# Plasma model

Warning

To be updated.

## Plasma particles

### 3d geometry

Warning

To be updated.

### Cylindrical geometry

Each plasma macro-particle is characterized by 7 quantities: transverse coordinate - \(r\), three components of the momentum - (\(p_r, p_\varphi, p_z\)), mass - \(M\), charge - \(q\), and ordinal number. Parameters of plasma macro-particles are initialized ahead of the beam (at \(\xi=0\)) and then advanced slice-by-slice according to equations

If a particle hits the wall at \(r=r_\text{max}\), it is returned to the simulation area to some near-wall location with zero momentum. Plasma current and charge density are obtained by summation over plasma macro-particles lying within a given radial interval:

where \(A\) is a normalization factor. The denominator in deposition appears since the contribution of a ‘’particle tube’’ to density and current depends on the macro-particle speed in the simulation window.

## Fields

The equations solved for the fields are Maxwell equations, which in the dimensionless variables take the form

Under the quasi-static assumption, the derivatives change as follows:

### 3d geometry

#### Transverse fields

In QSA, transverse Maxwell’s equations can be rewritten as:

with mixed boundary conditions.

#### Longitudinal fields

In QSA, equation for \(B_z\) can be rewritten as:

with Neumann boundary conditions (\(\partial B_z/ \partial r_\perp = 0\)).

In QSA, equation for \(E_z\) can be rewritten as:

with Dirichlet boundary conditions (\(E_z = 0\)).

### Cylindrical geometry

In cylindrical geometry Maxwell’s equations take the following form:

Here we neglect the components \(j_{br}`\) and \(j_{b\varphi}\) of the beam current and put \(j_{bz} = \rho_b\), since beam particles are assumed to move mostly in \(z\)-direction. To provide stability of the algorithm, we solve in finite differencesd the following equations on \(E_r\) and \(B_r\) :

where \(\tilde{E}_r\) and \(\tilde{B}_r\) are some predictions for fields \(E_r\) and \(B_r\). Subtraction ofthe fields (with or without the tildes) from both sides of the equalities does not produce a big error if the predictions are close to final fields. The boundary conditions for equations are those of a perfectly conducting tube of the radius \(r_\text{max}\):

where \(B_0\) is an external longitudinal magnetic field, if any (the presence of this field does not change the axial symmetry of the system).