\chapter{Introduction and Point Groups}


\section{Introduction}

 
Amorphous solids are homogeneous and isotropic because there
is no long range order or periodicity in their internal
atomic arrangement. In contrast, the crystalline state is characterised by a 
regular arrangement of atoms over large distances. Crystals are therefore
anisotropic -- their properties vary with
direction. For example, the interatomic spacing varies with
orientation within the crystal, as does the elastic response
to an applied stress. Note that crystals can be solid or liquid, and can have arbitrary shapes.

     Engineering materials are usually aggregates of many
crystals of varying sizes and shapes; these {\it  
polycrystalline } materials have properties which depend
on the nature of the individual crystals, but also on
aggregate properties such as the size and shape
distributions of the crystals, and the orientation
relationships between the individual crystals. The 
randomness in the orientation of the crystals is a measure of
{\it   texture}, which for reasons described later, has to be controlled in the
manufacture of transformer steels, uranium fuel rods and beverage cans. 

The crystallography of interfaces connecting adjacent crystals can determine the
deformation behaviour of the polycrystalline aggregate; it can also influence
the toughness through its effect on the degree of segregation of
impurities to such interfaces. 

\section{The Lattice}

Crystals have translational symmetry: it is possible in a crystal, 
to identify a regular set of points, known as the lattice
points, each of which has an identical environment. 
 
     The set of these lattice points constitutes a three
dimensional lattice. A {\it   unit cell } may be defined
within this lattice as a space--filling parallelepiped with origin at a
lattice point, and with its edges given by three
non-coplanar {\it  basis vectors } {\bf a}$_1$, {\bf a}$_2$ and {\bf a}$_3$, each
of  which represents translations between two adjacent lattice points.
The entire lattice can then be generated by stacking unit
cells in three dimensions.
Any vector representing a translation between lattice
points is called a {\it   lattice vector}.
 
    The unit cell defined above has lattice points located at
its corners. Since these are shared with seven other such
cells, and since each cell has eight corners, there is only
one lattice point per unit cell. Such a unit cell is {\it  
primitive } and has the lattice symbol {\it   P}.

Non--primitive unit cells can have two or more lattice points,
in which case, the additional ones will be located
at positions other than the corners of the cell. A cell with additional
lattice points located at the centres of all
its faces has the lattice symbol {\it   F}; such a cell
would contain four lattice points. Not all the faces of the
cell need to have face--centering lattice points; when a cell
containing two lattice points has the additional
point  located at the centre of the face defined by {\bf a}$_2$ and
{\bf a}$_3$, the lattice symbol is {\it   A } and the cell is said
to be {\it   A}-centred. {\it   B}-centred and {\it  
C}-centred cells have the additional lattice point
located on the face defined by {\bf a}$_3$ \& \ {\bf a}$_1$  or
{\bf a}$_1$ \& \ {\bf a}$_2$ respectively. A unit cell with two lattice points
can alternatively have the additional lattice point at the
body-centre of the cell, in which case the lattice symbol is
{\it   I}. The lattice symbol {\it   R } is for a trigonal
cell; the cell is usually defined such that it contains three
lattice points. 

   The basis vectors {\bf a}$_1$, {\bf a}$_2$ and {\bf a}$_3$ define the
unit cell; their magnitudes  $a_1$, $a_2$ and  $a_3$ 
respectively, are  the {\it   lattice parameters } of the
unit cell.  The angles $\widehat{{\bf a_1}~{\bf a_2}}$, $\widehat{{\bf a_2}~{\bf a_3}}$
and $\widehat{{\bf a_3}~{\bf a_1}}$ are conventionally labelled $\gamma$, $\alpha$
and $\beta$ respectively. 

Our initial choice of the basis vectors was
arbitrary since there are an infinite number of lattice vectors which
could have been used in defining the unit cell. The preferred choice includes
small basis vectors which are as equal as possible, provided the shape of the
cell  reflects the essential symmetry of the lattice.

\section{Bravais Lattices}

The number of ways in which points can be arranged regularly
in three dimensions, such that  the stacking of unit cells
fills space, is not limitless; Bravais showed in 1848 that
all possible arrangements can be represented by just fourteen
lattices. 

