br For an isolated particle n where n is
For an isolated particle, 0 = 0 = n0, where n0 is the concentration of charged particles in unperturbed plasma; besides, the thickness r0 of the perturbed region for that particle is not known in advance. The charge (the potential) of the particle in the steady-state case is determined by the charge balance equation : where is the total current density of the emitted electrons. The ion current density on the surface , as well as the surface potential are determined through solving a system of differential equations in the perturbed region. The required value r0 is found by the optimization method (described in Ref. ) from the minimal value of the objective function
This approach allows to determine the size of the perturbed region and obtain the distribution of the plasma parameters in this region, as well as the expression for the density of the ion current onto the particle surface where and are the normalized concentration and the ion velocity on the particle surface. In a plasma crystal, the thickness r0 may exceed the radius of the Wigner–Seitz cell where is the concentration of dust particles. In this case, the particle cannot be regarded as isolated, so the following ion concentration should be chosen as the optimized parameter at the boundary of the Wigner–Seitz cell: for the negative charge of the dust particles and 0 < 0 for their positive charge. The quasi-neutrality of the Wigner–Seitz cell is thus maintained.
The balance equations of particle charge and energy The balance equations of charge and energy on the particle surface  were solved simultaneously to determine the dust particle charge taking into account the secondary, ion-electron, photoelectron and thermal field emission processes: where and are the densities of ion, electron, resonant photon, thermal-field electron and calcitonin gene related peptide currents onto the particle surface (the wall factor W is omitted to simplify the expression); is the temperature of the dust particle surface; and are the secondary-electron emission coefficients for the elastically backscattered and the true secondary electrons, averaged over the electron energy distribution function (EEDF) ; γ is the coefficient of the potential ion-electron emission; Y is the quantum yield for the photoelectric effect; is the probability of an electron escaping from the rough surface of the dust particle without a repeated collision ; α and α are the accommodation coefficients of atoms and ions ; σ0 is the Stefan–Boltzmann constant; a0 is the integrated absorptivity of a dust particle (the emissivity coefficient); ε, εγ, εδ and ε are the kinetic energy quantities of ions and secondary electrons during ion-electron emission, and of true secondary electrons and of photoelectrons; is the excitation energy of the resonant levels. The factor κ is equal to unity if the charge of the dust particle is negative and to if it is positive, as the emitted electrons experience additional deceleration. The temperature T takes the values and  for the respective processes and for the thermal field emission electrons. Table 1 lists the particle current densities and particle energies carried onto or from the surface of the dust particle that were used in the calculations.
Calculation results Fig. 1 shows the calculated curves of the normalized potential of a smooth dust particle surface versus τ=, with the above-listed emission processes factored in separately and together. The calculations were performed based on the orbital-motion-limited (OML) theory and on the proposed technique under conditions corresponding to the OML approximation. We examined aluminum oxide (Al2O3) particles in a neon (Ne) discharge under the following parameters:
These parameters were used to calculate the potentials excluding emission and including all types of emission (see Fig. 1). A comparison of the obtained data shows that the results of the calculation by the described technique are close to those described by the OML theory. It has been shown that the secondary, ion-electron emission and photoemission from the surface of the smooth particle reduce the absolute value of the potential of the dust particle surface in the whole considered electron temperature range. The effect of thermal field emission on the potential begins at / ≈ 300 and increases with an increase in the electron temperature.