IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 54, NO. 3, JUNE 2007 643 Calculation of Drift Velocities and Diffusion Coefficients for Xe+ Ions in Gaseous Xe/3He Mixtures J. A. S. Barata and C. A. N. Conde, Member, IEEE Abstract—The drift velocities and the longitudinal and trans- verse diffusion coefficients for Xe+ ions in gaseous Xe/3He mixtures, at atmospheric pressures, for Xe concentrations be- tween 1 and 30%, and reduced electric field strengths, from about 3 to 70 Td, corresponding to , from about 1 to 22.5 V cm 1Torr 1 at 300 K, are calculated by simulation using a de- tailed Monte Carlo computer programme. This programme uses a set of integral and differential elastic collision cross sections for Xe+ ions with neutral He and Xe atoms, previously calculated by the authors using a modified Tang-Toennies ion-atom interaction potential and the JWKB approximation to calculate the phase shifts. The calculated drift velocities of Xe+ ions depend strongly on the concentration of Xe in the mixtures. Index Terms—Atomic, molecular, ion, and heavy-particle col- lisions, diffusion coefficients, ion mobility, kinetic and transport theory of gases, Monte Carlo simulation. I. INTRODUCTION THE detection of neutrons is often made with He basedproportional counters. However, since helium has a low stopping power for the reaction products, heavy gases like Xe or Ar are frequently added to He, especially for position sen- sitive detectors based on multiwire detectors [1]. It is also well known, though sometimes neglected, that the drift of positive ions near the anode wires plays an important role in the forma- tion of proportional counter pulses [2]. Thus, to model and un- derstand the physics of avalanche based gaseous radiation detec- tors, a detailed understanding of the transport of positive ions in the gaseous media is required, namely their drift velocities and diffusion properties. In Xe/He based gas mixtures only single charge Xe and He ions should be considered, but since He ions will transfer their charge to the neutral Xe atoms in the mix- ture, only the drift of Xe ions needs to be considered in Xe/ He mixtures. In the present work we report calculated results, using a de- tailed three-dimensional Monte Carlo simulation method [3], Manuscript received July 3, 2006; revised January 8, 2007. This work was supported by FEDER through Project POCTI/FP/FNU/50240/2003 from Fun- dação para a Ciência e para a Tecnologia—Portugal, and carried out in Centro de Instrumentação (Unit 217/94), Departamento de Física da Universidade de Coimbra, Portugal, and in Departamento de Física da Universidade da Beira Interior, Portugal. J. A. S. Barata acknowledges travel support from Fundação Luso-Americana para o Desenvolvimento. J. A. S. Barata is with the Departamento de Física da Universidade da Beira Interior, P-6201-001 Covilhã, Portugal (e-mail:
[email protected]). C. A. N. Conde is with the Departamento de Física da Universidade de Coimbra, P-3004-516 Coimbra, Portugal (e-mail:
[email protected]). Digital Object Identifier 10.1109/TNS.2007.897821 for the drift velocities and diffusion coefficients of Xe ions in Xe/ He mixtures for Xe concentrations in the range 1–30% at standard pressure Torr, and temperature K and reduced electric fields, , expressed in townsend (Td), from about 3 to 70 Td ( is the neutral gas number density and V cm ). For these calculations we used a set of Xe - He [4] and Xe - Xe [5] elastic collision, integral and dif- ferential, cross sections at centre-of-mass energies in the 1 meV to 10 eV range, calculated before by the authors. II. THE MONTE CARLO METHOD As the Xe ions collide with the gas atoms, Xe or He, with centre-of-mass energy, , integral and differential elastic scattering cross-sections are required in order to determine the path of the ions through the gas, the time between collisions and the scattering angle in the centre-of-mass reference frame, together with the energy trans- ferred in each collision. For a given ion impact laboratory energy (ion velocity ) the real total collision frequency, , is the sum of the colli- sion frequencies with Xe and with He : (1) where and are the integral elastic cross section for collision of Xe ions with Xe and and are the gas number density of Xe and He in the mixture, and is the total gas number density. The paths between collisions of the ions under the influence of an uniform electric field are parabolic and their energies change continuously. The time, , between collisions, i.e., the duration of a free path, is calculated from the total collision frequency, , constant for a given free path, using the null collision technique [6], where: (2) and is given by (3) where is an uniform distribution random number in [0,1]. 