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\DOI{10.5802/crchim.266}
\datereceived{2023-05-31}
\daterevised{2023-07-28}
\dateaccepted{2023-10-12}
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%\makeatletter
%\def\TITREspecial{\relax}
%\def\cdr@specialtitle@english{French/Nordic Special Issue on Materials and Coordination Chemistry}
%%\def\cdr@specialtitle@french{Num\'ero sp\'ecial franco-nordique sur les mat\'eriaux et la chimie de coordination}
%\makeatother

\title{Parallel-mode EPR spectra of the  hexaaqua manganese(II) Ion in
tetrahedral symmetry}

\alttitle{Spectres RPE en mode parall\`{e}le de l'ion 
hexaaquamangan\`{e}se(II) en sym\'{e}trie t\'{e}tra\'{e}drique}

\author{\firstname{Margrete} \lastname{Juel Henrichsen}\CDRorcid{0009-0001-7190-7158}\IsCorresp}
\address{Department of Chemistry, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark}
\email[M. Juel Henrichsen]{mnr410@alumni.ku.dk}

\author{\firstname{Jesper} \lastname{Bendix}\CDRorcid{0000-0003-1255-2868}}
\addressSameAs{1}{Department of Chemistry, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark}

\author{\firstname{H{\o}gni} \lastname{Weihe}\CDRorcid{0000-0001-6194-1486}}
\addressSameAs{1}{Department of Chemistry, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark}

\shortrunauthors

\keywords{\kwd{Parallel-mode EPR}
\kwd{Manganese(II)}
\kwd{Hyperfine interaction}
\kwd{Cubic symmetry}
\kwd{Flip-flop transitions}}


\altkeywords{\kwd{RPE en mode parall\`{e}le}
\kwd{Mangan\`{e}se(II)} 
\kwd{Interaction  hyperfine}
\kwd{Sym\'{e}trie cubique}
\kwd{Transitions flip-flop}}


\begin{abstract}
Parallel-mode X-band EPR spectra of the manganese(II) hexaaqua ion as
substitutional impurity in Cs[Mg(H\tsub{2}O)\tsub{6}]AsO\tsub{4} are
presented and interpreted. In this lattice the aqua ion, considering
also the disposition of the hydrogen atoms, occupies a crystallographic
site with tetrahedral symmetry. This rare situation enables that the
line positions as well as the intensities in the parallel-mode EPR
spectra can be modelled with a simple three-parameter cubic-symmetry
spin Hamiltonian, including the isotropic Zeeman interaction ($g$)
pertinent to the $S=\sfrac{5}{2}$ electronic spin, the isotropic
hyperfine interaction ($A$), between the electronic spin and the
$I=\sfrac{5}{2}$ nuclear spin, and the cubic splitting ($a$) of the
$S=\sfrac{5}{2}$ manifold. A simple formalism, based on perturbation
theory to grossly account for the observed intensities, is presented.
We observe all parallel-mode-allowed hyperfine transitions associated
with all fine-structure transitions.
\end{abstract}

\begin{altabstract}
Les spectres RPE en bande X et en mode parall\`{e}le de  l'ion
hexaaquamangan\`{e}se(II) utilis\'{e}s comme dopant dans 
Cs[Mg(H\tsub{2}O)\tsub{6}]AsO\tsub{4} sont  pr\'{e}sent\'{e}s. Dans ce
r\'{e}seau, l'aqua-ion, compte tenu de la  disposition des atomes
d'hydrog\`{e}ne, occupe un site cristallographique  \`{a} sym\'{e}trie
t\'{e}tra\'{e}drique. Cette situation rare permet de  mod\'{e}liser les
positions des raies ainsi que les intensit\'{e}s des  spectres RPE en
mode parall\`{e}le \`{a} l'aide d'un simple hamiltonien de  spin \`{a}
trois param\`{e}tres \`{a} sym\'{e}trie cubique, comprenant 
l'interaction Zeeman isotrope ($g$) relative au spin \'{e}lectronique 
$S=\sfrac{5}{2}$, l'interaction hyperfine isotrope ($A$), entre le spin
\'{e}lectronique  et le spin nucl\'{e}aire $I=\sfrac{5}{2}$, et le
d\'{e}doublement cubique ($a$) de  l'ensemble $S=\sfrac{5}{2}$. Un
formalisme simple, bas\'{e} sur la th\'{e}orie des  perturbations, est
pr\'{e}sent\'{e} pour rendre compte des intensit\'{e}s  observ\'{e}es.
Nous observons toutes les transitions hyperfines  autoris\'{e}es en
mode parall\`{e}le, associ\'{e}es \`{a} toutes les  transitions de
structure fine.
\end{altabstract}
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\section{Introduction}
Parallel-mode EPR (Electron Paramagnetic Resonance) spectra are
recorded with the magnetic component of the microwave radiation
oscillating parallel to the applied magnetic field, in contrast to the
normal perpendicular-mode setting where the magnetic component of the
microwave radiation is perpendicular to the applied magnetic field.
Different selection rules are associated with these two instrumental
settings, and therefore the parallel-mode setting may induce
transitions inaccessible by perpendicular mode. The intensity and
overall appearance of the parallel-mode spectrum of a given spin system
is strongly dependent on the composition of the spin system, and the
magnitude of (as well as the ratio between) various parameters in the
effective Hamiltonian describing the spin system. These parameters
encompass the quantification of Zeeman terms, various anisotropy
parameters, exchange coupling constants, and hyperfine coupling
constants. Hence, the simple appearance of a parallel-mode spectrum,
i.e.,\ the number of lines, may reveal the intensity-giving
mechanism.

