Kinetic Determination of Thermionic Emission

Kinetic Determination of Thermionic Emission.—We have now found the pressure of an electron gas in equilibrium with a metal, at an 

Let us first find the number of molecules of a perfect gas, at pressure P, temperature T, striking a square centimeter of surface per second. Since the electrons act like a perfect gas, this calculation will apply to them as well as to an ordinary gas. The calculation is similar to that of Sec. 3, Chap. IV, where we found the pressure by the kinetic method. Consider the molecules whose momentum lies in the range dpx dpv dpz. As in Fig. IV-2, the number of such molecules crossing one square centimeter perpendicular to the x axis per second will be the number in a prism of base one centimeter, slant height along the direction of the velocity equal to p/m, or altitude equal to px/m. The volume of this prism is Ps/ra, and the number of such molecules per unit volume, by Eq. (2.4), Chap. IV, is 

Thus the number of molecules crossing the square centimeter per second, found by integrating over all values of pv and p., but only over positive values of p, is 

For equilibrium, the number of electrons leaving the metal per second must equal this value. Thus the electron current is this multiplied by the 

If No(dCp/dN) were zero, F f3.4) would have the form A TH T. This is the familiar formula for thermionic emission, and is one that shows good agreement with experiment, except that the experimental value of A does not agree with the theoretical value given in Eq. (3.4). Actually No(dCp/dN) is undoubtedly not zero, and the last term of Eq. (3.4) must 

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