Lattice Design in Particle Accelerators Bernhard Holzer, DESY

Home , Quadrupole magnet

Historical note:

... Particle acceleration where lattice design is not needed

N ntZ 2e 4 1 N(θ ) = i * πε 2 2 2 4 θ ( 8 0 ) r K sin ( / 2 ) N(θ)

Rutherford Scattering, 1906

Using radioactive particle sources: α-particles of some MeV energy

Historical note: II.) The „infancy of particle accelerators“

1923 Betatron, 1928 Cockroft-Walton Generator, 1929 Cyclotron 1932 Van de Graaf Generator, etc

Diagram of the first cyclotron by Lawrence and Livingstone

12 MV-Tandem van de Graaf Accelerator

1 1952: Courant, Livingston, Snyder: Theory of strong focusing in particle beams Ted Wilson in this school

Lattice design: design and optimisation of the principle elements of an accelerator ... the lattice cells

Lattice Design: Prerequisites

... I will start very easy ... and slowly the topic will become more and more difficult ... interesting

Lorentz Force: F = q* ( E + v × B )

neglecting electrical fields: F = q* ( v × B )

High energy accelerators circular machines

somewhere in the lattice we need a number of dipole magnets, that are bending the design orbit to a closed ring

2 In a constant external magnetic field the particle trajectory will be a part of a circle and ...... the centrifugal force will be equal to the Lorentz force

mv 2 mv e v B = → e * B = = p / ρ * * ρ ρ

→ B * ρ = p /e

p = momentum of the particle, ρ = curvature radius

B*ρ is called the “beam rigidity”

Example: Heavy Ion Storage Ring: TSR 8 dipole magnets of equal bending strength

Circular Orbit: α

ds α = ds ≈ dl α = B* dl ρ ρ B* ρ

The angle swept out in one revolution field map of a storage ring dipole magnet must be 2π, so

* Bdl p α = = 2π ...for a full circle → * Bdl = 2π * B* ρ q

Example HERA: * Bdl ≈ N * l * B = 2π p / q 920 GeV Proton storage ring number of dipole magnets N = 416 2π * 920 * 10 9 eV l = 8.8m B ≈ ≈ 5.15 Tesla m q = +1 e 416 * 3* 10 8 * 8.8m * e s

3 Focusing forces and particle trajectories:

Magnetic field in a quadrupole magnet

= − = − Bx g* z , Bz g* x

leads to a linear retrieving force on the particle.

Relating the fields to their optical effect: normalise to the particles momentum:

1 B dipole magnet : = ρ p / e

g 1 quadrupole lens : k = focal length : f := p / e k * l

Under the influence of the focusing and defocusing forces the differential equation of the particles trajectory can be developed:

2 x''+K * x = 0 K = −k + 1 / ρ hor. plane K = k vert. plane

Example:

HERA Ring: Circumference:

Circumference: C0 =6335 m Bending radius: ρ = 580 m Quadrupol Gradient: G= 110T/m

k = 33.64*10-3 /m2 1/ρ2 = 2.97 *10-6 /m2

the two storage rings of the HERA collider

For estimates in large accelerators the weak focusing term 1/ρ2 can ! in general be neglected ...

4 Single particle trajectories:

Equation of motion y' = + 0 y( s ) y0 * cos( K * s ) * sin( K * s ) y''+K( s )* y = 0 K = − + y' ( s ) y0 * K * sin( K * s ) y'0* cos( K * s )

y . y . Or written more convenient in matrix form: / = M * / y' 0s y' 00

Matrices of lattice elements 1 . cos( K * l ) sin( K * l )/ Hor. focusing Quadrupole Magnet = MQF K / / − K sin( K * l ) cos( K * l ) 0

Hor. defocusing Quadrupole Magnet 1 . cosh( K * l ) sinh( K * l )/ = MQD K / / K sinh( K * l ) cosh( K * l ) 0 1 s. Drift space M = / Drift 0 10

Periodic Lattices:

In the case of periodic lattices the transfer matrix can be expressed as a function of a set of periodic parameters α, β, γ

µ + α µ β µ . s+ L dt cos S sin S sin µ = * M( s ) = / β − γ µ µ − α µ / s ( t ) S sin cos( ) S sin 0 μ = phase advance per period:

For stability of the motion in periodic lattice structures it is required that

trace( M ) < 2

In terms of these new periodic parameters the solution of the equation of motion is y( s ) = ε * β ( s )* cos(Φ ( s ) − δ ) − ε y' ( s ) = * {}sin(Φ ( s ) − δ ) + α cos(Φ ( s ) − δ ) β

5 The new parameters α, β, γ can be transformed through the lattice via the matrix elements defined above.

β . C 2 − 2SC S 2 . β . / / / α / = − CC' SC'+S' C − SS' /* α / γ / 2 − 2 / γ / 0S C' 2S' C' S' 0 00

Question: What does that mean ????

... and here starts the lattice design !!!

Question: What does that mean ????

C S . 1 l . Most simple example: drift space M = / = / C' S' 0 0 10

x . 1 l . x . → x( l ) = x + l * x ' particle coordinates / = /* / 0 0 x' 0 0 10 x' 0 → = l 0 x' ( l ) x0'

transformation of twiss parameters:

β . 1 − 2l l 2 . β . / / / β = β − α + 2 γ α / = 0 1 − l /* α / ( s ) 0 2l * 0 l * 0 γ / / γ / 0l 0 0 1 0 00

Stability ...?

trace( M ) = 1 + 1 = 2 A periodic solution doesn‘t exist in a magnetic lattice built exclusively of drift spaces.

6 Lattice Design:

Arc: regular (periodic) magnet structure: bending magnets define the energy of the ring main focusing & tune control, chromaticity correction, multipoles for higher order corrections

Straight sections: drift spaces for injection, dispersion suppressors, low beta insertions, RF cavities, etc...... and the high energy experiments if they cannot be avoided

The FoDo-Lattice

A magnet structure consisting of focusing and defocusing quadrupole lenses in alternating order with nothing in between. (Nota bene: nothing = elements that can be neglected on first sight: drift, bending magnet, RF structures ... and especially experiments...)

QFQD QF

Starting point for the calculation: in the middle of a focusing quadrupole Phase advance per cell µ = 45°, calculate the twiss parameters for a periodic solution

7 Periodic solution of a FoDo Cell

β x

β y

Output of the optics program:

NR TYP LENGTH STRENGTH BETAX ALFAX PHIX BETAZALFAZPHIZ

0 IP 0 0.00E+00 11.611 0 0 5.295 0 0 1 QFH 0.25 -5.41E-01 11.228 1.514 0.0035 5.488 -0.78 0.007 2 QD 3.251 5.41E-01 5.4883 -0.781 0.0699 11.23 1.514 0.066 3 QFH 6.002 -5.41E-01 11.611 0 0.125 5.295 0 0.125 4 IP 6.002 0.00E+00 11.611 0 0.125 5.295 0 0.125

QX= 0.125 QZ= 0.125 0.125* 2π = 45°

... Can we understand, what the optics code is doing?

1 . cos( K * l ) sin( K * l )/ The action of a magnet on the beam = MQF K / is given by the transfer matrix: / − K sin( K * l ) cos( K * l ) 0

Input: strength and length of the FoDo elements 1 l . M = / K =+/- 0.54102 m-2 Drift 0 10 lq = 0.5 m ld = 2.5 m

The matrix for the complete cell is obtained by multiplication of the element matrices = M FoDo MQFH * M LD * MQD * M LD * MQFH

Putting the numbers in and multiplying out ...

