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![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/ce/IonizationFractionVsTemperatureHydrogen.png/330px-IonizationFractionVsTemperatureHydrogen.png)
The Saha ionization equation is an expression that relates the ionization state of a gas in thermal equilibrium to the temperature and pressure.[1][2] The equation is a result of combining ideas of quantum mechanics and statistical mechanics and is used to explain the spectral classification of stars. The expression was developed by Meghnad Saha in 1920.[3][4]
Equation
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Saha_Equation_Hydrogen.png/330px-Saha_Equation_Hydrogen.png)
For a gas at sufficiently high temperature (here measured in energy units, i.e. keV or J) and density, the thermal collisions of the atoms will ionize some of the atoms, making an ionized gas. When several or more of the electrons that are normally bound to the atom in orbits around the atomic nucleus are freed, they form an independent electron gas cloud co-existing with the surrounding gas of atomic ions and neutral atoms. In turn, this generates an electric field, where the motion of charges generates currents, making a localized magnetic field, and creates the state of matter called plasma.
The Saha equation describes the degree of ionization for any gas in thermal equilibrium as a function of the temperature, density, and ionization energies of the atoms. The Saha equation only holds for weakly ionized plasmas for which the Debye length is large. This means that the screening of the Coulomb interaction of ions and electrons by other ions and electrons is negligible. The subsequent lowering of the ionization potentials and the "cutoff" of the partition function is therefore also negligible.
For a gas composed of a single atomic species, the Saha equation is written [5]:
where:
- is the number density of atoms in the i-th state of ionization
- is the partition function of an atom in the i-th state of ionization
- is the number density of free electrons
- is the temperature of the gas
- is the Boltzmann constant
- is the mass of an electron
- is Planck's constant
- is the ionization energy required to remove the i-th electron from an ion in its ground state
The Saha equation can also be expressed in terms of pressure:
where
is the pressure of free electrons. As the electron pressure increases, the predicted ionization fraction decreases. This makes sense, as a higher pressure will result in more collisions and therefore more opportunities for atoms to recombine.
In cases where only one level of ionization is important, such as in a pure Hydrogen gas, we have . Defining the total density as , the Saha equation simplifies to:
where is the energy of ionization, is the degeneracy of states for the i-level ions, and is the thermal de Broglie wavelength of an electron:
Derivation
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c3/IonizationIllustration.png/330px-IonizationIllustration.png)
The Saha equation can be derived following the steps presented here (see external links). Let a, b, etc. denote different excited states while i, i+1, etc. are used to denote different stages of ionization.
Let be the number of atoms in ionization stage . From the Boltzmann distribution, the relative number of atoms in ionization stage and ionization stage is given by:
where is the statistical weight of ionization stage and is the momentum of the free electron produced during ionization. The exponent then represents the total change in energy during the ionization process.
It is necessary to account for all possible configurations of the electrons among the discrete energy levels of the atom, as well as the possible states of the resulting free electron. The statistical weight of an atom or ion is given by the partition function:
where is the energy required to excite the ion from the ground state to level , and is the number of degenerate states having energy .
