∎

11institutetext: John M. Ball 22institutetext: Heriot-Watt University and
Maxwell Institute for Mathematical Sciences, Edinburgh. 22email:
<EMAIL_ADDRESS>
# Slip and twinning in Bravais lattices

John M. Ball

###### Abstract

A unified treatment of slip and twinning in Bravais lattices is given,
focussing on the case of cubic symmetry, and using the Ericksen energy well
formulation, so that interfaces correspond to rank-one connections between the
infinitely many crystallographically equivalent energy wells. Twins are
defined to be such rank-one connections involving a nontrivial reflection of
the lattice across some plane. The slips and twins minimizing shear magnitude
for cubic lattices are rigorously calculated, and the conjugates of these and
other slips analyzed. It is observed that all rank-one connections between the
energy wells for the dual of a Bravais lattice can be obtained explicitly from
those for the original lattice, so that in particular the rank-one connections
for fcc can be obtained explicitly from those for bcc.

###### Keywords:

slip twinning rank-one connections

###### MSC:

74N15 74G65

††journal: Journal of Elasticity

In memoriam Jerry Ericksen, from whom I learnt so much.

##  1 Introduction

This paper gives a unified treatment of slip and twinning in Bravais lattices,
focussing on the case of cubic symmetry, and using the Ericksen energy well
formulation, so that interfaces correspond to rank-one connections between the
infinitely many crystallographically equivalent energy wells. Inevitably there
is a considerable overlap with other such treatments, most notably with that
of Pitteri & Zanzotto pitterizanzotto03  (especially Chapter 8).

A main contribution is that we calculate rigorously the slips and Type 1/Type
2 twins ${\mathbf{F}}={\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}$ that
minimize the shear magnitude $|{\mathbf{a}}|$ for cubic lattices, these
corresponding to well-known statements in the materials science literature.
This reduces to the minimization of products $G({\mathbf{p}})H({\mathbf{q}})$
of quadratic forms, where ${\mathbf{p}},{\mathbf{q}}\in{\mathbb{Z}}^{3}$ are
such that ${\mathbf{p}}\cdot{\mathbf{q}}=0$ for slip and
${\mathbf{p}}\cdot{\mathbf{q}}=2$ for Type 1/Type 2 twins. The principle of
minimizing $|{\mathbf{a}}|$ as a means of selecting preferred slips and twins
has been proposed for slips by Chalmers & Martius chalmers1952slip  and for
twins by Kiho kiho1954 ; kiho1958  and Jaswon & Dove jaswondove1957 . In fact
the Type 1/Type 2 twins minimize the shear magnitude among all rank-one
connections between the energy wells. We also calculate the conjugates to the
minimizing slips, together with certain other slips for bcc, showing that they
are all twins. Further, we determine all the slips for cubic lattices whose
conjugates are also slips.

An apparently new observation is that all rank-one connections for the dual of
a Bravais lattice can be obtained explicitly from those for the original
lattice. In particular the rank-one connections for fcc lattices can be
obtained explicitly in terms of those for bcc lattices (and vice versa).

The plan of the paper is as follows. In Section 2 we review standard material
on Bravais lattices. Here and throughout the paper we use a matrix
formulation, so that the lattice vectors
${\mathbf{b}}_{1},{\mathbf{b}}_{2},{\mathbf{b}}_{3}$ are written as the
columns of a matrix ${\mathbf{B}}$; at least for the author this makes
calculations easier to follow. We define lattice planes and the dual lattice,
largely following Ashcroft & Mermin ashcroftmermin . Then in Section 3 we
recall the Ericksen energy-well picture, whereby the macroscopic free-energy
density inherits an infinite family of crystallographically equivalent energy
wells from those of the lattice free-energy via the Cauchy-Born rule.

In Section 4 we review standard results for the existence of rank-one
connections as well as establishing (Theorem 4.1) the correspondence between
rank-one connections for Bravais lattices and their duals. In Section 5 we
give a characterization of slip systems in terms of rank-one connections not
involving lattice rotation (Theorem 5.1), and determine the slips that
minimize the shear magnitude (Theorem 5.2) for the simple cubic, bcc and fcc
lattices. In Section 6 we give a definition of twins in terms of interfaces
separating nontrivially reflected lattices, and determine (Theorem 6.1) the
Type 1/Type 2 twins minimizing the shear magnitude for the simple cubic, bcc
and fcc lattices. In Section 7 we address the problem of determining general
rank-one connections of minimum shear magnitude, showing in particular
(Corollary 3) that the twins in Theorem 6.1 minimize the shear amplitude among
all rank-one connections. Finally in Section 8 we determine (Theorem 8.1) all
slips for cubic lattices whose conjugates are also slips, and show (Theorems
8.2, 8.3) that the conjugates of all the specific slips discussed in Section 5
are twins of various types.

This paper focusses on cubic Bravais lattices, but extensions of the results
to noncubic lattices, multilattices and crystals undergoing martensitic phase
transformations would be valuable.

##  2 Bravais lattices

###  2.1 Definitions and basic properties

Denote by ${\mathbf{e}}_{i}$ the unit vector in the $i^{th}\rm$ coordinate
direction of $\mathbb{R}^{3}$. A Bravais lattice is an infinite lattice of
points in $\mathbb{R}^{3}$ generated by linear combinations with integer
coefficients of three linearly independent basis vectors
${\mathbf{b}}_{1},{\mathbf{b}}_{2},{\mathbf{b}}_{3}$. Representing these basis
vectors with respect to the orthonormal basis $\\{{\mathbf{e}}_{i}\\}$, and
letting ${\mathbf{B}}=({\mathbf{b}}_{1},{\mathbf{b}}_{2},{\mathbf{b}}_{3})$ be
the matrix with columns ${\mathbf{b}}_{i}$, so that
$B_{ij}={\mathbf{b}}_{j}\cdot{\mathbf{e}}_{i}$, we write the corresponding
Bravais lattice as

$\displaystyle{\mathcal{L}}({\mathbf{B}})$ $\displaystyle=$
$\displaystyle\\{m_{1}{\mathbf{b}}_{1}+m_{2}{\mathbf{b}}_{2}+m_{3}{\mathbf{b}}_{3}:m_{i}\in{\mathbb{Z}}\\}$
$\displaystyle=$
$\displaystyle\\{{\mathbf{B}}{\mathbf{m}}:{\mathbf{m}}\in{\mathbb{Z}}^{3}\\},$

where ${\mathbb{Z}}$ denotes the set of integers. All lattice sites
${\mathbf{p}}\in{\mathcal{L}}({\mathbf{B}})$ are equivalent, i.e.

${\mathbf{p}}+{\mathcal{L}}({\mathbf{B}})={\mathcal{L}}({\mathbf{B}}).$

We denote by $\mathbb{R}^{3\times 3}$ the space of real $3\times 3$ matrices,
by ${\mathbb{Z}}^{3\times 3}$ the set of $3\times 3$ matrices with integer
entries, and

$\displaystyle GL(3,\mathbb{R})$
$\displaystyle=\\{{\mathbf{F}}\in\mathbb{R}^{3\times 3}:\det{\mathbf{F}}\neq
0\\},$ $\displaystyle GL^{+}(3,\mathbb{R})$
$\displaystyle=\\{{\mathbf{F}}\in\mathbb{R}^{3\times
3}:\det{\mathbf{F}}>0\\},$ $\displaystyle GL(3,\mathbb{Z})$
$\displaystyle=\\{{\bm{\mu}\in{\mathbb{Z}}^{3\times 3}}:\det{\bm{\mu}}=\pm
1\\},$ $\displaystyle GL^{+}(3,\mathbb{Z})$
$\displaystyle=\\{{\bm{\mu}\in{\mathbb{Z}}^{3\times
3}}:\det{\bm{\mu}}=1\\}=SL(3,\mathbb{Z}),$ $\displaystyle{\rm O}(3)$
$\displaystyle=\\{{\mathbf{Q}}\in\mathbb{R}^{3\times
3}:{\mathbf{Q}}^{T}{\mathbf{Q}}={\mathbf{1}}\\},$ $\displaystyle{\rm SO}(3)$
$\displaystyle=\\{{\mathbf{Q}}\in{\rm O}(3):\det{\mathbf{Q}}=1\\}.$

For ${\mathbf{F}}\in\mathbb{R}^{3\times 3}$ we denote the Euclidean norm of
${\mathbf{F}}$ by $|{\mathbf{F}}|:=(\rm
tr\,{\mathbf{F}}^{T}{\mathbf{F}})^{\frac{1}{2}}$.

The following standard theorem (see, for example, Ericksen77
,(pitterizanzotto03, , Proposition 3.1)) characterizes the sets of basis
vectors that are equivalent in the sense that they generate the same Bravais
lattice.

###### Theorem 2.1

${\mathcal{L}}({\mathbf{B}})={\mathcal{L}}(\mathbf{C})$ if and only if

$\mathbf{C}={\mathbf{B}}{\bm{\mu}},\mbox{ for some {\boldmath$\mu$}}\in
GL(3,\mathbb{Z}).$

###### Proof

Let
${\mathbf{B}}=({\mathbf{b}}_{1},{\mathbf{b}}_{2},{\mathbf{b}}_{3}),\;\mathbf{C}=({\mathbf{c}}_{1},{\mathbf{c}}_{2},{\mathbf{c}}_{3})$.
If ${\mathcal{L}}({\mathbf{B}})={\mathcal{L}}({\bf C})$ then
${\mathbf{b}}_{i}=\mu_{ji}{\bf c}_{j}$ for some
$\bm{\mu}=(\mu_{ij})\in{\mathbb{Z}}^{3\times 3}$, so that ${\mathbf{B}}={\bf
C}{\bm{\mu}}$. Similarly ${\bf C}={\mathbf{B}}{\bm{\mu}}^{\prime}$ for some
${\bm{\mu}}^{\prime}\in{\mathbb{Z}}^{3\times 3}$. So
${\bm{\mu}}^{\prime}={\bm{\mu}}^{-1}$ and $\bm{\mu}\in GL(3,{\mathbb{Z}})$.

Conversely, if ${\mathbf{B}}={\bf C}{\bm{\mu}}$ then
${\mathbf{b}}_{i}={\mu}_{ji}{\bf c}_{j}$ and so
${\mathcal{L}}({\mathbf{B}})\subset{\mathcal{L}}(\bf C)$. Similarly
${\mathcal{L}}(\bf C)\subset{\mathcal{L}}({\mathbf{B}})$. ∎

###### Corollary 1

If ${\bf F}\in GL(3,\mathbb{R})$, then ${\mathcal{L}}({\bf
F}{\mathbf{B}})={\mathcal{L}}({\mathbf{B}})$ if and only if

${\bf F}={\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1}\mbox{ for some }{\bm{\mu}}\in
GL(3,{\mathbb{Z}}).$

The point group $P({\mathbf{B}})$ is

$\displaystyle P({\mathbf{B}})$ $\displaystyle=$
$\displaystyle\\{{\mathbf{Q}}\in{\rm
O}(3):\mathcal{L}({\mathbf{Q}}{\mathbf{B}})=\mathcal{L}({\mathbf{B}})\\}$ (1)
$\displaystyle=$ $\displaystyle\\{{\mathbf{Q}}\in{\rm
O}(3):{\mathbf{Q}}={\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1}\text{ for some
}{\bm{\mu}}\in GL(3,\mathbb{Z})\\}.$

###  2.2 Lattice planes and the dual lattice

A lattice plane is a plane
$\Pi({\mathbf{n}})=\\{{\mathbf{x}}\in\mathbb{R}^{3}:{\mathbf{x}}\cdot{\mathbf{n}}=k\\}$
with unit normal ${\mathbf{n}}$ such that
$\Pi({\mathbf{n}})\cap{\mathcal{L}}({\mathbf{B}})$ contains 3 non-collinear
points. Equivalently, $\Pi({\mathbf{n}})\cap{\mathcal{L}}({\mathbf{B}})$ is a
translate of a 2D Bravais lattice of the form

$\\{r_{1}{\mathbf{m}}_{1}+r_{2}{\mathbf{m}}_{2}:r_{1},r_{2}\in{\mathbb{Z}}\\},$

where ${\mathbf{m}}_{1},{\mathbf{m}}_{2}\in{\mathcal{L}}({\mathbf{B}})$ and
${\mathbf{m}}_{1},{\mathbf{m}}_{2}$ are linearly independent. (Taking without
loss of generality $k=0$ this can be proved by first choosing
${\mathbf{m}}_{1}$ to be a nonzero vector in
$\Pi({\mathbf{n}})\cap{\mathcal{L}}({\mathbf{B}})$ of minimum length, and then
${\mathbf{m}}_{2}$ a nonzero vector in
$\Pi({\mathbf{n}})\cap{\mathcal{L}}({\mathbf{B}})$ not parallel to
${\mathbf{m}}_{1}$ and of minimum length.)

Then, following Ashcroft & Mermin ashcroftmermin , there exists
${\mathbf{p}}\in{\mathcal{L}}({\mathbf{B}})$ with
${\mathbf{p}}\cdot{\mathbf{n}}>0$ such that

${\mathcal{L}}({\mathbf{B}})=\bigcup_{r\in{\mathbb{Z}}}\left(\left({\mathcal{L}}\left({\mathbf{B}}\right)\cap\Pi\left({\mathbf{n}}\right)\right)+r{\mathbf{p}}\right),$

so that ${\mathcal{L}}({\mathbf{B}})$ is the union of its intersection with a
family of planes parallel to $\Pi({\mathbf{n}})$ with interplane spacing
$d={\mathbf{p}}\cdot{\mathbf{n}}$.

The dual (or reciprocal) lattice of ${\mathcal{L}}({\mathbf{B}})$ is the set

$\\{{\mathbf{k}}\in\mathbb{R}^{3}:{\mathbf{k}}\cdot{\mathbf{p}}\in{\mathbb{Z}}\text{
for all }{\mathbf{p}}\in{\mathcal{L}}({\mathbf{B}})\\}.$

The vector ${\mathbf{k}}$ belongs to the dual lattice if and only if
${\mathbf{k}}\cdot{\mathbf{B}}{\mathbf{e}}_{i}\in{\mathbb{Z}}$ for each $i$,
which holds if and only if
${\mathbf{k}}=\sum_{i=1}^{3}r_{i}{\mathbf{B}}^{-T}{\mathbf{e}}_{i}$ for
${\mathbf{r}}\in{\mathbb{Z}}^{3}$. Hence the dual lattice is the Bravais
lattice $\mathcal{L}({\mathbf{B}}^{-T})$.

