text:
Are the Days of Car Ownership Coming to a Close?
Manos Legal Services > Bankruptcy > Are the Days of Car Ownership Coming to a Close?
Self-driving vehicles are coming much sooner than expected — and the general consensus is that car ownership will bid us farewell. After all, with a dominant ride-sharing model in place, upfront purchase prices and ongoing maintenance costs seem less appealing. On top of that is the surging trend around "access vs. ownership," where the car-owning value proposition is taking some real hits.
In the future, getting around will be much easier and cheaper, and the hassle of owning a vehicle may quickly outweigh the benefits. What's more, as driverless taxies become ubiquitous and accrue revenue 24/7, their prices will only drop.
A report published last year by RethinkX, an independent think tank based in San Francisco, estimates fully autonomous vehicles will make up 95 percent of passenger miles in the U.S. within a decade of the anticipated 2020 rollout. The study says that cars will be owned by companies like Lyft and Uber rather than individuals. This firm estimates the windfall of savings to be around $1 trillion for Americans each year. That's about $5,600 per household, "generating the largest infusion of consumer spending in history." What's more, that figure only accounts for improvements in general overhead costs and doesn't factor productivity increases and reduction losses due to traffic accidents. Talk about a bang for your buck.å
Researchers at consulting firm KPMG similarly predict that by 2030, households will no longer need sedans — only keeping larger vehicle models for road trips — and that fully autonomous vehicles will be used as ride-hailing services from the outset. Anticipating the upended car market, Waymo and General Motors' already have pilot ride-hailing programs in the works, set to launch in select urban and suburban areas. Likewise, Ford bought out their ridesharing company, Chariot, as part of their self-driving vehicle initiative. Further, public officials in Helsinki are pushing to make car ownership obsolete by 2025.
Don't believe how big a business this is? Just look at the rift between Google and Uber — and the nose-bleeding $245 million settlement over alleged "trade secrets" in 2016. Autonomous cars are big business…and all the tech firms want in.
But while logically sound, these forecasts focus heavily on the economic disruption caused by full automation, not so much the cultural implications. For instance, few consumer goods have profoundly molded our cultural ethos as much as the automobile, and the majority of Americans are still wary of self-driving cars — not to mention a slew of other important factors.
How Is Safe Safe Enough?
Critical to the success of automation is proving that it's safe. Although researchers say it'll reduce auto accidents by a whopping 90 percent, "better than human" might be a weak benchmark when you consider that people have a greater tolerance for deaths caused by humans than robots. For this reason, Mobileye CEO Amnon Shashua proposed a formula to ensure self-driving vehicles are virtually infallible — in effect, laying the groundwork for how policymakers and manufacturers can manage the deployment of wholesale driverless cars without constricting innovation.
The Home Away From Home
Considering many of us like to keep personal belongings in our car, the allure of the private car could hold sway for a lot of people — especially commuters who can get a jump-start on work without having to watch the road. It's not so implausible that consumers are willing to pay extra for the flexibility of being able to travel when, where and with whom they want — without compromise. An article published by Yale points out that while ride-sharing has soared in popularity, it hasn't directly dissuaded people from buying cars for this very reason.
The "shared autonomous" firms of the future would do well to ensure that they can "user integrate" consumer preferences whenever they pick up a new rider. Making the autonomous vehicle feel "owned" is a great way to bridge this gap, and it can be as simple as a Bluetooth integration and content sync.
What If You're Too Far Away?
Waymo and General Motors are set to test their pilot programs in select urban and suburban areas. After all, 81 percent of America's population lives in areas that are dubbed "urban or suburban." But what about rural settings, or even remote suburbs? A driverless cab might arrive in just minutes in a densely populated area, but it might not be as readily accessible for those living in areas where houses can be miles apart. The automobile has made it easier for people to live on the outskirts, enabling folks to leave at the drop of a hat. Those living in rural outposts may not be keen on relinquishing that level of control, even when driverless cabs are ubiquitous in cities.
Suburban Sprawl and Environmental Strain
On the other hand, more people might be apt to live in remote locales if a robot can schlep them to work every day. That could mean more emigration to rural areas and thus higher demand for driverless cars. But a policy brief from UC Davis warns that without proper oversite, self-driving technology could increase suburban sprawl, which was already spurred by the invention of the automobile.
This is especially true in places like California, where sprawl is a huge point of contention among developers and public officials. The policy brief reasons that if passengers can work or rest more during their commute, then they'll likely be willing to travel further. As a result, affordable housing developments will be on the rise. That's a win for housing, but a loss for a decreased eco-footprint.
Drivers Licenses on the Decline
In fairness, there's another cultural phenomenon that ought to be called out: Since 1983, fewer and fewer people have been getting their drivers licenses. To understand why this is happening, the Transportation Research Institute at the University of Michigan conducted a survey of 618 respondents, ages 18 to 39. They found the three main reasons people forewent getting their driver's licenses were busy schedules, the cost of owning and maintaining a car, and the ability to get rides easily from others.
It's important to note that predictions, however researched, shouldn't be confused with conclusions. The one thing we can say for sure, however, is that the culture of tomorrow's automobile won't be a carbon copy of what it is today.
Haden Kirkpatrick is the head of marketing strategy and innovation at Esurance, where he is responsible for all initiatives related to product and service innovation. Haden is an innovator writes about how smart technology—from IoT to machine learning to self-driving cars — will impact the insurance industry. To view Esurance's car insurance options, click here.


text:
Rivers of London author Ben Aaronovitch joins line-up for St Albans Literary Festival
Matt Adams
Ben Aaronovitch
A bestselling author who successfully fuses police procedural drama with supernatural folklore is the latest big name joining the line-up for this summer's St Albans Literary Festival.
Ben Aaronovitch will be appearing at Waterstones on the evening of Saturday July 9 to discuss his acclaimed supernatural police series Rivers of London, which sees the sixth instalment The Hanging Tree due later this year.
A cult supernatural crime series with one of the most diverse audiences in contemporary publishing, the series follows the adventures of Police Constable and apprentice wizard Peter Grant as he tries to maintain law and order amongst the more fantastical members of London's population.
Storming Sunday Times bestseller lists and accumulating sales of over 500,000 copies to date, the key to the success of the Rivers of London books are their unique blend of police action, supernatural mayhem and threads of London's fascinating hidden history woven through the very fabric of the plot.
Former Doctor Who and Casualty scriptwriter Ben was a huge success when he appeared in a panel at the first St Albans LitFest in 2014, and this is a fantastic opportunity to see him on his own.
It includes talks and workshops taking place in different sections of St Peter's and also in the churchyard, using the storyteller's circle opened there in the summer of 2014, as well as select locations around St Albans city centre.
The festival will be kicking off on Monday July 4 and runs until Sunday July 10, and promises a programme of exciting activities aimed at appealing to both adults and children. Confirmed guests include Human Rights lawyer Anthony Lester QC, graphic novelist Dave Gibbons, crime writers David Mark and Peter Robinson, and historical author Alison Weir, with more to be announced in the coming weeks.
St Albans LitFest will also support any fringe events taking place during the weekend, offering publicity and assistance if any organisation wants to run their own talk.
Run entirely by volunteers, the festival is supported by the Herts Advertiser, and editor Matt Adams said: "What a line-up there is for this year's festival! Don't risk disappointment and leave it too late to buy your tickets as many of these events are sure to be sell-outs."
If you'd like to get involved in the festival, or just want to find out more information as it becomes available, visit the Facebook site or visit the website.
Tickets for all announced events are now available from the LitFest website.

redpajama_set_name: RedPajamaCommonCrawl


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text:
\section{\label{sec:Introduction}Introduction}\enlargethispage{8pt}

The combination of intriguing fundamental physics with far-reaching
potential applications has made unconventional superconductors one of
the most studied classes of materials. Standing out among them are the
noncentrosymmetric superconductors (NCSCs)~\cite{Bauer2012}, whose
crystal structures lack an inversion symmetry. As a consequence, in NCSCs, the
strict symmetry-imposed requirements are relaxed, allowing
mixtures of spin-singlet and spin-triplet Copper pairing channels,
thus setting the scene for a variety of exotic properties, as e.g.,
upper critical fields beyond the Pauli limit, nodes in the superconducting
gaps, etc. (see Refs.~\cite{Bauer2012,Smidman2017,Ghosh2020} for an overview).
The degree of mixing in such combined pairings is related to the
strength of the antisymmetric spin-orbit coupling (ASOC) and to other
microscopic parameters, still under investigation. Currently,
NCSCs rank among the foremost categories of superconducting materials
in which to look for topological superconductivity (SC) or to realize the
Majorana fermions, pairs of the latter potentially acting as
noise-resilient qubits in quantum computing~\cite{Kim2018,Sun2015,Ali2014,Sato2009,Tanaka2010,Sato2017,Qi2011,Kallin2016}.

In general, the various types of NCSCs can be classified into two classes.
One consists of strongly correlated materials, as e.g., CePt$_3$Si~\cite{Bauer2004}, or
Ce(Rh,Ir)Si$_3$~\cite{Muro1998},
which belong to the heavy-fermion compounds. Owing to the strong
correlation and the interplay between $d$- and $f$-electrons, these materials
often exhibit rich magnetic and superconducting properties. Since
their superconductivity is most likely mediated by
spin fluctuations, this implies an unconventional (i.e., non phonon-related) pairing mechanism.
Conversely, the other class consists mainly of weakly correlated materials,
which are free of ``magnetic'' $f$-electrons, as e.g., LaNiC$_2$, La$_7$Ir$_3$,
CaPtAs, or Re$T$ ($T$ = 3$d$-5$d$ early transition metals)~\cite{Hillier2009,Barker2015,Shang2020,Singh2014,Singh2017,Shang2018a,Shang2018b}.
Obviously, their superconductivity
is not mediated by electrons' spin fluctuations.
Hence, they lend themselves as prototype parent systems where one can study the intrinsic pairing mechanisms in NCSCs.

Recently, superconductivity with broken time-reversal symmetry (TRS)
has become a hot topic in NCSCs. The main reason for this is the
discovery of TRS breaking in some
weakly-correlated NCSCs using
muon-spin relaxation ($\mu$SR).
Surprisingly,
the superconducting properties of the latter
largely resemble those of conventional superconductors, i.e., they
are clearly distinct from those of the above mentioned
strongly-correlated NCSCs. To date, only a handful of
NCSC families have been shown to exhibit TRS breaking in the superconducting state, including LaNiC$_2$~\cite{Hillier2009}, La$_7$(Rh,Ir)$_3$~\cite{Barker2015,Singh2018La7Rh3}, Zr$_3$Ir~\cite{Shang2020b}, CaPtAs~\cite{Shang2020}, and Re$T$~\cite{Hillier2009,Barker2015,Shang2020,Singh2014,Singh2017,Shang2018a,Shang2018b}. Except for the recently studied CaPtAs, where coexisting TRS breaking and superconducting gap nodes were observed below $T_c$, in most of the above cases the
superconducting properties evidence a conventional $s$-wave
pairing, characterized by a fully opened superconducting gap.
This leads to an interesting fundamental question: does the observed
TRS breaking in NCSCs originate from an unconventional superconducting mechanism
(i.e., from a pairing other than that mediated by phonons),
or it can occur also in presence of conventional pairing
(via some not yet understood mechanism)~\cite{Smidman2017,Ghosh2020}?
Why, among the many different NCSCs families, only a few exhibit
a broken TRS in the superconducting state, also remains an intriguing
open question.

