text:
\section{\label{sec:Intro}Introduction}
Micas are common rock-forming phyllosilicate minerals displaying monoclinic symmetry and comprising approximately 5-12\% of the continental crust \cite{almq2017}.  The perfect \{001\}
cleavage and platy habit of the mica group lead to preferred orientation of their grains.  In sedimentary rocks preferred
orientation arises as a depositional or diagenetic feature.  In
shales, composed predominantly of clay minerals, which are closely
related in structure to micas, such preferred orientation leads to
their fissility.  In metamorphic rocks preferred orientation of micas
(and other minerals) is a response to crystallisation (or
recrystallisation) in a stress field, causing the development of a
foliation (or schistosity) perpendicular to the direction of the
primary stress.  At the highest temperatures of metamorphism, and in
the presence of an anisotropic stress field, compositional
segregation, combined with foliation, may lead to the development of
very strongly layered gneisses (gneissosity).

The presence of fissility, foliation, or gneissosity causes anisotropy
in the velocity of seismic waves through the Earth.  This was
recognised in the nineteenth century and with the development of
improved instrumentation and processing techniques has become a tool
that is used both in exploration geophysics and deep Earth studies \cite{helb2005, roma2017, chan2021}.

The anisotropy of rocks can provide information on rock microstructure and crystal anisotropy bears on efforts to use inclusions of one mineral in another to obtain measures of pressure at the time of inclusion \cite{hea20}. Complete and accurate characterization of the elastic properties of rocks and the minerals
that compose them is therefore important to understanding the
composition and structure of the Earth.  On a microscopic scale,
knowledge of the elasticity of sheet silicates like micas provides
insight into the nature of bonding and acoustic wave behaviour in
layered materials.  Moreover, the use of micas in applications such as
flexible electronic devices also requires that the elastic properties
be known \cite{ma2016van,xu2018flexible}.

\subsection{The Micas}
\label{sec:Micas}

The silicates are, volumetrically, the most important mineral group in
the Earth's crust.  Their structures are based on tetrahedra,
{\it i.e.} sites in which each central cation (commonly Si or Al) is
coordinated by four oxygen atoms arrayed at the apices of a tetrahedron.
Sharing of the apical oxygens between adjacent tetrahedra, together
with substitution or larger cations on sites that arise between
tetrahedra give rise to a wide variety of silicate structures and
symmetries.

A key determinant of structure type is the number of oxygens per
tetrahedron that are shared with adjacent tetrahedra.  In the sheet
silicates three of the four oxygen atoms of each tetrahedral site are
shared, giving rise to the (Si, Al)-O tetrahedral sheet as the common
structural characteristic of these minerals.  Within the sheet
silicates the micas form a large and important group.  Their
distinguishing feature is a structure based on a double sheet of
tetrahedra between which is sandwiched a layer of octahedral sites
(coordination number six, see Fig. \ref{fig:micastruct}) in which four of
the six apices of each octahedron are coordinated to the oxygens of
the tetrahedra and two to sites that can be occupied by hydroxyl,
fluorine or chlorine.  There are two tetrahedral sheets per octahedral
sheet and the combined tetrahedral-octahedral-tetrahedral layer is
often called a TOT sheet.  The TOT sheets carry a net negative charge
that is compensated by large cations (K, Na, {\it etc}.) lying between them.
Various stacking sequences of the TOT sheets are possible but the most
common gives rise to monoclinic symmetry. Figure 1 was created using VESTA \cite{momma2011vesta} software along with data publicly available through the American Mineralogist database \cite{downs2003american} for biotite \cite{takeda1975mica} and muscovite \cite{guggenheim1987muscovite}.

\begin{figure}
\centering
\includegraphics[scale = 0.4]{MicaStructure.png}
\caption{Crystal structures of biotite and muscovite. \textit{a}) Biotite structure \cite{takeda1975mica} with two $T$ = Si, Al tetrahedral layer shown with light grey spheres which sandwich an octahedral layer. Mg (or Fe) octahedral layer shown with medium grey spheres. Interlayer potassium cations shown with black spheres.  b) Muscovite structure \cite{guggenheim1987muscovite} with interlayer cations shown with black spheres, tetrahedral cations shown with light grey spheres, and octahedral cations shown with medium grey spheres. }
\label{fig:micastruct}
\end{figure}

\subsection{Biotite and Muscovite}
\label{sec:biomusc}

Biotite (K$_2$(Mg,Fe)$_6$[Si$_6$Al$_2$O$_{20}$](OH,F,Cl)$_4$) is trioctahedral
with three octahedral sites per formula unit (six in the conventional
unit cell, which we use here) completely or almost completely filled
(see left panel of Fig. \ref{fig:micastruct}).  Muscovite (K$_2$Al$_4$[Si$_6$Al$_2$O$_{20}$](OH,F,Cl)$_4$) is
dioctahedral, meaning that one of every three octahedra is vacant
(see right panel of Fig. \ref{fig:micastruct}).  In biotite, substitution of trivalent
(Fe\textsuperscript{3+}, Al\textsuperscript{3+}) or quadrivalent
(Ti\textsuperscript{4+}) cations on octahedral sites can be
compensated by either octahedral vacancies or by replacement of Si by
Al on tetrahedral sites beyond the ideal one out of four and there can
be extensive solid solution towards siderophyllite (an end member with
Fe and Al mixing on octahedral sites).  Similarly, in muscovite
altervalent substitution on the octahedral sites can be compensated by
changing the composition of the tetrahedra, although such
substitutions are less common than in biotite.  The chemical diversity
of the micas accounts for their widespread occurrence in igneous,
metamorphic and sedimentary rocks of widely differing chemistry and
paragenesis.

\subsection{Elasticity of Micas}
\label{sec:elasmicas}

Despite its importance and utility, the elasticity of many micas remains merely estimated or unexplored. This is largely due to chemical variability and the low symmetry of these
crystals. While complete sets of elastic constants have been measured
for muscovite \cite{vaug1986,mcne1993}, analogous data for the biotite
group have not been reported. Experimentally-determined elastic
constants for biotite are limited to estimates obtained via ultrasonic
techniques for two phlogopite samples and a biotite sample of
unspecified composition with the symmetry approximated as hexagonal
rather than the true monoclinic \cite{alek1961}. This precludes direct
comparison with theoretical results for the full monoclinic elastic
constants tensor for phlogopite \cite{chhe2014} and muscovite
\cite{militzer2011first}, but differences in some constants among
these samples hint at the dependence of biotite elasticity on
composition.  This effect has not, however, yet been quantified.

In this paper, the results of Brillouin light scattering experiments
on natural crystals of biotite are reported.  The crystals were
selected {\it a priori} to be compositionally different, with the
composition later quantified by electron probe microanalysis.
Directional dependences of elastic wave velocities in the $ac$ and
$bc$ crystal planes were measured from Brillouin peak frequency shifts
and refractive indices determined from the Becke line test. These
velocities, together with sample densities, permitted twelve of
thirteen elastic constants and related mechanical properties to be
determined for each of the samples. The results show that the elastic
properties of biotite depend on octahedral site chemistry.  Comparison
is also made to new and published results for muscovite and to elastic
constants of phlogopite determined from \textit{first-principles}
calculations.

\section{\label{sec:Theory}Theory}
\subsection{Brillouin Light Scattering}
Brillouin spectroscopy is a technique used to probe thermally-excited acoustic phonons (elastic waves) in a medium via the inelastic scattering of light. For a $180^\circ$ backscattering geometry such as that used here, conservation of energy and momentum applied to the scattering process  \cite{dil1982,spez2014} yield the phonon velocity
\begin{equation}
v = \frac{\Omega}{q}= \frac{f\lambda_i}{2n}.
\label{brillouineqn}
\end{equation}
Here, $f= \Omega/2\pi = |f_s - f_i|$ is the Brillouin peak frequency shift and $q = 4\pi n/\lambda_i$ is the magnitude of the wavevector of the probed phonon, where $n$ is the refractive index of the target material and $\lambda_{i}$ the wavelength of the incident light.

\subsection{Elastic Waves} \label{elasticwaves}
Acoustic modes may be considered as sound waves in a crystal since their wavelengths are much greater than the primitive unit cell dimensions. The equation that describes the motion of these waves is

\begin{equation}
\rho \frac{\partial ^2 u_i}{\partial t^2} = C_{ijkl} \frac{\partial ^2 u_k}{\partial x_j \partial x_l},
\label{eq:eom}
\end{equation}
where $\rho$ is the density of the medium, $C_{ijkl}$ is the elastic stiffness tensor, $u_i(x_i,t)$ is the particle displacement as a function of position, $x_i$, and time, $t$, with $i,j,k,l = 1, 2, 3$ \cite{haye2012,every1980general}.

Assuming plane wave solutions to Eq. \ref{eq:eom} of the form $u= e^{i(\boldsymbol{k}\cdot\boldsymbol{r}- \omega t)}$, where $\boldsymbol{r}$ is the position vector, and $\boldsymbol{k}$ and $\omega$ are, respectively, the phonon wavevector and angular frequency, yields the secular equation

\begin{equation}
|\Gamma_{ik}-\rho v^2 \delta_{ik}| = 0,
\label{eq:Christoffel}
\end{equation}
where $\Gamma_{ik} = C_{ijkl} n_{j} n_{l}$ is the Christoffel matrix, $n_j$ and $n_l$ are direction cosines, and $\delta_{ik}$ is the Kronecker delta  \cite{haye2012}.

For monoclinic symmetry the elastic constants tensor takes the form \cite{Nye},

\begin{equation}
C_{ij} = \begin{bmatrix}
 C_{11} & C_{12} & C_{13} & 0 & C_{15} & 0 \\
 C_{12} & C_{22} & C_{23} &0 & C_{25} & 0 \\
 C_ {13} & C_{23} & C_{33} & 0 & C_{35} & 0 \\
  0 & 0 & 0 & C_{44} & 0 & C_{46}\\
  C_{15} & C_{25} & C_{35} & 0 & C_{55} & 0\\
  0 & 0 & 0 & C_{46} & 0 & C_{66}\\
\end{bmatrix}.
\label{eq:Monotensor}
\end{equation}
\noindent
where here the standard monoclinic orientation with $ a \neq b \neq c $ with the unique $b$-axis and $\beta \neq 90^\circ$ used. Figure \ref{fig:bioopt} shows the orientation for a monoclinic biotite structure. Voigt notation has been introduced to reduce the number of subscripts on the elastic constants from four to two by replacing each of the couples $ij$ and $kl$ with a single subscript via the scheme: 11{ $\rightarrow$ 1}; 22{ $\rightarrow$ 2}; 33{$\rightarrow$ 3}; 23, 32{$\rightarrow$ 4}; 13, 31{ $\rightarrow$ 5}; and 12, 21{ $\rightarrow$ 6} \cite{malgrange2014symmetry}.

For elastic waves propagating in the $ac$ plane (010) at angle $\theta$ to the crystallographic $c$-axis, $n_x=\sin\theta$, $n_y=0$,  and $n_z=\cos\theta$, and Eq. \ref{eq:Christoffel} becomes for the pure transverse (T) mode,
\begin{equation}
C_{eff}= C_{66}\sin^2\theta + C_{44}\cos^2\theta + 2 C_{46}\sin\theta\cos\theta,
\label{eq:puretransverse}
\end{equation}
and
\begin{equation}
C_{eff}^{\pm}= \frac{1}{2}(-b \pm \sqrt{b^2-4c}),
\label{eq:CeffEQsolve}
\end{equation}
where,
\begin{equation}
b= - [\sin^2{\theta}(C_{11}+C_{55})+2 \sin{\theta}\cos{\theta}(C_{15}+C_{35})
+\cos^2{\theta(C_{33}+C_{55})}], \nonumber
\end{equation}
and
\begin{multline*}
c=\sin^4{\theta}(C_{11}C_{55}-C^{2}_{15})+2\sin^{3}{\theta}\cos{\theta}(C_{11}C_{35}-C_{13}C_{15})\\+
\sin^{2}{\theta}\cos^{2}{\theta}(C_{11}C_{33}-C^{2}_{13}+2C_{15}C_{35}-2C_{13}C_{55})\\+
2\sin{\theta}\cos^{3}{\theta}(C_{33}C_{15}-C_{13}C_{35})+\cos^{4}{\theta}(C_{33}C_{55}-C^{2}_{35}),\\
\end{multline*}
with effective elastic moduli $C_{eff} = \rho v^2$.  The ``+" term in Eq. \ref{eq:CeffEQsolve} refers to the quasi-longitudinal (QL) mode and the ``-" term refers to the quasi-transverse (QT) mode.

