Beam Dynamics – 4

Beam Dynamics – 4

2021.5.20

K. Kubo

Contents

• Particle motion in ele-mag field

• Coordinate system

• Basics of transverse motion

• Betatron oscillation

• Weak-focus and Strong-focus

• Effects of extra field

• Longitudinal motion (synchrotron oscillation)

• Effects of synchrotron radiation to beam

• Polarization

• Use Hamiltonian formula ?

Longitudinal motion (synchrotron oscillation)

For describing transverse motion, we assumed constant longitudinal momentum (~ particle energy).

Now, we consider change of particle energy in circular accelerators.

Ignoring transverse oscillation.

Longitudinal Variables

• Longitudinal position, or time when the particle passes a certain location

• Longitudinal momentum, or energy

(𝑡, 𝛿):

𝑡: Time delay from design (reference)

𝛿 : Relative momentum deviation (Δ𝑝/𝑝

)

Velocity depends on energy Orbit length depends on energy

(Closed orbit depends on energy)

Higher energy particle has larger velocity.

Higher energy particle tends to have outer orbit.

→ Orbit length depends on particle energy

Sources of time change:

Time of one turn depends on energy

“Momentum compaction factor”

• Closed orbit depends of particle’s longitudinal momentum, then, length of 1-turn orbit also depends longitudinal momentum.

• Particle velocity depends on longitudinal momentum (if not ultra- relativistic)

• Then, time for 1-turn depends on longitudinal momentum.

• Linear part of dependence is called “momentum compaction factor”.

1 , 1 ,

0 0

0 =0 =



 



= 



 



= 



  

 

d dC C

d dT

T ^c

c

v C T

c m p

pc

c v

c = − ⁻²,  = ² + ^{2 2}, = /



Usually, higher energy particles go outer (longer) orbit → _c > 0 Large  for high energy → _c > 0

Small  for low energy (heavy particle (proton)) → _c < 0

(

)

(see next page)

0 0 0 0

1 1

=

= 



 



= 



 



= 



  

 

d dC C

d dT

T ^c

c

Time change Orbit length change

(

)

v C T

c m p

pc E

pc v

/

,

/ ^{2 2 2}

2

=

+

=

= ₀  E₀ /mc²

Mathematics Note

Momentum compaction – example 1

Uniform vertical magnetic field (orbit is circle)

constant

y = B

) 1

( 

   +



By

C p 1 1

0 0

 =



 



 

→

=

 

d dC

c C

Radius is proportional to momentum

0

1 ²

2 = − 

−

=  ⁻  ⁻

_{c c}

Momentum compaction – example 2

Uniform weak focusing



 



 − −

=

−

= (1 ) 1 ( ₀)

0 0

0

0  

 Kx B ^a B

B_y

0

1 0

) 1

(







 ₋

−

 +

 B a

p

y

(

)

a C

C

 −

= −

→ 

1 2

) (

2

0 0 0











1 ) 1 /(

1 − 

=

→ _c a

Vertical field is weaker for larger radius.

Radius increase more rapidly than momentum

(

)

𝜂_𝑐 = 𝛼_𝑐 − 𝛾⁻² => 0

Momentum compaction in high energy (strong focus) accelerator

• 𝛼

is usually positive (high energy particle takes outer orbit) but much smaller than 1.

• 𝜂

is negative at low energy, positive at high energy

0 < 𝛼

≪ 1

𝜂

= 𝛼

− 𝛾

ቊ < 0 (small 𝛾)

> 0 (large 𝛾)

Change of particle energy (longitudinal momentum)

• Acceleration in RF cavity (oscillating electric field)

• Energy loss by synchrotron radiation – energy loss in curved orbit

– Important for electron or positron accelerators – Not important for protons and heavy particles

– Here, for a while, we assume energy loss in one turn is constant (dependence on energy will be discussed later)

𝑝 → 𝑝 + (𝑒𝑉/𝑣) sin( 𝜔(𝑛𝑇

+ 𝑡) + 𝜓

)

𝑝 → 𝑝 − 𝑢

Charged particle

Electric field RF Cavity

Energy change (momentum change) in RF Cavity

𝐸_𝑧 𝑧, 𝑡′ = 𝐸₀ 𝑧 sin(𝜔𝑡^′ + 𝜓′)

Electric field:

𝑡^′ = 𝑡 + 𝑧/𝑣 (particle is at 𝑧 = 0 at time 𝑡)

(𝑡: delay time from reference, 𝑣: velocity ≈constant)

Energy change (charge 𝑞): Δ𝐸 = 𝑞∫ 𝐸_𝑧 𝑧, 𝑡′ 𝑑𝑧

Momentum change: Δ𝑝 = 𝑞∫ 𝐸_𝑧 𝑧, 𝑡^′ 𝑑𝑡^′ = 𝑞∫ 𝐸_𝑧 𝑧, 𝑡^{′ 𝑑𝑡′}

𝑑𝑧 𝑑𝑧

Δ𝑝 ≈ Δ𝐸/𝑣 = ^𝑞𝑉

𝑣 sin(𝜔𝑡 + 𝜓) (𝑉: constant voltage, 𝜓: constant phase) We assume velocity is

approximately constant and

deviation of velocity from design is very small.

