The interaction of relativistic particles with strong crystalline fields

Crystals present a uniquely simple environment for the investigation of strong electromagnetic fields. When energetic charged particles are incident on crystals close to major crystallographic directions, their electromagnetic interactions depend crucially on the kinematic conditions. The coherence of the crystalline field can produce very strong electric fields in the rest frame of the particle, exceeding the so-called Schwinger field or quantum critical field. In that domain, the radiation emission takes a substantial part of the electron energy and the “formation zone” changes character. In this review the theory appropriate to the different kinematics domains is described, concentrating on the effects occurring at extreme fields. Properties discussed include strong field synchrotron radiation, channeling radiation, bremsstrahlung, and photon interactions. Applications are given to radiation sources, bending of particle beams, and sources of polarized GeV photons.

Figure 1

A primitive cubic lattice with the main crystallographic axes indicated.

Figure 2

Synchrotron-radiation emission by an energetic electron traversing a magnetic field $B$. The typical emission angle $1∕\gamma$ makes photon emission from any point within the arc length from $a$ to $b$ indistinguishable by a detector. Therefore the distance $ab$ represents the formation length.

Figure 3

A schematic diagram showing the photon emission by an energetic lepton in the nuclear field. Angles are exaggerated for clarity.

Figure 4

A schematic drawing of the discrete nature of the scattering centers in a crystal and the resulting continuum approximation. The target atoms with atomic number $Z_2$ and distance $d$ along the string impose a curved trajectory on the penetrating particle with atomic number $Z_1$ through binary encounters over the transverse distance $r_{\perp }$. The resulting trajectory with entrance angle $\psi$ can be accurately described as if being the result of an interaction with a string of continuous charge distribution, i.e., the charges $Z_{2}e$ being “smeared” along the direction of motion $z$.

Figure 5

A channeled negatively charged particle moves near (a) an axis and (b) a plane. From 235.

Figure 6

(Color online) An example of a transverse potential in the continuum approximation for diamond along the ⟨110⟩ axis at room temperature. The main regions for channeled $e^{-}$ and $e^{+}$ are indicated. The Doyle-Turner approximation for the atomic potential has been used.

Figure 7

(Color online) Doughnut formation for $150\text{−}\mathrm{GeV}$ electrons after $\left(e^{-}\right)$ and during $\left(\gamma \right)$ their photon emission. The selected incident-polar-angle range is given below each plot of vertical vs horizontal exit angle, ${\theta }_{\mathrm{x},\mathrm{out}}\times {\theta }_{\mathrm{y},\mathrm{out}}∊[-150,150\mu \mathrm{rad}]\times [-150,150\mu \mathrm{rad}]$. The radiated energy is restricted to the interval $10–70\mathrm{GeV}$. Adapted from 137.

Figure 8

Illustration of Lorentz transformations between the laboratory and the rest frame for a radiating planar-channeled positron with $\gamma =3$. From 10.

Figure 9

High-energy channeling radiation for $6.7\mathrm{GeV}∕c$ positrons incident parallel to the (110) plane in a silicon crystal. The experimental result shown by open dots is compared to a calculated spectrum shown by a solid line. The calculation is on an absolute scale and enabled an identification of the erroneous assignment of the beam energy of $7\mathrm{GeV}$ such that the points are slightly modified compared to the original data (53). From 212.

Figure 10

An illustration of a section of the “Überall pancake” (hatched area, curvature neglected). The main part of the emitted radiation comes from reciprocal-lattice points for which the projection on the momentum vector ${\vec{p}}_1$ lie in the range given by Eq. (30). The reciprocal lattice is spanned by the primitive reciprocal-lattice vectors ${\vec{b}}_i$. From 217.

Figure 11

Two particle trajectories in a crystal lattice. Track (a) gives rise to the emission of channeling radiation while (b) leads to coherent bremsstrahlung. From 12.

Figure 12

Radiation from $150\text{−}\mathrm{GeV}$ electrons incident on $0.6\text{−}\mathrm{mm}$ Si with different angular intervals to the (110) plane as indicated. The enhacement is with respect to an amorphous target of the same material and thickness. The polar angle to the ⟨110⟩ axis is $8.2\mathrm{mrad}$. From 172.

Figure 13

Synchrotron radiation in a strong field, where (a) the power spectrum is rescaled by $\chi$ according to Eq. (56) and (b) the contribution from the spin; see text for details. The labels on the curves denote the value of $\chi$ which are in both cases $\chi =5^{-1}$ (dashed), $\chi =1$ (solid), $\chi =5$ (dash-dotted), $\chi =5^2$ (long-dashed), and $\chi =5^3$ (long-dash-dotted). Adapted from 209.

Figure 14

(Color online) The intensity $I$ of synchrotron radiation as a function of $\chi$, normalized to the classical value $I_{\mathrm{cl}}$ obtained for $\chi \to 0$. The upper part shows the region $0⩽\chi ⩽1$ on a linear scale while the lower part shows the region $0.01⩽\chi ⩽100$ on a logarithmic scale.

