Acceleration of heavy ions in the hall accelerator

The process of acceleration of heavy ions in the Hall accelerator is studied. Numerical solutions of one – dimensional hydrodynamic equations describing a three-component system-neutral particles, electrons and ions-are obtained. Ions move in a collisionless manner and are accelerated by a self-consistent electric field, electrons diffuse across the magnetic field. The self-consistent field is calculated using the Poisson equation, and it is shown that there is no singularity when the ion velocity coincides with the ion sound velocity. It is also shown that there is a critical magnetic field above which it is impossible to propagate the ion flow.


Introduction
In connection with the interest in plasma and ion engines, it is important to study various systems of electrons and ions designed to create accelerators. One of the issues that arise in the study of the extraction of ions from plasma is the possibility of exceeding the velocity of ion sound by the flow of heavy ions. In [1] it is shown that the rejection of the use of the exact terms of neutrality i.e., in an environment where the use of the Poisson equation for the electric field, there is no feature match of the flow velocity of the ion sound.

System of equations
Usually, electron Coulomb collisions are neglected in comparison with collisions with neutral atoms. The following equations accurately take into account Coulomb scattering and obtain their numerical solutions.
We present equations describing the behavior of a one-dimensional beam in a longitudinal electric field E z = d/dz in an axisymmetric magnetic field H r at H r >> H z . Equation of dynamics of electrons moving in diffusion mode: where  coll = n 0 V Te  tr + n i V Te  ie + n e V Te  ee is the collision frequency,  tr the transport cross section of the scattering by neutral particles,  ie and  ee the cross sections of Coulomb electron-ion and electronelectron collisions respectively, the thermal velocity of electrons, the temperature, the charge, the mass of electrons and the concentration of neutral atoms, respectively,  с is the cyclotron frequency. The ratio (1) is executed then, when the Larmor radius is less than the free run length. Continuity equation for ions: where n i , V i is the density and directed velocity of ions,  =  ie V Te , is the specific frequency of particle production during ionization of neutral atoms,  i is the ionization cross-section. The equation of pulses of cold tones can be represented as: where M -the mass of the ion, V A -the flow rate of neutral atoms. The flow conservation equation which takes into account the burnout of the neutral gas has the form: where Q -is gas flow, A -cross-sectional area. It is used further as a condition of constancy of the discharge current: here I dis -discharge current.
In conditions close to the real experiment [2], it is possible to put I dis /Ae = 610 18 cm -2 s -1 mass flow rate Q/AM = 410 18 cm -2 s -1 , the flow rate of atoms at the input V A = 10 4 cms -1 . Let T e = 6 eV. In this case the velocity of ion sound in the plasma of single ionized atoms Xe will be c s = (2T e /M) = 2.9510 5 cms -1 . Next, we will Express the density of particles n * in units n * where the discharge current is determined n * = I dis /(eAc s ) = 2.0410 13 cm -3 and the velocity in units c s . To determine the cross sections of collisions, we use the results of [3]. According to this work the scattering cross section of an electron on an atom has the form: where  -the energy of the electron in eV. In our case, the average energy 6 eV, so the scattering cross section  e0  e0  7.210 -15 cm 2 . The ionization cross section at  > I (Iionization potential) have the form: Since in our case ionization is produced only by particles of the Maxwell tail at  > I, then for the averaged cross section at T e 6 eV can be obtained: Given the frequency of ionization  ** = n * =  i V Te n * = 1.1210 5 s -1 . We also introduce the reduced scattering frequency  ** =  tr V Te n * = 2.1310 7 s -1 . Determine the dimensionless length: s = z( * /c s ) = z/l 0 , and l 0 = (c zv / * ) = 2.6 cm. The dimensionless potential U = e/T e ,  = n/n * - 3 dimensionless density,  = V/c s -dimensionless speed. We also introduce a dimensionless gas flow q = Q/(AMn * c s ). In these variables, equations (1-5) are given as: In equations (6)  i,e -the ratio of the Coulomb scattering cross section on ions and electrons to the scattering cross section on atoms. The potential U satisfies the Poisson equation: where dimensionless parameter: If (7) is neglected U ''/, the system is reduced to two equations having a singularity at  = 2 -1/2 (see [4,5]). In contrast to the work [4, 5] the magnetic field will be considered constant. Then we transform the system of equations (6-7). From (7) follows:

Solving a system of equations
We present the results of the solution of the system (8) under the following initial conditions: Coulomb collisions can be taken into account by means of summands  i,e = i/. The Coulomb section  с = 4(e 4 / 2 )ln, where ln   20. Numerical calculations show that the deviation from the neutrality condition exists only in a narrow region near the beam entry into the system and is small. Therefore, in (8) electron scattering will be taken into account by the total coefficient  =  i +  e . In accordance with the above formula T e = 6 eV,  с  1.410 -13 cm 2 , the ratio  с / Tr  20.Taking into account the scattering by electrons can be put the total coefficient  =  i +  e  50. The The dependence of the current coordinates: Curve I -for the discharge current I R = 1 A, curve II -for the discharge current I P = 4 A, curve III is for discharge current I P = 1 A when accounting for the Coulomb scattering of electrons.     For values of the parameter K < 35, this system has reasonable solutions for such values of the parameter s, which significantly exceed one. At K > 35.2 at s < 0.001 there is a singularity, which is explained by the presence of non-monotonic flow velocity behavior in the region of small s. Figure 4 shows the dependence of i(s) at s < 0.01, K = 35.175 at neglect of Coulomb collisions and at K = 35.176 the minimum speed reaches zero, which leads to the singularity of the solution. The locking of ions occurs on its own charge, the charge of electrons, unlike ions, varies monotonicallyslowly growing in this region of s values.

Conclusion
One of the main results of this work is the absence of a feature of the "sound point" type that appears when using the exact condition of neutrality (see [4], [5]). In this case, the real difference from this condition is small. The presence of a critical magnetic field above which the ion beam is locked is also revealed.