Crystalline Undulators

Abstract

The basic idea of a Crystalline Undulator (CU) is formulated, and the features that distinguish this device from conventional undulators based on the action of the magnetic field are stressed. Results of numerical calculations of the spectra of electromagnetic radiation emitted by ultra-relativistic electrons and positrons in periodically bent oriented crystals are presented. The interplay of different motions of the channeling particles, including the channeling oscillations and the undulator motion, in forming the emission spectrum is discussed. The results presented refer to two different schemes of the periodic bending: (i) the Large-Amplitude Long-Period, $d \ll a \ll \lambda _{\mathrm{u}}$, and (ii) the Small-Amplitude Short-Period, $d > a$. Wherever available the results of numerical simulations are compared with the experimental data or/and the results of calculations carried out by accompanying propagation of ultra-relativistic projectiles. In an exemplary case study, the brilliance of radiation emitted in a CU-LS by available positron beams is estimated. Intensity of CU radiation in the photon energy range $10^0-10^1$ MeV, which is inaccessible to conventional synchrotrons, undulators and XFELs, greatly exceeds that of laser-Compton scattering LSs and can be higher than predicted in the Gamma Factory proposal to CERN. The result of estimations is confirmed by means of accurate numerical simulations.

Appendix: Continuous Potential and Transverse Motion in a SASP Crystal

In this supplementary section, explicit formulae are derived that describe non-periodic and periodic parts of the continuous planar potential in a SASP bent crystal. The analytical and numerical analysis of the results obtained allow us to qualitatively explain the peculiar features in the motion of ultra-relativistic projectiles as well as in the radiative spectra. 

6.1.1 Continuous Potential

Consider a crystallographic plane which coincides with the (xz) Cartesian plane. For the sake of clarity, let us introduce a cosine periodic bending, $a\cos ({k_{\mathrm{u}}}z)$ with ${k_{\mathrm{u}}}=2\pi /\lambda _{\mathrm{u}}$, of the plane in the transverse y direction. The bending amplitude a and period $\lambda _{\mathrm{u}}$ satisfy the SASP bending condition

$$\begin{aligned} a < d \ll \lambda _{\mathrm{u}}, \end{aligned}$$

(6.11)

where d stands for the interplanar distance.

Similar to the procedure used for a straight plane (see Sect. 4.5.3), the continuous potential of a periodically bent plane one obtains summing up the potentials of individual atoms assuming that the latter is distributed uniformly along the plane:

$$\begin{aligned} \mathcal{U}_{\mathrm{pl}}(y,z) = \mathcal{N}\int w(\varDelta )\, \mathrm{d}\boldsymbol{\varDelta } \int \limits _{-\infty }^{\infty } \mathrm{d}z^{\prime } \int \limits _{-\infty }^{\infty } \mathrm{d}x^{\prime } \, U_{\mathrm{at}}(|\mathbf{r}-\boldsymbol{\varDelta } |)\,. \end{aligned}$$

(6.12)

Vector $\boldsymbol{\varDelta }$ stands for the displacement of an atom from its equilibrium position, characterized by the coordinates $(x^{\prime }, y^{\prime }, z^{\prime })$ with with $y^{\prime }=a\cos {k_{\mathrm{u}}}z^{\prime }$ (see illustrative Fig. 6.25) due to thermal vibrations.

Fig. 6.25

Supplementary figure illustrating the derivation of the continuous potential of a periodically bent crystallographic plane (the thick curve represents the bending profile). The atoms are displaced randomly from their equilibrium positions $(x^{\prime },y^{\prime },z^{\prime })$ due to to thermal vibrations (the vector $\boldsymbol{\varDelta }$ shows the position of a displaced atom “A”)

Expressing $U_{\mathrm{at}}$ in terms of its Fourier transform and using (4.28) one integrates over $x^{\prime }, z^{\prime }, \boldsymbol{\varDelta }$ and presents the planar potential in the form of a series:

$$\begin{aligned} \mathcal{U}_{\mathrm{pl}}(y,z)&= V_0(y; a) + \sum _{n=1}^{\infty } \cos (n{k_{\mathrm{u}}}z)\, V_n(y;a) \end{aligned}$$

