A compendium of photon emission rates, absorption cross sections and scattering cross sections

Abstract

We provide a compendium of the quantum mechanical equations for photon emission rates, photon absorption cross sections, and photon scattering cross sections. For each case, the different equations that apply for discrete or continuous electron states of the emitting, absorbing, or scattering material are given.

Appendices

Appendix 1: Continuous states

Ionization of materials and ion-electron recombination involve states in the continuous energy spectrum of a material. Transitions in materials which involve energy bands also involve states in an energy continuum. Equations for transitions involving continuous states, e.g., the Golden Rule, are often expressed in terms of densities of states in the energy scale. However, transition probabilities between energy bands in condensed materials are more commonly derived in terms of the quasiperiodic Bloch energy eigenstates $|n,{\varvec{k}}_e\rangle$,

$$\begin{aligned} \langle {\varvec{x}}|n,{\varvec{k}}_e\rangle= &\, {} \psi _n({\varvec{k}}_e,{\varvec{x}}) \nonumber \\= &\, {} \sqrt{\frac{V}{8\pi ^3}}\exp (\mathrm{i}{\varvec{k}}_e\cdot {\varvec{x}})u_n({\varvec{k}}_e,{\varvec{x}}), \end{aligned}$$

(95)

where V is the volume of the Wigner-Seitz cell and $u_n({\varvec{k}}_e,{\varvec{x}})$ are the periodic Bloch factors for the energy band $E_n({\varvec{k}}_e)$. The Bloch energy eigenfunctions $\psi _n({\varvec{k}}_e,{\varvec{x}})$ are periodically perturbed plane waves with normalization

$$\begin{aligned} \langle n',{\varvec{k}}'_e|n,{\varvec{k}}_e\rangle =\delta _{nn'}\delta ({\varvec{k}}_e-{\varvec{k}}'_e). \end{aligned}$$

(96)

We extract the factor $\sqrt{V/8\pi ^3}$ from the Bloch factors, because with this definition the property (96) of the Bloch energy eigenfunctions implies normalization of the periodic Bloch factors $\langle {\varvec{x}}|u_n({\varvec{k}}_e)\rangle =u_n({\varvec{k}}_e,{\varvec{x}})$ to the Wigner-Seitz cell,

$$\begin{aligned} \langle n',{\varvec{k}}_e|n,{\varvec{k}}_e\rangle _V\equiv & {} \int _V d^3{\varvec{x}}\,u^+_{n'}({\varvec{k}}_e,{\varvec{x}}) u_n({\varvec{k}}_e,{\varvec{x}}) \nonumber \\= &\, {} \delta _{nn'}, \end{aligned}$$

(97)

where the integration is over the Wigner-Seitz cell and both Bloch factors must refer to the same wave vector ${\varvec{k}}_e$ in the Brillouin zone.

The connections between densities of states in the energy scale and wave vector parametrizations for continuous states are encoded in the completeness relations, which involve sums over the discrete energy eigenvalues $E_j$ and the continuous parts C of the spectrum,

$$1= \sum _{j,\nu }|E_j,\nu \rangle \langle E_j,\nu | +\sum _{\nu }\int _{C} dE\,|E,\nu \rangle {\tilde{\varrho }}(E,\nu )\langle E,\nu |$$

(98)

$$\quad= \sum _{j,\nu }|E_j,\nu \rangle \langle E_j,\nu | +\sum _{n}\int _{{\mathcal {B}}} d^3{\varvec{k}}_e\,|n,{\varvec{k}}_e\rangle \langle n,{\varvec{k}}_e|.$$

(99)

The integral $\int _{{\mathcal {B}}} d^3{\varvec{k}}_e\ldots$ covers the first Brillouin zone. The energy integral $\int _{C} dE\ldots$ covers all the continuous energy eigenvalues, i.e., in a condensed material the integration domain $C=\sum _n\oplus \, C_n$ covers all the energy bands. The sum $\sum _{\nu }$ over degeneracy indices for the energy eigenstates $|E,\nu \rangle$ in the continuous part C of the spectrum can involve summations over discrete quantum numbers or integrations over continuous quantum numbers.

The contribution from the energy band $E_n({\varvec{k}}_e)$ to the density of states follows from

$$\begin{aligned} d^3{\varvec{k}}_e=d^2{\varvec{k}}_{e\Vert }dk_{e,\perp }\rightarrow d^2{\varvec{k}}_{e\Vert } \frac{dE}{|\partial E_n({\varvec{k}}_e)/\partial {\varvec{k}}_e|}, \end{aligned}$$