Parallel-mode EPR  spectra have been successfully recorded for a wide
variety of both simple and composite spin systems. In the following,
not exhaustive list, we group the systems into two classes, namely
those exhibiting no resolved hyperfine splittings, and those exhibiting
clear hyperfine splittings.

The former class is exemplified by the following works: the
parallel-mode setting was used to elucidate the electronic spin-quintet
($S=2$) ground state of the iron(II) ion in myoglobin~\cite{DEBRUNNER};
the ground state $S=4$ spin multiplet of the all-ferrous state of the
[4Fe--4S] cluster in nitrogenase was characterized by parallel-mode
EPR~\cite{MUNCK}; and several parallel-mode resonances were detected in
the ground $S=6$ state of a dodecanuclear chromium(III) spin
cluster~\cite{WINPENNY}.

For the latter class exhibiting clear hyperfine splittings,
parallel-mode EPR was applied: to determine the sign and magnitude of
the pseudo-axial anisotropy splitting in manganese(III) ions,
$S=2,I=\sfrac{5}{2}$ (with $I$ being the nuclear-spin quantum number),
in superoxide dismutase~\cite{CAMPBELL}; to study small splittings
arising from the Jahn--Teller distortion in the manganese(III) hexaaqua
ion~\cite{PIGGOTT1}; to observe and resolve the Mn--F superhyperfine
interaction in two fluorido complexes of manganese(III), namely
$\mathrm{MnF}_{6}^{3-}$ and $\mathrm{Mn(salen)F}_{2}^{-}$~\cite{PIGGOTT2};
and to characterize several cobalt(II) complexes with
$(S,I)=(\sfrac{3}{2},\sfrac{7}{2})$~\cite{MARTS}. Recently, parallel-mode
spectra of Bi as an impurity in silicon were presented and the
complementary information obtained from parallel-mode was used to aid
the interpretation of the perpendicular-mode spectra of two molecular
bismuth radicals with $(S,I)=(\sfrac{1}{2},\sfrac{9}{2})$~\cite{HAAK}.
Also recently, a detailed exposition of the parallel-mode EPR spectrum
of hydrogen atoms with $(S,I)=(\sfrac{1}{2},\sfrac{1}{2})$ and deuterium
atoms with $(S,I)=(\sfrac{1}{2},1)$ trapped in silsesquioxanes was
given~\cite{SANAKIS}. We note in passing that understanding this very
simple spin system is important for radio astronomy, as the splitting
of the 21~cm, 1420.4~MHz, line originating from hydrogen atoms in
galaxies might give information about interstellar magnetic
fields~\cite{WEILHA}.

For an isolated isotropic spin multiplet with spin quantum number $S$,
the parallel-mode spectrum has no intensity simply because the $2S+1$
projections $|S,M\rangle$ are exact eigenfunctions to the
$\hat{S}_z$ operator (cf. Equation~(\ref{eqn5}) below). Therefore, the
parallel-mode intensity observed in the nuclear spin--free spin systems,
as, e.g.,\ in refs~\cite{DEBRUNNER,MUNCK,WINPENNY},    is
induced by the anisotropic terms in the Hamiltonian. Parallel-mode
intensities in these spin systems are rather high and can be comparable
to the usual perpendicular-mode intensities.

However,  for an electronic spin interacting with a nuclear spin, the
situation is slightly different as here the hyperfine interaction
constitutes an additional intensity-giving mechanism. All the
parallel-mode lines in the class of systems exhibiting resolved
hyperfine splittings, quoted above, are hyperfine transitions between
the two components of an effective
non-Kramers~\cite{CAMPBELL,PIGGOTT1,PIGGOTT2} or a
genuine~\cite{MARTS,HAAK}  $S=\sfrac{1}{2}$ spin system. The
parallel-mode spectra of the manganese(III) systems all exhibit
$2I+1=6$ hyperfine lines connected to only a single fine-structure
transition, with $I=\sfrac{5}{2}$ appropriate for the \tsup{55}Mn
nucleus. In these systems, the intensity is induced by a small
splitting of a non-Kramers doublet. However, the parallel-mode spectrum
of Bi in silicon~\cite{HAAK} exhibits $2I=9$ lines, with
$I=\sfrac{9}{2}$ for \tsup{209}Bi. Likewise, spectrum of hydrogen and
deuterium exhibit $2I_{\mathrm{H}}=1$ and $2I_{\mathrm{D}}=2$ lines,
respectively, where $I_{\mathrm{H}}=\sfrac{1}{2}$ and
$I_{\mathrm{D}}=1$, respectively. In the bismuth, hydrogen, and
deuterium spectra, the intensity is induced solely by the hyperfine
interaction.

