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萨哈方程[Saha Equation] [2005-8-26]
iamet 发表在 Ω〖物理〗
| The Saha equation gives a relationship between free particles and those bound in atoms. 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
 (1)
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
   (2)
 (3)
 (4)
So the probability function, assuming indistinguishable particles, is
 (5)
The sums in the partition functions are actually integrals, since the particles have a continuous momentum distribution. Therefore, for  and 
 (6)
where i is either e or p. Using
 (7)
gives
 (8)
Let
 (9)
 (10)
 (11)
then
  
img]http://scienceworld.wolfram.com/physics/simg34.gif[/img]
img]http://scienceworld.wolfram.com/physics/simg34.gif[/img] (12)
Since electrons and protons are both fermions,
 (13)
and
 (14)
 (15)
The derivation is identical for  , except that the binding energy term is carried through and  , resulting in
 (16)
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 
 (17)
the result is
 
 (18)
Using the definitions
  (19)
  (20)
and taking the derivative of (18)
 
 (21)
The resulting relationship is
 (22)
Plugging in (14)-(16) into (22),
 (23)
Canceling and taking 
 (24)
 (25)
Defining the ionization fraction as
 (26)
then
 (27)
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