BAS, C

Dauwe, Charles

Djourelov, Nikolay

Laforest, Nicolas

PALACIO GOMEZ, CA

Ghent University Academic Bibliography

phys. stat. sol. (c) 4, No. 10, 3710–3713 (2007) / DOI 10.1002/pssc.200675732  © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim On the consistency between positron annihilation lifetime and Doppler broadening results in polypropylene N. Djourelov*1,2, C. Dauwe1, C. A. Palacio1,3, N. Laforest1,4, and C. Bas4 1 Department of Subatomic and Rad. Physics, Ghent University, Proeftuinstraat 86, 9000 Ghent, Belgium 2 Institute of Nuclear Research and Nuclear Energy, 72 Tzarigradsko Shoosse, 1784 Sofia, Bulgaria 3 Institute of Physics, University of Antioquia, A. A. 1226, Medellin, Colombia 4 University of Savoie, Batiment IUT, 73376 Le Bourget-du-lac Cedex, France Received 23 July 2006, accepted 3 March 2007 Published online 29 June 2007 PACS 71.60.+z, 78.70.Bj Polypropylene samples were measured as a function of the source exposure time at 20 K and as a function of the temperature (20-400 K) simultaneously by Positron Annihilation Lifetime (PAL) and Doppler Broadening (DB) spectroscopy. The peak shape parameter (S) measured by DB can be calculated as a sum of contributions of specific parameters corresponding to each positron state. The contributions are ob-tained by analysis of the PAL spectra. The confinement of positronium (Ps) in a free volume hole (FVH) determines the self annihilation shape parameter Ssa of the para-Ps, the pick-off annihilation lifetime τpo, and supposes FVH size-dependence of the pick-off shape parameter Spo. The ortho-Ps lifetime and shape parameters are used to compute Ssa. In this way, the temperature and time dependences of the S parameter are modelled. The fit of this model to the experimental S parameter shows a stronger dependence of the Spo on the FVH size (R) than the one as expected by taking into account only the Ps momentum. This re-sult suggests that the penetration of Ps into the walls of the FVH must be properly considered and that the penetration depth depends on R itself. © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction The positrons (e+) injected into a polymer may form positronium (Ps) which may get confined into free-volume holes (FVH). Ps exists in two states, triplet ortho-Ps (oPs) and singlet para-Ps (pPs) with lifetimes in vacuum τvo = 142 ns and τvp= 0.125 ns. It is widely reported that PAL spectra of various polymers can be fitted with three components but with unacceptable intensities ratio of pPs to oPs. Better fit gives a four-component analysis but usually requires a constraint on the intensities. A HPGe-detector is often used to detect the annihilation γ-line which is Doppler-broadened due to the an-nihilating pair momentum projection along the detection direction (DB spectroscopy). The extent of the broadening is usually evaluated by the peak shape (S) parameter which is an integral characteristic. S parameter can be calculated as a sum of contributions of specific parameters corresponding to each e+ state. The contributions can be obtained reliably as output of the PAL analysis. The simulated peak shape parameter is compared to the experimental one in order to investigate the consistency between the PAL and DB results.   2 Experimental Two slabs of size 10×10×2 mm3 of isotactic polypropylene (PP, 66% crystallinity) were attached to the source (1.5 MBq of 22NaCl in 7-µm-thick Kapton envelope) in sandwich geometry and mounted on the cold finger of a cryostat. The PAL measurements were carried out using a fast-fast coincidence system having a time resolution (full width at half maximum, FWHM) of 230 ps. For the   * Corresponding author: e-mail: nikdjour@inrne.bas.bg, Phone: +359 2 714 308, Fax: +359 2 9744978 phys. stat. sol. (c) 4, No. 10 (2007)  3711 www.pss-c.com © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim DB measurements a HPGe detector with energy resolution of G1/2 = 1.06 keV (FWHM of the 514-keV line of 85Sr) was used. The spectra were recorded at every 2 h with statistics ~ 9×105 for PAL and 8.5×106 counts in the 511-keV peak for DB. The PAL spectra were analyzed by 4 components by the LT code [1] with a constraint on the intensities either 3I1 = I4 or 3I1 = I3+I4. The constraints are based on the assumptions that the third component (C3) is either due to a positron state (A) or to a Ps state (B). Both analyses showed relatively high scattering of τ3= 0.5-0.9 ns and therefore we repeated the analyses with fixed τ3 = 0.7 ns. The experimental S parameter was calculated as the sum of the counts in the central region (|∆E| < ES = 0.7 keV) of the photo peak relative to the total peak counts. For more detailed analy-sis, the photo peak (-1.6 keV <∆E < 4.2 keV) also was decomposed into three equi-centered Gaussians taking into account the HPGe detector resolution.   