rotate DTBLrotate

11institutetext:  Department of Physics and Astronomy, Aarhus University, Ny
Munkegade 120, DK-8000 Aarhus C, Denmark
(11email<EMAIL_ADDRESS>22institutetext: Carnegie Observatories, Las
Campanas Observatory, Casilla 601, La Serena, Chile 33institutetext:
Observatories of the Carnegie Institution for Science, 813 Santa Barbara St.,
Pasadena, CA 91101, USA 44institutetext: Facultad de Ciencias Astronómicas y
Geofísicas, Universidad Nacional de La Plata, Paseo del Bosque S/N, B1900FWA,
La Plata, Argentina 55institutetext: Instituto de Astrofísica de La Plata
(IALP), CCT-CONICET-UNLP. Paseo del Bosque S/N, B1900FWA, La Plata, Argentina
66institutetext: Kavli Institute for the Physics and Mathematics of the
Universe(WPI), The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba
277-8583, Japan 77institutetext: Fundación Chilena de Astronomía, El Vergel
2252 #1501, Santiago, Chile 88institutetext: Hagler Institute for Advanced
Studies, Texas A&M University, College Station, TX 77843, USA 99institutetext:
DTU Space, National Space Institute, Technical University of Denmark,
Elektrovej 327, 2800 Kgs. Lyngby, Denmark 1010institutetext: George P. and
Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy,
Department of Physics and Astronomy, Texas A&M University, College Station, TX
77843, USA 1111institutetext: European Southern Observatory, Alonso de Córdova
3107, Casilla 19, Santiago, Chile 1212institutetext: Institute for Astronomy,
University of Hawai’i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
1313institutetext: Department of Physics and Astronomy, University of
Oklahoma, 440 W. Brooks, Rm 100, Norman, OK 73019 1414institutetext: Aix
Marseille Univ, CNRS, CNES, LAM, Marseille, France 1515institutetext:
Department of Physics, Florida State University, 77 Chieftain Way,
Tallahassee, FL, 32306, USA 1616institutetext: Instituto de Astronomia y
Ciencias Planetarias, Universidad de Atacama, Copayapu 485, Copiapo, Chile

# The Carnegie Supernova Project I. Optical spectroscopy of stripped-envelope
supernovae

M. D. Stritzinger 11 S. Holmbo 11 N. Morrell 22 M. M. Phillips 22 C. R. Burns
33 S. Castellón 22 G. Folatelli 445566 M. Hamuy 7788 G. Leloudas 99 N. B.
Suntzeff 1010 J. P. Anderson 1111 C. Ashall 1212 E. Baron  131311 S. Boissier
1414 E. Y. Hsiao 1515 E. Karamehmetoglu 11 F. Olivares 1616

(Received: February 20, 2022; Accepted: February 27, 2023)

We present 170 optical spectra of 35 low-redshift stripped-envelope core-
collapse supernovae observed by the Carnegie Supernova Project-I between 2004
and 2009. The data extend from as early as $-$19 days (d) prior to the epoch
of $B$-band maximum to $+$322 d, with the vast majority obtained during the
so-called photospheric phase covering the weeks around peak luminosity. In
addition to histogram plots characterizing the redshift distribution, number
of spectra per object, and the phase distribution of the sample, spectroscopic
classification is also provided following standard criteria. The CSP-I spectra
are electronically available and a detailed analysis of the data set is
presented in a companion paper being the fifth and final paper of the series.

###### Key Words.:

supernovae: general, individual: SN 2004ew, SN 2004ex, SN 2004fe, SN 2004ff,
SN 2004gq, SN 2004gt, SN 2004gv, SN 2005Q, SN 2005aw, SN 2005bf, SN 2005bj, SN
2005em, SN 2006T, SN 2006ba, SN 2006bf, SN 2006ep, SN 2006fo, SN 2006ir, SN
2006lc, SN 2007C, SN 2007Y, SN 2007ag, SN 2007hn, SN 2007kj, SN 2007rz, SN
2008aq, SN 2008gc, SN 2008hh, SN 2009K, SN 2009Z, SN 2009bb, SN 2009ca, SN
2009dp, SN 2009dq, SN 2009dt – techniques: spectroscopic

##  1 Introduction

This is the fourth in a series of five papers focusing on the presentation and
analysis of a sample of stripped-envelope core-collapse supernovae (SE SNe)
observed by the Carnegie Supernova Project I (hereafter CSP-I; Hamuy et al.
2006). SE SNe are the deaths of massive stars that have undergone prodigious
mass loss over their evolutionary lifetimes. Details associated with the
physics of mass loss and the explosion processes drives the diversity
characterizing the known spectroscopic SE SN subtypes. In Paper 1 of the
series a detailed discussion on the contemporary understanding of SE SNe is
presented along with the broadband optical and near-infrared (NIR) photometry
of the sample (see Stritzinger et al. 2018a). These data served as the basis
in Paper 2 to improve upon our ability to estimate host-reddening parameters
(see Stritzinger et al. 2018b), while the results of both works were used in
Paper 3 to characterize the photometric properties of the CSP-I sample and
estimate key explosion parameters via model comparisons (see Taddia et al.
2018).

SE SNe are classified into spectroscopic subtypes based on the presence and/or
absence of hydrogen (H i) and helium (He i) spectral features in their optical
spectra taken near the epoch of peak brightness (see Filippenko 1997; Gal-Yam
2017, for useful reviews). Within this framework a spectroscopic
classification sequence emerges following the nomenclature: Type
IIb$\rightarrow$Type Ib$\rightarrow$Type Ic; hereafter SNe IIb, Ib, and Ic.
The progression of this sequence reflects a higher degree of mass stripping
experienced by the progenitor stars (Filippenko 1997; Langer 2012; Yoon 2015;
Gal-Yam 2017; Prentice & Mazzali 2017; Shivvers et al. 2019), though other
factors such as the prevalence of 56Ni mixing in the SN ejecta and the
explosion energy do contribute (see Paper 3, and below). As we briefly
summarize in the following, significant diversity exists amongst the
observational properties of various SE SN subtypes, prompting some authors to
suggest various extensions to the classical classification system (e.g.,
Folatelli et al. 2014; Prentice & Mazzali 2017; Williamson et al. 2019). In
this work, we adopt the traditional nomenclature SNe IIb, Ib, Ic, and Ic-
broad-line (BL).

Turning first to SNe IIb, their spectra in the first days after explosion are
blue and featureless (see the case of SN 2009K below). However, around a week
past explosion they typically exhibit a prevalent H$\alpha$ $\lambda 6563$
feature in their optical spectra along with other (less prominent) Balmer
series features, very much akin to the spectra of similar epoch H-rich SNe II.
The H i features are associated with a thin shell of hydrogen material (on the
order of $\gtrsim 0.5$ M☉; Elmhamdi et al. 2006) surrounding the progenitor
stars. The low-$M_{H}$ shell differentiates SNe IIb from classical SNe II, and
ultimately leads to the diminishing strength of the hydrogen features
(particularly H$\alpha$) within days to weeks of the explosion. As the
brightness of SNe IIb evolves through maximum light, they undergo a dramatic
transition in their spectral properties with the disappearance of Balmer
features accompanied by the emergence of prevalent He i $\lambda\lambda$4472,
5876, 6678, 7065, and 7281 features. SN 1993J was the first object with a
spectral sequence that revealed a the metamorphosis from a SN II-like spectrum
to a SN Ib spectrum (Filippenko et al. 1993; Swartz et al. 1993). Today, SN
1993J is the archetype SN IIb (Matheson et al. 2000), and its progenitor was
likely a massive red supergiant star within a binary system (e.g., Maund et
al. 2004).

By definition SNe Ib show no signatures of hydrogen in their early spectra,
though they do typically show conspicuous He i lines that reach prevalence
within several weeks post maximum (Harkness et al. 1987a; Matheson et al.
2001). Similar to the other SE SN subtypes, the spectra of SNe Ib also contain
features from various intermediate-mass elements, including Ca ii H&K, Na i
$\lambda\lambda$5890, 5896 doublet (blended with He i $\lambda$5876), possibly
the Si ii $\lambda$6355 doublet blended with C ii $\lambda$6580 (and/or with a
multitude of Fe ii lines), O i $\lambda$7444, and the Ca ii NIR triplet. In
addition, a variety of iron lines are located at the blue end of the optical
spectrum, most notably the Fe ii multiplet 42, whose features are often used
as a proxy of the bulk ejecta velocity (Branch et al. 2002). SN 1984L served
as an early prototypical example of a SN Ib (Wheeler & Levreault 1985;
Harkness et al. 1987b), and today there is an assortment of data on dozens of
objects in the literature. Interestingly, several detailed case studies
combined with radiative transfer calculations suggested some SNe Ib
progenitors likely retain a hydrogen shell with $M_{H}$ $\sim$ 10${}^{-3}-0.2$
M☉ (e.g., Branch et al. 2002; James & Baron 2010; Hachinger et al. 2012).
Moreover, in addition to normal SNe Ib, a small number of objects are also
known to harbor weak helium lines that slowly emerge and strengthen over time.
Examples of such objects include SN 1999ex (Hamuy et al. 2002), SN 2007Y
(Stritzinger et al. 2018b), SN 2005bf (Folatelli et al. 2006), SN 2008ax
(Chornock et al. 2011; Taubenberger et al. 2011) and SN 2010as (Folatelli et
al. 2014). These objects have been referred to as intermediate and/or
transitional SNe Ib/c (e.g., Hamuy et al. 2002; Leloudas et al. 2011), while
Folatelli et al. (2014) suggested they should form their own spectroscopic
subtype, referring to them as flat-velocity SNe IIb.

Rounding out the end of the classical spectroscopic sequence of SE SNe are the
SNe Ic and SNe Ic-BL. These designations apply to a mixed bag of peculiar
transients, with the SNe Ic-BL being the subset of objects that exhibit broad
spectral features. In general, optical spectra of SNe Ic lack both hydrogen
and helium features, while prevalent features associated with ions of O i, Ca
ii, Fe ii, and notably Si ii, typically dominate the photospheric phase
spectrum. The broad spectral features seen in SNe Ic-BL are thought to be the
consequence of significant line blending produced by ejecta traveling at
relatively high velocities (i.e., $\gtrsim$ 30,000 km s-1. SN 1998bw was the
first SN Ic-BL associated with the afterglow of a long-duration gamma ray
burst (GRB; Galama et al. 1998; Iwamoto et al. 1998; Patat et al. 2001). In
other cases, SNe Ic-BL have been linked to less energetic X-ray flashes such
as SN 2006aj and XRF060218 (Campana et al. 2006; Mirabal et al. 2006; Modjaz
et al. 2006; Sollerman et al. 2006). Alternatively, some SNe Ic-BL such as SN
2002ap (e.g., Mazzali et al. 2002; Gal-Yam et al. 2002) and SN 2009bb (Pignata
et al. 2011) lack an association with high-energy emission, though they are
accompanied by evidence of central engine activity (e.g., Soderberg et al.
2010; Wang et al. 2017).

It is a matter of open discussion whether or not SNe Ic are devoid of helium
(or even hydrogen for that matter (see, e.g., Matheson et al. 2001; Branch et
al. 2006; Hachinger et al. 2012; Williamson et al. 2019), as even a small
amount of helium could be retained and remain transparent due to its high
ionization potential (see Piro & Morozova 2014). Indeed, Shahbandeh et al.
(2022) recently identified a non-negligible fraction of their SNe Ic sample
exhibit signatures of He i $\lambda$20581\. In addition, observations of SN
2016coi/ASASSN-16fp provide one such example where He i may account for some
features (Yamanaka et al. 2017; Prentice et al. 2018; Terreran et al. 2019),
and Yamanaka et al. (2017) suggested its classification could be revised to SN
Ib-BL. Prior to the discovery of SN 2016coi, the presence of helium was also
suggested in the SN Ic-BL objects SN 2009bb (Pignata et al. 2011) and SN
2012ap (Milisavljevic et al. 2015). These objects therefore appear to be
transitional objects between SNe Ic-BL and SNe Ib.

Signatures of circumstellar interaction (CSI) driven emission features
produced by the interaction between expanding SN ejecta and circumstellar
material (CSM) have also been observed in post-maximum spectra of some SE SNe
ranging from weeks to months post explosion (e.g., Tartaglia et al. 2021) to
even over a decade (e,g., Mauerhan et al. 2018). In the case of SN 2017dio,
early phase spectra presented by Kuncarayakti et al. (2018) exhibit prevalent
Balmer emission features superposed on a SN Ic spectrum. In other instances,
narrow helium lines (often with a P Cygni profile) are observed in the early
spectra of stripped SNe known as SNe Ibn (e.g. Pastorello et al. 2007; Foley
et al. 2007; Pastorello et al. 2008; Hosseinzadeh et al. 2017; Karamehmetoglu
et al. 2021). Finally, narrow carbon features have also been detected in a
handful of the so-called SNe Icn (e.g., Fraser et al. 2021; Gal-Yam et al.
2021).

