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Positron annihilation lifetime spectroscopy (PALS) is used to study the defect structure (namely open volume defects) of materials, and the technique has been used and refined for many years [34,35]. The lifetime of a positron is measured from a known starting time, until a characteristic 511keV photon from positron-electron annihilation is detected, re- peated over many measurements. Analysis of these lifetimes gives insight into the physical properties exhibited by the material and in some cases the defect size can related to the positron lifetime, for instance using the Tau-Eldrup model [36–41].

time within the material, ortho- and para- positronium characteristic lifetimes if positron- ium formation is possible within the the material, and a number of further lifetimes which come from defects. The lifetimes from defects inside a material can range from specific lifetimes, for example the di-vacancy in silicon has a positron lifetime of 320ps [42,43], or a range of lifetimes for materials with a distribution of different sized defects. Comparison of the presence of these lifetime characteristics in a material can reveal information on the internal structure of the material sample.

Typically there are two types of PALS measurement, bulk PALS [44] and pulsed beam PALS [45]. The two measurements differ in experimental set-up and the information gained from the results, however theformof the results is the same. Bulk PALS systems are much simpler than pulsed beam PALS and use as-emitted positrons from radioactive decay. The positrons are therefore high energy with a large distribution of energies, meaning that they will annihilate throughout the entire target and the bulk - hence the name, “bulk PALS”. Pulsed beam PALS systems control the implantation energy of the positrons and therefore probe only a selected region of the target, using a pulse of positrons which gives a specific start point for the lifetime measurement.

2.4.1 PALS Data

A PALS spectrum is a histogram of positron lifetimes as observed from positrons implanted into a particular material of interest. A typical histogram often contains 1₋3.5_×106

counts. The spectrum typically shows a Gaussian resolution function convoluted with a series of exponential decays for each present feature as mentioned previously.

The Gaussian resolution function is formed by the resolution in the detection system in the experiment convoluted with the temporal width of the positron pulse.

To extract information from a PALS spectra, curve fitting programs are used to fit the convolution of the resolution function and the positron lifetimes. The fitted exponential decays give the lifetimes of positrons inside the material. The intensity is the amplitude of the exponential decay. The generalised form of this is:

I0e−t

2_/2σ2

Figure 2.7: An example PALS spectra, reproduced from the work by Saarinen and Ranki [46]. Here the PALS spectra of float-zone (Fz) refined silicon and Czochralski (Cz) grown silicon are compared after irradiation with 2MeV electrons. From the original work: “Positrons annihilate in the as-grown sample with a single lifetime of 220 ps corre- sponding to delocalized positrons in the lattice. In the irradiated samples the experiments reveal vacancies with positron lifetimes of 250 ps (V–As pair in Cz Si:As sample doped with [As] = 1020 _cm−3_{) and 300 ps (divacancy in undoped FZ Si sample)”}

where_∗denotes the convolution. The coefficients are determined by fitting the spectrum, where I0e−t

2_/2σ2

is the instrument function, and the Iie−τit terms are the lifetimes (τi)

and corresponding intensities (λi). An example of this data and fitting process is shown

in figure 2.7.

The intensity correlates to the relative presence of that lifetime, and in the case of lifetimes arising from vacancies, correlates to the density of vacancies with that particular positron lifetime. It can be thought of as the relative concentration of defects, although a direct relationship between I and defect concentration is difficult to quantify.

2.4.2 Bulk PALS Systems

In a bulk PALS experiment, the material sample to be analysed is sandwiched between

22_{Na deposited on Kapton foil, and measurements are made using fast scintillation detec-}

tors configured to collect the start and stop signals. The ‘start’ signal for the positron lifetime is given by the 1.27keV photon which accompanies the birth of the positron from

the radioactive disintegration of a 22_{Na atom into} 22_{Ne. The positron is emitted 3.7 pi-}

coseconds before the 1.27keV photon, see figure 2.3. These systems have good resolution and can usually determine the intrinsic positron lifetimes of materials, even for metals, which can be 100ps or less.

Since the positrons emitted in the decay of22_{Na have a large energy distribution and}

are also of high energy, positrons that annihilate in the material will do so throughout the whole sample without the possibility for singling out a particular depth or region of interest within the sample. Additionally, since the22_{Na is deposited in some foil or other}

layer outside of the sample, the material of the foil itself will interact in some way with the emitted positrons, as radioactive decay is isotropic in emission direction. Positrons may annihilate within the foil that they are prepared on, which can add complications to analysing to the collected PALS spectra. The method also requires a lot of preparation since a new radioactive source needs to be deposited on a foil for each sample measured.

2.4.3 Pulsed Beam PALS

Pulsed beam PALS forms positrons from a source into a controllable beam, and implants the positrons into the target material in pulses at a well defined energy. The pulsing of the beam gives the start signal for the lifetime measurement, and the annihilation photon is again used as the stop signal. The pulsed beam system allows for depth profiling targets with PALS since the energy of the positron pulse is defined by the experimental apparatus. The simple model for the implantation depth of the positron beam is discussed in section

2.3.1.

The implantation distribution of the positrons is determined by the interaction energy and so the energy of the positron pulse can be tailored such that annihilating positrons sample regions of interest within the material being investigated. Preparation of samples by irradiation can be systematically investigated from surface through to bulk.

Difficulties in these systems typically occur in the timing resolution of the experiment, as this is determined by the temporal pulse width of the positrons. Trapping, bunching, and accelerating positrons to form a pulsed beam is difficult task and many solutions have been developed for this [47–50].

2.4.4 Resolution of Lifetimes

The ability to resolve different lifetimes in a PALS spectrum depends on the full-width at half-maximum (FWHM) of the resolution function in the experiment, as well as the number of counts in the spectra. It is important to note that the FWHM does not limit the experiment to only resolving lifetimes the same as or longer than the temporal width of the FWHM. Collecting more data can allow you to resolve lifetimes shorter than the inherent pulse width. However, in Poissonian statistics, the uncertainty scales as√n(the number of counts in the spectra), so diminishing returns will limit the practical benefit to collecting more data for higher resolution. The impact of resolution and spectra total counts was investigated by Yamawaki et al. [51].

2.4.5 The Tao-Eldrup Model

The relationship between positron lifetime and vacancy size was determined intially in a semi-empirical model, the Tao-Eldrup model, and later reconciled in a more theoretical framework relating to the quantum-mechanical size of the Ps wavefunction and in-material cavity where it annihilates [38,39]. This work has more recently been reconsidered and a fully quantum mechanical model was developed which agreed extremely well with the previous Tao-Eldrup model [34,52]. Figure 2.8 illustrates the relationship between pore dimension and positron lifetime for spherical or rectangular pores.

10-1 100 101 102 103 Pore dimension (nm) 0 20 40 60 80 100 120 140 160 P o si tr o n li fe ti m e (n s) Spherical model Rectangular model

Figure 2.8: TheTao-Eldrup model, reproduced by the work by Wada and Hyodo in [41]. Here the comparison between rectangular and spherical pores made, and shows the differ- ence in the two assumptions.

It can be seen that the Tao-Eldrup model is not applicable to all systems with positron lifetimes, and is un-physical for positron lifetimes below 400ps. For this reason it is reliably used to relate positron lifetimes in molecular or organic materials, but not metallic materials. The Tao-Eldrup model asymptotes to a lifetime of_≃142ns, which is the lifetime of ortho-positronium in vacuum.