0018-9499/$25.00 © 2007 IEEE 644 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 54, NO. 3, JUNE 2007 The type of collision is determined comparing a random number with the cumulative sum of the collision frequencies normalized to the total collision frequency, : The thermal velocity of the Xe or He gas atom for each col- lision is randomly sampled from a Maxwellian distribution. The angular differential elastic scattering cross-section, or , will determine the direction of the ion after the collision. The polar scattering angle is sampled in the interval , using the cumulative probability function for the scattering of an ion with energy : (4) and a similar expression for collisions with He. The azimuthal scattering angle is sampled from a uniform distribution in [0,2 ]. The centre-of-mass angles and are converted to laboratory angles through the standard formulae. III. CROSS SECTIONS When an atomic ion collides with an atom having the same nuclei as the ion, several curves for the potential energy must be taken into account, due to the possibility of resonant charge transfer from the ion to the neutral atom. For the interaction of Xe ions with Xe atoms, if we include the spin-orbit coupling of electronic angular momentum, the Xe ions show a spin-orbit splitting into a ground state and a metastable excited state, and the molecular ion, , has six low-lying molecular states [7], three doubly degenerate gerade-ungerade potential pairs: and . The two g-u pairs of molecular states I corre- spond to the interaction of the state ion with its parent atom, and the pair of molecular states II correspond to the interaction of the metastable ion with the parent atom. For the interaction of a Xe ion with a He atom, the Xe ion is either in its ground state or in the metastable ex- cited state. Such an ion interacts with the He atom in the ground state to produce molecular states and , with equal statistical weights, if the ion is in the ground state, or a molecular state, if the ion is in the metastable state. The elastic integral (Fig. 1) and differential collision cross sections were previously calculated by the authors for Xe ions with neutral He atoms [4] and for Xe ions with neutral Xe atoms [5], at centre-of-mass energies in the 1 mV to 10 eV range. For the calculation of the ion-atom interaction poten- tial energy curves , for each one of the six molec- ular states and the three molecular states (considering spin-orbit coupling), we have used a model based on universal damping functions for the dispersion coefficients, suggested by Fig. 1. Calculated integral elastic collision cross sections for scattering of Xe ( P ) ions by Xe [5] and by He [4]. Fig. 2. Variation of the average values of the ion energy, ", axial position, z, square of the radial position, r , and axial position quadratic deviation, with drift time t for Xe ions in a 1% Xe/99% He mixture (760 Torr and 300 K), for E=N = 5 Td, i.e., E = 1223 V/cm. Tang and Toennies [8] for closed-shell systems, and modified by Siska [9]. The phase shifts, for the calculation of scattering amplitudes, were obtained using the JWKB (Jeffreys-Wentzel-Kramers- Brillouin) semi-classical approximation [10]. IV. RESULTS The Monte Carlo simulation was performed, using the above cross-sections, to study the drift of Xe ions in gaseous mixtures for Xe concentrations between 1 and 30% at stan- dard pressure, Torr, and temperature K, under the influence of reduced electric fields, , in the 3 to 70 Td range, i.e., uniform electric fields from 734 V/cm to 17124 V/cm. We assume that all the Xe ions are in the ground state . A sample of 3000 ions is allowed to drift under the applied electric field , for a time long enough for the drift parameters to reach stable values (Fig. 2). This data allows the calculation of the main parameters describing the swarm of Xe ions in the gaseous mixture of , namely drift velocities and the BARATA AND CONDE: CALCULATION OF DRIFT VELOCITIES AND DIFFUSION COEFFICIENTS 645 Fig. 3. Simulation results for drift velocities V of Xe ions in pure He, in pure Xe [5], and in 1%, 5%, 10% and 30% Xe in a Xe= He mixture, at 300 K and 760 Torr, as a function of E=N . Fig. 4. Standard reduced mobilityK of Xe ions in pure He, in pure Xe [5], and in 1%, 5%, 10% and 30% Xe in aXe= Hemixture, at 300 K and 760 Torr, as a function of E=N . diffusion coefficients parallel (longitudinal, ) and perpendic- ular (transverse, ) to the direction of the electric field: (5) The calculated drift velocities for Xe ions in gaseous mixtures, for Xe concentrations between 1 and 30%, as a function of the reduced electric field are presented in Fig. 3. The calculated reduced mobility, (Fig. 4), is related to the average drift velocity , in , of the ions by: (6) where and are the neutral gas pressure in Torr and the tem- perature in K, respectively, and is the intensity of the electric field in . In Figs. 5 and 6 we present the Monte Carlo simulation results for the longitudinal and transverse diffusion coefficients for Xe ions in pure , pure Xe and in mixtures with Fig. 5. Simulation results for longitudinal D diffusion coefficients for Xe ions in pure He, in pure Xe [5], and in 1%, 5%, 10% and 30% Xe in aXe= He mixture, at 300 K and 760 Torr, as a function of E=N . Fig. 6. Simulation results for transverseD diffusion coefficients for Xe ions in pure He, in pure Xe [5], and in 1%, 5%, 10% and 30% Xe in a Xe= He mixture, at 300 K and 760 Torr, as a function of E=N . Xe concentrations ranging from 1% to 30%, as a function of the reduced electric field . In Table I we present part of the same data under numerical format for the drift parameters ( and ) for Xe ions in the ground state, drifting in mixtures under 3 Td, 10 Td and 50 Td reduced electric fields. The statis- tical accuracy of the data is about 1% for the drift velocities and 2% or 3% for the diffusion coefficients. Approximation rules