Beam Dynamics –2

Beam Dynamics –2

2021.05.06

K.Kubo

Transfer matrix summary

• Drift space (free space)

• Quadrupole field thin lens

• Dipole field – sector

– with edge effect:

• Determinant =1

1 𝐿 0 1

cosh 𝐾 𝑙 1

𝐾 sinh 𝐾 𝑙 𝐾 sinh 𝐾 𝑙 cosh 𝐾 𝑙 cos 𝐾 𝑙 1

𝐾 sin 𝐾 𝑙

−𝐾 sin 𝐾 𝑙 cos 𝐾 𝑙

, 1 0

±𝑘 1

cos 𝜃 𝜌 sin 𝜃

− 1

𝜌 sin 𝜃 cos 𝜃 , 1 𝜌𝜃 0 1

thin lens × sector × thin lens

Motion described by transfer matrix

• Transverse linear motion can be described by 4x4 Transfer Matrix. 2x2 Matrix without coupling

• Transfer matrix of

– drift(free) space – dipole magnet

– quadrupole magnet

All expressed by same form of quadrupole field

• Transfer matrix of any beam line can be calculated by

multiplying matrix of (short) components

𝑠 ₀ 𝑠 ₁ 𝑠 ₂ …

… 𝑠 _𝑖

𝑠 _𝑁

… 𝑠 _𝑁−1 𝑀 _0→1 𝑀 _1→2

𝑀 _2→3

𝑠 _𝑖+1 …

𝑀 _{𝑖→𝑖+1} 𝑀 _{𝑁−1→𝑁}

𝑀 _0→𝑁 = 𝑀 _{𝑁−1→𝑁} … 𝑀 _{𝑖→𝑖+1} 𝑀 _{𝑖−1→𝑖} … 𝑀 _1→2 𝑀 _0→1

Dividing beam line into small pieces, (using computer)) transfer matrix can be accurately calculated.

May be enough for describing motions in given beam line.

But, not enough for designing beam line. (?)

More intuitive(?

直観的に分かりやすい

) method is desirable.

Treat motions as oscillations

Transverse motion in circular

accelerator

Closed orbit

（閉軌道）

• After one turn, return to the same position and angle:

Closed Orbit

– Particles on this orbit are circulating the same orbit – Without errors, this should be the design orbit

• Closed orbit depends on particle energy

Design orbit

Closed orbit

• From here, for a while, x, y, etc. denote deviation from closed orbit

• （ or, closed orbit = design orbit ）

• Particles oscillate around the closed orbit

Closed orbit

Actual orbit

Particles oscillate around the closed orbit

Linear motion with 1 degree of freedom

0 )

(

'' + K s x = x

1 degree of freedom

• Ignore x-y coupling. (No skew quadrupole field)

• Consider motions in only x or y direction.

Linear motion

• Linear motion around closed orbit can be expressed by a differential equation

• K is strength of quadrupole field at location s.

• In a circular accelerator, K is periodic

• where L is circumference of the accelerator.

) ( )

( L s K s

K + =

Linear motion with 1 degree of freedom

0 )

(

'' + K s x = x

1 degree of freedom

• Ignore x-y coupling. (No skew quadrupole field)

• Consider motions in only x or y direction.

Linear motion

• Linear motion around closed orbit can be expressed by a differential equation

• K is strength of quadrupole field at location s.

• In a circular accelerator, K is periodic

• where L is circumference of the accelerator.

) ( )

( L s K s

K + =

Transfer Matrix of all “linear”

components (drift space, dipole field,

quadrupole field) have the same form as

quadrupole field)

If K is constant, the equation can be solved easily

Generally, K depends on s,

solution can be expressed as

Linear motion with 1 degree of freedom

0 )

(

'' + K s x = x

Generally, for a periodic function , differential equation

) ( )

( L s K s

K + =

) )

( cos(

) ( )

( s = a  s  s + 

₀

x beta-function

Hill’s equation has this form of solution. There are two cases:

,  , a are all real.  stable oscillation

,  , a are imaginary.  divergent.

