A non-conventional method of polymer PAL spectrum analysis by the LT computer program

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Radiation Physics and Chemistry 76 (2007) 291–296 www.elsevier.com/locate/radphyschem

A non-conventional method of polymer PAL spectrum analysis by the LT computer program Jerzy Kansya,, Takenori Suzukib a Institute of Material Science, Silesian University, Bankowa 12, 40-007 Katowice, Poland High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan

b

Abstract A new version of computer program (LT v.10) offers a new concept of polymer spectrum analysis. The theoretical model takes into account a few processes in which positron is involved: the process of slow formation and localization of positronium (Ps), delay formation of positronium from shallow trapped electrons and positron trapping in irradiation-induced centers. The model was applied to two series of spectra for LDPE and HDPE collected at constant temperature (much below the glass temperature) as a function of measurement time. The Ps internal relaxation time and time of localization of Ps in a free-volume center were determined. The results show that the trapping rate of positron is strictly correlated with the amount of the shallow trapped electrons. That suggests a coupling between the positron and electron-trapping centers. r 2006 Elsevier Ltd. All rights reserved. Keywords: Positron annihilation; PALS; Positronium formation; Radiation effect

1. Introduction The LT computer program (Kansy, 1996) has already found wide application in many laboratories, especially to the analysis of PAL spectra of polymers. The program can decompose a polymer spectrum into a few components of discrete, continuous or of mixed character. The program offers a few other options: The wellknown two-state simple positron trapping model, the simple three-state trapping model and the two-state diffusion-trapping model. Recently, a new option serving for polymer spectra analysis was introduced. The option combines two models based on a theoretical description of the mechanisms leading to the positronium (Ps) formation in polymers. The first model is a Corresponding author. Fax: +48 32 259 69 29.

E-mail address: [email protected] (J. Kansy).

continuation of ideas, which, for the first time, have been published by Dauwe et al. (1998). It takes into consideration a process of slow thermalization and slow trapping of Ps into free volume of polymer. The second model was inspired, on the one hand, by many experimental data, which show the role of irradiationinduced and shallow trapped electrons on the efficiency of Ps production (Hirade et al., 1998; Ito et al., 1999; Hirade et al., 2000) and, on the other hand, by the AMOC experiments devoted the studying of delayed formation of Ps (Suzuki et al., 2003a and Dauwe et al., 2003). It was shown (Suzuki et al., 2001, 2003b) that a presence of irradiation-induced radicals is an additional factor influencing the efficiency of Ps formation. This factor is also taken into account in the theoretical description of polymer spectrum. A discussion of mentioned models and an example of their application is the main topic of present paper. However, a few remarks about the possibility of distinguishing between

0969-806X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.radphyschem.2006.03.052

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the discrete and continuous character of a polymer spectrum are also included.

reasonable approximation we propose the following simple function of time, which satisfies the assumptions 1 and 2 (Kansy, et al. 2000),

2. Theoretical model

lo=p ðtÞ ¼ lo=p intr ðtÞ þ lpo ðtÞ,

Among several competitive theoretical models our discussion will be mostly concentrated on the one in which the present version of the program is based. We assume that the positron injected into polymer after a short period of order 1011 s loses its high energy through ionizing collisions in positron spur (Mogensen, 1974) and finally it is situated inside a spherical region containing many ion pairs, so-called blob (Stepanov et al., 2000). The positron can leave the blob as a free particle or, alternatively, after attracting an electron from interior of the blob, it forms a loosely bound pair e+–e. The most probable value of the initial binding energy of the pair is comparable to the intermolecular vibration energy (several tenths of eV), which corresponds to the average size of the pair E15–20 A˚ (Stepanov et al., 2000). Next the following processes take place: 2.1. Slow localization of positronium (SLP) By getting closer the electron and positron decrease their mutual Coulombic energy, which increases the kinetic energy of the species. This kinetic energy is absorbed in the environment, via excitations and vibrations. The process goes through a set of temporary states representing equilibrium between e+–e Coulombic forces, the polarization interactions and the efficiency of the kinetic energy absorption. As a result the e+–e wave function gets more and more localized (Stepanov et al., 2000) and the contact density between e+ and e gets higher and higher values. As a consequence, the intrinsic annihilation rate changes from almost 0 to its characteristic value lo=p intr ð1Þ equal to 1/142 ns1 for ortho–Ps or 1/0.125 ns1 for para–Ps. At the same time, mainly because of exchange repulsion between e inside the pair and the electrons in medium, the pair is being pushed into more and more empty space and its zero-point energy gradually decreases. Because the medium cannot absorb a large portion of energy at a time, the differences between the energy levels in the centers of free volume, successively occupied by Ps, should be of the order of energy of intramolecular vibration. This way the particle is localized slowly and the pick off annihilation rate gradually decreases from a value l(0) to an asymptotic value lð1Þ, where l(0) depends on the electron density in the polymer bulk and lð1Þ is a function of the final value of mean hole size occupied by Ps. A quantitative description of the processes described above is a very difficult quantum-mechanical task. As a