######  Theorem 2.2 (see (ashcroftmermin, , p90))

$\Pi({\mathbf{n}})=\\{{\mathbf{x}}:{\mathbf{x}}\cdot{\mathbf{n}}=0\\}$ is a
lattice plane if and only if there exists
${\mathbf{k}}\in{\mathcal{L}}({\mathbf{B}}^{-T})\setminus\\{0\\}$ with
${\mathbf{k}}$ parallel to ${\mathbf{n}}$, and then $d^{-1}$ is the minimum
length of such a vector ${\mathbf{k}}$.

###### Proof

We give a slightly different proof to that in ashcroftmermin  for the
convenience of the reader. If $\Pi({\mathbf{n}})$ is a lattice plane then
$d^{-1}{\mathbf{n}}\cdot{\mathbf{b}}\in{\mathbb{Z}}$ for all
${\mathbf{b}}\in{\mathcal{L}}({\mathbf{B}})$, so that
$d^{-1}{\mathbf{n}}\in{\mathcal{L}}({\mathbf{B}}^{-T})$, and there is no
shorter such dual lattice vector.

Conversely, if
${\mathbf{k}}\in{\mathcal{L}}({\mathbf{B}}^{-T})\setminus\\{{\mathbf{0}}\\}$
with ${\mathbf{k}}$ parallel to ${\mathbf{n}}$ then
${\mathbf{k}}\cdot{\mathbf{B}}{\mathbf{e}}_{i}=n_{i}\in{\mathbb{Z}}$ for each
$i$, and hence
${\mathbf{B}}^{T}{\mathbf{k}}=\sum_{i=1}^{3}n_{i}{\mathbf{e}}_{i}:={\mathbf{n}}\in{\mathbb{Z}}^{3}$.
Pick
${\mathbf{m}},{\mathbf{m}}^{\prime}\in{\mathbb{Z}}^{3}\setminus\\{{\mathbf{0}}\\}$
with ${\mathbf{m}}\cdot{\mathbf{n}}={\mathbf{m}}^{\prime}\cdot{\mathbf{n}}=0$
and ${\mathbf{m}},{\mathbf{m}}^{\prime}$ linearly independent, which is
possible because the vectors ${\mathbf{n}}\wedge{\mathbf{e}}_{i}$ are not all
parallel. Then
${\mathbf{B}}^{T}{\mathbf{k}}\cdot{\mathbf{m}}={\mathbf{B}}^{T}{\mathbf{k}}\cdot{\mathbf{m}}^{\prime}=0$,
so that
${\mathbf{k}}\cdot{\mathbf{B}}{\mathbf{m}}={\mathbf{k}}\cdot{\mathbf{B}}{\mathbf{m}}^{\prime}=0$
and $\Pi({\mathbf{n}})$ is a lattice plane.∎

###### Remark 1

Suppose ${\mathbf{k}}={\mathbf{B}}^{-T}{\mathbf{q}}$ is parallel to
${\mathbf{n}}$, where
${\mathbf{q}}=(q_{1},q_{2},q_{3})\in{\mathbb{Z}}^{3}\setminus\\{{\mathbf{0}}\\}$.
Then the formula for $d$ can be rewritten as

$\displaystyle
d^{-1}=\frac{|{\mathbf{B}}^{-T}{\mathbf{q}}|}{\text{gcd}(q_{1},q_{2},q_{3})},$
(2)

where $\text{gcd}(q_{1},q_{2},q_{3})$ is the greatest common divisor of the
$q_{i}$. Indeed

$\displaystyle\bar{\mathbf{q}}:=\frac{{\mathbf{q}}}{\text{gcd}(q_{1},q_{2},q_{3})}\in{\mathbb{Z}}^{3}$

and the $\bar{q}_{i}$ have no common factor. If
$\hat{\mathbf{q}}\in{\mathbb{Z}}^{3}$ is parallel to ${\mathbf{q}}$ we have
that $n\hat{\mathbf{q}}=m\bar{\mathbf{q}}$ for coprime integers $n\geqslant 1$
and $m$. Thus $n=1$ and so $|\bar{\mathbf{q}}|\leqslant|\hat{\mathbf{q}}|$ and
thus
$|{\mathbf{B}}^{-T}\hat{\mathbf{q}}|\geqslant|{\mathbf{B}}^{-T}\bar{\mathbf{q}}|$.

###  2.3 Cubic lattices

As examples we consider
(i) the simple cubic lattice, for which ${\mathbf{B}}={\mathbf{B}}_{\rm c}$,
where

$\displaystyle{\mathbf{B}}_{\rm c}={\mathbf{Q}}\in{\rm O}(3).$ (3)

Equivalently

$\displaystyle{\mathbf{B}}_{\rm c}^{T}{\mathbf{B}}_{\rm c}={\mathbf{1}}.$ (4)

Figure 1: (a) face-centred cubic lattice, (b) body-centred cubic lattice

(ii) the face-centred cubic (fcc) lattice (see Fig. 1 (a)) with the indicated
basis vectors, for which ${\mathbf{B}}={\mathbf{B}}_{\rm fcc}$, where

$\displaystyle{\mathbf{B}}_{\rm
fcc}=\frac{1}{2}{\mathbf{Q}}\left(\begin{array}[]{rrr}1&-1&\,\,0\\\
1&1&\,\,1\\\ 0&0&\,\,1\end{array}\right),{\mathbf{Q}}\in{\rm O}(3).$ (8)

Equivalently

$\displaystyle{\mathbf{B}}_{\rm fcc}^{T}{\mathbf{B}}_{\rm
fcc}=\frac{1}{4}\left(\begin{array}[]{ccc}2&0&1\\\ 0&2&1\\\
1&1&2\end{array}\right),$ (12)

(iii) the body-centred cubic (bcc) lattice (see Fig. 1 (b)), for which one
could take the basis vectors $\hat{\mathbf{b}}_{i}$ shown, but the usual
choice is ${\mathbf{B}}={\mathbf{B}}_{\rm bcc}$, where

$\displaystyle{\mathbf{B}}_{\rm
bcc}=\frac{1}{2}{\mathbf{Q}}\left(\begin{array}[]{rrr}-1&1&1\\\ 1&-1&1\\\
1&1&-1\end{array}\right),{\mathbf{Q}}\in O(3),$ (16)

for which

$\displaystyle{\mathbf{B}}_{\rm bcc}^{T}{\mathbf{B}}_{\rm
bcc}=\frac{1}{4}\left(\begin{array}[]{rrr}3&-1&-1\\\ -1&3&-1\\\
-1&-1&3\end{array}\right).$ (20)

Below we always take ${\mathbf{Q}}={\mathbf{1}}$ for these lattices.

###### Remark 2

It follows from Theorem 2.1 that for fcc or bcc lattices it is not possible to
choose lattice vectors such that ${\mathbf{B}}\in{\mathbb{Z}}^{3\times 3}$.
Taking the case of fcc, for example, this would imply that

$\displaystyle\frac{1}{2}\left(\begin{array}[]{rrr}1&-1&\,\,0\\\ 1&1&\,\,1\\\
0&0&\,\,1\end{array}\right){\bm{\mu}}\in{\mathbb{Z}}^{3\times 3}$ (24)

for some ${\bm{\mu}}\in GL(3,{\mathbb{Z}})$, so that
$\frac{1}{2}\mu_{3j}\in{\mathbb{Z}}$ for each $j$, and hence $\mu_{3j}$ is
even, implying that $\det{\bm{\mu}}$ is even, a contradiction.

###### Remark 3

We have that ${\mathbf{B}}_{\rm fcc}^{T}{\mathbf{B}}_{\rm
bcc}=\frac{1}{2}{\bm{\omega}}^{-1}$, where

$\displaystyle{\bm{\omega}}=\left(\begin{array}[]{rrr}0&0&\,\,1\\\
0&-1&\,\,1\\\ 1&0&\,0\end{array}\right)\in GL^{+}(3,\mathbb{Z}).$ (28)

Hence ${\mathbf{B}}_{\rm fcc}^{-T}=2{\mathbf{B}}_{\rm bcc}\bm{\omega}$,
proving the well-known result that the dual lattice to fcc is 2bcc.

The simple cubic, bcc and fcc lattices all have the same point group $P^{48}$
consisting of the 48 matrices ${\mathbf{Q}}\in{\rm O}(3)$ mapping
${\mathbb{Z}}^{3}$ to itself (equivalently the matrices with a single entry
$\pm 1$ in each row and column). We write

$\displaystyle P^{24}=P^{48}\cap{\rm SO}(3)$ (29)

for the 24 such matrices ${\mathbf{Q}}$ with $\det{\mathbf{Q}}=1$.

##  3 The Ericksen energy well picture

Suppose that the free energy per unit volume of a crystalline material with
atoms at the points of the Bravais lattice $\mathcal{L}(\mathbf{C})$, where
$\mathbf{C}\in GL^{+}(3,\mathbb{R})$, is given by
$\varphi(\mathbf{C})\geqslant 0$. For simplicity we assume that the
temperature is constant, so that there are no phase transformations.

Natural requirements on $\varphi$ are
(i) (frame-indifference)

$\varphi({\mathbf{Q}}\mathbf{C})=\varphi(\mathbf{C})\text{ for all
}{\mathbf{Q}}\in{\rm SO}(3),$

(ii) (invariance with respect to equivalent lattices)

$\varphi(\mathbf{C}{\bm{{\bm{\mu}}}})=\varphi(\mathbf{C})\text{ for all
}{\bm{{\bm{\mu}}}}\in GL^{+}(3,\mathbb{Z}).$

We assume further that $\varphi(\mathbf{C})=0$ iff
$\mathbf{C}={\mathbf{Q}}{\mathbf{B}}{\bm{\mu}}$ for ${\mathbf{Q}}\in{\rm
SO}(3)$, ${\bm{\mu}}\in GL^{+}(3,\mathbb{Z})$, so that the minimum value zero
is attained only for the unstrained lattice.

As is standard, we use the Cauchy-Born rule to relate the macroscopic free-
energy density $\psi$ to $\varphi$, thus defining an elastic free energy

$I({\mathbf{y}})=\int_{\Omega}\psi(D{\mathbf{y}}({\mathbf{x}}))\,d{\mathbf{x}}$

for a deformation ${\mathbf{y}}:\Omega\to\mathbb{R}^{3}$, where
$\Omega\subset\mathbb{R}^{3}$ is an open reference domain.

Choosing a reference configuration in which the crystal lattice is
$\mathcal{L}({\mathbf{B}})$, where ${\mathbf{B}}\in GL^{+}(3,\mathbb{R})$, we
assume that

$\psi({\mathbf{A}})=\varphi({\mathbf{A}}{\mathbf{B}}),\mbox{ for
}{\mathbf{A}}\in GL^{+}(3,\mathbb{R}).$

Thus $\psi\geqslant 0$ inherits from $\varphi$ the invariances
(i) $\psi({\mathbf{Q}}{\mathbf{A}})=\psi({\mathbf{A}})$ for all
${\mathbf{Q}}\in{\rm SO}(3)$,
(ii)
$\psi({\mathbf{A}}{\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1})=\psi({\mathbf{A}})$
for all ${\bm{\mu}}\in GL^{+}(3,{\mathbb{Z}})$,
and $\psi$ has zero set

$\displaystyle\psi^{-1}(0)=\bigcup_{{\bm{\mu}}\in GL^{+}(3,\mathbb{Z})}{\rm
SO}(3){\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1}.$ (30)

The energy wells ${\rm SO}(3){\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1}$ are not
all distinct, because ${\rm SO}(3){\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1}={\rm
SO}(3){\mathbf{B}}\tilde{\bm{\mu}}{\mathbf{B}}^{-1}$ iff

${\mathbf{B}}{\bm{\mu}}\tilde{\bm{\mu}}^{-1}{\mathbf{B}}^{-1}\in
P({\mathbf{B}})\cap{\rm SO}(3).$

But since $P({\mathbf{B}})$ is finite and $GL^{+}(3,\mathbb{Z})$ is infinite,
there are infinitely many distinct energy wells.

