# Site Split of Antiferromagnetic $\alpha$-Mn Revealed by 55Mn Nuclear
Magnetic Resonance

Masahiro Manago<EMAIL_ADDRESS>Gaku Motoyama  Department of
Physics and Materials Science, Shimane University, Matsue 690-8504, Japan
Shijo Nishigori  ICSR, Shimane University, Matsue 690-8504, Japan  Kenji
Fujiwara  Department of Physics and Materials Science, Shimane University,
Matsue 690-8504, Japan  Katsuki Kinjo  Shunsaku Kitagawa  Kenji Ishida
Department of Physics, Kyoto University, Kyoto 606-8502, Japan  Kazuto Akiba
Shingo Araki  Tatsuo C. Kobayashi  Graduate School of Natural Science and
Technology, Okayama University, Okayama 700-8530, Japan  Hisatomo Harima
Department of Physics, Kobe University, Kobe 657-8501, Japan

###### Abstract

The magnetic structure of antiferromagnetic $\alpha$-Mn has been unclarified
for almost 70 years since its magnetism was discovered. We measured the zero-
field nuclear magnetic resonance spectra of antiferromagnetic $\alpha$-Mn to
obtain further insight into magnetism below $T_{\text{N}}=95$ K. The site II
spectra split into two sites with five subpeaks owing to quadrupole
interaction, and this shows that the ordered moments at site II are slightly
tilted from the $[001]$ direction. The site III spectra revealed that this
site splits into four sites below $T_{\text{N}}$. These findings clearly
demonstrate that the antiferromagnetic $\alpha$-Mn symmetry is lower than
previously considered.

Contrary to most elemental metals, $\alpha$-Mn, the stable form of Mn metal at
ambient temperature and pressure, has unique crystallographic properties. It
crystallizes into a complicated cubic structure without an inversion center
($I\bar{4}3m$, No. 217, $T_{d}^{3}$) with four inequivalent Mn sites and 58
atoms in a unit cell. $\alpha$-Mn exhibits antiferromagnetic (AFM) ordering at
$T_{\text{N}}=95$ K at ambient pressure. Neutron diffraction measurements
revealed the commensurate and non-collinear magnetic structure in the AFM
state.[1, 2]. The AFM phase is suppressed by the hydrostatic pressure at $\sim
1.5$ GPa, and another magnetic phase emerges[3, 4, 5, 6]. This is a weak
ferromagnetic one with a small spontaneous magnetization and is suppressed at
4.2 GPa. A remarkable feature is the anomalous Hall effect in the pressure-
induced phase despite the absence of large spontaneous magnetization. This
effect is caused by the non-zero Berry curvature in the momentum space of non-
collinear antiferromagnets[7], which was observed in Mn3Sn and Mn3Ge[8, 9,
10]. A key factor of these rich properties in $\alpha$-Mn is the competition
of AFM interactions between neighbor Mn sites arising from its unique crystal
structure. Such competing interactions are also observed in magnetically-
frustrated systems $\beta$-Mn [11], Y(Sc)Mn2 [12], and Mn3P [13], where
application of hydrostatic pressure and atomic substitution can drastically
alter the magnetic states. The magnetism in $\alpha$-Mn has been studied for
several decades, however, it still remains to be a fundamental subject.

The magnetic structure of $\alpha$-Mn remains elusive even at ambient
pressure. Neutron diffraction studies revealed that the magnetic moment is
largest at site I, followed by sites II, III, and IV[14, 2]. Sites III and IV
split into two sites in the AFM state with a tetragonal distortion of the
crystal structure[1, 2]. The site I magnetic moment points to $[001]$, and the
site II moments slightly tilt from the $[001]$ axis, showing the non-collinear
orientation. A previous nuclear magnetic resonance (NMR) study[15],
conversely, revealed that the magnetic structure is more complicated than
expected: the NMR spectrum arising from site II split into two sites in the
AFM state, and this suggested that site III split into four sites. The recent
NMR results [16, 17] agree with the previous report. The site II split[15]
suggests that the structure in the AFM state is lower than tetragonal[18].
However, it has not been taken into consideration in the analysis of the
neutron study[2], and thus, the true magnetic structure has not been revealed.
Because of the complex crystal structure, the magnetic structure of
$\alpha$-Mn remains unclear for almost 70 years since the antiferromagnetism
of this system was first reported by neutron diffraction[19].