The timescales at which CSI occurs are largely a consequence of the
distribution and location of the CSM relative to the expanding ejecta, while
the ions associated with the emission are largely dependent on the chemical
composition of the CSM. With robust constraints on the emergence and duration
of the CSI features we are able to put tight constraints on the mass-loss
history of the progenitors during the pre-SN phase (months to centuries),
while objects exhibiting late-phase emission enable one to place constraints
over much longer evolutionary timescales. In addition to emission lines, CSI
produces additional energy deposition. This has been inferred through
signatures of optical broadband excesses (e.g., Sollerman et al. 2020), radio
emission (Maeda et al. 2021), and a combination of optical, radio and X-ray
emission (Pooley et al. 2019).

Finally, we note that with the advent of high-cadence surveys, a variety of
hydrogen-deficient SNe that exhibit rapid light-curve evolution have been
discovered. Examples studied to date include: SN 2002bj (Poznanski et al.
2010), SN 2005ek (Drout et al. 2013), SN 2010X (Kasliwal et al. 2010), iPTF
14gqr (De et al. 2018), SN 2019bkc (Chen et al. 2020; Prentice et al. 2020),
and SN 2019ehk (Nakaoka et al. 2021). Despite their similarities to some SE
SNe, no single explosion model can satisfactorily reproduce the main
observational properties of these fast-evolving cosmic explosions (see Chen et
al. 2020, for a discussion).

In the following we present 170 optical spectra of 35 SE SNe observed by the
CSP-I. The majority of these objects are rather normal (see Paper 3) and
enable us to perform a robust analysis on SE SN spectra (see Paper 5; Holmbo
et al. 2023). A key contribution of the CSP-I sample to the literature sample
is the quality of the spectra, characterized by high signal-to-noise that
often extends through 9000 Å, enabling an examination of the prominent Ca ii
NIR triplet. In Sect. 2 details concerning the facilities used by the CSP-I to
collect the data and the reduction processes are presented. In Sect. 3 the
spectra are presented, and their spectroscopic subtype are determined. We
conclude with a summary in Sect. 4.

To date, optical spectroscopic samples consisting of hundreds of SE SN spectra
have been presented by the Berkeley SN group (Matheson et al. 2001; Shivvers
et al. 2019), the CfA SN group (Modjaz et al. 2014; Liu et al. 2016), and the
(i)PTF collaboration (Fremling et al. 2018). Turning to longer wavelengths, we
also note that our follow-up project, dubbed CSP-II, recently released 75 NIR
spectra of 34 SE SNe observed between 2011 and 2015 (Shahbandeh et al. 2022).
The light curves and optical spectroscopy of the CSP-II SE SN sample will be
the subject of a future publication.

##  2 Observations

The CSP-I carried out five, a nine-month-long observing campaigns between 2004
and 2009 centered on the southern hemisphere summer. By completion of the
CSP-I, optical/NIR photometry and optical spectroscopy were obtained for
several hundred SNe of all types, including 34 SE SNe largely discovered by
targeted transient surveys (see Paper 1 for details). Table 1 contains a
compilation of key information of each object as well as SN 2009dq which was
not a part of the light curve sample presented in Paper 1. This includes: the
Internationcal Astronomical Union (IAU) name, the coordinates, the redshift
($z$) of the host galaxy, the spectral classification, the number of spectra
obtained for each object, the phase range of the spectra, and an estimate of
the time of $t(B)_{max}$. Given our constraints to follow relatively bright
SNe (i.e., peak apparent $m_{V}\leq 18$ mag) the targets are located in nearby
galaxies. As demonstrated in the top panel of Fig. 1, 34 of the objects have
heliocentric redshifts in the range $z=0.0046-0.0492$, while the luminous SN
2009ca (peak $M_{V}\approx-20$ mag) has a redshift of $z=0.0957$. Additional
panels in Fig. 1 contain histograms of the number of spectra obtained per SN
(middle panel) and the distribution of temporal phases with respect to
$t(B)_{max}$ of the first epoch of observation (bottom panel).

The CSP-I spectroscopy sample of SE SNe was obtained with a handful of
telescopes and instruments located in Chile. A journal of spectroscopic
observations is provided in Table LABEL:table2. This includes a listing of the
spectra, date of observations, the observational facilities, and the key
spectral parameters: restframe phase relative to the epoch of $B$-band
maximum, wavelength range, resolution, exposure time, and airmass. The vast
majority of observations were conducted with facilities at the Las Campanas
Observatory with the 2.5 m du Pont telescope and the 6.5 m Magellan Baade and
Clay telescopes. Most of our spectra were taken with the du Pont telescope
equipped with the Wide Field reimaging CCD (WFCCD) camera. A few spectra were
also taken with the Boller & Chivens (B&C) spectrograph. Spectroscopy
performed with the much larger-aperture Magellan telescopes made use of the
Low Dispersion Survey Spectrograph 3 (LDSS3) and the Inamori Magellan Areal
Camera and Spectrograph (IMACS). CSP-I also used the European Southern
Observatory (ESO) 3.5 m New Technology Telescope (NTT) equipped with the ESO
Multi-Mode Instrument (EMMI) and the 3.6 m telescope equipped with the ESO
Faint Object Spectrograph and Camera (EFOSC), both located at the La Silla
Observatory. In addition, the CSP-I also obtained three SE SN spectra with the
Ritchey-Chrétien (RC) Cassegrain spectrograph attached the 1.5 m telescope at
the Cerro Tololo Inter-American Observatory (CTIO), and a half dozen (mostly
late phase) spectra with the Gemini Multi-Object Spectrograph (GMOS) attached
to the 8.1 m Gemini-South telescope on Cerro Pachon.

The reduction procedures followed to obtain one-dimensional, flux-calibrated
spectra are described in detail by Hamuy et al. (2006). In short, a two-
dimensional spectral image is bias- and flat-field-corrected whereupon the SN
trace is extracted. A wavelength calibration solution determined from arc lamp
exposures is then applied to the one-dimensional extracted spectrum followed
by (when possible) the division of a telluric spectrum. Next the wavelength-
corrected (and telluric-corrected) one-dimensional spectrum is multiplied by a
nightly sensitivity function determined from observations of one or more flux
standard stars. If multiple science exposures are obtained for a given object
on the same night, the one-dimensional spectra are averaged. Finally,
prevalent cosmic rays are removed.

##  3 Results

###  3.1 Optical spectroscopy

The sample consists of 170 optical spectra of the 34 objects presented in
Paper 1, as well as two spectra of the Type IIb SN 2009dq. The data sample
covers a range of phases, extending from as early as $-$19 rest-frame days(d)
relative to $t(B)_{max}$ (SN 2009K) to $+$310 d (SN 2008aq).111Estimates of
$t(B)_{max}$ are presented in Paper 3. In the present paper, all temporal
phases are given in rest-frame days (d) relative to $t(B)_{max}$. The
histograms plotted in the middle and bottom panels of Fig. 1 indicate that an
average of two to three spectra were obtained per object and that for the
majority of objects the first spectrum was obtained around the time of the
$B$-band maximum. Four objects in the sample (i.e., SN 2007Y, SN 2008aq, SN
2009K and SN 2009bb) have a least one late-phase spectrum taken after $+$200
d. The nebular spectra of SN 2008aq and SN 2009K are presented for the first
time and those of SN 2007Y and SN 2009bb were previously presented by
Stritzinger et al. (2009) and Pignata et al. (2011), respectively. The final
extracted, one-dimensional spectra, normalized to their mean flux values are
plotted in Fig. 2.222The spectra can be downloaded electronically from the CSP
Pasadena-based web page at http://csp.obs.carnegiescience.edu/data/, and the
Weizmann Interactive Supernova Data Repository (WISeREP; Yaron & Gal-Yam 2012)
https://www.wiserep.org//object/. For presentation purposes, each plotted
spectrum was smoothed using a median filter with a width of 5 pixels, while in
a handful of objects widths of 7 pixels (SNe 2007kj and 2009dp) and 9 pixels
(SNe 2006bf and SN 2006ep) were adopted.

To assess the general quality of the flux calibration of our SE SN
spectroscopic sample, we compared the observed broadband colors of the SNe to
the observed colors obtained from synthetic photometry. The inferred synthetic
colors were computed for the entire WFCCD subset of spectra using the CSP-I
system response functions that define the natural photometric system of the
Swope telescope (see Stritzinger et al. 2011; Krisciunas et al. 2017).

The results of this exercise are plotted in Fig. 3, which shows the
comparisons between synthetic and observed ($g-r$), ($r-i$), and ($V-i$)
colors in the top panel. In the bottom panel the differences between the
observed and synthetic colors are plotted versus the corresponding synthetic
colors. Comparison of the observed and synthetic colors reveals a fairly good
agreement between the ($g-r$) colors, exhibiting average differences of
$+0.015$ mag (with an associated root-mean-square uncertainty rms $=0.101$
mag). Good agreement is also found between the observed and synthetic ($V-r$)
colors (not plotted in Fig. 3), which exhibit an average difference of
$-0.017$ mag (rms of 0.059 mag).

Less agreement is found between the observed and synthetic colors that include
the $i$ band. Specifically, the average difference between the observed versus
synthetic ($V-i$) and ($r-i$) colors amounts to $+$0.055 mag (rms $=0.078$
mag) and $+$0.072 mag (rms $=0.073$ mag), respectively. The systematic offsets
between the observed and synthetic ($V-i$) and ($r-i$) colors are evident in
the bottom panel of Fig. 3. As a first step towards identifying the culprit of
these offsets we eliminated spectra that contained prevalent telluric
absorption. The curated sample provided improved ($V-i$) and ($r-i$) color
offsets that decreased to $+$0.034 mag (rms $=0.064$ mag) and $+$0.055 mag
(rms $=0.072$ mag), respectively. Although the offsets decreased, they are not
entirely brought down to the levels found in the other color combinations that
do not include the $i$ band.

We further investigated the issue and conducted an additional sequence of
tests. First we examined whether poorly subtracted telluric absorption
features could be a problem. To this end, observed and synthetic broadband
colors were compared for the nearly dozen different spectrophotometric flux
standards observed over the course of the CSP-I with the du Pont telescope
equipped with WFCCD and which have had their telluric features removed
following our standard procedures as prescribed by Bessell (1999). This test
revealed average synthetic versus broadband color differences as follows:
($g-r$) $=+0.002$ mag (rms $=0.007$ mag), ($V-i$) $=+0.002$ mag (rms $=0.028$
mag), and ($r-i$) $=+0.006$ mag (rms $=0.024$ mag). The close agreement
between these color indices reveals that, at least in the case of
spectrophotometric standard star observations, our telluric removal technique
produces robust results.

We also considered the accuracy of the CSP-I $i$-band system response function
used to compute synthetic photometry as its measurement was not as robust as
those of the other bands (Rheault et al. 2014). Indeed, the $i$-band system
response function was only scanned a single time and suffered illumination
issues at red wavelengths due to the light source used. To this end, synthetic
colors were computed with the CSP-I system response functions using the atlas
of spectrophotometric Landolt standard stars (Stritzinger et al. 2005), which
have both Landolt (1992) and Smith et al. (2002) broadband photometry.
Comparison between the resulting synthetic colors to those inferred from the
broadband photometry of these stars yields average color differences as
follows: ($g-r$) $=-0.005$ mag (rms $=0.010$ mag), ($V-i$) $=-0.031$ mag (rms
$=0.018$ mag), and ($r-i$) $=-0.006$ mag (rms $=0.012$ mag).

Although we are unable to conclusively identify the culprit driving the
offsets with the $i$-band colors, we speculate it is linked to the accuracy of
the $i$-band system response function. Indeed, as noted by Rheault et al.
(2014), the $i$-band system response function proved to be more difficult to
measure and less accurate than the other bands due to the low illumination of
the light source used in the scanning process. In summary, we find good
agreement between the observed and synthetic colors that do not include the
$i$ band.

###  3.2 Spectroscopic classification

We now turn to the spectral classification of objects in the CSP-I SE SN
sample. If the spectrum of a given object lacked prevalent hydrogen and/or He
i lines, we turned to guidance from the spectral template comparison program
SNID (SuperNova IDentification; Blondin & Tonry 2007). In doing so, we made
use of the expanded set of SE SN templates presented by Liu & Modjaz (2014).
The spectroscopic classification of the sample is reported in Table 1 and
reveals a breakdown between subtypes as follows: eleven SNe IIb, twelve SNe
Ib, ten SNe Ic, and two SNe Ic-BL. As a caveat to these classifications, we
note that traditionally it would not be entirely possible to exclude a SN IIb
classification for the Type Ib SNe 2004ew, 2006ep, and 2008gc as their
earliest spectra are post-maximum (see Table 1). However, as demonstrated in
Paper 5, from pseudo-equivalent width measurements of certain spectral
features and/or through the use of principal component analysis, these objects
are confirmed to be SNe Ib. Similarly the distinction between some SNe Ib and
SNe Ic is not always clear cut as was found in the cases of SNe 2005em and
2009dt, whose spectral phase coverage is limited to only good spectra taken
around or before maximum (see Table 1). However, through the use of pEW
measurements of O i and/or PCA these two objects are found to be consistent
with a SN Ic classification.