Such accelerators do not work has solution with a form

“Hill’s equation”

with periodic function 

: Depend on s, common for all particles

: Constant, different for different particles

) cos(  

₀

 +

= a x

0 )

(

'' + K s x = x

 



 

 + − +

= cos( ) ' sin( )

2

' '  

₀

   

₀

 a 

x

  + 

 

 



 +

 −

 

 +

 





 



 − −

= ' ' '' sin( )

) cos(

' 4

' 2

'' ''

₃ ² _{0 0}

2

    



 





 



 a 

x

0 '

4 ' 2

''

₂

3

2

− + =

−   







 K ' ' '' 0

= +  

 



Motion is not solved yet.

①

②

③ Substituting ② and ③ into ① ,

and require coefficients of sine and cosine to be zero

0 '

4 ' 2

''

₂

3

2

− + =

−   







 K

0 '' ' '

= +  

 



( ) ' ' 0 '' 1

' '

=

=

+ 

 



 

   ' = const ( ) = 1

0 4 1

' 2

''

² ₂

= +

−

−  

 K

(this constant can be set as 1 without losing generality.

Because of another constant a,

 can be multiplied by any constant.) (1)

(2)

From (2)

(1) becomes

In principle,  can be solved if K is known.

(usually numerically solved)

(cannot generally be solved analytically)

Betatron oscillation

) )

( cos(

) ( )

( s = a  s  s +  ₀ x

( ^{sin( ( ) ) ( ) cos( ( ) )} )

) ) (

(

'   ₀    ₀

 ^{+ + +}

−

= s s s

s s a

x

ds d 

 2

− 1 ds 

d / /

1  = 

phase n

oscillatio

 :

Oscillation around closed orbit,

with s-dependent amplitude and phase advance

𝛽 : beta-function, represents s-dependence of amplitude

Beta-function

If

second-order differential equation with two boundary conditions

→ No ambiguity Ambiguous without periodicity

(linac, transport line)

≠ 0

In a region of 𝐾 = 0

𝛽 𝑠 = 1 + 𝛼 ² (0)

𝛽(0) 𝑠 ² − 2𝛼 0 𝑠 + 𝛽 0

𝛼 ≡ − 1 2

𝑑𝛽 𝑑𝑠

In a drift space, beta-function is a 2

^nd

order function of s.

(Downward-convex parabola)

𝛽𝛽 ^′′

2 − 𝛽 ^′2

4 − 1 = 0 → 𝛽𝛽 ^′′

2 − 𝛽 ^′2 4

′

= 𝛽𝛽 ^′′′

2 = 0 → 𝛽 ^′′′ = 0

Transfer-matrix can be written by betatron oscillation parameters,

M(s ₁ → s ₂ ) : expressed by

 (s ₁ ),  (s ₁ ),  (s ₁ ),  (s ₂ ),  (s ₂ ),  (s ₂ )

Some manipulations of betatron oscillation equations

) )

( cos(

) ( )

( s ⁼ a  s  s ⁺  ₀ x

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

) ) (

(

'   ₀    ₀

 ^{+ + +}

−

= s s s

s s a

x

( )























































sin ) ( /

1 )

( ' cos

) ( /

sin ) )

( sin(

cos ) )

( cos(

) )

( cos(

) )

( cos(

) (

1 1 1

1 1

2 1

1 2

0 1

2 0

1 2

0 1

2 0

2 2

2

s x s

x s

x

s a

s a

s a

s a

s x

−

−

−

=

+

− +

=

+ +

= +

=

(  =  ^{( s}

₂

⁾ −  ^{( s}

₁

⁾ )