In this paper, we report on the result of 55Mn zero-field NMR (ZF-NMR)
measurements on a high-quality $\alpha$-Mn sample at ambient pressure to
obtain additional insight into the AFM structure. It was identified that the
two types of moments at split site II are parallel with a 6% difference in
size, and the moments direction is slightly (6°) tilted from the $[001]$ axis,
in agreement with the previous neutron study. We clarified that site III split
into four sites in the AFM state, with the hyperfine fields varying in 2–3 T.
In addition, site IV also splits into more than two sites with complicated
spectra. These findings confirm that the AFM $\alpha$-Mn symmetry is lower
than that previously considered. This presents a key indicator for AFM
structure identification.

The $\alpha$-Mn sample was synthesized from the Pb-flux method in a horizontal
configuration with a temperature gradient[20] using Mn (99.999%) and Pb
(99.9999%), as in Ref. 6. A typical residual resistivity ratio between 2 and
300 K is $\sim 17$ for samples by this method[6], which is higher than the
previous values $\sim 2$[4, 3]. The sample was moderately crushed into powder
in an argon atmosphere with a typical grain size of $\sim 0.1$ mm. The sample
was inserted into the NMR coil with a typical diameter of 2 mm. The standard
spin-echo technique was used in the NMR measurements without applying the
external field. Spontaneous moments in the AFM state induce hyperfine fields
at the Mn sites, causing the Zeeman split of the 55Mn nucleus. The NMR
measurements were performed at a temperature of 1.5 or 2 K well below
$T_{\text{N}}$.

Figure 1: (Color online) Crystal structure of $\alpha$-Mn, drawn by VESTA[21].  Table 1:  The local symmetry and parameters of quadrupole interaction of four Mn sites in $\alpha$-Mn for the paramagnetic state. Quadrupole parameters are calculated using band calculation.  site | Wyckoff letter | symmetry |  $\nu_{\text{Q}}^{\text{calc}}$ (MHz) | $\eta^{\text{calc}}$
---|---|---|---|---
I | $2a$ | $\bar{4}3m$ | $0$ | -
II | $8c$ | $.3m$ | $2.520$ | $0$
III | $24g$ | $..m$ | $2.627$ | $0.882$
IV | $24g$ | $..m$ | $1.808$ | $0.851$

Figure 1 shows the crystal structure of $\alpha$-Mn. Table 1 summarizes the
local symmetries of the Mn sites for the paramagnetic state and calculated
parameters of the electric field gradient (EFG) $V_{ij}$ for 55Mn by band
calculation. These values are used for the NMR spectra analysis described
herein. The EFG tensor is diagonalized by rotating the principal axes such
that $\lvert V_{zz}\rvert\geq\lvert V_{yy}\rvert\geq\lvert V_{xx}\rvert$, and
they are typically represented by the quadrupole frequency
$\omega_{\text{Q}}\propto QV_{zz}$, and the asymmetric parameter $\eta=\lvert
V_{yy}-V_{xx}\rvert/\lvert V_{zz}\rvert$, where $Q$ is the nuclear quadrupole
moment. The total Hamiltonian of the 55Mn nucleus ($I=5/2$) is expressed as
follows:

$\mathcal{H}=-\gamma\hbar\bm{H}_{\text{hf}}\cdot\bm{I}+\frac{\hbar\omega_{\text{Q}}}{6}\left[3I_{z}^{2}-I(I+1)+\frac{\eta}{2}(I_{+}^{2}+I_{-}^{2})\right],$
(1)

where $\gamma$ is the nuclear gyromagnetic ratio [$\gamma/(2\pi)=10.554$
MHz/T], $\bm{H}_{\text{hf}}$ is the hyperfine field caused by the ordered
moment. Because site I in $\alpha$-Mn is located at the cubic symmetrical
site, the EFG tensor is zero, leading to $\omega_{\text{Q}}=0$. Under the
hyperfine field, each Mn site, except for site I, splits into five lines owing
to the finite $\omega_{\text{Q}}$. The three-fold symmetry at site II ensures
the asymmetric parameter is zero, and the $V_{zz}$ direction is along the
three-fold rotation axis. Sites III and IV possess the same local symmetry,
and they were identified[15] based on the difference between ordered moments
reported by neutron diffraction studies[1, 2].

