3 Non-linear Dynamics in Accelerators
61
The average of cos 2 can immediately be evaluated as 0.5 and defining the emittance
as:
x = < J x >,
(3.17)
we write
< x
2 > = β x · x .
(3.18)
Using a similar procedure (details and derivation in e.g. [3], and to a much lesser
extent in [1]) one can determine the moments
< p
2
x > = γ x · x ,
(3.19)
and
< x · p x > = − α x · x .
(3.20)
Using these expressions, the emittance becomes readily
x =
< x 2 >< p 2
x > − < x · p x > 2
(3.21)
Therefore, once the emittance is measured, the Courant-Snyder parameters are
determined by Eqs. (3.18), (3.19), and (3.20).
Since other definitions often refer to the treatment by Courant and Snyder, here
a quote from Courant himself in [9]:
Interlude 1
The invariant J is simply related to the area enclosed by the ellipse:
Area enclosed = 2πJ.
(3.22)
In accelerator and storage ring terminology there is a quantity called the emittance
which is closely related to this invariant. The emittance, however, is a property of
a distribution of particles, not a single particle. Consider a Gaussian distribution in
amplitudes. Then the (rms) emittance, , is given by:
(x rms )
2 = β x (s) · x .
(3.23)
In terms of the action variable, J , this can be rewritten
x = < J > .
(3.24)
where the bracket indicates an average over the distribution in J .