(100)

where $d^2{\varvec{k}}_{e\Vert }$ denotes an integration measure on the surface of constant energy $E_n({\varvec{k}}_e)$. This implies

$$\int _{{\mathcal {B}}} d^3{\varvec{k}}_e\,|n,{\varvec{k}}_e\rangle \langle n,{\varvec{k}}_e| \rightarrow \int _{C_n} dE\int _{E_n({\varvec{k}}_e)=E} d^2{\varvec{k}}_{e\Vert }\, \frac{|n,{\varvec{k}}_e\rangle \langle n,{\varvec{k}}_e|}{|\partial E_n({\varvec{k}}_e)/\partial {\varvec{k}}_e|}.$$

(101)

If we agree to use the continuous variables included in the integration measure $d^2{\varvec{k}}_{e\Vert }$ as degeneracy indices $\nu$, $\sum _\nu \rightarrow \int d^2{\varvec{k}}_{e\Vert }$, and set $|E,{\varvec{k}}_\Vert \rangle =|n,{\varvec{k}}_e\rangle$ for $E=E_n({\varvec{k}}_e)$, comparison of (101) with (98) tells us that the contribution from the energy band $E_n({\varvec{k}})$ to the partial density of states ${\tilde{\varrho }}(E,\nu )=\sum _n{\tilde{\varrho }}_n(E,\nu )$ is

$$\begin{aligned} {\tilde{\varrho }}_n(E,\nu )\equiv {\tilde{\varrho }}_n(E,{\varvec{k}}_\Vert )= \frac{1}{|\partial E_n({\varvec{k}}_e)/\partial {\varvec{k}}_e|}. \end{aligned}$$

(102)

The normalization of ${\tilde{\varrho }}(E,\nu )$ depends on the normalization of the continuous states $|E,\nu \rangle$. The expression (14) for energies in the continuous part of the spectrum,

$$\begin{aligned} \varrho (E,{\varvec{x}})dE= \sum _\nu \langle {\varvec{x}}|E,\nu \rangle {\tilde{\varrho }}(E,\nu )\langle E,\nu |{\varvec{x}}\rangle dE \end{aligned}$$

(103)

gives the local density of continuous states per volume V at location ${\varvec{x}}$ and with energies in the interval $[E,E+dE]$, i.e., scaling the states $|E,\nu \rangle$ with a factor $\xi$ scales ${\tilde{\varrho }}(E,\nu )$ by $|\xi |^{-2}$.

We can switch from the a priori continuous degeneracy indices $\nu _c$, which provide coordinates on the constant energy surface, to discrete indices $\nu _d$ through harmonic analysis,

$$\begin{aligned} \int d^2\nu _c\,|E,\nu _c\rangle {\tilde{\varrho }}(E,\nu _c)\langle E,\nu _c| =\sum _{\nu _d}|E,\nu _d\rangle {\tilde{\varrho }}(E,\nu _d)\langle E,\nu _d|. \end{aligned}$$

Both parametrizations will yield the same local density of states $\varrho (E,{\varvec{x}})$. The local density of states (per spin state) for nonrelativistic free electrons with plane wave states $|{\varvec{k}}_e\rangle$ is

$$\begin{aligned} \varrho _e(E,{\varvec{x}})= &\, {} \int _{E({\varvec{k}}_e)=E} d^2{\varvec{k}}_{e\Vert }\frac{|\langle {\varvec{x}}|{\varvec{k}}_e\rangle |^2 }{|\partial E({\varvec{k}}_e)/\partial {\varvec{k}}_e|} \nonumber \\= &\, {} \frac{1}{8\pi ^3} \int _{E({\varvec{k}}_e)=E}\frac{\textrm{d}^2{\varvec{k}}_{e\Vert }}{|\partial E({\varvec{k}}_e)/\partial {\varvec{k}}_e|} \nonumber \\= &\, {} \frac{\Theta (E)}{2\pi ^2\hbar ^3}\sqrt{2m^3E}, \end{aligned}$$

(104)

and this is independent of position. However, the local density of electron states (per spin state) in the energy band $E_n({\varvec{k}}_e)$ involves the Bloch factors,

$$\begin{aligned} \varrho _n(E,{\varvec{x}})=\frac{V}{8\pi ^3} \int _{E_n({\varvec{k}}_e)=E} d^2{\varvec{k}}_{e\Vert }\frac{|u_n({\varvec{k}}_e,{\varvec{x}})|^2 }{|\partial E({\varvec{k}}_e)/\partial {\varvec{k}}_e|}, \end{aligned}$$

(105)

where V is the volume of the Wigner-Seitz cell. The density of states (64) emerges from the local density of states (105) after integration over a Wigner-Seitz cell,

$$\begin{aligned} \varrho _{n,V}(E)=\int _V d^3{\varvec{x}}\,\varrho _n(E,{\varvec{x}}). \end{aligned}$$

(106)

Appendix 2: The quantum fields in the quantum optics Hamiltonian

The electron field $\Psi _s({\varvec{x}},t)$ and the photon field ${\varvec{A}}({\varvec{x}},t)$ in the Coulomb gauge Hamiltonian (1–6) are quantum fields in the Heisenberg picture which are related to the time-independent quantum fields of the Schrödinger picture through

$$\begin{aligned} \Psi _s({\varvec{x}},t)=\exp (\mathrm{i}Ht/\hbar )\Psi _s({\varvec{x}})\exp (-\,\mathrm{i}Ht/\hbar ), \end{aligned}$$