Relation of x, x’ at s

₁

and s

₂

: Phase advance between s

₁

and s

₂

Transfer-matrix written by betatron oscillation parameters

𝑥 𝑠 ₂ = 𝛽 ₂

𝛽 ₁ 𝑥 𝑠 ₁ cos 𝜓 − 𝛽 ₂ − 𝛽 ₁ 𝑥 ^′ 𝑠 ₁ − 𝛼 ₁ 1

𝛽 ₁ 𝑥 𝑠 ₁ sin 𝜓

= ^𝛽

²

𝛽

₁

cos 𝜓 + 𝛼 ₁ ^𝛽

²

𝛽

₁

sin 𝜓 𝑥 𝑠 ₁ + 𝛽 ₁ 𝛽 ₂ sin 𝜓 𝑥′(𝑠 ₁ )

( )

( ) ( ) ( )

 

 

 





 

 

 





− − +

− +

+

=





 





























 



sin cos cos

sin 1

sin sin

cos )

, (

2 2

1 1

2

1 2

1 2

1 2 1

1 2

1 2

s s M

(  =  ^{( s}

₂

⁾ −  ^{( s}

₁

⁾ )

Transfer matrix from s1 to s2

𝑥 ^′ (𝑠 ₂ ) = ⋯

Transfer-matrix written by betatron oscillation parameters

( )

( ) ( ) ( )

 

 

 





 

 

 





− − +

− +

+

=





 





























 



sin cos cos

sin 1

sin sin

cos )

, (

2 2

1 1

2

1 2

1 2

1 2 1

1 2

1 2

s s M

(  =  ( s

₂

) −  ( s

₁

) )

Inversely, change of the parameters can be given by transfer matrix (formula will be given later)

 ₁ ,  ₁ ,  ₁ →  ₂ ,  ₂ ,  ₂

Determinant of Transfer-matrix is 1

( )

( 1 ) sin ( ) cos ( cos sin ) ¹

sin sin

cos det

2 2

1 1

2

1 2

1 2

1 2 1

1 2

=

 

 

 





 

 

 





− − +

− +

+





 





























 



Determinant of Transfer-matrix is 1 for linear motion

Dividing the beam line into N short sections.

“Tune” in circular accelerator

Important parameter of circular accelerator Phase advance in one turn







 ( L + s ) − ( s ) = = 2

Does not depend on s.

（ This can be shown from is periodic function ）

y x 

 , are called “tunes”

ds d / /

1  = 

s ₁

s ₂

 is periodic:

(same at the same position at any turns)

Phase change in one turn does

not depend on position.

One-turn, multi-turn Transfer-matrix

( )

( ) ^{( )}

 

 





 

 





+ −

−

+

=

+   

















sin sin cos

1

sin sin

cos )

,

( L s s

₂

M

( )

( )

( ) ^{( )}

 

 





 

 





+ −

−

+

= +

=

+ 1 sin( ) cos( ) sin( )

) sin(

) sin(

) cos(

) , (

) ,

(

₂

  

















n n n

n n

n s

s L M s

s nL

M

ⁿ

1 turn

n turns

phase advance becomes n times

( 𝜑 = 2𝜋𝜈)

Stability Condition

( )

( ) ^{( )}

 

 





 

 





+ −

−

+

=

+   

















sin sin cos

1

sin sin

cos )

,

( L s s

₂

M 1-turn transfer-matrix

2 cos

2 )

( = 

 Tr M 

 is real → stable

2 ) cosh(

2 cos

2 )

( = = 

 Tr M  i 

 is (pure) imaginary → unstable

Absolute of Trace can be used for checking stability.

cos and sin include

exponentially divergent term

cos and sin are oscillating

Look trance of the matrix (= 2 cos 𝜑)

( )

( ) ^{( )}

 

 





 

 





+ −

−

+

=

+   

















sin sin cos

1

sin sin

cos )

,

( L s s

₂

M





 , , Should be either All real or

All pure imaginary

All components are real. (transformation of physical values.)

