Electron Paramagnetic Resonance Linewidths and Lineshapes for the Molecular Magnets Fe$_{\mathbf 8}$ and Mn$_{\mathbf 12}$
Kyungwha ${\mathrm Park^1}$
M.~A.\ ${\mathrm Novotny^2}$
N.~S.\ ${\mathrm Dalal^3}$
S.\ ${\mathrm Hill^{4}}$
and P.~A.\ ${\mathrm Rikvold^{1,5}}$
% insert abstract here We study theoretically Electron Paramagentic Resonance (EPR) linewidths for single crystals of the molecular magnets Fe$_8$ and Mn$_{12}$ as functions of energy eigenstates $M_s$, frequency, and temperature when a magnetic field along the easy axis is swept at fixed excitation frequency. This work was motivated by recent EPR experiments. %which revealed complex nonlinearities in the linewidths. % as functions of $M_s$, frequency, and temperature. To calculate the linewidths, we use density-matrix equations, including dipolar interactions and distributions of the uniaxial anisotropy parameter $D$ and the Land\'{e} $g$ factor. Our calculated linewidths agree well with the experimental data. We also examine the lineshapes of the EPR spectra due to local rotations of the magnetic anisotropy axes caused by defects in samples. Our preliminary results predict that this effect leads to asymmetry in the EPR spectra. %We find that for the Fe$_8$ %sample examined the distribution in $D$ and the dipolar interactions are %crucial to explain the linewidths, while for the Mn$_{12}$ sample %the distributions in $D$ and $g$ are more important than the %dipolar interactions. %the distribution in $D$ broadens the linewidths %with increasing absolute value of $2M_s$$-$$1$, while the distribution %in $g$ narrows the linewidths with increasing $M_s$. %The dipolar interactions broaden the linewidths with increasing %$M_s$. %Our calculated linewidths agree well with the experimental %data for various frequencies %when the standard deviations of $D$ and $g$, and the effective %distance between dipoles are estimated as 0.01$D$ (0.02$D$), %0 (0.008$g$), and 11 $\AA$ (14 $\AA$) for Fe$_8$ (Mn$_{12}$), %respectively.
paper05.tex
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\documentstyle[prb,aps,preprint,epsf]{revtex}
\begin{document}
\title{Electron Paramagnetic Resonance Linewidths and Lineshapes
for the Molecular Magnets Fe$_{\mathbf 8}$ and Mn$_{\mathbf 12}$}
% \draft command makes pacs numbers print
\draft
% repeat the \author\address pair as needed
\author{Kyungwha ${\mathrm Park^1}$ 
\and M.~A.\ ${\mathrm Novotny^2}$ 
\and N.~S.\ ${\mathrm Dalal^3}$ \and S.\ ${\mathrm Hill^{4}}$, 
and P.~A.\ ${\mathrm Rikvold^{1,5}}$}
\address{${\mathrm ^{1}}$School of Computational Science and 
Information Technology, Florida State University, Tallahassee, Florida 32306 \\
${\mathrm ^{2}}$Department of Physics and Astronomy, Mississippi 
State University, Mississippi State, Mississippi 39762 \\
${\mathrm ^{3}}$Department of Chemistry, Florida State University,
Tallahassee, Florida 32306 \\
${\mathrm ^{4}}$Department of Physics, 
University of Florida, Gainesville, Florida 32611 \\
${\mathrm ^{5}}$Center for Materials Research and 
Technology and Department of Physics, 
Florida State University, Tallahassee, Florida 32306}
%\author{Kyungwha Park}
%\address{School of Computational Science and 
%Information Technology, Florida State University, Tallahassee, Florida 32306}
%\author{M.~A.\ Novotny}
%\address{Department of Physics and Astronomy, Mississippi State University,
%Mississippi State, Mississippi 39762}
%\author{N.~S.\ Dalal}
%\address{Department of Chemistry, Florida State University, 
%Tallahassee, Florida 32306}
%\author{S. Hill}
%\address{Department of Physics, University of Florida, Gainesville, 
%Florida 32611}
%\author{P.~A.\ Rikvold}
%\address{Department of Physics, School of Computational Science and 
%Information Technology, and Center for Materials Research and 
%Technology, Florida State University, Tallahassee, Florida 32306}
\date{\today}
\maketitle
\begin{abstract}
% insert abstract here
We study theoretically Electron Paramagentic Resonance (EPR) 
linewidths for single crystals of
the molecular magnets Fe$_8$ and Mn$_{12}$ as functions of energy
eigenstates $M_s$, frequency, and temperature when a magnetic field 
along the easy axis is swept at fixed excitation frequency.
This work was motivated by recent EPR experiments.
%which revealed complex nonlinearities in the linewidths.
% as functions of $M_s$, frequency, and temperature. 
To calculate the linewidths, we use density-matrix equations, 
including dipolar interactions and distributions of the uniaxial 
anisotropy parameter $D$ and the Land\'{e} $g$ factor. Our 
calculated linewidths agree well with the experimental data. 
We also examine the lineshapes of the EPR spectra due to local
rotations of the magnetic anisotropy axes caused by defects in
samples. Our preliminary results predict that this effect leads to 
asymmetry in the EPR spectra.
%We find that for the Fe$_8$ 
%sample examined the distribution in $D$ and the dipolar interactions are 
%crucial to explain the linewidths, while for the Mn$_{12}$ sample 
%the distributions in $D$ and $g$ are more important than the 
%dipolar interactions.
%the distribution in $D$ broadens the linewidths 
%with increasing absolute value of $2M_s$$-$$1$, while the distribution 
%in $g$ narrows the linewidths with increasing $M_s$. 
%The dipolar interactions broaden the linewidths with increasing 
%$M_s$. 
%Our calculated linewidths agree well with the experimental 
%data for various frequencies 
%when the standard deviations of $D$ and $g$, and the effective 
%distance between dipoles are estimated as  0.01$D$ (0.02$D$), 
%0 (0.008$g$), and 11 $\AA$ (14  $\AA$) for Fe$_8$ (Mn$_{12}$), 
%respectively.
\end{abstract}
% insert suggested PACS numbers in braces on next line
\pacs{PACS numbers:75.45.+j,75.50.Xx,76.30.-v}

% body of paper here

%\begin{multicols}{2}

Molecular magnets such as ${\mathrm Mn_{12}}$
\cite{LIS80} and ${\mathrm Fe_8}$ \cite{WIEG84} have recently 
drawn much attention because of the macroscopic quantum tunneling 
of their magnetizations 
at low temperatures \cite{CHUD98} and their possible applications
to quantum computing.\cite{LEUE00-3} 
These materials consist of many identical clusters (see Fig. \ref{conf})
with the same magnetic properties and characteristic energies.  
Each cluster has many different species of ions and atoms, with a
total spin angular momentum in the ground state of $S$$=$$10$.  
The clusters have strong crystal-field anisotropy and, thus,
a well-defined easy axis, 
and the magnetic interaction between different clusters is weak.
%Many competing models have been proposed to 
%explain MQT in molecular magnets: 
%higher-order transverse anisotropy,\cite{HART96} 
%thermally-assisted quantum tunneling,\cite{FRIE96} 
%the Landau-Zener effect,\cite{LEUE00-2}
%and dipolar interactions with dynamic hyperfine fields.\cite{PROK98} 
%So far, the MQT in these materials has not been completely understood,
%and new experimental and theoretical approaches are needed.

