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Experimental investigation of a DC electric field on positron self-trapping in helium gas
Solid State Communications, Vol. 53, No. 1, pp. 6 3 - 6 6 , 1985. Printed in Great Britain.
0038-1098/85 $3.00 + .00 Pergamon Press Ltd.
EXPERIMENTAL INVESTIGATION OF A DC ELECTRIC FIELD ON POSITRON SELF-TRAPPING IN HELIUM GAS A.H. Ruttenberg,* R. Tawel and K.F. Canter Physics Department, Brandeis University, Waltham, MA 02254
(Received 29 June 1984 by J.M. Rowell) \ Positron lifetime spectra taken in helium gas at 5.5 K and 129 standard densities are compared with Monte Carlo simulations of Farazdel and Epstein. This comparison establishes that the energy threshold for positron self-trapping and the decay rate of the self-trapped state are not affected by an electric field of 52 V/cm. The comparison also suggests a small change in the positron diffusion prior to the self-trapping. THE POSITRON self trapped state in low temperature helium vapor has to date been only investigated by measurements of the temperature and densitydependence of the positron annihilation rate [ 1 - 4 ] . This rate, which is a direct measure of the overlap of the positron wave function with the surrounding helium, has been in excellent agreement with the helium cluster model [5] over a large range of temperatures and densities [ 6 - 8 ] . Although this should be sufficient to confirm the cluster model, an apparent disagreement between the observed [9] and predicted [10] effects of an externally applied electric field now warrants further investigation of the cluster model. The experiments reported here were carried out to see if the positroncluster complex is relatively immobile and impervious to weak electric fields as predicted by the cluster model. The electric field for these measurements was produced by a special arrangement of electrodes designed to enable future measurements with a pulsed electric field as well. The pulsed field experiments will serve as an important test of the hypothesis that the positrons slow down through a mobility edge prior to forming the cluster complex [11, 12]. However, in order for the pulsed field experiments to be of value, it is important that there first be no outstanding questions about the effect of a static electric field on the positron-cluster complex. Our method of inferring the effect of electric fields on the positrons is to observe the effect on a positron lifetime spectrum (PLS) which is a measure of the rate of annihilations of an ensemble of positrons emitted (essentially one at a time) by a 22Na source mounted inside of the sample chamber. The first experimental investigation of the effect of an electric field on positron annihilation in low temperature helium gas was carried out by Deshpande
and Roellig ( D - R ) [9]. It was difficult to quantitatively interpret their results at the time since an electric field has a large effect on the positron energy distribution and there were no full diffusion calculations available to separate out the effects of the electric field on positron energy loss from possible effects on the self trapping phenomenon. Subsequent Monte Carlo simulations of positron diffusion in the presence of an electric field were carried out by Farazdel and Epstein ( F - E ) [ 10] to see if the D - R data were consistent with the cluster model. The F - E calculations were able to simulate the delayed peak in the PLS by introducing an abrupt increase in the annihilation rate when the positrons slowed down to a threshold ER, typically 5 × 10 -3 eV. Unfortunately, there was some ambiguity in comparing the F - E simulations with other features of the D - R data when a DC electric field was applied. The F - E simulation had the option of introducing "sink-like" behaviour or "non sink-like" behaviour when the positrons reached En. The sink-like simulation was in keeping with the cluster model in that the positrons were no longer free to diffuse when they reached En. The non sink-like simulation corresponded to the positrons still having an increase in annihilation rate at ER but being able to diffuse as free particles below this threshold and to possible gain enough energy to go above En and have the lower free positron rate. The ambiguity arose in comparing apparent equilibrium annihilation rates, obtained from single exponential fits to the PLS past the delayed peak, corresponding to the simulated and experimental PLS. The experimentally determined equilibrium decay rate as measured by Deshpande and Roellig lay approximately halfway between those calculated using the sink-like and non sink-like simulations. Farazdel and Epstein attributed this unsatisfactory agreement with experiment to two things. Firstly this comparison was made only between apparent equilibrium decay rates, and secondly that there were large inhomogeneities in the experimental
*Present address: Physics Department, Princeton University, Princeton, N.J. 08544, USA. 63
64
EXPERIMENTAL INVESTIGATION OF A DC ELECTRIC FIELD
