Optimality of general lattice transformations with applications to the Bain strain in steel

Abstract

This article provides a rigorous proof of a conjecture by E. C. Bain in 1924 on the optimality of the so-called Bain strain based on a criterion of least atomic movement. A general framework that explores several such optimality criteria is introduced and employed to show the existence of optimal transformations between any two Bravais lattices. A precise algorithm and a graphical user interface to determine this optimal transformation is provided. Apart from the Bain conjecture concerning the transformation from face-centred cubic to body-centred cubic, applications include the face-centred cubic to body-centred tetragonal transition as well as the transformation between two triclinic phases of terephthalic acid.

1. Introduction

In his seminal article on ‘The nature of martensite’, Bain [1] proposed a mechanism that transforms the face-centred cubic (fcc) lattice to the body-centred cubic (bcc) lattice, a phase transformation most importantly manifested in low-carbon steels. Bain [1] writes

It is reasonable, also, that the atoms them- selves will rearrange [⋯ ] by a method that will require least temporary motion. [⋯ ] A mode of atomic shift requiring minimum motion was conceived by the author [⋯ ]

The key observation that led to his famous corres- pondence was that ‘If one regards the centers of faces as corners of a new unit, a body-centered structure is already at hand; however, it is tetragonal instead of cubic.’ He remarks that this is not surprising ‘as it is the only easy method of constructing a bcc atomic structure from the fcc atomic structure.’

Even though it is now widely accepted, his mechanism, which he illustrated with a model made of cork balls and needles (figure 1), was not without criticism from his contemporaries. In their fundamental paper, Kurdjumov & Sachs [2] wrote (free translation from German) that ‘nothing certain about the mechanism of the martensite transformation is known. Bain imagines that a tetragonal unit cell within the fcc lattice transforms into a bcc unit cell through compression along one direction and expansion along the two other. However a proof of this hypothesis is still missing.’¹ Interestingly, without being aware of it, the authors implicitly used the Bain mechanism in their derivation of the Kurdjumov and Sachs orientation relationships (see [3] for details).

Figure 1.

From E. C. Bain: the small models show the fcc and bcc unit cells; the large models represent 35 atoms in an fcc and bcc arrangement, respectively.

With the years passing and a number of supporting experimental results (for a discussion, see, for example, [4]), the Bain mechanism rose from a conjecture to a widely accepted fact. Nevertheless, for almost a century after Bain first announced his correspondence, a rigorous proof based on the assumption of minimal atom movement has been missing. Of course, the transformation from fcc to bcc is not the only instance where the determination of the transformation strain is of interest. The overall question remains the same: Which transformation strain(s), out of all the possible deformations mapping the lattice of the parent phase to the lattice of the product phase, require(s) the least atomic movement?

To provide a definite answer to this question, one first needs to quantify the notion of least atomic movement in such a way that it does not require additional input from experiments. Then, one needs to establish a framework that singles out the optimal transformation among the infinite number of possible lattice transformation strains. One way to appropriately quantify least atomic movement is the criterion of smallest principal strains as suggested by Lomer [5]. In his paper, Lomer compared 1600 different lattice correspondences for the β to α phase transition in uranium and concluded that only one of them involved strains of less than 10%. More recently, in [6], an algorithm is proposed to determine the transformation strain based on a similar minimality criterion (see remark 4.7) that also allows for the consideration of different sublattices. The present paper considers a criterion of least atomic movement in terms of a family of different strain measures and, for each such strain measure, rigorously proves the existence of an optimal lattice transformation between any two Bravais lattices.² As a main application, it is shown that the Bain strain is the optimal lattice transformation from fcc to bcc with respect to three of the most commonly used strain measures.

The structure of the paper is as follows: after stating some preliminaries in §2, we explore in more detail some mathematical aspects of lattices in §3. This section is mainly intended for the mathematically inclined reader and may be skipped on first reading without inhibiting the understanding of §4, which constitutes the main part of this paper. In this section, we establish a geometric criterion of optimality and prove the existence of optimal lattice transformations for any displacive phase transition between two Bravais lattices. Additionally, a precise algorithm to compute these optimal strains is provided. In the remaining sections, the general theory is applied to prove the optimality of the Bain strain in an fcc-to-bcc transformation, to show that the Bain strain remains optimal in an fcc to body-centred tetragonal (bct) transformation and, finally, to derive the optimal transformation strain between two triclinic phases of terephthalic acid. Similar to the fcc-to-bcc transition, this phase transformation is of particular interest as it involves large stretches, and thus the lattice transformation requiring least atomic movement is not clear.

2. Preliminaries

Throughout the paper, it is assumed that both the parent and product lattices are Bravais lattices (see definition 2.8) and that the transformation strain, i.e. the deformation that maps a unit cell of the parent lattice to a unit cell of the product lattice, is homogeneous.

The following definitions are standard and will be used throughout.

Definition 2.1 —

Let $\mathcal{R}\in \{\mathbb{Z},\mathbb{R}\}$ denote the set of integer or real numbers, respectively, and define

• — ${\mathcal{R}}^{3\times 3}:$ vector space of matrices with entries in $\mathcal{R}$.

• — ${\mathcal{R}}_{+}^{3\times 3}:={\mathcal{R}}^{3\times 3}\cap \{det>0\}:$ orientation-preserving matrices with entries in $\mathcal{R}$.

• — $\mathrm{GL}(3,\mathcal{R}):={\mathcal{R}}^{3\times 3}\cap \{A\in {\mathcal{R}}^{3\times 3}\text{ is invertible}\}$ (general linear group).

• — ${\mathrm{GL}}^{+}(3,\mathcal{R}):=\mathrm{GL}(3,\mathcal{R})\cap {\mathcal{R}}_{+}^{3\times 3}$: set of orientation-preserving invertible 3×3 matrices with entries in $\mathcal{R}$.

• — $\mathrm{SL}(3,\mathbb{Z}):={\mathrm{GL}}^{+}(3,\mathbb{Z})=\{A\in {\mathbb{Z}}^{3\times 3}:detA=1\}$ (special linear group).

• — $\mathrm{SO}(3)=\{R\in {\mathbb{R}}^{3\times 3}:R^{T}R=\mathbb{I},detR=1\}$ (group of proper rotations).

• — ${\mathcal{P}}^{24}\subset \mathrm{SO}(3)$ (symmetry group of a cube)—see lemma 2.2.

Further define the multiplication of a matrix F and a set of matrices $\mathcal{S}$ by $F.\mathcal{S}:=\{FS:S\in \mathcal{S}\}$.

Lemma 2.2 establishes a characterization of the group ${\mathcal{P}}^{24}$, i.e. the set of all rotations that map a cube to itself.

Lemma 2.2 —

Let Q=[−1,1]³ be the cube of side length 2 centred at 0 and define ${\mathcal{P}}^{24}=\{P\in \mathrm{SO}(3):PQ=Q\}$. Then, $|{\mathcal{P}}^{24}|=24$ and ${\mathcal{P}}^{24}=\mathrm{SO}(3)\cap \mathrm{SL}(3,\mathbb{Z})$.

Proof. —

Suppose that $P\in {\mathcal{P}}^{24}$ and let {e_i}_i=1,2,3 denote the standard basis of ${\mathbb{R}}^3$. By linearity, P maps the face centres of Q to face centres, i.e. for each i=1,2,3 there exists j∈{1,2,3} such that Pe_i=±e_j. Therefore, P_ki=Pe_i⋅e_k=±δ_kj∈{−1,0,1} and thus $P\in \mathrm{SL}(3,\mathbb{Z})$.

Conversely, if $P\in \mathrm{SO}(3)\cap \mathrm{SL}(3,\mathbb{Z})$ then its columns form an orthonormal basis and, because P has integer entries, the columns have to be in the set {±e_i}_i=1,2,3. Hence, P is of the form

$$P=\sum _{j=1}^3\pm e_{k_j}\otimes e_j$$

and Pe_i=±(e_i⋅e_j)e_kj. Thus, P maps face centres of Q to face centres and, by linearity, the cube to itself. Further, there are precisely six choices (3×2) for the first column ±e_k1, four choices (2×2) for ±e_k2 and two choices for ±e_k3. Thus, taking into account the determinant constraint, there are 24 elements in ${\mathcal{P}}^{24}$. ▪

Remark 2.3 —

Essentially, the same proof can be used to show that $\mathrm{SO}(n)\cap \mathrm{SL}(n,\mathbb{Z})={\mathcal{P}}^{N_n}$, where ${\mathcal{P}}^{N_n}$ with N_n=2ⁿ⁻¹n! denotes the symmetry group of an n-dimensional cube. As shown earlier, the value of N_n arises from having 2×n choices for the first column of Q, then 2×(n−1) choices for the second column of Q,…, and the last column of Q is already fully determined by the determinant constraint.

Definition 2.4 (pseudo-metric and metric). —

Let X be a vector space and x, y, z∈X. A map $d:X\times X\to [0,\mathrm{\infty })$ is a pseudo-metric if

• (i) d(x,x)=0,

• (ii) d(x,y)=d(y,x) (symmetry),

• (iii) d(x,z)≤d(x,y)+d(y,z) (triangle inequality).

If, in addition, d is positive definite, i.e. d(x,y)=0⇔x=y, then we call d a metric.

Definition 2.5 (matrix normsmatrix norms). —

For a given matrix $A\in {\mathbb{R}}^{3\times 3}$, we define the following norms.

• — Frobenius norm:

  $$|A|=\sqrt{\mathrm{Tr}(A^{T}A)}=\sqrt{\sum _{i,j=1}^3{A}_{ij}^2}.$$
• — Spectral norm:

  $$|A{|}_2=\underset{|x|=1}{sup}|Ax|=\sqrt{\underset{i=1,2,3}{max}{\mathrm{\lambda }}_i(A^{T}A)}=\underset{i=1,2,3}{max}{\nu }_i(A),$$

  where for i=1,2,3, ν_i(A) are the principal stretches/singular values of A and λ_i(A^TA) are the eigenvalues of A^TA.
• — Column max norm:

  $$\parallel A{\parallel }_{2,\mathrm{\infty }}=\underset{i=1,2,3}{max}|A{e}_i|=\underset{i=1,2,3}{max}|a_i|,$$

  where {a₁,a₂,a₃} are the columns of A.

Unless otherwise specified, here and throughout the rest of the paper, |⋅| always denotes the Euclidean norm if the argument is a vector in ${\mathbb{R}}^3$, and the Frobenius norm if the argument is a matrix in ${\mathbb{R}}^{3\times 3}$. Additionally, we henceforth denote by col[A]:={a₁,a₂,a₃} the columns of the matrix A and then write A=[a₁,a₂,a₃].

The proofs of the following statements are elementary and can be found in standard textbooks on linear algebra [7].

