1 Introduction

The interaction of relativistic charged particles in aligned crystals remains of growing interest for theorists and experimentalists since the first discoveries related to the possibility of its use as a powerful source of electromagnetic radiation based on the phenomena of coherent bremsstrahlung (CB) and channeling radiation (CR) [1,2,3,4,5,6,7,8,9]. Both types of the radiation have been in detail studied within classical and quantum theoretical models as well as in many dedicated experiments that has formed rather adequate formalism of general common description of coherent radiation processes by charged particles in crystals (see, for instance, in [10]). However, even today some features of particles scattering in crystals can be the origin of fine peculiarities to be observed in the emitted electromagnetic radiation. In this work we examine the influence of the asymmetry in angular scattering of axially channeled electrons on the accompanying channeling radiation.

So far the theory of electromagnetic radiation by electrons in crystals has been developed for the planar channeling regime within the approximations of both classical and quantum mechanics revealing very details of the beam motion in the crystal field characterised by strong redistribution of the beam in a phase space [4, 7, 8, 11,12,13,14,15,16,17]. Having known the behaviours of channeled beam evolution in the transverse space of motion, usually the calculations of radiation characteristics are limited just in getting the total radiation intensity. Its dependence on the angular distance from the forward direction, for which the maximum of radiation intensity is observed, is assumed to be azimuthally symmetric at a small distance and just formally estimated. However, new types of electromagnetic radiation can be observed at well defined angular parameters of channeled particles (for instance, see in [18]).

As regards the theory of CR at axial channeling the calculations are mostly performed within the limits of classical electrodynamics (CED), i.e. one calculates the trajectories of channeled particles in the potentials of crystal axes followed by the numerical integration of the derived CED expressions [2, 19,20,21,22,23]. Some details of the beam dynamics can be additionally included in the calculations of axial CR ones applied a well known “quasi-classical theory” ([6] and Refs. therein).

In many papers the quantum electrodynamics (QED) estimations of CR at axial channeling have been obtained (see, for instance, in [9, 24]). These approximations are based on studying the transverse wave functions of channeled particles for the stationary Schrödinger equation and well describe CR for single crystal axes (the single-string approximation). However, taking into account the crystal periodicity in a plane perpendicular to the crystal axes allows the \(\psi \) -function of the transverse motion of a channeled particle to be presented as a sum of Bloch functions [25,26,27,28], which satisfy the Dirac equation (the many-beam approximation). The latter may result in appearance of some singularities for the angular and energy distributions of CR. For electrons and positrons the quantum mechanical description of the projectile motion at channeling conditions becomes mostly applicable for low and moderate particle energies (\(\le \,\)100 MeV), as well as for hyper-relativistic ones (\(\ge \,\)10 GeV) [26].

In our previous works [14,15,16,17, 29] we have developed a new method to figure the Bloch wave functions for planar and axial crystal orientations, the details of which is given in [30]. The present work aims in deducing the analytical expressions for spectral-angular CR distributions in thin crystals that demonstrate distinctive features of these dependences to be, in our opinion, of a special interest for future experimental studies.

2 CR probability matrix element

In QED the radiation probability is defined by the expression

$$\begin{aligned} dw_{if} = \frac{2 \pi }{\hbar } \mid M_{if}\mid ^2 \delta (E) d\varrho _f\, , \end{aligned}$$
(1)

which, in our case, is determined by the wave functions of the quantum states for channeled projectile motion described by Dirac equation in so-called continuous potential formed by the system of crystal planes and axes. The potential is strongly related to the crystal orientation with respect to the projectile initial momentum. In this definition \(d\varrho _f = 1/(2\pi )^3 d^3\kappa \) is the density of the final quantum states f, and \(M_{if}\) is the matrix element for the \(i\rightarrow f\) transition accompanied by CR

$$\begin{aligned} M_{if} = - e \int \mathbf {A}^{*}\mathbf {J}_{fi}d^3\mathbf {r}\, , \end{aligned}$$
(2)

where the vector \(\mathbf {A}= \sqrt{2\pi \hbar c^2/\omega }\, {\varvec{\epsilon }}_{\varvec{\kappa }} e^{\mathbf {i}\varvec{\kappa }\mathbf {r}}\) represents the wave function of a CR-photon with the polarisation vector \({\varvec{\epsilon }}_{\varvec{\kappa }}\), \(\mathbf{J}_{fi} = \Psi _f \varvec{\gamma } \Psi _i\) is the current operator, namely, the probability density for the channeled particle transition \(i\rightarrow f\) with the Dirac \(\varvec{\gamma }\)-matrix.