The fourteen \htmladdnormallink{Bravais lattices}{http://www.msm.cam.ac.uk/phase-trans/2003/Lattices/bravais.html} can be categorised
into seven {\it   crystal systems } (cubic, tetragonal,
orthorhombic, trigonal, hexagonal, monoclinic and triclinic, Table~\ref{tab:systems});
the cubic system contains for example, the cubic-{\it  P},
cubic-{\it  F } and cubic-{\it  I } lattices. Each crystal system can be
characterised uniquely by a set of defining symmetry elements, which any crystal
within that system must possess as a minimum requirement (Table~\ref{tab:systems}).      The Bravais lattices are illustrated in Fig.~\ref{fig:Bravais}.

\begin{table}[!ht]
\caption{The crystal systems. The symmetry axes monad, diad, triad, tetrad and hexad involve rotations of 360$^\circ$, 180$^\circ$, 120$^\circ$, 90$^\circ$ and 60$^\circ$ respectively. These particular rotations leave the lattice in a state that cannot be distinguished from its starting configuration. $a_i$ are the magnitudes of the basis vectors ${\bf a}_i$.}
\begin{center}
\begin{tabular}{lclr}\midrule
System   &  \multicolumn{2}{c}{Conventional unit cell}  &   Defining symmetry \\[3pt]\midrule
Triclinic  &  ${a_1} \neq {a_2} \neq {a_3}$ & $\alpha \neq \beta \neq \gamma$   &  monad \\[3pt]
Monoclinic  &  ${a_1} \neq {a_2} \neq {a_3}$ &  $ \alpha = \gamma,~~ \beta \geq 90^\circ $ &  1 diad \\[3pt]
Orthorhombic  &  ${a_1} \neq {a_2} \neq {a_3}$ &   $\alpha = \beta = \gamma = 90^\circ $ &  3 diads \\[3pt]
Tetragonal  &  ${a_1} = {a_2} \neq {a_3}$ &   $\alpha = \beta = \gamma = 90^\circ $ &  1 tetrad \\[3pt]
Trigonal  &  ${a_1} = {a_2} = {a_3}$ &   $\alpha = \beta = \gamma \neq 90^\circ $ &  1 triad \\[3pt]
Hexagonal  &  ${a_1} = {a_2} \neq {a_3}$ &  $ \alpha = \beta = 90^\circ, \gamma = 120^\circ $ &  1 hexad \\[3pt]
Cubic  &  ${a_1} = {a_2} = {a_3}$ &   $\alpha = \beta = \gamma = 90^\circ $ &  4 triads\\[3pt]\midrule
\end{tabular}
\end{center}
\label{tab:systems}
\end{table}


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{Introduction/Bravais.eps}
\caption{The fourteen three-dimensional Bravais lattices.}
\label{fig:Bravais}
\end{center}
\end{figure} 



\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{Introduction/projections.eps}
\caption{Projection of the three-dimensional shapes of the fourteen
Bravais lattices on to the  {\bf a}$_1$-{\bf a}$_2$ plane. The numbers indicate the
coordinates of lattice points relative to the {\bf a}$_3$ axis; the unlabelled
lattice points are by implication located at coordinates 0
and 1 with respect to the {\bf a}$_3$ axis.}
\label{fig:Bravais2}
\end{center}
\end{figure} 
\section{ Directions }
 
Any vector {\bf u} can be represented as a linear combination of
the basis vectors ${\bf a_i}$ of the unit cell ($i = 1, 2, 3$):
\begin{equation} {\bf u} = u_1{\bf a_1} + u_2{\bf a_2} + u_3{\bf a_3} \end{equation}
and the scalar quantities $u_1$, $u_2$ and $u_3$ are the
components of the vector {\bf u} with respect to the basis
vectors {\bf a}$_1$, {\bf a}$_2$ and {\bf a}$_3$.  Once  the unit cell is
defined, any direction {\bf u} within the lattice can be
identified uniquely by its components $[u_1~u_2~u_3]$,
and the components are called the Miller indices of that
direction and are by convention enclosed in square brackets (Fig.~\ref{fig:indexing}).


\begin{figure}[!ht]
\centering
\subfigure(a){
\includegraphics[width=0.25\textwidth]{Introduction/112.eps}}
\subfigure(b){
\includegraphics[width=0.25\textwidth]{Introduction/020.eps}}
\subfigure(c){
\includegraphics[width=0.35\textwidth]{Introduction/11.eps}}\\
\caption{ (a,b) Miller indices for
directions and planes. (c) Notice that in non-cubic systems, a direction with indices identical to those of a plane is not necessarily normal to that plane.}
\label{fig:indexing}
\end{figure}



It is sometimes the case  that the properties along two or more
different directions are identical. These directions are said
to be {\it   equivalent } and the crystal is said to possess
{\it   symmetry}. For example, the $[1~0~0]$ direction for a
cubic lattice is equivalent to the $[0~1~0]$, $[0~0~1]$,
$[0~\overline 1~0]$, $[0~0~\overline 1]$ and $[\overline 1~0~0]$ directions;
the bar on top of the number implies that the index is negative.  