In assigning classifications to SNe IIb no differentiation was made between
objects thought to be characterized by extended or compact progenitors since
such detailed subtyping is reliant on radio emission limits and/or detections
(e.g., Chevalier & Soderberg 2010). In the case of SN 2009bb, previous papers
by our group and collaborators have noted it exhibits broad spectral features
akin to other SNe Ic-BL (Pignata et al. 2011) and that it displays signatures
of a relativistic outflow possibly linked to a central engine (Soderberg et
al. 2010). SN 2009ca is also a noteworthy object as its peak luminosity
reached a value of at least $5\times 10^{43}$ ergs s-1, which is reminiscent
of SN 1992ar (Clocchiatti et al. 2000). This value is more than a factor of 10
larger compared to the average peak luminosity values determined from the
literature sample of normal SNe Ic and a factor of 3 higher compared to the
brightest of the gamma-ray bursts associated SNe Ic-BL, such as SN 2012bz
(Schulze et al. 2014). Indeed, SN 2009ca appears to be a super-luminous
supernova (see Taddia et al. 2018).

##  4 Summary

This paper presents 170 optical spectra of 35 SE SNe observed by CSP-I. The
majority of the spectra were taken with facilities at the Las Campanas
Observatory before or within a week after the epoch of $B$-band maximum.
Nebular spectra are also presented for a subset of five objects. In general
the spectra are of high quality and often cover the spectral range to around
9000 Å. In the companion paper presented by Holmbo et al. (2023, i.e., Paper
5) these data are used to construct mean template spectral sequences for the
IIb, Ib, and Ic SE SN subtypes, study key line diagnostics, and perform a
robust principle component analysis, leading to a fast and accurate method for
classifying SE SN subtypes using only a single spectrum taken at either early
or post-maximum epochs.

###### Acknowledgements.

The CSP has received support from the National Science Foundation (USA) under
grants AST–0306969, AST–0607438, AST–1008343, AST–1613426, AST–1613455, and
AST-1613472. The Aarhus supernova group is funding in part by a research
Project 1 grant from the Independent Research Fund Denmark (IRFD grant numbers
8021-00170B and 10.46540/2032-00022B), and by a VILLUM FONDEN Experiment
(grant number 28021). GL is supported by a Villum Young Investigator
fellowship (grant number 19054) from the VILLUM FONDEN. This research has made
use of the NASA/IPAC Extragalactic Database (NED), which is operated by the
Jet Propulsion Laboratory, California Institute of Technology, under contract
with the National Aeronautics and Space Administration. Based on observations
collected with the Magellan Clay and Baade 6.5-m telescopes and and the du
Pont 2.5-m telescope located on Las Campanas Observatory, Chile; the European
Organization for Astronomical Research in the Southern Hemisphere (ESO),
Chile, Programs 076.A-0156, 078.D-0048, 080.A-0516, 082.A-0526, and
380.D-0272; the 1.5 m telescope at CTIO, Chile; and the Gemini Observatory,
Cerro Pachon, Chile (Gemini Programs GS-2008B-Q-8-77).

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Table 1: Spectroscopic sample of CSP-I SE SNe.

SN | $\alpha(2000)$ | $\delta(2000)$ | Redshifta | Spectral | No. of | Phase | $t(B)_{\rm max}$
---|---|---|---|---|---|---|---
|  |  | $z$ | Type | Spectra | Range | JD+2450000
2004ew | 02:05:06.17 |  $-$55:06:31.6 | 0.0218 | Ib | 4 |  $+26\cdots+$81 | $3276.11\pm 2.03$
2004ex | 00:38:10.17 |  $+$02:43:16.9 | 0.0176 | IIb | 5 |  $-4\cdots+$41 | $3306.69\pm 0.03$
2004fe | 00:30:11.27 |  $+$02:05:23.5 | 0.0179 | Ic | 4 |  $+9\cdots+$35 | $3316.79\pm 0.11$
2004ff | 04:58:46.19 |  $-$21:34:12.0 | 0.0227 | IIb | 3 |  $+14\cdots+$30 | $3312.60\pm 0.03$
2004gq | 05:12:04.81 |  $-$15:40:54.2 | 0.0065 | Ib | 3 |  $-5\cdots+$24 | $3357.90\pm 0.02$
2004gt | 12:01:50.37 |  $-$18:52:12.7 | 0.0055 | Ic | 4 |  $-2\cdots+$49 | $3360.37\pm 0.08$
2004gv | 02:13:37.42 |  $-$00:43:05.8 | 0.0199 | Ib | 1 |  $-7\cdots-$7 | $3365.27\pm 0.31$
2005Q | 01:30:03.51 |  $-$42:40:48.4 | 0.0224 | IIb | 2 |  $-1\cdots+$7 | $3406.10\pm 0.04$
2005aw | 19:15:17.44 |  $-$54:08:24.9 | 0.0095 | Ic | 5 |  $+7\cdots+$23 | $3456.48\pm 1.37$
2005bf | 10:23:57.27 |  $-$03:11:28.6 | 0.0189 | Ib | 9 |  $-7\cdots+$52 | $3474.79\pm 0.28$
2005bj | 16:49:44.74 |  $+$17:51:48.7 | 0.0222 | IIb | 2 |  $+1\cdots+$4 | $3471.89\pm 1.37$
2005em | 03:13:47.71 |  $-$00:14:37.0 | 0.0260 | Ic | 1 |  $+0\cdots+$0 | $3648.97\pm 1.36$
2006T | 09:54:30.21 |  $-$25:42:29.3 | 0.0081 | IIb | 11 |  $+0\cdots+$70 | $3779.61\pm 0.01$
2006ba | 09:43:13.40 |  $-$09:36:53.0 | 0.0191 | IIb | 3 |  $+4\cdots+$28 | $3820.92\pm 1.36$
2006bf | 12:58:50.68 |  $+$09:39:30.1 | 0.0239 | IIb | 5 |  $+7\cdots$+33 | $3817.46\pm 2.03$
2006ep | 00:41:24.88 |  $+$25:29:46.7 | 0.0151 | Ib | 3 |  $+19\cdots+$34 | $3985.41\pm 0.05$
2006fo | 02:32:38.89 |  $+$00:37:03.0 | 0.0207 | Ib | 3 |  $+2\cdots+$18 | $4003.17\pm 1.36$
2006ir | 23:04:35.68 |  $+$07:36:21.5 | 0.0200 | Ic | 2 |  $+19\cdots+$43 | $3999.07\pm 2.03$
2006lc | 22:44:24.48 |  $-$00:09:53.5 | 0.0162 | Ib | 1 |  $+1\cdots+$1 | $4041.33\pm 0.04$
2007C | 13:08:49.30 |  $-$06:47:01.0 | 0.0056 | Ib | 9 |  $-1\cdots+$92 | $4115.25\pm 1.37$
2007Y | 03:02:35.92 |  $-$22:53:50.1 | 0.0046 | Ib | 12 |  $-12\cdots+$271 | $4162.99\pm 0.01$
2007ag | 10:01:35.99 |  $+$21:36:42.0 | 0.0207 | Ib | 2 |  $+7\cdots+$10 | $4163.29\pm 1.42$
2007hn | 21:02:46.85 |  $-$04:05:25.2 | 0.0273† | Ic | 4 |  $+10\cdots+$56 | $4351.63\pm 1.41$
2007kj | 00:01:19.58 |  $+$13:06:30.6 | 0.0179 | Ib | 5 |  $-4\cdots+$40 | $4380.91\pm 0.03$
2007rz | 04:31:10.84 |  $+$07:37:51.5 | 0.0130 | Ic | 2 |  $+7\cdots+$32 | $4437.31\pm 2.03$
2008aq | 12:50:30.42 |  $-$10:52:01.4 | 0.0080 | IIb | 12 |  $-5\cdots+$308 | $4531.15\pm 0.02$
2008gc | 02:10:36.63 |  $-$53:45:59.5 | 0.0492† | Ib | 6 |  $+9\cdots+$117 | $4744.26\pm 1.37$
2008hh | 01:26:03.65 |  $+$11:26:26.5 | 0.0194 | Ic | 3 |  $+2\cdots+$31 | $4791.50\pm 2.03$
2009K | 04:36:36.77 |  $-$00:08:35.6 | 0.0117 | IIb | 8 |  $-18\cdots+$282 | $4867.22\pm 0.01$
2009Z | 14:01:53.61 |  $-$01:20:30.2 | 0.0248 | IIb | 8 |  $-6\cdots+$41 | $4876.94\pm 0.01$
2009bb | 10:31:33.87 |  $-$39:57:30.0 | 0.0099 | Ic-BL | 18 |  $-1\cdots+$308 | $4920.05\pm 0.02$
2009ca | 21:26:22.20 |  $-$40:51:48.6 | 0.0957† | Ic-BL | 3 |  $+3\cdots+$16 | $4926.17\pm 2.03$
2009dp | 20:26:52.69 |  $-$18:37:04.2 | 0.0232 | Ic | 3 |  $+2\cdots+$34 | $4949.81\pm 2.03$
2009dq | 10:08:49.94 |  $-$67:01:57.3 | 0.0047† | IIb | 2 |  $-9\cdots+$14 | $4960.50\pm 2.00$
2009dt | 22:10:09.27 |  $-$36:05:42.6 | 0.0104 | Ic | 2 |  $-6\cdots-$16 | $4958.31\pm 1.38$

aafootnotetext: Heliocentric redshifts were retrieved from NED, or if not,
then as determined from host-galaxy emission lines in optical spectra. The
latter of these are indicated by a †.

Figure 1: Breakdown of the sample. Top panel: Distribution of redshift ($z$)
for the sample, as obtained from host-galaxy redshift measurements provided by
the NASA/IPAC Extragalactic Database (NED). Middle panel: Distribution of the
number of spectra per SN for the sample. Bottom panel: Temporal phase
distribution of the first epoch of observation for the CSP-I SE SN
spectroscopy sample.

$\begin{array}[]{cc}\includegraphics[width=195.12767pt]{SN2004ew.pdf}&\includegraphics[width=195.12767pt]{SN2004ex.pdf}\\\
\includegraphics[width=195.12767pt]{SN2004fe.pdf}&\includegraphics[width=195.12767pt]{SN2004ff.pdf}\\\
\end{array}$

Figure 2: Optical spectroscopy of the CSP-I SE SN sample. Each spectrum has
been normalized to its mean flux values and plotted in units of $f_{\lambda}$.
For presentation purposes additive constants have also been applied. The phase
of each spectrum relative to the epoch of the $B$-band maximum is provided
next to each spectrum and the spectroscopic subtype is provided below the SN
designation at the top right. Vertical dashed blue lines indicate rest
wavelengths of He i $\lambda\lambda$4472, 5876, 6678, 7065, 7281 lines,
vertical dashed red lines are H$\gamma$, H$\beta$, and H$\alpha$, and the
vertical gray line indicates the location of a prominent telluric feature.

$\begin{array}[]{cc}\includegraphics[width=195.12767pt]{SN2004gq.pdf}&\includegraphics[width=195.12767pt]{SN2004gt.pdf}\\\
\includegraphics[width=195.12767pt]{SN2004gv.pdf}&\includegraphics[width=195.12767pt]{SN2005Q.pdf}\\\
\end{array}$

Figure 2: Continued.

$\begin{array}[]{cc}\includegraphics[width=195.12767pt]{SN2005aw.pdf}&\includegraphics[width=195.12767pt]{SN2005bf.pdf}\\\
\includegraphics[width=195.12767pt]{SN2005bj.pdf}&\includegraphics[width=195.12767pt]{SN2005em.pdf}\\\
\end{array}$

Figure 2: Continued.

$\begin{array}[]{cc}\includegraphics[width=195.12767pt]{SN2006T.pdf}&\includegraphics[width=195.12767pt]{SN2006ba.pdf}\\\
\includegraphics[width=195.12767pt]{SN2006bf.pdf}&\includegraphics[width=195.12767pt]{SN2006ep.pdf}\\\
\end{array}$

Figure 2: Continued.

$\begin{array}[]{cc}\includegraphics[width=195.12767pt]{SN2006fo.pdf}&\includegraphics[width=195.12767pt]{SN2006ir.pdf}\\\
\includegraphics[width=195.12767pt]{SN2006lc.pdf}&\includegraphics[width=195.12767pt]{SN2007C.pdf}\\\
\end{array}$

Figure 2: Continued.

$\begin{array}[]{cc}\includegraphics[width=195.12767pt]{SN2007Y.pdf}&\includegraphics[width=195.12767pt]{SN2007ag.pdf}\\\
\includegraphics[width=195.12767pt]{SN2007hn.pdf}&\includegraphics[width=195.12767pt]{SN2007kj.pdf}\\\
\end{array}$

Figure 2: Continued.