(107)

$$\begin{aligned} {\varvec{A}}({\varvec{x}},t)=\exp (\mathrm{i}Ht/\hbar ){\varvec{A}}({\varvec{x}})\exp (-\,\mathrm{i}Ht/\hbar ). \end{aligned}$$

(108)

Canonical quantization implies the anticommutation relations for electron operators

$$\begin{aligned} \{\Psi _s({\varvec{x}}),\Psi _{s'}^+({\varvec{x}}')\}=\delta ({\varvec{x}}-{\varvec{x}}'),\quad \{\Psi _s({\varvec{x}}),\Psi _{s'}({\varvec{x}}')\}=0, \end{aligned}$$

and commutation relations for photon operators (with $E_{i}({\varvec{x}})=-\,{\dot{A}}_{i}({\varvec{x}},t)|_{t\rightarrow 0}$)

$$\begin{aligned}{}[A_{i}({\varvec{x}}),E_{j}({\varvec{x}}')] =-\,\frac{\mathrm{i}\hbar }{\epsilon _0}\delta _{ij}^\perp ({\varvec{x}}-{\varvec{x}}'), \end{aligned}$$

(109)

$$\begin{aligned}{}[A_{i}({\varvec{x}}),A_{j}({\varvec{x}}')]=0,\quad [E_{i}({\varvec{x}}),E_{j}({\varvec{x}}')]=0, \end{aligned}$$

(110)

where

$$\begin{aligned} \delta _{ij}^\perp ({\varvec{x}}) =\frac{1}{(2\pi )^3} \int d^3{\varvec{k}}\left( \delta _{ij} -\frac{k_i k_j}{{\varvec{k}}^2}\right) \exp (\mathrm{i}{\varvec{k}}\cdot {\varvec{x}}) \end{aligned}$$

(111)

is the transverse $\delta$ function.

The mode expansion of $A_{i}({\varvec{x}})$ contains the photon operators $a^+_\alpha ({\varvec{k}})$ which create photons with momentum $\hbar {\varvec{k}}$ and polarization ${\varvec{\epsilon }}_\alpha ({\varvec{k}})$,

$$\begin{aligned} {\varvec{A}}({\varvec{x}})=\sqrt{\frac{\hbar \mu _0 c}{8\pi ^3}} \int \frac{\textrm{d}^3{\varvec{k}}}{\sqrt{2k}}\sum _{\alpha =1}^2 {\varvec{\epsilon }}_\alpha ({\varvec{k}})\Big ( a_\alpha ({\varvec{k}}) \exp (\mathrm{i}{\varvec{k}}\cdot {\varvec{x}}) +a_{\alpha }^+({\varvec{k}}) \exp (-\,\mathrm{i}{\varvec{k}}\cdot {\varvec{x}})\Big ). \end{aligned}$$

(112)

The electron creation operators $\Psi ^+_s({\varvec{x}})$ create, e.g., single-electron states with spinor components $\psi _s({\varvec{x}},t)$ through

$$\begin{aligned} {\varvec{|}}\psi (t){\varvec{\rangle }}=\sum _s\int d^3{\varvec{x}}\,\Psi ^+_s({\varvec{x}}){\varvec{|}}0{\varvec{\rangle }}\psi _s({\varvec{x}},t), \end{aligned}$$

(113)

or general many-electron states,

$$\begin{aligned} {\varvec{|}}\psi (t){\varvec{\rangle }}=\sum _{s,s',\ldots }\int d^3{\varvec{x}}\int d^3{\varvec{x}}' \,\Psi ^+_s({\varvec{x}})\Psi ^+_{s'}({\varvec{x}}')\ldots {\varvec{|}}0{\varvec{\rangle }} \times \psi _{s,s',\ldots }({\varvec{x}},{\varvec{x}}',\ldots ,t). \end{aligned}$$

(114)

The many-electron states (114) are usually approximated through products of orthonormalized single-electron states,

$$\begin{aligned} \psi _{s_1,s_2,\ldots ,s_N}({\varvec{x}}_1,{\varvec{x}}_2,\ldots ,{\varvec{x}}_N,t) \rightarrow \psi _{n_1,s_1}({\varvec{x}}_1,t)\times \psi _{n_2,s_2}({\varvec{x}}_2,t)\ldots \psi _{n_N,s_N}({\varvec{x}}_N,t). \end{aligned}$$

(115)