electric field. In this paper we report new experimental PLS measurements carried out with improved electric field uniformity and with a more detailed comparison with the simulated spectrum. The electronic setup used to obtain the lifetime spectra is well described elsewhere [4, 11 ]. Briefly, two plastic scintillator-photomultiplier tubes are brought close to a sample chamber containing a 22Na source and the helium gas. The emission of the positron into the gas is detected by observing the co-emitted 1.28 MeV gamma-ray, which starts a time to amplitude converter (TAC). The decay of the positron is signalled by the 0.511 MeV annihilation gamma-ray, which stops the TAC. Pulse heights are binned by a 1024 channel multichannel analyzer. The sample chamber temperature regulation system was the same as that used for precision PLS measurements near the vapor-liquid critical point of He-4 [4]. The sample chamber, however, was modified to allow a larger internal diameter (21 mm) for the electrode and insulator assembly (Fig. 1) used to produce the electric field. The reduced wall thickness of 1.2 mm contributed to a reduced temperature accuracy as compared to that of the critical point experiments. A discussion of the possible systematic errors will be given later. In designing the electrode assembly our design criteria was to maintain a uniform electric field magnitude over a large volume of the sample chamber. The electric field effect on the PLS is proportional to E 2, thus field direction is unimportant as long as it does not vary on the scale of the mean free path of the positron (typically 103 Angstroms). We chose to produce an alternating electric field direction by stacking pairs of closely spaced ground and high voltage plates as shown in Fig. 1. The sample chamber itself was held at ground. Macor, a machinable ceramic, was used as a combined spacer-insulator. The close spacing of the plates helped to minimize fringing effects. The brass lips on the ground electrodes further reduced fringing. Plates were composed of a fine mesh (50 lines per inch) with a 92% transmission. The design was a compromise between excessive attenuation of the positrons as they Cw°P'Ier ~ M Mesh ° F i I f rmlt HighVoltage , at , acor/~ j 9 5 Y ° edUnio 'y \
~ii!!!!!!!!!i i";~ !ii)ii i i!!~" 100 rail
Fig. 1. Section of the electrode arrangement used to produce an electric field which is uniform in magnitude but alternating in direction. 1 mill = 2.54 x 10 -3 cm.
Vol. 53, No. 1
passed through a number of meshes, and the degree of field uniformity which we wanted. The field was calculated with a relaxation technique to solve the laplace equation. The results of that calculation are indicated by the stippled region in Fig. 1, which represents the region in which the field was 95% uniform. We estimate that 80% of the positron annihilations in the gas occurred in a region of > 95% field homogeneity. This is a considerable improvement in the homogeneity reported by D - R [91. The lifetime spectrum can be resolved into several components. The prompt component represents the rapid annihilation of the fraction ( ~ 50%) of the positrons which annihilate through collisions with the sample chamber walls or formation of para-positronium. Those positrons which form ortho-positronium have a longer lifetime, and appear as a relatively long lived pure exponential (100 ns lifetime). The background is a flat contribution to the lifetime spectrum and originates from uncorrelated start and stop pulses. The remainder of the spectrum is the slow positron lifetime spectrum, produced by positrons which do not have energy to form positronium, and thus can only lose energy via elastic collisions with the helium. In our analysis we neglect the prompt region, as it can not be accurately subtracted. The magnitude of the background and ortho-positronium exponential decay components are determined by a chi-squared minimization fitting to a region of the lifetime spectrum past the point where the slow positron annihilation component has a significant contribution. These components are then subtracted from the spectra. Our measured experimental conditions were 5.5 K at a density of 129Am (Am = amagat = 2.69 x 1019 cm -3) and fields of 0 and 51.6 V/cm, the quoted parameters for the F - E simulation where the relevant corresponding electric field-gas density ratio was 0.4 V/cm-Am. However, upon comparison of our spectra with the F - E simulation we noted that our delayed peak position was off by about 10%. The time scale of the PLS for free positrons is proportional to the inverse density, so that time multiplied by density is a constant. Also the peak position in F - E simulation was an empirically determined feature, i.e. En was adjusted to give the best agreement between previous experimental [2] and simulation spectra. In view of this, we normalized our spectra by stretching them 10% so that the zero field peak position matched the simulation. A confirmation that we were justified in doing this is that the decay constant after the peak is in good agreement with that of the simulation. The stretching would, assuming our time scaling was accurate, correspond to the density of 140 amagat, or 0.37 V/cm-Am. If our pressure measurements were accurate this would correspond to a
Vol. 53, No. 1
EXPERIMENTAL INVESTIGATION OF A DC ELECTRIC FIELD
