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Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/1
Transverse beam motion
Transverse Beam Motion:• bending & focusing
• emittance
• FODO cell & tune
• equation of motion & closed orbit
• Twiss (transport) matrices
• lattice & stability requirements
• Liouville’s theorem
• adiabatic damping
• beam distributions & control
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/2
Transverse beam motion
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/3
Bending magnet
Coordinate system
displacement &divergence fromdesign (= idealorbit) in horizontal(vertical) plane:x (y) and x’ =dx/ds (y’ = dy/ds)
(E.Wilson)
N-pole
S-pole
Bg
gx
y
Bendmagnet
Designorbit
F
vs
B
s
dipole(bending)magnet
a particle, withdesign beammomentum, isdescribed by acircular path witha equilibriumbetween thecentripetal ¢rifugal forces.
Beam
constant force inx and 0 force in y
or y
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/4
Bending
magnetic rigidity:
(E.Wilson)
(E.Wilson)
A particle bending in a dipole:
dtdspdtdp
dtpd
dtdseB
Bve
/
||epB
dtpdBve
/
/
]GeV[3356.3]Tm[ pB
BBl
BlB
/
2/
2/)2/sin(
8/8/
))2/cos(1(2 l
s
sagitta:
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/5
Focusing
)/()/1( dzdBBk z )/()/1( dzdBBk z
lkxdzdBBlxx z )/(/'
Principal focusing elements - quadrupoles
Field 0 on axis & rises linearly with distance to axis.Quadrupole focusing in one plane & defocusing inother plane (e.g. in Fig. x-focusing & z-defocusing).
Quadrupole characterized by its gradient dBz /dx:
Angular deflection (of particle passing quadrupoleof length l):
Comparison with a converging lens in optics ( x’= x / f ):
(focal length of a quadrupole)
(E.Wilson)
k 0 (k > 0) focusing (defocusing) in x-plane.
(E.Wilson)
klf /1
)/()/1( dzdBBk z
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/6
Focusing
At bottom of gutter divergence maximal whereasdisplacement minimal. At maximum displacementvice versa. Below displacement-divergence picture:
(E.Wilson)
(E.Wilson)
Area of ellipse(”phase space”) :
(mm rad)
: emittance
x = (s)
x’ = / (s)
Analogy for infinite long quadrupole is a ”gutter”:
NB! property ofaccelerator (gutter),
of particle beam.
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/7
Alternating-gradient focusing
to achieve focusingsufficient that particlestend to be closer to axisin D lenses than in Flenses (optimalspacing 2 focal lengths)
(E.Wilson)Alternating-gradient focusing
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/8
Tune & FODO cell
Betatronenvelopes ofa FODO cell
(E.Wilson)
(s): beta-function
(s): phaseadvance
basic focusing structure: a FODO cell = 1 focusing +1 defocusing quadrupole + arbritrary # of dipoles
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/9
The equation of motion
Angular deflection of particle passing a short focusingquadrupole (length ds; strength k) at displacement z:
dzdB
Bkdszkdz x1where,'
equivalent to 2nd order linear equation with a periodiccoefficient k(s), ”Hill’s equation”:
0)()(
1'' 2 xsks
xFor vertical plane, for horizontal:
0)('' zskz
NB! change of sign for k(s) between focusing & defocusing
(defocusing invertical plane)
Solution: (assume k(s) periodic for 1 turn around ring)simple harmonic motion with restoring constant k(s)(varying with s).
0)(cos)( ssxNote: amplitude component (s) depends on positions along accelerator and also phase (s) doesn’tadvance linear neither with time nor distance.However both these functions must have sameperiodicity as the lattice and are linked by condition:
/or,1' ds
00 )(cos)(2
)(')(sin)(
' ss
sss
x
Differentiation gives:
’(s) = 0 ellipse with area (semi-axis in x-direction & / in x’-direction. an invariant of motion.
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/10
The equation of motion
In older constant-gradient accelerators (containingonly dipoles) simple harmonic a good approximation :
equation)(wave0202
2
2
2
2
zds
zdzkds
zd
solution:12'sin2sin 00 zszz
identify as local wavelength of betatron oscillation.
Define tune Q as # of turns around phase space ellipseduring a full turn around accelerator. In constant-gradient machine = 2 R / after one full turn.
QRorRQ ,2Approximately true for alternating-gradient machinesas well. Note: Q determines & hence beam size.
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/11
Closed orbitZero betatronamplitude
x
x
Closed orbit
• In general particles executing betatron oscillationshave a finite amplitude• One particle will have zero amplitude & follow anorbit which closes on itself = closed orbit• In an ideal machine this passes down the axis; inreality might need to displace the beam (usuallyhorizontally) not to interact with beam coming in
opposite direction (in aone ring collider likeTevatron) or introducecrossing angles at IP’s (in2 ring colliders like LHCusing kicker magnets.
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/12
Twiss Matrix
Influence of each accelerator element can be givenas a matrix particle’s displacement & divergencechange by an acclerator section: a single matrix M.