##  4 Interfaces

###  4.1 Rank-one connections

We are interested in possible planar interfaces between distinct constant
gradients on the energy wells, that is in pairs of distinct matrices
${\mathbf{F}},{\mathbf{G}}\in\psi^{-1}(0)$ with

$\displaystyle{\mathbf{F}}-{\mathbf{G}}={\mathbf{a}}\otimes{\mathbf{n}},\;\;{\mathbf{a}},{\mathbf{n}}\in\mathbb{R}^{3},|{\mathbf{n}}|=1.$
(31)

We can assume that ${\mathbf{G}}={\mathbf{1}}$, so that we are interested in
the ${\bm{\mu}}\in GL^{+}(3,\mathbb{Z})$ with
${\mathbf{M}}={\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1}\not\in{\rm SO}(3)$ such
that

$\displaystyle{\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}={\mathbf{R}}{\mathbf{M}}\text{
for some
}{\mathbf{a}},{\mathbf{n}}\in\mathbb{R}^{3},|{\mathbf{n}}|=1,{\mathbf{R}}\in{\rm
SO}(3).$ (32)

Note that since ${\mathbf{M}}$ is independent of the scale of
${\mathcal{L}}({\mathbf{B}})$ (that is it is the same for
${\mathcal{L}}(t{\mathbf{B}})$ for any $t\neq 0$) the rank-one connections are
also independent of scale. If (32) holds then since $\det{\mathbf{M}}=1$ we
have that

$\displaystyle{\mathbf{a}}\cdot{\mathbf{n}}=0,$ (33)

and thus

$\displaystyle|{\mathbf{a}}|^{2}=|{\mathbf{M}}|^{2}-3.$ (34)

We denote by $0<\lambda_{1}\leqslant\lambda_{2}\leqslant\lambda_{3}$ the
eigenvalues of the positive definite symmetric matrix
${\mathbf{M}}^{T}{\mathbf{M}}$, which satisfy
$\lambda_{1}\lambda_{2}\lambda_{3}=1$, and by $\hat{\mathbf{e}}_{i}$ the
corresponding orthonormal eigenvectors, so that ${\mathbf{M}}^{T}{\mathbf{M}}$
has spectral decomposition

$\displaystyle{\mathbf{M}}^{T}{\mathbf{M}}=\lambda_{1}\hat{\mathbf{e}}_{1}\otimes\hat{\mathbf{e}}_{1}+\lambda_{2}\hat{\mathbf{e}}_{2}\otimes\hat{\mathbf{e}}_{2}+\lambda_{3}\hat{\mathbf{e}}_{3}\otimes\hat{\mathbf{e}}_{3}.$
(35)

A necessary and sufficient condition for (32) to hold (see e.g. (j32, , Prop.
4), (j56, , Theorem 2.1), Khachaturyan83 , (p37, , Lemma 1)) is that
$\lambda_{2}=1$, or (since ${\mathbf{M}}^{T}{\mathbf{M}}\neq{\mathbf{1}}$)
that ${\mathbf{M}}^{T}{\mathbf{M}}$ has an eigenvalue equal to one. Then
$0<\lambda_{1}<1=\lambda_{2}<\lambda_{3}=\lambda_{1}^{-1}$. An equivalent
condition is that

$\displaystyle|{\mathbf{M}}|^{2}=|{\mathbf{M}}^{-1}|^{2},$ (36)

which follows from the identity

$\displaystyle\det({\mathbf{M}}^{T}{\mathbf{M}}-{\mathbf{1}})=(\det{\mathbf{M}})^{2}-(\det{\mathbf{M}})^{2}|{\mathbf{M}}^{-1}|^{2}+|{\mathbf{M}}|^{2}-1.$
(37)

(This implies in particular, as observed in (pitterizanzotto03, , Remark 8.4),
that there is a rank-one connection to ${\rm SO}(3){\mathbf{M}}$ iff there is
a rank-one connection to ${\rm SO}(3){\mathbf{M}}^{-1}$, which can be verified
directly.)

There are then exactly two distinct such conjugate (or reciprocal) rank-one
connections

$\displaystyle{\mathbf{1}}+{\mathbf{a}}_{+}\otimes{\mathbf{n}}_{+}$
$\displaystyle=$ $\displaystyle{\mathbf{R}}_{+}{\mathbf{M}},$
$\displaystyle{\mathbf{1}}+{\mathbf{a}}_{-}\otimes{\mathbf{n}}_{-}$
$\displaystyle=$ $\displaystyle{\mathbf{R}}_{-}{\mathbf{M}}.$

These can be calculated explicitly by comparing the spectral decomposition
(35) with the relation

$\displaystyle{\mathbf{M}}^{T}{\mathbf{M}}=({\mathbf{1}}+{\mathbf{n}}\otimes{\mathbf{a}})({\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}})$
(38)

that follows from (32) (see e.g. (j32, , Prop. 4)). A straightforward
calculation then shows that (up to a change of sign)

$\displaystyle{\mathbf{n}}_{\pm}=\frac{1}{\sqrt{1+\lambda_{1}}}(\pm\sqrt{\lambda_{1}}\hat{\mathbf{e}}_{1}+\hat{\mathbf{e}}_{3}),\;\;{\mathbf{a}}_{\pm}=\frac{1-\lambda_{1}}{\sqrt{1+\lambda_{1}}}\left(\mp\frac{1}{\sqrt{\lambda_{1}}}\hat{\mathbf{e}}_{1}+\hat{\mathbf{e}}_{3}\right).$
(39)

From (34) or (39) we have that $|{\mathbf{a}}_{+}|=|{\mathbf{a}}_{-}|$. Also
$\displaystyle{\mathbf{n}}_{+}\cdot{\mathbf{n}}_{-}=\frac{1-\lambda_{1}}{1+\lambda_{1}}$,
so that the planes corresponding to the two rank-one connections are never
orthogonal.

It also follows from (39) that if we know one rank-one connection
${\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}={\mathbf{R}}{\mathbf{M}}$ then
the conjugate rank-one connection
${\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}=\tilde{\mathbf{R}}{\mathbf{M}}$
is given by

$\displaystyle\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}=\frac{1}{4+|{\mathbf{a}}|^{2}}(2{\mathbf{n}}-{\mathbf{a}})\otimes(2{\mathbf{a}}+|{\mathbf{a}}|^{2}{\mathbf{n}}).$
(40)

From (40) we have that

$\displaystyle\tilde{\mathbf{a}}\otimes\tilde{\mathbf{a}}=(\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}})(\tilde{\mathbf{n}}\otimes\tilde{\mathbf{a}})=\frac{|{\mathbf{a}}|^{2}}{4+|{\mathbf{a}}|^{2}}(2{\mathbf{n}}-{\mathbf{a}})\otimes(2{\mathbf{n}}-{\mathbf{a}}),$
(41)

from which, recalling from (33),(34) that ${\mathbf{a}}\cdot{\mathbf{n}}=0$
and $|\tilde{\mathbf{a}}|^{2}=|{\mathbf{a}}|^{2}$, we deduce that

$\displaystyle
2{\mathbf{n}}\otimes{\mathbf{n}}-{\mathbf{a}}\otimes{\mathbf{n}}=2\frac{\tilde{\mathbf{a}}}{|\tilde{\mathbf{a}}|}\otimes\frac{\tilde{\mathbf{a}}}{|\tilde{\mathbf{a}}|}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}.$
(42)

Hence we obtain the relations

$\displaystyle(-{\mathbf{1}}+2{\mathbf{n}}\otimes{\mathbf{n}})({\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}})$
$\displaystyle=$
$\displaystyle\left(-{\mathbf{1}}+2\frac{\tilde{\mathbf{a}}}{|\tilde{\mathbf{a}}|}\otimes\frac{\tilde{\mathbf{a}}}{|\tilde{\mathbf{a}}|}\right)({\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}})$
(43)
$\displaystyle(-{\mathbf{1}}+2\tilde{\mathbf{n}}\otimes\tilde{\mathbf{n}})({\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}})$
$\displaystyle=$
$\displaystyle\left(-{\mathbf{1}}+2\frac{{\mathbf{a}}}{|{\mathbf{a}}|}\otimes\frac{{\mathbf{a}}}{|{\mathbf{a}}|}\right)({\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}),$
(44)

where (44) is obtained from (43) by interchanging ${\mathbf{a}},{\mathbf{n}}$
and $\tilde{\mathbf{a}},\tilde{\mathbf{n}}$. Thus
$\tilde{\mathbf{R}}{\mathbf{R}}^{T}$ can be expressed as the product of two
$180^{\circ}$ rotations in the two ways

$\displaystyle\tilde{\mathbf{R}}{\mathbf{R}}^{T}$ $\displaystyle=$
$\displaystyle(-{\mathbf{1}}+2\tilde{\mathbf{n}}\otimes\tilde{\mathbf{n}})\left(-{\mathbf{1}}+2\frac{{\mathbf{a}}}{|{\mathbf{a}}|}\otimes\frac{{\mathbf{a}}}{|{\mathbf{a}}|}\right)$
$\displaystyle=$
$\displaystyle\left(-{\mathbf{1}}+2\frac{\tilde{\mathbf{a}}}{|\tilde{\mathbf{a}}|}\otimes\frac{\tilde{\mathbf{a}}}{|\tilde{\mathbf{a}}|}\right)(-{\mathbf{1}}+2{\mathbf{n}}\otimes{\mathbf{n}}).$

We will see that there are always ${\bm{\mu}}$ such that
${\mathbf{M}}^{T}{\mathbf{M}}$ has an eigenvalue one. However this is not true
for general ${\bm{\mu}}$.

###  4.2 Correspondence between rank-one connections for Bravais lattices and
their duals

It turns out that the rank-one connections for the dual of a Bravais lattice
can be obtained explicitly in terms of the rank-one connections for the
original lattice (and vice versa). On account of Remark 3 this implies that
the rank-one connections for fcc can be obtained explicitly in terms of those
for bcc, which are a bit easier to calculate due to the more symmetric form of
the matrix ${\mathbf{B}}_{\rm bcc}$.

Note that the energy wells for the dual lattice
${\mathcal{L}}({\mathbf{B}}^{-T})$ of the Bravais lattice
${\mathcal{L}}({\mathbf{B}})$ are given by ${\rm
SO}(3){\mathbf{B}}^{-T}{\bm{\mu}}^{T}{\mathbf{B}}^{T}$ for ${\bm{\mu}}\in
GL^{+}(3,{\mathbb{Z}})$.

###### Theorem 4.1

The rank-one connection

$\displaystyle{\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}={\mathbf{R}}{\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1},$
(46)

for ${\mathbf{R}}\in{\rm SO}(3)$ and ${\bm{\mu}}\in GL^{+}(3,{\mathbb{Z}})$
holds iff

$\displaystyle{\mathbf{1}}+{\mathbf{R}}^{T}{\mathbf{n}}\otimes{\mathbf{R}}^{T}{\mathbf{a}}={\mathbf{R}}^{T}{\mathbf{B}}^{-T}{\bm{\mu}}^{T}{\mathbf{B}}^{T}.$
(47)

Furthermore, if ${\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}$ is
the conjugate of ${\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}$ then
${\mathbf{1}}+{\mathbf{R}}^{T}\tilde{\mathbf{n}}\otimes{\mathbf{R}}^{T}\tilde{\mathbf{a}}$
is the conjugate of
${\mathbf{1}}+{\mathbf{R}}^{T}{\mathbf{n}}\otimes{\mathbf{R}}^{T}{\mathbf{a}}$.

###### Proof

Taking the transpose of (46) and pre- and post-multiplying by
${\mathbf{R}}^{T}$ and ${\mathbf{R}}$ respectively, we see that (46) and (47)
are equivalent.

The statement about the conjugates follows from (40), or more directly by
noting that

$\displaystyle({\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}})({\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}})^{-1}={\mathbf{R}}\in{\rm
SO}(3),$ (48)

where ${\mathbf{R}}\neq{\mathbf{1}}$, implies that

$\displaystyle({\mathbf{1}}+{\mathbf{R}}^{T}{\mathbf{n}}\otimes{\mathbf{R}}^{T}{\mathbf{a}})^{-1}({\mathbf{1}}+{\mathbf{R}}^{T}\tilde{\mathbf{n}}\otimes{\mathbf{R}}^{T}\tilde{\mathbf{a}})={\mathbf{R}}^{T}{\mathbf{R}}^{T}{\mathbf{R}}\neq{\mathbf{1}},$
(49)

as required. ∎

From Remark 3 we thus obtain

###### Corollary 2

The rank-one connection

$\displaystyle{\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}={\mathbf{R}}{\mathbf{B}}_{\rm
fcc}{\bm{\mu}}{\mathbf{B}}_{\rm fcc}^{-1}$ (50)

for ${\mathbf{R}}\in{\rm SO}(3)$ and ${\bm{\mu}}\in GL^{+}(3,{\mathbb{Z}})$
holds iff

$\displaystyle{\mathbf{1}}+{\mathbf{R}}^{T}{\mathbf{n}}\otimes{\mathbf{R}}^{T}{\mathbf{a}}={\mathbf{R}}^{T}{\mathbf{B}}_{\rm
bcc}\tilde{\bm{\mu}}{\mathbf{B}}_{\rm bcc}^{-1},$ (51)

where $\tilde{\bm{\mu}}={\bm{\omega}}{\bm{\mu}}^{T}{\bm{\omega}}^{-1}$ and
${\bm{\omega}}$ is given by (28).

##  5 Slip

###  5.1 Slip systems and lattice invariant shears

A slip system $(\Pi({\mathbf{n}}),{\mathbf{b}})$ consists of a lattice plane
$\Pi({\mathbf{n}})$ with unit normal ${\mathbf{n}}$ and interplane spacing
$d$, and a nonzero lattice vector ${\mathbf{b}}\in{\mathcal{L}}({\mathbf{B}})$
(the Burgers vector) with ${\mathbf{b}}\cdot{\mathbf{n}}=0$. If
$(\Pi({\mathbf{n}}),{\mathbf{b}})$ is a slip system so is
$(\Pi({\mathbf{n}}),-{\mathbf{b}})$, which corresponds to an opposite
direction of shear on the same lattice plane, and when counting slip systems
we identify $(\Pi({\mathbf{n}}),{\mathbf{b}})$ with
$(\Pi({\mathbf{n}}),-{\mathbf{b}})$.

Equivalently

${\mathcal{L}}({\mathbf{B}})=({\mathcal{L}}({\mathbf{B}})\cap\\{{\mathbf{x}}\cdot{\mathbf{n}}\leqslant
0\\})\cup(({\mathcal{L}}({\mathbf{B}})\cap\\{{\mathbf{x}}\cdot{\mathbf{n}}>0\\})+{\mathbf{b}}),$

so that if the part of ${\mathcal{L}}({\mathbf{B}})$ in the half-space
$\\{{\mathbf{x}}\cdot{\mathbf{n}}>0\\}$ is rigidly displaced by ${\mathbf{b}}$
then ${\mathcal{L}}({\mathbf{B}})$ is restored.

###### Theorem 5.1

The following are equivalent:
$\rm(i)$
${\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}={\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1},$
where ${\mathbf{a}},{\mathbf{n}}\in\mathbb{R}^{3}$, $|{\mathbf{n}}|=1$, and
${\bm{\mu}}\in GL^{+}(3,{\mathbb{Z}})$.
$\rm(ii)$ ${\bm{\mu}}={\mathbf{1}}+{\mathbf{p}}\otimes{\mathbf{q}}$, where
${\mathbf{p}},{\mathbf{q}}\in{\mathbb{Z}}^{3}\setminus\\{{\mathbf{0}}\\}$,
${\mathbf{p}}\cdot{\mathbf{q}}=0$, and

$\displaystyle{\mathbf{n}}=\tau\frac{{\mathbf{B}}^{-T}{\mathbf{q}}}{|{\mathbf{B}}^{-T}{\mathbf{q}}|},\;\;{\mathbf{a}}=\tau{\mathbf{B}}{\mathbf{p}}|{\mathbf{B}}^{-T}{\mathbf{q}}|,$
(52)

where $\tau=\pm 1$.
$\rm(iii)$ $(\Pi({\mathbf{n}}),{\mathbf{b}})$ is a slip system with interplane
spacing $d$ and Burgers vector ${\mathbf{b}}=d{\mathbf{a}}$.

