THEORY OF METAL SURFACES: WORK FUNCTION 1223 high-coverage-limit the net importance of (6v)2 terms in d work functions for Li deposited on various faces of Reference 19 gives also a second, much larger, valu and R of r for each of these metals, which leads to somewhat J. C. Riviere, in Solid State Sunface Science, edited less satisfactory results for bulk properties. The work by M. Green(Dekker, New York, 1969), Vol. 1 BC. Herring and M. H. Nichols, Rev. Mod. Phys than those corresponding to the preferred set, giving an 21,185(1949) even greater discrepancy with experiment It might be mentioned that T. Schneider [Phys. Status Cu and Ag: D. E. Eastman, Phys. Rev. B 2, 1 32, 323(1969) has shown second-o 970); Au: E. E. Huber, Appl. Phys. Letters 8, 169 ce-A, the bulk contribution to the work func n, by 0.1-0.2 eV for the alkalis and by 0. 3 and 0. 8ev There is some theoretical evidence suggesting the for Al and Mg, respectively. The second-order effect existence of occupied surface states in the noble metal of 6v on the surface term Ao must be determined, how this could of course also affect Ad. See F. Forstmann ever, before any conclusions can be drawn concerning and J. B. Pendry, Z. Physik 235, 75 (1970) PHYSICAL REVIE W B VOlUME 3, NUMBER 4 15 FEBRUARY 1971 Some Formal Aspects of a Dynamical Theory of Diffusion University of llinois, Urbana, llinois 61801 eceived 2 October 1970) A classical dynamical theory of diffusion is presented in which the reaction coordinate and its to the Kac equation is shown to be exact for classical statistics, and the jump rate is calcu- lated accordingly. The jump rate can be expressed in terms of the vector R and the dynam matrix; this leads to a frequency factor different from that obtained from the reaction-rate theory, and an indication that the system does not jump through the relaxed saddle-point con- iguration. The self-diffusion isotope effect is also considered. It is shown explicitly that the migration energy is mass independent, and that the effect may be expressed simply in terms f the reaction coordinate. In terms of the phonons, the isotope effect depends on a weighted verage of the fraction of the energy carried by the jumping atom in each mode, Quantum corrections are discussed and a connection is made between the isotope effect and thermomi- gration . INTRODUCTION quilibrium configurations The troubling aspect of rate theory is that been increasing interest in recent on prop years in formulating the theory of diffusion jump saddle-point configuration. In a quantum-mechani- rates in a way that avoids some of the conceptual cal theory it would not only be impossible to treat difficulties of the absolute -rate theory. The rate theory was applied to diffusion by wert and Zener positions and velocities independently, but, more significantly, it would be completely inappropriate plicitly how the motions of many atoms are involved as has been pointed out by Flynn and Stoneham, ate and elegantly formulated by vineyard to show ex to speak of the properties of the intermediate in the jump process. This can be seen in Vine Attempts to circumvent these difficulties have yards form for the preexponential or frequency been made by several authors. These"dyi factor cal"theories view the jump process as resulting special kind of fluctuation f and attempt to calculate the frequency of such fluc tuations. In the earlier work, fluctuations were n which v and vl are the normal-mode frequencies considered that carried the system to a definite of the equilibrium and saddle-point configurations configuration, e.g., the relaxed saddle-point con respectively, with the unstable "mode"of the sad figuration. Glydehas shown that, for this case, dle point left out of the product in the denominator. the dynamical and rate theories have the same The many-body nature is also apparent in the iden- formal content. More recently, Flynn has con- tification of the migration energy with the potentia sidered fluctuations in a reaction coordinate, made energy difference of the relaxed saddle-point and up of a linear combination of particle displacements