The Particle Beam feature can be used to release nonlaminar particle beams with specified emittance values. The classification of a particle beam as laminar or nonlaminar depends on the distribution of transverse position and velocity within the beam. Further discussion of laminar and nonlaminar beams can be found in
Ref. 10.
If at least one Particle Beam node is present, additional global variables are defined for certain beam properties. Quantities that indicate the distribution of transverse beam position and velocity are defined in a coordinate system that is centered at the average position of beam particles,
qav. In 2D, the transverse direction is orthogonal to the average particle velocity
vav; in 3D, the two transverse directions are orthogonal to
vav and to each other.
Let the transverse displacement from the beam center be denoted x, and the transverse velocity be expressed using the dimensionless variable
, which is the ratio of transverse velocity to axial velocity. For 3D, the following discussion can be extended to consider two distinct transverse displacement components in orthogonal directions, called for example
x1 and
x2.
Where the brackets represent an arithmetic mean over all particles. In addition, the 4-rms emittance ε4,rms is frequently reported because it corresponds to the area of an ellipse if the distribution of particles in phase space is uniform:
The Particle Beam node includes built-in options for releasing distributions of particles in velocity space by sampling from uniform or Gaussian distributions in each transverse velocity direction. These distributions can be upright if the initial value of the Twiss parameter
α is set to
0. In this case, for the elliptical distributions of particles in phase space, the semimajor and semiminor axes of the ellipse are initially parallel to the
x- and
-axes. The initial values of the Twiss parameters must fulfill the Courant–Snyder condition,
Given the initial value of the Twiss parameter β, the 1-rms beam emittance
ε and the Twiss parameter
α, the initial distribution of particles in phase space depends on the option selected from the
Sampling from phase space ellipse list.
where a and
b are the semi-major and semi-minor axis of the ellipse in physical space,
and
are the envelope angles which are related to the Twiss parameters according to:
The amplitude of A2 can be resolved into two components:
As described in Ref. 11, start by defining two uniformly-distributed random numbers
and
, then define:
with the definition of A depends on the type of distribution and follows later. Now define two additional uniformly-distributed random numbers
and
then the relative initial positions are the particles are given by:
The definition of A changes depending on the requested distribution:
where rc is the beam centroid,
t1 and
t2 are the two tangent vectors on the surface,
n is the surface normal, and
V is the velocity magnitude.
When the Beam symmetry is
Symmetric, the same initial values of the Twiss parameter
β, Twiss parameter
α and beam emittance
ε are applied to each of the two transverse directions. When the
Beam symmetry is
Asymmetric, distinct values of these parameters can be assigned to each transverse direction.
When the Particle release specification property is
Specify current, the
Beam symmetry is
Symmetric, the
Transverse velocity distribution specification is
Specify emittance and Twiss parameters and the
Emittance specification is
Specify brightness, the
Brightness is entered instead of the emittance. In this case, the 1
−rms emittance is computed from the
Brightness B (SI unit: A/m
2) and the
Release current magnitude I (SI unit: A) using
When the Transverse velocity distribution specification is set to
Specify phase space ellipse dimensions, the Twiss parameters are computed from the
Maximum transverse displacement,
xm, the
Maximum relative transverse velocity,
x'm and the
Rotation angle,
θ using the following
The Twiss parameter α is computed by first defining a rotation matrix