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Positron mobilities in isooctane, n-hexane and hexafluorobenzene
Radiation Physics and Chemistry 58 (2000) 451±455
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Positron mobilities in isooctane, n-hexane and hexa¯uorobenzene C.L. Wang*, Y. Kobayashi, K. Hirata National Institute of Materials and Chemical Research, Tsukuba, Ibaraki 305-8565, Japan
Abstract Positron mobilities in three nonpolar liquids, isooctane (2, 2, 4-trimethylpentane), n-hexane and hexa¯uorobenzene, were measured by observing the drift velocity of free positrons in the presence of an external electric ®eld. The Doppler shift of the 511 keV annihilation line was measured as a function of the electric ®eld up to 26 kV cmÿ1 at room temperature. The free positron mobilities in isooctane and n-hexane were determined to be 69 2 3 cm2 Vÿ1 sÿ1 and 53 2 3 cm2 Vÿ1 sÿ1, respectively, whereas a much smaller value of 0 2 7 cm2 Vÿ1 sÿ1 was obtained for hexa¯uorobenzene. 7 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Positron mobility measurements provide useful information on the behavior of positrons in various substances. The positron mobility is related to positron diusion, positron scattering by impurities (Mills and Karl, 1993) and in some cases to the nature of the positron state (Iakubov and Pogosov, 1996). Knowledge of the positron mobility is also important for a better understanding of the spur mechanism of positronium (Ps) formation in molecular substances (Mogensen, 1995). Some eorts have been made to determine the positron mobility in nonpolar liquids (Linderoth et al., 1982; Heinrich and Schiltz, 1982) and polymers (Brusa et al., 1995; Zheng et al., 1998) by measuring the Doppler shift of the 511 keV annihilation g-ray in the presence of an external electric ®eld. Since the amount of the Doppler shift is comparable to the long-term drift
* Corresponding author. Fax: +81-298-61-4709. E-mail address: [email protected] (C.L. Wang).
of the spectrometer, Brusa et al. (1995) used two additional g-ray sources for drift compensation. However, as we see below, the use of the reference sources introduces additional statistical error in the Doppler shift obtained. Furthermore, available positron mobility data for hexane reported by Linderoth et al. (1982) and Heinrich and Schiltz, (1982) are much dierent from each other. In this work, we propose a way of improving the precision of the measurement and give the positron mobility data in several neat liquids, which allow us to discuss the factors that in¯uence Ps formation in these nonpolar liquids (Jacobsen et al., 1982; Billard et al., 1998). 2. Experiment As received nonpolar liquids used in this work were isooctane (i-C8H18), hexane (n-C6H14) purchased from Wako Pure Chemical, and hexa¯uorobenzene (C6F6) purchased from Tokyo Chemical Industry, Ltd. A glass tube containing the liquids and the positron source, i.e., about 0.74 MBq (20 mCi) 22Na deposited
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 1 9 9 - 7
452
C.L. Wang et al. / Radiation Physics and Chemistry 58 (2000) 451±455
between two 0.8 mg cm2 thick Kapton foils, was kept in between a pair of electrodes to which a high voltage (V ) was applied. The distance between the electrodes was 6 mm. Prior to the measurements, all the samples were degassed to remove dissolved air. The energy pro®le of the positron annihilation g-line was recorded with an ORTEC high purity Ge detector connected with an ORTEC 92X Spectrum Master. The distance between the detector and the source was 20 cm. The energy resolution of the spectrometer was 1.1 keV at 511 keV, when the dead time was kept around 10% and the count rate was about 110 sÿ1. Each Doppler spectrum was collected for 20 min at room temperature with a channel width of 41.3 eV. The positron mobility m can be evaluated from the positron drift velocity vd under an external electric ®eld F = V/d (d = 6 mm is the distance between the two electrodes) by the relation m vd =F
1
The drift velocity vd is related to the Doppler shift, DE, of the centroid of the positron annihilation g-line at F relative to that at the zero ®eld by DE vd E511 2c