(1)

where    1  exp t=trelax ,   lpo ðtÞ ¼ lð1Þ þ ½lð0Þ  lð1Þ exp t=tlocal ,

lo=p

intr ðtÞ

¼ lo=p

intr ð1Þ

trelax is the time of internal relaxation of Ps, and tlocal the time needed for Ps localization. At the beginning the processes run simultaneously. However, if the holes are sufficiently large, the Ps internal relaxation should be finished before the Ps final localization. Probably the relaxation process is completed only if the size of empty space accessible for the e+–e pair is sufficient to form a Ps atom in its ground state, whereas the Ps localization is continued until the Ps zero-point energy reaches the lowest possible value. For large holes tlocal/trelax is expected to be much grater than 1. Thus the p- or o-Ps component in the PAL spectrum is  Z t  C o=p; Ps ¼ lðtÞ exp  lo=p ðtÞ dt . (2) 0

The right-hand side of Eq. (2) can be expanded into a series of exponential curves:  1 X I kj t , (3) exp  C o=p; Ps ¼ tkj t k;j¼0 kj where sk rj expðs  rÞ; s ¼ lo=p ð1Þtrelax , k! j! r ¼ ½lð0Þ  lð1Þtlocal

I kj ¼

and   t1 kj ¼ lo=p intr ð1Þ þ lð1Þ þ k=trelax þ j=tlocal . In practice, the summation in Eq. (3) can be limited to a few terms only. This way the spectrum has a multiexponential character. Each exponent is described by its intensity Ikj and a time coefficient tkj. The reciprocal  1of the longest time coefficient t1 ¼ 1=142 þ l ð 1 Þ ns is 00 the annihilation rate of the ortho-Ps localized in a freevolume cavity. The coefficient t00 is much longer than all the other coefficients tkj, therefore the ortho-Ps component is easily detected from polymer spectrum. In the 3discrete-component conventional analysis t00 is usually denoted as t3 and I00 as I3. Although t3 has well-defined physical meaning, I3 cannot be interpreted straightforward. It depends on the efficiency of ortho-Ps formation, but also on other factors such as lð0Þ, lð1Þ, trelax and tlocal.

ARTICLE IN PRESS J. Kansy, T. Suzuki / Radiation Physics and Chemistry 76 (2007) 291–296

c+ (0) = 1 - ISLP c+

+

c+t

t

+

1 κ 4

cp

p

1

¼ ltþ ctþ þ m cþ ,

In the stochastic meaning.

(4d)

co

cþ ¼ c0þ expðltÞ,  c0þ m   t  exp lþ t  expðltÞ , ctþ ¼ l  ltþ 1   c0þ k   exp lp t  expðltÞ , cp ¼ 4 l  lp

o

co ¼

3 4 c0þ k

l  lo

½expðlo tÞ  expðltÞ,

(4a)

(4b)

ð5Þ

where l ¼ lþ þ m þ k and c0þ ¼ cþ ð0Þ. The respective spectrum components for free e+, trapped e+ para-Ps and ortho-Ps originated from DFP are Bþ ¼ lþ cþ ;

