J. Phys. Chem. B 1999, 103, 4555-4558
4555
Extension of the Equation for the Annihilation Lifetime of ortho-Positronium at a Cavity Larger than 1 nm in Radius Kenji Ito,*,† Hiroshi Nakanishi,‡,§ and Yusuke Ujihira† Research Center for AdVanced Science and Technology, The UniVersity of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan, Department of Pathobiology, UniVersity of Washington, Seattle, Washington, and Molecumetics, Ltd., 2023 120th AVe. NE, BelleVue, Washington 98005 ReceiVed: September 11, 1998; In Final Form: December 1, 1998
Evaluations of cavity concentrations and their size distributions in industrially important materials by positron annihilation lifetime methods require a well calibrated ortho-positronium (o-Ps) lifetime standard curve as a function of pore size. A semiempirical equation with a fitting parameter obtained by Nakanishi and Jean has been successfully utilized in various porous and polymeric materials to estimate pore sizes up to approximately 1 nm in radius. However, as experimental data accumulate recently, it has become clear that the equation no longer yields a good correlation between o-Ps lifetimes and the pore size for the pore radius larger than 1 or o-Ps lifetime longer than ∼20 ns. Therefore, we assumed for the larger pore that o-Ps behaves more like a quantum particle, bouncing back and forth between the energy barriers as the potential well becomes large. An equation derived from this hypothesis gives excellent fitting with experimental data despite a crude approximation used in our model. More importantly, this approach has produced a simple analytical relationship between the o-Ps lifetime and the pore radius. Agreement between the estimated o-Ps wave packet size at thermal energy (∼1.6 nm) and the fitted parameter RPs (∼1 nm) lends credence to our hypothesis that o-Ps behaves more like a quantum particle as a pore size becomes larger.
1. Introduction Pick-off annihilation rates of positrons and ortho-positronium (o-Ps: the triplet bound-state of a positron and an electron) are mainly determined by the electron density at their annihilation sites.1 Their lifetimes, inverse of the annihilation rates, become longer when positrons or o-Ps are localized at spaces with lower electron density, such as atomic vacancies, voids, free volumes, pores and cavities in various solid materials. Therefore, positrons can be used as a unique probe to investigate average sizes of the free volume, size distribution, and free volume concentration in a sample by measuring o-Ps annihilation lifetimes.2 Since it has been demonstrated that the lifetime of the o-Ps correlates well with free volume sizes (0.5-0.8 nm in diameter) in various polymeric materials,2,3 a large number of positron annihilation lifetime measurements applied to the characterization of free volumes in polymers have been reported within the last 10 years. Due to the physical importance of microscopic hole properties in industrial applications, there have been increased interests in obtaining information on size, distributions, and concentration of microcracks, nanocavities, and pores in industrial materials. Additionally, nondestructive analytical methods are required in many cases. The positron annihilation technique is the only method which satisfies these conditions and has a great potential to become a routine method to study the hole property. Some of the applications are the evaluations of the o-Ps lifetimes against average pore radii in zeolites,4 porous glasses,5 temperature and pressure dependence of free volume in polymeric materials,6 anisotropy and distributions of free volume in polymers,7,8 and studies on the volume reduction mechanisms in sintering processes of ceramics.9 †
The University of Tokyo. University of Washington. § Molecumetics, Ltd. ‡