Figure 2:  Zero-field NMR spectrum of $\alpha$-Mn originating from site I
measured at 1.54 K in the AFM state. The intensity was multiplied by
$f^{-1.5}$ to cancel the frequency ($f$) dependence of the emf and the skin
effect (see text).

Figure 2 shows the ZF-NMR spectrum at site I. The NMR intensity was multiplied
by $f^{-1.5}$ to cancel the frequency ($f$) dependence of the emf ($f^{2}$)
induced by nuclear moments and the skin effect ($f^{-0.5}$), which limits the
effective sample volume. This is essential for obtaining accurate intensity
ratios of signals especially for sites III and IV shown later. A sharp peak of
0.2 MHz width was observed at 199.0 MHz. The absence of the quadrupole split
is consistent with the local cubic symmetry of site I, although the crystal
structure should slightly distort below $T_{\text{N}}$ [1, 2]. The frequency
is in agreement with previous reports[15, 16] and the width is approximately
five times sharper than the previous report[15], showing good sample quality.
A minor broad peak was identified at 198.7 MHz, and this will be discussed
later.

Figure 3: (Color online) (a) Zero-field NMR spectrum of $\alpha$-Mn from site
II measured at 1.54 K. The previous result by Yamagata and Asayama[15] is
shown with a dashed line for comparison. (b), (c) The analysis of the
quadrupole split of site II for (b) the lower and (c) the higher lines. The
experimental result was fitted with the summation of five Lorentz functions
with the same width (0.4 MHz), as shown by the dashed lines. The hyperfine
field $\mu_{0}H_{\text{hf}}$ and its direction $\theta$ were obtained from the
peak frequencies by numerical diagonalization, as shown by the vertical solid
orange lines. $\delta\theta\equiv\theta-54.7$° indicates the angle of the
hyperfine field from the $[001]$ direction.

Figure 3(a) shows the ZF-NMR spectra at site II. Two main peaks were observed
in the ranges of 142–145 MHz and 151–154 MHz with five split lines caused by
the quadrupole interaction. Because splitting is determined by the direction
of hyperfine fields, similar splitting in the two main peaks is the first
direct evidence that moments at two Mn II sites are almost parallel. The
frequencies agree with previous reports[15, 16], and the two peaks are well
separated in this result.

Figures 3(b) and (c) show the site II spectra analysis. The quadrupole split
frequency depends on the angle $\theta$ between the magnetic field and the EFG
maximum principal axis $V_{zz}$. Within the first-order perturbation with
respect to the EFG, the frequency for the $m-1\leftrightarrow m$ transition is
as follows:

$\nu_{m}=[\gamma/(2\pi)]H_{\text{hf}}+(m-1/2)\frac{\nu_{\text{Q}}}{2}(3\cos^{2}\theta-1),$
(2)

where $\nu_{\text{Q}}=\omega_{\text{Q}}/(2\pi)$. The quadrupole splitting
almost vanishes if $\theta=\cos^{-1}(1/\sqrt{3})\simeq 54.7$° (so-called the
magic angle). This angle will be realized when the magnetic moment is along
the $[001]$ because the $V_{zz}$ direction is along the $[111]$ at site II.
The spectral shape was reproduced with the summation of five Lorentz functions
of 0.4 MHz width, as shown by dashed lines in Figs. 3(b) and (c). The best fit
of peak frequencies was obtained with $\theta\simeq 61$° or $49$°, which
corresponds to the angle
$\delta\theta\equiv\lvert\theta-54.7\text{\textdegree}\rvert\sim 6$° from the
$[001]$ directions. This value is in excellent agreement with the previous
neutron diffraction study by Lawson et al. showing that the moment at site II
is in a non-collinear structure tilted by $6$° from the $[001]$ to $[110]$
directions[2]. The angle somewhat differs from the report by Yamada et al.
[1]. Although it could not be distinguished from the NMR whether $\theta\simeq
61$° or $49$°, i.e., whether the site II moment is tilted away from $[111]$ or
toward $[111]$, the neutron results indicate the latter. The present NMR
result shows that the directions of ordered moments at site II are similar
with only a difference of $\sim 6$% in moment size.

Figure 4: (Color online) Zero-field NMR spectrum of $\alpha$-Mn originating
from site III measured at 2 K. The vertical lines show the calculated
frequencies using the EFG parameters in Table 1.