For propagation in the $bc$ crystallographic plane (100), the effective moduli are roots of a cubic equation having the form
\begin{equation} \label{eq: bcplaneeqn}
C_{eff}^3  - \alpha C_{eff}^2 + \beta C_{eff}  - \gamma = 0,
\end{equation}
where
\begin{equation*}
\alpha =  [ (C_{66} + C_{22}) \sin^2\theta + (C_{55}+C_{33})\cos^2\theta +C_{44} ],
\end{equation*}
\begin{multline*}
\beta=\sin^4 \theta (C_{44}C_{66} + C_{22}C_{44} + C_{22}C_{66} - C_{46}^2 )\\ +
\sin^2 \theta \cos^2 \theta (C_{44}C_{66} - C_{23}^2 - C_{25}^2  - 2 C_{23}C_{44}- 2C_{46}(C_{25} + C_{35})\\ +
C_{44}C_{55} + C_{33}C_{66} + C_{22}(C_{33}+C_{55}) - C_{46}^2)\\ +
\cos^4 \theta (C_{33} C_{44} - C_{35}^2 +C_{33}C_{55} + C_{44}C_{55}),\\
\end{multline*}
and
\begin{multline*}
\gamma = [\sin^6 \theta ( C_{22}C_{44}C_{66} - C_{22}C_{46}^2 ) \\+
\sin^4 \theta \cos^2 \theta ( 2 C_{22} C_{35}C_{46} + C_{22}C_{44}C_{55} \\ +
  C_{22}C_{33}C_{66} - C_{25}^2 C_{44} - C_{23}^2 C_{66}
 + 2 C_{25}C_{23}C_{46} + 2 C_{23}C_{46}^2 - 2 C_{44}C_{23}C_{66}) \\  +
 \cos^4 \theta \sin^2 \theta ( C_{33}C_{55}C_{22} - C_{35}^2 C_{22} - C_{33}C_{46}
  + C_{33}C_{44}C_{66}\\ - C_{25}^2C_{33}  - C_{23}^2 C_{55} +
  2 C_{25}C_{35}C_{44} - 2 C_{25} C_{33}C_{46}\\
 + 2 C_{25}C_{23}C_{35} + 2 C_{23}C_{35}C_{46} - 2 C_{23}C_{44}C_{55} ) \\ +
 \cos^6 \theta (C_{44}C_{55}C_{33} - C_{35}^2 C_{44}) ].\\
 \end{multline*}
The roots of Eq. \ref{eq: bcplaneeqn} are known from the theory of algebraic equations \cite{mens1997} and are given by
\begin{equation*}
M^{i} = X^{i} + \frac{\alpha}{3},  \hspace{5mm} i = 1,2,3
\end{equation*}
where
\begin{equation*}
X^{i} = 2 \sqrt[3]{R} \cos( {\frac{\phi}{3} + (i-1)\frac{2 \pi}{3}}) , \hspace{5mm} i =1,2,3,
\end{equation*}
with
\begin{equation*}
R = \sqrt{(\frac{1}{3} (\frac{\alpha^2}{3} - \beta  ))^3},
\end{equation*}
and
\begin{equation*}
\phi = \arccos \Big ( \frac{{\frac{2}{27} \alpha^3 - \frac{1}{3}\alpha \beta + \gamma  }}{2 R} \Big ).
\end{equation*}

\section{\label{sec:Experiment}Experimental Details}
\subsection{Samples}

\subsubsection{General Physical Characteristics}
The biotite samples used in this study were natural crystals deliberately selected to have a range of colours to maximize the likelihood that they possessed substantially different levels of major cations Fe and Mg.  Platelets with areas of $\sim1$ cm$^{2}$ and thicknesses of a few hundred $\mu$m were cleaved from these larger bulk samples to allow loading onto the sample stage and to expose a pristine surface for Brillouin light scattering experiments.  For the purposes of comparison, two muscovite samples were also analyzed.

\subsubsection{Chemical Composition}
Chemical analysis was carried out on fragments mounted in epoxy cement with the \{001\} cleavage plane orientated vertically and polished to a final grade of 0.25 micron using diamond abrasive.  The samples were carbon coated and examined initially using a JEOL JSM-7100F Scanning Electron Microscope (SEM).  Reconnaissance chemical analysis by energy dispersive x-ray spectroscopy (Thermo\texttrademark) allowed us to confirm the identification of the mica species and identify potential sites for analysis by electron microprobe.

Following SEM investigation the samples were analysed using a JEOL
JXA-8230 Electron Probe Microalalyser (EPMA) operating at an
accelerating voltage of 15kV and beam current of 20nA in wavelength
dispersive (WDS) mode.  The $K_\alpha$ emission lines were counted
using a TAP diffracting crystal for Si, Al, Mg, and Na, a LIFH crystal for Fe, Ti and Mn, a PETL crystal for K, Ca, and Cl and a LDE1 crystal for F.  Counts were collected for the emission peak and for positions on both the low and high wavelength side of the peak.  Total background counting time was the same as that for the emission peak.  The following standards were used: Si, Na albite; Al, pyrope; K K-feldspar; Ca, Mg diopside; Fe almandine; Ti rutile; Mn rhodenite; Cl tugtupite; F apatite.  A secondary standard (Astimex biotite) was analysed intermittently to test for instrument drift.

Estimated detection limits (3$\sigma$, weight percent) for those elements expected to be present in trace to minor quantities are as follows: Mn, Ti 0.015; Na 0.02, Ca 0.006; Cl 0.01; F 0.07. Table \ref{tab:EDX} presents the mean of the analyses for each sample as weight percent oxide recalculated to 22 oxygen atoms and assigned to the tetrahedral (T), octahedral (O) or large cation (X) sites.  The number of analyses per sample was as
follows: Muscovite $\#$2, one, Biotite $\#$1, six, and ten analyses for each of
Biotite $\#$3, $\#$2 and Muscovite $\#$1.

\begin{table}
  \caption{Chemical composition (weight percent oxides) and calculated
    formulae (atoms per formula unit, apfu) of muscovite and biotite samples determined using electron probe
    microanalysis.  Overall atomic percentages of Fe and Mg are presented in the last two rows and are used throughout the paper to identify the samples.}
  \centering \footnotesize
\begin{tabular}{lccccc} \hline\hline
Analyte & Biotite \#1 & Biotite \#2 & Biotite \#3 & Muscovite \#1 & Muscovite \#2 \\
\hline
SiO$_2$	&	31.1855	&	35.0984	&	39.7346	&	48.8039	&	47.3331 \\
TiO$_2$	&	0.10572	&	2.41863	&	1.16121	&	0.64852	&	0.2562 \\
Al$_2$O$_3$	&	17.405	&	14.7722	&	16.8896	&	31.2563	&	37.382 \\
FeO	&	36.612	&	24.8768	&	3.94769	&	7.0886	&	1.7884 \\
MnO	&	0.60836	&	0.31048	&	0.04437	&	0.23769	&	0.0356 \\
MgO	&	0.34392	&	7.65661	&	23.9435	&	1.04965	&	0.8496 \\
CaO	&	0.2015	&	0.17557	&	0.02239	&	0.0457	&	0.0448 \\
Na$_2$O	&	0.41982	&	0.36057	&	0.15724	&	0.43438	&	0.6551 \\
K$_2$O	&	7.27324	&	8.89984	&	10.0847	&	8.5955	&	9.1772 \\
Cl	&	0.17004	&	0.09353	&	0.02054	&	0.02558	&	0.0241 \\
F	&	1.73391	&	1.3567	&	2.13393	&	0.39759	&	0.179 \\
Total	&	95.2906	&	95.4269	&	97.2366	&	98.4102	&	97.6443 \\ \hline
Ion	&	apfu	&	apfu	&	apfu	&	apfu	&	apfu \\ \hline
Si	&	5.16	&	5.48	&	5.46	&	6.39	&	6.10 \\
Al	&	2.84	&	2.52	&	2.54	&	1.61	&	1.90 \\
Al	&	0.55	&	0.2	&	0.19	&	3.21	&	3.77 \\
Ti	&	0.01	&	0.28	&	0.12	&	0.06	&	0.02 \\
Fe	&	5.06	&	3.25	&	0.45	&	0.78	&	0.19 \\
Mn	&	0.09	&	0.04	&	0.01	&	0.03	&	0.00 \\
Mg	&	0.08	&	1.78	&	4.9	&	0.20	&	0.16 \\
Ca	&	0.04	&	0.03	&	0.00	&	0.01	&	0.01 \\
Na	&	0.13	&	0.11	&	0.04	&	0.11	&	0.16 \\
K	&	1.53	&	1.77	&	1.77	&	1.43	&	1.51 \\
Cl	&	0.05	&	0.02	&	0.00	&	0.01	&	0.01 \\
F	&	0.91	&	0.67	&	0.93	&	0.16	&	0.07 \\ \hline
Site	&	Total	&	Total	&	Total	&	Total	&	Total \\ \hline
T	&	8.00	&	8.00	&	8.00	&	8.00	&	8.00 \\
O	&	5.79	&	5.56	&	5.68	&	4.28	&	4.16 \\
X	&	1.70	&	1.91	&	1.81	&	1.55	&	1.68 \\
OH	&	0.95	&	0.69	&	0.93	&	0.17	&	0.08 \\
Total	&	16.44	&	16.16	&	16.42	&	14.00	&	13.92 \\
Fe	At\% (Cations)	&	30.78	&	20.11	&	2.74	& 5.57 &	1.36 \\
\hline
\bf{Fe At\% (Overall)} & \bf{12.7} & \bf{8.1} & \bf{1.1} & \bf{2.1} & \bf{0.5} \\
\bf{Mg At\% (Overall)} & \bf{0.2} & \bf{4.5} & \bf{12.3} & \bf{0.5} & \bf{0.4} \\
\hline \hline
\end{tabular}
\label{tab:EDX}
\end{table}

\begin{table}
\caption{Principal refractive indices of biotites and muscovite determined using the Becke line test.}
\centering \footnotesize
\begin{tabular}{cccccc c}
\hline
Sample  &  Colour    &  $n_{\alpha}$  &  $n_{\beta}$  &  $n_{\gamma}$  & $n_{avg}$ & $2V$ \\ [0.5ex] \hline
Biotite & Green & 1.551 & 1.665 & 1.665 & 1.613 & $\ll20^{\circ}$ \\
12.7\% Fe, 0.2\% Mg & & & & & & \\ \hline
Biotite & Green & 1.575 & 1.657 & 1.657 & 1.629 & $\ll10^{\circ}$ \\
8.1\% Fe, 4.5\% Mg & & & & & & \\ \hline
Biotite & Pale Brown & 1.544 & 1.597 & 1.597 & 1.579 &Too small \\
1.1\% Fe, 12.3\% Mg & & & & &  &to measure \\ \hline
Muscovite & Very Pale & 1.543 & 1.567 & 1.618 & 1.576& $\sim 35^{\circ}-40^{\circ}$ \\
2.1\% Fe, 0.5\% Mg & Brown-Green & & & & & \\ \hline
Muscovite & Pale Brown & 1.558 & 1.603 & 1.610 & 1.590 &-- \\
0.5\% Fe 0.4\% Mg & & & & & & \\ \hline
Biotite (Ref. \cite{coat1944}) & Green-Grey &1.570 & 1.640&1.640 & 1.620 & -- \\ \hline
Biotite (Ref. \cite{bilg1956}) & -- &1.575 & 1.617 &1.621 & 1.604 & 30$^\circ$\\ \hline
Biotite (Ref. \cite{hutt1947}) & Dark Brown &1.57 & 1.64&1.64 & 1.620 & -- \\ \hline
Biotite (Ref. \cite{lars1938}) & -- &1.586 & 1.643 &1.643 & 1.623 & 0$^\circ$ - 8$^\circ$ \\ \hline
Biotite (Ref. \cite{nock1939}) & Blue-Green &1.582 & 1.625&1.625 & 1.610 & -- \\ \hline
Biotite (Ref. \cite{pagl1940}) & Violet-Brown &1.544 & 1.583&1.585 & 1.571 & -- \\

