id
stringlengths 18
42
| text
stringlengths 0
8.25M
| added
stringlengths 24
24
| created
stringlengths 20
24
| source
stringclasses 4
values | original_shard_dir
stringclasses 163
values | original_shard_idx
int64 0
311k
| num_tokens
int64 1
1.44M
|
|---|---|---|---|---|---|---|---|
proofpile-arXiv_065-135 |
\section{Introduction}
\label{sec:intro}
Interactions of cosmic-ray particles with detector materials can produce radioactive isotopes that create backgrounds for experiments searching for rare events such as dark matter interactions and neutrinoless double beta decay. Silicon is a widely used detector material because it is available with very high purity, which leads to low intrinsic radioactive backgrounds. In particular, solid-state silicon-based detector technologies show promise because their eV-scale energy thresholds~\cite{PhysRevLett.123.181802,Abramoff:2019dfb,Agnese:2018col} provide sensitivity to scattering events between atoms and ``low-mass'' dark matter particles with masses below 1\,GeV/c$^{2}$~\cite{Essig:2015cda}.
Three prominent low-mass dark matter efforts that employ silicon detectors are DAMIC~\cite{aguilararevalo2020results}, SENSEI~\cite{Abramoff:2019dfb}, and SuperCDMS~\cite{PhysRevD.95.082002}. All three use the highest-purity single-crystal silicon as detector substrates~\cite{VONAMMON198494}, with sensors fabricated on the surfaces for readout of charge or phonons and installed in low-background facilities to reduce the event rate from environmental backgrounds.
A primary challenge in these rare-event searches is to distinguish potential signal events from the much higher rate of interactions due to conventional sources of radiation, both from the terrestrial environment and in the detector materials. A variety of mitigation strategies are used to minimize backgrounds; nevertheless, a nonzero residual background expectation is generally unavoidable. Beta-emitting radiocontaminants in the bulk and on the surfaces of the detectors are especially challenging in the search for dark matter because the decay products can produce energy signals that are indistinguishable from the expected signal. Both DAMIC and SuperCDMS have investigated these detector backgrounds (see, e.g., Refs.~\cite{Aguilar-Arevalo:2015lvd,aguilararevalo2020results,PhysRevD.95.082002,Orrell:2017rid}), and they have identified $^3$H~(tritium), $^{32}$Si (intrinsic to the silicon) and $^{210}$Pb (surface contamination) as the leading sources of background for future silicon-based dark matter experiments. Unlike for $^{32}$Si, there are not yet any direct measurements of the tritium background in silicon; current estimates are based on models that have yet to be validated.
Tritium and other radioactive isotopes such as $^7$Be~and $^{22}$Na~are produced in silicon detectors as a result of cosmic-ray exposure, primarily due to interactions of high-energy cosmic-ray neutrons with silicon nuclei in the detector substrates~\cite{cebrian,Agnese:2018kta}.
The level of background from cosmogenic isotopes in the final detector is effectively determined by the above-ground exposure time during and following detector production, the cosmic-ray flux, and the isotope-production cross sections. The neutron-induced production cross sections for tritium, $^7$Be, and to a lesser extent $^{22}$Na, are not experimentally known except for a few measurements at specific energies. There are several estimates of the expected cross sections; however, they vary significantly, leading to large uncertainties in the expected cosmogenic background for rare-event searches that employ silicon detectors. To address this deficiency, we present measurements of the integrated isotope-production rates from a neutron beam at the Los Alamos Neutron Science Center (LANSCE) ICE HOUSE facility \cite{lisowski2006alamos, icehouse}, which has a similar energy spectrum to that of cosmic-ray neutrons at sea level. This spectral-shape similarity allows for a fairly direct extrapolation from the measured beam production rates to the expected cosmogenic production rates. While the spectral shape is similar, the flux of neutrons from the LANSCE beam greater than \SI{10}{MeV} is roughly \num{5E8} times larger than the cosmic-ray flux, which enables production of measurable amounts of cosmogenic isotopes in short periods of time. Our measurement will allow the determination of acceptable above-ground residency times for future silicon detectors, as well as improve cosmogenic-related background estimates and thus sensitivity forecasts.
We begin in Sec.~\ref{sec:isotopes} with a discussion of radioisotopes that can be cosmogenically produced in silicon, and we identify those most relevant for silicon-based dark matter searches: $^3$H, $^7$Be, and $^{22}$Na. For these three isotopes, we review previous measurements of the production cross sections and present the cross-section models that we use in our analysis. Section~\ref{sec:exposure} introduces our experimental approach, in which several silicon targets---a combination of charge-coupled devices (CCDs) and wafers---were irradiated at LANSCE. In Sec.~\ref{sec:counting} and Sec.~\ref{sec:production_rates} we present our measurements and predictions of the beam-activated activities, respectively. These results are combined in Sec.~\ref{sec:cosmogenic_rates} to provide our best estimates of the production rates from cosmogenic neutrons. In Sec.~\ref{sec:alternate} we evaluate other (non-neutron) production mechanisms and we conclude in Sec.~\ref{sec:discussion} with a summarizing discussion.
\section{Cosmogenic Radioisotopes}
\label{sec:isotopes}
\begin{table}[t]
\centering
\begin{tabular}{c c c c}
\hline
Isotope & Half-life & Decay & Q-value \\
& [yrs] & mode & [keV]\\
\hline
\vrule width 0pt height 2.2ex
$^3$H & 12.32\,$\pm$\,0.02 & $\beta$- & 18.591\,$\pm$\,0.003 \\
$^7$Be & 0.1457\,$\pm$\,0.0020 & EC & 861.82\,$\pm$\,0.02\\
$^{10}$Be & (1.51\,$\pm$\,0.06)$\times$10$^6$ & $\beta$- & 556.0\,$\pm$\,0.6\\
$^{14}$C & 5700\,$\pm$\,30 & $\beta$- & 156.475\,$\pm$\,0.004\\
$^{22}$Na & 2.6018\,$\pm$\,0.0022 & $\beta$+ & 2842.2\,$\pm$\,0.2\\
$^{26}$Al & (7.17\,$\pm$\,0.24)$\times$10$^5$ & EC & 4004.14\,$\pm$\,6.00\\
\hline
\end{tabular}
\caption{List of all radioisotopes with half-lives $>$\,30 days that can be produced by cosmogenic interactions with natural silicon. All data is taken from NNDC databases \cite{dunford1998online}. \protect\footnotemark[1]}
\footnotetext{Unless stated otherwise, all uncertainties quoted in this paper are at 1$\sigma$ (68.3\%) confidence.}
\label{tab:rad_isotopes}
\end{table}
Most silicon-based dark matter experiments use high-purity ($\gg$\,99\%) natural silicon (92.2\% $^{28}$Si, 4.7\% $^{29}$Si, 3.1\% $^{30}$Si \cite{meija2016isotopic}) as the target detector material. The cosmogenic isotopes of interest for these experiments are therefore any long-lived radioisotopes that can be produced by cosmic-ray interactions with silicon; Table~\ref{tab:rad_isotopes} lists all isotopes with half-lives greater than 30 days that are lighter than $^{30}$Si + n/p. None of them have radioactive daughters that may contribute additional backgrounds. Assuming that effectively all non-silicon atoms present in the raw material are driven out during growth of the single-crystal silicon boules used to fabricate detectors, and that the time between crystal growth and moving the detectors deep underground is typically less than 10 years, cosmogenic isotopes with half-lives greater than 100 years (i.e., $^{10}$Be, $^{14}$C, and $^{26}$Al) do not build up sufficient activity~\cite{reedy2013cosmogenic, caffee2013cross} to produce significant backgrounds. Thus the cosmogenic isotopes most relevant to silicon-based rare-event searches are tritium, $^7$Be, and $^{22}$Na. Tritium is a particularly dangerous background for dark matter searches because it decays by pure beta emission and its low Q-value (\SI{18.6} {\keV}) results in a large fraction of decays that produce low-energy events in the expected dark matter signal region. $^7$Be~decays by electron capture, either directly to the ground state of $^7$Li (89.56\%) or via the \SI{477}{\keV} excited state of $^7$Li (10.44\%). $^7$Be~is not a critical background for dark matter searches, because it has a relatively short half-life (\SI{53.22}{\day}); however, the \SI{54.7}{\eV} atomic de-excitation following electron capture may provide a useful energy-calibration tool. $^{22}$Na~decays primarily by positron emission (90.3\%) or electron capture (9.6\%) to the 1275 keV level of $^{22}$Ne. For thin silicon detectors $^{22}$Na~can be a significant background as it is likely that both the \SI{1275}{\keV} $\gamma$ ray and the \SI{511}{\keV} positron-annihilation photons will escape undetected, with only the emitted positron or atomic de-excitation following electron capture depositing any energy in the detector. Note that compared to $^3$H, the higher $\beta^+$ endpoint (\SI{546}{keV}) means that a smaller fraction of the $^{22}$Na~decays produce signals in the energy range of interest for dark matter searches.
\subsection{Tritium Production}
\begin{figure}[t!]
\centering
\includegraphics[width=\columnwidth]{si_h3_crosssections.pdf}
\caption{Experimental measurements (magenta error bars) \cite{QAIM1978150, Tippawan:2004sy, benck2002secondary} and model estimates (continuous curves) of neutron-induced tritium production in silicon. Measurements of the proton-induced cross section \cite{goebel1964production, kruger1973high} are also shown for reference (gray error bars).}
\label{fig:si_3h_cross_sections}
\end{figure}
Tritium production in silicon at sea-level is dominated by spallation interactions of high-energy cosmogenic neutrons with silicon nuclei. Tritium is a pure $\beta$ emitter and it is therefore not possible to directly measure the production cross section using conventional methods that rely on $\gamma$-ray~detectors to tag the reaction products. There are three previous experimental measurements of the neutron-induced tritium production cross section in silicon (shown in Fig.~\ref{fig:si_3h_cross_sections}), which either extracted tritium from a silicon target and measured the activity in a proportional counter \cite{QAIM1978150} or measured the triton nuclei ejected from a silicon target using $\Delta E-E$ telescopes \cite{Tippawan:2004sy,benck2002secondary}. The proton-induced cross section is expected to be similar to that of neutrons so we also show previous measurements with proton beams \cite{goebel1964production, kruger1973high}. While these measurements provide useful benchmarks at specific energies, they are insufficient to constrain the cosmogenic production cross section across the full range of relevant neutron energies (from $\sim$10\,MeV to a few GeV).
For this reason, previous estimates of tritium production in silicon dark matter detectors have relied on estimates of the cross section from calculations and simulations of the nuclear interactions or compiled databases that combine calculations with experimental data \cite{martoff1987limits, zhang2016cosmogenic, agnese2019production}. The production of tritons due to spallation is difficult to model, because the triton is a very light nucleus that is produced not only during the evaporation or de-excitation phase but also from coalescence of nucleons emitted during the high-energy intra-nuclear cascade stage \cite{leray2010improved, leray2011results, filges2009handbook}. Due to large variations among the predictions of different cross-section models, we consider several models for comparison to our experimental results and extraction of cosmogenic production rates. Shown in Fig.~\ref{fig:si_3h_cross_sections} are the semi-empirical formulae of Konobeyev and Korovin (K\&K) \cite{konobeyev1993tritium} (extracted from the commonly used ACTIVIA code \cite{back2008activia}) and results from nuclear reaction calculations and Monte Carlo simulations that are performed by codes such as TALYS \cite{koning2008talys}, INCL \cite{boudard2013new} and ABLA \cite{kelic2008deexcitation}.\footnote{The Konobeyev and Korovin ($^3$H), and Silberberg and Tsao ($^7$Be, $^{22}$Na) cross sections were obtained from the ACTIVIA code package \cite{activia2017}, the TALYS cross sections were calculated using TALYS-1.9 \cite{talys1.9}, and the INCL cross sections were calculated using the INCL++ code (v6.0.1) with the ABLA07 de-excitation model \cite{mancusi2014extension}. The default parameters were used for all programs. We note that the TALYS models are optimized in the \SI{1}{\keV} to \SI{200}{\MeV} energy range though the maximum energy has been formally extended to \SI{1}{\GeV} \cite{koning2014extension}.} We also compared effective cross sections (extracted through simulation) from built-in physics libraries of the widely used Geant4 simulation package \cite{agostinelli2003geant4,allison2016recent} such as INCLXX \cite{boudard2013new,mancusi2014extension}, BERTINI \cite{bertini1963low, guthrie1968calculation, bertini1969intranuclear, bertini1971news}, and Binary Cascades (BIC) \cite{folger2004binary}.\footnote{We used Geant4.10.3.p02 with physics lists QGSP\_INCLXX 1.0 (INCL++ v5.3), QGSP\_BERT 4.0, and QGSP\_BIC 4.0.}
\subsection{$^7$Be~Production}
$^7$Be~is produced as an intermediate-mass nuclear product of cosmogenic particle interactions with silicon. The neutron-induced production cross section has been measured at only two energies \cite{ninomiya2011cross}, as shown in Fig.~\ref{fig:si_7be_cross_sections}. Although the neutron- and proton-induced cross sections are not necessarily the same, especially for neutron-deficient nuclides such as $^7$Be~and $^{22}$Na~\cite{ninomiya2011cross}, there are a large number of measurements with protons
that span the entire energy range of interest \cite{otuka2014towards, zerkin2018experimental}, which we show in Fig.~\ref{fig:si_7be_cross_sections} for comparison.\footnote{We have excluded measurements from Ref.~\cite{rayudu1968formation}, because there are well-known discrepancies with other measurements \cite{ michel1995nuclide, schiekel1996nuclide}.} For ease of evaluation, we fit the proton cross-section data with a continuous 4-node spline, hereafter referred to as ``$^{\text{nat}}$Si(p,x)$^7$Be Spline Fit''.
As with tritium, we also show predictions from different nuclear codes and semi-empirical calculations, including the well-known Silberberg and Tsao (S\&T) semi-empirical equations \cite{silberberg1973partial,silberberg1973partial2, silberberg1977cross, silberberg1985improved, silberberg1990spallation, silberberg1998updated} as implemented in the ACTIVIA code. We note that the model predictions for the $^7$Be~production cross section in silicon vary greatly, with significantly different energy thresholds, energy dependence, and magnitude. $^7$Be~is believed to be produced predominantly as a fragmentation product rather than as an evaporation product or residual nucleus \cite{michel1995nuclide}, and fragmentation is typically underestimated in most theoretical models \cite{michel1995nuclide, titarenko2006excitation}. We note that unlike for the tritium cross-section models, there is a significant difference between the predictions obtained by evaluating the INCL++ v6.0.1 model directly versus simulating with Geant4 (INCL++ v5.3), probably due to updates to the model.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{si_be7_crosssections.pdf}
\caption{Experimental measurements (magenta error bars) \cite{ninomiya2011cross} and model estimates (continuous curves) of the neutron-induced $^7$Be~production cross section in silicon. Measurements of the proton-induced cross section \cite{otuka2014towards, zerkin2018experimental} are also shown for reference (gray error bars).}
\label{fig:si_7be_cross_sections}
\end{figure}
\subsection{$^{22}$Na~Production}
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{si_na22_crosssections.pdf}
\caption{Experimental measurements (magenta and pink error bars) \cite{michel2015excitation, hansmann2010production, yashima2004measurement, sisterson2007cross, ninomiya2011cross} and model estimates (continuous curves) of the neutron-induced $^{22}$Na~production cross section in silicon. Measurements of the proton-induced cross section \cite{otuka2014towards, zerkin2018experimental} are also shown for reference (gray error bars).}
\label{fig:si_22na_cross_sections}
\end{figure}
$^{22}$Na~is produced as a residual nucleus following cosmogenic interactions with silicon. Compared to tritium and $^7$Be, the production of $^{22}$Na~is the best studied. Measurements of the neutron-induced cross section were carried out by Michel et.\ al.\ using quasi-monoenergetic neutrons between 33 and 175 MeV, with TALYS-predicted cross sections used as the initial guess to unfold the experimentally measured production yields \cite{michel2015excitation, hansmann2010production}. These, along with six other data points between 66 and 370 MeV \cite{yashima2004measurement, sisterson2007cross, ninomiya2011cross}, are shown in Fig.~\ref{fig:si_22na_cross_sections}. Proton-induced cross-section measurements\footnote{Similar to $^7$Be, we have excluded measurements from Ref.~\cite{rayudu1968formation}.} \cite{otuka2014towards, zerkin2018experimental} span the entire energy range of interest and are significantly larger than the measured neutron-induced cross sections. As before, we also show the predicted cross sections from Silberberg and Tsao, TALYS, INCL++ (ABLA07) and Geant4 models. In order to compare the existing neutron cross-section measurements to our data, we use a piecewise model that follows the measurements in Refs.~\cite{michel2015excitation, hansmann2010production} below 180\,MeV and follows the TALYS model at higher energies. This model is hereafter referred to as ``Michel-TALYS'' (see Fig.~\ref{fig:si_22na_cross_sections}). $^{22}$Na~can also be produced indirectly through the production of the short-lived isotopes $^{22}$Mg, $^{22}$Al, and $^{22}$Si, which eventually decay to $^{22}$Na, but for the models considered the total contribution from these isotopes is $<$ \SI{1}{\percent}, and is ignored here.
\section{Beam Exposure}
\label{sec:exposure}
To evaluate the production rate of cosmogenic isotopes through the interaction of high-energy neutrons, we irradiated silicon charge-coupled devices (CCDs) and silicon wafers at the LANSCE neutron beam facility. Following the irradiation, the CCDs were readout to measure the beam-induced $\beta$ activity within the CCD active region, and the $\gamma$ activity induced in the wafers was measured using $\gamma$-ray spectroscopy. In this section we describe the details of the targets and beam exposure, while in Sec.~\ref{sec:counting} we present the measurement results.
\subsection{CCDs}
\label{sec:ccds}
The irradiated CCDs were designed and procured by Lawrence Berkeley National Laboratory (LBNL)~\cite{ccdtech} for the DAMIC Collaboration.
CCDs from the same fabrication lot were extensively characterized in the laboratory and deployed underground at SNOLAB to search for dark matter~\cite{Aguilar-Arevalo:2016zop, PhysRevD.94.082006}.
The devices are three-phase scientific CCDs with a buried $p$-channel fabricated on a \SI{670}{\micro\meter}-thick $n$-type high-resistivity (10--20\,\si{\kilo\ohm\cm}) silicon substrate, which can be fully depleted by applying $>$\,\SI{40}{\volt} to a thin backside contact.
The CCDs feature a 61.44$\times$30.72\,mm$^2$ rectangular array of 4096$\times$2048 pixels (each 15$\times$15 \si{\micro\meter\squared}) and an active thickness of \SI{661 \pm 10}{\micro\meter}.
By mass, the devices are $>$\,\SI{99}{\percent} elemental silicon with natural isotopic abundances. Other elements present are oxygen ($\sim$\,\SI{0.1}{\percent}) and nitrogen ($<$\,\SI{0.1}{\percent}) in the dielectrics, followed by phosphorous and boron dopants ($<$\,\SI{0.01}{\percent}) in the silicon.
