\documentclass[12pt]{article}
\usepackage{wrapfig}
\usepackage{amsmath}
\usepackage{graphicx}
\message{
Copyright, 1995-2017, all rights reserved, Mark Jarrell (Dept.of
Physics and Atronomy, Louisiana State University, LA 70803). This
material may not be reproduced for profit, modified or published in any
form (this includes electronic redistribution) without the prior written
permission of the author listed above.
}
\title{Chapter 2: Crystal Structures and Symmetry}
\author{Laue, Bravais}
\input{../defs}
\def\baselinestretch{1.4}
\bdo
\maketitle
\large
\tableofcontents
\pagebreak
A theory of the physical properties of solids would be
practically impossible if the most stable elements were not
regular crystal lattices. The N-body problem is reduced to
manageable proportions by the existence of translational
symmetry. This means that there exist a set of basis vectors
($\a$,$\b$,$\c$) such that the atomic structure remains invariant under
translations through any vector which is the sum of integral
multiples of these vectors. As shown in Fig.~\ref{fig:basisvecs}
this means that one may go from any location in the lattice to
an identical location by following path composed of integral
multiples of these vectors.
\begin{figure}[htb]
\centerline{\includegraphics[width=6.0in,keepaspectratio,clip=true]{basisvecs.pdf}}
\caption[]{\em{One may go from any location in the lattice to
an identical location by following path composed of integral
multiples of the vectors $\a$ and $\b$.}}
\label{fig:basisvecs}
\end{figure}
Thus, one may label the locations of the "atoms"\footnote{we will
see that the basic building blocks of periodic structures can be more
complicated than a single atom. For example in NaCl, the basic building
block is composed of one Na and one Cl ion which is repeated in a cubic
pattern to make the NaCl structure}. which compose the lattice with
\beq
\r_n=n_1\a+n_2\b+n_3\c
\eeq
where $n_1,n_2,n_3$ are integers. In this way we may construct any periodic
structure.
\section{Lattice Types and Symmetry}
\subsection{Two-Dimensional Lattices}
These structures are classified according to their symmetry.
For example, in 2d there are 5 distinct types. The lowest symmetry is an
oblique lattice, of which the lattice shown in Fig.~\ref{fig:basisvecs}
is an example if $a\neq b$ and $\alpha$ is not a rational fraction
of $\pi$.
\begin{figure}[htb]
\centerline{\includegraphics[height=3.0in,keepaspectratio,clip=true]{2dlattices.pdf}}
\caption[]{\em{Two dimensional lattice types of higher symmetry.
These have higher symmetry since some are invariant under rotations
of $2\pi/3$, or $2\pi/6$, or $2\pi/4$, etc. The centered lattice
is special since it may also be considered as lattice composed of
a two-component basis, and a rectangular unit cell (shown with a
dashed rectangle).}}
\label{fig:2dlattices}
\end{figure}
Notice that it is invariant only under rotation of $\pi$ and
$2\pi$. Four other lattices, shown in Fig.~\ref{fig:2dlattices} of higher
symmetry are also possible, and called special lattice types (square,
rectangular, centered, hexagonal).
A Bravais lattice is the common name for a distinct lattice
type. The primitive cell is the parallel piped (in 3d) formed by
the primitive lattice vectors which are defined as the lattice vectors
which produce the primitive cell with the smallest volume
(${\bf{a}}\cdot ({\bf{c}}\times{\bf{c}})$). Notice that the
primitive cell does not always capture the symmetry as well as
a larger cell, as is the case with the centered lattice type.
The centered lattice is special since it may also be considered as lattice
composed of a two-component basis on a rectangular unit cell (shown with
a dashed rectangle).
\begin{figure}[htb]
\centerline{\includegraphics[height=3.2in,keepaspectratio,clip=true]{cuo2.pdf}}
\caption[]{\em{A square lattice with a complex basis composed of one Cu
and two O atoms (c.f.~cuprate high-temperature superconductors).}}
\label{fig:cuo2}
\end{figure}
To account for more complex structures like molecular solids,
salts, etc., one also allows each lattice point to have structure
in the form of a basis. A good example of this in two dimensions
is the CuO$_2$ planes which characterize the cuprate high temperature
superconductors (cf.\ Fig.~\ref{fig:cuo2}). Here the basis is composed
of two oxygens and one copper atom laid down on a simple square
lattice with the Cu atom centered on the lattice points.
