Stochastic matrix description of glass transition in ternary chalcogenide systems Gerardo G. Naumis a,b , Richard Kerner a, * a Laboratoire GCR, Universit e Pierre et Marie Curie±CNRS URA 769, Tour 22, Bo ^ õte 142-4 Place Jussieu, 75005 Paris, France b Instituto de F  õsica, Universidad Nacional Aut onoma de M exico, Apdo. Postal 20-364, 01000 M exico D.F., Mexico Received 22 December 1997 Abstract A method using stochastic transfer matrices is applied to description of microcluster growths in typical ternary chal- cogenide glasses (Ge±As±Se and Ge±Sb±Se systems) during glass transition. By ®nding the unit eigenvalue and eigen- vector of the stochastic matrix, an analytical relation between the glass transition temperature and the concentration of atomic species is found. In the chalcogen-rich region this formula reproduces the experimentally observed modi®ed Gibbs±DiMarzio equation, which is a semi-empirical rule that relates the average coordination number with the glass transition temperature, using a parameter b which is ®xed by the experimental data. The present approach allows us to predict a b close to the observed one. Ó 1998 Published by Elsevier Science B.V. All rights reserved. PACS: 64.70.Pf; 61.43.Fs; 81.10.Fq 1. Introduction There is still no consensus about which thermodynamical and structural factors are important in deter- mining the glass transition temperature [1] (T g ), although the subject has considerable practical and tech- nological importance. Furthermore, this problem must be intimately related to the question of how the glasses do form [1], i.e., how the complex structures forming the glass network grow as one decreases the temperature of a liquid. In particular, for chalcogenide glasses, attention has been devoted to correla- tions of T g with other physical or chemical properties [2], since such glasses have electric and infrared trans- mission properties that make them useful in several technological applications [3]. Among these chalcogenide glasses, ternary systems such as Ge x As y Se 1xy and Ge x Sb y Se 1xy have been extensively stud- ied [3±5] not only for their applications, but also for testing the Phillips constraint theory [6], since the bonding numbers (valencies) of Ge, As, Sb and Se (4, 3, 3, and 2, respectively) allow the realization of the rigidity threshold (attained when the average coordination number, hri 4x  3y  21  x  y  is 2.4) in many dierent chemical compounds. Also, since T g is one of the most important parameters for a description of the glassy state, much work has been done on ternary glasses to determine the dependence Journal of Non-Crystalline Solids 231 (1998) 111±119 * Corresponding author. Tel.: +33-1 44 27 72 86; fax: +33-1 44 27 72 87; e-mail: rk@ccr.jussieu.fr. 0022-3093/98/$19.00 Ó 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 4 1 7 - 7