Betatron Motion with Coupling of Horizontal and Vertical Degrees … · 2016-06-17 · Operated by...

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Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 1

Betatron Motion with Coupling of Horizontal and Vertical Degrees of

Freedom – Part II

USPAS, Fort Collins, CO, June 13-24, 2016

S.A. Bogacz, G.A. Krafft, S. DeSilva and R. Gamage

Jefferson Lab and Old Dominion University



Outline

  Practical Examples:

  Vertex-to-plane adapter for electron cooling (Fermilab)

  Spin Rotator for Figure-8 Collider ring

  Ionization cooling channel for Neutrino Factory and Muon Collider

  Generalized vertex-to-plane transformer insert

 V. Lebedev, A. Bogacz, ‘Betatron Motion with Coupling of Horizontal and Vertical Degrees of Freedom’, 2000, http://dx.doi.org/10.1088/1748-0221/5/10/P10010



Thomas Jefferson National Accelerator Facility Lecture 9 - Coupled Betatron Motion II 3 USPAS, Fort Collins, CO, June 13-24, 2016

Axisymmetric rotational distribution - Twiss functions

v Fermilab electron cooling The electron beam distribution

is axially symmetric, and uncoupled at the cathode:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

00

00

00

00

0000

0000

1

βα

αγ

βα

αγ

εTBΞ

where 0/ PmkTr ccT =ε is the thermal emittance of the beam

Electrostatic accelerator Solenoid

Gaussian distribution

Rotational distribution




♦ At the exit of the solenoid the electron beam distribution is still axially symmetric

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

Φ−

Φ+Φ

Φ

Φ−Φ+

==

000

002

00

000

0002

0

00

00

1

βαβ

αβγβ

ββα

βαβγ

εTB

Tin ΦΞΦΞ

where

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

Φ−

Φ=

10001000100001

Φ

♠ cPeB 02/=Φ is the rotational focusing strength of the solenoid edge

♠ B is the solenoid magnetic field.




♦ The eigen-vectors of the rotational distribution:

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−

+−

=

βα

ββα

β

22

22

ˆ

ii

i

i

1v ,

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−

+−

=

βα

ββα

β

22

22

ˆ 2

i

ii

i

v

♠ It corresponds to u = 1/2, 2/21 πνν ==

♦ Then, the matrix V̂ is

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

−

−

=

ββα

βα

β

βββα

βββα

ββ

21

21

0021

21

00

V̂




• 4D-emmitance conservation:

2

21 Tεεε =

• Rotational emittance estimate

( ) ( )Φ=Φ=Φ== 02 βεθε Trot rrrr

♦ Comparing left and right hand sides of the equation

TTin UVVUΞ ˆ

/10000/10000/10000/1

ˆˆ

2

2

1

1

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

ε

ε

ε

ε

♠ One obtains

.21

,21

,12

,12

0

1

020

22

01

020

21

20

2

0

20

2

0

0

0

β

ε

ββ

εε

εβββ

εε

β

αα

β

ββ

β

β

Φ⎯⎯ →⎯

Φ+Φ+=

Φ⎯⎯ →⎯Φ−Φ+

=

Φ+=

Φ+=

>>Φ

>>Φ

TT

TT



Spin rotators for Figure-8 Collider Ring

-10000

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

10000

-20000 -15000 -10000 -5000 0 5000 10000 15000 20000x [cm]

z [cm]

Figure-8 Collider Ring - Footprint

total ring circumference ~1000 m

60 deg. crossing




Spin Rotator - Ingredients…

Arc end 4.4 0 8.8 0

Spin rotator ~ 46 m

BL = 11.9 Tesla m BL = 28.7 Tesla m

320 230

15

0

0.15

-0

.15

BETA

_X&Y

[m]

DIS

P_X&

Y[m

]

BETA_X BETA_Y DISP_X DISP_Y




Locally decoupled solenoid pair

solenoid 4.16 m solenoid 4.16 m decoupling quad insert

BL = 28.7 Tesla m

M = C

- C 0 0

17.9032 0

15

0

5 0

BETA

_X&Y

[m]

BETA_1X BETA_2Y BETA_1Y BETA_2X


Hisham Sayed, PhD Thesis ODU, 2011



solenoid 4.16 m solenoid 4.16 m decoupling quad insert

BL = 28.7 Tesla m

M = C

- C 0 0

17.9032 0

15

0

1 -1

BETA

_X&Y

[m]