To explain this mechanism one has to assume that after a long irradiation the sample contains some amount of shallow trapped electrons and also some amount of positron traps (validity of the latter will be discussed later). The free positron, which has left the blob, diffuses through the material losing its kinetic energy and (1) annihilates with the rate l+ or (2) is captured by a trapping center and then annihilates with the rate ltþ , or (3) after meeting on its way a shallow trapped electron, forms a Ps (Suzuki et al., 2003a). Ps is created from a thermalized e+ and a trapped e. Presumably, a great fraction of positronium binding energy (UPs) is used for tearing off the electron from its trap (like in Ore process). As a consequence, the initial kinetic energy of Ps is much smaller than UPs. Therefore, we assume, that, in this case, both Ps formation and its localization are short processes and the para/ortho-Ps annihilation rate can be approximated by its asymptotic value, i.e. lp=o ¼ lp=o intr ð1Þ þ lð1Þ. Let c+, ctþ , cp and co denote fractions1 of free positrons, trapped positrons, para-Ps and ortho-Ps at an instant t. Then the kinetics of the DPF process is expressed, according to Fig. 1, by

dt

dco ¼ lo co þ 34kcþ , dt

where m -is the trapping rate positrons into the positron traps and k is the rate of Ps production due to the trapped electrons. The solutions of Eqs. (4a–d) are

2.2. Delayed formation of positronium (DFP)

dctþ

(4c)

3 κ 4

Fig. 1. The scheme of processes involved in DFP model.

dcþ ¼ lþ cþ  mcþ  kcþ , dt

dcp ¼ lp cp þ 14kcþ , dt

Btþ ¼ ltþ ctþ ;

Bp ¼ lp cp ;

Bo ¼ lo co . (6)

These components are shown in Fig. 2. In the calculations the data determined on basis of an experiment (Section 4) has been used. The delay in para-Ps formation is manifested by the shifted maximum of the para-Ps component. Again the ortho-Ps component is well separated from the other ones. For enough time the long-living component can be considered as an exponential function with the lifetime l1 o . This lifetime has a direct physical meaning and it is equal to t00 (in the SLP model) or t3 (in the conventional three-component decomposition). 100000

Intensity x 106+1



293

10000

e+

1000 o -Ps 100 p -Ps 10 1 -0.5

0

0.5

1

1.5

2

time [ns] Fig. 2. The shapes of spectrum components predicted by DFP model. e+ denotes the sum of free and trapped positron components (i.e. the sum of Bþ þ Btþ see Eq. (7)), p-Ps—the para-Ps component and o-Ps—the ortho-Ps component. The calculations were performed with parameters specified in Table 1 for HDPE.

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In general, the shape function describing a polymer PAL spectrum is the sum of components given by Eqs. (2) and (6),   F ðtÞ ¼ I SLP 14C p;Ps þ 34C o;Ps   ð7Þ þ ð1  I SLP Þ Bþ þ Btþ þ Bo þ Bp . F(t) contains eight parameters: ISLP, tð0Þ  l1 ð0Þ (which 1 t t is assumed equal to l1 þ ), tð1Þ  l ð1Þ, tþ  1=lþ , trelax, tlocal, m and k. The parameters have to be determined by fitting the model to experimental data.

Table 1 The time-independent parameters of the SLP and DFP theoretical models determined from PAL spectra for both series of investigated polymers.

HDPE LDPE

The samples used in this study were commercially available HDPE (crystallinity 66%) and LDPE (crystallinity 43%). The HDPE sample was cooled to 80 K. In this temperature the PAL spectra were measured one by one, each for 1 h. At 68 h the sample was exposed to visible light from xenon lamp and the measurement was continued at 80 K for next 3 h in the light (Ito et al., 1999). The spectra of LDPE were measured in darkness, one by one, at 30 K for 21 h—each spectrum for 1 h (Shantarovich et al., 2003). PAL measurements were carried out using conventional ‘‘fast–fast’’ coincidence system (Suzuki et al., 1995). A positron source 22Na of 1 MBq deposited between two kapton films of thickness 7 mm and sandwiched by two pieces of sample was placed in a cooling system (CW303) made by Iwanti Co. Ltd. The data were analyzed with LT v.10 program. The SLP and DFP models (Section 2) were enclosed directly into the program code. All the spectra for a given polymer were analyzed together. The parameters ISLP, tð1Þ, t(0), trelax, tlocal, and ttþ as well as the prompt parameters and source contribution (12%, 0.386 ns) were common for all spectra in the series. Only m and k could receive unconstrained values for each spectrum in the series. The resolution curve was composed from two gaussians. Their average width (FWHM) was about 300 ps.

