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Temperature variations of average o-Ps lifetime in porous media
Radiation Physics and Chemistry 58 (2000) 719±722
www.elsevier.com/locate/radphyschem
Temperature variations of average o-Ps lifetime in porous media T. Goworek*, K. Ciesielski, B. JasinÂska, J. Wawryszczuk Institute of Physics UMCS, 20-031 Lublin, Poland
Abstract Modi®cation of the Tao±Eldrup model is proposed in order to extend its usefulness to the case of porous media. The modi®cation consists in the transition from spherical to capillary geometry and in inclusion of pick-o annihilation from the excited states of a particle in the well. Approximated equations for pick-o constant in these states are given. The model was tested by observing the temperature dependences of o-Ps lifetime in various media. In the case of silica gels and Vycor glass with narrow pores, the model seems to work well, while for larger pores in Vycor unexpectedly long lifetimes appear in the range of lowest temperatures. 7 2000 Elsevier Science Ltd. All rights reserved.
In positron studies of voids in solids the model of in®nitely deep rectangular potential with spherical symmetry is in common use. The pick-o rate is given by a very popular equation (Eldrup et al., 1981): R 1 R lpo lb 1 ÿ sin 2p R0 2p R0
1
where lb is the o-Ps lifetime in the bulk, R is the well radius, R0=R+DR, and DR approximates the penetration of Ps wave function into the bulk in real well with ®nite depth. Two parameters of the model: lb and DR are of empirical character. The lifetime of o-Ps is, according to the accepted model, determined exclusively by geometry. From the experimental lpo we deduce R, which is only the radius of an ``equivalent
* Corresponding author. Tel.: +48-81-537-6225; fax: +4881-537-6191. E-mail address: [email protected] (T. Goworek).
sphere'' giving the same lifetime as that observed in the experiment. In the framework of this model there is no place for temperature dependence of lpo, except for the small eect of dilatation. Constancy of lpo is justi®ed in the case of small voids, when the spacing of levels (if more than one exists) is much larger than kT.
1. Modi®ed model for large voids With increasing void dimensions more levels appear in the potential well and their energies decrease, falling into the range of kT at room temperature when R 1 2 nm. In such a case, assuming thermal equilibrium, we can expect the lpo averaged over the populated states lpo
X X lipo gi exp ÿEi =kT = gi exp ÿEi =kT i
2
i
where l ipo is the decay rate from i-th level, gi, Ei is the
0969-806X/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 6 X ( 0 0 ) 0 0 2 4 6 - 2
720
T. Goworek et al. / Radiation Physics and Chemistry 58 (2000) 719±722
statistical weight and energy of the i-th level, respectively. Boltzmann distribution of Ps over the levels is assumed. Changes in the population of levels lead to the dependence of lifetime on the temperature. As a rule, the higher Ei, is the larger l ipo, thus with the rise of temperature the o-Ps lifetime should decrease (dilatation gives an opposite eect). Experimental veri®cation of Eq. (2) requires a medium with large free volumes, e.g. a porous one. In physical chemistry the pore structure is usually approximated as a bundle of capillaries. Thus, instead of spherical symmetry as in Eq. (1), we should apply a cylindrical one. The radial wave functions of a particle in the cylindrical potential are Bessel Jm(r ) ones, and lnm po lb
Xnm Xnm R=R0
Xnm J 2m rrdr= J 2m rrdr 0
3
where Xnm is the n-th node (zero crossing point) of Jm(r ) function. Calculation of integrals in Eq. (3) gives:
2
6 4 lnm po R lb 1 ÿ
R R0
2 J 2m
Xnm R R0
ÿ
Jm r1 r ÿ Xnm
dJm dr
Xnm
1, 21 lpo 9:75lb
1, 22 lpo 17:4lb
DR R0
3 3DR 1ÿ 4R0
DR R0
DR R0
3
3
5
then one obtains o-Ps decay rates for the three lowest levels (n = 1; m = 0,21,22):
Fig. 1. Relative deviation d of the approximated lifetimes (Eqs. 6±8) from the exact values for the three lowest states in the cylindrical well. Solid line, (n,m )=(1,0); dashed line, (1,2 1); dash dot line, (1, 2 2). The parameter DR is assumed 0.19 nm.
