Complete treatment of single-photon emission in planar channeling

TL;DR

This paper derives approximate quantum solutions for ultrarelativistic particles in a one-dimensional periodic potential and compares the resulting photon emission spectra with semi-classical predictions, highlighting differences at high energies.

Contribution

It provides a full quantum treatment of single-photon emission in planar channeling and compares it with semi-classical methods, revealing energy-dependent discrepancies.

Findings

Quantum calculations differ from semi-classical results at high energies.

Semi-classical method agrees well at lower energies with shifted harmonic peaks.

Quantum approach captures effects neglected in semi-classical approximation.

Abstract

Approximate solutions of the Dirac equation are found for ultrarelativistic particles moving in a periodic potential, which depends only on one coordinate, transverse to the largest component of the momentum of the incoming particle. As an example we employ these solutions to calculate the radiation emission of positrons and electrons trapped in the planar potential found between the (110) planes in Silicon. This allows us to compare with the semi-classical method of Baier, Katkov and Strakhovenko, which includes the effect of spin and photon recoil, but neglects the quantization of the transverse motion. For high-energy electrons, the high-energy part of the angularly integrated photon energy spectrum calculated with the found wave functions differs from the corresponding one calculated with the semi-classical method. However, for lower particle energies it is found that the angularly integrated emission energy spectra obtained via the semi-classical method is in fairly good agreement with the full quantum calculation except that the positions of the harmonic peaks in photon energy and the photon emission angles are shifted.

Equations

$$\chi =\frac{e\sqrt{|(F^{\mu \nu }{p}_{\nu }{)}^2|}}{m^3}.$$

$$\psi (x)=\frac{1}{\sqrt{2\epsilon }}{e}^{i(p_{x}x+p_{z}z-\epsilon t)}U(y),$$

$$U(y)=\sqrt{\varepsilon+m}\left(\begin{array}[]{c}\boldsymbol{s}\\ \frac{\boldsymbol{\sigma}\cdot\tilde{\boldsymbol{p}}}{\varepsilon+m}\boldsymbol{s}\end{array}\right)I(y),$$

$$\frac{\mathbf{\sigma }\cdot \mathbf{p}}{\epsilon +m+e\phi (y)}\simeq \frac{\mathbf{\sigma }\cdot \tilde{\mathbf{p}}}{\epsilon +m},$$

$$\left[-\frac{1}{2\epsilon }\frac{d^2}{d{y}^2}-e\phi (y)\right]I(y)=\frac{{\epsilon }^2-p_x^2-p_z^2-m^2}{2\epsilon }I(y).$$

$$I(y)=e^{i{k}_{B}y}{u}_{k_B}(y),$$

$$\left[-\frac{1}{2\epsilon }\left(\frac{d^2}{d{y}^2}+2i{k}_B\frac{d}{dy}-k_B^2\right)+q\phi (y)\right]u_{k_B}(y)$$

$$=\frac{{\epsilon }^2-p_x^2-p_z^2-m^2}{2\epsilon }{u}_{k_B}(y).$$

$$u_{k_B}(y)=\sum _j{c}_j{e}^{ij{k}_{0}y}.$$

$$E_n=\frac{{\epsilon }^2-p_x^2-p_z^2-m^2}{2\epsilon },$$

$$\sum _j\frac{1}{2\epsilon }{\left[j{k}_0+k_B\right]}^2{\delta }_{j,l}{c}_j$$

$$+\sum _j{c}_j\frac{1}{d_p}\int q\phi (y)e^{i(j-l)k_{0}y}dy$$

$$=\sum _j\frac{{\epsilon }^2-p_x^2-p_z^2-m^2}{2\epsilon }{\delta }_{j,l}{c}_j,$$

$$U(y)=\sum _j{c}_j{\mathbf{S}}_j{e}^{i(j{k}_0+k_B)y},$$

$$\boldsymbol{S}_{j}=\sqrt{\varepsilon+m}\left(\begin{array}[]{c}\boldsymbol{s}\\ \frac{\boldsymbol{\sigma}\cdot\boldsymbol{p}_{j}}{\varepsilon+m}\boldsymbol{s}\end{array}\right),$$

$${\mathbf{p}}_j=\left(p_x+E_n-\frac{(j{k}_0+k_B{)}^2}{2\epsilon },j{k}_0+k_B,p_z\right).$$

$$⟨p_y⟩={\int }_{-\infty }^{\infty }{\psi }^{†}(x)\left(-i\frac{d}{dy}\right)\psi (x)d^{3}x,$$

