Importance of starting time for defect analysis using positron annihilation lifetime measurements

Conventional positron annihilation lifetime measurements have focused on determining the lifetime and relative intensity of each lifetime component deduced by multi-component analysis or the “mean” lifetime deduced by single-component analysis. So far, little attention has been paid towards the starting time (T0) on the spectrum’s time axis. When analysing lifetime spectra with multiple components using an exponential function with a single-component, there is a difference between the experimental data and the fitted spectrum. Compensating for this difference causes a shift in the T0 value in the fitted spectrum. This study examines the shifts in T0 (ΔT0) in positron lifetime spectrum analyses of metal samples with defects. We conducted simulations of trapping models and experiments with shot-peened stainless steel, verifying that ΔT0 changed depending on the defect concentration even when almost all positrons were trapped (full trap). Therefore, we propose ΔT0 as a new parameter for evaluating positron annihilation lifetime measurements.


Background of research
Positron annihilation lifetime measurement [1][2][3][4] is a highly sensitive method used for probing open volume defects in metals and free volumes in polymers. It is used to analyse atomic vacancies and nanometer scale holes in various structural and functional materials.
When positrons are incident on a metal sample without defects, they are all annihilated in the bulk and the positron lifetime spectrum only has a single-component. If the defect, i.e. a single vacancy, is introduced into the metal, some of the positrons will be captured by the defect and have a longer lifetime, causing the positron lifetime spectrum to have two components. In such cases, the lifetime and relative intensity of each component can be obtained by analysing the lifetime spectrum using an exponential function with two components. Applying a trapping model 5) to the analysis results enables the lifetime, which is correlated with the defect size, and trapping rate, which is proportional to the defect concentration, to be obtained. However, multi-component analyses of positron lifetime spectra are often difficult. For example, various kinds of defects in the sample, such as dislocations, vacancies or voids, can produce several positron lifetime components of only slightly different lifetimes or high defect concentrations can lead to the shortest lifetime component with an unresolvably short lifetime, as in the case of shot-peened stainless steel to be reported in this paper. When multi-component analyses are difficult, the spectrum is sometimes analysed in terms of an exponential function with a single-component. However, such single-component analyses limit the information that can be obtained. Assuming that the positron lifetime is continuously distributed, i.e. following a log-normal distribution, they can identify the distribution's width and "mean" lifetime, but this information is hardly sufficient.
Conventional positron annihilation lifetime measurements have focused on the lifetime and intensity of each component deduced by multi-component analysis or "mean" lifetime deduced by single-component analysis, and little attention has been paid towards the starting time (T 0 ) on the spectrum's time axis. However, as shown in Fig. 1, when analysing a positron lifetime spectrum with multiple components using an exponential function with a single-component, there is a difference between the spectrum and fitted data. To compensate this difference, T 0 is predicted to shift in the analysis result. In the presence of one-type defects, The T 0 shift is expected to increase from 0 ps, assume a maximum at a certain defect concentration and then return to 0 ps with increasing defect concentration. Yet to accurately measure the change in T 0 (ΔT 0 ), care must be taken to prevent the time axis shifting from measurement to measurement.
Herein, we used two-state, i.e. bulk and dislocations, and three-state, i.e. bulk, dislocations and vacancies, trapping models to simulate the positron lifetime spectrum. Then, we analysed the positron lifetime spectra obtained using an exponential function with a single-component to obtain the "mean" lifetime and ΔT 0 . We also measured the positron annihilation lifetimes of SUS304 stainless steel samples with defects introduced by shot-peening using a system that produces little shift in the time axis (T 0 ) of the positron lifetime spectrum, 6) comparing these with simulated results obtained under the same conditions.

Simulation evaluation
Using Excel VBA, we created positron lifetime spectra that duplicates the trapping model via Monte Carlo simulations, and analysed the same. Figure 2 shows a block diagram for the simulations, which model the trapping of positrons by one type of defect and their annihilation by two states, i.e. in the bulk and by dislocations. In a similar manner, we also simulated positron annihilation by three states, i.e. in the bulk and by two types of defect (dislocations and vacancies). Figure 3 shows an example of a simulated positron lifetime spectrum.
We used the trapping model to create two-state positron lifetime spectra with bulk and dislocation components and three-state positron lifetime spectra with bulk, dislocation and vacancy components, with trapping rates (given by the product of the specific trapping rate and the defect concentration) ranging from 10 6 to 10 13 s −1 . Each lifetime was based on the approximate value for pure steel 7) : 100 ps for positrons annihilated in the bulk, 150 ps for dislocations and 200 ps for vacancies. In addition, the full width at half maximum (FWHM) time resolution was 150 ps, total positron count was 200 000, channel width was 10 ps/ch, and background was 2 counts/ch. In the three-state model, the trapping rates for dislocations and vacancies were the same.