The proof uses the following lemma.

###### Lemma 1

The matrix ${\mathbf{c}}\otimes{\mathbf{d}}\in{\mathbb{Z}}^{3\times 3}$ for
some ${\mathbf{c}},{\mathbf{d}}\in\mathbb{R}^{3}$ if and only if
${\mathbf{c}}\otimes{\mathbf{d}}={\mathbf{p}}\otimes{\mathbf{q}}$ for some
${\mathbf{p}},{\mathbf{q}}\in{\mathbb{Z}}^{3}$.

###### Proof

We only have to prove the necessity, and may assume
${\mathbf{c}}\otimes{\mathbf{d}}$ is a nonzero matrix of integers. Thus some
$c_{i}\neq 0$ and $c_{i}d_{1},c_{i}d_{2},c_{i}d_{3}$ are integers, so that
$\displaystyle{\mathbf{d}}=\frac{1}{c_{i}}{\mathbf{m}}$ for some
${\mathbf{m}}\in{\mathbb{Z}}^{3}$. Similarly
$\displaystyle{\mathbf{c}}=\frac{1}{d_{j}}{\mathbf{r}}$ for some $j$ and
${\mathbf{r}}\in{\mathbb{Z}}^{3}$. Hence
$\displaystyle{\mathbf{c}}\otimes{\mathbf{d}}=\frac{1}{c_{i}d_{j}}{\mathbf{r}}\otimes{\mathbf{m}}$.
We can write
${\mathbf{r}}=k{\mathbf{r}}^{\prime},{\mathbf{m}}=l{\mathbf{m}}^{\prime}$,
where $k,l$ are integers and neither
$(r_{1}^{\prime},r_{2}^{\prime},r_{3}^{\prime})$ nor
$(m_{1}^{\prime},m_{2}^{\prime},m_{3}^{\prime})$ have common factors. Thus

${\mathbf{c}}\otimes{\mathbf{d}}=\frac{r}{s}{\mathbf{r}}^{\prime}\otimes{\mathbf{m}}^{\prime},$

where $\displaystyle\frac{r}{s}$ is a rational number expressed in its lowest
terms. Suppose $s\neq\pm 1$, so that $s$ has a prime factor $S$. For some
$r_{i}^{\prime}$, $S$ does not divide $r_{i}^{\prime}$. Hence $S$ divides
$m_{j}^{\prime}$ for all $j$, a contradiction. Hence $s=\pm 1$, giving the
result.∎

Proof of Theorem 5.1
(i)$\Rightarrow$(ii). (i) implies that
${\mathbf{1}}+{\mathbf{B}}^{-1}{\mathbf{a}}\otimes{\mathbf{B}}^{T}{\mathbf{n}}={\bm{\mu}}$.
Hence, by Lemma 1,
${\mathbf{a}}\otimes{\mathbf{n}}={\mathbf{B}}{\mathbf{p}}\otimes{\mathbf{B}}^{-T}{\mathbf{q}}$
for ${\mathbf{p}},{\mathbf{q}}\in{\mathbb{Z}}^{3}$, and
${\mathbf{a}}\cdot{\mathbf{n}}={\mathbf{p}}\cdot{\mathbf{q}}=0$, giving (ii).

(ii)$\Rightarrow$(iii). We can suppose that the $q_{i}$ have no common factor,
so that by Remark 1,
$\displaystyle{\mathbf{n}}=\tau\frac{{\mathbf{B}}^{-T}{\mathbf{q}}}{|{\mathbf{B}}^{-T}{\mathbf{q}}|}$
is the normal to a lattice plane $\Pi({\mathbf{n}})$ with interplane spacing
$\displaystyle d=\frac{1}{|{\mathbf{B}}^{-T}{\mathbf{q}}|}$. Also
${\mathbf{b}}=\tau{\mathbf{B}}{\mathbf{p}}\in{\mathcal{L}}({\mathbf{B}})$ with
${\mathbf{b}}\cdot{\mathbf{n}}=0$, and so ${\mathbf{b}}=d{\mathbf{a}}$, giving
(iii).

(iii)$\Rightarrow$(i). By Theorem 2.2 and Remark 1 we know that
$\displaystyle{\mathbf{n}}=\frac{{\mathbf{B}}^{-T}{\mathbf{q}}}{|{\mathbf{B}}^{-T}{\mathbf{q}}|},d=\frac{1}{|{\mathbf{B}}^{-T}{\mathbf{q}}|}$,
for some ${\mathbf{q}}\in{\mathbb{Z}}^{3}\setminus\\{{\mathbf{0}}\\}$. Also
${\mathbf{b}}={\mathbf{B}}{\mathbf{p}}$ for some
${\mathbf{p}}\in{\mathbb{Z}}^{3}\setminus\\{{\mathbf{0}}\\}$, where
${\mathbf{b}}\cdot{\mathbf{n}}=d{\mathbf{B}}{\mathbf{p}}\cdot{\mathbf{B}}^{-T}{\mathbf{q}}=d{\mathbf{p}}\cdot{\mathbf{q}}=0$.
Then

${\mathbf{1}}+d^{-1}{\mathbf{b}}\otimes{\mathbf{n}}={\mathbf{1}}+{\mathbf{B}}{\mathbf{p}}\otimes{\mathbf{B}}^{-T}{\mathbf{q}}={\mathbf{B}}({\mathbf{1}}+{\mathbf{p}}\otimes{\mathbf{q}}){\mathbf{B}}^{-1},$

giving (i).∎

Thus a slip system corresponds to a lattice invariant shear giving a rank-one
connection between ${\rm SO}(3)$ and ${\rm
SO}(3){\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1}$ without lattice rotation. The
shear can correspond to a shear band, or in the case of rigid slip to a
‘microshear’ over just one pair of adjacent lattice planes. Note that Theorem
5.1 shows that the slip systems for any Bravais lattice are in 1-1
correspondence with those for the simple cubic lattice (having deformation
gradient ${\mathbf{1}}+{\mathbf{p}}\otimes{\mathbf{q}}$). For the use of such
rank-one connections in continuum theories of plasticity see e.g. Ortiz &
Repetto ortizrepetto1999 .

###  5.2 Slips of minimum shear magnitude

Since if ${\bm{\mu}}={\mathbf{1}}+{\mathbf{p}}\otimes{\mathbf{q}}$ ,

${\mathbf{M}}={\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1}={\mathbf{1}}+{\mathbf{B}}{\mathbf{p}}\otimes{\mathbf{B}}^{-T}{\mathbf{q}}$

we have that
$|{\mathbf{a}}|^{2}=d^{-2}|{\mathbf{b}}|^{2}=|{\mathbf{B}}{\mathbf{p}}|^{2}|{\mathbf{B}}^{-T}{\mathbf{q}}|^{2}.$
Hence, to minimize the shear magnitude $|{\mathbf{a}}|$ we need to minimize
$|{\mathbf{B}}{\mathbf{p}}|^{2}|{\mathbf{B}}^{-T}{\mathbf{q}}|^{2}$ subject to
${\mathbf{p}},{\mathbf{q}}\in{\mathbb{Z}}^{3}\setminus\\{0\\}$ with
${\mathbf{p}}\cdot{\mathbf{q}}=0$.

Minimizing the shear magnitude is the criterion proposed by Chalmers & Martius
chalmers1952slip , who give a heuristic justification in terms of energetics,
pointing out that the criterion of minimizing the magnitude of the Burgers
vector ${\mathbf{b}}$ does not determine the preferred slip planes. A review
of different possible criteria is given in cordier2002 .

In the following theorem we give the slips of minimum shear magnitude for the
simple cubic, bcc and fcc lattices, reproducing in our framework classical
results for the slip systems for these lattices (see, for example,
jackson2012handbook ). See below for further comments.

###### Theorem 5.2

The slips ${\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}$ minimizing
$|{\mathbf{a}}|$ are given by:
$\rm(a)$ Simple cubic:
${\mathbf{a}}\otimes{\mathbf{n}}=\pm{\mathbf{e}}_{i}\otimes{\mathbf{e}}_{j}$,
where $i\neq j$, for which $|{\mathbf{a}}|^{2}=1$.
$\rm(b)$ bcc:

$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}$
$\displaystyle=\frac{1}{2}({\mathbf{e}}_{1}+{\mathbf{e}}_{2}+{\mathbf{e}}_{3})\otimes({\mathbf{e}}_{j}-{\mathbf{e}}_{k}),$
$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}$
$\displaystyle=\frac{1}{2}({\mathbf{e}}_{j}+{\mathbf{e}}_{k}-{\mathbf{e}}_{i})\otimes({\mathbf{e}}_{j}-{\mathbf{e}}_{k}),$
$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}$
$\displaystyle=\pm\frac{1}{2}({\mathbf{e}}_{j}+{\mathbf{e}}_{k}-{\mathbf{e}}_{i})\otimes({\mathbf{e}}_{i}+{\mathbf{e}}_{k}),$

where $i,j,k\in\\{1,2,3\\}$ are distinct, for which
$|{\mathbf{a}}|^{2}=\frac{3}{2}$.
$\rm(c)$ fcc:

$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}$
$\displaystyle=\frac{1}{2}({\mathbf{e}}_{j}-{\mathbf{e}}_{k})\otimes({\mathbf{e}}_{1}+{\mathbf{e}}_{2}+{\mathbf{e}}_{3}),$
$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}$
$\displaystyle=\frac{1}{2}({\mathbf{e}}_{j}-{\mathbf{e}}_{k})\otimes({\mathbf{e}}_{j}+{\mathbf{e}}_{k}-{\mathbf{e}}_{i}),$
$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}$
$\displaystyle=\pm\frac{1}{2}({\mathbf{e}}_{i}+{\mathbf{e}}_{k})\otimes({\mathbf{e}}_{j}+{\mathbf{e}}_{k}-{\mathbf{e}}_{i}),$

where $i,j,k\in\\{1,2,3\\}$ are distinct, for which
$|{\mathbf{a}}|^{2}=\frac{3}{2}$.

###### Remark 4

Without loss of generality we may take for simple cubic
${\mathbf{a}}\otimes{\mathbf{n}}={\mathbf{e}}_{1}\otimes{\mathbf{e}}_{2}$, for
bcc
${\mathbf{a}}\otimes{\mathbf{n}}=\frac{1}{2}({\mathbf{e}}_{1}+{\mathbf{e}}_{2}+{\mathbf{e}}_{3})\otimes({\mathbf{e}}_{1}-{\mathbf{e}}_{2})$,
and for fcc
${\mathbf{a}}\otimes{\mathbf{n}}=\frac{1}{2}({\mathbf{e}}_{1}-{\mathbf{e}}_{2})\otimes({\mathbf{e}}_{1}+{\mathbf{e}}_{2}+{\mathbf{e}}_{3})$,
since all the other cases are given by
${\mathbf{P}}{\mathbf{a}}\otimes{\mathbf{n}}{\mathbf{P}}^{-1}$ for suitable
${\mathbf{P}}\in P^{24}$, where $P^{24}$ is defined in (29).

###### Proof

(a) Since ${\mathbf{B}}_{\rm c}={\mathbf{1}}$ we have
$|{\mathbf{a}}|^{2}=|{\mathbf{p}}|^{2}|{\mathbf{q}}|^{2}\geqslant 1$, and
$|{\mathbf{a}}|^{2}=1$ for $|{\mathbf{p}}|^{2}=|{\mathbf{q}}|^{2}=1$, giving
the result.

(b) For bcc

${\mathbf{B}}_{\rm bcc}=\frac{1}{2}\left(\begin{array}[]{rrr}-1&1&1\\\
1&-1&1\\\ 1&1&-1\end{array}\right),\;\;{\mathbf{B}}_{\rm
bcc}^{-1}=\left(\begin{array}[]{ccc}0&1&1\\\ 1&0&1\\\
1&1&0\end{array}\right),$

and we have to minimize $\frac{1}{4}G({\mathbf{p}})H({\mathbf{q}})$, subject
to ${\mathbf{p}},{\mathbf{q}}\in{\mathbb{Z}}^{3}\setminus\\{0\\}$ and
${\mathbf{p}}\cdot{\mathbf{q}}=0$, where

$\displaystyle
G({\mathbf{p}})=(p_{2}+p_{3}-p_{1})^{2}+(p_{3}+p_{1}-p_{2})^{2}+(p_{1}+p_{2}-p_{3})^{2},$
$\displaystyle
H({\mathbf{q}})=(q_{2}+q_{3})^{2}+(q_{3}+q_{1})^{2}+(q_{1}+q_{2})^{2}.$

The minimum value of $H({\mathbf{q}})$ is 2, with minimizers
${\mathbf{q}}=\pm{\mathbf{e}}_{i}$ and
${\mathbf{q}}={\mathbf{e}}_{i}-{\mathbf{e}}_{j},i\neq j$, while the minimum
value of $G({\mathbf{p}})$ is 3, with minimizers
${\mathbf{p}}=\pm{\mathbf{e}}_{i}$ and
${\mathbf{p}}=\pm({\mathbf{e}}_{1}+{\mathbf{e}}_{2}+{\mathbf{e}}_{3})$. Hence
the minimizers of $\frac{1}{4}G({\mathbf{p}})H({\mathbf{q}})$, subject to
${\mathbf{p}},{\mathbf{q}}\in{\mathbb{Z}}^{3}\setminus\\{0\\}$ and
${\mathbf{p}}\cdot{\mathbf{q}}=0$ are given by

$\displaystyle\pm{\mathbf{p}}\otimes{\mathbf{q}}\in\\{{\mathbf{e}}_{i}\otimes{\mathbf{e}}_{j},{\mathbf{e}}_{i}\otimes({\mathbf{e}}_{j}-{\mathbf{e}}_{k}),({\mathbf{e}}_{1}+{\mathbf{e}}_{2}+{\mathbf{e}}_{3})\otimes({\mathbf{e}}_{i}-{\mathbf{e}}_{j}),i,j,k\text{
distinct}\\}.$