$$⟨p_y⟩=\sum _j{\left|c_j\right|}^2(j{k}_0+k_B),$$

$$S_{i\to f}^{(1)}$$

$$d{P}_{i\to f}^{(1)}={\left|S_{i\to f}^{(1)}\right|}^2\frac{d^3{p}_f}{(2\pi {)}^3}\frac{d^{3}k}{(2\pi {)}^3}.$$

$$S_{i\to f}^{(1)}=$$

$$\times \delta (p_{x,i}-k_x-p_{x,f})\delta (p_{z,i}-k_z-p_{z,f})$$

$$\times \delta (k_{B,i}-k_y-k_{B,f}-n_B{k}_0),$$

$${\mathcal{M}}_{i\to f}=e\sqrt{\frac{4\pi }{2\omega }}\frac{1}{2\sqrt{{\epsilon }_f{\epsilon }_i}}\sum _j{c}_{n_B+j,f}^{\ast }{c}_{j,i}{\bar{\mathbf{S}}}_{n_B+j,f}/{ϵ}_k{\mathbf{S}}_{j,i},$$

$$d{W}_{i\to f}^{(1)}$$

$$\Delta =E_n+\frac{p_z^2+m^2}{2\epsilon }.$$

$$f(\theta )$$

$$=E_{n_f}-E_{n_i}+\frac{p_{z,f}^2+m^2}{2{\epsilon }_f}-\frac{p_{z,i}^2+m^2}{2{\epsilon }_i}$$

$$+p_{x,f}-p_{x,i}+\omega$$

$$\simeq E_{n_f}-E_{n_i}+\frac{m^2}{2{\epsilon }_f}-\frac{m^2}{2{\epsilon }_i}$$

$$+\frac{\omega {\theta }^2}{2}\left(1+\frac{\omega {\text{sin}}^2\phi }{{\epsilon }_f}\right).$$

$${\theta }_0=\sqrt{\frac{2\left(E_{n_i}-E_{n_f}\right)+\frac{m^2}{{\epsilon }_i}-\frac{m^2}{{\epsilon }_f}}{\omega \left(1+\frac{\omega {\text{sin}}^2\phi }{{\epsilon }_f}\right)}},$$

$${\omega }_{\text{th,q}}={\epsilon }_i\left(1-\frac{1}{1+\frac{2{\epsilon }_i\left(E_{n_i}-E_{n_f}\right)}{m^2}}\right).$$

$$\frac{d^2{I}^{\text{Cl}}}{d\omega d\Omega }=\frac{e^2{\omega }^2}{4{\pi }^2}{\left|{\int }_{-\infty }^{\infty }\left(\mathbf{n}-\mathbf{v}\right)e^{i(kx)}dt\right|}^2,$$

$$\frac{d^{2}I}{d\omega d\Omega }$$

Full text

I Introduction

Under certain circumstances, when a high-energy charged particle enters a crystalline medium the particle dynamics is not dominated by the scattering on single atoms, but rather by the coherent scattering on many atoms resulting in a smooth, bound motion along crystal axes or planes (Lindhard, 1965). This motion leads to radiation emission called channeling radiation. This has been studied both experimentally (Bak et al., 1985, 1988; Swent et al., 1979; Andersen et al., 1981, 1982; Klein et al., 1985; Alguard et al., 1979; Andersen et al., 2012; Uggerhøj, 2005; Wistisen et al., 2018, 2014, 2017a) and theoretically (Kumakhov, 1976, 1977; Andersen et al., 1981; Sáenz et al., 1981). Channeling radiation from high-energy electrons/positrons represents one of the few experimental realizations of non-perturbative and non-linear problems in quantum electrodynamics, where the field strength experienced by the particle in its rest frame approaches the Schwinger field $E_{\text{cr}}=1.3\times 10^{16}$ V/cm. The only other experiments where this has been realized were the SLAC laser experiment (Bula et al., 1996) and the recently reported experiments (Cole et al., 2018; Poder et al., 2018), although with smaller quantum non-linearity parameter. The quantum nonlinearity parameter is the ratio of the field strength experienced by a charged particle in its rest frame and the Schwinger field strength. While the quantum treatment of radiation emission in a laser field can be fully treated using the Volkov state (Ritus, 1985; Boca and Florescu, 2009; Dinu and Torgrimsson, 2018; Seipt and Kämpfer, 2011; Mackenroth and Di Piazza, 2011, 2013; King, 2015), no such full treatment has been presented for channeling radiation. The theory of channeling radiation so far has consisted of a quantum approach involving wave functions at low particle energies, but which neglects spin, photon recoil, and non-dipole transitions in the emission process, because these effects are not important at low energies as compared to the effect of the quantization of the energy levels in the potential. Such an approach can be found, for example, in Refs. (Andersen et al., 1982; Lervig et al., 1967; Andersen and Augustyniak, 1977). For higher particle energies, the theory relies on the semi-classical operator method by Baier et al. (Baier and Katkov, 1968), which then includes the effects of spin and photon recoil, but neglects the quantization of the transverse motion. The semi-classical method is relatively easy to implement numerically and this explains why this method is often employed for numerical calculations in crystal channeling, and crystalline undulators (Korol et al., 2013; Bandiera et al., 2015a; Guidi et al., 2012).

