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\title{Course C6: Crystallography \\ Interfaces, Orientation Relationships}
\author{H. K. D. H. Bhadeshia}
\begin{document}
\maketitle

\section{Introduction}

Much of the science and technology of polycrystalline materials depends on the nature of interfaces between crystals and the relative orientations of adjacent crystals. The corrosion resistance of a boundary depends on how coherent it is, the continuity of deformation depends on the relative orientations of the adjacent crystals $\ldots$ the list is seemingly endless!

 Atoms in the boundary between crystals must in general be displaced from positions
they would occupy in the undisturbed crystal, but it is now well established that many
interfaces have a periodic structure. In such cases, the misfit between the crystals
connected by the boundary is not distributed uniformly over every element of the
interface; it is periodically localised into discontinuities which separate
patches of the boundary where the fit between the two crystals is good or perfect.
When these discontinuities are well separated, they may individually be recognised as interface
dislocations which separate coherent patches in the boundary, which is macroscopically said to be semi--coherent. The simple example of a symmetrical tilt boundary illustrates this.

\section{Symmetrical Tilt Boundary}

The misorientation $\theta$ between grains can be described in terms of
dislocations (Fig.~\ref{fig:structure}). Inserting an edge dislocation of   Burgers vector  {\bf b} is like forcing a wedge into the lattice, so that each dislocation is associated with a small change in the orientation of the lattice on either side of the extra half plane. If the spacing of
dislocations is $d$, then
\begin{equation}\theta \simeq \frac{b}{d}.\end{equation}

\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.58\linewidth}
\raisebox{-5cm}{\includegraphics[width=0.98\linewidth]{structure.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.42\linewidth}
      \caption{ Structure of a
symmetrical tilt boundary.  The definition of a tilt boundary is that the unit boundary normal is perpendicular to the axis of rotation which generates one crystal from the other; a symmetrical tilt boundary has the additional property that the lattice of one crystal can be generated from the other by reflection across the boundary plane.
}  \label{fig:structure}
    \end{minipage}
  \end{center} \end{figure} 

This gives a model for the energy of grain boundaries, based on the
density of dislocations in the boundary ($\propto d^{-1}$). The
variation in energy as a function of misorientation is illustrated in
Fig.~\ref{fig:energy}. The model is not valid for large misorientations where the
dislocation spacing becomes comparable with the magnitude of the Burgers
vector, since the dislocation cores then begin to overlap.

\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.62\linewidth}
\raisebox{-5cm}{\includegraphics[width=0.98\linewidth]{energy.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.35\linewidth}
      \caption{The interfacial energy as
a function of the misorientation for a tilt boundary. The dashed curve
represents actual measurements with cusps at coincidence site lattices. }  \label{fig:energy}
    \end{minipage}
  \end{center} \end{figure} 

\section{Coincidence Site Lattices}

For high--misorientation boundaries, the predicted spacings of dislocations may turn out to be so small that the misfit is highly localised with respect to the boundary, and
the dislocation model of the interface has only formal significance (it is often said that the dislocations get so close to each other that their cores overlap). The arrangement of atoms in such incoherent boundaries may be  haphazard, with little correlation of atomic positions across the boundary.

  On the other hand, it is unreasonable to assume that all high--angle boundaries have the disordered structure. There is clear experimental evidence which shows that certain high--angle boundaries exhibit the characteristics of low-energy coherent or semi--coherent interfaces; for example, they exhibit strong faceting, have very low mobility in pure materials and the boundary diffusion coefficient may
be abnormally low. These observations imply that at certain special relative crystal orientations, which would usually be classified as high--angle orientations, it is possible to obtain boundaries which have a distinct structure -- they contain regions of good fit, which occur at regular intervals in the boundary plane, giving a pattern
of good fit points in the interface. It is along these points that the two crystals connected by the boundary match exactly.

