End-members-
calculate the number and permutations of unique end members for sublattice model used in CALPHAD method.
Case 1 All the sublattices are identical (FCC structure belongs to this case)
number of unique endmembers recurrence formula for a(m,n) n>=m a(m,n) = C(n,1)*a(m-1,1)+...+C(n,k)*a(m-k,k)+...C(n,m)*a(0,m) n<m a(m,n) = C(n,1)*a(m-1,1)+...+C(n,k)*a(m-k,k)+...C(n,m)*a(0,m)
Permutations of unique endmembers are also calculated in recurrence formula and coorrespond to the calculation of number. e.g. for permutations p(m,n), the C(n,1)*a(m-1,1) term in the first sublattice is one of the n constituents, the rest of the m-1 sublattices is the permutations of p(m-1,1)
Case 2 Two kinds of sublattices in symmetric position (BCC structure belongs to this case)
Suppose we have 2m sublattices (2 kinds of sublattices in symmetric position, each kind have m sublattices) and n constituents in each sublattices, and the number of unique endmembers for this structure is b(2m,n). The formula of general term of b(2m,n) can be written into a ecurrence formula as follows: b(2m,n) = a(2,a(m,n))
Suppose p(m:m,n) are the permutations of structure with 2m sublattices (2 kinds of sublattices in symmetric position, each kind have m sublattices) and n constituents in each, the strategy to obtain p(m:m,n) is as follows: p(m:m,n) are the permutations of 2 sublattices, in each sublattices is the permutations of unique endmembers in m identical sublattices and n constituents in each.