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Positron physics

As the antimatter counterpart to the electron, the positron has the opposite charge and magnetic moment, and the same mass (511.0034 keV/c² ) and spin as the electron. The positron is stable in vacuum (average lifetime 10²¹ years), whereas in condensed matter it typically remains only a short time (10^-10 sec) before annihilating with an electron. Being anti-electrons, positrons are identical to electrons in all respects except charge. Thus they behave in solids in ways that are identical in many respects. The thermalization processes and consequent implantation profiles of positrons and electrons are very similar, and once they are implanted in a solid (at least in metals and semiconductors), the scattering processes that determine the motions of electrons and positrons are similar.

A very significant feature of positrons, however, is that they are distinguishable from electrons. There is no way to follow the diffusion history of a particular electron implanted in a target-it becomes lost in the sea of identical electrons in the solid. In the case of positrons however, it is possible to follow the history of each positron after it has thermalized but before it annihilates. The influence on the positron diffusion of such material properties as internal fields, impurity and defect distributions, and spatial changes in composition such as occur in microelectronic devices and in layered structures such as heterostructures, can all in principle be measured.

A further important feature specific to positrons is their positive charge. Because of it, positrons can participate in many processes not available to electrons. They can trap at negatively charged lattice defects such as monovacencies and other small open-volume defects, at impurities, and in image-potential induced (ie. extrinsic) surface states both at external surfaces and at internal surfaces bounding large open-volume defects such as voids. Moreover because the surface-dipole contribution to the electron work function is repulsive in the case of the positron work function, this work function is negative for many materials. Thus positrons are reemitted into the vacuum from the surfaces of these materials, or emitted into the interior of a large open-volume defect such as a void. Finally, a positron can bind to an electron in a hydrogen-like atom called positronium. Although a positronium atom cannot exist in the interior of a metal because the electron density is too large, it can exist inside an insulator, and can be emitted from the surface of any material.

Finally, the anti-matter nature of positrons gives rise to a number of microscope signals that have no analog in electron microscopes. We have already mentioned the reemission of positrons from surfaces, and the formation of positronium. In addition, the annihilation of positrons with electrons provides a unique signal. Both the energy and angular distributions of the annihilation gamma-rays can be measured and provide information about the electronic environment of the positron at the point of annihilation. In addition, the rate at which annihilation occurs can be measured and provides detailed information about whether the annihilating positron is freely diffusing through a lattice, or is bound at some type of defect or impurity.

Some of the processes by which positrons may interact with condensed matter are illustrated in the figure on the right. The positrons may be back-scattered from the surface, or they may enter the solid where they are quickly thermalized (10^-11 secs) by conduction electron scattering, including plasmon and electron-hole pair excitations, and finally by phonon scattering. The mean implantation depth is found to vary approximately as E^1.6. In typical small-laboratory beams (with 0 < E < 50 keV) this ranges from a few angstroms up to a few microns. The positrons may penetrate quite deeply, as the probability of annihilation is small unless the velocity of the positron relative to that of the electrons is low.

The positrons then diffuse through the solid, with typical diffusion lengths in relatively defect-free materials on the order of 1000 angstroms. In the course of this diffusion the positrons may undergo free annihilation (100 picoseconds), or encounter an open-volume defect in which trapping and subsequent annihilation (200-400 picoseconds) may occur. Thus the presence of defects tends to increase the positron lifetime, while reducing the diffusion length. Some fraction of the positrons may diffuse back to the surface where they meet one of four possible fates: (i) they may reflect off the surface potential, due to their wave-like nature (the DeBroglie wavelength for a thermal positron is approximately 75 angstroms at 300K), (ii) they may become trapped in, and subsequently annihilate from, a surface state (~500 picoseconds), (iii) they may bind with an electron to form positronium (Ps), which has a binding energy of 6.8 eV, or (iv) they may be reemitted into the vacuum with a well-defined energy which is characteristic of the material. Those materials which reemit thermal positrons are said to have a negative work function. Those positrons which failed to fully thermalize before returning to the surface may be reemitted as epi-thermal positrons or Ps. The Ps atom may exist in either the singlet state which decays predominantly into two gamma-rays with a lifetime of approximately 125 psec, or the triplet state which decays (in vacuum) predominantly into three gamma-rays with a lifetime of approximately 140 nsec. Note that Ps cannot be formed in the bulk of metals, as the high electron density effectively screens out the Coulomb attraction.