Recently, multi-frequency Electron Paramagnetic Resonance (EPR)
measurements\cite{HILL98,MACC01,PERE98} on single-crystals of the
molecular magnets Fe$_8$ and Mn$_{12}$ showed interesting
results in the linewidths as functions of the energy level $M_s$,
excitation frequency, and temperature when a magnetic field along 
the easy axis was swept with the excitation 
frequency fixed. Figures~\ref{FWHM}(a) and (b) show the 
experimental linewidths as a function of the value of $M_s$ from which the
spin system is excited. For example, $M_s$$=$5 in Fig.~\ref{FWHM}(a)
denotes the transition $M_s$$=$5$\rightarrow$4. 
The experimental results are that (i) the linewidths are about 200 G 
to 1400 G at $T$$=$10 K for ${\mathrm Fe_8}$
[Figs.~\ref{FWHM}(a) and (b)], and about 
1000 G to 2000 G at $T$$=$25 K for ${\mathrm Mn_{12}}$;
(ii) the linewidths increase {\it non-linearly} as a function of 
the absolute value of the energy eigenstate $M_s$
[Figs.~\ref{FWHM}(a) and (b)];
(iii) the linewidths attain a minimum at 
$M_s$$=$1 and $M_s$$=$0 
%$M_s$$=$1$\rightarrow$0 and $M_s$$=$0$\rightarrow$-1
[Fig.~\ref{FWHM}(a)]; (iv) the linewidths decrease with increasing
frequency (compare the linewidth for $M_s$$=$5
in Fig.~\ref{FWHM}(a) with that in Fig.~\ref{FWHM}(b)). 
%These experimental results are different
%from a previous measurement of the EPR linewidths
%as a function of $M_s$ for Fe$_8$ and Fe$_4$ \cite{BARR00},
%in the sense that 

In this paper, we present the theoretical EPR linewidths for
single crystals of Fe$_8$ and Mn$_{12}$,
using density-matrix equations with the assumption that the uniaxial 
crystal-field anisotropy parameter $D$ and the Land{\'e} $g$ factor 
are randomly distributed around their mean values 
(``$D$-strain'' and ``$g$-strain'' effects \cite{PILB})
due to possible random defects and impurities in the samples.
%The calculated linewidths are compared with the experimental
%data by adjusting the standard deviations of $D$ and $g$, and the
%strength of the dipolar interactions. 
We find that the calculated linewidths \cite{PARK01PRB} agree well with 
recent experimental data \cite{MACC01,PARK01PRB} as functions of the energy
level $M_s$ and the frequency.
%Our work \cite{PARK01PRB} is distinguished from the previous 
%speculation on the EPR linewidths as a function of $M_s$
%for Fe$_8$ and Fe$_4$ \cite{BARR00}, in the sense that
%our systematic calculations show that the $D$-strain effect 
%leads to a linear $M_s$ dependence of the linewidths, that
%the dipolar interactions\cite{MUKH01} were needed along with the $D$-strain 
%to explain the linewidths for our examined Fe$_8$ sample, and
%that we also calculated the linewidths for Mn$_{12}$.
We also briefly consider the effect of the temperature on the linewidths.
Finally we present a preliminary analysis of how
local rotations of the magnetic anisotropy axes, caused by defects 
in the samples, affect the lineshapes.

To obtain the linewidths for ${\mathrm Fe_8}$, 
we consider a single-spin system with $S$$=$$10$ in a weak oscillating
transverse field. We choose the easy axis to be the $z$-axis.  
Since the ${\mathrm Fe_8}$ clusters have an approximate $D_2$ symmetry,
the lowest-order ground-state single-spin Hamiltonian is \cite{AABB}
\begin{eqnarray}
{\cal H}_{0} &=& -D S_z^2 - E(S_x^2 - S_y^2) 
- g \mu_B H_z S_z ~, \label{eq:ham}
\end{eqnarray}
where $D$$=$$0.288k_B$ and the transverse 
anisotropy parameter, $E$$=$0.043$k_B$.\cite{MACC01}
Here $S_{\alpha}$ is the $\alpha$-th component of 
the spin operator, $g$ is 
the Land\'{e} $g$-factor ($\approx$2), $\mu_B$ is the Bohr magneton,
and $H_z$ is the longitudinal static applied field.
When $H_z$ is large enough, the eigenvalue of $S_z$, $M_s$, is a good
quantum number.
Next we introduce an interaction $V(t)$$=$$V_0 \cos(\omega t)$
between the spin system and an
oscillating transverse field $H_x$ with angular frequency 
$\omega$$\equiv$$ 2 \pi f$, where $V_0$$\propto$$H_x S_x$.
%\begin{eqnarray}
%V(t) &=& \frac{V_0}{2}(e^{i \omega t}+e^{-i \omega t})~, 
%~~~~V_0 ~\propto~  H_x S_x ~. 
%\label{eq:inter}
%\end{eqnarray}
We treat $V(t)$ as a small perturbation to ${\cal H}_0$.
%Besides the interaction $V(t)$, the spin system interacts 
%with the environment, such as the thermal fluctuations of the clusters.  
%Through this spin-phonon coupling, 
%the spin system can relax to re-establish thermal equilibrium.  
%Since the times of interest are much longer than the correlation 
%time of the phonon heat bath, the heat bath is considered always to be 
%in thermal equilibrium. Thus we can integrate out all degrees of freedom
%of the bath to obtain an equation of motion for the 
%density matrix $\rho(t)$ of the spin system \cite{BLUM}:
The interaction between the spin system and the surrounding
environment can be understood by density-matrix equations.\cite{BLUM}
%\begin{eqnarray}
%\frac{d \rho(t)_{m^{\prime},m}}{dt} &=& \frac{i}{\hbar} 
%[\rho(t),{\cal H}_{0}+V(t)]_{m^{\prime},m} \nonumber \\
%& & + \delta_{m^{\prime},m} \sum_{n\neq m} \rho_{n,n} W_{mn} 
%- \gamma_{m^{\prime},m} \rho_{m^{\prime},m} \;.
%\label{eq:GME} 
%%\gamma_{m^{\prime},m} &=& \frac{W_m + W_{m^{\prime}}}{2}~, ~~~~
%%W_m ~=~ \sum_{k \neq m} W_{km}~
%\end{eqnarray}

