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Positronium states in the pores of silica gel
Chemical Physics 230 Ž1998. 305–315
Positronium states in the pores of silica gel T. Goworek ) , K. Ciesielski, B. Jasinska, J. Wawryszczuk ´ Institute of Physics UMCS, Pl. Marii Curie Sklodowskiej 1, 20-031 Lublin, Poland Received 21 September 1997
Abstract The lifetime spectra of positrons annihilating in silica gels with pore radii from 2 to 10 nm were measured, and the results compared with the predictions made by the Tao–Eldrup model extended to include the annihilation of positronium from the excited states in the potential well. The long-lived parts of the spectra were analysed as a sum of 1–3 components with continuous l distributions Žgaussian in the logarithmic scale.. The 1p state population in Si40 is demonstrated; the decay constant of ortho-Ps in it agrees quite well with the model estimate and the growth of intensity with temperature resembles that expected from the Boltzmann distribution. In one-component fit the lifetimes at the maximum of l distribution at a room temperature for various pore radii were found also to be very close to the calculated lifetime averaged over all states of the positronium particle in the rectangular potential well. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Positron annihilation methods w1x are becoming popular as a tool to study free volumes in solids. Up to now attention has mainly been focused on subnanometer objects, such as microvoids in amorphous solids w2x, in particular the reconstruction of void size distributions. Positron studies of the media containing larger free volumes, like porous materials, fine powders etc., have been predominantly oriented toward the chemical interactions of positrons with the gases filling the medium or with molecular layers adsorbed on the pore surface w3x, rarely to the relation between annihilation characteristics and pore size w4–6x. This paper represents an attempt to obtain more information on the Ps states in porous media and to compare the experimental results with model estimates. A short note with partial results for the )
Corresponding author. E-mail: [email protected]
Si40 silica gel in a limited range of temperatures has been published earlier w7x. The energetic positron entering matter loses its energy by producing ion–electron pairs and, at the end of its track, being already thermalized, annihilates with one of the medium electrons. It can annihilate directly as a free particle or the annihilation is preceded by the formation of a bound state with an electron from the ionization track. This hydrogen-like structure is called positronium ŽPs. and exists in two spin states: S s 0 Žsinglet, para-positronium, p-Ps. or S s 1 Žtriplet, ortho-positronium, o-Ps., with distinctly different annihilation properties. According to the parity conservation rules, p-Ps decays into two 511 keV gamma quanta with a mean lifetime in vacuum of ts s 125 ps, while the unperturbed o-Ps decays into three quanta with a continuous energy spectrum extending from 0 to 511 keV; its lifetime in vacuum is t t s 142 ns. Higher-order processes with the production of four or more quanta can be
0301-0104r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 8 . 0 0 0 6 8 - 8
T. Goworek et al.r Chemical Physics 230 (1998) 305–315
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neglected. In the condensed media the intrinsic decay of Ps competes with the pick-off process, i.e. annihilation of eq bound in Ps with a strange electron picked from the surrounding medium and having antiparallel spin orientation. The decay rates of p-Ps and o-Ps are thus
l1 s ls q l po ,
l3 s lt q l po
Ž 1.
where ls s 1rts and lt s 1rt t , and l po is the pick-off rate usually assumed identical for both paraand ortho-states. In liquids and bulk solids, the pickoff process shortens the o-Ps lifetime by approximately two orders of magnitude; the p-Ps lifetime remains almost unaffected. The pick-off annihilation is a two-quantum process; thus the f 3g fraction of o-Ps decaying via 3g annihilation is f 3g s
lt lt q l po
.
Ž 2.
2. Pick-off annihilation model In solids the Ps atom, except in rare cases of delocalization w8x, is trapped in the region of low electron density. It can be a defect of void type, e.g. vacancy w9x, neighbourhood of small admixture molecule w10x, or natural free volume between the molecules existing in a perfect crystal w11x and also the pore in an amorphous solid. A simple model describing the pick-off annihilation of o-Ps trapped in the perfectly spherical void of radius R was proposed by Tao w12x and modified by Eldrup et al. w13x. In that model one assumes that the void represents a rectangular infinite potential well for Ps. The well radius R 0 is assumed larger than that of the void; the density of electrons between R and R 0 assures l po s la Ž la , the spin-averaged Ps decay rate. when the overlap of Ps wavefunction with the electron cloud around the void closes to unity. In molecular crystals and polymers these assumptions are not strictly fulfilled: the well is shallow, the potential is not rectangular, the voids are often not spherical. Thus, the Tao–Eldrup equation
ž
l po s la 1 y
R
1 q
R0
2p
sin 2p
R R0
/
Ž 3.
is only an approximation of real l po versus R relation Žas emphasized by these authors.. Other versions of the model, using a finite height of the potential barrier, are not yet in common use, at least for solids w6,14,15x. It seems that among various materials the porous ones, like silica gels, are closest to the suppositions of the classic Tao model. The positronium, formed in the bulk, is ejected into the vacuum gaining energy equal to its negative work function. The height of the potential barrier for Ps is equal to this work function. The barrier shape resembles the rectangular one when the ‘transition region’ Ž R, R 0 . is narrow comparing to R. The value of D R s R 0 y R is usually assumed 0.166 nm Žchosen empirically for small voids in the organic media., thus much smaller than typical pore radius. The Ps work function for silica is of the order of eV. According to Sferlazzo et al. w16x it is y3.3 eV, while Morinaka et al. w17x give the values of y0.8 and y3.27 eV for Ps formed in the bulk and on the surface, respectively. On the other hand, the spacing of levels of the particle in the well, decreasing like 1rR 2 , falls in the silica gels into the range of tens of meV, thus is two orders smaller than the well depth and the infinite barrier can be assumed as a good approximation. However, when the level spacings are small, of the same order as thermal energy kT or less, description of the annihilation processes becomes more complicated, as the population of levels lying above the lowest one Ž1s. cannot be neglected ŽEq. Ž3. relates to the annihilation from the 1s state.. Note that Ps, which managed to thermalize in the bulk, again becomes a ‘hot’ particle after injection into the pore and undergoes the thermalization process once more. In this paper we do not discuss the mechanism of Ps thermalization, paying attention only to o-Ps in the state not too distant from thermal equilibrium. An estimate of the time needed to thermalize Ps in fine powders is given by Fox and Canter w18x; that time is of the order of 30 ns. A similar estimate by Chang et al. w19x for o-Ps in aerosil gives the time to complete thermalization as f 60 ns. The dimensions of free volumes in both of these media Žjudging from the o-Ps lifetime. are larger than those of pores in the silica gels investigated by us, so we can expect the thermalization time to be not longer than that obtained by the Fox
T. Goworek et al.r Chemical Physics 230 (1998) 305–315
or Chang estimates, and thus comparable to the expected o-Ps lifetimes. The pores in silica gel are not spherical or cylindrical, but following the common practice for similar irregularly shaped free volumes Že.g. in polymers., we will hereafter use idealized geometries, spherical or, eventually, cylindrical. Thus, the radius R discussed throughout this paper is an ‘effective’ or ‘equivalent’ one. The energies of a particle in spherical and infinitely long cylindrical wells are, respectively, Ensph l s
"2
X n2l
2 m Ps R 02
Encyl ms
;
"2
Zn2m
2 m Ps R 02
q Ea
y1
la Ns
X nl
HX
nl RrR 0
jl2
2
Ž r . r d r q lt
Ž 5. y1
tncyl m s la Nc
Znm
HZ
nm RrR 0
Jm2 Ž r . rd r q lt
Fig. 1. The o-Ps lifetime in the lowest levels Ž n, l . of the particle in a spherical well of radius R. Ža. 1s, Žb. 1p, Žc. 1d, Žd. 2s, Že. 1f.