\hline
\end{tabular}
\label{tab:EDS1}
\end{table}
\subsubsection{Refractive Indices}
Fig. \ref{fig:bioopt} shows the optical orientation of biotite, with
the crystallographic {\it b} axis chosen coincident with the two fold
symmetry axis.  Refractive indices of the biotite samples were
obtained using the Becké line test \cite{ness2012}. Polarized light
was provided by a Nikon Eclipse i50 Pol polarizing light microscope
and Cargille optical liquids of known refractive indices and graduated
at intervals of 0.002 were used as immersion liquids.  Measurements of
refractive indices $\gamma$ and $\beta$ were made on sample fragments
that were placed on the perfect \{001\} cleavage.  The remaining refractive
index $\alpha$ was determined by mounting each sample on a spindle
stage with cleavage plane perpendicular to the stage
surface. Refractive index values determined using this approach are
presented in Table \ref{tab:EDS1} and are considered accurate to $\pm$
0.001.

\begin{figure}
  \centering
  \includegraphics[scale = 0.5]{Biotite_Structure_Monoclinic_Crystal.png}
  \caption{ \textbf{a}) $a$, $b$ and $c$ are the crystallographic axes, $\beta$ and $\gamma$ are the principal refractive indices. Incident laser beam shown with $\theta_i$ indicating angle from $c$-axis. Not shown principle refractive index $\alpha$.
  \textbf{b)} Biotite structure (unique $b$ axis and $\beta \neq 90^\circ$) \cite{takeda1975mica} viewed down from \textit{b}. Projection of unit cell shown along with the Cartesian coordinate system, z $\parallel$ $c$, x $\parallel$ ${a^*}$ and y $\parallel$ $b$. Note: $b \parallel b^*$. }
  \label{fig:bioopt}
\end{figure}

\subsubsection{Density Determination}
Mass densities for the biotite samples used in this work were required to determine elastic constants.  The equation $\rho = Ax^{B} + C$, with $A=-0.141$, $B=-0.523$, and $C=3.085$, was found to yield a good empirical fit to published density versus $x =$ [Fe]/[Mg] data for biotites of known composition \cite{coat1944,bilg1956,hutt1947,lars1938,nock1939,pagl1940}. This was used, together with the composition data obtained from electron probe microanalysis, to estimate the density for the samples of this work. Figure \ref{fig:density} shows the fitting results with $\displaystyle{\chi^2}$ quoted in the legend.  The uncertainty in density determined using this approach is $\sim5$\%.  It is important to note that the empirical equation above should not be used to estimate the density of a biotite for which [Fe]/[Mg] lies outside the range of those of the samples studied in present work as it may yield inaccurate or unphysical values, particularly for small [Fe]/[Mg] values.

The density of muscovite was taken as fixed and set equal to the average of two previously published values: $(2832 + 2844)/2$ kg$\cdot$m$^{-3}$ = 2838 kg$\cdot$m$^{-3}$ \cite{vaug1986,mcne1993}.

\begin{figure}
  \centering
  \includegraphics[scale = 0.4]{DensityPlot.png}
  \caption{Mass density versus $x =$ [Fe]/[Mg] data for biotites of known composition \cite{coat1944,bilg1956,hutt1947,lars1938,nock1939,pagl1940}. The dashed vertical lines indicate [Fe]/[Mg] ratios for the biotites of the present work.}
  \label{fig:density}
\end{figure}

\subsection{Brillouin Light Scattering Apparatus}
Brillouin spectra were obtained under ambient conditions utilizing a $180^\circ$ backscattering geometry.  Light of wavelength $\lambda_{i} = 532$ nm and power of 60 mW from a Nd:YV$O_4$ single mode laser was incident on the target sample at angles $5^{\circ} \lesssim \theta_{i} \lesssim 75^{\circ}$, corresponding to probed phonon propagation directions ranging from $\sim3^{\circ}$ to $\sim37^{\circ}$ from the crystallographic $c$-axis.  Focusing of incident light onto the sample and collection of scattered light was accomplished using a high-quality anti-reflection-coated camera lens of focal length $f=5$ cm and $f/\# = 5.6$.  After exiting this lens, the scattered light was processed by a spatial filter (40 cm lens - 450 $\mu$m-diameter pinhole - 20 cm lens) and subsequently frequency-analyzed by an actively-stabilized 3+3 pass tandem Fabry-Perot interferometer (JRS Scientific Instruments).  The free spectral range of the interferometer was set to 40 GHz and the finesse was $\sim100$. The light transmitted by the interferometer was incident on a pinhole of diameter 700 um and detected by a low-dark count rate ($\lesssim 1$ s$^{-1}$)  photomultiplier tube where it was converted to an electrical signal and sent to a computer for storage and display. A schematic diagram of the apparatus can be found in Ref. \cite{andr2018}.

\section{\label{sec:Results}Results \& Discussion}

\subsection{Spectra}
Figure \ref{fig:figure4.1} shows a series of Brillouin spectra collected from the Fe-rich biotite crystal containing 12.7\% Fe and 0.2\% Mg.   Three sets of Brillouin doublets were observed and attributed to T, QT, and QL acoustic modes due to the similarity of the frequency shifts to those of muscovite \cite{vaug1986,mcne1993}.  The frequency shifts of all of these peaks showed significant variation with angle of incidence, reflecting the expected large elastic anisotropy of biotite. Spectra of other samples were qualitatively similar, although the T peaks in the sample with 8.1\% Fe and 4.5\% Mg were noticeably weaker than for the other samples.

\begin{figure}
\centering
\includegraphics[scale = 0.5]{Brillouin_FeRich_AC_Spectra_2.jpg}
  \caption{Brillouin spectra of an Fe-rich biotite crystal (12.7\% Fe, 0.2\% Mg).  The probed phonons propagated in the (010) plane.  Angles of incidence are indicated. T, QT, and QL refer to peaks due to pure transverse, quasi-transverse, and quasi-longitudinal acoustic modes, respectively.}
  \label{fig:figure4.1}
\end{figure}

\subsection{Elastic Wave Velocities}
Figure \ref{velvsdir} shows the T, QT, and QL mode velocities in the $ac$ and $bc$ crystallographic planes as measured from the $c$-axis.  These velocities were calculated from Eq. \ref{brillouineqn} using the associated Brillouin peak frequency shifts and, because birefringence effects were negligible (no obvious peak splitting), the average refractive index for each sample \cite{ness2012}.
For propagation directions near the $c$-axis, the T and QT velocities are relatively low and increase with increasing angle away from the $c$-axis in both the $ac$ and $bc$ planes. The QL mode velocity remains relatively constant for angles close to the $c$-axis and increases for propagation directions greater than $\approx 20-25^\circ$. Studies on muscovite  \cite{vaug1986} and phlogopite \cite{chhe2014} show a similar dependence of sound velocities on propagation direction over common ranges.

It can also be seen from Figure \ref{velvsdir} that the elastic wave velocities for biotite, nearly without exception, increase with decreasing Fe concentration (or, equivalently, increasing Mg concentration), and approach those of muscovite for the biotite sample with the lowest Fe concentration. The same type of behaviour was observed in ultrasonic studies on muscovite, phlogopite, and a biotite of unknown composition, where, for nearly all propagation directions for which measurements were made, the velocities for the phlogopites and the biotite were lower than the corresponding velocities for muscovite \cite{alek1961}. This may be due in part to the relatively large atomic mass of Fe compared to Mg and Al, resulting in a lower vibrational frequency and therefore a lower velocity, but it could also be due to other factors such as differences in chemical bonding.

\begin{figure}[H]
\centering
\includegraphics[scale=0.32]{Velocity_All.jpg}
  \caption{Quasi-longitudinal (top), quasi-transverse (center), and pure transverse (bottom) mode velocities versus direction ({\it i.e.}, angle from the crystallographic $c$-axis) in the $ac$ and $bc$ crystallographic planes. $\circ$ -  muscovite 2.1\% Fe, 0.5\% Mg; $\square$ - muscovite 0.5\% Fe, 0.4\% Mg; $\blacksquare$ - biotite 1.1\% Fe, 12.3\% Mg; $\blacktriangle$ - biotite 8.1\% Fe, 4.5\% Mg; $\blacklozenge$ -  biotite 12.7\% Fe, 0.2\% Mg.  The vertical and horizontal error bars are approximately the size of the symbols.}
\label{velvsdir}
\end{figure}

\begin{landscape}

\begin{table}
\scriptsize
\vspace{30mm}

\caption{Elastic constants (in GPa) and selected elastic constant ratios for biotite and muscovite obtained in the present work and for biotite, muscovite, and phlogopite obtained in previous studies.}

\begin{tabular}{c  c  c  c  c  c  c  c  c  c  c  c c c c c c}
\hline
Sample & $C_{11}$ &$C_{22}$ & $C_{33}$ & $C_{44}$ & $C_{55}$ & $C_{66}$ & $C_{12}$ & $C_{13}$ & $C_{15}^{a}$ & $C_{23}$ & $C_{25}^{a}$ & $C_{35}^{a}$ & $C_{46}^{a}$ & $C_{22}/C_{11}^{b}$ & $C_{55}/C_{44}^{b}$ & $C_{23}/C_{13}^{b}$ \\
 \hline
Biotite  & 178 & 176  & 42 & 4.4 & 11 & 77 & 24 & 14 & -9.3 & 22 & 27 & 1.8 & -7.7 & 0.99 & 2.50 & 1.57 \\
12.7\% Fe, 0.2\% Mg & $\pm$ 5.4 & $\pm$ 3.6 & $\pm$ 0.3 & $\pm$ 0.2 & $\pm$ 0.8 & $\pm$ 1.3 & - & $\pm$ 6.9 & $\pm$ 4.6 & $\pm$ 5.2 & $\pm$ 1.5& $\pm$ 0.9& $\pm$ 3.8 & $\pm$ 0.05 & $\pm$ 0.30 & $\pm$ 1.14 \\ \hline

Biotite  & 183 & 183 & 48 & 7.9 & 16 & 75 & 33 & 21 & -12 & 22 & 27 & -2.7 & -6.2 & 1.00 & 2.02 & 1.04 \\
8.1\% Fe, 4.5\% Mg & $\pm$ 5.4 & $\pm$ 5.5 & $\pm$ 0.6 & $\pm$ 0.2 & $\pm$ 0.8 & $\pm$ 1.7 & - & $\pm$ 4.7 & $\pm$ 2.1 & $\pm$ 4.5 & $\pm$ 3.5& $\pm$ 1.3& $\pm$ 2.2 & $\pm$ 0.06 & $\pm$ 0.15 & $\pm$ 0.44  \\ \hline

Biotite  & 178 & 171 & 52 & 12.2 & 17 & 76 & 26 & 28 & -13 & 22 & 22 & -8.5 & -6.2 & 0.96 & 1.39 & 0.78 \\
1.1\% Fe, 12.3\% Mg & $\pm$ 5.2 & $\pm$ 3.5 & $\pm$ 1.2 & $\pm$ 0.3 & $\pm$ 0.5 & $\pm$ 1.5 & - & $\pm$ 5.4 & $\pm$ 1.6 & $\pm$ 5.1 & $\pm$ 2.5& $\pm$ 1.9& $\pm$ 3.4& $\pm$ 0.05 & $\pm$ 0.08 & $\pm$ 0.33  \\ \hline

Muscovite & 182 & 182 & 66 & 13.8 & 17 & 71 & 40 & 22 & -2.3 & 24 & 18 & -3.8 & 1.3 & 1.00 & 1.23 & 1.09 \\
2.1\% Fe, 0.5\% Mg & $\pm$ 4.0 & $\pm$ 3.8 & $\pm$ 1.1 & $\pm$ 0.5 & $\pm$ 0.6 & $\pm$ 1.4 & - & $\pm$ 2.2 & $\pm$ 3.3 & $\pm$ 2.7 & $\pm$ 3.4& $\pm$ 1.7& $\pm$ 2.0 & $\pm$ 0.04 & $\pm$ 0.09 & $\pm$ 0.23  \\ \hline