$\begin{array}[]{cc}\includegraphics[width=195.12767pt]{SN2007rz.pdf}&\includegraphics[width=195.12767pt]{SN2008aq.pdf}\\\
\includegraphics[width=195.12767pt]{SN2008gc.pdf}&\includegraphics[width=195.12767pt]{SN2008hh.pdf}\\\
\end{array}$

Figure 2: Continued.

$\begin{array}[]{cc}\includegraphics[width=195.12767pt]{SN2009K.pdf}&\includegraphics[width=195.12767pt]{SN2009Z.pdf}\\\
\includegraphics[width=195.12767pt]{SN2009bb.pdf}&\includegraphics[width=195.12767pt]{SN2009ca.pdf}\\\
\end{array}$

Figure 2: Continued.

$\begin{array}[]{cc}\includegraphics[width=195.12767pt]{SN2009dp.pdf}&\includegraphics[width=195.12767pt]{SN2009dq.pdf}\\\
\includegraphics[width=195.12767pt]{SN2009dt.pdf}\end{array}$

Figure 2: Continued.

Figure 3: Quality of the spectral flux calibration. Top panel: Comparison
between observed and synthetic colors ($g-r$) indicated with black dots,
$(r-i$) with red squares, and ($V-i$) with blue triangles. Synthetic colors
are computed using WFCCD spectra and the CSP-I system response functions (see
Krisciunas et al. 2017, and references therein). Bottom panel: Residual
between the observed and synthetic colors vs. synthetic colors. The dashed
line has a slope of unity.

