# On the Anti-Correlation between Duration and Redshift in Gamma-ray Bursts.

Nicole Lloyd-Ronning Los Alamos National Lab, Los Alamos, NM, USA 87545
Department of Math, Enginnering, & Science, University of New Mexico, 4000
University Dr., Los Alamos, NM, USA 87544  Jarrett Johnson  Los Alamos
National Lab, Los Alamos, NM, USA 87545  Roseanne M. Cheng Los Alamos National
Lab, Los Alamos, NM, USA 87545  Ken Luu  San Francisco State University, San
Francisco, CA 94132  Los Alamos National Lab, Los Alamos, NM, USA 87545
Phoebe Upton Sanderbeck  Los Alamos National Lab, Los Alamos, NM, USA 87545
Lailani Kenoly  Los Alamos National Lab, Los Alamos, NM, USA 87545  Celia
Toral  Cornell University, Ithaca, NY, 14850  Los Alamos National Lab, Los
Alamos, NM, USA 87545

###### Abstract

For gamma-ray bursts (GRBs) with durations greater than two seconds (so-called
long GRBs), the intrinsic prompt gamma-ray emission appears, on average, to
last longer for bursts at lower redshifts. We explore the nature of this
duration-redshift anti-correlation, describing systems and conditions in which
this cosmological evolution could arise. In particular, we explore its
dependence on metallicity of a massive star progenitor, as we can securely
count on the average stellar metallicity to increase with decreasing redshift.
Although higher metallicity/lower redshift stars lose mass and angular
momentum through line-driven winds, in some cases these stars are able to form
more extended accretion disks when they collapse, potentially leading to
longer duration GRBs. We also examine how this duration-redshift trend may
show up in interacting binary models composed of a massive star and compact
object companion, recently suggested to be the progenitors of radio bright
GRBs. Under certain conditions, mass loss and equation of state effects from
higher metallicity, lower redshift massive stars can decrease the binary
separation. This can then lead to spin-up of the massive star and allow for a
longer duration GRB upon the massive star’s collapse. Finally, the duration-
redshift trend may also be supported by a relatively larger population of
small-separation binaries born in situ at low redshift.

##  1 Introduction

There is strong evidence that most long gamma-ray bursts (GRBs), those with
prompt gamma-ray durations lasting more than about two seconds, are associated
with the collapse of a massive star. Observationally, there exist definitive
associations of Type Ibc supernovae with long GRBs (Galama et al., 1998;
Hjorth et al., 2003; Hjorth & Bloom, 2012). Theoretically, such progenitors
provide the necessary energy reservoir over the relevant timescales (Woosley,
1993; MacFadyen & Woosley, 1999a; Woosley & MacFadyen, 1999; Woosley & Bloom,
2006; Woosley & Heger, 2006; Kumar et al., 2008).

However, the details of GRB progenitors are still far from resolved. The
central engine is believed to be a highly spinning black hole-disk
system111There are also viable models with a neutron star as the central
engine (e.g, Lin et al., 2020). that launches a very relativistic jet.
Internal dissipation processes in the jet produce the prompt gamma-ray
emission, while the subsequent afterglow (emitted across the electromagnetic
spectrum) results from the jet front’s interaction with the external medium.
But which massive stars are capable of producing such powerful jets and why
(i.e. what are the key parameters that allow certain massive stars to create a
GRB)? One key approach to answering this question is to examine the evolution
of GRB properties over cosmic time, and attempt to understand that evolution
in the context of particular progenitor systems. For example, the evolution of
GRB jet opening angle with redshift (Lü et al., 2012; Laskar et al., 2014,
2018a, 2018b; Lloyd-Ronning et al., 2019b; Chakraborty et al., 2022) connects
GRBs with massive stars in a unique way. Lloyd-Ronning et al. (2020) showed
that the smaller jet opening angles at higher redshifts can be quantitatively
explained by a massive star’s envelope collimating the jet, and is consistent
with evolving IMF models reported in the literature (e.g. Davé, 2008).

In this vein, we examine the nature of the evolution of GRB prompt duration
over redshift. Taking care to account for selection effects, Lloyd-Ronning et
al. (2019a) found a statistically significant anti-correlation between
intrinsic duration (that is, duration corrected for cosmological time
dilation) and redshift in a sample of 376 long GRBs. Lloyd-Ronning et al.
(2019b) confirmed this in a sub-sample of radio bright GRBs with isotropic
equivalent energies above $10^{52}$ ergs. Recently Chakraborty et al. (2022)
further confirmed this trend, employing different techniques to account for
data selection effects that could have artificially produced this intrinsic
duration-redshift anti-correlation.

How might this cosmological evolution of GRB prompt duration arise? We might,
at first glance, expect that it is the lower metallicity stars (at high
redshift) that would produce longer duration GRBs; higher metallicity massive
stars (at lower redshifts) lose more mass and angular momentum (through line-
driven winds) relative to lower metallicity stars (e.g. Puls et al., 2008).
Under the assumption that higher angular momentum systems are capable of
producing longer duration GRBs (Proga & Begelman, 2003; Janiuk & Proga, 2008;
Lloyd-Ronning, 2022), we would then naively expect to see, on average, longer
duration GRBs at higher redshifts. That the opposite trend is seen in the data
suggests this expectation warrants closer examination.

In this paper, we qualitatively consider different long GRB progenitor
scenarios in which this unexpected correlation could arise, focusing
particularly on the role metallicity plays in determining the properties of
these systems and their post-collapse GRB jets. We first examine the presence
of this correlation in the context of a single star collapse. We then consider
progenitor systems with a massive star collapsing in an interacting binary
system, and examine whether the evolution of binary properties with redshift
can account for the observed duration-redshift anti-correlation. Our paper is
organized as follows. In §2, we discuss the data and summarize the evidence
for this correlation. Then, in §3, we describe how single star progenitors
might produce this duration-redshift trend. Recognizing the strong assumptions
necessary in single star progenitor models for long GRBs, §4 is devoted to
exploring how this trend may show up in binary progenitor models for long
GRBs. In §5, we present discussion and summarize our conclusions.

Figure 1: Intrinsic gamma-ray duration (corrected for cosmological time
dilation) as a function of redshift for 376 GRBs with redshift measurements.
The data are from Lloyd-Ronning et al. (2019b) and binned every 20 GRBs, for
clarity.

##  2 Observed Correlation

Lloyd-Ronning et al. (2019a) looked at a sample of 376 long GRBs with measured
redshifts, and corrected the durations for cosmological time dilation to get
the intrinsic duration: $T_{\rm int}=T_{90}/(1+z)$, where $T_{90}$ is the
measured duration of $90\%$ of the photon counts and $z$ is redshift. They
found $T_{\rm int}$ appears on average to decrease with redshift - that is,
gamma-ray bursts’ intrinsic prompt durations are on average longer at lower
redshifts. It is important to consider whether this correlation has a true
physical origin or is a result of observational selection effects. For
example, for a given luminosity, a high redshift GRB will have a lower flux
and some of the emitted flux could in principle be “shifted” beneath the
detector noise, making the GRB time profile appear shorter than it really is
(i.e. with an apparent smaller duration; a recent example of this effect in
the context of ultra-long duration GRBs is found in Huang et al., 2022).
However, GRBs are not standard candles in their luminosities and in fact have
much scatter in this variable; in other words, a higher redshift GRB does not
necessarily imply a lower flux, and therefore this effect is substantially
mitigated (Lee et al., 2000). In fact, many studies have suggested GRBs are
brighter at high redshifts, which would further mitigate (if not eliminate)
this selection effect. Nonetheless, taking a conservative approach to account
for such biases using Lyndell-Bell (Lynden-Bell, 1971) and Efron-Petrosian
(Efron & Petrosian, 1992) non-parametric methods, Lloyd-Ronning et al. (2019a)
found a statistically significant ($\sim 4\sigma$) anti-correlation between
intrinsic duration and redshift: $T_{\rm int}\propto(1+z)^{-0.8\pm 0.25}$.
This correlation is shown in Figure 1, with data points binned (every 20 GRBs)
for clarity.

This correlation shows up even more prominently in certain sub-samples of long
GRBs. Lloyd-Ronning et al. (2019b) found this correlation exists with even
higher statistical significance in radio bright GRBs (that is, those GRBs for
which radio follow-up was attempted and an afterglow was detected; note that
in their analysis, they examined only those GRBs with isotropic equivalent
energies above $10^{52}$ ergs). In their radio bright sub-sample consisting of
79 GRBs, they found $T_{\rm int}\propto(1+z)^{-1.4\pm 0.3}$. Recently,
Chakraborty et al. (2022) re-analyzed this correlation with an updated, larger
sample of radio bright and radio dark GRBs, using a variety of approaches to
account for selection effects. Even with fairly stringent and conservative
assumptions about the biases, they found the correlation still exists, with
$T_{\rm int}\propto(1+z)^{-1.3\pm 0.3}$ for radio-bright GRB sample and
$T_{\rm int}\propto(1+z)^{-1.2\pm 0.4}$ for the radio dark sample.

The presence of this correlation, if indeed physical, can offer important
clues to the progenitor systems of long GRBs. Because the duration of a GRB
(in the context of a black hole-disk central engine) is roughly proportional
to the amount of mass in the disk divided by its accretion rate (i.e. $T_{\rm
int}\sim M/\dot{M}$), longer duration GRBs at lower redshifts imply these
systems have more massive accretion disks, lower accretion rates, or both. Of
course, extrapolating from the final state of the progenitor star (or stars)
to the black hole-disk system that produces the GRB jet is not straightforward
(Halevi et al., 2022). However, we might expect general trends to hold - for
example, a more extended massive star and/or one with an overall higher
angular momentum reservoir may be able to produce a longer-lived jet upon
collapse, leading to a longer duration GRB (see the recent arguments in Lloyd-
Ronning, 2022).

