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size=2 face=Arial>Alexey,</FONT></DIV></DIV>
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size=2 face=Arial>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>
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<DIV
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size=2 face=Arial>------------</FONT></DIV></DIV>
<DIV>
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size=2 face=Arial>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
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size=2 face=Arial>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>
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size=2 face=Arial>--Kirk</FONT></DIV></DIV>
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<DIV style="FONT: 10pt tahoma">
<DIV style="BACKGROUND: #f5f5f5">
<DIV style="font-color: black"><B>From:</B> <A 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 title=burov@fnal.gov href="mailto:burov@fnal.gov">Alexey
Burov</A> </DIV>
<DIV><B>Cc:</B> <A 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
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size=2 face=Arial>Alexey,</FONT></DIV></DIV>
<DIV>
<DIV
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size=2 face=Arial></FONT> </DIV></DIV>
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size=2 face=Arial>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
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size=2 face=Arial></FONT> </DIV></DIV>
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size=2 face=Arial>It is not clear why parameters epsilon1 and epsilon2 are
called “emittances”, since they are not invariants.</FONT></DIV></DIV>
<DIV>
<DIV
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size=2 face=Arial></FONT> </DIV></DIV>
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size=2 face=Arial>And, I don’t know what indices 1 and 2 refer
to.</FONT></DIV></DIV>
<DIV>
<DIV
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size=2 face=Arial></FONT> </DIV></DIV>
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size=2 face=Arial>Etc.</FONT></DIV></DIV>
<DIV>
<DIV
style="FONT-STYLE: normal; DISPLAY: inline; FONT-FAMILY: 'Calibri'; COLOR: #000000; FONT-SIZE: small; FONT-WEIGHT: normal; TEXT-DECORATION: none"><FONT
size=2 face=Arial></FONT> </DIV></DIV>
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size=2 face=Arial>If Valery or Alex care to enlighten me, that would be most
welcome.</FONT></DIV></DIV>
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size=2 face=Arial></FONT> </DIV></DIV>
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size=2 face=Arial>--Kirk</FONT></DIV></DIV>
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size=2 face=Arial></FONT></DIV>
<DIV style="FONT: 10pt tahoma">
<DIV style="BACKGROUND: #f5f5f5">
<DIV style="font-color: black"><B>From:</B> <A 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 title=kirkmcd@Princeton.EDU
href="mailto:kirkmcd@Princeton.EDU">Kirk T McDonald</A> </DIV>
<DIV><B>Cc:</B> <A 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: #000000; 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 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 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: 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><PRE wrap=""><FIELDSET class=mimeAttachmentHeader></FIELDSET>
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