Modeling of ionization produced by fast charged particles in gases

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Abstract

A computer modeling of ionization is necessary for the simulation of gaseous detectors of fast charged particles. The interactions of the incident particle with matter are well described by the photoabsorption ionization (PAI) model, which is based on the relation between the energy deposited by the fast charged particle in a medium and the photoabsorption cross-section of this medium. Some modification of the PAI model energy-transfer cross-section allows to distinguish the interactions with different atomic shells and to determine the energy of the primary photoelectrons and possible atomic relaxation cascades. Further simulation of paths and absorption of secondary particles results in a realistic reproduction of the space distributions and amount of initial ionization.

Introduction

Detectors based on the registration of ionization produced by fast charged particles in gases are widely used in high-energy physics experiments. Their main role is the detection of the position of the track and the time of its passage without the particle being absorbed or any noticeable influence on its further movement being made. The amount of ionization deposited in the sensitive volume of the detector can also be measured and gives information about the particle charge and velocity. Despite the wide use of gas-filled detectors their computer modeling still represents a difficult problem. The large variety of phenomena and microscopic processes involved is difficult to simulate realistically in a detailed way or to reproduce reliably in a phenomenological way. The existing models are approximate and based on combining microscopic modeling of some phenomena and well-established phenomenological or generalized features of the others.

When the incident particle passes through matter, it transfers a part of its energy to atoms through inelastic collisions with them. This energy is dissipated in matter by emission of a series of electrons and photons, which ionize other atoms and so on. The multiplication of liberated electrons ends when electron and photon energies become smaller than the minimal ionization potential of the matter. After that the liberated electrons remain free for some significant time. Together with ions, they may be called initial ionization. The modeling of their number and initial position is a goal of this research. We do not consider here what happens with the initial ionization later. Many important characteristics of the proportional chambers can be deduced just from the amount and space position of the initial ionization, while the other more detailed characteristics can be obtained after simulation of the other processes. The complete models, such as GARFIELD [1], can include the simulation of the drift of electrons and ions to chamber electrodes with various additional effects such as attachment, recombination, diffusion, fluorescence, avalanche amplification in the vicinity of the wires, space charge, charge induction at electrodes, influence of magnetic field. The complete simulation is a challenging problem. This paper is devoted to modeling the initial ionization only. It describes the model implemented in the latest version of the computer program HEED [2] developed in 2003–2005.1 The abbreviation HEED stands for high-energy electrodynamics. The name was prompted by the title of the book “High Energy Electrodynamics in Matter” written by Akhiezer and Shulga [3] and indicates the relation between the ionization energy losses as well as Cherenkov and transition radiation (formerly also generated by one of the versions of this program) and the classical electrodynamic properties of media, crossed by high-energy particles. The former version of this program was used as a component of GARFIELD, and also together with other software packages in various important studies and developments [4], [5], [6], [7], [8], [9], [10], [11]. The former version was written in Fortran-77, the new one is made in C++ and based on an improved physical model and arbitrary geometry. The new C++ version contains an interface package which allows it to be called from Fortran in approximately the same way as the old program. Calculations presented in this paper are made by the new C++ version.

There is a wide consensus that the rate of ionization processes occurring when a fast charged particle travels through a medium depends in a certain way on the cross-section of ionization of these atoms by real photons, and also on the dielectric permeability of this medium [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. The dielectric permeability is a function of the photoabsorption cross-section of this medium. Following the work by Allison and Cobb [16] the corresponding theory is called the photoabsorption ionization (PAI) model. The model gives the cross-section of the energy transfers from the particle to the medium. At the practical application of this model for the description of the signals from gaseous detectors one usually assumes that the amount of ionization created after each energy transfer is approximately proportional to the transferred energy with some fluctuations. This approximation is usually sufficient for the practical calculations in which the little space scattering of ionization around the interaction point is not significant. Because of the small practical range of δ-electrons with an energy up to a few keV, and also because of the small probability of fluorescence, we can consider all the energy to be absorbed and converted into ionization at the point of interaction, except for chambers with a high position resolution or when investigating the processes that can depend on the density of the initial ionization (for example, the space-charge effect around anode wires of the proportional chamber). However, the ultimate spatial resolution of cathode strip and micropattern chambers (see, for example, Refs. [23], [24]) is much smaller than the typical range of electrons of keV energies. The range of electrons and the fluorescent photon yield and range also play a role when studying the performance of transition radiation detectors and X-ray detectors.

In order to simulate the range of electrons and the fluorescent photons, one has to determine their energies. The energies of the emitted particles depend on the atomic shell that absorbs a portion of the transferred energy. Assuming that the transferred energy is absorbed by a single atomic electron, we conclude that the photoelectron should carry the transferred energy minus the binding energy of the given shell. Then the vacancy left by the knockout photoelectron can be filled with the emittance of fluorescence photons and secondary autoionization (Auger) electrons. The secondary electrons and photons are as a rule absorbed elsewhere, sometimes with emittance of new secondary products and so on. Since the original PAI cross-section gives the probabilities of interactions of the incident particle with the total atom, rather than with a single electron or a particular atomic shell, we have to modify or replace it by the cross-sections for individual shells. These partial cross-sections should, however, take into account the presence of the dielectric medium, in which all shells “participate” together. “Intuitive” separation based on deduction of the shell number from the transferred energy (choosing the shell with binding energy less than the transferred energy and nearest to it) is not always justified, in particular, in some gas mixtures and in wide gas layers. Although there are many research papers and computer programs devoted to or including the detailed calculation of ionization effects (starting from sixties of the last century see Refs. [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36]), the full PAI model with separated shells and relaxation cascades for fast charged particles and arbitrary gas mixture has not been consistently developed and tested yet. The purpose of this work is to develop such a model and to test it. We will call it the photoabsorption ionization and relaxation (PAIR) model.

Section snippets

Cross-section of energy transfers from incident particle to medium

The differential cross-section for the transfer of the energy E in a single collision of the incident particle with an atom, normalized per one atom of the absorbing media (see Ref. [16], and also Refs. [13], [14], [15], [18], [19], [20], [21], [22]), is expressed bydσdE=αβ2πσγ(E)Eln1(1-β2ε1)2+β4ε22+1Ncβ2-ε1|ε|2θ+σγ(E)Eln2mec2β2E+1E20Eσγ(E1)dE1where me is the electron mass, βc is the velocity of the incident particle, α=1/137 is the fine structure constant, σγ(E) is the atomic photoabsorption

Numbers of primary clusters

The integral over E of Eq. (7), multiplied by the electron density, gives the number of energy transfers per unit length traveled by the incident particle. This number is practically identical to the number of primary ionization clusters measured in many experiments and frequently referred to as the specific primary ionization. This parameter is practically important because it determines the time resolution of various triggers (see, for instance, Refs. [7], [24]) and also used for many other

Conclusions

The PAI model reproduces quite well the numbers of primary clusters, the amplitude spectra and the relativistic dependencies of ionization in gaseous detectors. The separation of atomic shells, the simulation of relaxation cascades and absorption of secondary particles, which can be called the PAIR model, allows one to describe also the space distribution of ionization. The results of modeling agree well with experimental data. The models can be used for research and development of gaseous

Acknowledgements

The author would like to thank for support of this work, stimulating discussions, and help A.A. Vorobyov, R. Veenhof, P. Nevski, O.E. Prokofiev, and M.B. Trzhaskovskaya.

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