ImperfectionsEric Prebys, FNAL

Dipole Error (or Correction) Recall our generic transfer matrix

If we use a dipole to introduce a small bend Θ at one point, it will in general propagate as

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 4 - Imperfections 2

Remember this one forever

Example: Local Correction (“Three Bump”) Consider a particle going down a beam line. By using a

combination of three magnets, we can localize the beam motion to one area of the line

We require

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 4 - Imperfections 3

1

2

3

11,

22 ,

33 ,

1223

231213

Local Bumps are an extremely powerful tool in beam tuning!!

Controls Example From Fermilab “Acnet” control system

The B:xxxx labels indicateindividual trim magnet powersupplies in the FermilabBooster

Defining a “MULT: N” willgroup the N following magnet power supplies

Placing the mouse over them and turning a knobon the control panelwill increment theindividual currents accordingto the ratios shown in green

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 4 - Imperfections 4

Closed Orbit Distortion (“cusp”) We place a dipole at one point in a ring which bends

the beam by an amount Θ. The new equilibrium orbit will be defined by a trajectory

which goes once around the ring, through the dipole, and then returns to its exact initial conditions. That is

Recall that we can express the transfer matrix for a complete revolution as

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 4 - Imperfections 5

0

00

1

0

0

0

0

0

0

0

0

MI

MIM

x

x

x

x

x

x

x

x

JJ

IJ

J

1

2

sincoscos

cossincos

sin2

1

sincossin2

1

sincossin2

1

sin2

1

sin2

sin2

2sin2cos2sin)(2cos2sin)(

2sin)(2sin)(2cos),(

111

2

IJ

JIJJ

JMI

JMI

JIM

J

J

JJJJ

J

e

e

eeee

ess

sssCs

Plug this back in

We now propagate this around the ring

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 4 - Imperfections 6

cossin

cos

sin2

0

sincoscos

cossincos

sin2

1

0

0

0

0

x

x

Quadrupole Errors We can express the matrix for a complete revolution at a point as

If we add focusing quad at this point, we have

We calculate the trace to find the new tune

For small errors

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 4 - Imperfections 7

2sin)(2cos2sin)(

2sin)(2sin)(2cos)(

ss

sssM

00000

0000

00

000

2sin)(2cos2sin)(2cos1

2sin)(

2sin)(2sin)(

2sin)(2cos

11

01

2sin)(2cos2sin)(

2sin)(2sin)(2cos)(

ssf

s

sf

ss

fss

sssM

00 2sin)(2

12cos)(

2

12cos s

fs M

f

s

sf

)(

4

1

2sin)(2

12cos2sin22cos2cos 00000

Total Tune Shift The focal length associated with a local anomalous gradient is

So the total tune shift is

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 4 - Imperfections 8

dsB

B

fd

1

dsB

sBs

)(

)(4

1