In general, the causes behind the
TRS breaking at the onset of superconductivity are mostly unknown.
In particular, the $\alpha$-Mn-type noncentrosymmetric Re$T$ ($T$ = Ti, Nb, Zr, and Hf) superconductors have been widely studied
and demonstrated to show a superconducting state with broken TRS~\cite{Singh2014,Singh2017,Shang2018a,Shang2018b}. Yet, TRS seems to be
preserved in the isostructural (but Re-free) Mg$_{10}$Ir$_{19}$B$_{16}$
and Nb$_{0.5}$Os$_{0.5}$~\cite{Acze2010,SinghNbOs}.
Further, depending on synthesis protocol, Re$_3$W is either a
centro- (hcp-Mg-type) or a noncentrosymmetric ($\alpha$-Mn-type)
superconductor, yet neither is found to break
TRS~\cite{Biswas2012}.
In case of binary Re-Mo alloys, depending on the
Re/Mo ratio, the compounds can exhibit up to four different crystal structures, including both cen\-tro\-sym\-me\-tric and non\-cen\-tro\-sym\-me\-tric cases.
Most importantly, all these alloys become superconductors at low temperatures~\cite{Shang2019}. A comparative $\mu$SR study of Re-Mo alloys,
covering all the different crystal structures, reveals that the
spontaneous magnetic fields occurring below $T_c$ (an indication of TRS breaking) were
only observed in elementary rhenium and in Re$_{0.88}$Mo$_{0.12}$.
By contrast, TRS was preserved in the Re-Mo alloys with a lower
Re-content (below $\sim 88$\%), independent of their centro- or noncentrosymmetric crystal structures~\cite{Shang2020ReMo}.
Since both pure rhenium and Re$_{0.88}$Mo$_{0.12}$ have a simple (hcp-Mg-type) centrosymmetric structure, this strongly
suggests that a noncentrosymmetric structure and the accompanying ASOC effects are not essential in realizing the TRS breaking in Re$T$ superconductors.
The $\mu$SR results regarding the Re-Mo family, as well as
other Re-free $\alpha$-Mn-type superconductors, clearly imply
that not only the Re presence, but also its amount are crucial for the appearance and the extent of TRS breaking in the Re$T$ superconductors.
How these results can be understood within a more general framework requires further experimental and theoretical investigations.

This short review article focuses mostly on the experimental study of Re-based
binary superconductors. In Sec.~\ref{sec:musr_intro}, we discuss the basic principles of our probe of choice, the $\mu$SR, here used
to detect TRS breaking and to characterize the superconducting properties.
Section~\ref{sec:Re_based_SC} describes the possible crystal
structures and superconducting transition temperatures of Re$T$ binary
alloys. In  Sec.~\ref{sec:Hc2_fields}, we focus on
the upper critical fields and the
order parameter in Re$T$ superconductors. Section~\ref{sec:TRS_breaking} discusses
the TRS breaking in Re$T$ superconductors and its possible origins.
Finally, in the last section, we outline some possible future research directions.

\begin{figure*}[!thp]
	\centering
	\includegraphics[width=0.8\textwidth,angle=0]{Fig1.pdf}

	\caption{\label{fig:musr_basics}Principle of the time-differential $\mu$SR experiment.
		\textbf{(A)} An incoming polarized muon (with spin $\boldsymbol{S_\mu}$
		antiparallel to momentum $\boldsymbol{p_\mu}$) is first detected by a
		(thin) muon detector, which starts the electronic clock.
		In the sample, the muon spin precesses in the internal/external
		field until the muon decays into a positron.
		This is emitted preferentially along the muon-spin direction
		and hits one of the positron detectors [here forward (F) or backward (B)],
		whose signal stops the clock.
		The gray curve depicts the anisotropic positron-emission pattern at the
		moment of muon implantation. The pattern rotates rigidly with the muon
		spin (initially pointing towards the B detector) at an angular
		frequency $\omega_{\mu} = \gamma_{\mu}B_\mathrm{ext}$.
		\textbf{(B)} Detected positron counts in the F and B
		detectors as a function of time after ca.\ $10^{7}$ events.
		Inset: The asymmetry signal, obtained as the normalized difference
		between the F and B counts, eliminates the inessential exponential
		decay and highlights the signal decay (here assumed to be Gaussian)
		reflecting the nature of the sample.
	}
\end{figure*}

\section{Muon-spin relaxation and rotation}
\label{sec:musr_intro}\enlargethispage{8pt}

Initially considered as an ``exotic'' technique, over the years muon-spin
rotation, relaxation, and resonance (known as $\mu$SR),
has become one of the most
powerful methods to study the magnetic and superconducting properties
of matter. This follows from a series of fortunate circumstances, related
to the muon properties as a fundamental particle. Most notably, these
include the 100\% initial muon-spin polarization, following the two-body
decay from pions, and the subsequent preservation of such information
through the weak decay into positrons.
In the search for unconventional superconductivity, as well as for
TRS breaking effects, the very high sensitivity of the $\mu$SR technique
to tiny magnetic fields is especially important~\cite{Amato1997}.
Below we briefly outline the basics of the $\mu$SR technique and direct
the reader to other references for more detailed information
\citep{Blundell1999,Brewer2003,Yaouanc2011}.

\vspace{5mm}
\subsection{Principles of the $\mu$SR technique}
Central to the $\mu$SR method is the availability of polarized positive
muon ($\mu^{+}$) beams, obtained by collecting the muons produced in
the two-body decay of positive pions,
$\pi^{+} \rightarrow \mu^{+} + \nu_{\mu}$ (with $\nu_{\mu}$ the muon
neutrino), decaying at rest in the laboratory frame.
Since pions have no intrinsic angular momentum and neutrinos have
a fixed helicity (relative orientation of spin and linear momentum),
the resulting muon beam is 100\% spin polarized, with the muon spins
directed antiparallel to the linear momentum (see \textbf{Figure~\ref{fig:musr_basics}A}).
Having an energy of $\sim$4.12\,MeV, muons can penetrate a sample between
0.1\,mm and 1\,mm, depending on the sample density.
Once implantated,
the monoenergetic muons decelerate
within 100\,ps through ionization processes (which do not interact
with the muon spin) and finally come to rest at an interstitial site,
practically without loss of their initial spin polarization.
From this moment, if subject to magnetic interactions, the muon-spin
polarization $\boldsymbol{P}(t)$ evolves with time (the muon spin
precesses around the local magnetic field), thus providing important
information on the sample's magnetism. The detection of the $P(t)$
evolution is made possible by the parity-violating weak-decay interaction
$\mu^{+} \rightarrow e^{+} + \nu_{e} + \bar\nu_{\mu}$
($e^{+}$, $\nu_{e}$, and $\bar\nu_{\mu}$ are the positron, electron neutrino,
and muon antineutrino, respectively), which implies a preferential
emission of positrons along the muon-spin
direction at the time of decay (see \textbf{Figure~\ref{fig:musr_basics}A}, which
depicts also the anisotropic positron-emission pattern).
Thus, by detecting the spatial distribution of positrons as a
function of time, one can determine the time evolution of the muon-spin
polarization $\boldsymbol{P}(t)$.

A schematic diagram of a time-differential $\mu$SR experiment is shown
in \textbf{Figure~\ref{fig:musr_basics}A}. The incoming muon triggers
a clock that defines the starting time $t_0$. Once implanted, the muon
spin precesses about the local magnetic field $\boldsymbol{B}(r)$ with
a Larmor frequency $\omega_{\mu} = \gamma_{\mu}B(r)$, where
$\gamma_{\mu}/2\pi = 135.53$\,MHz/T is the muon gyromagnetic ratio.
The clock stops when, after a mean lifetime of 2.197\,$\mu$s, the muon
decays into a positron $e^+$, registered as an event by one of
the positron detectors. The measured time intervals for ca.\ 10--50
millions of such events are stored in a histogram, given by
(see \textbf{Figure~\ref{fig:musr_basics}B}):
\begin{equation}
	\label{eq:counts}
	N(t) = N_0\exp(-t/\tau_{\mu})[1 + A_0P(t)] + C.
\end{equation}
Here, the exponential factor accounts for the radioactive muon decay,
$N_0$ is the initial count rate at time $t_0$, while $C$ is a
time-independent background (due to uncorrelated start and stop counts).
As shown in the inset of \textbf{Figure~\ref{fig:musr_basics}B},
unlike the inessential exponential decay, the physical information in
a $\mu$SR experiment is contained in the $A(t) = A_0P(t)$ term (often
known as the $\mu$SR spectrum).
Here, $A_0$ is the so-called initial asymmetry (typically 0.3,
depending on the detector's solid angle and efficiency),
while $P(t)$ is the muon-spin depolarization function, here given by
the projection of $\boldsymbol{P}(t)$ on the unit vector describing
the detector.
Since $P(t)$ represents the autocorrelation function of the
muon spin $\boldsymbol{S}$, i.e.,
$P(t) = \left< \boldsymbol{S}(t) \boldsymbol{S}(0)\right> /S(0)^2$, it
depends on the average value, the distribution, and the time
evolution of the internal magnetic fields, thus reflecting
the physics of the
magnetic interactions in the sample under study
To access the $\mu$SR signal we need to remove the extrinsic
decay factor by combining the positron counts from pairs of
opposite-lying detectors, for instance, $N_F$ and $N_B$
(for forward and backward), and obtain the asymmetry
\tcr{
	$A(t) = [N_F(t) - \alpha N_B(t)]/ [N_F(t) + \alpha N_B(t)]$. Clearly, $A(t)$
	behaves as a normalized detector ``contrast'', proportional to $A_0$. The parameter $\alpha$ is introduced to take into account the different efficiencies  of the positron-detectors and needs to be determined by calibration.
}

\vspace{5mm}
\subsection{Transverse-field $\mu$SR}
\label{ssec:TF_muSR}
Depending on the reciprocal orientation of the external magnetic field
$\boldsymbol{B}$ with respect to the initial muon-spin direction
$\boldsymbol{S}(0)$, in a $\mu$SR experiment, two different configurations
are possible. (i) In transverse-field (TF) $\mu$SR the externally
applied field $\boldsymbol{B}$ is perpendicular to
$\boldsymbol{S}(0)$ and the muon spin precesses
around $\boldsymbol{B}$
(see \textbf{Figure~\ref{fig:musr_basics}A}). (ii) In a
longitudinal field (LF) configuration the applied field is parallel to
$\boldsymbol{S}(0)$, generally implying only a relaxing $\mu$SR signal.

Although, in principle, the TF scheme shown in \textbf{Figure~\ref{fig:musr_basics}A}
works fine, strong transverse fields perpendicular to the muon
momentum ($\boldsymbol{p_{\mu}}$)
would deviate the muon beam too much from its original path. The Lorentz
force can be zeroed by applying the field along the muon momentum.
At the same time, to maintain the transverse geometry, the initial
muon spin is rotated by $90^{\circ}$ (in the $x$ or $y$ direction) by
using a so-called Wien filter, consisting of crossed electric and magnetic
fields. Such a configuration is also known as transverse muon-spin mode,
while \textbf{Figure~\ref{fig:musr_basics}A} plots the longitudinal
muon-spin mode (i.e., $\boldsymbol{p_{\mu}} \parallel \boldsymbol{S_{\mu}}$).