Ionizing particles produce charge in the CCD active region; e.g., a fast electron or $\beta$ particle will produce on average one electron-hole pair for every \SI{3.8}{\eV} of deposited energy. The ionization charge is drifted by the applied electric field and collected on the pixel array. The CCDs are read out serially by moving the charge vertically row-by-row into the serial register (the bottom row) where the charge is moved horizontally pixel-by-pixel to the output readout node.
Before irradiation, the charge-transfer inefficiency from pixel to pixel was $< 10^{-6}$~\cite{ccdtech}, the dark current was $<$\SI{1}{e^- \per pixel \per \hour}, and the uncertainty in the measurement of the charge collected by a pixel was $\sim$2\,$e^-$ RMS. Further details on the response of DAMIC CCDs can be found in Sec.~IV of Ref.~\cite{PhysRevD.94.082006}.
Even after the significant increase in CCD noise following irradiation (e.g., due to shot noise associated with an increase in dark current), the CCD can still resolve most of the tritium $\beta$-decay spectrum.
Irradiation generates defects in silicon devices that can trap charges and negatively impact the performance of CCDs. Fully depleted devices are resilient to irradiation damage in the bulk silicon because the ionization charge is collected over a short period of time, which minimizes the probability of charge being trapped by defects before it is collected.
For this reason LBNL CCDs have been considered for space-based imaging where the devices are subjected to high levels of cosmic radiation~\cite{snap}.
Measurements at the LBNL cyclotron demonstrated the remarkable radiation tolerance of the CCDs proposed for the SNAP satellite, which follow the same design principles and fabrication process as the DAMIC CCDs.
For the measurements presented in this paper, there is a trade-off between activation rate and CCD performance.
Higher irradiation leads to a higher activity of radioisotopes in the CCD and hence a lower statistical uncertainty in the measurement.
On the other hand, higher irradiation also decreases the CCD performance, which needs to be modeled and can thus introduce significant systematic uncertainty.
The two most relevant performance parameters affected by the irradiation are the charge-transfer inefficiency (CTI) and the pixel dark current (DC).
Ref.~\cite{snap} provides measurements of CTI and DC after irradiation with 12.5 and \SI{55}{MeV} protons.
Following irradiation doses roughly equivalent to a LANSCE beam fluence of $2.4\times10^{12}$ neutrons above \SI{10}{\MeV}, the CCDs were still functional with the CTI worsened to $\sim$\,$10^{-4}$ and asymptotic DC rates (after days of operation following a room-temperature anneal) increased to $\sim$\SI{100}{e^- \per pixel \per \hour}.
These values depend strongly on the specific CCD design and the operation parameters, most notably the operating temperature.
Considering the available beam time, the range of estimated production rates for the isotopes of interest, and the CCD background rates, we decided to irradiate three CCDs with different levels of exposure, roughly corresponding to $2.4\times10^{12}$, $1.6\times10^{12}$, and $0.8\times10^{12}$ neutrons above \SI{10}{MeV} at the LANSCE neutron beam. Furthermore, we used a collimator (see Sec.~\ref{sec:lansce_beam}) to suppress irradiation of the serial register at the edge of the CCDs by one order of magnitude and thus mitigate CTI in the horizontal readout direction. Following the beam exposure, we found that the least irradiated CCD had an activity sufficiently above the background rate while maintaining good instrumental response and was therefore selected for analysis in Sec.~\ref{sec:ccd_counting}.
The CCDs were packaged at the University of Washington following the procedure developed for the DAMIC experiment.
The CCD die and a flex cable were glued onto a silicon support piece such that the electrical contact pads for the signal lines are aligned.
The CCDs were then wedge bonded to the flex cable with \SI{25}{\micro\meter}-thick aluminum wire.
A connector on the tail of the flex cable can be connected to the electronics for device control and readout.
Each packaged device was fixed inside an aluminum storage box, as shown in Fig.~\ref{fig:CCDphoto}. The CCDs were kept inside their storage boxes during irradiation to preserve the integrity of the CCD package, in particular to prevent the wire bonds from breaking during handling and to reduce any possibility of electrostatic discharge, which can damage the low-capacitance CCD microelectronics.
To minimize the attenuation of neutrons along the beam path and activation of the storage box, the front and back covers that protect each CCD were made from relatively thin (0.5\,mm) high-purity aluminum (alloy 1100).
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{CCD_photo.pdf}
\caption{Photograph of the CCD package inside its aluminum storage box. Left: Package before wire bonding. Right: After wire bonding, with aluminum frame to keep the CCD package fixed in place.}
\label{fig:CCDphoto}
\end{figure}
\subsection{Wafers}
In addition to the CCDs, we exposed several Si wafers, a Ge wafer, and two Cu plates to the neutron beam. These samples served both as direct targets for activation and measurement of specific radioisotopes, and as witness samples of the neutron beam. In this paper, we focus on the Si wafers; however, the Ge wafer and Cu plates were also measured and may be the subject of future studies.
A total of eight Si wafers (4 pairs) were used: one pair matched to each of the three CCDs (such that they had the same beam exposure time) and a fourth pair that served as a control sample. The eight wafers were purchased together and have effectively identical properties. Each wafer was sliced from a Czochralski-grown single-crystal boule with a 100-mm diameter and a resistivity of $>$\SI{20}{\ohm\cm}. The wafers are undoped, were polished on one side, and have a $\langle$100$\rangle$ crystal-plane alignment. The thickness of each individual wafer is \SI{500 \pm 17}{\micro\meter} (based on information from the vendor). The control sample was not exposed to the neutron beam and thus provides a background reference for the gamma counting. Note that because the wafers were deployed and counted in pairs, henceforth we distinguish and refer to only pairs of wafers rather than individual wafers. The (single) Ge wafer is also \SI{100}{\milli\meter} in diameter and undoped, with a thickness of \SI{525 \pm 25}{\micro\meter}, while the Cu plates have dimensions of $114.7 \times 101.6 \times$ \SI{3.175}{\milli\meter}.
\subsection{LANSCE Beam Exposure}
\label{sec:lansce_beam}
\begin{figure*}
\centering
\includegraphics[width=0.32\textwidth]{config1-pers.pdf}
\includegraphics[width=0.32\textwidth]{config2-pers.pdf}
\includegraphics[width=0.32\textwidth]{config3-pers.pdf}
\caption{Geant4 renderings of the three setups used to position targets in the neutron beam, with the beam passing from right to left.
Aluminum (Al) boxes holding the CCDs (yellow) were held in place by an Al rack (dark gray).
For the initial setup (left), the Al box is made transparent to show the positioning of the CCD (red), air (grey), and other structures (light brown).
The other targets include pairs of Si wafers (green), a Ge wafer (blue), and Cu plates (copper brown).
The polyethylene wafer holder (purple) is simplified to a rectangle of the same thickness and height as the actual object, with the sides and bottom removed.
All targets were supported on an acetal block (light gray).}
\label{fig:g4rendering}
\end{figure*}
The samples were irradiated at the LANSCE WNR ICE-HOUSE II facility~\cite{icehouse} on Target 4 Flight Path 30 Right (4FP30R). A broad-spectrum (0.2--800 MeV) neutron beam was produced via spallation of 800 MeV protons on a tungsten target. A 2.54-cm (1") diameter beam collimator was used to restrict the majority of the neutrons to within the active region of the CCD and thus prevent unwanted irradiation of the serial registers on the perimeter of the active region. The neutron fluence was measured with $^{238}$U foils by an in-beam fission chamber~\cite{wender1993fission} placed downstream of the collimator. The beam has a pulsed time structure, which allows the incident neutron
energies to be determined using the time-of-flight technique (TOF)---via a measurement between the proton beam pulse and the fission chamber signals~\cite{lisowski2006alamos,wender1993fission}.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=\columnwidth]{InBeamLayout_cropped.jpg}
\end{center}
\caption{Layout of the samples as placed in the beam during the final irradiation setup (cf.\ Fig.~\ref{fig:g4rendering} right). The beam first passes through the cylindrical fission chamber (far right) and then through the samples (from right to left): 3~CCDs in Al boxes (with flex cables emerging at the top), 3~pairs of Si wafers, 1~Ge wafer, and 2~Cu plates.}
\label{Fig:CCDlayout}
\end{figure}
The beam exposure took place over four days between September 18$^{\mathrm{th}}$ and 22$^{\mathrm{nd}}$, 2018. On Sept.\,18, CCD\,1 was placed in the beam line at 18:03 local time, located closest to the fission chamber, along with a pair of Si wafers, one Ge wafer, and one Cu plate placed downstream (in that order; cf.\ Fig.~\ref{fig:g4rendering} left). The front face of the Al box containing CCD\,1 was \SI{260}{\mm} from the face of the fission chamber. At 17:16 on Sept.\,20, CCD\,2 was added directly downstream from CCD\,1, along with another pair of Si wafers. The front face of the Al box for CCD\,2 was \SI{14.3}{\mm} from the front face of CCD\,1. At 09:11 on Sept.\,22, CCD\,3 was added downstream with an equidistant spacing relative to the other CCDs, along with another pair of Si wafers and a second Cu plate. Figure~\ref{fig:g4rendering} shows schematics of these three exposure setups, while Fig.~\ref{Fig:CCDlayout} shows a photograph of the final setup in which all three CCDs were on the beam line. The exposure was stopped at 08:00 on Sept.\,23, and all parts exposed to the beam were kept in storage for approximately seven weeks to allow short-lived radioactivity to decay prior to shipment for counting.
\subsection{Target Fluence}
The fluence measured by the fission chamber during the entire beam exposure is shown in Fig.~\ref{fig:lanscebeamenergy}, with a total of \num{2.91 \pm 0.22 E12} neutrons above 10 MeV. The uncertainty is dominated by the systematic uncertainty in the $^{238}$U(n, f) cross section used to monitor the fluence, shown in Fig.~\ref{fig:fission_cs}. Below 200 MeV the assumed LANSCE cross section and various other experimental measurements and evaluations \cite{lisowski1991fission, carlson2009international, tovesson2014fast, marcinkevicius2015209} agree to better than 5\%. Between 200 and 300 MeV there are only two measurements of the cross section \cite{lisowski1991fission, miller2015measurement} which differ by 5--10\%. Above \SI{300}{\MeV} there are no experimental measurements. The cross section used by the LANSCE facility assumes a constant cross section above \SI{380}{\MeV} at roughly the same value as that measured at \SI{300}{\MeV} \cite{miller2015measurement}. This is in tension with evaluations based on extrapolations from the $^{238}$U(p, f) cross section that recommend an increasing cross section to a constant value of roughly \SI{1.5}{\barn} at 1 GeV \cite{duran2017search,carlson2018evaluation}. We have used the LANSCE cross section and assumed a 5\% systematic uncertainty below \SI{200}{\MeV}, a 10\% uncertainty between 200 and \SI{300}{\MeV}, and a constant 20\% uncertainty between 300 and \SI{750}{\MeV}. The uncertainty in the neutron energy spectrum due to the timing uncertainty in the TOF measurement (FWHM $\sim$ \SI{1.2}{\nano\second}) is included in all calculations but is sub-dominant (2.5\%-3.5\%) for the estimates of isotope production rates.
While the nominal beam diameter was set by the 1" collimator, the cross-sectional beam profile has significant tails at larger radii. At the fission chamber approximately 38.8\% of neutrons fall outside a 1" diameter, as calculated with the beam profile provided by LANSCE.
Additionally the beam is slightly diverging, with an estimated cone opening angle of 0.233\degree. A Geant4 \cite{agostinelli2003geant4,allison2016recent} simulation that included the measured beam profile and beam divergence, the measured neutron spectrum, and the full geometry and materials of the targets, mounting apparatus, and fission chamber, was used to calculate the neutron fluence through each material, accounting for any attenuation of the neutrons through the targets.
To reduce computational time, a biasing technique was used to generate neutrons. Instead of following the beam profile, neutrons were generated uniformly in a \SI{16}{\cm}$\times$\SI{16}{\cm} square in front of the fission chamber, covering the entire cross-sectional area of the setup. After running the Geant4 simulation, each event was assigned a weight which is proportional to the intensity of the beam at the simulated neutron location, as obtained from the two-dimensional beam profile supplied by LANSCE. This allows reuse of the same simulation results for different beam profiles and alignment offsets. A total of \num{5.5 E10} neutrons above 10 MeV were simulated for each setup and physics list.
At this level of statistics, the statistical uncertainties in the simulation are sub-dominant to the total neutron fluence uncertainty.
The simulations show that each CCD receives about \SI{83}{\percent} of the whole beam. To assess the uncertainty in the neutron fluence due to misalignment of the beam with the center of the CCDs, the profile of the beam was reconstructed by measuring the dark current rate in the CCDs as a function of position (see Sec.~\ref{sec:ccd_counting}). The beam misalignment is calculated to be about $-2.3$\,mm in the $x$ direction and $+0.5$\,mm in the $y$ direction, which when input into the Geant4 simulation yields a systematic uncertainty in the neutron fluence of less than 1\%. The total neutron fluence ($>$ \SI{10}{\MeV}) through each CCD and its Si-wafer matched pair is listed in Table~\ref{tab:neutron_fluences}; corresponding energy spectra are shown in Fig.~\ref{fig:lanscebeamenergy} (the spectral shape of the fluence through each Si-wafer pair is very similar to that of the corresponding CCD and has been omitted for clarity).
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{neutron_flux_targets.pdf}
\caption{Comparison of the LANSCE 4FP30R/ICE II neutron beam with sea-level cosmic-ray neutrons. The black data points and left vertical axis show the number of neutrons measured by the fission chamber during the entire beam exposure used for this measurement. Uncertainties shown are statistical only (see main text for discussion of systematic uncertainties). The colored markers show the simulated fluence for each of the CCDs in the setup. For comparison, the red continuous line and the right vertical axis show the reference cosmic-ray neutron flux at sea level for New York City during the midpoint of solar modulation \cite{gordon2004measurement}}.
\label{fig:lanscebeamenergy}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fission_cs.pdf}
\end{center}
\caption{Experimental measurements (circles) \cite{lisowski1991fission, tovesson2014fast, miller2015measurement} and evaluations (squares) \cite{carlson2009international, marcinkevicius2015209, duran2017search, carlson2018evaluation} of the $^{238}$U(n, f) cross section. The cross section assumed by the LANSCE facility to convert the fission chamber counts to a total neutron fluence is shown by the black line, with the shaded grey band indicating the assumed uncertainty.}
\label{fig:fission_cs}
\end{figure}
\begin{table}
\centering
\begin{tabular}{c c c}
\hline
Target & Exposure time & Neutrons through target \\
& [hrs] & ($> 10$ MeV)\\
\hline
\vrule width 0pt height 2.2ex
CCD 1 & 109.4 & \num{2.39 \pm 0.18 E12}\\
Wafer 1 & 109.4 & \num{2.64 \pm 0.20 E12}\\
\hline
\vrule width 0pt height 2.2ex
CCD 2 & 62.7 & \num{1.42 \pm 0.11 E12}\\
Wafer 2 & 62.7 & \num{1.56 \pm 0.12 E12}\\
\hline
\vrule width 0pt height 2.2ex
CCD 3 & 22.8 & \num{5.20 \pm 0.39 E11}\\
Wafer 3 & 22.8 & \num{5.72 \pm 0.43 E11}\\
\hline
\end{tabular}
\caption{Beam exposure details for each CCD and its Si-wafer matched pair.}
\label{tab:neutron_fluences}
\end{table}
\section{Counting}
\label{sec:counting}
\subsection{Wafers}
\label{ssec:wafer_counting}
\begin{table*}[ht]
\centering
\begin{tabular}{ccccc}
\hline
& Wafer 0 & Wafer 1 & Wafer 2 & Wafer 3 \\
\hline
\vrule width 0pt height 2.2ex
Si areal density [atoms/cm$^2$] & \multicolumn{4}{c}{\num{4.99 \pm 0.17 e21}~~~~~~~~~~~~~~~~~~~~~} \\
Beam to meas.\ time [days] & - & \num{184.107} & \num{187.131} & \num{82.342} \\
Ge counting time [days] & \num{7.000} & \num{1.055} & \num{3.005} & \num{7.000} \\
\hline
\vrule width 0pt height 2.2ex
Measured $^7$Be~activity [mBq] & $<$\num{40} & \num{161 \pm 24} & \num{75 \pm 12} & \num{149 \pm 12}\\
Decay-corrected $^7$Be~activity [mBq] & - & \num{1830 \pm 270} & \num{870 \pm 140} & \num{437 \pm 34}\\
Beam-avg.\ $^7$Be~cross section [cm$^2$] & - & \num{0.92 \pm 0.16 E-27} & \num{0.74 \pm 0.13 E-27} & \num{1.01 \pm 0.12 E-27}\\
\hline
\vrule width 0pt height 2.2ex
Measured $^{22}$Na~activity [mBq] & $<$\num{5.1} & \num{606 \pm 29} & \num{370 \pm 16} & \num{139.5 \pm 6.3}\\
Decay-corrected $^{22}$Na~activity [mBq] & - & \num{694 \pm 33} & \num{424 \pm 19} & \num{148.2 \pm 6.6}\\
Beam-avg.\ $^{22}$Na~cross section [cm$^2$] & - & \num{6.23 \pm 0.60 E-27} & \num{6.44 \pm 0.61 E-27} & \num{6.15 \pm 0.58 E-27}\\
\hline
\end{tabular}
\caption{Gamma-counting results for the Si-wafer pairs. Measured activities are corrected for isotope decay that occurred during the beam exposure, as well as between the end of the beam exposure and the time of the gamma counting. Uncertainties are listed at 1$\sigma$ (68.3\%) confidence while upper limits quoted for the unirradiated pair (``Wafer 0'') represent the spectrometer's minimum detectable activity (Currie MDA with a 5\% confidence factor~\cite{currie}) at the corresponding peak energy.}
\label{tab:wafer_counting}
\end{table*}
The gamma-ray activities of the Si-wafer pairs (including the unirradiated pair) were measured with a low-background counter at Pacific Northwest National Laboratory (PNNL). Measurements were performed using a Canberra Broad Energy Ge (BEGe) gamma-ray spectrometer (model BE6530) situated within the shallow underground laboratory (SUL) at PNNL \cite{aalseth2012shallow}. The SUL is designed for low-background measurements, with a calculated depth of \SI{30}{\meter} water equivalent.
The BEGe spectrometer is optimized for the measurement of fission and activation products, combining the spectral advantages of low-energy and coaxial detectors, with an energy range from \SI{3}{\keV} to \SI{3}{\MeV}.
The detector is situated within a lead shield (200\,mm), lined with tin (1\,mm) and copper (1\,mm). It is equipped with a plastic scintillator counter \cite{burnett2017development, burnett2014cosmic, burnett2012development, burnett2013further} to veto cosmic rays, which improves sensitivity by further reducing the cosmic-induced detector background by 25\%.
The detector was operated with a Canberra Lynx MCA to provide advanced time-stamped list mode functionality.