##  3 Single Star Models

A single massive star collapsing to a black hole-disk central engine - the so-
called collapsar model (Woosley, 1993; MacFadyen & Woosley, 1999b; Woosley &
MacFadyen, 1999) \- has for some time been the canonical model for long gamma-
ray bursts. As long as the star can retain enough angular momentum to launch a
jet (with enough momentum to punch through any existing stellar envelope),
this progenitor provides the necessary overall energy budget to power a long
GRB. Therefore, we first want to consider how the duration-redshift anti-
correlation might arise in isolated stellar systems, particularly as a
function of metallicity (again, because this securely increases with
decreasing redshift). As mentioned in the introduction, we might naively
expect that, because higher metallicity stars lose more mass and angular
momentum compared to lower metallicity stars (Woosley & Heger, 2006), they
would produce on average shorter duration GRBs, opposite to the trend we see
in the data (although see Maeder (2002) who show that anisotropic -
specifically, polar enhanced - mass loss leads to an increase in stellar
rotational velocity; and indeed Olivier et al. (2022) show observations of an
80 $M_{\odot}$ star undergoing such bipolar mass loss).

However, higher metallicity stars are also, on average, less dense/have larger
radii than low metallicty stars. This is established observationally in dwarf
stars (Houdebine, 2008)222Although see Boyajian et al. (2012) who do not find
a strong dependence on metallicity for many physical properties of low-mass
stars.. In the case of high mass stars, although secure measurements of
metallicity and radius are more difficult to come by, Farrell et al. (2022)
have shown through a number of detailed numerical simulations of massive star
evolution, that lower metallicity stars are indeed more compact due to lower
opacity in the stellar envelope as well as lower CNO abundances in the burning
regions of the star. Sanyal et al. (2017) explore the metallicity dependence
of envelope inflation in massive stars and find, consistent with expectations,
that, for a given mass, low metallicity stars haven smaller radii relative to
high metallicity stars (see their Figures 2 and 11).333We note the results of
Chun et al. (2018) whose simulations of red supergiants with 1D stellar
evolution codes show that the final radii of these stars’ hydrogen envelopes
are largely independent of metallicity (over the range of Z=0.004 to 0.04).
However, GRB models (and observations of their coincident Type Ic supernovae)
generally require a stripped hydrogen envelope, and so these models may not
apply to our study. Xin et al. (2022) make an estimate of a $15M_{\odot}$
star’s radius dependence on metallicity, modelling the star with the stellar
evolution code MESA, and also see a significant increase in the stellar radius
with metallicity, over the range of metallicities they consider ($Z_{\odot}$
between $0.001$ and $0.4$). Tout et al. (1996); Maccarone et al. (2004) found
that the radius of a star is proportional to metallicity roughly to the $1/8$
power.

What are implications of this for single star progenitors of long GRBs (and,
in particular, for the lifetime of the jets they create and therefore the
timescale of the prompt emission)? We can get a handle on the duration of the
GRB jet by estimating the interval over which stellar material feeds the
central engine upon collapse. In other words, the free-fall time, given by
$t_{ff}\approx 4298{\rm s}\ (R^{3}/M)^{1/2}$, can be used as a rough estimate
for the lifetime of a star’s disk. As long as there is sufficient angular
momentum to launch the GRB jet, simply the difference in the stellar radius at
different metallicities can therefore qualitatively explain the trend we see
in the data. We represent this in Figure 2, which shows the free fall time as
a function of stellar radius, for stars of different masses. Imposed on the
plot are a set of three stellar evolution simulations we ran with the 1D
stellar evolution code MESA (Paxton et al., 2011, 2013, 2015, 2018, 2019;
Paxton, 2021)444We use release r21.12.1. Our models consist of three 25 solar
mass stars of different metallicities ranging from $Z=0.0001$ to $Z=0.1$,
where $Z$ is the mass fraction of “heavy” elements (heavier than helium)
relative to the total mass of the gas. The MESA equation of state (EOS) is a
blend of the OPAL (Rogers & Nayfonov, 2002), SCVH (Saumon et al., 1995),
FreeEOS (Irwin, 2004), HELM (Timmes & Swesty, 2000), PC (Potekhin & Chabrier,
2010), and Skye (Jermyn et al., 2021) EOSes. Radiative opacities are primarily
from OPAL (Iglesias & Rogers, 1993, 1996), with low-temperature data from
Ferguson et al. (2005) and the high-temperature, Compton-scattering dominated
regime by Poutanen (2017). Electron conduction opacities are from Cassisi et
al. (2007) and Blouin et al. (2020). Nuclear reaction rates are from JINA
REACLIB (Cyburt et al., 2010), NACRE (Angulo et al., 1999) and additional
tabulated weak reaction rates Fuller et al. (1985); Oda et al. (1994);
Langanke & Martínez-Pinedo (2000). Screening is included via the prescription
of Chugunov et al. (2007). Starting with the pre-main sequence model option in
MESA, we evolve and measure the radius at about two million years, close to
the end of stellar lifetime, with a range of spatial and temporal resolution
parameters until convergence. As expected, and consistent with the results
quoted in the previous paragraph, there is a clear trend of larger radius for
higher metallicity.

We note that Table 2 of Woosley & Heger (2006) also shows a general trend
among all of their models of a smaller iron core mass with increasing
metallicity. This means not only is the disk more extended in higher
metallicity stars but can contain more mass; there is therefore a larger
reservoir of fuel to feed the black hole and sustain the jet for a longer
amount of time.

Metallicity has also been shown to slow accretion rate (Wei & Liu, 2022). The
accretion rate (due to fallback) is estimated as $\dot{M}\sim dM_{r}/dt_{ff}$,
where $M_{r}$ is the mass contained in a radius $r$. Related to our arguments
above, a higher metallicity (more extended, less dense) star will have a
longer fallback accretion timescale and therefore lower accretion rate (see
also equation 11 of Suwa & Ioka, 2011). Additionally, higher stellar opacity
leads to larger radiation pressure that can stall accretion (Fukushima et al.,
2018). Toyouchi et al. (2019) show how metallicity decreases the accretion
rate even in the case of super-Eddington accretors555However, GRBs are known
to be extremely super-Eddington accretors (more than 10 to 15 orders of
magnitude above the Eddington rate), and it is not clear exactly how strong of
an effect metallicity will have in this case..

The arguments above suggest that higher metallicity isolated stars can in
principle produce longer duration GRBs provided these systems have enough
angular momentum to launch a jet. This, of course, is a strong assumption
(see, e.g., Proga & Begelman (2003) and Janiuk & Proga (2008) for discussions
on the difficulty of single stars retaining enough angular momentum upon
collapse to produce a long duration GRB), and has led to some authors
considering massive stars collapsing in an interacting binary system as a way
to increase or sustain the angular momentum of the massive star upon collapse.
We now discuss this correlation in the context of that scenario.

Figure 2: Free fall time as a function of stellar radius for stars of
different masses. The plotted symbols show the radius of an evolved 25 solar
mass star for different metallicities (marked on the plot), using the MESA
stellar evolution code. As expected stars of larger metallicity have larger
radii near the end of their evolution. We find the radius depends on
metallicity roughly as: $R\propto Z^{1/8}$, consistent with Tout et al. (1996)
and Maccarone et al. (2004)

.

##  4 Interacting Binary Star Models

Recently, Lloyd-Ronning (2022) examined the particular case of a massive star
collapsing in a tidally locked system with a black hole companion666Their work
was motivated by the apparent dichotomy between radio bright and dark GRBs -
as discussed above, several works have shown that the radio bright population
has on average significantly longer prompt gamma-ray durations and isotropic
equivalent energy compared to the radio dark population. In their model,
Lloyd-Ronning (2022) show these systems can explain the longer duration gamma-
ray emission, higher isotropic energy, and presence of radio emission in the
radio bright sub-population of GRBs.. Observationally, we know such systems
exist as HMXBs. As discussed in Tauris & van den Heuvel (2006), these systems
(as observed in our own galaxy) have massive star companions greater than 20
solar masses and with radii of 10 to 20 solar radii. They appear to almost
fill their critical Roche lobes. Their orbital periods are $<10$ days and they
have low eccentricity. Motivated by this, we consider how the $T_{\rm
int}-(1+z)$ anti-correlation may arise in systems that consist of a massive
star in a tidally locked binary with a black hole companion.

All of the points made in the previous section relevant to single star models
will also apply to the massive star in our binary. However, here we consider
how the interaction of the massive star with its companion affects the state
of the star at the end of its life. In particular, we consider the angular
momentum of the massive star right before collapse and relate this to the
duration of the GRB jet. We then examine how this depends on the metallicity
of the system, in order to understand how these systems (and their GRB
durations) might evolve over redshift.

The net angular momentum (denoted by $J$) in a binary system is given by:

$\begin{split}&J_{\rm binary}=J_{\rm orb}+J_{*}+J_{\rm BH}=\\\
&\left[\frac{aG(M_{*}M_{\rm BH})^{2}}{(M_{*}+M_{\rm
BH})}\right]^{1/2}+I_{*}\omega_{*}+I_{\rm BH}\omega_{\rm BH}\end{split}$ (1)

where $a$ is the binary separation, $G$ is the gravitational constant, $M_{*}$
is the mass of the massive star, $M_{BH}$ is the mass of the black hole,
$\omega_{*}$ is the rotational rate of the massive star, $\omega_{BH}$ is the
rotational rate of the black hole, $I_{*}$ is the moment of inertial of the
massive star and $I_{BH}$ is the moment of inertia of the black hole. We
approximate the moments of inertia of each object as $I\approx\alpha MR^{2}$,
where $\alpha$ is a constant of order unity and $R$ is the radius of the
spinning object (massive star or black hole).

