Download Particle accelerator exercises set 2

Particle accelerator exercises set 2
Erik Adli
October 2014 (rev. 27. Oct 2014)
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Envelope equation
In exercise set 1 we calculated the evolution of the beta function along an accelerator lattice,
as a function of the lattice focusing properties, quantified by K(s). An alternative way of
describing the evolution of the transverse beam size is by use of the envelope equation
σx00 + K(s)σx (s) =
ε2x
.
σx3
Show that by substituting the expression for the rms beam size
σx (s) =
q
εx βx (s)
into the envelope equation, we recover the differential equation for the beta function βx (s),
as presented in the lectures,
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β(s)β 00 − β 02 + K(s)β 2 (s) = 1.
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2
CLIC beam parameters
This exercise will give some familiarity with the beam parameters required for linear colliders.
Many of the parameters, for example the emittances, are at the extreme of what is possible to
achieve with today’s technology. Assume the following bunch parameters at the interaction
point for a 3 TeV Compact LInear Collider (CLIC). Assume that the bunch has a Gaussian
charge distribution both in the transverse dimensions and in the longitudinal dimension.
N = 3.7 × 109 particles/bunch,
E = 1.5 TeV,
σE /E = 0.35%,
εN,x = 660 nm,
εN,y = 20 nm,
σx = 40 nm,
σy = 1 nm,
σz = 44 µm.
a) Calculate the following bunch parameters :
• The peak current
• The peak beam density
• The particle kinetic energy (in x,y and z)
• The particle temperature (in x,y and z)
• The beta functions at the interaction point
b) In CLIC trains with 312 micro-bunches with N particles collide with a repetition rate of
50 Hz. The micro-bunches are spaced 0.5 ns apart. Calculate the following beam parameters
• The total energy of one bunch train
• The average beam current
• The average beam power
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Fields from a relativistic particle
a) Calculate the electric field originating from a relativistic particle, as observed in the laboratory frame, by performing a Lorentz transformation of the field in the particle rest frame
E=
q r
.
4πε0 r3
Show that the electric field is compressed into a flat field with an opening angle of the order
of the inverse of the Lorentz factor, 1/γ.
b) Show that when the velocity approaches the speed of light (γ → ∞) the fields originating
from the particle can be written as
E=
and
1 2qr
δ(z − ct)
4πε0 r2
1
B = ẑ × E.
c
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4
Relativistic space charge cancellation
Assume a cylinder beam with length L and radius a, and uniformly distributed total charge
N e as shown in Figure 1. The beam is traveling with velocity v.
a) Calculate the charge density, ρ, in the laboratory frame.
b) In accelerator physics we usually work with beam lengths as measured in the laboratory
frame. Does the beam length, as measured in the laboratory frame, change as the beam is
accelerated to relativistic velocities? What happens to the beam length as measured in the
beam rest frame? Hint: inside an accelerating cavity the head and the tail of the beam are
experiencing the same acceleration simultaneously in the laboratory frame.
Figure 1: Cylinder beam
In the following you may assume that the electric field points mainly in the radial direction
(Er Es ). This is true if either the beam is very long with respect to its width, or if the
bunch is traveling at relativistic velocities (as was shown in the previous exercise).
c) Use Gauss’s law inside the cylinder to show that the (radial) electric field Er inside the
cylinder is
ρ
Er =
r.
2ε0
d) Calculate the electric field at a radius r outside the cylinder.
e) Use Ampere’s law to show that the (azimuthal) magnetic field Bφ , inside the cylinder is
Bφ =
ρ v
r.
2ε0 c2
f) Calculate the magnetic field at a radius r outside the cylinder.
g) Combine the electric and magnetic field terms and show that the electromagnetic force on
a particle inside the cylinder is suppressed by a factor 1/γ 2 , when observed in the laboratory
frame.
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5
Skin depth
Assume a cavity with an electromagnetic field oscillating at an angular frequency ω. If the
cavity walls were perfectly conducting, the fields would not penetrate into walls and no power
would be dissipated. This cavity would have an infinite Q factor, assuming the cavity was
fully closed. Real cavities have a finite conductance σ. In this case the fields will penetrate
into the cavity walls. The fields will exponentially decay over a length of scale δskin , the
metal skin depth, which depends on frequency. We will derive an expression for the skin
depth.
Figure 2: A metal wall starts at z = 0. An electromagnetic field oscillates with frequency ω
at z < 0.
Assume a cavity wall with conductivity σ, starting at z = 0 and continuing infinitely far into
the z-axis. See Figure 2. Let the magnetic field component parallel to the cavity wall be
Hx = H0 exp(−iωt).
We calculate the currents and fields inside the metal in order to derive the skin depth.
a) Let j be the current inside the metal. Use the Maxwell equation and Ohms law,
j = σE,
to derive a differential equation for the magnetic field, B = µH. Assume that the permeability
µ is constant and that the displacement current ∂D/∂t is negligible. Show that this equation
is
∂B
1 2
=
∇B
∂t
µσ
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b) We seek to find how the amplitude of the magnetic field decays inside the metal, i.e. we
seek a solution along z of the form
Hx (z) = h(z) exp(−iωt)
The field is continuous on the boundary of the metal (Hx,z=0− = Hx,z=0+ ). Show that h(z)
has as solution
h(z) = H0 exp((i − 1)z/δskin )
where
s
δskin =
2
,
µσω
is the metal skin depth.
c) Copper has conductivity σ = 6 × 107 Ω−1 m−1 . Calculate the skin depth for a f = 12 GHz
copper cavity.
d) It can be shown that the quality factor Q for the fundamental mode of a pill-box cavity
with length L and radius R (see exercise set 1) is
Q=
RL
δskin (R + L)
Assume a pill-box with a geometrical shape factor of R/L = 1. What is the fundamental
mode quality factor for such a pill box cavity, with fundamental mode frequency 12 GHz?
