LONGITUDINAL DYNAMICS - - PowerPoint PPT Presentation

LONGITUDINAL DYNAMICS

Boris Vodopivec

• Joël Le DuFF

summary

• Radio-Frequency Acceleration and Synchronism
• Properties of Radio-Frequency cavities
• Principle of Phase Stability and Consequences
• Synchronous linear accelerator
• The Synchrotron
• RF cavities for Synchrotron
• Energy-Phase Equations in a Synchrotron
• Phase space motions

Bibliography : Old Books

• M. Stanley Livingston

High Energy Accelerators (Interscience Publishers, 1954) J.J. Livingood Principles of cyclic Particle Accelerators (D. Van Nostrand Co Ltd , 1961)

• M. Stanley Livingston and J. B. Blewett

Particle Accelerators (Mc Graw Hill Book Company, Inc 1962) K.G. Steffen High Energy optics (Interscience Publisher, J. Wiley & sons, 1965)

• H. Bruck

Accelerateurs circulaires de particules (PUF, Paris 1966)

• M. Stanley Livingston (editor) The development of High Energy Accelerators

(Dover Publications, Inc, N. Y. 1966) A.A. Kolomensky & A.W. Lebedev Theory of cyclic Accelerators (North Holland Publihers Company, Amst. 1966)

• E. Persico, E. Ferrari, S.E. Segre

Principles of Particles Accelerators (W.A. Benjamin, Inc. 1968) P.M. Lapostolle & A.L. Septier Linear Accelerators (North Holland Publihers Company, Amst. 1970) A.D. Vlasov Theory of Linear Accelerators (Programm for scientific translations, Jerusalem 1968)

Bibliography : New Books

• M. Conte, W.W. Mac Kay An Introduction to the Physics of particle Accelerators

(World Scientific, 1991)

• P. J. Bryant and K. Johnsen The Principles of Circular Accelerators and Storage Rings

(Cambridge University Press, 1993)

• D. A. Edwards, M. J. Syphers An Introduction to the Physics of High Energy Accelerators

(J. Wiley & sons, Inc, 1993)

• H. Wiedemann

Particle Accelerator Physics (Springer-Verlag, Berlin, 1993)

• M. Reiser

Theory and Design of Charged Particles Beams (J. Wiley & sons, 1994)

• A. Chao, M. Tigner

Handbook of Accelerator Physics and Engineering (World Scientific 1998)

• K. Wille

The Physics of Particle Accelerators: An Introduction (Oxford University Press, 2000) E.J.N. Wilson An introduction to Particle Accelerators (Oxford University Press, 2001)

Methods of Acceleration

1_ Electrostatic Field

Energy gain : W=n.e(V2-V1) limitation : Vgenerator =Σ Vi

2_ Radio-frequency Field

Synchronism :

L=vT/2

v=particle velocity T= RF period Wideroe structure Electrostatic accelerator

The advantage of Resonant Cavities

• Considering RF acceleration, it is obvious that when particles get high

velocities the drift spaces get longer and one loses on the efficiency. The solution consists of using a higher operating frequency.

• The power lost by radiation, due to circulating currents on the electrodes,

is proportional to the RF frequency. The solution consists of enclosing the system in a cavity which resonant frequency matches the RF generator frequency.

• Each such cavity can be independently

powered from the RF generator.

• The electromagnetic power is now

constrained in the resonant volume.

• Note however that joule losses will
• ccur in the cavity walls (unless made
• f superconducting materials)

ωRF

Ez r

J

• u

H r r

The Pill Box Cavity

2 2 2

= ∂ ∂ − ∇ t A A μ ε ) ( H

• u

E A =

( )

kr J Ez =

( )

kr J Z j H

1

− =

θ

Ω = = = = 377 62 , 2 2 Z a c k λ ω λ π

e

t jω

}

Ez Hθ From Maxwell’s equations one can derive the wave equations : Solutions for E and H are oscillating modes, at discrete frequencies, of types TM or TE. For l<2a the most simple mode, TM010, has the lowest frequency ,and has only two field components:

The Pill Box Cavity (2)

The design of a pill-box cavity can be sophisticated in order to improve its performances:

• A nose cone can be introduced in
• rder to concentrate the electric

field around the axis,

• Round shaping of the corners

allows a better distribution of the magnetic field on the surface and a reduction of the Joule losses. A good cavity is a cavity which efficiently transforms the RF power into accelerating voltage.