The many-particle state in a material can then be thought of as the product of a many-electron state of the form (114,115) with corresponding many-particle states for the pertinent nuclei. The expectation value of the kinetic electron energy operator ${\mathcal {H}}_e$ (2) for the many-particle state of the material then generates the sum of the kinetic energy densities of all the fundamental nonrelativistic electrons, and the expectation value of the potential operator for the electrons, i.e., the sum of relevant terms from (5),

$$\begin{aligned} {\mathcal {V}}_e({\varvec{x}})=\int d^3{\varvec{x}}'\,{\mathcal {H}}_{ee}({\varvec{x}},{\varvec{x}}') +\,2\sum _{b=\textrm{nuclei}}\int d^3{\varvec{x}}'\,{\mathcal {H}}_{eb}({\varvec{x}},{\varvec{x}}'), \end{aligned}$$

(116)

generates an effective single-electron potential. We use the Schrödinger picture in (116), such that the wave functions in our many-particle quantum states are time-dependent but the operators are time-independent, see (114).

The Coulomb terms in (5) automatically generate the exchange interaction terms in ${\varvec{\langle }}{\mathcal {V}}_e({\varvec{x}}){\varvec{\rangle }}$ through Fermi statistics, but for realistic potentials in materials we would also have to include first order relativistic corrections like spin-orbit coupling. Furthermore, instead of summation over bare nuclei, we may decide to allocate core electrons to the nuclei and not count them separately toward the electronic states in the material, in the interest of efficiency. However, these details do not impact the spectroscopic formulae in this paper. The important observation is that even in a complex material we are still considering fundamental electrons which are moving in a potential, e.g., a periodic lattice potential in a solid material, and that couple to photons through the minimal coupling terms ${\mathcal {H}}_{e\gamma }$ (6) where $m=m_e$ is the electron mass. Furthermore, these electrons are described on the quantum mechanical level by the Hamiltonian (8) with an effective potential $V(\textbf{x})$, such that (7) still holds. Equation (8) and the transition between velocity and length form would not hold anymore if we have to take into account relativistic corrections to the kinetic energy, e.g., for deep core electrons in materials with heavy atoms.

The representation of the ${\varvec{x}}$-space electron operators in terms of free ${\varvec{k}}_e$-space electron operators is

$$\begin{aligned} \Psi _s({\varvec{x}})=\frac{1}{\sqrt{2\pi }^3}\int d^3{\varvec{k}}_e\,\exp (\mathrm{i}{\varvec{k}}_e\cdot {\varvec{x}}) a_s({\varvec{k}}_e). \end{aligned}$$

(117)

The corresponding representation for electrons in periodic potentials is

$$\begin{aligned} \Psi _s({\varvec{x}})=\sqrt{\frac{V}{8\pi ^3}}\sum _n\int _{{\mathcal {B}}} d^3{\varvec{k}}_e\, \exp (\mathrm{i}{\varvec{k}}_e\cdot {\varvec{x}}) u_n({\varvec{k}}_e,{\varvec{x}})a_{n,s}({\varvec{k}}_e), \end{aligned}$$

(118)

where n labels the energy bands, V is the volume of the Wigner-Seitz cell, $u_n({\varvec{k}}_e,{\varvec{x}})$ is the Bloch factor with normalization (46), and the integration is over the Brillouin zone ${\mathcal {B}}$.

However, in both cases we are still dealing with electrons which couple to photons through the quantum optics interaction (6). The inversion of (118),

$$\begin{aligned} a_{n,s}({\varvec{k}}_e)=\sqrt{\frac{V}{8\pi ^3}}\int d^3{\varvec{x}}\, \exp (-\,\mathrm{i}{\varvec{k}}_e\cdot {\varvec{x}}) u_n^+({\varvec{k}}_e,{\varvec{x}})\Psi _s({\varvec{x}}), \end{aligned}$$

(119)

provides the unitary transformation from the free electron operators in ${\varvec{k}}_e$-space to the operators $a_{n,s}({\varvec{k}}_e)$ for electrons in energy bands,

$$\begin{aligned} a_{n,s}({\varvec{k}}_e)=\frac{\sqrt{V}}{8\pi ^3}\int d^3{\varvec{x}}\int d^3{\varvec{k}}'_e\, \exp [\mathrm{i}({\varvec{k}}'_e-{\varvec{k}}_e)\cdot {\varvec{x}}] u_n^+({\varvec{k}}_e,{\varvec{x}})a_s({\varvec{k}}'_e). \end{aligned}$$

(120)

The different ${\varvec{k}}_e$-space operators correspond to different convenient representations of electron operators depending on whether the electrons are moving freely or in a periodic potential.

