Transverse Beam Motion - helsinki.fi · Transverse beam motion ... called lattice, has strong influence on design & aperture of magnets ... MADX to get a,b,c & d and print out E(s)

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


Transverse beam motion


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 &centrifugal forces.

Beam

constant force inx and 0 force in y

or y


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:


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


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.


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


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


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.


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.


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.


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


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:


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


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)


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:


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


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


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


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


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


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 !!



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Transverse Beam Motion - helsinki.fi · Transverse beam motion ... called lattice, has strong influence on design & aperture of magnets ... MADX to get a,b,c & d and print out E(s)