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<DIV><FONT size=2 face=Arial>Alex,</FONT></DIV>
<DIV><FONT size=2 face=Arial></FONT> </DIV>
<DIV>"Leaving the vector-potential out of the calculations" changes the partial
emittances.</DIV>
<DIV> </DIV>
<DIV><FONT size=2 face=Arial>I now don't believe this is true.</FONT></DIV>
<DIV><FONT size=2 face=Arial></FONT> </DIV>
<DIV><FONT size=2 face=Arial>Hiding behind words like "symplectic" does not add
insight to this issue.</FONT></DIV>
<DIV><FONT size=2 face=Arial></FONT> </DIV>
<DIV><FONT size=2 face=Arial>--Kirk</FONT></DIV>
<DIV style="FONT: 10pt Tahoma">
<DIV><BR></DIV>
<DIV style="BACKGROUND: #f5f5f5">
<DIV style="font-color: black"><B>From:</B> <A
title="mailto:burov@fnal.gov CTRL + Click to follow link"
href="mailto:burov@fnal.gov">Alexey Burov</A> </DIV>
<DIV><B>Sent:</B> Thursday, March 10, 2011 7:22 PM</DIV>
<DIV><B>To:</B> <A
title="mailto:kirkmcd@Princeton.EDU CTRL + Click to follow link"
href="mailto:kirkmcd@Princeton.EDU">Kirk T McDonald</A> </DIV>
<DIV><B>Cc:</B> <A
title="mailto:map-l@lists.bnl.gov CTRL + Click to follow link"
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><BR></DIV>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">
<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
size=2 face=Arial>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
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>
<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
size=2 face=Arial>------------</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
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
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
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>
<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
size=2 face=Arial>--Kirk</FONT></DIV></DIV>
<DIV>
<DIV style="FONT: 10pt tahoma">
<DIV style="BACKGROUND: rgb(245,245,245)">
<DIV><B>From:</B> <A title=kirkmcd@Princeton.EDU
href="mailto:kirkmcd@Princeton.EDU" moz-do-not-send="true">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"
moz-do-not-send="true">Alexey Burov</A> </DIV>
<DIV><B>Cc:</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">
<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
size=2 face=Arial>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
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
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
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
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
size=2 face=Arial>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
size=2 face=Arial>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
size=2 face=Arial>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
size=2 face=Arial>--Kirk</FONT></DIV></DIV>
<DIV>
<DIV style="FONT: 10pt tahoma">
<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 6:02 PM</DIV>
<DIV><B>To:</B> <A title=kirkmcd@Princeton.EDU
href="mailto:kirkmcd@Princeton.EDU" moz-do-not-send="true">Kirk T McDonald</A>
</DIV>
<DIV><B>Cc:</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">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"
moz-do-not-send="true">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: 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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