The **Einstein coefficients** describe the probability of absorption or emission of a photon by an atom or molecule. The Einstein $A$ coefficient is related to the rate of [[#spontaneous emission]], and the Einstein $B$ coefficients are related to the [[#absorption]] and [[#stimulated emission]]. ![[einstein_coefficents.jpg|align:center]] | Einstein Coefficient | Atomic Process | Reaction | Units | | :------------------: | :------------------------------------: | :----------------------------------: | :---------------------------------------------: | | $A_{21}$ | <nobr>[[#Spontaneous Emission]]</nobr> | $E_2 \rightarrow E_1 + h\nu$ | $\pu{s^{-1}}$ | | $B_{12}$ | [[#Absorption]] | $h\nu + E_1 \rightarrow E_2$ | $\pu{s^{-1}\,(\text{specific intensity})^{-1}}$ | | $B_{21}$ | [[#Stimulated Emission]] | $h\nu + E_2 \rightarrow E_1 + 2h\nu$ | $\pu{s^{-1}\,(\text{specific intensity})^{-1}}$ | > [!note] > The coefficients are in units of [[Intensity#Specific Intensity|specific intensity]] since these processes depend on the photon "density". > - $A$ does not depend on nearby light (spontaneous) > - $B$ does depend on nearby light *(B = Buddies)* With these coefficients defined, we can use the volume number density of particles in each energy state ($n_{1}$, $n_{2}$) with the [[Intensity#Specific Mean Intensity|specific mean intensity]] to identify the transition rate (per unit volume) at which the processes occur. > [!example] For example... > $ > \left[ A_{ij} n_{i} J_{\nu} \right] \equiv \left[ \frac{1}{{\rm second} \cdot {\rm intensity}} \cdot \frac{\rm particles}{{\rm volume}} \cdot \frac{\rm intensity}{1} \right] = \left[ \frac{\rm particles}{{\rm second} \cdot {\rm volume}} \right] = \left[ \frac{\rm transitions}{\rm volume} \right] > $ ## Spontaneous Emission The **"Einstein $A$ coefficient for spontaneous emission"** ($A_{21}$) represents the probability (per unit time) of an electron "spontaneously" decaying from a higher energy level to a lower one. In terms of number density of the two states... $ \left( \frac{\mathrm{d} n_{1}}{\mathrm{d} t} \right)_{\rm spontaneous} = A_{21} n_{2} \hspace{2cm} \left( \frac{\mathrm{d} n_{2}}{\mathrm{d} t} \right)_{\rm spontaneous} = - A_{21} n_{2} $ ## Stimulated Emission The **"Einstein $B$ coefficient for stimulated emission"** ($B_{21}$) represents the probability (per unit time) of an electron transitioning from a higher energy level to a lower one $ \left( \frac{\mathrm{d} n_{1}}{\mathrm{d} t} \right)_{\rm stimulated} = B_{21} n_{2} J_{\nu} \hspace{2cm} \left( \frac{\mathrm{d} n_{2}}{\mathrm{d} t} \right)_{\rm stimulated} = - B_{21} n_{2} J_{\nu} $ ## Absorption The **"Einstein $B$ coefficient for absorption"** ($B_{12}$) represents the probability (per unit time) of an electron moving to a higher level from a lower level through the absorption of a photon. $ \left( \frac{\mathrm{d} n_{1}}{\mathrm{d} t} \right)_{\rm absorption} = - B_{12} n_{1} J_{\nu} \hspace{2cm} \left( \frac{\mathrm{d} n_{2}}{\mathrm{d} t} \right)_{\rm absorption} = B_{12} n_{1} J_{\nu} $ ## Detailed Balancing In thermal equilibrium, the rate of transitions from ($1 \to 2$) should be equal to the rate of transitions ($2 \to 1$). Meaning, if we look specifically at a single energy level, the total inflow and outflow of particles should be equal to zero. $ \begin{align} (\text{Transitions } \uparrow)_{12} +(\text{Transitions } \downarrow)_{21} &= 0 \\ \\ \left[ \left( \frac{\mathrm{d} n_{1}}{\mathrm{d} t} \right)_{\rm absorption} \right] + \left[ \left( \frac{\mathrm{d} n_{1}}{\mathrm{d} t} \right)_{\rm spontaneous} + \left( \frac{\mathrm{d} n_{1}}{\mathrm{d} t} \right)_{\rm stimulated} \right]&= 0 \\ \\ \Big[ - B_{12} n_{1} J_{\nu} \Big] + \Big[ A_{21} n_{2} + B_{21} n_{2} J_{\nu} \Big] &= 0 \\ \\ n_{2} \Big[ A_{21} + B_{21} J_{\nu} \Big] &= n_{1} \Big[ B_{12} J_{\nu} \Big] \\ \frac{n_{2}}{n_{1}} &= \frac{B_{12} J_{\nu}}{A_{21} + B_{21} J_{\nu}} \end{align} $ From the [[Statistical Mechanics#Maxwell-Boltzmann]] distribution, we can re-express the ratio number densities ($n_{i}$) in terms of the energy ($E_{i}$) and the degeneracy ($g_{i}$) of the states. $\frac{n_{i}}{n} = \frac{1}{Z} \left( g_{i} \, e^{- E_{i} / k_{\rm B} T} \right) \hspace{1cm} \Rightarrow \hspace{1cm} \frac{n_{2}}{n_{1}} = \frac{g_{2}}{g_{1}} \; e^{- (E_{2} - E_{1}) / k_{\rm B} T}$ Applying this to the balancing equation where $(E_{2} - E_{1}) = h \nu$... $ \frac{n_{2}}{n_{1}} = \frac{B_{12} J_{\nu}}{A_{21} + B_{21} J_{\nu}} = \frac{g_{2}}{g_{1}} \; e^{- h \nu / k_{\rm B} T} \hspace{1cm} \Rightarrow \hspace{1cm} \begin{aligned}[t] J_{\nu} &= \frac{A_{21} \left( \frac{g_{2}}{g_{1}} \; e^{- h \nu / k_{\rm B} T} \right)}{B_{12} - B_{21}\left( \frac{g_{2}}{g_{1}} \; e^{- h \nu / k_{\rm B} T} \right)} \\ \\ J_{\nu} &= \frac{\left(\frac{A_{21}}{B_{21}}\right)}{\left(\frac{g_{1} B_{12}}{g_{2} B_{21}}\right) e^{h \nu / k_{\rm B} T} - 1} \\ \end{aligned} $ For a thermal process, we can take $J_{\nu}$ as the [[Blackbody Radiation#Blackbody Specific Intensity]] ($B_{\nu}$). $B_{\nu} = \frac{2 h \nu^{3}}{c^{2}} \left(\frac{1}{e^{h \nu / k_{\rm B} T} - 1}\right)$ Comparing the two expressions, we can find the **Einstein Coefficient Relationships**, such that if we know one of the numbers, then we know all three. $J_{\nu} = B_{\nu} \hspace{1cm} \Rightarrow \hspace{1cm} \boxed{\frac{A_{21}}{B_{21}} = \left(\frac{2 h \nu^{3}}{c^{2}}\right)} \hspace{1cm} , \hspace{1cm} \boxed{\frac{B_{12}}{B_{21}} = \frac{g_{2}}{g_{1}}}$ > [!note] > Though these relationships were derived while in thermal equilibrium, they still hold true outside of equilibrium as well. - Cian