Richardson-Dushman equation.


The Richardson-Dushman equation is (from Hawkes and Kasper, eqn 44.12a,b)

 js = A T^2 exp(-W/kT),

where the Richardson’s constant is

 A = 4 pi m e k^2 h^-3 = 1.202 10^6 SI units = 120.2 A/cm^2/K^2 = 1202 mA/mm^2/K^2,

and the current density js is the saturation current density (in SI units, A/m^2), and W is the work function of the material of the cathode.


In the present programs the user is asked to enter:

either js in units of mA/mm^2

or T in degrees K, W in eV and A in units of A/cm^2/K^2.

Using these units, js is given by 10.A T^2 exp(-W*11604/T).


It seems the Richardson-Dushman equation is not often relied upon and that Richardson’s constant A is often treated as an empirical constant.


It is certainly treated as an empirical constant in the present program.


In practice (see Szilagyi) js is approximately 50 mA/mm^2 for tungsten (work function 4.2eV) at 2900K, which implies that A = 12 A/cm^2/K^2. Other authors refer to tungsten operating at 1950K with A = 55 A/cm^2/K^2, which would give js = 0.03 mA/mm^2. For lanthanum hexaboride Szilagyi quotes a value for js of approximately 500 mA/mm^2 at 1900K. Other authors refer to impregnated cathodes having W = 1.95eV and operating at 1300K with A = 45 A/cm^2/K^2, which would give js = 21 mA/mm^2.


It is because of this confusion and uncertainty that the program allows the user to enter the value of js instead of the values of T, W and A.