A semiempirical equation was proposed by Nakanishi and Jean10,11 using a simple quantum mechanical model of Tao,12,13 in order to correlate the o-Ps lifetime with the average pore size. It is used successfully to study physicochemical properties in wide varieties of condensed matters.2,3 This equation, however, fails to account for the lifetimes of the o-Ps at the pore radius larger than 1 nm. One source for this deviation is a contribution from the intrinsic o-Ps annihilation process which becomes significant as the o-Ps lifetime becomes larger. Neither inclusion of this process and the use of more sophisticated potential wells14,15 nor inclusion of the Ps annihilation process from the excited states,16 however, produced a satisfactory analytical expression between the lifetime and the pore size for practical analysis. Here we introduce quantum particle nature of the o-Ps to overcome this deficiency and to extend the previous equation for the o-Ps in the large cavities. A simple model for the o-Ps in the larger pore is proposed by assuming a distribution of an o-Ps particle with its effective particle radius as a fitting parameter. 2. Quantitative Estimation of Cavity Size from Ps Lifetime Using a Quantum Mechanical Model The free volume model for positron annihilation was proposed by Brandt et al.17,18 to explain the observed o-Ps lifetimes in highly periodic molecular solids and polymers introducing a delocalized Ps state analogous to a positron in the Bloch state. For less periodic solids and polymers, a simple correlation curve can be obtained using a rigid spherical potential of radius Rc with an electron layer of thickness ∆R. The model was initially suggested by Tao12,13 and Eldrup19 as shown in Figure 1a, as a natural extension of the bubble model in solution proposed by Ferrell.20 Using this simple model, the pick-off annihilation rate
10.1021/jp9831841 CCC: $18.00 © 1999 American Chemical Society Published on Web 04/22/1999
4556 J. Phys. Chem. B, Vol. 103, No. 21, 1999
Ito et al. equation R λR ) λ2γ + λ3γ )
[
21-
Figure 1. Schematic diagrams of annihilation models for orthopositronium (o-Ps) in a free space. (a) Infinite spherical square well potential with an electron layer of thickness ∆R. (b) A proposed model for the o-Ps annihilation in a large pore. RPs represents a quantum radius of o-Ps, where the annihilation process can be described by eq 1. See text for details.
Figure 2. Annihilation lifetimes of o-Ps measured in various porous material as a function of average pore radius ((O) oxides, data listed in Table 1, (]) molecular solids and organic liquids from ref 31, ([) porous polymers from ref 32). The solid line is a correlation curve between o-Ps lifetime and a pore radius calculated from eq 1. The dashed line is calculated from eq 2 with a contribution from the o-Ps intrinsic annihilation. R of o-Ps, λ2γ , in a rigid spherical potential is given in the unit of -1 ns by
[
R λ2γ )21-
2πR 1 R + sin R + ∆R 2π R + ∆R
(
)]
(1)
where ∆R is fitted to be 0.166 nm for zeolites and other porous materials.10,11 The annihilation rate of o-Ps inside the electron layer is assumed to be 2 ns-1, which is a spin averaged annihilation rate of para-Ps ( p-Ps: the singlet state of Ps) and o-Ps, or the annihilation rate of Ps anion.21,22 An excellent correlation (R ) 0.965) is found between “Bondi” free volumes in various liquids and polymers and the cavity volumes obtained from eq 1.23 For the pore radius less than 1 nm, eq 1 correlates o-Ps lifetimes with pore sizes very well. However, the o-Ps lifetime deviates from the values obtained from eq 1 for the pore radius larger than 1 nm as shown in Figure 2 with a solid line. This discrepancy is obviously due to the omission of the contribution from 3γ annihilation which becomes increasingly significant as the lifetime approaches 140 ns. A logical way to incorporate this effect is to add a 3γ annihilation rate to eq 1 so that the
R 1 2πR 1 + + sin (2) R + ∆R 2π R + ∆R 140
(
)]
reaches 140 ns (o-Ps lifetime in a vacuum) asymptotically as R becomes large. This should hold true even for o-Ps annihilating at small cavities with observed o-Ps lifetimes of ∼3 ns although the contribution from the 3γ process is almost negligible (∼2%). Equation 2, however, overestimates o-Ps lifetimes at R larger than 1 nm as shown in Figure 2 by a dashed line. Varying λ3γ as a parameter to fit the experimental values of o-Ps lifetimes did not produce a desired curvature. Shantarovich et. al. has used a finite potential well to estimate o-Ps lifetimes from probabilities of finding o-Ps outside the cavity and by adding a contribution from the intrinsic o-Ps annihilation.14,15 Their graphical solution, however, did not reproduce the o-Ps lifetime data well for the large pores. To obtain a better fitting with the experimental data for the large pore size, an additional potential well at the pore wall was introduced. It appears that one set of parameters cannot describe the data universally well by this approach. The other conflicting issue was the prediction of a constant o-Ps lifetime beyond a certain pore radius much less than the asymptotic value of 140 ns (about 60 ns beyond 2.5 nm in radius14,15). This observation led us to introduce a following model. 