Electromagnetism Ionization

Saha Equation

The Saha equation gives a relationship between free particles and those bound inatoms. To derive the Saha equation, choose a consistent set of energies. Also choose E = 0 when the electron velocity is zero, so for n = 1. Ignore

the energy of the higher n levels, since if an electron has enough energy to reach n = 2, it needs only 1/4 more energy to ionize completely, by the Bohr energy equation

Let be the probability that the gas has electrons out of N

particles in a given ensemble. The partition functions for each class of particles are

So the probability function, assuming indistinguishable particles, is

The sums in the partition functions are actually integrals, since the particles have a continuous momentum distribution. Therefore, for and

where i is either e or p. Using

gives

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

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Let

then

Since electrons and protons are both fermions,

and

The derivation is identical for , except that the binding energy term is

carried through and , resulting in

We want to find the most probable state, so we should differentiate (2). However, because is monotonic, will have a maximum at the

same place as f(x). Taking the log of (5) and using Stirling's approximation and

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

Page 2 of 4Saha Equation -- from Eric Weisstein's World of Physics

9/26/2006http://scienceworld.wolfram.com/physics/SahaEquation.html

the result is

Using the definitions

and taking the derivative of (18)

The resulting relationship is

Plugging in (14)-(16) into (22),

Canceling and taking ,

Defining the ionization fraction as

then

Ionization

© 1996-2006 Eric W. Weisstein

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

Page 3 of 4Saha Equation -- from Eric Weisstein's World of Physics

9/26/2006http://scienceworld.wolfram.com/physics/SahaEquation.html