Betatron Oscillations

Modern accelerators have lattices which are composed of successive regions of
constant values of K (which might include curvature). The matrix for each region is
one of the forms 4.3 to 4.5. Then we can find the one period matrix M(s) by matrix
multiplication, and find µ from the trace of that matrix, and α, β, γ by using 4.13. On
the other hand, to find α, β, γ at every point in the lattice by this method is tedious and
time consuming. We will develop better means to calculate these functions, but first we
need to learn more about them.
Let us attempt a solution to 4.1 in the phase-amplitude form
4.22
where w and ψ are real, and w is periodic. Then
4.23 y ′ + Ky = [ w ′ ± i2 ( w ′ ψ ′ + w ψ ′ )− w ( ψ ′) 2 + Kw]exp( iψ)= 0
The exponent is not zero, in general, so both the real and imaginary parts of the bracket
must vanish.
y = ws ()exp[±iψ( s)]
19 Jun 2007 Accelerators: Theory and
Applications
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