We consider the case that the frequency $\omega$
is fixed while $H_z$ is quasi-statically varied to induce a resonance.  
With the selection rule $\Delta M_s$$=$$\pm 1$, 
solving the density-matrix equations for the population change with time
in the level $M_s$ due to $V(t)$, 
to first order in $V_0$ near resonance,
%[$\omega \approx ({\cal E}_{M_s} - {\cal E}_{M_s-1})/\hbar$],
provides the power absorption between the levels $M_s$ and $M_s-1$:
\cite{BLUM}
%the power absorbed between the $M_s$ 
%and the $M_s-1$ level is
%\begin{eqnarray}
%\frac{d{\cal E}}{dt} 
%&=&{\cal E}_{M_s} [\dot{\rho}_{M_s,M_s}]_{\rm rad} 
%+  {\cal E}_{M_s-1} [\dot{\rho}_{M_s-1,M_s-1}]_{\rm rad}~,
%\end{eqnarray}
%where ${\cal E}_{M_s}$ is the energy of the $M_s$ level, and 
%$[\dot{\rho}_{M_s,M_s}]_{\rm rad}$ is the change with time
%of the population in the level $M_s$ caused by the first term in 
%Eq.~(\ref{eq:GME}). We solve Eq.~(\ref{eq:GME}) 
%for $[\dot{\rho}_{M_s,M_s}]_{\rm rad}$ up 
%to first order in $V_0$ near resonance 
%[$\omega \approx ({\cal E}_{M_s} - {\cal E}_{M_s-1})/\hbar$]. In this limit, 
%%among the many off-diagonal density-matrix elements,
%only $\rho(t)_{M_s-1,M_s}$ 
%%($=\rho(t)_{M_s,M_s-1}^{\star}$) 
%contributes to $[\dot{\rho}_{M_s,M_s}]_{\rm rad}$. 
%The matrix element $\rho(t)_{M_s-1,M_s}$
%is obtained by solving Eq.~(\ref{eq:GME}). Consequently, only 
%two neighboring energy levels are involved for each resonance peak.   
%An increase of the population in one of the two energy levels results
%in a decrease of the population in the other.
%%$[ \dot{\rho}_{M_s,M_s} ]_{\rm rad}$$=$$ 
%%- [ \dot{\rho}_{M_s+1,M_s+1} ]_{\rm rad}$.  
%So the power absorption becomes \cite{BLUM} 
\begin{eqnarray}
\!\!\frac{d{\cal E}}{dt} \! &=& \! 
\frac{({\cal E}_{M_s-1} \!\! - \!\! {\cal E}_{M_s})}{\hbar^2} {\tilde V}^2 
\Delta (\rho_{M_s,M_s} \!\! - \!\! \rho_{M_s-1,M_s-1}),  
\label{eq:dedt}\\
\!\!\Delta \!&\equiv&\! \frac{\hbar^2 \gamma_{M_s-1,M_s}}
{ (g \mu_B)^2 ( H_z - H_{\rm res})^2 + ( \hbar \gamma_{M_s-1,M_s} )^2 },
\nonumber \\ 
%\! \!  {\tilde V} &\! \equiv \! &|\langle M_s|V_0|M_s-1 \rangle|,
\!\!H_{\rm res} \!&\equiv&\!
\frac{\hbar \omega - D(2 M_s - 1)}{g \mu_B} \;.
\label{eq:res_field} 
\end{eqnarray}
Here ${\mathcal E}_{M_s}$ is the energy of the level $M_s$,
${\tilde V}$$\equiv$$|\langle M_s|V_0|M_s-1 \rangle|$,
$\rho(t)$ is the density matrix of the spin system. 
The subscripts represent eigenstates of 
the longitudinal part of ${\cal H}_0$, $\rho(t)_{m^{\prime} m}$
$=$$\langle m^{\prime}| \rho(t) | m \rangle$, 
$\gamma_{m^{\prime},m}$$=$$(W_m + W_{m^{\prime}})/2$,
$W_m$$=$$\sum_{k \neq m} W_{km}$, and
$W_{km}$ is the transition rate \cite{LEUE00}
from the $m$-th to the $k$-th eigenstate,
%$\Delta(H_z)$$=$$\frac{\hbar^2 \gamma_{M_s-1,M_s}}{ (g \mu_B)^2 
%( H_z - H_{\rm res})^2 + ( \hbar \gamma_{M_s-1,M_s} )^2 }$, 
$H_{\rm res}$ is the resonant field, 
and $\hbar \gamma_{M_s-1,M_s}/g \mu_B$ gives a natural linewidth
which is about 5 to 50 G at 10 K and increases as $M_s$ decreases.
%As $M_s$ decreases, the resonant field increases. 
%The linewidth determined by $\gamma_{Ms-1,Ms}$
%is on the order of several to several tens of gauss
%below several tens of kelvin, and it increases as $M_s$ decreases
%because of the corresponding shorter lifetimes.
However, the experimentally observed linewidths are much larger 
than the natural linewidths and {\em decrease\/} as $M_s$ decreases 
%until $M_s$$=$$1 \rightarrow 0$ and $M_s$$=$$0 \rightarrow -1$ transitions
%(hereafter $M_s$$=$$1,0$), 
%and are 
%to a thousand gauss in the same temperature range 
[Figs.~\ref{FWHM}(a) and (b)].

To resolve these large discrepancies, we first assume that 
$D$ and $g$ are independent random variables
with Gaussian distributions centered at $0.288k_B$ and $2.00$, with
standard deviations $\sigma_D$ and $\sigma_g$, respectively.  
%Such distributions can be caused by impurities or crystalline
%defects in the sample, which result in different
%clusters seeing slightly different values of $D$ and $g$.
%(``$D$-strain'' and ``$g$-strain'' \cite{PILB}).
Then we calculate the average power absorption at a fixed frequency 
and $T$$=$$10~{\mathrm K}$ by integrating over Eq.~(\ref{eq:dedt}) 
using {\texttt{Mathematica}} \cite{MATH} to obtain 
the linewidth as a function of $M_s$.
The lineshapes depend on the magnitudes of $\sigma_D$ and $\sigma_g$,
compared to the natural linewidth determined by
$\gamma_{M_s+1,M_s}$. If $\sigma_D$ and $\sigma_g$ are much larger
than (comparable to) the natural linewidth, then the lineshape is 
Gaussian (Lorentzian).
%because the $D$ and $g$ values are distributed
%with a Gaussian density. 
At intermediate values of $\sigma_D$ and 
$\sigma_g$, the absorption lineshapes are neither Gaussian nor Lorentzian.
Our numerical calculations show that 
the distribution in $D$ narrows the linewidths linearly 
with decreasing absolute value of $2M_s$$-$$1$
[Figs.~\ref{FWHM}(c) and (d)], with a
slight rounding close to the linewidth minimum ($M_s$$=$1 and $M_s$$=$0)
[Fig.~\ref{FWHM}(c)].
%where the $D$-strained linewidths approach the intrinsic
%lifetime broadening.
%flatten out at $M_s$$=$$1,0$, and then rise again for $M_s < 0$. 
On the other hand,
the distribution in $g$ broadens the linewidths with 
decreasing $M_s$ because the resonant field increases 
with decreasing $M_s$ [see Eq. (\ref{eq:res_field})].\cite{HILL98,PARK01PRB} 
%(Although Fig.~\ref{FWHM}(f) is for
%Mn$_{12}$, the trend is the same for Fe$_8$.) 
For small-$M_s$ transitions, the lineshapes are close to a Lorentzian
because the natural linewidths are not very small compared to
the measured linewidths.
%These trends can be predicted 
%from the expression for the resonant field $H_{\mathrm res}$ 
%in Eq. (\ref{eq:res_field}) 
%because the intrinsic linewidths are substantially smaller than the
%measured linewidths (with the exception of the lowest absolute
%$M_s$ transitions). However, the density-matrix 
%equation (\ref{eq:GME}) is needed to understand the temperature dependence 
%of the linewidths. 