Since muons are uniformly implanted in the sample, they can detect
the coexistence of different domains, characterized by distinct
$P_i(t)$ functions, whose amplitudes $A_i$ represent a measure of
the respective \emph{volume fractions}. In case of superconductors,
one can thus extract the SC volume fraction. More importantly, in a
TF-$\mu$SR experiment one can directly probe the SC flux-line lattice (FLL).
In this case, at the onset of superconductivity, the muon-spin
precession in a TF field loses coherence, reflecting the magnetic
field modulation (i.e., broadening) due to the FLL.
The shape of the field distribution arising from
the FLL can be analyzed and eventually used to extract the magnetic
penetration depth $\lambda$ and the coherence length $\xi$
\cite{Sonier2000}. In many type-II superconductors, the simple relation,
$\sigma_\mathrm{sc}^2 / \gamma_{\mu}^2 = 0.00371 \Phi_0^2/\lambda_\mathrm{eff}^4$,
connects the muon-spin depolarization rate in the superconducting
phase, $\sigma_\mathrm{sc}$, with the effective magnetic penetration
depth, $\lambda_\mathrm{eff}$ (here $\Phi_0$ is the magnetic flux
quantum)~\cite{Barford1988,Brandt2003}.
In case of superconductors with relatively low upper critical
fields, the effects of the overlapping vortex cores with increasing field ought to be
considered when extracting the magnetic penetration depth $\lambda_\mathrm{eff}$ from $\sigma_\mathrm{sc}$.
Since $\lambda(T)$ is sensitive to the low-energy excitations, its
evolution with temperature is intimately related to the structure
of the superconducting gap. Hence, $\mu$SR allows us to directly
study the \emph{symmetry} and \emph{value} of the superconducting gap.

More in detail, in a TF-$\mu$SR experiment, the time evolution of
the asymmetry can be modeled by:
\begin{equation}
	\label{eq:TF_muSR}
	A_\mathrm{TF}(t) = \sum\limits_{i=1}^n A_i \cos(\gamma_{\mu} B_i t + \phi) e^{- \sigma_i^2 t^2/2} +	A_\mathrm{bg} \cos(\gamma_{\mu} B_\mathrm{bg} t + \phi).
\end{equation}
Here $A_{i}$, $A_\mathrm{bg}$ and $B_{i}$, $B_\mathrm{bg}$
are the asymmetries and local fields sensed by implanted muons in the
sample and the sample holder, $\gamma_{\mu}$
is the muon gyromagnetic ratio, $\phi$ is a shared initial phase, and $\sigma_{i}$
is the Gaussian relaxation rate of the $i$th component. The number of
required components is material dependent, typically in the $1 \leq n \leq 5$ range. In general, for superconductors with a large Ginzburg-Landau parameter
$\kappa$ ($\gg$ 1), the magnetic penetration depth is much larger than the coherence length. Hence, the field profiles of
each fluxon overlap strongly, implying a narrow field distribution. Consequently, a single-oscillating component
is sufficient to describe $A(t)$. In case of a small $\kappa$ ($\geq 1/\sqrt{2}$),
the magnetic penetration depth is comparable with the coherence length.
The rather small penetration depth implies fast-decaying fluxon
field profiles and a broad field distribution, in turn requiring
multiple oscillations to describe $A(t)$. The choice of $n$ can be
determined from the fast-Fourier-transform (FFT) spectra of the
TF-$\mu$SR, which is normally used to evaluate the goodness of the fits.
In case of multi-component oscillations,
the first term in Equation~\eqref{eq:TF_muSR} describes the field distribution
as the sum of $n$ Gaussian relaxations~\cite{Maisuradze2009}:
\begin{equation}
	\label{eq:TF_muSR_2}
	p(B) = \gamma_{\mu} \sum\limits_{i=1}^n \frac{A_i}{\sigma_i} \, \exp\!\left[-\frac{\gamma_{\mu}^2(B-B_i)^2}{2\sigma_i^2}\right].
\end{equation}
The first- and second moments of the field distribution in the sample
can be calculated by:
\begin{equation}

	\label{eq:2nd_moment}
	\langle B   \rangle = \sum \limits_{i=1}^n \frac{A_i B_i}{A_\mathrm{tot}}
	\quad \textrm{and} \quad
	\langle B^2 \rangle = \frac{\sigma_\mathrm{eff}^2}{\gamma_\mu^2} = \sum\limits_{i=1}^{n}%
	\frac{A_i}{A_\mathrm{tot}} \left[\frac{\sigma_i^2}{\gamma_{\mu}^2} + (B_i - \langle B \rangle)^2 \right],
\end{equation}
where $A_\mathrm{tot} = \sum_{i=1}^n A_i$. The total Gaussian relaxation
rate $\sigma_\mathrm{eff}$ in Equation~\eqref{eq:2nd_moment}
includes contributions from both a temperature-independent relaxation,
due to nuclear moments ($\sigma_\mathrm{n}$), and a temperature-dependent
relaxation, related to the FLL in the superconducting state ($\sigma_\mathrm{sc}$).
The $\sigma_\mathrm{sc}$ values are then extracted by subtracting
the nuclear contribution following
$\sigma_\mathrm{sc}$ = $\sqrt{\sigma_\mathrm{eff}^{2} - \sigma^{2}_\mathrm{n}}$.

To get further insights into the superconducting gap value and its
symmetry, the temperature-dependent superfluid density $\rho_\mathrm{sc}(T)$
[proportional to $\lambda_\mathrm{eff}^{-2}(T)$] is often analyzed by
using a general model:
\begin{equation}
	\label{eq:rhos}
	\rho_\mathrm{sc}(T) = \frac{\lambda_0^2}{\lambda_\mathrm{eff}^2(T)} =  1 + 2\, \Bigg{\langle} \int^{\infty}_{\Delta_{k}} \frac{E}{\sqrt{E^2-\Delta_{k}^2}} \frac{\partial f}{\partial E} \mathrm{d}E \Bigg{\rangle}_\mathrm{FS}.
\end{equation}
Here, $\lambda_0$ is the effective magnetic penetration depth
in the 0-K limit, $f = (1+e^{E/k_\mathrm{B}T})^{-1}$ is the Fermi function and $\langle \rangle_\mathrm{FS}$ represents an average over the Fermi surface~\cite{Tinkham1996}.
$\Delta_{k}(T) = \Delta(T) A_{k}$ is an angle-dependent gap
function, where $\Delta$ is the maximum gap value and $A_{k}$ is
the angular dependence of the gap, equal to 1, $\cos2\phi$, and $\sin\theta$
for an $s$-, $d$-, and $p$-wave model, respectively, with $\phi$
and $\theta$ being the azimuthal angles.
The temperature dependence of the gap is assumed to follow
the relation
$\Delta(T) = \Delta_0 \mathrm{tanh} \{1.82[1.018(T_\mathrm{c}/T-1)]^{0.51} \}$~\cite{Tinkham1996,Carrington2003}, where $\Delta_0$ is the 0-K gap value.

\vspace{5mm}
\subsection{Zero-field $\mu$SR}
\label{ssec:ZF_muSR}
A particular case of LF, is that of zero-field (ZF) $\mu$SR, characterized
by the absence of an external magnetic field. In this configuration
the frequency of the $\mu$SR signal is exclusively proportional to
the internal magnetic field, making it possible to determined the size
of the ordered moments and, hence, the magnetic order parameter.
Unlike various techniques, which require an external field to polarize
the probe, $\mu$SR is unique in its capability of studying materials
unperturbed by externally applied fields and in accessing their
spontaneous magnetic fields. Due to the large muon magnetic moment
($\mu_{\mu} = 8.89\,\mu_\mathrm{N}$), ZF-$\mu$SR can sense
even very small internal fields ($\sim 10^{-2}$\,mT), hence
probing local magnetic fields of either nuclear or electronic nature.
In addition, since the muon is an elementary spin-1/2 particle,
it acts as a purely magnetic probe, i.e., free of quadrupole interactions.
All these features, make ZF-$\mu$SR an ideal technique for detecting
TRS breaking in the superconducting state. This
corresponds to the appearance (at the onset of SC) of spontaneous
magnetic moments, whose magnitude can be very small, often lacking
a proper magnetic order. As we show further on, in case of TRS breaking,
we expect the appearance of an additional enhancement of
relaxation below $T_c$,
reflecting the occurrence of these weak spontaneous fields.
During the ZF-$\mu$SR measurements, to exclude the possibility of stray
magnetic fields (typically larger than the weak internal fields), the
magnets are quenched before starting the measurements, and an active
field-nulling facility is used to compensate for stray fields down to 1\,$\mu$T.

If the amplitudes of the local fields reflect
a Gaussian distribution with zero average
(a rather common cir\-cum\-stance),
the $\mu$SR signal consists of overlapping oscillations with
different frequencies. While at short times the spin dephasing is limited,
at long times it becomes relevant and gives rise to a so-called
Kubo-Toyabe (KT) relaxation function~\cite{Yaouanc2011,Kubo1967}.
Two different models are frequently used to analyze the
ZF-$\mu$SR data:
\begin{equation}
	\label{eq:KT_and_electr}
	A_\mathrm{ZF} = A_\mathrm{s}\left[\frac{1}{3} + \frac{2}{3}(1 -
	\sigma_\mathrm{ZF}^{2}t^{2} - \Lambda_\mathrm{ZF} t)\,
	\mathrm{e}^{\left(-\frac{\sigma_\mathrm{ZF}^{2}t^{2}}{2} - \Lambda_\mathrm{ZF} t\right)} \right] + A_\mathrm{bg},
\end{equation}
or
\begin{equation}
	\label{eq:KT_and_electr_2}
	A_\mathrm{ZF} = A_\mathrm{s}\left[\frac{1}{3} + \frac{2}{3}(1 -
	\sigma_\mathrm{ZF}^{2}t^{2})\,
	\mathrm{e}^{-\frac{\sigma_\mathrm{ZF}^{2}t^{2}}{2}} \right] \mathrm{e}^{-\Lambda_\mathrm{ZF} t}  + A_\mathrm{bg}.
\end{equation}
Equation~\eqref{eq:KT_and_electr} is also known as a combined Gaussian-
and Lorentzian Kubo-Toyabe function, with the additional exponential
relaxation describing the electronic contributions present in
many real materials.
In polycrystalline samples, the 1/3-non-relaxing and the
2/3-relaxing components of the asymmetry cor\-re\-spond to the powder
average of the internal fields with respect to the initial
muon-spin direction (statistically, with a 1/3 probability, the directions of the
muon spin and of the local field coincide). Clearly, in case of single crystals, such prefactors might be different.
The $\sigma_\mathrm{ZF}$ and $\Lambda_\mathrm{ZF}$
represent the zero-field Gaussian and Lorentzian relaxation rates,
respectively. Typically, $\Lambda_\mathrm{ZF}$ shows an almost
temperature-independent behavior. Hence, an increase of $\sigma_\mathrm{ZF}$
across $T_{c}$ can be attributed to the spontaneous magnetic fields
which break the TRS, as e.g., in Re$T$~\cite{Singh2014,Singh2017,Shang2018b}.
In case of diluted nuclear moments, $\sigma_\mathrm{ZF}$ is
practically zero, hence, the TRS breaking is reflected in an
increase of $\Lambda_\mathrm{ZF}$ below $T_c$, as e.g.,
in Zr$_3$Ir and CaPtAs~\cite{Shang2020,Shang2020b}.

