Lecture 2 - Transverse motion ACCELERATOR PHYSICS Lecture 2 Transverse Motion TT 2012 E. J. N. Wilson

Need for Accelerators

Weak focusing in a synchrotron The Cosmotron magnet Vertical focusing comes from the curvature of the field lines when the field falls off with radius ( positive n-value) Horizontal focusing from the curvature of the path The negative field gradient defocuses horizontally and must not be so strong as to cancel the path curvature effect

Gutter

Transverse ellipse

Cyclotron Magnetic rigidity Constant revolution frequency

Vertical Focusing People just got on with the job of building them. Then one day someone was experimenting Figure shows the principle of vertical focusing in a cyclotron In fact the shims did not do what they had been expected to do Nevertheless the cyclotron began to accelerate much higher currents

Equation of motion in a cyclotron Non relativistic Cartesian Cylindrical

Cyclotron orbit equation For non-relativistic particles (m = m0) and with an axial field Bz = -B0 The solution is a closed circular trajectory which has radius and an angular frequency Take into account special relativity by And increase B with g to stay synchronous!

Cyclotron focusing – small deviations See earlier equation of motion If all particles have the same velocity: Change independent variable and substitute for small deviations Substitute To give

Cyclotron focusing – field gradient From previous slide Taylor expansion of field about orbit Define field index (focusing gradient) To give horizontal focusing

Cyclotron focusing – betatron oscillations From previous slide - horizontal focusing: Now Maxwell’s Determines hence In vertical plane Simple harmonic motion with a number of oscillations per turn: These are “betatron” frequencies Note vertical plane is unstable if

Components of a synchrotron

Transverse coordinates

Strong focussing

Fields and force in a quadrupole No field on the axis Field strongest here (hence is linear) Force restores Gradient Normalised: POWER OF LENS Defocuses in vertical plane Fig. cas 10.8

SPS

Equation of motion in transverse co-ordinates Hill’s equation (linear-periodic coefficients) where at quadrupoles like restoring constant in harmonic motion Solution (e.g. Horizontal plane) Condition Property of machine Property of the particle (beam) e Physical meaning (H or V planes) Envelope Maximum excursions

Twiss Matrix All such linear motion from points 1 to 2 can be described by a matrix like: To find elements first use notation We know Differentiate and remember Trace two rays one starts “cosine” The other starts with “sine” We just plug in the “c” and “s” expression for displacement an divergence at point 1 and the general solutions at point 2 on LHS Matrix then yields four simultaneous equations with unknowns : a b c d which can be solved

Twiss concluded Can be simplified if we define the “Twiss” parameters: Giving the matrix for a ring (or period)

Alternating gradients

The lattice

Beam sections after,pct

Physical meaning of Q and beta

Example of Beam Size Calculation Emittance at 10 GeV/c