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Kirk, <br>
<br>
<font face="Arial" size="2"> epsilon1 and epsilon2 are called
emittances,<font face="Arial"> </font></font>because they are
emittances =<font size="2"> <big>phase volumes for two eigenvectors
of the 4D matrix of beam second moments (sigma matrix).</big></font><big><font
size="2"><big> </big></font></big>They are invariant under any
symplectic transformations. <br>
<br>
You may convince yourself that this result is correct in a following
numerical way: generate a numerical ensemble of electrons born at
magnetized cathode, inside a solenoid. Propagate them along that
solenoid, until they will come out, in a free space. Calculate
sigma-matrix there, find its eigenvectors. In the basis of its
eigenvectors, sigma-matrix is diagonal with 2 emittances on its main
diagonal. See Eqs. (3.26, 3.27) of that paper. Indices 1 and 2
relate to 2 eigenmodes of the sigma-matrix. Generally, they are not
just x and y planar modes, because the beam state is x-y coupled. <br>
<br>
Alexey.<br>
<br>
On 3/10/2011 5:15 PM, Kirk T McDonald wrote:
<blockquote cite="mid:98FEE0D2849743468CEB0449A30CF803@mumu30"
type="cite">
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small; font-weight: normal; text-decoration: none;"><font
face="Arial" size="2">Alexey,</font></div>
</div>
<div>
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small; font-weight: normal; text-decoration: none;"> </div>
</div>
<div>
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font-family: 'Calibri'; color: rgb(0, 0, 0); font-size:
small; font-weight: normal; text-decoration: none;"><font
face="Arial" size="2">I understand your hope to “avoid
long discussion”, as the Lededev/Bogacz paper is more or
less incomprehensible to me.</font></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;"> </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;"><font
face="Arial" size="2">It is not clear why parameters
epsilon1 and epsilon2 are called “emittances”, since
they are not invariants.</font></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;"> </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;"><font
face="Arial" size="2">And, I don’t know what indices 1
and 2 refer to.</font></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;"> </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;"><font
face="Arial" size="2">Etc.</font></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;"> </div>
</div>
<div>
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font-family: 'Calibri'; color: rgb(0, 0, 0); font-size:
small; font-weight: normal; text-decoration: none;"><font
face="Arial" size="2">If Valery or Alex care to
enlighten me, that would be most welcome.</font></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;"> </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;"><font
face="Arial" size="2">--Kirk</font></div>
</div>
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<div style="background: none repeat scroll 0% 0% rgb(245,
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 6:02 PM</div>
<div><b>To:</b> <a moz-do-not-send="true"
title="kirkmcd@Princeton.EDU"
href="mailto:kirkmcd@Princeton.EDU">Kirk T McDonald</a>
</div>
<div><b>Cc:</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;">Kirk,<br>
<br>
they are not invariant. To avoid long discussion here,
please have a look at Lebedev-Bogacz paper:<br>
<a moz-do-not-send="true" 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">
<div dir="ltr">
<div style="font-family: 'Arial'; color: rgb(0, 0, 0);
font-size: 10pt;">
<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>
<div style="font-style: normal; display: inline;
font-family: 'Calibri'; color: rgb(0, 0, 0);
font-size: small; font-weight: normal;
text-decoration: none;">
<div style="font: 10pt tahoma;">
<div> </div>
<div style="background: none repeat scroll 0% 0%
rgb(245, 245, 245);">
<div><b>From:</b> <a title="burov@fnal.gov"
href="mailto:burov@fnal.gov"
moz-do-not-send="true">Alexey Burov</a> </div>
<div><b>Sent:</b> Thursday, March 10, 2011 5:33
PM</div>
<div><b>To:</b> <a title="map-l@lists.bnl.gov"
href="mailto:map-l@lists.bnl.gov"
moz-do-not-send="true">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:
<blockquote
cite="mid:468B48A3C96B4BA3AA66387F9E650168@mumu30"
type="cite">
<div dir="ltr">
<div style="font-family: 'Arial'; color: rgb(0,
0, 0); font-size: 10pt;">
<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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