Ž 4.
where X n l and Zn m are the nodes of Bessel functions jl (r) and Jm(r), respectively; m Ps is the Ps mass, n s 1, 2, 3, . . . ; l s 0, 1, 2, . . . ; m s 0, " 1, " 2, . . . , etc; Ea is the energy related to the axial motion of particle Žnot quantized.. The values of X n l and Zn m for 10 lowest states are listed in Table 1. The o-Ps lifetimes can be calculated the same way as in the Tao–Eldrup model:
tnsph l s
307
Ž 6.
where the normalizing factors are: Ns s 1rH0X n l jl2 Ž r . r 2 d r and Nc s 1rH0Z n m Jm2 Ž r . rd r Žaxial motion does not influence the penetration of wavefunction into the electronic layer.. The diagrams in
Figs. 1 and 2 show the lifetime of o-Ps as a function of the pore radius R for both pore geometries. As a rule, the higher the level, the shorter the o-Ps lifetime in it. The extended model described here transforms in the limit of small R or 0 K temperature into the commonly used Eldrup–Tao model ŽEq. Ž3... It would be interesting to observe experimentally the effects of o-Ps annihilation from states other than the lowest one. As can be seen from Fig. 2, distinct differences of lifetimes in particular states can be observed for R ( 5 nm; for bigger pores the term lt becomes dominant and all lifetimes approach the vacuum value 142 ns. On the other hand, at small R, less than 1 nm, the energies in the well are so large that the second level population is negligible at all accessible temperatures. The most appropriate medium to study the population of excited levels in the well seems thus to be the silica gel Si40 ŽMerck, the average R is assumed here as 2.0 nm.; some measurements were also performed for Si60, Si100
Table 1 The nodes of Bessel functions jl and Jm for lowest l and m values n, l
Xn l
n, < m <
Zn < m <
1, 0 1, 1 1, 2 2, 0 1, 3 2, 1 1, 4 2, 2 1, 5 3, 0
3.142 4.493 5.763 6.283 6.988 7.725 8.183 9.095 9.356 9.425
1, 0 1, 1 1, 2 2, 0 1, 3 2, 1 1, 4 2, 2 3, 0 1, 5
2.405 3.832 5.136 5.520 6.380 7.016 7.588 8.417 8.654 8.771
Fig. 2. The o-Ps lifetime in the lowest levels Ž n, " m. of the particle in an infinite cylindrical well of radius R. Ža. Ž1,0.; Žb. Ž1,1.; Žc. Ž1,2.; Žd. Ž2,0.; Že. Ž1,3..
T. Goworek et al.r Chemical Physics 230 (1998) 305–315
308
and Si200 Žaverage pore radii are 3.5, 6 and 9.5 nm w20x, respectively..
3. Experimental The positron source 22 Na in a Kapton envelope was placed between two layers of silica gel, and the sandwich was pressed together in a copper container. Before use the samples were dried during 4 h at 470 K. The container was placed in a chamber where the pressure Žf 0.5 Pa. was maintained by continuous pumping. The temperature of the samples was controlled in the range of 95–575 K with 0.2 K accuracy. The lifetime spectrum was registered by a conventional fast–slow spectrometer with plastic Pilot U scintillators. The ‘stop’ energy window was opened wide to transmit 80% of the energy range and thus to obtain a good efficiency for registration of 3g events. At this setting the resolution was f 300 ps, FWHM; the channel definition was 0.260 ns. In order to keep the net count to the background ratio as high as possible, the source was of very low activity Ž100 kBq.. At the 100 ns delay the net counting rate was still f 9 times higher than the background. With our source activity and counter geometry 2.2 = 10 6 coincidences were collected during a 1-day run. In the silica gels there is no single pore radius but rather a continuous bell-shaped distribution; thus we should also expect a continuous distribution of lifetimes. In such a case the LT program w21x seems best suited to process the data; it fits the sum of discrete components or continuous l distributions of gaussian shape in the logarithmic l scale. Each component with the finite distribution width s is described by the function: N Ž t. s I
I
`
'2p s H0 exp y
ln2 Ž ltp . 2s 2
long-lived part. Thus in the majority of cases analysis was limited to the range beginning from the delay Ž10–40 ns.; the end point of the analysed range was always 1025 ns. The uncertainty limits of fitted parameters given in this paper are statistical only.
4. Results and discussion 4.1. Testing Tao–Eldrup model at large radii The main object of our study was silica gel Si40 ŽMerck.. At the 260 ps channel definition it was not possible to resolve p-Ps and free annihilation components which gave one averaged lifetime Žt 1 s 0.49 ns.. The spectrum also contained a weak Ž2.4%. component with t 2 s 1.9 ns and the long-lived part with the total intensity I L exceeding 30%. The temperature dependence of I L intensity is shown in Fig. 3. The I L values were determined by calculating the area under the long-lived component and needed not be identical with actual o-Ps formation probability. Contrary to the short-lived components annihilating almost entirely via two-quantum annihilation; the long-lived components decay in substantial part via a three-quantum process Žsee Eq. Ž2.., and different efficiencies of registration of two- and three-quantum spectra can distort the proportion of particular intensities.
exp Ž yl t . d l
Ž 7. where I is the relative intensity of the component, tp is the lifetime in the maximum of distribution Žnote that l p s 1rtp does not coincide exactly with the average lav .. The short-lived components were assumed discrete. In this work we are only interested in the
Fig. 3. The intensity of the long-lived part of the positron lifetime spectrum I L for Si40 as a function of temperature.
T. Goworek et al.r Chemical Physics 230 (1998) 305–315
The internal pore surface was much larger than the outer surface of silica grains. Thus we assumed that o-Ps annihilates entirely inside the pores; Ps in the space between the grains was neglected. At very low temperatures only the 1s ground state in the well should be populated, and the long-lived part of spectrum should be described by one component with certain distribution of l, i.e. by two parameters: the l value at the maximum of distribution l3p s 1rt 3p and the width of distribution s 3 . The lowest temperature obtainable in our experimental set-up is 95 K; at 100 K the population 1p, next to state 1s, should only be f 2%, according to the Boltzmann distribution. Such an admixture cannot measurably distort the observed l3p rate. However, it was noticed that the lifetime spectra of positrons in Si40 registered at 95–113 K contain at the beginning of the time scale a certain amount of short-lived species, leading in the one-component fit to anomalous broadening of s 3 and increase of l3p ŽFig. 4.. The l3p and s 3 values stabilize when the beginning of fitting range Žspectrum cut-off point. Dc exceeds
309
50 ns, i.e. for the cut-off points from 70 to 220 ns the fitted values of t 3p and s 3 are identical, in the limits of error. The 70 ns delay can be the measure of thermalization time. The parameter values describing the long-lived component derived from the spectra at 95 and 113 K are:
t 3p s Ž 83.1 " 0.4 . ns,
s 3 s 0.22 " 0.02 .
This lifetime value, as well as those quoted in other parts of this paper, is very close to those expected from spherical model for ‘effective R’ values identical to R determined by nitrogen desorption or mercury intrusion methods. For spherical void with R s 2.0 nm, Eq. Ž3. gives a 78 ns lifetime while, for a cylinder having the same radius, the expected lifetime is 95 ns. Thus, in the calculations, we will use the spherical model with R taken from the literature. The width of l 3 distribution is larger than expected. Data about the width of distribution of pore radii, given in the literature w20,22,23x, are not too precise; the upper limit of the expected s 3 value based on these data is s 3 s 0.16 Žprocessing the spectrum with s 3 fixed at 0.16 gives a minor change of lifetime, t 3p s 81 ns.. However, the 0.16 limit value follows from the distribution of pore radii only; additional spread of level energies Žand consequently of l. can be the result of nonsphericity of voids. The spectroscopic symbols s, p, d, etc., relate to the states in the spherical potential. Thus in our case the terms ‘1s-like’, ‘1p-like’ should be more appropriate; however, for brevity, the classic spectroscopic terminology is used below. This result indicates that, at sufficiently low temperature, when only the ground state in the well is populated, the Tao–Eldrup model describes quite well the lifetime of o-Ps. 4.2. Temperature Õariation of the positron lifetime spectrum in Si40
Fig. 4. The variation of the fitted lifetime and of the distribution width with the change of spectrum cut-off point Žbeginning of the analysed time range.. Si40, one-component fit. v, Spectrum registered at 113 K; B, at 95 K.
The lifetime spectra in Si40 were registered in a wide range of temperatures. In the model of rectangular well Žfinite or infinite. limited to the 1s ground state there is no place for temperature variation of lifetime. Evident shortening of the lifetime with rise in temperature proofs the population of higher states with larger decay rates.