  If we consider the good fit points to correspond to lattice points in the
interface which are common to both crystals, then the following procedure allows us
to deduce the pattern and frequency of these points for any given orientation
relationship.
If the two lattices (with a common origin) are notionally allowed to interpenetrate and fill all space, then there
may exist lattice points (other than the origin) which are common to both
the crystals. The set of these coincidence points forms a coincidence site lattice (CSL), and
the fractions of lattice points which are also coincidence sites is a
rational fraction $1/\Sigma$. $\Sigma$ is thus the reciprocal density of
coincidence sites relative to ordinary lattice sites. The value of $\Sigma$ is a
function of the relative orientation of the two grains and not of the orientation
of the boundary plane. The boundary simply intersects the CSL and will contain
regions of good fit which have a periodicity corresponding to the periodicity of a
planar net of the CSL along which the intersection occurs.
Boundaries parallel to low-index planes of the CSL are two dimensionally periodic
with relatively small repeat cells, and those boundaries with a high planar
coincidence site density should have a relatively low energy.
 
\section{Example of CSL}
\label{section:CSL}

  The axis--angle pair describing the orientation  relationship between the two
grains (A and B) of austenite is given by:
$$\hbox{axis of rotation parallel to} \qquad [1~1~2]_A$$
$$\hbox{right--handed angle of rotation}\qquad 180^\circ$$
 
 Show that a CSL with $\Sigma = 3$  can be formed by allowing the two lattices
to notionally interpenetrate. Also show that 1 in 3 of the sites in an interface
parallel to $(0~\overline 2~1)_A$  are coincidence sites.

  Because of the centrosymmetric nature of the austenite lattice, a rotation of
180$^\circ$  about $[1~1~2]_A$, which generates the B grain from A, is equivalent
to a reflection of the A lattice about $(1~1~2)_A$. We may therefore imagine that
B is generated by reflection of the A lattice about $(1~1~2)_A$  as the
mirror plane.
The stacking sequence of $(1~1~2)_A$  is $\ldots ABCDEFABCDEFABCDEF\ldots$ and
grain A is represented below as a stack of $(1~1~2)$ planes, B being generated by
reflecting the A lattice about one of the $(1~1~2)_A$  planes.

\begin{eqnarray}
&\overbrace{A~B~C~D~E~F~A~B~C~D~E~F~A~B~C~D~E~F~A~B~C~D~E~F~A~B~C~D~E~F}^{\hbox{Grain
A}} &\\\nonumber
& &
\hfill [1~1~2]_A \rightarrow \\ \nonumber
&\underbrace{A~B~C~D~E~F~A~B~C~D~}_{\hbox{Grain
A}}\underbrace{C~B~A~F~E~D~C~B~A~F~E~D~C~B~A~F~E~D~C~B}_{\hbox{Grain B}} &\\\nonumber
\end{eqnarray}


  The first sequence represents a stack of $(1~1~2)_A$  planes, as do the first
9 layers of the second sequence. The remainder of the second sequence represents
a stack of $(1~1~2)_B$  planes (note that B lattice is obtained by reflection
of A about the 9th layer of the second sequence).
Since the two sequences have the same origin, comparison of the first sequence
with the B part of the second sequence amounts to allowing the two crystals
to interpenetrate in order to identify coincidences.
Clearly, every 3rd layer of grain B coincides exactly with a layer from the
A lattice (dashed vertical lines), giving $\Sigma =3$.

  A boundary parallel to $(1~1~2)_A$  will be fully coherent; at least 1 in 3 of the
sites in any other boundary, such as $(0~\overline 2~1)_A$  will be coincidence
sites.