If we assume our system is tidally locked (with zero eccentricity)777This is
an arguably justifiable assumption; the tidal locking timescale is given by
(Gladman et al., 1996):
$\displaystyle t_{lock}\approx$ $\displaystyle 10^{3}\ {\rm yr}\
(r/10^{12}cm)^{6}(\rho_{star}/10^{3}gcm^{-3})$ (2)
$\displaystyle(\omega_{o}/10^{-5}s^{-1})(Q/100)(M_{BH}/10M_{\odot})^{2}(k_{2}/0.1)$
where $\rho_{star}$ is the average density of the star, $\omega_{o}$ is its
initial rotational velocity, $Q$ is the so-called dissipation function of the
star, and $k_{2}$ is the tidal love number. Quantities like $Q$ and $k_{2}$
are highly unknown (Gladman et al., 1996) and usually taken to be around $\sim
100$ and of order unity, respectively., the orbital frequency
$\omega_{orbit}=\omega_{*}=\omega_{BH}=\sqrt{GM_{tot}/a^{3}}$, where
$M_{tot}=M_{BH}+M_{*}$. In this special case the total angular momentum in the
system can be written as:

$\begin{split}&J_{\rm binary}=\left[\frac{aG(M_{*}M_{\rm
BH})^{2}}{(M_{*}+M_{\rm BH})}\right]^{1/2}+\\\ &\left[\frac{G(M_{*}+M_{\rm
BH})}{a^{3}}\right]^{1/2}\left[\alpha_{\rm BH}M_{\rm BH}R_{\rm
BH}^{2}+\alpha_{*}M_{*}R_{*}^{2}\right]\end{split}$ (3)

Again, here our goal is to consider how the angular momentum of the massive
star (inextricably linked to the intrinsic duration of the prompt GRB) depends
on the parameters of this interacting binary system - particularly metallicity
- and how this could lead to the evolution of GRB prompt duration as a
function of redshift.

Just as in the single star case, a higher metallicity massive star in such a
system will lose more mass due to line driven winds, compared to one of low
metallicity. This can lead to the binary widening (Broekgaarden et al., 2022),
and decrease the binary interaction (lessening the potential spin-up of the
massive star)888although we note again that mass loss could be anisotropic as
discussed in Maeder (2002), which can lead to spin up of the massive star. As
mentioned in the previous section, such anisotropic mass loss is observed in a
massive star in an interacting binary system closely related to the ones we
consider here (Olivier et al., 2022).. Therefore, as in the single star case,
we might at first glance expect higher metallicity systems to have lower
angular momentum and therefore shorter duration GRBs. However, this ignores a
number of other subtle factors that can decrease the binary separation, and
lead to spin-up of the massive star. For example, a higher metallicity massive
star in a binary system may also be more likely to undergo Roche lobe
overflow, while these stars are less compact, and this can serve to tighten
our binary. Drag forces by ambient gas can also play an important role and
lead to a decrease in the binary separation. We now discuss scenarios in which
(ultimately metallicity-dependent) changes in mass and radius affect the
binary system and play a role in the duration of a GRB.

###  4.1 Evolution of Binary Separation

How the separation between two stars in a binary system evolves over the
lifetime of each star is a complicated problem with many possible outcomes,
depending on the physical properties the stars, the initial binary separation,
the nature of each star’s mass loss, and the system’s environment.

It is crucial to point out that we observe a number of HMXB systems (similar
to the systems we consider here) which are indeed undergoing a decrease in
their orbital separation - for example, Cen X-3 (Kelley et al., 1983), SMC X-1
(Levine et al., 1993), LMC X-4 (Levine et al., 2000), and seven additional
HMXBs (Falanga et al., 2015), all with orbital decay timescales on the order
of about a half a million years (see Table 3 of Falanga et al., 2015).

For the idealized case of two point-like masses, we can write the normalized
change in binary separation as (Postnov & Yungelson, 2014):

$\displaystyle\frac{\dot{a}}{a}$ $\displaystyle=$ (4)
$\displaystyle-2\frac{\dot{M_{*}}}{M_{*}}\left[1+(\alpha-1)\frac{M_{*}}{M_{BH}}-\frac{\alpha
M_{*}}{2M_{tot}}\right]+\ 2\frac{\dot{J}_{orb}}{J_{orb}}$

where $\dot{M}_{*}<0$ (i.e. the star is losing mass). Note that $\dot{M}_{*}$
and $\dot{J}_{orb}$ are not independent variables. The parameter $\alpha$ is
the fraction of the lost mass from $M_{*}$ that leaves the system. That is,
for $\alpha=0$, all the mass lost from our star stays in the binary system;
for $\alpha=1$, all of it leaves the binary system.

In the case of conservative mass transfer ($\alpha=0$ and $\dot{J}_{orb}=0$),
when all of the massive star’s lost mass is deposited onto the black hole, the
ratio of initial to final binary separation is given by (Postnov & Yungelson,
2014):

$\frac{a_{f}}{a_{i}}=\left(\frac{M_{BH,i}M_{*,i}}{M_{BH,f}M_{*,f}}\right)^{2}$
(5)

where $i$ and $f$ denote initial and final states, respectively. For a massive
star that is more massive than the black hole companion, as we reasonably
assume is the case here (Heger et al., 2003; Ramachandran, 2022)999We note
that if the binary system was born in situ (rather than through dynamical
capture), the black hole companion must be the result of the more massive star
of the two original stars in the binary. However, we assume the death of the
first massive star left a remnant black hole that is less massive than the
second massive star (the one we consider here as the GRB progenitor)., the
binary orbit will shrink as $M_{*}$ transfers mass to the black hole. As a
result, the system’s angular frequency $\omega$ increases, and (because the
system is tidally locked) the massive star’s rotation rate increases. As
mentioned above, this provides a larger angular momentum reservoir which can
lead to a longer duration GRB (Lloyd-Ronning, 2022). In such a scenario, more
mass loss would therefore lead to a longer duration GRB. However, conservative
mass loss applies primarily in the the case of Roche lobe overfill. In
general, mass loss from winds (either isotropic mass loss or pole-dominated
anisotropic mass loss)is not conservative and some mass will escape the
system.

In the case when all of the mass directly leaves the system (perfect non-
conservative mass loss with $\alpha=1$ or the so-called Jeans mode), the
binary separation will increase in order to conserve angular momentum.
However, there are situations of non-conservative mass loss that lead to
reduced orbital separation. For example, in the case when accretion from the
massive star onto the black hole exceeds the Eddington limit, matter can be
expelled from the system by radiation pressure. In this case, angular momentum
is depleted from the accretor and the orbital separation can exponentially
shrink with mass loss (see equation 43 of Postnov & Yungelson, 2014):

$\frac{a_{f}}{a_{i}}=\left(\frac{M_{BH,i}}{M_{BH,f}}\right)\left(\frac{M_{*,i}}{M_{*,f}}\right){\rm
exp}\left(-2\frac{M_{*,i}-M_{*,f}}{M_{BH}}\right)$ (6)

Hence, in a realistic system there are a number competing effects when we
consider the role of metallicity driven mass loss as it affects our binary
system and ultimately the spin of the massive star.

Finally, we note that drag forces can also have an important effect on the
system’s evolution. Schrøder et al. (2021) showed that, under the assumption
of non-spinning stars, line driven winds can either tighten or widen the
binary depending on the mass ratio of the binary, as well as the ratio of the
wind velocity to orbital velocity. They found that for slower winds (relative
to the orbital velocity) and higher mass companions, drag forces serve to
shrink the binary separation (see their Figure 1). Interestingly, Falanga et
al. (2015) report inferred wind speeds for the massive stars in 10 HMXBs on
the order of $1000km/s$ (see their Table 9). Meanwhile, the orbital velocities
for our closely separated, tightly bound binaries are about an order of
magnitude higher than this, satisfying the necessary velocity ratio condition
that leads to binary separation decrease.101010In our systems, we are
typically considering a massive star with a compact object companion whose
mass is expected to be less than that of the massive star and so - although
drag forces may play an important role in the binary separation evolution -
the Schrøder et al. (2021) calculations may not apply to our models.

####  4.1.1 The Effect of Stellar Expansion

As mentioned in the previous section on single star progenitors, higher
metallicity stars are more extended in radius. A more extended star can have a
higher moment of inertia and therefore - for a given rotation rate - higher
angular momentum. Therefore, as the star expands, the binary separation will
decrease (in order to conserve angular momentum in the system as a whole).
Levine et al. (1993, 2000) derive an expression for this change in orbital
period $P_{orb}$ as a function of the change in the moment of inertia of an
expanding star:

$\frac{\dot{P}}{P}=\frac{3\dot{a}}{2a}\approx\frac{\omega_{BH}d{\rm
ln}I_{*}/dt}{\omega(\mu a^{2}/3I_{*}-1)}$ (7)

where $\mu$ is the binary reduced mass111111Recall, in our scenario,
$\omega_{BH}\sim\omega$ (the orbital angular frequency), which simplifies the
equation. This equation only accounts for radial expansion and assumes the
mass stays constant. As mass is lost, the moment of inertia will decrease
accordingly. However, if the normalized mass lost to winds is less than twice
the normalized change in radius due to stellar expansion,

$\Delta M/M<2\Delta R/R$ (8)

then the star’s expansion will “win”, and serve to decrease the separation in
our binary system. Again, in our tidally locked model, this leads to spin-up
of our massive star.

The rate of mass loss due to line-driven winds is a complicated and unresolved
problem. However, it is reasonable to expect the condition in equation 8
holds, while stellar winds are often weaker than simple models predict. We
refer the reader to the review by Smith (2014) for an in-depth look at this
issue, and also Fullerton et al. (2006) who showed, observationally, that
stellar models tend to overestimate mass loss due to winds by a factor of 10
or more, when realistic conditions (such as clumping) are not considered.

Figure 3: Upper Panel: Range of $\beta$ values that accommodate the trend seen
in the data, in the context of a massive star collapsing in an interacting
binary. The parameter $\beta$ describes how the binary orbit shrinks as a
function of mass loss, where we assume mass loss depends on metallicity to the
1/2 power. Lower Panel: Same as the upper panel, but here we assume mass loss
depends on metallicity to the 1/4 power.

In short, there is a complicated interplay of many different factors related
to changes in mass and stellar radius that can serve to either increase or
decrease the binary separation. In the next section, we focus on the
consequences of the cases in which binary separation decreases with
(metallicity-driven) mass loss and relate this to the observed anti-
correlation between GRB duration and redshift.

###  4.2 Reproducing the Anti-correlation

As discussed above, there exist various scenarios in which the binary
separation will decrease for systems consisting of a massive star with a
compact object companion. In our model, where the tidal locking timescale is
relatively short compared to other timescales in the system (e.g. orbital
decay time, lifetime of star), the decreased separation will lead to a more
rapidly spinning massive star, which is capable of producing a longer duration
gamma-ray burst. We would like to understand how the increase in average
metallicity over cosmic time may affect properties of interacting binary
systems, and how this might lead to the observed anti-correlation between
prompt GRB and redshift.