3. Modification of the Semiempirical Equation for Large Pores o-Ps lifetimes in various porous materials are plotted against average pore radii using the experimental data from our laboratory24 and others4,5,13,25-32 as shown in Figure 2. A strong correlation is apparent although there is some scattering depending on characteristics of materials. Positrons experience vastly complex and diverse thermalization processes and electronic environments before annihilating at cavities in each material. Thus, it is almost impossible to take into account all interactions between positrons/o-Ps and characteristic electronic environment in a simple model. It is also questionable to construct a more detailed model due to the inaccuracy in measurements of positron lifetimes and/or average pore size and shape. Therefore, we used a naively simple model to obtain improved correlation between the o-Ps annihilation lifetime at a large pore as a function of the pore radius using the available data for oxides reported by the positron annihilation technique as listed in Table 1. As a cavity becomes larger, the o-Ps particle is no longer represented by a standing wave function in a potential well but rather by a Gaussian wave packet scattering back and forth between the energy barriers at the cavity wall.33 We, therefore, assume two regions for o-Ps annihilation in a large cavity; one is near the cavity wall where o-Ps can annihilate with an electron by a pick-off process, and the other at the center of the cavity where o-Ps mainly annihilates via an intrinsic 3γ process. A parameter RPs is introduced as shown in Figure 1b to define the boundary of the near-the-wall region. The o-Ps annihilation rate is then given by a sum of two fractional rates at the two annihilation regions; one is a pick-off annihilation of o-Ps with electrons on the cavity wall, and the other is an intrinsic 3γ annihilation process of o-Ps in a vacuum at the middle of the cavity, i.e., R + λ3γ)(1 - f(R)) + λ3γf(R) ) λR ) (λ2γ R (1 - f(R)) + λ3γ (3) λ2γ
Annihilation Lifetime of ortho-Positronium
J. Phys. Chem. B, Vol. 103, No. 21, 1999 4557
TABLE 1: ortho-Positronium Lifetime and Pore Radius in Oxides τ (ns)
sample
I (%)
pore surface radius area (nm) (m2/g) ref
silica gel 55a 4.1 almina gel 62a 4.4 silica gel (Davidson 32 27 1.25 grade 07) silica gel (KSK-2) 71 ( 7 15 ( 1 7.0 silica gel (KSK-2.5) 67 ( 7 15 ( 1 5.16 silica gel (KSS-3) 56 ( 6 20 ( 1 3.54 silica gel (KSS-4) 47 ( 5 20 ( 1 2.34 silica gel (KSM-5) 35 ( 4 18 ( 1 1.61 silica gel (KSM-6s) 20 ( 2 5(1 1.16 silica gel (KSM-6p) 22 ( 2 10 ( 1 1.12 silica gel (S-3) 70 ( 7 15 ( 1 6.0 silica gel (S-4) 39 ( 4 15 ( 1 1.75 silica gel (Davidson ∼27a ∼26a 1.25 grade 07) silica gel (Davidson ∼59a ∼15a 2.35 grade 81) silica gel (Davidson ∼84a ∼10a 6.75 grade 59) silica gel 12.3 ( 1 17 ( 2 0.8 silica gel 31 ( 2 12 ( 2 1.5 silica gel 70 ( 4 18 ( 2 5.0 silica (porous 53.1 ( 2 3.8 ( 0.2 3.4 vycor glass) zeolite (13X) 4.42 ( 0.25 21.8 0.65 zeolite (4A) 3.76 ( 0.10 25.4 0.55 zeolite (13X) 8.0 ∼9 0.65 zeolite (4A) 5.0 0.55 silica gel 42.2 1.25 silica gel 45.5 2.0 silica gel 69.4 2.5 silica gel 76.3 4.5 silica gel 100 5.0 silica gel 93.5 7.5 zirconia 33.6 10.2 1.91 zirconia (Tosoh 122.7 4.38 25.3 2YSZ) zirconia (Tosoh 128.0 2.81 27.9 3YSZ) zirconia (Tosoh 130.2 1.76 33.4 8YSZ) silica (porous 117 9.11 glass) zeolite 37.7 3.59 1.37 (MCM-41) zeolite 26.9 2.32 1.26 (MCM-48)
145 135 600
25 25 26
338 376 522 650 715 624 527 346 540 600
27 27 27 27 27 27 27 27 27 13
600
13
340
13
840 1000 300 154
28 28 28 5
28 28 4 4 600 30 675 30 500 30 450 30 425 30 300 30 185.4 24 16.5 24 15.3 24 13.3 24 24 1786.8 24 1260.8 24
a The values were estimated by the extra or interpolation of the available data in the literature.
