phase-space description of charged particle beams
TRANSCRIPT
Phase-space Description of Charged Particle Beams
Fall, 2017
Kyoung-Jae Chung
Department of Nuclear Engineering
Seoul National University
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Basic properties of electron and proton
Electron (e)β’ Mass (me) = 9.11x10-31 kgβ’ Charge (qe) = -1.6x10-19 C (-e)β’ Rest energy = 0.511 MeVβ’ Electrons are relativistic when they have kinetic energy above about 100
keV.
Proton (p, H+)β’ Mass (mp) = 1.67x10-27 kg mi = Amp (A: atomic mass number)β’ Charge (qp) = +1.6x10-19 C (+e) qi = Z*qp (Z*: charge state of the ion)β’ Rest energy = 938.27 MeVβ’ Because of the high rest energy, we can use Newtonian dynamics to predict
the motion of ions in many applications.
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Beam
Although single charged particles may be useful for some physics experiments, we need large numbers of energetic particles for most applications. A flux of particles is a beam when the following two conditions hold:
β’ The particles travel in almost the same direction.β’ The particles have a small spread in kinetic energy.
A beam is an ordered flow of charged particles. A disordered set of particles, such as a thermal plasma, is not a beam.
The degree of order in a flow of particles is called coherence.
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Charged particle beam parameters
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Description of particle dynamics
The central issue in beam physics is the solution of collective problems involving large numbers of particles.
Collective physics is a science of approximation.
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Particle dynamics
The special theory of relativity states that the inertia of a particle observed in a frame of reference depends on the magnitude of its speed in that frame.
The inertia of a particle is proportional to πΎπΎ. The apparent mass is ππ = πΎπΎππ0. The particle momentum, a vector quantity, equals ππ = πΎπΎππ0ππ. The equation of motion:
The kinetic energy equals the total energy minus the rest energy:
Newtonian dynamics describes the motion of low-energy particles when ππ βͺππ0ππ2.
πΎπΎ =1
1 β π£π£/ππ 2=
11 β π½π½2
ππππππππ =
ππ(πΎπΎππ0ππ)ππππ = ππ
ππ = πΎπΎ β 1 ππ0ππ2
Lorentz factor
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Transfer matrix
Most beam transport devices, such as charged particle lenses and bending magnets, apply transverse forces that are linearly proportional to the distance of a particle from a preferred axis.
To specify the orbit of the particle in the x-direction, we must give its position, x, and velocity, vx. The convention in charged-particle optics is to represent particle orbits in terms of their angle relative to the main axis, rather than the transverse velocity. In the limit that π£π£π₯π₯ βͺ π£π£π§π§, the angle is
If the x-directed forces in the device are linear, then we can express the exit vector as a linear combination of the entrance vector components:
The quantities amn depend on the distribution of forces. Without acceleration, the determinant of the transfer matrix equals unity.
π₯π₯β² =πππ₯π₯ππππ
βπ£π£π₯π₯π£π£π§π§
π₯π₯1π₯π₯1β²
=ππ11 ππ12ππ21 ππ22
π₯π₯0π₯π₯0β²
Transfer matrixExit vector Entrance vector
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Transfer matrix
If a particle travels through linear device A and then through device B, the final orbit vector is
The orbit vector transformation from any combination of one-dimensional focusing elements is a single transfer matrix, the product of the individual matrices of all the elements.
The particle orbit vector for a two-dimensional focusing system is ππ = [π₯π₯, π₯π₯π₯,π¦π¦,π¦π¦π₯]. A 4Γ4 matrix represents the effect of a general linear focusing element or system.
In many practical devices, such as quadrupole lens arrays, the forces in the π₯π₯and π¦π¦ directions are independent. Then, we can calculate motion in π₯π₯ and π¦π¦separately using individual 2Γ2 matrices.
π₯π₯2π₯π₯2β²
= ππ11 ππ12ππ21 ππ22
ππ11 ππ12ππ21 ππ22
π₯π₯0π₯π₯0β²
ππππ = π©π© π¨π¨ππππ = π©π©π¨π¨ ππππ = πͺπͺππππ
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Phase dynamics
Despite differences in geometry, all radio-frequency accelerators use a traveling electromagnetic wave to accelerate charged particles. For ion acceleration, the axial component of the electric field on the axis has the form:
Axial variation of the electric field:
The wave can accelerate particles to high energy only if they stay within the region of accelerating electric field. In other words, the particles must remain at about the same phase of the accelerating wave. This means that the wave phase-velocity must increase to match the velocity of the accelerating particles.
πΈπΈπ§π§ ππ, ππ = πΈπΈ0sin[ππ ππ ππ β ππππ]
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Synchronous particle
The wave accelerates particles with phase in the range 0 < ππ < ππ and decelerates particles in the phase range βππ < ππ < 0 .
πππ£π£π π ππππππ
=πππΈπΈ0 sinπππ π
ππ0
We can define conditions where a particle stays at a constant phase. A particle with this property is a synchronous particle β its phase is the synchronous phase, πππ π .
Figure shows that the synchronous particle experiences a constant axial electric field, πΈπΈπ§π§π π = πΈπΈ0 sinπππ π .
The velocity of the synchronous particle changes as:
The accelerating structure must vary along its length so that the wave number is ππ ππ =
πππ£π£π π ππ
Conditions for having synchronous particles in an accelerator
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Non-synchronous particle
Under some conditions non-synchronous particles have stable oscillations about the synchronous particle position, πππ π .