We parameterize the binary system’s separation decrease as function of donor
star’s mass loss as follows:

$\frac{a_{f}}{a_{i}}\propto(\frac{\Delta M_{*}}{M_{*}})^{-\beta}$ (9)

where $\beta>0$ parameterizes the extent to which the orbit shrinks as a
function of mass loss. Once again, when considering how equation 9 describes
separation as a function of cosmic time, we will consider the variables’
dependencies on metallicity.

As discussed in the previous section, mass loss from both winds and Roche lobe
overflow will depend on metallicity. Several groups have parameterized the
former relationship, showing that mass loss from winds is $\Delta M\propto
Z^{1/2}$ (Vink et al., 2001; Yoon et al., 2006; Woosley & Heger, 2006). Mass
loss due to RLOF is not as well parameterized and depends on a number of
factors in the system; however, given that the radius of the star depends on
metallicity roughly to the $1/8$ power as discussed above (Tout et al., 1996;
Maccarone et al., 2004), then the mass loss dependence on metallicity due to
RLOF can be estimated as $\Delta M\propto Z^{2/8}$ to $Z^{3/8}$ (the former
considers only surface mass loss). Substituting these relationships into
equation 9, we find then that $\frac{a_{f}}{a_{i}}\propto Z^{-\beta/2}$ to
$Z^{-\beta/4}$.

Because the angular momentum of the massive star is inversely proportional to
binary separation under the tidal locking assumptions of our model,
$J_{*}\propto a_{f}^{-3/2}$, we find a relationship between the star’s angular
momentum and its metallicity (in this special case of an interacting binary):

$J_{*}\propto Z^{3\beta/4}$ (10)

Equation 10 is considering the angular momentum of the star as it depends on
binary separation in a tidally locked system. At the same time, the star’s
winds will carry away angular momentum and we need to consider the importance
of that depletion in these arguments. Assuming that the mass lost from winds
is much less than the total mass in the system (Fullerton et al., 2006; Smith,
2014), we can estimate the change in angular momentum due to winds as $\Delta
J_{wind\ loss}\approx\Delta M_{*}R_{*}^{2}\omega_{*}$, and that gained from
spin up as $\Delta J_{spin\ up}\approx M_{*}R_{*}^{2}\omega_{*}(\Delta(a)/a)$.
The net gain in angular momentum of the star (due to orbital separation
decrease) is then $\Delta J_{*}=J_{spin\ up}-J_{wind\ loss}$. The assumption
that angular momentum loss due to winds is negligible compared to the gain due
to decreasing binary separation and spin-up of the star, amounts to the
following condition:

$\frac{\Delta J_{\rm wind\ loss}}{\Delta J_{\rm spin\ up}}\approx(\frac{\Delta
M}{M})(\frac{a}{\Delta a})<1$ (11)

or

$\frac{\Delta a}{a}>\frac{\Delta M}{M}$ (12)

Falanga et al. (2015) provide estimates of mass loss and observed orbital
period change in the types of binary systems we are considering. From the data
in their Tables 1, 3, and 9, we see that the condition in equation 12 above
generally holds - in other words, the fractional change in binary separation
is larger than the fractional mass loss in these systems and we can invoke
equation 10 going forward. Then, using $T_{dur}\propto J_{*}^{7/4}$ (Lloyd-
Ronning, 2022) and equation 10, we find:

$T_{\rm int}\propto Z^{21\beta/16}$ (13)

Now consider $Z$’s dependence on redshift. Figure 2 of Fynbo et al. (2006) and
Figure 14 of Madau & Dickinson (2014) show how cosmic metallicity increases
with decreasing redshift. Although there is ample scatter in the data, we can
approximate the relationship roughly as $Z\propto(1+z)^{-0.4}$. Substituting
that relation into equation 13 leads to:

$T_{\rm int}\propto(1+z)^{-21\beta/40}\approx(1+z)^{-0.5\beta}$ (14)

assuming mass loss is roughly proportional to $Z^{1/2}$. For mass loss
proportional to $Z^{1/4}$, we find:

$T_{\rm int}\propto(1+z)^{-21\beta/80}\approx(1+z)^{-0.25\beta}$ (15)

As discussed in section 2, the observed anti-correlation between duration and
redshift for the larger GRB sample (that is, not separating into radio bright
and dark sub-samples) is approximately parameterized as
$T_{dur}\propto(1+z)^{-0.8\pm 0.25}$. We conservatively choose constraints on
$\beta$ within the $2\sigma$ error bars and find this implies
$1\lesssim\beta\lesssim 3$ for $\Delta M\propto Z^{1/2}$ and
$2\lesssim\beta\lesssim 6$ for $\Delta M\propto Z^{1/4}$. These two
metallicity dependencies and our model parameter spaces are shown in the two
panels of Figure 3.

##  5 Discussion and Conclusions

In this paper we have examined scenarios in which the average intrinsic prompt
duration of GRBs increases with decreasing redshift, as the data suggest. As
metallicity is a variable we can securely count on increasing over cosmic time
(Langeroodi et al., 2022), we have in particular linked this cosmological
evolutionary trend to the metallicity of the system. We have considered both
single massive stars and massive stars collapsing in interacting binary
systems as our GRB progenitors.

Our main results are summarized as follows:

  * •

Single massive star GRB progenitors may produce the duration-redshift anti-
correlation as a result of higher metallicity stars being able to form more
extended disks with lower accretion rates. We present 1D stellar evolution
simulations that support this picture. Provided there is enough angular
momentum to launch a jet, the more extended disk of a higher metallicity star
allows for longer duration GRB jets at lower redshift.

  * •

Massive stars in interacting binary systems may produce the duration-redshift
anti-correlation through metallicity-dependent mass loss scenarios (as well as
metallicity-dependent equations of state) that lead to a decrease in the
binary separation for higher metallicity, low-redshift systems. This can
produce a more pronounced spin-up of the massive star and therefore longer
duration GRBs at lower redshifts. Our key assumption, when connecting the
binary separation to the angular momentum of the massive star, is that the
tidal locking timescale is shorter than other timescales in our system’s
evolution, so that the binary separation sets the rotation rate of the massive
star ($\omega_{*}=\sqrt{GM_{tot}/a^{3}}$).

Both our single star and binary star pictures have made a number of
simplifying assumptions. For example, in our binary scenario, we’ve implicitly
assumed a larger angular momentum resevoir present in the massive star at the
end of its life will lead to a more rapidly spinning black hole-disk system
(produced upon the star’s collapse), and therefore a longer GRB. There are
many complicated steps connecting the spin of the massive star at the end of
its life to the angular momentum of the black hole-disk system after collapse
(Qin et al., 2018). Going forward we plan to examine our models through more
detailed simulations of these systems, particularly the end-of-life state of
massive stars in interacting binaries. We will be able to test the range of
values that are valid for our parameter $\beta$, while also including effects
of the stellar equation of state and the star’s expansion. We would also like
to further examine the dichotomy between radio bright and radio dark GRBs,
particularly the conjecture that GRBs with radio afterglows, which have longer
prompt gamma-ray duration, are the result of massive stellar collapse in such
interacting binary systems (Lloyd-Ronning, 2022). As the data improve and we
get a better handle on how to divide the radio bright and dark sub samples,
this and similar correlations may be key to unlocking any fundamental
differences in the underlying progenitor systems of GRBs.

We note again that we are only considering black hole-disk central engines,
and not magnetar-driven engines for our GRBs. In GRBs with magnetar central
engines, the GRB can be initially powered by accretion onto a newly born
neutron star (e.g. Bernardini et al. (2013); Bernardini (2015); extended
emission (like a plateau phase) may result from the spin down power of the
magnetar (Zhang & Mészáros, 2001). It is not immediately clear whether these
systems would be able to produce the anti-correlation between duration and
redshift, although many of the arguments we make in this paper may apply to
these systems as well.

Finally, evolution of the population statistics of interacting binaries may
also explain the duration-redshift anti-correlation. If there exists a larger
population of interacting binary systems (of the type we discuss in §4 above)
at lower redshifts relative to high redshifts, this could account for the
trend we see in the data. Indeed, Liu et al. (2021) predict there are fewer
close binaries at high redshift, although they examine only PopIII stars in
their study. On the other hand, Moe et al. (2019) and Bate (2019) suggest he
close binary fraction is anti-correlated with metallicity. The latter in
particular explore the dependence of fragmentation on metallicity and find -
for low mass stars - there is a higher close binary fraction at lower
metallicity (although Kuruwita & Haugbølle (2022), who explore binary
separation in situ, suggest disc fragmentation is not a dominant pathway for
binaries of close separation). A closer look at the viability of this pathway
to explain the duration-redshift trend we see in the data requires detailed
and informed population synthesis modelling, which we explore in a future
paper.

##  6 Acknowledgements

We thank Anne Noer Kolborg and Krystal Ruiz-Rocha for helpful conversations
related to cosmic metallicity evolution, as well as Angana Chakrobarty and
Maria Dainotti for useful conversations related to the anti-correlation
between duration and redshift. We thank Gabriel Casabona and Shane Larson for
insightful conversations on population synthesis. We thank the MESA community
and developers for their help and the many excellent resources in support of
this code. This work was supported by the US Department of Energy through the
Los Alamos National Laboratory. Los Alamos National Laboratory is operated by
Triad National Security, LLC, for the National Nuclear Security Administration
of U.S. Department of Energy (Contract No. 89233218CNA000001). Research
presented in this article was supported by the Laboratory Directed Research
and Development program of Los Alamos National Laboratory under project number
20230115ER.