With respect to the momenta of electrons moving in a periodic potential, we note that the equation for free electrons, $\langle {\varvec{k}}'_e|\textbf{p}|{\varvec{k}}_e\rangle =\hbar {\varvec{k}}_e\delta ({\varvec{k}}_e-{\varvec{k}}'_e)$ gets modified to

$$\begin{aligned} \langle n,{\varvec{k}}'_e|\textbf{p}|n,{\varvec{k}}_e\rangle =\delta ({\varvec{k}}_e-{\varvec{k}}'_e) \left( \hbar {\varvec{k}}_e +\int _V d^3{\varvec{x}}\, u^+_n({\varvec{k}}_e,{\varvec{x}}) \frac{\hbar }{\mathrm{i}}\frac{\partial }{\partial {\varvec{x}}}u_n({\varvec{k}}_e,{\varvec{x}})\right) , \end{aligned}$$

(121)

where the remaining integration on the right hand side is over the Wigner-Seitz cell. The limitation of translation symmetry to lattice vectors due to the periodic lattice potential implies that the energy eigenstates are not momentum eigenstates anymore. However, $\textbf{p}$ is still the first-quantized momentum operator for the electrons in the Bloch energy eigenstates.

We gave the Schrödinger picture electron states with their time-dependence in Eqs. (113,114). However, the states only enter for time $t=0$ into the transition matix elements, because the unperturbed time evolution operators $U_0(t,0)$ from the external states transform the full time evolution operator in the scattering matrix elements into the time evolution operator of the interaction picture, evaluated between states at time $t=0$.

Note that the bold-face kets in (113,114) correspond to Fock space states, which appear in the derivations of the transition amplitudes reported in the body of this review, but not in the final results. The kets in the transition amplitudes in Sects. 2–4 correspond to the wave functions of the first quantized theory that enter into the matrix elements, e.g., $\psi _s({\varvec{x}})=\langle {\varvec{x}}|\psi _s\rangle$.

Appendix 3: Atomic recoil

Recoils are usually neglected in transition matrix elements involving discrete atomic states. Recoil effects will be more prominent for transitions involving free atoms than for atoms which are bound into a molecule or a lattice. We will therefore discuss recoils between states of free atoms.

In a description of single electrons moving in an effective potential $V({\varvec{x}}_e-{\varvec{x}}_p)$ created by the nucleus at position ${\varvec{x}}_p$ (the notation is motivated by exactness of these considerations for hydrogen atoms) and a radially symmetric distribution of the other electrons, the eigenstates can be written in the form

$$\begin{aligned} \langle {\varvec{x}}_e,{\varvec{x}}_p|{\varvec{K}},n\rangle =\frac{1}{\sqrt{2\pi }^3} \exp \left( \mathrm{i}{\varvec{K}}\cdot \frac{m_p{\varvec{x}}_p+m_e{\varvec{x}}_e}{m_p+m_e}\right) \psi _n({\varvec{x}}_e-{\varvec{x}}_p). \end{aligned}$$

(122)

Here $\hbar {\varvec{K}}$ is the center-of-mass momentum and n represents the remaining set of orbital and spin quantum numbers of this effective two-particle problem. The mass $m_p\gg m_e$ includes the mass of the nucleus. The energy of the state (122) is

$$\begin{aligned} E_n({\varvec{K}})=\frac{\hbar ^2{\varvec{K}}^2}{2(m_p+m_e)}+\hbar \omega _n. \end{aligned}$$

(123)

Scattering matrix elements for transitions $|{\varvec{K}}_i,n_i\rangle \rightarrow |{\varvec{K}}_f,n_f\rangle$ due to electron-photon coupling then involve transition matrix elements

$$\begin{aligned}{} & {} \langle {\varvec{K}}_f,n_f|\exp (-\,\mathrm{i}{\varvec{k}}\cdot \textbf{x}_e) {\varvec{\epsilon }}^+_\gamma ({\varvec{k}})\cdot \frac{\textbf{p}_e}{m_e} |{\varvec{K}}_i,n_i\rangle \nonumber \\{} & \quad{} = \int \frac{\textrm{d}^3{\varvec{x}}_p}{(2\pi )^3}\int d^3{\varvec{x}}_e\, \exp \left( \mathrm{i}({\varvec{K}}_i-{\varvec{K}}_f)\cdot \frac{m_p{\varvec{x}}_p+m_e{\varvec{x}}_e}{m_p+m_e}\right) \nonumber \\{} & {} \quad \quad\times \psi ^+_f({\varvec{x}}_e-{\varvec{x}}_p)\exp (-\,\mathrm{i}{\varvec{k}}\cdot {\varvec{x}}_e) \nonumber \\{} & {} \quad\quad \times {\varvec{\epsilon }}^+_\gamma ({\varvec{k}})\cdot \left( \frac{\hbar }{\mathrm{i}m_e} \frac{\partial }{\partial {\varvec{x}}_e}+\frac{m_e\hbar {\varvec{K}}_i}{m_p+m_e}\right) \psi _i({\varvec{x}}_e-{\varvec{x}}_p). \end{aligned}$$

(124)

With regard to the photon terms, this is formulated for the case of emission of a real or virtual photon through this matrix element, but the argument of negligibility of atomic recoils given below is equally valid for the substitutions which correspond to photon absorption, ${\varvec{\epsilon }}^+_\gamma ({\varvec{k}})\rightarrow {\varvec{\epsilon }}_\gamma ({\varvec{k}})$, $\exp (-\,\mathrm{i}{\varvec{k}}\cdot \textbf{x}_e)\rightarrow \exp (\mathrm{i}{\varvec{k}}\cdot \textbf{x}_e)$.