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{ge_counting.pdf}
\caption{Spectral comparison of the gamma-counting results for the Si-wafer pairs. Inspection of the full energy range (top panel) reveals two peaks in the irradiated samples (1, 2, and 3) at \SI{478}{\keV} (bottom left) and \SI{1275}{\keV} (bottom right) that are not present in the unirradiated sample (0), corresponding to $^7$Be\ and $^{22}$Na\ activated by the LANSCE neutron beam, respectively.}
\label{fig:ge_counting}
\end{figure*}
Each wafer pair was measured independently, with wafer pair 3 and the unexposed wafer pair 0 counted for longer periods because their expected activities were the lowest. Table~\ref{tab:wafer_counting} shows the gamma-counting details, and Fig.~\ref{fig:ge_counting} shows the measured gamma-ray spectra. Spectral analysis was performed using the Canberra Genie 2000 Gamma Acquisition \& Analysis software (version 3.4) and all nuclear data were taken from the Evaluated Nuclear Data File (ENDF) database \cite{chadwick2011endf} hosted at the National Nuclear Data Center by Brookhaven National Laboratory. Compared to the unirradiated wafer-pair spectrum, the only new peaks identified in the spectra of the irradiated wafer pairs are at 478 and \SI{1275}{\keV}, corresponding to $^7$Be~(10.44\% intensity per decay) and $^{22}$Na~(99.94\% intensity per decay), respectively (cf.\,Fig.\,\ref{fig:ge_counting}). Note that each of the irradiated wafer pairs also has a significant excess at \SI{511}{\keV}, corresponding to positron-annihilation photons from $^{22}$Na\ decays, and an associated sum peak at \SI{1786}{\keV} ($= 511 +$ \SI{1275}{\keV}).
The $^7$Be\ and $^{22}$Na\ activities in each wafer pair were calculated using the 478 and \SI{1275}{\keV} peaks, respectively. The measured values listed in Table~\ref{tab:wafer_counting} include the detector efficiency and true-coincidence summing corrections for the sample geometry and gamma-ray energies considered (calculated using the Canberra In Situ Object Counting Systems, or ISOCS, calibration software \cite{venkataraman1999validation}). The activity uncertainties listed in Table~\ref{tab:wafer_counting} include both the statistical and systematic contributions, with the latter dominated by uncertainty in the efficiency calibration ($\sim$\SI{4}{\percent}). Each measured activity is then corrected for isotope decay that occurred during the beam exposure, as well as between the end of the beam exposure and the time of the gamma counting.
To compare among the results of the different wafer pairs, we divide each decay-corrected activity by the total number of incident neutrons and the number of target Si atoms to obtain a beam-averaged cross section (also listed in Table~\ref{tab:wafer_counting}). The values are in good agreement for both $^7$Be\ and $^{22}$Na\ (even if the common systematic uncertainty associated with the neutron beam fluence is ignored), which serves as a cross-check of the neutron-beam exposure calculations. The lack of any other identified peaks confirms that there are no other significant long-lived gamma-emitting isotopes produced by high-energy neutron interactions in silicon. Specifically, the lack of an identifiable peak at \SI{1808.7}{\keV} allows us to place an upper limit on the produced activity of $^{26}$Al at the minimum detectable activity level of \SI{12}{\milli\becquerel} (Currie MDA with a 5\% confidence factor~\cite{currie}), i.e.\ at least 58$\times$ lower than the $^{22}$Na\ activity in wafer pair 1.
\subsection{CCDs}
\label{sec:ccd_counting}
Images from CCD\,3 were acquired at The University of Chicago in a custom vacuum chamber. Prior to counting, the CCD was removed from the aluminum transport box and placed in a copper box inside the vacuum chamber. Images taken were 4200 columns by 2100 rows in size, with 52 rows and 104 columns constituting the ``overscan'' (i.e., empty pixel reads past the end of the CCD pixel array). These overscan pixels contain no charge and thus provide a direct measurement of the pixel readout noise. A total of 8030 post-irradiation images with \SI{417}{\sec} of exposure were acquired, for a total counting time of 38.76 days. Data were taken in long continuous runs of many images, with interruptions in data taking for testing of the CCD demarcating separate data runs.
Background data were taken prior to shipment to the LANSCE facility for neutron irradiation. These background data consist of the combined spectrum from all radioactive backgrounds in the laboratory environment, including the vacuum chamber, the intrinsic contamination in the CCD, and cosmic rays. A total of 1236 images were acquired using the same readout settings as post-irradiation images, but with a longer exposure of \SI{913}{\sec}, for a total counting time of 13.06 days.
CCD images were processed with the standard DAMIC analysis software~\cite{PhysRevD.94.082006}, which subtracts the image pedestal, generates a ``mask'' to exclude repeating charge patterns in the images caused by defects, and groups pixels into clusters that correspond to individual ionization events. The high dark current caused by damage to the CCD from the irradiation (see Fig.~\ref{fig:darkcurrentprofile}) necessitated a modification to this masking procedure because the average CCD pixel values were no longer uniform across the entire CCD, as they were before irradiation. The images were therefore split into 20-column segments which were treated separately for the pedestal subtraction and masking steps.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{dark_current_profile.pdf}
\caption{Post-irradiation dark-current profile for CCD\,3, obtained from the median pixel values across multiple images. The elevated number of dark counts in the center of the CCD shows the effect of the neutron damage on the CCD.}
\label{fig:darkcurrentprofile}
\end{figure}
Simulations of $^3$H{}, $^{22}$Na{}, and $^7$Be{} decays in the bulk silicon of the CCD were performed using a custom Geant4 simulation, using the Penelope Geant4 physics list, with a simplified geometry that included only the CCD
and the surrounding copper box. Radioactive-decay events were simulated according to the beam profile, assumed to be proportional to the dark current profile (shown in Fig. ~\ref{fig:darkcurrentprofile}). The CCD response was simulated for every ionization event, including the stochastic processes of charge generation and transport that were validated in Ref.~\cite{PhysRevD.96.042002}.
To include the effects of noise and dark current on the clustering algorithm, simulated ``blank'' images were created with the same noise and dark-current profile as the post-irradiation data. The simulated ionization events were pixelated and added onto the blank images, which were then processed with the standard DAMIC reconstruction code to identify clusters.
The increase in the vertical (row-to-row) charge transfer inefficiency (CTI) observed in the post-irradiation data was simulated with a Poissonian kernel, which assumes a constant mean probability, $\lambda$, of charge loss for each pixel transfer along a column~\cite{janesick}. We assume a dependence of $\lambda$ as a function of column number that is proportional to the dark current profile. The total effect of CTI on a particular cluster depends on the number of vertical charge transfers $n$. The continuous CCD readout scheme, chosen to optimize the noise while minimizing overlap of charge clusters, results in a loss of information about the true number of vertical charge transfers for each cluster. For every simulated cluster we therefore pick a random $n$ uniformly from 1 to 2000 to simulate events distributed from the bottom row to the top row of the CCD and apply the Poissonian kernel. We determined the maximum value of $\lambda$ near the center of the CCD to be $9\times10^{-4}$ by matching the distribution of the vertical spread of clusters in the simulation to the data.\footnote{The data from CCD\,1 and CCD\,2, which experienced significantly higher neutron irradiation than CCD\,3, were discarded from the analysis because the vertical CTI could not be well described with a Poissonian kernel. We suspect that the CTI in these CCDs is dominated by the effect of charge traps introduced by the neutron irradiation. During the readout procedure these traps are filled with charge from ionization clusters. The charge is then released on the time scale of milliseconds, corresponding to $\sim$25 vertical transfers. This effect is difficult to model and results in considerable loss of charge from clusters in these two CCDs.}
The identified clusters in the background data acquired prior to irradiation at LANSCE were also introduced on simulated blank images to include the effect of dark current, defects, and CTI on the background spectrum in the activated region of the CCD.
The post-irradiation energy spectrum was fit using a model that includes components for the CCD background, $^{22}$Na{} decays, and $^3$H{} decays. $^7$Be{} was excluded from the fit because the decay does not produce a significant contribution to the total energy spectrum, even if the activity were many times the value we expect based on the wafer measurement.
We constructed a binned Poissonian log-likelihood as the test statistic for the fit, which was minimized using Minuit \cite{James:1994vla} to find the best-fit parameters.
Due to the relatively low statistics in the background template compared to post-irradiation data, statistical errors were corrected using a modified Barlow-Beeston method \cite{BARLOW1993219}, allowing each bin of the model to fluctuate by a Gaussian-constrained term with a standard deviation proportional to the bin statistical uncertainty.
The data spectrum was fit from 2 to \SI{25}{\kilo\eV} to contain most of the $^3$H{} spectrum, while excluding clusters from noise at low energies.
A \SI{2}{\kilo\eV}-wide energy region around the copper K-shell fluorescence line at \SI{8}{\kilo\eV} was masked from the fit because it is not well-modeled in the simulation.
This peak-like feature is more sensitive to the details of the energy response than the smooth $^3$H{} spectrum. We have verified that including this K-shell line in the fit has a negligible effect on the fitted $^3$H\ activity.
The background rate for the fit was fixed to the pre-irradiation value, while keeping the amplitude of the $^{22}$Na{} spectrum free.
This choice has a negligible impact on the $^3$H{} result because the background and $^{22}$Na{} spectra are highly degenerate within the fit energy range, with a correlation coefficient of 0.993.
Figure~\ref{fig:finalfitresults} shows the measured energy spectrum and the best-fit result ($\chi^2$/NDF=104/87).
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{plot_final_fit.pdf}
\caption{Data spectrum and best-fit model with the spectral components stacked in different colors. The spectrum was fit from 2 to \SI{25}{\keV} with the shaded region around the \SI{8}{\keV} copper K-shell fluorescence line excluded from the fit. The rise in the spectrum below \SI{18}{\keV} from $^3$H{} decay is clearly visible above the nearly flat background and $^{22}$Na{} spectrum.}
\label{fig:finalfitresults}
\end{figure}
After the fit was performed, the activities were calculated by dividing the fitted counts by the cumulative data exposure. This number was corrected for the isotope-specific event detection efficiency obtained from the simulation for the energy region of interest.
Systematic errors were estimated from a series of fits under different configurations, including varying the energy range of the fit, varying the energy response and charge transfer parameters within their uncertainties, and floating versus constraining the amplitudes of the background and/or $^{22}$Na{} components in the fit.
The best estimate for the tritium activity in CCD\,3 (after correcting for radioactive decay) is $45.7 \pm 0.5 $ (stat) $\pm 1.5 $ (syst) \si{\milli\becquerel}.
The precision of the $^{22}$Na\ measurement in the CCDs is limited because the relatively flat $^{22}$Na{} spectrum is degenerate with the shape of the background spectrum. Unfortunately, there are no features in the CCD spectrum at low energies that can further constrain the $^{22}$Na{} activity. Further, the damage to the CCD renders the spectrum at higher energies unreliable because events with energies $>$\SI{50}{\kilo\eV} create large extended tracks where the effects of CTI, dark current, and pileup with defects become considerable, preventing reliable energy reconstruction.
Notably, characteristic full-absorption $\gamma$ lines are not present in the CCD spectrum because $\gamma$ rays do not deposit their full energy in the relatively thin CCDs. As a cross-check of the post-irradiation background rate, we separately fit the first and last 400 columns of the CCD (a region mostly free of neutron exposure) and found values consistent with the pre-irradiation background to within $\sim$\SI{7}{\percent}. Constraining the background to within this range has a negligible effect on the fitted tritium activity but leads to significant variation in the estimated $^{22}$Na\ activity, which dominates the overall systematic uncertainty. The best estimate for the $^{22}$Na~activity in CCD\,3 is $126 \pm 5 $ (stat) $ \pm 26 $ (syst) \si{\milli\becquerel}. This is consistent with the more precise measurement of the $^{22}$Na~activity in the silicon wafers, which corresponds to a CCD\,3 activity of \SI{88.5 \pm 5.3}{\milli\becquerel}.
\section{Predicted Beam Production Rate}
\label{sec:production_rates}
If the neutron beam had an energy spectrum identical to that of cosmic-ray neutrons, we could simply estimate the cosmogenic production rate by scaling the measured activity by the ratio of the cosmic-ray neutrons to that of the neutron beam. However the beam spectrum falls off faster at higher energies than that of cosmic rays (see Fig.~\ref{fig:lanscebeamenergy}). Thus we must rely on a model for the production cross sections to extrapolate from the beam measurement to the cosmogenic production rate.
We can evaluate the accuracy of the different cross-section models by comparing the predicted $^3$H, $^7$Be, and $^{22}$Na~activity produced by the LANSCE neutron beam irradiation to the decay-corrected measured activities. We note that measurements of the unirradiated targets confirm that any non-beam related isotope concentrations (e.g. due to cosmogenic activation) are negligible compared to the beam-induced activity.
For a given model of the isotope production cross section $\sigma(E)$ [cm$^2$], the predicted isotope activity, $P$ [Bq], produced by the beam (correcting for decays) is given by
\begin{linenomath*}
\begin{align}
\label{eq:beam_act}
P = \frac{n_a}{\tau} \int S(E) \cdot \sigma(E)~dE
\end{align}
\end{linenomath*}
where $n_a$ is the areal number density of the target silicon atoms [\si{\atoms\per \cm\squared}], $\tau$ is the mean life [\si{\second}] of the isotope decay, and $S(E)$ is the energy spectrum of neutrons [\si{\neutrons \per \MeV}]. The second column of Table~\ref{tab:trit_pred} shows the predicted activity in CCD 3, $P_\text{CCD3}$, for the different $^3$H~cross-section models considered. The corresponding numbers for $^7$Be~and $^{22}$Na~in Wafer 3 ($P_\text{W3})$ are shown in Tables~\ref{tab:ber_pred} and \ref{tab:sod_pred} respectively. The uncertainties listed include the energy-dependent uncertainties in the LANSCE neutron beam spectrum and the uncertainty in the target thickness.
\begin{table*}[t!]
\centering
\begin{tabular}{cccccc}
\hline
Model & Predicted LANSCE & Ejected & Implanted & Predicted LANSCE & Measured/Predicted\\
& $^3$H~produced act. & activity & activity & $^3$H~residual act. & $^3$H~residual activity\\
& $P_\text{CCD3}$ [\si{\milli\becquerel}] & $E_\text{CCD3}$ [\si{\milli\becquerel}] & $I_\text{CCD3}$ [\si{\milli\becquerel}] & $R_\text{CCD3}$ [\si{\milli\becquerel}] & \\
\hline
K\&K (ACTIVIA) & \num{40.8 \pm 4.5} & & &\num{41.5 \pm 5.6} & \num{1.10 \pm 0.15}\\
TALYS & \num{116 \pm 16} & \num{46.70 \pm 0.12} & \num{53.8 \pm 2.1} & \num{123 \pm 17} & \num{0.370 \pm 0.053} \\
INCL++(ABLA07) & \num{41.8 \pm 4.8} & & & \num{42.5 \pm 5.9} & \num{1.07 \pm 0.15}\\
GEANT4 BERTINI & \num{13.0 \pm 1.5} & \num{3.354 \pm 0.072} & \num{3.699 \pm 0.045} & \num{13.3 \pm 1.6} & \num{3.43 \pm 0.42}\\
GEANT4 BIC & \num{17.8 \pm 1.8} & \num{4.995 \pm 0.084} & \num{6.421 \pm 0.059} & \num{19.2 \pm 2.0} & \num{2.38 \pm 0.26}\\
GEANT4 INCLXX & \num{42.3 \pm 5.1} & \num{20.65 \pm 0.11} & \num{16.94 \pm 0.10} & \num{38.5 \pm 4.6} & \num{1.19 \pm 0.15}\\
\hline
\end{tabular}
\caption{Predicted $^3$H~activity in CCD 3 based on different cross-section models. The second column lists the total predicted activity produced in the CCD. The third and fourth columns list the activity ejected and implanted respectively with listed uncertainties only due to simulation statistics. The fifth column shows the final predicted residual activity calculated from the second, third, and fourth columns, including systematic uncertainties due to the geometry. For models without ejection and implantation information we use the average of the other models---see text for details. The final column shows the ratio of the experimentally measured activity to the predicted residual activity.}
\label{tab:trit_pred}
\end{table*}
\begin{table*}[t!]
\centering
\begin{tabular}{cccccc}
\hline
Model & Predicted LANSCE & Ejected & Implanted & Predicted LANSCE & Measured/Predicted\\
& $^7$Be~produced act. & activity & activity & $^7$Be~residual act. & $^7$Be~residual act.\\
& $P_\text{W3}$ [\si{\milli\becquerel}] & $E_\text{W3}$ [\si{\milli\becquerel}] & $I_\text{W3}$ [\si{\milli\becquerel}] & $R_\text{W3}$ [\si{\milli\becquerel}] & \\
\hline
S\&T (ACTIVIA) & \num{408 \pm 46} & & & \num{405 \pm 49} & \num{1.08 \pm 0.16}\\
TALYS & \num{294 \pm 41} & & & \num{292 \pm 42} & \num{1.50 \pm 0.25}\\
INCL++(ABLA07) & \num{141 \pm 21} & & & \num{140 \pm 22} & \num{3.12 \pm 0.55}\\
$^{\text{nat}}$Si(p,x)$^7$Be Spline Fit & \num{518 \pm 68} & & & \num{514 \pm 72} & \num{0.85 \pm 0.14}\\
GEANT4 BERTINI & \num{0.99 \pm 0.20} & $<0.33$ & \num{0.64 \pm 0.14} & \num{1.63 \pm 0.43} & \num{268 \pm 74} \\
GEANT4 BIC & \num{1.27 \pm 0.24} & $<0.33$ & \num{0.61 \pm 0.16} & \num{1.98 \pm 0.50} & \num{221 \pm 59}\\
GEANT4 INCLXX & \num{21.6 \pm 3.0} & \num{3.59 \pm 0.85} & \num{3.42 \pm 0.38} & \num{21.4 \pm 3.1} & \num{20.4 \pm 3.4}\\
\hline
\end{tabular}
\caption{Predicted $^7$Be~activity in Wafer 3 based on different cross-section models. See Table~\ref{tab:trit_pred} caption for a description of the columns. Upper limits are 90\% C.L.}
\label{tab:ber_pred}
\end{table*}
\begin{table*}[t!]