6
1ÿ
3DR 4R0
1ÿ
3DR : 4R0
7
8
Fig. 1 shows the relative deviation of approximated lpo from the exact one. For Re2 nm, when the upper states play a signi®cant role, the deviation does not exceed 5%, which is acceptable for such a rough model as that discussed here. Applying Eq. (2) one needs to know Ei. The energies of a particle in the in®nitely high cylindrical well of radius R are
3 2nR0 Xnm R Xnm R Xnm R Jm1 J 2m1 Jm 7 R0 R0 R0 Xnm R 5 : 2 J m1 Xnm
The Jm functions are not too handy in calculations, thus for DR<
0 3:85lb l 1, po
Enm
h 2 X 2nm Ea 2mPs R2 ÿ
4
9
where mPs is the mass of positronium, Ea is the energy related to axial motion. It is of the order kT/2; not quantized, but the same for all levels. Thus, in Eq. (2) we have rather the average energies. The energy Ea adds (on average) a constant to the exponent and does not change the relative population of levels; with good approximation we can introduce into Eq. (2) the ®rst term from Eq. (9). The quantized energy (both in cylindrical and spherical symmetry) is inversely proportional to the well radius square. It means that the population of levels is extremely sensitive to the accepted value of R. In the Tao±Eldrup model the well radius is assumed R0, while the real one is by DR smaller. If the model is consistently used with R0 as radius the calculated energies are decreased. On the other hand, transition from ®nite to in®nite well depth at the same radius leads to increasing the energies of the states. Both factors, i.e. R increase and well shallowing act in opposite directions. Although the energy shifts due to these factors are not identical, the use of R0 in Eq. (9) instead of R0ÿDR is somehow justi®ed, keeping in mind the approximate character of the model. In our earlier paper (Ciesielski et al., 1998) we
T. Goworek et al. / Radiation Physics and Chemistry 58 (2000) 719±722
721
have shown that the best ®t of Eq. (2) to the experimental data is obtained when DR = 0.19 nm (preserving the traditional value of lb=2 nsÿ1).
2. Samples and apparatus Samples of porous Vycor glass of various pore diameter were prepared by changing the time of liquation with subsequent leaching (5 h at 908C) of the samples (Haller, 1965). Silica gels were commercially available samples (Merck). The positron lifetime spectra were recorded using a standard fast±slow spectrometer with the energy window in the STOP channel widely open in order to register the most of 3g decays. The resolution was 280 ps (FWHM), the time base was 2 ms wide, typical spectrum contained (2±4) 106 coincidences. The spectra were analysed using the LT program (Kansy, 1996) in which the long lived components were assumed as continuous log-Gaussian distributions of l. The reciprocal of l at the peak of distribution, tp, was used as a measure of the lifetime. The measurements were performed at a pressure 10.5 Pa in order to avoid ortho±para conversion in atmospheric oxygen.
3. Experimental results and discussion The positron lifetime spectra were measured in the range 110±420 K. The results for silica gels Si40 and
Fig. 2. Temperature dependence of o-Ps lifetime in silica gels Si40 and Si60. Solid curves represent the result of calculations according to Eq. (2) for R = 1.85 nm and R = 3.5 nm, respectively.
Fig. 3. The o-Ps lifetime in Vycor glass (sample No.1) as a function of temperature.
Si60 are shown in Fig. 2. In the case of large pores positron needs a relatively long time to be thermalized, thus the whole initial part of the spectrum was neglected and one long lived component was ®tted to the truncated spectrum beginning at 40 ns delay. It is seen that the model curves representing Eq. (2) for cylindrical geometry and DR = 0.19 nm ®t quite well to the experiment. It would be interesting to see how the temperature dependences of lifetime spectra change at the transition from one-level to multi-level structure of quantized states in the well. Therefore, a set of ®ve Vycor glasses with dierent pore radii (all below 2 nm) was studied. Sample No. 1 was leached without previous liquation, thus the narrowest possible pores were produced. According to the model, the lifetime 8 ns corresponds to the capillary radius 0.56 nm. At this radius we cannot expect a measurable population of the next level above the ground state and no lifetime variation with temperature should be observed. The experiment
Fig. 4. Temperature dependence of o-Ps lifetime in Vycor glass with average pore radii 1.55, 1.31 and 1.25 nm (from top to bottom). The solid line represents the result of calculations according to Eq. (2) for R = 1.31 nm, as determined by LN method. One long lived component ®t.