have been proposed to calculate ion mo- bilities and diffusion coefficients for mixtures of gases, from the available data for the transport coefficients for each pure gas [11]–[13]. The mean relative collision energy between the ions and the species is different in the gaseous mixtures and in the pure gas of species at the same values of gas number density , temperature and reduced electric field . Therefore, the collision frequency of the ions with neutral species in the mixture and the collision frequency of the ions with species in the pure gas are also different. The earliest approximation rule is due to Blanc [11], and states that the cross sections in the binary collision regime are just added. The Blanc’s law is precise only at very low values of 646 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 54, NO. 3, JUNE 2007 TABLE I MONTE CARLO SIMULATION RESULTS FOR THE DRIFT OF XE IONS, STATE P , IN PURE HE, IN PURE XE AND IN 1%, 5%, 10% AND 30% XE IN A Xe= He MIXTURE, AT 300 K AND 760 TORR, FOR REDUCED ELECTRIC FIELDS E/N OF 3, 10, AND 50 TD Fig. 7. Monte Carlo simulation values for the reduced mobilities K of Xe ions as a function of the Xe percentage in the Xe/ He mixture for a reduced electric fieldE=N of 3 Td (�) (left) and 50 Td ( ) (right); and corresponding calculations according to Blanc’s law (—). reduced electric fields , where the ions have approx- imately thermal energies (the energy that the ions obtain from the electric field is insignificant compared to the thermal energy) and the mean ion relative energy is the same in the mixture and in the single gas. Under this condition, the ratio of collision fre- quencies become all unity and for the particular binary mixture the Blanc’s law can be written as [13, pp. 174]: (7) which gives the reduced mobility in the mixture in terms of the reduced mobilities in pure gases of the species , at the same gas number density, and , and is the corresponding molar fraction in the mixture. In order to use the above expression (7) knowledge of the mobility of the drifting Xe ion both in pure He and in pure Xe is needed. The mobilities of Xe ions in pure He and in pure Xe, calculated in this work from Monte Carlo simulation, for K and Torr, were used to compare in Fig. 7 the calculations of Blanc’s law with the simulation re- sults for the drift of Xe ions in 1%, 5%, 10% and 30% Xe in gaseous mixtures under the influence of a uniform re- duced electric field of 3 Td and 50 Td. In the plot of the reduced mobility as a function of the Xe percentage in the mix- ture for Td (Fig. 7), we conclude that there is a good agreement between the reduced mobilities calculated from the Monte Carlo simulation and those predicted from Blanc’s rule. For stronger fields (50 Td) there are discrepancies. V. DISCUSSION AND CONCLUSIONS The drift velocity and diffusion coefficients of the Xe ions in mixtures show a strong dependence on the concentra- tion of Xe and can vary by more than two orders of magnitude. In the case of a heavy ion moving through a mixture of a light gas and a heavy gas, having the same mass as the ion, the heavy gas is very effective in causing the ion to lose energy, because the energy loss during collisions is maximum for equal masses. For low reduced electric fields, the ion reduced mobility obeys Blanc’s rule. However, for strong fields the calculated reduced mobilities depart from this rule, mainly due to the fact that the mean ion energy is larger than the thermal energy. The deviations from Blanc’s law have been the subject of recent discussions [14]. It has been suggested [15], [16] that the data for pure gases to be used in gas mixtures should be for the same mean energies rather than for the same . At low reduced electric fields , as it was expected, the ion-gas diffusion occurs at the same rate in directions parallel and perpendicular to the electric field, as shown by the fact that then . Finally, we must point out that Xe ions may easily combine with Xe and other neutral atoms, leading to the formation of and other molecular ions. However, the time that the Xe ions take to drift across the m, or so, distance for signifi- cant contribution to pulse formation in a Proportional Counter, is about 1 s (assuming a typical drift velocity of cm/s, as shown in Table I). As shown by Vitols and Oskam [17] the reaction rate for the formation is cm s : which means that, for Xe at atmospheric pressure, the time con- stant for exponential transformation of Xe atomic ions into molecular ions is 12.5 ns. However, at 0.1 atm this time increases to 1.25 s, and at 0.01 atm to 125 s. Therefore, de- pending on the Xe partial pressure, and the travelled distance, Xe ions drift mostly as atomic ions or as molecular ions. Our group has calculated the drift velocities of Xe atomic ions in Xe [5] and measured experimentally [18] the drift velocity of BARATA AND CONDE: CALCULATION OF DRIFT VELOCITIES AND DIFFUSION COEFFICIENTS 647 xenon molecular ions at atmospheric pressures and found evi- dence that molecular Xe ions drift about 40% faster than atomic ions. In the above discussion we have not taken into consideration the formation of in three body processes like: which are known to occur for Ar with reaction rates similar to those for pure Xe, i.e., cm s [19]. 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