2
where E and vd are in the direction connecting the sample to the detector. To measure the Doppler shift DE as a function of electric ®eld F, the following two methods were attempted. In the ®rst method each energy spectrum was calibrated using two reference g-lines from 133Ba Eg 356 keV) and 137Cs Eg 662 keV). The g-ray spectra were collected following the sequence (0, F1; 0, F2;. . .,0, Fi;. . ., 0, Fj;. . .), where the order of Fi . . . Fj was preset randomly and the measurements at each ®eld were repeated 8±15 times. The individual relative shift under the ®eld Fi was determined for each pair of the microspectra (0, Fi) and the average shift for a series of (0, Fi) pairs was calculated. Details of this method are described in (Brusa et al., 1995; Zheng et al., 1997). A typical standard deviation of about 4 eV in the average shift is rather large (see Appendix A), because the total number of counts in one microspectrum was not enough to determine the centroids of the three peaks with good precision and also the large statistical errors in the reference peaks were transmitted to the peak energy of the 511 keV line after calibration. In the second method we did not use the reference sources to avoid any error transmission. The g-ray spectra were collected following the sequence (0, F; 0, F;......; 0, F ). Ten pairs of the spectra at (0, F ) were collected; then the spectra at 0 and F were added, respectively, to improve the statistics. After the constant backgrounds in the higher and lower energy regions
Fig. 1. Energy shift of the 511 keV line versus external electric ®eld for isooctane. Dierent symbols show the data points obtained in two dierent runs for the same sample.
were subtracted, the peak position was evaluated. When the total number of the counts of the spectrum was 1.2±1.5 M, the error for the peak position of the 551 keV line was only about 1±2 eV in this method (see Appendix A), signi®cantly lower than that in the ®rst method. This means that the eect of the longterm drift is almost compensated by repeated measurements at 0 and F even without calibration by reference sources. The measured Doppler shift DE versus ®eld for isooctane was well reproducible and less ¯uctuated. We therefore, measured DE versus electric ®eld for nhexane and hexa¯uorobenzene only by the second method. 3. Results and discussion The measured Doppler shift for the three liquids are
Fig. 2. Energy shift of the 511 keV line versus external electric ®eld for n-hexane. Symbols as in Fig. 1.
C.L. Wang et al. / Radiation Physics and Chemistry 58 (2000) 451±455
Fig. 3. Energy shift of the 511 keV line versus external electric ®eld for hexa¯uorobenzene. Symbols as in Fig. 1.
plotted against external electric ®eld in Figs. 1±3. DE for isooctane and n-hexane exhibits substantial changes as a function of the applied electric ®eld, indicating high positron mobilities in the two liquids. The data in Figs. 1±3 were ®tted to Eqs. (1) and (2) by least-square analysis and the overall positron mobilities in the three liquids were determined as listed in Table 1. One may argue that in the above treatment the contribution of immobile positrons which annihilate in the Kapton foil or annihilate as Ps in the liquids was not taken into consideration. Provided f is the fraction of these immobile positrons, the Doppler shift for the ``free'' positrons can be written as DEt
DE 1ÿf
3
Here, we assume the ratio between the total Ps yield and the o-Ps yield is 4:3, i.e., f = (4/3)I3, where I3 is the intensity of the long-lived o-Ps component in the positron lifetime spectrum, although this 4/3 ratio may slightly vary from one liquid to another as a result of
453
