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diff --git a/content/know/concept/einstein-coefficients/index.pdc b/content/know/concept/einstein-coefficients/index.pdc new file mode 100644 index 0000000..37141f2 --- /dev/null +++ b/content/know/concept/einstein-coefficients/index.pdc @@ -0,0 +1,409 @@ +--- +title: "Einstein coefficients" +firstLetter: "E" +publishDate: 2021-07-11 +categories: +- Physics +- Optics +- Quantum mechanics + +date: 2021-07-11T18:22:14+02:00 +draft: false +markup: pandoc +--- + +# Einstein coefficients + +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: + +$$\begin{aligned} + u(\omega_0) + = \frac{A_{21}}{B_{21} \big( (g_1 B_{12} / g_2 B_{21}) \exp\!(\hbar \omega_0 \beta) - 1 \big)} + = \frac{\hbar \omega_0^3}{\pi^2 c^3} \frac{1}{\exp\!(\hbar \omega_0 \beta) - 1} +\end{aligned}$$ + +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*. +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. + + +## Electric dipole approximation + +In fact, we can analytically calculate the Einstein coefficients, +if we make a mild approximation. +Consider the Hamiltonian of an electron with charge $q = - e$: + +$$\begin{aligned} + \hat{H} + &= \frac{\vec{P}{}^2}{2 m} - \frac{q}{2 m} (\vec{A} \cdot \vec{P} + \vec{P} \cdot \vec{A}) + \frac{q^2 \vec{A}{}^2}{2m} + q \phi +\end{aligned}$$ + +With $\vec{A}(\vec{r}, t)$ the magnetic vector potential, +and $\phi(\vec{r}, t)$ the electric scalar potential. +We reduce this by fixing the Coulomb gauge $\nabla \!\cdot\! \vec{A} = 0$, +such that $\vec{A} \cdot \vec{P} = \vec{P} \cdot \vec{A}$, +and by assuming that $\vec{A}{}^2$ is negligibly small. +This leaves us with: + +$$\begin{aligned} + \hat{H} + &= \frac{\vec{P}{}^2}{2 m} - \frac{q}{m} \vec{P} \cdot \vec{A} + q \phi +\end{aligned}$$ + +The last term is the Coulomb interaction +between the electron and the nucleus. +We can interpret the second term, involving the weak $\vec{A}$, as a perturbation $\hat{H}_1$: + +$$\begin{aligned} + \hat{H} + = \hat{H}_0 + \hat{H}_1 + \qquad \quad + \hat{H}_0 + \equiv \frac{\vec{P}{}^2}{2 m} + q \phi + \qquad \quad + \hat{H}_1 + \equiv - \frac{q}{m} \vec{P} \cdot \vec{A} +\end{aligned}$$ + +Suppose that $\vec{A}$ is oscillating sinusoidally in time and space as follows: + +$$\begin{aligned} + \vec{A}(\vec{r}, t) = \vec{A}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t) +\end{aligned}$$ + +The corresponding perturbative electric field $\vec{E}$ points in the same direction: + +$$\begin{aligned} + \vec{E}(\vec{r}, t) + = - \pdv{\vec{A}}{t} + = \vec{E}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t) +\end{aligned}$$ + +Where $\vec{E}_0 = i \omega \vec{A}_0$. +Let us restrict ourselves to visible light, +whose wavelength $2 \pi / k \approx 10^{-6} \:\mathrm{m}$. +By comparison, the size of an atomic orbital is on the order of $10^{-10} \:\mathrm{m}$, +so we can ignore the dot product $\vec{k} \cdot \vec{r}$. +This is the **electric dipole approximation**: +the radiation is treated classicaly, +while the electron is treated quantum-mechanically. + +$$\begin{aligned} + \vec{E}(\vec{r}, t) + \approx \vec{E}_0 \exp\!(- i \omega t) +\end{aligned}$$ + +Next, we want to convert $\hat{H}_1$ +to use the electric field $\vec{E}$ instead of the potential $\vec{A}$. +To do so, we rewrite the momemtum $\vec{P} = m \: \dv*{\vec{r}}{t}$ +and evaluate this in the [Heisenberg picture](/know/concept/heisenberg-picture/): + +$$\begin{aligned} + \matrixel{2}{\dv*{\vec{r}}{t}}{1} + &= \frac{i}{\hbar} \matrixel{2}{[\hat{H}_0, \vec{r}]}{1} + = \frac{i}{\hbar} \matrixel{2}{\hat{H}_0 \vec{r} - \vec{r} \hat{H}_0}{1} + \\ + &= \frac{i}{\hbar} (E_2 - E_1) \matrixel{2}{\vec{r}}{1} + = i \omega_0 \matrixel{2}{\vec{r}}{1} +\end{aligned}$$ + +Therefore, $\vec{P} / m = i \omega_0 \vec{r}$, +where $\omega_0 = (E_2 - E_1) / \hbar$ is the resonance frequency of the transition, +close to which we assume that $\vec{A}$ and $\vec{E}$ are oscillating. +We thus get: + +$$\begin{aligned} + \hat{H}_1(t) + &= - \frac{q}{m} \vec{P} \cdot \vec{A} + = - i q \omega_0 \vec{r} \cdot \vec{A}_0 \exp\!(- i \omega t) + \\ + &= - q \vec{r} \cdot \vec{E}_0 \exp\!(- i \omega t) + = - \vec{p} \cdot \vec{E}_0 \exp\!