\subsection{Three-Dimensional Lattices}
\begin{figure}[htb]
\centerline{\includegraphics[height=3.0in,keepaspectratio,clip=true]{3dcubic.pdf}}
\caption[]{\em{Three-dimensional cubic lattices. The primitive lattice
vectors ($\a$,$\b$,$\c$) are also indicated. Note that the primitive
cells of the centered lattice is not the unit cell commonly drawn.}}
\label{fig:3dcubic}
\end{figure}
The situation in three-dimensional lattices can be more complicated.
Here there are 14 lattice types (or Bravais lattices). For example
there are 3 cubic structures, shown in Fig.~\ref{fig:3dcubic}. Note that the
primitive cells of the centered lattice is not the unit cell commonly drawn.
In addition, there are triclinic, 2 monoclinic, 4 orthorhombic ...
Bravais lattices, for a total of 14 in three dimensions.
\section{Point-Group Symmetry}
The use of symmetry can greatly simplify a problem.
\subsection{Reduction of Quantum Complexity}
If a Hamiltonian is invariant under certain symmetry operations, then we
may choose to classify the eigenstates as states of the symmetry operation
and H will not connect states of different symmetry.
As an example, imagine that a symmetry operation $R$ leaves $H$
invariant, so that
\beq
RHR^{-1} = H \;\;\;{\rm{then}}\;\;\;\left[H,R\right]=0
\eeq
Then if $|j>$ are the eigenstates of $R$, then $\sum_j |j> =\sum_k *\,.
\label{eq:symeqn2}
\eeq
If we recall that $R_{ik}=**=R_{ii}\delta_{ik}$ since $|k>$ are
eigenstates of $R$, then Eq.~\ref{eq:symeqn2} becomes
\beq
\left(R_{ii}-R_{jj}\right)H_{ij}=0\,.
\eeq
So, $H_{ij}=0$ if $R_i$ and $R_j$ are different eigenvalues of
$R$. Thus, when the states are
classified by their symmetry, the Hamiltonian matrix becomes Block
diagonal, so that each block may be separately diagonalized.
\subsection{Symmetry in Lattice Summations}
As another example, consider a Madelung sum in a two-dimensional
square centered lattice (i.e. a 2d analog of NaCl). Here we want
to calculate
\beq
\sum_{ij}\frac{\pm}{p_{ij}}\,.
\label{eq:2dmade}
\eeq
This may be done by a brute force sum over the lattice, i.e.
\beq
\lim_{n\to\infty} \sum_{i=-n,n\,j=-n,n}\frac{(-1)^{i+j}}{p_{ij}}\,.
\eeq
Or, we may realize that the lattice has some well defined operations
which leave it invariant. For example, this lattice in invariant
under inversion $(x,y)\to(-x,-y)$, and reflections about the
x $(x,y)\to(x,-y)$ and y $(x,y)\to(-x,y)$ axes, etc. For these
reasons, the eight points highlighted in Fig.~\ref{fig:2dsymm}(a)
\begin{figure}[htb]
\centerline{\includegraphics[height=2.5in,keepaspectratio,clip=true]{2dsymm.pdf}}
\caption[]{\em{Equivalent points and irreducible wedge for the 2-d
square lattice. Due to the symmetry of the 2-d square lattice, the
eight patterned lattice sites all contribute an identical amount to
the Madelung sum calculated around the solid black site. Due to this
symmetry, the sum can be reduced to the irreducible wedge (b) if the
result at each point is multiplied by the degeneracy factors indicated.}}
\label{fig:2dsymm}
\end{figure}
all contribute an identical amount to the sum in Eq.~\ref{eq:2dmade}. In fact
all such interior points have a degeneracy of 8. Only special points
like the point at the origin (which is unique) and points along the
symmetry axes (the xy and x axis, each with a degeneracy of four)
have lower degeneracies. Thus, the sum may be restricted to the
irreducible wedge, so long as
the corresponding terms in the sum are multiplied by the appropriate
degeneracy factors, shown in Fig.~\ref{fig:2dsymm}(b). An appropriate
algorithm to calculate both the degeneracy table, and the sum
\ref{eq:2dmade} itself are:
\begin{verbatim}
c First calculate the degeneracy table
c
do i=1,n
do j=0,i
if(i.eq.j.or.j.eq.0) then
deg(i,j)=4
else
deg(i,j)=8
end if
end do
end do
deg(0,0)=1
c
c Now calculate the Madelung sum
c
sum=0.0
do i=1,n
do j=0,i
p=sqrt(i**2+j**2)
sum=sum+((-1)**(i+j))*deg(i,j)/p
end do
end do
\end{verbatim}
By performing the sum in this way, we saved a factor of 8! In fact, in three-
dimensions, the savings is much greater, and real band structure calculations
always make use of the point group symmetry to accelerate the calculations.