0.707 8.206. M = / FoDo − 0.061 0.7070

8 The transfer matrix for 1 period gives us all the information that we need !

= → 1.) is the motion stable? trace( M FoDo ) 1.415 < 2

2.) Phase advance per cell

1 cos(µ ) = * trace( M ) = 0.707 cosµ + α sinµ β sinµ . 2 M( s ) = / → − γ sinµ cos(µ ) − α sinµ 0 1 µ = acos( * trace( M )) = 45° 2

3.) hor β-function 4.) hor α-function M( 1,2 ) β = = 11 611 M(1,1 ) − cos( µ ) µ . m α = = 0 sin( ) sin( µ )

Some perls of wisdom about lattice cells:

1.) think first ... L * a first estimate of the FoDo parameters can and should be done before we run our optics codes. * we can learn a lot without doing to many too sophisticated calculations * the optic codes will never tell us whether a lattice cell is technically feasible * at the very beginning we have to define parameters that lead at least to a stable periodic solution

2.) some rules of „thumb“... to start with: the tune Q s+ L dt phase advance per cell (i.e. )period: µ = * s β ( t ) Tune := phase advance of the machine in µ 1 ds Q := N * = * * units of 2π 2π 2π β ( s )

1 2πR → Q ≈ * = R 2π β β

The tune is roughly given by the mean bending radius of the circular accelerator divided by the mean β-function

9 3.) can we do it a little bit easier ? We can: the thin lens approximation

1 . cos( K * l ) sin( K * l )/ M = K / Matrix of a focusing quadrupole magnet: QF / − K sin( K * l ) cos( K * l ) 0

If the focal length f is much larger than the length of the quadrupole magnet,

= 1 >> f lQ klQ

= → the transfer matrix can be aproximated using klQ const , lQ 0

1 0. M = / 1 1/ f 0

FoDo in thin lens approximation

Calculate the matrix for a half cell, starting in the middle of a foc. quadrupole:

= M halfCell MQD / 2 M lD MQF / 2

1 0. . 1 0. lD indicates the length of the drift / 1 lD / M = * /* − which is now just half the cell length halfCell 1 ~ 1/ 1 ~ 1/ f 0 0 1 0 f 0 = lD L / 2

l . and starting at the middle of a 1 − D ~ l / f D quadrupole the focal length of the M = / half quad is halfCell − l l D ~ 1 + D ~/ ~ f 2 f 0 f = 2 f

For the second half cell set f -f

10 FoDo in thin lens approximation l . l . 1 + D ~ l / 1 − D ~ l / f D f D Matrix for the complete FoDo cell: M = /* / − l l − l l D ~ 1 − D ~/ D ~ 1 + D ~/ f 2 f 0 f 2 f 0

2l 2 l . − D + D / Multiplying out we get ... 1 ~ 2lD (1 ~ ) f 2 f / M = l 2 l l 2 / 2( ~D − ~D ) 1 − 2 ~D / f 3 f 2 f 2 0

Now we know, that the phase advance is related to the transfer matrix by 1 4l 2 2l 2 cos µ = trace ( M ) = 1 * ( 2 − ~D ) = 1 − ~D 2 2 f 2 f 2

After some beer and with a little bit of trigonometric gymnastics

cos( x ) = cos2 ( x ) − sin2 ( x / 2 ) = 1 − 2 sin2 ( x ) 2 2

We can calculate the phase advance as a function of the FoDo parameter

2 2 2l cos( µ ) = 1 − 2sin ( µ / 2 ) = 1 − ~D f 2 L µ = ~ = Cell sin( / 2 ) lD / f ~ 2 f L sin( µ / 2 ) = Cell 4 f

Example: 45-degree Cell

LCell = lQF + lD + lQD +lD = 0.5m+2.5m+0.5m+2.5m = 6m

-2 -1 1/f = k*lQ = 0.5m*0.541 m = 0.27 m

L sin( µ / 2 ) ≈ Cell = 0.405 Remember: 4 f Exact calculation yields: → µ ≈ 47 .8° µ = 45 ° → β ≈ 11 .4m β = 11 .6m

11 Stability in a FoDo structure 2l 2 l . − D + D / 1 ~ 2lD (1 ~ ) f 2 f / M = transfer matrix for the complete cell l 2 l l 2 / 2( ~D − ~D ) 1 − 2 ~D / f 3 f 2 f 2 0

4l 2 Stability requires: trace( M ) < 2 trace( M ) = 2 − ~D < 2 f 2

L → f > cell 4

For stability the focal length has to be larger than a quarter of the cell length !!

