# Chemical Sciences: A Manual for CSIR-UGC National Eligibility Test for Lectureship and JRF/Atomic spectral line

In physics, atomic spectral lines are of two types:

• An emission line is formed when an electron makes a transition from a particular discrete energy level of an atom, to a lower energy state, emitting a photon of a particular energy and wavelength. A spectrum of many such photons will show an emission spike at the wavelength associated with these photons.
• An absorption line is formed when an electron makes a transition from a lower to a higher discrete energy state, with a photon being absorbed in the process. These absorbed photons generally come from background continuum radiation and a spectrum will show a drop in the continuum radiation at the wavelength associated with the absorbed photons.

The two states must be bound states in which the electron is bound to the atom, so the transition is sometimes referred to as a "bound–bound" transition, as opposed to a transition in which the electron is ejected out of the atom completely ("bound–free" transition) into a continuum state, leaving an ionized atom, and generating continuum radiation.

A photon with an energy equal to the energy difference between the levels is released or absorbed in the process. The frequency ${\displaystyle \nu }$ at which the spectral line occurs is related to the photon energy ${\displaystyle E}$ by Planck's law ${\displaystyle E=h\nu }$ where ${\displaystyle h}$ is Planck's constant.

## Emission and absorption coefficients

The emission of atomic line radiation may be described by an emission coefficient ${\displaystyle \epsilon }$  with units of energy/time/volume/solid angle. ε dt dV dΩ is then the energy emitted by a volume element ${\displaystyle dV}$  in time ${\displaystyle dt}$  into solid angle ${\displaystyle d\Omega }$ . For atomic line radiation:

${\displaystyle \epsilon ={\frac {h\nu }{4\pi }}n_{2}A_{21}\,}$

where ${\displaystyle n_{2}}$  is the density of emitting atoms, ${\displaystyle A_{21}}$  is the Einstein coefficient for spontaneous emission, which is fixed for any two energy levels. By Kirchhoff's law, the absorption characteristics in a region of space are closely tied to its emission characteristics, so we must mention the absorption coefficient as well. The absorption coefficient ${\displaystyle \kappa }$  will have units of 1/length and κ' dx gives the fraction of intensity absorbed for a light beam at frequency ν while traveling distance dx. The absorption coefficient is given by:

${\displaystyle \kappa '={\frac {h\nu }{4\pi }}~(n_{1}B_{12}-n_{2}B_{21})\,}$

where ${\displaystyle I_{\nu }}$  is the spectral intensity of radiation at (and near) frequency ${\displaystyle \nu }$ , ${\displaystyle n_{1}}$  is the density of absorbing atoms, and ${\displaystyle B_{12}}$  and ${\displaystyle B_{21}}$  are the Einstein coefficients for absorption and induced emission respectively. Like the coefficient ${\displaystyle A_{21}}$ , these are also constant for any two energy levels.

In the case of local thermodynamic equilibrium, the densities of the atoms, both excited and unexcited, may be calculated from the Maxwell-Boltzmann distribution, but for other cases, (e.g. lasers) the calculation is more complicated.

The above equations have ignored the influence of the spectral line shape. To be accurate, the above equations need to be multiplied by the (normalized) spectral line shape, in which case the units will change to include a 1/Hz term.

## The Einstein coefficients

In 1916, Albert Einstein proposed that there are essentially three processes occurring in the formation of an atomic spectral line. The three processes are referred to as spontaneous emission, induced emission and absorption and with each is associated an Einstein coefficient which is a measure of the probability of that particular process occurring.

### Spontaneous emission

Spontaneous emission is the process by which an electron "spontaneously" (i.e. without any outside influence) decays from a higher energy level to a lower one. The process is described by the Einstein coefficient ${\displaystyle A_{21}}$  which gives the probability per unit time that an electron in state 2 with energy ${\displaystyle E_{2}}$  will decay spontaneously to state 1 with energy ${\displaystyle E_{1}}$ , emitting a photon with an energy ${\displaystyle E_{2}-E_{1}=h\nu }$ . If ${\displaystyle n_{i}}$  is the number density of atoms in state i  then the change in the number density of atoms in state 1 per unit time due to spontaneous emission will be:

${\displaystyle \left({\frac {dn_{1}}{dt}}\right)_{A_{21}}=A_{21}n_{2}}$

### Stimulated emission

Stimulated emission (also known as induced emission) is the process by which an electron is induced to jump from a higher energy level to a lower one by the presence of electromagnetic radiation at (or near) the frequency of the transition. The process is described by the Einstein coefficient ${\displaystyle B_{21}}$  which gives the probability per unit time per unit energy density of the radiation field, that an electron in state 2 with energy ${\displaystyle E_{2}}$  will decay to state 1 with energy ${\displaystyle E_{1}}$ , emitting a photon with an energy ${\displaystyle E_{2}-E_{1}=h\nu }$ . The change in the number density of atoms in state 1 per unit time due to induced emission will be:

${\displaystyle \left({\frac {dn_{1}}{dt}}\right)_{B_{21}}=B_{21}n_{2}I(\nu )}$

where ${\displaystyle I(\nu )}$  is the spectral intensity of the radiation field at the frequency of the transition.

Stimulated emission is one of the fundamental processes that led to the development of the laser.