temperature of ~.3 K. The most likely candidate for the error wds in the temperature measurement, possibly due in part to the decreased thickness of the sample chamber wall. Figure 2 shows our normalized results as comcompared to the F - E simulation. Note that there is a striking match between the simulated 0.4 V/cm-Am sink-like spectrum and our experimental spectrum for the region past the peak. The result unambiguously favors the sink-like over the non sink-like simulation. The fact that the F - E simulation for the electric field and no-field spectra were carried out with the same ER and equilibrium decay rate (i.e. the cluster rate ?~c) means that we have also experimentally established that E n and ~ are not affected by the electric field. While F - E chose to compare their simulation with experiment by comparison of the decay rate after the peak, we chose not to for two reasons. First, measuring the decay rate after the peak is ill-defined since one must be able to establish that the choice of the region to fit is at sufficiently long time after t = 0 that the positions have truly reached thermal equilibrium. The F - E simulation is based on the fact that the decay rate increases spontaneously below E n but to a rate corresponding to that of positrons in the self-trapped cluster complex which is unaffected by a weak field. The apparent difference in the decay rates, with and without
04 Volts/cm-Amagat
E _J
' 0
10'o 0
' 2000
3o ~O0
J 4000
TIME (ns-Amagot)
Fig. 2. Experimental lifetime spectra (points) with background and ortho-positronium components subtracted. The spectra were obtained nominally at 5.5 K and 140 Am, but were stretched to give a delayed peak position agreeing with the 129 Am simulations of Farazdel and Epstein (ref. 10). Solid curves: sink-like simulations. Dashed curves: non sink-like simulation.
65
an applied electric field is an illusion due to the superposition of the pre-En decay rate, and the post E R decay rate. Second, there is a practical difficulty in measuring that decay rate for the electric field lifetime spectra. Since the peak broadens as the field is increased, there is a diminishing region of the spectrum which can be fitted to the exponential, with a corresponding increase in the statistical error. The simplicity of the assumptions of the Monte Carlo simulation, and the degree to which the experimental results match is additional confirmation of the cluster model. What has not been resolved, however, is whether as Azbel and Platzman [12] propose, there exists two distinct energy thresholds, a "mobility edge" Ee below which the positron is immobile due to localization and a separate energy threshold En below which the annihilation cross section increases due to cluster formation. The F - E simulation only had one of these parameters, namely ER. The principal way in which the introduction of an electric field affects a spectrum is to broaden the positron energy distribution and thus allow some positrons to reach ER at a time earlier than the no-field case. A mobility edge threshold could thus be added to the F - E simulation in a simple manner. Allow positrons to diffuse, but when they reach a threshold Ec, remove the electric field term in the energy loss-per-collision calculation. At the lower threshold energy ER, increase the annihilation cross section to the equilibrium cluster state annihilation rate. Such a modification in the simulation might improve still further the agreement with the experimental data. In particular, there is a small but noticeable difference in the onset region of the peaks in the F - E simulation and the experimental electric field spectrum shown in Fig. 2. There is also a small disagreement in the zero field spectra as well, but this is due to the fact that the F - E simulations do not include convoluting the experimental timing resolution (typically 1 nsec) into the lifetime spectra. However in the 0.4 V/cm-Am case, there is a disagreement that can not be attributed to resolution, i.e. in the F - E simulation a small fraction of positrons reach cluster state ~ 2 nsec earlier than is observed experimentally. Whether the introduction of a mobility edge into the simulation would account for this discrepancy is an important question to be answered since it would be a valuable clue as to the existence of a positron mobility edge in helium. We are presently carrying out pulsed field experiments, as suggested by Azbel and Platzman, to further investigate the possibility of a mobility edge. In summary we have shown that an electric field of 51.6 V/cm has no effect on the enhanced annihilation rate (k~) of positrons that have reached the energy threshold E n where the enhancement occurs. We have
66
EXPERIMENTAL INVESTIGATION OF A DC ELECTRIC FIELD
also shown that the electric field has no effect on ER. The impervious nature of the enhanced annihilation rate to the presence of the electric field is good support for the cluster model since the massive cluster (~ 100 atomic helium masses) is immobile and the positron is too tightly bound (E R ~ 1 eV) [6, 7] to be affected by 51.6 V/cm (= 5.16 x 10-9 V/A). Similarly this field is too weak to have any effect on the quantum mechanical multiple scattering of the positron in the extended state over a coherence length (i.e. mean free path) of less than 103 A. We have also observed a small departure in the experimental lifetime spectrum from the semiclassical simulation that might suggest a precurser transition in the positron mobility prior to the transition to the cluster state. Acknowledgments - Discussions with A. Farazdel are greatly appreciated. One of us (A.H.R.)wishes to acknowledge the support of the Mazer Undergraduate Research Program. This work was supported in part by NSF Grant DMR-8109509.