Let and represent the partition functions of the atom in ionization stages and . We then have:
where is the statistical weight of the free electron. To account for the possible states of the free electrons, we must integrate over a unit cell in phase space. An electron has two possible spin states, so two electrons can exist within a quantum cell in phase space:
where is Planck's constant. To obtain the final expression of Saha's equation, we must now integrate over all possible momenta. Due to symmetry, we can perform the integral over momentum using :
The integral over volume gives the number of electrons, :
In general, we have:
Using we find:
Finally, this can be rearranged to produce the given form of the Saha equation:
Stellar atmospheres
[edit]In the early twenties Ralph H. Fowler (in collaboration with Charles Galton Darwin) developed a new method in statistical mechanics permitting a systematic calculation of the equilibrium properties of matter. He used this to provide a rigorous derivation of the ionization formula which Saha had obtained, by extending to the ionization of atoms the theorem of Jacobus Henricus van 't Hoff, used in physical chemistry for its application to molecular dissociation. Also, a significant improvement in the Saha equation introduced by Fowler was to include the effect of the excited states of atoms and ions. A further important step forward came in 1923, when Edward Arthur Milne and R.H. Fowler published a paper in the Monthly Notices of the Royal Astronomical Society, showing that the criterion of the maximum intensity of absorption lines (belonging to subordinate series of a neutral atom) was much more fruitful in giving information about physical parameters of stellar atmospheres than the criterion employed by Saha which consisted in the marginal appearance or disappearance of absorption lines. The latter criterion requires some knowledge of the relevant pressures in the stellar atmospheres, and Saha following the generally accepted view at the time assumed a value of the order of 1 to 0.1 atmosphere. Milne wrote:
Saha had concentrated on the marginal appearances and disappearances of absorption lines in the stellar sequence, assuming an order of magnitude for the pressure in a stellar atmosphere and calculating the temperature where increasing ionization, for example, inhibited further absorption of the line in question owing to the loss of the series electron. As Fowler and I were one day stamping round my rooms in Trinity and discussing this, it suddenly occurred to me that the maximum intensity of the Balmer lines of hydrogen, for example, was readily explained by the consideration that at the lower temperatures there were too few excited atoms to give appreciable absorption, whilst at the higher temperatures there are too few neutral atoms left to give any absorption. ..That evening I did a hasty order of magnitude calculation of the effect and found that to agree with a temperature of 10000° [K] for the stars of type A0, where the Balmer lines have their maximum, a pressure of the order of 10−4 atmosphere was required. This was very exciting, because standard determinations of pressures in stellar atmospheres from line shifts and line widths had been supposed to indicate a pressure of the order of one atmosphere or more, and I had begun on other grounds to disbelieve this.[7]
Stellar coronae
[edit]Saha equilibrium prevails when the plasma is in local thermodynamic equilibrium, which is not the case in the optically-thin corona. Here the equilibrium ionization states must be estimated by detailed statistical calculation of collision and recombination rates.
The early universe
[edit]Equilibrium ionization, described by the Saha equation, explains evolution in the early universe. After the Big Bang, all atoms were ionized, leaving mostly protons and electrons. According to Saha's approach, when the universe had expanded and cooled such that the temperature reached about 3,000 K, electrons recombined with protons forming hydrogen atoms. At this point, the universe became transparent to most electromagnetic radiation. That 3,000 K surface, red-shifted by a factor of about 1,000, generates the 3 K cosmic microwave background radiation, which pervades the universe today.
See also
[edit]References
[edit]- ^ Alexander A. Fridman (2008). Plasma Chemistry. Cambridge, UK: Cambridge University Press. p. 94. ISBN 978-0-521-84735-3.
- ^ Chen, Francis F. (2016). Introduction to Plasma Physics and Controlled Fusion. p. 2. Bibcode:2016ippc.book.....C. doi:10.1007/978-3-319-22309-4. ISBN 978-3-319-22309-4.
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ignored (help) - ^ Saha, Megh Nad (1920). "LIII.Ionization in the solar chromosphere". Philosophical Magazine. Series 6. 40 (238): 472–488. doi:10.1080/14786441008636148.
- ^ Saha, M. N. (1921). "On a Physical Theory of Stellar Spectra". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 99 (697): 135–153. Bibcode:1921RSPSA..99..135S. doi:10.1098/rspa.1921.0029.
- ^ Carroll, Bradley; Ostlie, Dale (2007). An Introduction to Modern Astrophysics (2nd ed.). Pearson Addison-Wesley. pp. 213–215. ISBN 0-8053-0402-9.
- ^ Aller, Lawrence H. (1953). Astrophysics: the atmospheres of the sun and stars. Ronald Press Co.
- ^ "Biographical Memoirs: Meghnad Saha".
External links
[edit]- Derivation & Discussion by Hale Bradt
- A detailed derivation from the University of Utah Physics Department
- Lecture notes from the University of Maryland Department of Astronomy
- Lecture notes from the Penn State University Department of Astronomy and Astrophysics