\begin{figure}
\includegraphics{fig01}
\caption{Illustration of the hexaaqua manganese(II) ion in strict
tetrahedral symmetry. Blue: Manganese, Red: Oxygen, White: Hydrogen.
The circumscribed tetrahedron whose sides straddle the hydrogen atoms
has been drawn to emphasize the $\mathrm{T}_{\mathrm{d}}$ lattice-point
symmetry of the hexaaqua ion.\label{FIG1}}
\end{figure}

We are not aware of any reports on parallel-mode spectra of an
$S>\sfrac{1}{2}$ spin system exhibiting all the possible allowed
fine-structure transitions. Therefore,   in this contribution we
present parallel- as  well as perpendicular-mode spectra of Mn\tsup{2+} in
strict tetrahedral symmetry  (Figure~\ref{FIG1}) where all metal
\mbox{centers}	 are magnetically and crystallographically equivalent.


This paper is organized as follows. In Section~\ref{sec2} we summarize
the experimental details; in Section~\ref{sec3} we present a
first-order perturbational treatment of the parallel-mode intensities
for a spin system where an electronic spin $S$ is interacting with a
nuclear spin $I$ via an isotropic hyperfine interaction. Although the
treatment is fairly general, it is aimed at the present spin system,
namely Mn\tsup{2+} with $S=\sfrac{5}{2}$ and $I=\sfrac{5}{2}$; and in
Section~\ref{sec4} we present and discuss single-crystal
perpendicular-mode and parallel-mode EPR spectra of manganese(II)
hexaaqua ions in perfect tetrahedral symmetry. 

\section{Experimental}\label{sec2}

\subsection{Synthesis}
The synthesis of single crystals of
Cs[Mg(H\tsub{2}O)\tsub{6}]AsO\tsub{4} was performed largely as
described in refs.~\cite{REDETERMINATION,FERRARI}, but with some
modifications in order to produce large single crystals.

Initially, two solutions were prepared. Solution 1: to 90~ml of water
were added 3.92~g of 75\% arsenic acid (20.7~mmol) and 2.77~g
(20.4~mmol) of sodium acetate trihydrate. Solution 2: to 43~ml of
water were added 2.0~ml (30~mmol) of glacial acetic acid, 3.34~g
(10.0~mmol) cesium carbonate, and 4.20~g (20.0~mmol) of magnesium
acetate tetrahydrate. To solution 2 were finally added 5~mg
(0.02~mmol) or 50~mg (0.20~mmol) of manganese(II) acetate
tetrahydrate to achieve a doping level of 0.1\% or 1\%, respectively.

Solution 1 (2~ml) and solution 2 (1~ml) were mixed in a small test
tube resulting in a clear solution. The small test tube was placed in a
larger test tube containing 4~ml pyridine. The large test tube was
stoppered and left for five days. Large octahedrally shaped crystals
were separated from a viscous gel by filtration and washed with water.
Crystals with edge length up to 1.5~mm could be obtained by this slow
vapor diffusion of pyridine into the solution. A microcrystalline
powder could be produced by slow addition of ammonia (1~M).


\subsection{X-ray crystallography}
The identity of the single crystals as well as the microcrystalline
powder was verified with single-crystal X-ray diffraction as well as
X-ray powder diffraction (see supporting information). The
single-crystal X-ray diffraction studies were performed at 100~K on
a Bruker D8 VENTURE diffractometer equipped with a Mo K$\alpha$
high-brilliance I$\mu$S radiation source ($\lambda = 0.71073$~\AA), a
multilayer X-ray mirror and a PHOTON 100 CMOS detector, and an Oxford
Cryosystems low-temperature device. The instrument was controlled with
the APEX3 software package using SAINT~\cite{APEX}. Final cell
constants were obtained from least squares fits of several thousand
strong reflections. Intensity data were corrected for absorption using
intensities of redundant reflections with the program SADABS 
\cite{SADABS}. Powder X-ray diffraction studies were performed at room temperature on
a Bruker D8 advanced equipped with a Cu K$\alpha$ source ($\lambda =
1.5406$~\AA).