Lemma 2.6 (properties of matrix norms). —

Both the Frobenius norm and the spectral norm are unitarily equivalent, i.e.

$$|RAS|=|A|$$ 2.1

for any R,S∈SO(3). Furthermore, both norms are submultiplicative, i.e. given $A,B,C\in {\mathbb{R}}^{3\times 3}$ then |ABC|≤|A||B||C|and thus in particular if |A||C|≠0 then

$$|B|\ge \frac{|ABC|}{|A||C|}.$$ 2.2

Further, the spectral norm is compatible with the Euclidean norm on ${\mathbb{R}}^3,$ i.e.

$$|Ax|\le |A{|}_2|x|.$$ 2.3

The following sets will be of particular importance when proving the optimality of lattice transformations.

Definition 2.7 (${\mathrm{SL}}^k(3,\mathbb{Z})$). —

For $k\in \mathbb{N}$ define

$${\mathrm{SL}}^{\mathrm{k}}(3,\mathbb{Z}):=\{A\in \mathrm{SL}(3,\mathbb{Z}):|A_{mn}|\le k\hspace{0.17em}\mathrm{\forall }\hspace{0.17em}m,n\in \{1,2,3\}\}$$

and

$${\mathrm{SL}}^{-k}(3,\mathbb{Z}):=\{A\in \mathrm{SL}(3,\mathbb{Z}):|{(A^{-1})}_{mn}|\le k\hspace{0.17em}\mathrm{\forall }\hspace{0.17em}m,n\in \{1,2,3\}\}.$$

Clearly, ${\mathrm{SL}}^{\hspace{0.17em}j}(3,\mathbb{Z})\subset {\mathrm{SL}}^k(3,\mathbb{Z})\hspace{0.17em}\mathrm{\forall }\hspace{0.17em}0\le j\le k$ and $|{\mathrm{SL}}^{-k}(3,\mathbb{Z})|=|{\mathrm{SL}}^k(3,\mathbb{Z})|$ for all $k\in \mathbb{Z}$. For example, we have $|\mathrm{S}{\mathrm{L}}^1(3,\mathbb{Z})|=3480,$ $|\mathrm{S}{\mathrm{L}}^2(3,\mathbb{Z})|=67\hspace{0.17em}704,$ $|\mathrm{S}{\mathrm{L}}^3(3,\mathbb{Z})|=640\hspace{0.17em}824,$ $|\mathrm{S}{\mathrm{L}}^4(3,\mathbb{Z})|=2\hspace{0.17em}597\hspace{0.17em}208,$ $|\mathrm{S}{\mathrm{L}}^5(3,\mathbb{Z})|=10\hspace{0.17em}460\hspace{0.17em}024$ and $|\mathrm{S}{\mathrm{L}}^6(3,\mathbb{Z})|=28\hspace{0.17em}940\hspace{0.17em}280$.

In the following, we recall some basic definitions and results from crystallography.

Definition 2.8 (Bravais lattice [8, ch. 3]). —

Let $F=[f_1,f_2,f_3]\in \mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R}),$ where col[F]={f₁,f₂,f₃} are the columns of F. We define the Bravais lattice $\mathcal{L}(F)$ generated by F as the lattice generated by col[F], i.e.

$$\mathcal{L}(F):=\mathrm{col}[F.{\mathbb{Z}}_{+}^{3\times 3}].$$

Thus, by definition, a Bravais lattice is ${\mathrm{span}}_{\mathbb{Z}}\{f_1,f_2,f_3\}$ together with an orientation.

Definition 2.9 (primitive, base-, body- and face-centred unit cells). —

Let $\mathcal{L}=\mathcal{L}(F)$ be generated by $F\in \mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R})$. We call the parallelepiped spanned by col[F] with one atom at each vertex a primitive unit cell of the lattice. We call a primitive unit cell with additional atoms in the centre of the bases a base-centred unit cell; we call a primitive unit cell with one additional atom in the body centre a body-centred (bc) unit cell; and we call a primitive unit cell with additional atoms in the centre of each of the faces a face-centred (fc) unit cell.

Remark 2.10 —

For any lattice generated by a base-, body- or face-centred unit cell, there is a primitive unit cell that generates the same lattice. Table 1 gives the lattice vectors that generate the equivalent primitive unit cell for a given base-centred (C), body-centred (I) or face-centred (F) unit cell spanned by the vectors $\{a,b,c\}\in {\mathbb{R}}^3$.

For our purposes, all unit cells that generate the same lattice are equivalent and, in order to keep the presentation as simple as possible, we will always work with the primitive description of a lattice. However, we note that often in the literature the unit cell is chosen such that it has maximal symmetry.

For example, for a face-centred cubic lattice, the unit cell would be chosen as face-centred and spanned by $\mathrm{col}[\mathbb{I}]=\{e_1,e_2,e_3\}$, so that it has the maximal ${\mathcal{P}}^{24}$ symmetry. However, if one considers primitive unit cells that span the same fcc lattice, then the one with maximal symmetry is given by the last entry in table 1 and thus spanned by col[F], where

$$F=\frac{1}{2}\left(\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right)$$

and has only sixfold symmetry.

Table 1.

Lattice vectors of a primitive unit cell that generates the same lattice.

primitive (P)	base-centred (C)	body-centred (I)	face-centred (F)
{a,b,c}	$\left\{{\displaystyle \frac{a-b}{2}},{\displaystyle \frac{a+b}{2}},c\right\}$	$\left\{{\displaystyle \frac{-a+b+c}{2}},{\displaystyle \frac{a-b+c}{2}},{\displaystyle \frac{a+b-c}{2}}\right\}$	$\left\{{\displaystyle \frac{b+c}{2}},{\displaystyle \frac{a+c}{2}},{\displaystyle \frac{a+b}{2}}\right\}$

Lemma 2.11 (identical lattice bases [8, result 3.1]). —

Let $\mathcal{L}(F)$ be generated by F= [f₁,f₂,f₃] and $\mathcal{L}(G)$ be generated by G=[g₁,g₂,g₃]. Then

$$\mathcal{L}(F)=\mathcal{L}(G)\iff G=F\mu \iff g_i=\sum _{j=1}^3{\mu }_{ji}{f}_j,$$

for some $\mu \in \mathrm{SL}(3,\mathbb{Z})$. The same result holds for a face-, base- and body-centred unit cell.

Definition 2.12 (lattice transformation). —

Given two lattices ${\mathcal{L}}_0=\mathcal{L}(F)$ and ${\mathcal{L}}_1=\mathcal{L}(G)$ generated by $F,G\in \mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R}),$ we call any matrix $H\in \mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R})$ such that $H.{\mathcal{L}}_0={\mathcal{L}}_1$ a lattice transformation from ${\mathcal{L}}_0$ to ${\mathcal{L}}_1$.

Remark 2.13 —

In the terminology of Ericksen (see, for example, [9], p. 62ff.), if ${\mathcal{L}}_0={\mathcal{L}}_1$ (i.e. G=Fμ), then the matrices H in definition 2.12 are precisely all the orientation-preserving elements in the global symmetry group of F. Additionally, the matrices H that are also rotations constitute the point group of the lattice. We point out that, in this terminology, ${\mathcal{P}}^{24}$ is the point group of any cubic lattice.

We end this section by defining the atom density of the lattice $\mathcal{L}(F)$ and relating it to the determinant of F.

Definition 2.14 (atom density). —

For a given lattice $\mathcal{L}$, we define the atom density $\rho (\mathcal{L})$ by

$$\rho (\mathcal{L}):=\underset{N\to \mathrm{\infty }}{lim}\frac{\mathrm{\#}\{Q_N\cap \mathcal{L}\}}{N^3},$$

where Q_N=[0,N]³ is the cube with side-length N and # counts the number of elements in a discrete set. Thus, $\rho (\mathcal{L})$ is the average number of atoms per unit volume.

Lemma 2.15 —

Let $\mathcal{L}=\mathcal{L}(F)$ be generated by $F\in \mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R}),$ then

$$\rho (\mathcal{L})=\frac{1}{detF}.$$

In particular, a transformation H=GF⁻¹ from ${\mathcal{L}}_0=\mathcal{L}(F)$ to ${\mathcal{L}}_1=\mathcal{L}(G)$ does not change the atom density if and only if it is volume preserving, i.e. $detH=1$. Further, the atom density is well defined.

Proof. —

Denote by $\mathcal{U}\subset {\mathbb{R}}^3$ the unit cell spanned by col[F], so that the volume of $\mathcal{U}$ is given by $|\mathcal{U}|=detF$. Taking n distinct points $x_i\in \mathcal{L}$, we find that $|\bigcup _{i=1}^n(x_i+\mathcal{U})|=$ $ndetF$, because all elements are disjoint (up to zero measure). Let l denote the side length of the smallest cube that contains $\mathcal{U}$. Further, as in definition 2.14, let Q_N=[0,N]³ denote a cube of side-length N and define $Q_N^{\pm }={[\mp 2l,N\pm 2l]}^3$. Then

$$Q_N^{-}\subset \bigcup _{x\in \mathcal{L}\cap Q_N}(x+\mathcal{U})\subset Q_N^{+},$$

and thus, by taking the volumes of the sets, we obtain

$${(N-4l)}^3\le \mathrm{\#}\{Q_N\cap \mathcal{L}\}detF\le {(N+4l)}^3.$$

Dividing by N³ and taking the limit $N\to \mathrm{\infty }$ yields the result. Because this limit exists for all sequences $N\to \mathrm{\infty }$, the density is well defined. ▪

3. Metrics and equivalence on matrices and lattices

Definition 3.1 (equivalent matrices and lattices). —

We define an equivalence relation $\sim$ between matrices F,G in $\mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R})$ by

$$F\sim G:\iff \mathrm{\exists }R\in \mathrm{SO}(3):G=RF,$$

so that the equivalence class [F] of F is given by $[F]=\{G\in {\mathbb{R}}_{+}^{3\times 3}:F\sim G\}$. We denote the quotient space, i.e. the space of all equivalence classes, by

$$\overline{\mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R})}:=\{[F]:F\in \mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R})\}.$$

Furthermore, we define an equivalence relation $\sim$ between two lattices ${\mathcal{L}}_0$ and ${\mathcal{L}}_1$ by

$${\mathcal{L}}_0\sim {\mathcal{L}}_1:\iff \mathrm{\exists }R\in \mathrm{SO}(3):{\mathcal{L}}_1=R.{\mathcal{L}}_0.$$

We are now in a position to define a metric on the quotient spaces.

