साहा समीकरण

साहा समीकरण का विकाश सुप्रसिद्ध भारतीय खगोलविज्ञानी (एस्ट्रोफिजिसिस्ट्) मेघनाद साहा ने 1920 में किया था। इसके द्वारा वर्णपट्ट के आधार पर तारों के वर्गीकरण की व्याख्या की गई है। For a gas composed of a single atomic species, the Saha equation is written:

${\displaystyle {\frac {n_{i+1}n_{e}}{n_{i}}}={\frac {2}{\Lambda ^{3}}}{\frac {g_{i+1}}{g_{i}}}\exp \left[-{\frac {(\epsilon _{i+1}-\epsilon _{i})}{k_{B}T}}\right]}$

where:

• ${\displaystyle n_{i}\,}$ is the density of atoms in the i-th state of ionization, that is with i electrons removed.
• ${\displaystyle g_{i}\,}$ is the degeneracy of states for the i-ions
• ${\displaystyle \epsilon _{i}\,}$ is the energy required to remove i electrons from a neutral atom, creating an i-level ion.
• ${\displaystyle n_{e}\,}$ is the electron density
• ${\displaystyle \Lambda \,}$ is the thermal de Broglie wavelength of an electron

${\displaystyle \Lambda \ {\stackrel {\mathrm {def} }{=}}\ {\sqrt {\frac {h^{2}}{2\pi m_{e}k_{B}T}}}}$

• ${\displaystyle m_{e}\,}$ is the mass of an electron
• ${\displaystyle T\,}$ is the temperature of the gas
• ${\displaystyle k_{B}\,}$ is the Boltzmann constant
• ${\displaystyle h\,}$ is Planck's constant

In the case where only one level of ionization is important, we have ${\displaystyle n_{1}=n_{e}}$ and defining the total density n  as ${\displaystyle n=n_{0}+n_{1}}$, the Saha equation simplifies to:

${\displaystyle {\frac {n_{e}^{2}}{n-n_{e}}}={\frac {2}{\Lambda ^{3}}}{\frac {g_{1}}{g_{0}}}\exp \left[{\frac {-\epsilon }{k_{B}T}}\right]}$

where ${\displaystyle \epsilon }$ is the energy of ionization.

The Saha equation is useful for determining the ratio of particle densities for two different ionization levels. The most useful form of the Saha equation for this purpose is

${\displaystyle {\frac {Z_{i}}{N_{i}}}={\frac {Z_{i+1}Z_{e}}{N_{i+1}N_{e}}}}$,

where Z denotes the partition function. The Saha equation can be seen as a restatement of the equilibrium condition for the chemical potentials:

${\displaystyle \mu _{i}=\mu _{i+1}+\mu _{e}\,}$