4. Results and discussion The fitted values of the time independent parameters, determined for the series of HDPE and LDPE spectra are shown in Table 1. These parameters were assumed common for all spectra of a series. According to the expectation (see remarks at the end of Section 2), the pickoff lifetime of localized positron tð1Þ is very close to t3 determined from the conventional three-component analyses (Ito et al., 1999; Shantarovich et al., 2003). This means that the information of the pickoff ortho-Ps lifetime can be

trapping rate [ns-1]

3. Experimental and numerical analysis of spectra

18 16 14 12 10 8 6 4 2 0

tð1Þ (ns)

t(0) (ns)

ISLP (%)

ttþ (ns)

trelax (ns)

tlocal (ns)

1.30 1.32

0.277 0.334

36.7 48.8

0.376 0.450

0.046 0.076

0.147 0.121

in light

in darkness (a) 1.LDPE

(c)

(b)

2. HDPE

0

20

40 elapsed time [h]

(d)

60

80

Fig. 3. The rate k of Ps production from e+ and a trapped e (solid symbols) in LDPE (a) and HDPE (d), and the positron trapping rate m (opened symbols) in LDPE (b) and in HDPE (c).

obtained independently on the employed theoretical model. However, the probability of Ps formation cannot be determined from the conventional analysis. According to the discussed models, I3 is a very complex function of many factors. Instead, SLP model gives ISLP, i.e. the probability of Ps formation in the blob. The probability of Ps formation due to the DFP process is proportional to k (Figs. 3a, d) and can be determined by integration of Bp and Bo components (Eqs. (6)). Because k increases with the number of trapped electrons in the sample, the efficiency of delayed Ps formation is a function of the irradiation dose (elapsed time). Because at t ¼ 0 the contact density e+–e was expected small, the pickoff lifetime t(0) was assumed equal to the free positron lifetime. The shorter value of t(0) in HDPE seems to be a consequence of higher density of HDPE than density of LDPE. The Ps internal relaxation time (trelax) and the Ps localization time (tlocal) have finite values comparable with para–Ps intrinsic lifetime. According to expectation trelax otlocal . In the theoretical model it was assumed that e+ could annihilate not only from its ‘‘free’’ state but also from a trapped state (Fig. 1). The fits of the model to the experimental data confirmed this assumption. Obtained

ARTICLE IN PRESS J. Kansy, T. Suzuki / Radiation Physics and Chemistry 76 (2007) 291–296

values of the trapping rate (m) versus the elapsed time (t) are shown in Figs. 3b (for LDPE) and Fig. 3c (for HDPE). Fig. 3a and d show the Ps formation rate k as a function of t for LDPE and HDPE, respectively. The low values of k for samples just after cooling (t ¼ 1 h) and after the illumination (t468 h) as well as the increasing of k with t confirm that k reflects the amount of shallow trapped electrons. One can show a strong linear correlation between m and k. The correlation suggests that the probability of positron trapping and the density of trapped electrons are coupled. Maybe the electrons and positrons are trapped in the same region, containing some radicals generated by the source irradiation. When the amount of trapped electrons elevates, the penetration of the damaged regions by the diffusing positron increases, because the trapped electrons attract the positron towards those regions. On the contrary, when the density of shallow trapped electrons is negligible then the positron diffuses randomly and the probability of positron trapping is small because of small probability of penetration the damaged regions by the positron. It is commonly accepted that the longest component of polymer spectrum is smeared because of a distribution of Ps pick off lifetime. That reflects a distribution of sizes of free-volume cavities in polymer. To confirm the prediction a test was carried out with the series of HDPE spectra. All the spectra were added to obtain a spectrum of a very high statistics (about 40 million counts). Then the long living part (t44 ns) of the spectrum was analyzed. According to the conclusion of Section 2, this part of the spectrum should exclusively contain information related to the ortho-Ps annihilation from its localized state. The analysis was performed twice. In the first fitting a single exponential function was used. In the second fitting a log-normal distribution of exponential functions was assumed. The results of these analyses are shown in Table 2. The fitting of lifetime distribution was only slightly better than the fitting of a single exponential curve. In conclusion it can be state that, at least of the discussed case, the ortho-Ps lifetime distribution was negligible and could be neglected in the analysis of the HDPE spectra.

