gamma$ process very time consuming. A possible way to improve this is to
use the stacked target activation method (e.g., (Morrell et al., 2020) for
measurements of cross sections at energies lower than the provided beam
energy. If there are $\gamma$ activity setups available, simultaneous
measurements of cross sections at a series of energies would be possible.

Many thanks to Dr. B. Mei for helpful discussions on the reaction rates. This
work was supported in part by the National Natural Science Foundation of China
(Nos.11575018, U1867210, U1832211, 11961141004, 11922501, and 11790322) and by
the National Key R & D program of China (No. 2016YFA0400504).

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##  Appendix A  The derivation of cross section

The activation method is constituted by the irradiation and the residual
measurements of the experimental target. The rate of change in the number of
radioactive nuclei is given by the difference of production and decay rate,

$\frac{dN(t)}{dt}=\sigma(E)N_{s}I(t)-\lambda N(t)\;,\qquad 0<t<t_{b}$ (A1)

where $N(t)$ and $\lambda$ are the number and the decay constant of the object
nucleus, $\sigma(E)$ the cross section of the reaction at the bombarding
energy $E$, $N_{s}$ the number of target nuclei, and $I(t)$ is the beam
intensity at a time $t$. $t_{b}$ is the irradiation time.

The number of reaction products is:

$N(t_{b})=N_{s}\sigma(E)e^{-\lambda t_{b}}\int_{0}^{t_{b}}I(t)e^{\lambda
t}dt\;,$ (A2)

In the present work, a waiting time $t_{w}$ was needed to release the vacuum,
dismount the target and place the target in the position for off-line
measurement. Then the targets were measured for $t_{m}$ (measurement time).
The decayed $\gamma$ rays emitted from the target is thus

$n_{\gamma}=N(t_{b})e^{-\lambda t_{w}}(1-e^{-\lambda
t_{m}})\varepsilon_{\gamma}\eta_{\gamma}\;,$ (A3)

where $\varepsilon_{\gamma}$ and $\eta_{\gamma}$ are the detection efficiency
and the gamma intensity, respectively.

Deduced from Eq. A2 and Eq. A3, the cross section of the reaction,
$\sigma(E)$, is

$\sigma(E)=\frac{n_{\gamma}}{N_{s}e^{-\lambda t_{b}}e^{-\lambda
t_{w}}(1-e^{-\lambda
t_{m}})\varepsilon_{\gamma}\eta_{\gamma}\int_{0}^{t_{b}}I(t)e^{\lambda
t}dt}\;,$ (A4)

For the special case of a constant flux, $I(t)$ = $I_{0}$, the above equations
can be solved analytically. Eq. A4 can be rewritten to

$\sigma(E)=\frac{\lambda n_{\gamma}}{N_{s}I_{0}e^{-\lambda
t_{w}}(1-e^{-\lambda t_{m}})(1-e^{-\lambda
t_{b}})\varepsilon_{\gamma}\eta_{\gamma}}\;$ (A5)