possible radiation chemical reactions (Stepanov et al., 2000). By using the I3(F ) data reported by us (Wang et al., 1998; Stepanov et al., 2000), the fraction f was estimated and the free positron mobilities for the three liquids were deduced (Table 1). Here, we did not take into account of the source component of the Kapton foil, because the estimation of the intensity of this component is still under debate (Saoucha, 1999). We also neglected the in¯uence of the solid-angle of source±detector geometry (Mills and Karl, 1993). Proper account of these factors may further increase the mobilities by 10±15%. From Table 1 we see that there is no simple correlation between the positron mobility and the electron mobility. For example, the electron mobility in isooctane is nearly two orders of magnitude higher than that in n-hexane, whereas the corresponding ratio of the free positron mobilities is only 1.3. It is also seen in the table that there is no correlation between I3 and the positron mobility m or the electron mobility mÿ : This is not so surprising in view of the complex nature of the Ps formation process in liquids. In the ®nal stage of the slowing-down process, an energetic positron loses its last part (0500 eV) of kinetic energy by generating tens of electron±ion pairs, which together with the positron after thermalization constitute a terminal positron spur or a positron blob, where Ps formation occurs as a result of recombination of the positron and one of the electrons. In the framework of the diusion±recombination model (DRM) (Stepanov et al., 2000), the Ps yield is proportional to the overlap of the spatial distributions of the positron and the blob electrons. In the simplest case of no positron and electron trapping, the Ps formation probability can be written as (Stepanov et al., 2000)
2rc PPs 1 ÿ exp ÿ 1=2 p r0
4
where rc n0 Rep 1 De =Dp and r0 abl2 ap2 1=2 ,
Table 1 Measured positron mobility m , and electron mobility mÿ (Lide, 1998) and o-Ps formation probability I3 (Wang et al., 1998; Stepanov et al., 2000). Data for air±saturated n-hexane, n-pentane and n-decane are taken from Linedroth et al. (1982) Samples
m a (cm2 Vÿ1 sÿ1)
m b (cm2 Vÿ1 sÿ1)
mÿ (cm2 Vÿ1 sÿ1)
I3 (%)
isooctane n-Hexane Hexa¯uorobezene n-Hexanec n-Pentanec n-Decanec
3722 3022 022
6923 5323 027 8.5 6 11.5
4.5 0.07ÿ0.09 0.01 0.13 0.08 ±
43.420.3 40.220.2 54.120.2 25.0 28.0 23.5
a
Overall mobility. Free positron mobility. c Air-saturated.
b
454
C.L. Wang et al. / Radiation Physics and Chemistry 58 (2000) 451±455
and n0 is the initial number of the ion±electron pairs, Rep is the reaction radius of electron±positron recombination; De and Dp are the diusion coecients and related to the mobilities me and mp by De me kB T=e, Dp mp kB T=e, respectively; abl2 and ap2 are the dispersions of the spatial distributions of the intrablob pairs and the positron at the end of thermalization. Based on the mÿ =m data in Table 1 and the r0 value obtained from the electric-®eld dependence of I3 (Stepanov et al., 2000), we obtain n0 Rep 180±290 AÊ for isooctane and n-hexane. A reasonable value of the number of the electron±ion pairs in the blob, n0 130, gives a positron±electron reaction radius Rep 13±9 AÊ, which is the minimum distance for a positron and an electron to react with each other to form Ps. The reaction radius of 3±9 AÊ seems to be rather small in light of the high free positron mobilities in the two liquids. Perhaps the dissipation of the excess energy released upon positron±electron recombination through activation of molecular excitations in the liquids is not very ecient and Ps formation is somewhat limited. Finally, it should be mentioned that there is a correlation between the positron mobility and r0 abl2 ap2 1=2 : As deduced from the variation of the o-Ps yield versus external electric ®eld, r0 in isooctane and n-hexane with higher positron mobilities is between 200 and 600 AÊ, whereas a value of only 70 AÊ was obtained for C6F6 (Stepanov et al., 2000). Here one should note that the Ps formation probability is very high in C6F6. Probably the overlap of the spatial distributions of the positron and the localized electrons in C6F6 is so large that they can recombine with a high probability. This picture is consistent with the idea that Ps is formed as a result of electron pick up from C6Fÿ 6 by the positron (Ito, 1988) and can also account for the very weak electric ®eld dependence of Ps formation in C6F6 (Stepanov et al., 2000).