\subsection{EPR spectroscopy}
Room-temperature EPR spectra were recorded on a Bruker Elexsys
spectrometer equipped with an ER4116DM dual-mode cavity. The crystal
was rotated around $\langle 110\rangle$ axis, and hence the magnetic
field vector sweeps a plane containing one four-fold (S\tsub{4}),
one two-fold (C\tsub{2}), and two three-fold (C\tsub{3}) axes.  The
power used for perpendicular-mode measurements was 6.325~mW, and for
parallel mode 200~mW. Modulation amplitude was in perpendicular-mode
1~G, and for parallel-mode a larger modulation amplitude of 4~G was
used due to the low intensity. At room temperature we observed no
saturation effects nor multiple quantum transitions~\cite{BERTAINA,SOROKIN}.


\section{Theory}\label{sec3}
Eigenvalues and eigenvectors pertinent to the EPR spectra of the
tetrahedral symmetric manganese(II) hexaaqua ion are obtained by
numerical diagonalization of the matrix representation of the
spin Hamiltonian~\cite{GRIFFITH}
{\begin{equation}\label{SPINHAM}
\hat{H} = g\mu_{\mathrm{B}} \boldsymbol{B}\boldcdot\hat{\boldsymbol{S}} +
A\hat{\boldsymbol{S}}\boldcdot\hat{\boldsymbol{I}} +
\frac{a}{6}\left(\hat{S}_x^4+\hat{S}_y^4+\hat{S}_z^4-\frac{707}{16}\right),
\end{equation}}\noindent
which is applicable to our spin system with $S=\sfrac{5}{2}$ and
$I=\sfrac{5}{2}$. In Equation~(\ref{SPINHAM}) $\boldsymbol{B}$,
$\hat{\boldsymbol{S}}=(\hat{S}_x,\hat{S}_y,\hat{S}_z)$, and
$\hat{\boldsymbol{I}}=(\hat{I}_x,\hat{I}_y,\hat{I}_z)$   are the
magnetic field vector, the electron spin operator, and the nuclear spin
operator, respectively. $g$ is the electron spin $g$ factor, $A$ is the
hyperfine coupling constant, and $a$ quantifies the cubic splitting of
the $S=\sfrac{5}{2}$ manifold into a quartet and a doublet. This
splitting is $3a$ with the quartet higher in energy for $a>0$. The
linewidths of the spectra presented below did not warrant the inclusion
of terms accounting for the nuclear Zeeman effect and nuclear
quadrupole splittings.

Fitting of spectral traces of both perpendicular-mode and parallel-mode
spectra was performed with a locally developed software as described
earlier\unskip\break~\cite{MOR,PILIGKOS}.

\begin{figure*}
\includegraphics{fig02}
\caption{Perpendicular-mode (9.6~GHz) density plot of the angular
dependence of the 30 allowed transitions of Mn\tsup{2+}(0.1\%) in
Cs[Mg(H\tsub{2}O)\tsub{6}]AsO\tsub{4}. Red and blue colors represent 
positive and negative derivative intensities, respectively. The crystal
was rotated around a C\tsub{2} axis with the magnetic field vector
sweeping a plane containing molecular S\tsub{4}, C\tsub{3},  and
C\tsub{2} axes. These special orientations are labelled on the right
ordinate axis. The solid black lines are computed resonance magnetic
fields based on Equation~(\ref{SPINHAM}) and the parameters in
Table~\ref{TAB1}.\label{FIG2}}
\vspace*{-4pt}
\end{figure*}


