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 1


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.

  • $\begingroup$ 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? $\endgroup$
    – Connor
    Commented Oct 7, 2021 at 13:48

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