where f(R) is a probability of o-Ps found inside of the sphere with radius (R - Ra) (see Figure 1b), i.e., fraction of o-Ps free from the pick-off annihilation process with electrons from the cavity wall. Next, the o-Ps probability fraction can be estimated by an integration,
f(R) )
R - Ra
∫02π∫0π∫0R + ∆RF(ξ)ξ2 sin θ dξ dθ dφ
3 4π
(4)
where F(ξ) is a radial probability density function of o-Ps in the spherical cavity and ξ is a normalized radial variable, i.e., 0 e ξ e 1. Since F(ξ) is not known, we assume that f(R) is given by power b of the radial fraction as a first approximation, i.e.,
f(R) )
(
)
R - Ra R + ∆R
b
(5)
where b is a fitting parameter and ∆R is equal to 0.166 nm. If
Figure 3. Annihilation lifetimes of the o-Ps measured in various porous materials as a function of average pore radius (see the caption of Figure 2 for detailed illustration of the data). The dashed line is a correlation curve calculated from eq 1. A solid line shows the correlation curve calculated from the extended equation, eq 6, where fitting parameters are determined as b ) 0.55 and Ra ) 0.8 with a correlation coefficient of 0.962, where ∆R is fixed at 0.166 nm.
F(ξ) is uniform over the cavity space, b should equal 3. Here, (R - Ra) defines the boundary of the spherical volume where o-Ps wave packets do not interact with the electron layer at the surface of the cavity wall. Thus, the o-Ps annihilation rate at the cavity with radius R can now be simplified as:
λ ) R
{(
Ra λ2γ 1-
R λ2γ
(
+ λ3γ
))
R - Ra R + ∆R
b
+ λ3γ (R g Ra)
(6a)
(R < Ra)
(6b)
Two variables Ra and b are then fitted with the experimental data4,5,13,24-30 listed in Table 1. The best fitting is obtained at parameter values Ra ) 0.8 nm and b ) 0.55, reproducing experimental o-Ps lifetimes in the large pores with the correlation coefficient of 0.962 as shown in Figure 3 with a solid line. Now let us consider the physical meaning of these two parameters. Inserting eq 5 into eq 4 with b ) 0.55 gives, F(ξ) ∝ ξ-3.55, which does not make physical sense at the origin. This may be due to the fact that the o-Ps probability density function F cannot be simply represented by eq 5. Using a Gaussian function, F(ξ) ) e-bξ2, may be a better approximation. Nonetheless f(R) implies that o-Ps populates significantly more at the center of the cavity for the R slightly larger than the Ra. Then, as R becomes larger, the o-Ps density becomes more uniformly distributed over the cavity. This would again conflict with the o-Ps population calculated from the double potential model with a lower potential bottom at the cavity wall used by a Russian group.14,15 RPs ()Ra + ∆R) may be considered as a size of o-Ps wave packet or an effective o-Ps quantum radius. Considering a particle moving with its thermal energy in a force-free region, Ps in a large pore can be represented by plane wave functions u(x) ∝ eipx/p, where p is the momentum of o-Ps. These plane waves represent particles having a definite momentum, but a complete absence of localization of the particle in space. To describe a particle which is confined in a certain spatial region, a wave packet can be formed by superposing plane waves of different momenta. Therefore, the wave packet for a free Ps which is confined along the x axis is given by a superposition
4558 J. Phys. Chem. B, Vol. 103, No. 21, 1999
Ito et al.