In the next step, we substitute center of mass and relative coordinates

$$\begin{aligned} {\varvec{X}}=\frac{m_p{\varvec{x}}_p+m_e{\varvec{x}}_e}{m_p+m_e},\quad {\varvec{x}}={\varvec{x}}_e-{\varvec{x}}_p, \end{aligned}$$

(125)

$$\begin{aligned} {\varvec{x}}_e={\varvec{X}}+\frac{m_p}{m_p+m_e}{\varvec{x}}, \end{aligned}$$

(126)

$$\begin{aligned} \frac{\partial }{\partial {\varvec{x}}_e} = \frac{\partial }{\partial {\varvec{x}}}+\frac{m_e}{m_p+m_e}\frac{\partial }{\partial {\varvec{X}}}, \end{aligned}$$

(127)

use dipole approximation in the atomic state matrix element,

$$\begin{aligned} \exp \left( -\,\mathrm{i}{\varvec{k}}\cdot \frac{m_p}{m_p+m_e}{\varvec{x}}\right) \rightarrow 1, \end{aligned}$$

(128)

and use orthogonality of the initial and final states, $\langle {\varvec{K}}_f,n_f|{\varvec{K}}_i,n_i\rangle =0$, to find in dipole approximation

$$\begin{aligned}{} & {} \langle {\varvec{K}}_f,n_f|\exp (-\,\mathrm{i}{\varvec{k}}\cdot \textbf{x}_e) {\varvec{\epsilon }}^+_\gamma ({\varvec{k}})\cdot \frac{\textbf{p}_e}{m_e} |{\varvec{K}}_i,n_i\rangle \nonumber \\{} &\quad {} =\frac{1}{(2\pi )^3}\int d^3{\varvec{X}}\, \exp [\mathrm{i}({\varvec{K}}_i-{\varvec{K}}_f-{\varvec{k}})\cdot {\varvec{X}}] \int d^3{\varvec{x}}\, \psi ^+_f({\varvec{x}}){\varvec{\epsilon }}^+_\gamma ({\varvec{k}})\cdot \frac{\hbar }{\mathrm{i}m_e}\frac{\partial }{\partial {\varvec{x}}}\psi _i({\varvec{x}}) \nonumber \\{} & \quad{} =\delta ({\varvec{K}}_i-{\varvec{K}}_f-{\varvec{k}})\int d^3{\varvec{x}}\, \psi ^+_f({\varvec{x}}){\varvec{\epsilon }}^+_\gamma ({\varvec{k}})\cdot \frac{\hbar }{\mathrm{i}m_e}\frac{\partial }{\partial {\varvec{x}}}\psi _i({\varvec{x}}). \end{aligned}$$

(129)

Inclusion of the center of mass motion of the atom multiplies the standard transition matrix element with a momentum conserving $\delta$ function. The squares of these extra factors cancel in the calculation of observables from the squares $|S_{fi}|^2$ of scattering matrix elements, since the final center of mass momentum $\hbar {\varvec{K}}_f$ adds an integration $\int d^3{\varvec{K}}_f\ldots$, the square of the momentum conserving $\delta$ function contributes a factor $({\mathcal {V}}/8\pi ^3)\delta ({\varvec{K}}_i-{\varvec{K}}_f-{\varvec{k}})$, and the fixed initial center of mass momentum contributes an elementary ${\varvec{K}}$ space volume $8\pi ^3/{\mathcal {V}}$, such that the only contribution from inclusion of the center of mass motion is to shift ${\varvec{K}}_i\rightarrow {\varvec{K}}_f={\varvec{K}}_i-{\varvec{k}}$. This shifts energy conservation for spontaneous emission, $E_i({\varvec{K}}_i)>E_f({\varvec{K}}_f)$, from $ck=\omega _{if}=[E_i({\varvec{K}})-E_f({\varvec{K}})]/\hbar$ to

$$\begin{aligned} ck= &\, {} \frac{E_i({\varvec{K}}_i)-E_f({\varvec{K}}_f)}{\hbar } \nonumber \\= &\, {} \frac{\hbar }{2}\frac{2{\varvec{K}}_i\cdot {\varvec{k}}-{\varvec{k}}^2}{m_p+m_e} +\omega _i-\omega _f. \end{aligned}$$

(130)