The wave function of a channeled particle in the ith quantum state can be written in the form \(\Psi _i = \sqrt{\big (E_{i \Vert } + mc^2\big )\big /2 E_{i\Vert }}u_i\psi _i(\mathbf {r})\), where [31]

$$\begin{aligned} \psi _i(\mathbf {r}) = \phi _i(\mathbf {r}_{\perp }) \exp (\mathbf {i}\mathbf {p}_{i\Vert }\mathbf {r}_{\Vert }/\hbar ),\, u_i = \left( \begin{array}{c} w \\ \frac{\varvec{\sigma }\hat{\mathbf {p}}c}{c^2 m + E_{i\Vert }} \end{array} \right) \end{aligned}$$
(3)

Here w is the 2d-spinor normalised by the condition \(w^{+}w = 1\), \(\varvec{\sigma }\) are the Pauli matrixes. The transverse wave function \(\phi _i(\mathbf {r}_{\perp })\) describes the ith quantum state of a relativistic channeled particle of the mass \(\gamma m\) (with the relativistic Lorentz factor \(\gamma = E_{\Vert }/m c^2\)) by the Schrödinger equation [4, 5, 8, 31]

$$\begin{aligned} \hat{H}\phi _i(\mathbf {r}_{\perp }) = \left( \frac{{\hat{\mathbf {p}}}\; ^2_{\perp }}{2 m\gamma _i} + U(\mathbf {r}_{\perp })\right) \phi _i(\mathbf {r}_{\perp }) = E_{i\perp }\phi _i(\mathbf {r}_{\perp }) \end{aligned}$$
(4)

Besides, because of the crystal field periodicity, the transverse wave function \(\phi _i(\mathbf {r}_{\perp }) \) can be represented by Bloch functions [25].

Successfully, having the longitudinal momentum of a channeled particle directed along the z-axis, which is coaxial with the crystal channeling chain, the matrix element for the CR probability is calculated as follows

$$\begin{aligned} M_{if}= & {} - e \sqrt{\frac{2\pi \hbar c^2}{\omega }} \int (\mathbf {\alpha }^{0}_{if}(\mathbf {r}_{\perp }){\varvec{\epsilon }}_{\varvec{\kappa }}) \nonumber \\&\times \, e^{\mathbf {i}\varvec{\kappa _{\perp }}\mathbf {r}_{\perp }} e^{\mathbf {i}\kappa _z z} \exp (\mathbf {i} \Delta p_{if} z/\hbar ) d\mathbf {r}_{\perp } dz \end{aligned}$$
(5)

with

$$\begin{aligned} \mathbf {\alpha }^{0}_{if}(\mathbf {r}_{\perp }) = \frac{c}{E_{i \Vert }} \phi ^{*}_f(\mathbf {r}_{\perp }) \hat{\mathbf {p}}_i \phi _i(\mathbf {r}_{\perp })\, \end{aligned}$$
(6)

Integration by z, this expression is deduced to much simpler expression

$$\begin{aligned} M_{if} = - e \sqrt{\frac{2\pi \hbar c^2}{\omega }} \delta _{\Delta p_{if} z/\hbar ,\kappa _z} (\mathbf {\alpha }_{if}(\varvec{\kappa _{\perp }}){\varvec{\epsilon }}_{\varvec{\kappa }}) \end{aligned}$$
(7)

with the angular operator

$$\begin{aligned} \mathbf {\alpha }_{if}(\varvec{\kappa _{\perp }}) = \int \mathbf {\alpha }^{0}_{if}(\mathbf {r}_{\perp }) e^{\mathbf {i}\varvec{\kappa _{\perp }}\mathbf {r}_{\perp }} d\mathbf {r}_{\perp } \end{aligned}$$
(8)

