Several metallic including for example elements from groups 5 (V, Nb, Ta) and 6 (Cr, Mo, W) have a body-centered cubic (BCC) ground state structure.
The face-centered cubic (FCC) lattice is one of the most common crystal structures for metallic elements including e.g., the late transition metals from group 10 (Ni, Pd, Pt) and 11 (Cu, Ag, Au).
The construction of force constants requires accurate reference data. Density functional theory (DFT) calculations are one of the most common source for such data.
Optimization and machine learning¶
- Compressive sensing
Compressive sensing (CS), also known as compressive sampling, is an efficient method for constructing sparse solutions for linear systems.
- Cross validation
Cross validation (CV) is commonly employed to evaluated the transferability and accuracy of linear problems.
The least absolute shrinkage and selection operator (LASSO) is a method for performing variable selection and regularization in problems in statistics and machine learning.
- Kernel ridge regression
Regularization, is commonly used in machine learning to combat overfitting and for solving underdetermined systems.
Crystal symmetry and clusters¶
- Crystal symmetry operation
A crystal symmetry operation for a specific lattice means that the lattice is invariant under this operation. An operation comprises translational and rotational components.
A cluster is defined as a set of points on a lattice.
- Cluster size
The size of a cluster (commonly refered to as the cluster radius) is defined as the average distance to the geometrical center of the cluster.
- Cluster space
The set of clusters into which a structure can be decomposed.
Cutoffs define the longest allowed distance between two atoms in a cluster for each order.
An orbit is defined as a set of symmetry equivalent clusters
- Orientation family
An orientation family is a subgroup to an orbit, which contains those clusters that are oriented identically in the lattice.
- Irreducible parameters
Many elements of the force constant matrices are related to each other by symmetry operations. The irreducible set up of parameters is obtained by applying all symmetry operations allowed by the space group of the ideal lattice and the sum rules.
- Sum rules
In order for a force constant potential to fulfill translational invariance certain constraints are imposed on the force constants. These constraints are commonly referred to as sum rules.