Accelerator Physics Transverse motion

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Accelerator PhysicsTransverse motion

• Mats Lindroos

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Co-ordinates
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2 particles in a dipole field
Two particles in a dipole field, with different
initial angles
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The horizontal displacement of the second
particle with respect to the first
The second particle oscillates around the first.
This type of oscillation forms the basis of all
transverse motion in an accelerator.
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Dipole
B? magnetic rigidity.
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Magnetic Rigidity
The force evB on a charged particle moving with
velocity v in a dipole field of strength B is
equal to its mass multiplied by its
acceleration towards the centre of its circular
path.
where ? radius of curvature of the path
remember p momentum mv
B? is called the magnetic rigidity, and if we put
in all the correct units we get - B? kG.m
33.356p (if p is in GeV/c ) B? T.m
3.3356p (if p is in GeV/c )
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Quadrupoles
Magnetic field
A quadrupole has 4 poles, (2 North and 2 South)
arranged symmetrically around the beam. No
magnetic field along the central axis
By
Bx
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This example is a Focusing Quadrupole (QF) It
focuses the beam horizontally and defocuses
vertically Rotating the poles by 90 degrees we
get a Defocusing Quadrupole (QD)
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FODO Cell
We will study the FODO cell in detail during the
tutorial!
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As the particles move around the accelerator or
storage ring, whenever their divergence (angle)
causes them to stray too far from the central
trajectory the quadrupoles focus then back
towards the central trajectory. This is rather
like a ball rolling around a circular gutter.
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We characterize the position of the particle in
this transverse motion by two things- Position
or displacement from central path, and angle with
respect to central path.
x displacement x angle dx/ds
This example is for a constant restoring force
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Hills equation
These transverse oscillations are called Betatron
Oscillations, and they exist in both horizontal
and vertical planes. The number of such
oscillations/turn is Qx or Qy. (Betatron Tune)
(Hills Equation) describes this motion
If the restoring force (K) is constant in s
then this is just SHM
Remember s is just longitudinal displacement
around the ring
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K varies strongly with s.
Therefore we need to solve Hills equation for K
varying as a function of s
What happens to the motion of the ball in the
circular gutter if we allow the shape of the
gutter to vary?
The phase advance and the amplitude modulation of
the oscillation are determined by the shape of
the gutter. The overall oscillation amplitude
will depend on the initial conditions, i.e. how
the motion of the ball started.
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Hills equation
To solve it, try
? and ?0 are constants, which depend on the
initial conditions. ?(s) the amplitude
modulation due to the changing focusing
strength. ?(s) the phase advance, which also
depends on focusing strength.
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This will give us
If we plot x .v. x as f goes from 0 to 2p we get
an ellipse, which is called a phase space ellipse
x
x
The area of this ellipse is pe
NB This area does not depend on b, a or g
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x
x
As we move around the machine the shape of this
ellipse will change as b changes under the
influence of the quadrupoles However the area of
the ellipse (pe) will not change
e is called the transverse emittance and is
determined by the initial beam conditions
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x
x
The projection of this ellipse onto the x axis
gives the physical transverse beam
size Therefore the variation of b(s) around the
ring will tell us how the transverse beam size
will vary
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x
emittance
beam
x

• To be rigorous we should define the emittance
  slightly differently.
• Observe all the particles at single position on
  one turn and measure both their position and
  angle
• A large number of points on our phase space
  plot, each corresponding to a pair of x,x values
  for each particle.

The emittance is the area of the ellipse which
contains all (or certain percentage) of the
points or particles
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This conservation of emittance is an illustration
of Liouvilles Theorem.
Liouvilles theorem states that if x is the
transverse position and px is the transverse
mometum, then for a group of particles-
constant
Transverse velocity
emittance (e)
NB bv/c, gm/m0
bge is the normalised emittance and it is
conserved even during longitudinal acceleration e
is only conserved if there is no longitudinal
acceleration
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Matrix formalism
As we move around the ring will vary under
the influence of the dipoles, quadrupoles and
empty (drift) spaces. Express this modification
in terms of a TRANSPORT MATRIX (M). If we know
x1 and x1 at some point s1, After the next
element in the accelerator ring at s2 -
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Drift space - No magnetic fields, length L
x2 x1 L.x1
x1
x1
L
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Quadrupole of length, L
deflection
x1
x2
x1
x2
Remember By K.x and the deflection due to the
magnetic field is
(Provided L is small)
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Define focal length of the quadrupole as f
Define f as positive for a focusing effect then-
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Multiply together our drift space and our
quadrupole matrices to form Transport Matrices,
to describe larger sections of our ring These
matrices will move our particles from one point
(x(s1),x(s1)) on our phase space plot to another
(x(s2),x(s2))
The elements of this matrix are fixed by the
elements through which the particles pass from s1
to s2. However we can also express (x,x) as
solutions to Hills equation-
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These special functions are called TWISS
PARAMETERS
Remember m is the total betatron phase advance
over one complete turn.
Number of betatron oscillations/turn
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It is also interesting to note that b(s) and Q or
m are related
Where m Df over a complete turn
But
Over one complete turn
Increasing the focusing strength decreases the
size of the beam envelope (b) and increases Q and
vice versa.
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Tune correction
What happens if we change the focusing strength
slightly?
This matrix relates the change in the tune to the
change in strength of the quadrupoles. We can
invert this matrix to calculate change in
quadrupole field needed for a given change in tune