The indices of directions of the same form are
conventionally enclosed in special brackets, {\em e.g.}  $\langle1~0~0\rangle$. The
number of equivalent directions within the form is called the
multiplicity of that direction, which in this case
is 6.

\section{ Planes }
 
If a  plane intersects the {\bf a}$_1$, {\bf a}$_2$ and {\bf a}$_3$ axes at
distances $x_1$, $x_2$ and $x_3$ respectively, relative to
the origin, then the Miller indices of that plane are given
by $(h_1~h_2~h_3)$ where:
\begin{equation}h_1 = \phi a_1/x_1,~~~    h_2 = \phi a_2/x_2,~~~    h_3 = \phi a_3/x_3\end{equation}
$\phi$  is a scalar which clears the numbers $h_i$ off
fractions or common factors. Note that $x_i$ are negative
when measured in the $-{\bf a_i}$ directions. The
intercept of the plane with an axis may occur at 
$\infty$, in which case the plane is parallel to that axis
and the corresponding Miller index will be zero (Fig.~\ref{fig:indexing}).
 
  Miller indices for planes are by convention written using
round brackets: $(h_1~h_2~h_3)$ with braces being used
to indicate planes of the same form: $\{h_1~h_2~h_3\}$.

\section{Weiss Zone Law}

This law states that if a direction $[u_1~u_2~u_3]$ lies in a plane $(h_1~h_2~h_3)$ then
\begin{equation}
u_1h_1 + u_2h_2 + u_3h_3 = 0 \label{eqn:Weiss}
\end{equation}
and the law applies to any crystal system. We shall prove this when we deal with the reciprocal lattice.


\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.22\linewidth}
\raisebox{-4cm}{\includegraphics[width=0.98\linewidth]{Introduction/zone.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.65\linewidth}
      \caption{ A {\em zone} refers to a set of planes which share a common direction, which in turn is known as a zone axis. The direction illustrated would satisfy the Weiss law for all the planes shown. For example, the  planes $(1~1~1)$, $(1~1~2)$ and $(1~1~0)$ all belong to the zone $[1~\overline 1~0]$.
}  \label{fig:Weiss}
    \end{minipage}
  \end{center} \end{figure} 
\section{Symmetry}
  
 
Although the properties of a crystal can be anisotropic, there may be 
different directions along which they are identical. These directions
are said to be {\it  equivalent } and the crystal is said to possess {\it 
symmetry}.

  That a particular edge of a  cube cannot be distinguished
from any other  is a measure of its symmetry; an
orthorhombic parallelepiped has lower symmetry, since its
edges can be distinguished by length.

Some symmetry operations are illustrated in Fig.~\ref{fig:symmetry}; in essence, they transform a
spatial arrangement into another that is
indistinguishable from the original. The rotation of a cubic
lattice through 90$^\circ$ about an axis along the edge of the
unit cell is an example of a symmetry operation, since the
lattice points of the final and original lattice coincide in
space and cannot consequently be distinguished.

  We have implicitly  encountered  translational
symmetry when defining the lattice; since the environment of
each lattice point is identical, translation between lattice
points  has the effect of shifting the origin. 


\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.3\linewidth}
\raisebox{-7.5cm}{\includegraphics[width=0.98\linewidth]{Introduction/symmetry_operations.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.5\linewidth}
      \caption[caption]{Some symmetry operations. The mirror plane involves reflection symmetry, whereas the glide plane is a combination of reflection and translation parallel to the mirror. Translational symmetry is intrinsic to all lattices whereas the other elements may or may not exist for all lattices. For example, the triclinic lattice has no mirror or glide plane.\\\hspace{\textwidth} \\\hspace{\textwidth}  Note that the translations involved in the screw axis and glide plane are rational fractions of the repeat distance along the translation direction. The screw axis illustrated involves a rotation of 180$^\circ$ ($\equiv$ diad) combined with a translation parallel to the axis. }  \label{fig:symmetry}
    \end{minipage}
  \end{center} \end{figure} 


\section{ Symmetry Operations }
 
An object possesses an {\it n}-fold axis of rotational
symmetry if it coincides with itself upon rotation about the
axis through an angle 360$^\circ$/{\it n}. The possible angles
of rotation, which are consistent with the translational
symmetry of the lattice, are 360$^\circ$, 180$^\circ$, 120$^\circ$, 90$^\circ$
and 60$^\circ$ for values of {\it n } equal to 1, 2, 3, 4 and 6
respectively. A five--fold axis of rotation does not preserve
the translational symmetry of the lattice and hence is forbidden. A
one-fold axis of rotation is called a monad and the terms
diad, triad, tetrad and hexad correspond to {\it n } = 2,
3, 4 and 6 respectively.