Experimental method
Shot-peening 8) involves blasting to harden the material's surface (<200 μm) by introducing many dislocations. Herein, we used the shot-peening machine shown in Fig. 4 and SUS304 stainless steel samples that had undergone "Bright Annealing" to remove defects. We fixed each sample onto a sample stage on top of a rotating stage to ensure accurate shot-peening. Table I lists the conditions used. We prepared nine different samples by shot-peening for different times, i.e. 0, 12, 36, 72, 120, 240, 480, 960 and 1920 s.
We used a PSA-TypeL-II 9) system (Toyo Seiko Co. Ltd.) for the positron annihilation lifetime measurements. This system uses the anti-coincidence (AC) method [10][11][12] and, as shown in Fig. 5, it can measure the positron annihilation lifetime of a single specimen. The sample is placed on a stage with a 22 Na positron source (∼0.8 MBq) enclosed in 7.5 μm  Simulating a two-state positron lifetime spectrum. Herein, T sg is the time spread due to finite instrumental resolution, which is given by a random number that follows a single-Gaussian distribution. In addition, T bulk , T trp and T def are the times when positrons are annihilated in the bulk, trapped by defects and annihilated by defects, respectively. These are given by random numbers that follow exponential distributions based on the assumed positron lifetime in the bulk, positron trapping rate and positron lifetime in defects. T ◇ and T □ correspond to the times when positrons are annihilated in the bulk and by defects, respectively. Finally, T ○ is the sum of T ◇ and T □ . Three-state positron lifetime spectra were simulated in a similar manner. thick Kapton ® . The positron lifetime is measured based on the time interval between the initial 1.27 MeV γ-ray, emitted at almost the same time that the positron is created by the 22 Na source, and the 0.511 MeV γ-ray emitted when the positrons are annihilated. For positrons emitted towards the sample, the aforementioned time interval is recorded directly. Positrons emitted in the opposite direction are detected by a system consisting of a plastic scintillator and photomultiplier tube, and the detection signals are used for AC processing. In this way, we can obtain lifetime data only for positrons that are annihilated in the sample and in the source. In this system, the locations of the source, sample stage, two γ-ray detectors and positron detector are all fixed. Consequently, the time axis rarely shifts from measurement to measurement. In fact, the standard deviation of T 0 over five measurements of the SUS304 sample without shot-peening (repeatedly placing the sample and recording 1 M counts) was 0.28 ps.
The measured positron lifetime spectrum was analysed using a single-component, and the "mean" positron lifetime and ΔT 0 were obtained. As for the resolution, the single-Gaussian component was set to be free, whereas the source component lifetime (Kapton ® lifetime) and intensity were fixed as 380 ps and 22%, respectively. The somewhat high intensity of this source component is due to that positrons annihilated in the source are not eliminated by AC processing. 11) The IPALM 9) software included with the PSA system was used for the analysis. Figure 6 shows a single-component analysis of the simulated positron lifetime spectra, based on assuming that the time resolution is single-Gaussian and treating its half-width as free. Here, the right-hand axis shows "−ΔT 0 " since the shifts ΔT 0 from the starting time set by the simulation were negative. The "mean" positron lifetime increased with the trapping rate before reaching a plateau. In the two-state (bulk and dislocations) model, the lifetime increased from 100.2 to 150.9 ps before becoming almost constant. In the three-state (bulk, dislocations and vacancies) model, it increased from 99.5 to 175.2 ps before becoming almost constant. Meanwhile, the −ΔT 0 values initially increased and then decreased. In the two-state model, they increased from 0.4 to 12.5 ps before decreasing. In the three-state model, they increased from 0.4 to 19.3 ps before decreasing. The −ΔT 0    values at a trapping rate of 10 13 s −1 were 0.2 ps (almost 0 ps) in the two-state model and 2.1 ps in the three-state model. The incomplete recovery of ΔT 0 in the three-state model is likely due to the multi-component nature of the positron lifetime spectrum. As it was assumed that the positron trapping rates due to dislocations and vacancies were the same, the positron lifetime spectrum never becomes singlecomponent no matter how high the defect (dislocation and vacancy) concentrations are. Interestingly, although the "mean" positron lifetime was almost constant above a trapping rate of 10 11 s −1 , as shown by the dashed line in the figure, −ΔT 0 decreased. Figure 7 shows the results of analysing the positron lifetime spectra obtained from the SUS304 samples after shot-peening. Herein, −ΔT 0 is the shift from the T 0 value of the sample that was shot-peened for 0 s. Starting from 102.8 ps, the "mean" positron lifetime monotonically increased with the shot-peening time, reaching 169.9 ps at a time of around 240 s before becoming almost constant. Meanwhile, −ΔT 0 increased between 0 and 36 s, then decreased between 36 and 1920 s. When the shot-peening time was more than 240 s, the "mean" positron lifetime was almost constant, but −ΔT 0 monotonically decreased.

