Courant–Snyder parameters
In accelerator physics, the Courant–Snyder parameters (frequently referred to as Twiss parameters or CS parameters) are a set of quantities used to describe the distribution of positions and velocities of the particles in a beam.
[1] When the positions along a single dimension and velocities (or momenta) along that dimension of every particle in a beam are plotted on a phase space diagram, an ellipse enclosing the particles can be given by the equation: where
Three sets of parameters can be calculated for a beam, one for each orthogonal direction, x, y, and z.
[2] The use of these parameters to describe the phase space properties of particle beams was popularized in the accelerator physics community by Ernest Courant and Hartland Snyder in their 1953 paper, "Theory of the Alternating-Gradient Synchrotron".
[3] They are also widely referred to in accelerator physics literature as "Twiss parameters" after British astronomer Richard Q. Twiss, although it is unclear how his name became associated with the formulation.
[4] When simulating the motion of particles through an accelerator or beam transport line, it is often desirable to describe the overall properties of an ensemble of particles, rather than track the motion of each particle individually.
[2]: 155 By Liouville's Theorem it can be shown that the density occupied on a position and momentum phase space plot is constant when the beam is only affected by conservative forces.
The area occupied by the beam on this plot is known as the beam emittance, although there are a number of competing definitions for the exact mathematical definition of this property.
[5] In accelerator physics, coordinate positions are usually defined with respect to an idealized reference particle, which follows the ideal design trajectory for the accelerator.
The direction aligned with this trajectory is designated "z", (sometimes "s") and is also referred to as the longitudinal coordinate.
Two transverse coordinate axes, x and y, are defined perpendicular to the z axis and to each other.
[6] In addition to describing the positions of each particle relative to the reference particle along the x, y, and z axes, it is also necessary to consider the rate of change of each of these values.
In most cases, x' and y' are both much less than 1, as particles will be moving along the beam path much faster than transverse to it.
When an ellipse is drawn around the particle distribution in phase space, the equation for the ellipse is given as: "Area" here is an area in phase space, and has units of length * angle.
Some sources define the area as the beam emittance
It is also possible to define the area as a specific fraction of the particles in a beam with a 2 dimensional gaussian distribution.
As this ellipse is an instantaneous plot of the positions and velocities of the particles at one point in the accelerator, these values will vary with time.
The relationship between the three parameters is given by:[2]: 160 In addition to treating the CS parameters as an empirical description of a collection of particles in phase space, it is possible to derive them based on the equations of motion of particles in electromagnetic fields.
is the focusing coefficient, which has units of length−2, and is only nonzero in a quadrupole field.
[6] (Note that x is used throughout this explanation, but y could be equivalently used with a change of sign for k. The longitudinal coordinate, z, requires a somewhat different derivation.)
is periodic, for example, as in a circular accelerator, this is a differential equation with the same form as the Hill differential equation.
is the "betatron phase" which is dependent on the value of
The amplitude is decomposed into a position dependent part
, such that:[6] (It is important to remember that ' continues to indicated a derivative with respect to position along the direction of travel, not time.)
Given these equations of motion, taking the average values for particles in a beam yields:[2]: 163 These can be simplified with the following definitions: giving: These are the CS parameters and emittance in another form.
Combined with the relationship between the parameters, this also leads to a definition of emittance for an arbitrary (not necessarily Gaussian) particle distribution:[2]: 163 The advantage of describing a particle distribution parametrically using the CS parameters is that the evolution of the overall distribution can be calculated using matrix optics more easily than tracking each individual particle and then combining the locations at multiple points along the accelerator path.
passes through an empty space of length L, the values