Figure 4 shows the ZF-NMR spectra at the site III region. We identified 17
lines between 17–33 MHz, showing that the four inequivalent sites exist in
this frequency region with two sites almost overlapping between 25–30 MHz.
This result is the first direct experimental evidence that site III splits
into four sites owing to magnetic ordering, as suggested by the NMR spectra
analysis [15]. The experimental result was reproduced using the EFG parameters
shown in Table 1, as shown by the vertical lines in Fig. 4. The overlapped
peaks between 25–30 MHz imply that the direction of ordered moments are close
for these two sites, as in site II, because the quadrupole splitting width is
determined by the angle between the hyperfine field and the $V_{zz}$
direction. These sites may not be resolved by neutron diffraction. By
contrast, the other two sites at approximately 21 and 31 MHz are quite
different in internal field size. Such a significant difference in some parts
of site III is atypical and may represent key properties for understanding its
magnetism.

Figure 5:  Zero-field NMR spectrum of $\alpha$-Mn of site IV at 2 K.

Figure 5 shows the ZF-NMR spectra of site IV in the 3–10 MHz range. At least
16 lines were resolved in this range. We consider that site IV also splits
into four sites because of the identical local symmetry to site III in the
paramagnetic state. Although detailed analysis of the site IV result is
difficult, this demonstrates that the site IV hyperfine field is approximately
0.5 T and significantly smaller than that of site III.

The AFM $\alpha$-Mn symmetry has been considered as tetragonal
$I_{P}\bar{4}2^{\prime}m^{\prime}$, which is the subgroup of $I\bar{4}2m$ [1,
2, 22]. However, the present NMR results indicate that the symmetry is lower
than this space group. There is a possibility that orthorhombic distortion
occurs, resulting in that the $\bar{4}$ local symmetry is broken at site I,
and the site II splits into two sites[15, 16]. More detailed symmetry
consideration was recently reported by H. Fukazawa et al.[23], which is
consistent with our scenario. A problem is the signal intensity ratio of the
split site II in the previous[15, 16] and present studies: the higher-
frequency peak is stronger, as shown in Fig. 3(a), while the 1:1 split is
expected for the orthorhombic symmetry. Because the present NMR measurements
cannot exactly identify the crystal structure, precise X-ray or neutron
diffraction measurements are required to determine the space group in the AFM
$\alpha$-Mn.

The site II spectral shape is an atypical quadrupole-splitting signal because
the central peak is frequently the strongest among the split lines, however,
the satellite peaks are stronger for site II. This spectrum discrepancy is
probably caused by strong magnetic fluctuations. The spin-echo intensity
$M(2\tau)$ decays as the time interval $\tau$ between the first and second
pulse oscillating fields increases:

$M(2\tau)=M(0)\exp(-2\tau/T_{2}).$ (3)

The values of the spin-echo decay rate $1/T_{2}$ differ in different lines if
the NMR line splits owing to quadrupole interaction. In the simplest case,
$1/T_{2}$ originating from magnetic fluctuations at the $m-1\leftrightarrow m$
transition will be as follows:[24]

$\frac{1}{T_{2}}=\left[I(I+1)-m(m-1)\right]\frac{1}{T_{1}},$ (4)

where $1/T_{1}$ is the nuclear spin-lattice relaxation rate, and its value
does not depend on where it is measured. Thus, $1/T_{2}$ is fastest at the
central peak ($-1/2\leftrightarrow 1/2$ transition). If the decay time $T_{2}$
is comparable to or shorter than $\tau$ because of strong fluctuations, the
intensity will be much weaker than that extrapolated to $\tau\to 0$. This
effect is the strongest at the central peak.

For site II, a rough estimation of $T_{2}$ was obtained from the $\tau$
dependence of the spectrum (not shown), and $T_{2}\sim 30$ µs at the second
satellite peaks at 1.54 K. Because the spectrum in Fig. 3 was measured by
$2\tau=26$ µs, the central peak may be weaker than the satellite peaks. The
double-horn shape identified in the recent NMR study[16] can be ascribed to
this strong-fluctuation effect. The value of $T_{2}\sim 30$ µs ($1/T_{2}\sim
30$ ms-1) is comparable to the nearly-AFM system $\beta$-Mn, where $1/T_{2}$
exhibits anomalous temperature dependence $1/T_{2}\sim T^{1/2}$ in the
paramagnetic state [25, 26]. The atypical spectral shape at site II in
$\alpha$-Mn suggests that strong spin fluctuations remain in the AFM state
significantly below $T_{\text{N}}$.

