Quantitative relation between the thermionic contrast of metal surfaces and their degree of monocrystallization

For better understanding the peculiarities of work function, a simple model is devised to calculate the effective work functions (ϕ+ and ϕe) for positive-ionic and electronic emissions from polycrystalline surfaces, which have a work function range from the maximum (ϕmax) to the minimum (ϕmin). Analysis of the theoretical results thus obtained and also of experimental data published to date enables us to find the quantitative relation between the thermionic contrast (Δϕ* ≡ ϕ+ − ϕe) and the degree of monocrystallization (δm), thereby yielding the three formulae of (1) Δϕ* = c for 0 < δm ≲ 1/2 (polycrystal), (2) Δϕ* = 4 cδm (1 − δm) for l/2 ≲ δm ≲ 1 (polycrystal), and (3) Δϕ* = 0 for δm = 1 (monocrystal). For a given surface consisting of a number of patchy faces (i), δm corresponds to the largest among its fractional surface areas (Fi) having different values of local work function (ϕi). In a typical case of tungsten, the constant of c is evaluated theoretically to be 0.53 ± 0.09 eV, which well agrees with 0.59 ± 0.06 eV determined experimentally by many workers and also which satisfies the essential condition of Δϕ* ≦ c < ϕmax − ϕmin ≈ 0.8–1.0 eV. Our theoretical model is quite simple, but it is very useful for (1) evaluating both ϕ+ and ϕe with an uncertainty of less than ±0.1 eV, (2) finding the quantitative relation between Δϕ* and δm for actual surfaces of both poly- and monocrystals, and also (3) getting a substantial clue as to the problem how the effective work functions are governed by the surface characteristics of both Fi and ϕi.