\chapter{Crystallography of Martensitic Transformations}

\section{Introduction}

Martensitic transformations, and displacive transformations in general, remain the only way of making bulk nanostructured materials for engineering applications. The transformation is diffusionless and hence can occur at incredibly low temperatures ({\em e.g.} $< 4\,$K), or grow at the speed of sound in the material (a few thousand ${\rm m\,s^{-1}}$). There is no composition change when it grows to consume the parent phase.

There are, however, a number of peculiarities associated with martensitic transformations,  which are important in the design of alloys. The interface plane between austenite and martensite is known as the {\em habit plane} and measurements show that the crystallographic indices of that plane are irrational and strange (Table~\ref{tab:habit}).
\begin{table}[!ht]
\caption{Habit plane indices for martensite. The quoted indices are
approximate because the habit planes are in general irrational.}
\begin{center}
\begin{tabular}{lc}
\midrule
Composition /wt.\% & Approximate habit plane indices \\[3pt]\midrule
Low-alloy steels, Fe-28Ni & $\{1~1~1\}_\gamma$ \\
Plate martensite in Fe-1.8C & $\{2~9~5\}_\gamma$ \\ 
Fe-30Ni-0.3C & $\{3~15~10\}_\gamma$ \\
Fe-8Cr-1C & $\{2~5~2\}_\gamma$ \\
\midrule
\end{tabular}
\end{center}
\label{tab:habit}
\end{table}
Similarly, the orientation relationship between the parent $(\gamma$) and product $(\alpha')$.
The formation of martensite involves the coordinated movement of atoms. Whereas the most densely packed planes tend to be parallel, and close-packed directions within those planes tend to be parallel, there are subtle discrepancies which make the orientations irrational. A typical approximate orientation relationship, in this case with $\alpha'$ being body-centred cubic and $\gamma$ face-centred cubic, is given {\em approximately} by 
\begin{eqnarray}
 \{1~1~1\}_\gamma & \vert\,\vert & \{0~1~1\}_{\alpha'} \\ \nonumber
 \langle1~0~\overline 1\rangle_\gamma & \vert\,\vert &  \langle1~1~\overline 1 \rangle_{\alpha'}\end{eqnarray}
 
Another difficulty is that the $\alpha'/\gamma$ interface has to be glissile, that is, it must be able to move without diffusion. This requires that the interface must contain no more than one set of dislocations. Any more than one array of line vectors can lead to interference between the dislocations which renders the interface sessile. In crystallographic terms, this means that there must be at least one line fully coherent in the $\alpha'/\gamma$ interface. The transformation strain relating the two lattices must therefore at a minimum be an {\em invariant line strain}.


\section{The Shape Deformation}
     
 During martensitic transformation, the pattern in which the atoms in the parent crystal are arranged is {\it deformed} into that appropriate for martensite,   there must be a corresponding   change in the macroscopic shape of the crystal undergoing transformation. The dislocations responsible for the deformation are in the $\alpha'/\gamma$ interface, with Burgers vectors such that in addition to deformation they also cause the change in crystal structure. The deformation is such that an initially flat surface   becomes uniformly tilted about the line formed by the intersection of the interface plane with the free surface (Fig.~\ref{fig:IPS}). Any scratch traversing the transformed region is similarly deflected though the scratch remains connected at the
$\alpha'/\gamma$ interface. 
\begin{figure}[!ht]
  \begin{center}
    \begin{minipage}[t]{0.42\linewidth}
\raisebox{-4cm}{\includegraphics[width=0.98\linewidth]{Martensite/IPS.eps}} 
    \end{minipage}\hfill
    \begin{minipage}[t]{0.55\linewidth}
      \caption{The measured shape deformation due to martensitic transformation is an invariant-plane strain 
with a large shear component ($\simeq 0.22$) and a small dilatational
strain ($\simeq 0.03$) directed normal to the habit plane. 
}  \label{fig:IPS}
    \end{minipage}
  \end{center}  
\end{figure}

\section{Bain Strain}

We now consider the nature of the strain necessary to transform the 
f.c.c. lattice of $\gamma$ into the b.c.c. lattice of $\alpha'$. Such a
strain was proposed by Bain in 1924 and hence is known as the \lq Bain
Strain' (Fig.~\ref{fig:Bain}). There is a compression along the
$z$ axis and a uniform expansion along the $x$ and $y$ axes. 


\begin{figure}[!ht]
\centering
\subfigure(a){
\includegraphics[width=0.6\textwidth]{./Martensite/correspondence_1.eps}}\\
\subfigure(b){
\includegraphics[width=0.25\textwidth]{./Martensite/correspondence_2.eps}}
\subfigure(c){
\includegraphics[width=0.25\textwidth]{./Martensite/correspondence_3.eps}}
\caption{The Bain strain. Some of the iron atoms at the face-centres of the austenite cell are omitted for clarity. (a) The lattice correspondence for formation of martensite  from austenite, showing a single carbon atom in an octahedral interstice on the $[001]_\gamma$ axis. (b) Tetragonal unit cell outlined in austenite. (c) Lattice  deformation (compression along \textit{c}-axis) to form martensite with an appropriate \textit{c}/\textit{a} ratio.}
\label{fig:Bain}
\end{figure}