\section{R\lowercase{e}-based superconductors}
\label{sec:Re_based_SC}
In this section, we review the different phases of the binary Re$T$ alloys.
These are obtained when rhenium reacts with various
early transition metals (see \textbf{Figure~\ref{fig:structure}A}) and
show rich crystal structures. Representative examples are shown in
\textbf{Figures~\ref{fig:structure}C--F}, including the hexagonal hcp-Mg- ($P6_3/mmc$, No.\ 194), cubic $\alpha$-Mn- ($I\overline{4}3m$, No.\ 217), tetragonal $\beta$-CrFe- ($P4_2/mnm$, No.\ 136), and cubic bcc-W-type ($Im\overline{3}m$, No.\ 229).
Among these the cubic $\alpha$-Mn-type structure is noncentrosymmetric, while the rest are centrosymmetric~\cite{Okamoto1996}.
Besides the above cases, a few other crystal structures have also been reported, including the
cubic CsCl- ($Pm$-$3m$, No.\ 221), cubic Cr$_3$Si- ($Pm$-$3n$, No.\ 223), and trigonal Mn$_{21}$Zn$_{25}$-type ($R$-$3c$, No.\ 167)~\cite{Okamoto1996}.
As for the pure elements listed in \textbf{Figure~\ref{fig:structure}A}, both Re and Os
have an hcp-Mg-type structure, and show superconductivity below 2.7 and 0.7 K, respectively~\cite{Shang2018b,Roberts1976}; while V, Nb, Mo, Ta, and W all adopt a bcc-W-type structure, and become superconductors at
$\sim$ 5.4, 9.0, 1.0, 4.5, and 0.015 K, respectively~\cite{Roberts1976}. Unlike the above cases, Ti, Zr, and Hf can form either high-temperature bcc-W-type or low-temperature hcp-Mg-type structures, with $T_c$ $\sim$ 0.4, 0.6, and 0.13 K, respectively~\cite{Roberts1976}.

\begin{figure*}[tb]
	\centering
	\includegraphics[width=0.9\textwidth]{Fig2.png}
	\caption{\label{fig:structure} Crystal structures of rhenium
		transition-metal (Re$T$) superconductors.
		\textbf{(A)} List of 3$d$, 4$d$, and 5$d$ early transition metals,
		which can react with rhenium to form different crystal structures.
		\textbf{(B)} Binary phase diagram for the typical case
		of Re-Mo alloys (data adopted from Ref.~\cite{Okamoto1996}).
		\textbf{(C--F)} Unit cells of four most representative
		crystal structures of Re$T$ binary compounds. Among these the
		cubic $\alpha$-Mn type ($I\overline{4}3m$, No.\ 217) in \textbf{(D)}
		is noncentrosymmetric, while the hexagonal hcp-Mg ($P6_3/mmc$, No.\ 194), tetragonal $\beta$-CrFe ($P4_2/mnm$, No.\ 136), and cubic bcc-W ($Im\overline{3}m$, No.\ 229) are centrosymmetric.
		The atomic coordinates for each structure can be found in Refs.~\cite{Shang2019,Cenzual1986}.
	}
\end{figure*}

For $T$ = Ti 3$d$ metal, the known binary compounds are Re$_{24}$Ti$_5$, Re$_6$Ti, and ReTi~\cite{Murray1982,Philip1957}.
The former two adopt a noncentroysmmetric $\alpha$-Mn-type structure and become
superconductors below $T_c = 6$\,K~\cite{Shang2018a,Roberts1976,Singh2018}, while the latter one crystallizes in a cubic CsCl-type structure. To date, no detailed physical properties have been reported for ReTi. For $T$ = V, superconductivity has been reported in hcp-Mg-type Re$_{0.9}$V$_{0.1}$ ($T_c = 9.4$\,K), $\beta$-CrFe-type Re$_{0.76}$V$_{0.24}$ ($T_c = 4.5$\,K), and bcc-W-type Re$_{0.6}$V$_{0.4}$ ($T_c = 2.2$\,K)~\cite{Roberts1976,Jorda1986}. Also for them, to date a microscopic study of their SC is still missing. The cubic Cr$_3$Si-type Re$_{0.71}$V$_{0.29}$ has also been synthesized, but its physical properties were never characterized~\cite{Eremenko1990}.

For $T$ = Zr 4$d$ metal, the $\alpha$-Mn-type Re$_{24}$Zr$_{5}$ ($T_c = 5$\,K) and Re$_6$Zr ($T_c = 6.7$\,K) have been investigated via both nuclear quadrupole resonance and $\mu$SR techniques~\cite{Singh2014,Matano2016}. Except for the $\alpha$-Mn-type Re-Zr alloys, the MgZn$_2$-type Re$_2$Zr (similar to hcp-Mg-type) and  Mn$_{21}$Zn$_{25}$-type Re$_{25}$Zr$_{21}$ have been synthesized~\cite{Okamoto1996}. Re$_2$Zr exhibits a $T_c$ value of $\sim 6$--7\,K~\cite{Roberts1976,Giorgi1970}, while Re$_{25}$Zr$_{21}$ has not been studied. For $T$ = Nb, depending on Re/Nb concentration, four different solid phases including hcp-Mg-, $\alpha$-Mn-, $\beta$-CrFe-, and bcc-W-type have been reported. On the Re-rich side, the hcp-Mg-type Re-Nb alloys are limited to less than 3\% Nb concentration~\cite{Okamoto1996}, whereas many $\alpha$-Mn-type Re-Nb binary alloys have been grown and widely studied by various techniques~\cite{Shang2018b,Cirillo2015,chen2013ReNb,Karki2011,Lue2011}, with the highest $T_c$ reaching 8.8\,K in Re$_{24}$Nb$_5$ (denoted as Re$_{0.82}$Nb$_{0.18}$ in the original paper~\cite{Shang2018b}). At intermediate Re/Nb values, for example, in $\beta$-CrFe-type Re$_{0.55}$Nb$_{0.45}$, $T_c$s in the range of 2 to 4\,K~\cite{Roberts1976} have been reported, but no microscopic studies yet. As for the Nb-rich side (Nb concentration larger than 60\%), here the Re-Nb alloys exhibit the same structure as that of pure Nb, but much lower
$T_c$ values than Nb~\cite{Okamoto1996,Roberts1976}. For $T$ = Mo, the binary
Re-Mo phase diagram (see \textbf{Figure~\ref{fig:structure}B}) covers also four different solid phases~\cite{Okamoto1996}. The binary Re-Mo alloys have been characterized by different techniques and all of them shown to become superconductors at low temperature~\cite{Shang2019,Shang2020ReMo}.
The $T_c$ varies nonmonotonically upon changing the Mo concentration,
giving rise to three distinct superconducting regions. On the Re-rich side,
the first SC region shows the highest $T_\mathrm{c} \sim 9.4$\,K in
the hcp-Mg-type Re$_{0.77}$Mo$_{0.23}$. The same material but with an
$\alpha$-Mn-type structure can also be grown, with a $T_\mathrm{c}$ value about 1\,K lower than the hcp-Mg-type. In the second superconducting region, where the alloys adopt a $\beta$-CrFe-type structure, the superconducting transition temperature $T_\mathrm{c} \sim 6.3$\,K is almost independent of Mo content. Finally, on the Mo-rich side, all Re-Mo alloys display a cubic bcc-W-type structure and form a third superconducting region with the highest $T_\mathrm{c}$ reaching 12.4\,K in Re$_{0.4}$Mo$_{0.6}$.

For $T$ = Hf 5$d$ metal, the Re-Hf alloys show a similar phase diagram to Re-Zr.
With only $\sim$3\% Hf substitution, $T_c$ increases from $< 3$\,K to 7.3\,K in the hcp-Mg-type Re-Hf alloys~\cite{Roberts1976}. Both the $\alpha$-Mn-type Re$_6$Hf and the MgZn$_2$-type Re$_2$Hf become superconductors below $T_c \sim 6$\,K~\cite{Singh2017,Roberts1976,Giorgi1970,chen2016ReHf,Singh2016ReHf}, whereas the
physical properties of Mn$_{21}$Zn$_{25}$-type Re$_{25}$Hf$_{21}$
remain largely unknown. On the Hf-rich side, the bcc-W-type alloys exhibit
relatively low $T_c$s, e.g., $T_c = 1.7$\,K for Hf$_{0.875}$Re$_{0.125}$~\cite{Roberts1976}.
For $T$ = Ta, although the
four different structures shown in \textbf{Figure~\ref{fig:structure}C--F} can be synthesized,
only the $\alpha$-Mn-type Re-Ta alloys have been well studied. For example, Re$_3$Ta and Re$_{5.5}$Ta show $T_c$ values of 4.7\,K and 8\,K, respectively~\cite{Barker2018,Arushi2020}.
On the Ta-rich side, the bcc-W-type Re-Ta alloys become superconducting at $T_c < 3.5$\,K, lower than the $T_c$ of pure Ta~\cite{Mamiya1970}. We note that in case of the $\beta$-CrFe-type Re-Ta alloys, no superconducting transition has been observed down to 1.8\,K in either Re$_{0.5}$Ta$_{0.5}$ or Re$_{0.6}$Ta$_{0.4}$.
For $T$ = W, the Re-W alloys show a very similar phase diagram to Re-Mo in \textbf{Figure~\ref{fig:structure}B}. As the W concentration increases, the highest $T_c$ values
reach $\sim 8$, 9, 6, and 5\,K in the hcp-Mg-, $\alpha$-Mn-, $\beta$-CrFe-, and the bcc-W-type alloys, respectively~\cite{Okamoto1996,Roberts1976}.
Among them, only the hcp-Mg- and the $\alpha$-Mn-type Re$_3$W have
been investigated~\cite{Biswas2012,Biswas2011}.
Finally, in case of $T$ = Os, the Re-Os alloys show a rather
monotonous phase diagram, since only hcp-Mg-type compounds
with $T_c$ values below 2\,K can be synthesized~\cite{Okamoto1996,Roberts1976}.

