Lifetime free positron

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. 

Positron annihilation lifetime spectroscopy (PALS) provides a method for studying changes in free volume and defect concentration in polymers and other materials [1,2]. A positron can either annihilate as a free positron with an electron in the material or capture an electron from the material and form a bound state, called a positronium atom. Pnra-positroniums (p-Ps), in which the spins of the positron and the electron are anti-parallel, have a mean lifetime of 0.125 ns. Ortho-positroniums (o-Ps), in which the spins of the two particles are parallel, have a mean lifteime of 142 ns in vacuum. In polymers find other condensed matter, the lifetime of o-Ps is shortened to 1-5 ns because of pick-off of the positron by electrons of antiparallel spin in the surrounding medium. 

In solids the free positron lifetime r lies in the approximate range 100-500 ps and is dependent upon the electron density. Following implantation, the positrons are able to diffuse in the solid by an average distance L+ = (D+t)1//2, where D+ is the diffusion coefficient. This quantity is usually expressed in cm2 s-1 and is of order unity for defect-free metallic moderators at 300 K (Schultz and Lynn, 1988). The requirement of very low defect concentration arises because the value of D+ is otherwise dramatically reduced owing to positron trapping at such sites. 

Fig. 6.5. Examples of positron lifetime spectra for (a) argon and (b) xenon gases. The argon data are for a density of 6.3 amagat at 297 K. The channel width is 1.92 ns. In (a), (i) shows the raw data, (ii) shows the signal with background removed, (iii) shows the free-positron component and (iv) shows the fitted ortho-positronium component. In (b), the spectrum for xenon is for room temperature and 9.64 amagat and has a channel width of 0.109 ns. The inset shows the fast components as extracted and discussed by Wright et al. (1985).

When an electric field was applied across the chamber some positrons annihilated prematurely, following field-induced drift to one of the electrodes. In this case the free-positron component of the lifetime spectrum was field dependent the maximum drift time, rmd, was given by the end-point of the lifetime spectrum and was due to thermalized positrons which had traversed the entire drift length l. The drift speed was then v+ = 1/rmd and the mobility could be found from... 

Positron chemistry is a specific field which aims at determining which solutes react with e+ to form a bound-state, comparing the related constants and studying the effects of temperature [3, 5, 6, 13, 18, 25, 46-48]. The results are scarce, because the bound-states can only be characterized through AC or DB experiments, which are less used than PALS as it seems, the lifetimes of all e+ bound-states known are very close to those of the free positrons, so that PALS cannot sufficiently distinguish these two states and is therefore unable to provide useful information. 

Theoretical arguments are twofold. On one hand, one may expect that e+ also gets solvated over a time comparable with r . Mobility of solvated particles drastically drops and they simply do not have enough time to meet each other during the free-positron lifetime ( 0.5 ns). Really, corresponding diffusion displacement of e+ is smaller than e+ thermalization... 

The lifetimes of the positron-molecule states rM (see Table 14.2) are considerably longer than that of the free positrons (rc+= 400 ps). rM is longest for PsF, which has the smallest number of electrons in its shell, and, hence, the lowest electron density, and decreases with increasing number of electrons. Seeger and Banhart [25] give an upper limit for the lifetime of positron states where no o-Ps is involved ... 

The linewidth of annihilation from the free-positron state is Doppler-broadening measurements. In lifetime measurements the PsF component hides beneath the o-Ps component which has a similar lifetime. This is a case where the two-dimensional data analysis shows its great advantage As the Doppler broadening of each positron state is determined in its own time regime even positron states with similar features may be seperated from each other. Moreover, a tentative fitting procedure with only the three positron states as in pure water did not come to a satisfactory result with the AMOC histogram of the NaF solution. 

Haidar, B., Singru, R.M., Maurya, K.K., Chandra, S. (1996) Temperature dependence of positron-annihilation lifetime, free volume, conductivity, ionic mobility, and number of charge carries in a polymer electrolyte polyethylene oxide complexed with NH4CIO4 . Phys. Rev. B. 54, 7143. 

This technique, firstly applied to metals and ceramics, has become a popular tool in polymers science for the determination of free volume [4,6-8] and starts to be applied to carbonaceous materials [9-12], Positron studies of porous materials have been predominantly oriented towards the chemical interaction of positrons with gases filling the porosity or with molecular layers adsorbed on the pore surface. Few studies have focused in the relation between annihilation characteristics with pore size and pore size distribution. Only in same cases, the annihilation time and the pore size have been directly related, and most of these studies have been carried out with silica gels [5,13,14], although other materials like porous resins (XADS) [15] have also been studied. In all these studies, it has been observed that the lifetime of positrons (t) increases with pore width. 

