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 (?⁺ and ?^e) for positive-ionic and electronic emissions from polycrystalline surfaces, which have a work function range from the maximum (?_max) to the minimum (?_min). 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 (Δ?^* ≡ ?⁺ − ?^e) and the degree of monocrystallization (δ_m), thereby yielding the three formulae of (1) Δ?^* = c for 0 < δ_m ? 1/2 (polycrystal), (2) Δ?^* = 4 cδ_m (1 − δ_m) for l/2 ? δ_m ? 1 (polycrystal), and (3) Δ?^* = 0 for δ_m = 1 (monocrystal). For a given surface consisting of a number of patchy faces (i), δ_m corresponds to the largest among its fractional surface areas (F_i) having different values of local work function (?_i). 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 Δ?^* ≦ c < ?_max − ?_min ≈ 0.8-1.0 eV. Our theoretical model is quite simple, but it is very useful for (1) evaluating both ?⁺ and ?^e with an uncertainty of less than ±0.1 eV, (2) finding the quantitative relation between Δ?^* and δ_m 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 F_i and ?_i.