\centering
\begin{tabular}{cccccc}
\hline
Model & Predicted LANSCE & Ejected & Implanted & Predicted LANSCE & Measured/Predicted\\
& $^{22}$Na~produced act. & activity & activity & $^{22}$Na~residual act. & $^{22}$Na~residual act.\\
& $P_\text{W3}$
[\si{\milli\becquerel}] & $E_\text{W3}$ [\si{\milli\becquerel}] & $I_\text{W3}$ [\si{\milli\becquerel}] & $R_\text{W3}$ [\si{\milli\becquerel}] & \\
\hline
S\&T (ACTIVIA) & \num{295 \pm 29} & & & \num{295 \pm 29} & \num{0.502 \pm 0.054}\\
TALYS & \num{209 \pm 18}& & & \num{208 \pm 18} & \num{0.711 \pm 0.070}\\
INCL++(ABLA07) & \num{207 \pm 21} & & & \num{206 \pm 21} & \num{0.718 \pm 0.081}\\
Michel-TALYS & \num{151 \pm 14} & & & \num{151 \pm 14} & \num{0.98 \pm 0.10}\\
GEANT4 BERTINI & \num{97 \pm 11} & $< 0.88$ & $<0.008$ & \num{96 \pm 11} & \num{1.54 \pm 0.18}\\
GEANT4 BIC & \num{393 \pm 40} & $<2.0$ & $<0.02$ & \num{392 \pm 40} & \num{0.378 \pm 0.042}\\
GEANT4 INCLXX & \num{398 \pm 40} & $<2.0$ & $<0.03$ & \num{398 \pm 40} & \num{0.373 \pm 0.041}\\
\hline
\end{tabular}
\caption{Predicted $^{22}$Na~activity in Wafer 3 based on different cross-section models. See Table~\ref{tab:trit_pred} caption for a description of the details. Upper limits are 90\% C.L.}
\label{tab:sod_pred}
\end{table*}
\begin{figure}
\centering
\includegraphics[width=0.75\columnwidth]{tritium_ejection_implantation.pdf}
\caption{Schematic diagram showing triton ejection and implantation. The filled circles indicate example triton production locations while the triton nuclei show the final implantation locations. Production rate estimates include trajectories (a) and (b), while counting the tritium decay activity in the CCD measures (a) and (c).}
\label{fig:trit_ejec_schematic}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{transmatices-logcolor-altstyle.pdf}
\caption{Shown are the activities [mBq] of $^3$H (left), $^7$Be (middle), and $^{22}$Na (right) produced and implanted in various volumes (i.e., $T_{ij}\cdot P_j$) as predicted by the GEANT4 INCLXX model. CCD\,1, CCD\,2, CCD\,3 are the CCDs, with CCD\,1 being closest to the fission chamber. Box\,1, Box\,2, and Box\,3 are the aluminum boxes that contain CCD\,1, CCD\,2, and CCD\,3, respectively. Si\,1, Si\,2, Si\,3, and Ge are the silicon and germanium wafers downstream of the CCDs. World represents the air in the irradiation room.}
\label{fig:transmat}
\end{figure*}
\subsection{Ejection and Implantation}
Light nuclei, such as tritons, can be produced with significant fractions of the neutron kinetic energy. Due to their small mass, these nuclei have relatively long ranges and can therefore be ejected from their volume of creation and implanted into another volume. The situation is shown schematically in Fig.~\ref{fig:trit_ejec_schematic}. While we would like to estimate the total production rate in the silicon targets, what is actually measured is a combination of the nuclei produced in the target that are not ejected and nuclei produced in surrounding material that are implanted in the silicon target. The measured activity therefore depends not only on the thickness of the target but also on the nature and geometry of the surrounding materials.
The residual activity, $R_i$, eventually measured in volume $i$, can be written as
\begin{align}
\label{eq:transfer}
R_i = \sum_j T_{ij} \cdot P_j
\end{align}
where $P_j$ is the total activity produced in volume $j$ (see Eq.~\ref{eq:beam_act}) and $T_{ij}$ is the transfer probability---the probability of a triton produced in $j$ to be eventually implanted in $i$. Because the ejection and implantation of light nuclei is also an issue for dark matter detectors during fabrication and transportation, we have also explicitly factored the transfer probability into ejected activity ($E_i$) and activity implanted from other materials ($I_i$) to give the reader an idea of the relative magnitudes of the two competing effects:
\begin{align}
\label{eq:ejection}
E_i &= (1 - T_{ii})\cdot P_i\\
\label{eq:implantation}
I_i &= \sum_{j \neq i} T_{ij} \cdot P_j\\
R_i &= P_i - E_i + I_i
\end{align}
For nuclear models that are built-in as physics lists within Geant4, explicit calculations of transfer probabilities are not necessary, because the nuclei produced throughout the setup are propagated by Geant4 as part of the simulation. For the TALYS model, which does calculate the kinematic distributions for light nuclei such as tritons but is not included in Geant4, we had to include the propagation of the nuclei separately. Since the passage of nuclei through matter in the relevant energy range is dominated by electromagnetic interactions, which are independent of nuclear production models and can be reliably calculated by Geant4, we used TALYS to evaluate the initial kinetic energy and angular distributions of triton nuclei produced by the LANSCE neutron beam and then ran the Geant4 simulation starting with nuclei whose momenta are drawn from the TALYS-produced distributions. For the remaining models which do not predict kinematic distributions of the resulting nuclei, we simply used the average and standard deviation of the transfer probabilities from the models that do provide this information. As an example, the transfer matrix (expressed in terms of activity $T'_{ij} = T_{ij}\cdot P_j$) from the Geant4 INCLXX model for all three isotopes of interest is shown in Fig.~\ref{fig:transmat}. The uncertainties are calculated by propagating the statistical errors from the simulations through Eqs.~(\ref{eq:transfer}), (\ref{eq:ejection}), and (\ref{eq:implantation}). Additionally we have evaluated a 1\% systematic uncertainty on ejection and implantation of $^3$H{} and $^7$Be~due to the uncertainty in the target thicknesses.
\subsubsection{Tritium}
The model predictions for the ejected and implantated activity of tritons in CCD 3 are shown in the third and fourth columns of Table~\ref{tab:trit_pred}. One can see that depending on the model, 25\%--50\% of the tritons produced in the CCDs are ejected and there is significant implantation of tritons from the protective aluminum boxes surrounding the CCDs.
Due to the similarity of the aluminum and silicon nucleus and the fact that the reaction Q-value for triton production only differs by \SI{5.3}{MeV}, at high energies the production of tritons in aluminum is very similar to that of silicon. In Ref.~\cite{benck2002secondary}, the total triton production cross section as well as the single and double differential cross sections for neutron-induced triton ejection were found to be the same for silicon and aluminum, within the uncertainty of the measurements. This led the authors to suggest that results for aluminum, which are more complete and precise, can also be used for silicon. We show all existing measurements for neutron- and proton-induced triton production in aluminum \cite{benck2002fast, otuka2014towards, zerkin2018experimental} in Fig.~\ref{fig:al_3h_cross_sections} along with model predictions. Comparison to Fig.~\ref{fig:si_3h_cross_sections} shows that all models considered have very similar predictions for aluminum and silicon.
This similarity in triton production, as well as the similar stopping powers of aluminum and silicon, leads to a close compensation of the triton ejected from the silicon CCD with the triton implanted into the CCD from the aluminum box. If the material of the box and CCD were identical and there was sufficient material surrounding the CCD, the compensation would be exact, with no correction to the production required (ignoring attenuation of the neutron flux). In our case, the ratio of production to residual tritons is predicted to be \num{0.985 \pm 0.078}, based on the mean and RMS over all models with kinematic information, and we apply this ratio to the rest of the cross-section models.
\subsubsection{$^7$Be}
Due to the heavier nucleus, the fraction of ejected $^7$Be~nuclei is expected to be smaller than for tritons. As listed in Table~\ref{tab:ber_pred}, the Geant4 INCLXX model predicts that $\sim17\%$ of $^7$Be~produced in the silicon wafers is ejected. For the BIC and BERTINI models, the predicted production rates in silicon are roughly 400 times smaller than our measurement and within the statistics of our simulations we could only place upper limits on the fraction ejected from the wafers at roughly 30\%. We chose to use Wafer 3 for our estimation because it has the largest amount of silicon upstream of the targets, allowing for the closest compensation of the ejection through implantation. However, for $^7$Be~there is also a contribution of implantation from production in the $\sim$\num{0.5}" of air between the wafer targets, which varies between \SIrange[range-phrase = --]{0.4}{0.6}{\milli\becquerel} for the different models. Because this is significant compared to the severely underestimated production and ejection in silicon for the BERTINI and BIC models, the ratio of the production to residual activity is also greatly underestimated and we have therefore chosen to not use the BERTINI and BIC models for estimations of the $^7$Be~production rate from here onward. For all models without kinematic information we have used the ratio of production to residual $^7$Be~activity from the Geant4 INCLXX model, i.e. \num{1.008 \pm 0.046}.
\subsubsection{$^{22}$Na}
As seen in the third and fourth columns of Table~\ref{tab:sod_pred}, both the ejection and implantation fraction of $^{22}$Na~nuclei are negligible due to the large size of the residual nucleus and no correction needs to be made to the predicted production activity.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{al_vs_si_h3_crosssections_2.pdf}
\caption{Experimental measurements (data points) and model estimates (continuous lines) of the neutron-induced tritium production in aluminum. Measurements of the proton-induced cross section are also shown for reference. For direct comparison, we also show the corresponding model predictions for silicon (dashed lines) from Fig.~\ref{fig:si_3h_cross_sections}.}
\label{fig:al_3h_cross_sections}
\end{figure}
\subsection{Comparison to Experimental Measurements}
The ratio of the experimentally measured activities to the predictions of the residual activity from different models are shown in the final column of Tables~\ref{tab:trit_pred}, \ref{tab:ber_pred}, and \ref{tab:sod_pred} for $^3$H{}, $^7$Be{}, and $^{22}$Na{} respectively. For tritium, it can be seen that the predictions of the K\&K and INCL models are in fairly good agreement with the measurement, while the TALYS model overpredicts and the Geant4 BERTINI and BIC models underpredict the activity by more than a factor of two. For $^7$Be, the best agreement with the data comes from the S\&T model and the spline fit to measurements of the proton-induced cross section. We note that the proton cross sections do slightly overpredict the production from neutrons, as found in Ref.~\cite{ninomiya2011cross}, but the value is within the measurement uncertainty. For $^{22}$Na, there is good agreement between our measured activity and the predictions from the experimental measurements of the neutron-induced activity by Michel et al. \cite{michel2015excitation, hansmann2010production}, extrapolated at high energies using the TALYS model. For comparison, the use of the proton-induced production cross section (shown in Fig.~\ref{fig:si_22na_cross_sections}) leads to a value that is roughly 1.9$\times$ larger than our measured activity.
If we assume that the energy dependence of the cross-section model is correct, the ratio of the experimentally measured activity to the predicted activity is the normalization factor that must be applied to each model to match the experimental data. In the next section we will use this ratio to estimate the production rates from cosmic-ray neutrons at sea level.
\section{Cosmogenic Neutron Activation}
\label{sec:cosmogenic_rates}
The isotope production rate per unit target mass from the interaction of cosmic-ray neutrons, $P'$ [\si{\atoms\per\kg\per\second}], can be written as
\begin{linenomath*}
\begin{align}
P' = n \int \Phi(E) \cdot \sigma(E)~dE,
\end{align}
\end{linenomath*}
where $n$ is the number of target atoms per unit mass of silicon [atoms/kg], $\sigma(E)$ is the isotope production cross section [cm$^2$], $\Phi(E)$ is the cosmic-ray neutron flux [\si{\neutrons\per\cm\squared\per\second\per\MeV}], and the integral is evaluated from 1\,MeV to 10\,GeV.\footnote{The TALYS cross sections only extend up to 1 GeV \cite{koning2014extension}. We have assumed a constant extrapolation of the value at 1\,GeV for energies $>$1\,GeV.} While the cross section is not known across the entire energy range and each of the models predicts a different energy dependence, the overall normalization of each model is determined by the comparison to the measurements on the LANSCE neutron beam. The similar shapes of the LANSCE beam and the cosmic-ray neutron spectrum allow us to greatly reduce the systematic uncertainty arising from the unknown cross section.
There have been several measurements and calculations of the cosmic-ray neutron flux (see, e.g., Refs.~\cite{hess1959cosmic, armstrong1973calculations, ziegler1996terrestrial}). The intensity of the neutron flux varies with altitude, location in the geomagnetic field, and solar magnetic activity---though the spectral shape does not vary as significantly---and correction factors must be applied to calculate the appropriate flux \cite{desilets2001scaling}. The most commonly used reference spectrum for sea-level cosmic-ray neutrons is the so-called ``Gordon'' spectrum \cite{gordon2004measurement} (shown in Fig.~\ref{fig:lanscebeamenergy}), which is based on measurements at five different sites in the United States, scaled to sea level at the location of New York City during the mid-point of solar modulation. We used the parameterization given in Ref.~\cite{gordon2004measurement}, which agrees with the data to within a few percent. The spectrum uncertainties at high energies are dominated by uncertainties in the spectrometer detector response function ($<4$\% below 10 MeV and 10--15\% above 150 MeV). We have assigned an average uncertainty of 12.5\% across the entire energy range.
\begin{table}[t!]
\centering
\begin{tabular}{ccc}
\hline
Model & Predicted & Scaled \\
& cosmogenic $^3$H & cosmogenic $^3$H \\
& production rate & production rate\\
& [\si{\atoms\per\kilogram\per\dayshort}] & [\si{\atoms\per\kilogram\per\dayshort}] \\
\hline
K\&K (ACTIVIA) & \num{98 \pm 12} & \num{108 \pm 20} \\
TALYS & \num{259 \pm 33} & \num{96 \pm 18}\\
INCL++(ABLA07) & \num{106 \pm 13} & \num{114 \pm 22}\\
G4 BERTINI & \num{36.1 \pm 4.5} & \num{124 \pm 22}\\
G4 BIC & \num{42.8 \pm 5.4} & \num{102 \pm 17}\\
G4 INCLXX & \num{110 \pm 14} & \num{130 \pm 23}\\
\hline
\end{tabular}
\caption{Predicted $^3$H\ production rates (middle column) from sea-level cosmic-ray neutron interactions in silicon for different cross-section models. The final column provides our best estimate of the production rate for each model after scaling by the ratio of the measured to predicted $^3$H~activities for the LANSCE neutron beam.}
\label{tab:trit_cosmic}
\end{table}
\begin{table}[t!]
\centering
\begin{tabular}{ccc}
\hline
Model & Predicted & Scaled \\
& cosmogenic $^7$Be & cosmogenic $^7$Be \\
& production rate & production rate\\
& [\si{\atoms\per\kilogram\per\dayshort}] & [\si{\atoms\per\kilogram\per\dayshort}] \\
\hline
S\&T (ACTIVIA) & \num{8.1 \pm 1.0} & \num{8.7 \pm 1.6}\\
TALYS & \num{4.17\pm 0.52} & \num{6.2 \pm 1.3}\\
INCL++(ABLA07) & \num{2.81 \pm 0.35} & \num{8.8 \pm 1.9}\\
$^{\text{nat}}$Si(p,x)$^7$Be Spl. & \num{9.8 \pm 1.2} & \num{8.3 \pm 1.7}\\
G4 INCLXX & \num{0.411 \pm 0.052} & \num{8.4 \pm 1.7}\\
\hline
\end{tabular}
\caption{Predicted $^7$Be\ production rates (middle column) from sea-level cosmic-ray neutron interactions in silicon for different cross-section models. The final column provides our best estimate of the production rate for each model after scaling by the ratio of the measured to predicted $^7$Be~activities for the LANSCE neutron beam.}
\label{tab:ber_cosmic}
\end{table}
\begin{table}[t!]
\centering
\begin{tabular}{ccc}
\hline
Model & Predicted & Scaled \\
& cosmogenic $^{22}$Na & cosmogenic $^{22}$Na\\
& production rate & production rate\\
& [\si{\atoms\per\kilogram\per\dayshort}] & [\si{\atoms\per\kilogram\per\dayshort}] \\
\hline
S\&T (ACTIVIA) & \num{86 \pm 11} & \num{43.2 \pm 7.1}\\
TALYS & \num{60.5 \pm 7.6} &\num{43.0 \pm 6.8}\\
INCL++(ALBA07) & \num{60.0 \pm 7.5} & \num{43.1 \pm 7.2}\\
Michel-TALYS & \num{42.8 \pm 5.4} & \num{42.0 \pm 6.8}\\
G4 BERTINI & \num{28.0 \pm 3.5} & \num{43.0 \pm 7.3}\\
G4 BIC & \num{115 \pm 14} & \num{43.4 \pm 7.2}\\
G4 INCLXX & \num{116 \pm 15} & \num{43.1 \pm 7.1}\\
\hline
\end{tabular}
\caption{Predicted $^{22}$Na\ production rates (middle column) from sea-level cosmic-ray neutron interactions in silicon for different cross-section models. The final column provides our best estimate of the production rate for each model after scaling by the ratio of the measured to predicted $^{22}$Na~activities for the LANSCE neutron beam.}
\label{tab:sod_cosmic}
\end{table}
The predicted production rates per unit target mass for the cross-section models considered are shown in the second columns of Tables~\ref{tab:trit_cosmic}, ~\ref{tab:ber_cosmic}, and~\ref{tab:sod_cosmic} for $^3$H, $^7$Be, and $^{22}$Na~respectively. Scaling these values by the ratio of the measured to predicted activities for the LANSCE neutron beam, we obtain our best estimates for the neutron-induced cosmogenic production rates per unit target mass, shown in the corresponding final columns. The spread in the values for the different cross-section models is an indication of the systematic uncertainty in the extrapolation from the LANSCE beam measurement to the cosmic-ray neutron spectrum. If the LANSCE neutron-beam spectral shape was the same as that of the cosmic-ray neutrons, or if the cross-section models all agreed in shape, the central values in the final column of each table would be identical.
Our best estimate of the activation rate of tritium in silicon from cosmic-ray neutrons is \mbox{$(112 \pm 15_\text{exp} \pm 12_\text{cs} \pm 14_\text{nf})$} \si{\atomstrit\per\kg\per\day}, where the first uncertainty listed is due to experimental measurement uncertainties (represented by the average uncertainty on the ratio of the measured to predicted activities from the LANSCE beam irradiation for a specific cross-section model), the second is due to the uncertainty in the energy dependence of the cross section (calculated as the standard deviation of the scaled cosmogenic production rates of the different models), and the third is due to the uncertainty in the sea-level cosmic-ray neutron flux. Similarly, the neutron-induced cosmogenic activation rates for $^7$Be\ and $^{22}$Na\ in silicon are \mbox{$(8.1 \pm 1.3_\text{exp} \pm 1.1_\text{cs} \pm 1.0_\text{nf})$} \si{\atomsber\per\kg\per\day} and \mbox{$(43.0 \pm 4.7_\text{exp} \pm 0.4_\text{cs} \pm 5.4_\text{nf})$} \si{\atomssod\per\kg\per\day}.
\section{Activation from other particles}
\label{sec:alternate}
In addition to activity induced by fast neutrons, interactions of protons, gamma-rays, and muons also contribute to the total production rate of $^3$H, $^7$Be~and $^{22}$Na. In the following subsections we describe the methods we used to estimate the individual contributions using existing measurements and models. In some cases experimental data is very limited and we have had to rely on rough approximations based on other targets and related processes.
