Phase-space Description of Charged Particle Beams

Phase-space Description of Charged Particle Beams

Fall, 2017

Kyoung-Jae Chung

Department of Nuclear Engineering

Basic properties of electron and proton

 Electron (e)

• Mass (m_e) = 9.11x10^-31 kg

• Charge (q_e) = -1.6x10^-19 C (-e)

• Rest energy = 0.511 MeV

• Electrons are relativistic when they have kinetic energy above about 100 keV.

 Proton (p, H⁺)

• Mass (m_p) = 1.67x10^-27 kg  m_i = Am_p (A: atomic mass number)

• Charge (q_p) = +1.6x10^-19 C (+e)  q_i = Z*q_p (Z*: charge state of the ion)

• Rest energy = 938.27 MeV

• Because of the high rest energy, we can use Newtonian dynamics to predict the motion of ions in many applications.

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:

• The particles travel in almost the same direction.

• 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.

Charged particle beam parameters

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.

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 𝛾𝛾. The apparent mass is 𝑚𝑚 = 𝛾𝛾𝑚𝑚₀.

 The particle momentum, a vector quantity, equals 𝒑𝒑 = 𝛾𝛾𝑚𝑚₀𝒗𝒗.

 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 𝑇𝑇 ≪ 𝑚𝑚₀𝑐𝑐².

𝛾𝛾 = 1

1 − 𝑣𝑣/𝑐𝑐 ² = 1

1 − 𝛽𝛽²

𝑑𝑑𝒑𝒑

𝑑𝑑𝑑𝑑 = 𝑑𝑑(𝛾𝛾𝑚𝑚₀𝒗𝒗)

𝑑𝑑𝑑𝑑 = 𝑭𝑭

𝑇𝑇 = 𝛾𝛾 − 1 𝑚𝑚₀𝑐𝑐²

Lorentz factor

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, v_x. 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 𝑣𝑣_𝑥𝑥 ≪ 𝑣𝑣_𝑧𝑧, 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 a_mn depend on the distribution of forces. Without acceleration, the determinant of the transfer matrix equals unity.

𝑥𝑥

^′

= 𝑑𝑑𝑥𝑥 𝑑𝑑𝑑𝑑 ≈

𝑣𝑣

_𝑥𝑥

𝑣𝑣

_𝑧𝑧

𝑥𝑥

₁

𝑥𝑥

₁^′

= 𝑎𝑎

₁₁

𝑎𝑎

₁₂

𝑎𝑎

₂₁

𝑎𝑎

₂₂

𝑥𝑥

₀

𝑥𝑥

₀^′

Transfer matrix

Exit vector Entrance vector

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 𝒙𝒙 = [𝑥𝑥,𝑥𝑥𝑥,𝑦𝑦,𝑦𝑦𝑥]. A 4×4 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 𝑥𝑥 and 𝑦𝑦 directions are independent. Then, we can calculate motion in 𝑥𝑥 and 𝑦𝑦 separately using individual 2×2 matrices.

𝑥𝑥

₂

𝑥𝑥

₂^′

= 𝑏𝑏

₁₁

𝑏𝑏

₁₂

𝑏𝑏

₂₁

𝑏𝑏

₂₂

𝑎𝑎

₁₁

𝑎𝑎

₁₂

𝑎𝑎

₂₁

𝑎𝑎

₂₂

𝑥𝑥

₀

𝑥𝑥

₀^′

𝒙𝒙

_𝟐𝟐

= 𝑩𝑩 𝑨𝑨𝒙𝒙

_𝟎𝟎

= 𝑩𝑩𝑨𝑨 𝒙𝒙

_𝟎𝟎

= 𝑪𝑪𝒙𝒙

_𝟎𝟎

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.

𝐸𝐸

_𝑧𝑧

𝑑𝑑, 𝑑𝑑 = 𝐸𝐸

₀

sin[𝑘𝑘 𝑑𝑑 𝑑𝑑 − 𝜔𝜔𝑑𝑑]

Synchronous particle

 The wave accelerates particles with phase in the range 0 < 𝜙𝜙 < 𝜋𝜋 and decelerates particles in the phase range −𝜋𝜋 < 𝜙𝜙 < 0 .

𝑑𝑑𝑣𝑣

_𝑠𝑠

𝑑𝑑

𝑑𝑑𝑑𝑑 = 𝑞𝑞𝐸𝐸

₀

sin 𝜙𝜙

_𝑠𝑠

𝑚𝑚

₀

 We can define conditions where a particle stays at a constant phase. A particle with this property is a synchronous particle – its phase is the synchronous phase, 𝜙𝜙_𝑠𝑠.