Lemma and definition 3.2 (induced metric). —

Any pseudo-metric $d:\mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R})\times \mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R})\to [0,\mathrm{\infty })$ with the property

$$\begin{array}{}\text{(}\ast \text{)}& d(F,G)=0\iff G=T^{\ast }F\text{ for some }{T}^{\ast }\in \mathrm{SO}(3)\text{(*)}\end{array}$$

naturally induces a metric $\overline{d}$ on $\overline{\mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R})}$ via

$$\overline{d}([F],[G])=\underset{R,S\in \mathrm{SO}(3)}{min}d(RF,SG).$$

Proof. —

The quantity $\overline{d}$ is clearly well defined, so that in particular $\overline{d}([F],[G])=\overline{d}(F,G)$ and we may henceforth drop the [⋅] in the arguments of $\overline{d}$. We first show that $\overline{d}$ is a metric on $\overline{\mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R})}$. Positivity and symmetry are obvious from the definition. For definiteness note that if $\underset{R,S\in \mathrm{SO}(3)}{min}d(RF,SG)=0,$ then, by (*), we have S*G=T*R*F for some R*,S*,T*∈SO(3) and thus $F\sim G$. It remains to show the triangle inequality. We have

where we have used the triangle inequality, symmetry of d and (*). ▪

Example 3.3 —

The family of maps $d_r:\mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R})\times \mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R})\to [0,\mathrm{\infty }),\hspace{0.17em}r\in \mathbb{R}\mathrm{\setminus }\{0\}$ given by

$$d_r(F,G)=|{(F^{T}F)}^{r/2}-{(G^{T}G)}^{r/2}|$$

are pseudo-metrics such that (*) holds. In particular, each of them induces a metric ${\overline{d}}_r$ on the quotient space $\overline{\mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R})}$.

Proof. —

Positivity is obvious. The triangle inequality follows from the corresponding property of the Frobenius norm, i.e.

$$\begin{array}{rl}{d}_r(F,H)& =|{(F^{T}F)}^{r/2}-{(H^{T}H)}^{r/2}|\\ & \le |{(F^{T}F)}^{r/2}-{(G^{T}G)}^{r/2}|+|{(G^{T}G)}^{r/2}-{(H^{T}H)}^{r/2}|=d_r(F,G)+d_r(G,H)\end{array}$$

and the property (*) follows from

$$\begin{array}{rl}{d}_r(F,G)=0& \iff {(F^{T}F)}^{r/2}={(G^{T}G)}^{r/2}\iff {(F{G}^{-1})}^T(F{G}^{-1})=\mathbb{I}\\ & \iff F{G}^{-1}\in \mathrm{SO}(3)\iff F=T^{\ast }G\text{ for some }{T}^{\ast }\in \mathrm{SO}(3).\end{array}$$

 ▪

Remark 3.4 —

The metric d₂ has previously been used in [8], ch. 3, where it was defined as the distance between the metric C_F=F^TF=(f_i⋅f_j)_ij of a set of lattice vectors col[F]={f₁,f₂,f₃} and the metric C_G=G^TG=(g_i⋅g_j)_ij of a set of lattice vectors col[G]={g₁,g₂,g₃}. The use of the term ‘metric’ in [8] is not to be confused with the use of ‘metric’ in this paper.

4. Optimal lattice transformations

This section embodies the main part of this paper. We first establish what we mean by an optimal transformation from one lattice to another and then, for a family of such criteria, show the existence of optimal transformations between any two Bravais lattices.

Lemma 4.1 —

Let ${\mathcal{L}}_0=\mathcal{L}(F)$ and ${\mathcal{L}}_1=\mathcal{L}(G)$ be generated by $F,G\in \mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R})$. Then, all possible lattice transformations from ${\mathcal{L}}_0$ to ${\mathcal{L}}_1$ are given by $H_{\mu }=G\mu F^{-1},\mu \in \mathrm{SL}(3,\mathbb{Z})$. In particular, the lattices coincide if and only if there exists $\mu \in \mathrm{SL}(3,\mathbb{Z})$ such that $G\mu F^{-1}=\mathbb{I}$ and they are equivalent, i.e. ${\mathcal{L}}_0\sim {\mathcal{L}}_1,$ if and only if there exists $\mu \in \mathrm{SL}(3,\mathbb{Z})$ such that GμF⁻¹∈SO(3).

Proof. —

Let $H_{\mu }=G\mu F^{-1},\mu \in \mathrm{SL}(3,\mathbb{Z})$. Then

$$\begin{array}{rl}{H}_{\mu }.{\mathcal{L}}_0& =H_{\mu }.\mathcal{L}(F)=H_{\mu }.\mathrm{col}[F.{\mathbb{Z}}_{+}^{3\times 3}]=\mathrm{col}[H_{\mu }F.{\mathbb{Z}}_{+}^{3\times 3}]\\ & =\mathrm{col}[G\mu F^{-1}F.{\mathbb{Z}}_{+}^{3\times 3}]=\mathrm{col}[G\mu .{\mathbb{Z}}_{+}^{3\times 3}]=\mathrm{col}[G.{\mathbb{Z}}_{+}^{3\times 3}]={\mathcal{L}}_1,\end{array}$$

where we used that μ is invertible over $\mathbb{Z}$, so that $\mu .{\mathbb{Z}}_{+}^{3\times 3}={\mathbb{Z}}_{+}^{3\times 3}$. Thus, H_μ is a lattice transformation from ${\mathcal{L}}_0$ to ${\mathcal{L}}_1$. For the reverse direction, we know by lemma 2.11 that $\mathcal{L}(F)=\mathcal{L}(F^{\prime })$ if and only if $F^{\prime }=F{\mu }^{\prime },{\mu }^{\prime }\in \mathrm{SL}(3,\mathbb{Z})$ and $\mathcal{L}(G)=\mathcal{L}(G^{\prime })$ if and only if $G^{\prime }=G{\mu }^{″},{\mu }^{″}\in \mathrm{SL}(3,\mathbb{Z})$, so that all possible generators for ${\mathcal{L}}_0$ are given by $F{\mu }^{\prime },{\mu }^{\prime }\in \mathrm{SL}(3,\mathbb{Z})$, and all possible generators for ${\mathcal{L}}_1$ are given by $G{\mu }^{″},{\mu }^{″}\in \mathrm{SL}(3,\mathbb{Z})$. Thus, any lattice transformation from ${\mathcal{L}}_0$ to ${\mathcal{L}}_1$ is given by

$$H_{{\mu }^{\prime }}^{{\mu }^{″}}=G{\mu }^{″}{{\mu }^{\prime }}^{-1}{F}^{-1},{\mu }^{\prime },{\mu }^{″}\in \mathrm{SL}(3,\mathbb{Z}).$$

But, by the group property, we may set $H_{\mu }:=H_{{\mu }^{\prime }}^{{\mu }^{″}}$ with $\mu ={\mu }^{″}{{\mu }^{\prime }}^{-1}\in \mathrm{SL}(3,\mathbb{Z})$. ▪

Definition 4.2 (d-optimal lattice transformations). —

Given two lattices ${\mathcal{L}}_0=\mathcal{L}(F)$ and ${\mathcal{L}}_1=\mathcal{L}(G)$ with $F,G\in {\mathrm{GL}}^{+}(3,\mathbb{R})$ and a metric d, we call a lattice transformation $H_{\mu }=G\mu F^{-1},\hspace{0.17em}\mu \in \mathrm{SL}(3,\mathbb{Z})$ (cf. lemma 4.1) d-optimal if it minimizes the distance to $\mathbb{I}$ with respect to d, i.e. $H_{min}=G{\mu }_{min}{F}^{-1},$ where

$${\mu }_{min}={\arg \hspace{0.17em}\min }_{\mu \in \mathrm{SL}(3,\mathbb{Z})}d(H_{\mu },\mathbb{I}).$$ 4.1

If d is a pseudo-metric satisfying (*), we call a transformation d-optimal if it is optimal with respect to the induced metric $\overline{d}$ on the quotient space $\overline{\mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R})},$ i.e. the d-optimal transformation is the one that is d-closest to being a pure rotation and maps ${\mathcal{L}}_0$ to a lattice in the equivalence class $[{\mathcal{L}}_1]:=\{\mathcal{L}:\mathcal{L}\sim {\mathcal{L}}_1\}$ of ${\mathcal{L}}_1$.

Example 4.3 —

For the pseudo-metrics d_r from example 3.3, the explicit expressions for the distance in (4.1) read

$$d_r(H,\mathbb{I})=|{(H^{T}H)}^{r/2}-\mathbb{I}|={\left(\sum _{i=1}^3{({\nu }_i^r-1)}^2\right)}^{1/2},$$ 4.2

where ν_i, i=1,2,3, are the principal stretches/singular values of H. The quantities ${(H^{\mathrm{T}}H)}^{r/2}-\mathbb{I}$ are clearly measures of principal strain and are known as the Doyle–Ericksen strain tensors (see [10], p. 65). For r=1, it is simple to verify that

$$d_1(H,\mathbb{I})=\mathrm{dist}(H,\mathrm{SO}(3))=\underset{R\in \mathrm{SO}(3)}{min}|H-R|.$$

Remark 4.4 —

For a general metric d as in definition 4.2, the optimal transformation $H_{min}$ between $\mathcal{L}(F)$ and $\mathcal{L}(G)$ is unchanged under actions of the point groups of both lattices, i.e.

$$H_{min}=G\hspace{0.17em}{\arg \hspace{0.17em}\min }_{\mu \in \mathrm{SL}(3,\mathbb{Z})}d(G\mu F^{-1},\mathbb{I})F^{-1}=(PG)\hspace{0.17em}{\arg \hspace{0.17em}\min }_{\mu \in \mathrm{SL}(3,\mathbb{Z})}d((PG)\mu {(QF)}^{-1},\mathbb{I}){(QF)}^{-1}$$

for any P in the point group of $\mathcal{L}(G)$ and Q in the point group of $\mathcal{L}(F)$. Throughout the rest of the paper, we use only pseudo-metrics satisfying (*). In this case, the notion of optimality is invariant not only under actions of the respective point groups, but also under rigid body rotations of the product lattice. Thus, definition 4.2 returns an equivalence class $[H_{min}]\in \overline{\mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R})}$ of d-optimal transformations. By the polar decomposition theorem, we may, and henceforth always will, pick the symmetric representative

$${\overline{H}}_{min}:=\sqrt{H_{min}^T{H}_{min}}\in {\mathbb{R}}_{\mathrm{sym}}^{3\times 3},$$

i.e. the pure stretch component of the transformation $H_{min}$. Note that, in general, the set of minimizing equivalence classes $\{[H_{min}]:H_{min}\text{ is }d-\text{ optimal}\}$ may contain more than one element. In such a case, different regions of the parent lattice may transform according to any of these optimal strains, giving rise to, for example, twinning.