2. 3. 4. 5.
6. 7. 8. 9.
10. 11.
REFERENCES 1.
K.F. Canter and L.O. Roetlig, Phys. Rev. Lett. 25,328 (1970).
12.
Vol. 53, No. 1
K.F. Canter, J.D. McNutt and L.O~ Rbellig, Phys. Rev. AI2,375 (1974). P. Hautoj/irvi, K. Rysts61~i, P. Tuovinen, A. Vehanen and P. Jauho, Phys. Rev. Lett. 38, 842 (1977). T.-P. Chen and K.F. Canter, SolidState Comm. 47, 903 (1983). L.O. Roellig and T.M. Kelly, Phys. Rev. Lett. 15, 746 (1965); L.O. Roellig, Positron Annihilation, A.T. Stewart and L.O. Roellig, eds., (Academic Press, NY, 1967), pp 127-141. M. Manninen and P. Hautoj~irvi, Phys. Rev. B17, 2129 (1978). M.J. Stott and E. Zaremba, Phys. Rev. Lett. 38, 1493 (1977). I.T. Iakubov and A.G. Khrapak, Rep. Prog. Phys. 45,697 (1982). A.S. Deshpande, L.O. Roellig, 2nd International Positron Annihilation Conference, Kingston, Ontario, Queens University, 5.107 ( 1971); A.S. Deshpande, PhD Dissertation, Wayne State University, 1972. A. Farazdel and I.R. Epstein, Phys. Rev. A17, 577 (1978). K.F. Canter, M. Fishbein, R.A. Fox, K. Gyasi, and J.F. Steinman, Solid State Comm. 34,773 (1980). M. Ya. Azbel and P.M. Platzman, Solid State Comm. 39,679 (1981).
0038-1098/85 $3.00 + .00 Pergamon Press Ltd.
EXPERIMENTAL INVESTIGATION OF A DC ELECTRIC FIELD ON POSITRON SELF-TRAPPING IN HELIUM GAS A.H. Ruttenberg,* R. Tawel and K.F. Canter Physics Department, Brandeis University, Waltham, MA 02254
(Received 29 June 1984 by J.M. Rowell) \ Positron lifetime spectra taken in helium gas at 5.5 K and 129 standard densities are compared with Monte Carlo simulations of Farazdel and Epstein. This comparison establishes that the energy threshold for positron self-trapping and the decay rate of the self-trapped state are not affected by an electric field of 52 V/cm. The comparison also suggests a small change in the positron diffusion prior to the self-trapping. THE POSITRON self trapped state in low temperature helium vapor has to date been only investigated by measurements of the temperature and densitydependence of the positron annihilation rate [ 1 - 4 ] . This rate, which is a direct measure of the overlap of the positron wave function with the surrounding helium, has been in excellent agreement with the helium cluster model [5] over a large range of temperatures and densities [ 6 - 8 ] . Although this should be sufficient to confirm the cluster model, an apparent disagreement between the observed [9] and predicted [10] effects of an externally applied electric field now warrants further investigation of the cluster model. The experiments reported here were carried out to see if the positroncluster complex is relatively immobile and impervious to weak electric fields as predicted by the cluster model. The electric field for these measurements was produced by a special arrangement of electrodes designed to enable future measurements with a pulsed electric field as well. The pulsed field experiments will serve as an important test of the hypothesis that the positrons slow down through a mobility edge prior to forming the cluster complex [11, 12]. However, in order for the pulsed field experiments to be of value, it is important that there first be no outstanding questions about the effect of a static electric field on the positron-cluster complex. Our method of inferring the effect of electric fields on the positrons is to observe the effect on a positron lifetime spectrum (PLS) which is a measure of the rate of annihilations of an ensemble of positrons emitted (essentially one at a time) by a 22Na source mounted inside of the sample chamber. The first experimental investigation of the effect of an electric field on positron annihilation in low temperature helium gas was carried out by Deshpande