3 Results and discussion Both pPs and oPs confined in FVH may annihilate via self annihilation (sa) or by pick-off (po) annihilation with an electron from the surroundings. The branching ratio of the self annihilation is given by β(η,τpo,τsa) = (η/τsa)/(η/τ sa+ 1/τpo), where η is the Ps contact density [2] and τ sa is either τvp for pPs or τvo for oPs. From the typical values τpo ~ 2-3 ns and η ~ 0.5-1 [3] in polymers follows β(η,τpo,τvo) ≈ 0. The intensity of pPs self annihilation can be taken as AIsa = β(η,τ4,τvp)I1 for the case of PAL analysis A and BIsa = [β(η,τ3,τvp)I3 + β(η,τ4,τvp)I4] /3 for analysis B. The fraction, Fsa, of the self annihilation of pPs localized in FVH can also be obtained as the intensity of the narrow Gaussian in the decomposition of the annihilation pair momentum distribution measured by DB technique [2].    Fig. 1 Intensities, AIsa and BIsa, of the self annihilation of pPs as obtained from PAL analyses A and B, versus the intensity, Fsa, of the narrow Gaussian in the decom-position of the DB annihilation peak for the temperature ramp 20-400 K. The Ps contact density is taken as η = 0.7 for the calculation of the self annihilation branching ratio, β.     The intensities, AIsa and BIsa, versus Fsa are plotted in Fig. 1. Only AIsa values agree satisfactorily with Fsa which means that the component C3 should not be considered as due to Ps confined in FVH. The sample temperature (T) schedule together with τpo = τ4, and Ic = I3 and IPs= I1+I4 from the PAL analysis A are presented in Fig. 2. The two short lifetimes were found being τ1 ~ 130 ps and τ2 ~ 300 ns.  On the assumption that C3 is due to e+ trapping (or e+ bound to molecule state) and by taking into account the pPs self annihilation the S parameter can be presented as  S = (IPs/4)[βSsa + (1-β)Spo] + (1 - Ic - IPs)S+ + IcSc + (3IPs/4)Spo, (1) where β = β(η,τpo,τvp), and the indexes ‘+’ and ‘c’ denote the free e+ annihilation and trapped e+ annihila-tion, and Ic and IPs are the fraction of trapped e+ and Ps yield. Generally, S+, Sc, Ssa and Spo may vary with T. Free e+ and trapped e+ annihilations occur in the occupied volume of the polymer and the correspond-ing annihilation characteristics may change similarly to the thermal expansion of the occupied volume. Therefore, it is admissible to consider linear dependence on T, i.e. S+ = S+(0)(1-α+T) and Sc = Sc(0)(1-αcT).   The following correlation has been found θ1/2(R) = 0.424/(R + δR), where θ1/2, here in keV, is the FWHM of the Ps linear momentum distribution (DPs(R)) represented by a narrow Gaussian in the DB peak decomposition, R is in nm, and δR = 0.1656 nm [4]. From the experimental τpo we calculate R with 3712 N. Djourelov et al.: On the consistency between PAL and Doppler broadening © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-c.com the help of the well-known Tao-Eldrup relation, after which we evaluate θ1/2(R) and finally the peak shape parameter Ssa(R) is obtained from  Ssa(R) = erf{2.35Es/[(θ1/2(R))2 + (G1/2)2]0.5}. (2)  The pick-off annihilation occurs between the positron of Ps ( +Pse ) and e- of the surroundings of the FVH where Ps is confined. The momentum of the annihilating pair is the sum of the individual momenta of the annihilating +Pse  and e-. The momentum of +Pse  in the laboratory system is composed of the +Pse  momentum in respect to the Ps center-of-mass (internal +Pse  momentum) and the Ps linear momentum in the laboratory system. Consequently, we can present the momentum distribution (MD) of the pick-off annihilation as Dpo(R) = DPs(R)⊗D, where D represents the convolution of the internal +Pse  MD with MD of e- involved in the pick-off annihilation. The last MD will be independent on R if the Ps penetration (into the outer e- shells of the atoms from the FVH wall) depth is considered independent on R. For esti-mation of Spo we may consider D as a Gaussian with FWHM of K1/2 and we obtain an equation similar to Eq. (2)  Spo(R) = erf{2.35Es/[(θ1/2(R))2 + (K1/2)2 + (G1/2)2]0.5}. (3)  The Ps penetration depth may increase with the decrease of FVH size. In this case, we have to con-sider K1/2 as dependent on R. In order to simplify the calculation procedure, for parameterization of Spo(R) we used a more common function which fairly well interpolates also Eq. (3)   Spo(R) = Spo(∞ ){1-exp[-(R+a)/b]}, (4) where Spo(∞ ), a, and b are parameters. By using Eq. (2) for calculation of Ssa and Eq. (4) for parameteri-zation of Spo together with the experimental intensities Ic and IPs, we fitted Eq. (1) to the experimental S parameter. The best fit (shown in Fig. 3 as fit 1) gave the following set of parameters [S+(0) = 0.487, α+ = 1.0×10–4 K-1, Sc(0) = 0.522, αc = 0.8×10–6 K–1, Spo(∞) = 0.65, a = 0.112 nm, b = 0.245 nm].  Fig. 2 Temperature schedule, pick-off annihilation lifetime τpo, and intensities IPs and Ic for the PAL analysis A vs. the time.   