Muscovite &  180 & 183 & 56 & 17.3 & 23 & 77 & 26 & 22 & -12 & 19 & 29 & -2.7 & -7.2 & 1.02 & 1.33 & 0.86 \\
0.5\% Fe, 0.4\% Mg & $\pm$ 3.6 & $\pm$ 3.6 & $\pm$ 1.2 & $\pm$ 0.5 & $\pm$ 0.6 & $\pm$ 1.3 & - & $\pm$ 1.9 & $\pm$ 2.9 & $\pm$ 4.6 & $\pm$ 4.5& $\pm$ 1.0 & $\pm$ 2.2 & $\pm$ 0.04 & $\pm$ 0.07 & $\pm$ 0.28  \\ \hline

Muscovite (Composition Unknown) & 181.0 & 178.4 & 58.6 & 16.5 & 19.5 & 72.0 & 48.8 & 25.6 & -14.2 & 21.2 & 1.1 & 1.0 & -5.2 & 0.99 & 1.18 & 0.83\\
\cite{vaug1986} & $\pm$ 1.2 & $\pm$ 1.3 & $\pm$ 0.6 & $\pm$ 0.6 & $\pm$ 0.5 & $\pm$ 0.7 & $\pm$ 2.5 & $\pm$ 1.5 & $\pm$ 0.8 & $\pm$ 1.8 & $\pm$ 3.7 & $\pm$ 0.6 & $\pm$ 0.9 & $\pm$ 3.7 & $\pm$ $0.6^{c}$ & $\pm$ $0.9^{c}$  \\ \hline

Muscovite (Composition Unknown) & 176.5 & 179.5 & 60.9 & 15.0 & 13.1 & 70.7 & 47.7 & 20.0 & -1.2 & 23.0 & 11.1 & -0.7 & 0.7 & 1.02 & 0.87 & 1.15 \\
\cite{mcne1993} & $\pm$ 1.1 & $\pm$ 1.3 & $\pm$ 0.6 & $\pm$ 0.3 & $\pm$ 0.2 & $\pm$ 0.6 & $\pm$ 1.2 & $\pm$ 1.1 & $\pm$ 0.6 & $\pm$ 8 &$\pm$ 5.3 & $\pm$ 0.5 & $\pm$ 0.5 & $\pm$ 0.01 & $\pm$ $0.07^{c}$ & $\pm$ $0.12^{c}$ \\ \hline

Phlogopite (DFT-GGA) & 181.2 & 184.7 & 62.1 & 13.5 & 20.0 & 67.9 & 47.6 & 12.2 &-15.7 & 12.1 & -4.9 & -1.2 & -5.9 & 1.02 & 1.48 & 0.99 \\
\cite{chhe2014} \\ \hline

Muscovite (DFT-GGA) & 180.9 & 170.0 & 60.3 & 18.4 & 23.8 & 70.5 & 53.4 & 27.4 &-14.7 & 23.5 & 1.4 & -1.0 & -1.8 & 1.06 & 1.29 & 0.86 \\
\cite{militzer2011first} \\ \hline

Phlogopite (DFT-LDA) & 199.5 & 201.2 & 82.2 & 17.0 & 25.3 & 72.4 & 54.1 & 25.4 &-13.1 & 24.4 & -4.5 & -2.8 & -6.4 & 1.01 & 1.49 & 0.96 \\
\cite{chhe2014} \\ \hline

Muscovite (DFT-LDA) & 194.3 & 188.0 & 91.1 & 25.2 & 30.5 & 71.3 & 68.1 & 43.2 &-14.3 & 39.5 & 1.7 & 1.1 & -0.6 & 1.03 & 1.21 & 0.91 \\
\cite{militzer2011first} \\

\hline
\multicolumn{17}{l}{$^{a}$Elastic constant is zero for hexagonal symmetry.}\\
\multicolumn{17}{l}{$^{b}$Ratio is unity for hexagonal symmetry.}\\ \multicolumn{17}{l}{$^{c}$Uncertainties in ratios calculated by the authors of the present work using standard error formulae.}
\end{tabular}
\label{tab:elasticconstants}
\end{table}
\end{landscape}

\begin{table}[t]
\centering
\small
\caption{Voigt and Reuss bulk ($K_X$) and shear moduli ($G_X$) (in GPa) for biotite and muscovite obtained in the present work and for biotite, muscovite, and phlogopite obtained in previous studies.}
\begin{tabular}{l c  c  c  c} \hline \hline
Sample & K$_V$ & K$_R$ & G$_V$ & G$_R$\\ \hline
Biotite - 12.7\% Fe, 0.2\% Mg & 57.3 $\pm$ 4.2 & 35.3 $\pm$ 2.8  & 41.0 $\pm$ 3.0 & 9.44 $\pm$ 0.8 \\
Biotite - 8.1\% Fe, 4.5\% Mg  & 63.0 $\pm$ 4.3 & 42.0 $\pm$ 2.9 & 43.0 $\pm$ 3.0 & 16.0 $\pm$ 1.1 \\
Biotite - 1.1\% Fe, 12.3\% Mg  & 61.4 $\pm$ 4.1 & 43.0 $\pm$ 2.9 & 42.9 $\pm$ 2.7 & 19.4 $\pm$ 1.2 \\
Muscovite - 2.1\% Fe, 0.5\% Mg & 66.9 $\pm$ 5.0 & 43.3 $\pm$ 3.2 & 52.0 $\pm$ 3.9 & 25.0 $\pm$ 1.9\\
Muscovite - 0.5\% Fe, 0.4\% Mg & 61.0 $\pm$ 4.8 & 45.8 $\pm$ 3.6 & 46.9 $\pm$ 3.7 & 26.9 $\pm$ 2.1\\
Muscovite (Comp Unknown) \cite{vaug1986} & 67.7 $\pm$ 1.5 & 48.7 $\pm$ 1.1 & 43.1 $\pm$ 0.94 & 27.6 $\pm$ 0.6 \\
Muscovite (Comp Unknown) \cite{mcne1993} & 66.5 $\pm$ 2.6 & 49.0 $\pm$ 1.8 & 42.0 $\pm$ 1.6 & 23.0 $\pm$ 0.9  \\
Phlogopite (DFT-GGA)\cite{chhe2014} & 61.0 $\pm$ 4.7 & 43.0 $\pm$ 8.2 & 43.3 $\pm$ 1.3 & 24.1 $\pm$ 2.9\\
Muscovite (DFT-GGA) \cite{militzer2011first} & 86.1 & 73.1 & 47.0 & 38.2 \\
Phlogopite (DFT-LDA) \cite{chhe2014} & 77.2 $\pm$ 5.6 & 62.9 $\pm$ 7.2 & 47.6 $\pm$ 1.1 & 31.2 $\pm$ 2.0 \\
Muscovite (DFT-LDA) \cite{militzer2011first} & 68.8 & 50.1 & 43.0 & 31.0 \\ \hline \hline
\end{tabular}
\label{tab:bulkshearmoduli}
\end{table}

\subsection{Elastic Constants}
\subsubsection{Determination of Elastic Constants}
Table \ref{tab:elasticconstants} and Fig.~\ref{fig:Cefftrends} give the elastic constants for the biotite and muscovite samples of the present work along with values obtained in previous studies. All of the constants for the samples of the present work except $C_{12}$ were obtained using a custom non-linear least-squares fitting routine written in MATLAB and based on the Levenberg-Marquardt algorithm. This routine executed a global fit of the expressions for $C_{eff}=\rho v_{m}^{2}(\theta, C_{ij})$, where $m$ = T, QT, and QL, given in Sec. \ref{elasticwaves} to experimental data by minimizing the square of the difference between experimental and calculated $\rho v_{m}^{2}(\theta, C_{ij})$ values for all three modes and measured propagation directions in the $ac$ and $bc$ planes simultaneously through adjustment of the $C_{ij}$.  Initial guesses of elastic constants were those determined for biotite with assumed hexagonal symmetry and muscovite found in the literature \cite{vaug1986,mcne1993,alek1961}.  Embedded in the fitting routine was the constraint that the elastic constants, when appropriately combined, satisfy the conditions for elastic stability ({\it i.e.}, that the elastic energy be positive) for a crystal with monoclinic symmetry \cite{mouhat2014necessary}.  These are that the diagonal elements of the elastic constants tensor, $C_{ii}$, be greater than zero and that

\begin{multline}
    (C_{11}C_{25}^2C_{33} + C_{15}^2(-C_{23}^2 + C_{22}C_{33})\\ - 2C_{11}C_{23}C_{25}C_{35} - C_{12}^2C_{35}^2 + C_{11}C_{22}C_{35}^2\\ + 2C_{15}(C_{13}C_{23}C_{25} - C_{12}C_{25}C_{33} - C_{13}C_{22}C_{35} \\+ C_{12}C_{23}C_{35}) + C_{11}C_{23}^2C_{55} + C_{12}^2C_{33}C_{55}\\ - C_{11}C_{22}C_{33}C_{55} + C_{13}^2(-C_{25}^2 + C_{22}C_{55}) \\+ 2C_{12}C_{13}(C_{25}C_{35}-C_{23}C_{55}))(C_{46}^2 - C_{44}C_{66}) > 0,
\label{stabcon1}
\end{multline}

\begin{multline}
C_{44}(-C_{11}C_{25}^2 C_{33} + C{15}^2(C_{23}^2 - C_{22}C_{33})\\ + 2C_{11}C_{23}C_{25}C_{35} + C_{12}^2C_{35}^2 - C_{11}C_{22}C_{35}^2\\ - 2C_{15}(C_{13}C_{23}C_{25} - C_{12}C_{25}C_{33} - C_{13}C_{22}C_{35}\\ + C_{12}C_{23}C_{35}) - C_{11}C_{23}^2C_{55} - C_{12}^2C_{33}C_{55}\\ + C_{11}C_{22}C_{33}C_{55} + C_{13}^2(C_{25}^2 - C_{22}C_{55})\\ + 2C_{12}C_{13}(-C_{25}C_{35} + C_{23}C_{55})) > 0,
\label{stabcon2}
\end{multline}

\begin{equation}
-C_{44}(C_{13}^2C_{22} - 2C_{12}C_{13}C_{23} + C_{12}^2C_{33} + C_{11}(C_{23}^2 - C_{22}C_{33})) > 0,
\label{stabcon3}
\end{equation}

\begin{equation}
C_{11}C_{22}C_{33} - C_{13}^2C_{22} + 2C_{12}C_{13}C_{23} - C_{11}C_{23}^2 - C_{12}^2C_{33}  > 0,
\label{stabcon4}
\end{equation}
and
\begin{equation}
C_{11}C_{22} - C_{12}^2 >0.
\label{stabcon5}
\end{equation}

In all cases, the quality of the fits was excellent as can be seen for Fe-rich and Fe-poor biotite in Fig. \ref{fig:Ceff_Fit_Fe_Mg}. The standard error of regression (SER) for the highest- and lowest-quality least-squares fits have been calculated as points of reference. The best fit (SER = 1.28 GPa) was obtained for Fe-rich biotite (12.7\% Fe, 0.2\% Mg) and the lowest-quality fit (SER = 1.69) was obtained for the biotite sample with 8.1\% Fe and 4.5\% Mg. Fig. \ref{fig:ResidualPlot} shows a representative series of residual plots for the Fe-poor (1.1\% Fe, 12.3\% Mg) sample.  The uncertainties in the elastic constants were estimated by averaging the difference between individual best-fit constants and corresponding members of second and third sets of elastic constants obtained by increasing and decreasing, respectively, all experimental $\rho v_{m}^{2}(\theta, C_{ij})$ values by 2\% and re-executing the fitting routine.