To grossly account for the main features, i.e.,\ the number of
lines and their intensities in the parallel-mode  
X-band EPR spectra, we wrote down the first-order corrected wave functions
for this system. To do so we chose the $z$-axis in the direction of the
magnetic field and approximated the Hamiltonian as
{\begin{equation}\label{HAMAPPROX}
\hat{H} \approx \hat{H}^{(0)} + \hat{H}^{(1)}  ,
\end{equation}}\noindent
where $\hat{H}^{(0)} = g\mu_{\mathrm{B}}B\hat{S}_z +
A\hat{S}_z\hat{I}_z$ and $\hat{H}^{(1)} =
\sofrac{A}{2}(\hat{S}_+\hat{I}_- \ubreak + \hat{S}_-\hat{I}_+)$. In this
discussion of the parallel-mode intensities we are ignoring the last
term of Equation~(\ref{SPINHAM}). In the present case with cubic
manganese(II), this is fully justified, as the energetic consequence of
the $a$ term is very much smaller compared to the hyperfine interaction
and the Zeeman term at field values corresponding to resonances at
X-band. However, in Section~\ref{sec4} we use the full Hamiltonian
Equation~(\ref{SPINHAM}) to reproduce the experimental spectra. The
direct-product functions $|M,m\rangle$ with $M$ and $m$ being electron
spin and nuclear spin components, respectively, are eigenfunctions to
$\hat{H}^{(0)}$. With $\hat{H}^{(1)}$ as a perturbation we obtain the
following expression for the non-normalized first-order corrected
wavefunctions
{\begin{eqnarray}
&&
|M,m {\mathop{\rangle}}_{\mathrm{c}} = |M,m {\mathop{\rangle}} \nonumber\\
&&\quad
 -\, \left(\frac{A}{2}\right)
 \frac{{\mathop{\langle}} M+1,m-1|\hat{S}_+\hat{I}_- |M,m {\mathop{\rangle}}}
 {E(M+1,m-1)-E(M,m)} |M+1,m-1 {\mathop{\rangle}} \nonumber\\ 
 &&\quad -\, \left(\frac{A}{2}\right)
\frac{{\mathop{\langle}} M-1,m+1|\hat{S}_-\hat{I}_+ |M,m {\mathop{\rangle}}}
 {E(M-1,m+1)-E(M,m)} |M-1,m+1 {\mathop{\rangle}}  .\nonumber\\
\label{APPROX1}
\end{eqnarray}}\noindent
At field values corresponding to resonances at X-band, the denominators
have absolute values roughly equal to $h\nu$. Therefore, we approximate
{\begin{eqnarray}
&&
|M,m {\mathop{\rangle}}_{\mathrm{c}} = |M,m {\mathop{\rangle}}
\nonumber\\
&&\quad -\, \left(\frac{A}{2h\nu}\right)
 {\mathop{\langle}} M+1,m-1|\hat{S}_+\hat{I}_-|M,m {\mathop{\rangle}}
 |M+1,m-1 {\mathop{\rangle}} \nonumber\\ 
&&\quad +\, \left(\frac{A}{2h\nu}\right)
 {\mathop{\langle}} M-1,m+1|\hat{S}_-\hat{I}_+ |M,m {\mathop{\rangle}}
 |M-1,m+1 {\mathop{\rangle}}  .\nonumber\\
\label{APPROX2}
\end{eqnarray}}\noindent

The intensity of the transitions in parallel mode is now computed
as the squared expectation value of $g\hat{S}_z$, as
{\begin{equation}\label{eqn5}
I (|M,m {\mathop{\rangle}}_{\mathrm{c}} \rightarrow |M',m' 
{\mathop{\rangle}}_{\mathrm{c}} ) =
|({\mathop{\langle}} M',m'|_{\mathrm{c}})
g\hat{S}_z (| M,m {\mathop{\rangle}}_{\mathrm{c}})|^2.
\end{equation}}\noindent
Inspection of the $(2S+1)(2I+1)$ first-order corrected wavefunctions
reveals that only $(2S)(2I)$ transitions have non-vanishing
parallel-mode intensity in the $g\approx 2$ field region. These are
those for which $M'=M \pm 1$ concerted with $m' \mp 1$. Hence,
{\begin{eqnarray}
&&I (|M,m {\mathop{\rangle}}_{\mathrm{c}}
\rightarrow
|M+1,m-1 {\mathop{\rangle}}_{\mathrm{c}}) 
\nonumber\\ 
&&\quad =\left(
\frac{gA}{2h\nu}
\right)^2
[(S-M)(S+M+1)]
[(I+m)(I-m+1)]  . \nonumber\\
\label{INTENS}
\end{eqnarray}}\noindent
Hence, each $\Delta M = +1$ transition consists of $2I$ hyperfine lines
with relative intensity given by the last factor of Equation~(\ref{INTENS})  
and a common scaling factor given by the first two factors of
Equation~(\ref{INTENS}).

For $S>\sfrac{1}{2}$ the factor containing $S$ influences the relative
intensities of all the possible resonances. The smallest and largest
value of this factor is $2S$ and $S(S+1)+x$, respectively, with $x=0$
and $x=\sfrac{1}{4}$ for $S$ being integer and half-integer,
respectively. Similarly, the smallest and largest value of the factor
containing $I$ is $2I$ and $I(I+1)+y$, respectively, with $y=0$ and
$y=\sfrac{1}{4}$ for $I$ being integer and half-integer,
respectively. Hence, the ratio of the largest to smallest intensity is
{\begin{equation}
\frac{I(\mathrm{largest})}{I(\mbox{smallest})} =
\frac{[S(S+1)+x][I(I+1)+y]}{(2S)(2I)}  .
\end{equation}}\noindent
In the present case with Mn\tsup{2+}, 
$(S,I)=(\sfrac{5}{2},\sfrac{5}{2})$ we have the intensity ratio
$I(\mathrm{largest})/I(\mathrm{smallest})=81/25$.

The intensities for the allowed perpendicular-mode transitions are in
stark contrast to those for parallel mode. For perpendicular mode, the
intensities are obtained by computing the squared expectation values of
$g\hat{S}_x$, and the result is
{\begin{equation}
I(|M,m {\mathop{\rangle}}\rightarrow |M+1,m {\mathop{\rangle}}) =
\left(\frac{g}{2}\right)^2[(S-M)(S+M+1)]  .
\end{equation}}\noindent
These theoretical perpendicular-mode intensities do not contain the
factor $(A/h\nu)^2$, and unless $A$ is comparable to $h\nu$,
intensities of parallel-mode spectra are significantly less than those
of perpendicular-mode spectra. In the present case with the parameters
determined below (see Section~\ref{sec4}), this factor is less than 0.001.

