Rutherford backscattering spectrometry
Rutherford backscattering spectrometry is an analytical technique used in materials science. Sometimes referred to as high-energy ion scattering spectrometry, RBS is used to determine the structure and composition of materials by measuring the backscattering of a beam of high energy ions impinging on a sample.
Geiger–Marsden experiment
Rutherford backscattering spectrometry is named after Lord Rutherford, a physicist sometimes referred to as the father of nuclear physics. Rutherford supervised a series of experiments carried out by Hans Geiger and Ernest Marsden between 1909 and 1914 studying the scattering of alpha particles through metal foils. While attempting to eliminate "stray particles" they believed to be caused by an imperfection in their alpha source, Rutherford suggested that Marsden attempt to measure backscattering from a gold foil sample. According to the then-dominant plum-pudding model of the atom, in which small negative electrons were spread through a diffuse positive region, backscattering of the high-energy positive alpha particles should have been nonexistent. At most small deflections should occur as the alpha particles passed almost unhindered through the foil. Instead, when Marsden positioned the detector on the same side of the foil as the alpha particle source, he immediately detected a noticeable backscattered signal. According to Rutherford, "It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you."Rutherford interpreted the result of the Geiger–Marsden experiment as an indication of a Coulomb collision with a single massive positive particle. This led him to the conclusion that the atom's positive charge could not be diffuse but instead must be concentrated in a single massive core: the atomic nucleus. Calculations indicated that the charge necessary to accomplish this deflection was approximately 100 times the charge of the electron, close to the atomic number of gold. This led to the development of the Rutherford model of the atom in which a positive nucleus made up of N positive particles of charge +e, or protons, was surrounded by N orbiting electrons of charge -e to balance the nuclear charge. This model was eventually superseded by the Bohr atom, incorporating some early results from quantum mechanics.
If the energy of the incident particle is increased sufficiently, the Coulomb barrier is exceeded and the wavefunctions of the incident and struck particles overlap. This may result in nuclear reactions in certain cases, but frequently the interaction remains elastic, although the scattering cross-sections may fluctuate wildly as a function of energy and no longer be calculable analytically. This case is known as "Elastic Backscattering Spectrometry". There has recently been great progress in determining EBS scattering cross-sections, by solving Schrödinger's equation for each interaction. However, for the EBS analysis of matrices containing light elements, the utilization of experimentally measured scattering cross-section data is also considered to be a very credible option.
Basic principles
We describe Rutherford backscattering as an elastic, hard-sphere collision between a high kinetic energy particle from the incident beam and a stationary particle located in the sample. Elastic in this context means that no energy is absorbed by the internal states of the incident particle or the stationary particle during the collision, and the energy loss of the incident particle is due only to the recoil of the "stationary" particle. Nuclear interactions are generally not elastic, since a collision may result in a nuclear reaction, with the release of considerable quantities of energy. Nuclear reaction analysis is useful for detecting light elements. However, this is not Rutherford scattering.Considering the kinematics of the collision, 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. 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, 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.
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 . 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 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. For ERD the scattering angle in the lab frame of reference is quite different from that in the centre of mass frame of reference.
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, 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.
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 the kinetic energy of incident ions as they penetrate into the sample, so that backscattering off interior nuclei occurs with a lower "effective" incident energy. Similarly backscattered ions lose energy to electrons as they exit the sample. 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 has units of energy per unit length. It is generally given in thin film units, that is eV / 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.
Instrumentation
An RBS instrument generally includes three essential components:- An ion source, usually alpha particles or, less commonly, protons.
- A linear particle accelerator capable of accelerating incident ions to high energies, usually in the range 1-3 MeV.
- A detector capable of measuring the energies of backscattered ions over some range of angles.
Two-stage systems, or "tandem accelerators", start with a source of He− ions and position the positive terminal at the center of the acceleration tube. A stripper element included in the positive terminal removes electrons from ions which pass through, converting He− ions to He++ ions. The ions thus start out being attracted to the terminal, pass through and become positive, and are repelled until they exit the tube at ground. This arrangement, though more complex, has the advantage of achieving higher accelerations with lower applied voltages: a typical tandem accelerator with an applied voltage of 750 kV can achieve ion energies of over 2 MeV.
Detectors to measure backscattered energy are usually silicon surface barrier detectors, a very thin layer of P-type silicon on an N-type substrate forming a p-n junction. Ions which reach the detector lose some of their energy to inelastic scattering from the electrons, and some of these electrons gain enough energy to overcome the band gap between the semiconductor valence and conduction bands. This means that each ion incident on the detector will produce some number of electron-hole pairs which is dependent on the energy of the ion. These pairs can be detected by applying a voltage across the detector and measuring the current, providing an effective measurement of the ion energy. The relationship between ion energy and the number of electron-hole pairs produced will be dependent on the detector materials, the type of ion and the efficiency of the current measurement; energy resolution is dependent on thermal fluctuations. After one ion is incident on the detector, there will be some dead time before the electron-hole pairs recombine in which a second incident ion cannot be distinguished from the first.
Angular dependence of detection can be achieved by using a movable detector, or more practically by separating the surface barrier detector into many independent cells which can be measured independently, covering some range of angles around direct back-scattering. Angular dependence of the incident beam is controlled by using a tiltable sample stage.