310
T. Goworek et al.r Chemical Physics 230 (1998) 305–315
At 100 K, in thermal equilibrium, o-Ps mainly populates the 1s ground state. With a rise in temperature the population of the first excited state in the well should also rise. The shape of the long-lived part of the spectrum depends on the average dwelling time t D of o-Ps particle in definite state: Ža. if t D is much longer than the o-Ps lifetime we should observe a separate component for the annihilation from each state, and Žb. if t D is much shorter than the o-Ps lifetime one component exists, with the lifetime being the reciprocal of l averaged over all engaged states. Condition Žb. is necessary to achieve thermal equilibrium. Distinguishing cases Ža. and Žb. by analysis of the spectrum is not easy when the decay curves are not exactly exponential but rather represent continuous and not too narrow l distributions produced by the distribution of R. Below we discuss both cases. 4.2.1. Case (a) Let us assume that at moderate temperatures the long-lived part of the spectrum consists of two components: one describing the decay from the 1s ground state, and the other from the 1p state. The parameters t 3p and s 3 found at 95–113 K and at the spectrum cut-off point exceeding 70 ns were assumed as characterizing the 1s component and thus fixed. The fitting parameters were t4p and I4 of possible 1p component. The intensity I4 does not relate to the whole spectrum; it denotes the share of the fourth component in the long-lived part of the spectrum, i.e. I3 q I4 s 1 Ž I3 is intensity of the ‘83 ns’ component.. The value of s4 was assumed, in analogy to the 1s state, to be equal to 0.22; changing it in moderate limits has no substantial influence on the fitted values Žexcept the experimental points at the lowest temperatures.. Two-component analysis of the long-lived part of the spectrum reveals the component with the lifetime t4p f 43 ns. Variation of parameters describing that new component with the change of spectrum cut-off point Dc was tested. The results of two-component fit in the range of 133–223 K are shown in Fig. 5 for the spectrum cut-off point varying from 10 to 40 ns. Change in cut-off point has little influence on the I4 intensity, but the t4p value for small delays is shorter than that for delays above 40 ns. This effect is large at 133 K but disappears rapidly with increase in temperature; at
Fig. 5. The lifetime t4p from the two-component analysis of the Si40 lifetime spectrum as a function of the spectrum cut-off point for several temperatures.
223 K it is almost invisible. When the cut-off point is shifted upwards, the fitted values of t4p tend to an asymptotic value which is very close to 43 ns. This form of t4p versus Dc dependence suggests the existence of additional relatively short-lived componentŽs.. Thus, the spectra were analysed from Dc s 10 ns also as a sum of three components. As before, the longest-lived component was fixed Žt 3p s 83 ns; s 3 s 0.22., the intermediate one had a fixed width Ž s4 s 0.22., and other parameters were left free. This kind of spectrum processing restores the t4p value f 43 ns in the low temperature range and demonstrates the existence of short-lived species with t 5p s 8.1 " 1.0 ns and intensity I5 s 5.1 " 1.0%. The 8 ns component is located in the time range where Ps is certainly far from thermalization; thus the component cannot be ascribed to a definite state of particle in the well, and it also does not need to be exponential Žthe width s5 is sometimes unrealistically large, reaching 0.6.. The numbers given above are only an indication that ‘something goes on’ at the beginning of spectrum.
T. Goworek et al.r Chemical Physics 230 (1998) 305–315
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The final results of two- and three-component fitting are shown in Fig. 6. ŽFor temperatures - 100 K, where the intensity of the 43 ns component is very low, its separation from the remainder of the spectrum was impossible and only two-component analysis was done.. In the broad range of 105–233 K the lifetime t4p obtained in three-component fit is constant, the average value
t4p s Ž 42.5 " 0.7 . ns is very close to that expected from the model for a 1p state in the spherical well Ž45.2 ns.. In the case of capillaries the expected lifetime would be 63.9 ns. Above 240 K the t4p lifetime begins to decrease, probably due to the increasing participation of the levels lying still higher than 1p. The shape of the l spectrum fitted by LT program is shown in Fig. 7 for several temperatures. Fig. 6 shows that I4 rises with temperature but, in the range of 100–220 K, it is systematically higher than expected from the Boltzmann population of well levels. In addition to inadequacies of the model, there can be at least two sources of discrepancy. If the efficiencies of registration of 3g and 2g I4 will be disevents are different, the ratio I3rI torted; i.e. lower efficiency for 3g decays would reflect in overestimated experimental I4 values. The increased intensity of 1p component can also be
Fig. 6. The lifetime t4p Ža. and intensity S4 Žb. as a function of temperature for the Si40 samples. `, Relate to the two-component analysis of the long-lived part of the spectrum with Dc s 20 ns; v, ', three-component analysis with Dc s10 ns; , calculated S4 in the case of the Boltzmann population of excited states; - - - -, population of the 1p state only.
Fig. 7. The shape of the l distribution as fitted to the spectrum of Si40 for several temperatures; two-component fit. In the case of T s100 K, where the 43 ns component is almost vanished, s4 was not fixed.
a result of incomplete Ps thermalization w24x. Note that I4 intensity cannot be calculated precisely; the population of excited states depends exponentially on their energies and these in turn depend on the square of the radius. The result of model calculation is very sensitive to the assumed pore dimension. The radii taken from nitrogen or mercury methods need not be identical with ‘effective radii’ acting in the pick-off process. We accepted them as giving t 3p and t4p close to the experimental values, but it is only a convenient supposition. If we accept e.g. the R value 1.8 nm, then at 200 K the calculated equilibrium population of 1p state will be 13.6%, i.e. half of that experimentally observed. 4.2.2. Case (b) If the spectrum contains one long-lived component only, averaged over-all populated states, one can try to explain the results of analysis described in the previous paragraph as forced by excessive number of constraints: we fix t 3p , thus the program seeks the part of the spectrum lacking to the average. However, for single component, we should observe a s 3 value similar as at low temperatures Ž s 3 s 0.22.. The experimental result for room temperature is s 3 s 0.47; for T s 208 K s 3 s 0.40. These are enormous widths of l distribution Žrising with the temperature., not acceptable if we have really single
312
T. Goworek et al.r Chemical Physics 230 (1998) 305–315
component — its halfwidth extends from 35 to 107 ns ŽFig. 8.. The lifetime t 3p found for room temperature is 59.9 ns. This result can be compared with an averaged lifetime calculated from the model. For simplicity the decay constants related to particular states were assumed as discrete with l corresponding to the R value at the maximum of pore size distribution; the population of the states was assumed to be Boltzmann-like. The calculated lifetime ²t 3 : is longer than the experimental one, again suggesting incomplete thermalization. The variances of one- and two-component fits are identical. The result is inconclusive. The population of states seems to be not too distant from Boltzmann law, however, the width of l distribution in one-component fit is not acceptable if we really have one component. Cases Ža. and Žb. are the extremities; the spectrum which we observe probably belongs to an intermediate case. The o-Ps lifetime measurements were continued up to 573 K. With the rise in temperature the population of levels lying higher than 1p should play an increasing role. In such a complex case we decided to determine the average lifetime only, i.e. to analyse the long-lived part of the spectrum as one-component with l distribution. The experimental results are shown in Fig. 9 together with the calculated lifetime ²t 3 : averaged over all states. The experimental data and model calculations are not directly comparable as t 3p and average lifetime ²t 3 : are not identical; the LT program fits the l
Fig. 8. The shape of the l distribution as fitted to the spectrum of Si40 at 253 K. , Two-component fit; - - - -, one-component fit.