Since the operator equation \(\mathbf {i}\hbar \hat{\mathbf {p}}_{\perp } = - m\gamma [\hat{H}_{\perp },\mathbf {r}_{\perp }]\), which defines the wave function \(\phi _i(\mathbf {r}_{\perp })\) for the ith quantum state of a channeled particle (9), the angular operator can be presented as a function of new matrix elements \(X_{if{\perp }}\) and \(Z_{if}\)

$$\begin{aligned} \mathbf {\alpha }_{if}(\varvec{\kappa _{\perp }}) = \left( -\mathbf {i}\frac{\Omega _{if}}{c}X_{if{\perp }}, \beta Z_{if}\right) \, , \end{aligned}$$
(9)

where \(\hbar \Omega _{if} = \varepsilon _i - \varepsilon _f\), \(\varepsilon _{l}\) is the transverse energy of the lth state of a channeled particle, \(\beta = \text {v}/c\),

$$\begin{aligned} X_{if{\perp }}&= \int e^{\mathbf {i}\varvec{\kappa _{\perp }}\mathbf {r}_{\perp }} \phi ^{*}_f(\mathbf {r}_{\perp })\mathbf {r}_{\perp }\phi _i(\mathbf {r}_{\perp }) d\mathbf {r}_{\perp } , \nonumber \\ Z_{if}&= \int e^{\mathbf {i}\varvec{\kappa _{\perp }}\mathbf {r}_{\perp }} \phi ^{*}_f(\mathbf {r}_{\perp })\phi _i(\mathbf {r}_{\perp }) d\mathbf {r}_{\perp } \end{aligned}$$
(10)

Applying well developed math techniques of the angular analysis for radiation polarisation [32, 33], we can easily prove that the components of polarisation vector \({\varvec{\epsilon }}_{\varvec{\kappa }}\) of Eq. (7) are defined by

$$\begin{aligned} {\varvec{\epsilon }}_{\varvec{\kappa }}&= \frac{1}{\sqrt{2}} \Big (\Lambda \cos \Theta \cos \Phi + \mathbf {i}\sin \Phi \, , \nonumber \\&\quad -\, \mathbf {i}\cos \Phi + \Lambda \cos \Theta \sin \Phi , -\Lambda \sin \Theta \Big )\,, \end{aligned}$$
(11)

where \( \Phi \) and \( \Theta \) are the azimuthal and polar angles, respectively, and \(\Lambda \) is the radiation (photon) helicity.

3 Matrix transition elements for periodic potentials

Let define the wave function \(\phi _i(\mathbf {r}_{\perp })\) in a form of Bloch function in the cartesian coordinate system

$$\begin{aligned} \phi _{i,i_z}(x, y) = e^{-\frac{\mathbf {i}\pi i_z (x+y)}{5 a_p}} \sum _{m_i,n_i} e^{-\mathbf {i}\left( \frac{4\pi m_i x}{a_p} + \frac{4\pi n_i y}{a_p}\right) }C_{i,i_z}^{m_i,n_i}\,, \end{aligned}$$
(12)

where \(C_{i,i_z}^{m_i,n_i}\) are the Fourier components of the wave function, and \(m_i, n_i\) are the Fourier components numbers for the ith energetic band of channeled electron transverse motion, \(a_p\) is the lattice constant. The \(i_z\) index corresponds to the internal point (numbered) within the ith energetic band.

The wave functions presented in the form (12) allow performing the calculations without the limits of dipole approximation. In this case for \(\mathbf {\alpha }_{if}(\mathbf {r}_{\perp })\) we can get

$$\begin{aligned}&\mathbf {\alpha }_{if}(\mathbf {r}_{\perp }) = -\, C_{i,i_z}^{m_i,n_i} \nonumber \\&\quad \times \left( \frac{\Omega _{if}}{c}F_x(\kappa _x,\kappa _y),\, \frac{\Omega _{if}}{c}F_y(\kappa _x,\kappa _y),\, -\beta F_z(\kappa _x,\kappa _y)\right) , \nonumber \\ \end{aligned}$$
(13)

