l3 beam phasespace - northern illinois universitynicadd.niu.edu › ... › slides ›...
Post on 24-Jun-2020
9 Views
Preview:
TRANSCRIPT
Charged par*cle beams and bunches
• defini'on, • phase space and emi0ance, • trace space, • beam matrix and its evolu'on, • phase space, Liouville’s theorem
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 1
trajectory of a single par*cle
• classical mechanics use where posi'on
canonical momentum. • form a set of canonical-‐conjugate variables
• alterna've descrip'on use divergence but are not canonical conjugates.
2
(x,p)x ⌘ (x, y, z)p ⌘ (p
x
, py
, pz
)(x,p)
x
0 ⌘ p
x
/p
z
y0 ⌘ py/pz(x, x0)
phase space
trace space
PHYS 790-‐D Special topics in Beam Physics, Fall 2014
how to characterize a beam?
• Each par'cle in the beam have the following possible a0ributes: – posi'on, momentum, – charge, mass, – spin.
• We will exclusively consider single-‐species beam so that only posi'on and momentum are need for each par'cle in the beam.
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 3
how to characterize a beam?
• Consider a beam with N par'cles • each are characterized by 2 vectors – posi'on – momentum
• so we have 6N scalars! • instead it is sufficient for most purposes to describe the beam with macroscopic quan''es (beam size, divergence,…)
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 4
x ⌘ (x, y, z)p ⌘ (p
x
, py
, pz
)
phase space portraits
• example of the un-‐damped pendulum,
• trajectories in phase space are curves with constant energy
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 5
[see J. Buon’s lecture “beam phase space and emi0ance”]
unstable fixed points
separatrix
unstable trajectories
stable trajectories
phase space portraits
• phase space portraits allows to “track” the trajectory of an object in the phase space,
• phase-‐space portraits difficult to interpret for an ensemble of par'cle
• instead use Poincare sec'on that shows intersec'ons of the phase space trajectories with a plane
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 6
[see J. Buon’s lecture “beam phase space and emi0ance”]
phase space…
• phase space portraits, and Poicare maps are most oben referred to as “phase spaces” in beam physics (no dis'nc'on)
• Also for non periodic system, phase spaces oben display the par'cle coordinates (for the full beam ensemble) at a given axial loca'on along the accelerator’s beamline.
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 7
example of phase space in beam physics
• asdsad
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 8 [s
ee J. Buo
n’s lecture “be
am phase sp
ace and em
i0ance”]
Courtesy of A. Seymour (NIU, 2014)
trace space snapshot
Poincare (stroboscopic) map
phase space characteriza*on
• restrain our discussion to or actually , we can write the phase space density distribu'on as with
• let’s define the moments:
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 9
(x, px
)(x, x0)
F (x, x0)Z Z +1
�1F (x, x0)dxdx0 = 1
hxi =Z Z +1
�1xF (x, x0)dxdx0
hx0i =Z Z +1
�1x
0F (x, x0)dxdx0
hxx0i =Z Z +1
�1xx
0F (x, x0)dxdx0
hx2i =Z Z +1
�1x
2F (x, x0)dxdx0
hx02i =Z Z +1
�1x
02F (x, x0)dxdx0
2nd order moments
1st order
mom
ents
phase space characteriza*on
• for simplicity we assume the 1st-‐order moments are zero: and
• we define the beam (or covariance) matrix as
• the matrix is definite posi've and
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 10
hx0i = 0 hxi = 0
⌃ =
hx2i hxx0ihxx0i hx02i
�
|⌃| � 0
beam matrix
• oben parameterized in term of Courant-‐Snyder parameters :
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 11
⌃ = "2x
�x
�↵x
�↵x
�x
�betatron func'on betatron slope
(↵x
,�x
)
�x
⌘ 1 + ↵2x
�x
determinant=1
"
2x
⌘ hx2ihx2i � hxx0i2
propaga*on of beam matrix
• ABCD formalism (see Lecture 1):
• can be applied to the beam matrix
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 12
accelerator or op'cal beamline with n components
1
X0 = (x0, x00) Xf = (xf , x
0f )
2 n
Xf = RnRn�1...R3R3R1X0
propaga*on of beam matrix
• first recognize that with and is the transpose.
• so that
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 13
⌃ = hXXX eXeXeXiX
X
X = (x, x0) eXXX
fXfXfXf = ]RX0X0X0 = fX0X0X0eR
XfXfXf = RX0X0X0
⌃f = R⌃0eR
emi?ance conserva*on
• Since the determinant and so that the determinant of the beam matrix is a conserved quan'ty ! geometric emi?ance is conserved • this is actually a consequence of Liouville’s theorem (simple case though)
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 14
|R| = 1
|⌃f | = |R⌃0eR| = |⌃0|
emi?ances
• The canonical emi0ance is a conserved quan'ty for linear system dominated by single-‐par'cle dynamics
• Some'me the normalized emi0ance is used:
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 15
"
2n,x
= (��)2⇥hx2ihx02i � hxx0i2
⇤= (��"
x
)2
"
2n,c
=1
m
2c
2
⇥hx2ihp2
x
i � hxpx
i2⇤
emi0ances have the dimension of length
normalized emi?ance
• star'ng from • the normlized emi0ance is defined by wri'ng and assuming
• this is OK for beam with small spread in energy (or longitudinal momentum), in some type of, e.g., electron source this approxima-‐'on does not always hold.
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 16
"
2n,c
=1
m
2c
2
⇥hx2ihp2
x
i � hxpx
i2⇤
p
x
= x
0p
z
pz = hpi = mc��
back to the phase-‐space density distribu*on
• we have assume the phase space can be described by a density distribu'on
• It is convenient to approximate the beam distribu'ons by analy'cal func'on
• BUT a beam is a ensemble of N par'cles. • use Klimontovich’s distribu'on
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 17
F (x, x0)
F (x, x0) =1
N
NX
i=1
�(x� xi, x0 � x
0i)
Dirac’s func'on (two-‐dimensional here)
back to the phase-‐space density distribu*on (cnt’d)
• Then the second order moment are
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 18
hx2i = 1
N
NX
i=1
x
2i hx02i = 1
N
NX
i=1
x
02i
hxx0i = 1
N
NX
i=1
x
0ixi
concept of “equivalent beams”
• the “rms” sta's'cal descrip'on of a beam provide an “equivalent beam descrip'on”
• The par'cle distribu'on is view as “contained” within an ellipse with equa'on:
• other descrip'ons provide an analy'cal form for .
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 19
F (x, x0)
�
x
x
2 + 2↵x
xx
0 + �
x
x
02 = "
x
top related