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# Measurement of nuclear spin relaxation time in lanthanum aluminate for
development of polarized lanthanum target

K. Ishizaki<EMAIL_ADDRESS>Nagoya University, Furocho,
Chikusa, Nagoya, 464-8602, Japan  Hiroshima University, Kagamiyama, Higashi-
hiroshima, 739-8527, Japan  Yamagata University, Koshirakawa, Yamagata
990-8560, Japan  KMI, Nagoya University, Furocho, Chikusa, Nagoya, 464-8602,
Japan  Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka,
567-0047, Japan  The University of Tokyo Hospital, Hongo, Bunkyo, Tokyo,
113-8655, Japan  H. Hotta  I. Ide  M. Iinuma  T. Iwata  M. Kitaguchi  H. Kohri
D. Miura  Y. Miyachi  T. Ohta  H. M. Shimizu  H. Yoshikawa  M. Yosoi

###### Abstract

The nuclear spin-lattice relaxation time ($T_{1}$) of lanthanum and aluminum
nuclei in a single crystal of lanthanum aluminate doped with neodymium ions is
studied to estimate the feasibility of the dynamically polarized lanthanum
target applicable to beam experiments. The application of our interest is the
study of fundamental discrete symmetries in the spin optics of epithermal
neutrons. This study requires a highly flexible choice of the applied magnetic
field for neutron spin control and favors longer $T_{1}$ under lower magnetic
field and at higher temperature. The $T_{1}$ of ${}^{139}{\rm La}$ and
${}^{27}{\rm Al}$ was measured under magnetic fields of $0.5$-$2.5$ T and at
temperatures of $0.1$-$1.5$ K and found widely distributed up to 100 h. The
result suggests that the $T_{1}$ can be as long as $T_{1}\sim$ 1 h at $0.1$ K
with a magnetic field of $0.1$ T, which partially fulfills the requirement of
the neutron beam experiment. Possible improvements to achieve a longer $T_{1}$
are discussed.

###### keywords:

solid polarized target , NMR , T-violation

##  1 Introduction

The feasibility of the polarized nuclear target of ${}^{139}{\rm La}$ is
explored for the study of the spin-related correlation terms in compound
nuclear states induced by polarized epithermal neutrons, which introduces an
enhanced sensitivity to the breaking of spatial and time-reversal symmetries.
Enhanced parity-nonconservation (PNC) effects are observed in the compound
resonances of medium heavy nuclei and the enhancement factor reaches $10^{6}$
compared with the PNC effect in the nucleon-nucleon interaction [1, 2]. The
enhancement mechanism is further studied for ${}^{139}{\rm La}$ by observing
the neutron spin dependence and angular distribution of $\gamma$-rays in the
reaction ${}^{139}{\rm La}({\rm n},\gamma){}^{140}{\rm La}$. These studies are
conducted to quantitatively estimate the possible enhancement of the time-
reversal violation (T-violation) in the optical behavior of the neutron spin
on transmission through a polarized target [4, 5, 6].

To polarize the nuclear spins of 139La, two well-known techniques, static
nuclear polarization (SNP), and dynamic nuclear polarization (DNP), are
applicable. In SNP, preparing an environment of a huge magnetic field and an
ultra-low temperature makes it possible to obtain high nuclear polarization in
thermal equilibrium. In the case of a hydrogen deuteride (HD) target [7, 8,
9], for example, the nuclear relaxation time is approximately a few hours
before the aging of the HD target and longer than several months after aging.
This is a special case because, generally, SNP requires a long aging time to
achieve high thermal polarization. In the case of 139La, a magnetic field of
17 T and temperature of 0.01 K can provide a polarization of approximately
59%. One candidate target for the beam experiments is a metal target, which
has been used in some beam experiments [3, 10, 11], because metal targets have
a considerably shorter relaxation time than the insulating materials. However,
keeping high polarization under a low external magnetic field is difficult
during the beam experiments.

The dynamic nuclear polarization (DNP) is a method to dynamically polarize
nuclei by transferring the polarization of the paramagnetic centers to the
nuclear polarization via a double spin-flip process induced by microwave
irradiation, and it is appropriate to fulfill the requirements of the
T-violation experiment. Lanthanum aluminate doped with neodymium ions
La1-xNd${}^{3+}_{x}$AlO3 has been studied as the candidate material for the
polarized lanthanum target after the dynamical enhancement of 139La and 27Al
was successfully observed at the Paul Scherrer Institute (PSI). The achieved
polarization was approximately 50% in the 139La spin ($I$ = 7/2) with the
dopant content of 0.03 mol% $(x=3\times 10^{-4})$ at a magnetic field of 2.35
T and a temperature lower than 0.3 K [12]. The LaAlO3 has an additional
advantage that the 139La in LaAlO3 can avoid the decrease of the vector
polarization, which is caused by the mixing of the eigenstates of the nuclear
spin by the nuclear quadrupole interaction with the electric field gradient
[13]. Here, we mention that the La2Mg3(NO3)${}_{12}\cdot$ 24H2O is not an
appropriate compound because its lanthanum content is small, although it is a
well known lanthanum compound to which the DNP can be applied.

The nuclear spin-lattice relaxation time ($T_{1}$) of 139La is the remaining
key parameter to be studied for the estimation of the feasibility and
applicability of LaAlO3 to the T-violation experiment. It was measured above 1
K [14], but additional measurement below 1 K is necessary for practical
installation to the beam experiment, which requires a long-term stability to
maintain the ${}^{139}{\rm La}$ polarization with the magnetic field as low as
0.1 T required for the neutron spin control.

Generally, the measurements of $T_{1}$ need sufficiently large NMR signals.
Because the thermal NMR signals of 139La are likely weak because of the
existence of eight sublevels, the $T_{1}$ measurements may be difficult
depending on the measurement conditions. For the LaAlO3 crystal, however,
studies on the $T_{1}$ of 27Al, which is easily measurable, help in the
evaluation of the $T_{1}$ of 139La under the target condition because the
nuclear relaxation usually originates from the paramagnetic dopant.

In this paper, we report the temperature and magnetic-field dependence of
$T_{1}$ of 139La and 27Al in the Nd-doped LaAlO3 crystal over a temperature
range of 0.1–1.5 K and magnetic fields from 0.5 to 2.5 T. We estimate the
$T_{1}$ of 139La under low magnetic fields that are desirable in T-violation
experiments.

##  2 Experiment

###  2.1 Cryostat

We used two types of refrigerators. One is a 3He-4He dilution refrigerator
(DRS) manufactured by Leiden Cryogenics B.V. [15]. This refrigerator covers
the temperature range above 6 mK and is equipped with a superconducting
solenoid (NbTi and Nb3Sn) produced by JASTEC Co., Ltd. [16] in Japan. This
magnet can generate a maximum magnetic field of 17 T. The other is a 4He
pumping cryostat, which is referred to as a storage cryostat (SC), produced by
Oxford Instruments [17]. This cryostat can provide an environment at a
temperature of 1.5 K under magnetic fields up to 2.5 T. The SC is normally
used for transporting polarized HD targets from Osaka University to SPring-8
[9].

At 0.1 K and 0.5 K, we excited the superconducting solenoid of the DRS and
performed nuclear magnetic resonance (NMR) measurements with a sweep of the
magnetic field. The temperature of the mixing chamber was stabilized by
switching a heater mounted there. To determine the sample temperature, we used
the readout from a carbon resistance thermometer, which was attached to the
mixing chamber. In the field sweep, the induced Eddy current generally leads
to an uncertainty in temperature. The uncertainty was approximately 0.01 K in
the experiments.

The NMR measurement was also performed in the SC refrigerator using the same
method as that in the DRS. The uncertainty of the SC temperature was less than
approximately 0.05 K. The temperature of the sample was monitored by reading a
RuO2 resistance thermometer attached to the Cold-finger2, as shown in Fig. 1.

Figure 1: LaAlO3 crystal installed at the bottom of the storage cryostat (SC).
The height of the SC is approximately 1.6 m, and the bottom part has the
lowest temperature of 1.5 K.

###  2.2 NMR Systems

We used a portable NMR system [18, 19], which had been developed for a
polarized HD target [9, 20], to measure the relaxation times of 139La and
27Al.

In the NMR measurements with the DRS, two coaxial cables were used to form a
connection between a pickup coil and a tuning circuit placed outside the
cryostat. One was a semi-flexible coaxial cable (MULTI-FLEX 141 produced by
Suhner) with a length of 2 m and a diameter of 4.2 mm. This cable was used in
the range of 293 – 301 K. The capacitance of the cable was 95.0 pF/m. The
other was a semi-rigid coaxial cable (SC-119/50-SCN-CN produced by Coax Co.,
Ltd.) with a length of 2.5 m and a diameter of 1.19 mm. The semi-rigid cable
was used inside the cryostat. The capacitance of the cable is 95.2 pF/m. The
measured self-inductance of the pickup coil holding the LaAlO3 crystal was
5.40 $\mu$H. In the measurements with the SC, a semi-rigid coaxial cable
(EZ47-AL-TP produced by EZ Form Cable Corporation) with a length of 1 m and a
diameter of 1.19 mm was used instead of the SC-119/50-SCN-CN coaxial cable.

In all the NMR measurements, the strength of the applied magnetic field was
measured by reading the current supplied to the superconducting magnet. The
uncertainty of the magnetic field strength was approximately 1%. The
continuous-wave radio frequency (CW-RF) applied through the pickup coil was
appropriately attenuated to avoid affecting the $T_{1}$ measurements.

###  2.3 Sample

The sample is a LaAlO3 crystal with Nd-dopant content of 0.03 mol% ($x=3\times
10^{-4}$) with dimensions of 1.5 $\times$ 1.5 $\times$ 1.5 cm3 and features
twin domains [21]. The Nd content was analyzed using inductively coupled
plasma mass spectrometry (ICP-MS) [22]. The results confirmed that the amount
of paramagnetic impurities including Ir, W, and Mo, which are typical
materials used to build the crucible, is much less than the amount of Nd. The
sample was used with the twin structure still contained.

Cold-Finger2 was attached to Cold-Finger1, which was thermally connected to
the mixing chamber, to install the LaAlO3 crystal at the bottom of the DRS, as
shown in Fig. 2. The crystal was wound with eight turns of a Teflon-coated
silver wire 0.3 mm in diameter to detect the NMR signal, as shown in Fig. 3.
The wire was fixed with a kapton tape to maintain the shape of the pickup
coil. The surfaces of the sample were kept in contact with the surface of the
Cold-Finger2 using Apiezon-N grease to increase the cooling efficiency.

Figure 2: LaAlO3 crystal installed at the bottom of the DRS. The height of the
DRS refrigerator is approximately 2.5 m, and the mixing chamber, where 3He and
4He are mixed, has the lowest temperature.  Figure 3: LaAlO3 crystal attached
to Cold-finger2 and the pickup coil. In the experiment, the crystal was
covered by a thin plate, which was attached to the kapton surface at the front
of this figure to fold the sample.

