VOLUME 47, NUMBER 23 PHYSICAL REVIEW LETTERS 7 DECEMBER 1981 Normal-Mode Analysis by Quench-Echo Techniques: Localization in an Amorphous Solid Sidney R. Nagel The James Franck Institute and Department of Physics, The University of Chicago, Chicago, Ittinois 60637 A. Rahman Argonne National Laboratory, Argonne, Illinois 6043'9 Gary S. Grest'" Department of Physics, Purdue University, Fest Lafayette, Indiana 47907 (Received 13 October 198l) A new technique is presented for investigating the normal modes of an amorphous solid with use of molecular-dynamics simulations. This method, based on the quench echo, can be used even in cases where the dynamical matrix is too large to be diagonalized. This technique has been applied to study the onset of localization in a Lennard-Jones frozen fluid. PACS numbers: 63. 20. Pw, 63. 50.+x In a recent paper' we have described a new physical phenomenon, the quench echo, which was observed in molecular-dynamics simulations of solids. This echo occurs in the kinetic energy of a solid which has been prepared by two succes- sive quenches of the velocity; when the quenches are separated by an interval t„ the echo appears in the form of a deep minimum in the kinetic ener- gy at a time t, after the last quench. In the present paper we show that this phenome- non can be developed into a new and powerful tool for studying the eigenvectors and eigenfrequencies of normal modes in large complex solids, even in cases where the dynamical matrix is too large to be diagonalized on present-day computers. ' The behavior of the eigenvectors in an amorphous solid has enabled us to examine the onset of pho- non localization. %e find that at high frequencies the phonons are all very well localized. As the frequency is lowered they become more spread out and extend over the entire sample. The sample we have studied is a 500-particle amorphous solid obtained by rapid cooling from the liquid state. Periodic boundary conditions were used and the particles interacted via a Len- nard-Zones potential: V(r) =4e[(o/r)" — (o/r)'] with a cutoff at ~ =2.50. This glass was well equilibrated for 200000bt [bt is the time step for each iteration: bt =0. 0lr = 0. Olo(M/e)"', M being the mass of the particles] at a reduced tempera- ture T*=ksT/~ =0. 11 and at a fixed number densi- ty p+ 0 95 For a harmonic solid we have derived' an ex- pression for the kinetic energy (or instantaneous temperature) as a function of time after an arbi- trary number of quenches. Each quench instan- taneously sets every velocity to zero so that the kinetic energy, and therefore T*, is also instan- taneously zero. Subsequent to the quench, some of the potential energy of the system is converted to kinetic energy as time passes. After several quenches separated by the intervals t„t». .. , t„, the kinetic energy can be expressed as T = f, D (&)A'(&u) cos'~t, cos'~t, ~ ~ ~ cos'&ut„sin'~t dry, where D(cu) is the density of normal modes at Iwhere we have replaced the random coefficients frequency ~ andA(~) is proportional to the aver- A'(&u) by their average value, zA'. If ~/~D«, age amplitude of the modes at that frequency. The &2v/~D, where &u~ is the highest frequency of a time t is measured from the instant of the last mode in the solid, then D(n&/t, ) is nonzero only quench. If the intervals are all equal to t, then for n =1 and only one mode exists (except for ac- f "D (~)As(~) cos2+~t sjn2~t d~ cidental degeneracies which would not be expected in a glass). Therefore after many quenches which for large à can be approximated by separated by the same interval ty (or multiples of it, mt, ) all the modes in the solid are completely As D — sin — t drained of energy except the one with ~ =~/t, . 1981 The American Physical Society 1665