722
T. Goworek et al. / Radiation Physics and Chemistry 58 (2000) 719±722
0.71 and 1.56 nm (0.71 nm pores are not visible with the classic LN method). In our model it corresponds to the lifetimes 12.6 and 60.5 ns (in the low temperature limit). The calculated curve for 0.71 nm pores goes close to the experimental points, while the curve for 1.56 nm is evidently shifted upwards and its slope in the range 110±210 K does not re¯ect, like in onecomponent ®t, the experimental trend. It is to be noted that a weak short lived component can be explained alternatively as the result of incomplete thermalization of Ps and overpopulation of high states in the ®rst nanoseconds of o-Ps life.
4. Conclusions
Fig. 5. Temperature dependence of o-Ps lifetime in Vycor glass with average pore radius from LN method 1.31 nm (sample No. 3). The long lived part of the spectrum assumed as two-component one. Solid lines are calculated from Eq. (2) assuming the pore radii as found by SAXS method (0.71 and 1.56 nm).
reveals a small decrease of the lifetime with rising temperature (Fig. 3), however, the changes are not signi®cant. The samples Nos. 2±5 were annealed (before leaching) during 8, 24, 48 and 96 h; measurements of pore size distribution by desorption of liquid nitrogen gave the average pore radii of 1.25, 1.31, 1.37 and 1.55 nm, respectively. All the experimental curves were similar, shifting towards longer lifetimes with the pore size increase. Typical data, collected for the samples No. 2, 3, 5 are shown in Fig. 4. The lifetime decreases systematically with the temperature, changing the slope which is largest at low temperature end of the ®gure. It is seen how sensitive the lifetime is on the pore radius: the dierence of the average radii for the neighbouring curves No. 2 and 3 is several percent only (0.06 nm). The model curve shown in Fig. 4 is calculated for sample No. 3 (R = 1.31 nm). It follows the tendency indicated by the experiment in the region above 250 K only. At low temperatures the lifetimes, instead of approaching the saturation with T 4 0, increase sharply. An attempt to split the long lived component into two did not bring any new information. Analysing the whole spectrum it was possible to single out an additional component, but it was a short lived one, of a very low intensity (for sample No. 3 at low temperature t4p 1 14 ns, I4 1 2%) and the whole temperature variation was retained in the longest lived part (Fig. 5). The measurements of pore sizes by small angle X-ray scattering (SAXS) gave for sample No. 3 two peaks of the distribution of radii:
The model of o-Ps in the capillary, taking the annihilation from excited states of a particle in the well into consideration, is consistent with the experimental data for silica gels and for Vycor glass of a small pore diameter. Anomalous t vs T dependence in Vycor already appears at R = 1.25 nm and does not change with R increase. It would be interesting to perform similar measurements for radii lying in the transition region from normal t(T ) dependence (R = 0.6 nm) to anomalous one (R = 1.25 nm). The average pore radii determined by various classic methods show a relatively large scatter of results, making a more accurate comparison of the annihilation data impossible with the proposed model.
Acknowledgements The authors wish to thank Prof. A.L. Dawidowicz (Department of Chemical Physics, UMCS) for preparation of Vycor samples, and Dr. S. Pikus (Department of Crystallography, UMCS) for SAXS measurements. This work was not supported by KBN grants.
References Ciesielski, K., Dawidowicz, A.L., Goworek, T., Jasinska, B., Wawryszczuk, J., 1998. Positronium lifetimes in porous Vycor glass. Chem. Phys. Lett. 289, 41. Eldrup, M., Lightbody, D., Sherwood, J.N., 1981. The temperature dependence of positron lifetimes in solid pivalic acid. Chem. Phys. 63, 51. Haller, W., 1965. Rearrangement kinetics of the liquid-liquid immiscible microphases in alkali borosilicate melts. J. Chem. Phys. 42, 686. Kansy, J., 1996. Microcomputer program for analysis of positron annihilation lifetime spectra. Nucl. Instrum. Methods A374, 235.