###  2.4 Procedure

To obtain the $T_{1}$ from the time dependence of many NMR spectra, we should
evaluate the height and width of each peak as accurately as possible. Each
peak of 139La was individually analyzed by fitting with a linear combination
of Lorentzian and linear functions as follows:

$\begin{split}f_{i}(B)=I_{i}\frac{w^{2}}{(B-B_{i})^{2}+w^{2}}+a(B-B_{i})+b,\end{split}$
(1)

where $B_{i}$ is the centroid of each signal peak, $I_{i}$ is the height of
the signal peak, $w$ is a parameter related to the peak width, and $a$ and $b$
are parameters of the linear function for evaluating the background.

In the 27Al NMR measurements, the peaks are not completely separated, because
the homogeneity of the magnetic field was insufficient. In this case, a
combination of multi Gaussian and linear functions is used as the fitting
function, which can be expressed as follows:

$\begin{split}f(B)=\sum_{i}I_{i}\exp\left(-\frac{(B-B_{i})^{2}}{2\sigma^{2}}\right)+a(B-B_{5})+b,\end{split}$
(2)

where $B_{i}$ ($B_{1}<B_{2}<\cdot\cdot\cdot<B_{9}$) is the centroid of each
signal peak, $I_{i}$ is the height of the signal peak, $\sigma$ is the
standard deviation, and $a$ and $b$ are parameters of the linear function for
evaluating the background. The Lorentzian fitting for the 27Al NMR spectrum
was less accurate than in the Gaussian case.

The relaxation rate $\Gamma\,(=1/T_{1})$ is obtained by fitting the following
simple exponential function to the time dependence of the evaluated
intensities:

$I(t)=(I_{0}-I_{\rm eq})\exp(-\Gamma t)+I_{\rm eq},$ (3)

where $I_{0}$ and $I_{\rm eq}$ are the evaluated intensity of the initial
polarization and that at thermal equilibrium, respectively.

The measurements were performed as follows.

  1. 1.

The sample is cooled down to a temperature suitable for the $T_{1}$
measurements.

  2. 2.

To prepare the initial polarization, the crystal is aged for more than half a
day under a magnetic field higher than that at the thermal equilibrium or
without a magnetic field.

  3. 3.

The NMR circuit is tuned to a frequency selected for the NMR detection.

  4. 4.

The magnetic field is changed to the vicinity of the field corresponding to
the 139La or 27Al resonance.

  5. 5.

Some NMR data are acquired by sweeping the field at a speed of 0.034 T/min.

  6. 6.

The magnetic field is changed to the condition for the $T_{1}$ measurement.
Subsequently, the crystal is left for some time.

  7. 7.

Steps 4 to 6 are repeated for a long term.

We measured the buildup and decay toward thermal equilibrium of the NMR
signals as shown in Fig. 4.

Figure 4: Sequence of the $T_{1}$ measurement with CW-NMR for (a) the
”buildup” method and (b) the ”decay” method. The NMR spectra were obtained
when $B_{0}$ was equal to $B_{\rm m}$. While the ”Waiting”, we waited for the
change of the nuclear polarization at $B_{\rm w}$. Thus, the measurements in
the sequence provided the $T_{1}$ at $B_{\rm w}$.

We measured the $T_{1}$ of 139La at four conditions and that of 27Al at seven
conditions, as presented in Table 1.

Table 1: Summary of the data set. The terms ”buildup” and ”decay” indicate the method of $T_{1}$ measurements. $f$ is the RF frequency of the NMR measurement. Nuclei |  $T$ [K] |  $B$ [T] |  $f$ [MHz] | Method
---|---|---|---|---
${}^{139}\mathrm{La}$ | 0.5 | 2.5 | 16.1 | decay
| 0.5 | 1.0 | 5.7 | decay
| 0.5 | 0.5 | 5.6 | decay
| 0.1 | 0.75 | 5.6 | buildup
${}^{27}\mathrm{Al}$ | 1.5 | 2.1 | 23.8 | buildup
| 1.5 | 1.0 | 11.0 | decay
| 0.5 | 2.5 | 28.2 | decay
| 0.5 | 1.0 | 11.8 | decay
| 0.5 | 0.5 | 9.1 | decay
| 0.1 | 1.6 | 16.8 | buildup
| 0.1 | 0.76 | 16.4 | buildup

##  3 Results

###  3.1 NMR spectra

Fig. 5 shows the dispersive and absorptive NMR spectra for 27Al($I=5/2$) at
0.5 K with the DRS and at 1.5 K with the SC. Nine peaks are observed. On the
absorptive spectrum of seven central peaks, Eq. (2) was used as the fitting
function. To reduce the complexity of the fitting, we assumed that the widths
of all the peaks and the spacings between the neighboring peaks were
identical. In contrast, peak 1 and peak 9, we individually fitted a linear
combination of Gaussian and linear functions.

Figure 5: Typical 27Al NMR spectra. The spectra (a) and (b) were obtained in
the DRS at 0.5 K. The spectra (c) and (d) were obtained in the SC at 1.5 K.

Consequently, the peak width is obtained as $\sigma=(2.12\pm 0.01)\times
10^{-4}$ T, which is almost consistent with the field homogeneity of
approximately 10-4 in the DRS. The distances between neighboring peaks among
the seven peaks were obtained as (7.86$\pm$0.01)$\times$10-4 T. The distances
between peak 1 and peak 2 and between peak 8 and peak 9 were
(2.42$\pm$0.01)$\times$10-3 T and (2.38$\pm$0.01)$\times$10-3 T, respectively,
which were approximately three times larger than those between the neighboring
peaks. The $\sigma$ of the seven peaks and those of peak 1 and peak 9 were
almost equal. From similar analyses of the NMR spectra, we confirmed that all
widths are independent of the elapsed time.

Figure 6: 139La NMR spectrum at approximately 5 T and 0.5 K. (a) : Absorptive
part of 139La NMR. (b) : Fit to ”Peak 3” that corresponds to the transition
$-1/2\leftrightarrow 1/2$. (c) : Fit to ”Twin 1”.

The (c) and (d) spectra in Fig. 5 were obtained in the SC at 1.5 K. Because
the superconducting solenoid of the SC has better homogeneity than that of the
DRS2500 refrigerator, the peaks are well separated from the neighboring peaks.
Only the central peak is fitted by a Gaussian with a linear function.

For the peak identification in 139La($I=7/2$), we measured broad NMR spectra
at 5 T at 0.5 K with high sensitivity separately from the systematic $T_{1}$
measurements. Fig. 6 clearly shows five peaks, indicated by 1, 2, 3, 4, and 5,
are separated by the same distance.

The LaAlO3 crystal has a distorted perovskite structure with a three fold
rotation axis ($C_{3}$ axis) at temperatures lower than 813 K [23]. Both the
La and Al sites have an electric-field gradient with uniaxial symmetry. If a
nucleus with a spin $I$ and a substate $m$ is in the electric-field gradient
with the external magnetic field $B_{0}$, the resonance frequency of the
transitions m+1$\leftrightarrow$m is

$\nu_{m+1\leftrightarrow
m}=\nu_{L}-\nu_{Q}\left(\frac{3\cos^{2}\theta-1}{2}\right)\left(m+\frac{1}{2}\right),$
(4)

where $\nu_{L}$ = $\gamma B_{0}/2\pi$ is the Larmor frequency of the nucleus,
$\nu_{Q}$ is the quadrupole splitting frequency, and $\theta$ is the angle
between the symmetrical axis of the electric-field gradient and direction of
$B_{0}$ [12].

Generally, single crystals of LaAlO3 naturally contain twin domain structures,
which are commonly referred to as twinning domains. These domains have a
symmetrical axis of the inner electric-field gradient, and their orientations
are different from each other. The angle between the two axes is calculated to
be approximately $\arccos(1/3)$ from the structural phase transition of the
crystal, which naturally occurs in the cooling process immediately after the
crystal growth. As a result, the symmetric axis in one domain is parallel to
the direction of $B_{0}$, whereas the other domain’s axis is at an angle of
$\arccos(1/3)$ from the direction of $B_{0}$. To distinguish them, henceforth,
the former is referred to as the ”primary domain” and the latter is the
”secondary domain” in this paper.

It becomes possible to identify the observed peaks in Fig. 5 and Fig. 6 by
comparing the parameters on the peaks to the characteristics listed in Tables
2 and 3. In the 139La spectra, Table 3 lists the value of the peak parameters,
which are useful for identifying the peaks listed in Table 4. From the
consistency of the values of $B_{i}-B_{3}$ with Table 3, the seven observed
peaks are successfully identified as indicated in Fig. 6. The other six peaks
of 139La are not observed in the spectrum, because of the weakness of the NMR
signals.