4. Conclusion In summary, we presented positron mobility data for three neat, nonpolar liquids, isooctane, n-hexane and hexa¯uorobenzene. We found that the free positron mobilities for isooctane and hexane are 69 2 3 cm2 Vÿ1 sÿ1 and 53 2 3 cm2 Vÿ1 sÿ1, signi®cantly larger than 0 2 7 cm2 Vÿ1 sÿ1 for C6F6. There is no simple correlation between I3 and the positron mobility or the electron mobility. The positron mobility data allowed us to estimate some positron blob parameters in the three liquids and revealed peculiar feature of Ps formation in C6F6.
Acknowledgements We wish to thank Dr. S.V. Stepanov and Dr. M. Eldrup for their helpful discussions. We also thank the Science and Technology Agency (STA) and the Agency of Industrial Science and Technology (AIST) for ®nancial supports.
Appendix A Here we brie¯y discuss the in¯uence of the total number of counts in a Doppler spectrum on the peak position P. Let us assume that P is given by X ci ni 1A P X ci where ci is the count at nith channel. If Dcj being the ¯uctuation at njth channel, it contributes to the error of P by the following amount: X ci ni Dcj nj X ci ni DPj X ÿ 2A ci ci Dcj In the case that Dcj =cj 1Eq. (2A) becomes ÿ DPj nj ÿ P Dcij A
3A
P P where A ci and B ci ni : We further assume that the probability distribution function of ci is of Gaussian-type with the standard deviation c1=2 i : Substituting Dci c1=2 into Eq. (3A), we obtain i DP
X
DPi2
1=2
4A
This equation gives an error of DP 0:081 channel (3.34 eV) for a Doppler spectrum containing a total count of 0.12 M, a typical number for a spectrum recorded for 20 min. After calibrating the 511 keV peak using two additional reference g-ray sources and evaluating the average Doppler shift for 8±15 pairs of (0, F ) ®elds, we obtain a large statistical error of about 4 eV in the average Doppler shift at ®eld F. In the second method, if the total number of counts is 1.2±1.5 M, the error in the Doppler shift is approximately 1±2 eV (details are shown in Figs. 1±3). References Billard, I., Goulet, T., Jay-Gerin, J.P., Bonnenfant, A., 1998. On the in¯uence of electron mobilities on the yields of formation of positronium in liquids: new I3 measurements and Monte Carlo simulations. J. Chem. Phys. 108, 2408. Brusa, R.S., Naia, M.D., Margoni, D., Zecca, A., 1995.
C.L. Wang et al. / Radiation Physics and Chemistry 58 (2000) 451±455 Positron mobility in polyethylene in the 64±400 K temperature range. Appl. Phys. A 60, 447. Heinrich, F., Schiltz, A., 1982. Positron implantation in molecular liquids at high electric ®elds. In: Jain, P.C., Singru, R.M., Gopinathan, K.P. (Eds.), Positron Annihilation. World Scienti®c, Singapore, p. 705. Iakubov, I.T., Pogosov, V., 1996. Positron states in noble liquids: anity, eective mass, and mobility. Phys Rev. B 53, 13362. Ito, Y., 1988. Radiation chemistry: intraspur eects and positronium formation mechanisms. In: Schrader, D.M., Jean, Y.C. (Eds.), Positron and Positronium Chemistry: Studies in Physics and Theoretical Chemistry. Elsevier, New York, p. 120. Jacobsen, F.M., Mogensen, O.E., Trumpy, G., 1982. Correlation of positronium yield with excess electron mobility in liquid neopentane. Chem. Phys. 69, 71. Lide, D.R. (Ed.), 1998. CRC handbook of Chemistry and Physics, 78th ed. CRC Press, Boca Raton, New York. Linderoth, S., MacKenzie, I.K., Tanigawa, S., Stone, R.J., 1982. Positron mobility in some liquid hydrocarbons. In:
