Distribution of Magnon Modes in a Disordered Magnetic Chain

Numerical methods for the calculation of the frequency spectrum of the linearized magnon modes in disordered magnetic chains are outlined. The distribution for the ferromagnetic chain is obtained from the Sturm-sequence algorithm introduced by Dean in the analogous calculation for the one-dimensional crystal lattice. The distribution for the antiferromagnetic chain cannot be calculated by Dean's method. For this case the Sturm sequence obtained from the amplitudes is appropriate (Rosenstock-McGill method). Applications are made to two systems: an amorphous ferromagnetic chain, where comparison is made with the analytic theory of Montgomery, Krugler, and Stubbs, and to an antiferromagnetic chain with parameters chosen to resemble those of ${\mathrm{Cu}}_x{\mathrm{Cr}}_{1-x}{\mathrm{Cl}}_2$. The results of numerical studies on chains of 50 000 spins are reported.