www.elsevier.com/locate/radphyschem
Temperature variations of average o-Ps lifetime in porous media T. Goworek*, K. Ciesielski, B. JasinÂska, J. Wawryszczuk Institute of Physics UMCS, 20-031 Lublin, Poland
Abstract Modi®cation of the Tao±Eldrup model is proposed in order to extend its usefulness to the case of porous media. The modi®cation consists in the transition from spherical to capillary geometry and in inclusion of pick-o annihilation from the excited states of a particle in the well. Approximated equations for pick-o constant in these states are given. The model was tested by observing the temperature dependences of o-Ps lifetime in various media. In the case of silica gels and Vycor glass with narrow pores, the model seems to work well, while for larger pores in Vycor unexpectedly long lifetimes appear in the range of lowest temperatures. 7 2000 Elsevier Science Ltd. All rights reserved.
In positron studies of voids in solids the model of in®nitely deep rectangular potential with spherical symmetry is in common use. The pick-o rate is given by a very popular equation (Eldrup et al., 1981): R 1 R lpo lb 1 ÿ sin 2p R0 2p R0
1
where lb is the o-Ps lifetime in the bulk, R is the well radius, R0=R+DR, and DR approximates the penetration of Ps wave function into the bulk in real well with ®nite depth. Two parameters of the model: lb and DR are of empirical character. The lifetime of o-Ps is, according to the accepted model, determined exclusively by geometry. From the experimental lpo we deduce R, which is only the radius of an ``equivalent
* Corresponding author. Tel.: +48-81-537-6225; fax: +4881-537-6191. E-mail address: [email protected] (T. Goworek).
sphere'' giving the same lifetime as that observed in the experiment. In the framework of this model there is no place for temperature dependence of lpo, except for the small eect of dilatation. Constancy of lpo is justi®ed in the case of small voids, when the spacing of levels (if more than one exists) is much larger than kT.
1. Modi®ed model for large voids With increasing void dimensions more levels appear in the potential well and their energies decrease, falling into the range of kT at room temperature when R 1 2 nm. In such a case, assuming thermal equilibrium, we can expect the lpo averaged over the populated states lpo
X X lipo gi exp ÿEi =kT = gi exp ÿEi =kT i
2
i
where l ipo is the decay rate from i-th level, gi, Ei is the
0969-806X/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 6 X ( 0 0 ) 0 0 2 4 6 - 2
720
T. Goworek et al. / Radiation Physics and Chemistry 58 (2000) 719±722
statistical weight and energy of the i-th level, respectively. Boltzmann distribution of Ps over the levels is assumed. Changes in the population of levels lead to the dependence of lifetime on the temperature. As a rule, the higher Ei, is the larger l ipo, thus with the rise of temperature the o-Ps lifetime should decrease (dilatation gives an opposite eect). Experimental veri®cation of Eq. (2) requires a medium with large free volumes, e.g. a porous one. In physical chemistry the pore structure is usually approximated as a bundle of capillaries. Thus, instead of spherical symmetry as in Eq. (1), we should apply a cylindrical one. The radial wave functions of a particle in the cylindrical potential are Bessel Jm(r ) ones, and lnm po lb
Xnm Xnm R=R0
Xnm J 2m rrdr= J 2m rrdr 0
3
where Xnm is the n-th node (zero crossing point) of Jm(r ) function. Calculation of integrals in Eq. (3) gives:
2
6 4 lnm po R lb 1 ÿ
R R0
2 J 2m
Xnm R R0
ÿ
Jm r1 r ÿ Xnm
dJm dr
Xnm
1, 21 lpo 9:75lb
1, 22 lpo 17:4lb
DR R0
3 3DR 1ÿ 4R0
DR R0
DR R0
3
3
5
then one obtains o-Ps decay rates for the three lowest levels (n = 1; m = 0,21,22):
Fig. 1. Relative deviation d of the approximated lifetimes (Eqs. 6±8) from the exact values for the three lowest states in the cylindrical well. Solid line, (n,m )=(1,0); dashed line, (1,2 1); dash dot line, (1, 2 2). The parameter DR is assumed 0.19 nm.