Let ππ and π£π£ be the axial position and velocity of a non-synchronous particle. We define the small quantities:
Then, we obtain
The instantaneous acceleration of the non-synchronous particle is:
The general phase equations for non-relativistic particles:
ππ2ππππππ2
=πππ£π£ππππ
=πππΈπΈ0 sinππ
ππ0
ππ2βππππππ2
=πππΈπΈ0 sinππ
ππ0βππ2πππ π ππππ2
=πππΈπΈ0ππ0
sinππ β sinπππ π
ππ = πππ π β ππβππ/π£π£π π
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Non-synchronous particle
If πΈπΈ0 and π£π£π π are almost constant during an axial oscillation of a non-synchronous particle, then we obtain a non-linear differential equation as following:
For small oscillations about the phase of the synchronous particle, βππ βͺ πππ π , the above equation reduces to:
If cosπππ π > 0, the axial oscillations of non-synchronous particles are stable. The conditions for synchronized particle acceleration are cosπππ π > 0 and sinπππ π >
0, or
The conditions for synchronized particle deceleration are cosπππ π > 0 and sinπππ π <0, or
ππ2ππππππ2
β βπππππΈπΈ0ππ0π£π£π π
sinππ β sinπππ π
ππ2βππππππ2
β βπππππΈπΈ0ππ0π£π£π π
cosπππ π βππ
0 < πππ π < ππ/2 (acceleration)
βππ/2 < πππ π < 0 (deceleration)
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Configuration space vs phase space
Lamina phase flow is the foundation for theories of collective behavior.
(π₯π₯(ππ),π¦π¦(ππ), ππ(ππ))
(π₯π₯,π¦π¦, ππ, π£π£π₯π₯, π£π£π¦π¦, π£π£ππ)
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Examples of phase space description
The trajectories of particles accelerated by a constant axial electric field πΈπΈππ:
Trajectories of particles in a linear focusing force πΉπΉπ₯π₯ = βπππ₯π₯:
ππ ππ = ππ0 + π£π£π§π§0ππ + βπππΈπΈπ§π§ ππ0 ππ2/2
π£π£π§π§ ππ = π£π£π§π§0 + βπππΈπΈπ§π§ ππ0 ππ
π₯π₯ ππ = π₯π₯0 cos(ππππ + ππ)
ππ = ππ/ππ0
π£π£π₯π₯ ππ = βπ₯π₯0ππ sin(ππππ + ππ)
300 keV protons with a betatron wavelength of 0.3 m
Protons in an electric field of 105 V/m
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Phase space description for relativistic particles
The relativistic equations of motion
Normalization:
πππ₯π₯ππππ
=πππ₯π₯πΎπΎππ0
=πππ₯π₯
ππ0 1 + πππ₯π₯2/ππ02ππ2
πππππ₯π₯ππππ
= βπππ₯π₯
πΎπΎππ0ππ2 2 = πππππ₯π₯ 2 + ππ0ππ2 2
ππππππππ =
πππ₯π₯1 + πππ₯π₯2
πππππ₯π₯ππππ = βπΌπΌππ
where, πΌπΌ = πππ₯π₯02/ππ0ππ2
ππ = ππ/( βπ₯π₯0 ππ) ππ = π₯π₯/π₯π₯0 πππ₯π₯ = πππ₯π₯/ππ0ππ
πΌπΌ = 0.5
πΌπΌ = 5
πΌπΌ = 5
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Conservation of phase space volume
Conservation of the phase-space volume occupied by a particle distribution is a fundamental theorem of collective physics. (Liouvilleβs theorem: the principle of incompressiblity of a phase fluid)
Same area
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Maxwell distribution
Particles in an isotropic Maxwell distribution are in thermal equilibrium. They have a spread in kinetic energy.
For a single species in thermal equilibrium with itself (e.g. electrons), in the absence of time variation, spatial gradients, and accelerations, the Boltzmann equation reduces to
Then, we obtain the Maxwell-Boltzmann velocity distribution
ππ π£π£ =2ππ
ππππππ
3π£π£2πππ₯π₯ππ β
πππ£π£2
2ππππ
ππ ππ =2ππ
ππππππ
1/2πππ₯π₯ππ β
ππππππ
The mean speed
οΏ½Μ οΏ½π£ =8ππππππππ
1/2
οΏ½ππππππππ ππ
= 0
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Displaced Maxwell distribution
Charged particle beams are non-isotropic and usually almost monoenergetic. In a sense, the primary goal of beam technology is to create non-Maxwelliandistributions and to preserve them over time scales set by the application.
A common assumption used in beam theory is that the particles have a Maxwell distribution when observed in the beam rest frame. The transformed distribution observed in the stationary frame of the accelerator is called a displaced Maxwell distribution.
For example, consider a nonrelativistic ion beam extracted from a plasma source with ion temperature ππππ. The beam is axially bunched passing through a radio-frequency quadrupole accelerator. The beam emerges from the accelerator with kinetic energy πΈπΈ0. We can represent the exit beam distribution in the stationary frame as
ππ π£π£π₯π₯,π£π£π¦π¦ ,π£π£π§π§ ~ πππ₯π₯ππ βππππ(π£π£π₯π₯2 + π£π£π¦π¦2)
2πππππππππ₯π₯ππ β
ππππ(π£π£π§π§ β π£π£0)2
2ππππππβ²
π£π£0 =2πΈπΈ0ππππ
1/2