Mathematics Note

𝐸 = 𝑝²𝑐² + 𝑚²𝑐⁴

→ 𝑑𝐸

𝑑𝑝 = 𝑝𝑐²

𝑝²𝑐² + 𝑚²𝑐⁴ = 𝑣

→ Δ𝑝 ≈ Δ𝐸/𝑣 (if 𝑣 is almost constant)

Reference particle (design)

• Momentum changes every turn at accelerating cavity.

at n-th turn :

• Loss by synchrotron radiation in one turn

• Total change in every turn (any n) should be zero

– Time for 1-turn should be integer times RF period.

– Energy loss (by synchrotron radiation) is compensated every turn.

( U₀ is negligibly small in proton, or heavier particle accelerator)

𝑝 → 𝑝 + 𝑒𝑉/𝑣 sin( 𝜔𝑛𝑇

+ 𝜓

)

𝜔𝑇

= 2𝜋ℎ (ℎ = integer)

𝑒𝑉 sin 𝜓

= 𝑈

(Energy loss/turn)

𝑝 → 𝑝 − 𝑈

/𝑣

time Time for 1-turn of reference particle, RF oscillate h times.

Acceleration voltage

Compensate energy loss or acceleration in the turn Same phase at every turn

Acceleration by RF Cavity

Momentum deviation from reference

• Momentum changes every turn at accelerating cavity.

at n-th turn :

• Loss by synchrotron radiation in one turn

• Change in one turn

𝑝 → 𝑝 + 𝑒𝑉/𝑣 sin( 𝜔𝑛𝑇

+ 𝜔𝑡 + 𝜓

)

(𝑒𝑉 sin 𝜓

= 𝑈

) 𝑝 → 𝑝 − 𝑈

/𝑣

≈ 𝑝 + 𝑒𝑉/𝑣 sin 𝜓

+ 𝑒𝑉/𝑣 𝜔𝑡 cos 𝜓

𝑝 → 𝑝 + 𝑒𝑉𝜔 cos 𝜓

𝑣 𝑡

(linear approximation)

Accelerating RF frequency and

“”reference” particle energy

• Time for 1-turn of reference particle is integer times RF period.

• If RF frequency is slightly changed, length of reference orbit should be changed, then, different closed orbit

becomes “reference”.

– Energy of reference particle is changed.

h T f =

0 0 0

0 1

= =



 



− 

 =



 



 



 



= 

→

  

 

d df f

f h d

d h f

c

f f

c

− 



→  ¹

0 0 0

0

E E E p

p

p −  −

 

SKIP

Longitudinal Motion

Time of one turn depend on particle momentum.

Momentum change depend on time.

𝑡 → 𝑡 + 𝜂

𝛿 (𝛿 = (𝑝 − 𝑝

)/𝑝

)

𝑝 → 𝑝 + 𝑒𝑉𝜔 cos 𝜓

𝑣 𝑡

Synchrotron oscillation (linear approximation)

• Momentum change per turn

• Time change per turn

• Two equations express harmonic oscillation:

• Stability condition：

𝑑𝛿

𝑑𝑛 = 𝑒𝑉

𝑝₀𝑣sin( 𝜔𝑡 + 𝜓₀) − 𝑈₀

𝑝₀𝑣 = 𝑒𝑉

𝑝₀𝑣 sin( 𝜔𝑡) cos 𝜓₀ + cos( 𝜔𝑡) sin 𝜓₀ − 𝑈₀ 𝑝₀𝑣

≈ 𝑒𝑉𝜔 cos 𝜓₀

𝑝₀𝑣 𝑡 𝜔𝑡 << 1



_c dn T

dt

 0

 ₂

2 2 2

2 2

, = −



−

= dn

t d dn

t

d Ω² = −𝑒𝑉𝑇₀𝜔𝜂_𝑐 cos 𝜓₀

𝑝₀𝑣 Ω² > 0 ⇔ 𝑒𝑉𝜂_𝑐 cos 𝜓₀ < 0

loss/turn) (Energy

sin ₀ U₀ eV  =

(𝛿 = (𝑝 − 𝑝₀)/𝑝₀)

Accelerating voltage



< 0 stable U



> 0 stable

t

𝑒𝑉 sin 𝜓

= 𝑈

Ω

> 0 ⇔ 𝑒𝑉𝜂

cos 𝜓

< 0

Transition

• Stability condition:

• In proton (or heavier particle) synchrotron, during acceleration

– Low energy: 

< 0

– At certain energy: 

= 0 – High energy 

> 0

• “Transition”: Stable phase is suddenly changed

• At transition, oscillation frequency is 0:

• Need special care.

