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diff --git a/source/know/concept/einstein-coefficients/index.md b/source/know/concept/einstein-coefficients/index.md new file mode 100644 index 0000000..ff44888 --- /dev/null +++ b/source/know/concept/einstein-coefficients/index.md @@ -0,0 +1,341 @@ +--- +title: "Einstein coefficients" +date: 2021-07-11 +categories: +- Physics +- Optics +- Quantum mechanics +- Two-level system +- Laser theory +layout: "concept" +--- + +The **Einstein coefficients** quantify +the emission and absorption of photons by a solid, +and can be calculated analytically from first principles +in several useful situations. + + +## Qualitative description + +Suppose we have a ground state with energy $E_1$ containing $N_1$ electrons, +and an excited state with energy $E_2$ containing $N_2$ electrons. +The resonance $\omega_0 \equiv (E_2 \!-\! E_1)/\hbar$ +is the frequency of the photon emitted +when an electron falls from $E_2$ to $E_1$. + +The first Einstein coefficient is the **spontaneous emission rate** $A_{21}$, +which gives the probability per unit time +that an excited electron falls from state 2 to 1, +so that $N_2(t)$ obeys the following equation, +which is easily solved: + +$$\begin{aligned} + \dv{N_2}{t} = - A_{21} N_2 + \quad \implies \quad + N_2(t) = N_2(0) \exp(- t / \tau) +\end{aligned}$$ + +Where $\tau = 1 / A_{21}$ is the **natural radiative lifetime** of the excited state, +which gives the lifetime of an excited electron, +before it decays to the ground state. + +The next coefficient is the **absorption rate** $B_{12}$, +which is the probability that an incoming photon excites an electron, +per unit time and per unit spectral energy density +(i.e. the rate depends on the frequency of the incoming light). +Then $N_1(t)$ obeys the following equation: + +$$\begin{aligned} + \dv{N_1}{t} = - B_{12} N_1 u(\omega_0) +\end{aligned}$$ + +Where $u(\omega)$ is the spectral energy density of the incoming light, +put here to express the fact that only photons with frequency $\omega_0$ are absorbed. + +There is one more Einstein coefficient: the **stimulated emission rate** $B_{21}$. +An incoming photon has an associated electromagnetic field, +which can encourage an excited electron to drop to the ground state, +such that for $A_{21} = 0$: + +$$\begin{aligned} + \dv{N_2}{t} = - B_{21} N_2 u(\omega_0) +\end{aligned}$$ + +These three coefficients $A_{21}$, $B_{12}$ and $B_{21}$ are related to each other. +Suppose that the system is in equilibrium, +i.e. that $N_1$ and $N_2$ are constant. +We assume that the number of particles in the system is constant, +implying that $N_1'(t) = - N_2'(t) = 0$, so: + +$$\begin{aligned} + B_{12} N_1 u(\omega_0) = A_{21} N_2 + B_{21} N_2 u(\omega_0) = 0 +\end{aligned}$$ + +Isolating this equation for $u(\omega_0)$, +gives following expression for the radiation: + +$$\begin{aligned} + u(\omega_0) + = \frac{A_{21}}{(N_1 / N_2) B_{12} - B_{21}} +\end{aligned}$$ + +We assume that the system is in thermal equilibrium +with its own black-body radiation, and that there is no external light. +Then this is a [canonical ensemble](/know/concept/canonical-ensemble/), +meaning that the relative probability that an electron has $E_2$ compared to $E_1$ +is given by the Boltzmann distribution: + +$$\begin{aligned} + \frac{\mathrm{Prob}(E_2)}{\mathrm{Prob}(E_1)} + = \frac{N_2}{N_1} + = \frac{g_2}{g_1} \exp(- \hbar \omega_0 \beta) +\end{aligned}$$ + +Where $g_2$ and $g_1$ are the degeneracies of the energy levels. +Inserting this back into the equation for the spectrum $u(\omega_0)$ yields: + +$$\begin{aligned} + u(\omega_0) + = \frac{A_{21}}{(g_1 / g_2) B_{12} \exp(\hbar \omega_0 \beta) - B_{21}} +\end{aligned}$$ + +Since $u(\omega_0)$ represents only black-body radiation, +our result must agree with [Planck's law](/know/concept/plancks-law/): + +$$\begin{aligned} + u(\omega_0) + = \frac{A_{21}}{B_{21} \big( (g_1 B_{12} / g_2 B_{21}) \exp(\hbar \omega_0 \beta) - 1 \big)} + = \frac{\hbar \omega_0^3}{\pi^2 c^3} \frac{1}{\exp(\hbar \omega_0 \beta) - 1} +\end{aligned}$$ + +This gives us the following two equations relating the Einstein coefficients: + +$$\begin{aligned} + \boxed{ + A_{21} = \frac{\hbar \omega_0^3}{\pi^2 c^3} B_{21} + \qquad \quad + g_1 B_{12} = g_2 B_{21} + } +\end{aligned}$$ + +Note that this result holds even if $E_1$ is not the ground state, +but instead some lower excited state below $E_2$, +due to the principle of [detailed balance](/know/concept/detailed-balance/). +Furthermore, it turns out that these relations +also hold if the system is not in equilibrium. + +A notable case is **population inversion**, +where $B_{21} N_2 > B_{12} N_1$ such that $N_2 > (g_2 / g_1) N_1$. +This situation is mandatory for lasers, where stimulated emission must dominate, +such that the light becomes stronger as it travels through the medium. + + +## Coherent light + +In fact, we can analytically calculate the Einstein coefficients in some cases, +by treating incoming light as a perturbation +to an electron in a two-level system, +and then finding $B_{12}$ and $B_{21}$ from the resulting transition rate. +We need to make the [electric dipole approximation](/know/concept/electric-dipole-approximation/), +in which case the perturbing Hamiltonian $\hat{H}_1(t)$ is given by: + +$$\begin{aligned} + \hat{H}_1(t) + = - q \vec{r} \cdot \vec{E}_0 \cos(\omega t) +\end{aligned}$$ + +Where $q = -e$ is the electron charge, +$\vec{r}$ is the position operator, +and $\vec{E}_0$ is the amplitude of +the [electromagnetic wave](/know/concept/electromagnetic-wave-equation/). +For simplicity, we let the amplitude be along the $z$-axis: + +$$\begin{aligned} + \hat{H}_1(t) + = - q E_0 z \cos(\omega t) +\end{aligned}$$ + +This form of $\hat{H}_1$ is a well-known case for +[time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/), +which tells us that the transition probability from $\Ket{a}$ to $\Ket{b}$ is: + +$$\begin{aligned} + P_{ab} + = \frac{\big|\!\matrixel{a}{H_1}{b}\!\big|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_{ba} - \omega) t / 2 \big)}{(\omega_{ba} - \omega)^2} +\end{aligned}$$ + +If the nucleus is at $z = 0$, +then generally $\Ket{1}$ and $\Ket{2}$ will be even or odd functions of $z$, +meaning that $\matrixel{1}{z}{1} = \matrixel{2}{z}{2} = 0$ +(see also [Laporte's selection rule](/know/concept/selection-rules/)), +leading to: + +$$\begin{gathered} + \matrixel{1}{H_1}{2} = - E_0 d^* + \qquad + \matrixel{2}{H_1}{1} = - E_0 d + \\ + \matrixel{1}{H_1}{1} = \matrixel{2}{H_1}{2} = 0 +\end{gathered}$$ + +Where $d \equiv q \matrixel{2}{z}{1}$ is a constant, +namely the $z$-component of the **transition dipole moment**. +The chance of an upward jump (i.e. absorption) is: + +$$\begin{aligned} + P_{12} + = \frac{E_0^2 |d|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2} +\end{aligned}$$ + +Meanwhile, the transition probability for stimulated emission is as follows, +using the fact that $P_{ab}$ is a sinc-function, +and is therefore symmetric around $\omega_{ba}$: + +$$\begin{aligned} + P_{21} + = \frac{E_0^2 |d|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2} +\end{aligned}$$ + +Surprisingly, the probabilities of absorption and stimulated emission are the same! +In practice, however, the relative rates of these two processes depends heavily on +the availability of electrons and holes in both states. + +In theory, we could calculate the transition rate $R_{12} = \ipdv{P_{12}}{t}$, +which would give us Einstein's absorption coefficient $B_{12}$, +for this specific case of coherent monochromatic light. +However, the result would not be constant in time $t$, +so is not really useful. + + +## Polarized light + +To solve this "problem", we generalize to (incoherent) polarized polychromatic light. +To do so, we note that the energy density $u$ of an electric field $E_0$ is given by: + +$$\begin{aligned} + u = \frac{1}{2} \varepsilon_0 E_0^2 + \qquad \implies \qquad + E_0^2 = \frac{2 u}{\varepsilon_0} +\end{aligned}$$ + +Where $\varepsilon_0$ is the vacuum permittivity. +Putting this in the previous result for $P_{12}$ gives us: + +$$\begin{aligned} + P_{12} + = \frac{2 u |d|^2}{\varepsilon_0 \hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2} +\end{aligned}$$ + +For a continuous light spectrum, +this $u$ turns into the spectral energy density $u(\omega)$: + +$$\begin{aligned} + P_{12} + = \frac{2 |d|^2}{\varepsilon_0 \hbar^2} + \int_0^\infty \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2} u(\omega) \dd{\omega} +\end{aligned}$$ + +From here, the derivation is similar to that of +[Fermi's golden rule](/know/concept/fermis-golden-rule/), +despite the distinction that we are integrating over frequencies rather than states. + +At sufficiently large $t$, the integrand is sharply peaked at $\omega = \omega_0$ +and negligible everywhere else, +so we take $u(\omega)$ out of the integral and extend the integration limits. +Then we rewrite and look up the integral, +which turns out to be $\pi t$: + +$$\begin{aligned} + P_{12} + = \frac{|d|^2}{\varepsilon_0 \hbar^2} u(\omega_0) \int_{-\infty}^\infty \frac{\sin^2\!\big(x t \big)}{x^2} \dd{x} + = \frac{\pi |d|^2}{\varepsilon_0 \hbar^2} u(\omega_0) \:t +\end{aligned}$$ + +From this, the transition rate $R_{12} = B_{12} u(\omega_0)$ +is then calculated as follows: + +$$\begin{aligned} + R_{12} + = \pdv{P_{12}}{t} + = \frac{\pi |d|^2}{\varepsilon_0 \hbar^2} u(\omega_0) +\end{aligned}$$ + +Using the relations from earlier with $g_1 = g_2$, +the Einstein coefficients are found to be as follows +for a polarized incoming light spectrum: + +$$\begin{aligned} + \boxed{ + B_{21} = B_{12} = \frac{\pi |d|^2}{\varepsilon_0 \hbar^2} + \qquad + A_{21} = \frac{\omega_0^3 |d|^2}{\pi \varepsilon_0 \hbar c^3} + } +\end{aligned}$$ + + +## Unpolarized light + +We can generalize the above result even further to unpolarized light. +Let us return to the matrix elements of the perturbation $\hat{H}_1$, +and define the polarization unit vector $\vec{n}$: + +$$\begin{aligned} + \matrixel{2}{\hat{H}_1}{1} + = - \vec{d} \cdot \vec{E}_0 + = - E_0 (\vec{d} \cdot \vec{n}) +\end{aligned}$$ + +Where $\vec{d} \equiv q \matrixel{2}{\vec{r}}{1}$ is +the full **transition dipole moment** vector, which is usually complex. + +The goal is to calculate the average of $|\vec{d} \cdot \vec{n}|^2$. +In [spherical coordinates](/know/concept/spherical-coordinates/), +we integrate over all directions $\vec{n}$ for fixed $\vec{d}$, +using that $\vec{d} \cdot \vec{n} = |\vec{d}| \cos(\theta)$ +with $|\vec{d}| \equiv |d_x|^2 \!+\! |d_y|^2 \!+\! |d_z|^2$: + +$$\begin{aligned} + \Expval{|\vec{d} \cdot \vec{n}|^2} + = \frac{1}{4 \pi} \int_0^\pi \int_0^{2 \pi} |\vec{d}|^2 \cos^2(\theta) \sin(\theta) \dd{\varphi} \dd{\theta} +\end{aligned}$$ + +Where we have divided by $4\pi$ (the surface area of a unit sphere) for normalization, +and $\theta$ is the polar angle between $\vec{n}$ and $\vec{d}$. +Evaluating the integrals yields: + +$$\begin{aligned} + \Expval{|\vec{d} \cdot \vec{n}|^2} + = \frac{2 \pi}{4 \pi} |\vec{d}|^2 \int_0^\pi \cos^2(\theta) \sin(\theta) \dd{\theta} + = \frac{|\vec{d}|^2}{2} \Big[ \!-\! \frac{\cos^3(\theta)}{3} \Big]_0^\pi + = \frac{|\vec{d}|^2}{3} +\end{aligned}$$ + +With this additional constant factor $1/3$, +the transition rate $R_{12}$ is modified to: + +$$\begin{aligned} + R_{12} + = \pdv{P_{12}}{t} + = \frac{\pi |\vec{d}|^2}{3 \varepsilon_0 \hbar^2} u(\omega_0) +\end{aligned}$$ + +From which it follows that the Einstein coefficients for unpolarized light are given by: + +$$\begin{aligned} + \boxed{ + B_{21} = B_{12} = \frac{\pi |\vec{d}|^2}{3 \varepsilon_0 \hbar^2} + \qquad + A_{21} = \frac{\omega_0^3 |\vec{d}|^2}{3 \pi \varepsilon_0 \hbar c^3} + } +\end{aligned}$$ + + + +## References +1. M. Fox, + *Optical properties of solids*, 2nd edition, + Oxford. +2. D.J. Griffiths, D.F. Schroeter, + *Introduction to quantum mechanics*, 3rd edition, + Cambridge. |