The next question is then, could we do the same thing for a more
complicated system (fcc in 3d?). To do this, we need some way of
classifying the symmetries of the system that we want to apply.
Group theory allows us to learn the consequences of the symmetry
in much more complicated systems.
A group $S$ is defined as a set $\{E, A, B, C \cdots\}$ which is
closed under a binary operation $*$ (ie. $A*B\in S$) and:
\begin{itemize}
\item the binary operation is associative $(A*B)*C = A*(B*C)$
\item there exists an identity $E\in S$ : $E*A = A*E = A$
\item For each $A\in S$, there exists an
$A^{-1}\in S$ : $AA^{-1} = A^{-1} A = E$
\end{itemize}
In the point group context, the operations are inversions,
reflections, rotations, and improper rotations (inversion
rotations). The binary operation is any combination of these; i.e.\
inversion followed by a rotation.
In the example we just considered we may classify the
operations that we have already used. Clearly we need $2!2^2$ of
these (ie we can choose to take (x,y) to any permutation of (x,y)
and choose either $\pm$ for each, in D-dimensions, there would be
$D!2^D$ operations). In table.~\ref{tab:2dsymmopps},
all of these operations are identified
\begin{table}[htb]
\begin{tabular}{|l|l|}\hline
Operation & Identification \\\hline
$(x,y)\to(x,y)$ & Identity \\
$(x,y)\to(x,-y)$ & reflection about x axis \\
$(x,y)\to(-x,y)$ & reflection about y axis \\
$(x,y)\to(-x,-y)$ & inversion \\
$(x,y)\to(y,x)$ & reflection about $x=y$ \\
$(x,y)\to(y,-x)$ & rotation by $\pi/2$ about z \\
$(x,y)\to(-y,-x)$ & inversion-reflection \\
$(x,y)\to(-y,x)$ & inversion-rotation \\\hline
\end{tabular}
\caption[]{\em{Point group symmetry operations for the two-dimensional
square lattice. All of the group elements are self-inverting except
for the sixth and eight, which are inverses of each other.}}
\label{tab:2dsymmopps}
\end{table}
The reflections are self inverting as is the inversion and
one of the rotations and inversion rotations. The set is clearly
also closed. Also, since their are 8 operations, clearly the interior
points in the irreducible wedge are 8-fold degenerate (w.r.t. the
Madelung sum).
This is always the case. Using the group operations one may
always reduce the calculation to an irreducible wedge. They the degeneracy
of each point in the wedge may be determined: Since a group operation
takes a point in the wedge to either itself or an equivalent point
in the lattice, and the former (latter) does (does not) contribute
the the degeneracy, the degeneracy of each point times the number of
operations which leave the point invariant must equal the number of
symmetry operations in the group. Thus, points with the lowest
symmetry (invariant only under the identity) have a degeneracy of the
group size.
