The Twiss Parameters and Particle Phase Advance From the analytic solu - PDF document

USPASThe Twiss Parameters and Particle
USPASThe Twiss Parameters and Particle Phase Advance From the analytic solution, we find the identi

fy the following important relations: d
fy the following important relations: dsss=0)(1exclusively by magnetic lattice, 21'21+=Š=

uation is for a single particle, if le,
uation is for a single particle, if le, i.e, highest emittance (u’maxmax The whole story is more

complicated.the beam line, the particle
complicated.the beam line, the particle executes oscillation on the ellipse.al space, u, is “quasi-

harmonic”.)(cos()(=ssuThe symmetric
harmonic”.)(cos()(=ssuThe symmetric and periodic FODO analytically determine the beta functions.

We enforce periodicity by ters be equal
We enforce periodicity by ters be equal to the outputTwiss periodicity we can derive:•The shape o

f the beta function.•The value of the be
f the beta function.•The value of the beta function through the FODO cell.•The optimum phase advance

of the FODO cell in order to minimizeL
of the FODO cell in order to minimizeLLQF HalfDriftQDDriftQF HalfFODOMMMMMM=Comparing the Twiss

representation ŠŠŠ+Š=22222FODO21)21(2)2
representation ŠŠŠ+Š=22222FODO21)21(2)21(221fLfLfLfLLfLMŠ+ŠŠ+=)sin(cossin1cossin)sin(cos

ooooooooM must be zero (symmetric m
ooooooooM must be zero (symmetric must be an extremum at ends. •There must be another extremum

in •In between the quads, m values in t
in •In between the quads, m values in the center of the FODOFODOxxWhat does the beam envelope in

the (x,y) plane look like?ture as small
the (x,y) plane look like?ture as small as possible, wewould ) to be a small as possible. So we 0=

+ ddResult:==+=3.766180.1251min Resul
+ ddResult:==+=3.766180.1251min Result for ==902 LR4min=yBeam Pipe(**Derivation**).02665.

12minmax==LL293.02707.12minmax==LL
12minmax==LL293.02707.12minmax==LLPeriodic Focusing ChannelsIn an accelerator, we often deal w

ith periodic, closed, lattices. These in
ith periodic, closed, lattices. These include rings and synchrotrons.period theoflength theis L

where),()(0)()("ppLsKsKsusKu+==+For a p
where),()(0)()("ppLsKsKsusKu+==+For a periodic, closed, system, we must have periodic solutions:)()

()()(ppLssLss+=+=Does the particl
()()(ppLssLss+=+=Does the particle return to the same phase positiu(s)=u(s+Lp)? Particle “T

une”in a Ring==)(2121sdstotal of
une”in a Ring==)(2121sdstotal of “phase space oscillations”a Though the Twiss parameters retu

rn to the same values on every turn, a a
rn to the same values on every turn, a advances”every turn, meaning it shows up on a different part

of the ul approximation for finding the
of the ul approximation for finding the average beta function from the tune, or conversely the tune

fromthe aveave122)(21sing”approx
fromthe aveave122)(21sing”approximation. It is useful aveRA closed, periodic system is

a speciainitial condition to calculate
a speciainitial condition to calculate the beta function, we must require the initialand end perio

dic boundary conditions.(ooo)Incoming
dic boundary conditions.(ooo)Incoming Distribution Beam Ellipse(Lattice EllipseMatch shape)()(

"2+ =+ususKuSo far we have considered on
"2+ =+ususKuSo far we have considered only the linear, homogeneous Hill’s equation (we set all highe

r order terms to zero). In reality, some
r order terms to zero). In reality, some lattices have a mismatched beam will dilutein 210122

02r'/R'a1012202r'/R'a1012
02r'/R'a1012202r'/R'a1012202r'/R'a1012202r'/R'a1012202r'/R'a101

2202r'/R'aReview of Transport Matr
2202r'/R'aReview of Transport MatricesMatrix for tracking (u,u’) through a lattice with K(s),

in terms of :Specific application to a
in terms of :Specific application to a periodic lattice:.Š+ŠŠ+=)sin(cossin1cossin)sin(cos

ooooooooMŠ=cossin1sincosFODO
ooooooooMŠ=cossin1sincosFODOMŠŠ+=sincossinsinsincosPeriodicMAnd for a FODO wit

h symmetry points at the end-points:rfe
h symmetry points at the end-points:rfect magnetic lattices. In reality there is no such thing. A m

ach•Magnet misalignments –offset and rol
ach•Magnet misalignments –offset and roll•Magnet strength errors•Magnet field imperfections•etc, etc

…There is a dBy Closed Orbit Distortio
…There is a dBy Closed Orbit Distortion due to Dipole ErrorThe new equilibrium trajectory (closed

orbit) of a particle in the presence of
orbit) of a particle in the presence of a dipole error at location soco(s)=(s)2sin(iicos((s)

+i)iuco(s)=(so)(s)2sin(And in the p
+i)iuco(s)=(so)(s)2sin(And in the presence of many This is called the “Closed Orbit Instabilit

y”, and occurs whenever the “Integer Res
y”, and occurs whenever the “Integer Resonance”Condition is met: r the closed orbit in thepresenc

e of 2sin(iicos((s)+i)iAs dipole
e of 2sin(iicos((s)+i)iAs dipole errors affect the referepresented as a matrix: =

14(s1)K(s1)ds1This gives us an easy
14(s1)K(s1)ds1This gives us an easy way to measure the all deviation k(s), then measure the per

iodicity gives the quad whenever we hav
iodicity gives the quad whenever we have a harmonic system, the danger of exciting a resonance ex“d

riving terms”exist in accelerators:•Col
riving terms”exist in accelerators:•Collective Effects•etc, etc..Resonance excitation between wind

gusts and natural frequency ofthe bridg
gusts and natural frequency ofthe bridgeA tune diagram is a convenient way toThis is a tune “work