of plane waves,
ψ(x) ∝
∫-∞∞eip x/pφ(px) dpx x
(7)
which is a Fourier transformation of the wave function φ(px). The momentum distribution of the ideal gas at temperature T is represented by a Maxwell-Boltzmann distribution with a Gaussian function,
φ(px) ) e-px/γ 2
2
(8)
where γ is given by (2MPsE)1/2 ) (2MPskT)1/2, and MPs is the mass of o-Ps. Substituting eq 8 into eq 7, the wave packet ψ(x) is now given by
ψ(x) ∝
∫-∞∞eip x/pe-p x
x
2/γ2
dpx ) γxπe-γ x /4p 2 2
2
(9)
The amplitude of the wave packet falls to 1/e of its maximum value at ( 2p/(2MPskT)1/2, which is about 1.6 nm at room temperature. This value may be considered as an effective radius of thermal Ps in a large pore. The agreement is rather good, suggesting the validity of the present model that Ps should be considered as a quantum particle in a spherical cavity with its radius larger than 1 nm. Zero-point energy E0 of o-Ps in an infinite spherical potential with a radius R is given by
E0 )
π2p2 2MPsR2
(10)
R ) 1.0 nm gives an o-Ps zero-point energy of 0.19 eV.11 The zero-point energy of o-Ps can be also estimated from a fwhm (θ1/2) of a narrow component in angular correlation of annihilation radiation (ACAR) experiments by assuming an infinite spherical potential,11 and was found to be between 0.05 and 0.20 eV in silica aerogels having o-Ps lifetimes longer than 10 ns by Nagashima et al.34 4. Conclusion Evaluations of cavity concentrations and their size distributions in industrially important materials by positron annihilation methods require a well calibrated o-Ps lifetime standard curve as a function of pore size. It is imperative for a universal acceptance of the positron annihilation method as a nondestructive analytical method. Previously determined semiempirical eq 1 has been successfully utilized in various porous and polymeric materials to estimate pore sizes up to approximately 1 nm in radius. However, as experimental data accumulate it has become clear that eq 1 no longer yields a good correlation between o-Ps lifetimes and the pore size if the pore radius is larger than 1 nm or the o-Ps lifetime is longer than ∼20 ns. The obvious correction by including the o-Ps intrinsic 3γ annihilations as given in eq 2 fails to reproduce the experimental data satisfactorily as shown by a dashed line in Figure 2. We, therefore, hypothesized that o-Ps behaves more like a quantum particle, scattering back and forth between the energy barriers as the potential well becomes large when pore size becomes larger than a certain radius, RPs. This model leads us to eq 6 with fitted parameters Ra ) 0.8 nm and b ) 0.55. This equation gives
excellent fitting with the experimental data as indicated by the correlation coefficient of 0.962 despite the crude approximation used in our model. More importantly, this approach has produced a simple analytical relationship between the observed o-Ps lifetime and the pore radius given by eq 6. A good agreement between the estimated o-Ps wave packet size at thermal energy (∼1.6 nm) and the fitted parameter RPs (∼1 nm) lends credence to our hypothesis that o-Ps behaves more like a quantum particle as the pore size becomes larger. References and Notes (1) Positron and Positronium