dimer
Eo=A
-|V|=B
H = [A B;B A]
energy eigen values satisfy:
0 = (A-E)^2-B^2
solve for E:
will be + and -
ยฑ
%%
โE total = Erep - B
minimized when:
โโE total /โd=0
then, plug back in to โE total
to find at equilibrium
Lennard-Jones potential
energy/atom = 2ษโ'[(๐/a)ยนยฒ *A12- (๐/a)โถ *A6]
to minimize set dervative to 0
โenergy/โatom = 2ษโ'[-12/a(๐/a)ยนยฒ *A12- 6/a(๐/a)โถ *A6]
12*A12*(๐/a)โถ = 6*A6
(๐/a)โถ = A6/2A12
a = ๐(2*A12/A6)^1/6
%%
to find total energy per atom
plug in values
Cohesive energy per atom:
Ecoh = -ษ/2*A6^2/A12
Bulk Modui = Ecoh/[(# of atoms)*V^3]
Bfcc = (16ษ/๐^3)(A6/sqrt(2))(A6/A12)^3/2
Bbcc = (8ษ/๐^3)(A6/sqrt(2))(A6/A12)^3/2
Bm= (#ofatoms)*(4ษ/๐^3)(A6/sqrt(2))(A6/A12)^3/2
Tensors
transformation rule:
%2nd---
๐'ij = aik ajl ๐kl
strain = ษ'ij =1/2(eij + eji)
aij = e'i * ej
Cijkl
11 = 1
22 = 2
33 = 3
23 or 32 = 4
13or 31 = 5
12 or 21 = 6
a matrix for:
primed to unprimed
%four-fold---
