2.1.1

Atomistic Local Strain

It is not too difficult to define local strain when the reference state is known: one
may carry out a least-square fitting procedure outlined in (6.34) to get a best fit of
the affine transformation matrix J to the relative positions of atoms nearby. The
problem is harder when there is no given reference system, for example a dislocation
would cause lattice mis-registry that makes the original reference hard to keep trackof. What we need is a geometrical characterization of the local atomic environmentsthat works in much the same way as the coordination number, but provides strain
information when the measure is applied to a piece of deformed yet perfect lattice.
Because there is no reference frame, it can at best output the matrix invariants ofthe strain tensor, e.g., the dilatational component
1ηm ≡ Trη,
3and the shear component,sηs ≡1Tr(η − ηm I)2 ,2(2.22)(2.23)

calibrated to the cases of hydrostatic dilatation ηxx = ηyy = ηzz = ηm , and pure shearηxy = ηyx = ηs .
Consider 3 × 3 matrix,
Mi ≡Xqij qTij ,(2.24)

j∈Ni

defined at each atom i, where qij is the relative position between atom j and i,
qij ≡ qj − qi ,(2.25)and j ∈ Ni is the set of i’s neighbors, whose definition is to be chosen. Two possible
choices are,NiR : {j ∈ NiR | |qij | ≤ R} ,(2.26)

where R is a cutoff distance, usually picked to be between the first and second nearest

34

neighbors at the material’s perfect crystalline state; or,NiC ≡ NiRi , with |NiRi | = C,

(2.27)

i.e., we sort j according to |qij | and pick only the nearest C atoms, where C is theordinary coordination number of the crystal, like 12 for fcc, etc. Anyway, Ni must be
chosen such that,
Mi0 = d0 I,(2.28)

at the material’s perfect crystalline equilibrium. Satisfying (2.28) may seem quite
impossible at the first glance, but actually materials with cubic or Td symmetry
automatically do, including fcc, bcc, hcp (perfect c/a ratio), diamond cubic, andNaCl, ZnS, CsCl binary structures.
Consider how local strain and rotation influences (2.24) of a perfect crystal. Fora given local affine transformation J,
qij → q0ij = Jqij .

(2.29)

Therefore, if neighbors do not enter or leave Ni ,
Mi =Xj∈Ni
q0ij q0Tij = JX

qij qTij  JT = JM0i JT .

(2.30)j∈Ni

But if requirement (2.28) is satisfied, there would be
Mi = J(d0 I)JT = d0 JJT .

(2.31)As the local Lagrangian strain at atom i is defined to be,1η = (JT J − I),
235(2.32)

the local strain hydrostatic invariant is,1111ηm = Trη = (TrJT J − 3) = (TrJJT − 3) = (d0−1 TrMi − 3),
3666(2.33)

and the local strain shear invariant is,sηs=s=s=s=s

1
Tr(η − ηm I)223 2
1
Trη 2 − ηm
22
13 2
Tr(JT J − I)2 − ηm
82
TrI 3 2
11
TrJT JJT J − TrJT J +− ηm
8482

11TrI 3 2
TrJJT JJT − TrJJT +− ηm
8482s
13 2=
Tr(JJT − I)2 − ηm
82=s=s=3 212Tr(d−10 Mi − I) − ηm821 −112Tr(d−1(d TrMi − TrI)20 Mi − I) −824 0s= d−10TrMi 21Tr(Mi −I) .
83(2.34)

After obtaining {Mi } for all atoms, one may also compute system average hMi
and evaluate local deviations from hMi. The above scheme (2.24) to (2.34) works
extremely well in practice, allowing one to directly visualize microstructures andtheir strain fields with color-encoding. Successful examples include point defects,dislocations, cracks, grain boundaries, stacking faults and even kinks.

2.2

Thermodynamic Ensembles and Fluctuations

This section deals with the thermodynamic state of an atomistic simulation which caneither be molecular dynamics (MD) or Monte Carlo (MC) [7]. In MD, one numerically36