When a positron is implanted into a metal it quickly thermalizes. From the incident energy down to the Fermi energy, the dominant means of energy loss is conduction electron scattering. At lower energies phonon scattering dominates. It is in this regime where temperature begins to play a large role. The diffusion of the positron is also determined by phonons, with the diffusion constant D⁺ ~ T^-1/2 . The thermalization time is typically an order of magnitude less than the average lifetime, thus the positron spends the majority of its life diffusing in thermal equilibrium through the metal. The positron is very sensitive to changes in the local electronic environment, and in the course of its diffusion it samples a relatively large volume of the material. Due to its positive charge, it has a high probability of trapping and subsequently annihilating in open-volume defects. This forms the basis of the more familiar condensed matter probes (ACAR, DBS, and PALS), which derive their information from the annihilation radiation. These techniques provide information about the electron density and momentum distribution, as well as the type and concentration of open-volume defects. One may also use reemitted positrons and positronium (RPS,REPELS) to gain information about the material. These studies are motivated by the information about the solid which may be gained from the energy distribution of the emitted positrons. The energy-loss processes involved in inelastic positron emission provide information about the density of states in the unperturbed, i.e. positron-less, system. In addition, the positron work function itself is dependent upon such properties as the crystal face, temperature, intrinsic stress, and the presence of adsorbates. Thus one may also gain information about these properties.

The more widely-applied slow positron techniques ACAR, DBS, and PALS are alike in that they make use of annihilation radiation to provide information about the processes through which the positron interacts with the sample. They share the advantage that the information is carried out of the sample by the gamma-rays; thus one may probe deeply into the bulk of the sample. In ACAR one measures the Doppler shift-induced deviation from collinearity of the two gamma-rays. This deviation is proportional to the center-of-mass momentum of the positron-electron pair. Since the time spent by positrons in condensed matter is so short, there is, on average, only one positron in a sample at any given time. Thus the positron resides near the bottom of its own band in a delocalized Bloch state. It therefore makes a contribution to the pair's momentum which is negligible except at very low temperatures. A major application of ACAR has been to exploit this property to map Fermi surfaces in metals and alloys with high precision.

A major application of positron techniques is the study of defects. Positrons have been shown to be very sensitive to changes in the local electronic environment. Despite the fact that the DeBroglie wavelength of a positron at 300K is an order of magnitude larger than typical inter-core spacings, the positron has a relatively high probability of becoming localized or trapped in vacancies. In fact, the threshold defect density for positron trapping is typically of order 1 ppm. When trapped in a vacancy, the positron's overlap with the core electrons (which have relatively large momenta) decreases relative to that with the conduction electrons. Thus the size of the electron momentum-induced Doppler shift is reduced. This results in a decrease in the deviation from collinearity of the gamma-rays. Thus by measuring the angular deviation, one can derive information about defects in the material. In addition to the reduced angular deviation, the width of the energy distribution of the 511 keV gamma-rays is correspondingly narrowed for trapped positrons. This forms the basis of DBS. This technique has the advantage that it is relatively easily implemented on a small laboratory scale using readily available high-resolution single crystal photon detectors. Conversely, ACAR requires a large, cumbersome apparatus, and a very intense source of positrons. This is due to the stringent angular resolution requirements, which are necessitated by the small angular deviations involved, typically a few mrad.