SPS Lattice

Transformation matrix in terms of the Twiss parameters en detail

Transformation of the coordinate vector (x,x´) in a magnet 1 . x( s ) . x . cos( K * l ) sin( K * l )/ / = 0 / = K / M / MQF 0 x' / x´( s ) 0 0 − K sin( K * l ) cos( K * l ) 0

General solution of the equation of motion

x( s ) = ε * β( s )* cos(φ( s ) + ϕ ) ε x' ( s ) = − * {}α( s )cos(φ( s ) + ϕ )+ sin(φ( s ) + ϕ ) β( s ) Transformation of the coordinate vector (x,x´) expressed as a function of the twiss parameters

β . S (cosφ + α sinφ ) β β sinφ / β S S 0 / M = 0 0→ S (α − α )cosφ − (1 + α α )sinφ β / 0 S 0 S 0 (cosφ − α sinφ )/ β β β S S 0 S 0

12 Transfer matrix for half a FoDo cell:

l . 1 − D ~ l / f D = / l M halfCell − D lD + lD / ~2 1 ~/ f f 0 L

β . Compare to the twiss (cos φ + α sin φ ) ββ sin φ / β 0 / parameter form of M M = 0 ( α − α )cos φ − ( 1 + α α ) sin φ β / where Φ denotes the 0 0 0 (cos φ − α sin φ )/ ββ β phase advance through 0 0 the half cell

In the middle of a foc (defoc) quadrupole of the FoDo we allwayshaveα=0, and the half cell will lead us from βmax to βmin

∨ . β ∨ / cos φ βˆ β sin φ / C S . βˆ M = / = / 0 − βˆ / C' S' 1 φ φ sin ∨ cos / ∨ / βˆ β β 0

µ l L sin = ~D = Solving for βmax and βmin and remembering that …. 2 f 4 f

µ ~ + µ S' βˆ 1 + l / f 1 sin + = = D = 2 (1 sin )L ∨ ~ C 1 − l / f µ βˆ = 2 ! β D 1 − sin µ 2 → sin µ ∨ 2 − S l ∨ (1 sin )L = βˆ β = ~2 = D f µ β = 2 C' 2 ! sin sin µ 2

The maximum and minimum values if the β-function are solely determined by the phase advance and the length of the cell. Longer cells lead to larger β

typical shape of a proton bunch in the HERA FoDo Cell

()ZX, , Y

13 Optimisation of the FoDo Phase advance: Beam dimension

In both planes a gaussian particle distribution is assumed given by the beam emittance ε and the β-function

σ = εβ

measured beam size at the HERA IP

In general proton beams are „round“ in the sense that ε ≈ ε x y So for highest aperture we have to minimise the β-function in both planes:

2 = ε β + ε β r x x y y

typical beam envelope, vacuum chamber and pole shape in a foc. Quadrupole lens in HERA

Optimising the FoDo phase advance 2 = ε β + ε β r x x y y

µ µ ∨ (1 + sin )* L (1 − sin )* L search for the phase advance µ that results in βˆ + β = 2 + 2 a minimum of the sum of the beta’s sinµ sinµ

∨ 2L d βˆ + β = ( 2L ) = 0 sin µ dµ sinµ

20 20 18 L µ = → µ = ° 16 2 * cos 0 90 β ges ()µ sin µ 14 12

10 10 0 36 72 108 144 180 0 µ 180

14 Nota bene: electron beams are typicalle flat, εy ≈ 2 … 10 % εy optimise only βhor µ + d d L(1 sin ) ( βˆ ) = 2 = 0 → µ ≈ 76° dµ dµ sin µ

