wilson lab tour guide orientation 11 december 2006 classe 1 focusing and bending wilson lab tour...

Wilson Lab Tour Guide Orientation 11 December 2006

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1Focusing and Bending

Wilson LabTour GuideOrientation

M. Forster

Mike Forster11 December 2006

Focusing and BendingOr

How to Build Your Own Storage Ring


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Lorentz Force Equation

)x( BvE qF

Wikipedia.org


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Make a Ring

• Electrostatic plates or Dipoles?

B

E

)x( BvE qF )x( BE cqF BE c


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Magnet Choices

• Permanent Magnets– Fixed field – Ferrites typically up to .05-.10 T– Rare earth alloys typically up to

1.2 T or more

• Iron-Copper Electromagnets– Copper coils around iron

laminations

• Superconducting Magnets– Cryogenically cooled– Conventional or Superferric


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Beam Coordinates

x

y

s

ReferenceParticleTrajectory

ActualParticleTrajectory


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Describing the Motion

ysx BevF RmvF s

r

2

),,(),,(

1syxB

p

e

syxR y

...!3

1

!2

1)( 3

3

32

2

2

0 xdx

Bdx

dx

Bdx

dx

dBBxB yyy

yy


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Multipole Expansion

...!3

1

!2

1)( 3

3

32

2

2

0 xdx

Bdx

dx

Bdx

dx

dBBxB yyy

yy

...!3

1

!2

1)( 3

3

32

2

2

0 xdx

Bd

p

ex

dx

Bd

p

ex

dx

dB

p

eB

p

exB

p

e yyyyy

...!3

1

!2

11 32 oxmxkxR

Dipole

Quadrupole

Sextupole

Octupole

The field around the beam can be seen as a sum of multipoles.


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Focusing Methods

• Dipole Beam Loss– Horizontally stable– Vertical unstable

• Weak Focusing– n = - (dB/B)/(dr/r)– Stability in both planes

requires 0<n<1– Gains vertical focusing at the

expense of horizontal

• Combined Function Magnets– More than one multipole term– Focusing to Bending Ratio is

fixed

B


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Focusing Methods

• Strong or Alternate Gradient Focusing (Christofilos 1950, Courant et al. 1952)

– First used operationally at Cornell in 1954

– Alternating combined function magnets

"If you can't do two things together, you just do one after the other - that's all there is to it!" Ernest Courant, Brookhaven


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Quadrupoles

• Hyperbolic shaped poles

• Focusing in one transverse plane, defocusing in the other

From J. Crittenden


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From circe.lnl.gov

Good field region


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Trajectory Equations

0)()()(''

)(

1)()(

)(

1)(''

2

sysksy

p

p

sRsxsk

sRsx

skBkskAksx

skBskAsx

coshsinh)('

sinhcosh)(

First assume on energy particles: p/p = 0.

Look at horizontal motion in a horizontally defocusing quad (k > 0).

Solutions are:

Use x0 and x’0 as initial conditions for the constants of integration.


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Particle Trajectories and Transfer Matrices

skxskxksx

skk

xskxsx

cosh'sinh)('

sinh'

cosh)(

00

00

Which can be put into matrix notation as:

0

0

'coshsinh

sinh1

cosh

)('

)(

x

x

skskk

skk

sk

sx

sx

M

M is the transfer matrix for a defocusing quadrupole.

Build up a toolbox of transfer matrices for various elements!


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Trajectory Tracking

• Through a section of elements:

– X1 = Mdrift·MQF·Mdrift·MQD·X0

• Or build up to a complete revolution of the ring• Combine transverse planes

0

0

0

0

'

'

0

0

'

'

y

y

x

x

B

A

y

y

x

x


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Dispersion

• Realistically p/p 0.• Only significant when 1/R 0, i.e. in a bend.• Solve for a special trajectory, (s) and ’(s) , when p/p = 1.• A particle with momentum offset p/p will have a horizontal

position of x(s) + (s) (p/p)• Likewise, the angle will be: x’(s) + ’(s) (p/p)

0)()()(''

)(

1)()(

)(

1)(''

2

sysksy

p

p

sRsxsk

sRsx


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Beta Functions and Betatron Oscillations

0)()()('' sxsksx

Need to describe a beam of particles.

Defines motion of transverse oscillation about the orbit called the betatron oscillation.

Trial solution: x(s) = A u(s) cos( (s) + )

After constants of integration are worked out:

))(cos()()( sssx


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Beta Functions and Betatron Oscillations

(s) is the beta function or amplitude function.

is the emittance.

(s) is the phase.

The square root of (s) defines transverse size of beam at any point s.

))(cos()()( sssx

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

)('

ssss

sx


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Courant-Snyder Parameters or Twiss Functions

2

)(')(

ss

)(

)(1)(

2

s

ss

)(s

)(')()(')()(2)()( 22 sxssxsxssxs

These functions combine to describe an ellipse in x-x’ phase space:


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Beam Phase Space

• Particles can be characterized by their position, x, and angle, x’ about the reference orbit.

• The area of the ellipse remains constant and equals the emittance. (Satisfies Liouville’s Theorem.)

x’

x

/ .

/

/

.

/


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From D. Robin USPAS lecture


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Beam cross section

• Electrons and positrons are well approximated by a Gaussian charge distribution:

2

2

2

2

22exp

2),(

yxyx

yxNeyx

• The standard is to use the phase ellipse of the particle at 1 to define the beam emittance.

• Generally look for 10 clearance for good beam lifetime.


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Phase Space Propagation• Twiss parameters can also be propagated through optics

elements with matrices.• Magnet focusing strengths are set to achieve the desired twiss

properties at arbitrary points in the ring.• Typically done in “cells” like FODO cells:

– Focusing Quad, drift or bend, Defocusing Quad, drift or bend

– Designed so Twiss functions at the end of the cell match the beginning

From J.Rossbach CERN Accelerator School lecture


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Periodicity

• FODO Cells• Many large accelerators combine magnet power supplies• Independent magnet control at CESR• Flexibility has contributed to long term viability


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Tunes

• The amount of phase advance through one complete revolution is call the Tune.

)(2

1

s

dsQ

• Integer tunes are trouble.• In fact, a lot of other tunes are trouble:

m Qx + n Qy = p Where m, n, p are integers.

• These are optical resonances• |m| + |n| is the order of the resonance


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Resonances

• Linear Resonances (Integer and half integer)• Non-linear resonances

– Coupling Resonance– Synchro-Betatron Resonance

• Tune Plane

From A. Temnykh talk at Frascati 2005


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Chromatic Errors

• Focusing errors due to energy differences



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Chromatic Errors

• Correct with sextupoles



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Insertion Regions

Match twiss conditions at ends and insert:• RF Cavities• Wigglers• Separators• Transfer Lines• Detector


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Interaction Point Focusing

• SCIR quads• Permanent magnet quads


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One beam will not make luminosity

• Horizontal Separators

• Horizontal tune of 10

• Pretzel Orbit

• Lab ingenuity

J. Crittenden 2004


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Detector Solenoid and Compensation

• Solenoid coupling

• Compensation with: tilted SCIR quads– Fixed 4.5 degree rotation

• SCIR and normal conducting skew quads• Superconducting Anti-solenoids

• Work to reduce any coupling between horizontal and vertical motion

0

0

0

0

'

'

?

?

'

'

y

y

x

x

B

A

y

y

x

x


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Cesr Optics


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Measurement and Correction


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See the real stuff!

http://www.lns.cornell.edu/~hoff/LECTURES/07S_488/index.html

wilson lab tour guide orientation 11 december 2006 classe 1 focusing and bending wilson lab tour...

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