\section{Upper critical field and nodeless superconductivity}
\label{sec:Hc2_fields}
As mentioned in the introduction, due to the mixture of singlet-
and triplet paring, some NCSCs may ex\-hib\-it relatively high
upper critical fields, often very close to or even exceeding
the Pauli limit, as e.g.,
CePt$_3$Si~\cite{Bauer2004}, Ce(Rh,Ir)Si$_3$~\cite{Kimura2007,Sugitani2006},
and recently (Ta,Nb)Rh$_2$B$_2$~\cite{Carnicom2018}.
Therefore, the upper critical field can provide valuable clues about
the nature of superconductivity.
To investigate the temperature evolution of the upper critical
field $H_{c2}(T)$, in general, the temperature- (or field-) dependent
electrical resistivity $\rho$, magnetic susceptibility $\chi$, and specific heat $C/T$ at various magnetic fields (or at various temperatures) are measured~\cite{Shang2018a,Shang2018b,Shang2020ReMo}.
As an example, \textbf{Figure~\ref{fig:Hc2_gap}A} shows the $H_{c2}(T)$ for Re$_{24}$Nb$_5$ ($\alpha$-Mn-type) and Re$_{0.4}$Mo$_{0.6}$ (bbc-W-type) versus
the normalized temperature $T/T_c(0)$. To obtain the upper
critical field in the zero-temperature limit, $H_{c2}(0)$, the Werthamer-Helfand-Hohenberg (WHH) or the Ginzburg-Landau (GL) models are usually applied when analyzing the
$H_{c2}(T)$ data of Re$T$ superconductors. Both models can adequately describe single-gap superconductors. Here, in case of Re$_{24}$Nb$_5$ and Re$_{0.4}$Mo$_{0.6}$, the WHH model (solid line in \textbf{Figure~\ref{fig:Hc2_gap}A}) reproduces the data very well and gives $\mu_0H_{c2}(0)$ = 15.6\,T, and 3.08\,T, respectively. \textbf{Figure~\ref{fig:Hc2_gap}B} summarizes the $\mu_0H_{c2}(0)$ values of the Re$T$ and $\alpha$-Mn-type NbOs$_2$ superconductors. As discussed in Section~\ref{sec:Re_based_SC}, most of the previous studies focused exclusively on $\alpha$-Mn-type Re$T$
superconductors, the physical properties of the other Re$T$
superconductors being practically neglected  and requiring
further studies. Unlike other Re$T$, all Re-Mo alloys belonging to four
different structures have been studied via macro- and
microscopic techniques~\cite{Shang2019,Shang2020ReMo}. The $\mu_0H_{c2}(0)$ of centrosymmetric Re-Mo alloys, including hcp-Mg-, $\beta$-CrFe-, and bcc-W-type, are far away from the Pauli limit $\mu_0 H_\mathrm{P} = 1.86\,T_{c}$ (indicated by a dashed line in \textbf{Figure~\ref{fig:Hc2_gap}B}). Conversely, the $\alpha$-Mn-type Re$T$ and
NbOs$_2$ both exhibit large upper critical fields, very close to or even exceeding the Pauli limit, despite their different $T_c$ values. For example, $\mu_0 H_{c2}(0) =  15.6$ and 16.5\,T for Re$_{24}$Nb$_5$ and Re$_{5.5}$Ta,
while their $\mu_0 H_P(0)$ are 16.4 and 14.9\,T, respectively.
The hcp-Mg-type Re$_3$W also exhibits a relatively high $H_{c2}$, as
determined from electrical resistivity data. However, its
$H_{c2}$ value might be
overestimated since, e.g., at 9\,T, no zero resistivity could be observed down
to 2\,K. Therefore, other bulk techniques, including magnetization- or heat capacity measurements are required to determine the intrinsic $H_{c2}$.
In general, it would be interesting to know the $H_{c2}$ values of other centrosymmetric Re$T$ superconductors. Overall, the upper critical fields in \textbf{Figure~\ref{fig:Hc2_gap}B} indicate the possibility of singlet-triplet mixing
in the noncentrosymmetric $\alpha$-Mn-type superconductors.

\begin{figure*}[tb]
	\centering
	\includegraphics[width=0.9\textwidth]{Fig3.png}
	\caption{\label{fig:Hc2_gap} Upper critical field and superconducting energy gap. \textbf{(A)} The upper critical field $H_\mathrm{c2}$, as determined from electrical resistivity-, heat capacity-, and magnetic susceptibility measurements, as a function of the reduced superconducting transition temperature $T_c$/$T_c$(0) for Re$_{24}$Nb$_{5}$ and Re$_{0.4}$Mo$_{0.6}$. Solid-lines represent fits to the Werthamer-Helfand-Hohenberg (WHH) model. \textbf{(B)} Zero-temperature $H_\mathrm{c2}$ versus the superconducting transition temperature $T_c$ for $\alpha$-Mn-type NbOs$_2$ and all Re$T$ superconductors.
		The shaded region in \textbf{(B)} marks the noncentrosymmetric $\alpha$-Mn type superconductors, while the dashed line indicates the Pauli
		limit (i.e., $\mu_0 H_\mathrm{P} = 1.86 T_c$).
		\textbf{(C)} Superfluid density vs reduced temperature $T_c$/$T_c$(0) for Re$_{24}$Nb$_{5}$ and Re$_{0.4}$Mo$_{0.6}$. Lines are fits to a fully-gapped $s$-wave model.
		The insert shows the TF-$\mu$SR spectra for Re$_{0.4}$Mo$_{0.6}$ measured
		in a field of 60\,mT in the normal- (16\,K) and the superconducting state (1.5\,K).
		Solid lines are fits to Equation~\ref{eq:TF_muSR}.
		\textbf{(D)} Zero-temperature superconducting energy gap $\Delta_0$ (in k$_\mathrm{B}T_c$ units) as a function of $T_c$ for Re$T$ and $\alpha$-Mn-type NbOs$_2$ and TaOs superconductors. Here, the dashed line
		represents the BCS superconducting gap in the weak-coupling
		limit (i.e., 1.76\,$k_\mathrm{B}T_c$). Data were taken from Refs.~\cite{Singh2014,Singh2017,Shang2018a,Shang2018b,Biswas2012,Shang2020ReMo,Singh2018,chen2016ReHf,Singh2016ReHf,Arushi2020,Biswas2011,Singh2019NbOs2,SinghTaOs,mayoh2017ReZr}.
			}
\end{figure*}

Transverse-field $\mu$SR represents one of the most powerful techniques to investigate the superconductivity at a microscopic level.
To illustrate this, in the inset of \textbf{Figure~\ref{fig:Hc2_gap}C}
we show two typical TF-$\mu$SR spectra for bcc-W-type Re$_{0.4}$Mo$_{0.6}$ in the normal and the superconducting states. Below $T_c$, the fast decay induced by FLL (encoded into $\sigma_\mathrm{sc}$) is clearly visible, while the slow decay in the normal state is attributed to the randomly oriented nuclear magnetic moments. By comparing the two spectra, one can also determine
the superconducting volume fraction of a superconductor.
As an example, the main panel of \textbf{Figure~\ref{fig:Hc2_gap}C} shows the normalized superfluid density calculated from $\sigma_\mathrm{sc}(T)$, which is proportional to $[\lambda(T)/\lambda(0)]^{-2}$ (see details in Section~\ref{ssec:TF_muSR}),
as a function of the reduced temperature $T/T_\mathrm{c}(0)$ for Re$_{24}$Nb$_5$ and Re$_{0.4}$Mo$_{0.6}$~\cite{Shang2018b,Shang2020ReMo}. The low-$T$ superfluid density is practically independent of temperature, clearly suggesting a lack of low-energy excitations and a fully-gapped superconductivity. Contrarily, such excitations exist in case
of nodes in the superconducting gap, implying a temperature-dependent
superfluid density
below $\sim T_c/3$. As shown by solid lines in \textbf{Figure~\ref{fig:Hc2_gap}C}, the $\rho_\mathrm{sc}(T)$ of Re$T$ superconductors is
described very well by a fully-gapped $s$-wave model (see Equation~\ref{eq:rhos}). The other $\alpha$-Mn-type Re$T$, TaOs, and NbOs$_2$ exhibit similar temperature-invariant superfluid densities~\cite{Singh2014,Singh2017,Shang2018a,Biswas2012,Singh2018,Arushi2020,Singh2019NbOs2,SinghTaOs}.
Although Re$T$ alloys adopt different crystal structures (i.e., centrosymmetric or noncentrosymmetric, see \textbf{Figure~\ref{fig:structure}C--F}) and have different $T_c$ values, they regularly exhibit low-$T$ superfluid densities which are independent of temperature~\cite{Shang2020ReMo}.
Except for $T$ = Mo (and for some $\alpha$-Mn structures), a systematic
microscopic study of superconductivity in other Re$T$ superconductors
is still missing. Clearly, it would be interesting to know if
their SC behavior is similar to that of Re-Mo alloys.
The nodeless SC scenario in Re$T$ alloys is also supported by other
techniques, as the electronic specific heat, the magnetic penetration
depth measured via the tunnel-diode-oscillator-based technique,
or the point-contact Andreev reflection~\cite{Shang2018a,Shang2018b,Shang2020ReMo,chen2016ReHf,Singh2016ReHf,Biswas2011,mayoh2017ReZr,Pang2018,Parab2019}. In addition, some studies have found evidence of
two-gap superconductivity in Re$_{0.82}$Nb$_{0.18}$ and Re$_6$Zr~\cite{Cirillo2015,Parab2019}.

\textbf{Figure~\ref{fig:Hc2_gap}D} summarizes the zero-temperature superconducting energy gap value for Re$T$ and $\alpha$-Mn-type NbOs$_2$ and TaOs superconductors as a function of their critical temperatures. Most of them
exhibit a $\Delta_0$/k$_\mathrm{B}T_c$ ratio larger than 1.76 (see dashed line in \textbf{Figure~\ref{fig:Hc2_gap}D}), the value expected for a weakly coupled BCS superconductor, which indicates a moderately strong coupling in these superconductors. In addition, the specific-heat discontinuity at $T_c$ (i.e., $\Delta C/\gamma T_c$) is larger than the conventional BCS value of 1.43, again indicating an enhanced electron-phonon coupling~\cite{Shang2018a,Shang2018b,Shang2020ReMo,chen2016ReHf,Singh2016ReHf,Biswas2011,mayoh2017ReZr}.
As mentioned above, it is worth noting that the superconducting parameters of all the other Re$T$ materials (except for $\alpha$-Mn-type
and $T$ = Mo) are missing, prompting further research efforts in this direction.