Transit Time Factor

t g V t E Ez ω ω cos cos = =

vt z=

Oscillating field at frequency ω and which amplitude is assumed to be constant all along the gap: Consider a particle passing through the middle of the gap at time t=0 : The total energy gain is:

dz v z g eV W

g g

∫ = Δ

− 2 / 2 /

cosω

eVT eV W = = Δ 2 / 2 / sin θ θ

angle transit v g ω θ =

factor time transit T

( 0 < T < 1 )

Transit Time Factor (2)

( )

dz e z E e W

t j g z e ω

∫ ℜ = Δ

( )

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∫ ℜ = Δ

−

dz e z E e e W

v z j g z j e

p

ω ψ

( )

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∫ ℜ = Δ

− g v z j z j j e

dz e z E e e e W

i p

ω ψ ψ

( )

φ

ω

cos ∫ = Δ

g v z j z

dz e z E e W

( ) ( )

∫ ∫ =

g z g t j z

dz z E dz e z E T

ω

p

v z t ψ ω ω − =

i p ψ

ψ φ − =

Ψp is the phase of the particle entering the gap with respect to the RF. and considering the phase which yields the maximum energy gain: Consider the most general case and make use of complex notations: Introducing:

Let’s consider a succession of accelerating gaps, operating in the 2π mode, for which the synchronism condition is fulfilled for a phase Φs .

For a 2π mode, the electric field is the same in all gaps at any given time.

s V e s eV Φ = sin ˆ

is the energy gain in one gap for the particle to reach the next gap with the same RF phase: P1 ,P2, …… are fixed points.

Principle of Phase Stability

If an increase in energy is transferred into an increase in velocity, M1 & N1 will move towards P1(stable), while M2 & N2 will go away from P2 (unstable).

Transverse Instability

< ∂ ∂ ⇒ > ∂ ∂ z z E t V

Longitudinal phase stability means : The divergence of the field is zero according to Maxwell :

. > ∂ ∂ ⇒ = ∂ ∂ + ∂ ∂ ⇒ = ∇ x E z E x E E

x z x

defocusing RF force External focusing (solenoid, quadrupole) is then necessary

A Consequence of Phase Stability

The Traveling Wave Case

( ) ( )

0 cos

t t v z v k kz t E E

RF RF z

− = = − =

ϕ

ω ω

velocity particle v velocity phase v = =

ϕ

The particle travels along with the wave, and k represents the wave propagation factor.

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − =

0 cos

φ ω ω

ϕ

t v v t E E

RF RF z 0 cos

If synchronism satisfied:

φ

ϕ

E E and v v

z =

=

where φ0 is the RF phase seen by the particle.

Multi-gaps Accelerating Structures: A- Low Kinetic Energy Linac (protons,ions)

Mode π L= vT/2 Mode 2π L= vT = βλ

In « WIDEROE » structure radiated power ∝ ω CV

In order to reduce the radiated power the gap is enclosed in a resonant volume at the operating

• frequency. A common wall

can be suppressed if no circulating current in it for the chosen mode. ALVAREZ structure

CERN Proton Linac

The Synchrotron

The synchrotron is a synchronous accelerator since there is a synchronous RF phase for which the energy gain fits the increase of the magnetic field at each

• turn. That implies the following operating conditions:

B e P B h V e

r RF s

⇒ = = Φ = Φ Φ ρ ρ, R ω ω sin

^

Energy gain per turn Synchronous particle RF synchronism Constant orbit Variable magnetic field

If v = c, ωr hence ωRF remain constant (ultra-relativistic e- )

Energy ramping is simply obtained by varying the B field:

v B R e r T B e turn p B e dt dp eB p ′ = ′ = Δ ⇒ ′ = ⇒ = ρ π ρ ρ ρ 2 ) (

Since:

p v E c p E E Δ = Δ ⇒ + = 2 2 2 2

• The number of stable synchronous particles is equal to the harmonic

number h. They are equally spaced along the circumference.

• Each synchronous particle satifies the relation p=eBρ. They have the

nominal energy and follow the nominal trajectory.

The Synchrotron (2)

( ) ( )

φ ρ π

s s turn

V e RB e W E sin ˆ ' 2 = = Δ = Δ

The Synchrotron (3)

During the energy ramping, the RF frequency increases to follow the increase of the revolution frequency : hence :

) , (

s RF r

R B h ω ω ω = =

) ( ) ( 2 2 1 ) ( ) ( 2 1 2 ) ( ) ( t B s R r t s E ec h t RF f t B m e s R t v h t RF f π π π = ⇒ > < = =

Since , the RF frequency must follow the variation of the B field with the law : which asymptotically tends towards when B becomes large compare to (m0c2 / 2πr) which corresponds to v c (pc >> m0c2 ). In practice the B field can follow the law:

2 2 2 2 c p c m E + =

2 1 2 ) ( 2 ) / 2 ( 2 ) ( 2 ) (

⎭ ⎬ ⎫ ⎩ ⎨ ⎧

+ = t B ecr c m t B s R c h t RF f π

R c fr π 2 →

t B t B t B 2 2 sin ) cos 1 ( 2 ) ( ω ω = − =

Longitudinal Dynamics

It is also often called “ synchrotron motion”. The RF acceleration process clearly emphasizes two coupled variables, the energy gained by the particle and the RF phase experienced by the same particle. Since there is a well defined synchronous particle which has always the same phase φs, and the nominal energy Es, it is sufficient to follow

