Diffusion positronium

The sizes and concentration of the free-volume cells in a polyimide film can be measured by PALS. The positrons injected into polymeric material combine with electrons to form positroniums. The lifetime (nanoseconds) of the trapped positronium in the film is related to the free-volume radius (few angstroms) and the free-volume fraction in the polyimide can be calculated.136 This technique allows a calculation of the dielectric constant in good agreement with the experimental value.137 An interesting correlation was found between the lifetime of the positronium and the diffusion coefficient of gas in polyimide.138,139 High permeabilities are associated with high intensities and long lifetime for positron annihilation. 

PALS is based on the injection of positrons into investigated sample and measurement of their lifetimes before annihilation with the electrons in the sample. After entering the sample, positron thermalizes in very short time, approx. 10"12 s, and in process of diffusion it can either directly annihilate with an electron in the sample or form positronium (para-positronium, p-Ps or orto-positronium, o-Ps, with vacuum lifetimes of 125 ps and 142 ns, respectively) if available space permits. In the porous materials, such as zeolites or their gel precursors, ort/zo-positronium can be localized in the pore and have interactions with the electrons on the pore surface leading to annihilation in two gamma rays in pick-off process, with the lifetime which depends on the pore size. In the simple quantum mechanical model of spherical holes, developed by Tao and Eldrup [18,19], these pick-off lifetimes, up to approx. 10 ns, can be connected with the hole size by the relation ... 

In contrast to the case for metals, positronium can be formed in the bulk of many insulators and molecular crystals, and any positronium which subsequently diffuses to the surface can be emitted into the vacuum with a kinetic energy < — Ps, where (f)Ps is the positronium work function. Its value can be expressed in terms of the binding energy of the positronium when in the solid, EB, and the positronium chemical potential, /xPs, as (Schultz and Lynn, 1988)... 

Perhaps of more general applicability for the study of the properties of positronium is its production by the desorption of surface-trapped positrons and by the interaction of positrons with powder samples. According to equation (1.15) it is energetically feasible for positrons which have diffused to, and become trapped at, the surface of a metal to be thermally desorbed as positronium. The probability that this will occur can be deduced (Lynn, 1980 Mills, 1979) from an Arrhenius plot of the positronium fraction versus the sample temperature, which can approach unity at sufficiently high temperatures. The fraction of thermally desorbed positronium has been found to vary as... 

Another model of positronium formation, the so-called spur model, was originally developed by Mogensen (1974) to describe positronium formation in liquids, but it has found some applications to dense gases. The basic premise of this model is that when the positron loses its last few hundred eV of kinetic energy, it creates a track, or so-called spur, in which it resides along with atoms and molecules (excited or otherwise), ions and electrons. The size of the spur is governed by the density and nature of the medium since these, loosely speaking, control the thermalization distances of the positron and the secondary electrons. It is clear that electrostatic attraction between the positron and electron(s) in the spur can result in positronium formation, which will be in competition with other processes such as ion-electron recombination, diffusion out of the spur and annihilation. 

Assuming that all positrons in the swarm have energies below the positronium formation threshold and that only elastic collisions and annihilation are possible, the speed distribution may be derived theoretically as the solution of the following diffusion equation (Orth and Jones, 1969) ... 

Brandt, W. and Paulin, R. (1968). Positronium diffusion in solids. Phys. Rev. Lett. 21 193-195. 

Eldrup, M., Vehanen, A., Schultz, P.J. and Lynn, K.G. (1985). Positronium formation and diffusion in crystalline and amorphous ice using a variable-energy positron beam. Phys. Rev. B 32 7048-7064. 

Positronium diffusion free state vs bubble state. The basic equation that has been widely used in Ps chemistry literature for the Ps diffusion-... 

Principles and Applications of Positron and Positronium Chemistry 5.5.2 Diffusion-recombination stage... 

The difference in the annihilation ratio for positronium at the surface and in the sample is used to measure the effective range of positronium. Implantation profiles for a range of incident energies (density 1 g/cm3) were calculated. In this simulation the fraction that stops within a diffusion length of the surface can reach it and annihilates into 3 photons the remainder annihilate 10% into three photons and 90% into two photons, as shown in Figure 7.4. The fractions are chosen as an example. A short effective range appears as a sharp transition from surface measurement to bulk measurement value. 

For a fixed temperature the diffusion length is constant, while the introduction of pores permits the motion of positronium to the surface from deeper in the sample. For isolated and closed pores this change in range is determined by the size of the pores. When the pores connect, the length of the connected chain becomes the dominant value. As percolation is reached, this length rises sharply. 

For atoms with a finite lifetime the diffusion length and the diffusion coefficient are linked via the lifetime of the particle. Here, we refer to the mean depth from which positronium can leave the sample as the escape depth. 

In addition to the effective diffusion length, which determine the curvature in the depth dependent data, the amount of positronium that is formed plays a role. Two effects contribute, one of which is energy dependent [26]. 