DIS

P_X&

Y[m

]


Locally decoupled solenoid pair


Hisham Sayed, PhD Thesis ODU, 2011



Universal Spin Rotator - Optics

4.4 0 8.8 0

Spin rotator ~ 46 m

BL = 11.9 Tesla m BL = 28.7 Tesla m

3 7 42 8 8

M o n S e p 0 6 1 7 : 4 4 : 4 5 2 0 1 0 O p t i M - M A I N : - C : \ W o r k i n g \ E L I C \ M E I C \ O p t i c s \ 5 G e V E l e c t e . R i n g \ h a l f _ r i n g _ i n _ s t r a i g h t _ 1 2

30

0

1-

1

BE

TA

_X

&Y

[m

]

DI

SP

_X

&Y

[m

]

B E T A _ XB E T A _ YD I S P _ XD I S P _ Y

374 288

30

0

1 -1

BETA

_X&Y

[m]

DIS

P_X&

Y[m

]


5 GeV




Ionization Cooling in an Axially Symmetric Channel

v A single-particle phase-space trajectory along the beam orbit can be expressed as:

( ) ,)(ˆ)(ˆRe)(ˆ ))((22

))((11

2211 sisi esess µψµψ εε +−+− += vvx

♦ One can rewrite the above equations in the following compact form

)()(ˆ)(ˆ sss aVx =

where

⎥⎦⎤

⎢⎣⎡ ʺ−ʹʺ−ʹ= )(ˆ),(ˆ),(ˆ),(ˆ)(ˆ 2211 sssss vvvvV

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

+

+

+

=

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

)(

222

222

111

111

ssss

s

µψε

µψε

µψε

µψε

a




♦ In the case of axially symmetric focusing the eigen-vectors reduce to:

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−

+−

=

βα

ββα

β

22

22

ˆ

ii

i

i

1v ,

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−

+−

=

βα

ββα

β

22

22

ˆ 2

i

ii

i

v

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

−

−

=

ββα

βα

β

βββα

βββα

ββ

21

21

0021

21

00

V̂

♠ here we used that u = 1/2, ν1 = ν2 =π/2




v Cooling Description ♦ Ionization cooling due to energy loss in a thin absorber can be

described as:

δθθθ ⊥⊥⊥ −≡Δ

−=Δpp ,

♠ here the longitudinal energy restoration by immediate re-acceleration is assumed

♦ Using canonical variables the above cooling equation can be written as:

inout xRRx ˆ

1000010000100001

ˆ 1−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−=

δ

δ



♦ Canonical variables

.2

,2

xRyp

yRxp

y

x

+ʹ=

−ʹ= PceBR s /= - longitudinal magnetic field

♦ Relation between geometrical and canonical variables Rxx =ˆ ,

where

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

≡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

≡

1002010002100001

,,ˆ

R

Ry

x

pypx

y

x

y

x Rxx

θ

θ

,

A ‘cap’ denotes transfer matrices and vectors related to the canonical variables.

Hamiltonian formulation - equations of motion




♦ Employing amplitude vector representation: )()(ˆ)(ˆ sss aVx = , one can rewrite the cooling equation as:

inout aVRRaV ˆ

1000010000100001

ˆ 1−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−=

δ

δ

and finally

inout aVRRVa ˆ

1000010000100001

ˆ 11 −−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−=

δ

δ




♦ Carrying out the above calculation explicitly one obtains:

inout

RR

RR

RR

RR

aa

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−

−

+−−

+−−

+−

−

−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

21

21

21

21

21

21

21

21

1000010000100001

βαα

β

αββ

α

αββ

α

βαα

β

δ

♦ 2D emittances after cooling are given by the following formula:

( )[ ] ( )[ ]

( )[ ] ( )[ ] )(sin1cos211

)(sin1cos2112

21222

2

2211

221

43

21

δδφβφαεεδβεε

δδφβφαεεδβεε

ORRaa

ORRaa

outout

outout

+−−++−=+≡ʹ

++−+−−=+≡ʹ

where 2121 ψψµµφ +++=




v Two alternative descriptions of cooling

♦ After passing through a thin absorber one computes ♠ a new 4D phase space ♠ new partial emittances ♠ new beta-functions

