Einstein coefficients

Emission and absorption coefficients

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 A₂₁ (s⁻¹), 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 E₂ − E₁ = hν. Due to the energy-time uncertainty principle, the transition actually produces photons within a narrow range of frequencies called the spectral linewidth. If ${\displaystyle n_i}$ is the volumetric number density of atoms in state i, then the change in the number density of atoms in state 2 per unit time due to spontaneous emission will be $${\displaystyle {\left(\frac{d{n}_2}{dt}\right)}_{\text{spontaneous}}=-A_{21}{n}_2.}$$

The same process results in an increase in the population of state 1: $${\displaystyle {\left(\frac{d{n}_1}{dt}\right)}_{\text{spontaneous}}=A_{21}{n}_2.}$$

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. From the thermodynamic viewpoint, this process must be regarded as negative absorption. The process is described by the Einstein coefficient ${\displaystyle B_{21}}$ (m³ J⁻¹ s⁻²), which gives the probability per unit time per unit energy density of the radiation field per unit frequency 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 E₂ − E₁ = hν. The change in the number density of atoms in state 1 per unit time due to induced emission will be $${\displaystyle {\left(\frac{d{n}_1}{dt}\right)}_{\text{neg. absorb.}}=B_{21}{n}_2\rho (\nu ),}$$ where ${\displaystyle \rho (\nu )}$ denotes the spectral energy density of the isotropic radiation field at the frequency of the transition (see Planck's law).

Stimulated emission is one of the fundamental processes that led to the development of the laser. Laser radiation is, however, very far from the present case of isotropic radiation.

Absorption

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 ${\displaystyle B_{12}}$ (m³ J⁻¹ s⁻²), which gives the probability per unit time per unit energy density of the radiation field per unit frequency that an electron in state 1 with energy ${\displaystyle E_1}$ will absorb a photon with an energy E₂ − E₁ = hν 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 $${\displaystyle {\left(\frac{d{n}_1}{dt}\right)}_{\text{pos. absorb.}}=-B_{12}{n}_1\rho (\nu ).}$$

Units

Different authors use different units for frequency-dependent quantities. In particular, the spectral energy density can be expressed per unit of frequency, angular frequency, energy (e.g., electronvolt), or wavelength. Consequently, the ${\displaystyle B_{12}}$ and ${\displaystyle B_{21}}$ coefficients will have different units. This is sometimes indicated (e.g., ${\displaystyle B_{21}^{\omega }}$), but more often silently omitted. The ${\displaystyle A_{21}}$ coefficient remains in the units of 1/time.

Detailed balance

The Einstein coefficients are fixed probabilities per time 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, thermodynamic equilibrium will be valid universally.

At thermodynamic 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 (valid only at equilibrium) requires that the change in time of the number of atoms in level 1 due to the above three processes be zero: $${\displaystyle 0=A_{21}{n}_2+B_{21}{n}_2\rho (\nu )-B_{12}{n}_1\rho (\nu ).}$$

Along with detailed balancing, at temperature T 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 Boltzmann distribution we have for the number of excited atomic species i: $${\displaystyle \frac{n_i}{n}=\frac{g_i{e}^{-E_i/kT}}{Z},}$$ where n is the total number density of the atomic species, excited and unexcited, k is the Boltzmann constant, T is the temperature, ${\displaystyle g_i}$ is the degeneracy (also called the multiplicity) of state i, and Z is the partition function. From Planck's law of black-body radiation at temperature T we have for the spectral radiance (radiance is energy per unit time per unit solid angle per unit projected area, when integrated over an appropriate spectral interval) at frequency ν $${\displaystyle {\rho }_{\nu }(\nu ,T)=F(\nu )\frac{1}{e^{h\nu /kT}-1},}$$ where $${\displaystyle F(\nu )=\frac{2h{\nu }^3}{c^2},}$$ where ${\displaystyle c}$ is the speed of light and ${\displaystyle h}$ is the Planck constant.

Substituting these expressions into the equation of detailed balancing and remembering that E₂ − E₁ = hν yields $${\displaystyle A_{21}{g}_2{e}^{-h\nu /kT}+B_{21}{g}_2{e}^{-h\nu /kT}\frac{F(\nu )}{e^{h\nu /kT}-1}=B_{12}{g}_1\frac{F(\nu )}{e^{h\nu /kT}-1},}$$ or $${\displaystyle A_{21}{g}_2(1-e^{-h\nu /kT})+B_{21}{g}_{2}F(\nu )e^{-h\nu /kT}=B_{12}{g}_{1}F(\nu ).}$$

The above equation must hold at any temperature, so from ${\displaystyle T\to \mathrm{\infty }}$ one gets $${\displaystyle B_{21}{g}_2=B_{12}{g}_1,}$$ and from ${\displaystyle T\to 0}$ $${\displaystyle A_{21}{g}_2=B_{21}{g}_{2}F(\nu ).}$$

Therefore, the three Einstein coefficients are interrelated by $${\displaystyle \frac{A_{21}}{B_{21}}=F(\nu )}$$ 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 ${\displaystyle A_{21}}$ and ${\displaystyle B_{12}}$, involving Planck's law.

