Issue 49

A.V. Vakhrushev et alii, Frattura ed Integrità Strutturale, 49 (2019) 370-382; DOI: 10.3221/IGF-ESIS.49.37

   

   

1 2

  ij r

        i i i U r F  

 

MEAM U r

ij 





i

1, 2, , , N 

(1)

,

j i 

i

i

 

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 , i i      , 0

    



i 

i i A E

0

  

F

(2)

,

i

i

i i A E

i

i

i A – is the empirical parameter of the potential field; 0 i

E – the value of the energy of sublimation; i  –

where

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 k

  0

   

   

3

i 

i   i



     ,

  k



i 

G

t

,

(3)

i

i

i

0  i



k

1

1, 2, 3 k  correspond to the p, d, f electron orbitals of the i -th atom;   k i t

where the indices

– weight coefficients of the

0 i  – background electron density of the initial structure;   k i 

– parameters characterizing the deviation of the

model;

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

 

k A j j 

t

S

ij

0,

  k

i j 



t

(4)

,

    0, k j t

i

2



  0

A

j 

S

ij

i j 

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

   

   

ikj C C 

ij   

r r

, k i j  

ikj

min,

c

S f 

f

,

(5)

 

 

ij

c

c



max, C C  ikj

r

ikj

min,

 r r   2 2 2 ik jk 2 2 r r r r ij ik ij jk ij r 2 4 2   ij r 

4

 

C

(6)

1 2

,

ikj

372