• ther particles with respect to that particle. So let’s

introduce the following reduced variables: revolution frequency : Δfr = fr – frs particle RF phase : Δφ = φ - φs particle momentum : Δp = p - ps particle energy : ΔE = E – Es azimuth angle : Δθ = θ - θs

First Energy-Phase Equation

θ R

∫ = Δ − = Δ ⇒ = dt with h hf f

r r RF

ω θ θ φ

For a given particle with respect to the reference one:

( ) ( )

dt d h dt d h dt d

r

φ φ θ ω 1 1 − = Δ − = Δ = Δ

Since:

s r rs s

dp d p ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ω ω η

• ne gets:

( )

φ ηω φ ηω ω &

rs s s rs s s rs

h R p dt d h R p E − = Δ − = Δ

and

c p E E

2 2 2 2

+ =

p R p v E

s rs s

Δ = Δ = Δ ω

Second Energy-Phase Equation

The rate of energy gained by a particle is:

π ω φ 2 sin ˆ

r

V e dt dE=

The rate of relative energy gain with respect to the reference particle is then:

) sin (sin ˆ 2

s r

V e E φ φ ω π − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Δ &

leads to the second energy-phase equation:

( )

s rs

V e E dt d φ φ ω π sin sin ˆ 2 − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Δ

( )

( )

E T dt d E T T E E T T E T E

rs rs r rs r r

Δ = Δ + Δ = Δ + Δ ≅ Δ & & & & &

Expanding the left hand side to first order:

Equations of Longitudinal Motion

( )

s rs

V e E dt d φ φ ω π sin sin ˆ 2 − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Δ

( )

φ ηω φ ηω ω &

rs s s rs s s rs

h R p dt d h R p E − = Δ − = Δ

deriving and combining

( ) 0

sin sin 2 ˆ = − + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡

s rs s s

V e dt d h p R dt d φ φ π φ ηω

This second order equation is non linear. Moreover the parameters within the bracket are in general slowly varying with time…………………

Small Amplitude Oscillations

( ) 0

sin sin cos

2

= − Ω +

s s s

φ φ φ φ& &

(for small Δφ) 2

= Δ Ω + φ φ

s

& &

s s s rs s

p R V e h π φ ηω 2 cos ˆ

2=

Ω

γ < γtr η > 0 0 < φs < π/2 sinφs > 0 γ > γtr η < 0 π/2 < φs < π sinφs > 0 with Let’s assume constant parameters Rs, ps, ωs and η:

( )

φ φ φ φ φ φ φ Δ ≅ − Δ + = −

s s s s

cos sin sin sin sin

Consider now small phase deviations from the reference particle: and the corresponding linearized motion reduces to a harmonic oscillation: stable for and Ωs real

2>

Ωs

Large Amplitude Oscillations

For larger phase (or energy) deviations from the reference the second order differential equation is non-linear:

( )

sin sin cos

2

= − Ω +

s s s

φ φ φ φ& &

(Ωs as previously defined) Multiplying by and integrating gives an invariant of the motion:

φ &

( )

I

s s s

= + Ω − φ φ φ φ φ sin cos cos 2

2 2

&

which for small amplitudes reduces to:

( )

I

s

= Δ Ω + 2 2

2 2 2

φ φ &

(the variable is Δφ and φs is constant) Similar equations exist for the second variable : ΔE∝dφ/dt

Large Amplitude Oscillations (2) ( ) ( ) ( ) ( )

s s s s s s s s

φ φ π φ π φ φ φ φ φ φ sin cos cos sin cos cos 2

2 2 2

− + − Ω − = + Ω − &

( ) ( )

s s s s m m

φ φ π φ π φ φ φ sin cos sin cos − + − = +

Second value φm where the separatrix crosses the horizontal axis: Equation of the separatrix: When φ reaches π-φs the force goes to zero and beyond it becomes non

• restoring. Hence π-φs is an extreme

amplitude for a stable motion which in the phase space( ) is shown as closed trajectories.

φ φ Δ Ω ,

s

&

Energy Acceptance

From the equation of motion it is seen that reaches an extremum when , hence corresponding to . Introducing this value into the equation of the separatrix gives:

φ &

= φ& &

s

φ φ =

( ) { }

s s s

φ π φ φ tan 2 2 2

2 2 max

− + Ω = &

That translates into an acceptance in energy: This “RF acceptance” depends strongly on φs and plays an important role for the electron capture at injection, and the stored beam lifetime.

( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧− = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Δ φ η π β

s s s

G E h V e E E ˆ

2 1 max

m

( ) ( )

[ ]

φ π φ φ φ

s s s s

G sin 2 cos 2 − + =

RF Acceptance versus Synchronous Phase

As the synchronous phase gets closer to 90º the area of stable motion (closed trajectories) gets

• smaller. These areas are
• ften called “BUCKET”.

The number of circulating buckets is equal to “h”. The phase extension of the bucket is maximum for φs =180º (or 0°) which correspond to no acceleration . The RF acceptance increases with the RF voltage.