  All of the Bravais lattices have a {\it  centre of
symmetry}, Fig.~\ref{fig:inversion}. An observer at the centre of symmetry
sees no difference in arrangement between the directions
$[u_1~u_2~u_3]$  and $[-u_1~-u_2~-u_3]$. The
centre of symmetry is such that inversion through that point
produces an identical arrangement but in the opposite sense.
A {\it  rotoinversion } axis of symmetry rotates a point
through a specified angle and then inverts it through the
centre of symmetry such that the arrangements before and
after this combined operation are in coincidence. For
example, a three-fold inversion axis involves a rotation
through 120$^\circ$ combined with an inversion, the axis being
labelled $\overline 3$.


\begin{figure}[!ht]
\centering
\subfigure(a){
\includegraphics[width=0.25\textwidth]{./Introduction/molecule.eps}} \hskip 2cm
\subfigure(b){
\includegraphics[width=0.19\textwidth]{././Introduction/inversion.eps}}
\caption{Illustration of centre of symmetry. (a) A molecule with the centre of symmetry identified by the dot. (b) The common point where  the arrows intersect is the centre of symmetry.}
\label{fig:inversion}
\end{figure}



 A rotation operation can also be combined with a translation
parallel to that axis to generate a {\it  screw axis } of
symmetry. The magnitude of the translation is a fraction of
the lattice repeat distance along the axis concerned. A
$3_1$ screw axis would rotate a point through 120$^\circ$ and
translate it through a distance $t/3$, where $t$ is the magnitude
of the shortest lattice vector along the axis. A
$3_2$  operation involves a rotation through 120$^\circ$ followed
by a translation through $2t/3$ along the axis. For a
right--handed screw axis, the sense of rotation is
anticlockwise when the translation is along the positive
direction of the axis.

  A plane of {\it  mirror symmetry } implies  arrangements which are mirror
images. Our left and right hands are (approximately)
mirror images. The operation of a $\overline 2$ axis
produces a result which is equivalent to a reflection through
a mirror plane normal to that axis.

 The operation of a {\it  glide plane } combines a
reflection with a translation parallel to the plane, through
a distance which is half the lattice repeat  in the
direction concerned. The translation may be parallel to a
unit cell edge, in which case the glide is {\it  axial};
the term {\it  diagonal } glide refers to
translation along a face or body diagonal of the unit
cell. In the latter case, the translation is through a
distance which is half the length of the diagonal concerned,
except for {\it  diamond } glide, where it is a quarter of
the diagonal length.

\section{Crystal Structure} 
 
Lattices are regular arrays of imaginary  points in space. A real crystal has
atoms associated with these points. Consider the projection of the primitive cubic cell illustrated in Fig.~\ref{fig:CuZn}a. In the sequence Fig.~\ref{fig:CuZn}b-e, a pair of atoms (Cu at 0,0,0 and Zn at $\frac{1}{2},\frac{1}{2},\frac{1}{2}$), which also is known as the {\em motif}, is placed at each lattice point of Fig.~\ref{fig:CuZn}a in order to build up the crystal structure of $\beta$--brass (Fig.~\ref{fig:CuZn}f). 
\begin{equation}
\hbox{lattice} + \hbox{motif} = \hbox{crystal structure}
\end{equation}

\begin{figure}[!ht]
\centering
\subfigure(a){
\includegraphics[width=0.25\textwidth]{./Introduction/CuZn_1.eps}}
\subfigure(b){
\includegraphics[width=0.25\textwidth]{./Introduction/CuZn_2.eps}}
\subfigure(c){
\includegraphics[width=0.25\textwidth]{./Introduction/CuZn_3.eps}}\\
\subfigure(d){
\includegraphics[width=0.25\textwidth]{./Introduction/CuZn_4.eps}}
\subfigure(e){
\includegraphics[width=0.25\textwidth]{./Introduction/CuZn_5.eps}}
\subfigure(f){
\includegraphics[width=0.25\textwidth]{./Introduction/CuZn_6.eps}}\\
\caption{Building up the crystal structure of $\beta$-brass by placing a motif consisting of an appropriate pair of Cu and Zn atoms at each lattice point of a primitive cubic lattice. (a) represents the primitive lattice with the motif consisting of a pair of distinct atoms that have yet to be placed. (b-e) The motif is placed at each lattice point.  (f) The three-dimensional representation of the final structure.}
\label{fig:CuZn}
\end{figure}


