Penning Traps

A Penning trap confines charged particles with a homogeneous axial magnetic field and a quadrupole electric field. Three variants are provided, differing in the shape of the magnetic field: uniform, a magnetic bottle, and an asymmetric configuration.

All three carry an electrostatic potential in addition to the magnetic field, so the generated code includes φ and the components of the electric field.

Uniform Magnetic Field

ElectromagneticFields.PenningTrapUniform — Module

Penning trap with uniform magnetic field in (x,y,z) coordinates. Based on Yanyan Shi, Yajuan Sun, Yulei Wang, Jian Liu, Study of adaptive symplectic methods for simulating charged particle dynamics, Journal of Computational Dynamics 6, 429-448, 2019.

The covariant components of the vector potential are given by

\[A (x,y,z) = B_0 \, ( 0, x, 0)^T ,\]

resulting in the magnetic field with covariant components

\[B (x,y,z) = B_0 \, ( 0, 0, 1)^T ,\]

and the electrostatic potential given by

\[\varphi (x,y,z) = - E_0 \, ( x^2 / 2 + y^2 / 2 - z^2) ,\]

resulting in the electric field with covariant components

\[E (x,y,z) = E_0 \, ( x, y, - 2 z)^T .\]

Parameters:

B₀: B-field strength
E₀: E-field strength

source

using ElectromagneticFields

equ = PenningTrapUniform.init()

Penning trap with uniform magnetic field in (x,y,z) coordinates with
  B₀ = 100.0
  E₀ = 10.0

PenningTrapUniform.@code()

The magnetic field is homogeneous along $z$, while the electric field is linear in all three coordinates and pulls the particle back towards the mid-plane:

t = 0.0
x = [0.1, 0.2, 0.3]

B(t, x), [E₁(t, x), E₂(t, x), E₃(t, x)]

(100.0, [1.0, 2.0, -6.0])

Magnetic Bottle

ElectromagneticFields.PenningTrapBottle — Module

Penning trap with magnetic bottle in (x,y,z) coordinates. Based on Yanyan Shi, Yajuan Sun, Yulei Wang, Jian Liu, Study of adaptive symplectic methods for simulating charged particle dynamics, Journal of Computational Dynamics 6, 429-448, 2019.

The covariant components of the vector potential are given by

\[A (x,y,z) = B_0 / 2 \, ( -y , x, 0)^T - B_1 \, (xz, yz, (x^2 + y^2)/2 - z^2)^T,\]

resulting in the magnetic field with covariant components

\[B (x,y,z) = B_0 \, ( 0, 0, 1)^T - B_1 \, ( yz^2 - y^3 / 6, x^3 / 6, xyz )^T ,\]

and the electrostatic potential given by

\[\varphi (x,y,z) = - E_0 \, ( x^2 / 2 + y^2 / 2 - z^2) ,\]

resulting in the electric field with covariant components

\[E (x,y,z) = E_0 \, ( x, y, - 2 z)^T .\]

Parameters:

B₀: B-field strength
Bₚ: B-field perturbation strength
E₀: E-field strength

source

PenningTrapBottle.init()

Penning trap with magnetic bottle in (x,y,z) coordinates with
  B₀ = 100.0
  Bₚ = 200.0
  E₀ = 10.0

Asymmetric Magnetic Field

ElectromagneticFields.PenningTrapAsymmetric — Module

Penning trap with asymmetric magnetic field in (x,y,z) coordinates. Based on Yanyan Shi, Yajuan Sun, Yulei Wang, Jian Liu, Study of adaptive symplectic methods for simulating charged particle dynamics, Journal of Computational Dynamics 6, 429-448, 2019.

The covariant components of the vector potential are given by

\[A (x,y,z) = B_0 / 2 \, ( -y , x-z/6, y / 6)^T + B_1 / 2 \, (z^2 - y^2, z^2 - x^2, y^2 - x^2)^T,\]

resulting in the magnetic field with covariant components

\[B (x,y,z) = B_0 \, ( 1/3, 0, 1)^T + B_1 \, ( y-z, x+z, y-x )^T ,\]

and the electrostatic potential given by

\[\varphi (x,y,z) = - E_0 \, ( x^2 / 2 + y^2 / 2 - z^2) ,\]

resulting in the electric field with covariant components

\[E (x,y,z) = E_0 \, ( x, y, - 2 z)^T .\]

Parameters:

B₀: B-field strength
Bₚ: B-field perturbation strength
E₀: E-field strength

source

PenningTrapAsymmetric.init()

Penning trap with asymmetric magnetic field in (x,y,z) coordinates with
  B₀ = 100.0
  Bₚ = 50.0
  E₀ = 10.0

These fields have no dedicated plotting routine. They are simple enough to sample and plot directly — see Plotting by Hand for how to do that.