Imperfections
8
Imperfections Eric Prebys, FNAL
description
Imperfections. Eric Prebys, FNAL. Dipole Error (or Correction). Remember this one forever. 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. Example: Local Correction (“Three Bump”). - PowerPoint PPT Presentation
Transcript of Imperfections
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
xx
xx
xx
xx
JJIJ
J
1
2
sincoscoscossincos
sin21
sincossin21
sincossin21
sin21sin2
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
cossincos
sin2
0sincoscos
cossincossin21
0
0
0
0
xx
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)(
ssss
sM
00000
0000
00
000
2sin)(2cos2sin)(2cos12sin)(
2sin)(2sin)(2sin)(2cos
1101
2sin)(2cos2sin)(2sin)(2sin)(2cos
)(
ssf
s
sfss
fssss
sM
00 2sin)(212cos)(
212cos s
fs M
fs
sf
)(41
2sin)(212cos2sin22cos2cos 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
dsBB
fd
1
dsBsBs
)()(41