\section{Results and discussion}\label{sec4}
In order to identify the crystal orientation, perpendicular-mode spectra
were recorded prior to the parallel-mode spectra.


\begin{figure*}
\includegraphics{fig03}
\caption{Experimental (solid black traces) perpendicular-mode (9.6~GHz)
spectra for three selected  magnetic field orientations from Figure~\ref{FIG1}:
$\boldsymbol{B}\parallel \mathrm{S}_4$, $\boldsymbol{B}\parallel
\mathrm{C}_3$, and $\boldsymbol{B}\parallel \mathrm{C}_2$.  Linewidths of
the resonances are in the range 7--10~G. The result of a
simultaneous fit of the three perpendicular-mode spectra to
Equation~(\ref{SPINHAM}) is shown with red solid lines.\label{FIG3}}
\vspace*{.5pc}
\end{figure*}


\begin{table*}
\caption{Room temperature spin-Hamiltonian parameters (second, third,
and fourth columns) for Mn\tsup{2+} as a six-coordinate substitutional
impurity in five different cubic lattices (first column). The fifth
column lists the metal-ligand bond length of the host lattice. Data for
perpendicular- and parallel-mode are indicated with an ``$\perp$'' and
``$\parallel$'' superscript, respectively\label{TAB1}} 
\begin{tabular}{lccccc}
\thead
Host & $g$ & $a$ ($10^{-4}~\mathrm{cm}^{-1}$) & $A$
($10^{-4}~\mathrm{cm}^{-1}$)  & M--L distance (\AA) & Reference \\
\endthead
KMgF\tsub{3}$^{a}$ & 2.000 & \08.0  &  ${-}91.2$  & 1.994 &  \cite{OGAWA}\\
Cs[Mg(H\tsub{2}O)\tsub{6}]AsO\tsub{4}$^{\perp}$ & 1.998 & \08.2  &  ${-}88.7$  &  2.064  & This work\\
Cs[Mg(H\tsub{2}O)\tsub{6}]AsO\tsub{4}$^{\parallel}$  & 1.997 & \08.2  &   ${-}88.5$  &  2.064 & This work\\
MgO$^{a}$ & 2.000 & 18.3  &  ${-}81.4$  &  2.107 &  \cite{RUBIOEPR,LOW}\\
CaO$^{a}$ & 2.001 & \05.6   &  ${-}80.8$  &  2.405 & \cite{RUBIOEPR}\\
SrO$^{a}$ & 2.001 & \03.0   &  ${-}78.2$  &  2.562 & \cite{RUBIOEPR}
\botline
\end{tabular}
\vspace*{.5pc}
\end{table*}



\subsection{Perpendicular-mode EPR} 


Single-crystal perpendicular-mode spectra of
Cs[Mg\tsub{0.999}Mn\tsub{0.001}(H\tsub{2}O)\tsub{6}]AsO\tsub{4} are
presented in Figure~\ref{FIG2} as a density plot. The spectrum extent
on the field axis is roughly 535~G. Forbidden lines of the type
$|\Delta M|=1$, $\Delta m=\pm1$ observed in other
studies~\cite{RUBIOEPR,WERTZ} of cubic Mn(II) systems were not visible
here. The peak-to-peak linewidths of 7--10~G are significantly broader
compared to linewidths in spectra of other cubic manganese(II)
systems~\cite{RUBIOEPR}. A source of the broad lines is the direct
dipole--dipole interaction between the electron spin and the 12 protons
of the coordinated water molecules. The magnitude of this dipole--dipole
interaction can be estimated by using the classical expression for the
interaction energy between two magnetic dipoles separated by the
distance $R$. This interaction energy amounts to $1.5 \times
10^{-4}~\mathrm{cm}^{-1}$ for each proton at a distance $R=2.611$~{\AA} from
the metal center. For each proton, this interaction energy translates
into approximately 1.5~G, and hence is likely the explanation for the
rather broad lines. The angular variation of the resonance magnetic
field   for each line does not exceed 36~G, i.e.,  about 6\% of the full
extent of the spectrum. Spectra for three selected orientations of the
magnetic field are shown in Figure~\ref{FIG3}. To determine the
parameters of Equation~(\ref{SPINHAM}), the three spectral traces were
simultaneously fitted to theoretical spectra based on 
Equation~(\ref{SPINHAM})~\cite{MOR,PILIGKOS}. The values obtained for
parameters $g$, $a$, and $A$ are collected in Table~\ref{TAB1}. The
calculated spectra are virtually identical to the experimental spectra.
The energy level diagram in $g\approx 2$ field region pertinent to the
magnetic field  parallel to an S\tsub{4} axis is shown in
Figure~\ref{FIG4}. The 30 allowed transitions are indicated with
vertical arrows. By comparing Figure~\ref{FIG4} with Figure~\ref{FIG2}
we see that the 5 transitions at approximately 3200~G correspond to
$\Delta M = 1$ and $m=\sfrac{5}{2}$, whereas the 5 transitions just
below 3700~G correspond to $\Delta M = 1$ and $m=\sfrac{-5}{2}$.



