Betatron Motion with Coupling of Horizontal and Vertical Degrees …casa.jlab.org/publications/viewgraphs/USPAS2013/L_9_CouplingII.pdf · Betatron Motion with Coupling of Horizontal

Operated by JSA for the U.S. Department of Energy

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 10-21, 2013

Alex Bogacz, Geoff Krafft and Timofey Zolkin




Outline

Practical Examples:

Spin Rotator for Figure-8 Collider ring

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

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


http://dx.doi.org/10.1088/1748-0221/5/10/P10010

http://dx.doi.org/10.1088/1748-0221/5/10/P10010

http://dx.doi.org/10.1088/1748-0221/5/10/P10010



Lecture 9 Coupled Betatron Motion II 3 USPAS, Fort Collins, CO, June 10-21, 2013

Axisymmetric rotational distribution Twiss functions

Fermilab electron cooling

The electron beam distribution

is axially symmetric, and

uncoupled at the cathode:

00

00

00

00

00

00

00

00

1

T

BΞ

where 0/ PmkTr ccT 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

000

00

2

00

000

000

2

0

0

0

0

0

1

T

B

T

in ΦΞΦΞ

where

100

0100

010

0001

Φ

cPeB 02/ is the rotational focusing strength of the

solenoid edge

B is the solenoid magnetic field.





The eigen-vectors of the rotational distribution:

2

2

2

2

ˆ

ii

i

i

1v ,

2

2

2

2

ˆ2

i

ii

i

v

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

Then, the matrix V̂ is

2

1

2

1

00

2

1

2

1

00

V̂





4D-emmitance conservation:

2

21 T

Rotational emittance estimate

0

2

Trot rrrr

Comparing left and right hand sides of the equation

TT

in UVVUΞ ˆ

/1000

0/100

00/10

000/1

ˆˆ

2

2

1

1

One obtains

.21

,21

,12

,12

0

1

0

2

0

22

0

1

0

2

0

21

2

0

2

0

2

0

2

0

0

0

TT

T

T




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 20000

x [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

0.1

5

-0.1

5

BE

TA

_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

BE

TA

_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

1

5

0

1

-1

BE

TA

_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

BE

TA

_X

&Y

[m]

DIS

P_X

&Y

[m]


5 GeV





Ionization Cooling in an Axially Symmetric Channel

A single-particle phase-space trajectory along the

beam orbit can be expressed as:

,)(ˆ)(ˆRe)(ˆ))((

22

))((

112211 sisi

esess vvx

One can rewrite the above equations in the following compact form

)()(ˆ)(ˆ sss aVx

where

)(ˆ),(ˆ),(ˆ),(ˆ)(ˆ 2211 sssss vvvvV

))(sin(

))(cos(

))(sin(

))(cos(

)(

222

222

111

111

s

s

s

s

sa





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

2

2

2

2

ˆ

ii

i

i

1v ,

2

2

2

2

ˆ2

i

ii

i

v

2

1

2

1

00

2

1

2

1

00

V̂

here we used that u = 1/2, 1 = 2 = /2





Cooling Description

Ionization cooling due to energy loss in a thin absorber can be

described as:

p

p ,

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

1000

0100

0010

0001

ˆ 1




Canonical variables

.2

,2

xR

yp

yR

xp

y

x

PceBR s / - longitudinal magnetic field

Relation between geometrical and canonical variables

Rxx̂ ,

where

1002

0100

0210

0001

,,ˆ

R

R

y

x

p

y

p

x

y

x

y

xRxx

,

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

1000

0100

0010

0001

ˆ 1

and finally

inout aVRRVa ˆ

1000

0100

0010

0001

ˆ 11





Carrying out the above calculation explicitly one obtains:

inout

RR

RR

RR

RR

aa

2

1

2

12

1

2

12

1

2

12

1

2

1

1000

0100

0010

0001

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

)(sin1cos211

)(sin1cos211

2

212

22

2

2

211

22

1

43

21

ORRaa

ORRaa

outout

outout

where 2121





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





If cooling effect of one absorber is sufficiently small one

can perform averaging over betatron phases. That

yields

R

R

1

1

22

11

ds

dp

p

R

ds

d

ds

dp

p

R

ds

d

0

2

2

0

1

1

11

11





Canonical momentum of a single particle

2ˆ

0000

0000

0100

1000

ˆˆ

0000

0000

0100

1000

ˆ 21aVaVxxMTT

xy ypxp





Second order moments of the Gaussian distribution

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

0

,2

,4

41

,

,

21

21

21

222

21

21

22

yx

xy

yx

yx

ppxy

Mypxp

pp

ypxp

yx





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:

04

1

43

2

2

2R

ds

d .

The standard recipe determines the alpha-function:

ds

d

2

1




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

15

0

50

BE

TA

_X

&Y

[m]

DIS

P_X

&Y

[m]

BETA_X BETA_Y

7.20

0.5

0P

HA

SE

_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

0P

HA

SE

_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) = /2





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

7

0

2-2

BE

TA

_X

&Y

[m]

DIS

P_X

&Y

[m]


betatron phase adv/cell (h/v) = /2





‘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

x

Px





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

2

2

2

2

2

1

0

0

1

1

F

F

i

i

i

ii

rotation

Rotational eigen-vectors

2

2

F

F

i

2

2

F

Fi





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

2

0

22

0

2

0

21TT

Lattice implementation – Twiss functions, beam sizes etc.





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

1 = 7.14 10-3 cm, 2 = 3.24 10-8 cm

2.62647 0

1

0

5

0

BE

TA

_X

&Y

[m]


2.62647 0

1

0

1

-1

Beta

tro

n s

ize X

&Y

[cm

]

Alp

haX

Y[-

1,

+1]

Ax Ay AlphaXY 2.62647 0

0.5

0

0.5

0

PH

AS

E/(

2*P

I)

Q_1 Q_2 Teta1 Teta2

2.62647 0

1

0

5

0

BE

TA

_X

&Y

[m]





Summary

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)

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).

Betatron Motion with Coupling of Horizontal and Vertical Degrees …casa.jlab.org/publications/viewgraphs/USPAS2013/L_9_CouplingII.pdf · Betatron Motion with Coupling of Horizontal

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