where we have used the following nominations

$$\begin{aligned}&F_x(\kappa _x,\kappa _y) = 16 a_p \sin \left( \frac{\kappa _y a_p}{4}\right) (-1)^{m_f+m_i+n_f+n_i} \nonumber \\&\qquad \times \frac{\left( \cos \left( \frac{\kappa _x a_p}{4}\right) \left( \kappa _x a_p + 4\pi (m_f - m_i)\right) - 4\sin \left( \frac{\kappa _x a_p}{4}\right) \right) }{\left( \kappa _x a_p + 4\pi (m_f - m_i)\right) ^2 \left( \kappa _y a_p + 4\pi (n_f - n_i)\right) }\, , \nonumber \\&F_y(\kappa _x,\kappa _y) = 16 a_p \sin \left( \frac{\kappa _x a_p}{4}\right) (-1)^{m_f+m_i+n_f+n_i} \nonumber \\&\qquad \times \frac{\left( \cos \left( \frac{\kappa _y a_p}{4}\right) \left( \kappa _y a_p + 4\pi (n_f - n_i)\right) - 4\sin \left( \frac{\kappa _y a_p}{4}\right) \right) }{\left( \kappa _y a_p + 4\pi (n_f - n_i)\right) ^2 \left( \kappa _x a_p + 4\pi (m_f - m_i)\right) }\, , \nonumber \\&F_z(\kappa _x,\kappa _y) \nonumber \\&\quad =\frac{64 \sin \left( \frac{\kappa _x a_p}{4}\right) \sin \left( \frac{\kappa _y a_p}{4}\right) (-1)^{m_f + m_i + n_f + n_i}}{\left( \kappa _x a_p + 4\pi (m_f - m_i)\right) \left( \kappa _y a_p + 4\pi (n_f - n_i)\right) } \end{aligned}$$
(14)

At such definitions we will be able to perform precise analytical calculations of CR via rather routine work. However, it will not essentially change the quantitative data for the forward CR. On the contrary, huge difference between the energy of emitted CR photon and the longitudinal projectile energy suggests all transformations to be done within the dipole approach, i.e.

$$\begin{aligned} \kappa _x a_p \ll 1\, , \, \sin \left( \frac{\kappa _x a_p}{4}\right)\Rightarrow & {} \, \frac{\kappa _x a_p}{4}\, , \nonumber \\ \kappa _y a_p \ll 1\, , \, \sin \left( \frac{\kappa _y a_p}{4}\right)\Rightarrow & {} \, \frac{\kappa _y a_p}{4}\, \end{aligned}$$
(15)

Hence, the above introduced matrix elements can be rewritten in the following way

$$\begin{aligned} \mathbf {\alpha }_{if}(\mathbf {r}_{\perp }) = - \left( \frac{\Omega _{if}}{c}\kappa _y,\, \frac{\Omega _{if}}{c}\kappa _x,\, -\beta \kappa _x \kappa _y\right) \langle x y\rangle _{if}\, \end{aligned}$$
(16)

with

$$\begin{aligned} \langle x y\rangle _{if}&= \int \phi ^{*}_f(x,y) x y \phi _i(x,y) dx dy\, \nonumber \\&=-C_{i,i_z}^{m_i,n_i}C_{f,f_z}^{m_f,n_f}\frac{(-1)^{m_f+m_i+n_f+n_i}a_p^2}{16 \pi ^2 (m_f-m_i)(n_f-n_i)} \end{aligned}$$
(17)

Finally, substituting in the formula (7) for the matrix element of CR probability \(M_{if}\) the polarisation vector \({\varvec{\epsilon }}_{\varvec{\kappa }}\) from Eq. (11) and the transition matrix element \(\alpha _{if}\) from Eq. (16), we can get the square of the scalar function \((\mathbf {\alpha }_{if}\cdot {\varvec{\epsilon }}_{\varvec{\kappa }}^{*})\)

$$\begin{aligned} \mid (\mathbf {\alpha }_{if}\cdot {\varvec{\epsilon }}_{\varvec{\kappa }}^{*})\mid ^2\; =\; \frac{\omega ^2 \langle x y\rangle _{if}^2}{8 c^2} F({ \Theta }, { \Phi }, \omega ) \end{aligned}$$
(18)

with the angular function F

$$\begin{aligned}&F({ \Theta }, { \Phi }, \omega ) = \sin ^2{ \Theta } \Big (\beta ^2\omega ^2\sin ^4{ \Theta }\sin ^2(2{ \Phi }) \nonumber \\&\quad +\, \Omega _{if}^2(3 + 2\sin ^2{ \Theta }\cos (4{ \Phi }) + \cos (2\Theta ))\nonumber \\&\quad - \, 4\beta \Omega _{if}\omega \cos { \Theta }\sin ^2{ \Theta }\sin ^2(2{ \Phi })\Big ) \end{aligned}$$
(19)

