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In
accelerator physics Accelerator physics is a branch of applied physics, concerned with designing, building and operating particle accelerators. As such, it can be described as the study of motion, manipulation and observation of relativistic charged particle beam ...
, 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. 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 In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usual ...
diagram, an ellipse enclosing the particles can be given by the equation: :\gamma x^2 + 2 \alpha x x' + \beta x'^2 = \epsilon where x is the position axis and x' is the velocity axis. In this formulation, \alpha, \beta, and \gamma are the Courant–Snyder parameters for the beam along the given axis, and \epsilon is the
emittance Emittance may refer to: *Beam emittance In accelerator physics, emittance is a property of a charged particle beam. It refers to the area occupied by the beam in a position-and-momentum phase space. Each particle in a beam can be described by ...
. Three sets of parameters can be calculated for a beam, one for each
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
direction, x, y, and z.


History

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 Hartland Sweet Snyder (1913, Salt Lake City – 1962) was an American physicist who along with Robert Oppenheimer calculated the gravitational collapse of a pressure-free sphere of dust particles as described by Einstein's general relativity, and f ...
in their 1953 paper, "Theory of the Alternating-Gradient Synchrotron". 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.


Phase space area description

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. 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.


Coordinates

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. 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. This is typically given as a rate of change with respect to the longitudinal coordinate (x' = dx/dz) rather than with respect to time. 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. Given this assumption, it is possible to use the small angle approximation to express x' and y' as angles rather than simple ratios. As such, x' and y' are most commonly expressed in milliradians.


Ellipse equation

When an ellipse is drawn around the particle distribution in phase space, the equation for the ellipse is given as: :\gamma x^2 + 2 \alpha x x' + \beta x'^2 = area "Area" here is an area in phase space, and has units of length * angle. Some sources define the area as the beam emittance \epsilon, while others use \epsilon / \pi. It is also possible to define the area as a specific fraction of the particles in a beam with a 2 dimensional
gaussian distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu i ...
. The other three coefficients, \alpha, \beta, and \gamma, are the CS parameters. 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. Since there are only two independent variables, x and x', and the emittance is constant, only two of the CS parameters are independent. The relationship between the three parameters is given by: :\beta \gamma - \alpha^2 = 1


Derivation for periodic systems

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.


Equation of motion

In a
strong focusing In accelerator physics strong focusing or alternating-gradient focusing is the principle that, using sets of multiple electromagnets, it is possible to make a particle beam simultaneously converge in both directions perpendicular to the direction ...
accelerator, transverse focusing is primarily provided by
quadrupole A quadrupole or quadrapole is one of a sequence of configurations of things like electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure ref ...
magnets. The linear equation of motion for transverse motion parallel to an axis of the magnet is: :\frac = -k(z) x where k(z) is the ''focusing coefficient'', which has units of length−2, and is only nonzero in a quadrupole field. (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.) Assuming k(z) is periodic, for example, as in a circular accelerator, this is a differential equation with the same form as the
Hill differential equation In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation : \frac + f(t) y = 0, where f(t) is a periodic function by minimal period \pi . By these we mean that for all t :f(t+ ...
. The solution to this equation is a pseudo
harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...
: :x(z) = A(z) \cos(\phi(z) + \phi_o) where A(z) is the amplitude of oscillation, \phi(z) is the "betatron phase" which is dependent on the value of k(z), and \phi_o is the initial phase. The amplitude is decomposed into a position dependent part \beta and an initial value A_o, such that: :x(z) = A_o \sqrt \cos(\phi(z) + \phi_o) :x'(z) = A_o \frac \cos(\phi(z) + \phi_o) - A_o \frac \sin(\phi(z) + \phi_o) (It is important to remember that ' continues to indicated a derivative with respect to position along the direction of travel, not time.)


Particle distributions

Given these equations of motion, taking the average values for particles in a beam yields: :\langle x^2\rangle =A_o^2 \beta \langle \cos^2(\phi(z)+\phi_o)\rangle = \fracA_o^2\beta :\langle ^2 \rangle = A_o^2 \frac + A_o^2 \frac = (1 + (-\beta'/2)^2) \frac :\langle x x' \rangle = -\frac (-\beta'/2) These can be simplified with the following definitions: :\epsilon = \fracA_o^2 :\alpha = -\frac :\gamma = \frac giving: :\langle x^2\rangle = \epsilon \beta :\langle ^2 \rangle = \epsilon \gamma :\langle x x' \rangle = -\epsilon \alpha 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: :\epsilon^2 = \langle x^2 \rangle \langle x'^2 \rangle - \langle x x' \rangle^2


Properties

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. For example, if a particle distribution with parameters \alpha, \beta, and \gamma passes through an empty space of length L, the values \alpha(L), \beta(L), and \gamma(L) at the end of that space are given by: \begin \beta(L) \\ \alpha(L) \\ \gamma(L) \end = \begin 1 & -2L & L^2 \\ 0 & 1 & -L \\ 0 & 0 & 1 \end \begin \beta\\ \alpha \\ \gamma \end


See also

* Beam emittance * Beta function (accelerator physics) * Ray transfer matrix analysis


References

{{DEFAULTSORT:Courant-Snyder parameters Accelerator physics