Calculating $\displaystyle{\mathbf{B}}_{\rm
bcc}{\mathbf{p}}\otimes{\mathbf{B}}_{\rm bcc}^{-T}{\mathbf{q}}$ for these
possibilities gives the slip systems in the theorem.
(c) For fcc it suffices to note that by Corollary 2 the slips minimizing
$|{\mathbf{a}}|$ are given by ${\mathbf{1}}+{\mathbf{n}}\otimes{\mathbf{a}}$,
where ${\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}$ are the slips for bcc
minimizing $|{\mathbf{a}}|$.∎

For bcc metals the experimental determination of slip planes is not
straightforward. One can ask what the slips are that give the next lowest
value of $|{\mathbf{a}}|^{2}$, and whether such slips are observed. Since
$H({\mathbf{q}})$ is even, the next lowest value of
$G({\mathbf{p}})H({\mathbf{q}})$ is greater than or equal to 8, for which
$|{\mathbf{a}}|^{2}=2$, and the value 8 is achieved with
$G({\mathbf{p}})=4,H({\mathbf{q}})=2$ and

${\mathbf{p}}\otimes{\mathbf{q}}\in\\{\pm({\mathbf{e}}_{i}+{\mathbf{e}}_{j})\otimes{\mathbf{e}}_{k},({\mathbf{e}}_{i}+{\mathbf{e}}_{j})\otimes({\mathbf{e}}_{i}-{\mathbf{e}}_{j}),i,j,k\text{
distinct}\\},$

giving the slips

$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}\in\\{\pm{\mathbf{e}}_{k}\otimes({\mathbf{e}}_{i}+{\mathbf{e}}_{j}),{\mathbf{e}}_{k}\otimes({\mathbf{e}}_{i}-{\mathbf{e}}_{j}),i,j,k\text{
distinct}\\}.$ (53)

These slips are consistent with those described by Chalmers & Martius
chalmers1952slip , who noted that they had not been reported experimentally;
in a more recent survey Weinberger, Boyce & Bataille weinberger2013slip
appear to confirm this, stating that “It is clear from numerous experiments
that slip occurs in the closest packed $\langle 111\rangle$ direction and the
Burgers vector is $\frac{1}{2}\langle 111\rangle$”. The slips in (53) have
Burgers vector magnitude $|{\mathbf{b}}|^{2}=1$, whereas those in Theorem
5.2(b) have $|{\mathbf{b}}|^{2}=\frac{3}{4}$.

It is often stated (see, for example, jackson2012handbook ) that there are up
to 48 slip systems for bcc, consisting of the 12 systems given in Theorem 5.2
(identifying $\pm{\mathbf{a}}\otimes{\mathbf{n}}$) together with 12 having
$(112)$ slip planes given by

$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}\in\left\\{\pm\frac{1}{2}(\kappa_{1}{\mathbf{e}}_{i}+\kappa_{2}{\mathbf{e}}_{j}-{\mathbf{e}}_{k})\otimes(\kappa_{1}{\mathbf{e}}_{i}-\kappa_{2}{\mathbf{e}}_{j}+2{\mathbf{e}}_{k}),\right.$
$\displaystyle\left.i,j,k\text{ distinct},\kappa_{1}=\pm 1,\kappa_{2}=\pm
1\right\\},$

with $|{\mathbf{a}}|^{2}=\frac{9}{2}$, and 24 having $(123)$ slip planes,
given by

$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}\in\left\\{\pm\frac{1}{2}({\mathbf{e}}_{i}-\kappa_{1}{\mathbf{e}}_{j}-\kappa_{2}{\mathbf{e}}_{k})\otimes(3{\mathbf{e}}_{i}+2\kappa_{1}{\mathbf{e}}_{j}+\kappa_{2}{\mathbf{e}}_{k}),\right.$
$\displaystyle\left.i,j,k\text{ distinct},\kappa_{1}=\pm 1,\kappa_{2}=\pm
1\right\\},$

with $|{\mathbf{a}}|^{2}=\frac{21}{2}$. That these are slips can be checked by
verifying that ${\mathbf{a}}\cdot{\mathbf{n}}=0$ and
${\mathbf{B}}^{-1}{\mathbf{a}}\otimes{\mathbf{B}}^{T}{\mathbf{n}}\in{\mathbb{Z}}^{3}\otimes{\mathbb{Z}}^{3}$.
These extra slip systems have the same Burgers vector magnitude
$|{\mathbf{b}}|^{2}=\frac{3}{4}$ as those in Theorem 5.2(b), but considerably
higher values of shear magnitude.

For further discussion of the Chalmers & Martius criterion for bcc crystals
see chenmaddin .

##  6 Twinning

###  6.1 Twins

In this paper we define
${\mathbf{F}}={\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}$ to be a
(mechanical) twin if the lattices ${\mathcal{L}}({\mathbf{B}})$ and
${\mathbf{F}}{\mathcal{L}}({\mathbf{B}})$ on either side of the interface are
non-trivially reflected with respect to each other, so that
${\mathbf{F}}={\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}$ is not a slip and
satisfies for some unit vector ${\mathbf{m}}$

$\displaystyle{\mathbf{F}}{\mathcal{L}}({\mathbf{B}})=({\mathbf{1}}-2{\mathbf{m}}\otimes{\mathbf{m}}){\mathcal{L}}({\mathbf{B}})=(-{\mathbf{1}}+2{\mathbf{m}}\otimes{\mathbf{m}})\mathcal{L}({\mathbf{B}}),$
(56)

where for the second equality we have used
${\mathcal{L}}({\mathbf{B}})={\mathcal{L}}(-{\mathbf{B}})$. (Note that if
${\mathbf{F}}$ is a slip then (56) holds for any $180^{\circ}$ rotation
$-{\mathbf{1}}+2{\mathbf{m}}\otimes{\mathbf{m}}\in P({\mathbf{B}})$.) By
Corollary 1, (56) holds iff

$\displaystyle{\mathbf{F}}=(-{\mathbf{1}}+2{\mathbf{m}}\otimes{\mathbf{m}}){\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1}$
(57)

for some ${\bm{\mu}}\in GL^{+}(3,{\mathbb{Z}})$.

The definition given above of twins corresponds to that given by Christian
(christian2002, , p51), except that we require that ${\mathbf{F}}$ is not a
slip, and grew out of a discussion with R.D. James jamesprivate . It covers
the cases of Type 1 and Type 2 twins in the next section, but other
possibilities too; on the other hand it is less general than that used e.g. by
Pitteri & Zanzotto pitterizanzotto03 , who give (pp 28,29) references to
various different definitions. According to their terminology ‘nonconventional
twins’ correspond to rank-one connections which are neither Type 1 nor Type 2
twins. In Section 8 we give different examples of such ‘nonconventional twins’
which are, and which are not, twins as defined in this paper. For a related
discussion see chenetal13 .

###  6.2 Type 1, Type 2 and compound twins

We shall be interested in Type 1 twins, for which ${\mathbf{m}}={\mathbf{n}}$,
so that the lattices are non-trivially reflected with respect to the twin
plane, and thus

$\displaystyle{\mathbf{F}}=(-{\mathbf{1}}+2{\mathbf{n}}\otimes{\mathbf{n}}){\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1}$
(58)

for some ${\bm{\mu}}\in GL^{+}(3,{\mathbb{Z}})$, and in Type 2 twins, for
which ${\mathbf{F}}$ satisfies (57) with
$\displaystyle{\mathbf{m}}=\frac{{\mathbf{a}}}{|{\mathbf{a}}|}$. From (43) it
follows that if ${\mathbf{F}}={\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}$ is
a Type 1 twin, then its conjugate
${\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}$ satisfies

$\displaystyle{\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}=\left(-{\mathbf{1}}+2\frac{\tilde{\mathbf{a}}}{|\tilde{\mathbf{a}}|}\otimes\frac{\tilde{\mathbf{a}}}{|\tilde{\mathbf{a}}|}\right){\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1},$
(59)

and so (see (pitterizanzotto03, , p254)) is a Type 2 twin (unless it is a
slip, which can happen – see Theorem 6.1(a)). Conversely the conjugate to a
Type 2 twin, unless it is a slip, is a Type 1 twin. ${\mathbf{F}}$ is a
compound twin if it is both a Type 1 and Type 2 twin, so that

$\displaystyle{\mathbf{F}}=(-{\mathbf{1}}+2{\mathbf{n}}\otimes{\mathbf{n}}){\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1}=\left(-{\mathbf{1}}+2\frac{{\mathbf{a}}}{|{\mathbf{a}}|}\otimes\frac{{\mathbf{a}}}{|{\mathbf{a}}|}\right){\mathbf{B}}{\bm{\nu}}{\mathbf{B}}^{-1}$
(60)

for some ${\bm{\nu}}\in GL^{+}(3,{\mathbb{Z}})$. This holds iff

$\displaystyle(-{\mathbf{1}}+2{\mathbf{n}}\otimes{\mathbf{n}})\left(-{\mathbf{1}}+2\frac{{\mathbf{a}}}{|{\mathbf{a}}|}\otimes\frac{{\mathbf{a}}}{|{\mathbf{a}}|}\right)\in
P({\mathbf{B}}),$ (61)

or equivalently iff

$\displaystyle{\mathbf{B}}^{-1}\left({\mathbf{1}}-2{\mathbf{n}}\otimes{\mathbf{n}}-2\frac{{\mathbf{a}}}{|{\mathbf{a}}|}\otimes\frac{{\mathbf{a}}}{|{\mathbf{a}}|}\right){\mathbf{B}}\in{\mathbb{Z}}^{3\times
3}.$ (62)

We will use (62) to check for compound twins; other criteria are given in
(pitterizanzotto03, , Proposition 8.7), forclaz99 . Obviously the conjugate of
a compound twin is either a slip or a compound twin.

From (58), to find the Type 1 twins (and hence their conjugate Type 2 twins)
we need to find ${\bm{\mu}},{\mathbf{a}},{\mathbf{n}}$ such that
${\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1}\not\in SO(3)$ and

$\displaystyle
1+{\mathbf{a}}\otimes{\mathbf{n}}=(-{\mathbf{1}}+2{\mathbf{n}}\otimes{\mathbf{n}}){\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1}.$
(63)

Equivalently
${\bm{\mu}}=-{\mathbf{1}}+{\mathbf{B}}^{-1}(2{\mathbf{n}}-{\mathbf{a}})\otimes{\mathbf{B}}^{T}{\mathbf{n}},$
and hence, by Lemma 1,

$\displaystyle{\mathbf{B}}^{-1}(2{\mathbf{n}}-{\mathbf{a}})\otimes{\mathbf{B}}^{T}{\mathbf{n}}={\mathbf{p}}\otimes{\mathbf{q}}$
(64)

for some ${\mathbf{p}},{\mathbf{q}}\in\mathbb{Z}^{3}$, and
${\mathbf{p}}\cdot{\mathbf{q}}=2$. Thus we can suppose that

$\displaystyle{\mathbf{n}}=\frac{{\mathbf{B}}^{-T}{\mathbf{q}}}{|{\mathbf{B}}^{-T}{\mathbf{q}}|},\;2{\mathbf{n}}-{\mathbf{a}}=|{\mathbf{B}}^{-T}{\mathbf{q}}|{\mathbf{B}}{\mathbf{p}},$
(65)

so that

$\displaystyle|{\mathbf{a}}|^{2}=|{\mathbf{B}}{\mathbf{p}}|^{2}|{\mathbf{B}}^{-T}{\mathbf{q}}|^{2}-4.$
(66)

It follows from (40),(65),(66) that the conjugate Type 2 twin is given by

$\displaystyle{\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}={\mathbf{1}}+{\mathbf{B}}{\mathbf{p}}\otimes\left({\mathbf{B}}^{-T}{\mathbf{q}}-2\frac{{\mathbf{B}}{\mathbf{p}}}{|{\mathbf{B}}{\mathbf{p}}|^{2}}\right).$
(67)

###  6.3 Twins of minimum shear magnitude

From (66), in order to minimize $|{\mathbf{a}}|^{2}$ we have to minimize
$|{\mathbf{B}}{\mathbf{p}}|^{2}|{\mathbf{B}}^{-T}{\mathbf{q}}|^{2}$ among
${\mathbf{p}},{\mathbf{q}}\in{\mathbb{Z}}^{3}$ with
${\mathbf{p}}\cdot{\mathbf{q}}=2$ and
$|{\mathbf{B}}{\mathbf{p}}|^{2}|{\mathbf{B}}^{-T}{\mathbf{q}}|^{2}>4$. In the
following theorem we give the Type 1/Type 2 twins minimizing the shear
magnitude for the simple cubic, bcc and fcc lattices. Note that by Theorem 4.1
and Corollary 2 the Type 1 (respectively Type 2) twins for fcc are given by
${\mathbf{1}}-{\mathbf{n}}\otimes{\mathbf{a}}$, where
${\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}$ are the Type 2 (respectively
Type 1) twins for bcc. The twins identified, and their shear magnitude,
correspond to those given in Christian & Mahajan (christianmahajan1995, ,
Table 1), which summarizes earlier work of Jaswon & Dove jaswondove1956 ;
jaswondove1957 ; jaswondove1960 , Bilby & Crockerbilbycrocker1965 , Bevis &
Crocker beviscrocker1968 ; beviscrocker1969 .