Each term in M must be a function of (s) & (s)For simplicity introduce a new quantity
thenwe differentiate & remember
let’s trace 2 particles one starts
the other starts &Put displacement & divergence expressions for point1 & general solution for point 2. Get 4 equations with4 unknowns (a, b, c, d ). The ”general” solution: M =
)(')(
M)(')(
)(')(
1
1
1
1
2
2
sysy
sysy
dcba
sysy
w
2/1/1' w)cos( 0wy
)sin()cos('' 00 wwy
"cosine"00
"sine"2/0 12
sin'coscos''sin''1
sinsin'cos
212
1
1
2
2
1
21
2121
21121
2
wwww
ww
ww
wwwwww
wwwwww
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/13
Twiss Matrix (continued)
Looks more complicated but can now apply constraintfor 2 points seperated by 1 period (either turn/cell)
The ”periodic” solution:
w 2 , = 12
, = 1 + 2
The simplified ”periodic” solution:
NB! , , & functions of s, position along accelerator.
Stability of an alternating-gradient focusing accelerator:[M(s)]Nk shouldn’t diverge (N periods per turns & k turns)
0I)det(M'
YwhereY,MYyy
Qwwwwww 2,''', 122121
sin'cossin'1sinsin'cos
M2
22
2
www
wwwww
sincossinsinsincos
M
to obtain eigenvalues need to solve this
fractional part of Q
=
To simply even more, define new functions:
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/14
Stability & calculating twiss parameters
THEORY COMPUTATION(multiply elements)
Real hard numbersSolve to get Twiss parameters:
Values of , , & are local & apply to the pointchoosen as starting (& finishing) point. By choosingdifferent starting points, s-dependence can be traced.
dcba
sincossinsinsincos
M
iei sincos
da21MTr
21cos
Determinant equation gives:
given that
For stability must be real
01)(2 da
1&1cos
Calculation of twiss parameters:
sin/sin2/)(0sin/
2/)(MTrcos 21
cda
bda
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/15
The lattice
Pattern of bending & focusing magnets, called lattice,has strong influence on design & aperture of magnets& transverse focusing system, in turn, has importanteffects on almost all other systems in an accelerator.
Particlesmakebetatronoscillations
h(v)
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/16
Transport matrices
Drift length
Quadrupole ( l << f )
optical analogy x-l plane x-x’ plane
(E.Wilson)
'tan 1 x '101
' 1
1
2
2
xxl
xx
'101
' 1
1
2
2
xx
klxx
'1/101
' 1
1
2
2
xx
fxx
fx /
klx
thin quadrupole:
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/17
FODO Cell
Matrix representation from mid-F(D) to mid-F(D)
Must equal the Twiss matrix so22 21cos fL fL 2)2sin(
sin/)2sin(12L 0 (symmetry plane)
(E.Wilson)
sincossinsinsincos
21
21
2
212
21
12/101
101
1/101
101
12/101
M
2
2
2
2
2
fL
fL
fL
fLL
fL
fL
fL
f
As FODO pattern in horizontal plane becomes DOFOin vertical plane, the maximum & minimum values offor both plane is equal the z and x at a F quadrupole:
)2/sin(1)2/sin(1/ minmax
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/18
FODO cell (continued)
Generally get region ofstability (”shaded area”)by solving matrixproduct for a F & a Dquadrupole with focallengths f1 & f2 andrequiring |cos | 1.
klklkklkkl
cossinsin)/1(cosMF
101
M v
(E.Wilson)
(E.Wilson)
In practice use beam simulation programs likeMADX to get a,b,c & d and print out (s) & (s)
Also dipole magnets focuses (as a functionof entry angle ) in horizontal plane i.e.
cossin)1(sincos
Mh
Most dipoles have instead parallell end faces withentry angle /2 giving reduced horizontal focusingbut adding a focusing for vertical displacements.
quadrupole length lnot small comparedto focal length f inreality so twissmatrix for a F (D)quadrupole become klklk
klkklcoshsinh
sinh)/1(coshMD
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/19
Liouville’s theorem
1)(')(')()(')(')()()(
M1212
1212
sysysysysysysysy
J
(E.Wilson)
(E.Wilson)
• “area of contour whichencloses all beam particles inphase space is conserved”• area = is “emittance”• area same all round ringthough shape changes• Hamiltonian time independent
Jacobian determinant
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/20
Adiabatic damping
What happens when particle beam is accelerated?
xqdtdxmpT )/(
'xcmdsdq
dtdsm
dtdqmpT
01' pdxxbeam dimensionsshrink as 1/ p0(”adiabatic damping”)
*0 '' mcdxxpdxxmcdqpT
= [ mm mrad]: ”normalized” emittance
Canonical coordinates (used in the Hamiltonian):
Accelerator coordinates:
x’
closed orbit x
q
pT = p0 x
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/21
Liouville’s theorem
(E.Wilson)
The phase space ellipse beam parameters
Equation for the ellipse (”Courant-Snyder” invariant):
22 ')(')(2)( ysyysys
'& maxmax yy
axis of ellipse horizontal & vertical only at thequadrupoles; elsewhere also & get a meaning:
Liouville’s theorem not valid when space-charge forceswithin bunch are large or particles emit syncrotron light
beam size calculation example: SPS at 10 GeV
radm1020mradmm20 6
mm461020108m108 6beam x
Accelerators and Experiments 2008Transverse beam dynamics Kenneth Österberg II/22
Beam distributions
(E.Wilson)
(protons)2 2
)(electrons2
The effect of aperture e.g.vacuum chamber or
collimators/beam screens ?
e)(acceptancor22 ryA
(E.Wilson)
Emittance definition fordifferent beam types(note convention!!):
Betatron oscillations:
'& xx
Most beam distributionsin reality Gaussian
convention not always strictly followed !!