\subsection{Proton Induced Activity}
At sea level the flux of cosmic-ray protons is lower than that of cosmic-ray neutrons due to the attenuation effects of additional electromagnetic interactions in the atmosphere. To estimate the production rate from protons we have used the proton spectra from Ziegler \cite{ziegler1979effect, ziegler1981effect} and Diggory et.\ al.\ \cite{diggory1974momentum} (scaled by the angular distribution from the PARMA analytical model \cite{sato2016analytical} as implemented in the EXPACS software program \cite{expacs}), shown in Fig.~\ref{fig:alt_flux_comp}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{alt_flux_comparison.pdf}
\caption{Comparison of sea-level cosmic-ray fluxes of protons \cite{diggory1974momentum, ziegler1979effect, ziegler1981effect}, gamma rays \cite{expacs}, and neutrons \cite{gordon2004measurement}.}
\label{fig:alt_flux_comp}
\end{figure}
Experimental measurements of the proton-induced tritium production cross section have been made only at a few energies (see Fig.~\ref{fig:si_3h_cross_sections}). We have therefore based our estimates on the neutron cross-section models, scaled by the same factor used in Table~\ref{tab:trit_pred}. To account for possible differences between the proton- and neutron-induced cross sections, we have included a 30\% uncertainty based on the measured differences between the cross sections in aluminum (see Fig.~\ref{fig:al_3h_cross_sections}). Similar to the neutron-induced production, we have used the mean and sample standard deviation of the production rates calculated with all the different combinations of the proton spectra and cross-section models as our estimate of the central value and uncertainty, yielding a sea-level production rate from protons of \SI{10.0 \pm 4.5}{\atomstrit\per\kg\per\day}.
For $^7$Be~and $^{22}$Na, measurements of the proton cross section across the entire energy range have been made; we have used spline fits to the data with an overall uncertainty of roughly 10\% based on the experimental uncertainties (see Figs.~\ref{fig:si_7be_cross_sections}~and \ref{fig:si_22na_cross_sections}). Our best estimates for the $^7$Be~and $^{22}$Na~production rates from protons are \SI{1.14 \pm 0.14}{\atomsber\per\kg\per\day} and \SI{3.96 \pm 0.89}{\atomssod\per\kg\per\day}.
\begin{table*}[t!]
\centering
\begin{tabular}{cccc}
\hline
\vrule width 0pt height 2.2ex
Source & $^3$H~production rate & $^7$Be~production rate & $^{22}$Na~production rate \\
& [\si{\atoms\per\kilogram\per\day}] & [\si{\atoms\per\kilogram\per\day}] & [\si{\atoms\per\kilogram\per\day}] \\
\hline
Neutrons & \num{112 \pm 24} & \num{8.1 \pm 1.9} & \num{43.0 \pm 7.2}\\
Protons & \num{10.0 \pm 4.5} & \num{1.14 \pm 0.14} & \num{3.96 \pm 0.89}\\
Gamma Rays & \num{0.73 \pm 0.51} & \num{0.118 \pm 0.083} & \num{2.2 \pm 1.5}\\
Muon Capture & \num{1.57 \pm 0.92} & \num{0.09 \pm 0.09} & \num{0.48 \pm 0.11}\\
\hline
Total & \num{124 \pm 25} & \num{9.4 \pm 2.0} & \num{49.6 \pm 7.4}\\
\hline
\end{tabular}
\caption{Final estimates of the radioisotope production rates in silicon exposed to cosmogenic particles at sea level.}
\label{tab:final_cosmic_prod}
\end{table*}
\subsection{Gamma Ray Induced Activity}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{si_gamma_crosssections.pdf}
\caption{Estimated photonuclear cross-section models for production of $^3$H, $^7$Be, and $^{22}$Na. The dashed lines indicate the original models from TALYS while the solid lines indicate the models scaled to match yield measurements made with bremsstrahlung radiation \cite{matsumura2000target, currie1970photonuclear}.}
\label{fig:gamma_cs}
\end{figure}
The flux of high-energy gamma rays at the Earth's surface was obtained using the PARMA analytical model \cite{sato2016analytical} as implemented in the EXPACS software program \cite{expacs}. Similar to the neutron spectrum, we used New York City as our reference location for the gamma spectrum, which is shown in Fig.~\ref{fig:alt_flux_comp}.
Photonuclear yields of $^7$Be~and $^{22}$Na~in silicon have been measured using bremsstrahlung beams with endpoints ($E_0$) up to \SI{1}{\giga\eV} \cite{matsumura2000target}. We are not aware of any measurements of photonuclear tritium production in silicon, though there is a measurement in aluminum with $E_0 =$ \SI{90}{\MeV} \cite{currie1970photonuclear} which we assume to be the same as for silicon. The yields, $Y(E_0)$, are typically quoted in terms of the cross section per equivalent quanta (eq.q), defined as
\begin{align}
Y(E_0) = \frac{\displaystyle\int_0^{E_0} \sigma(k)N(E_0,k)dk}{\displaystyle \frac{1}{E_0}\int_0^{E_0} kN(E_0,k)dk}
\end{align}
where $\sigma(k)$ is the cross section as a function of photon energy $k$, and $N(E_0, k)$ is the bremsstrahlung energy spectrum.
To obtain an estimate for $\sigma(k)$, we assume a $1/k$ energy dependence for $N(E_0, k)$~\cite{tesch1971accuracy} and scale the TALYS photonuclear cross section models to match the measured yields of \SI{72}{\micro\barn \per \eqquanta} at $E_0 =$ \SI{90}{\MeV} for tritium and \SI{227}{\micro\barn \per \eqquanta} and \SI{992}{\micro\barn \per \eqquanta} at $E_0 =$ \SI{1000}{\MeV} for $^7$Be\ and $^{22}$Na , respectively (see Fig.~\ref{fig:gamma_cs}).
This corresponds to estimated photonuclear production rates of \SI{0.73}{\atomstrit\per\kilogram\per\day}, \SI{0.12}{\atomsber\per\kilogram\per\day}, and \SI{2.2}{\atomssod\per\kilogram\per\day}. Given the large uncertainties in the measured yields, the cross-section spectral shape, and the bremsstrahlung spectrum, we assume a $\sim 70\%$ overall uncertainty on these rates.
\subsection{Muon Capture Induced Activity}
The production rate of a specific isotope $X$ from sea-level cosmogenic muon capture can be expressed as
\begin{align}
P_\mu(X) = R_0 \cdot \frac{\lambda_c\text{(Si)}}{Q\lambda_d + \lambda_c\text{(Si)}}\cdot f_\text{Si}(X)
\end{align}
where $R_0 = \SI{484 \pm 52}{\muons\per\kg\per\day}$ is the rate of stopped negative muons at sea level at geomagnetic latitudes of about \SI{40}{\degree} \cite{charalambus1971nuclear}, the middle term is the fraction of muons that capture on silicon (as opposed to decaying) with the capture rate on silicon $\lambda_c$(Si) = \SI{8.712 \pm 0.018 E5}{\per\sec} \cite{suzuki1987total}, the decay rate of muons $\lambda_d$ = \SI{4.552E5}{\per\sec} \cite{tanabashi2018m}, and the Huff correction factor $Q = 0.992$ for bound-state decay \cite{measday2001nuclear}. The final term, $f_\text{Si}(X)$, is the fraction of muon captures on silicon that produce isotope $X$.
For $^{28}$Si the fraction of muon captures with charged particles emitted has been measured to be \SI{15 \pm 2}{\percent} with theoretical estimates \cite{lifshitz1980nuclear} predicting the composition to be dominated by protons ($f_\text{Si}(^1$H) = \SI{8.8}{\percent}), alphas ($f_\text{Si}(^4$He) = \SI{3.4}{\percent}), and deuterons ($f_\text{Si}(^2$H) = \SI{2.2}{\percent}). The total fraction of muon captures that produce tritons has not been experimentally measured\footnote{A direct measurement of triton production from muon capture in silicon was performed by the \href{http://muon.npl.washington.edu/exp/AlCap/index.html}{AlCap
Collaboration} and a publication is in preparation. }, but a lower limit can be set at \SI{7 \pm 4 e-3}{\percent} from an experimental measurement of tritons emitted above 24 MeV \cite{budyashov1971charged}.
Recent measurements of the emission fractions of protons and deuterons following muon capture on aluminum have found values of $f_\text{Al}(^1$H) = \SI{4.5 \pm 0.3}{\percent} and $f_\text{Al}(^2$H) = \SI{1.8 \pm 0.2}{\percent} \cite{gaponenko2020charged}, and those same data can be used to calculate a rough triton emission fraction of $f_\text{Al}(^3$H) = \SI{0.4}{\percent} \cite{gaponenkopersonal}. If one assumes the same triton kinetic energy distribution in silicon as estimated for aluminum \cite{gaponenko2020charged} and uses it to scale the value measured above 24 MeV, one obtains a triton production estimate of $f_\text{Si}(^3$H) = \SI{0.49 \pm 0.28}{\percent}. The production rate of tritons from muon capture is then estimated to be \SI{1.57 \pm 0.92}{\atomstrit\per\kg\per\day}.
The fraction of muon captures that produce $^{22}$Na~has been measured at $f_\text{Si}$($^{22}$Na) = \SI{0.15 \pm 0.03}{\percent} \cite{heisinger2002production}, corresponding to a production rate from muon captures of \SI{0.48 \pm 0.11}{\atomssod\per\kg\per\day}. To our knowledge there have been no measurements of the production of $^7$Be~through muon capture on silicon. We assume the ratio of $^7$Be~to $^{22}$Na~production is the same for muon capture as it is for the neutron production rates calculated earlier, with roughly \SI{100}{\percent} uncertainty, resulting in an estimated production rate from muon captures of \SI{0.09 \pm 0.09}{\atomsber\per\kg\per\day}.
\section{Discussion}
\label{sec:discussion}
The final estimates for the total cosmogenic production rates of $^3$H, $^7$Be, and $^{22}$Na~at sea level are listed in Table~\ref{tab:final_cosmic_prod}. These rates can be scaled by the known variations of particle flux with altitude or depth, location in the geomagnetic field, and solar activity, to obtain the total expected activity in silicon-based detectors for specific fabrication, transportation, and storage scenarios. The production rate at sea level is dominated by neutron-induced interactions, but for shallow underground locations muon capture may be the dominant production mechanism. For estimates of the tritium background, implantation of tritons generated in surrounding materials and ejection of tritons from thin silicon targets should also be taken into account.
Tritium is the main cosmogenic background of concern for silicon-based dark matter detectors. At low energies, 0--5\,keV,
the estimated production rate corresponds to an activity of roughly \SI{0.002} {\decays \per \keV \per \kg \per \day} per day of sea-level exposure. This places strong restrictions on the fabrication and transportation of silicon detectors for next-generation dark matter experiments. In order to mitigate the tritium background we are currently exploring the possibility of using low-temperature baking to remove implanted tritium from fabricated silicon devices.
Aside from silicon-based dark matter detectors, silicon is also widely used in sensors and electronics for rare-event searches due to the widespread use of silicon in the semiconductor industry and the availability of high-purity silicon. The relative contributions of $^3$H, $^7$Be, and $^{22}$Na~to the overall background rate of an experiment depends not only on the activation rate but also on the location of these components within the detector and the specific energy region of interest. The cosmogenic production rates determined here can be used to calculate experiment-specific background contributions and shielding requirements for all silicon-based materials.
\section{Acknowledgements}
We are grateful to John Amsbaugh and Seth Ferrara for designing the beamline holders, Larry Rodriguez for assistance during the beam time, and Brian Glasgow and Allan Myers for help with the gamma counting. We would also like to thank Alan Robinson and Andrei Gaponenko for useful discussions on production mechanisms from other particles. This work was performed, in part, at the Los Alamos Neutron Science Center (LANSCE), a NNSA User Facility operated for the U.S.\ Department of Energy (DOE) by Los Alamos National Laboratory (Contract 89233218CNA000001) and we thank John O'Donnell for his assistance with the beam exposure and data acquisition.
Pacific Northwest National Laboratory (PNNL) is operated by Battelle Memorial Institute for the U.S.\ Department of Energy (DOE) under Contract No.\ DE-AC05-76RL01830;
the experimental approach was originally developed under the Nuclear-physics, Particle-physics, Astrophysics, and Cosmology (NPAC) Initiative, a Laboratory Directed Research and Development (LDRD) effort at PNNL, while the application to CCDs was performed under the DOE Office of High Energy Physics' Advanced Technology R\&D subprogram. We acknowledge the financial support from National Science Foundation through Grant No.\ NSF PHY-1806974 and from the Kavli Institute for Cosmological Physics at The University of Chicago through an endowment from the Kavli Foundation. The CCD development work was supported in part by the Director, Office of Science, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
| 2024-02-18T23:39:40.337Z | 2020-11-24T02:21:07.000Z | algebraic_stack_train_0000 | 22 | 14,429 |
|
proofpile-arXiv_065-241 | \section{Motivation}
Chromium is considered as the archetypical itinerant antiferromagnet~\cite{1988_Fawcett_RevModPhys, 1994_Fawcett_RevModPhys}. Interestingly, it shares its body-centered cubic crystal structure $Im\overline{3}m$ with the archetypical itinerant ferromagnet $\alpha$-iron and, at melting temperature, all compositions Fe$_{x}$Cr$_{1-x}$~\cite{2010_Okamoto_Book}. As a result, the Cr--Fe system offers the possibility to study the interplay of two fundamental forms of magnetic order in the same crystallographic environment.
Chromium exhibits transverse spin-density wave order below a N\'{e}el temperature $T_{\mathrm{N}} = 311$~K and longitudinal spin-density wave order below $T_{\mathrm{SF}} = 123$~K~\cite{1988_Fawcett_RevModPhys}. Under substitutional doping with iron, the longitudinal spin-density wave order becomes commensurate at $x = 0.02$. For $0.04 < x$, only commensurate antiferromagnetic order is observed~\cite{1967_Ishikawa_JPhysSocJpn, 1980_Babic_JPhysChemSolids, 1983_Burke_JPhysFMetPhys_I}. The N\'{e}el temperature decreases at first linearly with increasing $x$ and vanishes around $x \approx 0.15$~\cite{1967_Ishikawa_JPhysSocJpn, 1976_Suzuki_JPhysSocJpn, 1978_Burke_JPhysFMetPhys, 1980_Babic_JPhysChemSolids, 1983_Burke_JPhysFMetPhys_I}. Increasing $x$ further, a putative lack of long-range magnetic order~\cite{1978_Burke_JPhysFMetPhys} is followed by the onset of ferromagnetic order at $x \approx 0.18$ with a monotonic increase of the Curie temperature up to $T_{\mathrm{C}} = 1041$~K in pure $\alpha$-iron~\cite{1963_Nevitt_JApplPhys, 1975_Loegel_JPhysFMetPhys, 1980_Fincher_PhysRevLett, 1981_Shapiro_PhysRevB, 1983_Burke_JPhysFMetPhys_II, 1983_Burke_JPhysFMetPhys_III}.
The suppression of magnetic order is reminiscent of quantum critical systems under pressure~\cite{2001_Stewart_RevModPhys, 2007_Lohneysen_RevModPhys, 2008_Broun_NatPhys}, where substitutional doping of chromium with iron decreases the unit cell volume. In comparison to stoichiometric systems tuned by hydrostatic pressure, however, disorder and local strain are expected to play a crucial role in Fe$_{x}$Cr$_{1-x}$. This conjecture is consistent with reports on superparamagnetic behavior for $0.20 \leq x \leq 0.29$~\cite{1975_Loegel_JPhysFMetPhys}, mictomagnetic behavior~\footnote{In mictomagnetic materials, the virgin magnetic curves recorded in magnetization measurements as a function of field lie outside of the hysteresis loops recorded when starting from high field~\cite{1976_Shull_SolidStateCommunications}.} gradually evolving towards ferromagnetism for $0.09 \leq x \leq 0.23$~\cite{1975_Shull_AIPConferenceProceedings}, and spin-glass behavior for $0.14 \leq x \leq 0.19$~\cite{1979_Strom-Olsen_JPhysFMetPhys, 1980_Babic_JPhysChemSolids, 1981_Shapiro_PhysRevB, 1983_Burke_JPhysFMetPhys_I, 1983_Burke_JPhysFMetPhys_II, 1983_Burke_JPhysFMetPhys_III}.
Despite the rather unique combination of properties, notably a metallic spin glass emerging at the border of both itinerant antiferromagnetic and ferromagnetic order, comprehensive studies addressing the magnetic properties of Fe$_{x}$Cr$_{1-x}$ in the concentration range of putative quantum criticality are lacking. In particular, a classification of the spin-glass regime, to the best of our knowledge, has not been addressed before.
Here, we report a study of polycrystalline samples of Fe$_{x}$Cr$_{1-x}$ covering the concentration range $0.05 \leq x \leq 0.30$, i.e., from antiferromagnetic doped chromium well into the ferromagnetically ordered state of doped iron. The compositional phase diagram inferred from magnetization and ac susceptibility measurements is in agreement with previous reports~\cite{1983_Burke_JPhysFMetPhys_I, 1983_Burke_JPhysFMetPhys_II, 1983_Burke_JPhysFMetPhys_III}. As the perhaps most notable new observation, we identify a precursor phenomenon preceding the onset of spin-glass behavior in the imaginary part of the ac susceptibility. For the spin-glass state, analysis of ac susceptibility data recorded at different excitation frequencies by means of the Mydosh parameter, power-law fits, and a Vogel--Fulcher ansatz establishes a crossover from cluster-glass to superparamagnetic behavior as a function of increasing $x$. Microscopic evidence for this evolution is provided by neutron depolarization, indicating an increase of the size of ferromagnetic clusters with $x$.
Our paper is organized as follows. In Sec.~\ref{sec:methods}, the preparation of the samples and their metallurgical characterization by means of x-ray powder diffraction is reported. In addition, experimental details are briefly described. Providing a first point of reference, the presentation of the experimental results starts in Sec.~\ref{sec:results} with the compositional phase diagram as inferred in our study, before turning to a detailed description of the ac susceptibility and magnetization data. Next, neutron depolarization data are presented, allowing to extract the size of ferromagnetically ordered clusters from exponential fits. Exemplary data on the specific heat, electrical resistivity, and high-field magnetization for $x = 0.15$ complete this section. In Sec.~\ref{sec:discussion}, information on the nature of the spin-glass behavior in Fe$_{x}$Cr$_{1-x}$ and its evolution under increasing $x$ is inferred from an analysis of ac susceptibility data recorded at different excitation frequencies. Finally, in Sec.~\ref{sec:conclusion} the central findings of this study are summarized.