 Figure shows that the synchronous particle experiences a constant axial electric field, 𝐸𝐸_{𝑧𝑧𝑠𝑠} = 𝐸𝐸₀ sin𝜙𝜙_𝑠𝑠.

 The velocity of the synchronous particle changes as:

 The accelerating structure must vary along its

length so that the wave number is

𝑘𝑘 𝑑𝑑 = 𝜔𝜔 𝑣𝑣

_𝑠𝑠

𝑑𝑑

Conditions for having synchronous particles in an accelerator

Non-synchronous particle

 Under some conditions non-synchronous particles have stable oscillations about the synchronous particle position, 𝑑𝑑_𝑠𝑠.

 Let 𝑑𝑑 and 𝑣𝑣 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:

𝑑𝑑

²

𝑑𝑑

𝑑𝑑𝑑𝑑

²

= 𝑑𝑑𝑣𝑣

𝑑𝑑𝑑𝑑 = 𝑞𝑞𝐸𝐸

₀

sin 𝜙𝜙 𝑚𝑚

₀

𝑑𝑑

²

∆𝑑𝑑

𝑑𝑑𝑑𝑑

²

= 𝑞𝑞𝐸𝐸

₀

sin 𝜙𝜙

𝑚𝑚

₀

− 𝑑𝑑

²

𝑑𝑑

_𝑠𝑠

𝑑𝑑𝑑𝑑

²

= 𝑞𝑞𝐸𝐸

₀

𝑚𝑚

₀

sin 𝜙𝜙 − sin 𝜙𝜙

_𝑠𝑠

𝜙𝜙 = 𝜙𝜙

_𝑠𝑠

− 𝜔𝜔∆𝑑𝑑/𝑣𝑣

_𝑠𝑠

Non-synchronous particle

 If 𝐸𝐸₀ and 𝑣𝑣_𝑠𝑠 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, ∆𝜙𝜙 ≪ 𝜙𝜙_𝑠𝑠, the above equation reduces to:

 If cos𝜙𝜙_𝑠𝑠 > 0, the axial oscillations of non-synchronous particles are stable.

 The conditions for synchronized particle acceleration are cos𝜙𝜙_𝑠𝑠 > 0 and sin𝜙𝜙_𝑠𝑠 >

0, or

 The conditions for synchronized particle deceleration are cos𝜙𝜙_𝑠𝑠 > 0 and sin𝜙𝜙_𝑠𝑠 <

0, or

𝑑𝑑

²

𝜙𝜙

𝑑𝑑𝑑𝑑

²

≅ − 𝜔𝜔𝑞𝑞𝐸𝐸

₀

𝑚𝑚

₀

𝑣𝑣

_𝑠𝑠

sin 𝜙𝜙 − sin 𝜙𝜙

_𝑠𝑠

𝑑𝑑

²

∆𝜙𝜙

𝑑𝑑𝑑𝑑

²

≅ − 𝜔𝜔𝑞𝑞𝐸𝐸

₀

𝑚𝑚

₀

𝑣𝑣

_𝑠𝑠

cos 𝜙𝜙

_𝑠𝑠

∆𝜙𝜙

0 < 𝜙𝜙

_𝑠𝑠

< 𝜋𝜋/2 (acceleration)

−𝜋𝜋/2 < 𝜙𝜙

_𝑠𝑠

< 0 (deceleration)

Configuration space vs phase space

 Lamina phase flow is the foundation for theories of collective behavior.

(𝑥𝑥(𝑑𝑑), 𝑦𝑦(𝑑𝑑), 𝑑𝑑(𝑑𝑑))

(𝑥𝑥, 𝑦𝑦, 𝑑𝑑, 𝑣𝑣

_𝑥𝑥

, 𝑣𝑣

_𝑦𝑦

, 𝑣𝑣

_𝑑𝑑

)

Examples of phase space description

 The trajectories of particles accelerated by a constant axial electric field 𝐸𝐸_𝑑𝑑:

 Trajectories of particles in a linear focusing force 𝐹𝐹_𝑥𝑥 = −𝑎𝑎𝑥𝑥:

𝑑𝑑 𝑑𝑑 = 𝑑𝑑₀ + 𝑣𝑣_𝑧𝑧0𝑑𝑑 + 𝑞𝑞𝐸𝐸_𝑧𝑧⁄𝑚𝑚₀ 𝑑𝑑²/2 𝑣𝑣_𝑧𝑧 𝑑𝑑 = 𝑣𝑣_𝑧𝑧0 + 𝑞𝑞𝐸𝐸_𝑧𝑧⁄𝑚𝑚₀ 𝑑𝑑

𝑥𝑥 𝑑𝑑 = 𝑥𝑥₀ cos(𝜔𝜔𝑑𝑑 + 𝜙𝜙)

