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Kirk,<br>
<br>
they are not invariant. To avoid long discussion here, please have
a look at Lebedev-Bogacz paper:<br>
<a class="moz-txt-link-freetext" href="http://iopscience.iop.org/1748-0221/5/10/P10010/pdf/1748-0221_5_10_P10010.pdf">http://iopscience.iop.org/1748-0221/5/10/P10010/pdf/1748-0221_5_10_P10010.pdf</a>,
<br>
the very end of it, pp. 21-23. You see that the 2 emittances of
e-beam born at the magnetized cathode, \epsilon_1 and \epsilon_2 may
differ by orders of magnitude. This is actual case for e-beam of our
e-cooler. <br>
<br>
Alexey.<br>
<br>
On 3/10/2011 4:42 PM, Kirk T McDonald wrote:
<blockquote cite="mid:608291C1C4744041A10D5279278A9353@mumu30"
type="cite">
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<div>Alexey,</div>
<div> </div>
<div>For the subspace (q,p) we have</div>
<div> </div>
<div>dq’ dp’ = J dq dp</div>
<div> </div>
<div>J = | dq’/dq dq’/dp |</div>
<div> | dp’/dq dp’/dp |</div>
<div> </div>
<div>Suppose p = m v + A (in units where e/c = 1)</div>
<div>and we transform</div>
<div>q’ = q</div>
<div>p’ = mv = p – A(q)</div>
<div> </div>
<div>Then the Jacobian is</div>
<div>J = | 1 0 |</div>
<div> | –dA/dq 1 | = 1</div>
<div> </div>
<div>It looks to me like the partial phase volumes are also
invariant under the “transformation” of neglecting the
vector potential.</div>
<div> </div>
<div>--Kirk</div>
<div> </div>
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245, 245);">
<div style=""><b>From:</b> <a moz-do-not-send="true"
title="burov@fnal.gov" href="mailto:burov@fnal.gov">Alexey
Burov</a> </div>
<div><b>Sent:</b> Thursday, March 10, 2011 5:33 PM</div>
<div><b>To:</b> <a moz-do-not-send="true"
title="map-l@lists.bnl.gov"
href="mailto:map-l@lists.bnl.gov">map-l@lists.bnl.gov</a>
</div>
<div><b>Subject:</b> Re: [MAP] Liouville's theorem and
electromagnetic fields</div>
</div>
</div>
<div> </div>
</div>
<div style="font-style: normal; display: inline; font-family:
'Calibri'; color: rgb(0, 0, 0); font-size: small;
font-weight: normal; text-decoration: none;">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:
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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
title="http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase"
href="http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase"
moz-do-not-send="true">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
title="http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf"
href="http://puhep1.princeton.edu/%7Emcdonald/examples/cylindrical.pdf"
moz-do-not-send="true">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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