In many of the original theoretical works on channeling radiation the connection between the Dirac equation and a Schrödinger like equation was seen but then various approximations were employed, such as a simplified potential yielding analytical solutions, the dipole approximation in the calculation of the radiation and/or the assumption that $\omega/\varepsilon$ is small, where $\omega$ is the energy of the emitted photon and $\varepsilon$ the initial energy of the radiating particle, see (Kumakhov and Wedell, 1977; Zhevago, 1978; Beloshitskii and Kumakhov, 1978; Bazylev et al., 1980; Kimball and Cue, 1985). See also (Shulga et al., 2017; Kozlov et al., 2010; Shul’ga et al., 2016; Piazza et al., 2017; Abdrashitov et al., 2018; Bandiera et al., 2015b; Wistisen et al., 2017b; Korol et al., 2016; Bandiera et al., 2013; Backe et al., 2013; Wienands et al., 2015; Wistisen et al., 2016; Kostyuk, 2013) for recent advancements on the subject of crystal channeling. In this paper, we present the calculation for planar channeling, based on wave functions that are approximate solutions of the Dirac equation in the realistic Doyle-Turner model of the periodic crystal potential and calculate the single-photon radiation emission without any of the mentioned approximations. Therefore, we now include all relevant quantum effects, which in some cases yield differences as compared to the semi-classical theory. In particular, the fact that the radiation emission stems from transitions between discrete bound states between the planes, is properly taken into account. We investigate an example of $20$ GeV planar channeled electrons as in this case one can have significant radiation emission from bound states with low quantum number. In this case the quantization of the motion is important and at the same time photon energies comparable to the electron energy are emitted, such that spin and photon recoil effects are also important. In addition, for very high particle energies, transitions from bound states with high to low quantum numbers become more likely. Therefore, the high-energy part of the spectrum is different from that obtained via the semi-classical model, as states with low quantum numbers are only approximately accounted for by the latter model. As an example, we will see this effect for planar channeled electrons with initial energy of $250$ GeV and $1$ TeV.

We use units where $\hbar=c=1$, $\alpha=e^{2}$, with $e$ being the positron charge, and the Feynman slash notation such that $\not{a}=a_{\mu}\gamma^{\mu}$, where $\gamma^{\mu}$ are the Dirac gamma matrices and $a^{\mu}$ an arbitrary four-vector. We adopt the metric tensor $\eta^{\mu\nu}=\text{diag}(+1,-1,-1,-1)$. Below, when the term ‘particle’ is employed, it will refer to either an electron or to a positron.

II Wave functions

In Ref. (Wistisen and Di Piazza, 2018) we discussed the Dirac equation in a parabolic potential in the regime where $\xi\ll\gamma$, where $\xi$ is the typical transverse momentum (here $p_{y}$) divided by the electron mass $m$ and $\gamma$ is the Lorentz gamma factor of the incoming particle. For channeling we have that $\xi=\gamma\theta_{c}$ where $\theta_{c}=\sqrt{2V_{0}/\varepsilon}$ is the critical angle for channeling (Lindhard, 1965), and so the requirement of validity reduces to $\theta_{c}\ll 1$ or $\varepsilon\gg V_{0}$. For Silicon $V_{0}=22.7$ eV and we are interested in ultra-relativistic particles with energy on the scale of GeV, such that this approximation is safely applicable. Note that at $\xi\gg 1$ the overall emission angle ($\sim 2\xi/\gamma$) is much larger than the instantaneous one ($\sim 1/\gamma$) and this corresponds to the regime where the local constant field approximation becomes applicable (Baier et al., 1998). When $\xi$ is small, however, the dipole approximation may be used (Baier et al., 1998). Another important parameter is the so-called quantum nonlinearity parameter already mentioned in the introduction and defined by

[TABLE]