Table 2:  Characteristics of nine peaks that correspond to the visible peaks in actual 27Al NMR spectra. The $B_{i}-B_{5}$ values were calculated from Eq.(4) and NMR parameters in Ref. [12]. (p) and (s) are the primary domain and secondary domain, respectively. See text for more details.  | Peak
---
index $i$
Transition | Domain |  $B_{i}-B_{5}$ [mT]
1 | $-5/2\leftrightarrow-3/2$ | (p) | -5.4
2 | $-3/2\leftrightarrow-1/2$ | (p) | -2.7
3 | $-5/2\leftrightarrow-3/2$ | (s) | -1.8
4 | $-3/2\leftrightarrow-1/2$ | (s) | -0.9
5 | $-1/2\leftrightarrow+1/2$ | (p), (s) | 0
6 | $+1/2\leftrightarrow+3/2$ | (s) | +0.9
7 | $+3/2\leftrightarrow+5/2$ | (s) | +1.8
8 | $+1/2\leftrightarrow+3/2$ | (p) | +2.7
9 | $+3/2\leftrightarrow+5/2$ | (p) | +5.4
Table 3:  Characteristics of seven peaks among thirteen ones that correspond to the visible peaks in actual 139La NMR spectra. (p) and (s) indicate the primary domain and secondary domain, respectively. The $B_{i}-B_{3}$ values were calculated from Eq.(4) and NMR parameters in Ref. [12].  | Peak
---
index $i$
Transition | Domain |  $B_{i}-B_{3}$ [T]
1 | $-5/2\leftrightarrow-3/2$ | (p) | -0.224
2 | $-3/2\leftrightarrow-1/2$ | (p) | -0.112
twin 1 | $-3/2\leftrightarrow-1/2$ | (s) | -0.037
3 | $-1/2\leftrightarrow+1/2$ | (p), (s) | 0
twin 2 | $+3/2\leftrightarrow+5/2$ | (s) | +0.037
4 | $+1/2\leftrightarrow+3/2$ | (p) | +0.112
5 | $+3/2\leftrightarrow+5/2$ | (p) | +0.224
Table 4: Characteristics of seven peaks observed in the NMR measurement of 139La. Peak index $i$ |  $B_{i}$ [T] |  $B_{i}-B_{3}$ [T] | Width [mT]
---|---|---|---
1 | 4.4505 | -0.217 | $1.1\pm 0.3$
2 | 4.5588 | -0.109 | $0.9\pm 0.0$
twin 1 | 4.6320 | -0.036 | $1.5\pm 0.0$
3 | 4.6677 | 0 | $1.0\pm 0.0$
twin2 | - | - | -
4 | 4.7774 | 0.110 | $1.1\pm 0.1$
5 | 4.8879 | 0.220 | $1.1\pm 0.1$

###  3.2 $T_{1}$ measurements of 139La NMR in the DRS2500 refrigerator

By confirming that the peak widths do not change with time, the peak heights
are used for the analysis of the time dependence of the NMR intensities. Fig.
7 (a) and (b) show the typical time dependence of the height of the central
peak. The values of $T_{1}$ can be obtained by fitting Eq. (3) to the time
dependence of the peak heights. The results of $T_{1}$ for the central peak
under various measurement conditions are summarized in Table 5. In the fitting
results as well as general long $T_{1}$ analysis, there are some large values
of $\chi^{2}/$ndf, because the fitting analysis does not contain the
systematic uncertainty. The dominant systematic effect is originated from a
long term instability of the ground line of the LC circuit in the NMR system.
Because of the sharp Q curve, tiny drifting and spiking of the ground level
causes huge systematic uncertainty. Generally, it is noted that such
systematic effects are practically unavoidable in the long term measurements
of the NMR, such as the long $T_{1}$ measurements.

Figure 7: Typical time dependence of the NMR peak heights for (a) 139La at 0.5 K and 0.5 T, (b) 139La at 0.1 K and 0.76 T, (c) 27Al at 0.5 K and 2.5 T, (d) 27Al at 0.1 K and 0.75 T, (e) 27Al at 1.5 K and 1.0 T, and (f) 27Al at 1.5 K and 2.0 T. Table 5:  Results of the $T_{1}$ measurements for 139La.  | Temperature
---
magnetic field
| Peak
---
index $i$
$T_{1}$ [h] |  $\chi^{2}/$ndf
0.5 K, 2.5 T | 3 | $1.87\pm 0.15$ | 52.8/8
0.5 K, 1.0 T | 3 | $1.82\pm 0.26$ | 10.8/11
0.5 K, 0.5 T | 3 | $1.21\pm 0.17$ | 3.4/6
0.1 K, 0.75 T | 3 | $316\pm 17$ | 24.2/11

###  3.3 $T_{1}$ measurements of 27Al NMR in the DRS2500 refrigerator

In the same manner as the 139La, the peak heights can be used as the peak
intensities in the analysis of the relaxation time, as in the case of 139La.
Fig. 7 (c) shows the height of the central peak in Fig. 5 (b) as a function of
time.

By fitting Eq. (3) to the time dependence of the peak heights, we can evaluate
$T_{1}$. The results for 0.5 K are summarized in Table 6. However, the data
for 0.1 K could not be analyzed with a simple exponential curve, because
$T_{1}$ was extremely large. The data analysis for 0.1 K will be discussed in
Sec. 4.1.

Table 6:  $T_{1}$ results for the 27Al NMR peaks.  | Temperature
---
magnetic field
| Peak
---
index $i$
$T_{1}$ [h] |  $\chi^{2}/$ndf
0.5 K, 2.5 T | 1 | $3.62\pm 1.08$ | 4.8/2
| 2 | $1.72\pm 0.68$ | 5.1/2
| 3 | $5.89\pm 1.50$ | 7.9/2
| 4 | $5.44\pm 0.90$ | 2.7/2
| 5 | $4.11\pm 0.36$ | 13.2/2
| 6 | $7.49\pm 1.16$ | 1.8/2
| 7 | $7.13\pm 1.84$ | 0.54/2
| 8 | - |
| 9 | - |
0.5 K, 1.0 T | 1 | $1.65\pm 0.27$ | 12.2/8
| 2 | $3.47\pm 0.57$ | 12.3/8
| 3 | $1.87\pm 0.17$ | 18.4/8
| 4 | $1.87\pm 0.11$ | 24.8/8
| 5 | $2.15\pm 0.09$ | 64.9/8
| 6 | $1.94\pm 0.14$ | 65.4/8
| 7 | $2.05\pm 0.24$ | 45.1/8
| 8 | $2.74\pm 0.41$ | 40.1/8
| 9 | $2.52\pm 0.50$ | 14.8/8
0.5 K, 0.5 T | 1 | $2.47\pm 0.42$ | 8.1/6
| 2 | $2.42\pm 0.22$ | 47.1/6
| 3 | $1.68\pm 0.12$ | 53.6/6
| 4 | $1.85\pm 0.07$ | 109.3/6
| 5 | $2.06\pm 0.05$ | 132.1/6
| 6 | $1.72\pm 0.06$ | 151.3/6
| 7 | $1.89\pm 0.12$ | 27.8/6
| 8 | $2.04\pm 0.16$ | 17.3/6
| 9 | $1.65\pm 0.24$ | 15.2/6
Table 7:  $T_{1}$ results at 1.5 K for 27Al.  | Temperature
---
magnetic field
| Peak
---
index $i$
$T_{1}$ [h] |  $\chi^{2}/$ndf
1.5 K, 1.0 T | 5 | $0.21\pm 0.01$ | 17.7/14
1.5 K, 2.0 T | 5 | $0.97\pm 0.06$ | 24.5/14

###  3.4 $T_{1}$ measurements of 27Al NMR using the SC

The time dependence of the peak height of the central peak at 1.5 K is shown
in Fig. 7 (e) and (f). As in the previous analysis, fitting Eq. (3) to the
peak heights yields the relaxation times at 1.5 K. The results for 1.0 T and
2.0 T are listed in Table 7.

##  4 Discussion

###  4.1 Estimation of $T_{1}$ at 0.1 K

As shown in the previous section, the analysis using Eq. (3) is not directly
applicable to the 27Al data at 0.1 K, because the NMR signals are not
saturated within the limited experimental period. To make a rough estimation,
we use the first approximation of Eq. (3) with respect to time as follows:

$I(t)=I_{0}+(I_{\rm eq}-I_{0})\Gamma t.$ (5)

Here, the $I_{\rm eq}$ at 0.1 K is treated as a fixed value and is converted
from the $I_{\rm eq}$ at 0.5 K and 1.0 T under the assumption of a Boltzmann
distribution. This assumption requires a sufficiently small difference in the
NMR sensitivity between the two experimental conditions. The sensitivity of
the used system must be checked in this experiment because the corresponding
NMR frequency, 16.8 MHz or 16.4 MHz, is different from that at 0.5 K and 1.0 T
as presented in Table 1. Consequently, we experimentally verified that the
sensitivity for 16.8 MHz, 16.4 MHz, and 11.8 MHz are nearly equivalent.
Therefore, the $I_{\rm eq}$ in the transition $m-1\leftrightarrow m$ can be
estimated as follows:

$\begin{split}&I_{\rm eq,m-1\leftrightarrow m}({\rm 0.1\,K},B_{0})\\\
&\,\approx I_{\rm eq,\,m-1\leftrightarrow m}({\rm 0.5\,K},{\rm
1.0\,T})\times\frac{{\rm 0.5\,K}}{{\rm 1.0\,T}}\times\frac{B_{0}{\rm\,T}}{{\rm
0.1\,K}}\end{split}$ (6)

The results of the data analysis are summarized in Table 8.

Table 8:  Estimation of $T_{1}$ at 0.1 K for 27Al.  | Temperature,
---
magnetic field
| Peak
---
index $i$
$T_{1}$ [h] |  $\chi^{2}/$ndf
0.1 K, 0.76 T | 1 | $53\pm 3$ | 2.7/3
| 2 | $40\pm 2$ | 5.9/3
| 3 | $78\pm 6$ | 5.8/3
| 4 | $71\pm 3$ | 6.1/3
| 5 | $54\pm 1$ | 10.1/3
| 6 | $81\pm 3$ | 11.0/3
| 7 | $86\pm 7$ | 7.3/3
| 8 | $45\pm 2$ | 6.6/3
| 9 | $49\pm 3$ | 0.3/3
0.1 K, 1.6 T | 1 | - |
| 2 | $546\pm 48$ | 4.1/2
| 3 | $293\pm 12$ | 5.9/2
| 4 | $339\pm 10$ | 4.5/2
| 5 | $397\pm 8$ | 13.8/2
| 6 | $363\pm 11$ | 19.3/2
| 7 | $388\pm 11$ | 2.6/2
| 8 | $356\pm 18$ | 1.7/2
| 9 | $235\pm 8$ | 19.7/2

###  4.2 Comparison of $T_{1}$ between the 139La and 27Al nuclei

Table 9 compares the $T_{1}$ in the $-1/2\leftrightarrow 1/2$ transitions of
the 139La and 27Al under the same conditions. All $T_{1}$ values of both the
nuclei at 0.5 K are roughly comparable, although the $T_{1}$ of 139La tends to
be slightly shorter than that of 27Al. However, the $T_{1}$ of 139La at 0.1 K
is approximately six times that of 27Al. The relation of the $T_{1}$ at 0.1 K
is substantially different from that at 0.5 K.

