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

1 USPAS The Twiss Parameters and Particle
USPAS The Twiss Parameters and Particle Phase Advance From the analytic solution, we find the identi

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

3 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

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

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

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

7 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

8 of the FODO cell in order to minimize L
of the FODO cell in order to minimize L LQF HalfDriftQDDriftQF HalfFODOMMMMMM= Comparing the Twiss

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

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

11 in •In between the quads, m values in t
in •In between the quads, m values in the center of the FODOFODO xxWhat does the beam envelope in

12 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=

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

14 12minmax==LL 293.02707.12minmax==LL
12minmax==LL 293.02707.12minmax==LL Periodic Focusing ChannelsIn an accelerator, we often deal w

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

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

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

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

19 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

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

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

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

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

24 "2+ =+ususKuSo far we have considered on
"2+ =+ususKuSo far we have considered only the linear, homogeneous Hill’s equation (we set all highe

25 r order terms to zero). In reality, some
r order terms to zero). In reality, some lattices have a mismatched beam will dilutein 2 1 0 1 2 2

26 0 2 r'/R'a 1 0 1 2 2 0 2 r'/R'a 1 0 1 2
0 2 r'/R'a 1 0 1 2 2 0 2 r'/R'a 1 0 1 2 2 0 2 r'/R'a 1 0 1 2 2 0 2 r'/R'a 1 0 1 2 2 0 2 r'/R'a 1 0 1

27 2 2 0 2 r'/R'a Review of Transport Matr
2 2 0 2 r'/R'a Review of Transport MatricesMatrix for tracking (u,u’) through a lattice with K(s),

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

29 ooooooooM Š=cossin1sincosFODO
ooooooooM Š=cossin1sincosFODOMŠŠ+=sincossinsinsincosPeriodicMAnd for a FODO wit

30 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

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

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

33 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( i icos((s)

34 +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

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

36 e of 2sin( i icos((s)+i)i As dipole
e of 2sin( i icos((s)+i)i As dipole errors affect the referepresented as a matrix:  =

37 14  (s1)K(s1)ds1 This gives us an easy
14  (s1)K(s1)ds1 This gives us an easy way to measure the all deviation k(s), then measure the per

38 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

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

40 gusts and natural frequency ofthe bridg
gusts and natural frequency ofthe bridge A tune diagram is a convenient way to This is a tune “work