Table 2 The results of fitting a single lifetime component and a distribution of lifetime components into the long-time part of the HDPE spectrum

Single lifetime Lifetime distribution

t (ns)

s (ns)

Fit variance

1.34 1.28

— 0.1870.2

0.994 0.979

295

5. Conclusions The combined models of slow Ps localization and delayed Ps formation well described polymer spectra at low temperatures. The decomposition of polymer spectra into three exponential functions is only a rough approximation. Such decomposition gives correct value of ortho-Ps pickoff lifetime but cannot give the absolute value of efficiency of Ps formation. The obtained results suggest that a great fraction of positrons, which have not formed Ps, annihilate from their trapped states. A strong correlation between the delay Ps formation rate and the rate of positron trapping suggest that the shallow electron traps (involved in the delayed Ps formation process) and the positron traps are interconnected. References Dauwe, C., Consolati, G., Kansy, J., Van Waeyenberge, B., 1998. A new view on positronium in polymers. Phys. Lett. A 238, 379–384. Dauwe, C., Van Waeyenberge, B., Balcaen, N., 2003. Positronium formation in poly(methyl methacrylate). Phys. Rev. B 68, 132202. Hirade, T., Wang, C.L., Maurer, F.H.J., Eldrup, M., Pedersen, N.J., 1998. Visible light effect on positronium formation in PMMA at low temperature. Abstract Book for the 35th Annual Meeting on Radioisotopes in the Physical Science and Industries, Tokyo, Japan, p. 89. Hirade, T., Maurer, F.H.J., Eldrup, M., 2000. Positronium formation at low temperatures: the role of trapped electrons. Radiat. Phys. Chem. 58, 465. Ito, Y., Hirade, T., Hamada, E., Suzuki, T., Ito, Y., 1999. The effect of visible light irradiation on positronium formation in polyethylene at low temperature. Acta Phys. Pol. A 95, 533–538. Kansy, J., 1996. Microcomputer program for analysis of positron annihilation lifetime spectra. Nucl. Instrum. Methods. A 374, 235–244. Kansy, J., Consolati, G., Dauwe, C., 2000. Positronium trapping in free volume of polymers. Radiat. Phys. Chem. 58, 427–431. Mogensen, O.E., 1974. Spur reaction model of positronium formation. J. Chem. Phys. 60, 998–1004. Shantarovich, V.P., Suzuki, T., He, C., Gustov, V.W., 2003. Inhibition of positronium formation by polar groups in polymers—relation with TSL experiments. Radiat. Phys. Chem. 67, 15–23. Stepanov, S.V., Wang, C.–L., Kobayashi, Y., Byakov, V.M., Hirata, K., 2000. What do we learn from the electric field effect on Ps formation in liquids? Radiat. Phys. Chem. 58, 403–409. Suzuki, T., Oki, Y., Numajiri, M., Miura, T., Kondo, K., Ito, Y., 1995. Positronium annihilation in molecular substances. Radiat. Phys. Chem. 45, 657. Suzuki, T., Kondo, K., Hamada, E., Chen, Z.Q., Ito, Y., 2001. Temperature and radiation effects on positronium formation. Radiat. Phys. Chem. 60, 535–540.

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Suzuki, N., Hirade, T., Saito, F., Hyodo, T., 2003a. Positronium formation reaction of trapped electrons and free positrons: delayed formation studied by AMOC. Radiat. Phys. Chem. 68, 647–649.

Suzuki, T., He, C., Kondo, K., Shantarovich, V., Ito, Y., 2003b. The influence of radiation on Ps formation in PE studied by coincidence Doppler-broadening spectroscopy. Radiat. Phys. Chem. 68, 489–492.