Fig. 9. Top: Temperature dependence of the t 3p lifetime in Si60; . repreone-component fit, Dc s 20 ns. The solid line Ž sents the calculated ²t 3 :. Bottom: The same data for Si40.
distribution which is gaussian in the logarithmic scale, thus not symmetrical with respect to tp . However, at low s the shift of average with respect to most probable t is to neglect. Experimental lifetimes are slightly different from the model predictions, a general tendency of faster than expected t 3p decrease with a rising temperature is visible, particularly at high temperatures. A partial explanation of this tendency can be the difference in efficiencies of 2g and 3g registration, mentioned in the previous section. If the short-lived, 3g poor, components are registered with higher efficiency, they enter the experimental average with the increased weight. However, only one-third of the longest-lived component, participating in t 3p and related to the 1p state, represents 3g decays; thus the presence of other shorter-lived species cannot cause as drastic a t 3p shortening. In the Tao–Eldrup model the D R parameter in the infinite potential well substitutes the actual penetration range of the Ps wave function outside the well of finite depth; in finite well, the penetration range depends on the distance of level from the well top. At moderate temperatures, when only the lowest states are engaged, and if the Ps work function is of the order of y3 eV w16x, the
T. Goworek et al.r Chemical Physics 230 (1998) 305–315
energies of populated levels are by two orders of magnitude smaller than the well depth and one cannot expect noticeable changes of the penetration range and of the equivalent D R. However, the results given in Refs. w17,25,26x suggest that the prevailing number of Ps atoms gains at emission from the silica the energy 0.8 eV and, at elevated temperatures, when the states up to 1f are populated Žat 570 K the population of 1f should be 8% of the total. the energy of 1f state Ž E1f ( 0.20 eV. in the infinite well becomes comparable to the actual well depth and the model can no longer be applicable. 4.3. Ortho-Ps lifetime in silica gels with large pores In the case of Si60 with R s 3.5 nm, even at 95 K the second and third levels in the spherical well are noticeably populated; according to the Boltzmann law f 32% of o-Ps atoms should annihilate from these states; thus, as in the previous section, we approximated the long-lived part of the spectrum by a single component Žwith l distribution.. The lifetime spectra were measured in the 113–420 K range and analysed from the Dc s 20 ns cut-off points; shifting that point to 40 or 60 ns gave the t 3p value differing by less than 0.6 ns, i.e. in the limits of error. The results of one-component fit are shown in Fig. 9. The continuous curve represents the average lifetime ²t 3 : calculated assuming, as previously, the discrete t i Žtaken from Eq. Ž5. for R s 3.5 nm. and the Boltzmann distribution over the states. The experimental t 3p values lie again below the calculated ²t 3 :; nevertheless, the tendency of changes of average lifetime with temperature seems to be much better reproduced. The s 3 value is about 0.2 at low temperatures, rising to 0.36 at 420 K. That broadening is difficult to explain in a pure one-component spectrum, suggesting a more complex structure of the spectrum. The variance of one-component fit for Si60 is close to unity in the whole range of temperatures Žat 420 K the variance is 1.02.. For Si40 the variance remains acceptable Ž( 1.15. up to f 400 K; it seems that the rise of s 3 compensates the effect of growth of the components related to the excited states; above 400 K that compensation mechanism ceases to be effective and the variation rises to 3 and higher. The lifetime spectra for Si100 and Si200 were
313
Fig. 10. The calculated average lifetime at room temperature ²t 3 : . and the experimental t 3p versus average pore radius Ž points for various silica gels. The R values and their uncertainties are taken from Ref. w20x. Dashed Ž — — — . line shows, for comparison, the lifetime of the 1s state Žstraight extension of Tao–Eldrup model, Eq. Ž3... The vacuum value of the lifetime in triplet state, 142 ns, is also marked ŽPPPPP..
measured only at room temperature and approximated by one component with the non-zero l distribution. Fig. 10 shows the experimental values of t 3p for various average pore radii and the calculated ²t 3 : versus R dependence. As before, in the model calculations, the discrete l and Boltzmann distribution were assumed. Only seven lowest levels were included in the calculation; for R ) 6 nm that number of levels is not sufficient and the ²t 3 : curve in Fig. 10 runs slightly higher than it would be in the case of more precise calculations. In spite of all approximations Žhypothetical sphericity of voids, the discrete l accepted in calculations, nonidentity of t 3p and ²t 3 :, Boltzmann distribution. the experimental points locate unexpectedly close to the ²t 3 : curve. These approximations and low number of experimental points prevent one from drawing fargoing conclusions; however, Fig. 10 seems to suggest that the pore dimensions in Si100 given in the literature are oversized. The o-Ps lifetime in Si200 is only several nanoseconds shorter than that for the particle in vacuum. 4.4. The D R Õalue Experimental data on o-Ps lifetime in the void coincide quite well with model expectations. Note that an important parameter strongly influencing the
314
T. Goworek et al.r Chemical Physics 230 (1998) 305–315
result of calculations is R 0 y R s D R. Its value Ž0.166 nm. accepted here was empirically chosen w9x to match the data in the range of sub-nanometer voids, for molecular crystals and liquids; the results of our measurements with silica gels seem to indicate that at this stage of study there is no necessity to modify D R when applying the Tao–Eldrup model to the case of large voids.
radii. to demonstrate more clearly constancy of l po for the definite state. For voids with R 0 5 nm, the average lifetime approaches the vacuum value of 142 ns. The experimental most probable lifetimes t 3p from the onecomponent fit at room temperature are very close to the calculated average ²t 3 :; that gives one the chance to use the positron lifetime spectroscopy as a method of estimation of the average pore radii, at least below 6 nm.
5. Conclusions Acknowledgements Processing the experimental data and comparison with the model required many simplifications, inevitable in the case of quite complex structures like silica gels. The model discussed in this paper does not pretend to describe quantitatively the o-Ps lifetimes in large voids; nevertheless, being relatively simple, it reproduces the experimental data rather well. Consistency of experiment and model requires to account the population of excited states. Decrease of lifetime with rising temperature is a direct consequence of the engagement of these states into the annihilation process. The spectrum structure is not definitively settled; there are arguments supporting the two-component character of the observed decay curve, while thermal equilibrium needs one component. All the considerations presented in this paper are valid for empty pores only. The presence of gases Žin particular of oxygen w27,28x. or adsorbed layers entering with Ps into chemical reactions can shorten all the lifetimes, adding in Eq. Ž1. a new term related to these processes. The idea of the model is to tie explicitly the lifetime with the dimensions of free volume. In the classic approach, as applied in the thermalization models w29x, positronium is treated as a particle interacting with the medium during collisions with the cavity walls and the pick-off annihilation rate per collision is assumed as constant. For definite cavity size, the pick-off rate should be temperature dependent Ž l po ; 'T .. In the quantum approach, upon lowering the temperature l po tends to the finite value, characteristic for the 1s state and for given void size. It would be advisable to extend the measurements toward lower temperatures Žor smaller
The authors wish to thank Dr. Y. Nagashima for making available the results of his work prior to publication. This work was not supported by the KBN committee grants. References w1x P. Hautojarvi, Positrons in Solids ŽSpringer, Berlin, 1979.. ¨ w2x Y.C. Jean, Mater. Sci. Forum 105-110 Ž1992. 309. w3x K.L. Cheng, Y.C. Jean, in: D.M. Schrader, Y.C. Jean ŽEds.., Positron and Positronium Chemistry ŽElsevier, Amsterdam, 1979. pp. 282-317. w4x K. Venkateswaran, K.L. Cheng, Y.C. Jean, J. Phys. Chem. 88 Ž1984. 2465. w5x K. Venkateswaran, K.L. Cheng, Y.C. Jean, J. Phys. Chem. 89 Ž1985. 3001. w6x V.P. Shantarovich, J. Radioanal. Nucl. Chem. 210 Ž1996. 365. w7x T. Goworek, K. Ciesielski, B. Jasinska, J. Wawryszczuk, Chem. Phys. Lett. 272 Ž1997. 91. w8x O.E. Mogensen, M. Eldrup, J. Glaciol. 21 Ž1978. 85. w9x M. Eldrup, N.J. Pedersen, J.N. Sherwood, Phys. Rev. Lett. 43 Ž1979. 1407. w10x T. Goworek, C. Rybka, J. Wawryszczuk, Phys. Stat. Sol. ŽB. 89 Ž1978. 253. w11x B. Jasinska, A.E. Koziol, T. Goworek, J. Radioanal. Nucl. ´ Chem. 210 Ž1996. 617. w12x S.J. Tao, J. Chem. Phys. 56 Ž1972. 5499. w13x M. Eldrup, D. Lightbody, J.N. Sherwood, Chem. Phys. 63 Ž1981. 51. w14x T.W. Zerda, G. Hoang, B. Miller, C.A. Quarles, G. Orcel, Mater. Res. Soc. Symp. Proc. 121 Ž1988. 653. w15x A. Domjan, A. Vertes, Mater. Sci. Forum 255-257 ´ K. Suvegh, ¨ ´ Ž1997. 251. w16x P. Sferlazzo, S. Berko, K.F. Canter, Phys. Rev. B 35 Ž1987. 5315. w17x Y. Morinaka, Y. Nagashima, Y. Nagai, T. Hyodo, T. Kurihara, T. Shidara, K. Nakahara, Mater. Sci. Forum 255-257 Ž1997. 689.
