22 Beam-based Correction and Optimization for Accelerators
s = 0
2L
β x
β y
Figure 1.9 Beta functions in a FODO cell.
moments are the center of the distribution defined as
y =
yρ(y, y
)dydy
,
y
=
y
ρ(y, y
)dydy
.
(1.79)
We assume a distribution centered on the reference orbit, i.e., with y =
y
= 0. The second-order moments are given by
Σ ≡
σ
2
y
σ yy
σ yy σ
2
y
=
ρ(y, y
)dydy
y
2
yy
yy
y
2
.
(1.80)
The second order moment matrix can be parametrized as follows,
σ
2
y
σ yy
σ yy σ
2
y
= rms
¯
β −¯ α
−¯ α ¯
γ
,
(1.81)
where the rms emittance is defined as
rms ≡
det(Σ),
(1.82)
and hence ¯
β¯ γ = 1 + ¯
α
2 is required. The second order moment matrix defines
an ellipse in the phase space, Y
T Σ
−1 Y = 1, with Y = (y, y
)
T , which can be
written as
y
2 + ( ¯
αy + ¯
βy
)
2 = ¯
ββ rms .
(1.83)
The area of the ellipse is the rms emittance. The ¯
β, ¯
α, and ¯
γ parameters
represent the shape and orientation of the ellipse in the same manner as
β, α, and γ for the phase space ellipse in Figure 1.6. If the beam distribution is Gaussian, the probability density function is given by ρ(y, y
) =
1
2ππrms exp(−
1
2 Y
T Σ
−1 Y). In this case, an enlarged ellipse with an area of
6σ rms will cover 95% of the particles in the beam [78].