Oscillator strengths

The oscillator strength ${\displaystyle f_{12}}$ is defined by the following relation to the cross section ${\displaystyle \sigma }$ for absorption: $${\displaystyle \sigma =\frac{e^2}{4{\epsilon }_0{m}_{e}c}{f}_{12}{\varphi }_{\nu }=\frac{\pi e^2}{2{\epsilon }_0{m}_{e}c}{f}_{12}{\varphi }_{\omega },}$$

where ${\displaystyle e}$ is the electron charge, ${\displaystyle m_e}$ is the electron mass, and ${\displaystyle {\varphi }_{\nu }}$ and ${\displaystyle {\varphi }_{\omega }}$ are normalized distribution functions in frequency and angular frequency respectively. This allows all three Einstein coefficients to be expressed in terms of the single oscillator strength associated with the particular atomic spectral line: $${\displaystyle \begin{array}{cc}{B}_{12}& =\frac{e^2}{4{\epsilon }_0{m}_{e}h\nu }{f}_{12},\\ B_{21}& =\frac{e^2}{4{\epsilon }_0{m}_{e}h\nu }\frac{g_1}{g_2}{f}_{12},\\ A_{21}& =\frac{2\pi {\nu }^2{e}^2}{{\epsilon }_0{m}_e{c}^3}\frac{g_1}{g_2}{f}_{12}.\end{array}}$$

Dipole approximation

The value of A and B coefficients can be calculated using quantum mechanics where dipole approximations in time dependent perturbation theory is used. While the calculation of B coefficient can be done easily, that of A coefficient requires using results of second quantization. This is because the theory developed by dipole approximation and time dependent perturbation theory gives a semiclassical description of electronic transition which goes to zero as perturbing fields go to zero. The A coefficient which governs spontaneous emission should not go to zero as perturbing fields go to zero. The result for transition rates of different electronic levels as a result of spontaneous emission is given as (in SI units): $${\displaystyle w_{i\to f}^{\text{s.emi}}=\frac{{\omega }_{if}^3{e}^2}{3\pi {\epsilon }_0\hslash c^3}{\left|⟨f|\overrightarrow{r}|i⟩\right|}^2=A_{if}}$$

For B coefficient, straightforward application of dipole approximation in time dependent perturbation theory yields (in SI units): $${\displaystyle w_{i\to f}^{\text{abs}}=\frac{u({\omega }_{fi})\pi e^2}{3{\epsilon }_0{\hslash }^2}{\left|⟨f|\overrightarrow{r}|i⟩\right|}^2=B_{if}^{\text{abs}}u({\omega }_{fi})}$$ $${\displaystyle w_{i\to f}^{\text{emi}}=\frac{u({\omega }_{if})\pi e^2}{3{\epsilon }_0{\hslash }^2}{\left|⟨f|\overrightarrow{r}|i⟩\right|}^2=B_{if}^{emi}u({\omega }_{if})}$$

Note that the rate of transition formula depends on dipole moment operator. For higher order approximations, it involves quadrupole moment and other similar terms.

Here, the B coefficients are chosen to correspond to ${\displaystyle \omega }$ energy distribution function. Often these different definitions of B coefficients are distinguished by superscript, for example, $B_{21}^f=\frac{B_{21}^{\omega }}{2\pi }$ where $B_{21}^f$ term corresponds to frequency distribution and $B_{21}^{\omega }$ term corresponds to ${\displaystyle \omega }$ distribution. The formulas for B coefficients varies inversely to that of the energy distribution chosen, so that the transition rate is same regardless of convention.

Hence, AB coefficients are calculated using dipole approximation as: $${\displaystyle \begin{array}{cc}{A}_{ab}& =\frac{{\omega }_{ab}^3{e}^2}{3\pi {\epsilon }_0\hslash c^3}{\left|⟨a|\overrightarrow{r}|b⟩\right|}^2\\ B_{ab}& =\frac{\pi e^2}{3{\epsilon }_0{\hslash }^2}{\left|⟨a|\overrightarrow{r}|b⟩\right|}^2\end{array}}$$ where ${\displaystyle {\omega }_{ab}=\frac{E_a-E_b}{\hslash }}$ and B coefficients correspond to ${\displaystyle \omega }$ energy distribution function.

Hence the following ratios are also derived: $${\displaystyle \frac{B_{12}}{B_{21}}=1}$$ and $${\displaystyle \frac{A_{if}}{B}=\frac{{\omega }_{if}^3\hslash }{{\pi }^2{c}^3}}$$

Derivation of Planck's law

It follows from theory that: $${\displaystyle \frac{d{N}_b}{dt}=-A_{ba}{N}_b-N_{b}u({\omega }_{ba})B_{ba}+N_{a}u({\omega }_{ba})B_{ab}=-N_b{w}_{b\to a}^{\text{s.emi}}-N_b{w}_{b\to a}^{\text{emi}}+N_a{w}_{a\to b}^{\text{abs}}}$$ where ${\displaystyle N_a}$ and ${\displaystyle N_b}$ are number of occupied energy levels of ${\displaystyle E_a}$ and ${\displaystyle E_b}$ respectively, where ${\displaystyle E_b>E_a}$. Note that from time dependent perturbation theory application, the fact that only radiation whose ${\displaystyle \omega }$ is close to value of ${\displaystyle {\omega }_{ba}}$ can produce respective stimulated emission or absorption, is used.

Where Maxwell distribution involving ${\displaystyle N_a}$ and ${\displaystyle N_b}$ ensures ${\displaystyle \frac{N_a}{N_b}=\frac{e^{-E_a\beta }}{e^{-E_b\beta }}=e^{{\omega }_{ba}\hslash \beta }}$

Solving for ${\displaystyle u}$ for equilibrium condition ${\displaystyle \frac{d{N}_b}{dt}=0}$ using the above equations and ratios while generalizing ${\displaystyle {\omega }_{ba}}$ to ${\displaystyle \omega }$, we get: $${\displaystyle u_{\omega }(\omega ,T)=\frac{{\omega }^3\hslash }{{\pi }^2{c}^3}\frac{1}{e^{\omega \hslash \beta }-1}}$$ which is the angular frequency energy distribution from Planck's law.