\section{Representation of Orientation Relationships}


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.9\linewidth]{coordinate.eps}
\caption{(a) Conventional face--centred cubic cell. (b) The same arrangement but represented with a body--centred tetragonal cell.}
\label{fig:coordinate}
\end{center}
\end{figure} 

  From Fig.~\ref{fig:coordinate}a, it is evident that the choice of basis vectors
{\bf a}$_i$  is arbitrary though convenient; Fig.~\ref{fig:coordinate}b illustrates an alternative
basis, a body--centred tetragonal (BCT) unit cell describing the same lattice. We label this as basis \lq B', consisting of basis vectors
{\bf b}$_1$, {\bf b}$_2$ and {\bf b}$_3$ which define the BCT unit cell.  It is obvious that
$[{\rm B};{\bf u}] = [0~2~1]$, compared with $[{\rm A};{\bf u}] = [1~1~1]$. The following
vector equations illustrate the relationships between the basis vectors of A and those of B
(Fig.~\ref{fig:coordinate}): 
\begin{eqnarray}{\bf a}_1 & =& 1{\bf b}_1  + 1{\bf b}_2 + 0{\bf b}_3 \\\nonumber
{\bf a}_2 &=& \overline 1{\bf b}_1  + 1{\bf b}_2 + 0{\bf b}_3\\\nonumber
{\bf a}_3 &=& 0{\bf b}_1  + 0{\bf b}_2 + 1{\bf b}_3\\\nonumber
\end{eqnarray}
These equations can also be presented in matrix form as follows:
\begin{equation}
 ({\bf a}_1~{\bf a}_2~{\bf a}_3) = ({\bf b}_1~{\bf b}_2~{\bf
b}_3) \times \left(\begin{array}{ccc}1 & \overline 1 & 0 \\ 1 & 1& 0\\ 0 & 0 & 1\end{array}\right)
\end{equation}

 This $3 \times 3$ matrix representing the co--ordinate transformation is denoted
(B~J~A) and transforms the components of vectors referred to the A basis to those referred to the B basis.
The first column of (B~J~A) represents the components of the basis
vector {\bf a}$_1$, with respect to the basis B, and so on.

  The components of a vector {\bf u} can now be transformed between bases using the matrix
{\rm (B~J~A)} as follows:
\begin{equation}[{\rm B};{\bf u}] = {\rm (B~J~A)}[{\rm A};{\bf u}] \label{eqn:a} \end{equation}
 Notice the juxtapositioning of like basis symbols.
If {\rm (A~J'~B)} is the transpose of {\rm (B~J~A)}, then
equation~\ref{eqn:a} can be rewritten as
\begin{equation}({\bf u};{\rm B}) = ({\bf u};{\rm A}){\rm (A~J'~B)}\nonumber\end{equation}
Writing {\rm (A~J~B)} as the inverse of {\rm (B~J~A)}, we obtain:
\begin{equation}[{\rm A};{\bf u}] = {\rm (A~J~B)}[{\rm B};{\bf u}]\nonumber \end{equation} and
\begin{equation}({\bf u};{\rm A}) = ({\bf u};{\rm B}){\rm (B~J'~A)}\nonumber \end{equation}
 
\subsection{Example: Co--ordinate transformations}

  Two adjacent grains of austenite are represented by bases \lq A' and \lq B'
respectively. The base vectors {\bf a}$_i$ of A and {\bf b}$_i$ of B
respectively define the FCC unit cells of the austenite grains concerned.
The lattice parameter of the austenite is $a_\gamma$ so that  $\vert{\bf a}_i\vert =
\vert{\bf b}_i\vert = a_\gamma$. The grains are orientated such that $[0~0~1]_A \vert\,\vert
[0~0~1]_B$, and $[1~0~0]_B$ makes an angle of 45$^\circ$ with both $[1~0~0]_A$ and
$[0~1~0]_A$.  Prove that if {\bf u} is a vector such that 
its components in crystal A are given by $[{\rm A};{\bf u}] =
[\sqrt2~~2\sqrt2~~0]$, then in the basis B,
$[{\rm B};{\bf u}] = [3~1~0]$.
Show that the magnitude of {\bf u} ({\em i.e.} $\vert {\bf u}\vert$)
does not depend on the choice of the basis.