###### Theorem 6.1

The Type $1$ and Type $2$ twins ${\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}$
minimizing $|{\mathbf{a}}|$ are given by:
$\rm(a)$ Simple cubic: the compound twins

$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}$
$\displaystyle=\frac{1}{5}(2{\mathbf{e}}_{j}-\kappa{\mathbf{e}}_{i})\otimes(2\kappa{\mathbf{e}}_{i}+{\mathbf{e}}_{j}),$

where $i\neq j,\,\kappa=\pm 1$, whose conjugates are the slips
$\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}=\kappa{\mathbf{e}}_{i}\otimes{\mathbf{e}}_{j}$
in Theorem $\ref{slipthm}(a)$, for which $|{\mathbf{a}}|^{2}=1$.
$\rm(b)$ bcc: the conjugate pairs

$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}$
$\displaystyle=\frac{1}{6}({\mathbf{e}}_{j}-{\mathbf{e}}_{k}+{\mathbf{e}}_{i})\otimes(2{\mathbf{e}}_{j}+{\mathbf{e}}_{k}-{\mathbf{e}}_{i}),$
$\displaystyle\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}$
$\displaystyle=\frac{1}{6}({\mathbf{e}}_{j}+{\mathbf{e}}_{k}-{\mathbf{e}}_{i})\otimes(2{\mathbf{e}}_{j}+{\mathbf{e}}_{i}-{\mathbf{e}}_{k}),$

and

$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}$
$\displaystyle=\frac{1}{6}({\mathbf{e}}_{1}+{\mathbf{e}}_{2}+{\mathbf{e}}_{3})\otimes(2{\mathbf{e}}_{i}-{\mathbf{e}}_{j}-{\mathbf{e}}_{k}),$
$\displaystyle\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}$
$\displaystyle=\frac{1}{6}({\mathbf{e}}_{i}-{\mathbf{e}}_{j}-{\mathbf{e}}_{k})\otimes(2{\mathbf{e}}_{i}+{\mathbf{e}}_{j}+{\mathbf{e}}_{k}),$

where $i,j,k$ are distinct, which are all compound twins, and for which
$|{\mathbf{a}}|^{2}=\frac{1}{2}$.
$\rm(c)$ fcc: the conjugate pairs

$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}$
$\displaystyle=\frac{1}{6}(-2{\mathbf{e}}_{j}-{\mathbf{e}}_{k}+{\mathbf{e}}_{i})\otimes({\mathbf{e}}_{j}-{\mathbf{e}}_{k}+{\mathbf{e}}_{i}),$
$\displaystyle\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}$
$\displaystyle=\frac{1}{6}(-2{\mathbf{e}}_{j}-{\mathbf{e}}_{i}+{\mathbf{e}}_{k})\otimes({\mathbf{e}}_{j}+{\mathbf{e}}_{k}-{\mathbf{e}}_{i}),$

and

$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}$
$\displaystyle=\frac{1}{6}(-2{\mathbf{e}}_{i}+{\mathbf{e}}_{j}+{\mathbf{e}}_{k})\otimes({\mathbf{e}}_{1}+{\mathbf{e}}_{2}+{\mathbf{e}}_{3}),$
$\displaystyle\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}$
$\displaystyle=\frac{1}{6}(2{\mathbf{e}}_{i}+{\mathbf{e}}_{j}+{\mathbf{e}}_{k})\otimes({\mathbf{e}}_{j}+{\mathbf{e}}_{k}-{\mathbf{e}}_{i}),$

where $i,j,k$ are distinct, which are all compound twins, and for which
$|{\mathbf{a}}|^{2}=\frac{1}{2}$.

###### Proof

(a) Since ${\mathbf{B}}={\mathbf{1}}$ we have to minimize
$|{\mathbf{p}}|^{2}|{\mathbf{q}}|^{2}$ among
${\mathbf{p}},{\mathbf{q}}\in{\mathbb{Z}}^{3}$ subject to
${\mathbf{p}}\cdot{\mathbf{q}}=2$ and
$|{\mathbf{p}}|^{2}|{\mathbf{q}}|^{2}>4$, since
$|{\mathbf{p}}|^{2}|{\mathbf{q}}|^{2}=4$ implies ${\mathbf{a}}=0$. The minimum
is given by $|{\mathbf{p}}|^{2}|{\mathbf{q}}|^{2}=5$ with minimizers

${\mathbf{p}}\otimes{\mathbf{q}}\in\\{(2{\mathbf{e}}_{j}-\kappa{\mathbf{e}}_{i})\otimes{\mathbf{e}}_{j},\;{\mathbf{e}}_{i}\otimes(2{\mathbf{e}}_{i}+\kappa{\mathbf{e}}_{j}),\;i\neq
j,\kappa=\pm 1\\}.$

From (65) we have that

${\mathbf{a}}\otimes{\mathbf{n}}=\frac{1}{|{\mathbf{q}}|^{2}}(2{\mathbf{q}}-|{\mathbf{q}}|^{2}{\mathbf{p}})\otimes{\mathbf{q}},$

which together with (40) gives the conjugate pairs in the statement. Note that
${\mathbf{a}}\otimes{\mathbf{n}}=\frac{1}{5}(2{\mathbf{e}}_{j}-\kappa{\mathbf{e}}_{i})\otimes(2\kappa{\mathbf{e}}_{i}+{\mathbf{e}}_{j})$
is not a slip because it does not belong to ${\mathbb{Z}}^{3\times 3}$, and is
a compound twin by the criterion (62), since
$2({\mathbf{e}}_{j}\otimes{\mathbf{e}}_{j}+{\mathbf{e}}_{i}\otimes{\mathbf{e}}_{i})\in{\mathbb{Z}}^{3\times
3}$.

(b) For bcc (see the proof of Theorem 5.2 (b)) we have to minimize
$G({\mathbf{p}})H({\mathbf{q}})$ subject to ${\mathbf{p}}\cdot{\mathbf{q}}=2$
and $G({\mathbf{p}})H({\mathbf{q}})>16$, where

$\displaystyle
G({\mathbf{p}})=(p_{2}+p_{3}-p_{1})^{2}+(p_{3}+p_{1}-p_{2})^{2}+(p_{1}+p_{2}-p_{3})^{2},$
$\displaystyle
H({\mathbf{q}})=(q_{2}+q_{3})^{2}+(q_{3}+q_{1})^{2}+(q_{1}+q_{2})^{2}.$

Since $H({\mathbf{q}})$ is even the minimum value is $\geqslant 18$ and we
show that the value 18 is attained. Also $G({\mathbf{p}})\geqslant 3$, so that
the possibilities are $G({\mathbf{p}})=9,H({\mathbf{q}})=2$,
$G({\mathbf{p}})=3,H({\mathbf{q}})=6$. But $G({\mathbf{p}})=9$ implies either
(i) that $p_{i}+p_{j}-p_{k}=\pm 3$, $p_{i}+p_{k}-p_{j}=p_{j}+p_{k}-p_{i}=0$
for $i,j,k$ distinct, so that $p_{k}=0$ and $2p_{i}=\pm 3$, which is
impossible, or (ii) that $p_{i}+p_{j}-p_{k}=2\kappa$,
$p_{k}+p_{i}-p_{j}=2\kappa^{\prime}$,
$p_{j}+p_{k}-p_{i}=\kappa^{\prime\prime}$ for $i,j,k$ distinct and
$\kappa,\kappa^{\prime},\kappa^{\prime\prime}=\pm 1$, giving
$2p_{k}=2\kappa^{\prime}+\kappa^{\prime\prime}$, again impossible. Hence we
need only consider the case $G({\mathbf{p}})=3$, $H({\mathbf{q}})=6$.

It is easy to check that $G({\mathbf{p}})=3=1^{2}+1^{2}+1^{2}$ iff

$\displaystyle\pm{\mathbf{p}}\in\\{{\mathbf{e}}_{i},{\mathbf{e}}_{1}+{\mathbf{e}}_{2}+{\mathbf{e}}_{3},i=1,2,3\\},$
(68)

and that $H({\mathbf{q}})=6=2^{2}+1^{2}+1^{2}$ iff

$\displaystyle\pm{\mathbf{q}}\in\\{{\mathbf{e}}_{i}+{\mathbf{e}}_{j},2{\mathbf{e}}_{i}-{\mathbf{e}}_{j},2{\mathbf{e}}_{i}-{\mathbf{e}}_{j}-{\mathbf{e}}_{k},i,j,k\text{
distinct}\\}.$ (69)

From (68),(69) we find that ${\mathbf{p}}\cdot{\mathbf{q}}=2$ iff

$\displaystyle{\mathbf{p}}\otimes{\mathbf{q}}\in\\{{\mathbf{e}}_{i}\otimes(2{\mathbf{e}}_{i}-{\mathbf{e}}_{j}),{\mathbf{e}}_{i}\otimes(2{\mathbf{e}}_{i}-{\mathbf{e}}_{j}-{\mathbf{e}}_{k}),$
$\displaystyle\mbox{
}({\mathbf{e}}_{1}+{\mathbf{e}}_{2}+{\mathbf{e}}_{3})\otimes({\mathbf{e}}_{i}+{\mathbf{e}}_{j}),i,j,k\text{
distinct}\\}.$

Calculating ${\mathbf{a}}\otimes{\mathbf{n}}$ from (65) for these
${\mathbf{p}}\otimes{\mathbf{q}}$ and using (40) we obtain the conjugate pairs
in the statement, none of which belong to ${\mathbb{Z}}^{3\times 3}$ and so
are not slips.

To show that the twins are compound we note that from (65) and the relations
$|{\mathbf{a}}|=\frac{1}{2}$, $|{\mathbf{B}}_{\rm
bcc}^{-T}{\mathbf{q}}|^{2}=6$, $|{\mathbf{B}}_{\rm
bcc}{\mathbf{p}}|^{2}=\frac{3}{4}$ that

$\displaystyle 2{\mathbf{B}}_{\rm
bcc}^{-1}\left({\mathbf{n}}\otimes{\mathbf{n}}+\frac{{\mathbf{a}}\otimes{\mathbf{a}}}{|{\mathbf{a}}|^{2}}\right){\mathbf{B}}_{\rm
bcc}$ $\displaystyle\mbox{
}=3{\mathbf{G}}^{-1}{\mathbf{q}}\otimes{\mathbf{q}}+24{\mathbf{p}}\otimes{\mathbf{G}}{\mathbf{p}}-8{\mathbf{p}}\otimes{\mathbf{q}}-8{\mathbf{G}}^{-1}{\mathbf{q}}\otimes{\mathbf{G}}{\mathbf{p}},$

where ${\mathbf{G}}={\mathbf{B}}_{\rm bcc}^{T}{\mathbf{B}}_{\rm bcc}.$ Since
${\mathbf{G}}^{-1},4{\mathbf{G}}\in{\mathbb{Z}}^{3\times 3}$, the twins are
compound by (62).

(c) The twins for fcc are obtained from those for bcc using Theorem 4.1 and
Corollary 2 as described above, which also imply that the twins are compound.∎

###### Remark 5

For the cubic lattices we consider it is not in general true that Type 1 and
Type 2 twins are compound. For example, we can take
${\mathbf{p}}=2{\mathbf{e}}_{1}-{\mathbf{e}}_{3},{\mathbf{q}}=2{\mathbf{e}}_{1}+{\mathbf{e}}_{2}+2{\mathbf{e}}_{3}$,
so that for the simple cubic lattice we obtain from (65)

${\mathbf{a}}\otimes{\mathbf{n}}=\frac{1}{9}(-14{\mathbf{e}}_{1}+2{\mathbf{e}}_{2}+13{\mathbf{e}}_{3})\otimes(2{\mathbf{e}}_{1}+{\mathbf{e}}_{2}+2{\mathbf{e}}_{3}),$

with conjugate

$\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}=\frac{1}{5}(2{\mathbf{e}}_{1}-{\mathbf{e}}_{3})\otimes(6{\mathbf{e}}_{1}+5{\mathbf{e}}_{2}+12{\mathbf{e}}_{3}).$

Then none of

${\mathbf{a}}\otimes{\mathbf{n}},\;\
\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}},\;2\left({\mathbf{n}}\otimes{\mathbf{n}}+\frac{{\mathbf{a}}\otimes{\mathbf{a}}}{|{\mathbf{a}}|^{2}}\right)$

belong to ${\mathbb{Z}}^{3\times 3}$ so that, by (62),
$1+{\mathbf{a}}\otimes{\mathbf{n}},{\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}$
are a pair of Type 1/Type 2 twins that are not compound. For the same
${\mathbf{p}},{\mathbf{q}}$ it can be checked that the corresponding pair of
conjugate bcc Type 1/Type 2 twins given by

$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}$ $\displaystyle=$
$\displaystyle\frac{1}{34}(57{\mathbf{e}}_{1}-9{\mathbf{e}}_{2}-45{\mathbf{e}}_{3})\otimes(3{\mathbf{e}}_{1}+4{\mathbf{e}}_{2}+3{\mathbf{e}}_{3}),$
(72) $\displaystyle\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}$
$\displaystyle=$
$\displaystyle\frac{3}{38}(-3{\mathbf{e}}_{1}+{\mathbf{e}}_{2}+3{\mathbf{e}}_{3})\otimes(23{\mathbf{e}}_{1}+24{\mathbf{e}}_{2}+15{\mathbf{e}}_{3}),$

are also not compound.

##  7 General rank-one connections minimizing shear magnitude

We ask what the rank-one connections
${\mathbf{F}}={\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}$ are between
${\mathbf{1}}$ and $\psi^{-1}(0)=\bigcup_{{\bm{\mu}}\in
GL^{+}(3,\mathbb{Z})}{\rm SO}(3){\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1}$ that
minimize $|{\mathbf{a}}|^{2}$. For the simple cubic lattice we have a complete
answer.

###### Theorem 7.1

For the simple cubic lattice the rank-one connections
${\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}={\mathbf{R}}{\bm{\mu}}$,
${\mathbf{R}}\in{\rm SO}(3)$, ${\bm{\mu}}\in GL^{+}(3,\mathbb{Z})\setminus{\rm
SO}(3)$ that minimize $|{\mathbf{a}}|$ are given by the slips in Theorem
$\ref{slipthm}(a)$ and the twins in Theorem $\ref{twinthm}(a)$, with the
minimum value $|{\mathbf{a}}|^{2}=1$.

###### Proof

If ${\bm{\mu}}\in GL^{+}(3,{\mathbb{Z}})$ then $|{\bm{\mu}}|^{2}\geqslant 3$
with $|{\bm{\mu}}|^{2}=3$ iff ${\bm{\mu}}\in{\rm SO}(3)$ (this follows either
by noting that each row and column must contain a nonzero entry, and that if
there is a single entry $=\pm 1$ then ${\bm{\mu}}$ is a rotation, or by use of
the AM$\geqslant$GM inequality applied to the squares of the singular values
of ${\bm{\mu}}$). Since $|{\bm{\mu}}|^{2}=3+|{\mathbf{a}}|^{2}$ is an integer,
the minimum value of $|{\mathbf{a}}|^{2}$ for a rank-one connection is
$\geqslant 1$, and so by Theorems 5.2, 6.1 the minimum value is
$|{\mathbf{a}}|^{2}=1$, with $|{\bm{\mu}}|^{2}=4$.