and Roellig ( D - R ) [9]. It was difficult to quantitatively interpret their results at the time since an electric field has a large effect on the positron energy distribution and there were no full diffusion calculations available to separate out the effects of the electric field on positron energy loss from possible effects on the self trapping phenomenon. Subsequent Monte Carlo simulations of positron diffusion in the presence of an electric field were carried out by Farazdel and Epstein ( F - E ) [ 10] to see if the D - R data were consistent with the cluster model. The F - E calculations were able to simulate the delayed peak in the PLS by introducing an abrupt increase in the annihilation rate when the positrons slowed down to a threshold ER, typically 5 × 10 -3 eV. Unfortunately, there was some ambiguity in comparing the F - E simulations with other features of the D - R data when a DC electric field was applied. The F - E simulation had the option of introducing "sink-like" behaviour or "non sink-like" behaviour when the positrons reached En. The sink-like simulation was in keeping with the cluster model in that the positrons were no longer free to diffuse when they reached En. The non sink-like simulation corresponded to the positrons still having an increase in annihilation rate at ER but being able to diffuse as free particles below this threshold and to possible gain enough energy to go above En and have the lower free positron rate. The ambiguity arose in comparing apparent equilibrium annihilation rates, obtained from single exponential fits to the PLS past the delayed peak, corresponding to the simulated and experimental PLS. The experimentally determined equilibrium decay rate as measured by Deshpande and Roellig lay approximately halfway between those calculated using the sink-like and non sink-like simulations. Farazdel and Epstein attributed this unsatisfactory agreement with experiment to two things. Firstly this comparison was made only between apparent equilibrium decay rates, and secondly that there were large inhomogeneities in the experimental
*Present address: Physics Department, Princeton University, Princeton, N.J. 08544, USA. 63
64
EXPERIMENTAL INVESTIGATION OF A DC ELECTRIC FIELD
electric field. In this paper we report new experimental PLS measurements carried out with improved electric field uniformity and with a more detailed comparison with the simulated spectrum. The electronic setup used to obtain the lifetime spectra is well described elsewhere [4, 11 ]. Briefly, two plastic scintillator-photomultiplier tubes are brought close to a sample chamber containing a 22Na source and the helium gas. The emission of the positron into the gas is detected by observing the co-emitted 1.28 MeV gamma-ray, which starts a time to amplitude converter (TAC). The decay of the positron is signalled by the 0.511 MeV annihilation gamma-ray, which stops the TAC. Pulse heights are binned by a 1024 channel multichannel analyzer. The sample chamber temperature regulation system was the same as that used for precision PLS measurements near the vapor-liquid critical point of He-4 [4]. The sample chamber, however, was modified to allow a larger internal diameter (21 mm) for the electrode and insulator assembly (Fig. 1) used to produce the electric field. The reduced wall thickness of 1.2 mm contributed to a reduced temperature accuracy as compared to that of the critical point experiments. A discussion of the possible systematic errors will be given later. In designing the electrode assembly our design criteria was to maintain a uniform electric field magnitude over a large volume of the sample chamber. The electric field effect on the PLS is proportional to E 2, thus field direction is unimportant as long as it does not vary on the scale of the mean free path of the positron (typically 103 Angstroms). We chose to produce an alternating electric field direction by stacking pairs of closely spaced ground and high voltage plates as shown in Fig. 1. The sample chamber itself was held at ground. Macor, a machinable ceramic, was used as a combined spacer-insulator. The close spacing of the plates helped to minimize fringing effects. The brass lips on the ground electrodes further reduced fringing. Plates were composed of a fine mesh (50 lines per inch) with a 92% transmission. The design was a compromise between excessive attenuation of the positrons as they Cw°P'Ier ~ M Mesh ° F i I f rmlt HighVoltage , at , acor/~ j 9 5 Y ° edUnio 'y \
~ii!!!!!!!!!i i";~ !ii)ii i i!!~" 100 rail
Fig. 1. Section of the electrode arrangement used to produce an electric field which is uniform in magnitude but alternating in direction. 1 mill = 2.54 x 10 -3 cm.