When $\chi$ becomes on the order of unity, the effects of particle spin and photon recoil become important in the radiation emission process. The quantum description of radiation emission using wave functions as seen in Refs. (Andersen et al., 1982; Lervig et al., 1967; Andersen and Augustyniak, 1977) is valid when $\chi\ll 1$, because spin effects and recoil are neglected, and when $\xi\ll 1$, because the magnetic field in the particle’s rest frame is neglected and only the dipole matrix element is calculated. In the present paper we treat the problem from the laboratory frame, and the field can therefore be described solely by an electrostatic potential $\varphi(y)$. We then assume that the largest component of the particle momentum is along the $x$ direction. In Ref. (Wistisen and Di Piazza, 2018) (and see Appendix A) we found that the positive-energy solutions of the Dirac equation, to leading order in $\xi/\gamma$, can be written as (we set the quantization volume $V=1$)

[TABLE]

and $U(y)$ is given by

[TABLE]

where $\tilde{\boldsymbol{p}}=\left(p_{x}-e\varphi(y),-i\frac{d}{dy},p_{z}\right)$, $\boldsymbol{s}$ is a two component vector describing the spin, which we can choose as either \left(\begin{array}[]{cc}1&0\end{array}\right)^{T} or \left(\begin{array}[]{cc}0&1\end{array}\right)^{T}, corresponding to spin-up and spin-down respectively. Note that in Eq. (3) we approximated

[TABLE]

where $\boldsymbol{p}=\left(p_{x},-i\frac{d}{dy},p_{z}\right)$, see also Appendix A. As already mentioned, we have taken $p_{x}$ to be the longitudinal direction, that is, in the initial state $p_{z}=0$, $p_{x}>0$ and $p_{x}\simeq\varepsilon\gg m\xi$, where $\varepsilon=m\gamma$ is the initial (constant) energy of the particle. The function $I(y)$ is the solution of the equation

[TABLE]

For $\varphi(y)$ we will use the Doyle-Turner model (Baier et al., 1998; Doyle and Turner, 1968; Avakian et al., 1982; Møller, 1995), chosen as symmetric around $y=0$. Clearly Eq. (5) corresponds to an eigenvalue problem in the form we are accustomed to from atomic physics. In a crystal the potential $\varphi(y)$ is periodic with the period of the inter planar distance which we will denote as $d_{p}$. Now, because of this periodicity the solution can be written as a Bloch wave such that

[TABLE]

and where $u_{k_{B}}(y)$ is also periodic with period $d_{p}$. The quantity $k_{B}$ is the Bloch momentum, which can be taken to be in the interval $0\leq k_{B}<k_{0}$, $k_{0}=\frac{2\pi}{d_{p}}$. It then follows from Bloch’s theorem that these solutions form an orthogonal and complete set of solutions of Eq. (5) (Ashcroft and Mermin, 1976) (see Appendix B for a proof that the resulting solutions of the Dirac equation are also orthonormal within our level of approximation). Now, we are interested in the solution $u_{k_{B}}(y)$ which is the nontrivial part of the wave function. Inserting $I(y)$ of Eq. (6) into Eq. (5) gives us the equation governing $u_{k_{B}}(y)$

[TABLE]

The periodicity of $u_{k_{B}}(y)$ implies that it can be written as the Fourier series

[TABLE]

To write any periodic function as a Fourier series, the sum includes infinitely many terms. However, for the numerical implementation we are restricted to reducing the series to a finite sum. When the number of basis vectors is increased, it is found that the lowest lying states do converge, and therefore one only needs enough basis elements, that the sum describing the states of interest has converged to a fixed degree of accuracy. To ensure normalization we should have $\sum_{j}\left|c_{j}\right|^{2}=1$ (see Appendix B). It is now clear that this is an eigenvalue problem where the quantized eigenvalue is

[TABLE]

where $n$ is the quantum number corresponding to the value of this energy in ascending order and where $n=0$ corresponds the ground state. This equation leads to a quantization of, e.g., $p_{x}$. That is, if $\varepsilon$ is fixed by the incoming energy of the particle, a larger quantum number $n$ corresponds to a smaller value of $p_{x}$ in order to accommodate for the larger transverse momentum in the $y$ direction. From this relation it is also clear that the quantity $2\varepsilon E_{n}$ is related to the square of the momentum in the $y$ direction, $p_{y}^{2}$. The coefficients $c_{j}$ are found by solving the matrix eigenvalue problem obtained by inserting Eq. (8) in Eq. (7), by multiplying by $e^{-ilk_{0}y}/d_{p}$, and by integrating over $y$ from $-d_{p}/2$ to $d_{p}/2$ to exploit orthogonality:

[TABLE]

where $q=e$ for a positron and $q=-e$ for an electron. With these results taken into consideration, we now see that we can write the function $U(y)$ in terms of the coefficients $c_{j}$ such that

[TABLE]

where

[TABLE]

and where

[TABLE]

In order to write the momentum in this form we have replaced the term with the potential by exploiting Eq. (7). Before calculating the radiation emission probability we will calculate the expectation value of the momentum in the $y$ direction, as this will provide insight on how the momentum relates to the quantum number $k_{B}$. By inserting our wave function from Eq. (2) in

[TABLE]

we find that this becomes simply

[TABLE]

to leading order in our approximation, see Appendix C.