As discussed in the introduction, the lack of inversion symmetry in the
NCSCs often induces an ASOC. This splits the Fermi surface by
lifting the degeneracy of the conduct{ion electrons, thus
	allowing admixtures of spin-singlet and spin-triplet pairing.
	In general, the strength of ASOC determines the degree of such an admixture and thus the superconducting properties of NCSCs~\cite{Bauer2012,Smidman2017}.
	A fully-gapped superconductor (i.e., dominated by spin-singlet
	pairing) can be tuned into a nodal superconductor (dominated by
	spin-triplet pairing) by increasing the strength of ASOC. Such mechanism has been successfully demonstrated, e.g., in weakly-correlated Li$_2$Pt$_3$B ($E_\mathrm{SOC}$/k$_\mathrm{B}T_c$ $\sim 831$)~\cite{yuan2006,NiShiyama2007}, CaPtAs ($E_\mathrm{SOC}$/k$_\mathrm{B}T_c$ $\sim 800$)~\cite{Shang2020,Xie2019}, and in strongly-correlated CePt$_3$Si ($E_\mathrm{SOC}$/k$_\mathrm{B}T_c$ $\sim 3095$) superconductors~\cite{Bauer2004,Samokhin2004}, all exhibiting a relatively large band splitting $E_\mathrm{SOC}$ compared to their superconducting energy scale k$_\mathrm{B}T_c$.
	In the $\alpha$-Mn-type Re$T$ alloys, the density of states (DOS) near the Fermi level
	is dominated by the 5$d$ orbitals of rhenium atoms, while contributions
	from the $d$ orbitals of $T$ atoms are negligible~\cite{Suetin2013,Winiarski2014,Mojammel2016}.
	Therefore, a possible enhancement of SOC due to 3$d$-(e.g., Ti, V)
	up to 5$d$-electrons (e.g., Hf, Ta, W, Os) will, in principle, neither increase the band splitting $E_\mathrm{SOC}$ nor
	affect the pairing admixture and thus the superconducting properties
	of $\alpha$-Mn-type Re$T$ superconductors. According to band-structure calculations,
	in Re$_6$Zr, the SOC-induced band splitting
	is about 30\,meV~\cite{Mojammel2016}, implying a very small ratio $E_\mathrm{SOC}$/k$_\mathrm{B}T_c$ $\sim 25$,
	comparable to that of fully-gapped Li$_2$Pd$_3$B, Mo$_3$P, and Zr$_3$Ir superconductors~\cite{Shang2020b,NiShiyama2007,Shang2019b}. Therefore, despite the relatively
	large SOC of rhenium atoms, its effects are too weak
	to significantly influence the bands near the Fermi level.
	This might explain why all the $\alpha$-Mn-type Re$T$ superconductors
	exhibit nodeless superconductivity, more consistent with a spin-singlet
	dominated pairing~\cite{Singh2014,Singh2017,Shang2018a,Shang2018b,Biswas2012,Barker2018}.
	However, we recall that often, due to the similar magnitude and same-sign of the order parameter on the spin-split Fermi surfaces, a possible mixed-pairing superconductor
	may be challenging to detect
	or to distinguish from a single-gap $s$-wave superconductor~\cite{Yip2014}. The almost spherical symmetry of the Fermi surface in these materials may also explain their BCS-like superconducting states~\cite{Winiarski2014}.
	As for the other centrosymmetric Re$T$ alloys, in most of them
	the Re and $T$ atoms occupy the same atomic positions in the unit cell.
	In this case, as the $T$-content increases, the contribution of $T$ $d$ orbitals to the DOS is progressively enhanced, at the expense of the Re $5d$ orbitals.
	Therefore, the chemical substitution of Re by another 3$d$, 4$d$,
	or 5$d$ $T$ metal (see \textbf{Figure~\ref{fig:structure}}),
	should significantly tune the SOC and, hence, the band splitting,
	an interesting hypothesis waiting for theoretical confirmation.
	However, even for $T$ = Hf, Ta, W, and Os, the maximum $E_\mathrm{SOC}$
	should still be comparable to that of $\alpha$-Mn-type Re$T$ alloys.
	Finally, irrespective of the strength of SOC, due to their
	centrosymmetric crystal structures, these compounds may exhibit
	either singlet- or triplet-pairing, but not an admixture of both.
	According to the TF-$\mu$SR results (see \textbf{Figure~\ref{fig:Hc2_gap}C}),
	despite a change in SOC and of the different crystal structures
	(see \textbf{Figure~\ref{fig:structure}C--F}), all Re$T$
	superconductors exhibit fully-gapped superconducting states.
	This finding strongly suggests that, in the Re$T$ superconductors,
	spin-singlet pairing is dominant.

\section{Time-reversal symmetry breaking}
\label{sec:TRS_breaking}
\begin{figure*}[tb]
	\centering
	\includegraphics[width=0.8\textwidth]{Fig4.png}
	\caption{\label{fig:ZF_muSR} ZF-$\mu$SR and evidence for TRS breaking.
		\textbf{(A)} ZF-$\mu$SR spectra for Re$_{24}$Nb$_{5}$ collected in the superconducting and normal states. Top: additional
		$\mu$SR data collected at 1.5\,K in a 15-mT longitudinal field, are also shown. The solid lines are fits using Equation~\ref{eq:KT_and_electr}.
		\textbf{(B)} Gaussian relaxation rate $\Delta\sigma_\mathrm{ZF}$
		vs $T$/$T_c$ for Re$_{24}$Nb$_5$, Re$_6$Zr, Re, and Re$_{0.77}$Mo$_{0.23}$
		--- here $\Delta\sigma_\mathrm{ZF}(T) = \sigma_\mathrm{ZF}(T) - \sigma_\mathrm{ZF}(T > T_c)$.
		While for the first three there is a clear increase of $\Delta\sigma_\mathrm{ZF}$
		across $T_c$ (hence a breaking of TRS), no changes occur in the last
		case (TRS is preserved).
		Data were taken from Refs.~\cite{Singh2014,Shang2018a,Shang2018b,Shang2020ReMo}.
	}
\end{figure*}

Owing to its very high sensitivity (see details in Section~\ref{ssec:ZF_muSR}),
ZF-$\mu$SR has been successfully used to search for spontaneous magnetic fields, reflecting the breaking of TRS in the superconducting states of different types of superconductors, as
e.g., Sr$_2$RuO$_4$, UPt$_3$, PrOs$_4$Sb$_{12}$, LaNiGa$_2$, LaNiC$_2$, La$_7$(Rh,Ir)$_3$, and $\alpha$-Mn-Re$T$~\cite{Hillier2009,Barker2015,Singh2014,Singh2017,Shang2018a,Shang2018b,Singh2018La7Rh3,Luke1993,Luke1998,Xia2006,Aoki2003,Schemm2014,Hillier2012}.
The latter three are typical examples of weakly-correlated NSCSs, to be contrasted with strongly-correlated NCSCs,
where either the TRS is broken by a coexisting long-range magnetic order,
or the tiny TRS-breaking signal is very difficult to detect due to the presence of strong magnetic fluctuations~\cite{Amato1997}. In the former case, the broken TRS is unrelated to
the superconductivity, while in the later case, a genuine
TRS breaking  effect is masked by the much faster muon-spin relaxation caused by magnetic fluctuations. Therefore, in general, a TRS breaking effect is
more easily (and reliably) detected in weakly-correlated- or non-magnetic
superconductors using $\mu$SR techniques.
Normally, in the absence of external fields, the onset of superconductivity does not imply changes in the ZF-$\mu$SR relaxation rate. However, in presence of a
broken TRS, the onset of a tiny spontaneous polarization
or of currents gives rise to associated (weak) magnetic fields, readily detected by ZF-$\mu$SR as an increase in the relaxation rate. Given the tiny size of such effects, the ZF-$\mu$SR measurements are usually performed in both the normal- and the superconducting state with a relatively high statistics, at least twice that of the TF-$\mu$SR spectra.
As an example, \textbf{Figure~\ref{fig:ZF_muSR}A} plots the ZF-$\mu$SR spectra of $\alpha$-Mn-type
Re$_{25}$Nb$_5$,
with the other Re$T$ superconductors showing a similar behavior. The ZF-$\mu$SR spectra collected below- and above $T_c$ (at 1.5 and 12\,K) exhibit small yet measurable differences.
The lack of any oscillations in the spectra, implies the non-magnetic nature of Re$T$ superconductors.
Further, longitudinal-field $\mu$SR measurements under a relatively small applied field (typically a few tens of mT) in the superconducting state are usually performed to check if the applied field can fully decouple the muon spins from the weak spontaneous magnetic fields, and thus exclude extrinsic effects.
In non-magnetic materials in the
absence of external magnetic fields, the muon-spin relaxation is mostly determined by the interaction between the muon spins and the randomly oriented nuclear magnetic moments.
Therefore, the spontaneous magnetic fields due to the
TRS breaking will be reflected in an additional increase of muon-spin relaxation.
The ZF-$\mu$SR asymmetry can be described by means of a Gaussian- or a
Lorentzian Kubo-Toyabe relaxation, or a combination thereof (see Equations~\ref{eq:KT_and_electr} and \ref{eq:KT_and_electr_2}).
\textbf{Figure~\ref{fig:ZF_muSR}B} summarizes the Gaussian relaxation
rate $\sigma_\mathrm{ZF}$ versus the reduced temperature $T/T_c$ for the
$\alpha$-Mn-type Re$_{24}$Nb$_5$, Re$_6$Zr, and Re$_{0.77}$Mo$_{0.23}$, and the hcp-Mg-type elementary Re. Above $T_c$, all the samples show a temperature-independent
$\sigma_\mathrm{ZF}$. Except for Re$_{0.77}$Mo$_{0.23}$, a small yet clear
increase of $\sigma_\mathrm{ZF}(T)$ below $T_c$ indicates the onset of spontaneous magnetic fields, which represent the signature of TRS breaking in the superconducting state~\cite{Singh2014,Shang2018b,Shang2020ReMo}.
The other $\alpha$-Mn-type superconductors, e.g., Re$_6$Ti, and Re$_6$Hf~\cite{Singh2017,Singh2018}, show similar $\sigma_\mathrm{ZF}(T)$ to Re$_{24}$Nb$_5$ and Re$_6$Zr, and thus the breaking of TRS in the superconducting state. At the same time, in the isostructural Re$_3$Ta, Re$_{5.5}$Ta, and Re$_3$W cases, there is no clear increase in $\sigma_\mathrm{ZF}(T)$ upon crossing $T_c$, implying a preserved TRS~\cite{Biswas2012,Barker2018,Arushi2020}.

Recently, the breaking of TRS and the presence of nodes in the SC gap, attributed to an admixture of singlet- and triplet paring,
has been reported in the noncentrosymmetric CaPtAs superconductor~\cite{Shang2020}.
In general, however, the breaking of TRS in the superconducting
state and a lack of space-inversion symmetry in
the crystal structure are independent events,
not necessarily occurring together. For instance, the unconventional
spin-triplet pairing is expected to break TRS below $T_c$, as
has been shown, e.g., in Sr$_2$RuO$_4$, UPt$_3$, and UTe$_2$ triplet superconductors~\cite{Ran2019,Luke1993,Luke1998,Xia2006,Schemm2014,Ishida1998,Tou1998,Mackenzie2003,Joynt2002}.
An $s + id$ spin-singlet state was proposed to account for the TRS breaking in some iron-based high-$T_\mathrm{c}$ superconductors~\cite{Lee2009}, where a nodal gap
is also expected. The frequent occurrence of TRS breaking in the fully-gapped
(i.e., dominated by spin-singlet pairing) Re$T$ superconductors
(see Section~\ref{sec:Hc2_fields}) is, therefore, rather puzzling.
A similarly surprising result was the report that elementary
rhenium also exhibits signatures of TRS breaking in its superconducting state
(see \textbf{Figure~\ref{fig:ZF_muSR}B}), with $\Delta\sigma_\mathrm{ZF}(T)$
being comparable to that of Re$_6$Zr~\cite{Shang2018b,Shang2020ReMo}.
Since elementary rhenium adopts a centrosymmetric hcp-Mg crystal
structure (see \textbf{Figure~\ref{fig:structure}C}), this indicates
that a lack of inversion symmetry and the accompanying ASOC effects are not
crucial factors for the occurrence of TRS breaking in Re$T$ superconductors.
Further on, a comparison of ZF-$\mu$SR measurements on Re-Mo alloys
with different Re/Mo contents, covering almost all the crystal structures
reported in \textbf{Figure~\ref{fig:structure}}, shows that only Re and Re$_{0.88}$Mo$_{0.12}$ exhibit a broken TRS in the superconducting state, while those with a
higher Mo-content ($\sim$23\%--60\%), including both the centrosymmetric- and noncentrosymmetric Re$_{0.77}$Mo$_{0.23}$, preserve the TRS. Considering the preserved TRS in Mg$_{10}$Ir$_{19}$B$_{16}$, and Nb$_{0.5}$Os$_{0.5}$~\cite{Acze2010,SinghNbOs},
all of which share the same
$\alpha$-Mn-type structure, this implies that TRS breaking in Re$T$ superconductors is clearly not related to the noncentrosymmetric crystal structure or to a
possible mixed pairing but, most likely, is due to the presence of rhenium and to its amount.
Such conclusion is further reinforced by the preserved TRS in many Re-based
superconductors, whose Re-content is below a certain threshold. Such cases include, e.g., Re$_3$W, Re$_3$Ta, Re-Mo (with Mo-content higher than 12\%)~\cite{Biswas2012,Shang2020ReMo,Barker2018}, the recently reported Re-B superconductors~\cite{Sharma2020}, and the diluted ReBe$_{22}$ superconductor~\cite{Shang2019c}.
Moreover, by comparing the ZF-$\mu$SR relaxation across various
Re$T$ superconductors, a clear positive correlation between
$\Delta\sigma_\mathrm{ZF}$ (i.e., spontaneous fields) and the size of
the nuclear magnetic moments $\mu_\mathrm{n}$ was identified~\cite{Shang2018b}. For instance, among the Re$T$ superconductors, Re$_{24}$Nb$_5$
shows the largest spontaneous fields below $T_c$
(see \textbf{Figure~\ref{fig:ZF_muSR}B}), a fact compatible with
the large nuclear magnetic moment of
niobium, practically twice that of rhenium (6.17 vs.\ 3.2\,$\mu_\mathrm{N}$).
However, the correlation between $\mu_\mathrm{n}$ and $\Delta\sigma_\mathrm{ZF}$
alone cannot explain TRS breaking, considering that elementary Nb itself,
despite having the highest $\mu_\mathrm{n}$, does not break TRS.
Clearly, the origin of such correlation is not yet understood and
it requires further experimental and theoretical studies.