𝜔𝜔 = 𝑎𝑎/𝑚𝑚₀

𝑣𝑣_𝑥𝑥 𝑑𝑑 = −𝑥𝑥₀𝜔𝜔sin(𝜔𝜔𝑑𝑑 + 𝜙𝜙)

300 keV protons with a betatron wavelength of 0.3 m

Protons in an electric field of 10⁵ V/m

Phase space description for relativistic particles

 The relativistic equations of motion

 Normalization:

𝑑𝑑𝑥𝑥

𝑑𝑑𝑑𝑑 = 𝑝𝑝_𝑥𝑥

𝛾𝛾𝑚𝑚₀ = 𝑝𝑝_𝑥𝑥

𝑚𝑚₀ 1 + 𝑝𝑝_𝑥𝑥²/𝑚𝑚₀²𝑐𝑐² 𝑑𝑑𝑝𝑝_𝑥𝑥

𝑑𝑑𝑑𝑑 = −𝑎𝑎𝑥𝑥

𝛾𝛾𝑚𝑚₀𝑐𝑐^{2 2} = 𝑐𝑐𝑝𝑝_𝑥𝑥 ² + 𝑚𝑚₀𝑐𝑐^{2 2}

𝑑𝑑𝑑𝑑

𝑑𝑑𝑇𝑇 = 𝑃𝑃_𝑥𝑥 1 + 𝑃𝑃_𝑥𝑥² 𝑑𝑑𝑃𝑃_𝑥𝑥

𝑑𝑑𝑇𝑇 = −𝛼𝛼𝑑𝑑

where, 𝛼𝛼 = 𝑎𝑎𝑥𝑥₀²/𝑚𝑚₀𝑐𝑐²

𝑇𝑇 = 𝑑𝑑/(𝑥𝑥₀⁄𝑐𝑐) 𝑑𝑑 = 𝑥𝑥/𝑥𝑥₀ 𝑃𝑃_𝑥𝑥 = 𝑝𝑝_𝑥𝑥/𝑚𝑚₀𝑐𝑐

𝛼𝛼 = 0.5

𝛼𝛼 = 5

𝛼𝛼 = 5

Conservation of phase space volume

 Conservation of the phase-space volume occupied by a particle distribution is a fundamental theorem of collective physics. (Liouville’s theorem: the principle of incompressiblity of a phase fluid)

Same area

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

𝑓𝑓 𝑣𝑣 = 2

𝜋𝜋 𝑚𝑚 𝑘𝑘𝑇𝑇

3 𝑣𝑣²𝑒𝑒𝑥𝑥𝑝𝑝 −𝑚𝑚𝑣𝑣² 2𝑘𝑘𝑇𝑇 𝑓𝑓 𝜀𝜀 = 2

𝜋𝜋 𝜀𝜀 𝑘𝑘𝑇𝑇

1/2 𝑒𝑒𝑥𝑥𝑝𝑝 − 𝜀𝜀 𝑘𝑘𝑇𝑇

 The mean speed

̅𝑣𝑣 = 8𝑘𝑘𝑇𝑇 𝜋𝜋𝑚𝑚

1/2

𝜕𝜕𝑓𝑓�

𝜕𝜕𝑑𝑑 _𝑐𝑐 = 0

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-Maxwellian

distributions 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 𝑇𝑇_𝑖𝑖. The beam is axially bunched passing through a radio- frequency quadrupole accelerator. The beam emerges from the accelerator with kinetic energy 𝐸𝐸₀. We can represent the exit beam distribution in the stationary frame as

𝑓𝑓 𝑣𝑣_𝑥𝑥,𝑣𝑣_𝑦𝑦,𝑣𝑣_𝑧𝑧 ~ 𝑒𝑒𝑥𝑥𝑝𝑝 −𝑚𝑚_𝑖𝑖(𝑣𝑣_𝑥𝑥² + 𝑣𝑣_𝑦𝑦²)

2𝑘𝑘𝑇𝑇_𝑖𝑖 𝑒𝑒𝑥𝑥𝑝𝑝 −𝑚𝑚_𝑖𝑖(𝑣𝑣_𝑧𝑧 − 𝑣𝑣₀)² 2𝑘𝑘𝑇𝑇_𝑖𝑖^′ 𝑣𝑣₀ = 2𝐸𝐸₀

𝑚𝑚_𝑖𝑖

1/2