Academic Training Lecture 2 : Beam Dynamics Equation of motion in transverse co-ordinates The lattice Meaning of Twiss parameters Twiss Matrix Effect of a drift length and a quadrupole Calculating the Twiss parameters FODO Cell Liouville’s theorem Phase stability Closed orbit of an ideal machine Dispersion Dispersion – from the “sine and cosine” trajectories Transition Synchrotron motion

The lattice

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

Meaning of Twiss parameters is either : Emittance of a beam anywhere in the ring Courant and Snyder invariant fro one particle anywhere in the ring

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 Matrix (continued) Writing The matrix elements are Above is the general case but to simplify we consider points which are separated by only one PERIOD and for which The “period” matrix is then If you have difficulty with the concept of a period just think of a single turn.

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

Effect of a drift length and a quadrupole

Calculating the Twiss parameters THEORY COMPUTATION (multiply elements) Real hard numbers Solve to get Twiss parameters:

FODO Cell Write down matrices from mid-F to mid-F Since they must equal the Twiss matrix:

Liouville’s theorem (Rough Version) “The area of a contour which encloses all the beam in phase space is conserved” This area = is the “emittance” It is the same all round the ring NOT TRUE: during acceleration in an electron machine where synchrotron emission damps

Phase stability E t

Bucket and pendulum   The “bucket” of synchrotron motion is just that of the rigid pendulum Linear motion at small amplitude Metastable fixed point at the top Continuous rotation outside

Closed orbit of an ideal machine In general particles executing betatron oscillations have a finite amplitude One particle will have zero amplitude and follows an orbit which closes on itself In an ideal machine this passes down the axis Closed orbit Zero betatron amplitude

Dispersion Low momentum particle is bent more It should spiral inwards but: There is a displaced (inwards) closed orbit Closer to axis in the D’s Extra (outward) force balances extra bends D(s) is the “dispersion function” Fig. cas 1.7-7.1C

Dispersion in the SPS This is the long straight section where dipoles are omitted to leave room for other equipment - RF - Injection - Extraction, etc The pattern of missing dipoles in this region indicated by “0” is chosen to control the Fourier harmonics and make D(s) small It doesn’t matter that it is big elsewhere

Dispersed beam cross sections These are real cross-section of beam The central and extreme momenta are shown There is of course a continuum between The vacuum chamber width must accommodate the full spread Half height and half width are:

Dispersion – from the “sine and cosine” trajectories The combination of diplacement, divergence and dispersion gives: Expressed as a matrix It can be shown that: Fulfils the particular solution of Hill’s eqn. when forced :

Transition - does an accelerated particle catch up - it has further to go Is a function of two, momentum dependent, terms  and R. and Using partial differentials to define a slip factor: This changes from negative to positive and is zero at ‘ transition’ when: GAMMA TRANSITION

Synchrotron motion Recall Elliptical trajectory for small amplitude Note that frequency is rate of change of phase From definition of the slip factor  Substitute and differentiate again But the extra acceleration is THUS

Synchrotron mtion (continued) This is a biased rigid pendulum For small amplitudes Synchrotron frequency Synchrotron “tune”

Summary of : Beam Dynamics Equation of motion in transverse co-ordinates The lattice Meaning of Twiss parameters Twiss Matrix Effect of a drift length and a quadrupole Calculating the Twiss parameters FODO Cell Liouville’s theorem Phase stability Closed orbit of an ideal machine Dispersion Dispersion – from the “sine and cosine” trajectories Transition Synchrotron motion