An Introduction to the Ion-Optics of Magnet Spectrometers

The 2-dimensional 2
case ( x, Θ )
Ellipse Area = π(det σ) 1/2
Emittance ε = (det σ) 1/2 is constant
for fixed energy & conservative
forces (Liouville’s Theorem)
Note: ε shrinks (increases) with
acceleration (deceleration);
Dissipative forces: ε increases in
gases; electron, stochastic, laser
cooling
σ = ⎛σ 11 σ 21 ⎫
⎝σ 21 σ 22 ⎭
σ −1 = 1/ε 2 ⎛σ 22 −σ 21 ⎫
⎝−σ 21 σ 11 ⎭
Exercise 1:
Show that:
2 dimensional cut x-Θ is shown
Real, pos. definite
symmetric σ Matrix
σσ −1 = ⎛1 0 ⎫
⎝ 0 1⎭
Inverse Matrix
= I (Unity Matrix)
2-dim. Coord. vectors
(point in phase space)
X =
⎛x ⎫
X T = (x Θ)
⎝Θ⎭
Ellipse in Matrix notation:
X T σ −1 X = 1
(6)
Exercise 2: Show that Matrix notation
is equivalent to known Ellipse equation:
σ 22 x 2 −2σ 21 x Θ + σ 11 Θ 2 = ε 2