0 cos 





e

Ω² = −𝑒𝑉𝑇₀𝜔𝜂_𝑐 cos 𝜓₀

𝑝₀𝑣 = 0

Oscillation amplitude of time and energy

• Two amplitudes are proportional

• Amplitude of time oscillation times particle velocity ( ~c at high energy) is amplitude of longitudinal position

oscillation = bunch length

• Ratio of bunch length and energy spread is proportional to – Square-root of momentum compaction

– Inverse of square-root of accelerating voltage





 A

eVT

E

A

T

0 0

0 0

cos

= −

SKIP

Synchrotron oscillation (linear approximation) summary

• Time change in 1-turn is proportional to momentum deviation （ Momentum compaction factor ）

• Momentum change in 1-turn is proportional to time deviation (Oscillating accelerating voltage)

• Time and momentum deviations: harmonic oscillations 𝑡 → 𝑡 + 𝜂

𝛿

𝑝 → 𝑝 + 𝑒𝑉𝜔 cos 𝜓

𝑣 𝑡

Oscillation

(above transition, 

> 0 )

Advanced (gaining energy)

Delayed (losing energy)

Higher energy (delaying)

Lower energy (advancing)

t



Synchrotron oscillation with large amplitude (no linear approximation)

• Momentum change per turn

• Time change per turn

• Two equations combined 𝑑𝛿

𝑑𝑛 = 𝑒𝑉

𝑝₀𝑣 sin( 𝜔𝑡 + 𝜓₀) − 𝑈₀ 𝑝₀𝑣



_c dn T

dt

 0

loss/turn) (Energy

sin ₀ U₀ eV  =

(𝛿 = (𝑝 − 𝑝₀)/𝑝₀)

𝑑²𝑡

𝑑𝑛² = 𝑇₀𝜂_𝑐 𝑒𝑉

𝑝₀𝑣 sin( 𝜔𝑡 + 𝜓₀) − 𝑈₀ 𝑝₀𝑣

𝑃₀ : design momentum 𝑣 : design velocity

Cannot be solved analytically?

