Plasma model

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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

\begin{gather} \frac{d \vec{p}}{d \xi} = \frac{d \vec{p}}{d t} \frac{d t}{d \xi} = \frac{q}{v_z - 1} \left( \vec{E} + \left[ \vec{v} \times \vec{B} \right] \right),\\ \frac{d r}{d \xi} = \frac{v_r}{v_z - 1}, \qquad \vec{v} = \frac{\vec{p}}{\sqrt{M^2+p^2}}. \end{gather}

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:

\[\vec{j} = A \sum_i \frac{q_i \vec{v}_i}{1-v_{z,i}}, \qquad \rho = A \sum_i \frac{q_i}{1-v_{z,i}},\]

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

\begin{gather} rot\,\vec{B} = \vec{j} + \vec{j}_b + \frac{\partial \vec{E}}{\partial t}, \qquad rot\,\vec{E} = - \frac{\partial \vec{B}}{\partial t}, \\ div\,\vec{B} = 0, \qquad div\,\vec{E} = \rho + \rho_b. \end{gather}

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

\[\frac{\partial}{\partial z} = - \frac{\partial}{\partial t} = \frac{\partial}{\partial \xi},\]

3d geometry

Transverse fields

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

\[ \begin{align}\begin{aligned}\Delta_\perp E_x &= \frac{\partial \rho}{\partial x} - \frac{\partial j_x}{\partial \xi}\\\Delta_\perp E_y &= \frac{\partial \rho}{\partial y} - \frac{\partial j_y}{\partial \xi}\\\Delta_\perp B_x &= \frac{\partial j_y}{\partial \xi} - \frac{\partial j_z}{\partial y}\\\Delta_\perp B_y &= \frac{\partial j_z}{\partial x} - \frac{\partial j_x}{\partial \xi}\end{aligned}\end{align} \]

with mixed boundary conditions.

Longitudinal fields

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

\[\Delta_\perp B_z = \frac{\partial j_x}{\partial y} - \frac{\partial j_y}{\partial x}\]

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

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

\[\Delta_\perp E_z = \frac{\partial j_x}{\partial x} - \frac{\partial j_y}{\partial y}\]

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

Cylindrical geometry

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

\begin{gather} \frac{1}{r} \frac{\partial}{\partial r} r E_r = \rho + \rho_b - \frac{\partial E_z}{\partial \xi}, \\ \frac{1}{r} \frac{\partial}{\partial r} r B_r = - \frac{\partial B_z}{\partial \xi},\\ \frac{1}{r} \frac{\partial}{\partial r} r (E_r - B_\varphi) = \rho - j_z, \\ \frac{\partial E_z}{\partial r} = j_r, \\ \frac{\partial B_z}{\partial r} = - j_\varphi, \\ E_\varphi = -B_r. \end{gather}

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 \(B_r\) :

\begin{gather} \frac{\partial}{\partial r} \frac{1}{r} \frac{\partial}{\partial r} r E_r - E_r = \frac{\partial (\rho + \rho_b)}{\partial r} - \frac{\partial j_r}{\partial \xi} - \tilde{E}_r, \\ \frac{\partial}{\partial r} \frac{1}{r} \frac{\partial}{\partial r} r B_r - B_r = \frac{\partial j_\varphi}{\partial \xi} - \tilde{B}_r, \end{gather}

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 \(r_\text{max}\):

\begin{gather} E_r (0) = B_r (0) = B_\varphi (0) = E_z (r_\text{max}) = B_r (r_\text{max}) = 0,\\ \int_0^{r_\text{max}} 2 \pi r B_z \, dr = \pi r_\text{max}^2 B_0, \end{gather}

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).