Appendix A

Kernel Matrices for EAMPotential

Here, we provide numerically derived K-matrices for the third shell EAM interaction in FCCgold. These matrices are utilized for the lattice governing equations, both dynamic (3.9) andquasi-static (3.106), in the harmonic approximation. They also serve as input informationin derivation of the lattice Green’s function (3.75) and the time history kernel (3.97).

The EAM interaction model is regarded as an extension of the nearest-neighbor modelrepresented by the matrices (3.126) on page 73. The unit cell numbering for the fcc latticestructure is shown in Figure 3.22 on page 72.

The parameters used for FCC gold are identical to those found in the work of Foileset al. (1986), which has been implemented in the Sandia simulation code Tahoe (2004).For a third shell fcc potential, each atom has 43 neighboring atoms; each of the 43 atomsin this test lattice was perturbed in such a manner that the resulting force on cell (0, 0, 0)

resulting from the perturbation could be determined. The K-matrices, which are given inunits of eV · A

−2, are as follows:

K0,0,0 =

−4.415218 0 00 −4.415218 00 0 −4.415218

K0,0,2 = K0,0,−2 =

−0.06714876 0 00 −0.06714876 00 0 0.0371758

K2,0,0 = K−2,0,0 =

0.03717614 0 00 −0.06714876 00 0 −0.06714876

K0,2,0 = K0,−2,0 =

−0.06714876 0 00 −0.03717614 00 0 −0.067114876

Nano Mechanics and Materials: Theory, Multiscale Methods and Applications W. K. Liu, E. G. Karpov and H. S. Park 2006 John Wiley & Sons, Ltd. ISBN: 0-470-01851-8

298 KERNEL MATRICES FOR EAM POTENTIAL

K−1,0,1 = K1,0,−1 =

0.5980579 0 −0.7103140 −0.1736703 0

−0.710314 0 0.5980579

K−1,0,−1 = K1,0,1 =

0.5980579 0 0.7103140 −0.1736703 0

0.710314 0 0.5980579

K0,−1,1 = K0,1,−1 =

−0.1736703 0 00 0.5980579 −0.7103140 −0.710314 0.5980418

K0,−1,−1 = K0,1,1 =

−0.1736703 0 00 0.5980579 0.7103140 0.710314 0.5980579

K1,1,0 = K−1,−1,0 =

0.5980418 0.710314 00.710314 0.5980418 0

0 0 −0.1736703

K1,−1,0 = K−1,1,0 =

0.5980418 −0.710314 0−0.710314 0.5980418 0

0 0 −0.1736703

K1,1,2 = K−1,−1,−2 =

−0.003171257 0.03711209 0.040077880.03711209 −0.003171257 0.040077880.04029503 0.04029503 0.07131029

K1,1,−2 = K−1,−1,2 =

−0.003171257 0.03711209 −0.040077880.03711209 −0.003171257 −0.04007788

−0.04029503 −0.04029503 0.07131029

K1,−1,2 = K−1,1,−2 =

−0.003171257 −0.03711247 0.04007788−0.03711209 −0.003171455 −0.040077880.04029503 −0.04029528 0.07131029

K1,−1,−2 = K−1,1,2 =

−0.003171257 −0.03711247 −0.04007829−0.03711209 −0.003171455 0.04007829−0.04029503 0.04029528 0.0713109

K−1,−2,1 = K1,2,−1 =

−0.003171455 0.04007829 −0.037112090.04029528 0.0713109 −0.04029503

−0.03711247 −0.04007829 −0.003171257

K−1,−2,−1 = K1,2,1 =

−0.003171455 0.04007829 0.037112470.04029528 0.0713109 0.040295280.03711247 0.04007829 −0.003171455

K1,−2,−1 = K−1,2,1 =

−0.003171257 −0.04007829 −0.03711247−0.04029503 0.0713109 0.04029528−0.03711209 0.04007829 −0.003171455

KERNEL MATRICES FOR EAM POTENTIAL 299

K1,−2,1 = K−1,2,−1 =

−0.003171257 −0.04007829 0.03711209−0.04029503 0.0713109 −0.040295030.03711209 −0.04007829 −0.003171257

K−2,−1,1 = K2,1,−1 =

0.0713109 0.04029528 −0.040295030.04007829 −0.003171455 −0.03711209

−0.04007829 −0.03711247 −0.003171257

K−2,−1,−1 = K2,1,1 =

0.0713109 0.04029528 0.040295280.04007829 −0.003171455 0.037112470.04007829 0.03711247 −0.003171455

K2,−1,−1 = K−2,1,1 =

0.07131029 −0.04029528 −0.04029528−0.04007788 −0.003171455 0.03711247−0.04007788 0.03711247 −0.003171455

K2,−1,1 = K−2,1,−1 =

0.07131029 −0.04029528 0.04029503−0.04007788 −0.003171455 −0.037112090.04007788 −0.03711247 −0.003171257