455
Jain, P.C., Singru, R.M., Gopinathan, K.P. (Eds.), Positron Annihilation. World Scienti®c, Singapore, p. 830; Also see Linderoth, S., MacKenzie, I.K., Tanigawa, S., 1985. Phys. Lett. A 107, 409. Mills Jr., A.P., Karl, N., 1993. Positron mobility in perylene. Phys. Rev. B 48, 7050. Mogensen, O.E., 1995. Positron Annihilation in Chemistry. Springer±Verlag, Berlin. Saoucha, A., 1999. Positron transmission into Kapton: the eective mass absorption coecient of relevance to positron lifetime experiments. J. Appl. Phys. 85, 1802. Stepanov, S.V., Wang, C.L., Kobayashi, Y., Byakov, V.M., Hirata, K., 2000 What do we learn from the E-®eld eect on positronium formation in liquids? 58, 403±409. Wang, C.L., Hirata, K., Kawahara, J., Kobayashi, Y., 1998. Electric-®eld dependence of positronium formation in liquids and polymers. Phys. Rev. B 58, 14864. Zheng, W., Kobayashi, Y., Hirata, K., Suzuki, T., 1998. Mobility of positrons in gamma-irradiated polypropylene and polyethylene. Radiat. Phys. Chem. 51, 269.
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Positron mobilities in isooctane, n-hexane and hexa¯uorobenzene C.L. Wang*, Y. Kobayashi, K. Hirata National Institute of Materials and Chemical Research, Tsukuba, Ibaraki 305-8565, Japan
Abstract Positron mobilities in three nonpolar liquids, isooctane (2, 2, 4-trimethylpentane), n-hexane and hexa¯uorobenzene, were measured by observing the drift velocity of free positrons in the presence of an external electric ®eld. The Doppler shift of the 511 keV annihilation line was measured as a function of the electric ®eld up to 26 kV cmÿ1 at room temperature. The free positron mobilities in isooctane and n-hexane were determined to be 69 2 3 cm2 Vÿ1 sÿ1 and 53 2 3 cm2 Vÿ1 sÿ1, respectively, whereas a much smaller value of 0 2 7 cm2 Vÿ1 sÿ1 was obtained for hexa¯uorobenzene. 7 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Positron mobility measurements provide useful information on the behavior of positrons in various substances. The positron mobility is related to positron diusion, positron scattering by impurities (Mills and Karl, 1993) and in some cases to the nature of the positron state (Iakubov and Pogosov, 1996). Knowledge of the positron mobility is also important for a better understanding of the spur mechanism of positronium (Ps) formation in molecular substances (Mogensen, 1995). Some eorts have been made to determine the positron mobility in nonpolar liquids (Linderoth et al., 1982; Heinrich and Schiltz, 1982) and polymers (Brusa et al., 1995; Zheng et al., 1998) by measuring the Doppler shift of the 511 keV annihilation g-ray in the presence of an external electric ®eld. Since the amount of the Doppler shift is comparable to the long-term drift
* Corresponding author. Fax: +81-298-61-4709. E-mail address: [email protected] (C.L. Wang).
