Accelerator Fundamentals Brief history of accelerator Accelerator building
Accelerator Fundamentals • Brief history of accelerator • Accelerator building blocks • Transverse beam dynamics • coordinate system Mei Bai BND School, Sept. 1 -6, 2015
Why Accelerator? • high energy particles are excellent probes in studying the micro-structure of matters – Wave length of a high energy particle • Discover new particles • Generate secondary beams for research in physics as well as other scientific fields, material science, biology, etc – Neutron source – Synchrotron radiation • Also widely used in industry as well as medical research and treatment such as hadron therapy Mei Bai BND School, Sept. 1 -6, 2015
Accelerator Timeline Mei Bai BND School, Sept. 1 -6, 2015
Legends of Accelerator Development John D. Cockcroft and Ernest Walton – Rolf Wideroe: a nuclear physicist – Invented concept of LINAC when he was a Ph. D student at RWTH – Invented the principle of Betatron Ernest Orlando Lawrence: a nuclear physicist – Invented/implemented cyclotron – Nobel Laureate in 1939 Ernest Courant: Accelerator physicist – Invented strong focusing principle Bruno Touschek: particle physicist – Invented Cockcroft-Walton Accelerator father of the 1 st e+e- collider (Ad. A) Simon van der Meer: particle physicist Mei Bai – Invented/implemented Stochastic Cooling, and Nobel Laureate in 1984 BND School, Sept. 1 -6, 2015
• Both electric field and magnetic field can be used to guide the particles path. • Magnetic field is more effective for high energy particles, i. e. particles with higher velocity. - For a relativistic particle, what kind of the electric field one needs to match the Lorentz force from a 1 Telsla magnetic field? Mei Bai BND School, Sept. 1 -6, 2015
• Dipoles: uniform magnetic field in the gap - Bending dipoles - Orbit steering • Quadrupoles - Providing focusing field to keep beam from being diverged • Sextupoles: - Provide corrections of chromatic effect of beam dynamics • Higher order multipoles Mei Bai BND School, Sept. 1 -6, 2015
• Two magnetic poles separated by a gap • homogeneous magnetic field between the gap • Bending, steering, injection, extraction g Mei Bai BND School, Sept. 1 -6, 2015
ρ • For synchrotron, bending field is proportional to the beam energy where p is the momentum of the particle and q is the charge of the particle Mei Bai BND School, Sept. 1 -6, 2015
Quadrupole • Magnetic field is proportional to the distance from the center of the magnet • Produced by 4 poles which are shaped as • Providing focusing/defoucsing to the particle – Particle going through the center: F=0 – Particle going off center Mei Bai BND School, Sept. 1 -6, 2015 y x
Quadrupole magnet • Theorem • Pick the loop for integral For the gap is filled with air, Mei Bai BND School, Sept. 1 -6, 2015
Sextupole • Focusing strength in horizontal plane: Place sextupole after a bending dipole where dispersion function is non zero Mei Bai BND School, Sept. 1 -6, 2015
Focusing from quadrupole x Δx’ s f • Required by Maxwell equation, a single quadrupole has to provide focusing in one plane and defocusing in the other plane Mei Bai BND School, Sept. 1 -6, 2015
Transfer matrix of a qudruploe • Thin lens: length of quadrupole is negligible to the displacement relative to the center of the magnet Mei Bai BND School, Sept. 1 -6, 2015
Transfer matrix of a drift space • Transfer matrix of a drift space x Δx’ s L Mei Bai BND School, Sept. 1 -6, 2015
Lattice • Arrangement of magnets: structure of beam line – Bending dipoles, Quadrupoles, Steering dipoles, Drift space and Other insertion elements • Example: – FODO cell: alternating arrangement between focusing and defocusing quadrupoles f -f L Mei Bai L One FODO cell BND School, Sept. 1 -6, 2015
FODO lattice • Net effect is focusing! Mei Bai BND School, Sept. 1 -6, 2015
FODO lattice • Net effect is focusing • Provide focusing in both planes! Mei Bai BND School, Sept. 1 -6, 2015
Curvilinear coordinate system • Coordinate system to describe particle motion in an accelerator • Moves with the particle Re fe re nc e or bi t Set of unit vectors: Mei Bai BND School, Sept. 1 -6, 2015
