Non‐Equilibrium Vacancy Concentrations in Metals Subject to Thermal Gradients and Electric Fields

Gradients of temperature and electric fields may be used to cause vacancy currents in metals. If vacancies are supposed to migrate between randomly distributed, fixed sources and sinks, a metal subject to an internal electric field E , parallel to a temperature gradient ∂ T /∂ x , should suffer a vacancy supersaturation given by q = C e r Lambda; 2 / k 2 T 4 [( E f ( E f + E m + E t ‐ 2 kT ) + E t ( E m ‐ 2 kT )) (∂ T /∂ x ) 2 ‐ Z e E T ( E f + E m ‐ kT ) ∂ T /∂ x ‐ Z e kT 3 ∂ E /∂ x + E t kT 2 ∂ 2 T /∂ x 2 ], where C e denotes the equilibrium vacancy concentration appropriate to the temperature T at any point x , and Lambda; is the mean free path of a vacancy during migration between source and sink. E f , E m , and E t are the energies for formation, motion, and transport of vacancies, Z e is the effective electrical charge carried by a vacancy when diffusing, and r is a geometrical constant (1/6 in f.c.c.).