Defects also tend to increase the positron's lifetime. Since the overlap with electronic wavefunctions is reduced at an open volume defect, the probability of annihilation is also reduced increasing the lifetime. Furthermore, in certain circumstances where the void is large enough and the electronic density is low enough, a positron may combine with an electron to form positronium in the void, tending to increase the lifetime towards the positronium vacuum value. This forms the basis of Positron Lifetime Spectroscopy (PALS) in which the lifetime of positrons is measured. In general several exponential components combine to form the lifetime distribution of positrons in a solid, each corresponding to a type of defect. The magnitude of each component is related to the size of the corresponding defect and the intensity of each component is related to the number of the corresponding defects. Thus, PALS provides information on both the size and number of defects.

Positrons that are implanted into a solid and wander back the surface before annihilation can be spontaneously emitted back into the vacuum. As will be explained below, the same surface dipole layer responsible for positive electron work-functions can, in certain cases, eject the oppositely charged positrons. The energies for electrons and positrons in metals are represented schematically in the figures to the right. In these figures, the energies are drawn approximately to scale. Note that upward-pointing arrows denote positive quantities, with downward-pointing arrows denoting negative quantities. Here the mean electrostatic potential energy in the metal's interior, or crystal zero, is defined as the zero of the Coulomb potential energy due to the nuclei and electron density of the infinite solid. The crystal zero is shifted in energy relative to the vacuum level by an amount equal to the surface dipole energy, D, which is the change in potential energy across the surface dipole or double-layer. The double-layer arises because of the spilling out of the delocalized conduction electron density into the vacuum beyond the surface. The surface dipole makes the dominant contribution to the electron work function.

The work function may be defined as the minimum energy required to remove a particle from the occupied state with the highest energy (neglecting thermal excitations) through a particular surface. The work function is the sum of two terms: the chemical potential, and the surface dipole potential. Note that the surface dipole serves to increase the work function, i.e. bind the electrons more tightly to the metal.

As the positron has the opposite charge as the electron, the surface dipole has the reverse effect on positrons as it does on electrons. That is, it tends to decrease the positron work function. In fact, if D is large enough to overcome the chemical potential, the work function can be negative. This is a mathematical manifestation of the fact the the positron ground state lies higher in energy than the vacuum level. Thus positrons may be spontaneously reemitted from the metal.

Positron reemision forms the basis of moderation (and remoderation) which is vital to the existence of slow positron beams. Most positrons used in research come from radioactive sources. They have both a broad energy distrubution extending, up to hundreds of keV, and a large angular distribution. It is impossible to form a well characterized beam from such a source of positrons using only electromagnetic fields. This is a consequence of Louiville's theorem which states that the volume in phase space of a system of particles cannot be altered by conservative forces alone. The phase space volume of a system of charged particles is the product of the angular divergence times the radius of their trajectories. Electromagnetic fields, being conservative, cannot alter this volume. Therfore one can make a small beam at the expense of a very divergent one, or a parallel beam at the expense of a very big one. Electron beams use apertures to get around this problem, making both angular and radial cuts and throwing away the vast majority of the electrons in the process.

This is not practical for positrons because of their low number to begin with. However, if a positron beam is focussed into a small spot (albiet with great angular divergence) onto a solid that happens to have a negative work-function the positrons quickly thermalize and those that make it back to the surface and are reemitted are reemitted over an area of roughly the same spot size, but biased very perpendicular to the surface because of the large perpendicular velocity kick they recieve from the negative work function. After thermalization the velocity parallel to the surface is governed by thermal processes with energies of order tenths of an eV. Typical work-functions may have energies of order several eV resulting in very perpendicular ejection. The majority of positrons annihilate in the solid. Typically only 1 in 100,000 positrons from a radioactive source ever make it back to the surface and are reemitted (because of the initially large implantation energy of tens or hundreds of keV), however subsequent implantation can result in ten or twenty percent reemission. The beam formed from the reemitted positrons, however, has both smaller angular divergence and a small radial size and is said to be brighter. This process may be repeated several times before the increase in brightness is overwhelmed by the shear loss of positron rate. Finding better moderating materials and developing better moderating techniques is the motivating factor behind a lot of positron research.