of the spectrometer, Brusa et al. (1995) used two additional g-ray sources for drift compensation. However, as we see below, the use of the reference sources introduces additional statistical error in the Doppler shift obtained. Furthermore, available positron mobility data for hexane reported by Linderoth et al. (1982) and Heinrich and Schiltz, (1982) are much dierent from each other. In this work, we propose a way of improving the precision of the measurement and give the positron mobility data in several neat liquids, which allow us to discuss the factors that in¯uence Ps formation in these nonpolar liquids (Jacobsen et al., 1982; Billard et al., 1998). 2. Experiment As received nonpolar liquids used in this work were isooctane (i-C8H18), hexane (n-C6H14) purchased from Wako Pure Chemical, and hexa¯uorobenzene (C6F6) purchased from Tokyo Chemical Industry, Ltd. A glass tube containing the liquids and the positron source, i.e., about 0.74 MBq (20 mCi) 22Na deposited
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 1 9 9 - 7
452
C.L. Wang et al. / Radiation Physics and Chemistry 58 (2000) 451±455
between two 0.8 mg cm2 thick Kapton foils, was kept in between a pair of electrodes to which a high voltage (V ) was applied. The distance between the electrodes was 6 mm. Prior to the measurements, all the samples were degassed to remove dissolved air. The energy pro®le of the positron annihilation g-line was recorded with an ORTEC high purity Ge detector connected with an ORTEC 92X Spectrum Master. The distance between the detector and the source was 20 cm. The energy resolution of the spectrometer was 1.1 keV at 511 keV, when the dead time was kept around 10% and the count rate was about 110 sÿ1. Each Doppler spectrum was collected for 20 min at room temperature with a channel width of 41.3 eV. The positron mobility m can be evaluated from the positron drift velocity vd under an external electric ®eld F = V/d (d = 6 mm is the distance between the two electrodes) by the relation m vd =F
1
The drift velocity vd is related to the Doppler shift, DE, of the centroid of the positron annihilation g-line at F relative to that at the zero ®eld by DE vd E511 2c
2
where E and vd are in the direction connecting the sample to the detector. To measure the Doppler shift DE as a function of electric ®eld F, the following two methods were attempted. In the ®rst method each energy spectrum was calibrated using two reference g-lines from 133Ba Eg 356 keV) and 137Cs Eg 662 keV). The g-ray spectra were collected following the sequence (0, F1; 0, F2;. . .,0, Fi;. . ., 0, Fj;. . .), where the order of Fi . . . Fj was preset randomly and the measurements at each ®eld were repeated 8±15 times. The individual relative shift under the ®eld Fi was determined for each pair of the microspectra (0, Fi) and the average shift for a series of (0, Fi) pairs was calculated. Details of this method are described in (Brusa et al., 1995; Zheng et al., 1997). A typical standard deviation of about 4 eV in the average shift is rather large (see Appendix A), because the total number of counts in one microspectrum was not enough to determine the centroids of the three peaks with good precision and also the large statistical errors in the reference peaks were transmitted to the peak energy of the 511 keV line after calibration. In the second method we did not use the reference sources to avoid any error transmission. The g-ray spectra were collected following the sequence (0, F; 0, F;......; 0, F ). Ten pairs of the spectra at (0, F ) were collected; then the spectra at 0 and F were added, respectively, to improve the statistics. After the constant backgrounds in the higher and lower energy regions
Fig. 1. Energy shift of the 511 keV line versus external electric ®eld for isooctane. Dierent symbols show the data points obtained in two dierent runs for the same sample.
were subtracted, the peak position was evaluated. When the total number of the counts of the spectrum was 1.2±1.5 M, the error for the peak position of the 551 keV line was only about 1±2 eV in this method (see Appendix A), signi®cantly lower than that in the ®rst method. This means that the eect of the longterm drift is almost compensated by repeated measurements at 0 and F even without calibration by reference sources. The measured Doppler shift DE versus ®eld for isooctane was well reproducible and less ¯uctuated. We therefore, measured DE versus electric ®eld for nhexane and hexa¯uorobenzene only by the second method. 3. Results and discussion The measured Doppler shift for the three liquids are
Fig. 2. Energy shift of the 511 keV line versus external electric ®eld for n-hexane. Symbols as in Fig. 1.
