In the study of matter, channelling is a process that guides the path of a charged particle through a crystal. When a charged particle hits a solid material, several events can occur, such as bouncing off atoms (elastic scattering), losing energy (inelastic energy-loss), releasing electrons (secondary-electron emission), creating light (electromagnetic radiation), or causing nuclear reactions. The likelihood of these events depends on how close the particle gets to the atoms in the material. If the material is uniform and the same in all directions, the particle's interactions do not depend on its movement direction. However, if the material is made of a single crystal (monocrystalline), the results of these events depend strongly on the direction the particle is moving compared to the crystal's structure. In such materials, the material stops the particle less effectively in certain directions. This effect is strongest along directions in the crystal with the greatest spacing between atoms, which vary depending on the crystal type, such as face-centered cubic (FCC) or body-centered cubic (BCC). When a charged particle enters a material close to these directions, it is more likely to travel through open paths in the crystal because electronic forces are stronger than nuclear forces. This phenomenon is called the "channelling" effect. It is connected to other effects that depend on the direction of movement, such as particle diffraction. These connections will be explained in more detail later.

History

The channeling effect was first discovered in early computer simulations in 1963 to explain unusual patterns in how ions moved through materials, which did not match existing theories. Scientists tested this prediction the next year by measuring how deep ions traveled through a special type of tungsten. The first experiments showing ions passing through crystals were conducted by a team from Oak Ridge National Laboratory, which demonstrated that the way ions spread out was determined by the crystal rainbow channeling effect.

Mechanism

From a basic, classical perspective, the channelling effect can be understood as follows: When a charged particle moves toward the surface of a single crystal and its direction is close to a major crystal direction (Fig. 1), it is likely to experience only small-angle scattering as it passes through the layers of atoms in the crystal. This allows the particle to remain within the same crystal "channel." The reason for this behavior is the continuous screened Coulomb repulsive force on both sides of the channel, created by rows or planes of atoms. This force is strongest in materials where atoms have a full or nearly full outer electron shell. In such cases, the Coulomb potential forms a long-range, slightly varying repulsive barrier that prevents large-angle scattering with nuclei and allows only small-angle scattering and inelastic collisions with the electron cloud on either side of the channel. This creates a path, or corridor, where the charged particle moves in an oscillatory motion within the channel’s width. If the particle’s direction is not aligned with a major crystal direction or plane ("random direction," Fig. 2), it is more likely to experience large-angle scattering, resulting in a shorter average penetration depth. If the particle’s momentum direction is close to a crystalline plane but not near major crystal axes, this is called "planar channelling." Channelling typically allows ions to penetrate deeper into the material, as observed in experiments and simulations (Figures 3-5).

Negatively charged particles, such as antiprotons and electrons, are attracted to the positively charged nuclei of the plane. After passing through the center of the plane, they are drawn back toward the plane, causing them to follow the direction of one crystalline plane.

Because crystalline planes contain a high density of atomic electrons and nuclei, channeled particles eventually experience high-angle Rutherford scattering or energy loss due to collisions with electrons, causing them to leave the channel. This process is called "dechannelling."

Positively charged particles, like protons and positrons, are repelled by the nuclei of the plane. After entering the space between two neighboring planes, they are repelled by the second plane. This causes positively charged particles to follow the direction between two neighboring crystalline planes but at the farthest possible distance from each plane. As a result, positively charged particles are less likely to interact with the nuclei and electrons of the planes (smaller "dechannelling" effect) and travel longer distances.

The same phenomenon occurs when a charged particle’s momentum direction is close to a major, high-symmetry crystal axis. This is called "axial channelling." Generally, axial channelling is more effective than planar channelling because it creates a deeper potential barrier.

As a particle moves through the channel, it continuously loses energy. Initially, this energy loss is dominated by interactions with electrons, which gradually reduce the particle’s kinetic energy in its main direction of travel while increasing it in the transverse direction. Eventually, this allows the particle to overcome the channeling potential barrier. Additionally, the particle’s oscillation causes an angle deviation. If this angle exceeds a critical threshold, the particle experiences large-angle scattering and collisions with nuclei, leading to dechannelling and significant energy loss.

At low energies, channelling effects in crystals are not observed because small-angle scattering at low energies requires larger impact parameters, which are greater than the distances between crystal planes. In this case, the particle’s diffraction is the dominant effect. At high energies, quantum effects and diffraction are less significant, and channelling becomes noticeable.

Applications

There are several important uses of channelling effects.

Channelling effects can help scientists study the structure of a crystal lattice and any changes in it, such as doping, in areas that X-rays cannot reach. This method can also help locate atoms that are not in their usual positions within the crystal. This is a special type of a technique called Rutherford backscattering/channelling (RBS-C), which is commonly used for analysis.

Channelling can also be used to focus ion beams extremely precisely, which can be helpful for examining very small structures at the atomic level.

At very high energies, such as tens of GeV, channelling has additional uses. It can help create high-energy gamma rays more effectively, and bent crystals can be used to guide particles from the outer part of a moving beam in a particle accelerator.

Classical channelling theory

The classical study of channelling assumes that interactions between ions and atomic nuclei in a crystal are not related events. The first detailed classical analysis was developed by Jens Lindhard in 1965. His model remains the standard reference for understanding channelling. Lindhard’s model describes how a continuous repulsive force is created by atomic nuclei arranged in lines or planes within a crystal. This force is an average of the individual Coulomb forces from charged nuclei, which are shielded by the surrounding electrons.