Discussion
Simple simulations and experiments using shot-peened samples showed that there is a T 0 shift (ΔT 0 ) in the analysis results when analysing multi-component positron lifetime spectra using an exponential function with a single-component. Therefore, to further examine the relation between ΔT 0 and defect conditions, we created more realistic positron lifetime spectra via simulation and compared them with the experimental results.

Comparison of experimental and simulated results
The experimental positron lifetime spectra include a source component from the Kapton ® enclosing the 22 Na source. Therefore, we first assumed that the positron lifetime spectra included source components and involved three states, i.e. bulk, dislocations and vacancies, simulating them under the below conditions. The positron lifetimes due to the bulk, dislocation, vacancy components, FWHM time resolution and channel width were the same as in the simulations in Fig. 4. We added a source component due to the 22 Na enclosure (Kapton ® ) with a positron lifetime of 380 ps and 20% intensity. The total count was 1 M and the background was 10 counts/ch. Again, the trapping rates for dislocations and vacancies were the same, and we considered 12 rates ranging from 10 6 to 10 13 s −1 . We analysed the resulting positron lifetime spectra and computed the "mean" lifetime and −ΔT 0 as a function of the trapping rate. The analyses were conducted under the same conditions as in the experiment shown in Fig. 7; except for the inclusion of the source component, whose lifetime and intensity were fixed at 380 ps and 20%.
Results of the above simulations are shown in Fig. 8, alongside the experimental results from Fig. 7. The lower horizontal axis shows the simulated trapping rate, whereas the upper axis shows the experimental shot-peening time. These have been scaled so that the variations in the "mean" positron lifetime for the experimental and simulated results are similar to each other. This figure shows 10 spectra with low trapping rates (<10 11 s −1 ), taken from the 12 simulated spectra. Figure 8 shows that the simulated results, despite being based on the simple assumption that the positron trapping rates due to dislocations and vacancies are the same, reproduced the changes in −ΔT 0 obtained in the experiments. The peak −ΔT 0 values were 15.5 ps and 9.9 ps in the simulations and experiments, respectively, but various factors can cause changes in ΔT 0 , just as it changed from 19.3 ps (Fig. 6) to 15.5 ps (Fig. 8) after the source component was added to the simulations. For this reason, a difference of a few picoseconds in the peak ΔT 0 value is unlikely to represent an essential difference between the annihilation processes in the simulations and experiments.   Figure 9 shows the relation between the "mean" positron lifetime and −ΔT 0 . The error bars represent three times the standard deviation (±3σ) seen in the corresponding analyses (for a total of about 1 M counts). For "mean" positron lifetimes above 160 ps, we observe little change in the "mean" lifetime compared with the error bars, but significant changes are observed in −ΔT 0 . This suggests an effective use of ΔT 0 in the evaluation of high defect concentrations with large trapping rates.