4 Angular and spectral distributions

As a result of performed calculations, for the radiation probability (1) we can use the following expression

$$\begin{aligned} dw_{if} = \frac{\alpha \hbar }{2\pi } \mid (\mathbf {\alpha }_{if}\cdot {\varvec{\epsilon }}_{\varvec{\kappa }}^{*})\mid ^2 \delta (\Delta E_{if} - \hbar \omega )\omega d\omega \, , \end{aligned}$$
(20)

where \(\Delta E_{if} \simeq \Omega _{if} + \beta \omega \cos { \Theta }\).

After integration of \(dI_{if} = \hbar \omega dw_{if}\) over the CR photon frequencies \(\omega \), we get the angular distribution of CR by axially channeled electrons

$$\begin{aligned} \frac{dI}{do} = \sum _{i,f}P_{\vartheta ,i_z}\frac{\alpha \hbar \, \Omega _{if}^6 \langle x y\rangle _{if}^2}{16 c^2\pi } F_{ang}({ \Theta }, \Phi )\, \end{aligned}$$
(21)

where

$$\begin{aligned}&F_{ang}({ \Theta }, { \Phi }) = \frac{\sin ^2{ \Theta }}{(1 - \beta \cos { \Theta })^7} \nonumber \\&\quad \times \Bigg [(3 + \cos 2{ \Theta } + 2\sin ^2{ \Theta }\cos 4{ \Phi }(1 - \beta \cos \Theta )^2\nonumber \\&\quad + \beta \Bigg (\beta -4(1 - \beta \cos { \Theta }) \frac{\cot { \Theta }}{\sin { \Theta }^2}\Bigg )\sin { \Theta }^4\sin 2{ \Phi }^2\Bigg ] \end{aligned}$$
(22)
Fig. 1
figure 1

CR angular distribution for 10 MeV electrons channeled in Si \(\langle 100 \rangle \) crystal (the arbitrary units have been used)

Full size image

Here we have taken into account the initial population \(P_{\vartheta ,i_z}\) of the ith energy band of the transverse electron motion (\(\vartheta \) is the angle of electron momentum relative to the crystal axes). All further numerical calculations were carried out for \(\vartheta = {\vartheta _c}/5\), where \(\vartheta _c\) is the critical channeling angle. It is important to underline that the populations of the above-barrier levels, which describe a quasi-channeling, is much smaller than those of the sub-barrier ones. Therefore, the contribution of these levels to the total radiation intensity is extremely small.

Figure 1 shows in arbitrary units an example of the CR angular distribution calculated for 10 MeV electrons channeled along the \( \langle 100 \rangle \) axes of a Si crystalFootnote 1. This distribution exhibits an emphasised structural dependence, which is tightly correlated to the crystal \(\langle 100 \rangle \) orientation and does not accordingly reveal an absolute maximum at the forward direction. For clarity, Fig. 2 shows a section of the angular distribution by the plane indicated in Fig. 1.

Fig. 2
figure 2

Arbitrary contour (level) plot of the CR angular distribution for 10 MeV electrons channeled in Si \(\langle 100 \rangle \) in a randomly selected transverse plane shown in Fig. 1

Full size image

Further integration of \(dI_{if} = \hbar \omega dw_{if}\) over allowed \({ \Theta }\) and \({ \Phi }\) angles results in the formula for spectral distribution of CR by axially channeled electrons

$$\begin{aligned} \frac{dI}{d\omega }= & {} \sum _{i,f}P_{\vartheta ,i_z}\frac{\alpha \hbar \, \langle x y\rangle _{if}^2}{16 c^4\beta ^5} F_{sp}\Bigg (\frac{\omega - \Omega _{if}}{\beta \omega }\Bigg ) \nonumber \\&\times \mathcal {H}(\omega _1) \mathcal {H}(\omega _2)\,, \end{aligned}$$
(23)

where \(\mathcal {H}(...)\) is the Heaviside function with the following arguments

$$\begin{aligned} \omega _1 = \omega -\frac{\Omega _{if}}{1 + \beta }\; ,\; \; \omega _2 = \frac{\Omega _{if}}{1 - \beta }-\omega \end{aligned}$$
(24)

and

$$\begin{aligned} F_{sp}(\omega )&= (\omega (1 + \beta ) - \Omega _{if})(\Omega _{if} - (1-\beta )\omega ) \nonumber \\&\quad \times (1 - \beta ^2)^2\omega ^4 + 2(3\beta - 1)\omega ^2\Omega _{if}^2 + \Omega _{if}^4\, \end{aligned}$$
(25)
Fig. 3
figure 3