\subsection{Group designations}
Point groups are usually designated by their Sch\"onflies point group symbol
described in table.~\ref{tab:schonflies}
\begin{table}[htb]
\begin{tabular}{|l|l|}\hline
Symbol & Meaning \\\hline
$C_j$ & (j=2,3,4, 6) j-fold rotation axis \\
$S_j$ & j-fold rotation-inversion axis \\
$D_j$ & j 2-fold rotation axes $\perp$ to a j-fold principle rotation axis\\
$T$ & 4 three-and 3 two-fold rotation axes, as in a tetrahedron\\
$O$ & 4 three-and 3 four-fold rotation axes, as in a octahedron\\
$C_i$ & a center of inversion \\
$C_s$ & a mirror plane \\ \\\hline
\end{tabular}
\caption[]{\em{The Sch\"onflies point group symbols. These give the
classification according to rotation axes and principle mirror planes.
In addition, their are suffixes for mirror planes (h: horizontal=perpendicular
to the rotation axis, v: vertical=parallel to the main rotation axis in
the plane, d: diagonal=parallel to the main rotation axis in the
plane bisecting the two-fold rotation axes).}}
\label{tab:schonflies}
\end{table}
As an example, consider the previous example of a square lattice. It is
invariant under
\begin{itemize}
\item rotations $\perp$ to the page by $\pi/2$
\item mirror planes in the horizontal and vertical (x and y axes)
\item mirror planes along the diagonal (x=y, x=-y).
\end{itemize}
The mirror planes are parallel to the main rotation axis which is itself
a 4-fold axis and thus the group for the square lattice is $C_{4v}$.
\section{Simple Crystal Structures}
\subsection{FCC}
\begin{figure}[htb]
\centerline{\includegraphics[height=2.5in,keepaspectratio,clip=true]{fcc.pdf}}
\caption[]{\em{The Bravais lattice of a face-centered cubic (FCC) structure.
As shown on the left, the fcc structure is composed of parallel planes
of atoms, with each atom surrounded by 6 others in the plane. The total
coordination number (the number of nearest neighbors) is 12. The principle
lattice vectors (center) each have length $1/\sqrt{2}$ of the unit cell length.
The lattice has four 3-fold axes, and three 4-fold axes as shown on the right.
In addition, each plane shown on the left has the principle 6-fold
rotation axis $\perp$ to it, but since the planes are shifted relative to
one another, they do not share 6-fold axes. Thus, four-fold axes are the
principle axes, and since they each have a perpendicular mirror plane, the
point group for the fcc lattice is $O_h$.}}
\label{fig:fcc}
\end{figure}
The fcc structure is one of the close packed structures, appropriate for
metals, with 12 nearest neighbors to each site (i.e., a coordination number
of 12). The Bravais lattice for the fcc structure is shown in Fig.~\ref{fig:fcc}
It is composed of parallel planes of nearest neighbors (with six nearest
neighbors to each site in the plane)
Metals often form into an fcc structure. There are two reasons
for this. First, as discussed before, the s and p bonding is
typically very long-ranged and therefore rather non-directional.
(In fact, when the p-bonding is short ranged, the bcc structure is
favored.) This naturally leads to a close packed structure.
Second, to whatever degree there is a d-electron overlap in the
transition metals, they prefer the fcc structure. To see this,
consider the d-orbitals shown in Fig.~\ref{fig:dorbitals} centered on one of
the face centers with the face the xy plane. Each lobe of the $d_{xy}$,
$d_{yz}$, and $d_{xz}$ orbitals points to a near neighbor.
The xz,xy,yz triplet form rather strong bonds. The $d_{x^2-y^2}$ and
$d_{3z^2-r^2}$ orbitals do not since they point away from the nearest
neighbors. Thus the triplet of states form strong bonding and anti-bonding
bands, while the doublet states do not split.
\begin{figure}[htb]
\centerline{\includegraphics[height=2.45in,keepaspectratio,clip=true]{dorbitals.pdf}}
\caption[]{\em{The d-orbitals. In an fcc structure, the triplet
of orbitals shown on top all point towards nearest neighbors; whereas,
the bottom doublet point away. Thus the triplet can form bonding
and antibonding states.}}
\label{fig:dorbitals}
\end{figure}
The system can gain energy by occupying the triplet bonding states,
thus many metals form fcc structures. According to Ashcroft and Mermin,
these include Ca, Sr, Rh, Ir, Ni, Pd, Pt, Cu, Ag, Au, Al, and Pb.