C.L. Wang et al. / Radiation Physics and Chemistry 58 (2000) 451±455
Fig. 3. Energy shift of the 511 keV line versus external electric ®eld for hexa¯uorobenzene. Symbols as in Fig. 1.
plotted against external electric ®eld in Figs. 1±3. DE for isooctane and n-hexane exhibits substantial changes as a function of the applied electric ®eld, indicating high positron mobilities in the two liquids. The data in Figs. 1±3 were ®tted to Eqs. (1) and (2) by least-square analysis and the overall positron mobilities in the three liquids were determined as listed in Table 1. One may argue that in the above treatment the contribution of immobile positrons which annihilate in the Kapton foil or annihilate as Ps in the liquids was not taken into consideration. Provided f is the fraction of these immobile positrons, the Doppler shift for the ``free'' positrons can be written as DEt
DE 1ÿf
3
Here, we assume the ratio between the total Ps yield and the o-Ps yield is 4:3, i.e., f = (4/3)I3, where I3 is the intensity of the long-lived o-Ps component in the positron lifetime spectrum, although this 4/3 ratio may slightly vary from one liquid to another as a result of
453
possible radiation chemical reactions (Stepanov et al., 2000). By using the I3(F ) data reported by us (Wang et al., 1998; Stepanov et al., 2000), the fraction f was estimated and the free positron mobilities for the three liquids were deduced (Table 1). Here, we did not take into account of the source component of the Kapton foil, because the estimation of the intensity of this component is still under debate (Saoucha, 1999). We also neglected the in¯uence of the solid-angle of source±detector geometry (Mills and Karl, 1993). Proper account of these factors may further increase the mobilities by 10±15%. From Table 1 we see that there is no simple correlation between the positron mobility and the electron mobility. For example, the electron mobility in isooctane is nearly two orders of magnitude higher than that in n-hexane, whereas the corresponding ratio of the free positron mobilities is only 1.3. It is also seen in the table that there is no correlation between I3 and the positron mobility m or the electron mobility mÿ : This is not so surprising in view of the complex nature of the Ps formation process in liquids. In the ®nal stage of the slowing-down process, an energetic positron loses its last part (0500 eV) of kinetic energy by generating tens of electron±ion pairs, which together with the positron after thermalization constitute a terminal positron spur or a positron blob, where Ps formation occurs as a result of recombination of the positron and one of the electrons. In the framework of the diusion±recombination model (DRM) (Stepanov et al., 2000), the Ps yield is proportional to the overlap of the spatial distributions of the positron and the blob electrons. In the simplest case of no positron and electron trapping, the Ps formation probability can be written as (Stepanov et al., 2000)
2rc PPs 1 ÿ exp ÿ 1=2 p r0
4
where rc n0 Rep 1 De =Dp and r0 abl2 ap2 1=2 ,
Table 1 Measured positron mobility m , and electron mobility mÿ (Lide, 1998) and o-Ps formation probability I3 (Wang et al., 1998; Stepanov et al., 2000). Data for air±saturated n-hexane, n-pentane and n-decane are taken from Linedroth et al. (1982) Samples
m a (cm2 Vÿ1 sÿ1)
m b (cm2 Vÿ1 sÿ1)
mÿ (cm2 Vÿ1 sÿ1)
I3 (%)
isooctane n-Hexane Hexa¯uorobezene n-Hexanec n-Pentanec n-Decanec
3722 3022 022
6923 5323 027 8.5 6 11.5
4.5 0.07ÿ0.09 0.01 0.13 0.08 ±
43.420.3 40.220.2 54.120.2 25.0 28.0 23.5
a
Overall mobility. Free positron mobility. c Air-saturated.