%Next, we consider the effect of the inter-cluster 
%dipolar interactions. 
%These make each cluster see a different 
%net magnetic field, which results in a broadening of the linewidths.
%When the thermal energy becomes comparable to (or smaller than) 
%the energy of the oscillating field, some energy levels are not 
%populated, so the dipolar interactions become temperature dependent.
Next, we consider the effect of the dipolar interactions between
different clusters. (There is no distribution in cluster size.)
The dipolar interactions narrow the linewidths
as $M_s$ decreases because the resonant
field becomes stronger for smaller-$M_s$ transitions, and the stronger
resonant fields lead to a more polarized system 
[Figs.~\ref{FWHM}(c) and (d)]. The details of this effect on the linewidths
at a {\it fixed} temperature were reported elsewhere.\cite{PARK01PRB}
The dipolar interactions give rise to a temperature dependence of
the linewidths and a shift in the positions of the resonance lines, 
because at low temperatures ($k_B T$$\ll$$\hbar \omega$)
high energy levels are not populated. Our study shows that 
a linewidth for a particular transition increases and then 
smoothly decreases with increasing 
temperature. The maximum linewidth as a function of 
temperature moves towards lower temperatures
for larger-$M_s$ transitions, which agrees with experiments.

The competition between $D$-strain, $g$-strain, and dipolar interactions
determines the overall features of the linewidth, as a function of $M_s$.
%are determined by the competition among $D$-strain, $g$-strain,
%and dipolar interactions.
We have varied $\sigma_D$, $\sigma_g$, and the effective distance between 
neighboring dipoles, $d$, within an experimentally acceptable range 
in order to fit the experimental data. 
%Those parameters are approximately independent of temperature 
%and EPR frequency.

For the ${\mathrm Fe_8}$ sample examined, the calculated linewidths
agree well with the experimental data at the measured frequencies
($f=68$, 89, 109, 113, 133, and 141~{GHz}) at $T$$=$$10~{\mathrm K}$,  
using $\sigma_D$$\approx$$0.01D$ and $d$$\approx$$12~{\mathrm \AA}$.
[Figs.~\ref{FWHM}(a) and (b)]
%Within acceptable ranges of $\sigma_D$, $\sigma_g$, and $d$,
%no other combination of parameter values is able to
%produce the same quality of fit to all of the frequencies 
%studied while other parameter values may produce good fits for
%a given frequency.
%Figure ~\ref{FWHM}(a) and (b) show the experimental and 
%theoretical linewidths for ${\mathrm Fe_8}$ at two frequencies.
As shown in Figs.~\ref{FWHM}(c) and (d), the $D$-strain effect
and the dipolar interactions are equally important for the 
linewidths of the sample, while the $g$-strain does not 
contribute significantly (not shown). 
%Since the distribution
%in $D$ does not depend on frequency or resonant field but
%the dipolar interactions do, for low frequencies the dipolar
%interactions contribute more to the linewidths than the 
%$D$-strain effect, as shown in Fig.~\ref{FWHM}(d).
%Although the linewidth is symmetric about $M_s$$=$1 and $M_s$$=$0
%due to the $D$-strain effect, a weak asymmetry in the linewidths
%about $M_s$$=$1 and $M_s$$=$0 is expected from  
%%The weak asymmetry of the linewidths about $M_s$$=$$1,0$, 
%%shown in Fig.~\ref{FWHM}(a), can be understood by 
%combining the $D$-strain effect with the dipolar interactions
%because
%%The $D$-strain produces linewidths symmetric about $M_s$$=$$1,0$, while 
%the dipolar interactions yield narrower linewidths with 
%smaller $M_s$ transitions.
%%which are observed at higher resonant fields. 
%%The distribution in $D$ does not give frequency-
%%or resonant-field dependent linewidths, but the dipolar interactions do. 
%%For example, Full Width at Half Maximum (FWHM) for $M_s$$=$$5$
%%at $f$$=$$68~{\mathrm GHz}$ is about $800~{\mathrm gauss}$ while that at
%%$f$$=$$133~{\mathrm GHz}$ is about $600~{\mathrm gauss}$.
%%We also considered distributions for other higher-order crystal-field 
%%anisotropy parameters ($E$ or fourth-order coefficients) and found that
%%they do not contribute substantially to the linewidths.
%%although a distribution of the effective transverse anisotropy 
%%plays an essential role to explain MQT in the recent tunneling model 
%%\cite{CHUD01}.

%Chudnovsky and Garanin \cite{CHUD01} recently considered 
%a transverse anisotropy in ${\mathrm Mn_{12}}$,
%due to dislocations, which provides one possible physical mechanism 
%to generate a distribution in $D$. 
For Mn$_{12}$, we perform a similar analysis as above, using
%We perform the same analysis as above for
%the ${\mathrm Mn_{12}}$ data ($f$$=$$169~{\mathrm GHz}$,
%$T$$=$$25~{\mathrm K}$, the field along the easy axis). 
%[Fig. \ref{FWHM}(c)]
%The molecular magnet ${\mathrm Mn_{12}}$ has tetragonal symmetry, so
the ground-state single-spin Hamiltonian\cite{AABB}
\begin{eqnarray}
{\cal H}_{0} &=& -D S_z^2 - C S_z^4 - g \mu_B H_z S_z 
\end{eqnarray}
with $D$$=$$0.55k_B$, $C$$=$$1.17 \times 10^{-3}k_B$,
and $g$$=$$1.94$.\cite{BARR97}
%The procedure of the analysis is the same as for Fe$_8$, 
%except for the modified resonant field.
%except for the following: the resonant field is modified to 
%\begin{eqnarray*}
%\! \! H_{\mathrm res} &\equiv& 
%\frac{\hbar \omega \!- \!D(2M_s-1) \! + \! C(4M_s^3-6M_s^2+4M_s-1)}{g \mu_B};
%\end{eqnarray*}
%%because of the fourth-order anisotropy
%dipoles are assumed to be 
%distributed on a bcc lattice; and the anisotropy axis is along the
%longer side of the sample. 
%The calculated linewidths [Fig.~\ref{FWHM}(c)] 
%are in a good agreement with the experimental
%data for $\sigma_D$$\approx$$0.02D$, $\sigma_g$$\approx$$0.008g$, 
%and $d$$\approx$$14~{\mathrm \AA}$. 
%For this sample, 
%the distribution in $D$ is wider than that for the ${\mathrm Fe_8}$ sample. 
Detailed analysis can be found in the literature.\cite{PARK01PRB}
For this sample, the $D$-strain and $g$-strain effects play 
significant roles in the linewidths, while the dipolar interactions
are not as important as for the Fe$_8$ sample.
%the $D$-strain effect is still dominant, 
%but a $g$-strain effect must be included as well to explain
%the linewidths for small $M_s$ transitions [Fig.~\ref{FWHM}(f)].
%The dipolar interactions do not play as significant a role as for the 
%${\mathrm Fe_8}$ sample [Fig.~\ref{FWHM}(f)]. 

Next we discuss how the distribution of the directions
of the magnetic anisotropy axes of clusters affect 
the EPR lineshapes. Because of defects in the samples,
each cluster sees a slightly different crystal field 
due to the surrounding clusters,
compared to the situation in a perfect crystal. We assume that
these slightly different crystal fields result in local rotations 
of the magnetic anisotropy axes of some of the clusters
by an angle $\theta$ from the crystal $c$ axis.
The angle $\theta$ is assumed to have a Gaussian distribution 
about zero with a small standard deviation.
Hereafter $a$, $b$, and $c$ denote the crystal axes,
while $x$, $y$, and $z$ denote the magnetic anisotropy axes 
of a single cluster.