### Photoabsorption

Absorption is the process by which a photon is absorbed by the atom, causing an electron to jump from a lower energy level to a higher one. The process is described by the Einstein coefficient ${\displaystyle B_{12}}$  which gives the probability per unit time per unit energy density of the radiation field, that an electron in state 1 with energy ${\displaystyle E_{1}}$  will absorb a photon with an energy ${\displaystyle E_{2}-E_{1}=h\nu }$  and jump to state 2 with energy ${\displaystyle E_{2}}$ . The change in the number density of atoms in state 1 per unit time due to absorption will be:

${\displaystyle \left({\frac {dn_{1}}{dt}}\right)_{B_{12}}=-B_{12}n_{1}I(\nu )}$

## Detailed balancing

The Einstein coefficients are fixed probabilities associated with each atom, and do not depend on the state of the gas of which the atoms are a part. Therefore, any relationship that we can derive between the coefficients at, say, thermal equilibrium will be valid universally. At equilibrium, we will have a simple balancing, in which the net change in the number of any excited atoms is zero, being balanced by loss and gain due to all processes. With respect to bound-bound transitions, we will have detailed balancing as well, which states that the net exchange between any two levels will be balanced. This is because the probabilities of transition cannot be affected by the presence or absence of other excited atoms. Detailed balance requires that the change in time of the number of atoms in level 1 due to the above three processes be zero:

${\displaystyle 0=A_{21}n_{2}+B_{21}n_{2}I(\nu )-B_{12}n_{1}I(\nu )\,}$

Along with detailed balancing, we may use our knowledge of the equilibrium energy distribution of the atoms, as stated in the Maxwell-Boltzmann distribution , and the equilibrium distribution of the photons, as stated in Planck's law of black body radiation to derive universal relationships between the Einstein coefficients.

From the Maxwell-Boltzmann distribution we have for the number of excited atomic specie i:

${\displaystyle {\frac {n_{i}}{n}}={\frac {g_{i}e^{-E_{i}/kT}}{Z}}}$

where n is the total density of the atomic specie, excited and unexcited, k is Boltzmann's constant, T is the temperature, ${\displaystyle g_{i}}$  is the degeneracy of state i, and Z is the partition function. From Planck's law of black body radiation we have for the spectral intensity at frequency ${\displaystyle \nu }$

${\displaystyle I(\nu )={\frac {F(\nu )}{e^{h\nu /kT}-1}}}$

where:

${\displaystyle F(\nu )={\frac {2h\nu ^{3}}{c^{2}}}}$

where ${\displaystyle c}$  is the speed of light and ${\displaystyle h}$  is Planck's constant. Note that in some treatments, the blackbody energy density is used rather than the intensity, in which case:

${\displaystyle F(\nu )={\frac {8\pi h\nu ^{3}}{c^{3}}}}$

Substituting these expressions into the equation of detailed balancing and remembering that ${\displaystyle E_{2}-E_{1}=h\nu }$  yields:

${\displaystyle A_{21}g_{2}e^{-h\nu /kT}+B_{21}g_{2}e^{-h\nu /kT}{\frac {F(\nu )}{e^{h\nu /kT}-1}}=B_{12}g_{1}{\frac {F(\nu )}{e^{h\nu /kT}-1}}}$

The above equation must hold at any temperature, so that the three Einstein coefficients are interrelated by:

${\displaystyle {\frac {A_{21}}{B_{12}}}={\frac {g_{1}}{g_{2}}}~F(\nu )}$

and

${\displaystyle {\frac {B_{21}}{B_{12}}}={\frac {g_{1}}{g_{2}}}}$

When this relation is inserted into the original equation, one can also find a relation between ${\displaystyle A_{12}}$  and ${\displaystyle B_{12}}$ , involving Planck's law.

## Oscillator strengths

Oscillator strength ${\displaystyle f_{12}}$  is defined by:

${\displaystyle a_{12}={\frac {\pi e^{2}}{m_{e}c}}\,f_{12}}$

where ${\displaystyle e}$  is the electron charge and ${\displaystyle m_{e}}$  is the electron mass. This allows all three Einstein coefficients to be expressed in terms of the single oscillator strength associated with the particular atomic spectral line:

${\displaystyle A_{21}={\frac {8\nu ^{2}\pi ^{2}e^{2}}{m_{e}c^{3}}}~{\frac {g_{1}}{g_{2}}}~f_{12}}$
${\displaystyle B_{12}={\frac {4\pi ^{2}e^{2}}{m_{e}h\nu c}}\,f_{12}}$
${\displaystyle B_{21}={\frac {4\pi ^{2}e^{2}}{m_{e}h\nu c}}~{\frac {g_{1}}{g_{2}}}~f_{12}}$

## References

• Chandrasekhar, S. (1960). Radiative Transfer. Dover Publications, Inc. New York. ISBN 0-486-60590-6.
• Condon, E.U. and Shortley, G.H. (1964). The Theory of Atomic Spectra. Cambridge University Press. ISBN 0-521-09209-4.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Rybicki, G.B. and Lightman, A.P. (1985). Radiative processes in Astrophysics. John Wiley & Sons, New York. ISBN 0-471-82759-2.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Shu, F.H. (1991). The Physics of Astrophysics - Volume 1 - Radiation. University Science Books, Mill Valley, CA. ISBN 0-935702-64-4.
• Robert C. Hilborn (1982). "Einstein coefficients, cross sections, f values, dipole moments, and all that". Am. J. Phys. 50, 982.