CR spectral distribution for 10 MeV electrons axially channeled in Si \(\langle 100 \rangle \) (the intensity is given in arbitrary units)

Full size image

An example of the total spectral distribution of CR calculated as the sum over all transitions between the energy bands of the transverse motion for 10 MeV electrons channeled in Si \(\langle 100 \rangle \) is shown in Fig. 3. The calculations demonstrate that, in contrast to the CR spectrum at planar channeling, the emission at axial channeling along the Si \(\langle 100 \rangle \) axis give out a smooth function up to \(\sim \)33 keV energy. This is might be a consequence of the much greater depth of the potential well of the crystal axes in comparison to that of the planes. The maximum radiation intensity is predicted at the photon energy \(\sim \)26.8 keV.

5 Axial CR polarisation

The dependence of the degree of polarisation CR on the inlet direction of relativistic electrons into the Si crystal relative to the \(\langle 100 \rangle \) axis was calculated in [35] at the axial channeling regime. A similar method has been applied to evaluate both the degree of polarisation and the angular directions of polarisation for a new type of radiation, i.e. diffracted channeling radiation (DCR), as a function of the emission angles [36].Footnote 2

Let’s present the circular polarisation vector \(\beta _\Lambda \) of a photon with the helicity \(\Lambda \) as

$$\begin{aligned} \beta _\Lambda = \frac{1}{\sqrt{2}}(\beta _2+\mathbf {i}\Lambda \beta _3)\, , \end{aligned}$$
(26)

where \(\beta _2\) and \(\beta _3\) are the vectors of the photon linear polarisation [32]. Then, for the radiation intensity \(I_\Lambda \) with the polarisation \(\beta _\Lambda \) proportional to the square of the matrix element of the transition probability \(i \rightarrow f\), i.e. \(I_\Lambda \propto |\alpha _{if}\beta _\Lambda |^2\), we obtain

$$\begin{aligned} I_\Lambda \propto \frac{1}{2} \left( (\alpha _{if}\beta _2)^2 + (\alpha _{if}\beta _3)^2\right) \end{aligned}$$
(27)

Thus, the radiation intensity does not depend on the photon helicity \(\Lambda \), and we defiine a real value of \(\alpha _{if}\) for CR in the case of axial channeling. This means that such radiation cannot disclose a circular polarisation [37] but only a linear one with the degree of polarisation defined as follows

$$\begin{aligned} P = \frac{|Me_1|^2 - |Me_2|^2}{|Me_1|^2 + |Me_2|^2}\,, \end{aligned}$$
(28)

where \(e_1\) and \(e_2\) are the components of the polarisation vector \(\beta _\Lambda \) perpendicular to the unit vector \(\varvec{\kappa }\) in the direction of photon emission. As known [32], the vectors of the photon linear polarisation \(\beta _2\) and \(\beta _3\) satisfy this condition by default.

Taking into account that in Eq. (27)

$$\begin{aligned} \alpha _{if}\beta _2 = \sqrt{2}\,\mathfrak {R}(\alpha _{if}\beta _\Lambda ),\,\,\alpha _{if}\beta _3 = \sqrt{2}\Lambda \,\mathfrak {I}(\alpha _{if}\beta _\Lambda ), \end{aligned}$$
(29)

where \(\mathfrak {R}(\alpha _{if}{\varvec{\epsilon }}_{\varvec{\kappa }}^{*})\) and \(\mathfrak {I}(\alpha _{if}{\varvec{\epsilon }}_{\varvec{\kappa }}^{*})\) are the real and imaginary parts of the complex function \((\alpha _{if}{\varvec{\epsilon }}_{\varvec{\kappa }}^{*})\), we reduce the formulas of the degree of polarisation for CR photons at axial channeling