\section{Experimental methods}
\label{sec:methods}
Polycrystalline samples of Fe$_{x}$Cr$_{1-x}$ for $0.05 \leq x \leq 0.30$ ($x = 0.05$, 0.10, 0.15, 0.16, 0.17, 0.18, 0.18, 0.19, 0.20, 0.21, 0.22, 0.25, 0.30) were prepared from iron (4N) and chromium (5N) pieces by means of radio-frequency induction melting in a bespoke high-purity furnace~\cite{2016_Bauer_RevSciInstrum}. No losses in weight or signatures of evaporation were observed. In turn, the composition is denoted in terms of the weighed-in amounts of starting material. Prior to the synthesis, the furnace was pumped to ultra-high vacuum and subsequently flooded with 1.4~bar of argon (6N) treated by a point-of-use gas purifier yielding a nominal purity of 9N. For each sample, the starting elements were melted in a water-cooled Hukin crucible and the resulting specimen was kept molten for about 10~min to promote homogenization. Finally, the sample was quenched to room temperature. With this approach, the imminent exsolution of the compound into two phases upon cooling was prevented, as suggested by the binary phase diagram of the Fe--Cr system reported in Ref.~\cite{2010_Okamoto_Book}. From the resulting ingots samples were cut with a diamond wire saw.
\begin{figure}
\includegraphics[width=1.0\linewidth]{figure1}
\caption{\label{fig:1}X-ray powder diffraction data of Fe$_{x}$Cr$_{1-x}$. (a)~Diffraction pattern for $x = 0.15$. The Rietveld refinement (red curve) is in excellent agreement with the experimental data and confirms the $Im\overline{3}m$ structure. (b)~Diffraction pattern around the (011) peak for all concentrations studied. For clarity, the intensities are normalized and curves are offset by 0.1. Inset: Linear decrease of the lattice constant $a$ with increasing $x$. The solid gray line represents a guide to the eye.}
\end{figure}
Powder was prepared of a small piece of each ingot using an agate mortar. X-ray powder diffraction at room temperature was carried out on a Huber G670 diffractometer using a Guinier geometry. Fig.~\ref{fig:1}(a) shows the diffraction pattern for $x = 0.15$, representing typical data. A Rietveld refinement based on the $Im\overline{3}m$ structure yields a lattice constant $a = 2.883$~\AA.
Refinement and experimental data are in excellent agreement, indicating a high structural quality and homogeneity of the polycrystalline samples. With increasing $x$, the diffraction peaks shift to larger angles, as shown for the (011) peak in Fig.~\ref{fig:1}(b), consistent with a linear decrease of the lattice constant in accordance with Vegard's law.
Measurements of the magnetic properties and neutron depolarization were carried out on thin discs with a thickness of ${\sim}0.5$~mm and a diameter of ${\sim}10$~mm. Specific heat and electrical transport for $x = 0.15$ were measured on a cube of 2~mm edge length and a platelet of dimensions $5\times2\times0.5~\textrm{mm}^{3}$, respectively.
The magnetic properties, the specific heat, and the electrical resistivity were measured in a Quantum Design physical properties measurement system. The magnetization was measured by means of an extraction technique. If not stated otherwise, the ac susceptibility was measured at an excitation amplitude of 0.1~mT and an excitation frequency of 1~kHz. Additional ac susceptibility data for the analysis of the spin-glass behavior were recorded at frequencies ranging from 10~Hz to 10~kHz. The specific heat was measured using a quasi-adiabatic large-pulse technique with heat pulses of about 30\% of the current temperature~\cite{2013_Bauer_PhysRevLett}. For the measurements of the electrical resistivity the samples were contacted in a four-terminal configuration and a bespoke setup was used based on a lock-in technique at an excitation amplitude of 1~mA and an excitation frequency of 22.08~Hz. Magnetic field and current were applied perpendicular to each other, corresponding to the transverse magneto-resistance.
Neutron depolarization measurements were carried out at the instrument ANTARES~\cite{2015_Schulz_JLarge-ScaleResFacilJLSRF} at the Heinz Maier-Leibniz Zentrum~(MLZ). The incoming neutron beam had a wavelength $\lambda = 4.13$~\AA\ and a wavelength spread $\Delta\lambda / \lambda = 10\%$. It was polarized using V-cavity supermirrors. The beam was transmitted through the sample and its polarization analyzed using a second polarizing V-cavity. While nonmagnetic samples do not affect the polarization of the neutron beam, the presence of ferromagnetic domains in general results in a precession of the neutron spins. In turn, the transmitted polarization with respect to the polarization axis of the incoming beam is reduced. This effect is referred to as neutron depolarization. Low temperatures and magnetic fields for this experiment were provided by a closed-cycle refrigerator and water-cooled Helmholtz coils, respectively. A small guide field of 0.5~mT was generated by means of permanent magnets. For further information on the neutron depolarization setup, we refer to Refs.~\cite{2015_Schmakat_PhD, 2017_Seifert_JPhysConfSer, 2019_Jorba_JMagnMagnMater}.
All data shown as a function of temperature in this paper were recorded at a fixed magnetic field under increasing temperature. Depending on how the sample was cooled to 2~K prior to the measurement, three temperature versus field histories are distinguished. The sample was either cooled (i)~in zero magnetic field (zero-field cooling, zfc), (ii)~with the field at the value applied during the measurement (field cooling, fc), or (iii)~in a field of 250~mT (high-field cooling, hfc). For the magnetization data as a function of field, the sample was cooled in zero field. Subsequently, data were recorded during the initial increase of the field to $+250$~mT corresponding to a magnetic virgin curve, followed by a decrease to $-250$~mT, and a final increase back to $+250$~mT.
\section{Experimental results}
\label{sec:results}
\subsection{Phase diagram and bulk magnetic properties}
\begin{figure}
\includegraphics[width=1.0\linewidth]{figure2}
\caption{\label{fig:2}Zero-field composition--temperature phase diagram of Fe$_{x}$Cr$_{1-x}$. Data inferred from ac susceptibility, $\chi_{\mathrm{ac}}$, and neutron depolarization are combined with data reported by Burke and coworkers~\cite{1983_Burke_JPhysFMetPhys_I, 1983_Burke_JPhysFMetPhys_II, 1983_Burke_JPhysFMetPhys_III}. Paramagnetic~(PM), antiferromagnetic~(AFM), ferromagnetic~(FM), and spin-glass~(SG) regimes are distinguished. A precursor phenomenon is observed above the dome of spin-glass behavior (purple line). (a)~Overview. (b) Close-up view of the regime of spin-glass behavior as marked by the dashed box in panel (a).}
\end{figure}
The presentation of the experimental results starts with the compositional phase diagram of Fe$_{x}$Cr$_{1-x}$, illustrating central results of our study. An overview of the entire concentration range studied, $0.05 \leq x \leq 0.30$, and a close-up view around the dome of spin-glass behavior are shown in Figs.~\ref{fig:2}(a) and \ref{fig:2}(b), respectively. Characteristic temperatures inferred in this study are complemented by values reported by Burke and coworkers~\cite{1983_Burke_JPhysFMetPhys_I, 1983_Burke_JPhysFMetPhys_II, 1983_Burke_JPhysFMetPhys_III}, in good agreement with our results. Comparing the different physical properties in our study, we find that the imaginary part of the ac susceptibility displays the most pronounced signatures at the various phase transitions and crossovers. Therefore, the imaginary part was used to define the characteristic temperatures as discussed in the following. The same values are then marked in the different physical properties to highlight the consistency with alternative definitions of the characteristic temperatures based on these properties.
Four regimes may be distinguished in the phase diagram, namely paramagnetism at high temperatures (PM, no shading), antiferromagnetic order for small values of $x$ (AFM, green shading), ferromagnetic order for larger values of $x$ (FM, blue shading), and spin-glass behavior at low temperatures (SG, orange shading). We note that faint signatures reminiscent of those attributed to the onset of ferromagnetic order are observed in the susceptibility and neutron depolarization for $0.15 \leq x \leq 0.18$ (light blue shading). In addition, a distinct precursor phenomenon preceding the spin-glass behavior is observed at the temperature $T_{\mathrm{X}}$ (purple line) across a wide concentration range. Before elaborating on the underlying experimental data, we briefly summarize the key characteristics of the different regimes.
We attribute the onset of antiferromagnetic order below the N\'{e}el temperature $T_{\mathrm{N}}$ for $x = 0.05$ and $x = 0.10$ to a sharp kink in the imaginary part of the ac susceptibility, where values of $T_{\mathrm{N}}$ are consistent with previous reports~\cite{1978_Burke_JPhysFMetPhys, 1983_Burke_JPhysFMetPhys_I}. As may be expected, the transition is not sensitive to changes of the magnetic field, excitation frequency, or cooling history. The absolute value of the magnetization is small and it increases essentially linearly as a function of field in the parameter range studied.
We identify the emergence of ferromagnetic order below the Curie temperature $T_{\mathrm{C}}$ for $0.18 \leq x$ from a maximum in the imaginary part of the ac susceptibility that is suppressed in small magnetic fields of a few millitesla. This interpretation is corroborated by the onset of neutron depolarization. The transition is not sensitive to changes of the excitation frequency or cooling history. The magnetic field dependence of the magnetization exhibits a characteristic S-shape with almost vanishing hysteresis, reaching quasi-saturation at small fields. Both characteristics are expected for a soft ferromagnetic material such as iron. For $0.15 \leq x \leq 0.18$, faint signatures reminiscent of those observed for $0.18 \leq x$, such as a small shoulder instead of a maximum in the imaginary part of the ac susceptibility, are interpreted in terms of an incipient onset of ferromagnetic order.
We identify reentrant spin-glass behavior below a freezing temperature $T_{\mathrm{g}}$ for $0.10 \leq x \leq 0.25$ from a pronounced maximum in the imaginary part of the ac susceptibility that is suppressed at intermediate magnetic fields of the order of 50~mT. The transition shifts to lower temperatures with increasing excitation frequency, representing a hallmark of spin glasses. Further key indications for spin-glass behavior below $T_{\mathrm{g}}$ are a branching between different cooling histories in the temperature dependence of the magnetization and neutron depolarization as well as mictomagnetic behavior in the field dependence of the magnetization, i.e., the virgin magnetic curve lies outside the hysteresis loop obtained when starting from high magnetic field.
In addition, we identify a precursor phenomenon preceding the onset of spin-glass behavior at a temperature $T_{\mathrm{X}}$ based on a maximum in the imaginary part of the ac susceptibility that is suppressed in small magnetic fields reminiscent of the ferromagnetic transition. With increasing excitation frequency the maximum shifts to lower temperatures, however at a smaller rate than the freezing temperature $T_{\mathrm{g}}$. Interestingly, the magnetization and neutron depolarization exhibit no signatures at $T_{\mathrm{X}}$.
\subsection{Zero-field ac susceptibility}
\begin{figure}
\includegraphics[width=1.0\linewidth]{figure3}
\caption{\label{fig:3}Zero-field ac susceptibility as a function of temperature for all samples studied. For each concentration, real part (Re\,$\chi_{\mathrm{ac}}$, left column) and imaginary part (Im\,$\chi_{\mathrm{ac}}$, right column) of the susceptibility are shown. Note the logarithmic temperature scale and the increasing scale on the ordinate with increasing $x$. Triangles mark temperatures associated with the onset of antiferromagnetic order at $T_{\mathrm{N}}$ (green), spin-glass behavior at $T_{\mathrm{g}}$ (red), ferromagnetic order at $T_{\mathrm{C}}$ (blue), and the precursor phenomenon at $T_{\mathrm{X}}$ (purple). The corresponding values are inferred from Im\,$\chi_{\mathrm{ac}}$, see text for details.}
\end{figure}
The real and imaginary parts of the zero-field ac susceptibility on a logarithmic temperature scale are shown in Fig.~\ref{fig:3} for each sample studied. Characteristic temperatures are inferred from the imaginary part and marked by colored triangles in both quantities. While the identification of the underlying transitions and crossovers will be justified further in terms of the dependence of the signatures on magnetic field, excitation frequency, and history, as elaborated below, the corresponding temperatures are referred to as $T_{\mathrm{N}}$, $T_{\mathrm{C}}$, $T_{\mathrm{g}}$, and $T_{\mathrm{X}}$ already in the following.
For small iron concentrations, such as $x = 0.05$ shown in Fig.~\ref{fig:3}(a), the real part is small and essentially featureless, with exception of an increase at low temperatures that may be attributed to the presence of ferromagnetic impurities, i.e., a so-called Curie tail~\cite{1972_DiSalvo_PhysRevB, 2014_Bauer_PhysRevB}. The imaginary part is also small but displays a kink at the N\'{e}el temperature $T_{\mathrm{N}}$. In metallic specimens, such as Fe$_{x}$Cr$_{1-x}$, part of the dissipation detected via the imaginary part of the ac susceptibility arises from the excitation of eddy currents at the surface of the sample. Eddy current losses scale with the resistivity~\cite{1998_Jackson_Book, 1992_Samarappuli_PhysicaCSuperconductivity} and in turn the kink at $T_{\mathrm{N}}$ reflects the distinct change of the electrical resistivity at the onset of long-range antiferromagnetic order.
When increasing the iron concentration to $x = 0.10$, as shown in Fig.~\ref{fig:3}(b), both the real and imaginary parts increase by one order of magnitude. Starting at $x = 0.10$, a broad maximum may be observed in the real part that indicates an onset of magnetic correlations where the lack of further fine structure renders the extraction of more detailed information impossible. In contrast, the imaginary part exhibits several distinct signatures that allow, in combination with data presented below, to infer the phase diagram shown in Fig.~\ref{fig:2}. For $x = 0.10$, in addition to the kink at $T_{\mathrm{N}}$ a maximum may be observed at 3~K which we attribute to the spin freezing at $T_{\mathrm{g}}$.
Further increasing the iron concentration to $x = 0.15$, as shown in Fig.~\ref{fig:3}(c), results again in an increase of both the real and imaginary parts by one order of magnitude. The broad maximum in the real part shifts to slightly larger temperatures. In the imaginary part, two distinct maxima are resolved, accompanied by a shoulder at their high-temperature side. From low to high temperatures, these signatures may be attributed to $T_{\mathrm{g}}$, $T_{\mathrm{X}}$, and a potential onset of ferromagnetism at $T_{\mathrm{C}}$. No signatures related to antiferromagnetism may be discerned. For $x = 0.16$ and 0.17, shown in Figs.~\ref{fig:3}(d) and \ref{fig:3}(e), both the real and imaginary part remain qualitatively unchanged while their absolute values increase further. The characteristic temperatures shift slightly to larger values.
For $x = 0.18$, 0.19, 0.20, 0.21, and 0.22, shown in Figs.~\ref{fig:3}(f)--\ref{fig:3}(j), the size of the real and imaginary parts of the susceptibility remains essentially unchanged. The real part is best described in terms of a broad maximum that becomes increasingly asymmetric as the low-temperature extrapolation of the susceptibility increases with $x$. In the imaginary part, the signature ascribed to the onset of ferromagnetic order at $T_{\mathrm{C}}$ at larger concentrations develops into a clear maximum, overlapping with the maximum at $T_{\mathrm{X}}$ up to $x = 0.20$. For $x = 0.21$ and $x = 0.22$, three well-separated maxima may be attributed to the characteristic temperatures $T_{\mathrm{g}}$, $T_{\mathrm{X}}$, and $T_{\mathrm{C}}$. While both $T_{\mathrm{g}}$ and $T_{\mathrm{X}}$ stay almost constant with increasing $x$, $T_{\mathrm{C}}$ distinctly shifts to higher temperatures.
For $x = 0.25$, shown in Fig.~\ref{fig:3}(k), the signature attributed to $T_{\mathrm{X}}$ has vanished while $T_{\mathrm{g}}$ is suppressed to about 5~K. For $x = 0.30$, shown in Fig.~\ref{fig:3}(l), only the ferromagnetic transition at $T_{\mathrm{C}}$ remains and the susceptibility is essentially constant below $T_{\mathrm{C}}$. Note that the suppression of spin-glass behavior around $x = 0.25$ coincides with the percolation limit of 24.3\% in the crystal structure $Im\overline{3}m$, i.e., the limit above which long-range magnetic order is expected in spin-glass systems~\cite{1978_Mydosh_JournalofMagnetismandMagneticMaterials}. Table~\ref{tab:1} summarizes the characteristic temperatures for all samples studied, including an estimate of the associated errors.
\subsection{Magnetization and ac susceptibility under applied magnetic fields}
\begin{figure*}
\includegraphics[width=1.0\linewidth]{figure4}
\caption{\label{fig:4}Magnetization and ac susceptibility in magnetic fields up to 250~mT for selected concentrations (increasing from top to bottom). Triangles mark the temperatures $T_{\mathrm{N}}$ (green), $T_{\mathrm{g}}$ (red), $T_{\mathrm{C}}$ (blue), and $T_{\mathrm{X}}$ (purple). The values shown in all panels correspond to those inferred from Im\,$\chi_{\mathrm{ac}}$ in zero field. \mbox{(a1)--(f1)}~Real part of the ac susceptibility, Re\,$\chi_{\mathrm{ac}}$, as a function of temperature on a logarithmic scale for different magnetic fields. \mbox{(a2)--(f2)}~Imaginary part of the ac susceptibility, Im\,$\chi_{\mathrm{ac}}$. \mbox{(a3)--(f3)}~Magnetization for three different field histories, namely high-field cooling~(hfc), field cooling (fc), and zero-field cooling (zfc). \mbox{(a4)--(f4)}~Magnetization as a function of field at a temperature of 2~K after initial zero-field cooling. Arrows indicate the sweep directions. The scales of the ordinates for all quantities increase from top to bottom.}
\end{figure*}
In order to justify further the relationship of the signatures in the ac susceptibility with the different phases, their evolution under increasing magnetic field up to 250~mT and their dependence on the cooling history are illustrated in Fig.~\ref{fig:4}. For selected values of $x$, the temperature dependences of the real part of the ac susceptibility, the imaginary part of the ac susceptibility, and the magnetization, shown in the first three columns, are complemented by the magnetic field dependence of the magnetization at low temperature, $T = 2$~K, shown in the fourth column.
For small iron concentrations, such as $x = 0.05$ shown in Figs.~\ref{fig:4}(a1)--\ref{fig:4}(a4), both Re\,$\chi_{\mathrm{ac}}$ and Im\,$\chi_{\mathrm{ac}}$ remain qualitatively unchanged up to the highest fields studied. The associated stability of the transition at $T_{\mathrm{N}}$ under magnetic field represents a key characteristic of itinerant antiferromagnetism, which is also observed in pure chromium. Consistent with this behavior, the magnetization is small and increases essentially linearly in the field range studied. No dependence on the cooling history is observed.
For intermediate iron concentrations, such as $x = 0.15$, $x = 0.17$, and $x = 0.18$ shown in Figs.~\ref{fig:4}(b1) to \ref{fig:4}(d4), the broad maximum in Re\,$\chi_{\mathrm{ac}}$ is suppressed under increasing field. Akin to the situation in zero field, the evolution of the different characteristic temperatures is tracked in Im\,$\chi_{\mathrm{ac}}$. Here, the signatures associated with $T_{\mathrm{X}}$ and $T_{\mathrm{C}}$ proof to be highly sensitive to magnetic fields and are suppressed already above about 2~mT. The maximum associated with the spin freezing at $T_{\mathrm{g}}$ is suppressed at higher field values.
In the magnetization as a function of temperature, shown in Figs.~\ref{fig:4}(b3) to \ref{fig:4}(d3), a branching between different cooling histories may be observed below $T_{\mathrm{g}}$. Compared to data recorded after field cooling (fc), for which the temperature dependence of the magnetization is essentially featureless at $T_{\mathrm{g}}$, the magnetization at low temperatures is reduced for data recorded after zero-field cooling (zfc) and enhanced for data recorded after high-field cooling (hfc). Such a history dependence is typical for spin glasses~\cite{2015_Mydosh_RepProgPhys}, but also observed in materials where the orientation and population of domains with a net magnetic moment plays a role, such as conventional ferromagnets.