$C_{12}$ could not be determined using the least-squares fitting procedure described above because it does not appear in the expressions for $C_{eff}$ given in Sec.\;\ref{elasticwaves}.  Estimates of this constant, however, were obtained by approximating the symmetry of biotite and muscovite as hexagonal, in which case $\displaystyle C_{12} = C_{11} - 2C_{66}$.  It is emphasized that this expression is used here only as a means to estimate $C_{12}$ and is not expected to hold in general for the samples of the present study due to the true symmetry being monoclinic.

Table \ref{tab:bulkshearmoduli} presents Voigt and Reuss bulk and shear moduli for the biotite and muscovite samples of the present work determined from the elastic constants.  Also included for the purposes of comparison are values of these quantities for muscovite and phlogopite from previously published studies \cite{vaug1986, mcne1993, chhe2014, militzer2011first}.

\begin{figure}[H]
\centering
\includegraphics[scale = 0.40]{Ceff_vs_cij.png}
  \caption{Elastic constants of biotite and muscovite obtained in the present work (open symbols) and previous studies (filled symbols). $\protect\square$ - biotite 12.7\% Fe, 0.2\% Mg; $\protect\medcirc$ - biotite 8.1\% Fe, 4.5\% Mg; $\protect\bigtriangleup$ - biotite 1.1\% Fe, 12.3\% Mg; $\protect\pentagon$ - muscovite 2.1\% Fe, 0.5\% Mg; $\protect\Diamond$ - muscovite 0.5\% Fe, 0.4\% Mg; $\protect\bigstar$ - muscovite \cite{mcne1993}; $\protect\vardiamondsuit$ - muscovite \cite{vaug1986}; $\protect\blacktriangleright$ - phlogopite \cite{chhe2014} DFT-GGA; $\protect\blacktriangleleft$ - phlogopite \cite{chhe2014} DFT-LDA.  Vertical error bars are approximately the size of the symbols for elastic constants of the present work.}
\label{fig:Cefftrends}
\end{figure}

\subsubsection{Monoclinic Character}
The non-zero values of $C_{15}$, $C_{25}$, $C_{35}$, and $C_{46}$, along with the fact that $C_{44} \neq C_{55}$, and $C_{13} \neq C_{23}$ (see Table \ref{tab:elasticconstants}) reaffirm the monoclinic character of biotite. $C_{15}$, $C_{25}$, $C_{35}$, and $C_{46}$ all differ significantly from zero with $C_{15}$ and $C_{46}$ having negative values for each of the three biotites and $C_{35}$ being negative for two of the samples.  Interestingly, while $C_{11} \simeq C_{22}$ for each of the biotite samples, $C_{44}$ is considerably smaller than $C_{55}$ for all three, with the ratio $C_{55}/C_{44}$ decreasing from about 2.5 to 1.4 as the Fe content decreases.  Similar behaviour is seen for $C_{23}/C_{13}$, which ranges from $\sim1.6$ for Fe-rich biotite to $\sim0.8$ for biotite with low Fe content.  Collectively, these results indicate a relatively high degree of elastic anisotropy, ostensibly concentrated in the shear properties, and highlight the deficiencies associated with the previously-invoked approximation of hexagonal symmetry \cite{alek1961}.

\begin{figure}
\centering
\includegraphics[scale=0.55]{Ceff_Fit_1.jpg}
  \caption{Effective elastic moduli, $C_{eff}=\rho v^{2}$, versus angle from the [001] direction for (i) Unfilled symbols - Fe-rich biotite (12.7\% Fe, 0.2\% Mg), and (ii) Filled symbols - Fe-poor biotite (1.1\% Fe, 12.3\% Mg). Upper plot - $bc$ plane; Lower plot - $ac$ plane. Solid ( -- ), dashed (-- --) and dash-dotted (-- $\cdot$ --) lines are best-fit curves for quasi-longitudinal, quasi-transverse, and pure transverse modes, respectively. $\circ$, $\bullet$ - Quasi-longitudinal mode; $\square$, $\blacksquare$ - Quasi-transverse mode; $\diamondsuit$, $\blacklozenge$ - Pure transverse mode.}
  \label{fig:Ceff_Fit_Fe_Mg}
\end{figure}

\subsubsection{Dependence on Primary Cation Concentration}
As can be seen in Table \ref{tab:elasticconstants} and Figure \ref{fig:Cefftrends}, the dependences of biotite elastic constants on primary cation concentration are varied. $C_{11}$, $C_{22}$, $C_{66}$, and $C_{23}$ are nearly independent of Fe (or Mg) concentration, taking on very similar values for all three samples of the present work (ranges: $C_{11}\sim3$\%, $C_{22}\sim7$\%, $C_{66}\sim3$\%, $C_{23}\sim0$\%).  The remaining constants show significant dependence on [Fe] (see Fig. \ref{fig:Cij_vs_Fe_Conc}). $C_{33}$, $C_{44}$, $C_{55}$, and $C_{13}$ all increase with decreasing Fe concentration, while $C_{15}$ and $C_{35}$ decrease with decreasing Fe concentration (see Table \ref{tab:Cij_vs_[Fe]_fits}. $C_{25}$ has the same value for the two biotites of higher Fe concentration, with the value for the sample with the smallest Fe concentration being lower by $\sim15\%$.  In contrast, $C_{46}$ is lowest for the Fe-rich biotite and takes on a common, $\sim20\%$ higher value for the two samples of lower Fe concentration.

The composition information in Table \ref{tab:EDX} shows that, by far, the most significant sample-to-sample difference among the biotite crystals is in Fe and Mg content, suggesting that the behaviour of the elastic constants is a manifestation of the structural response of the crystal to Fe-Mg substitution in the octahedral sheets, a well-known and common substitution mechanism in biotite. The structural changes that occur as a result of this substitution are functions of cation radius and include rotation of tetrahedra and flattening of octahedra in the tetrahedral and octahedral sheets, respectively \cite{cibi2005,hewi1975,tomb2002}. Elastic constants  $C_{11}$, $C_{22}$, and $C_{66}$ will be relatively insensitive to these changes due to the strong intralayer bonding within a tetrahedral-octahedral-tetrahedral (TOT) sheet. In contrast, the weak interlayer bonding between adjacent TOT sheets will, in general, result in the remaining constants displaying a stronger dependence on Fe concentration.  With the exception of $C_{23}$, this is precisely the behaviour that is observed.

\begin{figure}
\centering
\includegraphics[scale=0.49]{Residual_Biotite_MgRich.png}
  \caption{Residuals plots for Fe-poor biotite (1.1\% Fe, 12.3\% Mg).  Left panels - {\it ac} plane.  Right panels - {\it bc} plane.  The standard error of regression is 1.51 GPa.}
  \label{fig:ResidualPlot}
\end{figure}

\begin{table}[t]
\centering

\caption{Selected elastic constants ratios versus Fe concentration for biotite samples of the present work. C$_{ij}$ / C$_{ij}^{Fe-Poor}$ = A $\times$ [Fe] + B}.
\scriptsize
\begin{tabular}{ccccccccc} \hline \hline
Fit Parameter & $\displaystyle \frac{C_{33}}{C_{33}^{Fe-Poor}}$ & $\displaystyle \frac{C_{44}}{C_{44}^{Fe-Poor}}$ & $\displaystyle \frac{C_{55}}{C_{55}^{Fe-Poor}}$ & $\displaystyle \frac{C_{13}}{C_{13}^{Fe-Poor}}$ & $\displaystyle \frac{C_{15}}{C_{15}^{Fe-Poor}}$ & $\displaystyle \frac{C_{25}}{C_{25}^{Fe-Poor}}$ & $\displaystyle \frac{C_{35}}{C_{35}^{Fe-Poor}}$ & $\displaystyle \frac{C_{46}}{C_{46}^{Fe-Poor}}$ \\ \hline
A & -0.0167 & -0.0572 & -0.0345 & -0.0444 & -0.0249 & 0.0208 & -0.1060 & 0.0168 \\
B & 1.032 & 1.091 & 1.065 & 1.0747 & 1.062 & 0.991 & 1.150 & 0.995 \\ \hline \hline
\end{tabular}
\label{tab:Cij_vs_[Fe]_fits}
\end{table}
\subsubsection{Elastic Stability}
Fig.\,\ref{fig:stability} shows the value of the left-hand side of stability conditions Eqs.\,\ref{stabcon1}-\ref{stabcon5}, each normalized to the corresponding value at lowest Fe concentration, as a function of Fe concentration.  While the values of the left-hand sides of Eqs.\,\ref{stabcon4} and \ref{stabcon5} are nearly independent of Fe concentration, those of Eqs. \ref{stabcon1}-\ref{stabcon3} exhibit a dramatic decrease with increasing Fe concentration, being reduced, at the highest Fe concentration, to $\sim10$\% of the respective low-Fe concentration values.   Given that violation of one or more of the stability conditions results in an unstable crystal, the latter behaviour suggests that biotite becomes progressively less stable with increasing Fe content in the octahedral sheet. The same conclusion is reached when one considers the stability condition that the diagonal components of the elastic constants tensor be positive definite.  $C_{11}$, $C_{22}$, and $C_{66}$ are nearly independent of [Fe], but $C_{33}$, $C_{44}$, and $C_{55}$ show relatively large or very large decreases with increasing Fe concentration (see Table \ref{tab:elasticconstants} and Fig.\,\ref{fig:Cij_vs_Fe_Conc}).

The decrease in elastic stability with increasing Fe concentration is likely related to the accompanying increase in average octahedral cation radius.  In fact, synthetic trioctahedral micas of the form KR$_{3}^{2+}$AlSi$_{3}$O$_{10}$(OH)$_{2}$ with octahedral cation radius greater than that of Fe$^{2+}$ are not stable, the instability being attributed to the inability of the tetrahedral layer to further expand by rotation of tetrahedra, resulting in smaller tetrahedral layers on a larger octahedral layer \cite{haze1972}.  A second study shows that the stability is limited not only by tetrahedral rotation angle, but also by the absolute size of the octahedra \cite{tora1981}.  In the KMg$_{3}^{2+}$AlSi$_{3}$O$_{10}$(OH)$_{2}$ - KFe$_{3}^{2+}$AlSi$_{3}$O$_{10}$(OH)$_{2}$ system, the misfit between the tetrahedral and octahedral sheets decreases by substitution of large Fe$^{2+}$ or Mn$^{2+}$ ions for Mg$^{2+}$ ions. This induces variations of shared edge lengths in edge-shared octahedra which, in turn, increases the mutual repulsion between octahedral cations, resulting in a corresponding decrease in elastic stability \cite{tora1981}.  Previous studies have also shown that annite, an Fe-rich biotite, is much less stable than the Fe-poor biotite phlogopite \cite{eugs1962}.  Also consistent with this argument is that Mg silicates have, in general, a higher melting point than their Fe analogues \cite{gowe1957}. The results obtained in the present study are consistent with those of the above works and also show that the elastic stability exhibits a progressive decrease from Fe-poor (Mg-rich) biotite to Fe-rich (Mg-poor) biotite.

\begin{figure}
    \centering
    \includegraphics[scale = 0.45]{C_B_C_Mg_MicaPaper.png}
    \caption{Selected elastic constants for biotite samples of the present work, each normalized to the corresponding constant for Fe-poor biotite (1.1\% Fe, 12.3\% Mg) ({\it i.e.}, $C^B_{ij}/C^{Fe-Poor}_{ij}$, where $ij =$ 33, 44, 55, 13, 15, 25, 35, 46), versus Fe concentration. Dashed lines - lines of best-fit to experimental data.}
    \label{fig:Cij_vs_Fe_Conc}
\end{figure}

\subsubsection{Comparison to Theory}
Due to the similarity in compositions, the elastic constants of the Fe-poor (Mg-rich) biotite sample can be compared to those of phlogopite obtained from first-principles calculations based on density functional theory  \cite{chhe2014}.  As seen in Table \ref{tab:elasticconstants} and in Figure \ref{fig:Cefftrends}, most of the elastic constants for Fe-poor biotite are within $\leq10$\% of the corresponding values obtained for phlogopite using the generalized gradient approximation (GGA) or the local density approximation (LDA).  $C_{11}$, $C_{22}$, $C_{33}$, $C_{44}$, and $C_{55}$ show better agreement with values obtained using the GGA, while constants $C_{66}$, $C_{13}$, $C_{15}$, $C_{23}$, and $C_{46}$ compare very well with values obtained using the LDA.  Experimental values of $C_{25}$ and $C_{35}$ agree neither with the GGA nor the LDA values.  It is also interesting to note that the ratios $C_{22}/C_{11}$, $C_{55}/C_{44}$, and $C_{23}/C_{13}$ for phlogopite obtained from the GGA and LDA calculations are nearly identical to one another and, furthermore, are consistent with those for the Fe-poor biotite (see Table \ref{tab:elasticconstants}). Moreover, $C_{33}$, $C_{44}$, and $C_{55}$ for the Fe-poor biotite are smaller than the corresponding values for phlogopite obtained  using the GGA, while $C_{15}$ for the Fe-poor biotite is larger than the LDA-calculated value for phlogopite.  These behaviours ({\it i.e.}, $C^{Fe-poor}_{ii} \lesssim C^{Phlog}_{ii}$, $i = 3, 4, 5$, and $C^{Fe-poor}_{15} \gtrsim C^{Phlog}_{15}$) are as would be expected given the systematic dependence (strictly increasing or strictly decreasing) of these constants on Fe concentration as demonstrated in the present work, and the fact that elastic constants determined using the GGA and LDA tend to represent lower and upper bounds on the $C_{ij}$, respectively. We note that our results in Table \ref{tab:elasticconstants} also indicate that $C_{13}$, like $C_{33}$, $C_{44}$, and $C_{55}$, is a strictly decreasing function of [Fe] and while its behaviour does not mimic that of these other constants in the sense that $C^{Fe-poor}_{13} \nlesssim C^{Phlog}_{13}$ calculated using the GGA, the value of $C^{Phlog}_{13}$ obtained using the LDA is $\sim0.9\times C^{Fe-poor}_{13}$.

