Issue 49

A.V. Vakhrushev et alii, Frattura ed Integrità Strutturale, 49 (2019) 370-382; DOI: 10.3221/IGF-ESIS.49.37 372         1 , 1, 2, , , 2 MEAM i i i ij ij i i j i U r U r F r i N                    (1) where   i U r – is the potential of the i -th atom, the potential affects the type of interaction of atoms and the magnitude of the forces in the equations of motion; i F – the immersion function of the i -th atom located at a point in space with electron background density i  ; N – the number of elements of the nanosystem, atoms or nanoparticles;   ij ij r  – the value of the pair potential between the i -th and j -th atoms, remotely separated on ij r . The immersion function depends on the background electron density, has a variable form for different types of chemical elements of the periodic system and is written using the expression       0 0 ln , 0 , , 0 i i i i i i i i i i i A E F A E              (2) where i A – is the empirical parameter of the potential field; 0 i E – the value of the energy of sublimation; i  – background electron density; the index i indicates the ownership of a particular type of atom. The background electron density at the point of immersion is determined by the following functional dependence           2 0 3 0 0 1 , , k k i i i i i i k i i G t                    (3) where the indices 1, 2, 3 k  correspond to the p, d, f electron orbitals of the i -th atom;   k i t – weight coefficients of the model; 0 i  – background electron density of the initial structure;   k i  – parameters characterizing the deviation of the electron density from its ideal state, when all atoms are in the lattice sites. Different formulations are used to calculate the function   G  . The total background electron density i  contains partial contributions of individual densities of atomic orbitals. Atomic orbitals are divided into spherically symmetric s, which corresponds to electron density (0) i  , and angular p, d, f clouds, with distributions (1) (2) (3) , , i i i    . To determine the weights of the model of (3), the expression is used             0 0, 2 0 0, , k A j j ij i j k i k A j j ij i j t S t t S        (4) where   0, k j t are the parameters depending on the chemical type of the j -th element. Together with the MEAM potential, the shielding function is used, which is used to reduce the computational cost and reduce the potential error min, , max, min, , ikj ikj c ij ij c c k i j ikj ikj C C r r S f f r C C                      (5)   2 2 2 2 4 2 4 2 2 1 2 , ij ik ij jk ij ikj ij ik jk r r r r r C r r r       (6)