→ Trace 𝑀 = 𝑀

₁₁

+ 𝑀

₂₂

= 2 cos 𝜑 = 𝑟𝑒𝑎𝑙

→ 𝜑 is either real or pure imaginary

𝛼 sin 𝜑, 𝛽 sin 𝜑 should be real (𝑀

₁₁

− 𝑀

₂₂

, 𝑀

₁₂

are real)

→ if 𝜑 is real, sin 𝜑 real → 𝛼, 𝛽 real.

if 𝜑 is pure imaginary, sin 𝜑 ∶ pure imaginary → 𝛼, 𝛽: pureimaginary.

Are the parameters real or imaginary?

( )

( )

( ) ^{( )}

 

 





 

 





+ −

−

+

= +

=

+ 1 sin( ) cos( ) sin( )

) sin(

) sin(

) cos(

) , (

) ,

(

₂

  

















n n n

n n

n s

s L M s

s nL

M

ⁿ

n turns

Divergent (becomes infinitely large for infinitely large n ) if 𝜑 is imaginary.

2 ) cosh(

2 cos

2 )

( = = 

 Tr M  i 

𝑇𝑟𝑎𝑐𝑒(𝑀) can be used for checking stability

x a

^1/2

x'

a(1+

²

/)

^1/2

a/

^

x'=-(/)x

Betatron oscillation In phase space, (x-x’)

a(1+

²

/

^1/2

)

At a certain location, x-x’

of one particle at any turns are all on the same ellipse.

At a certain location, x-x’

of any particles with same a are all on the same

ellipse



 cos a

x =

(    )

 ^{sin cos}

' = − a +

x Same amplitude

Constant a, change 

Courant-Snyder Invariance

2 2

2 2 2

2

1 2 ' '

1 '

a x

xx x

x x

x  = + + + =



 

  +

+  







 



From betatron oscillation formula, the following is ocnstant

（ s independent, determined by initial condition ）

Invariance of each particle

Consider many particles in a beam

Emittance

2 2

2 x ' xx '

x  x −

 Invariant in linear motion

: Average of particles

ds x dx ds x

xx dx ds

x dx x

ds x x dx ds

d '

' '

' 2 '

2 '

) 2

( 

²

=

₂

+

₂

− +

x s ds K

x d ds

dx ds

x dx ' ( )

,

'

₂

2

= −

=

=

inserting , this becomes zero.

Conservation of Emitance can be also shown from that the determinant of transfer-matrix is 1. (next next page)

2

2

'

, ' '

, ' '

' ,

' K x

ds x dx x

ds x xx dx

ds K x dx ds xx

x dx = = − = = −



Meaning of Emittance ?

If particles are distributed uniformly in an ellipse, Emittance ~ Area of ellipse in phase space (x, x’)

x=ac

x’ = a/c x’ = ab/c

x’

x

Meaning of Emittance ?

For general distribution

Emittance = Area of ellipse of 1-sigma in phase space/

𝑥

²

𝑥

^′2

𝜀 𝑥

²

𝜀

𝑥′

²

Emittance is invariant, shown from Beam Matrix form

2 2

2 x ' xx '

x  x −



( ^' )

det ' '

'

det ₂ '

2

2 x x

x x x

xx

xx x

x 



 



= 

 





 



= 

 

) , ' (

' ) '

, ' (

'

'

1 2 )

( 2 2

1 2 )

( 2 2

1 2

s s x M

xx

xx s x

s x M

xx

xx

x

_T

s s

 





 



= 

 





 





Change of beam matrix from s1 to s2 can be expressed by transfermatrix

) det det

det ,

det (det

, 1 det

B A

AB A

A M

T = = 

=

Determinant unchanged because determinant of transfer-matrix is 1

→ Emittance is invariant

Emittance - oscillation parameters

2 0 0

2 0

2 2

0 2 2

2 sin  cos  sin  cos 

 = a a − a

Betatron oscillation

Emittance can be expressed as

a 、 

₀

are different for different particles (initial condition).

