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diff --git a/source/know/concept/einstein-coefficients/index.md b/source/know/concept/einstein-coefficients/index.md index 27d6013..179d866 100644 --- a/source/know/concept/einstein-coefficients/index.md +++ b/source/know/concept/einstein-coefficients/index.md @@ -19,16 +19,16 @@ 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 $\omega_0 \equiv (E_2 \!-\! E_1)/\hbar$ +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 $E_1$. +when an electron falls from $$E_2$$ to $$E_1$$. -The first Einstein coefficient is the **spontaneous emission rate** $A_{21}$, +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, +so that $$N_2(t)$$ obeys the following equation, which is easily solved: $$\begin{aligned} @@ -37,43 +37,43 @@ $$\begin{aligned} N_2(t) = N_2(0) \exp(- t / \tau) \end{aligned}$$ -Where $\tau = 1 / A_{21}$ is the **natural radiative lifetime** of the excited state, +Where $$\tau = 1 / A_{21}$$ is the **natural radiative lifetime** of the excited state, which gives the lifetime of an excited electron, before it decays to the ground state. -The next coefficient is the **absorption rate** $B_{12}$, +The next coefficient is the **absorption rate** $$B_{12}$$, which is the probability that an incoming photon excites an electron, per unit time and per unit spectral energy density (i.e. the rate depends on the frequency of the incoming light). -Then $N_1(t)$ obeys the following equation: +Then $$N_1(t)$$ obeys the following equation: $$\begin{aligned} \dv{N_1}{t} = - B_{12} N_1 u(\omega_0) \end{aligned}$$ -Where $u(\omega)$ is the spectral energy density of the incoming light, -put here to express the fact that only photons with frequency $\omega_0$ are absorbed. +Where $$u(\omega)$$ is the spectral energy density of the incoming light, +put here to express the fact that only photons with frequency $$\omega_0$$ are absorbed. -There is one more Einstein coefficient: the **stimulated emission rate** $B_{21}$. +There is one more Einstein coefficient: the **stimulated emission rate** $$B_{21}$$. An incoming photon has an associated electromagnetic field, which can encourage an excited electron to drop to the ground state, -such that for $A_{21} = 0$: +such that for $$A_{21} = 0$$: $$\begin{aligned} \dv{N_2}{t} = - B_{21} N_2 u(\omega_0) \end{aligned}$$ -These three coefficients $A_{21}$, $B_{12}$ and $B_{21}$ are related to each other. +These three coefficients $$A_{21}$$, $$B_{12}$$ and $$B_{21}$$ are related to each other. Suppose that the system is in equilibrium, -i.e. that $N_1$ and $N_2$ are constant. +i.e. that $$N_1$$ and $$N_2$$ are constant. We assume that the number of particles in the system is constant, -implying that $N_1'(t) = - N_2'(t) = 0$, so: +implying that $$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}$$ -Isolating this equation for $u(\omega_0)$, +Isolating this equation for $$u(\omega_0)$$, gives following expression for the radiation: $$\begin{aligned} @@ -84,7 +84,7 @@ $$\begin{aligned} We assume that the system is in thermal equilibrium with its own black-body radiation, and that there is no external light. Then this is a [canonical ensemble](/know/concept/canonical-ensemble/), -meaning that the relative probability that an electron has $E_2$ compared to $E_1$ +meaning that the relative probability that an electron has $$E_2$$ compared to $$E_1$$ is given by the Boltzmann distribution: $$\begin{aligned} @@ -93,15 +93,15 @@ $$\begin{aligned} = \frac{g_2}{g_1} \exp(- \hbar \omega_0 \beta) \end{aligned}$$ -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: +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, +Since $$u(\omega_0)$$ represents only black-body radiation, our result must agree with [Planck's law](/know/concept/plancks-law/): $$\begin{aligned} @@ -120,14 +120,14 @@ $$\begin{aligned} } \end{aligned}$$ -Note that this result holds even if $E_1$ is not the ground state, -but instead some lower excited state below $E_2$, +Note that this result holds even if $$E_1$$ is not the ground state, +but instead some lower excited state below $$E_2$$, due to the principle of [detailed balance](/know/concept/detailed-balance/). Furthermore, it turns out that these relations also hold if the system is not in equilibrium. A notable case is **population inversion**, -where $B_{21} N_2 > B_{12} N_1$ such that $N_2 > (g_2 / g_1) N_1$. +where $$B_{21} N_2 > B_{12} N_1$$ such that $$N_2 > (g_2 / g_1) N_1$$. This situation is mandatory for lasers, where stimulated emission must dominate, such that the light becomes stronger as it travels through the medium. @@ -137,38 +137,38 @@ such that the light becomes stronger as it travels through the medium. In fact, we can analytically calculate the Einstein coefficients in some cases, by treating incoming light as a perturbation to an electron in a two-level system, -and then finding $B_{12}$ and $B_{21}$ from the resulting transition rate. +and then finding $$B_{12}$$ and $$B_{21}$$ from the resulting transition rate. We need to make the [electric dipole approximation](/know/concept/electric-dipole-approximation/), -in which case the perturbing Hamiltonian $\hat{H}_1(t)$ is given by: +in which case the perturbing Hamiltonian $$\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}$$ -Where $q = -e$ is the