Synchrotron oscillation

Equations of motion

Change variables

Equations of motion become

𝑑𝛿

𝑑𝑛 = 𝑒𝑉

𝑝₀𝑣 sin( 𝜔𝑡 + 𝜓₀) − 𝑈₀ 𝑝₀𝑣

𝑑𝑡

𝑑𝑛 = 𝑇₀𝜂_𝑐𝛿

𝑥 = 𝜔𝑡 + 𝜓₀ 𝑝 = 𝜔𝑇₀𝜂_𝑐𝛿

𝑑𝑝

𝑑𝑛 = 𝑎 sin 𝑥 − 1/𝑏 𝑑𝑥

𝑑𝑛 = 𝑝

𝑎 = 𝑒𝑉𝑇₀𝜔𝜂_𝑐 𝑝₀𝑣

Synchrotron oscillation

𝑑𝑝

𝑑𝑛 = 𝑎 sin 𝑥 − 1/𝑏 𝑑𝑥

𝑑𝑛 = 𝑝

𝐻 𝑥, 𝑝 = 𝑝²/2 + 𝑎 cos 𝑥 + 𝑥/𝑏

Equations of motion

Introduce a function of 𝑥 and 𝑝

𝑑𝑝

𝑑𝑛 = − 𝜕𝐻

𝜕𝑥

𝑑𝑥

𝑑𝑛 = 𝜕𝐻

𝜕𝑝

Equations of motion can be expressed as

→ 𝑑𝐻

𝑑𝑛 = 𝑑𝑥 𝑑𝑛

𝜕𝐻

𝜕𝑥 + 𝑑𝑝 𝑑𝑛

𝜕𝐻

𝜕𝑝 = 0

𝐻

Hamiltonian for Synchrotron oscillation

𝑑𝑝

𝑑𝑛 = 𝑎 sin 𝑥 − 1/𝑏 𝑑𝑥

𝑑𝑛 = 𝑝

𝐻 𝑥, 𝑝 = 𝑝²/2 + 𝑎 cos 𝑥 + 𝑥/𝑏

Equations of motion

Introduce a function of 𝑥 and 𝑝

Equations of motion can be expressed as

→ 𝑑𝐻

𝑑𝑛 = 𝑑𝑥 𝑑𝑛

𝜕𝐻

𝜕𝑥 + 𝑑𝑝 𝑑𝑛

𝜕𝐻

𝜕𝑝 + 𝜕𝐻

𝜕𝑛 = 𝜕𝐻

𝜕𝑛

𝐻

n

Hamiltonian

Hamilton’s equations

Theorem of

Hamilton formula

𝑑𝑝

𝑑𝑛 = − 𝜕𝐻

𝜕𝑥

𝑑𝑥

𝑑𝑛 = 𝜕𝐻

𝜕𝑝

This is similar to Hamiltonian for a pendulum.

Same, if 1/b=0.

𝐻 𝑥, 𝑝 = 𝑝

/2 + 𝑎 cos 𝑥 + 𝑥/𝑏

𝐻 𝑥, 𝑝; 𝑛 = 𝑝²

2 + 𝑎 (cos 𝑥 + 𝑥/𝑏) = constant

Trajectories of H= constant

separatorix

Stable Unstable

Unstable

-p

𝐻 𝑥, 𝑝; 𝑛 = 𝑝²

2 + 𝑎 (cos 𝑥 + 𝑥/𝑏) = constant

Trajectories of H= constant

Stable

Unstable

-p

Synchrotron oscillation with large amplitude

Cannot be solved analytically But,

Various properties of the motion can be understood

by using Hamilton formula.

Effects of Synchrotron Radiation to

Beam

Synchrotron radiation

photon Charged particle Magnetic field

Here, effects of the radiation to the beam particles are discussed.

Synchrotron Radiation

• Electromagnetic waves are radiated when charged particles are accelerated.

• Electromagnetic waves radiated by beam particles bended by magnetic field (in accelerators) are called “synchrotron radiation”.

• Synchrotron radiation is important for high energy electron (positron) accelerators. Not for protons or heavier particle accelerators.

Radiation power：



 

 



 ⁴₂ ² ₄ ² , ₂

mc E m

B

P E 







Orbit radius

Particle mass

Energy Magnetic field

Electron, positron storage ring: repeat of

• Energy loss in synchrotron radiation

• Acceleration in RF cavities

RF cavities

Radiation Damping

Amplitudes of Synchrotron oscillation and betatron oscillation become small. (damped)

• Damping of synchrotron oscillation

– Higher energy particles lose more energy due to radiation

• Damping of betatron oscillation

– Acceleration in common direction makes angular divergence small.

(see next slides)

Radiation Damping of synchrotron oscillation (Rough discussion)

• Radiation power is proportional to E²B² , then, energy loss in 1-turn is (approximately) proportional to E² .

(Radiation is important only for ultra-relativistic particles.

Therefore, we use approximation

𝐸 ∝ 𝑝

• The higher the energy, the larger the energy loss.

Higher energy particles lose energy Lower energy particles gain energy

Slide for Synchrotron oscillation (linear approximation)

• Energy change per turn

𝑑𝛿

𝑑𝑛 = 𝑒𝑉

𝐸₀ sin( 𝜔𝑡 + 𝜓₀) − 𝑈₀

𝐸₀ = 𝑒𝑉

𝐸₀ sin( 𝜔𝑡) cos 𝜓₀ + cos( 𝜔𝑡) sin 𝜓₀ − 𝑈₀ 𝐸₀

≈ 𝑒𝑉𝜔 cos 𝜓₀ 𝐸₀ 𝑡

𝛿 ≡ 𝑝 − 𝑝₀

𝑝₀ ≈ 𝐸 − 𝐸₀ 𝐸₀

𝑒𝑉 sin 𝜓₀ = 𝑈₀ : Design energy loss/turn Should be replaced

（U₀ → U） Should be added

-(𝑈 − 𝑈₀)/𝐸₀

𝑈 ≈ 𝑈₀𝐸²/𝐸₀² ≈ 𝑈₀ 1 + 𝛿 ² ≈ 𝑈₀ + 2𝑈₀𝛿

Radiation Damping of synchrotron oscillation (Rough discussion)

• Energy change per turn

• Equation for energy, by combining with time change per turn

• Amplitude damps as

• Time for amplitude becomes 1/e ：damping time exp −𝑛𝑈₀/𝐸₀ = exp −(𝑡/𝑇₀)𝑈₀/𝐸₀ 𝑑𝛿

𝑑𝑛 = 𝑒𝑉

𝐸₀ sin( 𝜔𝑡 + 𝜓₀) − 𝑈

𝐸₀ ≈ 𝑒𝑉𝑇₀𝜔 cos 𝜓₀

𝐸₀ 𝑡 − 2𝑈₀ 𝐸₀ 𝛿 𝑑²𝛿

𝑑𝑛² = −Ω²𝛿 − 2𝑈₀ 𝐸₀

𝑑𝛿 𝑑𝑛



 