But $|{\bm{\mu}}|^{2}=4$ implies that each row and column of ${\bm{\mu}}$
contains an entry $\pm 1$ and that there is a single additional entry $\pm 1$.
Thus

${\bm{\mu}}={\mathbf{Q}}+\kappa{\mathbf{e}}_{k}\otimes{\mathbf{e}}_{j}$

for some ${\mathbf{Q}}\in P^{48}$ with $Q_{kj}=0$ and $\kappa=\pm 1$. Hence

${\bm{\mu}}={\mathbf{Q}}({\mathbf{1}}+\kappa^{\prime}{\mathbf{e}}_{i}\otimes{\mathbf{e}}_{j}),$

where
${\mathbf{e}}_{i}=\kappa\kappa^{\prime}{\mathbf{Q}}^{T}{\mathbf{e}}_{k}$,
$\kappa^{\prime}=\pm 1$, and thus ${\mathbf{e}}_{i}\cdot{\mathbf{e}}_{j}=0$,
from which it follows that $\det{\mathbf{Q}}=1$ and hence ${\mathbf{Q}}\in{\rm
SO}(3)$. The corresponding rank-one connections are then given by
${\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}={\mathbf{R}}{\mathbf{Q}}({\mathbf{1}}+\kappa^{\prime}{\mathbf{e}}_{i}\otimes{\mathbf{e}}_{j})$
and are thus those in Theorems 5.2, 6.1.∎

For bcc and fcc we have the following result.

###### Theorem 7.2

For bcc and fcc the squared magnitude $|{\mathbf{a}}|^{2}$ for any rank-one
connection
${\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}={\mathbf{R}}{\mathbf{B}}{\bm{\mu}}{\mathbf{B}}^{-1}\not\in{\rm
SO}(3)$, ${\bm{\mu}}\in GL^{+}(3,\mathbb{Z})$ is given by
$|{\mathbf{a}}|^{2}=\frac{m}{2}$, where $m$ is a positive integer.

###### Proof

By Theorem 4.1 it suffices to prove the result for bcc, for which we can write

$\displaystyle{\mathbf{B}}_{\rm
bcc}=-{\mathbf{1}}+\frac{1}{2}{\mathbf{v}}\otimes{\mathbf{v}},\;{\mathbf{v}}={\mathbf{e}}_{1}+{\mathbf{e}}_{2}+{\mathbf{e}}_{3}.$
(73)

Then, setting ${\mathbf{M}}={\mathbf{B}}_{\rm bcc}{\bm{\mu}}{\mathbf{B}}_{\rm
bcc}^{-1}$, ${\mathbf{G}}={\mathbf{B}}_{\rm bcc}^{T}{\mathbf{B}}_{\rm
bcc}={\mathbf{1}}-\frac{1}{4}{\mathbf{v}}\otimes{\mathbf{v}}$, and noting that
${\mathbf{G}}^{-1}={\mathbf{1}}+{\mathbf{v}}\otimes{\mathbf{v}}$, we calculate

$\displaystyle|{\mathbf{M}}|^{2}$ $\displaystyle=$ $\displaystyle\rm
tr\,({\mathbf{G}}^{-1}{\bm{\mu}}^{T}{\mathbf{G}}{\bm{\mu}})$ (74)
$\displaystyle=$
$\displaystyle|{\bm{\mu}}|^{2}+|{\bm{\mu}}{\mathbf{v}}|^{2}-\frac{1}{4}(|{\bm{\mu}}^{T}{\mathbf{v}}|^{2}+({\bm{\mu}}{\mathbf{v}}\cdot{\mathbf{v}})^{2}).$

We now note that

$\displaystyle|{\bm{\mu}}^{T}{\mathbf{v}}|^{2}+({\bm{\mu}}{\mathbf{v}}\cdot{\mathbf{v}})^{2}$
$\displaystyle=$ $\displaystyle
c_{1}^{2}+c_{2}^{2}+c_{3}^{2}+(c_{1}+c_{2}+c_{3})^{2}$ (75) $\displaystyle=$
$\displaystyle
2(c_{1}^{2}+c_{2}^{2}+c_{3}^{2}+c_{2}c_{3}+c_{3}c_{1}+c_{1}c_{2}),$

where $c_{i}$ denotes the sum of the entries in the $i^{\rm th}$ column of
${\bm{\mu}}$. Since ${\bm{\mu}}\in{\mathbb{Z}}^{3\times 3}$,
${\mathbf{v}}\in{\mathbb{Z}}^{3}$, and
$|{\mathbf{a}}|^{2}=|{\mathbf{M}}|^{2}-3$, the result follows from (74),
(75).∎

###### Corollary 3

For bcc and fcc the twins in Theorem $\ref{twinthm}(b),(c)$ minimize
$|{\mathbf{a}}|^{2}$.

###### Proof

The twins satisfy $|{\mathbf{a}}|^{2}=\frac{1}{2}$, the least possible value.∎

###### Remark 6

Corollary 3 leaves unresolved the possibility that some other rank-one
connection for bcc or fcc also gives the value
$|{\mathbf{a}}|^{2}=\frac{1}{2}$. This could be decided computationally, since
there are only finitely many possibilities for ${\bm{\mu}}$, for each of which
the existence of rank-one connections could be checked. Indeed, if
$|{\mathbf{M}}|^{2}=3+\frac{1}{2}=\frac{7}{2}$ then
$|{\bm{\mu}}|^{2}\leqslant|{\mathbf{B}}_{\rm
bcc}^{-1}|^{2}|{\mathbf{M}}|^{2}|{\mathbf{B}}_{\rm bcc}|^{2}=47.25$ so that
$|{\bm{\mu}}|^{2}\leqslant 47$. However we can get a better estimate by noting
that (74) can be rewritten as

$\displaystyle|{\mathbf{M}}|^{2}=\frac{1}{4}|{\bm{\mu}}|^{2}+\frac{1}{4}|{\bm{\mu}}{\mathbf{v}}|^{2}+\frac{1}{12}|3{\bm{\mu}}^{T}-{\bm{\mu}}^{T}{\mathbf{v}}\otimes{\mathbf{v}}|^{2}+\frac{1}{12}|3{\bm{\mu}}{\mathbf{v}}-({\bm{\mu}}{\mathbf{v}}\cdot{\mathbf{v}}){\mathbf{v}}|^{2}.$

Since ${\bm{\mu}}$ is nonsingular we have that
$|{\bm{\mu}}{\mathbf{v}}|^{2}\geqslant 1$. Also ${\bm{\mu}}^{T}$ is not a
rank-one matrix, so that
$|3{\bm{\mu}}^{T}-{\bm{\mu}}^{T}{\mathbf{v}}\otimes{\mathbf{v}}|^{2}\geqslant
1$. Hence $|{\bm{\mu}}|^{2}\leqslant 13-\frac{1}{3}$ and so
$|{\bm{\mu}}|^{2}\leqslant 12$.

##  8 Conjugates of slips

We first ask when the conjugate of a slip is a slip.

###### Theorem 8.1

For simple cubic, bcc and fcc the only slips whose conjugates are slips are
given by the conjugate pairs

$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}$
$\displaystyle=2\kappa{\mathbf{e}}_{i}\otimes{\mathbf{e}}_{j},$
$\displaystyle\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}$
$\displaystyle=({\mathbf{e}}_{j}-\kappa{\mathbf{e}}_{i})\otimes(\kappa{\mathbf{e}}_{i}+{\mathbf{e}}_{j})$

with $i\neq j$ and $\kappa=\pm 1$, for which $|{\mathbf{a}}|^{2}=4$.

###### Proof

We need to find the slips ${\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}$ such
that

$\displaystyle{\mathbf{Q}}$ $\displaystyle=$
$\displaystyle({\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}})({\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}})^{-1}$
$\displaystyle=$
$\displaystyle{\mathbf{1}}-\frac{2|{\mathbf{a}}|^{2}}{4+|{\mathbf{a}}|^{2}}\left({\mathbf{n}}\otimes{\mathbf{n}}+\frac{{\mathbf{a}}}{|{\mathbf{a}}|}\otimes\frac{{\mathbf{a}}}{|{\mathbf{a}}|}\right)+\frac{4}{4+|{\mathbf{a}}|^{2}}\left({\mathbf{n}}\otimes{\mathbf{a}}-{\mathbf{a}}\otimes{\mathbf{n}}\right)\in
P^{24},$

where we have used (40) and
$({\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}})^{-1}={\mathbf{1}}-{\mathbf{a}}\otimes{\mathbf{n}}$.

For this to be the case we need that

$\displaystyle\rm
tr\,{\mathbf{Q}}=\frac{12-|{\mathbf{a}}|^{2}}{4+|{\mathbf{a}}|^{2}}=m\in{\mathbb{Z}},$
(77)

so that $|{\mathbf{a}}|^{2}=\displaystyle 4\left(\frac{3-m}{1+m}\right)$,
giving the possibilities $m=0,1$ or 2.

The case $m=2$ is impossible since no ${\mathbf{Q}}\in P^{24}$ has $\rm
tr\,{\mathbf{Q}}=2$. The case $m=0$ gives $|{\mathbf{a}}|^{2}=12$ and

$\displaystyle{\mathbf{Q}}={\mathbf{1}}-\frac{3}{2}({\mathbf{n}}\otimes{\mathbf{n}}+{\bm{\nu}}\otimes{\bm{\nu}})+\sqrt{\frac{3}{2}}({\mathbf{n}}\otimes{\bm{\nu}}-{\bm{\nu}}\otimes{\mathbf{n}}),$
(78)

where ${\bm{\nu}}=\displaystyle\frac{{\mathbf{a}}}{|{\mathbf{a}}|}$. But this
is a rotation about ${\bm{\nu}}\wedge{\mathbf{n}}$ through an angle
$\displaystyle\pm\frac{\pi}{3}$, and there are no such elements of $P^{24}$.

Thus we only have to consider the case $m=1$, for which $|{\mathbf{a}}|^{2}=4$
and

$\displaystyle{\mathbf{Q}}={\mathbf{1}}-({\mathbf{n}}\otimes{\mathbf{n}}+{\bm{\nu}}\otimes{\bm{\nu}})+{\mathbf{n}}\otimes{\bm{\nu}}-{\bm{\nu}}\otimes{\mathbf{n}}.$
(79)

This is a rotation through an angle $\pm\pi/2$ about
${\bm{\nu}}\wedge{\mathbf{n}}$, and the only such rotations in $P^{24}$ are
those with axes ${\mathbf{e}}_{i}$, so that without loss of generality we can
suppose that

$\displaystyle{\mathbf{Q}}=\left(\begin{array}[]{ccc}0&-\kappa&0\\\
\kappa&0&0\\\ 0&0&1\end{array}\right),$ (83)

where $\kappa=\pm 1$, and
${\mathbf{n}}=\cos\theta\,{\mathbf{e}}_{1}+\sin\theta\,{\mathbf{e}}_{2}$,
${\bm{\nu}}=\kappa(\sin\theta\,{\mathbf{e}}_{1}-\cos\theta\,{\mathbf{e}}_{2})$.
For the simple cubic case we then require that

$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}=\kappa\left(\begin{array}[]{ccc}\sin\,2\theta&1-\cos
2\theta&0\\\ -1-\cos\,2\theta&-\sin\,2\theta&0\\\
0&0&\kappa\end{array}\right)\in{\mathbb{Z}}^{3\times 3},$ (87)

which holds iff $2\theta=k\frac{\pi}{2}$, giving the possibilities in the
statement. Similarly, for bcc we require that
${\mathbf{B}}^{-1}{\mathbf{a}}\otimes{\mathbf{B}}^{T}{\mathbf{n}}\in{\mathbb{Z}}^{3\times
3}$, which again holds iff $2\theta=k\frac{\pi}{2}$.∎

It is interesting to calculate the conjugates to the slips in Theorem 5.2 and
(53)- (5.2). For the simple cubic case we have already seen (Theorem 6.1(a))
that the rank-one connections conjugate to the slips in Theorem 5.2(a) are
compound twins. For bcc and fcc we can without loss of generality (using
Theorem 4.1) take the case of bcc, when we have the following result.

###### Theorem 8.2

The conjugates of the slips in Theorem $\ref{slipthm}(b)$ are twins which are
neither Type $1$ nor Type $2$.

###### Proof

By Remark 4 we need only consider the case

$\displaystyle{\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}}={\mathbf{1}}+\frac{1}{2}({\mathbf{e}}_{1}+{\mathbf{e}}_{2}+{\mathbf{e}}_{3})\otimes({\mathbf{e}}_{1}-{\mathbf{e}}_{2}),$
(88)

which by (40) has conjugate

$\displaystyle{\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}={\mathbf{1}}+\frac{1}{22}({\mathbf{e}}_{1}-3{\mathbf{e}}_{2}-{\mathbf{e}}_{3})\otimes(7{\mathbf{e}}_{1}+{\mathbf{e}}_{2}+4{\mathbf{e}}_{3}),$
(89)

and is not a slip by Theorem 8.1, since $|{\mathbf{a}}|^{2}\neq 4$. Then the
rotation (see (4.1))
${\mathbf{Q}}=\tilde{\mathbf{R}}{\mathbf{R}}^{T}=({\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}})({\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}})^{-1}$
is given by

$\displaystyle{\mathbf{Q}}=\frac{1}{11}\left(\begin{array}[]{rrr}6&\,\,9&2\\\
-7&6&-6\\\ -6&2&9\end{array}\right),$ (93)

and setting
${\mathbf{m}}=\frac{1}{\sqrt{22}}(3{\mathbf{e}}_{1}+2{\mathbf{e}}_{2}-3{\mathbf{e}}_{3})$
we find that

$\displaystyle(-{\mathbf{1}}+2{\mathbf{m}}\otimes{\mathbf{m}}){\mathbf{Q}}=\left(\begin{array}[]{rrr}0&0&-1\\\
1&0&0\\\ 0&-1&0\end{array}\right)\in P^{24},$ (97)

so that ${\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}$ is a twin
by (57). It is not a Type 1 or Type 2 twin because by computation neither
$(-{\mathbf{1}}+2\tilde{\mathbf{n}}\otimes\tilde{\mathbf{n}}){\mathbf{Q}}$ nor
$\displaystyle\left(-{\mathbf{1}}+2\frac{\tilde{\mathbf{a}}\otimes\tilde{\mathbf{a}}}{|\tilde{\mathbf{a}}|^{2}}\right){\mathbf{Q}}$
belong to $P^{24}$.∎

Next we look at the conjugates of the slips for bcc mentioned in (53)-(5.2).