Fig. 3 S parameter vs. the time (the black points). The fits are based on Eq. (1) and the output of PAL analysis B (see Fig. 2) by the use of Eq. (4) or Eq. (3) for Spo(R), correspondingly fit 1 and fit 2.   The change in Spo as a function of R corresponding to Eq. (4) is plotted as solid line in Fig. 4. In order to make the further discussion easier, first we will report some of the results obtained in a separate DB experiment on Kapton. We found that SKapton decreases linearly with T (20-400 K), and we determined αKapton= 4.4×10–6 K–1. In Kapton all e+ are localized by forming e+-molecule bound state (Me+) [5]. The e+ attachment and localization implies that the Me+ annihilation characteristics will be rather more sensi-tive to the chemistry of the nearest environment than to the macroscopic properties of the material. The phys. stat. sol. (c) 4, No. 10 (2007)  3713 www.pss-c.com © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim value of αKapton is ~ 4 times lower than the linear thermal expansion coefficient of Kapton (2×10–5 K–1). In our opinion, this indicates that the characteristic peak shape parameter of a Ме+ state is almost insensi-tive to T. The last is confirmed by the obtained very low value of αc for PP. Both PP and polyethylene show high positron diffusion constants [6] and the estimated diffusion length is ~100 nm which is of three orders longer than the typical interatomic distances. This indicates that free e+ are not localized and their wave functions feel relatively vast volume, consequently, their annihilation characteristics will be sensitive to the macroscopic properties. Therefore, the value of α+ for PP is in excellent agreement with the reported linear thermal expansion coefficient (~1×10-4K-1) of the occupied volume in polymers [7, 8].   We performed another fit (fit 2 in Fig. 3) in which for Spo(R) we used Eq. (3) with the assumption that K1/2 is independent on R. The best fit resulted in K1/2 = 1.73 keV and Spo(R) reproduced from Eq. (3) is plotted as dashed line in Fig. 4. However, the last fit shows obvious discrepancy in the low and high temperature regions. This implies that Ps wave function penetration depth increases with decreasing of R.   Fig. 4 Spo parameter as a function of R. The dashed line corresponds to the simplified model of Eq. (3) with the assumption that K1/2 is independent on R. The solid line takes the variation of the Ps penetration into the FVH wall into account according to Eq. (4).  The described fitting procedure can likewise be performed on the assumption that the additional com-ponent C3 is due to Ps – molecule (MPs) bound state and the output of the PAL analysis B. The fitting result has led to similar dependences for S+(T), and Spo(R) as already reported, and to constant SMPs.  4 Conclusions  We have shown that coupled DB and PAL analysis is a critical tool for models and hypotheses about e+ and Ps states in polymer. Based on the reported experimental results, it is impossible to distinguish if the component C3 is due to Me+ or MPs, however, we have shown that it should not be considered as due to Ps confined in FVH. The best fit of the simulated S parameters set to the experimen-tal one is obtained when the Ps wave function penetration depth is considered dependent on the FVH size. The experimentally determined change of Spo as a function of R could be a good base for a compari-son with a theoretical calculation of the momentum distribution of the annihilation pair for the oPs pick-off annihilation. The best fit also reveals that S+ decreases linearly with the temperature with a coeffi-cient close to the linear thermal expansion coefficient of the occupied volume, while Sc has been found being almost temperature independent.  Acknowledgements  This research was supported by the IAP program P5/01 of the Belgian Science Policy Office (BELSPO). The first author (N.D.) is grateful for the research fellowship awarded by BELSPO(FEDRA). References [1] J. Kansy, Nucl. Instrum. Methods Phys. Res. A 374, 235 (1996). [2] O. E. Mogensen, Positron Annihilation in Chemistry (Springer-Verlag, Berlin, 1995). [3] G. Consolati, J. Radioanal. Nucl. Chem. 210, 273 (1996). [4] H. Nakanishi and Y. C. Jean, in: Positron and Positronium Chemistry, Vol. 57, edited by D. M. Schrader and Y. C. Jean (Elsevier, Amsterdam, 1988), chap. 5. [5] R. S. Brusa, A. Dupasquier, E. Galvanetto, and A. Zecca, Appl. Phys. A 54, 233 (1992). [6] W. Zheng, Y. Kobayashi, K. Hirata, and T. Suzuki, Radiat. Phys. Chem. 51, 51 (1998). [7] P. Bandzuch, J. Kristiak, O. Sausa, and J. Zrubkova, Phys. Rev. B 61, 8784 (2000). [8] G.Consolati, Mater. Sci. Forum 363-365, 244 (2001).  

On the consistency between positron annihilation lifetime and Doppler broadening results in polypropylene

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On the consistency between positron annihilation lifetime and Doppler broadening results in polypropylene

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