  −

 n

E n U i n

0

exp 0

) 0 (

~ )

( 



0 0

0 /

~ T E U

z

𝑑𝑡

𝑑𝑛 ≈ 𝑇₀𝜂_𝑐𝛿

Number of turns in 𝜏_𝑧 : 𝐸₀/𝑈₀

→ Total radiation energy in 𝜏_𝑧 : 𝐸₀

Radiation Damping of betatron oscillation (Rough discussion)

• Lost momentum in radiation is almost parallel to particle’s momentum.

– Transverse components are also reduced. (Angle of each particle’s motion is not changed.)

• Acceleration is common for all particles

– Transverse momentum is not changed. Angle of particle motion is reduced.

• Every turn, angle of motion reduce as

Acceleration

p

A B

C

Radiation

Beam direction '

) / ( '

' x U₀ E₀ x x → −

Radiation Damping of betatron oscillation (Rough discussion) 2

Change of Courant-Snyder invariance (square of amplitude of oscillation) from

(change in one turn) Average of many turns

(

)

2 (1 )/ x 2 xx' x'

a = +  +  +

( )



( )

2 1 /

' ,

2 / '

0 0 2 2

2 2 2

2

E U

a a

a x

a xx

−

=





= +

−

= 

 

(

)

Does not depend on 𝛼, 𝛽

Radiation Damping of betatron oscillation (Rough discussion) 3

• Change of Courant-Snyder invariance per 1-turn

• Damping of Courant-Snyder invariance (n: turn number)

• Amplitude of betatron oscillation is square-root of this,

• Time for amplitude becomes 1/e ： Damping time

Δ 𝑎² = −𝑎²(𝑈₀/𝐸₀) → 𝑑(𝑎²)/𝑑𝑛 = −𝑎²(𝑈₀/𝐸₀)

) / exp(

) 0 ( )

( ² _{0 0}

2 n a nU E

a  −

)) 2

/(

exp(

) 0 ( )

(n a nU₀ E₀

a  −

)) 2

/(

exp(

) 0 ( )

(t a tU₀ E₀T₀

a  −

z y

x T E U 

 _, ~ 2 _{0 0} / ₀ ~ 2

Radiation Damping (Rough discussion) Summary

• Damping of synchrotron oscillation

– Higher energy particles lose more energy by radiation

• Damping of betatron oscillation

– Acceleration in common direction makes angular divergence small.

• Time for amplitude becomes 1/e ： Damping time

0 0

0

,_y ~ 2T E /U

x

0 0

0 /

~ T E U

z

Note on Radiation Damping (Rough discussion)

Damping of momentum spread and bunch length

Damping of transverse emittance

0 0

0

,_y ~ 2T E /U

x

0 0

0 /

~ T E U

z

𝑝/𝑝₀ − 1 ² ∝ 𝑧² ∝ exp( − 2𝑡/𝜏_𝑧)

) /

2 exp(

) 0 ( )

( _{, ,}

,y x y x y

x t







Note on Radiation Damping (some details)

Radiation power is proportional to E²B².

But we considered only dependence on E² . Dependence on B² can be important.

There are two effects

• Length in bending filed depends on x – Outer orbit is longer by factor

• Field strength (B) depends on x. （combined bend）

Our “Rough discussion” is to be corrected considering these effects.

(no more discussions here)

(

)

SKIP

Radiation excitation

• Quantum effect in synchrotron radiation: random fluctuation of particle energies → excite oscillations

• Excitation of synchrotron oscillation: Directly from random energy change

• Excitation of betatron oscillation: Through coupling

between transverse motion and longitudinal motion

(Dispersion).

Fluctuation of energy

• Quantum effect

• In radiation (quantum theory) – number of photons is finite

– Energy of each photon is finite

– Emission of photons are randomly occur Energy of each particle in radiation: 𝐸 → 𝐸′

Average of particles: ,

Rlative energy deviation: 𝛿 ≡ (𝐸 − 𝐸 )/ 𝐸 , ( 𝛿 is always 0) 𝑑 𝛿²

𝑑𝑠 = ℏ𝑟_𝑒𝑚𝑐³𝛾₀⁷ 𝜌³

ℏ: Planck constant 𝑟_𝑒: classical radius 𝑚: mass 𝑐: velocity of light 𝛾₀ = 𝐸 /𝑚𝑐² 𝜌: curvature radius

Property of synchrotron radiation １

• Expected number of photons with energy between u and u+du in unit time by a charged particle: n(u)du

• S is expressed by K (modified Bessel function)

• u_c is “critical energy”:

u du S u

u u mc

du r u n

c c

e 



 



= ₂ ⁻²  27

) 8

(





) 2 / 3

( c ³  u_c = 

( )

_

S ( )

8 3 9

3 /

 5

This is zero in classical theory.