The pseudo-metrics d_r are additionally invariant under rotations from the right, i.e. $d_r(H,\mathbb{I})=d_r(HS,\mathbb{I})$ for all S∈SO(3). For any such pseudo-metric, a rigid body rotation R of the parent lattice ${\mathcal{L}}_0$ results in an optimal transformation with stretch component $R{\overline{H}}_{min}{R}^T$, where ${\overline{H}}_{min}$ is the stretch component of the optimal transformation from ${\mathcal{L}}_0$ to $[{\mathcal{L}}_1]$. Note that $R{\overline{H}}_{min}{R}^T$ is simply ${\overline{H}}_{min}$ expressed in a different basis and in particular, even though the coordinate representation is different, the underlying transformation mechanism is unchanged.

Our main theorem says that a d_r-optimal lattice transformation always exists and lemma 4.5 will be the crucial tool in the proof.

Key lemma 4.5 —

Let H be a lattice transformation from ${\mathcal{L}}_0=\mathcal{L}(F)$ to ${\mathcal{L}}_1=\mathcal{L}(G)$ and consider a lattice vector $f\in {\mathcal{L}}_0$ that is transformed by H to g=Hf. Then

$${\nu }_{max}(H)\ge |g|/|f|\ge {\nu }_{min}(H),$$

where ${\nu }_{min}(H),$ ${\nu }_{max}(H)$ denote, respectively, the smallest and largest principal stretches/singular values of H. In particular, for any s>0,

$$\begin{array}{c}{\mathrm{d}}_s(H,\mathbb{I})={\left(\sum _{i=1}^3{({\nu }_i^s-1)}^2\right)}^{1/2}\ge \underset{i}{max}\frac{|H{f}_i{|}^s}{|f_i{|}^s}-1,\end{array}$$ 4.3

$$\begin{array}{c}{\mathrm{d}}_{-s}(H,\mathbb{I})={\left(\sum _{i=1}^3{({\nu }_i^{-s}-1)}^2\right)}^{1/2}\ge \underset{i}{max}\frac{|H^{-1}{g}_i{|}^s}{|g_i{|}^s}-1,\end{array}$$ 4.4

where col[F]={f₁,f₂,f₃} and col[G]={g₁,g₂,g₃}.

Proof. —

Consider the singular value decomposition of H=UDV , where D= diag(ν₁,ν₂,ν₃) and U,V ∈SO(3). Then,

$$|g|=|UDVf|=|DVf|\le \underset{i}{max}{\nu }_i(H)|Vf|={\nu }_{max}(H)|f|$$

and analogously for the lower bound. ▪

Theorem 4.6 —

Given two lattices ${\mathcal{L}}_0=\mathcal{L}(F)$ and ${\mathcal{L}}_1=\mathcal{L}(G)$ generated by $F,G\in \mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R})$, respectively, there exists a d_r-optimal lattice transformation $H_{{\mu }_{min}}=G{\mu }_{min}{F}^{-1},$ for any $r\in \mathbb{R}\mathrm{\setminus }\{0\}$. For s>0, all optimal changes of basis are contained in the finite compact sets

$$\begin{array}{c}{d}_s:\left\{\mu \in \mathrm{SL}(3,\mathbb{Z}):\parallel \mu {\parallel }_{2,\mathrm{\infty }}^s\le \frac{\parallel F{\parallel }_{2,\mathrm{\infty }}^s}{{\nu }_{min}^s(G)}(m_{0,s}+1)\right\},\end{array}$$ 4.5

$$\begin{array}{c}{d}_{-s}:\left\{\mu \in \mathrm{SL}(3,\mathbb{Z}):\parallel {\mu }^{-1}{\parallel }_{2,\mathrm{\infty }}^s\le \frac{\parallel G{\parallel }_{2,\mathrm{\infty }}^s}{{\nu }_{min}^s(F)}(m_{0,-s}+1)\right\},\end{array}$$ 4.6

where ${\nu }_{min}(A)$ denotes the smallest principal stretch/singular value of A and

$$m_{0,r}:=d_r(H_{\mathbb{I}},\mathbb{I})=d_r(G{F}^{-1},\mathbb{I}).$$

Proof. —

As the minimization is over the discrete set $\mathrm{SL}(3,\mathbb{Z})$ it suffices to show that the minimum is attained in a (compact) finite subset of $\mathrm{SL}(3,\mathbb{Z})$ given by (4.5) and (4.6). Let $H_{\mu }=G\mu F^{-1},\hspace{0.17em}\mu \in \mathrm{SL}(3,\mathbb{Z})$ be a lattice transformation from ${\mathcal{L}}_0=\mathcal{L}(F)$ to ${\mathcal{L}}_1=\mathcal{L}(G)$. Then, letting {e_i}_i=1,2,3 denote the standard basis vectors of ${\mathbb{R}}^{3\times 3}$,

$$H_{\mu }{f}_i=G\mu F^{-1}F{e}_i=G\mu e_i\text{ and }{H}_{\mu }^{-1}{g}_i=F{\mu }^{-1}{G}^{-1}G{e}_i=F{\mu }^{-1}{e}_i,$$

and thus, by using the key lemma 4.5 and (2.3), we obtain

$$\begin{array}{rl}{\mathrm{d}}_s(H_{\mu },\mathbb{I})& \ge \underset{i}{max}\frac{|G\mu e_i{|}^s}{|f_i{|}^s}-1\ge \frac{\parallel \mu {\parallel }_{2,\mathrm{\infty }}^s}{|G^{-1}{|}_2^s\parallel F{\parallel }_{2,\mathrm{\infty }}^s}-1=\frac{{\nu }_{min}^s(G)\parallel \mu {\parallel }_{2,\mathrm{\infty }}^s}{\parallel F{\parallel }_{2,\mathrm{\infty }}^s}-1\mathrm{and}\\ {\mathrm{d}}_{-s}(H_{\mu },\mathbb{I})& \ge \underset{i}{max}\frac{|F{\mu }^{-1}{e}_i{|}^s}{|g_i{|}^s}-1\ge \frac{\parallel {\mu }^{-1}{\parallel }_{2,\mathrm{\infty }}^s}{|F^{-1}{|}_2^s\parallel G{\parallel }_{2,\mathrm{\infty }}^s}-1=\frac{{\nu }_{min}^s(F)\parallel {\mu }^{-1}{\parallel }_{2,\mathrm{\infty }}^s}{\parallel G{\parallel }_{2,\mathrm{\infty }}^s}-1,\end{array}$$

where in the equality we have used that {ν_i(A⁻¹)}_i=1,2,3={(ν_i(A))⁻¹}_i=1,2,3. Thus, $d_r(H_{\mu },\mathbb{I})>d_r(H_{\mathbb{I}},\mathbb{I})$ for all μ in the complement of the respective sets given by (4.5) and (4.6) and, therefore, H_μ cannot be d_r-optimal. ▪

Remark 4.7 —

The distance $d_1(H,\mathbb{I})$ seems to be the most natural candidate to determine the transformation requiring least atomic movement. The quantities ν_i−1 measure precisely the displacement along the principal axes and thus their use is in line with the criterion of smallest principal strains as in, for example, [4,5,11]. The distance $d_2(H,\mathbb{I})$ seems natural from a mathematical perspective as the tensor H^TH corresponds to the flat metric induced by the deformation H and it has also been used to define the Ericksen–Pitteri neighbourhood of a lattice (see, for example, (2.17) in [12]). Finally, the distance $d_{-2}(H,\mathbb{I})$ has recently been used in [6] in order to avoid singular behaviour when considering sublattices.

In the following, we illustrate the differences between d₁,d₂ and d₋₂ through a simple but instructive one-dimensional example.

Example 4.8 (a comparison of different optimality conditions). —

We consider two atoms A,B that are originally at unit distance, i.e. |A−B|=1, and then move the atom B to its deformed position B′. Thus, H is simply a scalar quantity given by H=|A−B′|/|A−B|=|A−B′|.

It can be seen from table 2 that d₁ depends only on the distance between B and B′; an expansion by 100% has the same d₁ distance to $\mathbb{I}$ as a contraction to 0, i.e. moving A onto B. The metric d₂ penalizes expansions more than contractions; for example, an expansion by ≈ 141% has the same d₂ distance to 1 as a contraction to 0. The metric d₋₂ penalizes contractions significantly more than expansions; for example, an expansion by $\mathrm{\infty }$ has the same d₋₂ distance to $\mathbb{I}$ as a contraction to ≈70%, i.e. reducing the distance between A and B by ≈30%.

Table 2.

Comparison of different distances.

	B′−B=0.5,	B′−B=−0.5,	deformation y such that
r	H=|A−B′|=1.5	H=|A−B′|=0.5	$d_r(y,\mathbb{I})=d_r(x,\mathbb{I})$
1	$d_1(H,\mathbb{I})=0.5$	$d_1(H,\mathbb{I})=0.5$	y=2−x
2	$d_2(H,\mathbb{I})=1.25$	$d_2(H,\mathbb{I})=0.75$	$y=\sqrt{2-x^2}$
−2	$d_{-2}(H,\mathbb{I})=0.\overline{5}$	$d_{-2}(H,\mathbb{I})=3$	$y=\frac{1}{\sqrt{2-x^{-2}}}$

A remark on the computation of the optimal transformation

Theorem 4.6 provides the necessary compactness result to reduce the original minimization problem over the infinite set $\mathrm{SL}(3,\mathbb{Z})$ to a finite subset given by (4.5) and (4.6), respectively. To this end, it is worth noting that the smaller the deformation distance $m_{0,r}=d_r(G{F}^{-1},\mathbb{I})$ of the initial lattice basis the smaller the radius of the ball in $\mathrm{SL}(3,\mathbb{Z})$ that contains the optimal μ. However, in specific cases, where better estimates are available, it might be advantageous to start with an initial lattice basis that is not optimal.

Nevertheless, in order to explicitly determine the optimal transformations, one still needs to compare the distances $d_r(H_{\mu },\mathbb{I})$ for all elements contained in the finite sets given by (4.5) and (4.6), respectively. This can easily be carried out with any modern computer algebra program, and possible implementations can be found in appendices A and B.

In order to ensure that the solution of this finite minimization problem is correct one needs to verify that the difference Δ between the minimal and the second to minimal deformation distance is large compared with possible rounding errors (if any). The computations in §§4a and 4b for the Bain strain from fcc to bcc/bct are exact and thus without rounding errors. In §4c, regarding the optimal transformation in terephthalic acid, we find that Δ>0.015, which is large compared with machine precision.

(a). The Bain strain in fcc to bcc

Having established the general theory of optimal lattice transformations, we apply these results to prove the optimality of the Bain strain with respect to the three different lattice metrics d_r, r=−2,1,2, from the previous example. In these cases, we rigorously prove the optimality of the Bain strain first proposed in [1].

