Courant–Snyder parameters

In accelerator physics, the Courant Snyder parameters (frequently referred to as Twiss 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 diagram, an ellipse enclosing the particles can be given by the equation:

\gamma x^{2}+2\alpha xx'+\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 (CS) parameters for the beam along the given axis, and $\epsilon$ is the emittance. Three sets of parameters can be calculated for a beam, one for each orthogonal direction, x, y, and z.^[1]

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 in their 1953 paper, "Theory of the Alternating-Gradient Synchrotron".^[2] 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.^[3]

Derivation

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", 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.^[4]

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.^[1]^: 30 Given this assumption, is 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.

Equation of motion

In a strong focusing accelerator, transverse focusing is primarily provided by quadrupole magnets. The linear equation of motion for transverse motion parallel to an axis of the magnet is:

{\frac {d^{2}x}{dz^{2}}}=-k(z)x

where k(z) is the focusing coefficient, which has units of length^-2, and is only nonzero in a quadrupole field.^[4]

Properties

References

^ ^a ^b Wiedemann, Helmut (2007). Particle accelerator physics (3rd ed.). Berlin: Springer. pp. 158–161. ISBN 978-3-540-49043-2.
^ Courant, E.D.; Snyder, H.S. (April 2000). "Theory of the Alternating-Gradient Synchrotron". Annals of Physics. 3 (1–2): 1–48. doi:10.1006/aphy.2000.6012. Retrieved 12 January 2022.
^ Ruth, Ronald D. (August 2002). "An Introduction to Particle Accelerators An Introduction to Particle Accelerators , E. J. N. Wilson Oxford U. Press, New York, 2001. $90.00, $45.00 paper (252 pp.). ISBN 0-19-852054-9, ISBN 0-19-850829-8 paper". Physics Today. 55 (8): 52–52. doi:10.1063/1.1510283. Retrieved 12 January 2022.
^ ^a ^b Minty, Michiko G.; Zimmerman, Frank (2003). Measurement and Control of Charged Particle Beams. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 6–12. ISBN 3-540-44187-5.

[Wiedemann-1] Wiedemann, Helmut (2007). Particle accelerator physics (3rd ed.). Berlin: Springer. pp. 158–161. ISBN 978-3-540-49043-2.

[courantsnyder-2] Courant, E.D.; Snyder, H.S. (April 2000). "Theory of the Alternating-Gradient Synchrotron". Annals of Physics. 3 (1–2): 1–48. doi:10.1006/aphy.2000.6012. Retrieved 12 January 2022.

[3] Ruth, Ronald D. (August 2002). "An Introduction to Particle Accelerators An Introduction to Particle Accelerators , E. J. N. Wilson Oxford U. Press, New York, 2001. $90.00, $45.00 paper (252 pp.). ISBN 0-19-852054-9, ISBN 0-19-850829-8 paper". Physics Today. 55 (8): 52–52. doi:10.1063/1.1510283. Retrieved 12 January 2022.

[minty-4] Minty, Michiko G.; Zimmerman, Frank (2003). Measurement and Control of Charged Particle Beams. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 6–12. ISBN 3-540-44187-5.

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