III Single photon emission

We will now derive the single photon emission probability. The leading-order $S$-matrix element for the emission of a single photon by an electron moving inside the potential $\varphi(y)$ is given by

[TABLE]

and the emission probability is then

[TABLE]

By inserting our wave functions and integrating over the coordinates, which provide energy-momentum conservation delta functions, we have that

[TABLE]

where $n_{B}$ is the integer such that $0\leq k_{B,f}<k_{0}$, henceforth denoted as the first Brillouin zone (FBZ) (see Appendix D for details on this derivation). In the above expression of the $S$-matrix element, we defined the reduced matrix element

[TABLE]

where $\boldsymbol{S}_{j,i}$ corresponds to the initial state, $c_{j,i}$ is the coefficient with index $j$ corresponding to the initial state $i$. It is therefore seen that one must solve the matrix problem of Eq. (10) for $\varepsilon_{f}=\varepsilon_{i}-\omega$ several times, as the final state depends on the energy of the emitted photon $\omega$ and the Bloch momentum $k_{B,f}$ of the final electron (see Appendix D for additional details especially on why $\mathcal{M}_{i\rightarrow f}$ reduces to a single sum over $j$). We find that the rate corresponding to Eq. (17), when dividing by total interaction time $T_{\text{int}}$ arising from squaring the energy delta function, is given by

[TABLE]

As is usually the case when dealing with calculations concerning ultra-relativistic particles it is pertinent to consider cancellations between the large terms in the expression in the delta function, $\varepsilon_{f}+\omega-\varepsilon_{i}$, because the relevant transverse energies $E_{n}$, comparable to the potential depth, are much smaller than the whole energy of the particle (recall that the former are of the order of several eV, whereas the latter is of the order of GeV). For this it is useful to consider the quantity $\Delta$ which we define via the equation $\varepsilon=p_{x}+\Delta$. Inserting this into the equation for our eigenvalues, Eq. (9), we obtain that to leading order in $E_{n}/\varepsilon$,

[TABLE]

Defining $\boldsymbol{k}=\omega(\text{cos}\theta,\text{sin}\theta\text{cos}\varphi,\text{sin}\theta\text{sin}\varphi)$ we can rewrite the argument of the remaining energy delta function as (note that in the present problem it is convenient to use the $x$ axis as polar axis)

[TABLE]

where $\theta_{0}$ is the positive solution to $f(\theta)=0$. When considering bound states, the energies $E_{n}$ become nearly independent of $k_{B}$, as seen in Fig. (2). In this case one can isolate the emission angle as

[TABLE]

and the threshold for a given transition is given by the condition where the numerator vanishes, which gives us that

[TABLE]

IV Semi-classical method

In order to compare with the classical theory we will derive a formula for the emitted power in the case of a periodic transverse motion, and then in the end this can be turned into a semi-classical formula by comparing with the expression of Baier et al. [see Eq. (26) below]. Classically the emitted energy per unit frequency and solid angle can be written as (Jackson, 1999; Wistisen, 2014)

[TABLE]

where $k^{\mu}=(\omega,\boldsymbol{k})$ is the four-momentum of the emitted radiation, where $x^{\mu}=(t,\boldsymbol{r})$ is the four-position of the particle, $\boldsymbol{v}$ its velocity, and $\boldsymbol{n}=\boldsymbol{k}/\omega$. In the semi-classical formalism the same quantity is given by (Baier et al., 1998; Baier and Katkov, 1968; Wistisen, 2014)

[TABLE]

where $\varepsilon^{\prime}=\varepsilon-\omega$, $\omega^{\prime}=\omega\varepsilon/\varepsilon^{\prime}$, and $k^{\prime}=k\varepsilon/\varepsilon^{\prime}$ (this result holds in the case when the sum over final particle spins and photon polarizations and the average over initial spins are taken). As it was shown in Refs. (Wistisen, 2014, 2015) we need only to calculate the transverse components of the integrand for ultra-relativistic particles as the longitudinal component is suppressed by at least a factor of $1/\gamma$ in comparison. Therefore, for the classical formula we have to calculate