The first-principles calculations cited above yield an elastic constants tensor for trioctahedral phlogopite that is similar to that of dioctahedral muscovite, leading the authors to conclude that the elastic properties of micas are rather insensitive to octahedral site chemistry \cite{chhe2014}.  While it is true that the tensors are similar, the systematic changes with Fe concentration of several of the biotite elastic constants in the present work suggest that the octahedral site chemistry does in fact play an important role in determining the elastic properties of these micas.

\begin{figure}
\centering
\includegraphics[scale=0.4]{Stability_Plots.png}
  \caption{Left-hand side of elastic stability conditions (Eqs. 8-12) versus Fe concentration.  For each condition, the value of the left-hand side is normalized to that at lowest Fe concentration.  For example, for the condition given by Eq. \ref{stabcon1}, the left-most data point (at [Fe] = 1.1\%) has an ordinate of unity. The next point, at [Fe] = 8.1\%, is down 5 vertical "ticks" from the left-most point to an ordinate value of 0.50 and the third point, at the highest Fe concentration of 12.7\%, is down 9 vertical "ticks" from the left-most point to an ordinate value of 0.10.}
  \label{fig:stability}
\end{figure}

\subsubsection{Comparison to Muscovite}
The biotite elastic constants can be compared to those of muscovite determined in the present and previous studies (see Table \ref{tab:elasticconstants} and Fig.~\ref{fig:Cefftrends}) \cite{vaug1986,mcne1993}.  $C_{11}$, $C_{22}$, and $C_{66}$ values for biotites are very similar to those for muscovites.  This is not unexpected because, for both of these mica subgroups, the intralayer bonding is strong and the layers are composed of many of the same types of atoms arranged in a like fashion. $C_{23}$ also displays this behaviour, although the reason for its similarity to the corresponding value for muscovite is not obvious.  The values of $C_{33}$, $C_{44}$, and $C_{55}$ approach those of muscovite as Fe concentration decreases (or, equivalently, Mg concentration increases). The same general behaviour is observed for $C_{13}$.  It is difficult to make meaningful comparisons for $C_{15}$, $C_{25}$, $C_{35}$, and $C_{46}$ because of the very large variation in the values of these constants for muscovite.

The elastic anisotropy within the basal cleavage plane of biotite can be compared to that for muscovite via the ratios $C_{22}/C_{11}$ and $C_{55}/C_{44}$, both of which would be equal to unity if the basal plane displayed elastic isotropy. As Table \ref{tab:elasticconstants} shows, for all biotite and muscovite samples, $C_{22}/C_{11} \approx 1.0$.  In contrast, $C_{55}/C_{44}$ differs substantially from this value, being largest for Fe-rich biotite (2.5) and approaching values similar to those for muscovite as Fe content decreases. This result indicates that the basal plane anisotropy of Fe-rich biotite is strongest for the shear properties and is considerably higher than that of muscovite and decreases with decreasing Fe content in the octahedral sheet. \cite{vaug1986,mcne1993}

\section{\label{sec:Conclusion}Implications}
Brillouin light scattering spectroscopy was used to probe the elastic properties of natural biotite crystals with compositions quantified by electron probe microanalysis.  The systematic dependence of elastic wave velocities, elastic stiffness constants, elastic stability, and basal plane anisotropy on iron content suggests that biotite elasticity is a function of octahedral site chemistry and provides a means to estimate the elastic properties of most biotite compositions with known iron or magnesium content.  Moreover, the overall agreement between the experimentally-determined elastic constants of a Mg-rich biotite and those for phlogopite obtained from first-principles calculation based on density functional theory suggests that the latter approach holds promise in describing the elastic properties of biotites.

\section*{Acknowledgement}
\noindent
GTA acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) (RGPIN-2015-04306).

\vspace*{0.3cm}
\noindent
Additional data and code available upon request.

\bibliographystyle{elsarticle-num-names}

redpajama_set_name: RedPajamaArXiv


text:
Q: Replace part of a string with anchor tag I have a sentence: "This is a sentence with a link1, also it has link2"
It is in p tag.
I also have data when it starts and when it ends, respectively:
[{ start: 26, end: 31 }, { start: 45, end: 50 }]

Is it possible to replace these parts of the sentence with anchor tags having upper data available. As a result it would be a something like this:

This is a sentence with a link1, also it has link2

Thanks!

A: Building a string of HTML and/or using dangerouslySetInnerHTML is not the correct solution here. To solve this the "React way," consider that this structure:

This is a sentence with a link1, also it has link2

...could be generated with this React component (this is more verbose than necessary, just for clarity's sake):
const LinkText = () => {
  const parts = [
    'This is a sentence with a ',
    {'link1'},
    ', also it has ',
    {'link2'},
  ];
  return
{parts}
;
};

With that in mind, the question is this: How do we generate the parts array from a string and an array of link positions? The solution is to iterate over the array of link positions and, in each iteration, push onto the array 1) The "non-link" text between the previous link position's end (or the start of the string), and 2) an  containing the current link position's start and end. A simple, working implementation looks like this:

const LinkText = ({ text, linkPositions }) => {
  const parts = [];
  let currentIndex = 0;

  linkPositions.forEach(({ start, end }) => {
    if (currentIndex < start) {
      parts.push(text.slice(currentIndex, start));
    }
    parts.push({text.slice(start, end)});
    currentIndex = end;
  });

  if (currentIndex < text.length - 1) {
    parts.push(text.slice(currentIndex)); // Remaining text after the last link
  }

  return
{parts}
;
};

const text = 'This is a sentence with a link1, also it has link2';
const links = [{ start: 26, end: 31 }, { start: 45, end: 50 }];

ReactDOM.render(<LinkText text={text} linkPositions={links}/>, document.querySelector('div'));
a{color:blue}

A: The best way to do that is using IDs and JS.
So, for example, your HTML would look like this:

This is a sentence with a link1, also it has link2

And then your JS would call this:
document.getElementById("linkOne").href = "http://linkOne.com"
document.getElementById("linkTwo").href = "http://linkTwo.com"

A: You can achieve that with str.replace with a regex.
as @Dupocas said, React will escape html tags, so you will need to use dangerouslySetInnerHTML

const str = 'This is a sentence with a link1, also it has link2';
const result = str.replace(/link(\d)/g, (match, linkId) => {
  return `<a href="${linkId}">${match}</a>`;
});

console.log(result);

A: You can use this snippet to replace the text at the specific places into links. I have sorted it in a descending order based on the start since changing it from the end of the string is easier.

document.getElementById("convert").addEventListener("click", convertLinks);
addEventListener

function convertLinks() {
  var pTag = document.getElementById("test")

  var text = pTag.innerHTML

  posData = [{
    start: 26,
    end: 31
  }, {
    start: 45,
    end: 50
  }]

  posData = posData.sort(function(pos1, pos2) {
    return pos2.start - pos1.start
  })

  for (pos of posData) {
    text = replaceLink(text, pos)
  }

  pTag.innerHTML = text;
}

function replaceLink(text, pos) {
  return text.slice(0, pos.start) + "" + text.slice(pos.start, pos.end) + "" + text.slice(pos.end)
}
a {
  color: blue
}