Theorem 4.9 (Bain optimality). —

In a transformation from an fcc to a bcc lattice with no change in atom density, there are three distinct equivalence classes of d_r-optimal lattice transformations for r=1,2,−2. The stretch components are given by

$${\overline{H}}_{min}\in \{\mathrm{diag}(2^{-1/3},2^{1/6},2^{1/6}),\hspace{0.17em}\mathrm{diag}(2^{1/6},2^{-1/3},2^{1/6}),\hspace{0.17em}\mathrm{diag}(2^{1/6},2^{1/6},2^{-1/3})\},$$

i.e. the three Bain strains are the d_r-optimal lattice transformations in a volume-preserving fcc-to-bcc transformation for r=1,2,−2. The respective minimal metric distances are

$$\begin{array}{rl}{m}_{min,1}& =d_1(H_{min},\mathbb{I})=\sqrt{{(2^{-1/3}-1)}^2+2{(2^{1/6}-1)}^2}\approx 0.269,\\ m_{min,2}& =d_2(H_{min},\mathbb{I})=\sqrt{{(2^{-2/3}-1)}^2+2{(2^{1/3}-1)}^2}\approx 0.522,\\ m_{min,-2}& =d_{-2}(H_{min},\mathbb{I})=\sqrt{{(2^{2/3}-1)}^2+2{(2^{-1/3}-1)}^2}\approx \mathrm{0.656.}\end{array}$$

Proof. —

Let ${\mathcal{L}}_0$ denote the fcc lattice, where the fcc unit cell has unit volume, and let ${\mathcal{L}}_1$ denote the bcc lattice with the same atom density (figure 2). Then, ${\mathcal{L}}_0=\mathcal{L}(F)$ and ${\mathcal{L}}_1=\mathcal{L}(B)$, where

$$F=\frac{1}{2}\left(\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right)=[f_1,f_2,f_3]\mathrm{and}B=2^{-1/3}\frac{1}{2}\left(\begin{array}{ccc}-1& 1& 1\\ 1& -1& 1\\ 1& 1& -1\end{array}\right)=[b_1,b_2,b_3]$$ 4.7

and, in particular, $detF=detB=4^{-1}$. Let $H_{\mu }=B\mu F^{-1},\hspace{0.17em}\mu \in \mathrm{SL}(3,\mathbb{Z})$ denote the lattice transformation from ${\mathcal{L}}_0$ to ${\mathcal{L}}_1$ (cf. lemma 4.1). By definition, H_μ is optimal if μ satisfies (4.1). We first show the optimality with respect to d₁ and d₂. We may find an upper bound on the minimum by only considering $\mu \in \mathrm{S}{\mathrm{L}}^1(3,\mathbb{Z})$ (cf. definition 2.7). With the help of a computer, we find exactly 72 such μ's, and all corresponding deformations are (volume-preserving) Bain strains. To complete the proof, we employ our key lemma 4.5 to show that any $\mu \in \mathrm{SL}(3,\mathbb{Z})\mathrm{\setminus }\mathrm{S}{\mathrm{L}}^1(3,\mathbb{Z})$ cannot be optimal with respect to either d₁ or d₂. Let $\mu \in {\mathbb{Z}}^{3\times 3}$ be given by

$$\mu =\left(\begin{array}{ccc}{\alpha }_1& {\alpha }_2& {\alpha }_3\\ {\beta }_1& {\beta }_2& {\beta }_3\\ {\gamma }_1& {\gamma }_2& {\gamma }_3\end{array}\right).$$ 4.8

Then, b_μ,i=H_μf_i=Bμe_i=α_ib₁+β_ib₂+γ_ib₃ and, after dropping the index i, we obtain

$$|b_{\mu }{|}^2=({\alpha }^2+{\beta }^2+{\gamma }^2)|b_1{|}^2+2(\alpha \beta +\beta \gamma +\alpha \gamma )⟨b_1,b_2⟩,$$ 4.9

where we have used that |b_i|=|b_j| and 〈b_i,b_j〉=〈b_k,b_l〉 for all i≠j and k≠l. We compute |f_i|=2^−1/2, |b_i|²=3⋅2^−8/3 and 〈b_i,b_j〉=−2^−8/3. By (4.3), we estimate

$$\begin{array}{c}{d}_1(H_{\mu },\mathbb{I})\ge \frac{2^{-4/3}{(\rho (\alpha ,\beta ,\gamma ))}^{1/2}}{2^{-1/2}}-1,\end{array}$$ 4.10

$$\begin{array}{c}{d}_2(H_{\mu },\mathbb{I})\ge \frac{2^{-8/3}\rho (\alpha ,\beta ,\gamma )}{2^{-1}}-1,\end{array}$$ 4.11

where ρ(α,β,γ):=α²+β²+γ²+(α−β)²+(β−γ)²+(α−γ)². If $\mu \in \mathrm{SL}(3,\mathbb{Z})\mathrm{\setminus }\mathrm{S}{\mathrm{L}}^1(3,\mathbb{Z}),$ then ρ(α,β,γ)≥8 and thus

$$d_1(H_{\mu },\mathbb{I})\ge 2^{2/3}-1>0.5\gg m_{min,1}\text{ and }{d}_2(H_{\mu },\mathbb{I})\ge 2^{4/3}-1>1.5\gg m_{min,2}.$$

To show d₋₂-optimality, we consider H_μ=B(Fμ⁻¹)⁻¹ and use the ansatz (4.8) for μ⁻¹ instead of μ. We compute

$$|F{\mu }^{-1}{e}_i{|}^2=\frac{1}{4}{((\alpha +\beta )}^2+{(\beta +\gamma )}^2+{(\alpha +\gamma )}^2)$$

and we note that b_i=H_μFμ⁻¹e_i. Thus, by (4.4), we can estimate

$$d_{-2}(H_{\mu },\mathbb{I})\ge \frac{\frac{1}{4}\sigma (\alpha ,\beta ,\gamma )}{3\cdot 2^{-8/3}}-1=2\cdot 2^{2/3}-1>2.17\gg m_{min,-2},$$ 4.12

where σ(α,β,γ):=(α+β)²+(β+γ)²+(α+γ)² and we have used that σ(α,β,γ)≥6 for ${\mu }^{-1}\in \mathrm{SL}(3,\mathbb{Z})\mathrm{\setminus }\mathrm{S}{\mathrm{L}}^1(3,\mathbb{Z})$. Therefore, the d₋₂-optimal μ is contained in $\mathrm{S}{\mathrm{L}}^{-1}(3,\mathbb{Z})$. ▪

Figure 2.

Face-centred cubic and body-centred cubic unit cells with equal atom density.

Corollary 4.10 —

The three Bain strains remain the d_r-optimal lattice transformations, r=1,2,−2, from fcc to bcc if the volume changes by λ³, provided that λ>0.84 for r=1, λ>0.64 for r=2 and λ<1.19 for r=−2. The stretch components of the three optimal equivalence classes are given by

$${\overline{H}}_{min}^{\mathrm{\lambda }}\in \{\mathrm{\lambda }\mathrm{diag}(2^{-1/3},2^{1/6},2^{1/6}),\mathrm{\lambda }\mathrm{diag}(2^{1/6},2^{-1/3},2^{1/6}),\mathrm{\lambda }\mathrm{diag}(2^{1/6},2^{1/6},2^{-1/3})\}.$$

The minimal metric distances are given by

$$\begin{array}{rl}{m}_{min,1}^{\mathrm{\lambda }}& =d_1(H_{min}^{\mathrm{\lambda }},\mathbb{I})=\sqrt{{(2^{-1/3}\mathrm{\lambda }-1)}^2+2{(2^{1/6}\mathrm{\lambda }-1)}^2},\\ m_{min,2}^{\mathrm{\lambda }}& =d_2(H_{min}^{\mathrm{\lambda }},\mathbb{I})=\sqrt{{(2^{-2/3}{\mathrm{\lambda }}^2-1)}^2+2{(2^{1/3}{\mathrm{\lambda }}^2-1)}^2},\\ m_{min,-2}^{\mathrm{\lambda }}& =d_{-2}(H_{min}^{\mathrm{\lambda }},\mathbb{I})=\sqrt{{(2^{2/3}{\mathrm{\lambda }}^{-2}-1)}^2+2{(2^{-1/3}{\mathrm{\lambda }}^{-2}-1)}^2}.\end{array}$$

Proof. —

Replace $\mu \to \mathrm{\lambda }\mu$ in (4.8) in the proof of theorem 4.9. Then, (4.10)–(4.12), respectively, read

$$\begin{array}{rl}{d}_1(H_{\mu }^{\mathrm{\lambda }},\mathbb{I})& \ge \frac{2^{-4/3}({\mathrm{\lambda }}^2\rho {(\alpha ,\beta ,\gamma ))}^{1/2}}{2^{-1/2}}-1\ge \frac{2^{1/6}}{2}\mathrm{\lambda }\cdot \underset{\mathcal{S}}{inf}{\rho }^{1/2}-1,\\ d_2(H_{\mu }^{\mathrm{\lambda }},\mathbb{I})& \ge \frac{2^{-8/3}{\mathrm{\lambda }}^2\rho (\alpha ,\beta ,\gamma )}{2^{-1}}-1\ge 2^{-2/3}{\mathrm{\lambda }}^2\cdot \underset{\mathcal{S}}{inf}\rho -1,\\ d_{-2}(H_{\mu }^{\mathrm{\lambda }},\mathbb{I})& \ge \frac{\frac{1}{4}\sigma (\alpha ,\beta ,\gamma )}{3\cdot 2^{-8/3}{\mathrm{\lambda }}^2}-1\ge \frac{2^{2/3}}{3{\mathrm{\lambda }}^2}\cdot \underset{\mathcal{S}}{inf}\sigma -1.\end{array}$$

If as above $\mathcal{S}=\mathrm{SL}(3,\mathbb{Z})\mathrm{\setminus }\mathrm{S}{\mathrm{L}}^1(3,\mathbb{Z})$ then $\underset{\mathcal{S}}{inf}\rho =8$ and thus $d_1(H_{\mu }^{\mathrm{\lambda }},\mathbb{I})\ge m_{min,1}^{\mathrm{\lambda }}$ for λ>0.84 and $d_2(H_{\mu }^{\mathrm{\lambda }},\mathbb{I})\ge m_{min,2}^{\mathrm{\lambda }}$ for λ>0.64 and $\underset{\mathcal{S}}{inf}\sigma =6$ so that $d_{-2}(H_{\mu }^{\mathrm{\lambda }},\mathbb{I})\ge m_{min,-2}^{\mathrm{\lambda }}$ for λ<1.19. ▪

Remark 4.11 (relations between the minimal deformations for fcc to bcc). —

Let $\mathcal{L}(F)$ and $\mathcal{L}(B)$ be the fcc and bcc lattices, respectively. Let μ₀ be one of the optimal changes of lattice basis and $H_i=B{\mu }_i{F}^{-1},\hspace{0.17em}i=0,\dots ,71$, denote the 72 optimal lattice deformations associated with the optimal changes of basis ${\mu }_i\in \mathrm{SL}(3,\mathbb{Z})$ given by theorem 4.9. Then, all optimal H_i's and corresponding μ_i's are given by

$$H_{PQ}=P{H}_{0}Q=B{\mu }_{PQ}{F}^{-1},P,Q\in {\mathcal{P}}^{24},$$

where ${\mu }_{PQ}=B^{-1}PB{\mu }_0{F}^{-1}QF\in \mathrm{SL}(3,\mathbb{Z}).$ We note that the latter equation holds because ${\mathcal{P}}^{24}$ is the point group of both cubic lattices and thus B⁻¹PB and F⁻¹QF are contained in $\mathrm{SL}(3,\mathbb{Z})$. Because there are only three equivalence classes of optimal lattice transformations, the 72 optimal changes of lattice basis split into three sets of 24 μ_PQ's such that the 24 corresponding H_PQ's lie in the same equivalence class.