Average ＜＞ should be taken for these parameters.

It is easy to show emittance is unchanged by adding common constant to 

₀

.

𝑥 = 𝑎 𝛽 cos(𝜙 + 𝜙 ₀ ) 𝑥′ = − 𝑎

𝛽 sin(𝜙 + 𝜙 ₀ ) + 𝛼 cos(𝜙 + 𝜙 ₀ )

2 2

2 x ' xx '

x  x −



Emittance, beam size, matching

In special case, 

₀

distributes uniformly from 0 to 2, （ no

correlation between a and 

₀

and average can be taken independently ） , Same beam size at the same beta-function.

) )

( ( cos )

( )

( )

( ²  ^{2 2}   ₀

 _x s  x s = s a s + Beam size can be defined as

Generally, beam size may be different at different turns at the same location in a circular accelerator.

) ( 2

/ ) ( )

( s a

²

s s

x

 

 = =

This condition is called “matched”.

Beam parameters in matched condition

Bea msize  = 

emittance  = a

²

/ 2



= x 2



−

= ' xx

𝑥′ ² = 𝜀 1 + 𝛼 ²

𝛽 = 𝜀𝛾

𝑥

²

= 𝑎

²

/2 𝛽 𝑥𝑥′ = − 𝑎

²

/2 𝛼 𝑥′

²

= 𝑎

²

/2 1 + 𝛼

²

𝛽 = 𝑎

²

/2 𝛾

Average over particles Average of one particle for many turns (no relation with matching)

“Matching” means:

Particle distribution and beam line design are matched.

Parameters Twiss

called are

, ,  



Note on “matched beam”

For the same emittance beam, matched beam gives the minimum required aperture.

beam size:

If not matched, beam size at a certain location changes turn by turn.

The maximum beam size at s is larger than  (s ) )

 (s

Change of Twiss parameters

( ) ^



 





−

= −

 





 



= 

 

 













2 2

' '

' '

' xx x

xx x x

x x x

) , ( )

,

(

_{2 1}

1 1

1 1

1 2 2

2

2

2

M s s  M

^T

s s



 





−

= −

 

 





−

−

















 

 



  −  +



 

 

 , ¹

²

2 , 1

ds

d

M : transfer matrix

𝑥

𝑥′ 𝑥 𝑥′ = 𝑥 ² 𝑥𝑥′

𝑥𝑥′ 𝑥′ ²

𝑥 ₂ ² 𝑥 ₂ 𝑥 ₂ ^′

𝑥 ₂ 𝑥 ₂ ^′ 𝑥′ ₂ ² = 𝑥 ₂

𝑥 ₂ ^′ 𝑥 ₂ 𝑥 ₂ ^′

= 𝑀 𝑥 ₁

𝑥 ₁ ^′ 𝑥 ₁ 𝑥 ₁ ^′ 𝑀 ^𝑇 = 𝑀 𝑥 ₁ ² 𝑥 ₁ 𝑥 ₁ ^′

𝑥 ₁ 𝑥 ₁ ^′ 𝑥′ ₁ ² 𝑀 ^𝑇

Because

Betatron Oscillation Summary

) )

( cos(

) ( )

( s = a  s  s +  ₀ x

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

) ) (

(

'   ₀    ₀

 ^{+ + +}

−

= s s s

s s a

x

ds d 

 2

− 1

= ds

d  /

 = (  ^,  ^{( L} + ^{s )} =  ^,  ^{( s )} )

• Tune: Phase advance in one turn divided by 2

• Stability condition:

• Courant-Snyder invariance ：

• emittance ：

• matching ：

Particle distribution and beam line design are matched

( )

²

2

' /

) / 1

(  x +  x +   x

2 2

2 x ' xx '

x  x −



2 )

matrix -

Transfer turn

- 1

( 

Tr ds

d / /

1  = 