♦ Or one can compute everything relative to the unperturbed beta-functions

♠ Seems like more convenient approach, although partial emittances (actions) depend on betatron phases




v If cooling effect of one absorber is sufficiently small one can perform averaging over betatron phases. That yields

( )( )δβεε

δβεε

RR

+−≈Δ

−−≈Δ

11

22

11 ⇒

dsdp

pR

dsd

dsdp

pR

dsd

0

2

2

0

1

1

11

11

βεε

βεε

+−≈

−−≈




♦ Canonical momentum of a single particle

( )2

ˆ

0000000001001000

ˆˆ

0000000001001000

ˆ 21 εε −=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−

=−= aVaVxxMTT

xy ypxp




♦ Second order moments of the Gaussian distribution

(Note that for a single particle - 2/εε =rms and we use rms. emittances below)

( )

( )

( )

0

,2

,441

,

,

2121

21

222

21

2122

==

−=⇒−

=−=

++

==

+−==

+==

yx

xy

yx

yx

ppxy

Mypxp

pp

ypxp

yx

εεεε

εεβα

εεα

εεβ




v Beta-function for Particle Motion with Axial-symmetric Solenoidal Focusing

♦ Equation for the square root of the beta-function is similar to the

equation for Floque-function in the case of uncoupled motion:

( )0

41

4 3

2

2

2

=−+β

ββ R

dsd .

♠ The standard recipe determines the alpha-function:

dsdβ

α21

−=



Beta functions in axially symmetric FOFO cell

c1 L[cm]=130 B[kG]=38.1 Aperture[cm]=10 c2 L[cm]=80 B[kG]=-34.3 Aperture[cm]=10

7.20

150

50

BETA_X&Y[m]

DISP_X&Y[m]

BETA_X BETA_Y

7.20

0.5

0PHASE_X&Y

Q_X Q_Y

abs abs



20

Fri Mar 10 09:55:38 2006 OptiM - MAIN: - D:\Cooling Ring\SolRing\snake_new.opt

0.5

0PH

ASE

_X&Y

Q_X Q_Y

Periodic Cell - Optics

20

Sat Mar 04 23:06:09 2006 OptiM - MAIN: - D:\Cooling Ring\SolRing\snake_new.opt

70

2-2

BE

TA

_X

&Y

[m]

DIS

P_X

&Y

[m]


‘inward half-cell’ ‘outward half-cell’

betatron phase adv/cell (h/v) = π/2π




Periodic Cell – Magnets ‘inward half-cell’

solenoids: L[cm] B[kG] 22 105 22 105

‘outward half-cell’

dipoles: $L=20; => 20 cm $B= 25; => 25 kGauss $Ang=$L*$B/$Hr; => 0.4996 rad $ang=$Ang*180/$PI; => 28.628 deg

quadrupoles: L[cm] G[kG/cm] 8 1.79754 8 -0.3325 8 1.79754

20

Sat Mar 04 23:06:09 2006 OptiM - MAIN: - D:\Cooling Ring\SolRing\snake_new.opt

70

2-2

BE

TA

_X

&Y

[m]

DIS

P_X

&Y

[m]


betatron phase adv/cell (h/v) = π/2π




‘Snake’ cooling channel – Dispersion suppression

60

Fri Mar 10 10:15:57 2006 OptiM - MAIN: - D:\Cooling Ring\SolRing\end_snake_new.opt

70

2-2

BE

TA

_X

&Y

[m]

DIS

P_X

&Y

[m]

BETA_X BETA_Y DISP_X DISP_Y 60

Fri Mar 10 10:16:18 2006 OptiM - MAIN: - D:\Cooling Ring\SolRing\end_snake_new.opt

70

2-2

BE

TA

_X

&Y

[m]

DIS

P_X

&Y

[m]


periodic cells entrance cell exit cell




Muon Cooling Channel - Optics

100

Wed Jun 21 11:47:31 2006 OptiM - MAIN: - D:\Cooling Ring\Snake Channel\snake_100_1.opt

50

3-3

BE

TA

_X

&Y

[m]

DIS

P_X

&Y

[m]


n periodic PIC/REMEX cells (n=2) beam extension beam extension

RF cavity skew quad

disp. anti-suppr. disp. suppr.