$$\begin{aligned} P_{CR} = \frac{\mid \mathfrak {R}(\alpha _{if}\cdot {\varvec{\epsilon }}_{\varvec{\kappa }}^{*})\mid ^2 - \mid \mathfrak {I}(\alpha _{if}\cdot {\varvec{\epsilon }}_{\varvec{\kappa }}^{*})\mid ^2}{\mid \mathfrak {R}(\alpha _{if}\cdot {\varvec{\epsilon }}_{\varvec{\kappa }}^{*})\mid ^2 + \mid \mathfrak {I}(\alpha _{if}\cdot {\varvec{\epsilon }}_{\varvec{\kappa }}^{*})\mid ^2}\,, \end{aligned}$$
(30)

and the polarisation directions

$$\begin{aligned} \tan \phi _{CR} = \Lambda \frac{\mathfrak {I}(\alpha _{if}\cdot {\varvec{\epsilon }}_{\varvec{\kappa }}^{*})}{\mathfrak {R}(\alpha _{if}\cdot {\varvec{\epsilon }}_{\varvec{\kappa }}^{*})} \end{aligned}$$
(31)

Successfully, using the matrix element for the transition probability \(M_{if}\) from Eq. (12) and Eqs. (21), (28) we can calculate the degree and directions of polarisation for CR by axially channeled relativistic electrons. For the degree of polarisation we obtain

$$\begin{aligned}&P_{CR} \nonumber \\&\quad =\frac{\sin ^2 2\Phi (3+\beta (\cos 2\Theta -4\cos \Theta )^2-16\cos ^2 2\Phi (1-\beta \cos \Theta )^2}{\sin ^2 2\Phi (3+\beta (\cos 2\Theta -4\cos \Theta )^2+16\cos ^2 2\Phi (1-\beta \cos \Theta )^2},\nonumber \\ \end{aligned}$$
(32)

while for the directions of CR polarisation -

$$\begin{aligned} \phi _{CR} = \arctan \left( \frac{4(\beta \cos \Theta -1)\cot 2\Phi }{\Lambda (\beta (\cos 2\Theta +3)-4\cos \Theta )}\right) \end{aligned}$$
(33)

The angular dependence of the degree of polarisation calculated by Eq. (32) is presented in Fig. 4.

Fig. 4
figure 4

Arbitrary contour plot of the degree of polarisation for CR by 10 MeV electrons channeled along Si \(\langle 100 \rangle \) for a selected transverse plane shown in Fig. 1

Full size image

Comparison of Figs. 2 and 4 indicates that CR exhibits a degree of polarisation close to the maximum \(P_{CR} = -1\) in the regions of maximum intensity.

The map of the angular directions of polarisation CR generated by 10 MeV electrons channeled along Si \(\langle 100 \rangle \) axes has been drawn according the formula (33) and shown in Fig 5. The calculated scheme points out that the maximum radiation intensities can be observed when the CR polarisation planes are directed along the y-axis.

Fig. 5
figure 5

The map of angular directions of polarisation CR for photons with positive helicity \(\Lambda = 1\) (for \(\Lambda = -1\) the directions will be opposite) generated by 10 MeV electrons channeled along the \(\langle 100 \rangle \) axes in Si for selected transverse plane is shown in Fig. 1

Full size image

One can underline that, using the equation \(\Lambda ^2~=~1\), the scalar function \((\mathbf {\alpha }_{if}{\varvec{\epsilon }}_{\varvec{\kappa }}^{*})\) can be written in the form

$$\begin{aligned} (\mathbf {\alpha }_{if}{\varvec{\epsilon }}_{\varvec{\kappa }}^{*}) = e^{\mathbf {i} \varphi } \sqrt{\mathfrak {R}(\alpha _{if}{\varvec{\epsilon }}_{\varvec{\kappa }}^{*})^2 + \mathfrak {I}(\alpha _{if}{\varvec{\epsilon }}_{\varvec{\kappa }}^{*})^2}\, \, , \end{aligned}$$
(34)

where \(\varphi \) is the main value of the function

$$\begin{aligned} \varphi = \arctan {\left( \Lambda \frac{\mathfrak {I}(\alpha _{if}\cdot {\varvec{\epsilon }}_{\varvec{\kappa }}^{*})}{\mathfrak {R}(\alpha _{if}\cdot {\varvec{\epsilon }}_{\varvec{\kappa }}^{*})}\right) }\,, \end{aligned}$$
(35)

that allows the emission polarisation to be analysed in a typical way. Indeed, in this case the emission intensity does not depend on the emitted photons helicity \(\Lambda \). The latter essentially simplifies its calculation.