(6.13)

with

$$\begin{aligned} V_0(y;a)&= {\mathcal{N}\over \pi } \int \limits _{0}^{\infty } \mathrm{d}q \, \mathrm{e}^{-{q^2u_T^2 \over 2}}\, \cos (q y) \,J_0(q a) \widetilde{U}_{\mathrm{at}}(q) \end{aligned}$$

(6.14)

$$\begin{aligned} V_n(y;a) = {2 \mathcal{N}\over \pi } \int \limits _{0}^{\infty } \mathrm{d}q \, \mathrm{e}^{-{Q_n^2 u_T^2 \over 2}} J_n(q a)\, \widetilde{U}_{\mathrm{at}}\left( Q_n\right) \times \left\{ \begin{array}{l} \displaystyle (-1)^{n\over 2} \cos (q y)\\ \displaystyle (-1)^{n-1\over 2} \sin (q y) \end{array} \right. \end{aligned}$$

(6.15)

where the short-hand notation $Q_n^2=q^2 + (n{k_{\mathrm{u}}})^2$ is used. The upper line stands for even n values, the lower line—for the odd ones.

In the limit of a straight channel, $a=0$, the right-hand side of Eq. (6.13) reduces to that in (4.49). Indeed, taking into account that $J_0(0)=0$ and $J_n(0)=0$ for $n>0$, one notices that this leads to $V_{n}(y;0)\equiv 0$ for all terms defined by (6.15) whereas the term $V_{0}(y;0)$, Eq. (6.14), reduces to (6.13).

To determine the interplanar potential U(y, z), one uses Eq. (4.54) where the potentials $\mathcal{U}_{\mathrm{pl}}(y,z)$ of individual planes are to be inserted.

The integrals on the right-hand sides of Eqs. (6.14) and (6.15) can be evaluated explicitly for a number of analytic approximations for $U_{\mathrm{at}}$ which can be found in literature [9, 78,79,80,81,82,83]. For reference purposes, we present the explicit formulae derived within the framework of the Molière approximation (4.43):

$$\begin{aligned} V_n(y;a)&= (1+\delta _{n0}) \mathcal{N}Z e \sum _{i=1}^3 {\alpha _j \over \varGamma _{nj}}\, \mathrm{e}^{\gamma _{j}^2 u_T^2\over 2} \, \mathcal{T}_n(y;a,\varGamma _{nj}) \end{aligned}$$

(6.16)

Here $\delta _{n0}$ is the Kronecker symbol, $\varGamma _{nj}=\left( \gamma _j^2+(n{k_{\mathrm{u}}})^2\right) ^{1/2}$, $\mathcal{T}_n$ stands for the integral:

$$\begin{aligned} \mathcal{T}_n(y;a,\varGamma ) = \int \limits _0^{\pi /2} \mathrm{d}\theta \cos (n\theta ) \Bigl ( \mathcal{F}(y - a \cos \theta ;\varGamma ) + (-1)^n \mathcal{F}(y + a \cos \theta ;\varGamma ) \Bigr ) \end{aligned}$$

(6.17)

where

$$\begin{aligned} \mathcal{F}(Y;\varGamma ,u_T) = F(Y; \varGamma ,u_T) + F(-Y; \varGamma ,u_T) \end{aligned}$$

(6.18)

with $F(\pm Y; \varGamma ,u_T)$ defined as in (4.51).

For $n=0$, Eq. (6.16) reproduces the expression derived in Ref. [48].

Similar to the case of a straight crystal, the interplanar potential U(y, z) in a SASP bent crystal is obtained by summing up the potentials (6.13) of individual planes. For the electron channel, the result can be written in the form

$$\begin{aligned} U(y,z) = \sum _{n=0}^{\infty } \cos (n{k_{\mathrm{u}}}z)\, U_n(y) \end{aligned}$$

(6.19)

where

$$\begin{aligned} U_n(y) = V_n(y) + \sum _{k=1}^{K_{\max }} \Bigl ( V_n(y+kd) + V_n(y-kd) \Bigr ) + C_n\,. \end{aligned}$$

(6.20)

Here, y is the transverse coordinate with respect to an arbitrary selected reference plane, and the sum describes a balanced contribution from the neighboring planes. The constants $C_n$ can be chosen to satisfy the condition $U_n(0)=0$.

For positrons, the interplanar potential can be obtained from Eq. (6.20) by reversing the signs of the planar potentials and selecting the constants $C_n$ to ensure $V_n(\pm d/2)=0$. Similar summation schemes allow one to calculate the charge densities, nuclear and electronic, across the periodically bent channels.