Nonzero in quantum theory:

Essential for radiation excitation.

Property of synchrotron radiation 2

• Average radiation power (energy per time)

• Average number of photons

• Average energy of one photon

• Average of square of energy of one photon uc

du P u

n ^

8 3 ) 15

( =

=



N

uc

P du u n du

u un

u 15 3

/ 8 )

( /

)

( 0

0 = =

=





2 4 3

0 3

) 2

( 

 r mc 

du u un

P =



2 0

0 2 2

27 ) 11

( /

)

(u du n u du u_c

n u

u =





are zero in classical theory.

They are non zero in quantum theory.

, u2

u

) 2 / 3

( c ³  u_c = 

s

E

1 turn

E

Preparation for “Energy change by radiation”

Re-define reference energy and energy deviation

• Beam energy is not constant

• Low before acc. cavity

• High after acc. cavity

• Different reference energies at different locations ： E

• Average ： E

• Energy deviation

RF cavities

00 0)/ (E − E E

 

SKIP

Energy deviation change by radiation

Consider energy change in infinitesimal distance ds

• Average of energy change

• Average of change of ”energy deviation ” is zero.

– Change of E₀ is same as average of change of E

• Average of square of change of energy deviation is not zero

• Assume very small ds, emitted number of photon is 0 or 1 (ignore possibility of emitting two or more photons),

average of change of  ² is

) /

) (

(  E − E₀ E₀₀

(

)

u

un( ) / /

0 = 



( )

( )

c ds E

u P c

ds E

E u c

ds N u c ds N u

E E E

E E

E E

E

c 2 00 2

00 2 2

00 2 2

2 00 2

00 2

00 00

3 24 / 55

/

/ )

/ (

) /

/ (

= 

→

−

=



−



=



−



N





Ignore the term of ds²

SKIP

Considering energy change in infinitesimal distance ds Average change of  ² (square of energy deviation) is

Δ𝛿

= 𝑢

ሶN 𝐸

𝑑𝑠

𝑐 = 55 24 3

𝑃

𝑢

𝐸

𝑑𝑠

𝑐 ∝ 𝛾

𝜌

𝑃_𝛾 = 2𝑟_𝑒𝑚𝑐³ 3

𝛾⁴ 𝜌²

) 2 / 3

( c ³  u_c = 

Radiation excitation of synchrotron oscillation

• Consider change of oscillation amplitude by random energy change by radiation

• Square of amplitude is

• Second term is not changed in radiation.

• Then, change of is change of

• Change in 1-turn:

2 0 0

2 0

2 cos

E t T

A eV

c



 

 = + −

2

A  ²

c ds E

u

A P ₂^c

00 2

2

3 24

55 _

 =  =



uc

E P T dn

dA 

2 00

0 2

3 24

= 55

( )

_

= ⁰

0

3 0

7 0 3 2

7 0

3 1/ ^{e cT} (1/ )

e

c ds

T mc mc r

r u

P_    ^  

Radiation excitation of betatron oscillation Qualitative explanation

Radiation at non zero Dispersion

Closed orbit for  = 





=

= =

0

0) , ( )

, (

0  





s x s

x x ^C

Closed orbit for

 = 





+

=

− +

=

− =

= a

a a

s x s

x s

a x

x _x ^{C C x}

0

0

0) ( , )

, ( )

, (









 



On closed orbit

Off closed orbit

) ( )

( )

(

) ( )

( )

(

s y s

s y

s x s

s x

y x













+

=

+

=

Radiation excitation of betatron oscillation

• Divide position and angle into closed orbit and betatron oscillation

（same for y）

• Change of betatron oscillation part by energy deviation change



  

 x , x' ' x'

x = + = +

0 ' ,

0  =

=

x x









 = −   = − 



 x , x' '





Radiation excitation of betatron oscillation

• Divide position and angle into closed orbit and betatron oscillation

（same for y）

• Consider change of Courant-Snyder invariance by change of 

• In 1-turn



  

 x , x' ' x'

x = + = +

( )

(

)

2 = (1+ )/   + 2'+' 

a

= 0





(

)

u ds E P^{ c}

 H

H ₂

00

2 1

3 24

= 55



=

uc

E P T dn

da ₂ H 

00 0 2

3 24

= 55



 



 =



3 0

7

0 ( / )

c cT e

c ds

T mc u r

P  

 H

H 

(

)

2 (1  )/ x_ 2x_ x'_ x'_

a = + + +

Radiation Excitation Summary

• Particle energy randomly fluctuate (quantum effect)

• As direct effect, synchrotron oscillation is excited

• Through dispersion, betatron oscillation is excited c

ds E

u P c

ds E

u

2 00 2

00 2 2

3 24

55

 = =

 N 

 

0

3

) /

cT

(

 ds H





0

3

) /

1 (

cT

 ds

( ^{H  ( 1 + }

^{) / }

^{+ 2  ' +  '}

)

No excitation of vertical oscillation if orbit is in horizontal plane, without errors.

