Accelerator Physics Notes

]> Accelerator Physics Notes

Accelerator Physics Notes

Magnet properties

are explored in this Matlab script

Average ring beta

β_{avg} \approx \frac{R}{ν}

Average beta is about = ring radius over tune.

Dispersion relations

\frac{Δ p}{p_{o}} = - \frac{1}{α_{c}} \frac{Δ f_{rf}}{f_{rf}}

δx = η (\frac{Δ p}{p_{o}}) = - (\frac{Δ f_{rf}}{f_{rf}})

relation between change in orbit and change in rf frequency. Also, ΔE/E = β² (Δp/p), so above relations can be used to find the energy change.

Slip Factor (The other eta)

η (\frac{Δ p}{p}) = \frac{Δ L}{L} - \frac{Δ v}{v}

η \equiv \frac{1}{{γ_{t}}^{2}} - \frac{1}{γ^{2}}

The slip factor times the fractional change in momentum gives the fractional change in time it will take for that particle to reach the next rf cavity. There is a competition between the path length growing with momentum and velocity growing with momentum. Whether the slip factor is positive or negative matters a great deal in which slope of the rf wave should be used for phase stability. When the slip factor equals zero, the beam is at its transition energy.

Horizontal source size:

σ_{x} = \sqrt{ε_{h} β + η^{2} δ^{2}}

where δ is σE over E.

Bump kick amplitude relation to phase in 3 kick closed bump:

Θ_{1} \sqrt{β_{1}} : Θ_{2} \sqrt{β_{2}} : Θ_{3} \sqrt{β_{3}} = \sin Ψ_{32} : - \sin Ψ_{31} : \sin Ψ_{21}

Measuring beta at a quad:

δ ν = \frac{β}{4 π} Δ k

β = \frac{4 π Δ ν}{Δ k}

Sands Curly H: (Sands, p.131)

H (s) = \frac{1}{β} (η^{2} + {(β η' + α η)}^{2})

Sand's curly H is used to give the change in the "invariant" amplitude due to the emission of a photon of quantum energy. In other words, a high curly H means a higher contribution to emittance growth due to radiation.Here's curly H plotted for a chess lattice.

Twiss Parameters:

β = \frac{1}{ψ'}

α = \frac{- β'}{2}

γ = \frac{1 + α^{2}}{β}

Hamilton's Principle:

δ \int_{t 1}^{t 2} (T - U) d t = 0

where T is the kinetic energy and U is the potential energy.

Of all the possible paths along which a dynamical system may move from one point to another within a specified time interval (consistent with any constraints), the actual path followed is that which minimizes the time integral of the difference between the kinetic and potential energies.

Lagrangian:

L (q, \dot{q}, t) = T (q, \dot{q}, t) - U (q, t)

The lagrangian is a function of the generalized coordinates q, and their derivitives with respect to time, as well as time.

Hamiltonian:

H = \sum_{j}^{} ({q_{}^{.}}_{j} \frac{δ L}{δ {q_{}^{.}}_{j}}) - L

Where the q's are the generalized coordinates.

Symplectic relationship:

is explored in this Matlab script

Some formulas on this page were written in latex, then converted to MathML using mzlatex. A little further cleanup was also required.

Now Open Math Edit is used to generate the MathML on windows and then copied and pasted into this xml file.