6.1.2 Transverse Motion in a SASP Channel

The function y(t) describes the transverse motion of a particle with respect to the centerline of the channel. The equation of motion (EM) reads

$$\begin{aligned} \ddot{y} = - {1 \over m \gamma }\, {\partial U(y,z) \over \partial y}\,. \end{aligned}$$

(6.21)

In what follows we outline a perturbative solution of the EM.

Assuming the longitudinal coordinate z changes linearly with time, $z\approx ct$, one re-writes the potential (6.19) substituting z with ct. Then, the EM is written as follows:

$$\begin{aligned} \ddot{y} = {1 \over m \gamma } \left( f_0(y) + \sum _{n=1}^{\infty } \cos (n{\varOmega _{\mathrm{u}}}t)\, f_n(y) \right) \end{aligned}$$

(6.22)

with

$$\begin{aligned} {\varOmega _{\mathrm{u}}}= {k_{\mathrm{u}}}c = {2\pi c \over \lambda _{\mathrm{u}}}, \quad f_{0}(y) = - {\mathrm{d}U_{0} \over \mathrm{d}y}, \quad f_{n}(y) = - {\mathrm{d}U_{n} \over \mathrm{d}y}\,. \end{aligned}$$

(6.23)

The action of the time-independent force $f_{0}$ results in the channeling oscillations. At the same time, the projectile experiences local small-amplitude oscillations (the jitter-like motion) due to the driving forces $f_{n}\cos (n{\varOmega _{\mathrm{u}}}t)$ ($n=1,2,\dots$). In a SASP channel, the frequency ${\varOmega _{\mathrm{u}}}$ (and, respectively, its higher harmonics $n{\varOmega _{\mathrm{u}}}$) exceeds greatly the frequency of the channeling oscillations: ${\varOmega _{\mathrm{u}}}\gg \varOmega _{\mathrm{ch}}$. As a result, the EM (6.22) can be integrated following the perturbative procedure outlined in Ref. [84], Sect. 30, for the motion in a rapidly oscillating field. Namely, y(t) is represented as a sum

$$\begin{aligned} y(t) = Y(t) + \varXi (t) \end{aligned}$$

(6.24)

where $\varXi (t)$ stands for a small (but rapidly oscillating) correction to the smooth dependence Y(t) which describes the channeling oscillations. The function $\varXi (t)$ satisfies the equation in which the coordinate Y is treated as a parameter:

$$\begin{aligned} \ddot{\varXi } = \sum _{n=1}^{\infty } \cos (n{\varOmega _{\mathrm{u}}}t)\, {f_n(Y)\over m \gamma }\,. \end{aligned}$$

(6.25)

Its solution reads

$$\begin{aligned} \varXi (t, Y) = \sum _{n=1}^{\infty } \xi _n(Y) \cos (n {\varOmega _{\mathrm{u}}}t)\,. \end{aligned}$$

(6.26)

where

$$\begin{aligned} \xi _n(Y) = - {1 \over m \gamma \varOmega _{\mathrm{u}}^2} {f_n(Y) \over n^2} \end{aligned}$$

(6.27)

The presence of the term $\varXi (t, Y)$ modifies the EM for Y(t). In addition to the force $f_0=-\mathrm{d}U_0(Y)/\mathrm{d}Y$ due to the static potential, a ponderomotive force $f_{\mathrm{pond}}$ appears. It can be calculated as follows [84] (below, the overline denotes averaging over the period $2\pi /{\varOmega _{\mathrm{u}}}$ which is much smaller than the characteristic time of the channeling motion and, thus, does not affect the value of Y(t)):

$$\begin{aligned} f_{\mathrm{pond}}(Y)&= \overline{ \varXi (t, Y) \sum _{n=1}^{\infty } \cos (n{\varOmega _{\mathrm{u}}}t)\, {\mathrm{d}f_n(Y) \over \mathrm{d}Y} } = - {\mathrm{d}U_{\mathrm{pond}}\over \mathrm{d}Y} \end{aligned}$$

(6.28)

The ponderomotive potential, $U_{\mathrm{pond}}$, introduced here is defined as follows:

$$\begin{aligned} U_{\mathrm{pond}}(Y) = {\lambda _{\mathrm{u}}^2 \over 16\pi ^2 \varepsilon } \sum _{n=1}^{\infty } {f_n^2(Y) \over n^2} \end{aligned}$$

(6.29)

Note that the ponderomotive correction to the potential becomes smaller as the energy increases since $U_{\mathrm{pond}}\propto 1/\varepsilon$.