(b). Stability of the Bain strain

Theorem 4.9 showed that the Bain strain is optimal in an fcc-to-bcc phase transformation. In this section, we restrict our attention to r=1,2 and show that the Bain strain remains optimal for a range of lattice parameters in an fcc-to-bct phase transformation. This type of transformation is found, for example, in steels with higher carbon content. The strategy of the proof is to treat the bct phase as a perturbation of the bcc phase. To this end, for B as in (4.7), let the bct lattice be generated by

$$B_{\mathrm{AC}}=\mathrm{diag}(A,A,C)B=2^{-4/3}\left(\begin{array}{cc}-A& A\\ A& -A& A\\ C& -C\end{array}\right),$$ 4.13

so that C denotes the elongation (or shortening) of the bcc cell in the z-direction and A the elongation (or shortening) in the x- and y-direction. We note that, because ${\mathcal{P}}^{24}=\mathrm{SL}(3,\mathbb{Z})\cap \mathrm{SO}(3)$, the lattice $\mathcal{L}(B_{\mathrm{AC}})$ is equivalent to the lattices $\mathcal{L}(\mathrm{diag}(A,C,A)B)$ and $\mathcal{L}(\mathrm{diag}(C,A,A)B)$. Further, we define $m_i^{0,\mathrm{AC}}=d_i(\mathrm{diag}(2^{1/6}A,2^{1/6}A,2^{-1/3}C),\mathbb{I})$.

Proposition 4.12 provides the most important ingredient.

Proposition 4.12 (‘the first excited state’). —

In a volume-preserving transformation from an fcc to a bcc lattice, the second to minimal deformation distances are given by

$$m_1^1:=\underset{\begin{array}{c}\mu \in \mathrm{SL}(3,\mathbb{Z})\\ \mu \ne {\mu }_{min}\end{array}}{min}{d}_1(H_{\mu },\mathbb{I})\approx 0.70\text{ and }{m}_2^1:=\underset{\begin{array}{c}\mu \in \mathrm{SL}(3,\mathbb{Z})\\ \mu \ne {\mu }_{min}\end{array}}{min}{d}_2(H_{\mu },\mathbb{I})\approx \mathrm{1.64.}$$

In particular, all H_μ with $\mu \in \mathrm{SL}(3,\mathbb{Z})\mathrm{\setminus }\mathrm{S}{\mathrm{L}}^2(3,\mathbb{Z})$ have distance strictly larger than $m_r^1,$ r=1,2.

Proof. —

For brevity, let us call any deformation H_μ and the corresponding change of basis μ that has deformation distance $m_r^1,\hspace{0.17em}r=1,2$, an excited state. The proof follows along the same lines as the proof of theorem 4.9. First, we show with the help of a computer that the second to minimal deformation distance within $\mathrm{S}{\mathrm{L}}^2(3,\mathbb{Z})$ is given by the above, and by (4.10) and (4.11), respectively, applied on $\mathrm{SL}(3,\mathbb{Z})\mathrm{\setminus }\mathrm{S}{\mathrm{L}}^2(3,\mathbb{Z})$ we know that there cannot be any excited states in $\mathrm{SL}(3,\mathbb{Z})\mathrm{\setminus }\mathrm{S}{\mathrm{L}}^2(3,\mathbb{Z})$. ▪

Corollary 4.13 (‘the first excited state’ with volume change). —

In a transformation from an fcc to a bcc lattice with volume change λ³, the second to minimal deformation distances are given by

$$\begin{array}{rl}{m}_1^{1,\mathrm{\lambda }}& :=\underset{\begin{array}{c}\mu \in \mathrm{SL}(3,\mathbb{Z})\\ \mu \ne {\mu }_{min}\end{array}}{min}{d}_1(H_{\mu }^{\mathrm{\lambda }},\mathbb{I})=2^{-3/2}\sqrt{25\hspace{0.17em}{2}^{1/3}{\mathrm{\lambda }}^2-4\hspace{0.17em}{2}^{2/3}(4+\sqrt{17})\mathrm{\lambda }+24},\\ m_2^{1,\mathrm{\lambda }}& :=\underset{\begin{array}{c}\mu \in \mathrm{SL}(3,\mathbb{Z})\\ \mu \ne {\mu }_{min}\end{array}}{min}{d}_2(H_{\mu }^{\mathrm{\lambda }},\mathbb{I})=2^{-3}\hspace{0.17em}\sqrt{305\hspace{0.17em}{2}^{2/3}{\mathrm{\lambda }}^4-400\hspace{0.17em}{2}^{1/3}{\mathrm{\lambda }}^2+192}.\end{array}$$

In particular, all $H_{\mu }^{\mathrm{\lambda }}$ with $\mu \in \mathrm{SL}(3,\mathbb{Z})\mathrm{\setminus }\mathrm{S}{\mathrm{L}}^2(3,\mathbb{Z})$ have distance strictly larger than $m_r^{1,\mathrm{\lambda }},$ r=1,2.

Theorem 4.14 —

The Bain strain is a d₁- and d₂-optimal lattice transformation from fcc to bct with lattice parameters A,C in the range

$$\begin{array}{rl}& \{(A,C):\hspace{0.17em}C\ge A>0.75\hspace{0.17em}\text{ and }\hspace{0.17em}{m}_1^1-3^{3/2}|B_{\mathrm{AC}}-B|\ge m_1^{0,\mathrm{AC}}\}\text{ for }r=1,\\ & \{(A,C):\hspace{0.17em}C\ge A>0.75\hspace{0.17em}\text{ and }\hspace{0.17em}{m}_2^1-27|B_{\mathrm{AC}}^T{B}_{\mathrm{AC}}-B^{T}B|\ge m_2^{0,\mathrm{AC}}\}\text{ for }r=2\end{array}$$

(cf. figure 3). For C>A, the stretch components of the optimal lattice transformations are given by ${\overline{H}}_{min}^{\mathrm{AC}}$ in the set

$$\left\{\left(\begin{array}{ccc}{2}^{1/6}A& 0& 0\\ 0& 2^{1/6}A& 0\\ 0& 0& 2^{-1/3}C\end{array}\right),\left(\begin{array}{ccc}{2}^{1/6}A& 0& 0\\ 0& 2^{-1/3}C& 0\\ 0& 0& 2^{1/6}A\end{array}\right),\left(\begin{array}{ccc}{2}^{-1/3}C& 0& 0\\ 0& 2^{1/6}A& 0\\ 0& 0& 2^{1/6}A\end{array}\right)\right\}.$$

The respective minimal metric distances are

$$\begin{array}{rl}{m}_{min,1}^{\mathrm{AC}}& =d_1(H_{min}^{\mathrm{AC}},\mathbb{I})={(2{(2^{1/6}A-1)}^2+{(2^{-1/3}C-1)}^2)}^{1/2}=m_1^{0,\mathrm{AC}},\end{array}$$ 4.14

$$\begin{array}{rl}{m}_{min,2}^{\mathrm{AC}}& =d_2(H_{min}^{\mathrm{AC}},\mathbb{I})={(2{({(2^{1/6}A)}^2-1)}^2+{({(2^{-1/3}C)}^2-1)}^2)}^{1/2}=m_2^{0,\mathrm{AC}}\end{array}$$ 4.15

and are achieved by exactly 24 distinct $\mu \in \mathrm{SL}(3,\mathbb{Z})$. The case C=A corresponds to an fcc-to-bcc transformation with volume change λ³ with λ=A=C, and we refer to corollary 4.10.

Figure 3.

The range of A,C where the Bain strain remains d₁-optimal (a) and d₂-optimal (b). (Online version in colour.)

Example 4.15 —

A=C=1 recovers theorem 4.9. C>A corresponds to the fcc-to-bct transformation found in, for example, steels with higher carbon content. $C=\sqrt{2}A=2^{1/3}$ is the bct lattice that is contained in the fcc lattice, i.e. $d({\mathcal{L}}_0,{\mathcal{L}}_1)=0$.

Proof of theorem 4.14. —

We will show that precisely 24 of the 72 μ's that were optimal in the fcc-to-bcc transition remain optimal. Let us start from one of those optimal transformations from fcc to bcc given by, for example,

$${\mu }_0=\left(\begin{array}{ccc}1& 1& 1\\ 0& 1& 0\\ 0& 1& 1\end{array}\right).$$

We know that H^AC_μ0=B_ACμ₀F⁻¹ is optimal for A=C=1, where B_AC is given by (4.13). The deformation distance is $d_r(H_{{\mu }_0}^{\mathrm{AC}},\mathbb{I})=m_{min,r}^{\mathrm{AC}}$ with $m_{min,r}^{\mathrm{AC}}$ given by (4.14) and (4.15), respectively. Further there exist 24 different μ's and corresponding H^AC_μ's that have the same distance. This follows as in remark 4.11 with the only difference being that the point group of the bct lattice has only eight elements. Then, owing to the invariance of d₁ and d₂ under multiplication from the left or right by any rotation, the 24 matrices H^AC_μ trivially have the same distance to $\mathbb{I}$ and one may easily verify that these H^AC_μ's are equally split into the three equivalence classes as in the statement of the theorem.

