Einstein Coefficients

Kirchoff's Law, $j_{\nu}=\alpha_{\nu}B_{\nu}\left(T\right)$, states that for a material whose emission is solely a function of its temperature, its emission must equal its absorption in equilibrium. Whilst considering this, Einstein realised that this implied a statistical relationship between the emission and absorption of individual levels in the molecules of the material. He initialy considered just two energy levels, separated by an energy of $h\nu$, in a radiation field $\overline{J}=\int_{0}^{\infty}J_{\nu} \phi\left(\nu\right)d\nu$ where $J_{\nu}$ is the radiation at a particular frequency and $\phi\left(\nu\right)$, the line profile function, is normalised as $\int^{\infty}_{0} \phi\left(\nu\right)d\nu=1$.

Spontaneous Emission:

When in level 2 the molecule has a probability of spontaneously releasing a photon (of energy $h\nu$) and transitioning to level 1. This probability is denoted by $A_{21}$, the Einstein A coefficient and is usually given in units of probability per second of a molecule transitioning. Note that $A_{12}=0$ since if this were not the case it would imply the molecule was gaining energy from nothing.

Stimulated Emission

Spontaneous emission takes place in the absence of a radiation field, however, if one is present then the radiation can stimulate a molecule in level 2 to emit a photon (energy $h\nu$) and transition to level 1. This is stimulated emission and is denoted by $B_{21}\overline{J}$ and is also usually given as probability of a transition occurring for one molecule per second. $B_{21}$ is one of the Einstein B coefficients.

Absorption

The third method of transitioning also requires a radiation field and is where a molecule in level 1 absorbs a photon of energy $h\nu$ and thereby transitions to level 2. The probability of this occurring is $B_{12}\overline{J}$ again usually the probability of a transition occurring for one molecule per second. $B_{12}$ is the second Einstein B coefficient.

Clearly, if the system is in equilibrium then the number of molecules leaving one level must equal the number entering it so the three coefficients are linked by

$$n_1 B_{12}\overline{J}=n_2A_{21}+n_2B_{21}\overline{J}$$

where $n_1$ and $n_2$ are the number of molecules in level 1 and level 2 respectively. Note that this assumes that the density is sufficiently low that no collisions take place (which could otherwise induce transitions). This can also be re-arranged to give

$$\overline{J}=\frac{\frac{A_{21}}{B_{21}}}{\frac{n_1}{n_2}\frac{B_{12}}{B_{21}}-1}$$

The ratio of the level populations can be replaced by $\frac{n_1}{n_2}=\frac{g_1}{g_2}e^{\frac{h\nu}{kT}}$(the ratio of the level populations in thermodynamic equilibrium) thus giving

$$\overline{J}=\frac{\frac{A_{21}}{B_{21}}}{\frac{g_1B_{12}}{g_2B_{21}}e^{\frac{h\nu}{kT}}-1}$$

However, in thermodynamic equilibrium the radiation field will be equal to the Planck function (ie. $\overline{J}=\frac{2h\nu^3 / c^2}{e^{\frac{h\nu}{kT}}-1}$) which implies that

$$g_1B_{12}=g_2B_{21}$$ (2.21)

and

$$A_{21}=\frac{2h\nu^3}{c^2}B_{21}$$ (2.22)

where $g_1$ and $g_2$ are the degeneracies for level 1 and 2 respectively (the number of ways the angular momentum, J, may be orientated in space - equal to $2J+1$). The Einstein coefficients are properties of a particular molecule and as equations 2.23 and 2.24 contain only the coefficients and some physical constants they are independent of the conditions in which the molecules find themselves (ie. specifically, they hold even if the molecules are not in thermodynamic equilibrium).

Unfortunately there are two possible definitions of the Einstein B coefficients depending upon whether the emission is due to the local radiation intensity or the local energy density. Equation 2.24 above is for radiation intensity case, the equivalent equation for emission due to energy density is

$$A_{21}=\frac{8 \pi h \nu^3}{c^3}B_{21}$$

Care needs to be taken over which definition is being used to avoid problems caused by the factor of $\frac{4 \pi}{c}$ difference.

It is now possible to define an emission coefficient which can be seen as the energy per transition ($h\nu$) multiplied by a function that measures the rate of change of the upper level population with frequency ( $\frac{dn_u}{d\nu}$) multiplied by the probability of a molecule spontaneously changing from the upper to the lower level (ie. the Einstein A coefficient - A$_{ul}$) divided by $4\pi$ steradians. All this together yields

$$j_\nu=\frac{h \nu}{4 \pi}\frac{dn_u}{d\nu}A_{ul}$$ (2.23)

which is the energy density emitted at a particular frequency in all directions per unit volume per second. Similarly an absorption coefficient can be defined as

$$\alpha_\nu=\frac{h\nu}{c}\left(\frac{dn_l}{d\nu}B_{lu}-\frac{dn_u}{d\nu}B_{ul}\right)$$ (2.24)

Note that here stimulated emission is taken to be negative absorption. This is the amount of energy absorbed per unit volume per second.