6 Conclusions

Since the first works dedicated to the interaction of light relativistic particles in aligned crystals, the influence of the crystal structure on the interaction potential has been mostly studied in order to define various scattering mechanisms of the beams in crystals. These results obtained within the classical and quantum approaches allowed the life-time in the channeling bound stateFootnote 3 to be evaluated with a scope to calculate the collimation or radiation abilities of a beam-aligned crystal system (see, for instance, in [38]). For such calculations carried out for highly relativistic energies, in general, a complex field structure revealed at the fine diffraction pictures of particles scattering in crystals is not of a decisive importance. It is even less weight for the case of planar channeling, while at axial channeling we can observe very reach schemes of particles angular scattering. The latter should essentially contribute to the formation of a structured angular distribution of channeled particles for various quantum levels of the transverse bound motion that can be registered as fine angular and polarisation peculiarities in CR at axial channeling of MeV electrons.

In our work the use of the Bloch wave function formalism applied to the transverse motion of channeled particles has permitted performing analytical calculations of CR at axial channeling of relativistic electrons in thin crystals taking into account a strong angular redistribution of the beamFootnote 4. The method has become rather precise being applied for the interaction potential formed by the crystal axes constructed by fitting the measured electron form-factors.

Evaluating the possibility of observing the features of the angular distribution of channeled radiation calculated by us, it should be noted that it is usually believed that for low-energy particles the angular distributions of radiation will be quickly smeared by above-barrier particles arising due to incoherent scattering. An estimate of the dechanneling length for 10 MeV electrons at axial channeling in Si \(\langle 100 \rangle \) gives the value \(L_d\) = 0.215 \(\upmu \)m (see the expression in [18]). Obviously, this is rather short distance for 10 MeV electrons. However, during the time of flight of even this distance, electrons in any case generate CR, which can be registered experimentally; the emission of dechanneled electrons in the direction of the channeling axes is essentially suppressed. As seen from our expressions, the angular distribution profile is prescribed by the function \(F_{ang}({ \Theta }, \Phi )\). The intensity of CR can be increased if to use the electrons of higher energies, which are characterised by larger dechanneling length; for instance, we get \(L_d\) = 1.1 \(\upmu \)m for 50 MeV electrons channeled in Si \(\langle 100 \rangle \). As seen, even for a bit higher electron energy we deal with very thin crystals (\(\sim 1\, \upmu \)m) that proves the feasibility of recording the picture similar of one presented in Fig. 1.

Reported calculations based on the newly deduced formulas for the radiation intensity show that the CR angular distributions have a much more complex structure in comparison with the results obtained in the approximation of a single crystal axes. This structure is associated with the crystal symmetry in general and in the transverse plane as well. So, for example, for a Si \(\langle 100 \rangle \) crystal, four peaks should be observed in the angular distributions, located symmetrically about the crystal axis. The CR angular distribution geometry uniquely demonstrates an explicit dependence of the CR probability on the azimuthal angle.

The radiation angular distribution in the case of axial channeling has no axial-symmetry in the plane perpendicular to the crystal axis. Axially symmetric angular distribution is typically obtained in a single-string approximation (no interference between separated axes), while the influence of adjacent axes leads to a more complex structure. The estimates can be performed in dipole approximation. However, the Bloch-function technique enables getting precise analytical expressions for the matrix elements of CR. But, it should be underlined that the obtained matrix elements contain additional components for the wave vector of CR-photons, which make extra extended the analytical expressions and, in turn, significantly increase the determination time for the radiation distribution.

A fundamentally new result of our work points out that the radiation itself in the case of axial channeling in a crystal does not exhibit circular polarisation, although the polarisation of each individual photon is circular. Accordingly, the tilt angles of the polarisation planes depend on the photons helicity. It is also notable that even the numerical results are obtained for 10 MeV electrons, our conclusions should be valid for any crystals and particles at a wide range of energies (up to \(\sim \)1 GeV).