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# Definition: Richardson-Dushman Equation

The Richardson-Dushman equation relates the current density of a thermionic emission to the work function (W) and temperature (T) of the emitting material:

js = A T2 exp(-W/kT)

where

js is the current density of the emission (mA/mm2)
A is Richardson's constant. A = 4*πmek2/h3 ~ 1202 mA/mm2K2, where m is the mass of electron, e is elementary charge, and h is Plank's constant. In practice, A may be multiplied by a correction factor that depends on the material (see Table I below). Note that A varies from about 32 to 160 A cm-2 K-2 for pure (polycrystalline) metals and over a much greater range for oxide and composite surfaces.[1]
T is temperature (K)
W is the work function of the cathode material (J or eV)
k is the Boltzmann constant (1.3806488E-23 J K-1 or 8.6173324E-5 eV K-1) [*]

In Cathode Emissions in CPO, A is treated as an empirical constant. In fact, js may be entered directly.

Table 1: Work functions and Richardson's constants for various materials [1,2]
Material W (eV) A*b (A cm-2 K-2)
(b is material correction factor)
Molybdenum 4.15 55
Nickel 4.61 30
Tantalum 4.12 60
Tungsten 4.54 60
Barium 2.11 60
Cesium 1.81 160
Iriduim 5.40 170
Platinum 5.32 32
Rhenium 4.85 100
Thorium 3.38 70
Ba on W 1.56 1.5
Cs on W 1.36 3.2
Th on W 2.63 3.0
Thoria 2.54 3.0
BaO + SrO 0.95 ~10-2
Cs-oxide 0.75 ~10-2
TaC 3.14 0.3
LaB6 2.70 29
theoretical: 120.2 (b=1)

## References

• [1] Bernhard Wolf. Handbook of ion sources. CRC Press, 1995. ISBN 0849325021, 9780849325021. p.27,11 [*]
• [2] C. J. Smithells, Metals Reference Book, Vol III, Butterworths, London, 1967, 737ff.
• [3] Wikipedia:Thermionic Emission [*]
• [4] CODATA2010 http://physics.nist.gov/cuu/index.html
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