Equilibrium state

• Emittance becomes “equilibrium”, where excitation rate and damping rate are the same.

Large emittance: damping > excitation → emittance reducing Small emittance: damping < excitation → emittance increasing NOTE： Oscillation amplitude of each particle is changing. But distribution (many particles in beam) is almost constant.

0

damp ,

, excite

,

,  =



 

 +



 





dn d dn

d_{x y z} _{x y z}

const

,

excite ,

,

, , damp

, ,

 =



 





 



 





dn d

dn d

z y x

z y x z

y x



 

(

)

Effects of Synchrotron Radiation to Beam Summary

• Radiation Damping

• Radiation Excitation

– Quantum effect. Finite photon energy.

– Random energy change → excite synchrotron oscillation – Random energy change ＋ Dispersion → excite betatron

oscillation

• Equilibrium Emittance

– Equilibrium between damping and excitation

Spin (Polarization) mechanics

𝑑𝑃

𝑑𝑡 = − 𝑒

𝑚𝛾 1 + 𝐺𝛾 𝐵_⊥ + 1 + 𝐺 𝐵_∥ + 𝐺𝛾 + 𝛾 𝛾 + 1

𝐸 × 𝑣Ԧ

𝑐² × 𝑃

𝛾 = 1/ 1 − 𝑣/𝑐 ²,

𝐺 ≈ 0.00116 (electron), 2.79 (proton): anomalous magnetic moment 𝐵_⊥, 𝐵_∥ : magnetic field perpendicular, parallel to particle velocity 𝐸 : electric field

Spin (Polarization) mechanics

Polarization vector

𝑃

average of spin vector of particles in beam

(This is probably not a good definition.

Spin should be treated in quantum mechanics.

But, if there are many particles, this classical view may be OK.

Anyways, following equations are correct for accurately defined 𝑃.)Ԧ

𝑃

𝑑𝑃

𝑑𝑡 = − 𝑒

𝑚𝛾 1 + 𝐺𝛾 𝐵_⊥ × 𝑃

Polarization in Transverse magnetic field (bending field)

This has the same form of the equation of motion ( 𝑝: Ԧ momentum)

𝑑 Ԧ𝑝

𝑑𝑡 = − 𝑒

𝑚𝛾𝐵_⊥ × Ԧ𝑝

Orbit angle change by 𝜃 → polarization angle change by 1 + 𝐺𝛾 𝜃

Relative change: 𝐺𝛾𝜃 For example,

longitudinal polarization can be changed to transverse polarization.

𝑑𝑃

𝑑𝑡 = − 𝑒

𝑚𝛾 1 + 𝐺 𝐵_∥ × 𝑃

Polarization in Longitudinal magnetic field (solenoid field)

Δ𝜙 = න1 𝑣

𝑑𝜙

𝑑𝑡 𝑑𝑠 = 𝑒

𝑝 1 + 𝐺 න𝐵_∥ 𝑑𝑠 Rotate transverse component of

polarization vector.

B

Manipulation of Polarization

Set of bending (horizontal bending) and solenoid Change direction of polarization arbitrary

WIEN FILTER

B

E

Setting 𝐸_𝑦 = −𝑣_𝑧𝐵_𝑥, no orbit change

(B: x direction, E: y direction, v: z direction) 𝑑 Ԧ𝑝

𝑑𝑡 = 𝑒 𝐸 + Ԧ𝑣 × 𝐵

y

x

z v

𝑑𝑃

𝑑𝑡 = 𝑒𝐵_𝑥(1 + 𝐺)

𝑚𝛾² 𝑒Ԧ_𝑥 × 𝑃

Rotate polarization around x axis.

Practical only for low energy beam.

Contents

• Coordinate system

• Basics of transverse motion

• Betatron oscillation

• Weak-focus and Strong-focus

• Effects of extra (infinitesimal) field

• Longitudinal motion (synchrotron oscillation)

• Effects of synchrotron radiation to beam

• Polarization

• Hamiltonian formula ?