The Lindhard potential is expressed as:
$$ V(r) = Z_1 Z_2 e^2 left( frac{1}{r} – frac{1}{sqrt{r^2 + C^2 a^2}} right) $$
Here, $ r $ is the distance from a nucleus, $ C $ is a constant equal to 3, and $ a $ is the screening radius from the Thomas-Fermi model:
$$ a = frac{0.885 a_0}{(sqrt{Z_1} + sqrt{Z_2})^{2/3}} $$
$ a_0 $ is the Bohr radius (0.53 Å, the radius of the smallest orbit in a Bohr atom). The typical screening radius $ a $ is between 0.1 and 0.2 Å.

For axial channelling, if $ d $ is the distance between atoms in a row, the average potential along the row is:
$$ U_a(rho) = frac{Z_1 Z_2 e^2}{d} ln left( left( frac{Ca}{rho} right)^2 + 1 right) $$
$ rho $ is the distance from the atomic row. This potential represents a continuous force from a line of atoms with atomic number $ Z_2 $, spaced $ d $ apart.

The energy of channeled ions (atomic number $ Z_1 $) is:
$$ E = frac{p_parallel^2}{2M} + frac{p_perp^2}{2M} + U_a(rho) – U_{text{min}} $$
$ p_parallel $ and $ p_perp $ are the parallel and perpendicular components of the ion’s momentum relative to the atomic row. $ U_{text{min}} $ is the lowest potential in the channel, considering all atomic lines in the crystal.

Momentum components are:
$$ p_perp = p sin psi, quad p_parallel = p cos psi $$
$ psi $ is the angle between the ion’s path and the crystallographic axis.

Assuming no energy loss, $ frac{p_parallel^2}{2M} $ remains constant. The transverse energy conservation equation is:
$$ E_perp = frac{p^2 sin^2(psi)}{2M} + U_a(rho) – U_{text{min}} $$
This equation is also called the transverse energy conservation law. Since $ sin(psi) approx psi $ for small angles, this approximation is valid for ions well-aligned with the crystal axis.

Channelling occurs when an ion’s transverse energy is too low to overcome the potential barrier from ordered nuclei. The critical energy $ E_c $ is the transverse energy threshold for channelling:
$$ U_a(rho_c) – U_{text{min}} = E_c $$
Typical $ E_c $ values are a few tens of eV, as $ rho_c $ (the critical distance) is similar to the screening radius (0.1–0.2 Å). Ions with transverse energy below $ E_c $ are channeled.

For perfect alignment ($ psi_0 = 0 $), all ions with impact parameter $ rho < rho_c $ are de-channeled. The minimum fraction of de-channeled ions is:
$$ chi_{text{min}} = Nd(pi rho_c^2) $$
where $ Nd $ is the atom density (atoms/cm³). For a silicon crystal oriented along <110>, $ chi_{text{min}} = 1.35 times 10^{-2} $, matching experimental results.

Thermal vibrations of nuclei affect channelling, as discussed in references. The critical angle $ psi_c $ is defined as:
$$ psi_c = sqrt{frac{U(r_{text{min}})}{E}} $$
Using the Lindhard potential and assuming $ rho $ is the minimum approach distance:
$$ psi_c(rho) = sqrt{frac{Z_1 Z_2 e^2}{Ed}} left[ ln left( frac{C a}{rho} right)^2 + 1 right]^{1/2} $$
Typical critical angles are:
– Silicon <110>: 2.17°
– Tungsten <100>: 2.17°
– Tungsten <100>: 2.17°

For planar channelling, ions are confined between charge planes, creating a continuous planar potential $ U_p(y) $:
$$ U_p(y) = 2pi Z_1 Z_2 e^2 a N_d left( sqrt{left( frac{y}{a} right)^2 + C^2} – frac{y}{a} right) $$
Planar channelling has critical angles 2–4 times smaller than axial channelling, and $ chi_{text{min}} $ is 10–20% higher than axial channelling (vs. >99% for axial). A full discussion of planar channelling is available in references.

General literature

• J.W. Mayer and E. Rimini, Ion Beam Handbook for Material Analysis, (1977) Academic Press, New York
• L.C. Feldman, J.W. Mayer, and S.T. Picraux, Material Analysis by Ion Channelling, (1982) Academic Press, New York
• R. Hovden, H. L. Xin, and D. A. Muller, Physical Review B, Volume 86, Page 195415 (2012), arXiv: 1212.1154
• G. R. Anstis, D. Q. Cai, and D. J. H. Cockayne, Ultramicroscopy, Volume 94, Page 309 (2003).
• D. Van Dyck and J. H. Chen, Solid State Communications, Volume 109, Page 501 (1999).
• S. Hillyard and J. Silcox, Ultramicroscopy, Volume 58, Page 6 (1995).
• S. J. Pennycook and D. E. Jesson, Physical Review Letters, Volume 64, Page 938 (1990).
• M. V. Berry and Ozoriode, Journal of Physics A: Mathematical and General, Volume 6, Page 1451 (1973).
• M. V. Berry, Journal of Physics Part C: Solid State Physics, Volume 4, Page 697 (1971).
• A. Howie, Philosophical Magazine, Volume 14, Page 223 (1966).
• P. B. Hirsch, A. Howie, R. B. Nicholson, D. W. Pashley, and M. Whelan, Electron Microscopy of Thin Crystals (Butterworths, London, 1965).
• J. U. Andersen, Notes on Channeling, http://phys.au.dk/en/publications/lecture-notes/ Archived on May 28, 2019, at the Wayback Machine (2014).