# What does the Combined Notation $\left<\text{abc}\right>\{\text{def}\}$ mean in Crystallography?

I have come across some notation of the form $$\left<\text{abc}\right>\{\text{def}\}$$ in a paper on the Solid-Liquid Coexistence Molecular Dynamics simulation of single element materials, please see: https://doi.org/10.1016/j.actamat.2014.12.010

Specifically, the exact notation is $$\left<\text{012}\right>\{\text{100}\}$$, and is supposed to define part of the Solid-Liquid interface. In the manuscript the meaning is given thus: $$\hat{x}_1$$ the first simulation box parameter is parallel to the $$\left<\text{012}\right>$$ direction, and the plane of the Solid-Liquid interface is $$\{\text{100}\}$$.

I have never seen this notation before, is it common? Or is it specific to this paper?

Edit: To be clear, I have seen each section of the notation separately and I'm aware of what they mean, but I am not aware of any standard meaning in combination.

## 1 Answer

Standard crystallography notation:

[hkl] is a specific direction

<hkl> is a family of equivalent directions ([100] and [001] are equivalent in a cubic system for example, and are <100> directions).

(hkl) is a specific plane

{hkl} is a family of equivalent planes

In the paper, one relevant usage is:

For the slab shown in Fig. 2, the element is Cu, [$$a_{1}b_{1}c_{1}$$] = [$$012$$], [$$a_{2}b_{2}c_{2}$$] = [$$0\bar 2 1$$] and [$$a_{3}b_{3}c_{3}$$] = [$$001$$]; thus the orientation is denoted by <012>{100}...

This is a bit confusing (and Figure 2 more so). The solid/liquid interface is $$z$$ but marked as the [100] direction (in the {100} family for fcc copper), and they chose their axes in the $$x-y$$ (or $$x - x_{2}$$ per the figure) plane to be along [$$012$$] and [$$0\bar 2 1$$].

So, the {100} plane is the solid-liquid interface, and they are looking on (and set up the problem on) the solid side along particular crystal directions.

• So the idea is, the first part $⟨\text{hkl}⟩$ defines the crystal lattice directions in the plane, and the second part $\{\text{hkl}\}$ defines the plane normal? Oct 7 '21 at 13:48