[TABLE]

Since most of the radiation is emitted in the forward direction for ultra-relativistic particles, we may perform the small-angle expansion and so we write $n_{x}=1-\frac{\theta^{2}}{2}$ and $x(t)=\left\langle v_{x}\right\rangle t+\delta x(t)$, where we exploit the fact that the motion is quasi-periodic. For this reason, the quantity $\delta x(t)$ is a periodic function. By using the fact that $v^{2}$ is approximately conserved we have that $\left\langle v_{x}\right\rangle\simeq 1-\frac{1}{2\gamma^{2}}-\frac{\boldsymbol{v}_{\bot}^{2}}{2}$, see also Ref. (Jackson, 1999). Inserting this result in the above expression and canceling the large terms we obtain

[TABLE]

Now, if $\left\langle\boldsymbol{v}_{\perp}\right\rangle$ is zero, i.e., we have chosen the coordinate system where this is the case, we can exploit that $\boldsymbol{r}_{\perp}$ and $\delta x$ are periodic and write $\int_{-\infty}^{\infty}f(t)dt=\sum_{n}\int_{nT}^{(n+1)T}f(t)dt$, where $T$ is the period of the motion, and change variable at which point the Dirichlet kernel appears, which can be replaced by a sum of delta functions. Thus, we obtain

[TABLE]

where $\omega_{0}=\frac{2\pi}{T}$ and where we introduced the function

[TABLE]

Analogously, we introduce the quantity

[TABLE]

For the integral with the velocity, completely analogous steps can be taken and therefore we have that

[TABLE]

By inserting these quantities into the classical formula and using that the delta function squared gives us the delta function again with a factor of $\Delta\varphi/2\pi=T_{\text{int}}\omega_{0}/2\pi$. Therefore, we obtain that the emitted energy per unit time $T_{\text{int}}$ is given by

[TABLE]

where we used the delta function to integrate over the angle $\theta$ such that

[TABLE]

Note that for the sake of convenience we have introduced here a “classical” emission probability even though such a quantity has a meaning only within the quantum theory. By performing the appropriate substitutions of $\omega\rightarrow\omega^{\prime}$ and by putting in the front factors as in Eq. (26), we can obtain the semi-classical version of the result in the form

[TABLE]

where

[TABLE]

The threshold is therefore found to be at

[TABLE]

V Discussion of results

In Fig. (2) we show the energy bands for a positron with 100 MeV in the Doyle-Turner potential describing the (110) planes of Silicon. It is noted that the energy $E_{n}$ of bound states with small values of $n$ is almost independent of $k_{B}$. In Fig. (3) we show a plot of the expectation value of the transverse momentum $p_{y}$ and we see that the bound states, with $E_{n}<V_{0}$ have $\left\langle p_{y}\right\rangle\simeq 0$, and for the states above the barrier, the quantity $\left\langle p_{y}\right\rangle$ steadily increases with $E_{n}$. Also, it depends on $k_{B}$ in a way such that half of the FBZ describes particles going to the left and the other half particles going to the right.

In Fig. (4) we show the total power of a 250 GeV electron/positron, calculated with the LCFA, depending on the initial position $y_{0}$, when assuming the beam angular divergence is negligible, i.e. much smaller than the critical angle $\theta_{c}$. What can be seen from this figure is that in both cases of the electron and positron, particles which start out close to the plane (here $y_{0}=0$) have a larger radiation power. However for positrons, as they are repelled from the planes, this means a large oscillation amplitude, and therefore large $n$ quantum number, while for electrons, which are attracted to the planes, starting close to the plane means a small oscillation amplitude and therefore small $n$ quantum numbers radiate more for electrons. This, combined with the fact that electrons will generally have an emission spectrum distributed around larger photon energies, implies that all the quantum effects mentioned in the introduction may become important. In Fig. (5) we show the calculation of the photon emission spectra for a 20 GeV electron in the state with $n=20$. This value of $n$ was picked as states around this value of $n$ have the largest rate of emission. In the particular case seen in Fig. (5), we can see the influence of the quantum effects in the following way. The fact that the classical calculation differs from the semi-classical method of Baier et al. means that the effects of spin and of photon recoil are present, i.e., that $\chi$ is large enough that these effects are sizable. At the same time, we see that the calculations presented here differ from the semi-classical of Baier et al. because the quantum number, $n=20$, of this state is not large enough that the quantization of the motion can be completely neglected. In addition, we have that $\xi$ is on the order of unity, and therefore one can apply neither the constant field approximation nor the dipole approximation. We see however in Fig. (5) that the overall level of the semi-classical spectrum falls together with the quantum results obtained here, while the most noticeable difference is in the position of the thresholds, which should also be clear from Eq. (24) and Eq. (37), see (Wistisen and Di Piazza, 2018; Raicher et al., 2018) where this is also found for different field configurations. Here, it is seen that the threshold depends critically on the difference in energy between the quantized levels in the transverse potential, information which is not contained in the semi-classical method. In an experiment one would however only obtain an average over the spectra corresponding to different states, and therefore these details would likely be washed out.

