Blundell Thermal Physics · Problem 22.6

The chemical potential

Physics — Statistical Mechanics · saha-equation-ionization

Problem

(a) Consider the ionization of atomic hydrogen, governed by the equation

\mathrm{H} \rightleftharpoons \mathrm{p}^+ + \mathrm{e}^-, \tag{22.104}

where

\mathrm{p}^+

is a proton (equivalently a positively ionized hydrogen) and

\mathrm{e}^-

is an electron. Explain why

\mu_\mathrm{H} = \mu_\mathrm{p} + \mu_\mathrm{e}. \tag{22.105}

Using the partition function for hydrogen atoms from eqn 21.50, and using eqn 22.103, show that

-k_\mathrm{B}T \ln \frac{Z_1^\mathrm{p}}{N_\mathrm{p}} - k_\mathrm{B}T \ln \frac{Z_1^\mathrm{e}}{N_\mathrm{e}} = -k_\mathrm{B}T \ln \frac{Z_1^\mathrm{H}}{N_\mathrm{H}} \mathrm{e}^{\beta R}, \tag{22.106}

where

Z_1^x

and

N_x

are the single-particle partition function and number of particles for species

x

respectively, and where

R = 13.6\,\mathrm{eV}

. Hence show that

\frac{n_\mathrm{e} n_\mathrm{p}}{n_\mathrm{H}} = \frac{(2\pi m_\mathrm{e} k_\mathrm{B} T)^{3/2}}{h^3} \mathrm{e}^{-\beta R}, \tag{22.107}

where

n_x = N_x/V

is the number density of species

x

, stating any approximations you make. Equation 22.107 is known as the Saha equation. (b) Explain why charge neutrality implies that

n_\mathrm{e} = n_\mathrm{p}

and conservation of nucleons implies

n_\mathrm{H} + n_\mathrm{p} = n

, where

n

is the total number density of hydrogen (neutral and ionized). Writing

y = n_\mathrm{p}/n

as the degree of ionization, show that

\frac{y^2}{1-y} = \frac{\mathrm{e}^{-\beta R}}{n \lambda^3_\mathrm{th}}, \tag{22.108}

where

\lambda_\mathrm{th}

is the thermal wavelength for the electrons. Find the degree of ionization of a cloud of atomic hydrogen at

1000\,\mathrm{K}

and density

10^{20}\,\mathrm{m}^{-3}

. (c) Equation 22.108 shows that the degree of ionization goes up when the density

n

goes down. Why is that?