The location of an atom of  copper at each
lattice point of a cubic--{\it F } lattice, generates the   crystal
structure  of copper, with four copper atoms per unit cell (one per lattice
point), representing the actual arrangement of copper atoms in space. 

Consider a motif consisting of a pair of carbon atoms, with
coordinates $[0~0~0]$ and $[\frac{1}{4}~\frac{1}{4}~\frac{1}{4}]$ relative to a
lattice point. Placing this motif at each lattice point of
the cubic-{\it F } lattice generates the diamond crystal structure (Fig.~\ref{fig:diamond}),
with each unit cell containing 8 carbon atoms (2 carbon atoms per lattice
point). 

Fig.~\ref{fig:superalloy} shows how a Cubic--F cell changes into a Cubic--P cell when the nickel and aluminium atoms order in the classical $\gamma/\gamma'$ superalloy system. 


\begin{figure}[!ht]
\centering
\subfigure(a){
\includegraphics[width=0.4\linewidth]{Introduction/diamond_1.eps}}
\subfigure(b){
\includegraphics[width=0.35\linewidth]{Introduction/diamond_2.eps}}\\
\subfigure(c){
\includegraphics[width=0.48\linewidth]{Introduction/diamond_3.eps}}
\caption{(a) Projection of the cubic-F lattice. (b) Projection of cubic-F lattice with a motif of a pair of  carbon atoms at $0,0,0$ and $\frac{1}{4},\frac{1}{4},\frac{1}{4}$ placed at each lattice point.  (c) Perspective of the same structure illustrating the tetrahedral bonding of the carbon atoms.}
\label{fig:diamond}
\end{figure}

\begin{figure}[!ht]
\centering
\subfigure(a){
\includegraphics[width=0.3\textwidth]{Introduction/gamma.eps}}
\subfigure(b){
\includegraphics[width=0.29\textwidth]{Introduction/gamma_prime.eps}}
\caption{Nickel based superalloys. (a) The face-centred cubic crystal structure of disordered $\gamma$. (b) The primitive cubic crystal structure of $\gamma'$.}
\label{fig:superalloy}
\end{figure}

\section{Point Group Symmetry}

Consider a molecule such as that illustrated in Fig.~\ref{fig:water}. The symmetry operations on the molecule constitute a collection known as a {\em point group} because there is always one point in space that is left unchanged by every symmetry operation in that group. The point group symmetry of  the molecule is illustrated in Fig.~\ref{fig:water}a. When exposed to infrared light the molecules absorb energy and vibrate as illustrated in  Fig.~\ref{fig:water}b. These vibrations result in a characteristic spectrum showing peaks at certain energies which depend on the modes illustrated. Other molecules with the same point group symmetry,  such as sulphur tetrafluoride, will exhibit similar spectroscopic properties. The point group of a crystal is also the symmetry common to all of its {\em macroscopic} properties.



\begin{figure}[!ht]
\centering
\subfigure(a){
\includegraphics[width=0.25\textwidth]{Introduction/water_1.eps}}\hskip 1cm
\subfigure(b){
\includegraphics[width=0.5\textwidth]{Introduction/water_2.eps}}
\caption{(a) A molecule with point group symmetry $2m$ (diad and a mirror plane parallel to the diad). (b) Vibration modes (symmetrical stretch, asymmetrical stretch and bending) of the molecule when appropriately stimulated.}
\label{fig:water}
\end{figure}

Translations are therefore necessarily excluded in a {\em point} group. Rotations, mirror planes, centre of symmetry and inversion axes are permitted. There are 32 point groups in three dimensions, classified within the seven crystal classes (Table~\ref{tab:pg}). The triclinic, monoclinic and orthorhombic groups do not contain triads, tetrads or hexads. For those systems, when the point group symbol contains three elements (e.g. $2mm$), then the symbols are presented in the order of the symmetry elements parallel to the $x$, $y$ and $z$ axes respectively, as illustrated in Fig.~\ref{fig:order}a for an object in the orthorhombic class. 