If SOC can be ignored, an alternative mechanism, which can
account for the TRS breaking in Re$T$ superconductors in presence of
a fully-opened superconducting gap, is the internally-antisymmetric
nonunitary triplet (INT) pairing.
The INT pairing was originally proposed to explain the TRS breaking and nodeless SC in centrosymmetric LaNiGa$_2$~\cite{Hillier2012, Weng2016,Ghosh2020}
and noncentrosymmetric LaNiC$_2$~\cite{Hillier2009, Quintanilla2012},
both exhibiting a relatively weak SOC.
In case of INT pairing, the superconducting pairing function is antisymmetric with respect to the orbital degree of freedom, while remaining symmetric in the spin-
and crystal-momentum channels~\cite{Hillier2009,Hillier2012, Weng2016,Quintanilla2012}.
Since in Re$T$ superconductors, too, the SOC interaction is relatively
weak ($\sim30$\,meV, see Section~\ref{sec:Hc2_fields})~\cite{Mojammel2016} and since neither TRS breaking nor
the nodeless SC are related to the symmetry of Re$T$ crystal structures,
the effect of SOC to the observed TRS breaking is insignificant.
This could, therefore, explain why a lack of inversion symmetry (essential to SOC) is not a precondition for TRS breaking in Re$T$ superconductors.
Moreover, the occurrence of an INT state relies on the availability of a local-pairing mechanism driven by Hund's rules, e.g., by Ni $3d$-electrons
in LaNiC$_2$ and LaNiGa$_2$~\cite{Hillier2009,Hillier2012,Weng2016,Ghosh2020,Quintanilla2012}. Such local-pairing mechanism may also occur in Re$T$ superconductors,
since rhenium too can be magnetic~\cite{Yang2015,Kochat2017}.
This consideration is also in good agreement with the observation
that TRS breaking depends on Re content, but not on a noncentrosymmetric
crystal structure.

\section{Conclusion}
In this short review we focused on recent experimental studies of
Re$T$ superconductors, where time-reversal symmetry breaking
effects are often present and whose superconductivity
can, therefore, be considered as unconventional. Due to its high
sensitivity to the weak internal fields associated with TRS
breaking, $\mu$SR represents one of the key techniques in the search
for TRS-breaking effects in the superconducting state.
Nonetheless, in certain cases, the amplitude of the spontaneous magnetic fields
(the fingerprint of TRS breaking) may still be below the
resolution of the $\mu$SR technique ($\sim 10^{-2}$\,mT).
Hence, the future use of other techniques, e.g., based on the optical
Kerr effect~\cite{Kallin2016}, another very sensitive probe of
spontaneous fields in unconventional superconductors, remains crucial.
Due to their rich crystal structures, covering both centro- and
noncentrosymmetric cases, and the pervasive presence of superconductivity
at low temperatures, the nonmagnetic Re-based materials are
the ideal choice for investigating the origin of TRS breaking.
Here, we reviewed different cases of Re-containing superconductors, ranging
from elementary rhenium, to Re$T$ ($T$ = 3$d$-5$d$ early transition metals),
to the dilute-Re case of ReBe$_{22}$, all of which were
investigated through both macroscopic and microscopic techniques.
By a comparative study of Re$T$ with different $T$ metals mostly using $\mu$SR technique,
we could demonstrate the secondary role played by SOC and why the
spin-singlet pairing is dominant in Re$T$ superconductors.
This, however, brings up
the question of reconciling the occurrence of TRS breaking with a
fully-gapped SC state (spin-singlet pairing).
A possible solution to this is offered by the so-called INT model,
which requires an antisymmetric pairing function involving the orbital degree of freedom,
making it insensitive to the presence (or lack) of inversion symmetry
and SOC. Overall, the reported results suggest that the rhenium
presence and its amount are two key factors for the appearance and the
extent of TRS breaking in the Re-based superconductors.
These key observations, albeit important, demand new experimental
and theoretical investigations to further generalize them.

To date, as nearly all current
studies have focused exclusively on
$\alpha$-Mn-type Re$T$ superconductors (except for the Re-Mo series
considered here), the superconducting properties of most other Re$T$
alloys remain basically unexplored.
Hence, the synthesis and characterization of non-$\alpha$-Mn-type Re$T$
alloys, including the study of their electrical, magnetic, and
thermodynamic properties, is of clear interest.
Similarly, systematic $\mu$SR measurements, crucial for detecting
the presence of TRS breaking in Re-based superconductors,
are in high demand. For instance, although both Re-Zr and Re-Nb alloys
exhibit rich crystal structures and superconducting phase diagrams,
only their $\alpha$-Mn-type phase has been explored.
In addition, most of the original measurements were performed only
on polycrystalline samples. Hence, the synthesis of single crystals
will be essential in the comprehensive search for possible superconducting nodes and,
thus, for mixed singlet-triplet pairing.
Finally, it would be of interest to extend the $\mu$SR studies
on  elementary rhenium from the bulk- to its thin-film form,
where inversion symmetry is artificially broken. By checking if the
TRS breaking is maintained or not, will help us to further clarify the rhenium conundrum.

\section*{Conflict of Interest Statement}
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

\section*{Author Contributions}
The manuscript was prepared and written by both the authors.

\section*{Funding}
This work was supported by start funding from
East-China Normal University (ECNU), the Swiss National Science Foundation
(Grant No.\ 200021-169455) and the Sino-Swiss Science and
Technology Cooperation (Grant No.\ IZLCZ2-170075).

\section*{Acknowledgments}
We thank M. Shi\ for the fruitful discussion. We thank M.\ Medarde for the assistance during the electrical
resistivity and magnetization measurements, and D.\ J.\ Gawryluk
and E.\ Pomjakushina for synthesizing the materials. We acknowledge
the allocation of beam time at the Swiss muon source (S$\mu$S)
(DOLLY, GPS, and LTF spectrometers).

\vspace{8pt}
\emph{Note} This is the version of article after peer review but without editing, as submitted to \emph{Frontiers in Physics}. Frontiers is not responsible for any errors or omissions in this version of the manuscript.
The paper is available online at \url{10.3389/fphy.2021.651163}.

redpajama_set_name: RedPajamaArXiv


text:
Старостенко — украинская фамилия. Известные носители:

 Старостенко Александр Игоревич (р. 1961, г. Сумы) — композитор и поэт, популярный автор так называемого православного шансона. Протоиерей РПЦ.
 Старостенко, Александр Хрисанфович (1903—1973) — российский инженер, конструктор турбин, лауреат Сталинской и Ленинской премий. Доктор технических наук.
 Старостенко, Виталий Иванович (род. 1935) — советский геолог и геофизик, теоретик интерпретации потенциальных полей.
 Старостенко, Владимир Иванович (1948—2017) — российский хозяйственный и государственный деятель, 4-й министр путей сообщения России (с мая по сентябрь 1999).
 Старостенко, Дмитрий Валерьевич (род. 1973) — белорусский хоккеист.
 Старостенко, Пётр Захарович (1850—?) — член III Государственной думы от Воронежской губернии, крестьянин.

redpajama_set_name: RedPajamaWikipedia


In this day and age there is no question that the cell phone has become one of the most ubiquitous items found with people of all ages. With this surge in popularity comes along the huge industry which is mobile service providers. There are several major mobile carries which account for the majority of cell phone users. These plans are often pricey for the average individual and many people are beginning to explore other options which are becoming more common. One such option is Red Pocket Mobile which is a pre-paid plan which does not require users to be tied in to lengthy contracts which often have penalties associated with early termination. For many individuals, having a pre-paid plan which can be purchased on a month to month basis makes more financial sense than to be locked in to a long term contract with a major wireless provider. This is the primary reason plans such as Red Pocket Mobile are rapidly growing in popularity.


Please let us spend our own money on things that would employ American workers.
It's comforting to know that some Americans can still afford vacations.
The Euro socio-economic model that the current Administration & Congress seems to follow seems to rely on high unemployment, large governments that provide or supplement everything (including vacations). They also seem to blame the US of A for all their ills, something that Obama is proficient at as well.
Yeah, but it's a "poopy" vacation....hehehe!


text:
d his seventh goal of the playoffs, Braden Holtby made 32 saves and Washington built a lead and this time held on to beat
Girls go wild » Forum » Gefahr im Internet » d his seventh goal of the playoffs, Braden Holtby made 32 saves and Washington built a lead and this time held on to beat
#1 | d his seventh goal of the playoffs, Braden Holtby made 32 saves and Washington built a lead and this time held on to beat 21.03.2019 07:02
elaine95
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I have a 2018 Springdale 2600TB. Need to drain water heater. Can't figure out socket size. Please help!
Plastic plug is 15/16" anode rod is 1 1/16". Odd plastic plug is 7/8".
Thanks Chuckstet. I'll try that size tomorrow.
Mine uses a 27mm socket, wrote it on the back of the water heater panel with a Sharpie.
27mm equates to 1 1/32". I'll have to see if I have that size socket and weather or not it fits.
Thx everyone. Mine is a 1 1/16" anode rod. Got it out today with no problem!!


