Stoichiometric Constraints

The most flexible method of imposing stoichiometric constraints in CLEASE is to use linear systems of equations. Here, you can find a list of examples of how different constraints can be imposed. In CLEASE a linear system of equations with the structure shown below

The number of sublattice concentration is simply the length of the flattened version of basis_elements that is passed to the Concentration class. Therefore, if basis_element = [['Au', 'Cu'], ['Cu', 'X]] there will be to Cu concentrations you can restrict; one for each sublattice. The total number of sublattice concentrations in the example above is 4. Hence, all rows of the matrix has 4 columns. CLEASE has two types of constraints: equality and lower bound. Equality constraints are passed via A_eq and b_eq arguments in the Concentration class, and lower bound constraints are passed via A_lb and b_lb. For lower bound constraints, the equality sign in the figure is replaced by a larger or equal than-symbol. Note that upper bound constraints can trivially be converted to a lower bound constraint by multiplying the equation by -1. Finally, the example below shows how you can generate random concentrations satisfying your constraints. The list passed to the function is the number of sites in each sublattice.

>>> import numpy as np
>>> np.random.seed(0)  # Set a seed for consistent tests
>>> from clease.settings import Concentration

Binary System With One Basis

>>> basis_elements = [['Au', 'Cu']]

This is a system where we have the basis_elements=[['Au', 'Cu']].

1. Force the Au concentration to be equal to the Cu concentration

   >>> A_eq = [[1.0, -1.0]]
   >>> b_eq = [0.0]
   >>> conc = Concentration(basis_elements=basis_elements, A_eq=A_eq, b_eq=b_eq)
   >>> for i in range(10):
   ...     x = conc.get_random_concentration([20])
   ...     assert np.abs(x[0] - x[1]) < 1e-10
   

2. Force number of Au atoms to be larger than 12

   >>> A_lb = [[20, 0.0]]
   >>> b_lb = [12]
   >>> conc = Concentration(basis_elements=basis_elements, A_lb=A_lb, b_lb=b_lb)
   >>> for i in range(10):
   ...    x = conc.get_random_concentration([20])
   ...    assert round(20*x[0]) >= 12
   

Two sublattices

>>> basis_elements = [['Li', 'V'], ['O', 'F']]

1. Force the concentration of O to be twice the concentration of F

   >>> A_eq = [[0.0, 0.0, -1.0, 2.0]]
   >>> b_eq = [0.0]
   >>> conc = Concentration(basis_elements=basis_elements, A_eq=A_eq, b_eq=b_eq)
   >>> for i in range(10):
   ...     x = conc.get_random_concentration([18, 18])
   ...     assert abs(x[2] - 2*x[3]) < 1e-10
   

2. Li concentration larger than 0.2 and O concentration smaller than 0.7

   >>> A_lb = [[1.0, 0.0, 0.0, 0.0], [0.0, 0.0, -1.0, 0.0]]
   >>> b_lb = [0.2, -0.7]
   >>> conc = Concentration(basis_elements=basis_elements, A_lb=A_lb, b_lb=b_lb)
   >>> for i in range(10):
   ...    x = conc.get_random_concentration([18, 18])
   ...    assert x[0] >= 0.2 and x[2] < 0.7