The spin-glass character below $T_{\mathrm{g}}$ is corroborated by the field dependence of the magnetization shown in Figs.~\ref{fig:4}(b4) to \ref{fig:4}(d4), which is perfectly consistent with the temperature dependence. Most notably, in the spin-glass regime at low temperatures, mictomagnetic behavior is observed, i.e., the magnetization of the magnetic virgin state obtained after initial zero-field cooling (red curve) is partly outside the hysteresis loop obtained when starting from the field-polarized state at large fields (blue curves)~\cite{1976_Shull_SolidStateCommunications}. This peculiar behavior is not observed in ferromagnets and represents a hallmark of spin glasses~\cite{1978_Mydosh_JournalofMagnetismandMagneticMaterials}.
For slightly larger iron concentrations, such as $x = 0.22$ shown in Figs.~\ref{fig:4}(e1) to \ref{fig:4}(e4), three maxima at $T_{\mathrm{g}}$, $T_{\mathrm{X}}$, and $T_{\mathrm{C}}$ are clearly separated. With increasing field, first the high-temperature maximum associated with $T_{\mathrm{C}}$ is suppressed, followed by the maxima at $T_{\mathrm{X}}$ and $T_{\mathrm{g}}$. The hysteresis loop at low temperatures is narrower, becoming akin to that of a conventional soft ferromagnet. For large iron concentrations, such as $x = 0.30$ shown in Figs.~\ref{fig:4}(f1) to \ref{fig:4}(f4), the evolution of Re\,$\chi_{\mathrm{ac}}$, Im\,$\chi_{\mathrm{ac}}$, and the magnetization as a function of magnetic field consistently corresponds to that of a conventional soft ferromagnet with a Curie temperature $T_{\mathrm{C}}$ of more than 200~K. For the ferromagnetic state observed here, all domains are aligned in fields exceeding ${\sim}50$~mT.
\begin{table}
\caption{\label{tab:1}Summary of the characteristic temperatures in Fe$_{x}$Cr$_{1-x}$ as inferred from the imaginary part of the ac susceptibility and neutron depolarization data. We distinguish the N\'{e}el temperature $T_{\mathrm{N}}$, the Curie temperature $T_{\mathrm{C}}$, the spin freezing temperature $T_{\mathrm{g}}$, and the precursor phenomenon at $T_{\mathrm{X}}$. Temperatures inferred from neutron depolarization data are denoted with the superscript `D'. For $T_{\mathrm{C}}^{\mathrm{D}}$, the errors were extracted from the fitting procedure (see below), while all other errors correspond to estimates of read-out errors.}
\begin{ruledtabular}
\begin{tabular}{ccccccc}
$x$ & $T_{\mathrm{N}}$ (K) & $T_{\mathrm{g}}$ (K) & $T_{\mathrm{X}}$ (K) & $T_{\mathrm{C}}$ (K) & $T_{\mathrm{g}}^{\mathrm{D}}$ (K) & $T_{\mathrm{C}}^{\mathrm{D}}$ (K) \\
\hline
0.05 & $240 \pm 5$ & - & - & - &- & - \\
0.10 & $190 \pm 5$ & $3 \pm 5$ & - & - & - & - \\
0.15 & - & $11 \pm 2$ & $23 \pm 3$ & $30 \pm 10$ & - & - \\
0.16 & - & $15 \pm 2$ & $34 \pm 3$ & $42 \pm 10$ & $18 \pm 5$ & $61 \pm 10$ \\
0.17 & - & $20 \pm 2$ & $36 \pm 3$ & $42 \pm 10$ & $23 \pm 5$ & $47 \pm 2$ \\
0.18 & - & $22 \pm 2$ & $35 \pm 3$ & $42 \pm 10$ & $22 \pm 5$ & $73 \pm 1$ \\
0.19 & - & $19 \pm 2$ & $37 \pm 5$ & $56 \pm 10$ & $25 \pm 5$ & $93 \pm 1$ \\
0.20 & - & $19 \pm 2$ & $35 \pm 5$ & $50 \pm 10$ & $24 \pm 5$ & $84 \pm 1$ \\
0.21 & - & $14 \pm 2$ & $35 \pm 5$ & $108 \pm 5$ & $25 \pm 5$ & $101 \pm 1$ \\
0.22 & - & $13 \pm 2$ & $32 \pm 5$ & $106 \pm 5$ & $21 \pm 5$ & $100 \pm 1$ \\
0.25 & - & $5 \pm 5$ & - & $200 \pm 5$ & - & - \\
0.30 & - & - & - & $290 \pm 5$ & - & - \\
\end{tabular}
\end{ruledtabular}
\end{table}
\subsection{Neutron depolarization}
\begin{figure}
\includegraphics{figure5}
\caption{\label{fig:5}Remaining neutron polarization after transmission through 0.5~mm of Fe$_{x}$Cr$_{1-x}$ as a function of temperature for $0.15 \leq x \leq 0.22$ (increasing from top to bottom). Data were measured in zero magnetic field under increasing temperature following initial zero-field cooling (zfc) or high-field cooling (hfc). Colored triangles mark the Curie transition $T_{\mathrm{C}}$ and the freezing temperature $T_{\mathrm{g}}$. Orange solid lines are fits to the experimental data, see text for details.}
\end{figure}
The neutron depolarization of samples in the central composition range $0.15 \leq x \leq 0.22$ was studied to gain further insights on the microscopic nature of the different magnetic states. Figure~\ref{fig:5} shows the polarization, $P$, of the transmitted neutron beam with respect to the polarization axis of the incoming neutron beam as a function of temperature. In the presence of ferromagnetically ordered domains or clusters that are large enough to induce a Larmor precession of the neutron spin during its transit, adjacent neutron trajectories pick up different Larmor phases due to the domain distribution in the sample. When averaged over the pixel size of the detector, this process results in polarization values below 1, also referred to as neutron depolarization. For a pedagogical introduction to the time and space resolution of this technique, we refer to Refs.~\cite{2008_Kardjilov_NatPhys, 2010_Schulz_PhD, 2015_Schmakat_PhD, _Seifert_tobepublished}.
For $x = 0.15$, shown in Fig.~\ref{fig:5}(a), no depolarization is observed. For $x = 0.16$, shown in Fig.~\ref{fig:5}(b), a weak decrease of polarization emerges below a point of inflection at $T_{\mathrm{C}} \approx 60$~K (blue triangle). The value of $T_{\mathrm{C}}$ may be inferred from a fit to the experimental data as described below and is in reasonable agreement with the value inferred from the susceptibility. The partial character of the depolarization, $P \approx 0.96$ in the low-temperature limit, indicates that ferromagnetically ordered domains of sufficient size occupy only a fraction of the sample volume. At lower temperatures, a weak additional change of slope may be attributed to the spin freezing at $T_{\mathrm{g}}$ (red triangle).
For $x = 0.17$, shown in Fig.~\ref{fig:5}(c), both signatures get more pronounced. In particular, data recorded after zero-field cooling (zfc) and high-field cooling (hfc) branch below $T_{\mathrm{g}}$, akin to the branching observed in the magnetization. The underlying dependence of the microscopic magnetic texture on the cooling history is typical for a spin glass. Note that the amount of branching varies from sample to sample. Such pronounced sample dependence is not uncommon in spin-glass systems, though the microscopic origin of these irregularities in Fe$_{x}$Cr$_{1-x}$ remains to be resolved.
When further increasing $x$, shown in Figs.~\ref{fig:5}(c)--\ref{fig:5}(h), the transition temperature $T_{\mathrm{C}}$ shifts to larger values and the depolarization gets more pronounced until essentially reaching $P = 0$ at low temperatures for $x = 0.22$. No qualitative changes are observed around $x = 0.19$, i.e., the composition for which the onset of long-range ferromagnetic order was reported previously~\cite{1983_Burke_JPhysFMetPhys_II}. Instead, the gradual evolution as a function of $x$ suggests that ferromagnetically ordered domains start to emerge already for $x \approx 0.15$ and continuously increase in size and/or number with $x$. This conjecture is also consistent with the appearance of faint signatures in the susceptibility. Note that there are no signatures related to $T_{\mathrm{X}}$.
In order to infer quantitative information, the neutron depolarization data were fitted using the formalism of Halpern and Holstein~\cite{1941_Halpern_PhysRev}. Here, spin-polarized neutrons are considered as they are traveling through a sample with randomly oriented ferromagnet domains. When the rotation of the neutron spin is small for each domain, i.e., when $\omega_{\mathrm{L}}t \ll 2\pi$ with the Larmor frequency $\omega_{\mathrm{L}}$ and the time required for transiting the domain $t$, the temperature dependence of the polarization of the transmitted neutrons may be approximated as
\begin{equation}\label{equ1}
P(T) = \mathrm{exp}\left[-\frac{1}{3}\gamma^{2}B^{2}_{\mathrm{0}}(T)\frac{d\delta}{v^{2}}\right].
\end{equation}
Here, $\gamma$ is the gyromagnetic ratio of the neutron, $B_{\mathrm{0}}(T)$ is the temperature-dependent average magnetic flux per domain, $d$ is the sample thickness along the flight direction, $\delta$ is the mean magnetic domain size, and $v$ is the speed of the neutrons. In mean-field approximation, the temperature dependence of the magnetic flux per domain is given by
\begin{equation}\label{equ2}
B_{\mathrm{0}}(T) = {\mu_{0}}^{2} {M_{0}}^{2} \left(1 - \frac{T}{T_{\mathrm{C}}}\right)^{\beta}
\end{equation}
where $\mu_{0}$ is the vacuum permeability, $M_{0}$ is the spontaneous magnetization in each domain, and $\beta$ is the critical exponent. In the following, we use the magnetization value measured at 2~K in a magnetic field of 250~mT as an approximation for $M_{0}$ and set $\beta = 0.5$, i.e., the textbook value for a mean-field ferromagnet. Note that $M_{0}$ more than triples when increasing the iron concentration from $x = 0.15$ to $x = 0.22$, as shown in Tab.~\ref{tab:2}, suggesting that correlations become increasingly important.
Fitting the temperature dependence of the polarization for temperatures above $T_{\mathrm{g}}$ according to Eq.~\eqref{equ1} yields mean values for the Curie temperature $T_{\mathrm{C}}$ and the domain size $\delta$, cf.\ solid orange lines in Fig.~\ref{fig:5} tracking the experimental data. The results of the fitting are summarized in Tab.~\ref{tab:2}. The values of $T_{\mathrm{C}}$ inferred this way are typically slightly higher than those inferred from the ac susceptibility, cf.\ Tab.~\ref{tab:1}. This shift could be related to depolarization caused by slow ferromagnetic fluctuations prevailing at temperatures just above the onset of static magnetic order. Yet, both values of $T_{\mathrm{C}}$ are in reasonable agreement. The mean size of ferromagnetically aligned domains or clusters, $\delta$, increases with increasing $x$, reflecting the increased density of iron atoms. As will be shown below, this general trend is corroborated also by an analysis of the Mydosh parameter indicating that Fe$_{x}$Cr$_{1-x}$ transforms from a cluster glass for small $x$ to a superparamagnet for larger $x$.
\begin{table}
\caption{\label{tab:2}Summary of the Curie temperature, $T_{\mathrm{C}}$, and the mean domain size, $\delta$, in Fe$_{x}$Cr$_{1-x}$ as inferred from neutron depolarization studies. Also shown is the magnetization measured at a temperature of 2~K in a magnetic field of 250~mT, ${M_{0}}$.}
\begin{ruledtabular}
\begin{tabular}{cccc}
$x$ & $T_{\mathrm{C}}^{\mathrm{D}}$ (K) & $\delta$ ($\upmu$m) & $M_{0}$ ($10^{5}$A/m) \\
\hline
0.15 & - & - & 0.70 \\
0.16 & $61 \pm 10$ & $0.61 \pm 0.10$ & 0.84 \\
0.17 & $47 \pm 2$ & $2.12 \pm 0.15$ & 0.96 \\
0.18 & $73 \pm 1$ & $3.17 \pm 0.07$ & 1.24 \\
0.19 & $93 \pm 1$ & $3.47 \pm 0.02$ & 1.64 \\
0.20 & $84 \pm 1$ & $4.67 \pm 0.03$ & 1.67 \\
0.21 & $101 \pm 1$ & $3.52 \pm 0.03$ & 2.18 \\
0.22 & $100 \pm 1$ & $5.76 \pm 0.13$ & 2.27\\
\end{tabular}
\end{ruledtabular}
\end{table}
\subsection{Specific heat, high-field magnetometry, and electrical resistivity}
\begin{figure}
\includegraphics{figure6}
\caption{\label{fig:6}Low-temperature properties of Fe$_{x}$Cr$_{1-x}$ with $x = 0.15$. (a)~Specific heat as a function of temperature. Zero-field data (black curve) and an estimate for the phonon contribution using the Debye model (gray curve) are shown. Inset: Specific heat at high temperatures approaching the Dulong--Petit limit. (b)~Specific heat divided by temperature. After subtraction of the phonon contribution, magnetic contributions at low temperatures are observed (green curve). (c)~Magnetic contribution to the entropy obtained by numerical integration. (d)~Magnetization as a function of field up to $\pm9$~T for different temperatures. (e)~Electrical resistivity as a function of temperature for different applied field values.}
\end{figure}
To obtain a complete picture of the low-temperature properties of Fe$_{x}$Cr$_{1-x}$, the magnetic properties at low fields presented so far are complemented by measurements of the specific heat, high-field magnetization, and electrical resistivity on the example of Fe$_{x}$Cr$_{1-x}$ with $x = 0.15$.
The specific heat as a function of temperature measured in zero magnetic field is shown in Fig.~\ref{fig:6}(a). At high temperatures, the specific heat approaches the Dulong--Petit limit of $C_{\mathrm{DP}} = 3R = 24.9~\mathrm{J}\,\mathrm{mol}^{-1}\mathrm{K}^{-1}$, as illustrated in the inset. With decreasing temperature, the specific heat monotonically decreases, lacking pronounced anomalies at the different characteristic temperatures.
The specific heat at high temperatures is dominated by the phonon contribution that is described well by a Debye model with a Debye temperature $\mathit{\Theta}_{\mathrm{D}} = 460$~K, which is slightly smaller than the values reported for $\alpha$-iron (477~K) and chromium (606~K)~\cite{2003_Tari_Book}. As shown in terms of the specific heat divided by temperature, $C/T$, in Fig.~\ref{fig:6}(b), the subtraction of this phonon contribution from the measured data highlights the presence of magnetic contributions to the specific heat below ${\sim}$30~K (green curve). As typical for spin-glass systems, no sharp signatures are observed and the total magnetic contribution to the specific heat is rather small~\cite{2015_Mydosh_RepProgPhys}. This finding is substantiated by the entropy $S$ as calculated by means of extrapolating $C/T$ to zero temperature and numerically integrating
\begin{equation}
S(T) = \int_{0}^{T}\frac{C(T)}{T}\,\mathrm{d}T.
\end{equation}
As shown in Fig.~\ref{fig:6}(c), the magnetic contribution to the entropy released up to 30~K amounts to about $0.04~R\ln2$, which corresponds to a small fraction of the total magnetic moment only.
Insights on the evolution of the magnetic properties under high magnetic fields may be inferred from the magnetization as measured up to $\pm9$~T, shown in Fig.~\ref{fig:6}(d). The magnetization is unsaturated up to the highest fields studied and qualitatively unchanged under increasing temperature, only moderately decreasing in absolute value. The value of 0.22~$\mu_{\mathrm{B}}/\mathrm{f.u.}$ obtained at 2~K and 9~T corresponds to a moment of 1.46~$\mu_{\mathrm{B}}/\mathrm{Fe}$, i.e., the moment per iron atom in Fe$_{x}$Cr$_{1-x}$ with $x = 0.15$ stays below the value of 2.2~$\mu_{\mathrm{B}}/\mathrm{Fe}$ observed in $\alpha$-iron~\cite{2001_Blundell_Book}.
Finally, the electrical resistivity as a function of temperature is shown in Fig.~\ref{fig:6}(e). As typical for a metal, the resistivity is of the order of several ten $\upmu\Omega\,\mathrm{cm}$ and, starting from room temperature, decreases essentially linearly with temperature. However, around 60~K, i.e., well above the onset of magnetic order, a minimum is observed before the resistivity increases towards low temperatures.
Such an incipient divergence of the resistivity with decreasing temperature due to magnetic impurities is reminiscent of single-ion Kondo systems~\cite{1934_deHaas_Physica, 1964_Kondo_ProgTheorPhys, 1987_Lin_PhysRevLett, 2012_Pikul_PhysRevLett}. When magnetic field is applied perpendicular to the current direction, this low-temperature increase is suppressed and a point of inflection emerges around 100~K. This sensitivity with respect to magnetic fields clearly indicates that the additional scattering at low temperatures is of magnetic origin. Qualitatively, the present transport data are in agreement with earlier reports on Fe$_{x}$Cr$_{1-x}$ for $0 \leq x \leq 0.112$~\cite{1966_Arajs_JApplPhys}.
\section{Characterization of the spin-glass behavior}
\label{sec:discussion}
In spin glasses, random site occupancy of magnetic moments, competing interactions, and geometric frustration lead to a collective freezing of the magnetic moments below a freezing temperature $T_{\mathrm{g}}$. The resulting irreversible metastable magnetic state shares many analogies with structural glasses. Depending on the densities of magnetic moments, different types of spin glasses may be distinguished. For small densities, the magnetic properties may be described in terms of single magnetic impurities diluted in a nonmagnetic host, referred to as canonical spin-glass behavior. These systems are characterized by strong interactions and the cooperative spin freezing represents a phase transition. For larger densities, clusters form with local magnetic order and frustration between neighboring clusters, referred to as cluster glass behavior, developing superparamagnetic characteristics as the cluster size increases. In these systems, the inter-cluster interactions are rather weak and the spin freezing takes place in the form of a gradual blocking. When the density of magnetic moments surpasses the percolation limit, long-range magnetic order may be expected.
For compositions close to the percolation limit, so-called reentrant spin-glass behavior may be observed. In such cases, as a function of decreasing temperature first a transition from a paramagnetic to a magnetically ordered state occurs before a spin-glass state emerges at lower temperatures. As both the paramagnetic and the spin-glass state lack long-range magnetic order, the expression ‘reentrant’ alludes to the disappearance of long-range magnetic order after a finite temperature interval and consequently the re-emergence of a state without long-range order~\cite{1993_Mydosh_Book}.
The metastable nature of spin glasses manifests itself in terms of a pronounced history dependence of both microscopic spin arrangement and macroscopic magnetic properties, translating into four key experimental observations; (i) a frequency-dependent shift of the maximum at $T_{\mathrm{g}}$ in the ac susceptibility, (ii) a broad maximum in the specific heat located 20\% to 40\% above $T_{\mathrm{g}}$, (iii) a splitting of the magnetization for different cooling histories, and (iv) a time-dependent creep of the magnetization~\cite{2015_Mydosh_RepProgPhys}. The splitting of the magnetization and the broad signature in the specific heat were addressed in Figs.~\ref{fig:5} and \ref{fig:6}.