This document specifies the CSA SHOGI protocol for computers to play Shogi.
This protocol assumes there is one server computer acting as the judge, and two or more clients acting as shogi players.
First, the server sends a game condition to two clients which are connected; the game will start when, both the clients have agreed to the condition. While playing a game, the player to move sends his move to the server and the server measures the time consumed and also determines whether the move is legal.
Initially, the server starts the CSA SHOGI service by listening on TCP port 4081. When a client wishes to make use of the service, it establishes a TCP connection with the server. After the connection is established, the client and CSA SHOGI server exchange human-readable commands and responses (respectively) until the connection is closed or aborted. Commands and responses in the CSA SHOGI must consist of case-sensitive ASCII characters in 0x21..0x7f, LF (0x0a) and space (0x20). All messages are terminated by an LF. TAB must NOT be used.
Once the TCP connection has been opened by a CSA SHOGI client, the client must then identify and authenticate itself to the CSA SHOGI server by LOGIN command. (NOTE: the CSA SHOGI server does NOT issue any greeting). After successful authentication, the session enters a GAME-WAITING state. The client can close the session by LOGOUT command. Also, the server may disconnect the session.
The LOGIN command identifies the client to the server and carries the plain-text password authenticating this user.
The <username> consists of digits ('0'-'9'), lowercase or uppercase letters ('A'-'Z', 'a'-'z'), underscore ('_'), hyphen ('-'), and may be up to 32 characters long. The <password> may be up to 32 characters long. The <password> must not contain any space characters.
Here is the summary for the LOGOUT command that is only allowed in a GAME-WAITING state.
The server may send the GAME CONDITION to a client in a GAME-WAITING state, in the following format.
A GAME CONDITION is hierarchically structured by BEGIN and END pairs. The first line of a GAME CONDITION is 'BEGIN Game_Summary' and the last is 'END Game_Summary.' A game condition consists of three blocks; general information (described in 3.2.1), definition of position (described in 3.2.2), and definition of time control (described in 3.2.3).
-- whether the client receiving the message plays first or second.
-- string is a game id.
The definition of time control is enclosed by 'BEGIN Time' and 'END Time.' When the definition of time control is omitted, clients are allowed to use infinite time.
-- the initial time that each player can use in the whole game.
-- the maximum time that each player can use for each move after total time has been run out.
-- the duration before timing consumed time.
-- added time to the total time for each turn of the player.
-- semantics:specifies unit of digits used in other fields.
-- semantics: if non-zero, the value is recorded as the consumed time of a player when the player played a move in a shorter time than the value.
-- semantics: whether time less than one time unit will be rounded up.
The definition of the initial position is enclosed by 'BEGIN Position' and 'END Position', and is written in the CSA file format. This definition may contain moves in addition to position, and in such cases, the game starts with a position after all the moves applied.
Property 'N+' and 'N-' must not be used, with 'Name+' and 'Name+' fields in general information being used instead. Moreover, 'AL' must not be used for the representation of captured pieces.
When a client receives a GAME CONDITION, the client must send AGREE or REJECT to the server.
When the server receives AGREE from both the clients, the server sends a start message in the following format.
Then both the clients enter a GAME state.
Otherwise, if any client has rejected the game, the server sends a REJECT message in the following format to both the clients.
Then both the clients enter a GAME-WAITING state.
Clients must use any of the following commands in a GAME state, which is a subset of the ones defined in the CSA file format.
For each move sent by a client to move, the server sends a MOVE-CONFIRMATION to both clients. MOVE-CONFIRMATION is a line consisting of the move and the consumed time joined by a comma (',').
In the case of illegal moves, the move included in MOVE-CONFIRMATION must be truncated up to 7 characters long.
If the server detects a game-end, the server must send a GAME-RESULT to both clients. If the game was ended by a client command, a GAME-RESULT must be sent immediately after MOVE-CONFIRMATION. In the case of time-up, the server immediately sends a GAME-RESULT without waiting for a message from the client to move.
GAME-RESULT consists of two lines beginning with the sharp ('#') character. The first line shows the reason and the second line shows win or lose from the viewpoint of a client receiving the message. Clients enter a GAME-WAITING state after they have received a GAME-RESULT.
For example, if a player has resigned after thinking four time units, the player receives the following messages.
The other player also receives the following messages.
Here is an example of messages in a game.


Introducing the first commercial trapped ion quantum computer. By manipulating individual atoms, it has the potential to one day solve problems beyond the capabilities of even the largest supercomputers.
Our quantum cores use lasers pointed at individual atoms to perform longer, more sophisticated calculations with fewer errors than any quantum computer yet built. In 2019, leading companies will start investigating real-world problems in chemistry, medicine, finance, logistics, and more using our systems.
At the microscopic scales of quantum mechanics, processes are noisy. Near-term quantum computers are not inherently error-corrected; instead, noisiness limits the number of operations they can perform before their results are no longer meaningful.
Qubits are the basic building blocks of quantum computers. They are most powerful when they are directly connected, or entangled. (For an introduction to qubits and entanglement, see Ars Technica's excellent article.) Because our qubits are fully connected, we can run a logic gate between any pair of qubits.
Fewer errors allow for longer and more useful calculations. For example, if our quantum logic gates had 20% error, we could apply 3 gates before our circuit output became too noisy to be useful. With 1% error, we can apply more than sixty gates. With 0.1% error, we could apply almost 700 gates.
Atoms make better quantum computers.
The atmospheric pressure we experience on the surface of earth is about 760 torr. By comparison, the vacuum of space in near-earth orbit is ~10-6 torr. For quantum computing, we aim for 10-11 torr or better, to maximize isolation of our qubits from the environment.we dynamically deploy and trap atomic qubits on a silicon chip using electromagnetic fields. This allows our quantum cores to adjust their configuration in software, and scale to handle potentially hundreds of qubits without new hardware.
Unlike other systems that try to simulate the behavior of atomic particles, IonQ quantum cores compute using many identical ytterbium atoms.
Ytterbium is a silvery rare earth metal first discovered in the late 19th century. As of 2018, ytterbium is used to make some of the most accurate atomic clocks in the world. For example, PTB built a single-ion clock one hundred times more accurate than the best cesium clocks. To create our qubits, we ionize ytterbium 171, removing one of its outermost electrons. Want to learn more about how it all works? Drop us a line or consider studying at UMD's Joint Quantum Institute!As in ytterbium atomic clocks, isolating individual atoms reduces error and improves stability.
Precise lasers store information on our atomic qubits, perform logical operations, and connect them together in a quantum process called entanglement.
By way of example, with full connectivity, when testing the Berenstain-Vazirani algorithm with a 10-qubit oracle, in the worst case we need simply 10 two-qubit gates. In a grid topology, with each qubit having 4 neighbors, you'd need 16 two-qubit gates. With a ring topology, you'd need 19 two-qubit gates. Since each additional gate introduces noise and thereby error, the fewer gates required, the higher quality the calculation.With no fixed wires, the IonQ system can connect any two qubits with a single laser operation, increasing accuracy.
We're perfecting trapped ion quantum computing to solve the world's hardest problems.
Whether you're an expert in your field or a recent graduate, if you love collaborative, challenging work, we encourage you to apply.
Copyright © 2018 IonQ, Inc.


Airport Hotel Shanghai is invested by Dazhong Transportation (Group) CO.,LTD. Dazhong Airport Hotel is in shanghai which named "The pearl of Eastern" . The hotel located between the first and second terminal of Pudong International Airport by five minutes walking distance.
The hotel above the maglev station is composed by the South and North buildings, and which is adjacent the airport access point. The business area is more than 30,005 square meters, and the total number of rooms is 712. Of which: there are 273 rooms in the south building, which forms big round King-size beds and suits, and in-room facilities and service is at four-star standard; there are 439 rooms in the north building, forms all kinds of standard rooms and the King-size beds, also there are Merrylin Restaurant and Merrylin XiaoChu providing delicious dish and quality service for guests.
There are a variety of conference room providing guests various business activities and affairs in our hotel. The air bar, which is in the middle of the North-South corridor connecting the two buildings, provides a leisure place where guests can enjoy a look at the whole Pudong International Airport. Looking forward to your coming, our hotel will supple first-class equipments and services.


The 2 greatest aspects when looking to purchase a new mattress, is comfort and also support.When you lie on a mattress, it should do a few things. It should properly support your body. It should also sink and contour to your body in the right places; to ensure your spine stays aligned, as you rest through the night. Appropriate alignment from head to toe is essential for healthy and balanced rest and to reduce neck and back pain. While being supportive of your body, a suitable mattress must provide stress relief. Stress points are what causes you to move around through out the night, while you are attempting to sleep. Since memory foam does such a great job of being supportive and comfortable, they have became really popular lately.
This goes hand in hand with comfort and support. Just like a mattress should never be too hard, it should also never be too soft.In order to keep your body and spine in the correct alignment, your mattress should be as neutral as it can be.If a mattress is too rigid, it will irritate your stress points, causing pinched nerves and stopping adequate blood flow. If you find that a lot of the times you awake, with the sensation of having pins and needles in your body, this is a sign of a mattress that is too rigid. And if your mattress is too soft, this can create a sagging effect, similar to a hammock. When this occurs, you can find yourself with back pain, due to your spine not being properly aligned.This is why it's crucial to find a mattress that is close to neutral. You want your mattress soft in the right places, without being too soft. But you also want your mattress to be supportive, without being too rigid.


Snapchat is finally admitting that its app is hard to use if you're not 14. Even in its most basic version — before Snapchat became "Snap Inc." and got into AR and hardware and maps and restaurant reservations and television — the closed platform wasn't the most user-friendly. Beloved by younger users and feared by parents convinced their spawn were just using the app to send nudes — they weren't, probably — the earliest iteration was a combination of unfamiliar swiping motions, puzzlingly ephemeral content, and offered no way to find other users on the platform. (It's gotten slightly better, but only slightly.) That was just the way Snapchat worked. And if that didn't work for you … too bad.
Overhauling Snapchat likely isn't so much about improving the user experience as it is Snapchat trying to ensure its user base keeps using the app at all. Today's letter also revealed the company has been losing money, a lot of it, since going public. In 2017, Snapchat's third-quarter net losses rang in at $443.2 million compared to 2016's $124.2 million. And Spectacles, Snapchat's much-hyped smart glasses first released in limited number via mysterious vending machines, wound up a loss for the company, too. Snapchat significantly overestimated demand for its first piece of hardware and took a $39.9 million bath for it, according to today's report.
This leaves Snapchat, not even a year post-IPO, scrambling to change the very core of the business model it sold on Wall Street back in March. A move which could alienate the app's most loyal users: the teenagers who aren't down with Instagram and Facebook's habit of cribbing straight from Spiegel's app and wear mastering Snap's learning curve as a badge of honor. Plus, changing things up doesn't necessarily mean everybody else will come running to Snapchat either. While teens call BS, other older users have been more than happy to take up with Instagram and Facebook's streamlined and user-friendly clones. Ostensibly those users are already plenty comfortable there, so Snapchat has its work cut out for it.


Hi, Kim! After experiencing the Disney Vacation Club (DVC) tour at Walt Disney World previously, we actually decided to join DVC on a Disney cruise. Using our points to take a Disney cruise seemed just the extra reason we needed to make the final commitment.
The time of year you book. Ranging from Value Season to High Peak Season, the point values vary greatly.
The stateroom category you prefer. From standard, inside staterooms to deluxe family staterooms with a verandah, there's a huge disparity between categories, and there's even a difference among each category depending on the options you choose.
The number of people in your traveling party and their ages. There are set values for the first two adults and then separate values for others -- depending on the age(s) of the guest(s).
Your best resource is the Vacation and Reservation Points Chart for Disney Cruise Line which you can find in your DVC Membership Book or on-line at www.dvcmember.com. There you can calculate your necessary points based on your unique wishes and needs.
I hope this helps you plan. Have a beautiful day and a magical vacation! Bon voyage!