The activation energy for lattice
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I.
INTRODUCTION
EXPERtMENTALvalues for the lattice self-diffusion coefficient are available for only about 50 of the 80 or so elements which are considered to be either metallic or semimetallic in nature. Also, some of the data available appear to be unreliable due to extreme difficulty in obtaining results for highly reactive materials. Therefore, it is sometimes necessary to resort to phenomenological rules to obtain reasonable estimates for diffusion coefficients, ltj Various relationships have been suggested since 1922, f~,2~the majority assuming that diffusion obeys an Arrhenius equation, with the frequency factor approximately constant for all elements, and that the activation energy is a function of the melting temperature, Tin. It is often assumed that the frequency factor, Do, equals about 5 • 10 -5 m2/s and that the activation energy, Q = 141Tin J/mole. tEj Sherby and Simnad pl assumed that Do = l • 10 -4 m2/s and included factors for both valence and crystal structure in the term for activation energy, giving Q = RTm(Ko + V), where R is the gas constant, V is the valence of the element, and K0 = 14 for bcc elements, 17 for fcc and cph elements, and 21 for elements with a diamond structure. Le Claire 141 also recognized the importance of valence and suggested the relationship Q = RTm(K + 1.5V), where K = 13 for bcc elements, 15.5 for fcc and cph elements, and 20 for elements with more open covalently bonded structures. All of the proposed formulae have disadvantages. Those formulae which utilize only the actual melting temperatures of the elements t~l or any other single variable cannot predict the activation energies for the various crystal structures of allotropic elements. For the formulae which include the valence, there is considerable ambiguity over the choice of valence, particularly for the transition metals. Le Claire t4~ chose "the lowest valency commonly encountered in chemical combination," but ambiguity J.R. CAHOON, Professor, is with the Metallurgical Sciences Laboratory, Department of Mechanical and Industrial Engineering, University of Manitoba, Winnipeg, MB R3T 2N2, Canada. OLEG D. SHERBY, Professor Emeritus, is with the Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305 -2205. Manuscript submitted February 14, 1992. METALLURGICAL TRANSACTIONS A
still exists as to the choice of valence. Therefore, a formula which will predict the lattice diffusion coefficient and which can be applied unambiguously is desirable. Over the past 35 years or so, Engel and Brewer advanced and developed a theory which relates the valence of pure metals (the number of outer s + p electrons) to their crystal structure. The theory arose from the observation that the elements Na, Mg, and A1 have 1, 2, and 3 (s + p) electrons and exhibit bcc, cph, and fcc crystal structures, respectively. Therefore, the EngelBrewer theory proposes that all bcc structures have 1 to 1.5 (s + p) electrons, all cph structures have 1.5 to 2.5 (s + p) electrons, and all fcc structures have 2.5 to 3
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