T. Goworek et al.r Chemical Physics 230 (1998) 305–315 w18x R.A. Fox, K.F. Canter, J. Phys. B 11 Ž1978. L255. w19x T. Chang, M. Xu, X. Zeng, Phys. Lett. A 126 Ž1987. 189. w20x J.M. Drake, P. Levitz, J. Klafter, New J. Chem. 14 Ž1990. 77. w21x J. Kansy, Nucl. Instr. Meth. A 374 Ž1996. 235. w22x J. Goworek, W. Stefaniak, Coll. Surf. 57 Ž1991. 161. w23x J. Goworek, A. Nieradka, A. Da¸browski, Fluid Phase Equilibria, in press. w24x A.P. Mills Jr., E.D. Shaw, R.J. Chichester, D.M. Zuckerman, Phys. Rev. B 40 Ž1989. 2045. w25x Y. Nagashima, M. Kakimoto, T. Hyodo, K. Fujiwara, A.
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Ichimura, T. Chang, J. Deng, T. Akahane, T. Chiba, K. Suzuki, B.T.A. McKee, A.T. Stewart, Phys. Rev. A 52 Ž1995. 258. Y. Nagashima, private communication. R. Paulin, G. Ambrosino, J. Phys. ŽParis. 29 Ž1968. 263. V.I. Goldanskii, A.D. Mokrushin, A.O. Tatur, V.P. Shantarovich, Kinet. Katal. 13 Ž1972. 961. C. Dauwe, T. Van Hoecke, D. Segers, in: K. Tomala, E.A. ŽEds.., Condensed Matter Studies by Nuclear MethGorlich ¨ ods, Institute of Physics, Jagellonian University, Krakow, 1995.
Positronium states in the pores of silica gel T. Goworek ) , K. Ciesielski, B. Jasinska, J. Wawryszczuk ´ Institute of Physics UMCS, Pl. Marii Curie Sklodowskiej 1, 20-031 Lublin, Poland Received 21 September 1997
Abstract The lifetime spectra of positrons annihilating in silica gels with pore radii from 2 to 10 nm were measured, and the results compared with the predictions made by the Tao–Eldrup model extended to include the annihilation of positronium from the excited states in the potential well. The long-lived parts of the spectra were analysed as a sum of 1–3 components with continuous l distributions Žgaussian in the logarithmic scale.. The 1p state population in Si40 is demonstrated; the decay constant of ortho-Ps in it agrees quite well with the model estimate and the growth of intensity with temperature resembles that expected from the Boltzmann distribution. In one-component fit the lifetimes at the maximum of l distribution at a room temperature for various pore radii were found also to be very close to the calculated lifetime averaged over all states of the positronium particle in the rectangular potential well. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Positron annihilation methods w1x are becoming popular as a tool to study free volumes in solids. Up to now attention has mainly been focused on subnanometer objects, such as microvoids in amorphous solids w2x, in particular the reconstruction of void size distributions. Positron studies of the media containing larger free volumes, like porous materials, fine powders etc., have been predominantly oriented toward the chemical interactions of positrons with the gases filling the medium or with molecular layers adsorbed on the pore surface w3x, rarely to the relation between annihilation characteristics and pore size w4–6x. This paper represents an attempt to obtain more information on the Ps states in porous media and to compare the experimental results with model estimates. A short note with partial results for the )
Corresponding author. E-mail: [email protected]
Si40 silica gel in a limited range of temperatures has been published earlier w7x. The energetic positron entering matter loses its energy by producing ion–electron pairs and, at the end of its track, being already thermalized, annihilates with one of the medium electrons. It can annihilate directly as a free particle or the annihilation is preceded by the formation of a bound state with an electron from the ionization track. This hydrogen-like structure is called positronium ŽPs. and exists in two spin states: S s 0 Žsinglet, para-positronium, p-Ps. or S s 1 Žtriplet, ortho-positronium, o-Ps., with distinctly different annihilation properties. According to the parity conservation rules, p-Ps decays into two 511 keV gamma quanta with a mean lifetime in vacuum of ts s 125 ps, while the unperturbed o-Ps decays into three quanta with a continuous energy spectrum extending from 0 to 511 keV; its lifetime in vacuum is t t s 142 ns. Higher-order processes with the production of four or more quanta can be
0301-0104r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 8 . 0 0 0 6 8 - 8
T. Goworek et al.r Chemical Physics 230 (1998) 305–315
306
neglected. In the condensed media the intrinsic decay of Ps competes with the pick-off process, i.e. annihilation of eq bound in Ps with a strange electron picked from the surrounding medium and having antiparallel spin orientation. The decay rates of p-Ps and o-Ps are thus
l1 s ls q l po ,
l3 s lt q l po
Ž 1.
where ls s 1rts and lt s 1rt t , and l po is the pick-off rate usually assumed identical for both paraand ortho-states. In liquids and bulk solids, the pickoff process shortens the o-Ps lifetime by approximately two orders of magnitude; the p-Ps lifetime remains almost unaffected. The pick-off annihilation is a two-quantum process; thus the f 3g fraction of o-Ps decaying via 3g annihilation is f 3g s
lt lt q l po
.
Ž 2.
2. Pick-off annihilation model In solids the Ps atom, except in rare cases of delocalization w8x, is trapped in the region of low electron density. It can be a defect of void type, e.g. vacancy w9x, neighbourhood of small admixture molecule w10x, or natural free volume between the molecules existing in a perfect crystal w11x and also the pore in an amorphous solid. A simple model describing the pick-off annihilation of o-Ps trapped in the perfectly spherical void of radius R was proposed by Tao w12x and modified by Eldrup et al. w13x. In that model one assumes that the void represents a rectangular infinite potential well for Ps. The well radius R 0 is assumed larger than that of the void; the density of electrons between R and R 0 assures l po s la Ž la , the spin-averaged Ps decay rate. when the overlap of Ps wavefunction with the electron cloud around the void closes to unity. In molecular crystals and polymers these assumptions are not strictly fulfilled: the well is shallow, the potential is not rectangular, the voids are often not spherical. Thus, the Tao–Eldrup equation
ž
l po s la 1 y
R
1 q
R0
2p
sin 2p
R R0
/
Ž 3.