###### Theorem 8.3

$\rm(i)$ The conjugates of the slips for bcc with

$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}\in\\{\pm{\mathbf{e}}_{k}\otimes({\mathbf{e}}_{i}+{\mathbf{e}}_{j}),{\mathbf{e}}_{k}\otimes({\mathbf{e}}_{i}-{\mathbf{e}}_{j}),i,j,k\text{
distinct}\\}$ (98)

are Type $2$ twins.
$\rm(ii)$ The conjugates of the slips (5.2), (5.2) for bcc with $(112)$ and
$(123)$ slip planes are twins that are neither Type $1$ nor Type $2$.

###### Proof

(i) As in Remark 4 it suffices to consider the case
${\mathbf{a}}\otimes{\mathbf{n}}={\mathbf{e}}_{3}\otimes({\mathbf{e}}_{1}+{\mathbf{e}}_{2})$,
for which

$\displaystyle\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}=\frac{1}{3}({\mathbf{e}}_{1}+{\mathbf{e}}_{2}-{\mathbf{e}}_{3})\otimes(2{\mathbf{e}}_{3}+{\mathbf{e}}_{1}+{\mathbf{e}}_{2})$
(99)

and

$\displaystyle{\mathbf{Q}}=({\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}})({\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}})^{-1}=\frac{1}{3}\left(\begin{array}[]{rrr}2&-1&\,\,2\\\
-1&2&2\\\ -2&-2&1\end{array}\right).$ (103)

Then ${\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}$ is not a slip
by Theorem 8.1, and

$\displaystyle\left(-{\mathbf{1}}+2\frac{\tilde{\mathbf{a}}\otimes\tilde{\mathbf{a}}}{|\tilde{\mathbf{a}}|^{2}}\right){\mathbf{Q}}=\left(\begin{array}[]{rrr}0&\,\,1&0\\\
1&0&0\\\ 0&0&-1\end{array}\right)\in P^{24},$ (107)

so that ${\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}$ is a Type 2
twin.

(ii) For the $(112)$ slips it suffices to consider the case

$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}=\frac{1}{2}({\mathbf{e}}_{1}+{\mathbf{e}}_{2}-{\mathbf{e}}_{3})\otimes({\mathbf{e}}_{1}+{\mathbf{e}}_{2}+2{\mathbf{e}}_{3}),$
(108)

for which

$\displaystyle\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}=\frac{1}{34}(-{\mathbf{e}}_{1}-{\mathbf{e}}_{2}+7{\mathbf{e}}_{3})\otimes(7{\mathbf{e}}_{1}+7{\mathbf{e}}_{2}+2{\mathbf{e}}_{3})$
(109)

and

$\displaystyle{\mathbf{Q}}=({\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}})({\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}})^{-1}=\frac{1}{17}\left(\begin{array}[]{rrr}8&-9&-12\\\
-9&8&-12\\\ 12&12&-1\end{array}\right).$ (113)

Then ${\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}$ is not a slip
by Theorem 8.1, and to show that it is a twin we let
${\mathbf{m}}=\frac{1}{34}(3{\mathbf{e}}_{1}+3{\mathbf{e}}_{2}-4{\mathbf{e}}_{3})$,
so that

$\displaystyle(\\-1+2{\mathbf{m}}\otimes{\mathbf{m}}){\mathbf{Q}}=-{\mathbf{1}}+2{\mathbf{e}}_{3}\otimes{\mathbf{e}}_{3}\in
P^{24}.$ (114)

It is not a Type 1 or Type 2 twin because by computation neither
$(-{\mathbf{1}}+2\tilde{\mathbf{n}}\otimes\tilde{\mathbf{n}}){\mathbf{Q}}$ nor
$\displaystyle\left(-{\mathbf{1}}+2\frac{\tilde{\mathbf{a}}\otimes\tilde{\mathbf{a}}}{|\tilde{\mathbf{a}}|^{2}}\right){\mathbf{Q}}$
belong to $P^{24}$.

For the $(123)$ slips it suffices to consider the case

$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}=\frac{1}{2}({\mathbf{e}}_{1}+{\mathbf{e}}_{2}-{\mathbf{e}}_{3})\otimes({\mathbf{e}}_{1}+2{\mathbf{e}}_{2}+3{\mathbf{e}}_{3}),$
(115)

for which

$\displaystyle\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}=\frac{1}{58}(-5{\mathbf{e}}_{1}-3{\mathbf{e}}_{2}+13{\mathbf{e}}_{3})\otimes(7{\mathbf{e}}_{1}+10{\mathbf{e}}_{2}+5{\mathbf{e}}_{3})$
(116)

and

$\displaystyle{\mathbf{Q}}=({\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}})({\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}})^{-1}=\frac{1}{29}\left(\begin{array}[]{rrr}12&-24&-11\\\
-16&3&-24\\\ 21&16&-12\end{array}\right).$ (120)

Letting
${\mathbf{m}}=\frac{1}{\sqrt{29}}(3{\mathbf{e}}_{1}-4{\mathbf{e}}_{2}-2{\mathbf{e}}_{3})$
we find that

$\displaystyle(-{\mathbf{1}}+2{\mathbf{m}}\otimes{\mathbf{m}}){\mathbf{Q}}=\left(\begin{array}[]{rrr}0&\,\,0&\,\,1\\\
0&1&0\\\ -1&0&0\end{array}\right),$ (124)

so that ${\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}}$ is a twin,
and as for the $(112)$ case one can check that it is not of Type 1 or Type 2.∎

###### Remark 7

It is not in general true that the conjugate of a slip is a twin or a slip.
For example we can take

$\displaystyle{\mathbf{a}}\otimes{\mathbf{n}}=2({\mathbf{e}}_{1}+3{\mathbf{e}}_{2}+5{\mathbf{e}}_{3})\otimes({\mathbf{e}}_{1}-2{\mathbf{e}}_{2}+{\mathbf{e}}_{3}),$
(125)

which is a slip for both bcc and simple cubic, albeit for a high value of
$|{\mathbf{a}}|^{2}$. A calculation then shows that

$\displaystyle{\mathbf{Q}}=({\mathbf{1}}+\tilde{\mathbf{a}}\otimes\tilde{\mathbf{n}})({\mathbf{1}}+{\mathbf{a}}\otimes{\mathbf{n}})^{-1}=\frac{1}{211}\left(\begin{array}[]{rrr}129&114&-122\\\
94&-177&-66\\\ -138&-14&-159\end{array}\right),$ (129)

and ${\mathbf{Q}}{\mathbf{P}}$ cannot equal
$-{\mathbf{1}}+2{\mathbf{m}}\otimes{\mathbf{m}}$ for any ${\mathbf{m}}$
because the absolute values of the entries of ${\mathbf{Q}}$ are all
different, and the action of multiplying on the right by ${\mathbf{P}}$
permutes the columns of ${\mathbf{Q}}$ with possible changes of sign.

This shows in particular that there are rank-one connections which are neither
twins nor slips. Such rank-one connections without mirror symmetry across the
interface are observed for martensitic phase transformations, e.g. for ${\rm
LaNbO}_{4}$ (Jian & Wayman jianwayman1995 , Jian & James jianjames1997 ) and
for NiMnGa (Seiner et al. seiner2019 ).

Conjugates of slips are observed in ‘kink deformations’; see Inamura
inamura2019 , and for earlier work Starkey starkey1968 .

###### Acknowledgements.

This paper grew out of an invitation to give the 2019 Leçons Jacques-Louis
Lions at Sorbonne Université Paris, and was developed during visits to the
Hong Kong Institute for Advanced Study and the Hausdorff Institute of
Mathematics (funded by the Deutsche Forschungsgemeinschaft under Germany’s
Excellence Strategy – EXC-2047/1 – 390685813). The research forms part of the
project Mathematical theory of polycrystalline materials supported by the
EPSRC grant EP/V00204X. I am grateful to Sergio Conti, Duvan Henao, Tomonari
Inamura, Dick James, Kostas Koumatos, Michael Ortiz, Hanuš Seiner and Dave
Srolowitz for their interest and helpful suggestions, and to the referees for
careful reading which led to improvements in presentation.

## Conflict of interest

The author declares that he has no conflict of interest.

## References

  * (1) Ashcroft, N.W., Mermin, N.D.: Solid state physics.  New York: Holt, Rinehart and Winston (1976)
  * (2) Ball, J.M.: Mathematical models of martensitic microstructure.  Materials Science and Engineering A 78(1-2), 61–69 (2004)
  * (3) Ball, J.M., Carstensen, C.: Interaction of martensitic microstructures in adjacent grains.  In: Proceedings of International Conference on Martensitic Transformations, Chicago. TMS, Springer (2017).  ArXiv:1708.03286
  * (4) Ball, J.M., James, R.D.: Fine phase mixtures as minimizers of energy.  Arch. Ration. Mech. Anal. 100, 13–52 (1987)
  * (5) Bevis, M., Crocker, A.G.: Twinning shears in lattices.  Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 304(1476), 123–134 (1968)
  * (6) Bevis, M., Crocker, A.G.: Twinning modes in lattices.  Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 313(1515), 509–529 (1969)
  * (7) Bilby, B.A., Crocker, A.G.: The theory of the crystallography of deformation twinning.  Proceedings of the Royal Society of London. Series A. Mathematical and physical sciences 288(1413), 240–255 (1965)
  * (8) Chalmers, B., Martius, U.M.: Slip planes and the energy of dislocations.  Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 213(1113), 175–185 (1952)
  * (9) Chen, N.K., Maddin, R.: Slip planes and the energy of dislocations in a bodycentered cubic structure.  Acta Metallurgica 2(1), 49–51 (1954)
  * (10) Chen, X., Srivastava, V., Dabade, V., James, R.D.: Study of the cofactor conditions: Conditions of supercompatibility between phases.  Journal of the Mechanics and Physics of Solids 61(12), 2566–2587 (2013).  DOI https://doi.org/10.1016/j.jmps.2013.08.004.  URL https://www.sciencedirect.com/science/article/pii/S002250961300149X
  * (11) Christian, J.W.: The Theory of Transformation in Metals and Alloys Pergamon.  Elsevier, New York (2002)
  * (12) Christian, J.W., Mahajan, S.: Deformation twinning.  Progress in materials science 39(1-2), 1–157 (1995)
  * (13) Cordier, P.: Dislocations and slip systems of mantle minerals.  Reviews in Mineralogy and Geochemistry 51(1), 137–179 (2002)
  * (14) Ericksen, J.L.: Special topics in elastostatics.  In: C.S. Yih (ed.) Advances in Applied Mechanics, vol. 17, pp. 189–244. Academic Press (1977)
  * (15) Forclaz, A.: A simple criterion for the existence of rank-one connections between martensitic wells.  J. Elasticity 57, 281–305 (1999)
  * (16) Inamura, T.: Geometry of kink microstructure analysed by rank-1 connection.  Acta Materialia 173, 270–280 (2019).  DOI https://doi.org/10.1016/j.actamat.2019.05.023.  URL https://www.sciencedirect.com/science/article/pii/S1359645419303052
  * (17) Jackson, A.G.: Handbook of crystallography: for electron microscopists and others.  Springer Science & Business Media (2012)
  * (18) James, R.D.: Private communication
  * (19) Jaswon, M.A., Dove, D.B.: Twinning properties of lattice planes.  Acta Crystallographica 9(8), 621–626 (1956)
  * (20) Jaswon, M.A., Dove, D.B.: The prediction of twinning modes in metal crystals.  Acta Crystallographica 10(1), 14–18 (1957)
  * (21) Jaswon, M.A., Dove, D.B.: The crystallography of deformation twinning.  Acta Crystallographica 13(3), 232–240 (1960)
  * (22) Jian, L., James, R.: Prediction of microstructure in monoclinic LaNbO4 by energy minimization.  Acta Materialia 45(10), 4271–4281 (1997).  DOI 10.1016/S1359-6454(97)00080-3
  * (23) Jian, L., Wayman, C.M.: Electron back scattering study of domain structure in monoclinic phase of a rare-earth orthoniobate LaNbO4.  Acta metallurgica et materialia 43(10), 3893–3901 (1995)
  * (24) Khachaturyan, A.G.: Theory of Structural Transformations in Solids.  John Wiley (1983)
  * (25) Kihô, H.: The crystallographic aspect of the mechanical twinning in metals.  Journal of the Physical Society of Japan 9(5), 739–747 (1954).  DOI 10.1143/JPSJ.9.739.  URL https://doi.org/10.1143/JPSJ.9.739
  * (26) Kihô, H.: The crystallographic aspect of the mechanical twinning in Ti and $\alpha$-U.  Journal of the Physical Society of Japan 13(3), 269–272 (1958).  DOI 10.1143/JPSJ.13.269.  URL https://doi.org/10.1143/JPSJ.13.269
  * (27) Ortiz, M., Repetto, E.A.: Nonconvex energy minimization and dislocation structures in ductile single crystals.  Journal of the Mechanics and Physics of Solids 47(2), 397–462 (1999)
  * (28) Pitteri, M., Zanzotto, G.: Continuum models for phase transitions and twinning in crystals.  Chapman & Hall/CRC (2003)
  * (29) Seiner, H., Chulist, R., Maziarz, W., Sozinov, A., Heczko, O., Straka, L.: Non-conventional twins in five-layer modulated Ni-Mn-Ga martensite.  Scripta Materialia 162, 497–502 (2019).  DOI https://doi.org/10.1016/j.scriptamat.2018.12.020.  URL https://www.sciencedirect.com/science/article/pii/S1359646218307553
  * (30) Starkey, J.: The geometry of kink bands in crystals—a simple model.  Contributions to Mineralogy and Petrology 19(2), 133–141 (1968)
  * (31) Weinberger, C.R., Boyce, B.L., Battaile, C.C.: Slip planes in bcc transition metals.  International Materials Reviews 58(5), 296–314 (2013).  DOI 10.1179/1743280412Y.0000000015.  URL https://doi.org/10.1179/1743280412Y.0000000015