\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.22\linewidth}
\raisebox{-3.5cm}{\includegraphics[width=0.98\linewidth]{coordinate2.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.65\linewidth}
      \caption{Diagram illustrating the relation
between the bases A and B.}  \label{fig:coordinate2}
    \end{minipage}
  \end{center}  
\end{figure}


Referring to Fig.~\ref{fig:coordinate2}, and recalling that the matrix {\rm (B~J~A)} consists of three
columns, each column being the components of one of the basis vectors of A,
with respect to B, we have 

\begin{equation}
\begin{array}{cccc}
[{\rm B};{\bf a}_1] = & [\cos45 & -\sin45 & 0] \cr
[{\rm B};{\bf a}_2] = & [\sin45 &  \cos45 & 0] \cr
[{\rm B};{\bf a}_3] = & [0 & 0 & 1]
\end{array}
\qquad\hbox{and}\qquad {\rm (B~J~A)}=\left(\begin{array}{ccc}
\cos45 & \sin45 & 0\\
-\sin45& \cos45 & 0 \\ 
0 & 0 & 1
\end{array}\right)
\end{equation} 
Now, $[{\rm B};{\bf u}] = {\rm (B~J~A)}[{\rm A};{\bf u}]$,
and on substituting for $[{\rm A};{\bf u}] = [\sqrt2~~2\sqrt2~~0]$, we get $[{\rm B};{\bf
u}] = [3~1~0]$. Both  the  bases  A and B are orthogonal so that the magnitude of {\bf u}
can be obtained  using  the Pythagoras theorem. Hence, choosing components referred to the
basis B, we get:
\begin{equation}\vert {\bf u}\vert ^2  =  (3\vert {\bf b}_1 \vert )^2 + (\vert {\bf b}_2\vert )^2 =
10a^2_\gamma\end{equation}
 With respect to basis A,
\begin{equation}\vert {\bf u}\vert ^2 =  
(\sqrt2\vert {\bf a}_1 \vert )^2 + (2\sqrt2\vert {\bf a}_2\vert )^2  = 10a^2_\gamma\end{equation}
 Hence, $\vert {\bf u}\vert$  is invariant to the co--ordinate transformation. This is a
general result, since a vector is a physical entity, whose magnitude and
direction clearly cannot depend on the choice of a reference frame, a choice which is
after all, arbitrary.

  We  note  that  the components of {\rm (B~J~A)} are the cosines of angles between
${\bf b}_i$ and ${\bf a}_j$ and that ${\rm (A~J'~B)} = {\rm (A~J~B)}^{-1}$; a matrix with
these properties is described as orthogonal (see appendix). An orthogonal matrix
represents an axis transformation between like orthogonal  bases.

\subsection{General Rotation Matrix}

If we  represent an axis of rotation as a unit vector, then
only three independent quantities are needed to define a misorientation between grains: two components of the axis of rotation, and an angle of rotation.  It follows that a
rotation matrix must also have only three independent terms.  The components of any rotation matrix can be written in terms of a vector
${\bf u} = [u_1 ~u_2~u_3]$ which lies along the axis of rotation
(such that $u_1u_1+u_2u_2+u_3u_3 = 1$), and in terms of
the right--handed angle of rotation $\theta$ as follows:

\begin{equation} ({\rm Y}~{\rm J}~{\rm X}) =
\left(\begin{array}{ccc} u_1 u_1(1-m)+m & u_1 u_2(1-m)+u_3n & u_1 u_3(1-m)-u_2n \cr
u_1 u_2(1-m)-u_3n & ~u_2u_2(1-m)+m & u_2u_3(1-m)+u_1 n \cr
u_1 u_3(1-m)+u_2n & u_2u_3(1-m)-u_1 n & u_3u_3(1-m)+m \end{array}\right) \label{eqn:rotation}\end{equation}
where $m = \cos\theta$ and $n = \sin\theta$
The right--handed angle of rotation can be obtained from the fact that
\begin{equation}J_{11} + J_{22} + J_{33} =  1 + 2\cos\theta\end{equation}
and the components of the vector {\bf u} along the axis of rotation are given by
\begin{equation}\begin{array}{ccc}u_1  & = & {{(J_{23}-J_{32})}/{2\sin\theta}} \cr
u_2 & = & {{(J_{31}-J_{13})}/{2\sin\theta}} \cr
$$u_3 & = & {{(J_{12}-J_{21})}/{2\sin\theta}} \end{array}\end{equation}
 