is only an approximation of real l po versus R relation Žas emphasized by these authors.. Other versions of the model, using a finite height of the potential barrier, are not yet in common use, at least for solids w6,14,15x. It seems that among various materials the porous ones, like silica gels, are closest to the suppositions of the classic Tao model. The positronium, formed in the bulk, is ejected into the vacuum gaining energy equal to its negative work function. The height of the potential barrier for Ps is equal to this work function. The barrier shape resembles the rectangular one when the ‘transition region’ Ž R, R 0 . is narrow comparing to R. The value of D R s R 0 y R is usually assumed 0.166 nm Žchosen empirically for small voids in the organic media., thus much smaller than typical pore radius. The Ps work function for silica is of the order of eV. According to Sferlazzo et al. w16x it is y3.3 eV, while Morinaka et al. w17x give the values of y0.8 and y3.27 eV for Ps formed in the bulk and on the surface, respectively. On the other hand, the spacing of levels of the particle in the well, decreasing like 1rR 2 , falls in the silica gels into the range of tens of meV, thus is two orders smaller than the well depth and the infinite barrier can be assumed as a good approximation. However, when the level spacings are small, of the same order as thermal energy kT or less, description of the annihilation processes becomes more complicated, as the population of levels lying above the lowest one Ž1s. cannot be neglected ŽEq. Ž3. relates to the annihilation from the 1s state.. Note that Ps, which managed to thermalize in the bulk, again becomes a ‘hot’ particle after injection into the pore and undergoes the thermalization process once more. In this paper we do not discuss the mechanism of Ps thermalization, paying attention only to o-Ps in the state not too distant from thermal equilibrium. An estimate of the time needed to thermalize Ps in fine powders is given by Fox and Canter w18x; that time is of the order of 30 ns. A similar estimate by Chang et al. w19x for o-Ps in aerosil gives the time to complete thermalization as f 60 ns. The dimensions of free volumes in both of these media Žjudging from the o-Ps lifetime. are larger than those of pores in the silica gels investigated by us, so we can expect the thermalization time to be not longer than that obtained by the Fox
T. Goworek et al.r Chemical Physics 230 (1998) 305–315
or Chang estimates, and thus comparable to the expected o-Ps lifetimes. The pores in silica gel are not spherical or cylindrical, but following the common practice for similar irregularly shaped free volumes Že.g. in polymers., we will hereafter use idealized geometries, spherical or, eventually, cylindrical. Thus, the radius R discussed throughout this paper is an ‘effective’ or ‘equivalent’ one. The energies of a particle in spherical and infinitely long cylindrical wells are, respectively, Ensph l s
"2
X n2l
2 m Ps R 02
Encyl ms
;
"2
Zn2m
2 m Ps R 02
q Ea
y1
la Ns
X nl
HX
nl RrR 0
jl2
2
Ž r . r d r q lt
Ž 5. y1
tncyl m s la Nc
Znm
HZ
nm RrR 0
Jm2 Ž r . rd r q lt
Fig. 1. The o-Ps lifetime in the lowest levels Ž n, l . of the particle in a spherical well of radius R. Ža. 1s, Žb. 1p, Žc. 1d, Žd. 2s, Že. 1f.
Ž 4.
where X n l and Zn m are the nodes of Bessel functions jl (r) and Jm(r), respectively; m Ps is the Ps mass, n s 1, 2, 3, . . . ; l s 0, 1, 2, . . . ; m s 0, " 1, " 2, . . . , etc; Ea is the energy related to the axial motion of particle Žnot quantized.. The values of X n l and Zn m for 10 lowest states are listed in Table 1. The o-Ps lifetimes can be calculated the same way as in the Tao–Eldrup model:
tnsph l s
307
Ž 6.
where the normalizing factors are: Ns s 1rH0X n l jl2 Ž r . r 2 d r and Nc s 1rH0Z n m Jm2 Ž r . rd r Žaxial motion does not influence the penetration of wavefunction into the electronic layer.. The diagrams in
Figs. 1 and 2 show the lifetime of o-Ps as a function of the pore radius R for both pore geometries. As a rule, the higher the level, the shorter the o-Ps lifetime in it. The extended model described here transforms in the limit of small R or 0 K temperature into the commonly used Eldrup–Tao model ŽEq. Ž3... It would be interesting to observe experimentally the effects of o-Ps annihilation from states other than the lowest one. As can be seen from Fig. 2, distinct differences of lifetimes in particular states can be observed for R ( 5 nm; for bigger pores the term lt becomes dominant and all lifetimes approach the vacuum value 142 ns. On the other hand, at small R, less than 1 nm, the energies in the well are so large that the second level population is negligible at all accessible temperatures. The most appropriate medium to study the population of excited levels in the well seems thus to be the silica gel Si40 ŽMerck, the average R is assumed here as 2.0 nm.; some measurements were also performed for Si60, Si100
Table 1 The nodes of Bessel functions jl and Jm for lowest l and m values n, l
Xn l
n, < m <
Zn < m <
1, 0 1, 1 1, 2 2, 0 1, 3 2, 1 1, 4 2, 2 1, 5 3, 0
3.142 4.493 5.763 6.283 6.988 7.725 8.183 9.095 9.356 9.425
1, 0 1, 1 1, 2 2, 0 1, 3 2, 1 1, 4 2, 2 3, 0 1, 5
2.405 3.832 5.136 5.520 6.380 7.016 7.588 8.417 8.654 8.771
Fig. 2. The o-Ps lifetime in the lowest levels Ž n, " m. of the particle in an infinite cylindrical well of radius R. Ža. Ž1,0.; Žb. Ž1,1.; Žc. Ž1,2.; Žd. Ž2,0.; Že. Ž1,3..
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308
and Si200 Žaverage pore radii are 3.5, 6 and 9.5 nm w20x, respectively..
3. Experimental The positron source 22 Na in a Kapton envelope was placed between two layers of silica gel, and the sandwich was pressed together in a copper container. Before use the samples were dried during 4 h at 470 K. The container was placed in a chamber where the pressure Žf 0.5 Pa. was maintained by continuous pumping. The temperature of the samples was controlled in the range of 95–575 K with 0.2 K accuracy. The lifetime spectrum was registered by a conventional fast–slow spectrometer with plastic Pilot U scintillators. The ‘stop’ energy window was opened wide to transmit 80% of the energy range and thus to obtain a good efficiency for registration of 3g events. At this setting the resolution was f 300 ps, FWHM; the channel definition was 0.260 ns. In order to keep the net count to the background ratio as high as possible, the source was of very low activity Ž100 kBq.. At the 100 ns delay the net counting rate was still f 9 times higher than the background. With our source activity and counter geometry 2.2 = 10 6 coincidences were collected during a 1-day run. In the silica gels there is no single pore radius but rather a continuous bell-shaped distribution; thus we should also expect a continuous distribution of lifetimes. In such a case the LT program w21x seems best suited to process the data; it fits the sum of discrete components or continuous l distributions of gaussian shape in the logarithmic l scale. Each component with the finite distribution width s is described by the function: N Ž t. s I
I
`
'2p s H0 exp y
ln2 Ž ltp . 2s 2
long-lived part. Thus in the majority of cases analysis was limited to the range beginning from the delay Ž10–40 ns.; the end point of the analysed range was always 1025 ns. The uncertainty limits of fitted parameters given in this paper are statistical only.
4. Results and discussion 4.1. Testing Tao–Eldrup model at large radii The main object of our study was silica gel Si40 ŽMerck.. At the 260 ps channel definition it was not possible to resolve p-Ps and free annihilation components which gave one averaged lifetime Žt 1 s 0.49 ns.. The spectrum also contained a weak Ž2.4%. component with t 2 s 1.9 ns and the long-lived part with the total intensity I L exceeding 30%. The temperature dependence of I L intensity is shown in Fig. 3. The I L values were determined by calculating the area under the long-lived component and needed not be identical with actual o-Ps formation probability. Contrary to the short-lived components annihilating almost entirely via two-quantum annihilation; the long-lived components decay in substantial part via a three-quantum process Žsee Eq. Ž2.., and different efficiencies of registration of two- and three-quantum spectra can distort the proportion of particular intensities.
exp Ž yl t . d l
Ž 7. where I is the relative intensity of the component, tp is the lifetime in the maximum of distribution Žnote that l p s 1rtp does not coincide exactly with the average lav .. The short-lived components were assumed discrete. In this work we are only interested in the
Fig. 3. The intensity of the long-lived part of the positron lifetime spectrum I L for Si40 as a function of temperature.
T. Goworek et al.r Chemical Physics 230 (1998) 305–315
The internal pore surface was much larger than the outer surface of silica grains. Thus we assumed that o-Ps annihilates entirely inside the pores; Ps in the space between the grains was neglected. At very low temperatures only the 1s ground state in the well should be populated, and the long-lived part of spectrum should be described by one component with certain distribution of l, i.e. by two parameters: the l value at the maximum of distribution l3p s 1rt 3p and the width of distribution s 3 . The lowest temperature obtainable in our experimental set-up is 95 K; at 100 K the population 1p, next to state 1s, should only be f 2%, according to the Boltzmann distribution. Such an admixture cannot measurably distort the observed l3p rate. However, it was noticed that the lifetime spectra of positrons in Si40 registered at 95–113 K contain at the beginning of the time scale a certain amount of short-lived species, leading in the one-component fit to anomalous broadening of s 3 and increase of l3p ŽFig. 4.. The l3p and s 3 values stabilize when the beginning of fitting range Žspectrum cut-off point. Dc exceeds
309
50 ns, i.e. for the cut-off points from 70 to 220 ns the fitted values of t 3p and s 3 are identical, in the limits of error. The 70 ns delay can be the measure of thermalization time. The parameter values describing the long-lived component derived from the spectra at 95 and 113 K are:
t 3p s Ž 83.1 " 0.4 . ns,
s 3 s 0.22 " 0.02 .