The remaining 48 μ's that were optimal for fcc to bcc lead to a larger deformation distance. One of these non-optimal lattices is generated by

$${\tilde{B}}_{\mathrm{AC}}=2^{-4/3}\left(\begin{array}{cc}-A& 0\\ A& 0\\ -C& -C& -2C\end{array}\right)$$

with

$$\begin{array}{rl}& {(d_1({\tilde{B}}_{\mathrm{AC}}{F}^{-1},\mathbb{I}))}^2-{(m_{min,1}^{\mathrm{AC}})}^2=2^{-5/3}(C-A)(2^{1/6}\hspace{0.17em}{A}^2+2^{1/6}\hspace{0.17em}{C}^2-4+2^{3/2})>0\\ & \Leftarrow C>A\text{ and }{A}^2+C^2>\frac{2^{5/6}}{2-\sqrt{2}}\hspace{0.17em}\Leftarrow \hspace{0.17em}C>A>0.75\end{array}$$ 4.16

and

$$\begin{array}{rl}& {(d_2({\tilde{B}}_{\mathrm{AC}}{F}^{-1},\mathbb{I}))}^2-{(m_{min,2}^{\mathrm{AC}})}^2=2^{-4/3}(C-A)(A+C)(3A^2+3C^2-2\hspace{0.17em}{2}^{2/3})>0\\ & \Leftarrow \hspace{0.17em}C>A\text{ and }{A}^2+C^2>\frac{2^{8/3}}{3}\hspace{0.17em}\Leftarrow \hspace{0.17em}C>A>0.75,\end{array}$$ 4.17

which holds true for all C>A in the range under consideration. The remaining 47 non-optimal deformations $H_{\mu }^{\mathrm{AC}}$ are all ${\mathcal{P}}^{24}$ related and thus have the same distance; in particular, larger than $m_{min,r}^{\mathrm{AC}}$.

To show the minimality of the 24 H^AC_μ's, we make use of our result on the first excited state to compare their distance to $\mathbb{I}$ against all the remaining μ's that were non-optimal in the fcc-to-bcc transition. In particular, we need to show that

$$m_{min,i}^{\mathrm{AC}}<\underset{\mu \ne {\mu }_{min}}{min}{d}_r(H_{\mu }^{\mathrm{AC}},\mathbb{I}),r=1,2,$$

where $H_{\mu }^{\mathrm{AC}}=B_{\mathrm{AC}}\mu F^{-1}$ and ${\mu }_{min}$ is any of the 72 minimizing μ's from theorem 4.9. Let us set $H_{\mu }:=H_{\mu }^{11}$ and estimate using the properties of d_r (cf. example 3.3)

$$\begin{array}{rl}\underset{\mu \ne {\mu }_{min}}{min}{d}_r(H_{\mu }^{\mathrm{AC}},\mathbb{I})& \ge \underset{\mu \ne {\mu }_{min}}{min}(d_r(H_{\mu },\mathbb{I})-d_r(H_{\mu }^{\mathrm{AC}},H_{\mu }))\\ & \ge m_r^1-\underset{\mu \ne {\mu }_{min}}{max}{d}_r(H_{\mu }^{\mathrm{AC}},H_{\mu }),\end{array}$$ 4.18

where $m_r^1$ denotes the first excited state (cf. proposition 4.12). We estimate

$$\begin{array}{rl}{d}_1(H_{\mu }^{\mathrm{AC}},H_{\mu })& \le |H_{\mu }^{\mathrm{AC}}-H_{\mu }|\le |B_{\mathrm{AC}}-B||\mu F^{-1}|,\\ d_2(H_{\mu }^{\mathrm{AC}},H_{\mu })& \le |{H_{\mu }^{\mathrm{AC}}}^T{H}_{\mu }^{\mathrm{AC}}-H_{\mu }^T{H}_{\mu }|\le |B_{\mathrm{AC}}^T{B}_{\mathrm{AC}}-B^{T}B||\mu F^{-1}{|}^2\end{array}$$

and, with the help of a computer, we calculate $\underset{\mu \in \mathrm{S}{\mathrm{L}}^1(3,\mathbb{Z})}{max}|\mu F^{-1}|=3^{3/2}$. Thus, a sufficient condition that the Bain strain is d₁-optimal within $\mathrm{S}{\mathrm{L}}^1(3,\mathbb{Z})$ is that 0.75<A≤C satisfy

$$m_1^1-3^{3/2}|B_{\mathrm{AC}}-B|\ge m_{min,1}^{\mathrm{AC}},$$

yielding the area drawn in figure 3a. To exclude any $\mu \in \mathrm{SL}(3,\mathbb{Z})\mathrm{\setminus }\mathrm{S}{\mathrm{L}}^1(3,\mathbb{Z}),$ we replace b by b_AC=diag(A,A,C)b in (4.9) in the proof of theorem 4.9 and estimate |b_AC|≥A|b|. Concluding as in (4.10), we arrive at

$$\underset{\mu \in \mathrm{SL}(3,\mathbb{Z})\mathrm{\setminus }\mathrm{S}{\mathrm{L}}^1(3,\mathbb{Z})}{min}{d}_1(H_{\mu }^{\mathrm{AC}},\mathbb{I})\ge 2^{2/3}A-1,$$ 4.19

which needs to be larger than $m_{min,1}^{\mathrm{AC}}$. This holds true, for example, for A∈[0.85,1.7] and C∈[A,1.7]. The proof of the d₂-optimality proceeds analogously. To obtain d₂-optimality within $\mathrm{S}{\mathrm{L}}^1(3,\mathbb{Z})$, we need to satisfy

$$m_2^1-27|B_{\mathrm{AC}}^T{B}_{\mathrm{AC}}-B^{T}B|\ge m_{min,2}^{\mathrm{AC}}$$

for all 0.75<A≤C, which yields the area drawn in figure 3b. To exclude all elements in the complement of $\mathrm{S}{\mathrm{L}}^1(3,\mathbb{Z})$, we have to ensure that $2^{4/3}A-1>m_{min,2}^{\mathrm{AC}}$, which holds true, for example, for A∈[0.75,1.5] and C∈[A,1.5]. ▪

Remark 4.16 —

By employing corollaries 4.10 and 4.13, we may extend the range of optimal parameters A and C in theorem 4.14 (cf. figure 3) by shifting the reference point from A=C=1 to A=C=λ. This enables us to show that the Bain strain remains the d₁- and d₂-optimal lattice deformation from fcc to bct in a much larger range of lattice parameters C≥A. The shaded regions in figure 4 show the values of A,C such that the deformation $H_{min}^{\mathrm{AC}}$ remains d₁- and d₂-optimal.

Figure 4.

Extended d₁-optimality (a) and d₂-optimality (b) range for A and C. The dark-shaded regions are obtained by a perturbation argument about the fixed optimal transformations $H_{min}^{{\mathrm{\lambda }}_0}$, λ₀=0.9,1.1,1.3. The light-shaded regions are obtained by a perturbation argument about the (A,C)-dependent optimal transformation $H_{min}^{\mathrm{\lambda }(A,C)}$ with $\mathrm{\lambda }(A,C)=0.995\sqrt{AC}$. (Online version in colour.)

Proof. —

The main idea of the proof is to consider the element $H_{\mu }^{\mathrm{\lambda }}=\mathrm{\lambda }{H}_{\mu }$ which is optimal in an fcc-to-bcc transformation with volume change (cf. corollary 4.10) and ‘closest’ to H^AC_μ. Thus, in (4.18), we write

$$\underset{\mu \ne {\mu }_{min}}{min}{d}_r(H_{\mu }^{\mathrm{AC}},\mathbb{I})\ge \underset{\mu \ne {\mu }_{min}}{min}(d_r(H_{\mu }^{\mathrm{\lambda }},\mathbb{I})-d_r(H_{\mu }^{\mathrm{\lambda }},H_{\mu }^{\mathrm{AC}})),$$

and for d₁ we estimate

$$\underset{\mu \ne {\mu }_{min}}{min}{d}_1(H_{\mu }^{\mathrm{AC}},\mathbb{I})\ge m_1^{1,\mathrm{\lambda }}-\underset{\mu \ne {\mu }_{min}}{max}|\mathrm{\lambda }{H}_{\mu }-H_{\mu }^{\mathrm{AC}}|\ge m_1^{1,\mathrm{\lambda }}-\mathrm{\lambda }\underset{\mu \ne {\mu }_{min}}{max}|\mu F^{-1}||B-B_{(A/\mathrm{\lambda })(C/\mathrm{\lambda })}|.$$

To obtain the d₁-optimality in $\mathrm{S}{\mathrm{L}}^1(3,\mathbb{Z})$, we again use $\underset{\mu \in \mathrm{S}{\mathrm{L}}^1(3,\mathbb{Z})}{max}|\mu F^{-1}|=3^{3/2}$ so that $m_1^{1,\mathrm{\lambda }}-\mathrm{\lambda }{3}^{3/2}|B-B_{(A/\mathrm{\lambda })(C/\mathrm{\lambda })}|\ge m_{min,1}^{\mathrm{AC}}$. Condition (4.19) regarding d₁-optimality outside $\mathrm{S}{\mathrm{L}}^1(3,\mathbb{Z})$ remains unchanged. Both conditions combined yield the area shown in figure 4a. Analogously, in order to show d₂-optimality, we estimate

$$\begin{array}{rl}\underset{\mu \ne {\mu }_{min}}{min}{d}_2(H_{\mu }^{\mathrm{AC}},\mathbb{I})& \ge m_2^{1,\mathrm{\lambda }}-\underset{\mu \ne {\mu }_{min}}{max}|{\mathrm{\lambda }}^2{H}_{\mu }^T{H}_{\mu }-{H_{\mu }^{\mathrm{AC}}}^T{H}_{\mu }^{\mathrm{AC}}|\\ & \ge m_2^{1,\mathrm{\lambda }}-{\mathrm{\lambda }}^2\underset{\mu \ne {\mu }_{min}}{max}|\mu F^{-1}{|}^2|B^{T}B-B_{(A/\mathrm{\lambda })(C/\mathrm{\lambda })}^T{B}_{(A/\mathrm{\lambda })(C/\mathrm{\lambda })}|\end{array}$$

and again use $\underset{\mu \in \mathrm{S}{\mathrm{L}}^1(3,\mathbb{Z})}{max}|\mu F^{-1}|=3^{3/2}$ to show d₂-optimality within $\mathrm{S}{\mathrm{L}}^1(3,\mathbb{Z})$. The condition to be d₂-optimal outside $\mathrm{S}{\mathrm{L}}^1(3,\mathbb{Z})$ remains unchanged. Both conditions combined yield the area shown in figure 4b. ▪

Remark 4.17 —

The previous estimates can be iterated, i.e. instead of picking the $H_{\mu }^{\mathrm{\lambda }}$ that is closest to $H_{\mu }^{\mathrm{AC}}$ one may pick any $H_{\mu }^{A^{\prime }{C}^{\prime }}$ that is in the range indicated in figure 4 that is closest to $H_{\mu }^{\mathrm{AC}}$.