snake - footprint

0

50

100

150

200

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000

z [ c m ]

x [ c m ]




Vertex-to-plane transformer insert

SolenoidSkew-quadrupole system

Uncoupledaxial-symmetricdistribution

Rotational distribution

Flat distribution

xPx




♦ Eigen-vectors of the decoupled motion in the coordinate system rotated by 450

⎥⎦

⎤⎢⎣

⎡≡

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−

+−

⎯⎯ →⎯

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−

+−

⎯⎯ →⎯

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡+

− ==

2

222

1

1

1

1

1

1

.deg45

1

1

1

22

22

21

00

11

FF

βα

ββα

β

βαββαβ

βαβ

ααββ

i

i

i

ii

rotation

♠ Rotational eigen-vectors

⎥⎦

⎤⎢⎣

⎡

2

2

FFi

⎥⎦

⎤⎢⎣

⎡

2

2

FFi




♦ Focusing system with 450 difference between the horizontal and vertical betatron phase advances will transform the initial vertex distribution into the flat one

♦ The resulting 2D emittances are as follows

.1

,1 0

20

22

020

21ββ

εε

ββ

εε

Φ+Φ+=

Φ−Φ+= TT

♦ Lattice implementation – Twiss functions, beam sizes etc.




Ekin = 10 MeV, Tc = 0.2 eV, Rc = 0.5 cm, Bsol = 1 kG,

⇒ ε1 = 7.14⋅10-3 cm, ε2 = 3.24⋅10-8 cm

2.62647 0

1 0

5 0

BET

A_X

&Y[

m]


2.62647 0

1 0

1 -1

Bet

atro

n si

ze X

&Y[

cm]

Alp

haXY

[-1, +

1]

Ax Ay AlphaXY 2.62647 0

0.5

0

0.5

0

PHA

SE/(2

*PI)

Q_1 Q_2 Teta1 Teta2

2.62647 0

1 0

5 0

BET

A_X

&Y[

m]




Summary

v Relationships between the eigen-vectors, beam emittances and the beam ellipsoid in 4D phase space ♦ From the beam ellipsoid to the eigen-vectors (equivalence of both

pictures) v New parametrization of eigen-vectors in terms of generalized Twiss

functions ♦ Complete Weyl-like representation

♠ 10 independent parameters to fully describe the motion ♠ transport line ambiguities resolved

♦ Developed software based on this representation allows effective analysis of coupled betatron motion for both circular accelerators and transfer lines (OptiM).



Alex Bogacz

Ring Design with Imaginary γt




Perturbed FODO Cell

ΔGF1 ΔGF2

17.528 0

40

0

5 -5

BET

A_X

&Y[

m]

DIS

P_X&

Y[m

]


Arc Cell - Perturbed 900 FODO

2 x 900 FODO

ΔGF1 ΔGF2

M56 =Dρds∫

M56 – large, positive M56 – very small, still positive

17.528 0

40

0

5 -5

BET

A_X

&Y[

m]

DIS

P_X&

Y[m

] BETA_X BETA_Y DISP_X DISP_Y

γ =56

tSM




‘Missing Dipole’ Arc Cell - Negative M56 Cell

M56 =Dρds∫

26.2911 0

40

0

5 -5

BET

A_X

&Y[

m]

DIS

P_X&

Y[m

]


ΔGF ΔGF

M56 – small negative

26.2911 0

40

0

5 -5

BET

A_X

&Y[

m]

DIS

P_X&

Y[m

]


ΔGF ΔGF

M56 – always positive

900 FODO – Empty - 900 FODO




21.808 0

40

0

5 -5

BET

A_X

&Y[

m]

DIS

P_X&

Y[m

]


‘New’ Arc Cell - Negative M56 Cell

26.2911 0

40

0

5 -5

BET

A_X

&Y[

m]

DIS

P_X&

Y[m

]



M56 – always positive

M56 – negative




‘Missing Dipole’ vs Perturbed 900 FODO

2 x 900 FODO

M56 =Dρds∫

M56 – small, but negative

M56 – very small, still positive

17.528 0

40

0

5 -5

BET

A_X

&Y[

m]

DIS

P_X&

Y[m

]

BETA_X BETA_Y DISP_X DISP_Y 21.808 0

40

0

5 -5

BET

A_X

&Y[

m]