According to previous DNP studies with Nd-doped LaAlO3 crystals [12, 14], the
nuclear Zeeman systems of both 27Al and 139La strongly couple to the
electronic spin-spin reservoir (SSR) [24], and their thermal contact
contributes to the large enhancement of the nuclear polarization in the LaAlO3
crystals. In such a system, the nuclear relaxation is likely to occur through
the SSR. In Ref. [14], coupling strengths of a few minutes were reported for
some central peaks for 139La. If the SSR dominantly contributes to the nuclear
relaxation, the $T_{1}$ values of both nuclear species can be expected to be
almost identical, because the two Zeeman systems and the SSR can be considered
a single reservoir with a common spin temperature. The results at 0.5 K are
consistent with the nuclear relaxation process via the SSR. If the SSR is also
the dominant process at 1.5 K, the results for 27Al can be regarded as the
relaxation of 139La at 1.5 K, and the $T_{1}$ at 2.0 T is 1 h.

A possible explanation for the large difference in $T_{1}$ between 139La and
27Al at 0.1 K is the strong temperature dependence at approximately 0.1 K. In
the relaxation process through the SSR, the relaxation rate is proportional to
the factor $1-P_{0}^{2}$, where $P_{0}$ is the polarization of the
paramagnetic impurities [24] in thermal equilibrium and is expressed as

$P_{0}=\tanh\left(\frac{g_{\rm Nd^{3+}}\mu_{B}B_{0}}{2k_{B}T}\right).$ (7)

Here, $g_{\rm Nd^{3+}}$ is the effective g factor of Nd3+ in the LaAlO3
crystal, and its value in the primary domain is $2.12\pm 0.01$ [13]. The
simple calculation of $1-P_{0}^{2}$ shows that the relaxation time at 0.11 K
is approximately three times shorter than that at 0.10 K under a magnetic
field of 0.75 T. Therefore, the temperature uncertainty of 5% is not
negligible under this condition. The sample was removed from Cold-Finger2
after the NMR measurement of 27Al and mounted again for that of 139La. As the
temperature of 0.1 K was always monitored using a thermometer attached to the
mixing chamber, it is likely that the sample temperature is higher than 0.1 K.

The $T_{1}$ difference between the primary and secondary domains at 0.1 K and
above 0.75 T in Table 8 can be explained by the anisotropy of $g_{\rm
Nd^{3+}}$ ($g_{\parallel}=2.12,g_{\perp}=2.68$) because, in the secondary
domain, the symmetry axis is not parallel to the magnetic field and the
effective $g$ factor is larger than that in the primary domain [13].

Table 9:  Comparison of the $T_{1}$ for the transition $m_{I}:-1/2\leftrightarrow 1/2$ of 139La and 27Al in the Nd-doped sample under various conditions. | Temperature
---
magnetic field
$T_{1}$ of 139La [h] |  $T_{1}$ of 27Al [h]
0.5 K, 0.5 T | $1.21\pm 0.17$ | $2.06\pm 0.05$
0.5 K, 1.0 T | $1.82\pm 0.26$ | $2.15\pm 0.09$
0.5 K, 2.5 T | $1.87\pm 0.15$ | $4.11\pm 0.36$
0.1 K, 0.75 T | $316\pm 17$ | -
0.1 K, 0.76 T | - | $54\pm 1$

###  4.3 Feasibility of the T-violation searches with the Nd-doped LaAlO3
crystal

When we use the polarized 139La target in the search for the T-violation, it
is necessary to decrease the external magnetic field for canceling out the
neutron spin rotations with the pseudomagnetic effect, which is caused by the
interaction between an incident neutron and a polarized nucleus [24]. It
possibly causes significant systematic uncertainty and a reduction in the
sensitivity for the T-violation search. Previous studies have demonstrated
that the magnitude of the external magnetic field for canceling pseudo
magnetic rotation is approximately 0.23 T when using a 100 % polarized LaAlO3
target [25]. Considering the past achievement of approximately 50 %
polarization, it can be inferred that the magnetic field in the target
operation is approximately 0.1 T. Considering the cryogenic system in a high-
intensity neutron beamline, a realistic condition for the temperature is
approximately 0.1 K. The $T_{1}$ of 139La under the above target condition is
rather significant, but most NMR apparatuses including our system are not
generally adaptable for such a low magnetic field. Thus, we roughly estimated
$T_{1}$ at 0.1 K and 0.1 T from our experimental results.

We applied a simple relaxation model via the SSR [24], which is described as,

$\frac{1}{T_{1}}\propto C^{2}\frac{1}{T_{\rm
1ss}}\left(\frac{1}{B_{0}}\right)^{2}(1-P_{0}^{2}),\\\ $ (8)

where $T_{\rm 1ss}$, $C$, and $P_{0}$ are the relaxation time of the SSR, the
concentration of Nd3+, and the polarization of Nd3+ given by Eq. (7),
respectively. Using Eq. (8), the $T_{1}$ results for 27Al facilitate the
deduction of the ratios of $T_{\rm 1ss}$ to that under a specific condition
for various conditions:

$\begin{split}&r(B_{0},T)\coloneqq\frac{T_{\rm 1ss}({\rm
1.0\,T,0.5\,K})}{T_{\rm 1ss}(B_{0},T)}=\\\ &\frac{T_{1}({\rm
1.0\,T,0.5\,K})}{T_{1}(B_{0},T)}\frac{B_{0}^{2}}{({\rm
1.0\,T})^{2}}\frac{1-P_{0}({\rm
1.0\,T,0.5\,K})^{2}}{1-P_{0}(B_{0},T)^{2}}.\end{split}$ (9)

Fig. 8 shows $\log_{10}r(B_{0},T)$ as a function of the magnetic field.

Because the coupling strength of the SSR to the lattice system has not been
sufficiently studied in terms of the field dependence so far, we simply
extrapolate the $T_{\rm 1ss}$ from Fig. 8 here. Consequently, the field
dependence of the relaxation rate $1/T_{\rm 1ss}$ is approximately identified
as an exponential decay with respect to the magnetic field. As Fig. 8 shows
that the $T_{\rm 1ss}$ at 0.1 K and 0.1 T is approximately 100 times that at
0.5 K and 1.0 T, we can obtain the $T_{\rm 1}$ of 139La under the target
condition as

$\displaystyle T_{1}\approx T_{1}({\rm
1.0\,T,0.5\,K,\,La})\left(\frac{0.1\,{\rm T}}{1.0\,{\rm T}}\right)^{2}\times
0.01$ $\displaystyle\times\frac{1-P_{0}({\rm 1.0\,T,0.5\,K})^{2}}{1-P_{0}({\rm
0.1\,T,0.1\,K})^{2}}.$ (10)

$T_{1}$ is obtained as approximately 1 h. However, this analysis is based on
the results of 27Al at 0.1 K. Assuming that the large difference of $T_{1}$ at
0.1 K between 139La and 27Al in Table 9 is caused by a slight difference in
the sample temperature, it is expected that the $T_{1}$ of 27Al was
underestimated. Therefore, we can expect that $T_{1}\geq 1$ h under the target
condition.

There is still much room for improving the nuclear relaxation in terms of the
following three points. The first is to decrease $C$ in Eq. (8) because the
nuclear relaxation through the SSR depends on the square of $C$. If $C$ is
reduced by a factor of three compared to the present value, for example,
$T_{1}$ can be expected to become an order of a large magnitude. The second is
the preparation of a crystal with high quality that is almost free from
crystal defects such as oxygen vacancies. Such defects act as paramagnetic
centers, which accelerate the nuclear relaxation. The third is to use a
powerful dilution refrigerator to reduce the temperature to 0.05 K.

Figure 8: $T_{\rm 1ss}$ estimated from the 27Al data using the simple
relaxation model described by Eq. (8). The circles and squares are the
estimated $\log_{10}r$ at 0.5 K and 0.1 K, respectively.

##  5 Summary

To evaluate the possibility of using a LaAlO3 crystal doped with 0.03 mol% Nd
ions as the polarized La target, we conducted the first experiment to
systematically measure the nuclear spin-lattice relaxation time of 139La and
27Al in the crystal at temperatures of 0.1, 0.5, and 1.5 K under various
magnetic fields up to 2.5 T. Additionally, we estimated the relaxation time at
0.1 K under 0.1 T, which is a suitable condition for beam experiments.

The relaxation time of 139La at 0.5 K was 1.21$\pm$0.17 h, 1.82$\pm$0.26 h,
and 1.87$\pm$0.15 h for 0.5, 1.0, and 2.5 T, respectively, and that at 0.1 K
was 316$\pm$17 h for 0.75 T. These results at 0.5 K are comparable to those of
27Al and consistent with the nuclear relaxation, which is dominant through the
SSR. The significant difference at 0.1 K can be explained from the temperature
uncertainty of 5 % under the assumption of the SSR. We simply extrapolated the
relaxation time of the SSR with the results of 27Al. Consequently, we
estimated that the $T_{1}$ of 139La is greater than 1 h at 0.1 K and 0.1 T.

There is still much room for improving the value, particularly by reducing the
doping amount of the Nd ions, using a crystal with a low defect density, and
cooling down to 0.05 K using powerful cryogenics. An improvement by two orders
of magnitude will enable the use of the Nd-doped LaAlO3 crystal in T-violation
experiments. To improve the present nuclear relaxation, crystal growth and
powerful cryogenics are expected to be key factors. We will continue the
studies on the crystal growth of Nd-doped LaAlO3 crystals using the floating-
zone method [26]. This method is expected to avoid undesirable contamination
by impurities originating from the use of the crucible.

## Acknowledgement

The present study was conducted at the Research Center for Nuclear Physics
(RCNP), Osaka University with the approval of the project proposal
”Development of polarized target for new physics search via T-violation.” This
work was partly supported by the RCNP Collaboration Research Network program
as the project number COREnet-026. The studies on the target material were
supported by the Tohoku University IMR cooperative program (proposal No.
18G0034, 19K0081, and 19G0037). We are grateful to the staff of the Low
Temperature Center of Osaka University for supplying us with the required
liquid helium. We thank Dr. Patrick Hautle (PSI) and Prof. Taku Matsushita
(Nagoya Univ.) for their fruitful discussions. The present work was supported
in part by the Ministry of Education, Science, Sports, and Culture of Japan.

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