##  Appendix A Journal of spectroscopic observations

Table 2: Journal of spectroscopic observations. SN | UT Date | JD | Phase | Telescope | Instrument | Wavelength | Resolution | Exp. time | Airmass
---|---|---|---|---|---|---|---|---|---
|  | +2450000 |  |  |  | Range [Å] | FWHM [Å] | [s] |
2004ew | 2004 Oct 24.22 | 3302.72 | 26.04 | Clay | LDSS3 | $3600-9000$ | 4.2 | 300 | 1.12
| 2004 Nov 26.16 | 3335.66 | 58.28 | Clay | LDSS3 | $3600-9000$ | 4.2 | 300 | 1.15
| 2004 Dec 13.14 | 3352.64 | 74.90 | du Pont | WFCCD | $3800-9235$ | 3.0 | 900 | 1.20
| 2004 Dec 19.04 | 3358.54 | 80.67 | Clay | LDSS3 | $3600-9000$ | 4.2 | 500 | 1.11
2004ex | 2004 Oct 24.24 | 3302.74 |  $-$3.88 | Clay | LDSS3 | $3600-9000$ | 4.2 | 300 | 1.55
| 2004 Nov 17.18 | 3326.68 | 19.65 | du Pont | WFCCD | $3800-9235$ | 3.0 | 600 | 1.57
| 2004 Nov 26.12 | 3335.62 | 28.43 | Clay | LDSS3 | $3600-9000$ | 4.2 | 300 | 1.35
| 2004 Dec 04.09 | 3343.59 | 36.26 | du Pont | WFCCD | $3800-9235$ | 3.0 | 900 | 1.28
| 2004 Dec 09.09 | 3348.59 | 41.18 | du Pont | WFCCD | $3800-9235$ | 3.0 | 900 | 1.35
2004fe | 2004 Nov 16.06 | 3325.56 | 8.61 | CTIO 1.5m | CS60 | $3200-9617$ | 5.4 | 1200 | 1.18
| 2004 Nov 17.14 | 3326.64 | 9.68 | du Pont | WFCCD | $3800-9325$ | 3.1 | 600 | 1.32
| 2004 Nov 26.10 | 3335.60 | 18.48 | Clay | LDSS3 | $3600-9000$ | 4.2 | 450 | 1.26
| 2004 Dec 13.06 | 3352.56 | 35.14 | du Pont | WFCCD | $3800-9235$ | 3.0 | 900 | 1.29
2004ff | 2004 Nov 17.28 | 3326.78 | 13.86 | du Pont | WFCCD | $3800-9235$ | 3.0 | 600 | 1.02
| 2004 Nov 26.19 | 3335.69 | 22.57 | Clay | LDSS3 | $3600-9000$ | 4.2 | 450 | 1.03
| 2004 Dec 04.22 | 3343.72 | 30.43 | du Pont | WFCCD | $3800-9235$ | 3.0 | 600 | 1.01
2004gq | 2004 Dec 13.23 | 3352.73 |  $-$5.13 | du Pont | WFCCD | $3800-9235$ | 3.0 | 300 | 1.07
| 2004 Dec 19.19 | 3358.69 | 0.78 | Clay | LDSS3 | $3600-9000$ | 4.2 | 180 | 1.03
| 2005 Jan 11.16 | 3381.66 | 23.61 | CTIO 1.5m | CS60 | $3200-9615$ | 5.4 | 900 | 1.09
2004gt | 2004 Dec 19.30 | 3358.80 |  $-$1.56 | Clay | LDSS3 | $3600-9000$ | 4.2 | 180 | 1.58
| 2005 Jan 11.30 | 3381.80 | 21.31 | CTIO 1.5m | CS60 | $3200-9617$ | 5.4 | 1200 | 1.17
| 2005 Feb 04.32 | 3405.82 | 45.20 | du Pont | WFCCD | $3800-9235$ | 3.0 | 300 | 1.02
| 2005 Feb 08.33 | 3409.83 | 49.19 | du Pont | WFCCD | $3800-9235$ | 3.0 | 300 | 1.02
2004gv | 2004 Dec 19.09 | 3358.59 |  $-$6.55 | Clay | LDSS3 | $3600-9000$ | 4.2 | 300 | 1.18
2005Q | 2005 Feb 04.06 | 3405.56 |  $-$0.53 | du Pont | WFCCD | $3800-9235$ | 3.0 | 400 | 1.58
| 2005 Feb 12.05 | 3413.56 | 7.29 | du Pont | WFCCD | $3800-9235$ | 3.0 | 400 | 1.76
2005aw | 2005 Apr 03.33 | 3463.83 | 7.28 | du Pont | WFCCD | $3800-9235$ | 3.0 | 500 | 1.39
| 2005 Apr 07.39 | 3467.89 | 11.30 | du Pont | WFCCD | $3800-9235$ | 3.0 | 600 | 1.16
| 2005 Apr 12.40 | 3472.90 | 16.26 | du Pont | WFCCD | $3800-9235$ | 3.0 | 500 | 1.13
| 2005 Apr 15.39 | 3475.89 | 19.23 | du Pont | WFCCD | $3800-9235$ | 3.0 | 500 | 1.13
| 2005 Apr 19.40 | 3479.90 | 23.20 | du Pont | WFCCD | $3800-9235$ | 3.0 | 500 | 1.11
2005bf | 2005 Apr 07.07 | 3467.57 |  $-$7.08 | du Pont | WFCCD | $3800-9235$ | 3.0 | 900 | 1.12
| 2005 Apr 12.10 | 3472.60 |  $-$2.15 | du Pont | WFCCD | $3800-9235$ | 3.0 | 900 | 1.12
| 2005 Apr 15.13 | 3475.63 | 0.83 | du Pont | WFCCD | $3800-9235$ | 3.0 | 900 | 1.20
| 2005 Apr 19.01 | 3479.51 | 4.64 | du Pont | WFCCD | $3800-9235$ | 3.0 | 700 | 1.15
| 2005 May 06.97 | 3497.47 | 22.26 | Clay | LDSS3 | $3838-10000$ | 0.7 | 200 | 1.14
| 2005 May 11.97 | 3502.47 | 27.17 | Clay | LDSS3 | $6057-10000$ | 1.1 | 900 | 1.12
| 2005 May 31.16 | 3521.67 | 46.01 | du Pont | WFCCD | $3800-8125$ | 3.0 | 1200 | 1.19
| 2005 Jun 01.99 | 3523.49 | 47.79 | du Pont | WFCCD | $3800-8128$ | 3.0 | 1200 | 1.16
| 2005 Jun 06.04 | 3527.54 | 51.77 | du Pont | WFCCD | $3800-8128$ | 3.0 | 600 | 1.48
2005bj | 2005 Apr 12.35 | 3472.85 | 0.94 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $900$ | 1.47
| 2005 Apr 15.35 | 3475.85 | 3.88 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $800$ | 1.48
2005em | 2005 Oct 05.35 | 3648.85 |  $-$0.12 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $600$ | 1.21
2006T | 2006 Feb 13.26 | 3779.76 | 0.15 | ESO NTT | EMMI | $3200-10200$ | $2.1$ | $300$ | 1.03
| 2006 Feb 27.35 | 3793.85 | 14.12 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $600$ | 1.72
| 2006 Mar 05.22 | 3799.72 | 19.95 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $600$ | 1.06
| 2006 Mar 08.23 | 3802.73 | 22.93 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $900$ | 1.09
| 2006 Mar 14.21 | 3808.71 | 28.86 | Clay | LDSS3 | $3785-9972$ | $1.2$ | $400$ | 1.14
| 2006 Mar 22.19 | 3816.69 | 36.78 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $900$ | 1.09
| 2006 Mar 24.15 | 3818.65 | 38.73 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $900$ | 1.03
| 2006 Apr 02.10 | 3827.60 | 47.61 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $900$ | 1.01
| 2006 Apr 16.24 | 3841.74 | 61.63 | Baade | IMACS | $3842-9692$ | $2.0$ | $600$ | 2.09
| 2006 Apr 23.10 | 3848.60 | 68.43 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $900$ | 1.08
| 2006 Apr 25.05 | 3850.55 | 70.73 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $900$ | 1.02
2006ba | 2006 Mar 30.15 | 3824.65 | 3.66 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $900$ | 1.13
| 2006 Apr 16.07 | 3841.57 | 20.27 | Baade | IMACS | $3842-9692$ | $2.0$ | $900$ | 1.08
| 2006 Apr 24.04 | 3849.54 | 28.08 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $900$ | 1.07
2006bf | 2006 Mar 30.31 | 3824.81 | 7.18 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $900$ | 1.54
| 2006 Apr 02.32 | 3827.82 | 10.12 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $1200$ | 1.67
| 2006 Apr 22.25 | 3847.75 | 29.58 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $1000$ | 1.53
| 2006 Apr 23.23 | 3848.73 | 30.54 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $1000$ | 1.47
| 2006 Apr 25.28 | 3850.78 | 32.55 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $900$ | 2.04
2006ep | 2006 Sep 26.20 | 4004.70 | 19.00 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $1200$ | 1.74
| 2006 Oct 10.19 | 4018.69 | 32.79 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $1200$ | 1.74
| 2006 Oct 11.18 | 4019.68 | 33.76 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $1200$ | 1.73
2006fo | 2006 Sep 26.28 | 4004.78 | 1.58 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $1200$ | 1.16
| 2006 Sep 28.26 | 4006.76 | 3.52 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $1200$ | 1.18
| 2006 Oct 13.32 | 4021.82 | 18.27 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $1200$ | 1.28
2006ir | 2006 Oct 10.10 | 4018.60 | 19.14 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $600$ | 1.25
| 2006 Nov 03.14 | 4042.64 | 42.71 | ESO NTT | EMMI | $4012-10200$ | $3.3$ | $300$ | 1.53
2006lc | 2006 Nov 03.08 | 4042.58 | 1.23 | ESO NTT | EMMI | $3200-10200$ | $2.1$ | $300$ | 1.20
2007C | 2007 Jan 13.34 | 4113.84 |  $-$1.40 | du Pont | B&C | $3932-10142$ | $3.0$ | $300$ | 1.29
| 2007 Jan 18.28 | 4118.78 | 3.51 | du Pont | B&C | $3402-9614$ | $3.0$ | $900$ | 1.63
| 2007 Jan 31.36 | 4131.86 | 16.52 | ESO NTT | EMMI | $3200-5300$ | $2.1$ | $400$ | 1.10
| 2007 Feb 12.37 | 4143.87 | 28.46 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $400$ | 1.09
| 2007 Feb 19.38 | 4150.88 | 35.43 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $400$ | 1.13
| 2007 Feb 25.35 | 4156.85 | 41.37 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $400$ | 1.10
| 2007 Mar 04.27 | 4163.77 | 48.25 | ESO NTT | EMMI | $4000-10200$ | $3.3$ | $300$ | 1.09
| 2007 Mar 19.22 | 4178.72 | 63.12 | du Pont | B&C | $3419-9626$ | $3.0$ | $900$ | 1.11
| 2007 Apr 17.30 | 4207.80 | 92.04 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $600$ | 1.51
2007Y | 2007 Feb 19.04 | 4150.54 |  $-$12.39 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $600$ | 1.31
| 2007 Feb 25.02 | 4156.52 |  $-$6.44 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $600$ | 1.28
| 2007 Mar 04.05 | 4163.55 | 0.56 | ESO NTT | EMMI | $3200-10200$ | $2.1$ | $200$ | 1.76
| 2007 Mar 11.00 | 4170.50 | 7.47 | Baade | IMACS | $3789-10879$ | $2.0$ | $600$ | 1.42
| 2007 Mar 14.01 | 4173.51 | 10.47 | du Pont | B&C | $3429-9648$ | $3.0$ | $600$ | 1.62
| 2007 Mar 19.00 | 4178.50 | 15.44 | du Pont | B&C | $3419-9631$ | $3.0$ | $600$ | 1.60
| 2007 Mar 25.99 | 4185.49 | 22.39 | du Pont | B&C | $3400-9606$ | $3.0$ | $600$ | 1.73
| 2007 Apr 12.98 | 4203.48 | 40.30 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $800$ | 2.59
| 2007 Oct 21.32 | 4394.82 | 230.76 | Clay | LDSS3 | $4097-9979$ | $0.7$ | $900$ | 1.11
| 2007 Oct 23.30 | 4396.80 | 232.73 | Clay | LDSS3 | $4033-9988$ | $0.7$ | $1200$ | 1.06
| 2007 Nov 17.28 | 4421.78 | 257.60 | Baade | IMACS | $3809-10675$ | $2.0$ | $1800$ | 1.21
| 2007 Dec 01.21 | 4435.71 | 271.46 | ESO 3.6m | EFOSC | $3300-9260$ | $2.8$ | $1200$ | 1.09
2007ag | 2007 Mar 11.17 | 4170.67 | 7.23 | Baade | IMACS | $3789-10887$ | $2.0$ | $600$ | 1.59
| 2007 Mar 14.13 | 4173.63 | 10.13 | du Pont | B&C | $3429-9642$ | $3.0$ | $1200$ | 1.15
2007hn | 2007 Sep 18.20 | 4361.70 | 9.80 | du Pont | B&C | $3408-9614$ | $3.0$ | $1200$ | 1.43
| 2007 Oct 04.03 | 4377.53 | 25.21 | ESO 3.6m | EFOSC | $3300-9260$ | $2.8$ | $1200$ | 1.11
| 2007 Oct 05.05 | 4378.55 | 26.20 | Clay | LDSS3 | $4452-9303$ | $2.0$ | $1200$ | 1.10
| 2007 Nov 05.01 | 4409.51 | 56.34 | ESO 3.6m | EFOSC | $3300-9260$ | $2.7$ | $1200$ | 1.16
2007kj | 2007 Oct 03.15 | 4376.65 |  $-$4.18 | ESO 3.6m | EFOSC | $3300-9260$ | $2.8$ | $600$ | 1.36
| 2007 Oct 05.12 | 4378.63 |  $-$2.24 | Clay | LDSS3 | $4452-9303$ | $2.0$ | $600$ | 1.39
| 2007 Oct 16.18 | 4389.68 | 8.61 | du Pont | B&C | $3412-9624$ | $3.0$ | $900$ | 1.40
| 2007 Oct 21.13 | 4394.63 | 13.48 | du Pont | B&C | $3352-9563$ | $3.0$ | $1200$ | 1.35
| 2007 Nov 17.10 | 4421.60 | 39.98 | Baade | IMACS | $3809-10675$ | $2.0$ | $600$ | 1.46
2007rz | 2007 Dec 10.26 | 4444.76 | 7.35 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $700$ | 1.48
| 2008 Jan 04.12 | 4469.62 | 31.89 | ESO 3.6m | EFOSC | $3300-9260$ | $2.8$ | $600$ | 1.26
2008aq | 2008 Feb 29.31 | 4525.81 |  $-$5.30 | Baade | IMACS | $3743-10721$ | $2.0$ | $600$ | 1.06
| 2008 Mar 14.32 | 4539.82 | 8.60 | ESO 3.6m | EFOSC | $3300-9240$ | $2.7$ | $600$ | 1.12
| 2008 Mar 19.31 | 4544.81 | 13.55 | Clay | MagE | $3101-9344$ | $0.3$ | $1200$ | 1.13
| 2008 Mar 20.32 | 4545.82 | 14.55 | Clay | LDSS3 | $3629-9437$ | $2.0$ | $600$ | 1.19
| 2008 Mar 28.33 | 4553.83 | 22.50 | ESO NTT | EMMI | $3200-10200$ | $2.1$ | $300$ | 1.35
| 2008 Mar 31.29 | 4556.79 | 25.43 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $600$ | 1.18
| 2008 Apr 12.27 | 4568.77 | 37.33 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $600$ | 1.26
| 2008 Apr 26.22 | 4582.72 | 51.16 | Baade | IMACS | $4246-10374$ | $2.0$ | $600$ | 1.17
| 2008 Apr 29.19 | 4585.69 | 54.11 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $600$ | 1.13
| 2008 May 11.15 | 4597.65 | 65.98 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $600$ | 1.12
| 2008 May 22.15 | 4608.65 | 76.89 | Baade | IMACS | $4338-10868$ | $2.0$ | $600$ | 1.19
| 2009 Jan 10.28 | 4841.78 | 308.17 | Gemini-S | GMOS | $3900-8118$ | $1.4$ | $1800$ | 1.53
2008gc | 2008 Oct 14.32 | 4753.82 | 9.11 | ESO NTT | EFOSC | $3401-9210$ | $2.8$ | $1200$ | 1.26
| 2008 Oct 15.15 | 4754.65 | 9.90 | ESO NTT | EFOSC | $3351-9210$ | $2.7$ | $1200$ | 1.17
| 2008 Oct 20.22 | 4759.72 | 14.74 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $900$ | 1.10
| 2008 Oct 28.30 | 4767.80 | 22.43 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $1200$ | 1.25
| 2008 Nov 04.07 | 4774.57 | 28.89 | Clay | LDSS3 | $3636-9452$ | $2.0$ | $1000$ | 1.23
| 2009 Jan 30.04 | 4861.54 | 117.28 | Gemini-S | GMOS | $3902-8118$ | $1.4$ | $2400$ | 1.32
2008hh | 2008 Nov 23.12 | 4793.62 | 2.08 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $600$ | 1.36
| 2008 Dec 08.10 | 4808.60 | 16.78 | Baade | IMACS | $4023-10719$ | $2.0$ | $900$ | 1.41
| 2008 Dec 23.04 | 4823.54 | 31.43 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $1200$ | 1.35
2009K | 2009 Jan 17.09 | 4848.59 |  $-$18.42 | Clay | LDSS3 | $3629-9440$ | $2.0$ | $500$ | 1.15
| 2009 Jan 22.08 | 4853.58 |  $-$13.48 | Clay | LDSS3 | $3620-9428$ | $2.0$ | $300$ | 1.16
| 2009 Feb 08.13 | 4870.63 | 3.37 | Clay | LDSS3 | $3716-9438$ | $2.0$ | $500$ | 1.59
| 2009 Feb 11.07 | 4873.57 | 6.28 | Baade | IMACS | $4032-10122$ | $2.0$ | $500$ | 1.27
| 2009 Feb 25.08 | 4887.58 | 20.13 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $500$ | 1.61
| 2009 Mar 16.00 | 4906.50 | 38.83 | ESO NTT | EFOSC | $3300-9210$ | $2.7$ | $500$ | 1.57
| 2009 Mar 28.02 | 4918.53 | 50.71 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $500$ | 1.93
| 2009 Nov 17.33 | 5152.84 | 282.31 | Gemini-S | GMOS | $4099-7999$ | $1.4$ | $1350$ | 1.50
2009Z | 2009 Feb 08.35 | 4870.85 |  $-$5.95 | Clay | LDSS3 | $3716-9439$ | $2.0$ | $900$ | 1.20
| 2009 Feb 11.35 | 4873.85 |  $-$3.01 | Baade | IMACS | $4032-10120$ | $2.0$ | $800$ | 1.17
| 2009 Feb 15.37 | 4877.87 | 0.91 | Baade | IMACS | $4021-10121$ | $2.0$ | $600$ | 1.13
| 2009 Feb 24.35 | 4886.85 | 9.67 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $700$ | 1.13
| 2009 Feb 25.31 | 4887.81 | 10.61 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $900$ | 1.18
| 2009 Feb 26.37 | 4888.87 | 11.65 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $900$ | 1.14
| 2009 Mar 15.36 | 4905.86 | 28.22 | Baade | IMACS | $4026-10114$ | $2.0$ | $900$ | 1.20
| 2009 Mar 28.37 | 4918.87 | 40.92 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $900$ | 1.41
2009bb | 2009 Mar 28.19 | 4918.69 |  $-$1.35 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $700$ | 1.09
| 2009 Mar 29.10 | 4919.60 |  $-$0.45 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $900$ | 1.03
| 2009 Apr 03.14 | 4924.64 | 4.55 | du Pont | WFCCD | $3800-9235$ | $3.0$ | $700$ | 1.04
| 2009 Apr 07.00 | 4928.50 | 8.37 | Baade | IMACS | $4057-10139$ | $2.0$ | $400$ | 1.16
| 2009 Apr 15.31 | 4936.81 | 16.60 | Gemini-S | GMOS | $3904-8122$ | $1.4$ | $900$ | 1.03
| 2009 Apr 17.10 | 4938.60 | 18.37 | Clay | LDSS3 | $3714-9435$ | $2.0$ | $600$ | 1.03
| 2009 Apr 18.08 | 4939.58 | 19.34 | du Pont | B&C | $3450-9657$ | $3.0$ | $700$ | 1.02
| 2009 Apr 22.08 | 4943.58 | 23.30 | du Pont | B&C | $3360-9567$ | $3.0$ | $900$ | 1.03
| 2009 Apr 23.11 | 4944.61 | 24.32 | du Pont | B&C | $3362-9565$ | $3.0$ | $1000$ | 1.07
| 2009 Apr 26.31 | 4947.81 | 27.49 | Gemini-S | GMOS | $3904-8122$ | $1.4$ | $900$ | 1.03
| 2009 Apr 30.13 | 4951.64 | 31.28 | Clay | LDSS3 | $3702-9443$ | $2.0$ | $600$ | 1.18
| 2009 May 01.08 | 4952.58 | 32.21 | Clay | LDSS3 | $3721-9445$ | $2.0$ | $600$ | 1.05
| 2009 May 14.03 | 4965.53 | 45.04 | Baade | IMACS | $3974-10136$ | $2.0$ | $600$ | 1.04
| 2009 May 22.96 | 4974.46 | 53.88 | du Pont | B&C | $3360-9565$ | $3.0$ | $900$ | 1.02
| 2009 May 31.05 | 4982.55 | 61.89 | du Pont | B&C | $3394-9591$ | $3.0$ | $1200$ | 1.17
| 2010 Jan 09.21 | 5205.71 | 282.86 | Clay | LDSS3 | $3632-9441$ | $2.0$ | $1800$ | 1.33
| 2010 Jan 10.29 | 5206.79 | 283.93 | Gemini-S | GMOS | $4197-8424$ | $1.4$ | $4000$ | 1.02
| 2010 Feb 03.09 | 5230.59 | 307.50 | Baade | IMACS | $3975-10051$ | $2.0$ | $1800$ | 1.75
2009ca | 2009 Apr 07.42 | 4928.92 | 2.51 | Baade | IMACS | $4057-10139$ | $2.0$ | $900$ | 1.31
| 2009 Apr 17.35 | 4938.85 | 11.57 | Clay | LDSS3 | $3714-9435$ | $2.0$ | $700$ | 1.62
| 2009 Apr 22.37 | 4943.87 | 16.15 | du Pont | B&C | $3360-9567$ | $3.0$ | $900$ | 1.36
2009dp | 2009 Apr 30.33 | 4951.83 | 1.98 | Clay | LDSS3 | $3702-9445$ | $2.0$ | $600$ | 1.28
| 2009 May 01.37 | 4952.87 | 2.99 | Clay | LDSS3 | $3721-9445$ | $2.0$ | $600$ | 1.11
| 2009 Jun 02.28 | 4984.78 | 34.18 | Clay | MagE | $3181-9111$ | $0.2$ | $900$ | 1.12
2009dq | 2009 Apr 30.10 | 4951.60 |  $-$8.85 | Clay | LDSS3 | $3702-9422$ | $2.0$ | $400$ | 1.36
| 2009 May 23.05 | 4974.55 | 13.98 | du Pont | B&C | $3360-9563$ | $3.0$ | $600$ | 1.37
2009dt | 2009 Apr 30.36 | 4951.86 |  $-$6.38 | Clay | LDSS3 | $3702-9445$ | $2.0$ | $600$ | 1.44
| 2009 May 23.39 | 4974.89 | 16.41 | du Pont | B&C | $3360-9565$ | $3.0$ | $900$ | 1.06
Table 2: continued


#  Pairing effects on vorticity of incident neutron current at quasiparticle
resonance energies in $n$-$A$ elastic scattering

K. Mizuyama1,2, H. Dai Nghia3,4, T. Dieu Thuy3, N. Hoang Tung5,6, T. V. Nhan
Hao3,4 corresponding author<EMAIL_ADDRESS>1 Institute of Research and
Development, Duy Tan University, Da Nang 550000, Vietnam
2 Faculty of Natural Sciences, Duy Tan University, Da Nang 550000, Vietnam
3 Faculty of Physics, University of Education, Hue University, 34 Le Loi
Street, Hue City, Vietnam
4 Center for Theoretical and Computational Physics, University of Education,
Hue University, 34 Le Loi Street, Hue City, Vietnam
5 Department of Nuclear Physics and Nuclear Engineering, Faculty of Physics
and Engineering Physics, University of Science, Ho Chi Minh City 700000,
Vietnam
6 Vietnam National University, Ho Chi Minh City 700000, Vietnam

###### Abstract

In this study, we analyzed how the incident neutron current is affected by the
pairing effect in the neutron-nucleus scattering described within the
framework of Hartree-Fock-Bogoliubov theory by performing numerical
calculations in terms of current, vorticity, and circulation of the incident
neutron current. We found that the pairing effect on the incident neutron flux
is completely different between particle-type and hole-type quasiparticle
resonances. In the case of h-type quasiparticle resonance, the pairing acts to
prevent the neutron flux from entering the nucleus, reducing circulation. In
the case of p-type quasiparticle resonance, pairing acts to reduce circulation
at energies lower than the resonance energy, but at energies higher than the
resonance energy, the effect of pairing on the neutron flux is reversed and,
conversely, circulation is increased.