The intensity is the second important information which is obtained from PALS technique. The intensity is related with the amount of positrons which annihilates by a given mechanism (free positron, o-Ps, As we have seen, the component with a higher contribution to the total intensity is the first one (L). This component corresponds to the annihilation of positrons on the surface of the porosity. Table 4 shows that in the series of ACFs obtained from CO2 (series CFC) the intensity L increases with bum-off and surface area. However, I2 decreases. These results agree with positron annihilation at a surface level, which contribution increases with burn off. The ACF samples obtained from steam activation are more complex because the component which increases with burn-off is that corresponding to o-Ps annihilation (I2). Moreover, the lifetime of o-Ps (12) decreases with burn-off. 

Thus, the annihilation mechanism for each of the lifetime components of the spectra are of very different nature. The annihilation mechanism which increases with burn-off is different depending on the activating agent used. For the ACFs obtained from CO2 activation (CFC) the main annihilation mechanism corresponds to the annihilation of free positrons with surface electrons (L) in contrast to the case of the ACFs obtained from steam (CFS) where the mechanism that increases with burn-off is the o-Ps annihilation (I2) (Table 4). 

Typically, therefore, a PALS spectrum consists of a minimum of three components the short-lived p-Ps component with intensity 7i and lifetime ti = 125 ps a free positron annihilation component, with intensity I2 and lifetime T2 and the o-Ps component, with intensity I3 and lifetime T3. Theory predicts the ratio /3//1 = 3, but as discussed in Chapter 11, certain effects may lead to a decrease in this ratio. The theoretical basis for relating the o-Ps lifetime to free volume is based on a model proposed by Tao [1972], in which < -Ps is assumed to be trapped in a potential well of... 

A positron lifetime spectrum, as shown in Fig. 9, can usually be dissolved into two components. As indicated schematically in Fig. 10, the short-lived component with an associated lifetime and intensity can be attributed to the annihilation of the free positron, the annlhy.atlon of the products formed in the reaction of "hot" ortho Ps and the self annihilation of para-Ps, whereas the long-lived component displaying a lifetime Tj and intensity I2 is attributed to the annihilation of thermalized o-Ps. 

Positronium Annihilation Lifetime Spectroscopy. Positron annihilation lifetime spectroscopy (pals) is primarily viewed as techniqne to parameterize the imoccnpied volnme, or so-called free volume, of amorphous polymers. In vacuo, the ortho-positronium (o-Ps) has a well-defined lifetime T3 of 142 ns. This lifetime is cut short when o-Ps is embedded in condensed matter via the pick-oflT mechanism whereby o-Ps prematurely annihilates with one of the surroimding boimd electrons. The quantum mechanical probability of o-Ps pick-off annihilation depends on the electron density of the medium, or the size of the heterogeneity. Typically the heterogeneity is assiuned to be a spherical cavity (164,165) so that T3 can be easily related to an average radius R (Ro = R -i- AR) of the nanopore ... 

In a real system, the positron exists in different states. It may annihilate either with valence or conduction electrons of the bulk. These processes give rise to a bulk annihilation rate Ab. It may also be trapped in various defect states Dj where the electron density is smaller than in the bulk, i.e. a single vacancy, a cluster of vacancies, dislocations, impurities etc. Each defect state will be characterized by an annihilation rate Dj- In a vacancy-like defect the trapped-positron lifetime is increased compared to free positrons aimihilating in the bulk, as the electron density is locally reduced. Each defect state leads to a different lifetime Tdj = 1/Ap,. 

We have discussed mostly the temperature dependence of the positron mean lifetime tm. The temperature dependence of each separate lifetime component is more complex to review because the extraction of different lifetime components present in a spectrum is a delicate process. Using the two-state trapping model, most of the authors have limited the analysis of positron lifetime spectra to two lifetime components, after subtracting for source components and a low-intensity long component related to positronium formation at the surfaces. The first component, rj, is ascribed usually to annihilation of free positrons and the second, t2, to positrons trapped by defects. It should be pointed out that the relations frequently used to interpret the lifetime are based on the hypothesis that there is no positron detrapping, which is questionable at temperatures above 200 K. 