As a preliminary study, we examine the lineshapes for 
Mn$_{12}$ when the magnetic field is 
applied along the $c$ axis and the magnetic anisotropy
easy axis of a single cluster is tilted by $\theta$ from
the $c$ axis.
Then the single-spin Hamiltonian, in terms of the spin 
operators along the magnetic anisotrpy axes, becomes
\begin{eqnarray}
{\cal H}&=&-D S_z^2 - g \mu_B H S_z \cos \theta 
+ g \mu_B H S_x \sin \theta \:.
\label{eq:ham_hz_xyz} 
\end{eqnarray}
For simplicity, we drop the fourth-order anisotropy terms.
We also set $\psi=90^{\circ}$, where $\psi$ is the third of the 
Euler angles defined in literature,\cite{GOLD} which does not affect
the eigenvalues of ${\cal H}$.

Because of the local rotations of the magnetic anisotropy axes,
some of the clusters experience slightly different resonant fields
than those aligned with the crystal $c$ axis.
In contrast to the Gaussian distributions of the resonant field
due to the distributions in $D$ and $g$, this effect gives rise
to an asymmetry in the resonant field, which leads to an asymmetry
in the EPR lineshapes. Assuming that $\theta$ is small, we 
treat the terms proportional to $\sin \theta$ as small perturbations 
to the rest of the terms in Eq.~(\ref{eq:ham_hz_xyz}). 
Using second-order perturbation theory, we obtain the resonant fields
as a function of $\theta$, shown in Fig.~\ref{shape}. 
Each curve in Fig.~\ref{shape} is symmetric about $\theta$$=$0.
Figure~\ref{shape} shows that for transitions between 
$M_s$ and $M_s$$+$1 ($M_s$$=$$-4$$\rightarrow$$-3$) an asymmetry 
appears in the direction of {\it decreasing} field, while for transitions 
between $M_s$ and $M_s$$-$1 ($M_s$$=$3$\rightarrow$2, $M_s$$=$2$\rightarrow$1)
the asymmetry is in the direction of {\it increasing} field.
At the examined frequency ($f$$=$66.135 GHz)
the asymmetry effect is more significant for the small-$M_s$ transitions.
Based on the analytic form of the resonant field, we can obtain 
analytically the distribution of the resonant field
at a particular frequency for transitions between the levels 
$M_s$ and $M_s$$\pm$1, including Gaussian distributions 
of $D$, $g$, and $\theta$.\cite{PARKPRP} For simplicity, 
we have neglected the effects of natural linewidths, 
dipolar interactions, and temperature on the EPR lineshapes.

In conclusion, we have examined the EPR linewidths as functions of
$M_s$, frequency, and temperature for single crystals of the molecular 
magnets Fe$_8$ and Mn$_{12}$. We found that the distribution in $D$
is important to explain the linewidths for both of the molecular magnets,
and that the dipolar interactions are crucial to understand the temperature 
dependence of the linewidths. Our preliminary study of the EPR lineshapes
shows that the distribution of the directions of the magnetic 
anisotropy axes of clusters provides an asymmetry in the EPR spectra.

%the for the Fe$_8$ sample the dipolar interactions 
%must be considered as well as the $D$-strain effect to explain the 
%linewidths, while for the Mn$_{12}$ sample the $D$-strain and 
%the $g$-strain effects are more significant to the linewidths 
%than the dipolar interactions. 

%standard deviation 
%$\sigma_D$ varies between samples. Other effects, such as 
%$g$-strain effect and dipolar
%interactions can contribute to the linewidths, depending on the material.
%These results provide the soundness of the assumption of a
%tunneling mechanism \cite{CHUD01}.
%%imply that the assumption made in the tunneling
%%model \cite{CHUD01} may be correct.
%%A distribution in $D$ may also facilitate multi-photon transitions 
%%through virtual states which are necessary to implement a quantum algorithm
%%for the molecular magnets ${\mathrm Mn_{12}}$ and ${\mathrm Fe_8}$ 
%%\cite{LEUE00-3}. Additionally we speculate that the $D$-strain effect 
%%may contribute to the widths of resonant steps in magnetization hysteresis 
%%loops. A more detailed report \cite{SING}, including a quantitative study 
%%of ${\mathrm Mn_{12}}$ data for several frequencies and the temperature
%%dependence of the linewidths at fixed frequency for both molecular magnets, 
%%will be published elsewhere \cite{KPAR01B}. 

%This work was partially 
Funded by NSF Grants DMR-9871455, DMR-0120310, and 
DMR-0103290, and FSU-CSIT, FSU-MARTECH, and Research Corporation.

\begin{references}

\bibitem{LIS80}
T.\ Lis, Acta Crystallogr.\ B\ {\bf 36}, 2042 (1980).

\bibitem{WIEG84}
%K.\ Wieghart {\it et al.},
K.\ Wieghart, K.\ Pohl, I.\ Jibril, and G.\ Huttner,
Angew.\ Chem.\ Int.\ Ed.\ Engl.\ {\bf 23}, 77 (1984).

%\bibitem{VILL94}
%J.\ Villain {\it et al.},
%J.\ Villain, F.\ Hartman-Boutron, R.\ Sessoli, and A.\ Rettori,
%Europhys.\ Lett.\ {\bf 27}, 159 (1994);
%{\it Quantum Tunneling of Magnetization -- QTM '94\/},
%(NATO ASI Series. Series E, Applied Sciences, Vol.\ 301)
%edited by L.\ Gunther and B.\ Barbara
%(Kluwer, Dordrecht, 1995).

\bibitem{CHUD98}
E.~M.\ Chudnovsky and J.\ Tejada,
{\it Macroscopic Quantum Tunneling of the Magnetic Moment\/},
Cambridge Studies in Magnetism, Vol. 4
(Cambridge University Press, Cambridge, 1998) and references therein.

\bibitem{LEUE00-3}
M.~N.\ Leuenberger and D.\ Loss, Nature (London) {\bf 410}, 789 (2001).

%\bibitem{HART96}
%%F.\ Hartmann-Boutron {\it et al.},
%F.\ Hartmann-Boutron, P.\ Politi, and J.\ Villain, 
%Int.\ J.\ Mod.\ Phys.\ B {\bf 10}, 2577 (1996);
%%A.\ Fort {\it et al.},
%A.\ Fort, A.\ Rettori, J.\ Villain, D.\ Gatteschi, and R.\ Sessoli,
%Phys.\ Rev.\ Lett.\ {\bf 80}, 612 (1998).

%\bibitem{FRIE96}
%%J.\ Friedman {\it et al.},
%J.\ Friedman, M.~P.\ Sarachik, J.\ Tejada, and R.\ Ziolo,
%Phys.\ Rev.\ Lett.\ {\bf 76}, 3830 (1996);
%D.~A.\ Garanin and E.~M.\ Chudnovsky, Phys.\ Rev.\ B {\bf 56},
%11102 (1997);
%%F.\ Luis {\it et al.},
%F.\ Luis, J.\ Bartolom{\'{e}}, and F.\ Fern{\'{a}}ndez, 
%Phys.\ Rev.\ B {\bf 57}, 505 (1998);

%\bibitem{MIYA97}
%H.\ De Raedt {\it et al.},
%H.\ De Raedt, S.\ Miyashita, K.\ Saito, D.\ Garci\'{a}-Pablos, 
%and N.\ Garc{\'{\i}}a,
%Phys.\ Rev.\ B {\bf 56}, 11761 (1997);
%%\bibitem{DOBR97}
%V.~V.\ Dobrovitsky and A.~K.\ Zvezdin, Europhys.\ Lett.\ 
%{\bf 38}, 377 (1997).

