**Cancellation of Magnetic Fields in the Tevatron IPM Magnets**

David Harding

19 February 2003

17 February 2003

14 February 2003

It is proposed to install two or more Ionization Profile Monitors in the
Tevatron, using DC magnets to control the drift path of the electrons to
the detector electronics. These magnets must not disrupt the Tevatron
beam unacceptably. While the exact parameters of the magnet shave not
yet been chosen, their scale is known and can be used to make some
calculations regarding the sensitivity of the beam to various
configurations. Let me call this my toy magnet.

We use the following parameters:

B 0.3 Tesla (this is the highest field discussed to date)

l 0.4 meters (nominal active length of 0.1 m plus field shaping,
fringing)

β~0~ 100 meters (Beta function at the detector)

Bρ 500 meters (approximate for 150 GeV)

L 0.4 meters or 0.3 meters (center to center for two magnets or three
magnets)

C 0.009 T-m (strength dipole corrector running at 1 A)

ν 20.577

$$`<!-- -->`{=html}1) **Is any compensation required?** The nominal
integrated strength of the toy magnet is Bl=0.12 T-m. It produces a bend
θ of Bl/Bρ=0.24 milliradians which propagates around the ring as an
orbit distortion with amplitude

$\frac{\theta\sqrt{\beta(z)}\sqrt{\beta 0}}{2\text{sin}\left( \text{πν} \right)}$Where
in the arcs, the maximum β is 100 m; in the intersection regions it
stays under 400 m. The amplitude of the distortion is in the
neighborhood of 12 mm, or 25 mm at the worst. This is clearly
unacceptable.

Furthermore, the strength of the magnet is equivalent in strength one
corrector dipole running at 13 A, or a quarter of its full range.
Correction with existing magnets is not deemed suitable.

2\) **Can we compensate with a single additional magnet of opposite
polarity?** Bending left, then bending right will introduce a dog leg
into the beam. The beam will enter and leave at the same angle, but and
offset will be introduced which will propagate around the ring. We now
have two waves with the amplitude given above, not quite exactly out of
phase with each other. The phase advance between the magnets is L/β~0~.
The oscillation is then\
cos(ψ−πν-L/2β~0~)-cos(ψ−πν+L/2β~0~).\
Expanding, we get\
\[cos(ψ−πν) cos(L/2β~0~) + sin(ψ−πν) sin(L/2β~0~)\]\
- \[cos(ψ−πν) cos(L/2β~0~) - sin(ψ−πν) sin(L/2β~0~)\]\
The cosine expressions drop out, leaving 2sin(ψ−πν) sin(L/2β~0~). The
oscillation has the same character as the uncompensated case, but is
diminished in amplitude by 2sin(L/2β~0~). Using the small angle
approximation, the reduction is L/β~0~, or approximately a factor of
250. This results in an orbit distortion of amplitude 0.05 mm if not
corrected by the trim dipole(s) (0.1 mm at the IR).

An orbit distortion of 0.05 mm is probably tolerable. Reducing the
amplitude of the orbit distortion by a factor of 250 means that it would
only take trim dipoles running at 0.05 A, 0.1% of their range, to
correct the orbit distortion. If one chose to make the correction, it
would be different with the magnets on and off, a mild argument for
permanent magnets.

3\) **What is the effect of splitting the compensation?** Putting half
the correction on each side of the main magnet leads to three orbit
distortions superimposed on each other.\
-½cos(ψ−πν-L/β~0~) + cos(ψ−πν) - ½cos(ψ−πν+L/β~0~)\
Expanding, we get\
-½\[cos(ψ−πν) cos(L/β~0~) + sin(ψ−πν) sin(L/β~0~)\]\
+ \[cos(ψ−πν)\]\
- ½\[cos(ψ−πν) cos(L/β~0~) - sin(ψ−πν) sin(L/2β~0~)\]\
Canceling the sine terms and gathering the oscillating terms, we get\
cos(ψ−πν) (1-cos(L/β~0~))\
Again approximating, we get a reduction of ½(L/β~0~)^2^, a factor of
over 200,000. This is certainly acceptable.

4\) **How well must the compensating magnet(s) match the main magnet?**
Any mismatch between the compensating field and the main field will
induce an orbit distortion with an amplitude equal to the fractional
mismatch times the number calculated in #1 above and require correction
at the same level. A 10% error (highly improbable), would require
correction by trim dipoles at the 1.5 A level. A 1% error, which would
probably require some trimming of the system in the three-magnet case
magnet, would leave us with orbit distortions at the tenth millimeter
level, perhaps negligible. For a main magnet with integrated field of
0.12 T‑m, this corresponds to a system residual integrated field of
0.012 T‑m or 0.0012 T‑m. Two electromagnets should match each other to
0.1% without difficulty, buying another factor of ten in reduced orbit
distortion and eliminating the need for correction. Permanent magnets
would require trimming to reach that level of cancellation.