Lecture 9 - Statistical Mechanics (2/11/99)

Prev Atomic Spectra --- | --- Radiative Transfer Next

ASTR012 Reading:
Chapter 8-4, 8-7 (ZG4)

pages 34 - 38

The Sun and prominence from NASA's Skylab space station on December 19, 1973. (Courtesy NASA)
? Key Question: What determines the energy state of atoms in a gas?
! Key Principle: The Boltzmann and Saha Equations
# Key Problem: What fraction of hydrogen in the Solar Photosphere is ionized?


  1. States - The Boltzmann Distribution
  2. Ionization - The Saha Equation
  3. The Balmer Series - Boltzmann + Saha Equations
  4. The Solar Photosphere
  5. Radiative Transfer I - The Problem

Ionization States

The process of ionization and recombination and the ionization states of hydrogen and helium:

The Boltzmann Equation

The ratio of the number density (in atoms per m^3) of atoms in energy state B to those in energy state A is given by

NB / NA = ( gB / gA ) exp[ -(EB-EA/kT ]

where the g's are the statisical weights of each level (the number of states of that energy). Note for the energy levels of hydrogen

gn = 2 n2

which is just the number of different spin and angular momentum states that have energy En.

The Saha Equation

The textbook is a little confusing on this subject, as they just give in (8-31)

Ni+1 / Ni = A Ne-1 (kT)3/2 exp[ -i/kT ]

but don't say what A is! In class, we gave the full equation

Ni+1 / Ni = ( 2Qi+1 / NeQi ) ( 2mekT / h2 ) exp[ -i/kT ]

where Q is the partition function for the two ionization states (normalized to the ground state)

Q = g1 + n=2 gn exp[ En - E1 / kT ]

which numerically becomes

Ni+1 / Ni = ( 2.41 x 1021 m-3 / Ne ) ( 2Qi+1 / Qi ) T3/2 exp[ -i/kT ]

(note that for HII Q=1 and HI Q=2 so the partition funtion terms go to unity). We also used the fact that the electrons can be seen to form a gas

Pe = Ne k T

so we can write the Saha equation as

Ni+1 / Ni = ( 3.33 x 10-2 Pa / Pe ) ( 2Qi+1 / Qi ) T5/2 exp[ -i/kT ]

since the density and temperatures vary through most stellar atmospheres but the are roughly isobaric.

Note that the Saha equation is just an application of the Boltzmann equation to the case of ionization states! If you are trying to compute the occupancy of different energy states in the same ionization state (be H I, He I, He II, etc) then you need to use the Boltzmann equation. If you are trying to compute the relative fraction of atoms in one particular energy state of one particular ionization state to the total number in all energy and ionization states, then you need to use both equations!

For example, to calculate the fraction in the n=2 state of H_I to the total number density N, which can be either in the ionization state H I (any n) or H II. Since we found in class (and you can demonstrate easiy) that almost all of the H I atoms are in the n=1 ground state, N_1 ~ N_I, and thus

N2/N = ( N2/N1 ) / ( NI/N1 + NII/N1 ) ( N2/N2 / ( 1 + NII/NI )

on which you can use the Boltzmann equation (for N_2/N_1) and the Saha equation (for N_II/N_I).

Types of Spectra: Continuum, Emission, Absorption

Three types of spectra:

More Atomic Spectra in Outline:

  1. The Bohr Atom: Orbital Radii and Energy Levels
  2. Hydrogen Energy Levels and Line Series
  3. Ionization
  4. Electron Orbitals and Quantum Numbers

Prev Prev Lecture --- Next Next Lecture --- Index Astr12 Index --- Home Astr12 Home

smyers@nrao.edu Steven T. Myers