##  I Introduction

Quantum vortices have been introduced as quantized circulation in superfluids
and magnetic flux in superconductors to understand the properties of
superfluid helium, type II superconductors, Bose-Einstein condensation of
ultracold atoms, etc. feynmann ; Abrikosov ; Kennedy ; Lin ; Stringari ;
Salomaa , and have been successfully observed experimentally Matthews .
Superfluid phase transition is thought to be caused by the excitation of
quantum vortices. Also in nuclear physics, the coupling rotation of the
deformed nucleus and the intrinsic vorticity has been discussed as the
superfluidity phenomena Mikhailov ; Mikhailov2 ; Brink . The effect of
superfluid vortices in the interior of neutron stars has also been discussed
Elgaroy ; Sauls ; Baym ; Langlois .

Even if we consider it apart from the nature of superfluidity, a vortex is a
phenomenon that is easy to imagine intuitively and is very characteristic in
physics in general, as seen in typhoons and tornadoes. In nuclear physics,
vortex motion has often been proposed as one of the characteristic collective
modes of nuclei (toroidal mode) Semenko ; Ryezayeva . This is because vortex
motion has a unique topological structure and dynamic stability (as seen in
Kelvin’s circulation theorem, etc.), and can appear anywhere in a system where
there is a current.

Within the framework of Hartree-Fock-Bogoliubov (HFB) theory, the resonances
appearing in neutron-nucleus ($n$-$A$) scattering include shape resonances
formed by centrifugal barriers and quasiparticle resonances formed by pair-
correlation effects. There are two types of quasiparticle resonances: particle
and hole type quasiparticle resonances dobac ; kobayashi ; jost-hfb ; classres
. In jost-hfb ; classres , we discussed the formation conditions of the
resonances appearing in $n$-$A$ scattering and the effects of pair correlation
in terms of the S- and K- matrix poles. It is confirmed that the wave
functions of all types of resonances have metastable structures, except when
they are affected by Fano effects jost-fano  or correlations from independent
K-matrix poles. Since the width of the resonant state can be understood as the
inverse of the lifetime, it is consistent that the wave function of the
resonant state has a metastable structure and is observed as a peak with a
finite width in the cross section.

In $n$-$A$ scattering, resonances are observed as sharp peaks with small
widths at all cross sections given as a function of incident neutron energy.
It is known that the cross section satisfies the optical theorem when there is
no absorption effect on the potential in the fundamental equations such as the
Schrödinger equation and the HFB equation, which means that the neutron flux
current satisfies the continuity equation, i.e., the current is conserved.
According to Helmholtz’s theorem, the current can be divided into the current
with vortices and the current without vortices. Regardless of the conservation
laws of the current, vortices can generally always be present in the current,
unless the wavefunction (as the solution used to define the current) is
constrained to be vortex-free, or the system requires a vortex-free condition
sasaki .

Given the stability of vortices due to their special topological properties
and the metastable fundamental nature of resonance, the existence and the
contribution of vortices to the resonant state can be expected. The vortices
(vortex-like current) that can appear in the neutron flux of $n$-$A$
scattering may be different from what has been defined as a quantum vortex
(whose circulation can be quantized), but may characterize the resonances in
this system.

In this paper, we focus on the vortex (vortex-like current) of neutron flux in
$n$-$A$ scattering within the framework of HFB theory, and analyze its
relation to resonance. Especially for the quasiparticle resonances, the
relation with pairing correlations will be discussed.

##  II Theoretical and calculation background

In this paper, the HFB equation is solved by assuming the spherical symmetry.
We adopt the Woods-Saxon potential for the mean-field potential $U$ and pair
potential $\Delta$ with same parameters as in Ref.jost-hfb . The chemical
potential $\lambda=-1$ MeV is adopted in order to set the neutron-rich target
of the system.

The neutron current $j$ of the $n$-$A$ scattering system is defined as

$\displaystyle\mbox{\boldmath$j$}(\mbox{\boldmath$r$})=\mbox{ Im
}\psi^{(+)*}_{1}(\mbox{\boldmath$r$})\nabla\psi^{(+)}_{1}(\mbox{\boldmath$r$})$
(1)

by using the upper component of the HFB scattering wave function
$\psi^{(+)}_{1}(\mbox{\boldmath$r$})$. The vorticity $\omega$ for the neutron
current is defined by

$\displaystyle\mbox{\boldmath$\omega$}(\mbox{\boldmath$r$})$ $\displaystyle=$
$\displaystyle\nabla\times\mbox{\boldmath$j$}(\mbox{\boldmath$r$}).$ (2)
$\displaystyle=$
$\displaystyle\frac{1}{i}\nabla\psi^{(+)*}_{1}(\mbox{\boldmath$r$})\times\nabla\psi^{(+)}_{1}(\mbox{\boldmath$r$})$
(3)

The vorticity is a pseudovector which describes the strength and direction of
the local spinning motion. The sign of the vorticity is defined as positive
for counterclockwise spinning motion. The circulation $\Gamma$ which is
defined by

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\int_{S}d\mbox{\boldmath$S$}\cdot\mbox{\boldmath$\omega$}(\mbox{\boldmath$r$})$
(4)

is the total amount of vorticity in the area ($S$) enclosed by the closed
path. By the Stokes’ theorem, the circulation can be rewritten as

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\oint_{C}d\mbox{\boldmath$l$}\cdot\mbox{\boldmath$j$}(\mbox{\boldmath$r$})$
(5)

where $C$ represents the closed path to give the area $S$.

In this paper, the direction of the incident neutron is taken as the $z$-axis
in the system of n-A scattering (i.e. $k$ of
$\exp(i\mbox{\boldmath$k$}\cdot\mbox{\boldmath$r$})$ is chosen as
$\mbox{\boldmath$k$}=k\mbox{\boldmath$e$}_{z}$). Since the scattering behavior
of neutrons by spherical nuclei should be axially symmetric, we show the
numerical results of the current ($j$) and vorticity ($\omega$) in the $z$-$x$
plane, which is defined by setting $\varphi=0$ in spherical coordinates. To
correctly represent a plane wave
$\exp(i\mbox{\boldmath$k$}\cdot\mbox{\boldmath$r$})$ up to 20 MeV in a region
of 20 fm dynamic diameter, $l_{max}=20$ is adopted for the representation of
the partial wave components.

In the $z$-$x$ plane, the vorticity is represented as

$\displaystyle\mbox{\boldmath$\omega$}(\mbox{\boldmath$r$})$ $\displaystyle=$
$\displaystyle 2\mbox{\boldmath$e$}_{y}\mbox{ Im
}\left[(\mbox{\boldmath$e$}_{z}\cdot\nabla\psi_{1}^{(+)*}(\mbox{\boldmath$r$}))(\mbox{\boldmath$e$}_{x}\cdot\nabla\psi_{1}^{(+)}(\mbox{\boldmath$r$}))\right]_{\varphi=0}$
(6)

where $\mbox{\boldmath$e$}_{x}$, $\mbox{\boldmath$e$}_{y}$ and
$\mbox{\boldmath$e$}_{z}$ are the unit vectors for $x$-, $y$\- and $z$-axis,
respectively. Using the partial components of the scattering wave function,
$\mbox{\boldmath$e$}_{z}\cdot\nabla\psi_{1}^{(+)}(\mbox{\boldmath$r$})$ and
$\mbox{\boldmath$e$}_{x}\cdot\nabla\psi_{1}^{(+)}(\mbox{\boldmath$r$})$ are
represented as

$\displaystyle\mbox{\boldmath$e$}_{z}\cdot\nabla\psi_{1}^{(+)}(\mbox{\boldmath$r$})$
$\displaystyle=\frac{1}{r}\sum_{lj}(i)^{l}\frac{2j+1}{2}\frac{\sqrt{4\pi}}{2l+1}$
$\displaystyle\times\left[\frac{l+1}{\sqrt{2l+3}}Y_{l+1,0}(\widehat{\mbox{\boldmath$r$}})\left(\frac{\partial}{\partial
r}-\frac{l+1}{r}\right)\right.$ $\displaystyle\hskip
28.45274pt+\left.\frac{l}{\sqrt{2l-1}}Y_{l-1,0}(\widehat{\mbox{\boldmath$r$}})\left(\frac{\partial}{\partial
r}+\frac{l}{r}\right)\right]\psi_{1,lj}^{(+)}(r)$ (7)
$\displaystyle\mbox{\boldmath$e$}_{x}\cdot\nabla\psi_{1}^{(+)}(\mbox{\boldmath$r$})$
$\displaystyle=\frac{1}{r}\sum_{lj}(i)^{l}\frac{2j+1}{2}\frac{\sqrt{4\pi}}{2l+1}$
$\displaystyle\times\left[-\sqrt{\frac{(l+1)(l+2)}{2l+3}}Y_{l+1,1}(\widehat{\mbox{\boldmath$r$}})\left(\frac{\partial}{\partial
r}-\frac{l+1}{r}\right)\right.$ $\displaystyle\hskip
28.45274pt+\left.\sqrt{\frac{l(l-1)}{2l-1}}Y_{l-1,1}(\widehat{\mbox{\boldmath$r$}})\left(\frac{\partial}{\partial
r}+\frac{l}{r}\right)\right]\psi_{1,lj}^{(+)}(r).$ (8)

Note that the spin non-flip is supposed in these expressions.

We can obtain the Lippmann-Schwinger (LS) integral equation for the HFB
scattering wave function

$\displaystyle\begin{pmatrix}\psi_{1}^{(+)}(\mbox{\boldmath$r$})\\\
\psi_{2}^{(+)}(\mbox{\boldmath$r$})\end{pmatrix}=\begin{pmatrix}\psi_{0}^{(+)}(\mbox{\boldmath$r$})\\\
0\end{pmatrix}$ $\displaystyle+\int
d\mbox{\boldmath$r$}_{1}\begin{pmatrix}G_{0}^{(+)}(\mbox{\boldmath$r$},\mbox{\boldmath$r$}_{1};k_{1}(E))&0\\\
0&-G_{0}^{(+)}(\mbox{\boldmath$r$},\mbox{\boldmath$r$}_{1};k_{2}(E))\end{pmatrix}$
$\displaystyle\times\begin{pmatrix}0&\Delta(\mbox{\boldmath$r$}_{1})\\\
\Delta(\mbox{\boldmath$r$}_{1})&0\end{pmatrix}\begin{pmatrix}\psi_{1}^{(+)}(\mbox{\boldmath$r$}_{1})\\\
\psi_{2}^{(+)}(\mbox{\boldmath$r$}_{1})\end{pmatrix}$ (9)

by applying the two-potential formula, where $\psi_{0}^{(+)}$ is the Hartree-
Fock (HF) scattering wave function and the HF Green’s function which satisfies

$\displaystyle\left(\frac{\hbar^{2}k^{2}}{2m}+\frac{\hbar^{2}}{2m}\nabla^{2}-U\right)G_{0}^{(+)}(\mbox{\boldmath$r$},\mbox{\boldmath$r$}^{\prime};k)=\delta(\mbox{\boldmath$r$}-\mbox{\boldmath$r$}^{\prime}),$
(10)

$k_{1}(E)$ and $k_{2}(E)$ are defined by
$k_{1}(E)=\sqrt{\frac{2m}{\hbar^{2}}(\lambda+E)}$ and
$k_{2}(E)=\sqrt{\frac{2m}{\hbar^{2}}(\lambda-E)}$ as is introduced in
Ref.jost-hfb .