The net effect of the atomic recoil is therefore a frequency shift that in leading order in $\hbar |{\varvec{K}}_i|/(m_p+m_e)c$ and $\hbar \omega _{if}/(m_p+m_e)c^2$ takes the form (with scattering angle $\theta$)

$$\begin{aligned} ck=\omega _{if}+\frac{\hbar |{\varvec{K}}_i|\cos \theta }{(m_p+m_e)c}\omega _{if} -\frac{\hbar \omega _{if}^2}{2(m_p+m_e)c^2}. \end{aligned}$$

(131)

The frequency shift is very small in comparison with $\omega _{if}$ if the transition involves nonrelativistic atoms, $\hbar |{\varvec{K}}_i|\ll m_pc$. In cases of absorption or scattering, when we have photons in the initial state, we should also exclude high energy $\gamma$-rays to neglect atomic recoil effects. However, recall that we have already excluded photons beyond the soft X-ray regime through the use of dipole approximation. We also note that for photons below the hard X-ray regime, possible energy shifts from recoils of quasifree electrons are also suppressed by $\hbar |{\varvec{k}}|/m_ec$.

The observation that recoils yield frequency shifts, but do not generate extra factors in transition rates, also applies if the matrix element does not directly connect initial and final atomic states but includes intermediate virtual states. Integration over intermediate virtual center of mass momenta $\hbar {\varvec{K}}_v$ only reduces momentum conserving $\delta$ functions in $|S_{fi}|^2$ according to

$$\int d^3{\varvec{K}}_v\,\delta ({\varvec{K}}_v-{\varvec{K}}_f-{\varvec{k}}')\delta ({\varvec{K}}_i-{\varvec{K}}_v+{\varvec{k}}) =\delta ({\varvec{K}}_i-{\varvec{K}}_f+{\varvec{k}}-{\varvec{k}}').$$

(132)

Appendix 4: Photon emission from radiative electron–hole recombination between energy bands

Equations (47,50) apply to transitions between conduction bands, but not to a conduction electron filling a valence band hole. Electron–hole annihilation through interband transition requires special considerations because now we are effectively dealing with two particles in the initial state, viz. a conduction electron with wave vector ${\varvec{k}}_{e,i}$ and a valence band hole with wave vector $-\,{\varvec{k}}_{e,f}=\Delta {\varvec{k}}-{\varvec{k}}_{e,i}$. Interband electron–hole recombination is therefore akin to two-particle annihilation events that are characterized by annihilation cross sections.

We use subscripts c and v for labeling the conduction band and the valence band, respectively. The initial electron–hole state is then $a^+_{c,s}({\varvec{k}}_{e,i})a_{v,s}({\varvec{k}}_{e,f}){\varvec{|}}\Omega {\varvec{\rangle }}$, where the ground state ${\varvec{|}}\Omega {\varvec{\rangle }}$ corresponds to the filled Fermi volume in the Brillouin zone, $a^+_{c,s}({\varvec{k}}_{e,i})$ is an electron creation operator in the conduction band, and $c^+_{v,s}(-\,{\varvec{k}}_{e,f})=a_{v,s}({\varvec{k}}_{e,f})$ acts as a hole creation operator in the valence band [48, 49]. We also assume the same spin projection s for the electron and the hole because the operators (6) preserve spin projections.

The reduction in final state measure due to the initial state $a^+_{c,s}({\varvec{k}}_{e,i})a_{v,s}({\varvec{k}}_{e,f}){\varvec{|}}\Omega {\varvec{\rangle }}$, compared to the case in Sect. 2.4 of transition between conduction bands, $d^3{\varvec{k}} d^3{\varvec{k}}_{e,f}\rightarrow d^3{\varvec{k}}$, implies that integration over final states now only takes care of momentum conservation, but an energy-conserving $\delta$-function remains,

$$\begin{aligned}{} & {} \delta [ck+\omega _v({\varvec{k}}_{e,f})-\omega _c({\varvec{k}}_{e,i})] \delta ({\varvec{k}}+{\varvec{k}}_{e,f}-{\varvec{k}}_{e,i}) \nonumber \\{} & {} \quad \rightarrow \delta [c|{\varvec{k}}_{e,i}-{\varvec{k}}_{e,f}|+\omega _v({\varvec{k}}_{e,f})-\omega _c({\varvec{k}}_{e,i})], \end{aligned}$$

(133)

and this constrains the pairs of wave vectors in the Brillouin zone where electron–hole recombination generates single-photon emission as a purely radiative process. Indeed, we should expect kinematic constraints since the corresponding process for free particle-antiparticle pairs is forbidden by energy-momentum conservation. Single-photon emission from electron–hole recombination for generic combinations of wave vectors therefore requires phonon assistance, creation of Auger electrons, assistance through particle traps, or two-photon emission. These mechanisms are outside of the scope of the current review of leading order radiative processes.

For completeness, however, we give a formula that applies to the purely radiative single-photon emission from electron–hole recombination if the constraint (133) can be satisfied within the linewidth of the transition.