Therefore, the channeling oscillations $Y=Y(t)$ are described by the EM

$$\begin{aligned} \ddot{Y} = - {1 \over m \gamma } {\mathrm{d}U_{\mathrm{eff}} \over \mathrm{d}Y} \end{aligned}$$

(6.30)

where the total effective potential reads

$$\begin{aligned} U_{\mathrm{eff}}(Y) = U_0(Y) + U_{\mathrm{pond}}(Y)\,. \end{aligned}$$

(6.31)

The non-periodic potential $U_0$ and the ponderomotive term $U_{\mathrm{pond}}$ calculated within the Molière approximation for positron SASP Si(110) channel are presented in Fig. 6.26. The curves correspond to different bending amplitude as indicated. The right panel shows the dependence of the product $\varepsilon U_{\mathrm{pond}}$ (with $\varepsilon$ measured in GeV) which is independent on the projectile energy. In both panels, the vertical lines mark the (110)-planes in the straight crystal. 

The modification of $U_0$ with increase of the bending amplitude is clearly seen on the left panel in Fig. 6.26. In detail, this issue was discussed in Ref. [48]. Here, for the sake of consistency, we mention several features relevant to the topic of the current paper. For small and moderate amplitude values, $a \le 0.4$ Å, the major change in the potential is the decrease of the interplanar potential barrier. As the a values approach the $0.4 \dots 0.6$ Å range, the volume density of atoms becomes more friable leading to flattening of the potential maximum. For larger amplitudes, the potential changes in a more dramatic way as additional potential well appears. In the figure, this feature is clearly seen in the behavior of the the $U_0$ curve for $a=0.8$ Å: in addition to the “regular” channels centered at the midplanes, i.e., at $y/d=\dots , -1,0,1,\dots$, “complementary” channels appear centered at $y/d=\dots , -0.5, 0.5, \dots$. As a result, a positron can experience channeling oscillations moving in the channels of the two different types. Similar feature characterizes the electron channeling phenomenon at sufficiently large values of bending amplitude [48].

Fig. 6.26

Left. The non-periodic part $U_0(y)$ of the continuous interplanar potential for positrons in Si(110) calculated at different values of bending amplitude indicated in Å near the curves ($a=0$ stands for the straight crystal). Right. The ponderomotive correction $U_{\mathrm{pond}}$ multiplied by $\varepsilon$ in GeV calculated for various a and for fixed bending period $\lambda _{\mathrm{u}}=308$ microns. The potentials shown are evaluated for temperature 300 K by using the Molière atomic potentials. The vertical dashed lines mark the adjacent (110) planes in straight crystal (the interplanar spacing is $d=1.92$ Å)

Fig. 6.27

The effective interplanar potential $U_{\mathrm{eff}}$, Eq. (6.31), (solid lines) for a 250 MeV positron in Si(110) at different values of bending amplitude indicated in Å near the curves ($a=0$ stands for the straight crystal). The dashed curves show the continuous potential $U_{\mathrm{0}}$ without the ponderomotive corrections. The bending period is $\lambda _{\mathrm{u}}=308$ microns

Comparing the absolute values of the ponderomotive term $U_{\mathrm{pond}}$ (curves in the right panel of Fig. 6.26 correspond to the terms at $\varepsilon =1$ GeV) and those of $U_0$ one can state that for all bending amplitudes $U_{\mathrm{pond}}$ is a negligibly small correction to $U_0$ for the projectile energies above 1 GeV. For much lower energies, say for a few hundreds-MeV, the contribution of $U_{\mathrm{pond}}$ becomes more noticeable, reaching the eV range in the regions $2|y|/d \approx \dots , -1,0,1,\dots$. Thus, it should be accounted if an accurate integration of the EM (6.30) is desired. Figure 6.27 illustrates the change in the continuous interplanar Si(110) potential due to the ponderomotive correction. Solid curves who the corrected potentials, the dashed ones—the term $U_0$. The data refer to 250 MeV positron channeling in Si(110) bent periodically with the period $\lambda _{\mathrm{u}}=308$ microns; the values of bending amplitude (in Å) are indicated in the figure.