6
1ÿ
3DR 4R0
1ÿ
3DR : 4R0
7
8
Fig. 1 shows the relative deviation of approximated lpo from the exact one. For Re2 nm, when the upper states play a signi®cant role, the deviation does not exceed 5%, which is acceptable for such a rough model as that discussed here. Applying Eq. (2) one needs to know Ei. The energies of a particle in the in®nitely high cylindrical well of radius R are
3 2nR0 Xnm R Xnm R Xnm R Jm1 J 2m1 Jm 7 R0 R0 R0 Xnm R 5 : 2 J m1 Xnm
The Jm functions are not too handy in calculations, thus for DR<
0 3:85lb l 1, po
Enm
h 2 X 2nm Ea 2mPs R2 ÿ
4
9
where mPs is the mass of positronium, Ea is the energy related to axial motion. It is of the order kT/2; not quantized, but the same for all levels. Thus, in Eq. (2) we have rather the average energies. The energy Ea adds (on average) a constant to the exponent and does not change the relative population of levels; with good approximation we can introduce into Eq. (2) the ®rst term from Eq. (9). The quantized energy (both in cylindrical and spherical symmetry) is inversely proportional to the well radius square. It means that the population of levels is extremely sensitive to the accepted value of R. In the Tao±Eldrup model the well radius is assumed R0, while the real one is by DR smaller. If the model is consistently used with R0 as radius the calculated energies are decreased. On the other hand, transition from ®nite to in®nite well depth at the same radius leads to increasing the energies of the states. Both factors, i.e. R increase and well shallowing act in opposite directions. Although the energy shifts due to these factors are not identical, the use of R0 in Eq. (9) instead of R0ÿDR is somehow justi®ed, keeping in mind the approximate character of the model. In our earlier paper (Ciesielski et al., 1998) we
T. Goworek et al. / Radiation Physics and Chemistry 58 (2000) 719±722
721
have shown that the best ®t of Eq. (2) to the experimental data is obtained when DR = 0.19 nm (preserving the traditional value of lb=2 nsÿ1).
2. Samples and apparatus Samples of porous Vycor glass of various pore diameter were prepared by changing the time of liquation with subsequent leaching (5 h at 908C) of the samples (Haller, 1965). Silica gels were commercially available samples (Merck). The positron lifetime spectra were recorded using a standard fast±slow spectrometer with the energy window in the STOP channel widely open in order to register the most of 3g decays. The resolution was 280 ps (FWHM), the time base was 2 ms wide, typical spectrum contained (2±4) 106 coincidences. The spectra were analysed using the LT program (Kansy, 1996) in which the long lived components were assumed as continuous log-Gaussian distributions of l. The reciprocal of l at the peak of distribution, tp, was used as a measure of the lifetime. The measurements were performed at a pressure 10.5 Pa in order to avoid ortho±para conversion in atmospheric oxygen.
3. Experimental results and discussion The positron lifetime spectra were measured in the range 110±420 K. The results for silica gels Si40 and
Fig. 2. Temperature dependence of o-Ps lifetime in silica gels Si40 and Si60. Solid curves represent the result of calculations according to Eq. (2) for R = 1.85 nm and R = 3.5 nm, respectively.
Fig. 3. The o-Ps lifetime in Vycor glass (sample No.1) as a function of temperature.
Si60 are shown in Fig. 2. In the case of large pores positron needs a relatively long time to be thermalized, thus the whole initial part of the spectrum was neglected and one long lived component was ®tted to the truncated spectrum beginning at 40 ns delay. It is seen that the model curves representing Eq. (2) for cylindrical geometry and DR = 0.19 nm ®t quite well to the experiment. It would be interesting to see how the temperature dependences of lifetime spectra change at the transition from one-level to multi-level structure of quantized states in the well. Therefore, a set of ®ve Vycor glasses with dierent pore radii (all below 2 nm) was studied. Sample No. 1 was leached without previous liquation, thus the narrowest possible pores were produced. According to the model, the lifetime 8 ns corresponds to the capillary radius 0.56 nm. At this radius we cannot expect a measurable population of the next level above the ground state and no lifetime variation with temperature should be observed. The experiment
Fig. 4. Temperature dependence of o-Ps lifetime in Vycor glass with average pore radii 1.55, 1.31 and 1.25 nm (from top to bottom). The solid line represents the result of calculations according to Eq. (2) for R = 1.31 nm, as determined by LN method. One long lived component ®t.
