In condensed-matter physics, channelling (or channeling) is the process that constrains the path of a charged particle in a crystalline solid.[1] [2] [3]
Many physical phenomena can occur when a charged particle is incident upon a solid target, e.g., elastic scattering, inelastic energy-loss processes, secondary-electron emission, electromagnetic radiation, nuclear reactions, etc. All of these processes have cross sections which depend on the impact parameters involved in collisions with individual target atoms. When the target material is homogeneous and isotropic, the impact-parameter distribution is independent of the orientation of the momentum of the particle and interaction processes are also orientation-independent. When the target material is monocrystalline, the yields of physical processes are very strongly dependent on the orientation of the momentum of the particle relative to the crystalline axes or planes. Or in other words, the stopping power of the particle is much lower in certain directions than others. This effect is commonly called the "channelling" effect. It is related to other orientation-dependent effects, such as particle diffraction. These relationships will be discussed in detail later.
The channelling effect was first discovered in pioneering binary collision approximation computer simulations in 1963 in order to explain exponential tails in experimentally observed ion range distributions that did not conform to standard theories of ion penetration. The simulated prediction was confirmed experimentally the following year by measurements of ion penetration depths in single-crystalline tungsten.[4] First transmission experiments of ions channelling through crystals were performed by Oak Ridge National Laboratory group showing that ions distribution is determinated by crystal rainbow channelling effect.[5]
From a simple, classical standpoint, one may qualitatively understand the channelling effect as follows: If the direction of a charged particle incident upon the surface of a monocrystal lies close to a major crystal direction (Fig. 1), the particle with high probability will only do small-angle scattering as it passes through the several layers of atoms in the crystal and hence remain in the same crystal 'channel'. If it is not in a major crystal direction or plane ("random direction", Fig. 2), it is much more likely to undergo large-angle scattering and hence its final mean penetration depth is likely to be shorter. If the direction of the particle's momentum is close to the crystalline plane, but it is not close to major crystalline axes, this phenomenon is called "planar channelling". Channelling usually leads to deeper penetration of the ions in the material, an effect that has been observed experimentally and in computer simulations, see Figures 3-5.[6]
Negatively charged particles like antiprotons and electrons are attracted towards the positively charged nuclei of the plane, and after passing the center of the plane, they will be attracted again, so negatively charged particles tend to follow the direction of one crystalline plane.
Because the crystalline plane has a high density of atomic electrons and nuclei, the channeled particles eventually suffer a high angle Rutherford scattering or energy-losses in collision with electrons and leave the channel. This is called the "dechannelling" process.
Positively charged particles like protons and positrons are instead repelled from the nuclei of the plane, and after entering the space between two neighboring planes, they will be repelled from the second plane. So positively charged particles tend to follow the direction between two neighboring crystalline planes, but at the largest possible distance from each of them. Therefore, the positively charged particles have a smaller probability of interacting with the nuclei and electrons of the planes (smaller "dechannelling" effect) and travel longer distances.
The same phenomena occur when the direction of momentum of the charged particles lies close to a major crystalline, high-symmetry axis. This phenomenon is called "axial channelling". Generally, the effect of axial channeling is higher than planar channeling due to a deeper potential formed in axial conditions.
At low energies the channelling effects in crystals are not present because small-angle scattering at low energies requires large impact parameters, which become bigger than interplanar distances. The particle's diffraction is dominating here. At high energies the quantum effects and diffraction are less effective and the channelling effect is present.
There are several particularly interesting applications of the channelling effects.
Channelling effects can be used as tools to investigate the properties of the crystal lattice and of its perturbations (like doping) in the bulk region that is not accessible to X-rays.The channelling method may be utilized to detect the geometrical location of interstitials. This is an important variation of the Rutherford backscattering ion beam analysis technique, commonly called Rutherford backscattering/channelling (RBS-C).