b
454
C.L. Wang et al. / Radiation Physics and Chemistry 58 (2000) 451±455
and n0 is the initial number of the ion±electron pairs, Rep is the reaction radius of electron±positron recombination; De and Dp are the diusion coecients and related to the mobilities me and mp by De me kB T=e, Dp mp kB T=e, respectively; abl2 and ap2 are the dispersions of the spatial distributions of the intrablob pairs and the positron at the end of thermalization. Based on the mÿ =m data in Table 1 and the r0 value obtained from the electric-®eld dependence of I3 (Stepanov et al., 2000), we obtain n0 Rep 180±290 AÊ for isooctane and n-hexane. A reasonable value of the number of the electron±ion pairs in the blob, n0 130, gives a positron±electron reaction radius Rep 13±9 AÊ, which is the minimum distance for a positron and an electron to react with each other to form Ps. The reaction radius of 3±9 AÊ seems to be rather small in light of the high free positron mobilities in the two liquids. Perhaps the dissipation of the excess energy released upon positron±electron recombination through activation of molecular excitations in the liquids is not very ecient and Ps formation is somewhat limited. Finally, it should be mentioned that there is a correlation between the positron mobility and r0 abl2 ap2 1=2 : As deduced from the variation of the o-Ps yield versus external electric ®eld, r0 in isooctane and n-hexane with higher positron mobilities is between 200 and 600 AÊ, whereas a value of only 70 AÊ was obtained for C6F6 (Stepanov et al., 2000). Here one should note that the Ps formation probability is very high in C6F6. Probably the overlap of the spatial distributions of the positron and the localized electrons in C6F6 is so large that they can recombine with a high probability. This picture is consistent with the idea that Ps is formed as a result of electron pick up from C6Fÿ 6 by the positron (Ito, 1988) and can also account for the very weak electric ®eld dependence of Ps formation in C6F6 (Stepanov et al., 2000).
4. Conclusion In summary, we presented positron mobility data for three neat, nonpolar liquids, isooctane, n-hexane and hexa¯uorobenzene. We found that the free positron mobilities for isooctane and hexane are 69 2 3 cm2 Vÿ1 sÿ1 and 53 2 3 cm2 Vÿ1 sÿ1, signi®cantly larger than 0 2 7 cm2 Vÿ1 sÿ1 for C6F6. There is no simple correlation between I3 and the positron mobility or the electron mobility. The positron mobility data allowed us to estimate some positron blob parameters in the three liquids and revealed peculiar feature of Ps formation in C6F6.
Acknowledgements We wish to thank Dr. S.V. Stepanov and Dr. M. Eldrup for their helpful discussions. We also thank the Science and Technology Agency (STA) and the Agency of Industrial Science and Technology (AIST) for ®nancial supports.
Appendix A Here we brie¯y discuss the in¯uence of the total number of counts in a Doppler spectrum on the peak position P. Let us assume that P is given by X ci ni 1A P X ci where ci is the count at nith channel. If Dcj being the ¯uctuation at njth channel, it contributes to the error of P by the following amount: X ci ni Dcj nj X ci ni DPj X ÿ 2A ci ci Dcj In the case that Dcj =cj 1Eq. (2A) becomes ÿ DPj nj ÿ P Dcij A
3A
P P where A ci and B ci ni : We further assume that the probability distribution function of ci is of Gaussian-type with the standard deviation c1=2 i : Substituting Dci c1=2 into Eq. (3A), we obtain i DP
X
DPi2
1=2
4A
This equation gives an error of DP 0:081 channel (3.34 eV) for a Doppler spectrum containing a total count of 0.12 M, a typical number for a spectrum recorded for 20 min. After calibrating the 511 keV peak using two additional reference g-ray sources and evaluating the average Doppler shift for 8±15 pairs of (0, F ) ®elds, we obtain a large statistical error of about 4 eV in the average Doppler shift at ®eld F. In the second method, if the total number of counts is 1.2±1.5 M, the error in the Doppler shift is approximately 1±2 eV (details are shown in Figs. 1±3). References Billard, I., Goulet, T., Jay-Gerin, J.P., Bonnenfant, A., 1998. On the in¯uence of electron mobilities on the yields of formation of positronium in liquids: new I3 measurements and Monte Carlo simulations. J. Chem. Phys. 108, 2408. Brusa, R.S., Naia, M.D., Margoni, D., Zecca, A., 1995.
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