%\bibitem{PROK98}
%N.~V.\ Prokof'ev and P.~C.~E.\ Stamp,
%Phys.\ Rev.\ Lett.\ {\bf 80}, 5794 (1998);
%%T.\ Ohm, C.\ Sangregorio, and C.\ Paulsen,
%%Eur. Phys. J. B {\bf 6}, 195 (1998);
%%A.\ Cuccoli, A.\ Fort, A.\ Rettori, E.\ Adam, and J.\ Villain,
%%Eur. Phys. J. B {\bf 12}, 39 (1999);
%%W.\ Wernsdorfer {\it et al.},
%W.\ Wernsdorfer, T.\ Ohm, C.\ Sangregorio, R.\ Sessoli, D.\ Mailly,
%and C.\ Paulsen,
%Phys.\ Rev.\ Lett.\ {\bf 82}, 3903 (1999).

%\bibitem{FRIE96}
%J.~M.\ Hernandez, X.~X.\ Zhang, F.\ Luis, J.\ Tejada,
%J.~R.\ Friedman, M.~P.\ Sarachik, and R.\ Ziolo,
%Phys. Rev. B {\bf 55}, 5858 (1997).
%L.\ Thomas, F.\ Lionti, R.\ Ballou, D.\ Gatteschi, R.\ Sessoli, and
%B.\ Barbara, Nature {\bf 383}, 145 (1996).

\bibitem{HILL98}
%%S.\ Hill {\it et al.},
S.\ Hill, J.~A.~A.~J.\ Perenboom, N.~S.\ Dalal, T.\ Hathaway, T.\ Stalcup,
and J.~S.\ Brooks,
Phys.\ Rev.\ Lett.\ {\bf 80}, 2453 (1998).

\bibitem{MACC01}
%S.\ Maccagnano {\it et al.},
S.\ Maccagnano, R.\ Achey, E.\ Negusse, A.\ Lussier, M.~M.\ Mola,
S.\ Hill, and N.~S.\ Dalal, 
Polyhedron, {\bf 20}, 1441 (2001).

\bibitem{PERE98}
%J.~A.~A.~J.\ Perenboom {\it et al.},
J.~A.~A.~J.\ Perenboom, J.~S.\ Brooks, S.\ Hill, T.\ Hathaway,
and N.~S.\ Dalal,
Phys.\ Rev.\ B {\bf 58}, 330 (1998);
%E.\ del Barco {\it et al.},
E.\ del Barco, J.~M.\ Hernandez, J.\ Tejada, N.\ Biskup, R.\ Achey,
I.\ Rutel, N.\ Dalal, and J.\ Brooks,
{\it ibid.} {\bf 62}, 3018 (2000).

\bibitem{PILB}
J.~R.\ Pilbrow, 
{\it Transition Ion Electron Paramagnetic Resonance}
(Clarendon, Oxford, 1990).

\bibitem{PARK01PRB}
K.\ Park, M.~A.\ Novotny, N.~S.\ Dalal, S.\ Hill, and P.~A.\ Rikvold,
Phys. Rev. B (in press).

%\bibitem{BARR00}
%%A.~L.\ Barra {\it et al.},
%A.~L.\ Barra, D.\ Gatteschi, and R.\ Sessoli,
%Chem--Eur.\ J.\ {\bf 6}, 1608 (2000);
%%A.\ Bouwen {\it et al.},
%A.\ Bouwen, A.\ Caneschi, D.\ Gatteschi, E.\ Goovaerts,
%D.\ Schoemaker, L.\ Sorace, and M.\ Stefan,
%e-print cond-mat/0101255.

%\bibitem{MUKH01}
%The presence of the dipolar fields was mentioned 
%%in a zero-field millimeter wave experiment 
%for Fe$_8$ in
%%A.\ Mukhin {\it et al.}, 
%A.\ Mukhin, B.\ Gorshunov, M.\ Dressel, C.\ Sangregorio, and
%D.\ Gatteschi,
%Phys.\ Rev.\ B {\bf 63}, 214411 (2001).

%\bibitem{CHUD01}
%E.~M.\ Chudnovsky and D.~A.\ Garanin, e-print cond-mat/0105195; 
%e-print cond-mat/0105518.

%\bibitem{WWER00}
%W.\ Wernsdorfer, R.\ Sessoli, A.\ Caneschi,
%D.\ Gatteschi, and A.\ Cornia,
%Europhys. Lett. {\bf 50}, 552 (2000).
%A.\ Garg, pre-print cond-mat/0012157.

%\bibitem{ZHON00}
%Y.~C.\ Zhong, M.~P.\ Sarachik, J.\ Yoo, and D.~N.\ Hendrickson,
%Phys. Rev. B {\bf 62}, R9256 (2000);
%I.\ Chiorescu, R.\ Giraud, A.~G.~M.\ Jansen, A.\ Caneschi, and B.\ Barbara,
%Phys.\ Rev.\ Lett.\ {\bf 85}, 4807 (2000);
%A.~D.\ Kent, Y.~C.\ Zhong, L.\ Bokacheva, D.\ Ruiz, D.~N.\ Hendrickson,
%and M.~P.\ Sarachik,
%Europhys.\ Lett.\ {\bf 49}, 521 (2000);
%L.\ Bokacheva, A.~D.\ Kent, and M.~A.\ Walters,
%Phys.\ Rev.\ Lett.\ {\bf 85}, 4803 (2000).

\bibitem{AABB}
A.\ Abragam and B.\ Bleaney, 
{\it Electron Paramagnetic Resonance of Transition Ions}
(Clarendon, Oxford, 1970).

\bibitem{BLUM} 
K.\ Blum, 
{\it Density Matrix Theory and Applications\/}, 2nd edition
(Plenum, New York, 1996).

\bibitem{LEUE00} 
M.~N.\ Leuenberger and D.\ Loss,
Phys.\ Rev.\ B {\bf 61}, 1286 (2000);
%\bibitem{LEUE00-2}
%M.~N.\ Leuenberger and D.\ Loss,
%Phys.\ Rev.\ B 
{\bf 61}, 12200 (2000).

\bibitem{MATH}
S.\ Wolfram, {\it The Mathematica Book}, 3rd edition
(Wolfram Media/Cambridge University Press, New York, 1996).

%\bibitem{VANV48}
%J.~H.\ Van Vleck, Phys.\ Rev.\ {\bf 74}, 1168 (1948);
%M.\ McMillan and W.\ Opechowski,
%Can.\ J.\ Phys.\ {\bf 38}, 1168 (1960).

\bibitem{BARR97}
%A.~L.\ Barra {\it et al.},
A.~L.\ Barra, D.\ Gatteschi, and R.\ Sessoli,
Phys.\ Rev.\ B {\bf 56}, 8192 (1997).

%\bibitem{MIYA98}
%S.\ Miyashita, K.\ Saito, and H.\ De Raedt,
%Phys.\  Rev.\  Lett.\  {\bf 80}, 1525 (1998);
%K.\ Saito, S.\ Miyashita, and H.\ De Raedt,
%Phys.\  Rev.\  B {\bf 60}, 14553 (1999).

%\bibitem{SING}
%%It will include a quantitative study 
%%of ${\mathrm Mn_{12}}$ data for several frequencies and the temperature
%%dependence of the linewidths at fixed frequency for both molecular magnets.
%Extension of this model to a full lineshape analysis
%indicates that it might be possible to distinguish
%between the distributions due to various types of physical
%mechanisms, including edge dislocations.