If C≤A, following the proof of theorem 4.14, one finds that the optimal strain becomes

$${\overline{H}}_{min}=\left(\begin{array}{ccc}{2}^{1/6}\frac{A+C}{2}& \pm 2^{1/6}\frac{A-C}{2}& 0\\ \pm 2^{1/6}\frac{A-C}{2}& 2^{1/6}\frac{A+C}{2}& 0\\ 0& 0& 2^{-1/3}A\end{array}\right)+\text{ its }{\mathcal{P}}^{24}\text{ conjugates}$$

at least if A and C are in the regions specified in the statement of theorem 4.14 with C≥A replaced by A≥C.

(c). Terephthalic acid

Terephthalic acid is a material that has two triclinic phases (type I and II) which are very different from each other (cf. [13], p. 46 ff.). Thus, any lattice transformation necessarily requires large principal stretches and, unlike in the Bain setting, it is not clear what a good candidate for the optimal transformation would be. However, with the help of the proposed framework the d_r-optimal lattice transformation can easily be determined. The only required input parameters are the lattice parameters of the two triclinic unit cells (cf. [14]) listed in table 3.

Table 3.

Lattice parameters of the triclinic unit cells of terephthalic acid.

form	a/A°	b/A°	c/A°	α/°	β/°	γ/°
I	7.730	6.443	3.749	92.75	109.15	95.95
II	7.452	6.856	5.020	116.6	119.2	96.5

To apply our analysis, we first ought to convert the triclinic to the primitive description.

Lemma 4.18 (conversion from triclinic to primitive unit cell). —

The triclinic unit cell with lattice parameters a,b,c and α,β,γ generates up to an overall rotation the same lattice as

$$F=\left(\begin{array}{ccc}a& b\cos (\gamma )& c\cos (\beta )\\ 0& b\sin (\gamma )& c{\sin }^{-1}\gamma (\cos (\alpha )-\cos (\beta )\cos (\gamma ))\\ 0& 0& c{({\sin }^2\beta -{\sin }^{-2}\gamma {(\cos (\alpha )-\cos (\beta )\cos (\gamma ))}^2)}^{1/2}\end{array}\right).$$

Proof. —

Let col[F]={f₁,f₂,f₃}. It is easy to verify that |f₁|=a, |f₂|=b, |f₃|=c and that $\measuredangle (f_1,f_2)=\gamma$, $\measuredangle (f_1,f_3)=\beta$, $\measuredangle (f_2,f_3)=\alpha$. ▪

Example 4.19 (primitive cells of terephthalic acid). —

Application of lemma 4.18 to the lattice parameters in table 3 leads to

$$F_I=\left(\begin{array}{ccc}7.730& -0.668& -1.230\\ 0& 6.408& -0.309\\ 0& 0& 3.528\end{array}\right)\text{ and }{F}_{II}=\left(\begin{array}{ccc}7.452& -0.776& -2.449\\ 0& 6.812& -2.541\\ 0& 0& 3.570\end{array}\right)$$ 4.20

(all measures in A°).

Theorem 4.20 (optimal lattice transformations in terephthalic acid). —

The unique equivalence class of d₁- and d₂-optimal transformations between terephthalic acid form I and terephthalic acid form II has a stretch component given by

$${\overline{H}}_{min}=\left(\begin{array}{ccc}0.820& -0.125& -0.072\\ -0.125& 0.994& -0.146\\ -0.072& -0.146& 1.329\end{array}\right)$$ 4.21

with principal stretches ν₁=0.725, ν₂=1.033 and ν₃=1.385. The minimal distances are given by $m_{min,1}=d_1({\overline{H}}_{min},\mathbb{I})=0.474$ and $m_{min,2}=d_2({\overline{H}}_{min},\mathbb{I})=1.035$.

Proof. —

We apply theorem 4.6 with F=F_I and G=F_II. We calculate $\parallel F_I{\parallel }_{2,\mathrm{\infty }}=|F_I{e}_1|=7.730<7.8$, ${\nu }_{min}(F_{II})=3.076>3$, m_0,1=0.529<0.55 and m_0,2=1.197<1.2. Further, we note that if $\mu \in \mathrm{SL}(3,\mathbb{Z})\mathrm{\setminus }{\mathrm{SL}}^{k-1}(3,\mathbb{Z}),$ then $\parallel \mu {\parallel }_{2,\mathrm{\infty }}\ge \sqrt{k^2+1}$. Therefore, by (4.5), any d₁-optimal μ satisfies

$$\parallel \mu {\parallel }_{2,\mathrm{\infty }}\le \frac{\parallel F{\parallel }_{2,\mathrm{\infty }}}{{\nu }_{min}(G)}(m_{0,1}+1)<\frac{7.8}{3}(0.55+1)=4.03<\sqrt{4^2+1}$$

and is thus contained in $\mathrm{S}{\mathrm{L}}^3(3,\mathbb{Z})$ and, by (4.5), any d₂-optimal μ satisfies

$$\parallel \mu {\parallel }_{2,\mathrm{\infty }}^2\le \frac{\parallel F{\parallel }_{2,\mathrm{\infty }}^2}{{\nu }_{min}^2(G)}(m_{0,2}+1)<\frac{{7.8}^2}{3^2}(1.2+1)=14.872$$

and is thus also contained in $\mathrm{S}{\mathrm{L}}^3(3,\mathbb{Z})$. With the help of a computer, we check that, within the set $\mathrm{S}{\mathrm{L}}^3(3,\mathbb{Z}),$ the minimum is in both cases attained at

$${\mu }_{min}=\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 1& 1& -1\end{array}\right)$$

with corresponding ${\overline{H}}_{min}=\sqrt{H_{{\mu }_{min}}^T{H}_{{\mu }_{min}}}$ as in (4.21), $m_{min,1}=d_1({\overline{H}}_{min},\mathbb{I})=0.474$ and $m_{min,2}=d_2({\overline{H}}_{min},\mathbb{I})=1.035$. ▪

Remark 4.21 —

We have shown that the d₁- and d₂-optimal transformations from form I to form II of terephthalic acid are the same. However, the d₋₂-optimal transformation is different and given by

$${\overline{H}}_{min,-2}=\left(\begin{array}{ccc}0.852& -0.119& -0.018\\ -0.119& 0.950& -0.197\\ -0.018& -0.197& 1.346\end{array}\right)$$

with principal stretches ν₁=0.743, ν₂=0.977 and ν₃=1.429. As expected, the smallest principal stretch ν₁ is bigger than before, because d₋₂ penalizes contractions significantly more than expansions. To obtain the required analytical bounds, one calculates $\parallel F_{II}{\parallel }_{2,\mathrm{\infty }}=|F_{II}{e}_1|=7.452<7.46$, ${\nu }_{min}(F_I)=3.464>3.45$ and m_0,−2=1.080<1.1 to get $\parallel {\mu }^{-1}{\parallel }_{2,\mathrm{\infty }}^2<10$ (cf. (4.6)) and thus the optimal μ lies in $\mathrm{S}{\mathrm{L}}^{-2}(3,\mathbb{Z})$. We note that the calculated principal stretches differ from the ones in [6]—possibly owing to the use of sublattices.

5. Concluding remarks

This paper provides a rigorous proof for the existence of an optimal lattice transformation between any two given Bravais lattices with respect to a large number of optimality criteria. Furthermore, a precise algorithm and a graphical user interface (GUI) to determine this optimal transformation is provided (see appendices A and B). As possible applications, the optimal transformations in steels, i.e. the transformation from fcc to bcc/bct, and in terephthalic acid were determined. Through theorem 4.6 and with the help of the provided algorithm/program, one is able to rigorously determine the optimal phase transformation in any material undergoing a displacive phase transformation from one Bravais lattice to another.

If the parent or product phases are multi-lattices, the proposed framework is not a priori applicable. Nevertheless, one may still consider Bravais sublattices of these multi-lattices and proceed as before. The choice of these sublattices may come from physical consideration. However, in order to rigorously determine the optimal transformation between two given multi-lattices, one would need to measure the movement of all atoms consistently, i.e. one would need to take into account both the overall periodic deformation of the unit cell and the shuffle movement of atoms within the unit cell. Establishing such a criterion would be of great interest but lies beyond the scope of this paper.

Appendix A. MATHEMATICA

The following Mathematica code³ determines for a given $k\in \mathbb{N}$ the optimal lattice transformation $H:\mathcal{L}(F)\to \mathcal{L}(G)$ within the set $\{H_{\mu }=G\mu F^{-1}:\hspace{0.17em}\mu \in {\mathrm{SL}}^k(3,\mathbb{Z})\}$ for any given $F,G\in \mathrm{G}{\mathrm{L}}^{+}(3,\mathbb{R})$ and for any distance measure $d_r(H,\mathbb{I})$, r>0. The case r<0 is analogous.

For the transformation from fcc to bcc, F and G would be given by (4.7), and for the transformation from terephthalic acid I to II, F and G would be given by (4.20).

First, we generate the set $\mathrm{S}{\mathrm{L}}^{\mathrm{k}}(3,\mathbb{Z})$(=SL): SL = Select[Flatten[Table[{a,b,c,d,e,f,g,h,i}, {a,-k,k},{b,-k,k},{c,-k,k},{d,-k,k},{e,-k,k},{f,-k,k}, {g,-k,k},{h,-k,k},{i,-k,k}],8],Det[Partition[#,3]]==1&]; Next, we generate a list (=distlist) of all values of $d_r(H_{\mu },\mathbb{I})$ for $\mu \in {\mathrm{SL}}^k(3,\mathbb{Z})$:

Hmu = Function[mu,G.Partition[mu,3].Inverse[F]]; distr = Function[mu,Norm[SingularValueList[Hmu[mu]]^r-{1,1,1}]]; distlist = distr/@SL; Then, we generate a list(=poslist) of all the positions of μ's in $\mathrm{S}{\mathrm{L}}^{\mathrm{k}}(3,\mathbb{Z})$ that give rise to the minimal deformation distance:

poslist = Flatten[Position[distlist,RankedMin[distlist,1]],1]; Further, we calculate the minimal deformation distance m₀, the second to minimal deformation distance m₁ and return their numerical difference Δ=m₁−m₀ (=delta):

m0 = distlist[[poslist[[1]]]]; m1 = Sort[distlist][[Length[poslist]+1]]; delta = N[m1-m0] Finally, we return a list of all μ's that give rise to an optimal deformation H_μ and a list of all optimal H_μ's:

SL[[poslist]] Hmu/@SL[[poslist]]

Appendix B. Matlab

A GUI called ‘OptLat’ can either be found on Matlab File exchange⁴ (requires Matlab) or downloaded directly as a standalone Windows application.⁵

Data accessibility

Source codes for the Mathematica and Matlab applications are available online.

Authors' contributions

The results of the paper were obtained jointly by K.K. and A.M.

Competing interests

We have no competing interests.

Funding

The research of A.M. leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant no. 291053.