Figure 15

Emission of bremsstrahlung by a charged particle crossing a string of atoms. In the upper part, the deflection due to the field from the string is sufficiently small such that coherent superposition can take place over many atoms—this is the limit with an angle $\psi >{\Theta }_0$ to the axis. The lower part reflects the increased deflection when incident with an angle $\psi <{\Theta }_0$ to the axis, which results in a shorter distance for the coherent superposition. From 208.

Figure 16

(Color online) Synchrotron radiation in a strong field, where the solid line is the total spectrum according to Eq. (56), the dash-dotted line shows the contribution from the spin, and the dashed line is the contribution neglecting the spin; see text for details. The value of $\chi$ is set to 100.

Figure 17

Pair-production probability as a function of energy by a photon incident along the ⟨111⟩ axis for C, Si, Fe, and W and along ⟨110⟩ for Ge. The temperature, if different from $293\mathrm{K}$, is given in the brackets. The values are according to the constant field approximation (CFA). Adapted from 38, 50.

Figure 18

Enhancement over random incidence for trident production of four different crystals. The electron energy is $50\mathrm{GeV}$ and the horizontal scale is the fractional energy taken by the positron. Adapted from 133.

Figure 19

The photon splitting probability $W_{\gamma \to \gamma \gamma }$ in W along the ⟨111⟩ axis as a function of photon energy. Curve 1 is $W_{\gamma \to \gamma \gamma }$ in the strong field at $77\mathrm{K}$, curve 2 is $W_{\gamma \to \gamma \gamma }$ in the strong field at $293\mathrm{K}$, and curves $1^{\prime }$ and $2^{\prime }$ denote the random (amorphous) value. Adapted from 52.

Figure 20

Relative enhancement of photons for $15\text{−}\mathrm{GeV}$ positrons directed along the ⟨100⟩ axis in diamond. The curves show calculated values based on the theory of 131 with $L$ as an effective path length in $\mu \mathrm{m}$. A classical spectrum is shown by the dot-dashed line (CL). Adapted from 98.

Figure 21

Enhancement of radiation yield for $150\text{−}\mathrm{GeV}$ electrons incident within $9\mu \mathrm{rad}$ to the ⟨110⟩ axis of a $0.2\text{−}\mathrm{mm}$ Ge crystal. The dots are experimental points and the line represents the calculated value based on the CFA. Adapted from 169.

Figure 22

Pair-production yield for photons incident along the ⟨110⟩ axis of a $1.4\text{−}\mathrm{mm}$ Ge crystal cooled to $100\mathrm{K}$ (74, 76). The constant field approximation is shown by the solid. Adapted from 50.

Figure 23

Enhancement as a function of angle of pair-production yield for photons incident along the ⟨110⟩ axis of a $1.4\text{−}\mathrm{mm}$ Ge crystal cooled to $100\mathrm{K}$ (74, 76). The energy intervals are as follows: line 5, $120–150\mathrm{GeV}$; line 4, $90–120\mathrm{GeV}$; line 3, $60–90\mathrm{GeV}$; line 2, $40–60\mathrm{GeV}$; and line 1, $22–40\mathrm{GeV}$. The lines are calculated values from 43. Adapted from 50.

Figure 24

A schematical drawing of the setup used in NA43. The abbreviations are B1, B2, B8, and Tr6, magnetic dipoles; DC1-6, drift chambers; Sc1-2, Sc4, Sc6, Sc9, Sc10b, Sc11, scintillator counters; AKS, “anti-K-short” scintillator counter; HeI-III, helium vessels; and SSD, solid-state detector.

Figure 25

(Color online) Enhancement as a function of energy for an incident angle of $0–5\mu \mathrm{rad}$. A spectrum produced by $149\text{−}\mathrm{GeV}$ electrons and positrons incident around the ⟨110⟩ axis in $0.7\text{−}\mathrm{mm}$ diamond. Adapted from 137.

Figure 26

A schematical drawing of the setup used by NA46. The abbreviations are B1, B2, and B5, magnetic dipoles; TAG and CAL, lead glass calorimeters; LR, target foil; XTAL, crystal target; $\mu \mathrm{S}$, microstrip detector; MWPC, multiwire proportional chambers; and D, AD, T1-5, scintillator counters. From 64.

Figure 27

Angular distribution of particles produced in a $0.42\text{−}\mathrm{mm}$-thick Ge crystal by photons within an energy range of $83–133\mathrm{GeV}$: (a) the emerging electrons and (b) positrons. The dots are experimental points and the histograms are results from a Monte Carlo code based on pair production in strong crystalline fields. Adapted from 23.

Figure 28

$149\text{−}\mathrm{GeV}$ electrons on the (111) plane of $0.7\text{−}\mathrm{mm}$ diamond, $0.6\mathrm{mrad}$ to the ⟨110⟩ axis. (a) The radiation enhancement as a function of total radiated energy, (b) the photon multiplicity, and (c) the number of photons emitted as a function of the energy of the photon given that the sum of photon energies exceeds $100\mathrm{GeV}$. From 137.