20 20

βmax()µ 12

βmin()µ 8

red curve: βmax 4 blue curve: βmin 0.013 0 as a function of the phase advance µ 0 36 72 108 144 180 1 µ 180

Orbit distortions in a periodic lattice field error of a dipole/distorted quadrupole

ds * Bds → δ ( mrad ) = = ρ p / e ●● ●

the particle will follow a new closed trajectory, the distorted orbit:

β ( s ) 1 x( s ) = * * β ( ~s ) cos( φ( ~s ) − φ( s ) − πQ )d~s 2 sinπQ ρ( ~s )

* the orbit amplitude will be large if the β function at the location of the kick is large β(s~ ) indicates the sensitivity of the beam here orbit correctors should be placed in the lattice * the orbit amplitude will be large at places where in the lattice β(s) is large here beam position monitors should be installed

15 Chromaticity in the FoDo Lattice ∆p Definition ∆Q = ξ * p The chromaticity describes an optical error of quadrupole lenses: For a given magnetic field, i.e. gradient particles with smaller momentum will feel a stronger focusing force and vice versa. For small momentum errors ∆p/p the focusing parameter k can be written as g e k( p ) = = g* + ∆ p / e p0 p ∆ e ∆p → ∆ = − p k( p ) ≈ (1 − )* g = k + ∆k k k0 0 p p0 p

This describes a quadrupole error that leads to a tune shift of ...

1 − 1 ∆p ∆Q = **∆kβ ( s )ds = k β ( s )ds 4π 4π p 0

1 ξ contribution in the lattice ξ = − * β ( s )* k( s )ds 4π

1 Chromaticity in the FoDo Lattice ξ = − * β ( s )* k( s )ds 4π

$ µ µ 4 ∨ + − − 1 βˆ − β 1 1 ' L ( 1 sin ) L ( 1 sin ) ' ξ ≈ − N * = − N * * % 2 2 5 π π µ 4 f Q 4 f Q ' sin ' & 6

using some trigonometric transformations ... ξ can be expressed in a very simple form:

µ µ 1 1 2 L sin 1 1 L sin remember ... ξ = − N 2 = − N 2 4π f sin µ 4π f µ µ x x Q Q f sin cos sin x = 2sin cos Q 2 2 2 2 µ 1 1 L tan µ L ξ = − 2 putting ... sin = Cell 4π f µ Q f sin 2 4 fQ Q 2

Contribution of one FoDo Cell to the 1 µ ξ = − * tan chromaticity of the ring: Cell π 2

16 Resumé

p 1.) Dipole strength: * Bds = N * B * l = 2π 0 eff q

leff effective magnet length, N number of magnets

2.) Stability condition: Trace( M ) < 2

for periodic structures within the lattice / at least for the transfer matrix of the complete circular machine

cos µ + α( s )sinµ β ( s )sinµ . 3.) Transfer matrix for periodic cell M( s ) = / − γ ( s )sinµ cos( µ ) − α( s )sinµ 0

α,β,γ depend on the position s in the ring, µ (phase advance) is independent of s

1 0. / 1 = 1 f = 4.) Thin lens approximation: MQF 1/ Q / kQlQ fQ 0

focal length of the quadrupole magnet fQ = 1/(kQlQ) >> lQ

R 5.) Tune (rough estimate): Q ≈ β

R , β average values of radius and β-function

µ L 6.) Phase advance per FoDo cell sin = Cell (thin lens approx) 2 4 fQ

LCell length of the complete FoDo cell, fQ focal length of the quadrupole, µ phase advance per cell

L > Cell 7.) Stability in a FoDo cell fQ (thin lens approx) 4

µ µ + − (1 sin )LCell ∨ (1 sin )LCell 8.) Beta functions in a FoDo cell βˆ = 2 β = 2 sin µ sin µ (thin lens approx)

LCell length of the complete FoDo cell, µ phase advance per cell

17 Resumé periodic section („the arc“ in a large accelerator):

parameters of the lattice cell defined by 2 quadrupole strengths (sometimes even only by one if in series)

for a given tune (phase advance) the twiss parameters are fixed et vice versa. tune adjustments, variations of cell lengths, matching of twiss parameters creation of symmetry points in the lattice ...... need more degrees of freedom, i.e. additional free quadrupole lenses, varying drift lengths etc.