Examples of Analytical Mechanics for Beam Dynamics

May be skipped

Analytical mechanics (Lagrange Formula)

) , , , , , ,

(x₁ x₁ x₂ x₂ x_n x_n

L    

Lagrangian for motion of n degree of freedom

2 0

1





 



 



 



 



 







− 

 + 



 







= 



 





 + 



= 

→

2

1 2

1 2

1

0

t

t i

i i i

t

t i i i

t

t i

i i

i

dt x x

L dt

d x

x L x

L

dt x x

x L x

L













 

For any small change of

) ( , ), ( ),

( ₂

1 t x t x t

x  _n

) , , 2 , 1 (

0 i n

x L dt

d x

L

i i

  = = 



 







− 



→ 



 

 = dt x_i dxⁱ

( 





)

Basic principle

(Principle of least action)

Example of Lagrangian

= 0

−

−

 =



 







− 



 mx

dx dU x

L dt

d x

L 



) 2 (

2 U x

m x

L =  −

1 dimensional motion of non-relativistic particle in potential U

This is equation of motion

dx x dU

m

F = = −

𝜕𝐿

𝜕𝑥 − 𝑑 𝑑𝑡

𝜕𝐿

𝜕 ሶ𝑥 = −𝑘𝑥 − 𝑚 ሷ𝑥 = 0 𝐿 = 𝑚

2 ሶ𝑥² − 𝑘 2 𝑥²

𝐹 = 𝑚 ሷ𝑥 = 𝑘𝑥

Harmonic Oscillation

Hamilton formula

• Generalized momentum (canonical conjugate of generalized coordinate)

• Hamiltonian

• Hamilton equations

Hamiltonian is constant of motion if it does not explicitly include t

i

i x

p L



 

L x

p t

p x p

x p x H

i

i i n

n 



, , ; ) ,

, , ,

( _{1 1 2 2}

i i

i i

x H dt

dp p

H dt

dx



−

 =

=  ,

t H t

H p

H dt

dp x

H dt

dx dt

dH

i i

i i

i



= 

 + 



 





 + 



=



Example of Hamiltonian

) 2 (

2 U x

m x

L =  −

𝑑𝑝

𝑑𝑡 = −𝑑𝑈 𝑑𝑥

x x m

p L 

 =



 

U m p

L x p t

p x

H  − = ² +

2 ) 1

; ,

( 

1 dimensional motion of non-relativistic particle in potential U

Generalized Momentum

Hamiltonian (Equal to total energy)

Hamilton Equation

The 2^nd equation is equation of motion:

dx dU x

H dt

dp m

p p

H dt

dx = −



−

=

 =

=  ,

𝑈(𝑥) = 𝑘 2𝑥² 𝐻(𝑥, 𝑝; 𝑡) = 1

2𝑚𝑝² + 1

2𝑘𝑥² 𝑑𝑝

𝑑𝑡 = −𝑘𝑥

Harmonic Oscillation

Canonical Transformation

Canonical Transformation using Generating function '

), , ( )

,

(x p → X P H → H

For new variables

This is satisfied if

Total time derivative of any function can be added

This is satisfied by setting

because

Note on canonical transformation

Similar for 𝐺₂, 𝐺₃, 𝐺₄

Hamilton formula for betatron oscillation

Start from Hamiltonian

Hamiltonian equations

These give the equation of motion

Change variables by a generating function

Remember the equation for the beta-function

New Hamiltonian is

Change independent variable

Additional two minor steps

Change variables by a generating function

Variable change by a constant factor

(this is not canonical, but Hamilton equations are kept)

Then we have Hamiltonian

SKIP

Describe Betatron oscillation by Hamiltonian of normalized coordinate

• Normalized coordinate

Independent variable:

• Hamiltonian

• Hamilton equation

(harmonic oscillation)

2 per 1-turn

) )

( cos(

) ( )

(s = a







Describe Betatron oscillation by action-phase

• Generating function

• New Hamiltonian and Hamilton equation J

J

H(, ;) =





) ( /2) tan ,

( ²

1 X X

G = − (X,P) →(,J)

 





  ⁼

= 

 =

−

= J

H d

d H

d

dJ 0,

(J/ : Courant-Snyder Invariance)

Hamilton Formula for

charged particle in electro-magnetic field

- Basics

Lagrangian for a relativistic charged particle in electro-magnetic field L

( ) ^{r c q} ( ^{r A} )

mc 



 − − 

−

−

=

1 /



L

field the

of potential vector

:

field the

of potential scalaer

:

particle the

of ector position v

:

light of

velocity :

rest at

particle the

of mass :

A

dt r r d

r c m













 







 











=



− 



−

=

A B

t A E







 

 