Definitive distinction could however be observed if the emitted photon energy $\omega$ could be measured along with the emission angles $\theta$ and $\varphi$. This can be seen from Eq. (23) and Eq. (36), which show that also the emission angle depends on the level spacing. In addition, a qualitative difference appears: the semi-classical treatment predicts that $\theta$ depends only on $\omega$, while in the full calculation presented here, there is also a dependence on $\varphi$ when $\omega\sim\varepsilon-\omega$ i.e. when $\chi$ is no longer small. In Fig. (6) we show that while the same differences of the position of the thresholds can be seen here, the effect is very small for positrons with larger values of $n$ typically, as explained. However if we picked a positron in a lower lying state, differences in the spectrum comparable to those seen in Fig. (5) would be seen, however such low lying states have a low total radiated power for positrons, and therefore do not contribute much to the spectrum when averaged over initial conditions.

In Fig. (7) we show the calculation for a $250$ GeV electron in the $n=45$ state, which corresponds to the particles with the largest power, i.e. the peak in Fig. (4) for electrons. We compare to the the semi-classical, classical and the locally constant field approximation (LCFA) model. As expected, the LCFA approximation agrees with the semi-classical for large photon energies, where the formation length is short. It is seen that the classical calculation fails for large photon energies, as expected. For the very high energy part of the spectrum it is also observed that the full quantum calculation is different from the semi-classical result. From Eq. (24) it can be seen that larger values of the difference $E_{n_{i}}-E_{n_{f}}$ imply a larger value of the emitted energy threshold. Therefore, the high-energy part of the photon spectrum arises from transitions where the electron goes from the initial state with $n=45$ to a state with a low value of the quantum number $n$. Now, in the semi-classical method the wave function of both initial and final states are approximated in a way which is only valid when the corresponding discrete quantum numbers are large. Since this is not the case for the final state, the semi-classical result differs from the full quantum one. In Fig. (8) the same effect is seen for the case of a $1$ TeV electron in the $n=120$ state initially. In this case the effect is more pronounced, and far above the level of the Bethe-Heitler bremsstrahlung which is approximately $\omega dW_{\text{BH}}/d\omega\simeq 2.8\times 10^{-6}$ eV. Therefore, if one carries out a precise measurement of the high-energy part of the spectrum, the discrepancy between the two models could be experimentally tested. In Fig. (9) we show the spectra for a 1 TeV positron with a large $n$ quantum number and in this case, the semi-classical method works well.

VI Conclusion

In conclusion, we have shown how to find approximate solutions of the Dirac equation for describing the motion of relativistic electrons and positrons in a periodic potential which depends on one transverse coordinate, as compared to the direction of the largest momentum (here indicated as the $x$-direction). We have shown how to calculate the emission rate of a single photon from transitions between the corresponding quantum states exactly, that is, without the use of neither the dipole approximation nor the locally constant field approximation. Therefore, we have been able to calculate single-photon planar channeling radiation, with all relevant quantum effects included: the effects of electron/positron spin and photon recoil during the emission, which the semi-classical method also incorporates, but also the effects of the quantization of the transverse motion. For planar channeling, and in particular for positrons, we saw that the semi-classical approximation of Baier et al. (beyond the locally constant field approximation) is accurate in describing the energy distribution of the emitted photon, when integrated over angles. For electrons, differences are more noticeable. For low electron energies, a clear experimental measurement of such differences would require angular resolution of the emitted photons along with their energy. However, for higher-energy electrons a difference could potentially be detected experimentally, even in the angularly integrated emission spectrum for emitted photons with high energy.

VII Acknowledgments

T. W. is supported by the Alexander von Humboldt-Stiftung apart from the initial part of the project where funding was provided by the VILLUM FONDEN (research grant VKR023371).