722
T. Goworek et al. / Radiation Physics and Chemistry 58 (2000) 719±722
0.71 and 1.56 nm (0.71 nm pores are not visible with the classic LN method). In our model it corresponds to the lifetimes 12.6 and 60.5 ns (in the low temperature limit). The calculated curve for 0.71 nm pores goes close to the experimental points, while the curve for 1.56 nm is evidently shifted upwards and its slope in the range 110±210 K does not re¯ect, like in onecomponent ®t, the experimental trend. It is to be noted that a weak short lived component can be explained alternatively as the result of incomplete thermalization of Ps and overpopulation of high states in the ®rst nanoseconds of o-Ps life.
4. Conclusions
Fig. 5. Temperature dependence of o-Ps lifetime in Vycor glass with average pore radius from LN method 1.31 nm (sample No. 3). The long lived part of the spectrum assumed as two-component one. Solid lines are calculated from Eq. (2) assuming the pore radii as found by SAXS method (0.71 and 1.56 nm).
reveals a small decrease of the lifetime with rising temperature (Fig. 3), however, the changes are not signi®cant. The samples Nos. 2±5 were annealed (before leaching) during 8, 24, 48 and 96 h; measurements of pore size distribution by desorption of liquid nitrogen gave the average pore radii of 1.25, 1.31, 1.37 and 1.55 nm, respectively. All the experimental curves were similar, shifting towards longer lifetimes with the pore size increase. Typical data, collected for the samples No. 2, 3, 5 are shown in Fig. 4. The lifetime decreases systematically with the temperature, changing the slope which is largest at low temperature end of the ®gure. It is seen how sensitive the lifetime is on the pore radius: the dierence of the average radii for the neighbouring curves No. 2 and 3 is several percent only (0.06 nm). The model curve shown in Fig. 4 is calculated for sample No. 3 (R = 1.31 nm). It follows the tendency indicated by the experiment in the region above 250 K only. At low temperatures the lifetimes, instead of approaching the saturation with T 4 0, increase sharply. An attempt to split the long lived component into two did not bring any new information. Analysing the whole spectrum it was possible to single out an additional component, but it was a short lived one, of a very low intensity (for sample No. 3 at low temperature t4p 1 14 ns, I4 1 2%) and the whole temperature variation was retained in the longest lived part (Fig. 5). The measurements of pore sizes by small angle X-ray scattering (SAXS) gave for sample No. 3 two peaks of the distribution of radii:
The model of o-Ps in the capillary, taking the annihilation from excited states of a particle in the well into consideration, is consistent with the experimental data for silica gels and for Vycor glass of a small pore diameter. Anomalous t vs T dependence in Vycor already appears at R = 1.25 nm and does not change with R increase. It would be interesting to perform similar measurements for radii lying in the transition region from normal t(T ) dependence (R = 0.6 nm) to anomalous one (R = 1.25 nm). The average pore radii determined by various classic methods show a relatively large scatter of results, making a more accurate comparison of the annihilation data impossible with the proposed model.
Acknowledgements The authors wish to thank Prof. A.L. Dawidowicz (Department of Chemical Physics, UMCS) for preparation of Vycor samples, and Dr. S. Pikus (Department of Crystallography, UMCS) for SAXS measurements. This work was not supported by KBN grants.
References Ciesielski, K., Dawidowicz, A.L., Goworek, T., Jasinska, B., Wawryszczuk, J., 1998. Positronium lifetimes in porous Vycor glass. Chem. Phys. Lett. 289, 41. Eldrup, M., Lightbody, D., Sherwood, J.N., 1981. The temperature dependence of positron lifetimes in solid pivalic acid. Chem. Phys. 63, 51. Haller, W., 1965. Rearrangement kinetics of the liquid-liquid immiscible microphases in alkali borosilicate melts. J. Chem. Phys. 42, 686. Kansy, J., 1996. Microcomputer program for analysis of positron annihilation lifetime spectra. Nucl. Instrum. Methods A374, 235.