A simple fit of the data with the product of an exponential association and an exponential decay to estimate the escape depth, overestimates the escape depth by folding the positron implantation profile and diffusion into the fitting parameters [30], A more appropriate numerical fitting method based on the diffusion equation was used to take both the implantation profile and diffusion into account [31]. When it is applied to the 3-to-2 photon ratio data suitable absorbing boundary conditions need to be included. The results for the escape depth are shown in Figure 7.8 [30]. In addition to the diffusionlike motion of positronium in connected pores, positrons and positronium diffuse to the pores. 

The contribution of positron diffusion length (L+ = 10 nm [22]) was removed from the escape depth values. The diffusion constant in a material is a function of diffusion length and annihilation rate D = L2X. Here, the rates for positrons and positronium are similar (X 2 ns 1). Thus the measured combined effective diffusion length of positrons L+ and positronium escape Lesc is l eff = L2+ + L2esc [30],... 

Figure 7.8 Escape depth from which pores are connected to the surface versus porogen load. The contribution of positron diffusion was removed from the fitted diffusion length to obtain the positronium escape depth. At 50% porogen load the escape depth exceeds the thickness of the samples. The present model does not include this and becomes...

It is expected that the sensitivity of positrons and positronium to changes in the matrix material will be used extensively in the near future. For example, tantalum-silicon-nitride was found to be an effective diffusion barrier between copper and silicon dioxide [34]. The applicability to porous materials needs to be checked. Work has been carried out, for example, by Gidley et al. on TiN diffusion barriers [35],... 

Since the positron traps in vacancy like open volume and positronium in pores, it will be possible to selectively detect impurities next to vacancies [71]. For metal indiffusion experiments one can design test structures of silicon, metal, low-/ layer samples. Two detector coincident measurements would be performed as a function of temperature and time to observe the chemical signature of the metal in the low-/ layer. The effectiveness of diffusion barriers can be tested by depositing the barrier prior to the low-/ layer. 

These 3y o-Ps fraction profiles were analyzed in a manner similar to the models used for Ps diffusion in ice [31], and for Ps diffusion in low-k films [26]. In the previous models, the initial positron deposition profile is an exponential function with respect to the depth. Although an analog expression of the out diffusion probability can be given by the exponential, the deposition profile generally is not believed to be exponential. In the analysis decribed below, a Makhovian distribution was used to represent the initial positron deposition profile. Let F to be the fraction of o-Ps particles that diffuse out into the vacuum, proportional to the 3g/2g ratio presented in Figure 9. It is given by the initial positronium yield, fn, multiplied by the diffusion probability, J(E) ... 

For the porous MSQ film templated by the F88 triblock copolymer porogen, which has the largest molecular mass in the samples studied, die positronium diffusion length was found to be the shortest and the pore size the smallest. 

Other important examples which exhibit both confinement and diffusion in the classical dynamics of their cyclic collective coordinates are the positronium [20] and the excitonic [14] atom. Because of the comparable masses of the two particles in both cases the mean CM velocity as well as the diffusion constant are orders of magnitude larger than the corresponding values of the hydrogen atom. 

In the case of implantation depths below approx. 50 nm, Iq.ps is reduced - the lower the positron implantation depth, the lower is Iq-ps- This reduction of io Ps can happen if fewer free electrons are available for the ortho-positronium formation. This is the case for the shorter spur of the positron due to a lower kinetic energy or if free electrons diffuse to the surface. Additionally, this Io.p reduction can be interpreted in terms of an out-diffusion of ortho-positronium [19]. In addition to these arguments, which are specific to the PALS method itself, depth-de-pendent properties in the top 10 nm of the epoxy, such as different densities of electron acceptor groups, could also play their part in the observed decHne in Io.ps. 

Regarding the average lifetime of ortho-positronium to.ps for the unaged epoxy film on Au substrate in Fig. 29.2, Tq-ps is constant within the experimental scatter for positron implantation depths above approx. 20 nm. This reflects constant free-volume void sizes, independent of the sample depth. A shallow maximum around 150 nm and a decrease in x. p above 400 nm are not significant. On the other hand, for very low depths up to 20 nm, a significant increase in To-Ps is observed. This can be interpreted as the effect of a surface region in the order of less than 20 nm wide where the sizes of the free-volume voids increase [19]. However, a diffusion of ortho-positronium to the epoxy surface and into vacuum or an impediment of the positronium formation due to a reduced... 

It is necessary to mention that the positron, entering into the solid matter, forms a pair with the electron (positronium Ps), which itself (Ps) migrates to the solid phase (about 10 -10 s). Meeting with defects of crystal structure, the electron and positron annihilate. This method can be used for the determination of defect and size distribution in solids. The diffusion coefficient of Ps is 0.1 cm /s. 

A noteworthy conclusion in the publication cited is that the explanation of the deviation from a linear correlation may be the interaction between the iodine and the individual solvents, i.e., the formation of charge-transfer complexes. This would alter the diffusion coefficients, and additionally, positronium may react at different rates with free iodine and with iodine in the complex form. 

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