\begin{figure*}
\vspace*{-3pt}
\includegraphics{fig04}
\vspace*{-4pt}
\caption{Energy level diagram for $\boldsymbol{B}\parallel
\mathrm{S}_4$ in the $g\approx 2$ field region  obtained by numerical
diagonalization of the matrix representation of Equation~(\ref{SPINHAM}) with
the parameters from Table \ref{TAB1}.  The arrows indicate the allowed
perpendicular-mode (9.6~GHz) transitions with $\Delta M = 1$ and 
$\Delta m = 0$.\label{FIG4}}
\vspace*{-3pt}
\end{figure*}



\begin{figure*}
\includegraphics{fig05}
\vspace*{-4pt}
\caption{Parallel-mode (9.3~GHz) density plot of the angular dependence
of the 25 allowed transitions of Mn\tsup{2+}(1.0\%) in
Cs[Mg(H\tsub{2}O)\tsub{6}]AsO\tsub{4}. For each spectra 10 scans were
made, due to the low intensity. Color code and crystal orientations as
described in caption to Figure~\ref{FIG2}. The solid black lines are computed
resonance magnetic fields based on Equation~(\ref{SPINHAM}) and the
parameters in Table~\ref{TAB1}.\label{FIG5}}
\vspace*{-10pt}
\end{figure*}


\begin{figure*}
\vspace*{3pt}
\includegraphics{fig06}
\vspace*{3pt}
\caption{Experimental (solid black traces) parallel-mode (9.3~GHz)
spectra for three selected magnetic field orientations from Figure~\ref{FIG5}:
$\boldsymbol{B}\parallel \mathrm{S}_4$, $\boldsymbol{B}\parallel
\mathrm{C}_3$, and $\boldsymbol{B}\parallel \mathrm{C}_2$. Linewidths of
the resonances are in the range 17--26~G. The result  of a
simultaneous fit of the three parallel-mode spectra to
Equation~(\ref{SPINHAM}) is shown with red solid lines. The presences of a
small amount, 0.7\permil, of perpendicular-mode in the microwave
field is the source of the shoulders marked with a star (*). Relative
intensities of the 9 observed lines for $\boldsymbol{B}\parallel
\mathrm{C}_2$: 19, 64, 152, 207, 259, 263, 182, 80, 20.\label{FIG6}}
\vspace*{3pt}
\end{figure*}


\begin{figure*}
\vspace*{4pt}
\includegraphics{fig07}
\vspace*{4pt}
\caption{Energy level diagram for $\boldsymbol{B}\parallel
\mathrm{C}_2$ in the $g\approx 2$ field region  obtained as described
in the caption to Figure~\ref{FIG3}.  The arrows indicate the allowed
parallel-mode (9.3~GHz) transitions with $\Delta M = +1$ and  $\Delta m
= -1$. The number attached to each arrow represent the relative
intensity obtained from the product of the last two factors of
Equation~(\ref{INTENS}).\label{FIG7}}
\vspace*{3pt}
\end{figure*}



\subsection{Parallel-mode EPR}

In order to increase the signal-to-noise ratio we recorded the
parallel-mode spectra of a crystal\unskip\break containing 1\% Mn\tsup{2+}.
Single-crystal parallel-mode spectra of
Cs[Mg\tsub{0.99}Mn\tsub{0.01}(H\tsub{2}O)\tsub{6}]AsO\tsub{4} are
presented in Figure~\ref{FIG5} as a density plot. The spectrum extent
on the field axis is 790~G, i.e.,\ significantly larger compared to
Figure~\ref{FIG2}. The broader lines, 17--26~G, are a result of the
higher doping level~\cite{SKINNER}. The spectra exhibit 9 features
centered at roughly 2960, 3044, 3126, 3215, 3303, 3403, 3503, 3608, and
3722~G, and the features are composed of 1, 2, 3, 4, 5, 4, 3, 2, and 1,
respectively, separate, partly overlapping or fully overlapping
transitions (see also Figure~\ref{FIG7}). Experimental spectra for three
selected orientations of the magnetic field relative to the molecular
axes are shown as black traces in Figure~\ref{FIG6}. Due to the rather
large bandwidths combined with the small splittings, only 20 of the 25
possible transitions could be resolved for this high doping level (see
the spectrum with $B\parallel \mathrm{S}_4$ in Figure~\ref{FIG6}). The
spectrum with $B\parallel \mathrm{C}_2$, lowest trace in
Figure~\ref{FIG6}, exhibits only 9 lines as a result of overlapping
transitions (see also Figure~\ref{FIG7}). The relative intensity of each
line can be estimated as $h_{\mathrm{pp}} \times w_{\mathrm{pp}}^2$,
where $h_{\mathrm{pp}}$ and $w_{\mathrm{pp}}$ are the peak-to-peak
height and width, respectively. The result is given in the caption of
Figure~\ref{FIG6}. The corresponding theoretical intensities are
obtained by vertical additions of the relative intensities in
Figure~\ref{FIG7}. The result is 25, 80, 154, 224, 259, 224, 154, 80,
25. The agreement is fair considering that most of the lines in the
lowest trace of Figure~\ref{FIG6} are composite and not perfectly
overlapping lines.

