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One remark to Swann's paper: <br>
His theorem relates to the total emittance, not to the partial ones.
Partial emittances are sensitive to eA/c term. <br>
<br>
A possible way to get rid of eA/c inside solenoidal structures is to
make a fake 0-length edge of the solenoid at a place where
emittances are calculated; kicks from the edge solenoidal fields
have to be taken into account, of course. <br>
<br>
Alexey. <br>
<br>
On 3/10/2011 4:09 PM, Kirk T McDonald wrote:
<blockquote cite="mid:468B48A3C96B4BA3AA66387F9E650168@mumu30"
type="cite">
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<div>Folks,</div>
<div> </div>
<div>There is a technical question as to how we should be
calculating emittance for beams in electromagnetic fields.</div>
<div> </div>
<div>The formal theory of Liouville’s theorem is clear that
the invariant volume in phase space is to be calculated with
the canonical momentum</div>
<div>gamma m v + e A / c</div>
<div>and not the mechanical momentum m v.</div>
<div> </div>
<div>This is awkward in two ways:</div>
<div>1. We don’t always know the vector potential of our
fields</div>
<div>2. The vector potential is subject to gauge
transformations, so canonical momentum is not gauge
invariant.</div>
<div> </div>
<div>The second issue is disconcerting in that it suggests
that phase-space volume, and emittance, are not actually
invariant -- with respect to gauge transformations.</div>
<div> </div>
<div>Hence, it is useful to note a very old paper,</div>
<div>W.F.G. Swann, Phys. Rev. 44, 233 (1933)</div>
<div>which shows that the phase-space volume for a set of
noninteracting particles is the same whether or not the term
e A / c is included in the “momentum”.</div>
<div> </div>
<div>This result has the consequence that phase-space volume
(and emittance) is actually gauge invariant – although the
location of a volume element in space space is gauge
dependent.</div>
<div> </div>
<div>---------------</div>
<div>This suggests that we could simply calculate emittances
based only on the mechanical momentum, and avoid having to
worry about the accuracy of our model for the vector
potential.</div>
<div> </div>
<div>Of course, our calculations are actually of rms
emittance, which is a better representation of the “ideal”
emittance if the phase-space volume is more “spherical”, and
not elongated/twisted.</div>
<div> </div>
<div>It could be that the shape of the phase-space volume is
better for rms emittance calculation if the vector
potential, in some favored gauge, is included in the
calculation.....</div>
<div> </div>
<div>--Kirk</div>
<div> </div>
<div>PS I have placed Swann’s paper as DocDB 560</div>
<div><a moz-do-not-send="true"
title="http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase"
href="http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase">http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase</a></div>
<div>user = ionization pass = mucollider1</div>
<div> </div>
<div>See also the paper by Lemaitre that used Liouville’s
theorem for cosmic rays in the Earth’s atmosphere (using
mechanical momentum). This may well be the earliest paper
about particle beams and Liouville’s theorem.</div>
<div> </div>
<div>PPS Scott Berg notes that when one evaluates emittance
at a fixed plane in space, rather than at a fixed time, it
is better to use the “longitudinal” coordinates (E,t) rather
than (P_z,z).</div>
<div> </div>
<div>Is there any written reference that explains this “well
known” fact?</div>
<div> </div>
<div>How is this prescription affected by electromagnetic
fields?</div>
<div> </div>
<div>The vector potential of even a simple rf accelerating
cavity has an A_z component (which is zero on axis, but
nonzero off it).</div>
<div><a moz-do-not-send="true"
title="http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf"
href="http://puhep1.princeton.edu/%7Emcdonald/examples/cylindrical.pdf">http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf</a></div>
<div>Note that the vector potential is nonzero outside the
cavity, even though the E and B fields are zero there!</div>
<div> </div>
<div>Do we know how to include A_z in our longitudinal
emittance calculations?</div>
</div>
</div>
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