The channelling may even be used for superfocusing of ion beam, to be employed for sub-atomic microscopy.[7]
At higher energies (tens of GeV), the applications include the channelling radiation for enhanced production of high energy gamma rays,[8] [9] and the use of bent crystals for extraction of particles from the halo of the circulating beam in a particle accelerator.[10] [11]
The classical treatment of channelling phenomenon supposes that the ion - nucleus interactions are not correlated phenomena. The first analytic classical treatise is due to Jens Lindhard in 1965,[12] who proposed a treatment that still remains the reference one. He proposed a model that is based on the effects of a continuous repulsive potential generated by atomic nuclei lines or planes, arranged neatly in a crystal. The continuous potential is the average in a row or on an atomic plane of the single Coulomb potentials of the charged nuclei
eZ2
The proposed potential (named Lindhard potential) is:
V(r)=Z1Z2e2\left(
| 1 | |
| r |
-
| 1 | |
| \sqrt{r2+C2a2 |
r represents the distance from the nucleus,
C2
a=
| 0.885~a0 | |
| (\sqrt{Z1 |
| 2/3 | |
| +\sqrt{Z | |
| 2}) |
a0
Considering the case of axial channelling, if d is the distance between two successive atoms of an atomic row, the mean of the potential along this row is equal to:
| U | ||||
|
| d/2 | |
| \int | |
| -d/2 |
V\left(\sqrt{z2+\rho2}\right)~dz=
| |||||||||||||
| d |
~ln\left(\left(
| Ca | |
| \rho |
\right)2+1\right)
\rho
Z2
The energy of the channeled ions, having an atomic number
Z1
| E= |
| + | ||||||
| 2M |
| |||||||
| 2M |
+Ua(\rho)-U\rm
where
p\shortparallel
p\perp
U\rm
It therefore follows that the components of the momentum are:
p\perp=p\sin\psi,~~p\shortparallel=p\cos\psi
where
\psi
Neglecting the energy loss processes, the quantity
| |||||||
| 2M |
| E | ||||||||||
|
+Ua(\rho)-U\rm=
| p2\sin2(\psi) | |
| 2M |
+Ua(\rho)-U\rm
The equation is also known as the expression of the conservation of transverse energy. The approximation of
\sin(\psi) ≈ \psi
The channelling condition can now be considered the condition for which an ion is channeled if its transverse energy is not sufficient to overcome the height of the potential barrier created by the strings of ordered nuclei. It is therefore useful to define the "critical energy"
Ec
Ua(\rhoc)-U\rm=Ec
Typical
Ec
\rhoc
Ec
In the case of
\psi0=0
\rho<\rho\rm
\chi\rm=
| |||||||||
|
=Nd(\pi
| 2 | |
| \rho | |
| c |
)
where
\pi
| ||||
| r | ||||
| 0 |
\chi\rm
\chi\rm=1.35~10-2
Further considerations can be made by considering the thermal vibration motion of the nuclei: for this discussion, see the reference.[13]
The critical angle
\psi\rm
\psi\rm=\sqrt{
| U(r\rm) | |
| E |
Using the Lindhard potential and assuming the amplitude of thermal vibration
\rho
\psi\rm(\rho)=\sqrt{
| |||||||||||||
| Ed |
}\left[ln\left(
| Ca | |
| \rho |
\right)2+1\right]1/2
Typical critical angles values (at room temperature) are for silicon <110> 0.71 °, for germanium <100> 0.89 °, for tungsten <100> 2.17 °.
Similar consideration can be made for planar channelling. In this case, the average of the atomic potentials will cause the ions to be confined between charge planes that correspond to a continuous planar potential
U\rm(\rho)
U\rm(y)=Nd\rm\intV\left(\sqrt{y2+r2}\right)~2\pir~dr=2\piZ1Z
| 2aNd | |
| \rmp |
\left(\sqrt{\left(
| y | |
| a |
\right)2+C2}-
| y | |
| a |
\right)
where
Nd\rm
d\rm
\chi\rm