Syllabus and slides Lecture 1: Overview and history of Particle accelerators (EW) Lecture 2: Beam optics I (transverse) (EW) Lecture 3: Beam optics II

Syllabus and slidesSyllabus and slides

• Lecture 1: Overview and history of Particle accelerators (EW)• Lecture 2: Beam optics I (transverse) (EW)• Lecture 3: Beam optics II (longitudinal) (EW)• Lecture 4: Liouville's theorem and Emittance (RB)• Lecture 5: Beam optics and Imperfections (RB)• Lecture 6: Beam optics in linac (Compression) (RB)• Lecture 7: Synchrotron radiation (RB)• Lecture 8: Beam instabilities (RB)• Lecture 9: Space charge (RB)• Lecture 10: RF (ET)• Lecture 11: Beam diagnostics (ET)• Lecture 12: Accelerator Applications (Particle Physics) (ET)• Visit of Diamond Light Source/ ISIS / (some hospital if possible)

The slides of the lectures are available at

http://www.adams-institute.ac.uk/training

Dr. Riccardo Bartolini (DWB room 622)

R. Bartolini, John Adams Institute, 1 May 2013 1/28

Lecture 4: Emittance and Liouville’s theorem

Hill’s equations (recap)

More on transfer matrix formalism

Courant-Snyder Invariant

Emittance

Liouville’s theorem

2/28

Linear betatron equations of motion (recap)

In the magnetic fields of dipoles magnets and quadrupole magnets the coordinates of the charged particle w.r.t. the reference orbit are given by the

Hill’s equations

0)(2

2

ysKds

ydy

x

sB

BssK z

x

)(1

)(

1)(

2

x

sB

BsK z

z

)(1

)(

No periodicity is assumed but for a circular machine Kx, Kz and are periodic

These are linear equations (in y = x, z). They can be integrated.

weak focussing of a dipole

quadrupole focussing

R. Bartolini, John Adams Institute, 1 May 2013

3/28

Pseudo-harmonic oscillations (recap)

The solution can be found in the form

)s(cos)s()s(y yyy s

s yy

0

)'s(

'ds)s(

which are pseudo-harmonic oscillations

The beta functions (in x and z) are proportional to the square of the envelope of the oscillations

The functions (in x and z) describe the phase of the

oscillations


4/28

Principal trajectories (recap)The solutions of the Hill’s equation can be cast equivalently in the form of principal trajectories. These are two particular solutions of the homogeneous Hill’s equation

0)('' ysky

which satisfy the initial conditions

C(s0) = 1;C’(s0) = 0; cosine-like solution

S(s0) = 0;S’(s0) = 1; sine-like solution

The general solution can be written as a linear combination of the principal trajectories

)(')()( 00 sSysCysy


5/28

Principal trajectories vs pseudo harmonic oscillations

))s(sin)s((cos)s(

)s(C 00

)s(sin)s()s(S 0

We can express amplitude and phase functions

)s(C)s(S

)s(Sarctg)s(

00

and viceversa

)s(C)s(S

)s(C)s(S)s(S1)s(

00

200

2

0

or more simply2

0 )s(sin

)s(S1)s(

)s(sin

)s(cos)s(

)s('S

)s( 0

)s(cos)s()s(y

)s(cos)s()s(sin)s(

)s('y

in terms of the principal trajectories )s(S'y)s(Cy)s(y 00

Simple algebraic manipulations yield


6/28

Principal trajectories (recap)As a consequence of the linearity of Hill’s equations, we can describe the evolution of the trajectories in a transfer line or in a circular ring by means of linear transformations

)('

)(

)(')('

)()(

)('

)(

0

0

sy

sy

sSsC

sSsC

sy

sy

This allows the possibility of using the matrix formalism to describe the evolution of the coordinates of a charged particles in a magnetic lattice

)(')('

)()(

sSsC

sSsCM 21

C(s) and S(s) depend only on the magnetic lattice not on the particular initial conditions


7/28

Matrices of most common elementsTransfer lines or circular accelerators are made of a series of drifts and quadrupoles for the transverse focussing and accelerating section for acceleration.

Each of these element can be associated to a particular transfer matrix

10

d1M

)LKcosh()LKsinh(K

)LKsinh(K

1)LKcosh(

M

)L|K|cos()L|K|sin(|K|

)L|K|sin(|K|

1)L|K|cos(

M

1KL

01M

1L|K|

01M

Matrix of a drift space

Matrix of a focussing quadurpole

Matrix of a defocussing quadurpole

Thin lens approximationL 0, with KL finite


8/28

Matrix formalism for transfer lines

For each element of the transfer line we can compute, once and for all, the corresponding matrix. The propagation along the line will be the piece-wise composition of the propagation through all the various elements

L Q2Q1

s1 s2

2

1

1221 /1*/1

/1

1/1

01

10

1

1/1

01

fLf

LfL

f

L

fM

2121

11

*

1

ff

L

fff

Notice that it works equally in the longitudinal plane, e.g.