\bibitem{GOLD} H.\ Goldstein, {\it Classical Mechanics} 
(Addison-Wesley, Reading, 1980).

\bibitem{PARKPRP}
%K.~Park {\it et al.},
K.\ Park, M.~A.\ Novotny, N.~S.\ Dalal, S.\ Hill, and P.~A.\ Rikvold,
(unpublished). 

%[Fe$_8$(tacn)$_6$O$_2$(OH)$_{12}$]Br$_8$9H$_2$O where tacn=1, 4, 7 
%triazacyclononane, and (b) the Mn$_{12}$ cluster,
%[Mn$_{12}$O$_{12}$(CH$_3$COO)$_{16}$(H$_2$O)$_4$]2CH$_3$COOH$\cdot$4H$_2$O.

\end{references}

\begin{figure}
\begin{center}
\epsfxsize=4.0cm
\epsfysize=4.0cm
\epsfbox{fe8_6.eps}
%\vspace{0.3cm}
\epsfxsize=4.0cm
\epsfysize=3.8cm
\epsfbox{mn12_3.eps}
%\vspace{0.3cm}
\caption{Schematic diagram of the magnetic core of (a) the Fe$_8$ cluster
and (b) the Mn$_{12}$ cluster. Each Fe$_8$ cluster has D$_2$ symmetry, 
while each Mn$_{12}$ cluster has tetragonal symmetry. 
Both clusters have ground-state spin $S=10$
(for Fe$_8$, $S=(6-2)\times 5/2$, and for Mn$_{12}$, 
$S=8\times 2 - 4 \times 3/2$).}
\label{conf}
\end{center}
\end{figure}

 
\begin{figure}
\begin{center}
\epsfxsize=5.5cm
\epsfysize=3.5cm
\epsfbox{fig1ab.eps}
\vspace{0.3cm}
\epsfxsize=5.5cm
\epsfysize=3.5cm
\epsfbox{fig1cd.eps}
\vspace{0.3cm}
\caption{(a) and (b)
Experimental (filled circles) and theoretical (open circles) 
Full Width at Half Maximum (FWHM) vs $M_s$ for different frequencies 
at $T$$=$$10~{\mathrm K}$ for ${\mathrm Fe_8}$.
(c) and (d) Line broadening due to the $D$-strain (filled squares)
and dipolar interactions (open squares) for (a) and (b), respectively.
Here $\vec{H} \, || \, \hat{z}$.
%The filled and open circles denote 
%the experimental and calculated FWHM, respectively.
The fit is best when the standard deviation of $D$ is $0.01D$ and 
the effective distance between dipoles is $12~{\mathrm \AA}$.}
\label{FWHM}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\epsfxsize=3.4cm
\epsfysize=3.cm
\epsfbox{fig2a.eps}
\hspace{0.5cm}
\epsfxsize=3.4cm
\epsfysize=3.cm
\epsfbox{fig2b.eps}
\vspace{0.3cm}
\caption{The resonant field vs $\theta$
for the transitions (a) $M_s$$=$$-4$$\rightarrow$$-3$ 
and $M_s$$=$3$\rightarrow$2, and (b) $M_s$$=$2$\rightarrow$1 
at $f$$=$66.135 GHz for $\vec{H} \, || \, \hat{c}$.
Each curve is symmetric about $\theta$$=$0.}
\label{shape}
\end{center}
\end{figure}

%\end{multicols}

\end{document}
%
% ****** End of file template.aps ******






cond-mat0111589.extracted_bib
\bibitem{LIS80}
T.\ Lis, Acta Crystallogr.\ B\ {\bf 36}, 2042 (1980).


\bibitem{WIEG84}
%K.\ Wieghart {\it et al.},
K.\ Wieghart, K.\ Pohl, I.\ Jibril, and G.\ Huttner,
Angew.\ Chem.\ Int.\ Ed.\ Engl.\ {\bf 23}, 77 (1984).

%
\bibitem{VILL94}
%J.\ Villain {\it et al.},
%J.\ Villain, F.\ Hartman-Boutron, R.\ Sessoli, and A.\ Rettori,
%Europhys.\ Lett.\ {\bf 27}, 159 (1994);
%{\it Quantum Tunneling of Magnetization -- QTM '94\/},
%(NATO ASI Series. Series E, Applied Sciences, Vol.\ 301)
%edited by L.\ Gunther and B.\ Barbara
%(Kluwer, Dordrecht, 1995).


\bibitem{CHUD98}
E.~M.\ Chudnovsky and J.\ Tejada,
{\it Macroscopic Quantum Tunneling of the Magnetic Moment\/},
Cambridge Studies in Magnetism, Vol. 4
(Cambridge University Press, Cambridge, 1998) and references therein.


\bibitem{LEUE00-3}
M.~N.\ Leuenberger and D.\ Loss, Nature (London) {\bf 410}, 789 (2001).

%
\bibitem{HART96}
%%F.\ Hartmann-Boutron {\it et al.},
%F.\ Hartmann-Boutron, P.\ Politi, and J.\ Villain, 
%Int.\ J.\ Mod.\ Phys.\ B {\bf 10}, 2577 (1996);
%%A.\ Fort {\it et al.},
%A.\ Fort, A.\ Rettori, J.\ Villain, D.\ Gatteschi, and R.\ Sessoli,
%Phys.\ Rev.\ Lett.\ {\bf 80}, 612 (1998).

%
\bibitem{FRIE96}
%%J.\ Friedman {\it et al.},
%J.\ Friedman, M.~P.\ Sarachik, J.\ Tejada, and R.\ Ziolo,
%Phys.\ Rev.\ Lett.\ {\bf 76}, 3830 (1996);
%D.~A.\ Garanin and E.~M.\ Chudnovsky, Phys.\ Rev.\ B {\bf 56},
%11102 (1997);
%%F.\ Luis {\it et al.},
%F.\ Luis, J.\ Bartolom{\'{e}}, and F.\ Fern{\'{a}}ndez, 
%Phys.\ Rev.\ B {\bf 57}, 505 (1998);

%
\bibitem{MIYA97}
%H.\ De Raedt {\it et al.},
%H.\ De Raedt, S.\ Miyashita, K.\ Saito, D.\ Garci\'{a}-Pablos, 
%and N.\ Garc{\'{\i}}a,
%Phys.\ Rev.\ B {\bf 56}, 11761 (1997);
%%
\bibitem{DOBR97}
%V.~V.\ Dobrovitsky and A.~K.\ Zvezdin, Europhys.\ Lett.\ 
%{\bf 38}, 377 (1997).

%
\bibitem{PROK98}
%N.~V.\ Prokof'ev and P.~C.~E.\ Stamp,
%Phys.\ Rev.\ Lett.\ {\bf 80}, 5794 (1998);
%%T.\ Ohm, C.\ Sangregorio, and C.\ Paulsen,
%%Eur. Phys. J. B {\bf 6}, 195 (1998);
%%A.\ Cuccoli, A.\ Fort, A.\ Rettori, E.\ Adam, and J.\ Villain,
%%Eur. Phys. J. B {\bf 12}, 39 (1999);
%%W.\ Wernsdorfer {\it et al.},
%W.\ Wernsdorfer, T.\ Ohm, C.\ Sangregorio, R.\ Sessoli, D.\ Mailly,
%and C.\ Paulsen,
%Phys.\ Rev.\ Lett.\ {\bf 82}, 3903 (1999).