This is a sentence with a link1, also it has link2

<button id="convert"> Convert Links </button>

redpajama_set_name: RedPajamaStackExchange


text:
What Colors And Patterns Does A Shih Tzu Come In?
The name Shih Tzu translates to "lion dog" which is what these cute little toy doggies were bred to look like. If you have ever seen a Shih Tzu or interacted with one, you probably know that the adorable fluff balls are the furthest thing from the ferocious cats that inspired their development
Shih Tzus are widely celebrated and wildly popular for their endearing aesthetic. From their long fluffy coats to their large, curious eyes, it is hard not to fall in love with these dogs.
What many people do not realize is that there are up to 27 types of Shih Tzus based on coat type, color, and patterns. We shall be taking a look at some of these including the rare white coat and brindle pattern.
27. Long Coat
Image from Instagram:@sandracapitu
Shih Tzus only have one coat which is the long coat. The many short-haired Shih Tzus out there get that aesthetic as a result of grooming. Short-haired Shih Tzus may also be a result of mixed breeding with short-haired toy dog breeds.
The long coat on a Shih Tzu may either be loose and wavy or it may have a curly texture to it. The fur is usually a bit coarse to the touch. However, it is possible to get it silky smooth with special grooming.
26. Black
Image from Instagram:@blackshihtzu.baloo
All black Shih Tzus are completely covered by black fur. If there is a patch of white or other color like tan then the Shih Tzu no longer qualifies as a black variation.
The black fur in these Shih Tzus, however, is not always the same shade especially as the dog grows older. You may have some parts appearing as lighter or less intense shades of black especially on the face, ears, and paws.
25. Black & White
Image from Instagram:@felicia.blackshihtzu
Solid black and white Shih Tzus are not commonly found as a variation within this breed. They may have black as the main coat color covering majority of the body with white appearing as markings on different sections.
Some black and white Shih Tzus also have the two colors somewhat equally distributed over the coat.
It is only in very rare situations that you will find a black and white Shih Tzu where the white fur is dominant and the black fur appears as the markings.
Image from Instagram:@leo_the_black_shihtzu
Blue Shih Tzus may look like any other black SHIH Tzu especially from a distance. Even up close, it may be difficult for the inexperienced dog enthusiast to tell the two variations apart.
Blue Shih Tzus have a watered down shade of black fur. It is not blue in the typical blue sky or ocean shades. Instead, their coats have more of a powdery dark gray appearance with dark blue undertones.
23. Blue & White
Image from Instagram:@roc_and_roe
Blue and white Shih Tzus are in a lot of ways just like black and white Shih Tzus. They tend to have the same distribution patterns with mainly dark fur and white markings.
With blue and white Shih Tzus, the dark part of the coat has watered down black fur which appears a shade of blueish-silver.
The white fur is similar to that in black and white Shih Tzus. However, due to the lighter and more faded nature of the blue parts of the coat, there is less clear contrast between the two colors in blue and white Shih Tzus.
22. Silver
Image from Instagram:@dopelifeofjax
Silver Shih Tzus vary in shade intensity. Some have a very shiny silver coat while others may have a faded and almost ashy gray color.
It is also important to appreciate that Shih Tzus in general tend to have gray color changes to their coat even when they were born with a different-colored coat. The age of the Shih Tzu is therefore an important consideration when trying to identify true silver varieties.
21. Silver & White
Image from Instagram:@pfeiffdawg
The silver and white Shih Tzu combination is quite common among the bicolor variations. These doggies are predominantly white with silver markings often around the face.
The silver appears as a watered down shade of black. In some Shih Tzus, the silver may be so watered down that it almost blends in with the bright and shiny white sections of fur.
In other silver and white Shih Tzus, the silver has a bit more of a solid quality to it which creates distinct contrast between the two colors.
Image from Instagram:@hello.teddy.ziggy
Brindle Shih Tzus have a combination of a dark color like black or even blue and a lighter color which is usually a shade of brown.
The brown fur acts as a sort of base coat with the darker fur forming stripes across this base.
On long coats like the Shih Tzu's, this may be a bit difficult to appreciate. However, trimmed Shih Tzus or young puppies that still haven't grown their fur to full length demonstrate the brindle markings more clearly.
19. Brindle & White
Image from Instagram:@buster.pupper
Brindle and white Shih Tzu combine the most common white markings with the rarest brindle pattern in the Shih Tzu breed.
The brindle coat covers most of the body. The white markings occur as small and distinct patches usually around the head and limbs.
With very long-haired Shih Tzus, these white patches may be difficult to pick out especially if they are on lower parts of the body like the belly.
Image from Instagram:@dottie_and_fleur_empawrium
Gold or Golden Shih Tzus are not a mix between Shih Tzus and Golden Retrievers as many people think. Gold Shih Tzus are purebred dogs. They are a variation of the breed known for their brown fur.
The shade of brown varies from one Gold Shih Tzu to another. It may be a darker tan brown or a bright and shiny golden shade.
Gold Shih Tzus sometimes have a reddish-pink nose but may also have a black nose like other varieties.
17. Gold & White
Image from Instagram:@littleshihtzu
Gold and white Shih Tzus are a rare bicolor variation known for their refreshing combination of these two bright and beautiful colors. In most cases, the gold fur will cover most of the body.
In contrast, the white fur serves as markings limited to the chest, neck, face, and paws. It is only in very rare cases where the white fur covers majority of the body.
These Shih Tzus may also have a black mask cover with black fur over the snout.
16. Liver
Image from Instagram:@livershihtzu
Liver Shih Tzus have a very unique and polished beauty to them. There is something special about the chocolate-brown shade of fur that works very well with the long and glossy finish of the Shih Tzu coat.
The liver coat is present all over the body with no white markings allowed for this variation.
You may also notice that the nose on a liver Shih Tzu is not always black. It may have a chocolate brown shade very similar to the rest of the liver fur on the Shih Tzu.
15. Liver & White
Image from Instagram:@heylilbowie
Liver and white Shih Tzus are basically a variation of liver Shih Tzus where there are white markings anywhere on the body.
It could be a few white patches on the face with an entirely liver body or an almost even split between these two colors.
The black face mask is not very common in liver and white Shih Tzus. It may also be very difficult to distinguish especially if the rest of the face is liver colored.
14. Red
Image from Instagram:@skippytheredtzu
Red Shih Tzus are often bunched up with brown or golden varieties. However, they have a very distinct rusty quality to their shade of brown that sets them apart from other similar Shih Tzus.
The red fur is present all over the body. If there are white markings, the Shih Tzu is not considered a true red variation. However, black markings from the class black face mask are acceptable within this variety.
13. Red & White
Image from Instagram:@karen.m.luciano
Red and white Shih Tzus don't always look like Shih Tzus. In fact, you may be forgiven for mistaking them for a super fluffy Cavalier King Charles Spaniel that has a similar coloring and pattern.
The predominant color varies from one red and white Shih Tzu to the next. Some have mainly white fur with red markings on the face and ears. Others are mainly red with white patches on the face, neck, and limbs.
12. Black Gold & Silver
Image from Instagram:@stormyshihtzu
Shih Tzus also come in tricolor combinations in which case the black, gold, and silver. This is a very rare color combination that itself comes in many different patterns.
The most common combination is a black and gold body with white markings mainly around the face. It could also be a mainly gold Shih Tzu with white markings over the chest and a black face mask.
The most rare possibility is a sable black, gold, and silver Shih Tzu. Here, the black appears as tip parkings on both gold and silver fur.
11. Black Gold & White
Another common tricolor combination with Shih Tzus is black, gold, and white. The three solid colors contrast beautifully with each other. The gold fur in this case varies in shades from bright and brilliant tan shades to darker brown.
The white fur may either cover most of the body or appears as chest and limb markings.
The black fur in black, gold, and white Shih Tzus may just appear simply as a face mask over the snout and around the eyes.
10. Black White & Silver
Image from Instagram:@nupy_shih.tzu
Black, white, and silver Shih Tzus are another variation of the breed that prove that not all black fur is watered down where there is silver present.
In these cases, the dog is mainly silver and white. The white covers most of the body especially the torso and limbs. The shiny silver fur will appear as large patch markings usually on the face.
Finally, these black, white, and silver Shih Tzus will also have solid black fur as small markings almost predominantly on the face.
9. Silver Gold & White
Image from Instagram:@ishihtzunotpuppiesandclothes
The tricolor combination of silver, gold, and white is one of the most common varieties of the Shih Tzu based on coat colors.
These Shih Tzus are more often than not predominantly white. The silver and gold or tan patches act more as markings. These markings may appear anywhere on the body but are usually on the face and ears, back, and chest.
The colors complement each other perfectly for a truly beautiful Shih Tzu.
8. White
Image from Instagram:@shihtzu.white
White Shih Tzus are incredibly rare. This is a shame as they are some of the most beautiful dogs out there.
They have all white fur which is perfectly complemented by the thick, long, and shiny nature of Shih Tzu coats.
White Shih Tzus may either be leucistic or albino. The Leucistic variety have recessive genes that give them white fur but leave other surfaces like eyes and nose pads pigmented.
With albinism on the other hand, there is no pigment on the nose and eyes in addition to the skin and fur.
7. Black Markings
Image from Instagram:@blackandwhiteshihtzu
Black fur is considered a marking on Shih Tzus if it covers the least surface of any other colors present. It is common in tricolor combinations with other colors like gold, tan, silver, or white.
In some cases, the black markings on a Shih Tzu may only appear as a black face mask as shall be discussed shortly.
If the black fur covers a majority of the coat as is the case with black bicolor variations, then the second minority color is what is considered the marking.
6. Tan Markings
Image from Instagram:@raincross_shihtzu
Tan markings are not a very common occurrence in Shih Tzus. More often than not, they appear as patch markings with a minority coverage on tricolor Shih Tzus. This includes common varieties like the popular black, gold/tan, and white Shih Tzu.
The distribution of tan markings on Shih Tzus is determined by genetics. The patches can occur anywhere from the face and ears to the back and limbs.
5. White Markings
Image from Instagram:@primemoonshihtzuoficial
White markings are the most commonly found on Shih Tzu dogs that have more than one color. The white fur covers a minority of the dog's coat. These patches will usually be distributed on lower parts of the dog's body like their belly, chest, and paws.
The white markings may also be on the face.
Shih Tzus with white markings may either be bicolor or tricolor. With such a scope, it is no wonder that it is the most common pattern.
4. Merle
Image from Instagram:@frosty_the_merle_shih_tzu
Merle Shih Tzus are very rare. They have unique, mottled spots all over the body. These markings vary in color but are usually either black, silver, blue, and in very rare cases, brown or gold.
The markings are randomly scattered over predominantly white or grayish fur.
In some merle Shih Tzus, the genes responsible for the markings also affect eye and ear development and may cause visual or auditory impairment for the dog.
3. Sable
Image from Instagram:@napoleon_the_shihtzu
Sable Shi Tzus have black ends on all the fur over the entire body. The rest of the fur may be any other lighter color from brown or tan to silver and even white. The contrast between the lighter fur and black tips creates a beautiful color gradient that makes Sable Shih Tzus beautiful.
This may be a bit difficult to elicit especially with Shih Tzus that get dramatically short haircuts. In this case, the best way to identify a sable Shih is through fur that is rarely trimmed especially around the face.
2. Brindle
Image from Instagram:@mookietzu
The brindle pattern is arguably one of the rarest varieties in the Shih Tzu breed. It is characterized by black stripes on brown or tan fur.
Due to the long nature of Shih Tzu fur, it is very possible to confuse brindle markings for sable patterns. The key difference is that the latter has the black marking at the tip of all fur strands as opposed to an alternation between black and tan.
The brindle trait is as a result of the presence of the brindle gene. Fortunately, this gene is not associated with any specific health issues.
1. Black Mask
The black mask is a patch of black colored fur found as a distinct facial marking on the Shih Tzu.
It will usually cover the snout from the nose bridge to the chin and lower jaw. In some cases, this black mask may also appear around the eyes and extend to the bases of the ears.
Interestingly, black mask markings are not present in all Shih Tzus. They are also very difficult to elicit in black, blue, or silver Shih Tzu varieties.
What Is The Most Popular Shih Tzu Color? Tricolor combinations like silver, gold, and white or black, gold and white are among the most popular colors for Shih Tzus. These are also some of the most commonly found coat colors. Other popular alternatives are solid black and solid white which are loved for their uniqueness and their niche availability.
Do Shih Tzus Change Colors? Most Shih Tzus change color as they grow older. One of the most commonly seen changes is fading or lightening of the shades which is best appreciated on dark coats like black, liver, and red. Another common change seen with Shih Tzu is graying. This mostly happens when the dog is in its middle to elderly years.
What Color Is Shih Tzu Skin? Underneath all the fluff and color combinations that different Shih Tzus have, they are all pretty much the same in terms of skin color. Most Shih Tzus will have pink skin under their fur. The shades and intensity of pink will vary from one dog to another. This may also change within the same dog as they grow older.

redpajama_set_name: RedPajamaCommonCrawl


I'm sure we all know how this goes. You are given three characters and must choose one that you would wed, one you would bed, and one you would kill. Then you give three characters for the next person to choose amongst. We'll be doing this with ASOIAF/ GOT characters of course.
That's hard. Can I enter into a polygamous relationship with Daenerys and Arianne and bed them both?
Bed Craster. Then kill him in his sleep.
Why is this thread not called 'Wed, Bed, Behead'?
Behead: Gyles Rosby - he could be contagious.


text:
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In these terms of use the privacy policy is also included, where the management of the members/visitors/users personal data is explained, according to the current legal system (EU REGULATION 2016/679 OF THE EUROPEAN PARLIAMENT AND COUNCIL) and in addition according to the greek legislature concerning the Human protection from the Personal Data Processing (2472/1997), after the subjected changes.
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This sale is for a Meade Instruments Telescope AudioStar Talking Hand Controller. It's essentially identical to the AutoStar 497, but has the talking "Astronomer Inside" capability. The condition is only used several times and "Like New", excellent condition.
Shipped with USPS Priority Mail, $10 within the United States.


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Doc Thompson September 29, 2017
Watch: Bruce Willis will 'Die Hard' again – but new movie isn't exactly a sequel
Bruce Willis will return for a sixth "Die Hard" film, a follow-up to 2013's "A Good Day to Die Hard" that is being described as both a prequel and a sequel.
In "Die Hard: Year One," John McClane will be in New York both in the 1970s and in the present day. The movie was stuck in development for nearly two years, but it's finally in production.
How do you feel about yet another "Die Hard" movie? "The Morning Blaze with Doc Thompson" crew came up with some fun alternative titles to express their feelings when Lex Jurgen stopped by for Friday Hollywood Headlines.
To see more from Doc, visit his channel on TheBlaze and listen live to "The Morning Blaze with Doc Thompson" weekdays 6–9 a.m. ET, only on TheBlaze Radio Network.