1/1

0121 f

M33

sin1

s

s

mc

qVL

f

thin lens quadrupole associate to an RF cavity of voltage V and length L


9/28

Matrix formalism and analogy with geometric optics

Particle trajectories can be described with a matrix formalism analogous to that describing the propagation of rays in an optical system.

The magnetic quadrupoles play the role of focussing and

defocussing lenses, however notice that, unlike an optical lens,

a magnetic quadrupole is focussing in one plane and

defocussing in the other plane.

Magnetic field of a quadrupole and Lorentz force


10/28

A slightly more complicated example: the FODO lattice (I)

Consider an alternating sequence of focussing (F) and defocussing (D) quadrupoles separated by a drift (O)

The transfer matrix of the basic FODO cell reads

2

2

2 f4

L

f2

L1

f2

Lf4

L1L

f2

L1

102

L1

1f

101

102

L1

1f

101

M


11/28

In terms of the amplitude and phase function the transfer matrix will read

where 0 , 0 and the phase 0 are computed at the beginning of the segment of transfer line

We still have not assumed any periodicity in the transfer line.

If we consider a periodic machine the transfer matrix over a whole turn reduces to (put = the phase advance in one turn)

sin)s(cos)s()s(

sin))s(1(cos))s((

sin)s()sin(cos)s(

)s('S)s('C

)s(S)s(CM

0

0

00

000

ss0

sincossin

sinsincosM

00

00ss 00

Matrix elements from principal trajectories and optics functions

0

20

01

This is the Twiss parameterisation of the one turn map


12/28

Consider a circular accelerator with transfer matrix over one turn equal to M (one turn map). Using the Twiss parameterisation for M

JsinIcossincossin

sinsincosM

00

00

Stability of motion with the matrix formalism

00

00J

01 xMx 0n

n xMx

It can be proven that (see bibliography)

nsinncosnsin

nsinnsinncosM

00

00n

After n turns, the transformation of the particle coordinates will be given by the successive application of the one turn matrix n times

In order for the phase advance to be real and hence for the motion to be a stable oscillation, the one turn map must satisfy the condition

1|trM|2

1|cos|


13/28

Example: the FODO lattice (II)

Using the Twiss parameterisation of the matrix or the FODO cell we have

sincossin

sinsincos

f4

L

f2

L1

f2

Lf4

L1L

f2

L1

M

2

2

2

hence

2

2

f8

L1trM

2

1cos

The stability requires

1|f8

L1||cos|

2

2

4

Lf

In a similar way we can compute the optics functions at the beginning of the FODO cell.


14/28

Optics functions in a transfer line

While in a circular machine the optics functions are uniquely determined by the periodicity conditions, in a transfer line the optics functions are not uniquely given, but depend on their initial value at the entrance of the system.

We can express the optics function in terms of the principal trajectories as

0

0

0

22

22

SSC2C

SSSCCSCC

SCS2C

''''

''''

This expression allows the computation of the propagation of the optics function along the transfer lines, in terms of the matrices of the transfer line of each single element, i.e. also the optics functions can be propagated piecewise from

)(')('

)()(

sSsC

sSsCM 21


15/28

Examples

0

0

02

100

10

21

s

ss

In a drift space

The function evolve like a parabola as a function of the drift length.

In a thin focussing quadrupole of focal length f = 1/KL

The function evolve like a parabola in terms of the inverse of focal length

1

01

KLM

0

0

0

2 12)(

01

001

KLKL

KL

10

1 sM


16/28

Diamond LINAC to booster transfer line

Optics functions from the LINAC

(Twiss parameters of

the beam)

Booster optics functions at the injection point


17/28

Transfer line example: Diamond LTB


18/28

Betatron motion in phase space (recap)

The solution of the Hill’s equations

0y)s(Kds

ydy2

2

describe an ellipse in phase space (y, y’)

)s(cos)s()s(y

)s(cos)s()s(sin)s(

)s('y

area of the ellipse in phase space (y, y’) is /)y'yy2'y()s(A 222


19/28

Courant-Snyder invariant (I)

Hill’s equations have an invariant

0)(2

2

ysKds

ydy .consty'yy2'y)s(A 222

This invariant is the area of the ellipse in phase space (y, y’) multiplied by .

This can be easily proven by substituting the solutions y, y’

into A(s). You will get the constant

A(s) is called Courant-Snyder invariant

)s(cos)s()s(y

)s(cos)s()s(sin)s(

)s('y


20/28

Courant-Snyder invariant (II)

Whatever the magnetic lattice, the area of the ellipse stays constant (if the Hill’s equations hold)

At each different sections s, the ellipse of the trajectories may change orientation shape and size but the area is an invariant.