electron charge, -$\vec{r}$ is the position operator, -and $\vec{E}_0$ is the amplitude of +Where $$q = -e$$ is the electron charge, +$$\vec{r}$$ is the position operator, +and $$\vec{E}_0$$ is the amplitude of the [electromagnetic wave](/know/concept/electromagnetic-wave-equation/). -For simplicity, we let the amplitude be along the $z$-axis: +For simplicity, we let the amplitude be along the $$z$$-axis: $$\begin{aligned} \hat{H}_1(t) = - q E_0 z \cos(\omega t) \end{aligned}$$ -This form of $\hat{H}_1$ is a well-known case for +This form of $$\hat{H}_1$$ is a well-known case for [time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/), -which tells us that the transition probability from $\Ket{a}$ to $\Ket{b}$ is: +which tells us that the transition probability from $$\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}$$ -If the nucleus is at $z = 0$, -then generally $\Ket{1}$ and $\Ket{2}$ will be even or odd functions of $z$, -meaning that $\matrixel{1}{z}{1} = \matrixel{2}{z}{2} = 0$ +If the nucleus is at $$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$$ (see also [Laporte's selection rule](/know/concept/selection-rules/)), leading to: @@ -180,8 +180,8 @@ $$\begin{gathered} \matrixel{1}{H_1}{1} = \matrixel{2}{H_1}{2} = 0 \end{gathered}$$ -Where $d \equiv q \matrixel{2}{z}{1}$ is a constant, -namely the $z$-component of the **transition dipole moment**. +Where $$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} @@ -190,8 +190,8 @@ $$\begin{aligned} \end{aligned}$$ 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 $\omega_{ba}$: +using the fact that $$P_{ab}$$ is a sinc-function, +and is therefore symmetric around $$\omega_{ba}$$: $$\begin{aligned} P_{21} @@ -202,17 +202,17 @@ Surprisingly, the probabilities of absorption and stimulated emission are the sa In practice, however, the relative rates of these two processes depends heavily on the availability of electrons and holes in both states. -In theory, we could calculate the transition rate $R_{12} = \ipdv{P_{12}}{t}$, -which would give us Einstein's absorption coefficient $B_{12}$, +In theory, we could calculate the transition rate $$R_{12} = \ipdv{P_{12}}{t}$$, +which would give us Einstein's absorption coefficient $$B_{12}$$, for this specific case of coherent monochromatic light. -However, the result would not be constant in time $t$, +However, the result would not be constant in time $$t$$, so is not really useful. ## Polarized light To solve this "problem", we generalize to (incoherent) polarized polychromatic light. -To do so, we note that the energy density $u$ of an electric field $E_0$ is given by: +To do so, we note that the energy density $$u$$ of an electric field $$E_0$$ is given by: $$\begin{aligned} u = \frac{1}{2} \varepsilon_0 E_0^2 @@ -220,8 +220,8 @@ $$\begin{aligned} E_0^2 = \frac{2 u}{\varepsilon_0} \end{aligned}$$ -Where $\varepsilon_0$ is the vacuum permittivity. -Putting this in the previous result for $P_{12}$ gives us: +Where $$\varepsilon_0$$ is the vacuum permittivity. +Putting this in the previous result for $$P_{12}$$ gives us: $$\begin{aligned} P_{12} @@ -229,7 +229,7 @@ $$\begin{aligned} \end{aligned}$$ For a continuous light spectrum, -this $u$ turns into the spectral energy density $u(\omega)$: +this $$u$$ turns into the spectral energy density $$u(\omega)$$: $$\begin{aligned} P_{12} @@ -241,11 +241,11 @@ 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 $\omega = \omega_0$ +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. +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 $\pi t$: +which turns out to be $$\pi t$$: $$\begin{aligned} P_{12} @@ -253,7 +253,7 @@ $$\begin{aligned} = \frac{\pi |d|^2}{\varepsilon_0 \hbar^2} u(\omega_0) \:t \end{aligned}$$ -From this, the transition rate $R_{12} = B_{12} u(\omega_0)$ +From this, the transition rate $$R_{12} = B_{12} u(\omega_0)$$ is then calculated as follows: $$\begin{aligned} @@ -262,7 +262,7 @@ $$\begin{aligned} = \frac{\pi |d|^2}{\varepsilon_0 \hbar^2} u(\omega_0) \end{aligned}$$ -Using the relations from earlier with $g_1 = g_2$, +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: @@ -278,8 +278,8 @@ $$\begin{aligned} ## Unpolarized light We can generalize the above result even further to unpolarized light. -Let us return to the matrix elements of the perturbation $\hat{H}_1$, -and define the polarization unit vector $\vec{n}$: +Let us return to the matrix elements of the perturbation $$\hat{H}_1$$, +and define the polarization unit vector $$\vec{n}$$: $$\begin{aligned} \matrixel{2}{\hat{H}_1}{1} @@ -287,22 +287,22 @@ $$\begin{aligned} = - E_0 (\vec{d} \cdot \vec{n}) \end{aligned}$$ -Where $\vec{d} \equiv q \matrixel{2}{\vec{r}}{1}$ is +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 $|\vec{d} \cdot \vec{n}|^2$. +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 $|\vec{d}| \equiv |d_x|^2 \!+\! |d_y|^2 \!+\! |d_z|^2$: +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 $\vec{d}$. +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: $$\begin{aligned} @@ -312,8 +312,8 @@ $$\begin{aligned} = \frac{|\vec{d}|^2}{3} \end{aligned}$$ -With this additional constant factor $1/3$, -the transition rate $R_{12}$ is modified to: +With this additional constant factor $$1/3$$, +the transition rate $$R_{12}$$ is modified to: $$\begin{aligned} R_{12} |