Equation of motion x x s Δθ =Δ s/ ρ • Equation of motion in transverse plane Mei Bai BND School, Sept. 1 -6, 2015
Equation of motion Mei Bai BND School, Sept. 1 -6, 2015
Equation of motion Mei Bai BND School, Sept. 1 -6, 2015
Solution of equation of motion • Comparison with harmonic oscillator: A system with a restoring force which is proportional to the distance from its equilibrium position, i. e. Hooker’s Law: Where k is the spring constant Equation of motion: Amplitude of the sinusoidal oscillation Mei Bai BND School, Sept. 1 -6, 2015 Frequency of the oscillation
transverse motion: betatron oscillation • The general case of equation of motion in an accelerator Where k can also be negative Mei Bai For k > 0 For k < 0 BND School, Sept. 1 -6, 2015
Transfer matrix of a quadrupole • For a focusing quadrupole • For a defocusing quadrupole Mei Bai BND School, Sept. 1 -6, 2015
Hill's equation • In an accelerator which consists individual magnets, the equation of motion can be expressed as, • Here, k(s) is an periodic function of Lp, which is the length of the periodicity of the lattice, i. e. the magnet arrangement. It can be the circumference of machine or part of it. • Similar to harmonic oscillator, expect solution as • or: Mei Bai BND School, Sept. 1 -6, 2015
Hill’s equation: cont’d • with Hill’s equation Mei Bai is satisfied BND School, Sept. 1 -6, 2015
Betatron oscillation • Beta function : – Describes the envelope of the betatron oscillation in an accelerator • Phase advance: • Betatron tune: # of betatron oscillations in one orbital turn Mei Bai BND School, Sept. 1 -6, 2015
Phase space • In a space of x-x’, the betatron oscillation projects an ellipse X’ where X The are of the ellipse is Mei Bai BND School, Sept. 1 -6, 2015
Courant-Snyder parameters • The set of parameter (βx, αx and γx) which describe the phase space ellipse • Courant-Snyder invariant: the area of the ellipse in unit of Mei Bai BND School, Sept. 1 -6, 2015
Transfer Matrix of beam transport • Proof the transport matrix from point 1 to point 2 is Hint: Mei Bai BND School, Sept. 1 -6, 2015
One Turn Map • Transfer matrix of one orbital turn Stable condition Closed orbit: Mei Bai BND School, Sept. 1 -6, 2015
Stability of transverse motion • Matrix from point 1 to point 2 Stable motion requires each transfer matrix to be stable, i. e. its eigen values are in form of oscillation With Mei Bai BND School, Sept. 1 -6, 2015 and
Dispersion function • Transverse trajectory is function of particle momentum ρ ρ+Δρ Momentum spread Define Dispersion function Mei Bai BND School, Sept. 1 -6, 2015
Dispersion function • Transverse trajectory is function of particle momentum. Mei Bai BND School, Sept. 1 -6, 2015
Dispersion function: cont’d • In drift space and dispersion function has a constant slope In dipoles, and Mei Bai BND School, Sept. 1 -6, 2015
Dispersion function: cont’d For a focusing quad, and dispersion function oscillates sinusoidally For a defocusing quad, and dispersion function evolves exponentially Mei Bai BND School, Sept. 1 -6, 2015
Compaction factor The difference of the length of closed orbit between off-momentum particle and on momentum particle, i. e. Mei Bai BND School, Sept. 1 -6, 2015
Path length and velocity For a particle with velocity v, Transition energy : when particles with different energies spend the same time for each orbital turn • Below transition energy: higher energy particle travels faster • Above transition energy: higher energy particle travels slower Mei Bai BND School, Sept. 1 -6, 2015
Chromatic effect • Comes from the fact the focusing effect of an quadrupole is momentum dependent Particles with different momentum have different betatron tune - Higher energy particle has less focusing Chromaticity: tune spread due to momentum spread Tune spread momentum spread Mei Bai BND School, Sept. 1 -6, 2015
Chromaticity Transfer matrix of a thin quadrupole • Transfer matrix A Mei Bai BND School, Sept. 1 -6, 2015 B
Chromaticity Mei Bai BND School, Sept. 1 -6, 2015
Chromaticity Assuming the tune change due to momentum difference is small Mei Bai BND School, Sept. 1 -6, 2015
Chromaticity of a FODO cell βd βf β L L One FODO cell Mei Bai BND School, Sept. 1 -6, 2015