The three spectral traces were again simultaneously fitted to
theoretical spectra based on Equation~(\ref{SPINHAM}) in order to
determine the parameters of Equation~(\ref{SPINHAM}). The obtained
parameters are collected in Table~\ref{TAB1}. The theoretical spectra
are shown with red lines in Figure~\ref{FIG6}, and the agreement is
good. The most noticeable discrepancies are indicated with a ``*'' in
the lowest trace of  Figure~\ref{FIG6}. These weak shoulders result
from the presence of 0.7\permil\ of perpendicular-mode microwave field.


The energy level diagram in the $g\approx 2$ field region calculated
with the magnetic field  parallel to a C\tsub{2} axis is shown in
Figure~\ref{FIG7}. The 25 parallel-mode allowed transitions are
indicated with vertical arrows.


\section{Discussion}


The parameter values determined in the two independent fits of the
spectra in Figures~\ref{FIG3} and~\ref{FIG6} are consistent and are
collected in Table~\ref{TAB1}. In the table we have also collected EPR
data of \mbox{manganese(II)} ions as a substitutional impurity in other cubic
hosts wherein the metal ion is six-coordinated?

The cubic parameter $a$ of the hexaaqua ion is found to be positive in
agreement with all results found for eight-, six-, and tetra-coordinated
Mn\tsup{2+} in cubic lattices~\cite{ABRAGAMBLEANEY}.

The isotropic hyperfine coupling constant $A$ for the hexaaqua ion is   
slightly smaller but comparable in magnitude with the hyperfine
coupling constant found for the hexafluorido ion in KMgF\tsub{3} (see
Table~\ref{TAB1}). This is in agreement with water and fluoride having
similar ligand-to-metal donor properties~\cite{GLERUP}. The slightly
smaller value of $A$ for the aqua ion is also in agreement with the
Mn--OH\tsub{2} bond being more covalent compared to the Mn--F
bond~\cite{ABRAGAMBLEANEY}. By the same token, the magnitude of the
$A$ parameters for Mn\tsup{2+} in the oxide lattices (see Table~\ref{TAB1}) are
significantly smaller than that for the aqua ion.   This indicates that
the Mn--O bond is more covalent than the Mn--OH\tsub{2} bond despite the
fact that the Mn--O bond length is significantly longer than the
Mn--OH\tsub{2} bond length. Finally we notice that slightly larger and
anisotropic hyperfine coupling constants were found for
$\mathrm{Mn}(\mathrm{H}_{2}\mathrm{O})_{6}^{2+}$
as an impurity in hexaaqua zinc
hexafluorosilicate and in the ammonium zinc Tutton salt~\cite{INGRAM}.
\looseness=-1

In summary, we have presented and interpreted parallel-mode EPR spectra
of   the manganese(II) aqua ion as a substitutional impurity in
Cs[Mg(H\tsub{2}O)\tsub{6}]AsO\tsub{4}, wherein the aqua ion occupies a
crystallographic site with perfect tetrahedral symmetry. The
interpretation agrees with the crystal symmetry by the fact that only a
three-parameter cubic model excellently reproduces the experimental
spectra. To the best of our knowledge, this is the first report of
parallel-mode EPR spectrum of an $S>\sfrac{1}{2}$ spin system wherein
all the allowed fine-structure transitions with associated hyperfine
transitions are observed. Expressions based on perturbation theory were
derived to account for the relative intensities of the allowed
transitions.  All the allowed parallel-mode resonances are of the
flip-flop type, being \mbox{zero-quantum} coherences \cite{WEILBOOK}, i.e.,
energy but no angular momentum is transferred from the microwave field
to the spin system.

{\vspace*{-2pt}}

\section*{Declaration of interests} 

{\vspace*{-2pt}}

The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\back{}

{\vspace*{-2pt}}

\section*{Supplementary data}

{\vspace*{-2pt}}

Supporting information for this article is available on the journal's
website under \printDOI\ or from the author.

\CDRsupplementaryTwotypes{supplementary-material}{\cdrattach{crchim-266-suppl.pdf}}

{\vspace*{-2pt}}

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