(- i \omega t) +\end{aligned}$$ + +Where $\vec{p} \equiv q \vec{r} = - e \vec{r}$ is the electric dipole moment of the electron, +hence the name *electric dipole approximation*. + + +## Polarized light + +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 location of the nucleus of the atom has $z = 0$, +then generally $\ket{1}$ and $\ket{2}$ will be even or odd functions of $z$, +such that $\matrixel{1}{z}{1} = \matrixel{2}{z}{2} = 0$, leading to: + +$$\begin{gathered} + \matrixel{1}{H_1}{2} = - q E_0 V + \qquad + \matrixel{2}{H_1}{1} = - q E_0 V^* + \\ + \matrixel{1}{H_1}{1} = \matrixel{2}{H_1}{2} = 0 +\end{gathered}$$ + +Where $V \equiv \matrixel{1}{z}{2}$ is a constant. +The chance of an upward jump (i.e. absorption) is: + +$$\begin{aligned} + P_{12} + = \frac{q^2 E_0^2 |V|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2} +\end{aligned}$$ + +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 $\omega_{ba}$: + +$$\begin{aligned} + P_{21} + = \frac{q^2 E_0^2 |V|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2} +\end{aligned}$$ + +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 $R_{12} = \pdv*{P_{12}}{t}$, +which would give us Einstein's absorption coefficient $B_{12}$, +for this particular case of coherent monochromatic light. +However, the result would not be constant in time $t$. + +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: + +$$\begin{aligned} + u = \frac{1}{2} \varepsilon_0 E_0^2 + \qquad \implies \qquad + E_0^2 = \frac{2 u}{\varepsilon_0} +\end{aligned}$$ + +Putting this in the previous result gives the following transition probability: + +$$\begin{aligned} + P_{12} + = \frac{2 u q^2 |V|^2}{\varepsilon_0 \hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2} +\end{aligned}$$ + +For a continuous light spectrum, +this $u$ turns into the spectral energy density $u(\omega)$: + +$$\begin{aligned} + P_{12} + = \frac{2 q^2 |V|^2}{\varepsilon_0 \hbar^2} + \int_0^\infty \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2} u(\omega) \dd{\omega} +\end{aligned}$$ + +From here, we 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 $\pi t$: + +$$\begin{aligned} + P_{12} + = \frac{q^2 |V|^2}{\varepsilon_0 \hbar^2} u(\omega_0) \int_{-\infty}^\infty \frac{\sin^2\!\big(x t \big)}{x^2} \dd{x} + = \frac{\pi q^2 |V|^2}{\varepsilon_0 \hbar^2} u(\omega_0) \:t +\end{aligned}$$ + +From this, the transition rate $R_{12} = B_{12} u(\omega_0)$ +is then calculated as follows: + +$$\begin{aligned} + R_{12} + = \pdv{P_{2 \to 1}}{t} + = \frac{\pi q^2 |V|^2}{\varepsilon_0 \hbar^2} u(\omega_0) +\end{aligned}$$ + +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: + +$$\begin{aligned} + \boxed{ + B_{21} = B_{12} = \frac{\pi q^2 |V|^2}{\varepsilon_0 \hbar^2} + \qquad + A_{21} = \frac{\omega_0^3 q^2 |V|^2}{\pi \varepsilon \hbar c^3} + } +\end{aligned}$$ + + +## 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 $\vec{n}$: + +$$\begin{aligned} + \matrixel{1}{\hat{H}_1}{2} + = - q \matrixel{1}{\vec{r} \cdot \vec{E}_0}{2} + = - q E_0 \matrixel{1}{\vec{r} \cdot \vec{n}}{2} + = - q E_0 W +\end{aligned}$$ + +The goal is to obtain the average of $|W|^2$, +where $W \equiv \matrixel{1}{\vec{r} \cdot \vec{n}}{2}$. +In [spherical coordinates](/know/concept/spherical-coordinates/), +we integrate over all possible orientations $\vec{n}$ for fixed $\vec{r}$, +using that $\vec{r} \cdot \vec{n} = |\vec{r}| \cos\!(\theta)$: + +$$\begin{aligned} + \expval{|W|^2} + = \frac{1}{4 \pi} \int_0^\pi \int_0^{2 \pi} |\matrixel{1}{\vec{r}}{2}|^2 \cos^2(\theta) \sin\!(\theta) \dd{\varphi} \dd{\theta} +\end{aligned}$$ + +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 $\vec{p}$. +Evaluating the integrals yields: + +$$\begin{aligned} + \expval{|W|^2} + = \frac{2 \pi}{4 \pi} |V|^2 \int_0^\pi \cos^2(\theta) \sin\!(\theta) \dd{\theta} + = \frac{|V|^2}{2} \Big[ \!-\! \frac{\cos^3(\theta)}{3} \Big]_0^\pi + = \frac{|V|^2}{3} +\end{aligned}$$ + +With this additional constant factor $1/3$, +the transition rate $R_{12}$ is modified to: + +$$\begin{aligned} + R_{12} + = \frac{\pi q^2 |V|^2}{3 \varepsilon_0 \hbar^2} u(\omega_0) +\end{aligned}$$ + +From which it follows that the Einstein coefficients for unpolarized light are given by: + +$$\begin{aligned} + \boxed{ + B_{21} = B_{12} = \frac{\pi q^2 |V|^2}{3 \varepsilon_0 \hbar^2} + \qquad + A_{21} = \frac{\omega_0^3 q^2 |V|^2}{3 \pi \varepsilon \hbar c^3} + } +\end{aligned}$$ + + + +## 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. |