Vol. 53, No. 1
passed through a number of meshes, and the degree of field uniformity which we wanted. The field was calculated with a relaxation technique to solve the laplace equation. The results of that calculation are indicated by the stippled region in Fig. 1, which represents the region in which the field was 95% uniform. We estimate that 80% of the positron annihilations in the gas occurred in a region of > 95% field homogeneity. This is a considerable improvement in the homogeneity reported by D - R [91. The lifetime spectrum can be resolved into several components. The prompt component represents the rapid annihilation of the fraction ( ~ 50%) of the positrons which annihilate through collisions with the sample chamber walls or formation of para-positronium. Those positrons which form ortho-positronium have a longer lifetime, and appear as a relatively long lived pure exponential (100 ns lifetime). The background is a flat contribution to the lifetime spectrum and originates from uncorrelated start and stop pulses. The remainder of the spectrum is the slow positron lifetime spectrum, produced by positrons which do not have energy to form positronium, and thus can only lose energy via elastic collisions with the helium. In our analysis we neglect the prompt region, as it can not be accurately subtracted. The magnitude of the background and ortho-positronium exponential decay components are determined by a chi-squared minimization fitting to a region of the lifetime spectrum past the point where the slow positron annihilation component has a significant contribution. These components are then subtracted from the spectra. Our measured experimental conditions were 5.5 K at a density of 129Am (Am = amagat = 2.69 x 1019 cm -3) and fields of 0 and 51.6 V/cm, the quoted parameters for the F - E simulation where the relevant corresponding electric field-gas density ratio was 0.4 V/cm-Am. However, upon comparison of our spectra with the F - E simulation we noted that our delayed peak position was off by about 10%. The time scale of the PLS for free positrons is proportional to the inverse density, so that time multiplied by density is a constant. Also the peak position in F - E simulation was an empirically determined feature, i.e. En was adjusted to give the best agreement between previous experimental [2] and simulation spectra. In view of this, we normalized our spectra by stretching them 10% so that the zero field peak position matched the simulation. A confirmation that we were justified in doing this is that the decay constant after the peak is in good agreement with that of the simulation. The stretching would, assuming our time scaling was accurate, correspond to the density of 140 amagat, or 0.37 V/cm-Am. If our pressure measurements were accurate this would correspond to a
Vol. 53, No. 1
EXPERIMENTAL INVESTIGATION OF A DC ELECTRIC FIELD
temperature of ~.3 K. The most likely candidate for the error wds in the temperature measurement, possibly due in part to the decreased thickness of the sample chamber wall. Figure 2 shows our normalized results as comcompared to the F - E simulation. Note that there is a striking match between the simulated 0.4 V/cm-Am sink-like spectrum and our experimental spectrum for the region past the peak. The result unambiguously favors the sink-like over the non sink-like simulation. The fact that the F - E simulation for the electric field and no-field spectra were carried out with the same ER and equilibrium decay rate (i.e. the cluster rate ?~c) means that we have also experimentally established that E n and ~ are not affected by the electric field. While F - E chose to compare their simulation with experiment by comparison of the decay rate after the peak, we chose not to for two reasons. First, measuring the decay rate after the peak is ill-defined since one must be able to establish that the choice of the region to fit is at sufficiently long time after t = 0 that the positions have truly reached thermal equilibrium. The F - E simulation is based on the fact that the decay rate increases spontaneously below E n but to a rate corresponding to that of positrons in the self-trapped cluster complex which is unaffected by a weak field. The apparent difference in the decay rates, with and without
04 Volts/cm-Amagat
E _J
' 0
10'o 0
' 2000
3o ~O0
J 4000
TIME (ns-Amagot)
Fig. 2. Experimental lifetime spectra (points) with background and ortho-positronium components subtracted. The spectra were obtained nominally at 5.5 K and 140 Am, but were stretched to give a delayed peak position agreeing with the 129 Am simulations of Farazdel and Epstein (ref. 10). Solid curves: sink-like simulations. Dashed curves: non sink-like simulation.