Appendix A

The general (unnormalized) solution to the Dirac equation with potential energy $V(\boldsymbol{r})=-e\varphi(\boldsymbol{r})$ can be written as

[TABLE]

The Dirac equation then becomes

[TABLE]

[TABLE]

If we insert $\chi(\boldsymbol{r})$ from Eq. (40) in Eq. (39) we can obtain an equation for $\phi(\boldsymbol{r})$.

[TABLE]

Now let us multiply with $\left(\varepsilon+e\varphi+m\right)$, then

[TABLE]

now if we neglect the spin-field interaction terms and the field squared term $e^{2}\varphi^{2}$ this becomes

[TABLE]

The electron solution is then

[TABLE]

The expression from Eq. (3) is then obtained by setting $\phi(\boldsymbol{r})=\boldsymbol{s}e^{i(p_{x}x+p_{z}z)}I(y)$ and expanding $\left(\varepsilon-V(\boldsymbol{r})+m\right)^{-1}\simeq\left(\varepsilon+m\right)^{-1}\left(1+\frac{V(\boldsymbol{r})}{\varepsilon+m}\right)$, and keeping this correction term with the potential only on the $\sigma_{x}p_{x}$ term, as here this correction yields the leading order in $\xi/\gamma$ on $\sigma_{x}$.

Appendix B

The electron state can be written as (putting back in the volume factor)

[TABLE]

where

[TABLE]

where $\boldsymbol{p}_{j}=(p_{x}+E_{n}-\frac{(jk_{0}+k_{B})^{2}}{2\varepsilon},jk_{0}+k_{B},p_{z})$ and then

[TABLE]

Explicitly we have that $c_{j}=c_{j}(\varepsilon,k_{B},n)$. Now since both $k_{B}$ and $k_{B}^{\prime}$ obey that $0\leq k_{B}<k_{0}$ we have that $-k_{0}<k_{B}-k_{B}^{\prime}<k_{0}$ and therefore $k_{B}-k_{B}^{\prime}$ can never be an integer value of $k_{0}$ unless $k_{B}-k_{B}^{\prime}=0$, and therefore we can write

[TABLE]

[TABLE]

However the vector $\boldsymbol{c}$ is a normalized ($|\boldsymbol{c}|=1$), eigenvector of a hermitian matrix and the vectors corresponding to $n$ and $n^{\prime}$ have different eigenvalues of this matrix, and are therefore orthogonal, so

[TABLE]

Now consider

[TABLE]

and therefore

[TABLE]

There $\simeq$ refers only to the normalization. The states are exactly orthogonal, but in the normalization we neglect corrections which are suppressed by at least $\xi/\gamma$ compared to leading order. So finally

[TABLE]

Appendix C

Inserting our wave-functions into the expression from the paper, only the $y$ component is non-trivial:

[TABLE]

Now due to periodicity of the integrand we have that

[TABLE]

and so

[TABLE]

and setting $p_{z}=0$,

[TABLE]

and therefore

[TABLE]

And so within our level of approximation we have that

[TABLE]

Appendix D

Starting from

[TABLE]

we insert the wave functions in terms of the plane wave expansion to obtain

[TABLE]

The quantity $F=\int_{-\infty}^{\infty}e^{i(k_{B,i}-k_{y}-k_{B,f})y}e^{i(j-l)k_{0}y}dy$ can be rewritten by exploiting that $e^{i(j-l)k_{0}y}$ is periodic, and so we see that

[TABLE]

and change variable such that $y^{\prime}=y-nd_{p}$,

[TABLE]

Now the sum $\sum_{n=-\infty}^{\infty}e^{ind_{p}\left(k_{B,i}-k_{y}-k_{B,f}\right)}$ can be recognized as the Dirichlet kernel which can be replaced with the Dirac comb $\sum_{n=-\infty}^{\infty}2\pi\delta(\left[k_{B,i}-k_{y}-k_{B,f}\right]d_{p}-2\pi n)$. Only the delta function which has $k_{B,f}$ in the first Brillouin zone will contribute, due to the fact that integration limit on $k_{B,f}$ is from [math] to $k_{0}$, and therefore we must set $n=n_{B}$ such that $0\leq k_{B,f}<k_{0}$. Therefore we may use that

[TABLE]

This simplifies the summation over $l$ as the sinc function means that only the term obeying $n_{B}+(j-l)=0$ contributes. Therefore the S-matrix element becomes

[TABLE]

Now a useful expression for the quantity $\bar{\boldsymbol{S}}_{n_{B}+j,f}\not{\epsilon}^{*}\boldsymbol{S}_{j,i}$ may be derived. We will momentarily suppress the $j$ index, and so

[TABLE]

where we have defined

[TABLE]

[TABLE]