The Bain strain implies the following {\em rational} orientation relationship between
the parent and product lattices:
\begin{equation}
[0~0~1]_{fcc}  \vert\,\vert  [0~0~1]_{bcc} \qquad
[1~\overline 1~0]_{fcc}  \vert\,\vert  [1~0~0]_{bcc} \qquad
[1~1~0]_{fcc}  \vert\,\vert  [0~1~0]_{bcc}
\end{equation} but in fact, the experimentally observed orientation relationships are irrational. 

The second inconsistency with the Bain strain is that it is not the required invariant-line strain. 
In Fig.~\ref{fig:ellipse}a,b, the austenite is represented as a sphere which, as a
result of the Bain strain {\bf B}, is deformed into an ellipsoid of
revolution which represents the martensite. There are no lines which are
left undistorted or unrotated by {\bf B}. There are no lines in the
$(0~0~1)_{fcc}$ plane which are undistorted. The lines  $wx$ and $yz$
are undistorted but are rotated to the new
positions $w'x'$ and $y'z'$. Such  rotated lines are not invariant.
However, the combined effect of the Bain strain {\bf
B} and the rigid body  rotation {\bf R} is indeed an
invariant-line strain (ILS) because it brings $yz$ and $y'z'$ into
coincidence (Fig.~\ref{fig:ellipse}c). This is the reason why the observed irrational
orientation relationship  differs from that implied by the
Bain strain. The rotation required to  convert {\bf B} into an ILS
precisely corrects the Bain orientation into that which is observed
experimentally. 


\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.75\textwidth]{./Martensite/ellipsoids.eps}
\caption{(a) and (b) show the effect of the Bain strain on austenite, which when
{undeformed} is represented as a sphere of diameter \textit{wx}$\,{=}\,$\textit{yz} in
three-dimensions. The strain transforms it into an ellipsoid of
revolution. (c) Shows the ILS obtained by combining
the Bain strain with a rigid body rotation through an angle $\theta$. 
{\bf a}$_{\sf 1}$, {\bf a}$_{\sf 2}$ and {\bf a}$_{\sf 3}$ refer to [100]$_{\gamma}$, [010]$_{\gamma}$ and [001]$_{\gamma}$ axes respectively.}
\label{fig:ellipse}
\end{center}
\end{figure} 


 As can be seen from 
Fig.~\ref{fig:ellipse}c, there is no  rotation which can make {\bf B}
into an invariant-plane strain since this would require two
non-parallel invariant-lines. Thus, for the $fcc\rightarrow bcc$
transformation,  austenite cannot be transformed into martensite by a
homogeneous strain which is an invariant-plane strain. And yet, the observed shape
deformation leaves the habit plane undistorted and unrotated, {\em i.e.,} it is
an invariant-plane strain. 

The phenomenological theory of
martensite crystallography solves this remaining problem (Fig.~\ref{fig:theory}). The
Bain strain converts the structure of the parent phase into that of the
product phase. When combined with an appropriate rigid body rotation,
the net homogeneous lattice deformation ${\bf RB}$ is an
invariant-line strain (step $a$ to $c$ in Fig.~\ref{fig:theory}). However, the
observed shape deformation is an invariant-plane strain ${\bf P_1}$
(step $a$ to $b$ in Fig.~\ref{fig:theory}), but this gives the wrong crystal structure.
If  a second homogeneous shear ${\bf P_2}$ is combined with
${\bf P_1}$ (step $b$ to
$c$), then the correct structure is obtained but the wrong shape since 
$${\bf P_1 P_2 = RB}.$$
These discrepancies are all resolved if the shape changing effect
of ${\bf P_2}$ is cancelled macroscopically by an inhomogeneous
lattice-invariant deformation, which may be slip or twinning as
illustrated in Fig.~\ref{fig:theory}. 

The theory explains all the observed
features of the martensite crystallography. The orientation
relationship is predicted by deducing the rotation needed to change
the Bain strain into  an invariant-line
strain. The habit plane does not have rational indices because the
amount of lattice-invariant deformation needed to recover the correct
the macroscopic shape is not usually rational. The theory predicts a
substructure in plates of martensite (either twins or slip steps) as is
observed experimentally. The transformation goes to all the trouble of
ensuring that the shape deformation is macroscopically an
invariant-plane strain because this reduces the strain energy when
compared with the case where the shape deformation might be an
invariant-line strain. 

\
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.65\textwidth]{./Martensite/martensite_6.eps}
\caption{The phenomenological
theory of martensite crystallography.}
\label{fig:theory}
\end{center}
\end{figure} 