18 APPENDIX

Single particle trajectories:

y''+K * y = 0

The differential equation for the particle movement can be solved by the Ansatz ...

= ω + ω y a1 * cos( * s ) a2 * sin( * s )

= − ω ω + ω ω y' a1 * sin( * s ) a2 * cos( * s ) = − ω 2 ω − ω 2 ω y'' a1 * cos( * s ) a2 * sin( * s ) = −ω 2 * y

→ K = ω 2 , ω = K

So we get for the equation of motion in a storage ring

= + y( s ) a1 * cos( K * s ) a2 * sin( K * s )

Equation of motion

y s = a K s + a K s ( ) 1 * cos( * ) 2 * sin( * )

The parameters a1 and a2 refer to the individual particle and are determined by boundary conditions.

y = y → a = y (0 ) 0 1 0 y ' y'(0 ) = y ' → a = 0 0 2 K

resulting in y' y (s ) = y * cos( K * s ) + 0 * sin( K * s ) 0 K y s = −y K K s + y K s '( ) 0 * * sin( * ) '0* cos( * )

Or written more convenient in matrix form:

y . y . C S . / = M * / , M = / y y C S '0s '00 ' '0

19 Matrices of lattice elements

1 . cos( K * l ) sin( K * l )/ M = QF K / Hor. focusing − K sin( K * l ) cos( K * l ) 0 Quadrupole Magnet

1 . cosh( K * l ) sinh( K * l )/ Hor. defocusing M = QD K / Quadrupole Magnet K sinh( K * l ) cosh( K * l ) 0

1 s. M = / Drift space Drift 0 10

1 K = −k + in horizontal plane ρ 2

K = k in vertical plane

Transformation of the principal trajectories in terms of the Twiss parameters

General solution of the equation of motion x(s ) = ε * β(s )* cos( φ(s ) + ϕ ) (1) ε x'(s ) = − * {}α(s )cos( φ(s ) + ϕ ) + sin(φ(s ) + ϕ ) β(s ) Using theorems of trigonometric functions sin(a+b) = sin(a) cos(b)+cos(a) sin(b) ...

x(s ) = ε * β(s )* {cos φ(s )cos( ϕ ) − sinφ(s )sin(ϕ )} (2) x s = − ε {α s φ s ϕ − α s φ s ϕ + '( ) β(s )* ( )cos ( )cos( ) ( )sin ( )sin( ) + sinφ(s )cos( ϕ ) + cos φ(s )sin(ϕ )}

Set initial conditions: x(0)=x0 ,x´(0)=x´0,

β(0)= β0, α(0)= α0, Φ(0)=0 x − 1 α x (3) cos φ = 0 sinφ = ( x ′ β + 0 0 ) εβ ε 0 0 β 0 0

20 β(s ) Inserting in (1) x(s ) = {}cos φ(s ) +α sinφ(s ) x + β(s )β sinφ(s )x ′ β 0 0 0 0 0

1 x ′(s ) = {}(α −α(s ))cos φ(s ) −(1 + α α(s )sinφ(s ) x β β 0 0 0 (s ) 0

β + 0 {}cos φ(s ) − α(s )sinφ(s ) x ′ β(s ) 0

So again we have got a matrix that transforms the orbit vector (x0, x´0) into (x(s), x´(s))

x(s ) . x . / = M 0 / x ' x´(s )0 0 0

β . S (cosφ + α sinφ ) β β sinφ / β S S 0 / M = 0 (α − α )cosφ − (1 + α α )sinφ β / 0 S 0 S 0 (cosφ − α sinφ )/ β β β S S 0 S 0