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Home»Blog» Find Out the Best Way to Order Tramadol Online.
Find Out the Best Way to Order Tramadol Online.
Tramadol medicines can be considered as one of a kind medicines which have revolutionized The field of pain management. These medicines come under the category of opiate drugs which are also termed as narcotic painkillers. There are numerous medicines that are considered as narcotic painkillers which are highly addictive in nature, but on the other hand tramadol is one of the medicines which is considered as a very safe medicine worldwide. This is one among the few medicines which does not create dependency symptoms in the patient. Tramadol pills are analgesic in nature which means they have a very good pain relief component which helps to significantly reduce the pain. In this article, we will tell you about the history of the tramadol pills, variations that are offered and we will also let you know how you can order tramadol online.
Tramadol pills are available in the market since 1995, after been approved by food and Drugs Administration (FDA). There are various brand names under which Tramadol is available in the market but the most popular of them all is Ultram and Ultram ER (Extended release). Extended release version of Ultram was made available to the market after its approval from FDA in the year 2005. Tramadol medication has been certified by Medicine Research Institutes from all over the world, including World Health Organisation (WHO) which is the topmost Institute that governs worldwide medicine industry.
Since 1995 doctors all over the world have widely regarded Tramadol as one of the most popular pain killer medicine, and have prescribed these medicines to many patients who are dealing with moderate to acute pain in their day to day life. Patient who are regularly facing Joint, pain, back pain, muscle pain or arthritis are advised by doctors to take tramadol medication daily. In fact a study which was made by expert scientists in 2014, also recommended that consumption of tramadol can be useful to the patients prior to the operation, is doing this prove to help reduce post anaesthesia shivering, which commonly occurs in patients who are recovering from normal anaesthesia.
Prior to 1995, during the research and development years of tramadol, it was mostly used in a liquid form as an injection. But after 1995 tramadol has only been available in medicinal form. Tramadol pills or Ultram come in two variants 50 mg and 100 mg tablets. Tramadol pills also have to be taken orally just as any other medicines. These pills are meant to be taken with a glass full of water, and should not be taken or mixed with any other form of liquid other than normal water. It is always the doctor who suggests whether a patient should be taking tramadol 50mg or 100mg, after careful examination of the cause of pain. it is the amount of pain an individual is going through, that is the main criteria for the selection of dosage. Doctors have to be extremely careful while prescribing tramadol to the patients. Since tramadol pills fall into the category of narcotic medication, there are chances that any patient with history of alcohol abuse or medicine abuse will tend to misuse tramadol pills. Which is why before prescribing tramadol a doctor carefully examines the patient, and actively checks if the patient has any history of medicinal abuse or any addiction symptoms.
In case a doctor finds out that a patient has the history of alcohol or medicinal abuse, in that case tramadol pills won't be prescribed to such patients. Not doing so will hamper both the credibility of the doctor as well as the health of the patient is also compromised.
The best way for Patients who are dealing with moderate to acute pain is to visit the doctor and perform a complete health check up and then to Order Tramadol Online they can visit Supremepharmacyrx.com.
5 Simple Ways That Help You To Keep Spine Healthy>>


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Suche dni (dni kwartalne) – w tradycji Kościoła łacińskiego kwartalne dni postu obejmujące środę, piątek i sobotę, obchodzone na początku każdej z czterech pór roku, tj.:
 po I niedzieli wielkiego postu – Dni modlitw o ducha pokuty – od środy po pierwszej niedzieli wielkiego postu do soboty przed drugą niedzielą wielkiego postu (luty – marzec)
 po Zesłaniu Ducha Świętego – Dni modlitw o powołania do służby w Kościele – od środy po Zesłaniu Ducha Świętego do soboty przed Uroczystością Trójcy Przenajświętszej (maj – czerwiec). Do roku 1095 przypadały na okres od drugiej środy czerwca.
 po Święcie Podwyższenia Krzyża Świętego (14 września) – Dni modlitw za dzieci, młodzież i wychowawców (trzeci tydzień września).
 po Wspomnieniu św. Łucji (13 grudnia) – Dni modlitw o życie chrześcijańskie rodzin (trzeci tydzień adwentu w grudniu).

Obchody te winny służyć odnowie duchowej wiernych poprzez obchody liturgiczne, praktyki pokutne i akcje duszpasterskie.

Obchodzenie suchych dni nie jest obecnie wymagane prawem kanonicznym. Ich termin i formę może określić lokalny ordynariusz. Do momentu wejścia w życie konstytucji apostolskiej Pænitemini w 1966 roku, obowiązywał w te dni post ścisły.

Zobacz też
 Pokuta
 Przykazania kościelne
 Umartwienie

Przypisy

Bibliografia

Linki zewnętrzne
 Wrześniowe suche dni

Katolicyzm

redpajama_set_name: RedPajamaWikipedia


Premium Navy Spiced Rum based on more than 175 years of know how in skillful blending and selection of world class rum.
Black Barrel rum is infused with a secret mix of spices carefully selected and based on the founders knowledge and reputation as a world famous Pharmacist.
The Black Barrel Rum is then further aged for a long time in charred oak casks.
Hence the name Black Barrel.


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Montaillou è un comune francese di 34 abitanti situato nel dipartimento dell'Ariège nella regione dell'Occitania.

Montaillou è conosciuto per essere il soggetto del libro Storia di un paese: Montaillou. Un villaggio occitanico durante l'Inquisizione. (1294-1324) di Emmanuel Le Roy Ladurie. Tale saggio storico analizza in gran dettaglio il villaggio per un arco di tempo trentennale, a partire dal 1294, ed è basato su dei documenti redatti all'epoca da Jacques Fournier, futuro Papa Benedetto XII.

Società

Evoluzione demografica

Note

Altri progetti

Montaillou


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March's Advanced Organic Chemistry: Reactions, Mechanisms, and Structure
Michael B. Smith, Jerry March
Wiley, 16 de gen. 2007 - 2384 pàgines
The Sixth Edition of a classic in organic chemistry continues its tradition of excellence Now in its sixth edition, March's Advanced Organic Chemistry remains the gold standard in organic chemistry. Throughout its six editions, students and chemists from around the world have relied on it as an essential resource for planning and executing synthetic reactions.
The Sixth Edition brings the text completely current with the most recent organic reactions. In addition, the references have been updated to enable readers to find the latest primary and review literature with ease.
New features include:
More than 25,000 references to the literature to facilitate further research
Revised mechanisms, where required, that explain concepts in clear modern terms
Revisions and updates to each chapter to bring them all fully up to date with the latest reactions and discoveries
A revised Appendix B to facilitate correlating chapter sections with synthetic transformations
Michael B. Smith,Jerry March
Previsualització limitada - 2007
Previsualització no disponible - 2007
Michael B. Smith
Sobre l'autor (2007)
MICHAEL B. SMITH, PhD, is Professor of Chemistry at the University of Connecticut. His current research interests include studies towards the total synthesis of pancratistatin and related phenanthridone alkaloids; synthesis and structural verification of bioactive bacterial ceramides; the study of reactions associated with or facilitated by conducting polymers; and, the synthesis of dye-conjugates to target cancerous tumors.
Jerry March, PhD, (deceased) was a professor in the Department of Chemistry at Adelphi University, Garden City, New York.
Títol March's Advanced Organic Chemistry: Reactions, Mechanisms, and Structure
Autors Michael B. Smith, Jerry March
Edició 6, il·lustrada
Editor Wiley, 2007
Nre. de pàgines 2384 pàgines


Orlando, FL (May 18, 2006) -- PBS today announced the addition of seven PBS programs to its existing roster of primetime titles available for viewing via local free video-on-demand. With these additions - including a number of hallmark PBS series - PBS stations now can provide local cable operators a diverse package of ten VOD titles, available to viewers on-demand for free up to seven days after their broadcast premieres. The announcement was made by PBS President and CEO Paula Kerger at PBS' Annual Showcase meeting in Orlando.
In addition, there will be quarterly features (including selected episodes from NOVA, MASTERPIECE THEATRE and MYSTERY!).
Showcasing the strong local associations PBS stations have within their communities, these PBS on-demand titles will be the same localized broadcasts seen by viewers in their own markets. Local PBS stations work with their cable operators, who record the selected programs from the local PBS station broadcasts as they air. The cable operator then encodes the recorded programs and makes them available on free VOD shortly after broadcast.
Paula Kerger said, "As viewer habits shift, it's critical that local PBS stations make their content available on-demand in this way, allowing our stations to not only reach additional viewers at times convenient to them, but also to highlight the quality content and variety of services the stations offer their communities."
PBS is a private, nonprofit media enterprise that serves the nation's 348 public noncommercial television stations, reaching nearly 90 million people each week through on-air and online content. Bringing diverse viewpoints to television and the Internet, PBS provides high-quality documentary and dramatic entertainment, and consistently dominates the most prestigious award competitions. PBS is a leading provider of educational materials for K-12 teachers, and offers a broad array of other educational services. PBS' premier kids' TV programming and Web site, PBS KIDS Online (pbskids.org), continue to be parents' and teachers' most trusted learning environments for children. More information about PBS is available at pbs.org, one of the leading dot-org Web sites on the Internet.


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■ The Seas. 9 p.m. April 12. $5.
■ Chris Knight. 9 p.m. April 13. $20-$25.
■ Slightly Stoopid. 8 p.m. April 17. $20-$25.
■ Randy Rogers Band, 8 p.m. April 18. $14-$17.
■ Rusted Root. 9 p.m. April 21. $22-$25.
■ Mushroomhead. 8 p.m. April 22. $18-$20.
■ Victor Wooten. 9 p.m. April 25. $20-$25.
■ Give Into the Groove. 9 p.m. April 26. $5.
■ Twiztid, (Hed) PE. 7 p.m. April 29. $18-$20.
Tickets on sale soon:
Patterson Hood & The Downtown Rumblers. 8 p.m. June 9. The Southgate House Revival — The Sanctuary, Newport. $20. On sale at 8 a.m. April 5. Ticketweb.com.
The Shoe at Horseshoe Casino, Cincinnati. On sale at 10 a.m. April 5. 1-800-745-3000 or Ticketmaster.com.
■ Ke$ha. 8 p.m. June 8. $43.
■ Barenaked Ladies, Ben Folds Five, Guster. 6:30 p.m. July 6. $45.75.
■ Billy Idol. 8 p.m. June 14. $43.
■ Under The Sun , Smash Mouth, Sugar Ray, Gin Blossoms and more. 7:30 p.m. July 25. $38.
Keith Urban, Little Big Town, Dustin Lynch. On sale at 10 a.m. April 5. Ticketmaster.com.
■ 7 p.m. July 18. Riverbend Music Center, Cincinnati. $25-$54.75.
■ 7:30 p.m. Dec. 8. KFC Yum Center, Louisville. $29.50-$68.50.
Cirque du Soleil: Quidam. 7:30 p.m. June 19. KFC Yum Center, Louisville. $18-$90. On sale at 10 a.m. April 5. Ticketmaster.com.
Dwight Yoakam. 8 p.m. July 19. Horseshoe Southern Indiana, Elizabeth, Ind. $42-$77. On sale at 11 a.m. April 5. Ticketmaster.com.
Diamond Pub & Billiards, Louisville. On sale at 10 a.m. April 6. Ticketweb.com.
■ Arsis, Esoteric Burial, Oedipus Complex. 8 p.m. May 29. $10.
■ Skeletonwitch. 8 p.m. June 20. $10.
■ The Agonist. 8 p.m. July 17. $10.
■ Fates Warning. 7:30 p.m. Nov. 17. $18-$35.
Gaither Vocal Band. 7 p.m. June 14. Eastern Kentucky Expo Center, Pikeville. $24.50-$32.50. On sale at 10 a.m. April 8. Ticketmaster.com.
Eddie and Jennifer Montgomery talk about their betting strategy
J.D. Shelburne talks fashion, trifecta win, and possible Kentucky Derby bets
MORE MUSIC NEWS & REVIEWS
Weekend best bets: Dragons at the pool, urchins on the stage, pianos in the parks
Iowa governor's aide now says Tupac email not tied to ouster


Federal Reserve Chairman Janet Yellen said today that the economy has "recovered substantially" in recent years and hinted that an increase to the federal funds rate could come before year-end while cautioning that a delay could inadvertently lead to another recession.
Yellen pointed to several signs of economic progress — including steadily falling unemployment numbers and increases to the real GDP — that have moved the labor market closer to the Federal Open Market Committee's goal of maximum employment. She expressed her optimism that economic growth will continue in the months ahead. She added that she also anticipates the inflation rate — which is currently below the FOMC's target rate of 2 percent — will improve, and warned that waiting too long to raise interest rates could have negative economic consequences.