Chromaticity correction • Nature chromaticity is always negative and can be large and can result to large tune spread and get close to resonance condition • Solution: - A special magnet which provides stronger focusing for particles with higher energy: sextupole Mei Bai BND School, Sept. 1 -6, 2015
Sextupole • Focusing strength in horizontal plane: • where and , l is the magnet length • Tune change due to a sextupole: Mei Bai BND School, Sept. 1 -6, 2015
Chromaticity Correction • Sextupole produces a chromaticity with the opposite sign of the quadrupole! • It prefers to be placed after a bending dipole where dispersion function is non zero • Chromaticity after correction Mei Bai BND School, Sept. 1 -6, 2015
Chromaticity correction ξ=20 ξ=1 Mei Bai BND School, Sept. 1 -6, 2015
How to measure betatron oscillation? Excite a coherent betatron motion with a pulsed kicker X’ X Mei Bai Record turn – by – turn beam position BND School, Sept. 1 -6, 2015
How to measure betatron oscillation? Turn-by-turn beam position monitor data Kicker betatron tune is obtained by Fourier transform Tb. T beam position data Mei Bai BND School, Sept. 1 -6, 2015
Beam Position Monitor (BPM) • A strip line bpm response – Right electrode response – And left electrode response • Hence, • Let x=rcosθ Mei Bai BND School, Sept. 1 -6, 2015 b
Coherent betatron oscillation at RHIC Mei Bai BND School, Sept. 1 -6, 2015
How to measure betatron functions and phase advance? Mei Bai BND School, Sept. 1 -6, 2015
Effects of Errors - dipole errors - quadrupole errors
Closed orbit distortion • Dipole kicks can cause particle’s trajectory deviate away from the designed orbit s 0 - Dipole error - Quadrupole misalignment Assuming a circular ring with a single dipole error, closed orbit then becomes: BPM s Mei Bai BND School, Sept. 1 -6, 2015
Closed orbit: single dipole error Let’s first solve the closed orbit at the location where the dipole error is The closed orbit distortion reaches its maximum at the opposite side of the dipole error location Mei Bai BND School, Sept. 1 -6, 2015
Closed orbit distortion In the case of multiple dipole errors distributed around the ring. The closed orbit is Amplitude of the closed orbit distortion is inversely proportion to sinπQx, y - No stable orbit if tune is integer! Mei Bai BND School, Sept. 1 -6, 2015
Measure closed orbit Distribute beam position monitors around ring. Mei Bai BND School, Sept. 1 -6, 2015
Control closed orbit minimized the closed orbit distortion. Large closed orbit distortions cause limitation on the physical aperture Need dipole correctors and beam position monitors distributed around the ring Assuming we have m beam position monitors and n dipole correctors, the response at each beam position monitor from the n correctors is: Mei Bai BND School, Sept. 1 -6, 2015
Control closed orbit Or, To cancel the closed orbit measured at all the bpms, the correctors are then Mei Bai BND School, Sept. 1 -6, 2015
Quadrupole errors • Misalignment of quadrupoles - dipole-like error: kx - results in closed orbit distortion Gradient error: - Cause betatron tune shift - induce beta function deviation: beta beat Mei Bai BND School, Sept. 1 -6, 2015
Tune change due to a single gradient error • Suppose a quadrupole has an error in its gradient, i. e. Mei Bai BND School, Sept. 1 -6, 2015
Tune shift due to multiple gradient errors • In a circular ring with a multipole gradient errors, the tune shift is Mei Bai BND School, Sept. 1 -6, 2015
Beta beat • In a circular ring with a gradient error at s 0, the tune shift is s 0 s Unstable betatron motion if tune is half integer! Mei Bai BND School, Sept. 1 -6, 2015
Beta beat • In a circular ring with multiple gradient errors, Unstable betatron motion if tune is half integer! Beta beat wave varies twice of betatron tune around the ring Mei Bai BND School, Sept. 1 -6, 2015
Resonance condition • Tune change due to a single quadrupole error If , the above equation becomes and Qx can become a complex number which means the betatron motion can become unstable Mei Bai BND School, Sept. 1 -6, 2015
Resonance X’ x Integer resonance Mei Bai x Half Integer resonance BND School, Sept. 1 -6, 2015
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