We can describe Rutherford backscattering as an elastic (hard-sphere) collision between a high kinetic energy particle from the incident beam (the projectile) and a stationary particle located in the sample (the target). Elastic in this context means that no energy is either lost or gained during the collision.
Note that the "law" of the conservation of energy is not generally applicable for nuclear interactions, since in some circumstances a collision may result in a nuclear reaction, with the release of what can be very considerable quantities of energy. Nuclear reaction analysis (NRA) is very useful for detecting light elements. The energy conservation law still applies for NRA, but in the more general mass-energy form.
Considering the kinematics of the collision (that is, the conservation of momentum and kinetic energy), the energy E1 of the scattered projectile is reduced from the initial energy E0:
where k is known as the kinematical factor, and
where particle 1 is the projectile, particle 2 is the target nucleus, and is the scattering angle of the projectile in the laboratory frame of reference (that is, relative to the observer). The plus sign is taken when the mass of the projectile is less than that of the target, otherwise the minus sign is taken.
While this equation correctly determines the energy of the scattered projectile for any particular scattering angle (relative to the observer), it does not describe the probability of observing such an event. For that we need the differential cross-section of the backscattering event:
where and are the atomic numbers of the incident and target nuclei. This equation is written in the centre of mass frame of reference and is therefore not a function of the mass of either the projectile or the target nucleus.
Note that the scattering angle in the laboratory frame of reference is not the same as the scattering angle in the centre of mass frame of reference (although for RBS experiments they are usually very similar). However, heavy ion projectiles can easily recoil lighter ions which, if the geometry is right, can be ejected from the target and detected. This is the basis of the Elastic Recoil Detection (ERD, with synonyms ERDA, FRS, HFS) technique. RBS often uses a He beam which readily recoils H, so simultaneous RBS/ERD is frequently done to probe the hydrogen isotope content of samples (although H ERD with a He beam above 1 MeV is not Rutherford: see http://www-nds.iaea.org/sigmacalc). For ERD the scattering angle in the lab frame of reference is quite different from that in the centre of mass frame of reference.
Note also that heavy ions cannot backscatter from light ones: it is kinematically prohibited. The kinematical factor must remain real, and this limits the permitted scattering angle in the laboratory frame of reference. In ERD it is often convenient to place the recoil detector at recoil angles large enough to prohibit signal from the scattered beam. The scattered ion intensity is always very large compared to the recoil intensity (the Rutherford scattering cross-section formula goes to infinity as the scattering angle goes to zero), and for ERD the scattered beam usually has to be excluded from the measurement somehow.
The singularity in the Rutherford scattering cross-section formula is unphysical of course. If the scattering cross-section is zero it implies that the projectile never comes close to the target, but in this case it also never penetrates the electron cloud surrounding the nucleus either. The pure Coulomb formula for the scattering cross-section shown above must be corrected for this screening effect, which becomes more important as the energy of the projectile decreases (or, equivalently, its mass increases).
While large-angle scattering only occurs for ions which scatter off target nuclei, inelastic small-angle scattering can also occur off the sample electrons. This results in a gradual decrease in ions which penetrate more deeply into the sample, so that backscattering off interior nuclei occurs with a lower "effective" incident energy. The amount by which the ion energy is lowered after passing through a given distance is referred to as the stopping power of the material and is dependent on the electron distribution. This energy loss varies continuously with respect to distance traversed, so that stopping power is expressed as
For high energy ions stopping power is usually proportional to ; however, precise calculation of stopping power is difficult to carry out with any accuracy.
Stopping power (properly, stopping force) has units of energy per unit length. It is generally given in thin film units, that is eV /(atom/cm2) since it is measured experimentally on thin films whose thickness is always measured absolutely as mass per unit area, avoiding the problem of determining the density of the material which may vary as a function of thickness. Stopping power is now known for all materials at around 2%, see http://www.srim.org.
Read more about this topic: Rutherford Backscattering Spectrometry
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