Coherent potential approximation of random nearly isostatic kagome lattice

The kagome lattice has coordination number 4, and it is mechanically isostatic when nearest-neighbor sites are connected by central-force springs. A lattice of N sites has O(root N) zero-frequency floppy modes that convert to finite-frequency anomalous modes when next-nearest-neighbor (NNN) springs are added. We use the coherent potential approximation to study the mode structure and mechanical properties of the kagome lattice in which NNN springs with spring constant. are added with probability P = Delta z/4, where Delta z = z - 4 and z is the average coordination number. The effective medium static NNN spring constant.m scales as P-2 for P << kappa and as P for P >> kappa, yielding a frequency scale omega* similar to Delta z and a length scale l* similar to (Delta z)(-1). To a very good approximation at small nonzero frequency, kappa(m)(P,omega)/kappa(m)(P,0) is a scaling function of omega/omega*. The Ioffe-Regel limit beyond which plane-wave states become ill-defined is reached at a frequency of order omega*.