The fcc structure also explains why metals are ductile since
adjacent planes can slide past one another. In addition each
plane has a 6-fold rotation axis perpendicular to it, but since 2
adjacent planes are shifted relative to another, the rotation
axes perpendicular to the planes are 3-fold, with one along the each
main diagonal of the unit cell. There are also 4-fold axes through
each center of the cube with mirror planes perpendicular to it. Thus the
fcc point group is $O_h$. In fact, this same argument also applies
to the bcc and sc lattices, so $O_h$ is the appropriate group for
all cubic Bravais lattices and is often called the cubic group.
\subsection{HCP}
As shown in Fig.~\ref{fig:hcpsymmetry} the Hexagonal Close Packed
(HCP) structure is described by the $D_{3h}$ point group.
\begin{figure}[htb]
\centerline{\includegraphics[height=2.0in,keepaspectratio,clip=true]{hcpsymmetry.pdf}}
\caption[]{\em{The symmetry of the HCP lattice. The principle
rotation axis is perpendicular to the two-dimensional hexagonal lattices
which are stacked to form the hcp structure. In addition, there is a
mirror plane centered within one of these hexagonal 2d structures,
which contains three 2-fold axes. Thus the point group is $D_{3h}$.}}
\label{fig:hcpsymmetry}
\end{figure}
The HCP structure (cf.~Fig.~\ref{fig:fccvshcp}) is similar to the FCC
structure, but it does not correspond to a Bravais lattice (in fact there
are five cubic point groups, but only three cubic Bravais lattices). As
with fcc its coordination number is 12. The simplest way to
construct it is to form one hexagonal plane and then add two
identical ones top and bottom. Thus its stacking is ABABAB... of
the planes.
\begin{figure}[htb]
\centerline{\includegraphics[height=2.3in,keepaspectratio,clip=true]{fccvshcp.pdf}}
\caption[]{\em{A comparison of the FCC (left) and HCP (right) close
packed structures. The HCP structure does not have a simple Bravais
unit cell, but may be constructed by alternately stacking two-dimensional
hexagonal lattices. In contract, the FCC structure may be constructed
by sequentially stacking three shifted hexagonal two-dimensional
lattices.}}
\label{fig:fccvshcp}
\end{figure}
This shifting of the planes clearly disrupts the d-orbital
bonding advantage gained in fcc, nevertheless many metals form
this structure including Be, Mg, Sc, Y, La, Ti, Zr, Hf, Tc, Re, Ru, Os,
Co, Zn, Cd, and Tl.
\subsection{BCC}
Just like the simple cubic and fcc lattices, the body-centered cubic
(BCC) lattice (cf.~Fig.~\ref{fig:3dcubic}) has four 3-fold axes, 3 4-fold axes,
with mirror planes perpendicular to the 4-fold axes, and therefore belongs to
the $O_h$ point group.
The body centered cubic structure only has a coordination number
of 8. Nevertheless some metals form into a BCC lattice (Ba V Nb,
Ta W M, in addition Cr and Fe have bcc phases.) Bonding of p-orbitals is
ideal in a BCC lattice since the nnn lattice is simply composed of
two interpenetrating cubic lattices. This structure allows the next-nearest
neighbor p-orbitals to overlap more significantly than an fcc (or hcp)
structure would. This increases the effective coordination number by
including the next nearest neighbor shell in the bonding
(cf.~Fig.~\ref{fig:IL211}).
\begin{figure}[htb]
\centerline{\includegraphics[height=2.7in,keepaspectratio,clip=true]{IL211.pdf}}
\caption[]{\em{Absolute square of the radial part of the electronic
wavefunction. For the bcc lattice, both the 8 nearest, and 6 next
nearest neighbors lie in a region of relatively high electronic density.
This favors the formation of a bcc over fcc lattice for some elemental
metals (This figure was lifted from I\&L).}}
\label{fig:IL211}
\end{figure}
\edo
*