This lifetime value, as well as those quoted in other parts of this paper, is very close to those expected from spherical model for ‘effective R’ values identical to R determined by nitrogen desorption or mercury intrusion methods. For spherical void with R s 2.0 nm, Eq. Ž3. gives a 78 ns lifetime while, for a cylinder having the same radius, the expected lifetime is 95 ns. Thus, in the calculations, we will use the spherical model with R taken from the literature. The width of l 3 distribution is larger than expected. Data about the width of distribution of pore radii, given in the literature w20,22,23x, are not too precise; the upper limit of the expected s 3 value based on these data is s 3 s 0.16 Žprocessing the spectrum with s 3 fixed at 0.16 gives a minor change of lifetime, t 3p s 81 ns.. However, the 0.16 limit value follows from the distribution of pore radii only; additional spread of level energies Žand consequently of l. can be the result of nonsphericity of voids. The spectroscopic symbols s, p, d, etc., relate to the states in the spherical potential. Thus in our case the terms ‘1s-like’, ‘1p-like’ should be more appropriate; however, for brevity, the classic spectroscopic terminology is used below. This result indicates that, at sufficiently low temperature, when only the ground state in the well is populated, the Tao–Eldrup model describes quite well the lifetime of o-Ps. 4.2. Temperature Õariation of the positron lifetime spectrum in Si40
Fig. 4. The variation of the fitted lifetime and of the distribution width with the change of spectrum cut-off point Žbeginning of the analysed time range.. Si40, one-component fit. v, Spectrum registered at 113 K; B, at 95 K.
The lifetime spectra in Si40 were registered in a wide range of temperatures. In the model of rectangular well Žfinite or infinite. limited to the 1s ground state there is no place for temperature variation of lifetime. Evident shortening of the lifetime with rise in temperature proofs the population of higher states with larger decay rates.
310
T. Goworek et al.r Chemical Physics 230 (1998) 305–315
At 100 K, in thermal equilibrium, o-Ps mainly populates the 1s ground state. With a rise in temperature the population of the first excited state in the well should also rise. The shape of the long-lived part of the spectrum depends on the average dwelling time t D of o-Ps particle in definite state: Ža. if t D is much longer than the o-Ps lifetime we should observe a separate component for the annihilation from each state, and Žb. if t D is much shorter than the o-Ps lifetime one component exists, with the lifetime being the reciprocal of l averaged over all engaged states. Condition Žb. is necessary to achieve thermal equilibrium. Distinguishing cases Ža. and Žb. by analysis of the spectrum is not easy when the decay curves are not exactly exponential but rather represent continuous and not too narrow l distributions produced by the distribution of R. Below we discuss both cases. 4.2.1. Case (a) Let us assume that at moderate temperatures the long-lived part of the spectrum consists of two components: one describing the decay from the 1s ground state, and the other from the 1p state. The parameters t 3p and s 3 found at 95–113 K and at the spectrum cut-off point exceeding 70 ns were assumed as characterizing the 1s component and thus fixed. The fitting parameters were t4p and I4 of possible 1p component. The intensity I4 does not relate to the whole spectrum; it denotes the share of the fourth component in the long-lived part of the spectrum, i.e. I3 q I4 s 1 Ž I3 is intensity of the ‘83 ns’ component.. The value of s4 was assumed, in analogy to the 1s state, to be equal to 0.22; changing it in moderate limits has no substantial influence on the fitted values Žexcept the experimental points at the lowest temperatures.. Two-component analysis of the long-lived part of the spectrum reveals the component with the lifetime t4p f 43 ns. Variation of parameters describing that new component with the change of spectrum cut-off point Dc was tested. The results of two-component fit in the range of 133–223 K are shown in Fig. 5 for the spectrum cut-off point varying from 10 to 40 ns. Change in cut-off point has little influence on the I4 intensity, but the t4p value for small delays is shorter than that for delays above 40 ns. This effect is large at 133 K but disappears rapidly with increase in temperature; at
Fig. 5. The lifetime t4p from the two-component analysis of the Si40 lifetime spectrum as a function of the spectrum cut-off point for several temperatures.
223 K it is almost invisible. When the cut-off point is shifted upwards, the fitted values of t4p tend to an asymptotic value which is very close to 43 ns. This form of t4p versus Dc dependence suggests the existence of additional relatively short-lived componentŽs.. Thus, the spectra were analysed from Dc s 10 ns also as a sum of three components. As before, the longest-lived component was fixed Žt 3p s 83 ns; s 3 s 0.22., the intermediate one had a fixed width Ž s4 s 0.22., and other parameters were left free. This kind of spectrum processing restores the t4p value f 43 ns in the low temperature range and demonstrates the existence of short-lived species with t 5p s 8.1 " 1.0 ns and intensity I5 s 5.1 " 1.0%. The 8 ns component is located in the time range where Ps is certainly far from thermalization; thus the component cannot be ascribed to a definite state of particle in the well, and it also does not need to be exponential Žthe width s5 is sometimes unrealistically large, reaching 0.6.. The numbers given above are only an indication that ‘something goes on’ at the beginning of spectrum.
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311
The final results of two- and three-component fitting are shown in Fig. 6. ŽFor temperatures - 100 K, where the intensity of the 43 ns component is very low, its separation from the remainder of the spectrum was impossible and only two-component analysis was done.. In the broad range of 105–233 K the lifetime t4p obtained in three-component fit is constant, the average value
t4p s Ž 42.5 " 0.7 . ns is very close to that expected from the model for a 1p state in the spherical well Ž45.2 ns.. In the case of capillaries the expected lifetime would be 63.9 ns. Above 240 K the t4p lifetime begins to decrease, probably due to the increasing participation of the levels lying still higher than 1p. The shape of the l spectrum fitted by LT program is shown in Fig. 7 for several temperatures. Fig. 6 shows that I4 rises with temperature but, in the range of 100–220 K, it is systematically higher than expected from the Boltzmann population of well levels. In addition to inadequacies of the model, there can be at least two sources of discrepancy. If the efficiencies of registration of 3g and 2g I4 will be disevents are different, the ratio I3rI torted; i.e. lower efficiency for 3g decays would reflect in overestimated experimental I4 values. The increased intensity of 1p component can also be
Fig. 6. The lifetime t4p Ža. and intensity S4 Žb. as a function of temperature for the Si40 samples. `, Relate to the two-component analysis of the long-lived part of the spectrum with Dc s 20 ns; v, ', three-component analysis with Dc s10 ns; , calculated S4 in the case of the Boltzmann population of excited states; - - - -, population of the 1p state only.
Fig. 7. The shape of the l distribution as fitted to the spectrum of Si40 for several temperatures; two-component fit. In the case of T s100 K, where the 43 ns component is almost vanished, s4 was not fixed.
