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Kirk, <br>
<br>
the first question in that business should be: what are
sub-emittances for arbitrary coupled beam state? Correct answer is:
they are diagonal elements of the sigma-matrix in a basis of its
eigenvectors. These two (or 3 in 3D coupled case) values are
invariant under any symplectic transformations - that is why they
are so important. "Leaving the vector-potential out of the
calculations" changes the partial emittances. This non-symplectic
procedure is equivalent to neglect of a kick from solenoidal edge
fields - which may easily lead to severe errors. <br>
<br>
Alexey.<br>
<br>
On 3/10/2011 5:27 PM, Kirk T McDonald wrote:
<blockquote cite="mid:9511C40A71744557AB0AABA7FFB821DE@mumu30"
type="cite">
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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">Alexey,</font></div>
</div>
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small; font-weight: normal; text-decoration: none;"> </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">A further comment is the Swann’s
method does not show that “subemittances” are invariant
in TIME, but it seems to show that they are invariant
under leaving the vector potential out of their
calculation.</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">------------</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;"><font
face="Arial" size="2">Maybe Lebedev and Bogacz
considered an example in which two “subemittances”
evolved with time such that one increased and the other
decreased, whereby the “total” emittance remained
invariant.</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">This in no way precludes that
these subemittances would have the same (time-dependent)
values if the vector potential were ignored in their
calculation.</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>
<div>
<div style="font: 10pt tahoma;">
<div style="background: none repeat scroll 0% 0% rgb(245,
245, 245);">
<div style=""><b>From:</b> <a moz-do-not-send="true"
title="kirkmcd@Princeton.EDU"
href="mailto:kirkmcd@Princeton.EDU">Kirk T McDonald</a>
</div>
<div><b>Sent:</b> Thursday, March 10, 2011 6:15 PM</div>
<div><b>To:</b> <a moz-do-not-send="true"
title="burov@fnal.gov" href="mailto:burov@fnal.gov">Alexey
Burov</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;">
<div dir="ltr">
<div style="font-family: 'Arial'; color: rgb(0, 0, 0);
font-size: 10pt;">
<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">Alexey,</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">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>
<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">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>
<div>
<div style="font: 10pt tahoma;">
<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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