For crystal systems with higher order axes, the $z$ direction is assigned to that higher order axis, with the second symbol corresponding to equivalent {\em secondary} axes that are normal to $z$, and the third also normal to $z$ to equivalent {\em tertiary} directions passing between the secondary ones. Fig.~\ref{fig:order}b shows an example for an object in the tetragonal class, with the tetrad placed along the $z$ axis, the two mirror planes with normals along the $x$ and $y$ axes and the additional mirror planes generated by this symmetry also illustrated. 

In the case of the cubic system the triad is always the second symbol given that the defining symmetry is four triads.

\begin{figure}[!ht]
\centering
\subfigure(a){
\includegraphics[width=0.4\textwidth]{Introduction/2mm.eps}} \hfill
\subfigure(b){
\includegraphics[width=0.45\textwidth]{Introduction/4mm.eps}}
\caption{Convention for point group notation. (a) Groups without high order axes. (b) Groups with high order axes.}
\label{fig:order}
\end{figure}

Some further details on the notation are as follows:
\begin{itemize}
\item $m\equiv \overline 2$ represents a mirror plane.
\item 2, 3, 4, 6 are 2-fold, 3-fold, 4-fold and 6-fold rotation axes, respectively.
\item $\overline 1$, $\overline 2$, $\overline 3$ {\em etc.} are inversion axes; the one fold inversion axis is equivalent to a centre of symmetry; $\overline 2$ signifies a rotation of $360^\circ/2$ combined with an inversion through the centre point.
\item $\frac{4}{m}mm$ is a 4-fold axis with a mirror normal to it and four mirror planes containing it. The symbol is usually written without the space as $4/mmm$
\item 432 refers to a point group with  3-fold axes which are not parallel  to the $z$ axis, where a unit cell has $x$, $y$ and $z$ axes. This is limited to the  cubic system.
\end{itemize}

\clearpage  \label{page:pg}
\begin{table}[!ht]
\caption{Point group symmetries associated with the seven crystal classes.}
\begin{center}
\resizebox{0.99\textwidth}{!}{\begin{tabular}{lllp{5cm}}\midrule
Class & Non-centrosymmetric & Centrosymmetric &  \\[3pt]\midrule
Cubic & 23,  432, $\overline 4 3 m$ & $m\overline 3$, $m\overline 3m$&\multirow{4}{*}{$\left\}\begin{array}{c} \\ \\ \\ \\ \\ \end{array}\right.$ With high order axes}\\[3pt]
Hexagonal & 6, $\overline 6$,  622, $6mm$, $\overline 6 m2$& $6/m$, $6/mmm$ & \\[3pt]
Trigonal & 3,  32, $3m$ &$\overline 3$, $\overline 3m$& \\[3pt]
Tetragonal & $4,~\overline 4,~422,~ 4mm,~\overline 42m$ & $4/m,~~4/mmm$&\\[10pt]
Orthorhombic & $222,~2mm$ & $mmm$&\multirow{3}{*}{$\left\}\begin{array}{c} \\ \\ \\ \\ \end{array}\right.$ Without high order axes}\\[3pt]
Monoclinic & $2,~m$ & $2/m$ &\\[3pt]
Triclinic &$1$ & $\overline 1$&\\[3pt]\midrule
\end{tabular} }
\end{center}
\label{tab:pg}
\end{table}%

\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.57\linewidth}
\raisebox{-4.5cm}{\hskip 0.5cm
\subfigure(a){
\includegraphics[width=0.25\textwidth]{Introduction/gypsum.eps}} \hskip 1cm
\subfigure(b){
\includegraphics[width=0.25\textwidth]{Introduction/epsomite.eps}}
}  \end{minipage}\hfill
    \begin{minipage}[t]{0.40\linewidth}
\caption{(a) Shape of gypsum (${\rm CaSO_4.2H_2O}$) with point group symmetry $2/m$. (b) Shape of epsomite (${\rm MgSO_4.7H_2O}$) with point group symmetry 222. Note that the angle between the $(110)$ and $(1\overline 1 0)$ faces is 89.37$^\circ$C, because the lattice parameters $a$, $b$, and $c$ are 1.1866, 1.1998 and 0.6855 nm respectively.}
\label{fig:shape}
    \end{minipage}
 \end{center} \end{figure} 






