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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.


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,…)


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


[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


[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.


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:


(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


hx0i = 0 hxi = 0

⌃ =

hx2i hxx0ihxx0i hx02i

�

|⌃| � 0

beam matrix

•  oben parameterized in term of Courant-‐Snyder parameters :


⌃ = "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


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


⌃ = 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)


|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:


"

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.


"

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


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


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 .


F (x, x0)

�

x

x

2 + 2↵x

xx

0 + �

x

x

02 = "

x

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