Electron–hole recombination rates are normalized by the differential current density of conduction electrons

$$\begin{aligned} \frac{\textrm{d}{\varvec{j}}_e({\varvec{k}}_e)}{d^3{\varvec{k}}_e}=\frac{1}{V}\int _V d^3{\varvec{x}}\, \frac{\textrm{d}{\varvec{{J}}}_e({\varvec{k}}_e,{\varvec{x}})}{d^3{\varvec{k}}_e}, \end{aligned}$$

(134)

which arises from the ${\varvec{x}}$-dependent differential current density of conduction electrons,

$$\begin{aligned} d{\varvec{{J}}}_e({\varvec{k}}_e,{\varvec{x}})= &\, {} \psi _c^+({\varvec{k}}_e,{\varvec{x}})\frac{\hbar }{2\mathrm{i}m} \frac{\partial \psi _c({\varvec{k}}_e,{\varvec{x}})}{\partial {\varvec{x}}}d^3{\varvec{k}}_e \nonumber \\{} & {} -\,\frac{\hbar }{2\mathrm{i}m}\frac{\partial \psi _c^+({\varvec{k}}_e,{\varvec{x}})}{\partial {\varvec{x}}} \psi _c({\varvec{k}}_e,{\varvec{x}})d^3{\varvec{k}}_e, \end{aligned}$$

(135)

through averaging over the Wigner-Seitz cell. Note that the differential current densities $d{\varvec{{J}}}_e({\varvec{k}}_e,{\varvec{x}})/d^3{\varvec{k}}_e$ and $d{\varvec{j}}_e({\varvec{k}}_e)/d^3{\varvec{k}}_e$ have dimensions of velocities, e.g., units of cm/s.

Radiative electron–hole recombination through single-photon emission can then be characterized by a cross section

$$\begin{aligned} \sigma ({\varvec{k}}_{e,i},{\varvec{k}}_{e,f},{\varvec{\epsilon }}_\gamma )= &\, {} \frac{\alpha _S}{2\pi }\frac{[\omega _c({\varvec{k}}_{e,i})-\omega _v({\varvec{k}}_{e,f})]^2}{ |{\varvec{k}}_{e,i}-{\varvec{k}}_{e,f}|} \nonumber \\{} & {} \times \frac{|\langle u_v({\varvec{k}}_{e,f})|{\varvec{\epsilon }}^+_\gamma ({\varvec{k}}_{e,i}-{\varvec{k}}_{e,f}) \cdot \textbf{x}|u_c({\varvec{k}}_{e,i})\rangle _V|^2}{|d{\varvec{j}}_e({\varvec{k}}_{e,i})/d^3{\varvec{k}}_{e,i}|} \nonumber \\{} & {} \times \Delta _\Gamma [c|{\varvec{k}}_{e,i}-{\varvec{k}}_{e,f}|+\omega _v({\varvec{k}}_{e,f})-\omega _c({\varvec{k}}_{e,i})]. \end{aligned}$$

(136)

As explained in Eqs. (17) and (19–23), averaging over angles removes dependence of matrix elements on polarization vectors. Here this yields an angle averaged electron–hole recombination cross section from single-photon emission,

$$\begin{aligned} \sigma ({\varvec{k}}_{e,i},{\varvec{k}}_{e,f})= &\, {} \frac{\alpha _S}{6\pi }\frac{[\omega _c({\varvec{k}}_{e,i})-\omega _v({\varvec{k}}_{e,f})]^2}{ |{\varvec{k}}_{e,i}-{\varvec{k}}_{e,f}|} \nonumber \\{} & {} \times \frac{|\langle u_v({\varvec{k}}_{e,f})| \textbf{x}|u_c({\varvec{k}}_{e,i})\rangle _V|^2}{|d{\varvec{j}}_e({\varvec{k}}_{e,i})/d^3{\varvec{k}}_{e,i}|} \nonumber \\{} & {} \times \Delta _\Gamma [c|{\varvec{k}}_{e,i}-{\varvec{k}}_{e,f}|+\omega _v({\varvec{k}}_{e,f})-\omega _c({\varvec{k}}_{e,i})]. \end{aligned}$$

(137)

Integration of $\sigma ({\varvec{k}}_{e,i},{\varvec{k}}_{e,f})$ against the ${\varvec{k}}_{e,i}$-dependent differential conduction current density $|d{\varvec{j}}_e({\varvec{k}}_{e,i})/d^3{\varvec{k}}_{e,i}|$ and the ${\varvec{k}}_{e,f}$-dependent differential hole density $|d\rho _h({\varvec{k}}_{e,f})/d^3{\varvec{k}}_{e,f}|$ (which is ${\varvec{x}}$-independent if averaged over the Wigner-Seitz cell) then yields an estimate for the photon emission rate per volume for those photons that were generated through purely radiative single-photon electron–hole recombination, i.e., from recombinations that were not assisted through phonon processes or Auger excitations or trapping mechanisms, and that did not result from two-photon emission.