65
an applied electric field is an illusion due to the superposition of the pre-En decay rate, and the post E R decay rate. Second, there is a practical difficulty in measuring that decay rate for the electric field lifetime spectra. Since the peak broadens as the field is increased, there is a diminishing region of the spectrum which can be fitted to the exponential, with a corresponding increase in the statistical error. The simplicity of the assumptions of the Monte Carlo simulation, and the degree to which the experimental results match is additional confirmation of the cluster model. What has not been resolved, however, is whether as Azbel and Platzman [12] propose, there exists two distinct energy thresholds, a "mobility edge" Ee below which the positron is immobile due to localization and a separate energy threshold En below which the annihilation cross section increases due to cluster formation. The F - E simulation only had one of these parameters, namely ER. The principal way in which the introduction of an electric field affects a spectrum is to broaden the positron energy distribution and thus allow some positrons to reach ER at a time earlier than the no-field case. A mobility edge threshold could thus be added to the F - E simulation in a simple manner. Allow positrons to diffuse, but when they reach a threshold Ec, remove the electric field term in the energy loss-per-collision calculation. At the lower threshold energy ER, increase the annihilation cross section to the equilibrium cluster state annihilation rate. Such a modification in the simulation might improve still further the agreement with the experimental data. In particular, there is a small but noticeable difference in the onset region of the peaks in the F - E simulation and the experimental electric field spectrum shown in Fig. 2. There is also a small disagreement in the zero field spectra as well, but this is due to the fact that the F - E simulations do not include convoluting the experimental timing resolution (typically 1 nsec) into the lifetime spectra. However in the 0.4 V/cm-Am case, there is a disagreement that can not be attributed to resolution, i.e. in the F - E simulation a small fraction of positrons reach cluster state ~ 2 nsec earlier than is observed experimentally. Whether the introduction of a mobility edge into the simulation would account for this discrepancy is an important question to be answered since it would be a valuable clue as to the existence of a positron mobility edge in helium. We are presently carrying out pulsed field experiments, as suggested by Azbel and Platzman, to further investigate the possibility of a mobility edge. In summary we have shown that an electric field of 51.6 V/cm has no effect on the enhanced annihilation rate (k~) of positrons that have reached the energy threshold E n where the enhancement occurs. We have
66
EXPERIMENTAL INVESTIGATION OF A DC ELECTRIC FIELD
also shown that the electric field has no effect on ER. The impervious nature of the enhanced annihilation rate to the presence of the electric field is good support for the cluster model since the massive cluster (~ 100 atomic helium masses) is immobile and the positron is too tightly bound (E R ~ 1 eV) [6, 7] to be affected by 51.6 V/cm (= 5.16 x 10-9 V/A). Similarly this field is too weak to have any effect on the quantum mechanical multiple scattering of the positron in the extended state over a coherence length (i.e. mean free path) of less than 103 A. We have also observed a small departure in the experimental lifetime spectrum from the semiclassical simulation that might suggest a precurser transition in the positron mobility prior to the transition to the cluster state. Acknowledgments - Discussions with A. Farazdel are greatly appreciated. One of us (A.H.R.)wishes to acknowledge the support of the Mazer Undergraduate Research Program. This work was supported in part by NSF Grant DMR-8109509.
2. 3. 4. 5.
6. 7. 8. 9.
10. 11.
REFERENCES 1.
K.F. Canter and L.O. Roetlig, Phys. Rev. Lett. 25,328 (1970).
12.
Vol. 53, No. 1
K.F. Canter, J.D. McNutt and L.O~ Rbellig, Phys. Rev. AI2,375 (1974). P. Hautoj/irvi, K. Rysts61~i, P. Tuovinen, A. Vehanen and P. Jauho, Phys. Rev. Lett. 38, 842 (1977). T.-P. Chen and K.F. Canter, SolidState Comm. 47, 903 (1983). L.O. Roellig and T.M. Kelly, Phys. Rev. Lett. 15, 746 (1965); L.O. Roellig, Positron Annihilation, A.T. Stewart and L.O. Roellig, eds., (Academic Press, NY, 1967), pp 127-141. M. Manninen and P. Hautoj~irvi, Phys. Rev. B17, 2129 (1978). M.J. Stott and E. Zaremba, Phys. Rev. Lett. 38, 1493 (1977). I.T. Iakubov and A.G. Khrapak, Rep. Prog. Phys. 45,697 (1982). A.S. Deshpande, L.O. Roellig, 2nd International Positron Annihilation Conference, Kingston, Ontario, Queens University, 5.107 ( 1971); A.S. Deshpande, PhD Dissertation, Wayne State University, 1972. A. Farazdel and I.R. Epstein, Phys. Rev. A17, 577 (1978). K.F. Canter, M. Fishbein, R.A. Fox, K. Gyasi, and J.F. Steinman, Solid State Comm. 34,773 (1980). M. Ya. Azbel and P.M. Platzman, Solid State Comm. 39,679 (1981).