Figure 29

Differential enhancement of pair production for photons in the energy interval $80–100\mathrm{GeV}$ incident around the ⟨110⟩ axis in a germanium crystal cooled to $100\mathrm{K}$. Filled squares are results of the pair spectrometer and the line represents the theoretical values according to 146. The results are shown for the three angles 0, 2, and $4\mathrm{mrad}$ to the ⟨110⟩ axis on the (110) plane. Adapted from 139.

Figure 30

Polar-angle differences for $149\text{−}\mathrm{GeV}$ electrons, positrons, and their respective photons. The filled squares show the electron angle difference calculated from the angle at the moment of photon emission and the entry angle, i.e., ${\theta }_{\gamma }-{\theta }_{\mathrm{in}}$, and the filled dots show the same for the positron. The open squares show the electron angle difference calculated from the exit angle and the entry angle, i.e., ${\theta }_{\mathrm{out}}-{\theta }_{\mathrm{in}}$, and the open dots show the same for the positron. Adapted from 137.

Figure 31

An example of radiative capture for $149\text{−}\mathrm{GeV}$ electrons in $0.7\text{−}\mathrm{mm}$ diamond. Shown are exit angles ${\theta }_{\mathrm{out}}$ as a function of entry angles with respect to the ⟨110⟩ axis, ${\theta }_{\mathrm{in}}$, for an interval of radiated energy, $0-E∕3$. The full squares with error bars denote the data points, and the lines indicate ${\theta }_{\mathrm{out}}={\theta }_{\mathrm{in}}$, and the critical angle of the exiting electron, ${\psi }_1\left(\overline{E^{\text{exit}}}\right)$, where the average energy loss, $\overline{\Delta E}$, for each ${\theta }_{\mathrm{in}}$ bin has been used with $\overline{E^{\text{exit}}}=E-\overline{\Delta E}$. The critical angle for $149\text{−}\mathrm{GeV}$ electrons in this configuration is $30\mu \mathrm{rad}$. Adapted from 137.

Figure 32

(Color online) Power spectrum of radiated photons $\xi dN∕d\xi$ for $243\text{−}\mathrm{GeV}$ electrons aligned $0.3\mathrm{mrad}$ from the axis in $0.2\text{−}\mathrm{mm}$ W ⟨111⟩. Filled circles denote the experimental points and the lines (solid, including spin; dashed, excluding spin) the calculated values. Adapted from 136.

Figure 33

(Color online) The enhancement $\eta$ for radiation emission from $0.2\text{−}\mathrm{mm}$ W ⟨111⟩. The points with error bars are the experimental values, the dashed line is a least-squares fit by $\eta =a{E}^b$, and the solid line is a fit with $\eta =a{E}^{0.07}+1$, where $a$ and $b$ are free parameters. Adapted from 136.

Figure 34

Three examples of beamstrahlung power spectra where the horizontal scale is the fractional photon energy and the constant $C$ here is approximately equal to $1∕\chi$. Adapted from 87.

Figure 35

(Color online) The conversion probability of a photon in the geomagnetic field as a function of photon energy for energies in the region ${10}^{19}–{10}^{21}\mathrm{eV}(10–1000\mathrm{EeV})$. The dashed line represents the conversion probability for a magnetic field corresponding to the site of the southern part of the Pierre Auger Observatory and the solid line represents that corresponding to the northern part. Adapted from 227.

Figure 36

(Color online) Efficiency as a function of normalized length $L∕L_D$ at $7\mathrm{TeV}$ according to Eq. (96) for Ge (110) and Si (110). The solid line is for Si bent to $\theta =0.275\mathrm{mrad}$, the dash-dotted line for Ge bent to $\theta =0.275\mathrm{mrad}$, the dashed line for Si bent to $\theta =0.100\mathrm{mrad}$, and the dotted line for Ge bent to $\theta =0.100\mathrm{mrad}$. The values ${\eta }_F=3$ and ${\epsilon }_S=0.9$ were used (see text). Shown by a filled dot is the efficiency, Eq. (99), at optimum length, Eq. (98), for Si bent to an angle $\theta =0.275\mathrm{mrad}$ (230).

Figure 37

A schematical drawing of extraction of particles from the halo of a circulating beam by means of a bent crystal. From 176.

Figure 38

An outline of the setup used in NA59. The leftmost crystal produces energetic, linearly polarized photons by electrons incident with an angle $\theta =5\mathrm{mrad}$ to the ⟨001⟩ axis and an angle $\psi =70\mu \mathrm{rad}$ to the (110) plane. The emerging photons are converted into circularly polarized photons in the thick, central crystal aligned with an angle $2.3\mathrm{mrad}$ to the ⟨110⟩ axis. The last crystal acts as an analyzer where the pair-production rate depends on the polarization state for photons incident with an angle $3\mathrm{mrad}$ to the ⟨110⟩ axis.