The LS equation Eq.(9) can be expressed in the form of independent integral
equations for the upper and lower components, respectively, using the HFB
Green’s function $\mathcal{G}$ and the Dyson equation that it satisfies.

The upper component can be expressed in two ways as

$\displaystyle\psi_{1}^{(+)}(\mbox{\boldmath$r$})$
$\displaystyle=\left\\{\begin{array}[]{l}\psi_{0}^{(+)}(\mbox{\boldmath$r$})\\\
+\displaystyle{\int\int
d\mbox{\boldmath$r$}_{1}d\mbox{\boldmath$r$}_{2}\mathcal{G}_{11}^{(+)}(\mbox{\boldmath$r$},\mbox{\boldmath$r$}_{1})\Sigma_{0}(\mbox{\boldmath$r$}_{1},\mbox{\boldmath$r$}_{2})\psi_{0}^{(+)}(\mbox{\boldmath$r$}_{2})}\\\
\\\ \psi_{0}^{(+)}(\mbox{\boldmath$r$})\\\ +\displaystyle{\int\int
d\mbox{\boldmath$r$}_{1}d\mbox{\boldmath$r$}_{2}G_{0}^{(+)}(\mbox{\boldmath$r$},\mbox{\boldmath$r$}_{1};k_{1})\Sigma(\mbox{\boldmath$r$}_{1},\mbox{\boldmath$r$}_{2})\psi_{0}^{(+)}(\mbox{\boldmath$r$}_{2})},\end{array}\right.$
(16)

and the lower component is expressed in a form that is computed using the
upper component as

$\displaystyle\psi_{2}^{(+)}(\mbox{\boldmath$r$})=-\int
d\mbox{\boldmath$r$}_{1}G_{0}^{(+)}(\mbox{\boldmath$r$},\mbox{\boldmath$r$}_{1};k_{2})\Delta(\mbox{\boldmath$r$}_{1})\psi_{1}^{(+)}(\mbox{\boldmath$r$}_{1}),$
(17)

with $\Sigma_{0}$ and $\Sigma$ which are defined by

$\displaystyle\Sigma_{0}(\mbox{\boldmath$r$},\mbox{\boldmath$r$}^{\prime})$
$\displaystyle=$
$\displaystyle-\Delta(\mbox{\boldmath$r$})G_{HF}^{(+)}(\mbox{\boldmath$r$},\mbox{\boldmath$r$}^{\prime};k_{2})\Delta(\mbox{\boldmath$r$}^{\prime})$
(18) $\displaystyle\Sigma(\mbox{\boldmath$r$},\mbox{\boldmath$r$}^{\prime})$
$\displaystyle=$
$\displaystyle\Delta(\mbox{\boldmath$r$})\mathcal{G}_{22}^{(+)}(\mbox{\boldmath$r$},\mbox{\boldmath$r$}^{\prime})\Delta(\mbox{\boldmath$r$}^{\prime}),$
(19)

where $\mathcal{G}_{11}^{(+)}$ and $\mathcal{G}_{22}^{(+)}$ are the diagonal
components of the HFB Green’s function which is expressed in $4\times 4$
matrix form. It should be noticed that Eq.(18) is the first term to appear in
the Dyson series expansion of Eq.(19) like $\Sigma=\Sigma_{0}+\cdots$.

Of course, it is obvious that $\Sigma_{0}$ and $\Sigma$ would behave as the
non-local potential which originates from the pair correlation, and it is
important to analyze the role of its non-locality, but we will not discuss it
yet in this paper.

In the HFB theory, it is known that quasiparticle states exist symmetrically
in the positive and negative regions of quasiparticle energy. This is also
true for the resonances (or more precisely, the poles of the S-matrix
corresponding to the resonances) jost-hfb . There are two types of
quasiparticle resonances, particle-type (p-type) and hole-type (h-type), both
of which exist in both the positive and negative quasiparticle energy regions.

By performing the similar approximate calculations to derive the spectral
representation of the HFB Green function as shown in Ref.jost-fano , it is
possible to confirm that in the positive energy region ($E>0$), a particle-
type resonance behaves as a pole of $\mathcal{G}_{11}$ and a hole-type
resonance behaves as a pole of $\mathcal{G}_{22}$, i.e.

$\displaystyle\langle p|\mathcal{G}_{11}|p\rangle$ $\displaystyle\sim$
$\displaystyle\frac{1}{E-E_{qp}^{(p)}}$ (20) $\displaystyle\langle
h|\mathcal{G}_{22}|h\rangle$ $\displaystyle\sim$
$\displaystyle\frac{1}{E-E_{qp}^{(h)}}$ (21)

where $E_{qp}^{(p)}$ and $E_{qp}^{(h)}$ are the S-matrix poles for the
particle-type and hole-type quasiparticle resonance energy which are given as
the complex number. The detail expressions of $E_{qp}^{(p)}$ and
$E_{qp}^{(h)}$ are given in Refs.jost-hfb ; classres . This is the origin of
the metastable structure of the scattering wave function at the resonance
energy.

By inserting Eq.(16) into Eqs.(1), we can divide the current $j$ into three
terms as

$j$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$j$}_{0}+\mbox{\boldmath$j$}_{int}+\delta\mbox{\boldmath$j$}$
(22)

with

$\displaystyle\mbox{\boldmath$j$}_{0}$ $\displaystyle=$ $\displaystyle\mbox{Im
}\psi_{0}^{(+)*}\nabla\psi_{0}^{(+)}$ (23)
$\displaystyle\mbox{\boldmath$j$}_{int}$ $\displaystyle=$
$\displaystyle\mbox{Im
}\left[\delta\psi^{(+)*}\nabla\psi_{0}^{(+)}+\psi_{0}^{(+)*}\nabla\delta\psi^{(+)}\right]$
(24) $\displaystyle\delta\mbox{\boldmath$j$}$ $\displaystyle=$
$\displaystyle\mbox{Im }\delta\psi^{(+)*}\nabla\delta\psi^{(+)}$ (25)

where $\delta\psi^{(+)}$ is the second term of Eq.(16) which includes the pair
correlation, $\mbox{\boldmath$j$}_{0}$ is the HF current,
$\mbox{\boldmath$j$}_{int}$ is the current which represents the interference
between the HF scattering wavefunction $\psi^{(+)}_{0}$ and
$\delta\psi^{(+)}$, and $\delta\mbox{\boldmath$j$}$ is the current formed by
the scattered wave by the pair potential in the HF mean field
($\delta\psi^{(+)}$).

By using Eqs.(22), the vorticity is also divided into three terms as

$\omega$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\omega$}_{0}+\mbox{\boldmath$\omega$}_{int}+\delta\mbox{\boldmath$\omega$}$
(26)

with

$\displaystyle\mbox{\boldmath$\omega$}_{0}$ $\displaystyle=$
$\displaystyle\nabla\times\mbox{\boldmath$j$}_{0}$ (27) $\displaystyle=$
$\displaystyle 2\mbox{\boldmath$e$}_{y}\mbox{ Im
}\left[(\mbox{\boldmath$e$}_{z}\cdot\nabla\psi^{(+)*}_{0})(\mbox{\boldmath$e$}_{x}\cdot\nabla\psi^{(+)}_{0})\right]$
$\displaystyle\mbox{\boldmath$\omega$}_{int}$ $\displaystyle=$
$\displaystyle\nabla\times\mbox{\boldmath$j$}_{int}$ (28) $\displaystyle=$
$\displaystyle 2\mbox{\boldmath$e$}_{y}\mbox{ Im
}\left[(\mbox{\boldmath$e$}_{z}\cdot\nabla\delta\psi^{(+)*})(\mbox{\boldmath$e$}_{x}\cdot\nabla\psi^{(+)}_{0})\right.$
$\displaystyle\hskip
10.0pt-\left.(\mbox{\boldmath$e$}_{x}\cdot\nabla\delta\psi^{(+)*})(\mbox{\boldmath$e$}_{z}\cdot\nabla\psi^{(+)}_{0})\right]$
$\displaystyle\delta\mbox{\boldmath$\omega$}$ $\displaystyle=$
$\displaystyle\nabla\times\delta\mbox{\boldmath$j$}$ $\displaystyle=$
$\displaystyle 2\mbox{\boldmath$e$}_{y}\mbox{ Im
}\left[(\mbox{\boldmath$e$}_{z}\cdot\nabla\delta\psi^{(+)*})(\mbox{\boldmath$e$}_{x}\cdot\nabla\delta\psi^{(+)})\right].$

$\delta\mbox{\boldmath$j$}$ and $\delta\mbox{\boldmath$\omega$}$ are expected
to be smaller in absolute value (than $\mbox{\boldmath$j$}_{int}$ and
$\mbox{\boldmath$\omega$}_{int}$) because they are quantities that give
higher-order pairing contributions. However, the continuity equation for
$\delta\mbox{\boldmath$j$}$ and $\mbox{\boldmath$j$}_{int}$ is given by

$\displaystyle\nabla\cdot(\mbox{\boldmath$j$}_{int}+\delta\mbox{\boldmath$j$})=0$
(30)

since $j$ and $\mbox{\boldmath$j$}_{0}$ satisfy
$\nabla\cdot\mbox{\boldmath$j$}=0$ and $\nabla\cdot\mbox{\boldmath$j$}_{0}=0$,
respectively. Eq.(30) is shown to say
$\nabla\cdot\mbox{\boldmath$j$}_{int}\neq 0$ as long as
$\delta\mbox{\boldmath$j$}\neq 0$, $\nabla\cdot\mbox{\boldmath$j$}_{int}\neq
0$ implies that $\mbox{\boldmath$j$}_{int}$ contains the “gushing” or
“suctioning” motion of the current. If we define

$\displaystyle\mbox{\boldmath$j$}_{pair}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$j$}_{int}+\delta\mbox{\boldmath$j$}=\mbox{\boldmath$j$}-\mbox{\boldmath$j$}_{0}$
(31)

as the current represents the pairing contribution,
$\mbox{\boldmath$j$}_{pair}$ doesn’t include “gushing” or “suctioning” motion,
i.e. $\nabla\cdot\mbox{\boldmath$j$}_{pair}=0$. The vorticity which
corresponds to $\mbox{\boldmath$j$}_{pair}$
($\mbox{\boldmath$\omega$}_{pair}$) is also defined as

$\displaystyle\mbox{\boldmath$\omega$}_{pair}$ $\displaystyle=$
$\displaystyle\nabla\times\mbox{\boldmath$j$}_{pair}$ (32) $\displaystyle=$
$\displaystyle\mbox{\boldmath$\omega$}_{int}+\delta\mbox{\boldmath$\omega$}=\mbox{\boldmath$\omega$}-\mbox{\boldmath$\omega$}_{0}$
(33)

The approximated expression of $\mbox{\boldmath$\omega$}_{pair}$ near the
quasiparticle resonance energy can be represented as

$\displaystyle\mbox{\boldmath$\omega$}_{pair}$ $\displaystyle\sim$
$\displaystyle\left\\{\begin{array}[]{l}\mbox{[For particle-type quasiparticle
resonance]}\\\ \displaystyle{\frac{(E-\mbox{Re }E_{qp}^{(p)})\mbox{ Re
}\langle p|\Sigma_{0}^{(1)}|\psi^{(+)}_{0}\rangle}{(E-\mbox{Re
}E_{qp}^{(p)})^{2}+(\mbox{Im }E_{qp}^{(p)})^{2}}2\mbox{\boldmath$e$}_{y}\mbox{
Im
}\left[(\mbox{\boldmath$e$}_{z}\cdot\nabla\phi_{p}^{*})(\mbox{\boldmath$e$}_{x}\cdot\nabla\psi^{(+)}_{0})-(\mbox{\boldmath$e$}_{x}\cdot\nabla\phi_{p}^{*})(\mbox{\boldmath$e$}_{z}\cdot\nabla\psi^{(+)}_{0})\right]}\\\
+(\mbox{Other partial wave components of
}\mbox{\boldmath$\omega$}_{int})+\delta\mbox{\boldmath$\omega$}\\\ \\\
\mbox{[For hole-type quasiparticle resonance]}\\\
\displaystyle{-\frac{(\mbox{Im }E_{qp}^{(h)})^{2}}{(E-\mbox{Re
}E_{qp}^{(h)})^{2}+(\mbox{Im
}E_{qp}^{(h)})^{2}}\mbox{\boldmath$\omega$}_{0}+(\mbox{Other partial wave
components of
}\mbox{\boldmath$\omega$}_{int})+\delta\mbox{\boldmath$\omega$}}\end{array}\right.$
(40)

The first term of Eq.(40) can be obtained as a leading order term of
$\mbox{\boldmath$\omega$}_{int}$ by applying Eqs.(20) and (21) to Eq.(16). The
contribution of other partial wave components of
$\mbox{\boldmath$\omega$}_{int}$ and $\delta\mbox{\boldmath$\omega$}$ are
expected to be very small compared to that of the first term. From Eq.(40), we
can expect that the contribution of pairing to vorticity is quite different
between p-type and h-type quasiparticle resonance. In the case of p-type, the
positive and negative contributions of $\mbox{\boldmath$\omega$}_{pair}$ are
reversed depending on whether $E$ is greater or less than $E_{qp}$. In the
case of h-type, if we can ignore all but the first term (leading term), the
pairing effect leads to zero vorticity, i.e. the vortices disappear at
$E\sim\mbox{ Re }E_{qp}^{(h)}$.