With fundamental mode frequency 400 MHz? Use that the fundamental mode frequency is
related to the pill-box radius as
ω
R = 2.405
c
as calculated in exercise set 1.
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6
Longitudinal wake fields in linacs
A longitudinal wake affects the particle energy. We will study a N-particle model, assuming
ultra-relativistic particles with v = c, so that the rigid beam approximation and the
impulse approximation apply (see lecture slides). The particles travel centered through a
cavity, starting at beam line position s = 0 and with length L. We assume that no external
fields are put into the cavity. See Figure 3 for an illustration. The leading particle will lose
energy as it travels through the cavity. Part of this energy is lost in the resistive walls, part
of the energy generates a wake field. The integrated effect on a test particle following a
generating charge at a distance z is quantified by the wake function defined as1
1 ZL
dsEz |s=ct−z [V/C]
Wk (z) ≡
q1 0
where q1 is the charge of the leading particle. Wk is cosine-like because the leading particles
lose energy. We model here the wake function of the cavity as
Wk (z) = Ak cos(
ωk
z) exp(z/Qk ), z<0
c
(1)
where Ak is the amplitude of the wake field, ωk is the frequency and Qk is the Q-factor. Note
that by convention z < 0 means trailing the particle generating the wake field.
Figure 3: N bunches traveling through a cavity, in the s direction. The co-moving coordinate
z ≡ s − vt is defined in a frame following the beam travelling with speed v, and gives thus
the relative position inside the beam.
a) A particle experiences only half of the voltage it generates. This is stated mathematically
as
1
Vself = q1 Wk (z = 0− )
2
and is called the fundamental theorem of beam loading. The following exercises is meant
to give some familiarity with the fundamental theorem of beam loading. Assume negligible
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The time variable t is defined such that t = 0 when the generating particle enters the cavity, at s = 0.
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wall losses and ignore the exponential damping term (Qk → ∞). Let the N trailing particles
be at the maximum decelerating phase, at a distance zn = −2π ωck n where n = 1, 2, 3...N .
Particle n experiences the wake from particle 1 through particle n − 1. What is the voltage
experienced by particle n after it has travelled through the cavity, in units of Vself ?
b) The energy change of a particle is proportional to the experienced voltage. The energy in
an electromagnetic field is proportional to the square of the field amplitude. The field from
our N particles add coherently. Show that the total energy loss from N particles is indeed
N 2 times the energy loss of a single particle.
c) Let instead the distance between the particles be half of the wake function period, zn =
−π ωck n. Calculate the energy loss or gain for particle n. Calculate the total energy loss from
N particles.
d) The wake potential of a bunch passing through a cavity is the wake contribution of each
part of the bunch, multiplied with the bunch charge. Calculate the wake potential Vk,bunch (z)
from a single bunch with a Gaussian charge profile given by
λ(z 0 ) = √
1 z0
Q
exp(− ( )2 ) [C/m],
2 σz
2πσz
where Q is the total bunch charge. The cavity has a wake function as defined by Eq. (1).
Assume that the wake potential is calculated long after the bunch has passed, i.e. −z σz .
This assumption gives the multi-bunch wake potential, as opposed to the wake potential
inside the generating bunch.
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Transverse wake fields in linacs
We will study a 2-particle model. This model may represent two bunches, or the head and
the tail of the same bunch. The particles travel through the same cavity as in the previous
exercise. The amplitude of transverse wake fields are proportional to the offset of the driving
beam. We define the transverse wake function as (analogous for y)
W⊥,x
1 1 ZL
dsEx |s=ct−z [V/C/m].
≡
x q1 0
The differences with Wk is that W⊥ is normalized to the beam offset from the symmetry axis
x, and is sine-like since a particle cannot deflect itself. We model here the transverse wake
function as
ω⊥
W⊥ (z) = A⊥ sin( z) exp(z/Q⊥ ), z<0.
c
We assume that a linac is filled with cavities, interleaved with focusing lenses. The leading
particle undergoes betatron oscillations which we will approximate as a smooth sine-motion
with average beta function < β >. The equation of motion for the leading particle is the
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Hill’s equation with constant focusing, x00 + <β>
2 x(s) = 0 with solution
x(s) = B sin(s/ < β >)
a) The trailing particle experiences a transverse force due to the transverse wake field. Use
the definition of the wake function from the lecture slides to relate the wake function and the
this force. What is the equation of motion for the trailing particle?
b) By solving the transverse equation of motion, show that the transverse amplitude of the
trailing particle grows proportionally to the distance travelled, s, and is proportional to the
wake it experiences, W⊥ (z). 2
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The resonant growth may be mitigated, or even cancelled completely, if the betatron frequency of the
trailing particle is different from the leading particle. This can be achieved by varying the energy of the
two particles. BNS-damping, which you can find described in literature, is a mitigation technique based this
principle.
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Panofsky-Wenzel theorem
In general, the longitudinal and transverse wake fields in an accelerator are related. The so
called Panofsky-Wenzel theorem holds when a beam in an accelerator is modeled as a
rigid beam and can be written
∇ × (∆p) = 0.
where ∆p is the impulse a particle experiences due to wake fields. Show, by splitting this
equation in longitudinal and transverse components, that the Panofsky-Wenzel theorem implies that the transverse gradient of the longitudinal impulse equals the the longitudinal gradient of the transverse impulse.
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