In the following, the frequency dependence of the ac susceptibility will be analyzed by means of three different ways, namely the Mydosh parameter, power law fits, and the Vogel--Fulcher law, permitting to classify the spin-glass behavior in Fe$_{x}$Cr$_{1-x}$ and its change as a function of composition.
\begin{figure}
\includegraphics[width=0.97\linewidth]{figure7}
\caption{\label{fig:7}Imaginary part of the zero-field ac susceptibility as a function of temperature for Fe$_{x}$Cr$_{1-x}$ with $x = 0.15$ measured at different excitation frequencies $f$. Analysis of the frequency-dependent shift of the spin freezing temperature $T_{\mathrm{g}}$ allows to gain insights on the microscopic nature of the spin-glass state.}
\end{figure}
In the present study, the freezing temperature $T_{\mathrm{g}}$ was inferred from a maximum in the imaginary part of the ac susceptibility as measured at an excitation frequency of 1~kHz. However, in a spin glass the temperature below which spin freezing is observed depends on the excitation frequency $f$, as illustrated in Fig.~\ref{fig:7} for the example of Fe$_{x}$Cr$_{1-x}$ with $x = 0.15$. Under increasing frequency, the imaginary part remains qualitatively unchanged but increases in absolute size and the maximum indicating $T_{\mathrm{g}}$ shifts to higher temperatures. Analyzing this shift in turn provides information on the microscopic nature of the spin-glass behavior.
The first and perhaps most straightforward approach utilizes the empirical Mydosh parameter $\phi$, defined as
\begin{equation}
\phi = \left[\frac{T_{\mathrm{g}}(f_{\mathrm{high}})}{T_{\mathrm{g}}(f_{\mathrm{low}})} - 1\right] \left[\ln\left(\frac{f_{\mathrm{high}}}{f_{\mathrm{low}}}\right)\right]^{-1}
\end{equation}
where $T_{\mathrm{g}}(f_{\mathrm{high}})$ and $T_{\mathrm{g}}(f_{\mathrm{low}})$ are the freezing temperatures as experimentally observed at high and low excitation frequencies, $f_{\mathrm{high}}$ and $f_{\mathrm{low}}$, respectively~\cite{1993_Mydosh_Book, 2015_Mydosh_RepProgPhys}. Small shifts associated with Mydosh parameters below 0.01 are typical for canonical spin glasses such as Mn$_{x}$Cu$_{1-x}$, while cluster glasses exhibit intermediate values up to 0.1. Values exceeding 0.1 suggest superparamagnetic behavior~\cite{1993_Mydosh_Book, 2015_Mydosh_RepProgPhys, 1980_Tholence_SolidStateCommun, 1986_Binder_RevModPhys}.
\begin{figure}
\includegraphics[width=1.0\linewidth]{figure8}
\caption{\label{fig:8}Evolution of the Mydosh-parameter in Fe$_{x}$Cr$_{1-x}$. (a)~Schematic depiction of the five different sequences of magnetic regimes observed as a function of temperature for different $x$. The following regimes are distinguished: paramagnetic~(PM), antiferromagnetic~(AFM), ferromagnetic~(FM), spin-glass~(SG). A precursor phenomenon~(PC) may be observed between FM and SG. (b)~Mydosh parameter $\phi$ as a function of the iron concentration $x$, allowing to classify the spin-glass behavior as canonical ($\phi \leq 0.01$, gray shading), cluster-glass ($0.01 \leq \phi \leq 0.1$, yellow shading), or superparamagnetic ($\phi \geq 0.1$, brown shading). }
\end{figure}
\begin{table*}
\caption{\label{tab:3}Parameters inferred from the analysis of the spin-glass behavior in Fe$_{x}$Cr$_{1-x}$, namely the Mydosh parameter $\phi$, the zero-frequency extrapolation of the spin freezing temperature $T_\mathrm{g}(0)$, the characteristic relaxation time $\tau_{0}$, the critical exponent $z\nu$, the Vogel--Fulcher temperature $T_{0}$, and the cluster activation energy $E_{a}$. The errors were determined by means of Gaussian error propagation ($\phi$), the distance of neighboring data points ($T_\mathrm{g}(0)$), and statistical deviations of the linear fits ($\tau_{0}$, $z\nu$, $T_{0}$, and $E_{a}$).}
\begin{ruledtabular}
\begin{tabular}{ccccccc}
$x$ & $\phi$ & $T_\mathrm{g}(0)$ (K) & $\tau_{0}$ ($10^{-6}$~s) & $z\nu$ & $T_{0}$ (K) & $E_{a}$ (K) \\
\hline
0.05 & - & - & - & - & - & - \\
0.10 & $0.064 \pm 0.011$ & - & - & - & - & - \\
0.15 & $0.080 \pm 0.020$ & $9.1 \pm 0.1$ & $0.16 \pm 0.03$ & $5.0 \pm 0.1$ & $8.5 \pm 0.1$ & $19.9 \pm 0.8$ \\
0.16 & $0.100 \pm 0.034$ & $13.4 \pm 0.1$ & $1.73 \pm 0.15$ & $2.2 \pm 0.0$ & $11.9 \pm 0.1$ & $14.4 \pm 0.3$ \\
0.17 & $0.107 \pm 0.068$ & $18.3 \pm 0.1$ & $6.13 \pm 1.52$ & $1.5 \pm 0.1$ & $16.3 \pm 0.3$ & $12.8 \pm 0.9$ \\
0.18 & $0.108 \pm 0.081$ & $14.5 \pm 0.1$ & $1.18 \pm 0.46$ & $7.0 \pm 0.5$ & $16.9 \pm 0.5$ & $24.2 \pm 2.3$ \\
0.19 & $0.120 \pm 0.042$ & $14.2 \pm 0.1$ & $0.47 \pm 0.15$ & $4.5 \pm 0.2$ & $14.6 \pm 0.4$ & $16.3 \pm 1.4$ \\
0.20 & $0.125 \pm 0.043$ & $13.5 \pm 0.1$ & $1.29 \pm 0.34$ & $4.1 \pm 0.2$ & $13.6 \pm 0.3$ & $18.8 \pm 1.3$ \\
0.21 & $0.138 \pm 0.048$ & $9.5 \pm 0.1$ & $1.67 \pm 0.21$ & $4.7 \pm 0.1$ & $10.3 \pm 0.4$ & $12.0 \pm 1.3$ \\
0.22 & $0.204 \pm 0.071$ & $11.7 \pm 0.1$ & $2.95 \pm 0.80$ & $2.6 \pm 0.1$ & $11.3 \pm 0.4$ & $11.3 \pm 1.2$ \\
0.25 & $0.517 \pm 0.180$ & $2.8 \pm 0.1$ & $75.3 \pm 5.34$ & $1.8 \pm 0.1$ & - & - \\
0.30 & - & - & - & - & - & \\
\end{tabular}
\end{ruledtabular}
\end{table*}
As summarized in Tab.~\ref{tab:3} and illustrated in Fig.~\ref{fig:8}, the Mydosh parameter in Fe$_{x}$Cr$_{1-x}$ monotonically increases as a of function of increasing iron concentration. For small $x$, the values are characteristic of cluster-glass behavior, while for large $x$ they lie well within the regime of superparamagnetic behavior. This evolution reflects the increase of the mean size of ferromagnetic clusters as inferred from the analysis of the neutron depolarization data.
\begin{figure}
\includegraphics[width=1.0\linewidth]{figure9}
\caption{\label{fig:9}Analysis of spin-glass behavior using power law fits and the Vogel--Fulcher law for Fe$_{x}$Cr$_{1-x}$ with $x = 0.15$. (a)~Logarithm of the relaxation time as a function of the logarithm of the normalized shift of the freezing temperature. The red solid line is a power law fit allowing to infer the characteristic relaxation time $\tau_{0}$ and the critical exponent $z\nu$. Inset: Goodness of fit for different estimated zero-frequency extrapolations of the freezing temperature, $T_{\mathrm{g}}^{\mathrm{est}}(0)$. The value $T_{\mathrm{g}}(0)$ used in the main panel is defined as the temperature of highest $R^{2}$. (b)~Spin freezing temperature as a function of the inverse of the logarithm of the ratio of characteristic frequency and excitation frequency. The red solid line is a fit according to the Vogel--Fulcher law allowing to infer the cluster activation energy $E_{a}$ and the Vogel--Fulcher temperature $T_{0}$.}
\end{figure}
The second approach employs the standard theory for dynamical scaling near phase transitions to $T_{\mathrm{g}}$~\cite{1977_Hohenberg_RevModPhys, 1993_Mydosh_Book}. The relaxation time $\tau = \frac{1}{2\pi f}$ is expressed in terms of the power law
\begin{equation}
\tau = \tau_{0} \left[\frac{T_{\mathrm{g}}(f)}{T_{\mathrm{g}}(0)} - 1\right]^{z\nu}
\end{equation}
where $\tau_{0}$ is the characteristic relaxation time of a single moment or cluster, $T_{\mathrm{g}}(0)$ is the zero-frequency limit of the spin freezing temperature, and $z\nu$ is the critical exponent. In the archetypical canonical spin glass Mn$_{x}$Cu$_{1-x}$, one obtains values such as $\tau_{0} = 10^{-13}~\mathrm{s}$, $T_{\mathrm{g}}(0) = 27.5~\mathrm{K}$, and $z\nu = 5$~\cite{1985_Souletie_PhysRevB}.
The corresponding analysis is illustrated in Fig.~\ref{fig:9}(a) for Fe$_{x}$Cr$_{1-x}$ with $x = 0.15$. First the logarithm of the ratio of relaxation time and characteristic relaxation time, $\ln(\frac{\tau}{\tau_{0}})$, is plotted as a function of the logarithm of the normalized shift of the freezing temperature, $\ln\left[\frac{T_{\mathrm{g}}(f)}{T_{\mathrm{g}}(0)} - 1\right]$, for a series of estimated values of the zero-frequency extrapolation $T_{\mathrm{g}}^{\mathrm{est}}(0)$. For each value of $T_{\mathrm{g}}^{\mathrm{est}}(0)$ the data are fitted linearly and the goodness of fit is compared by means of the $R^{2}$ coefficient, cf.\ inset of Fig.~\ref{fig:9}(a). The best approximation for the zero-frequency freezing temperature, $T_{\mathrm{g}}(0)$, is defined as the temperature of highest $R^{2}$. Finally, the characteristic relaxation time $\tau_{0}$ and the critical exponent $z\nu$ are inferred from a linear fit to the experimental data using this value $T_{\mathrm{g}}(0)$, as shown in Fig.~\ref{fig:9}(a) for Fe$_{x}$Cr$_{1-x}$ with $x = 0.15$.
The same analysis was carried out for all compositions Fe$_{x}$Cr$_{1-x}$ featuring spin-glass behavior, yielding the parameters summarized in Tab.~\ref{tab:3}. Characteristic relaxation times of the order of $10^{-6}~\mathrm{s}$ are inferred, i.e., several order of magnitude larger than those observed in canonical spin glasses and consistent with the presence of comparably large magnetic clusters, as may be expected for the large values of $x$. Note that these characteristic times are also distinctly larger than the $10^{-12}~\mathrm{s}$ to $10^{-8}~\mathrm{s}$ that neutrons require to traverse the magnetic clusters in the depolarization experiments. Consequently, the clusters appear quasi-static for the neutron which in turn is a prerequisite for the observation of net depolarization across a macroscopic sample. The critical exponents range from 1.5 to 7.0, i.e., within the range expected for glassy systems~\cite{1980_Tholence_SolidStateCommun, 1985_Souletie_PhysRevB}. The lack of systematic evolution of both $\tau_{0}$ and $z\nu$ as a function of iron concentration $x$ suggests that these parameters in fact may be rather sensitive to details of microscopic structure, potentially varying substantially between individual samples.
The third approach uses the Vogel--Fulcher law, developed to describe the viscosity of supercooled liquids and glasses, to interpret the properties around the spin freezing temperature $T_{\mathrm{g}}$~\cite{1993_Mydosh_Book, 1925_Fulcher_JAmCeramSoc, 1980_Tholence_SolidStateCommun, 2013_Svanidze_PhysRevB}. Calculating the characteristic frequency $f_{0} = \frac{1}{2\pi\tau_{0}}$ from the characteristic relaxation time $\tau_{0}$ as determined above, the Vogel--Fulcher law for the excitation frequency $f$ reads
\begin{equation}
f = f_{0} \exp\left\lbrace-\frac{E_{a}}{k_{\mathrm{B}}[T_{\mathrm{g}}(f)-T_{0}]}\right\rbrace
\end{equation}
where $k_{\mathrm{B}}$ is the Boltzmann constant, $E_{a}$ is the activation energy for aligning a magnetic cluster by the applied field, and $T_{0}$ is the Vogel--Fulcher temperature providing a measure of the strength of the cluster interactions. As a point of reference, it is interesting to note that values such as $E_{a}/k_{\mathrm{B}} = 11.8~\mathrm{K}$ and $T_{0} = 26.9~\mathrm{K}$ are observed in the archetypical canonical spin glass Mn$_{x}$Cu$_{1-x}$~\cite{1985_Souletie_PhysRevB}.
For each composition Fe$_{x}$Cr$_{1-x}$, the spin freezing temperature $T_{\mathrm{g}}(f)$ is plotted as a function of the inverse of the logarithm of the ratio of characteristic frequency and excitation frequency, $\frac{1}{\ln(f/f_{0})}$, as shown in Fig.~\ref{fig:9}(b) for Fe$_{x}$Cr$_{1-x}$ with $x = 0.15$. A linear fit to the experimental data allows to infer $E_{a}$ and $T_{0}$ from the slope and the intercept. The corresponding values for all compositions Fe$_{x}$Cr$_{1-x}$ featuring spin-glass behavior are summarized in Tab.~\ref{tab:3}. All values of $T_{0}$ and $E_{a}$ are of the order 10~K and positive, indicating the presence of strongly correlated clusters~\cite{2012_Anand_PhysRevB, 2011_Li_ChinesePhysB, 2013_Svanidze_PhysRevB}. Both $T_{0}$ and $E_{a}$ follow roughly the evolution of the spin freezing temperature $T_{\mathrm{g}}$, reaching their maximum values around $x = 0.17$ or $x = 0.18$.
\section{Conclusions}
\label{sec:conclusion}
In summary, a comprehensive study of the magnetic properties of polycrystalline Fe$_{x}$Cr$_{1-x}$ in the composition range $0.05 \leq x \leq 0.30$ was carried out by means of x-ray powder diffraction as well as measurements of the magnetization, ac susceptibility, and neutron depolarization, complemented by specific heat and electrical resistivity data for $x = 0.15$. As our central result, we present a detailed composition--temperature phase diagram based on the combination of a large number of quantities. Under increasing iron concentration $x$, antiferromagnetic order akin to pure Cr is suppressed above $x = 0.15$, followed by the emergence of weak magnetic order developing distinct ferromagnetic character above $x = 0.18$. At low temperatures, a wide dome of reentrant spin-glass behavior is observed for $0.10 \leq x \leq 0.25$, preceded by a precursor phenomenon. Analysis of the neutron depolarization data and the frequency-dependent shift in the ac susceptibility indicate that with increasing $x$ the size of ferromagnetically ordered clusters increases and that the character of the spin-glass behavior changes from a cluster glass to a superparamagnet.
\acknowledgments
We wish to thank P.~B\"{o}ni and S.~Mayr for fruitful discussions and assistance with the experiments. This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under TRR80 (From Electronic Correlations to Functionality, Project No.\ 107745057, Project E1) and the excellence cluster MCQST under Germany's Excellence Strategy EXC-2111 (Project No.\ 390814868). Financial support by the Bundesministerium f\"{u}r Bildung und Forschung (BMBF) through Project No.\ 05K16WO6 as well as by the European Research Council (ERC) through Advanced Grants No.\ 291079 (TOPFIT) and No.\ 788031 (ExQuiSid) is gratefully acknowledged. G.B., P.S., S.S., M.S., and P.J.\ acknowledge financial support through the TUM Graduate School.
| 2024-02-18T23:39:40.791Z | 2020-07-22T02:11:10.000Z | algebraic_stack_train_0000 | 45 | 9,343 |
|
proofpile-arXiv_065-258 | "\\section{Introduction}\nWith technological advancements in the automotive industry in recent times(...TRUNCATED) | 2024-02-18T23:39:40.868Z | 2020-07-22T02:06:02.000Z | algebraic_stack_train_0000 | 50 | 7,840 |
|
proofpile-arXiv_065-297 | "\\section{Introduction}\n\\label{sec:start}\n\n\\subsection{The landscape}\\label{sect:landsc}\n\nI(...TRUNCATED) | 2024-02-18T23:39:40.993Z | 2021-05-07T02:21:39.000Z | algebraic_stack_train_0000 | 58 | 31,462 |
|
proofpile-arXiv_065-337 | "\\section{Introduction}\nFirst-principles calculations based on density functional \ntheory~\\cite{(...TRUNCATED) | 2024-02-18T23:39:41.175Z | 1996-09-20T17:30:30.000Z | algebraic_stack_train_0000 | 66 | 6,802 |
|
proofpile-arXiv_065-359 | "\\section{Why should we study rare charm decays? }\nAt HERA recent measurements of the charm produc(...TRUNCATED) | 2024-02-18T23:39:41.244Z | 1996-09-12T17:37:21.000Z | algebraic_stack_train_0000 | 71 | 5,262 |
|
proofpile-arXiv_065-513 | "\\section{Introduction}\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\nIt is we(...TRUNCATED) | 2024-02-18T23:39:41.757Z | 1996-09-06T02:43:49.000Z | algebraic_stack_train_0000 | 96 | 10,474 |
|
proofpile-arXiv_065-524 | "\\section*{Acknowledgments}\n\nWe are grateful to V.A. Kuzmin and M.A. Shifman for useful\ndiscussi(...TRUNCATED) | 2024-02-18T23:39:41.792Z | 1996-09-22T14:34:57.000Z | algebraic_stack_train_0000 | 100 | 109 |
|
proofpile-arXiv_065-547 | "\\section{Introduction}\n\nLate-type Low Surface Brightness galaxies (LSBs) are considered to be ve(...TRUNCATED) | 2024-02-18T23:39:41.852Z | 1996-09-12T21:33:37.000Z | algebraic_stack_train_0000 | 105 | 2,412 |
|
proofpile-arXiv_065-600 | "\\section{Introduction}\n\nIrrotational dust spacetimes have been widely studied, in\nparticular as(...TRUNCATED) | 2024-02-18T23:39:41.996Z | 1996-09-30T12:38:02.000Z | algebraic_stack_train_0000 | 115 | 3,577 |
End of preview. Expand
in Dataset Viewer.
README.md exists but content is empty.
Use the Edit dataset card button to edit it.
- Downloads last month
- 3