Figure 1: (Color online) The neutron currents (a)$j$,
(c)$\mbox{\boldmath$j$}_{0}$ and (e)$\mbox{\boldmath$j$}_{pair}$ at
$\epsilon=3.94$ MeV (a corresponding energy to the h-type $d_{3/2}$ resonance)
with $\lambda=-1.0$ MeV and $\langle\Delta\rangle=3.0$ MeV. The panels (b),
(d) and (f) are the corresponding vortices to the currents (a), (c) and (e),
respectively. The arrows length of the currents are represented by the same
length. The strength (absolute value) of the current which is normalized by
$k\left(=\sqrt{\frac{2m}{\hbar^{2}}(\lambda+E)}\right)$ (the absolute value of
the plane wave) is represented by color. The unit of the vorticity is fm2. The
black solid semicircle shows the size of the target nucleus given by
$r_{0}A^{\frac{1}{3}}=3.66$ fm.

##  III Numerical results

In Fig.1, the neutron currents ($j$, $\mbox{\boldmath$j$}_{0}$, and
$\mbox{\boldmath$j$}_{pair}$) and the vorticities ($\omega$,
$\mbox{\boldmath$\omega$}_{0}$ and $\mbox{\boldmath$\omega$}_{pair}$) at the
energy corresponding to the h-type $d_{3/2}$ resonance are shown in the left
and right panels, respectively.

As given in Eqs.(4) and (5), the circulation ($\Gamma$, $\Gamma_{0}$ and
$\Gamma_{pair}$) are calculated by the area integration of $\omega$,
$\mbox{\boldmath$\omega$}_{0}$ and $\mbox{\boldmath$\omega$}_{pair}$ or line
integration of $j$, $\mbox{\boldmath$j$}_{0}$, and
$\mbox{\boldmath$j$}_{pair}$. Since the current is symmetric about the z-axis,
the region of the upper half of $z$-$x$ plane $0\leq x\leq 10$ fm, $-10\leq
z\leq 10$ fm (the area showing current and vorticity in Fig.1) is adopted as
the integration area for the calculation of the circulation.

Figure 2: (Color online) The contour plot of $\Gamma$ as a function of
$\epsilon$ and $\langle\Delta\rangle$. The dotted curves represent the real
part of the S-matrix poles for the quasiparticle resonances which were shown
in Ref.jost-hfb .

In Fig.2, the contour plot of the circulation $\Gamma$ is shown as a function
of the energy $\epsilon$ and mean pairing gap $\langle\Delta\rangle$. The
black dotted curves are the real part of the S-matrix poles for the
quasiparticle resonances shown in Figs.6 and 7 in Ref.classres . The
curvilinear pattern in the contour plot of the circulation coincides with the
curve of the real part of the pole of the S-matrix of the quasiparticle
resonance, indicating that the circulation of the incident neutron flux in
$n$-$A$ scattering is greatly affected by the pairing effect at the
quasiparticle resonance energy.

Figure 3: (Color online) The circulation obtained by integrating the area of
the region $0\leq x\leq 10$ fm, $-10\leq z\leq 10$ fm (the area as shown in
Fig.1), or by line integrating this region in a counterclockwise direction.
$\Gamma$(red solid), $\Gamma_{0}$(black dash), $\Gamma_{pair}$(blue dot) and
$\delta\Gamma$(purple dot-dash) are the circulations which are corresponding
to $\omega$, $\mbox{\boldmath$\omega$}_{0}$, $\mbox{\boldmath$\omega$}_{pair}$
and $\delta\mbox{\boldmath$\omega$}$, respectively. The lower panel shows an
enlargement of the region $0\leq\epsilon\leq 2$ MeV to make it easier to see
the circulation for the p-type resonances.

In Fig.3, $\Gamma$, $\Gamma_{0}$ and $\Gamma_{pair}$ calculated with
$\langle\Delta\rangle=3.0$ MeV are plotted as a function of the energy
$\epsilon$, and $\delta\Gamma$ which is obtained from
$\delta\mbox{\boldmath$\omega$}$(or $\delta\mbox{\boldmath$j$}$) is also shown
together in order to see the contribution. In the previous section, we
expected that $\delta\mbox{\boldmath$\omega$}$ would be negligible as a
contribution to $\mbox{\boldmath$\omega$}_{pair}$ because it provides a
higher-order contribution of the pairing. The correctness of our expectation
is confirmed by the fact that $\delta\Gamma$, shown as a purple dotted curve
in Fig.3, is actually negligibly small. The characteristics of the p-type and
h-type quasiparticle resonances for $\mbox{\boldmath$\omega$}_{pair}$ shown in
Eq.(40) are clearly expressed in the energy dependence of the circulation
$\Gamma_{pair}$ in Fig.3.

As shown in Eq.(40), the $\Gamma_{pair}$ calculated from
$\mbox{\boldmath$\omega$}_{pair}$ takes the form of a Lorentz distribution and
acts to reduce $\Gamma$ near the h-type quasiparticle resonance. From the
current point of view, it can be seen from Fig.1 that the pairing effect,
$\mbox{\boldmath$j$}_{pair}$, occurs in the direction of counteracting the
current $\mbox{\boldmath$j$}_{0}$ in the nucleus, and acts to reduce the
vorticity $\omega$. In the case of the h-type quasiparticle resonances, the
pairing correlation works to create a flow along the surface while preventing
the incident neutron current from entering the target nucleus.

As indicated in Eq.(40), the effect of pairing on p-type quasiparticle
resonances is quite different. In the case of p-type quasiparticle resonance,
the pairing acts to reduce the circulation $\Gamma$ on the lower side of the
resonance energy (the real part of the S-matrix pole), but on the contrary, it
acts to increase the circulation $\Gamma$ in the energy region higher than the
resonance energy.

Figure 4: (Color online) The current $\mbox{\boldmath$j$}_{pair}$ and
vorticity $\mbox{\boldmath$\omega$}_{pair}$ at $\epsilon=1.45-1.16\times
10^{-2}$, $1.45$ and $1.45+1.16\times 10^{-2}$ MeV. $\epsilon=1.45$ MeV is the
energy at which $\Gamma_{pair}$ becomes zero (found near the real part of the
S-matrix pole of the p-type $f_{7/2}$ resonance), and $\pm 1.16\times 10^{-2}$
MeV is the imaginary part of the S-matrix pole (half width of the resonance).

Fig.4 shows the currents $\mbox{\boldmath$j$}_{pair}$ and their vorticity
$\mbox{\boldmath$\omega$}_{pair}$ for the p-type quasiparticle resonance at
$f_{7/2}$. Panels (a), (c), and (e) show the currents at
$\epsilon=1.45-1.16\times 10^{-2}$, $1.45$, and $1.45+1.16\times 10^{-2}$ MeV,
respectively, and panels (b), (d), and (f) show the corresponding vorticity,
where $1.45$ MeV is the quasiparticle energy (the real part of the S-matrix
pole of $f_{7/2}$), and $1.16\times 10^{-2}$ MeV is the half width of the
resonance (the imaginary part of the S-matrix pole).

In Fig.4, we can see the obvious vortex that the current creates. In panel
(a), there is a clockwise vortex in the $z>0$ region and a counterclockwise
vortex in the $z<0$ region. In panel (e), these vortices are reversed. Since
the vortices in the $z>0$ region are stronger in both cases, the vorticity is
negative in panel (a) and positive in panel (e). In panel (c), the weak
vortices are symmetrically located inside and outside the nucleus, so that the
overall vorticity is almost zero. In panel (c), the weak vortices are
symmetrically located inside and outside the nucleus, so that the overall
vorticity is almost zero. It should be noted that, since the current is
symmetric about the $z$-axis, the vortex is a torus-shaped vortex (like a
donut about the $z$-axis).

As shown in Ref.classres , the $f_{7/2}$ p-type quasiparticle resonance is a
resonant state that appears at energies lower than the centrifugal barrier,
and the behavior of the $f_{7/2}$ partial wave component of the scattering HFB
wave function is more like a bound state than a metastable state. Since
$\mbox{\boldmath$j$}_{int}$ is dominant in $\mbox{\boldmath$j$}_{pair}$
defined in Eq.(31) and $\mbox{\boldmath$j$}_{int}$ is defined in Eq.(24), the
vortices arise from the interference between the partial wave component of the
HFB wave function for $f_{7/2}$ and the HF scattering wave function.

By carefully observing the current flow in Fig.4, we can see that in panel
(a), the current is only allowed to enter the nucleus from the front ($z>0$
region, $\theta\sim 0$), while in panel (e), the current is allowed to enter
from the back region ($z<0$ region). Qualitatively, this is a common property
of p-type quasiparticle resonances, although the angle at which the current is
allowed to flow into the nucleus from back region and the formation of
vortices are different for $p_{3/2}$ and $f_{7/2}$.

In Ref.classres , we showed that when the independent K-matrix pole is located
near the quasiparticle resonance, the correlation between the K-matrix pole of
the quasiparticle resonance and the independent K-matrix pole causes the
K-matrix pole to disappear when the pairing gap $\langle\Delta\rangle$ becomes
larger than the critical gap $\langle\Delta\rangle_{c}$. We also showed that
the critical gap of the p-type $p_{3/2}$ is $\langle\Delta\rangle_{c}=4.08$
MeV.

Figure 5: (Color online) The pairing dependence of the circulations $\Gamma$
and $\Gamma_{pair}$ around the energy region for the p-type $p_{3/2}$
resonance. The solid blue, red and purple curves represent the circulation
with $\langle\Delta\rangle=2.0$, $3.0$ and $4.0$ MeV, respectively. The dotted
curves shows the circulation calculated every $0.2$ MeV.

Fig.5 shows how the circulations $\Gamma$ and $\Gamma_{pair}$ change when the
pairing gap is varied around the energy region of the p-type $p_{3/2}$
quasiparticle resonance. From the analysis of this figure, we found that the
energy for which $\Gamma=0$ is found almost at the same energy of the real
part of the S-matrix pole, and the energy for which $\Gamma_{pair}=0$ is found
almost at the same energy with the K-matrix pole when
$\langle\Delta\rangle<\langle\Delta\rangle_{c}$, but no longer exists when
$\langle\Delta\rangle>\langle\Delta\rangle_{c}$.

##  IV Summary

In this paper, we analyze the effect of pairing on incident neutron current in
$n$-$A$ scattering described within the framework of HFB theory. For this
purpose, the current was defined using the HFB scattering wave function and
its numerical calculation was performed. To analyze the effect of pairing, the
current was decomposed into HF part and pairing part, and the corresponding
vorticity and circulation were also calculated and analyzed.

It was found that the pairing effects on current, vorticity, and circulation
are quite different between p-type and h-type quasiparticle resonances. In the
case of the h-type quasiparticle resonances, the pairing has the effect of
preventing the neutron current from flowing into the nucleus, which reduces
vorticity and circulation. In the case of the p-type quasiparticle resonance,
the pairing acts to prevent neutron current from entering the nucleus in the
energy region below the resonance energy, while the pairing acts to allow
neutron current from the backward region to enter in the energy region above
the resonance energy. Vortices appear due to the presence of metastable
structures of a certain partial wave component, but the direction of the
vortices is reversed before and after the resonance energy. These features are
consistent with the results of Eq.(40), which shows how the S-matrix poles of
p-type and h-type quasiparticle resonances contribute to the vorticity.

We also found that the effect of the disappearance of the K-matrix pole due to
the correlation between the independent K-matrix pole near the quasiparticle
resonance and the K-matrix of the quasiparticle resonance (which has been
discussed in Ref.classres ) appears in the form of the disappearance of the
energy which satisfies $\Gamma_{pair}=0$.

In order to discuss the dynamic features and properties of vortices in quantum
systems in more detail, it is necessary to derive the fluid equation based on
the Schrödinger equation (including the HFB equation, etc.) and discuss it in
terms of the quantum pressure term that will appear in the fluid equation,
touching on the violation of Kelvin’s circulation theorem and the hydrodynamic
conservation law. However, we leave it for future work without touching on it
in this paper.

## Acknowledgments

This work is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under grant number “103.04-2019.329”. T. V. Nhan Hao
acknowledges the partial support of the Hue University under the Core Research
Program, Grant No. NCM.DHH.2018.09. Hoang Dai Nghia is funded by Vingroup JSC
and supported by the Master, PhD Scholarship Programme of Vingroup Innovation
Foundation (VINIF), Institute of Big Data, code VINIF.2021.ThS.32.

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