Comparison between the lifetime distribution width and ΔT 0
The positron lifetime spectrum of a metal is usually expressed as a sum of discrete lifetime components, but assuming a spectrum with a continuous positron lifetime distribution means the width of the distribution provides additional information about the positron annihilation process. When analysing positron lifetime spectra with multiple components using an exponential function with a singlecomponent, ΔT 0 reflects the difference between the actual and the fitted data, and can potentially show similar changes to those in the lifetime distribution width. We therefore compared the lifetime distribution width and ΔT 0 .
The IPALM 8) software included with the PSA system was used to analyse the lifetime distribution width. This analysis assumed that the distribution was log-normal and obeyed the following equation ⎛ where f (x) is the lifetime distribution, x is the positron lifetime and σ is the lifetime distribution width. We analysed the lifetime spectra from the simulation shown in Fig. 6 and obtained the corresponding lifetime distribution widths (Fig. 10).
The lifetime distribution width monotonically increases with the trapping rate before decreasing again. The lifetime distribution width at a trapping rate of 10 11 s −1 is approximately 0 ps and 27.8 ps for the two-and three-state model, respectively. However, it does not change much for trapping rates of more than 10 11 s −1 , whereas −ΔT 0 decreases significantly (Fig. 6). Therefore, although the lifetime distribution width and ΔT 0 are both parameters that reflect the positron lifetime spectrum's complexity, their defect dependencies are essentially different.
Based on the three-state trapping model, for trapping rates of more than 10 11 s −1 , the dislocation and vacancy components of the positron lifetime spectra both exceed relative intensities of 49%. For this reason, the lifetime distribution width and "mean" positron lifetime do not change significantly for such trapping rates. On the contrary, the relative intensity of the bulk component (lifetime < 5 ps) decreases from 2.04% to 0.02% as the trapping rate increases from 10 11 to 10 13 s −1 . Since ΔT 0 is a parameter that expresses the starting time of the lifetime spectrum, where short lifetimes make large contributions, slight changes in the bulk component can have a significant impact. Figure 11 plots the changes in the positron lifetime distribution widths of the SUS304 samples as a function of the shot-peening time, along with the "mean" positron lifetimes and −ΔT 0 values (taken from Fig. 7). The lifetime    (Fig. 7) for shot-peened SUS304. Based on the scaling of the horizontal axes in Fig. 8, shot-peening for about 800 s corresponds to a trapping rate of 10 11 s −1 (dashed line). distribution widths after shot-peening for 960 and 1920 s are approximately 33 ps. Based on the horizontal axis in Fig. 8, 800 s of shot-peening corresponds to a trapping rate of 10 11 s −1 (dashed line in Fig. 11). In the simulated positron lifetime spectra based on the two-state model (Fig. 10), the lifetime distribution width was 0 ps for trapping rates of 10 11 s −1 or higher. Therefore, the behaviour of the lifetime distribution width for shot-peened SUS304 is clearly different from that predicted by the two-state model simulation. In addition, −ΔT 0 decreases slightly for shot-peening times of approximately 800 s or more (dashed line in Fig. 11), indicating increased defect concentrations. It can therefore be concluded that, in SUS304 stainless steel with defects introduced by extended shot-peening, there are at least two types of defect (dislocations and vacancies), and their concentrations increase with the shot-peening time. In shotpeening, the aim is to introduce a sufficient concentration of dislocations, and ΔT 0 is useful for studying defects in shotpeened materials.

Conclusion
This research has found that ΔT 0 is a practically useful parameter, although it does not have a simple physical meaning. Similar to the lifetime distribution width, ΔT 0 reflects the complexity of the positron lifetime spectrum, but its defect dependency is different from that of the lifetime distribution width, and it is particularly useful for evaluating high defect concentrations.
In the positron annihilation lifetime measurement method, the suitable defect concentration range is said to be approximately 1-100 ppm, and defects with concentrations of 100 ppm or higher are understood to trap all the positrons (full trap). We found that ΔT 0 is sensitive to changes in the positron lifetime spectrum at high defect concentrations, where the "mean" positron lifetime becomes almost constant, and can be particularly useful for analysing defects in materials with defect concentrations of 100 ppm or higher. Based on the positron lifetime distribution widths and ΔT 0 values for SUS304 stainless steel after extended shotpeening, at least two types of defect were present, in increasing concentrations. In the future, we plan to consider using ΔT 0 for analysing defects in metals produced by fatigue tests and radiation.