a result of incomplete Ps thermalization w24x. Note that I4 intensity cannot be calculated precisely; the population of excited states depends exponentially on their energies and these in turn depend on the square of the radius. The result of model calculation is very sensitive to the assumed pore dimension. The radii taken from nitrogen or mercury methods need not be identical with ‘effective radii’ acting in the pick-off process. We accepted them as giving t 3p and t4p close to the experimental values, but it is only a convenient supposition. If we accept e.g. the R value 1.8 nm, then at 200 K the calculated equilibrium population of 1p state will be 13.6%, i.e. half of that experimentally observed. 4.2.2. Case (b) If the spectrum contains one long-lived component only, averaged over-all populated states, one can try to explain the results of analysis described in the previous paragraph as forced by excessive number of constraints: we fix t 3p , thus the program seeks the part of the spectrum lacking to the average. However, for single component, we should observe a s 3 value similar as at low temperatures Ž s 3 s 0.22.. The experimental result for room temperature is s 3 s 0.47; for T s 208 K s 3 s 0.40. These are enormous widths of l distribution Žrising with the temperature., not acceptable if we have really single
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component — its halfwidth extends from 35 to 107 ns ŽFig. 8.. The lifetime t 3p found for room temperature is 59.9 ns. This result can be compared with an averaged lifetime calculated from the model. For simplicity the decay constants related to particular states were assumed as discrete with l corresponding to the R value at the maximum of pore size distribution; the population of the states was assumed to be Boltzmann-like. The calculated lifetime ²t 3 : is longer than the experimental one, again suggesting incomplete thermalization. The variances of one- and two-component fits are identical. The result is inconclusive. The population of states seems to be not too distant from Boltzmann law, however, the width of l distribution in one-component fit is not acceptable if we really have one component. Cases Ža. and Žb. are the extremities; the spectrum which we observe probably belongs to an intermediate case. The o-Ps lifetime measurements were continued up to 573 K. With the rise in temperature the population of levels lying higher than 1p should play an increasing role. In such a complex case we decided to determine the average lifetime only, i.e. to analyse the long-lived part of the spectrum as one-component with l distribution. The experimental results are shown in Fig. 9 together with the calculated lifetime ²t 3 : averaged over all states. The experimental data and model calculations are not directly comparable as t 3p and average lifetime ²t 3 : are not identical; the LT program fits the l
Fig. 8. The shape of the l distribution as fitted to the spectrum of Si40 at 253 K. , Two-component fit; - - - -, one-component fit.
Fig. 9. Top: Temperature dependence of the t 3p lifetime in Si60; . repreone-component fit, Dc s 20 ns. The solid line Ž sents the calculated ²t 3 :. Bottom: The same data for Si40.
distribution which is gaussian in the logarithmic scale, thus not symmetrical with respect to tp . However, at low s the shift of average with respect to most probable t is to neglect. Experimental lifetimes are slightly different from the model predictions, a general tendency of faster than expected t 3p decrease with a rising temperature is visible, particularly at high temperatures. A partial explanation of this tendency can be the difference in efficiencies of 2g and 3g registration, mentioned in the previous section. If the short-lived, 3g poor, components are registered with higher efficiency, they enter the experimental average with the increased weight. However, only one-third of the longest-lived component, participating in t 3p and related to the 1p state, represents 3g decays; thus the presence of other shorter-lived species cannot cause as drastic a t 3p shortening. In the Tao–Eldrup model the D R parameter in the infinite potential well substitutes the actual penetration range of the Ps wave function outside the well of finite depth; in finite well, the penetration range depends on the distance of level from the well top. At moderate temperatures, when only the lowest states are engaged, and if the Ps work function is of the order of y3 eV w16x, the
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energies of populated levels are by two orders of magnitude smaller than the well depth and one cannot expect noticeable changes of the penetration range and of the equivalent D R. However, the results given in Refs. w17,25,26x suggest that the prevailing number of Ps atoms gains at emission from the silica the energy 0.8 eV and, at elevated temperatures, when the states up to 1f are populated Žat 570 K the population of 1f should be 8% of the total. the energy of 1f state Ž E1f ( 0.20 eV. in the infinite well becomes comparable to the actual well depth and the model can no longer be applicable. 4.3. Ortho-Ps lifetime in silica gels with large pores In the case of Si60 with R s 3.5 nm, even at 95 K the second and third levels in the spherical well are noticeably populated; according to the Boltzmann law f 32% of o-Ps atoms should annihilate from these states; thus, as in the previous section, we approximated the long-lived part of the spectrum by a single component Žwith l distribution.. The lifetime spectra were measured in the 113–420 K range and analysed from the Dc s 20 ns cut-off points; shifting that point to 40 or 60 ns gave the t 3p value differing by less than 0.6 ns, i.e. in the limits of error. The results of one-component fit are shown in Fig. 9. The continuous curve represents the average lifetime ²t 3 : calculated assuming, as previously, the discrete t i Žtaken from Eq. Ž5. for R s 3.5 nm. and the Boltzmann distribution over the states. The experimental t 3p values lie again below the calculated ²t 3 :; nevertheless, the tendency of changes of average lifetime with temperature seems to be much better reproduced. The s 3 value is about 0.2 at low temperatures, rising to 0.36 at 420 K. That broadening is difficult to explain in a pure one-component spectrum, suggesting a more complex structure of the spectrum. The variance of one-component fit for Si60 is close to unity in the whole range of temperatures Žat 420 K the variance is 1.02.. For Si40 the variance remains acceptable Ž( 1.15. up to f 400 K; it seems that the rise of s 3 compensates the effect of growth of the components related to the excited states; above 400 K that compensation mechanism ceases to be effective and the variation rises to 3 and higher. The lifetime spectra for Si100 and Si200 were
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Fig. 10. The calculated average lifetime at room temperature ²t 3 : . and the experimental t 3p versus average pore radius Ž points for various silica gels. The R values and their uncertainties are taken from Ref. w20x. Dashed Ž — — — . line shows, for comparison, the lifetime of the 1s state Žstraight extension of Tao–Eldrup model, Eq. Ž3... The vacuum value of the lifetime in triplet state, 142 ns, is also marked ŽPPPPP..
measured only at room temperature and approximated by one component with the non-zero l distribution. Fig. 10 shows the experimental values of t 3p for various average pore radii and the calculated ²t 3 : versus R dependence. As before, in the model calculations, the discrete l and Boltzmann distribution were assumed. Only seven lowest levels were included in the calculation; for R ) 6 nm that number of levels is not sufficient and the ²t 3 : curve in Fig. 10 runs slightly higher than it would be in the case of more precise calculations. In spite of all approximations Žhypothetical sphericity of voids, the discrete l accepted in calculations, nonidentity of t 3p and ²t 3 :, Boltzmann distribution. the experimental points locate unexpectedly close to the ²t 3 : curve. These approximations and low number of experimental points prevent one from drawing fargoing conclusions; however, Fig. 10 seems to suggest that the pore dimensions in Si100 given in the literature are oversized. The o-Ps lifetime in Si200 is only several nanoseconds shorter than that for the particle in vacuum. 4.4. The D R Õalue Experimental data on o-Ps lifetime in the void coincide quite well with model expectations. Note that an important parameter strongly influencing the
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result of calculations is R 0 y R s D R. Its value Ž0.166 nm. accepted here was empirically chosen w9x to match the data in the range of sub-nanometer voids, for molecular crystals and liquids; the results of our measurements with silica gels seem to indicate that at this stage of study there is no necessity to modify D R when applying the Tao–Eldrup model to the case of large voids.
radii. to demonstrate more clearly constancy of l po for the definite state. For voids with R 0 5 nm, the average lifetime approaches the vacuum value of 142 ns. The experimental most probable lifetimes t 3p from the onecomponent fit at room temperature are very close to the calculated average ²t 3 :; that gives one the chance to use the positron lifetime spectroscopy as a method of estimation of the average pore radii, at least below 6 nm.
5. Conclusions Acknowledgements Processing the experimental data and comparison with the model required many simplifications, inevitable in the case of quite complex structures like silica gels. The model discussed in this paper does not pretend to describe quantitatively the o-Ps lifetimes in large voids; nevertheless, being relatively simple, it reproduces the experimental data rather well. Consistency of experiment and model requires to account the population of excited states. Decrease of lifetime with rising temperature is a direct consequence of the engagement of these states into the annihilation process. The spectrum structure is not definitively settled; there are arguments supporting the two-component character of the observed decay curve, while thermal equilibrium needs one component. All the considerations presented in this paper are valid for empty pores only. The presence of gases Žin particular of oxygen w27,28x. or adsorbed layers entering with Ps into chemical reactions can shorten all the lifetimes, adding in Eq. Ž1. a new term related to these processes. The idea of the model is to tie explicitly the lifetime with the dimensions of free volume. In the classic approach, as applied in the thermalization models w29x, positronium is treated as a particle interacting with the medium during collisions with the cavity walls and the pick-off annihilation rate per collision is assumed as constant. For definite cavity size, the pick-off rate should be temperature dependent Ž l po ; 'T .. In the quantum approach, upon lowering the temperature l po tends to the finite value, characteristic for the 1s state and for given void size. It would be advisable to extend the measurements toward lower temperatures Žor smaller
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