The large $1/T_{2}$ was also observed at site I, and $1/T_{2}\simeq 40$ ms-1
was obtained at 1.6 K from the fitting with Eq. (3). The magnetic moment at
site I strongly fluctuates even significantly below $T_{\text{N}}$.
Conversely, $1/T_{2}$ is smaller at the minor peak measured at 198.75 MHz, and
$1/T_{2}\simeq 20$ ms-1. Although the minor peak origin remains unclear, the
relative intensity of this peak is $\sim 0.1$ times of the main peak for
$\tau\to 0$.

Table 2:  The hyperfine fields at the 55Mn sites at 2 or 1.5 K in the AFM state. The values for site IV are tentative; they strongly depend on the assignments of the spectra and the values of the quadrupole parameters $\nu_{\text{Q}}$ and $\eta$ used for the analysis.  site | hyperfine fields (T)
---|---
I | 18.86 |  |  |
II | 13.68 | 14.48 |  |
III | 1.995 | 2.476 | 2.607 | 2.938
IV | 0.43 | 0.47 | 0.47 | 0.69
Figure 6: (Color online) The hyperfine field against the ordered moment of
$\alpha$-Mn (left axis) and the field divided by the moment, or apparent
hyperfine coupling constant (right). The values of the ordered moments are
from Ref. 2.

The intensities of the hyperfine fields for the Mn sites are summarized in
Table 2. The relationship between the hyperfine fields and ordered moments
reported by Lawson et al.[2] is shown in Fig. 6. As asserted in the previous
paper[15], the hyperfine field is not proportional to the spontaneous moments,
which is significant at sites III and IV. Three possible mechanisms may cause
this nonlinear relationship[15]. The first is related to the analysis of the
neutron diffraction. The form factors for the Mn sites used in the analysis
are common to each other[1, 2]. If the actual form factors vary in different
sites, magnetic moments may differ from those obtained by the analysis. It
should be also noted that the sites II and III split in the present NMR
spectra are not considered in the analysis. The second is the site dependence
of the hyperfine coupling constant. The contribution of the $4s$ electron to
the ordered moments, or the $s$-$d$ mixing, may vary in different sites. The
hyperfine coupling constant is negative and large for the $d$ electron owing
to core polarization, whereas the $s$ electron provides a positive coupling
constant from the contact interaction. In metallic vanadium, cancellation of
the hyperfine field occurs between $s$ and $d$ electrons, leading to a small
51V NMR Knight shift reduction in the superconducting state despite the
$s$-wave symmetry[27, 28] However, this effect would not be a major
contribution for 55Mn because $d$ electrons play a significant role in
$\alpha$-Mn. The third is the hyperfine field from neighbor Mn sites. This
effect will be relatively larger as on-site moments are smaller.

In summary, ZF-NMR measurements were performed on $\alpha$-Mn in the AFM
state, and it was revealed that site II splits into two sites with slightly
tilted ($\sim 6$° from $[001]$ direction) magnetic moments. The two split
moments are almost parallel with similar moment size. The NMR spectra also
revealed that site III splits into four sites. More than two sites were
identified in the site IV region. These findings demonstrate that the AFM
$\alpha$-Mn symmetries are lower than previously proposed by neutron
measurements.

###### Acknowledgements.

This paper is dedicated to Prof. K. Asayama, who made a significant
contribution to the development of condensed matter physics based on NMR and
nuclear quadrupole resonance measurements. The authors would like to extend
their gratitude to H. Fukazawa and Y. Kohori for their insightful discussions.
One of the authors (K. F.) thanks K. Asayama for the stimulating
encouragements. K. F. and M. M. thank K. Miyoshi, M. Ohno, C. Akasaka, and W.
Irie for their experimental supports. This work was supported by the JSPS
KAKENHI Grant Numbers JP21H01042, JP20H00130, JP21K03448, and FY2021 Shimane
University Internal Competitive Grants. One of the authors (K. K.) was also
supported by and the JST SPRING (Grant Number JPMJSP2110).

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