This is true for the motion of a single particle !


21/28

Real beams – distribution function in phase space

A beam is a collection of many charged particles

The beam occupies a finite extension of the phase space and it is described by a distribution function such that

),z,p,y,p,x(x yx 1xd)s,x( 6 The beam distribution is characterised by the momenta of various orders

xd)s,x(x)s(x 6jj Average coordinates (usually zero)

xd)s,x()xx)(xx()xx()xx()s(R 6jjiijiij

The R-matrix also called -matrix describes the equilibrium properties of the beam giving the second order momenta of the distribution

R11 = bunch H size; R33 = bunch Y size; R55 bunch Z size; R66 = energy spread


22/28

Gaussian beams

In many cases the equilibrium beam distribution is a Gaussian distribution

ji1

ij )xx()xx(R2

1

3e

Rdet)2(

1)s,x(

Usually the three planes are independent hence in each plane

2

x'xx2'x

'xx

22

eRdet2

1)'x,x(

The isodensity curves are ellipses


23/28

-matrix for Gaussian beams

zzzyzx

yzyyyx

xzxyxx

with (assuming <x> = 0 and <x’> = 0)

2

2

xx'x'xx

'xxx

Direct computation using

xxxx

xxxxxx

The 6x6 –matrix can be partitioned into nine 2x2 submatrices

We can associate an ellipse with the Gaussian beam distribution. The evolution of the beam is completely defined by the evolution of the ellipse

2x

2x

2xxxxx )det(

The ellipse associated to the beam is chosen so that its Twiss parameters are those appearing in the distribution function, hence, e.g.

222x 'xx'xx xxx xx'x

yields

2

x'xx2'x 22

e2

1)'x,x(


24/28

Generic beams – rms emittanceFor a generic beam described by a distribution functions we can still compute the average size and divergence and the whole -matrix

2

2

xx'x'xx

'xxx

xxxx

xxxxxx

we associate to this distribution the ellipse which has the same second order momenta Rij and we deal with this distribution as if it was a Gaussian distribution

2x

2xxxxxx )det(

and since

The invariant of the ellipse will be 222x 'xx'xx

which is the rms emittance of the beam


25/28

Beam emittance and Courant-Snyder invariant

We have seen that the beam distribution can be associated to an ellipse containing 66% of the beam (one r.m.s.)

In this way the beam rms emittance is associated with the Courant Snyder invariant of the betatron motion

This links a statistical property of the beam (rms emittance) with single particle property of motion (the Courant-snyder invairnat)

In this way the Courant-Snyder invariant acquires a statistical significance as rms emittance of the beam. Hence the beam rms emittance is a conserved quantity also for generic beams.

This is valid as long as the Hill’s equations are valid or more generally the system is Hamiltonian. As such the conservation of the emittance is a manifestation of the general theorem of Hamiltonian system and statistical mechanics known as the Liouville theorem


26/28

Liouville’s theoremLiouville’s theorem: In a Hamiltonian system, i.e. n the absence of collisions or dissipative processes, the density in phase space along the trajectory is invariant’.

0f,Ht

f

t

p

p

f

t

q

q

f

t

f

dt

df n

1k

k

k

n

1k

k

k

Liouville theorem states that volume of 6D phase are preserved during the beam evolution (take f to be the characteristic function of the volume occupied by the beam). However if the Hamiltonian can be separated in three independent terms

),(H)p,y(H)p,x(H),,p,y,p,x(HH yxyx

The conservation of the phase space density occurs for the three projection on the

(x, px) plane (Horizontal emittance)

(y, py) plane (Vertical emittance)

(z, pz) = (, ) plane (longitudinal emittance)R. Bartolini, John Adams Institute, 1 May 2013

27/28

Beam emittance and Liouville’s theorem

The Courant-Snyder invariant is the area of the ellipse phase space.

The conservation of the area is a general property of Hamiltonian systems (any area not only ellipses !)

The invariance of the rms emittance is the particular case of a very general statement for Hamiltonian systems (Liouville theorem)

This is valid as long as the motion is Hamiltonian, i.e.

No damping effects, no quantum diffusion, due to emission of radiation

no scattering with residual gas, no beam beam collisions

no collective effects (e.g. interaction with the vacuum chamber, no self interaction)


28/28

Bibliography

E. Wilson, Introduction to Particle Accelerators

J. Rossbach and P. Schmuser, CAS Lecture 94-01

K. Steffen, CAS Lecture 85-19


Documents

Syllabus and slides Lecture 1: Overview and history of Particle accelerators (EW) Lecture 2: Beam optics I (transverse) (EW) Lecture 3: Beam optics II