Generally, three to four lifetime components are resolved in polymers, and their attribution is as follows. The shortest lifetime component ri with intensity h is attributed to contributions from free positron annihilation (inclusive of p-Ps lifetime). The intermediate lifetime component Z2 with intensity 12 is considered to be due to the annihilation of positrons trapped at defects present in the crystalline regions, or those trapped at the crystalline-amorphous interface boundaries. The longest-lived component T3 with intensity 1, is due to pick-off annihilation of the o-Ps in the free volume cavities present mainly in the amorphous regions of the polymer [42,43]. The simple model of a Ps atom in a spherical potential well of radius R leads to a correlation between o-Ps hfetime and R [70,128-130] ... 

The lifetimes of positrons trapped in small vacancy clusters were calculated using the LDA approach. As the lattice relaxation around the cluster has a relatively small influence on positron annihilation parameters, it was not included in the calculations. The results show that the lifetime of a positron trapped by a divacancy does not differ a lot from that trapped by a monovacancy (it increases by about 10 ps for Fe and 20 ps for Al). However, the lifetime increases rapidly when the cluster grows into a two-dimensional trivacancy and then into a three-dimensional tetravacancy [72]. For large clusters, the lifetime saturates at around 500 ps. The dependence of the positron lifetime in vacancy clusters on the free volume of the cluster expressed as the number of vacancies comprising the cluster is shown in Figure 4.32. 

Mean diffusion length at room temperature for the defect-free material. For the free positron lifetimes, we used the bulk Ufetimes calculated by GGA and the experimental ones shown in Table 4.17. If the non-negbgible defect concentration is included, the parameter L+ will be smaller. 

PAS Data Treatment. The lifetime spectra are resolved into two components, a long lifetime due to positrons trapped at defects (Td) and a short lifetime that mainly comes from free positrons in the matrix. For general interpretation, it can be considered that the positron lifetime (PL) technique is a well-established method for studying open-volume-type atomic defects and defect impurity interactions in metals and alloys. The lifetime of positrons trapped at vacancies, vacancy-impurity pairs, dislocations, microvoids, etc., is longer than that of free positrons in the perfect region of the same material. As a result of the presence of open-volume defects, the average positron lifetime observed in structural materials is found to increase with damage [127,128],... 

This equation differs from the conventional model (6) for positronium-for-ming media in having l/2.5r instead of l/2r as the left hand side term. This form has been dictated by the following considerations (a) The positron annihilation in polyimides reportedly (7) differs considerably from that observed in most polymers. It proceeds from the free or trapped positron states without the formation of positronium atoms (b) Positron lifetime spectra in all of the polyimides (PMDA, BFDA, BTDA and 6FDA-based polyimides etc.) investigated in this laboratory exhibit only two lifetime components. The shorter lifetime (r,) ranges from 100 to 300 picoseconds and arises firom free positron... 

Network properties and microscopic structures of various epoxy resins cross-linked by phenolic novolacs were investigated by Suzuki et al.97 Positron annihilation spectroscopy (PAS) was utilized to characterize intermolecular spacing of networks and the results were compared to bulk polymer properties. The lifetimes (t3) and intensities (/3) of the active species (positronium ions) correspond to volume and number of holes which constitute the free volume in the network. Networks cured with flexible epoxies had more holes throughout the temperature range, and the space increased with temperature increases. Glass transition temperatures and thermal expansion coefficients (a) were calculated from plots of t3 versus temperature. The Tgs and thermal expansion coefficients obtained from PAS were lower titan those obtained from thermomechanical analysis. These differences were attributed to micro-Brownian motions determined by PAS versus macroscopic polymer properties determined by thermomechanical analysis. 

A fascinating insight into the impact that modelling can make in polymer science is provided in an article by Miiller-Plathe and co-workers [136]. They summarise work in two areas of experimental study, the first involves positron annihilation studies as a technique for the measurement of free volume in polymers, and the second is the use of MD as a tool for aiding the interpretation of NMR data. In the first example they show how the previous assumptions about spherical cavities representing free volume must be questioned. Indeed, they show that the assumptions of a spherical cavity lead to a systematic underestimate of the volume for a given lifetime, and that it is unable to account for the distribution of lifetimes observed for a given volume of cavity. The NMR example is a wonderful illustration of the impact of a simple model with the correct physics. 

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