Chemistry; Schrader, D. M., Jean, Y. C., Eds.; Elsevier: Amsterdam, 1988. (2) Jean, Y. C. AdVances with Positron Spectroscopy of Solids and Surfaces, The Proceedings of NATO AdVanced Research Workshop, Varenna, Italy, 1993; pp 563-580. (3) For example, see: Ujihira, Y.; Tanaka, M.; Jean, Y. C. Radioisotopes 1993, 42, 43-56. Wang, S. J.; Wang, C. L.; Wang, B. J. Radioanal. Nucl. Chem. 1996, 210, 407-421. (4) Nakanishi, H.; Ujihira, Y. J. Phys. Chem. 1982, 86, 4446-4450. (5) Ito, Y.; Yamashina T.; Nagasaka, M. Appl. Phys. 1975, 6, 323326. (6) Wang, Y. Y.; Nakanishi, H.; Jean, Y. C.; Sandreczki, T. C. J. Polym. Sci. B 1990, 28, 1431-1441. (7) Jean, Y. C.; Nakanishi, H.; Hao, L. Y.; Sandreczki, T. C. Phys. ReV. B 1990, 42, 9705-9708. (8) Jean, Y. C.; Shi, H. J. Non-Crystal. Solids 1994, 172-174, 806814. (9) Yagi, Y.; Hirano, S.; Miyayama, M.; Ujihira, Y. Mater. Sci. Forum 1997, 255-257, 433-435. (10) Nakanishi, H.; Wang, S. J.; Jean, Y. C. In Positron Annihilation Studies of Fluids; Sharma, S. C., Ed.; World Scientific: Singapore, 1988; pp 292-298. (11) Nakanishi, H.; Jean, Y. C. Positron and Positronium Chemistry; Schrader, D. M., Jean, Y. C., Eds.; Elsevier: Amsterdam, 1988; Chapter 5, pp 159-192. (12) Tao, S. J. J. Chem. Phys. 1972, 56, 5499-5510. (13) Chuang, S. Y.; Tao, S. J. Can. J. Phys. 1973, 51, 820-829. (14) Shantarovich, V. P.; Yampolskii, Yu. P.; Kevdina, I. B. Khim. Vys. Energ. 1994, 28, 53-59. (15) Shantarovich, V. P. J. Radioanal. Nucl. Chem. 1996, 210, 357369. (16) Goworek, T.; Ciesielski, K.; Jasin´ska, B.; Wawryszczuk, J. Chem. Phys. 1998, 230, 305-315. (17) Positron Solid-State Physics; Brandt, W., Dupasquier, A., Eds.; North-Holland: Amsterdam, 1983. (18) Brandt, W.; Berko, S.; Walker, W. W. Phys. ReV. 1960, 120, 12891295. (19) Eldrup, M.; Lightbody, D.; Sherwood, J. N. Chem. Phys. 1981, 63, 51-58. (20) Ferrell, R. A. Phys. ReV. 1957, 108, 167-168. (21) Mills, A. P., Jr. Phys. ReV. Lett. 1981, 46, 717-720. (22) Ho, Y. K. J. Phys. B 1983, 16, 1503-1509. (23) Kobayashi, Y.; Haraya, K.; Hattori, S.; Sasuga, T. Polymer 1994, 35, 925-928. (24) Ito, K.; Yagi, Y.; Hirano, S.; Miyayama, M.; Kudo, T.; Kishimoto, A.; Ujihira, Y. J. Ceram. Soc. Jpn. 1999, 107, 123-127. (25) Gol’danskii, V. I.; Levin, B. M.; Mokrushin, A. D.; Kaliko, M. A.; Pervushina, M. N. Dokl. Akad. Nauk SSSR 1970, 191, 855-858. (26) Chuang, S. Y.; Tao, S. J. J. Chem. Phys. 1970, 52, 749-751. (27) Mikrushin, A. D.; Levin, B. M.; Gol’danskii, V. I.; Tsyganov, A. D.; Bardyshev, I. I. Russ. J. Phys. Chem. 1972, 46, 368-371. (28) Goldanskii, V. I.; Mokrushin, A. D.; Tatur, A. O.; Shantarovich, V. P. Appl. Phys. 1975, 5, 379-382. (29) Perkal, M. B.; Walters, W. B. J. Chem. Phys. 1970, 53, 190-198. (30) Hopkins, B.; Zerda, T. W. Phys. Lett. 1990, 145, 141-145. (31) Mogensen, O. E. Positron and Positronium Chemistry; SpringerVerlag: Berlin, 1995; pp 54-61. (32) Venkateswaran, K.; Cheng, K. L.; Jean, Y. C. J. Phys. Chem. 1984, 88, 2465-2469. (33) For example, see: Schiff, L. I. Quantum Mechanics, 3rd ed.; McGraw-Hill: New York, 1968; p 106. (34) Nagashima, Y.; Kakimoto, M.; Hyodo, T.; Fujiwara, K.; Ichimura, A.; Chang, T.; Deng, J.; Akahane, T.; Chiba, T.; Suzuki, K.; McKee, B. T. A.; Stewart, A. T. Phys. ReV. A 1995, 52, 258-265.