\section{Continuation of Example of CSL from Section~\ref{section:CSL}}


  We now consider a mathematical method  of determining the CSL formed
by allowing the lattices of crystals A and B to notionally interpenetrate; A and B are
assumed to be related by a transformation {\rm (A~S~A)} which deforms the A lattice into that
of {\rm B}; A and B need not have the same crystal structure or lattice parameters,
so that {\rm (A~S~A)} need not be a rigid body rotation.
 Consider a vector {\bf u} which is a lattice vector whose integral components do not have a common factor.
As a result of the transformation {\rm (A~S~A)}, {\bf u} becomes a new vector {\bf x} such that
\begin{equation}[{\rm A};{\bf x}] = {\rm (A~S~A)}[{\rm A};{\bf u}]\label{eqn:CSL}\end{equation}
 
Of course, {\bf x} does not necessarily have integral  components in the A basis
({\em i.e.,} it need not be a lattice vector of A). CSL vectors, on the other hand,
identify lattice points which are common to both A and B, and therefore are lattice
vectors of both crystals. It follows that CSL vectors have integral indices when
referred to either crystal. Hence, {\bf x} is only a CSL vector if it has integral
components in the basis A. We note that {\bf x} always has integral components in
B, because a lattice vector of A (such as {\bf u}) always deforms into a lattice
vector of B.

  The meaning of $\Sigma$ is that $1/\Sigma$ of the lattice sites of A or B
are common to both A and B. It follows that any primitive lattice vector of A or B, when
multiplied by $\Sigma$, must give a CSL vector. $\Sigma{\bf x}$ must therefore
always be a CSL vector and if equation~\ref{eqn:CSL} is multiplied by $\Sigma$, then we obtain
an equation in which the vector {\bf u} always transforms into a CSL vector:
\begin{equation}\Sigma[{\rm A};{\bf x}] = \Sigma{\rm (A~S~A)}[{\rm A};{\bf u}]\end{equation}
{\em i.e..} given that {\bf u} is a lattice vector of A, whose components have no common
factor,
$\Sigma{\bf x}$ is a CSL vector with integral components in either basis. This can
only be true if the matrix $\Sigma{\rm (A~S~A)}$ has elements which are all integral
since it is only then that $\Sigma[{\rm A};{\bf x}]$ has elements which are all
integral.

  It follows that if an integer $H$ can be found such that all the elements of the
matrix $H{\rm (A~S~A)}$ are integers (without a common factor), then $H$ is the
$\Sigma$ value relating A and B.

  Applying this to the problem considered in section~\ref{section:CSL}, the rotation matrix corresponding
to the rotation 180$^\circ$  about $[1~1~2]_A$  is given by (equation~\ref{eqn:rotation})
\begin{equation}{\rm (A~J~A)} =\frac{1}{3}\left(\begin{array}{ccc}\overline 2 & 1 & \overline 2\cr
1 & \overline 2 & 2\cr 2 & 2 & 1\end{array}\right) \end{equation}
and since 3 is the integer which when multiplied with {\rm (A~J~A)} gives a matrix
of integral elements (without a common factor), the $\Sigma$ value for this
orientation is given by $\Sigma =3$ .

Coincidence site lattices have in recent years become popular because of the advent of orientation imaging in the scanning electron microscope. The microscope is usually associated with software which calculates the $\Sigma$ value at every junction between two grains. The accuracy of the technique is limited so such $\Sigma$ maps should be taken with a pinch of salt, and large $\Sigma$ values are not very meaningful as representations of low--energy boundaries.

 \end{document}
