Coherent effects in crystals

Critical angle (Just a try)

The limiting angle for capture to planar or axial channeling (related to Lindhard's critical angle) fora a charged particle in a straight single crystals is $$\psi_L = \sqrt{\frac{2U_0}{\text{pv}}}$$ where p is momentum and v velocity. For planar channeling $U_0$ is the depth of the potential well and is proportional to $Z_i Z N d_p a_{TF}$ where Z is the crystal atomic number, $Z_i$ the beam particle atomic number, N the atomic number density, $d_p$ the planar spacing, and $a_{TF}\approx0.8853\;a_{\infty} Z^{-\frac{1}{3}}$, the TF screening parameter, At 1 TeV/c $\psi_L = 6.7\mu\text{rad}$ for the 110 plane in Si where $U_0 = 22.5$eV. The effective critical angle is smaller because of atomic vibrations including thermal effects. Characteristically $\psi_L$ is small compared to the typical beam divergence. Crystals must be aligned along planes or axes to see channeling. The critical angle is roughly equal to $\psi_{1/2}$, the HWHM of a transmission or alignment distribution of $\psi$, the particle angle relative to a crystal plane. However the distribution is not Gaussian.

For a bent crystal $$\psi_{Lb} = \psi_L \left(1-\frac{R_T}{R_m}\right)$$ where $R_m$ is the minimum radius of curvature and $R_T$ is the Tsyganov radius (see below).

Higher Z crystals, e.g. W, have larger $\psi_L$ but large dislocation-free crystals are not available whereas they are for Si.

Bent crystal channeling

As in a synchrotron, positive particles will stay in the channel of a gently bent crystal. At a small enough bending radius, the Tsyganov or critical radius $$R_T = \frac{\text{p} \beta c}{e E_c}$$ particles no longer channel. Here $E_c$ is the inter-atomic field at a distance where the particle trajectory no longer remains stable $\left(5.7 \text{GeV/e [cm]}\right)$ for Si(110). The equivalent magnetic field for relativistic channeling and a uniform bend with radius of curvature R is $$B\left[ T \right] = \frac{\text{p} \left[ \text{GeV/c} \right]}{0.3 R\left[ \text{m} \right]}$$

Equivalent fields up to $\mathcal{O}\left(1000T\right)$ are feasible. Practical problems that must be taken into account include bending crystals, unintended twists introduced in the process, nearby planes and axes, surface amorphous layers and miscuts in a crystal face relative to a plane.

From Fermilab website

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