%
\bibitem{FRIE96}
%J.~M.\ Hernandez, X.~X.\ Zhang, F.\ Luis, J.\ Tejada,
%J.~R.\ Friedman, M.~P.\ Sarachik, and R.\ Ziolo,
%Phys. Rev. B {\bf 55}, 5858 (1997).
%L.\ Thomas, F.\ Lionti, R.\ Ballou, D.\ Gatteschi, R.\ Sessoli, and
%B.\ Barbara, Nature {\bf 383}, 145 (1996).


\bibitem{HILL98}
%%S.\ Hill {\it et al.},
S.\ Hill, J.~A.~A.~J.\ Perenboom, N.~S.\ Dalal, T.\ Hathaway, T.\ Stalcup,
and J.~S.\ Brooks,
Phys.\ Rev.\ Lett.\ {\bf 80}, 2453 (1998).


\bibitem{MACC01}
%S.\ Maccagnano {\it et al.},
S.\ Maccagnano, R.\ Achey, E.\ Negusse, A.\ Lussier, M.~M.\ Mola,
S.\ Hill, and N.~S.\ Dalal, 
Polyhedron, {\bf 20}, 1441 (2001).


\bibitem{PERE98}
%J.~A.~A.~J.\ Perenboom {\it et al.},
J.~A.~A.~J.\ Perenboom, J.~S.\ Brooks, S.\ Hill, T.\ Hathaway,
and N.~S.\ Dalal,
Phys.\ Rev.\ B {\bf 58}, 330 (1998);
%E.\ del Barco {\it et al.},
E.\ del Barco, J.~M.\ Hernandez, J.\ Tejada, N.\ Biskup, R.\ Achey,
I.\ Rutel, N.\ Dalal, and J.\ Brooks,
{\it ibid.} {\bf 62}, 3018 (2000).


\bibitem{PILB}
J.~R.\ Pilbrow, 
{\it Transition Ion Electron Paramagnetic Resonance}
(Clarendon, Oxford, 1990).


\bibitem{PARK01PRB}
K.\ Park, M.~A.\ Novotny, N.~S.\ Dalal, S.\ Hill, and P.~A.\ Rikvold,
Phys. Rev. B (in press).

%
\bibitem{BARR00}
%%A.~L.\ Barra {\it et al.},
%A.~L.\ Barra, D.\ Gatteschi, and R.\ Sessoli,
%Chem--Eur.\ J.\ {\bf 6}, 1608 (2000);
%%A.\ Bouwen {\it et al.},
%A.\ Bouwen, A.\ Caneschi, D.\ Gatteschi, E.\ Goovaerts,
%D.\ Schoemaker, L.\ Sorace, and M.\ Stefan,
%e-print cond-mat/0101255.

%
\bibitem{MUKH01}
%The presence of the dipolar fields was mentioned 
%%in a zero-field millimeter wave experiment 
%for Fe$_8$ in
%%A.\ Mukhin {\it et al.}, 
%A.\ Mukhin, B.\ Gorshunov, M.\ Dressel, C.\ Sangregorio, and
%D.\ Gatteschi,
%Phys.\ Rev.\ B {\bf 63}, 214411 (2001).

%
\bibitem{CHUD01}
%E.~M.\ Chudnovsky and D.~A.\ Garanin, e-print cond-mat/0105195; 
%e-print cond-mat/0105518.

%
\bibitem{WWER00}
%W.\ Wernsdorfer, R.\ Sessoli, A.\ Caneschi,
%D.\ Gatteschi, and A.\ Cornia,
%Europhys. Lett. {\bf 50}, 552 (2000).
%A.\ Garg, pre-print cond-mat/0012157.

%
\bibitem{ZHON00}
%Y.~C.\ Zhong, M.~P.\ Sarachik, J.\ Yoo, and D.~N.\ Hendrickson,
%Phys. Rev. B {\bf 62}, R9256 (2000);
%I.\ Chiorescu, R.\ Giraud, A.~G.~M.\ Jansen, A.\ Caneschi, and B.\ Barbara,
%Phys.\ Rev.\ Lett.\ {\bf 85}, 4807 (2000);
%A.~D.\ Kent, Y.~C.\ Zhong, L.\ Bokacheva, D.\ Ruiz, D.~N.\ Hendrickson,
%and M.~P.\ Sarachik,
%Europhys.\ Lett.\ {\bf 49}, 521 (2000);
%L.\ Bokacheva, A.~D.\ Kent, and M.~A.\ Walters,
%Phys.\ Rev.\ Lett.\ {\bf 85}, 4803 (2000).


\bibitem{AABB}
A.\ Abragam and B.\ Bleaney, 
{\it Electron Paramagnetic Resonance of Transition Ions}
(Clarendon, Oxford, 1970).


\bibitem{BLUM} 
K.\ Blum, 
{\it Density Matrix Theory and Applications\/}, 2nd edition
(Plenum, New York, 1996).


\bibitem{LEUE00} 
M.~N.\ Leuenberger and D.\ Loss,
Phys.\ Rev.\ B {\bf 61}, 1286 (2000);
%
\bibitem{LEUE00-2}
%M.~N.\ Leuenberger and D.\ Loss,
%Phys.\ Rev.\ B 
{\bf 61}, 12200 (2000).


\bibitem{MATH}
S.\ Wolfram, {\it The Mathematica Book}, 3rd edition
(Wolfram Media/Cambridge University Press, New York, 1996).

%
\bibitem{VANV48}
%J.~H.\ Van Vleck, Phys.\ Rev.\ {\bf 74}, 1168 (1948);
%M.\ McMillan and W.\ Opechowski,
%Can.\ J.\ Phys.\ {\bf 38}, 1168 (1960).


\bibitem{BARR97}
%A.~L.\ Barra {\it et al.},
A.~L.\ Barra, D.\ Gatteschi, and R.\ Sessoli,
Phys.\ Rev.\ B {\bf 56}, 8192 (1997).

%
\bibitem{MIYA98}
%S.\ Miyashita, K.\ Saito, and H.\ De Raedt,
%Phys.\  Rev.\  Lett.\  {\bf 80}, 1525 (1998);
%K.\ Saito, S.\ Miyashita, and H.\ De Raedt,
%Phys.\  Rev.\  B {\bf 60}, 14553 (1999).

%
\bibitem{SING}
%%It will include a quantitative study 
%%of ${\mathrm Mn_{12}}$ data for several frequencies and the temperature
%%dependence of the linewidths at fixed frequency for both molecular magnets.
%Extension of this model to a full lineshape analysis
%indicates that it might be possible to distinguish
%between the distributions due to various types of physical
%mechanisms, including edge dislocations.


\bibitem{GOLD} H.\ Goldstein, {\it Classical Mechanics} 
(Addison-Wesley, Reading, 1980).


\bibitem{PARKPRP}
%K.~Park {\it et al.},
K.\ Park, M.~A.\ Novotny, N.~S.\ Dalal, S.\ Hill, and P.~A.\ Rikvold,
(unpublished). 

%[Fe$_8$(tacn)$_6$O$_2$(OH)$_{12}$]Br$_8$9H$_2$O where tacn=1, 4, 7 
%triazacyclononane, and (b) the Mn$_{12}$ cluster,
%[Mn$_{12}$O$_{12}$(CH$_3$COO)$_{16}$(H$_2$O)$_4$]2CH$_3$COOH$\cdot$4H$_2$O.