DIS

P_X&

Y[m

]






85.1122 0

40

0

5 -5

BET

A_X

&Y[

m]

DIS

P_X&

Y[m

]


Arc Optics

Perturbed 1200 FODO

77.8675 0

40

0

5 -5

BET

A_X

&Y[

m]

DIS

P_X&

Y[m

]


Perturbed 900 FODO

Negative M56 Lattice

16 bends

18 bends

2 x cell

2 x cell




P. McIntyre Texas A&M

Dual-dipole + Quad Cryomodule

Dual-dipole Quad

Arc Cell - Super-ferric Magnets

Bend: Lb = 120 cm (magnetic length) Lead ends: 2×22 cm B = 3.13 Tesla bend ang. = 8.12 deg. Sagitta = 2.1 cm

Sextupole: Ls = 10 cm S = 780 Tesla/m2

Quadrupole: Lq = 40 cm G = 10-37 Tesla/m Correctors (H/V): 20 cm BPM can: 20 cm

Magnet aperture radius (inj. at 285 MeV) 6σrms + 10 mm = 41 mm

39

Bend Sextupole Correctors BPM

Quad

17.2 4.4

30

0

3 -3

BET

A_X

&Y[

m]

DIS

P_X&

Y[m

]

BETA_X BETA_Y DISP_X DISP_Y Bend



Arc-Straight Optics

128.463 0

40

0

5 -5

BET

A_X

&Y[

m]

DIS

P_X&

Y[m

]


Arc (85 meters) Straight (43 meters)




injection extraction RF cavity

Crossing angle: 75 deg.

Figure-8 Booster for JLEIC

γ =56

tSM

M56 =Dρds∫

Ekin = 285 MeV – 7.062 GeV Ring circumference: 273 m (~2200/8)

41 USPAS, Fort Collins, CO, June 13-24, 2016



275.087 0

40

0

5 -5

BET

A_X

&Y[

m]

DIS

P_X&

Y[m

]


Inj. Arc Straight Straight Arc

= −56 98M cm

γ = =56

16.8tSM

i

= 275S m

Booster Lattice (8 GeV, γt = 16.8 i )



275.087 0

40

0

5 -5

BET

A_X

&Y[

m]

DIS

P_X&

Y[m

]


Arc Quadrupoles: Lq = 40 cm G = 12-65 Tesla/m

Arc Bends: Lb = 120 cm B = 3.13 Tesla bend ang. = 8.12 deg. Sagitta = 2.1 cm

Straight Quads: Lq = 40 cm GF = 12.6 Tesla/m GD = -24.5 Tesla/m

Booster Lattice (8 GeV, γt = 16.8 i )

Lattice configured with super-ferric magnets

43

Proton beam energy (total) GeV 1.2 - 8

Circumference m 275

Straights’ crossing angle deg 79.8

Arc length m 103 / 85

Straight section length m 43

Maximum hor. / ver. β functions m 22 / 22

Maximum hor. dispersion m 4.3

Hor. / ver. betatron tunes νx,y 7.87 / 5.85

Hor. / ver. natural chromaticities ξx,y -6.8 / -4.6

Momentum compaction factor α -3.6 10-3

Hor. / ver. normalized emittance εx,y µm rad 1 / 1

Maximum hor. / ver. rms beam size at inj. σx,y mm 5.1 / 5.1




  8 GeV Booster design based on imaginary γt lattice   Negative M56 lattice based on lightly perturbed 900 FODO

  Footprint fits into the same circumference of 275 meters (2200/8).

  Lattice configured with super-ferric 3 Tesla magnets (Texas A&M design)

  stronger bends (3.1 Tesla vs 2.7 Tesla)

  substantially lower magnet aperture (4.1 cm vs 5.2 cm)

  fewer number of bends (32 vs 36)

  same number of quads (10 extra arc quads and 10 fewer straight quads)

  Much lower betas (22 m vs 38 m), better tunability, less sensitivity to errors

  Lower dispersion tunes and natural chromaticity

  Superior control of chromaticity with 2 families of sextupoles in each plane

  Lower quadrupole strengths (overall)

Summary


Betatron Motion with Coupling of Horizontal and Vertical Degrees … · 2016-06-17 · Operated by...

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