Lattice Statics Green’s function method for modeling of defects in crystals- million atom model (MAM)
Application to extended defects- dislocations and cracks in crystals
Description of the Lattice statics GF method
This method is essentially a reformulation of Kanzaki approach that was originally developed for point defects. In this method, the Green's function is first calculated in reciprocal space. The real space Green's function is then calculated by taking its inverse transform. It is computationally much more efficient than a direct computer simulation method. Even for a million atom Born-van Karman model, the computation of the Green's function requires only a few minutes of CPU time.
For a review of lattice statics Green's function method, please see
1. V. K. Tewary, Green’s function method for lattice statics, Adv. Phys. 22, p757 (1973).
2. R. Thomson, S.J. Zhou, A.E. Carlsson, and V.K. Tewary, Lattice imperfections studied by use of lattice Green’s functions, Phys. Rev. B46, p 10613 (1992).
For an excellent review of lattice dynamics Green's function method and Born-van Karman model, please see
Maradudin A.A., Montroll E., Weiss G., and Ipatova I., The theory of lattice dynamics in the harmonic approximation. Solid State Physics, Supp. 3: New York: Academic Press; 1971.
In order to apply the Green's function method to defects, in solids, we need to solve the Dyson equation. For point defect or periodic defects, the Dyson equation can be solved by matrix partitioning technique. This technique has been used to calculate the lattice distortion due to grain boundary interfaces in solids, cracks in solids, and interfacial cracks in bi-material composite materials.
New method for extended defects- dislocations and cracks in crystals:
We are currently in the process of developing a defect space Fourier transform method for solving the Dyson's equations for dislocations. For a preliminary account, please see:
Lattice statics Green's function for modeling of dislocations in crystals by V.K. Tewary; J. Mater. Res., Proc. of the MRS Spring Meeting at San Francisco, 1998; To be published. Download the above paper in pdf.
Description of the Lattice statics GF method
The objective is to calculate lattice strains and elastic interaction between the defects.
Notation:
l, l’ - lattice sites (3D indices) indexing N atoms in a Born-van Karman supercell
f (
l, l’)- 3d matrix – force constants, obtained by the first and second derivatives of the interatomic potential.F(l) – Force on atom at l, obtained by the first derivative of the interatomic potential
u(l) - displacement of the atom at l
The introduction of a defect introduces force F(l) at site l, displaces its atom by u(l) and changes the force constant matrix from
f to f*.Total Energy associated with lattice strains:
W = -
S F(l) u(l) + (1/2) S f*( l, l’) u(l) u(l’)For equilibrium:
¶ W/ ¶ u(l) = 0
This gives
u(l) =
S G*( l, l’) F(l’),
G* = [
f*]-1 3N x 3N matrices,where
f* = f - D f,
and
D f denotes the change in force constants. The defect Green's function is given by the following Dyson's equation:G* = G + G
Df G*The displacement field is given by
u = G*F
or, equivalently, by the Kanzaki formula:
u = G F*(u)
where,
the Kanzaki force: F* = F +
Df (u) u.In the harmonic (non-linear) case,
Df (u) is independent of u.G is the perfect lattice (without defects) Green's function. We will also call it reference lattice Green's function.
G = [
f]-1
G and [
f] are 3N x 3N matrices. In computer simulation methods, this matrix is directly inverted. This puts a severe restriction on the computer simulation method since, for any realistic calculation, N should be taken to be at least of the order of a million.G for the perfect lattice is calculated by using the Fourier representation as given below
G(l,l’) = (1/N)
Sq G(q) exp[iq.(l-l’)],where
G(q) = [
f(q)]-1For notational convenience, we follow the usual custom of denoting the Fourier transforms of G and
f by the same symbols, the distinguishing feature being their argument q for the Fourier transform. G(q) and [f(q)] are 3 x 3 matrices. In order to calculate G(l,l'), we only need to invert N matrices of order 3x3 instead of one matrix of order 3Nx3N. This represents a major advantage of the Green's function method.Because of translation symmetry, G(l,l') depends upon l and l' only through their difference. We therefore label them by a single index l-l'. thus G(l) denotes G(0,l)=G(l,0).
The calculated values of G(l) for some fcc and bcc materials are given below.
FCC: Copper, Aluminum
BCC: Molybdenum, Iron
The Dyson's equation for point defects such as vacancy or interstitials , and periodic defects such as grain boudaries, surfcaes, and interfaces can be solved by matrix partitioning techniques. See the following references.
1. Green's function method for lattice statics by V.K. Tewary, Advances in Physics, 22, 757, (1973).
2. Lattice theories of dislocations by R. Bullough and V.K. Tewary, Dislocations in Solids; Editor- F.R.N. Nabarro, (North Holland, Amsterdam), 2, Chapter 5, (1979).
3. Theory of the void lattice in molybdenum by V.K. Tewary and R. Bullough, J. Phys. F (Metal Phys.), 3, L69, (1972).
4. Theory of the defect superlattices in crystals with application to void/ vacancy and nitrogen interstitials' lattice in tantalum and vanadium by V.K. Tewary, J. Phys. F (MetalPhys.), 3, 1275, (1973).
5. Activation energy of gas interstitials in bcc metals by E.J. Savino and V.K. Tewary, J. Phys. F(Metal Phys.), 3, 1919,(1973).
6. Change of phonon dispersion curves due to interstitials in aluminium by H.R. Schober, V.K. Tewary, and P.H. Dederichs, Zeits. Physik., B21, 255, (1975).
7. V.K. Tewary and R. Thomson, Lattice statics of interfaces and interfacial cracks in bimaterial solids, J. Mater. Res. 7, p1018 (1992).
8. Theory of chemically induced kink formation on cracks in silica- I: 3D crack Green's functions by R.M. Thomson, V.K. Tewary and K. Masudo-Jindo, J. Materials Res., 2, 619, (1987).
9. Theory of chemically induced kink formation on cracks in silica- II: Force law calculations by K. Masudo-Jindo, V.K. Tewary and R.M. Thomson, J. Materials Res., 2, 631, (1987).
10. Lattice statics Green's function method for calculation of atomistic structure of grain boundary interfaces in solids: Part I.- Harmonic theory by V.K. Tewary, E.R. Fuller Jr. and R.M. Thomson, J. Materials Res., 4, 309, (1989).
11. Lattice statics Green's function method for calculation of atomistic structure of grain boundary interfaces in solids: PartII.- Anharmonic theory by V.K. Tewary and E.R. Fuller Jr., J. Materials Res., 4, 309, (1989).
12. Theoretical study of fracture of brittle materials: atomistic calculations by K. Masuda-Jindo, V.K. Tewary and R. Thomson; Materials Science and Engineering; A 146, 273, (1991).
13. Atomic theory of fracture of brittle materials: Application to covalent semiconductors by K. Masuda-Jindo, V.K. Tewary and R. Thomson; J. Materials Res., 6, 1553, (1991).