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Fundamental of Accelerators Fundamental of Accelerators Fanglei Lin Center for Advanced Studies of Accelerators, Jefferson Lab 29th Annual Hampton University Graduate Studies Program HUGS 2014, Jefferson Lab, June 2-20, 2014 Definition of Beam Optics Beam Optics: the process of guiding a charged particle beam from A to B using magnets An array of magnets which accomplishes this is a transport system, or magnetic lattice Recall Lorentz Force on a particle d(γmυ) = q(E +υ × B) dt Energy changes from electric fields Direction changes (energy conservative) from magnetic fields 2 Force on a Particle in a Magnetic Field The simplest type of magnetic field is a constant field. A charged particle in a constant field executes a circular orbit, with radius ρ and frequency ω To find the direction of the force on the particle, use the right-hand-rule. B ω ρ What would happen if the initial velocity had v a component in the direction of the field ? 3 Dipole Magnets A dipole magnet gives a constant field B Field lines in a magnet run from the North to South Symbol convention: × : traveling into the page ● : traveling out of the page In the field shown, for a positively charged particle traveling into the page, the force is to the right. In an accelerator lattice, dipoles are used to bend the beam trajectory. The set of dipoles in a lattice defines the reference trajectory ᶩ p[GeV / c] ≈ 0.3B[T ]ρ[m] q[e] θ ρ l θ = 4 ρ Generating B Field from a Current A current in a wire generates a magnetic field B which curls around the wire: Winding many turns on a coil generates a strong uniform magnetic field Field strength is given by one of Maxwell’s equations: ∇ × B = J µ or Ampere circuital law B ⋅ dl = J.⋅ dS = I enc ∫C ∫∫s 5 µ Dipole Current to Field In an accelerator, we use current-carrying wires and metal cores of high µ (magnetic permeability) to set up a strong dipole field N turns of current generate a field perpendicular to the pole tip surface Relationship between B in the gap and I in the wire: B G B B ⋅ dl = ⋅ d.l + .dl = I enc = 2I coil ∫C ∫0 ∫ µ µ° iron µiron negligible, since µiron >> µo GB I coil = NI wire = 2µ° 6 Focusing Particles w/ Magnets Consider the optical analogy of focusing a ray light through a convex lens: The focusing angle depends on the the distance from the center. The farther off axis, the stronger the focusing effect. r θ = − f Now consider a magnetic lens l l Byl grl θ = − = = = ρ y ( p / e) / By p / e Bρ Here ∂By Focusing strength g = ∂r p Beam rigidity = (Bρ) e dipole 7 Quadrupole Magnets A quadrupole magnet imparts a force proportional to distance from the center. This magnet has 4 poles. Consider a positive particle traveling into the the page in the quadrupole magnetic field. This magnets is horizontally defocusing. What about in the vertical direction? Focusing ! A quadrupole which defocuses in one plane focuses in the other ! 8 Quadrupole Current to Field As with a dipole, we can use current-carrying wires wrapped around metal cores to create a quadrupole magnet The field lines are denser near the edges of the of the magnet, meaning the field is stronger there The strength of B is function of the distance to the center of magnet Relationship between focusing strength g and current I in the wire: R X B Br Biron By ⋅ dl = ⋅ d.r + .dliron + ⋅ d.x = I enc ∫C ∫0 ∫ ∫0 µ µ° iron µiron µ° ⊥ negligible, since =0, since By x µiron >> µo gR 2 I enc = 9 2µ° Focusing Using Arrays of Quadrupoles Quadrupoles focus in one plane while defocusing in the other. So, how can this be used to provide net focusing in an accelerator? Consider again the optical analogy of two lenses, with focal length f1 and f2, separated by a distance d 1 1 1 d = + − The combined fcombined is f combined f1 f 2 f1 f 2 if f1 = -f2 , the net effect is focusing(positive) 1 d = f combined f1 f 2 To focus particles in both planes in an accelerator, one need arrays of quadruple magnets ! 10 Other Types of Magnets Examples of magnets: In general, poles are 360°/2n apart The skew version of the magnet is obtained by rotating the upright magnet by 180°/2n Combined function magnet: bend and focus simultaneously 11 Trajectories and Phase Space In general, in an accelerator we assume the dipoles define the nominal particle trajectory, and we solve for lateral deviations from that trajectory Lateral x s deviation Position along At any point along the trajectory, trajectory each particle can be represented by its position in “phase space” dx x′ ≡ ds We would like to solve for x(s) We assume: x Both transverse planes are independent no coupling All particles independent from each other no space charge effects 12 Transfer Matrices The simplest magnetic lattice consists of quadrupoles and the spaces in between them (drift). We can express each of these as a linear operation in phase space. Quadrupole: x = x(0) x 1 0 x(0) ⇒ = 1 1 − 1 x'= x'(0) − x(0) x' x'(0) f f Drift: = + x(s) x(0) sx'(0) x(s) 1 s x(0) x ↑ ⇒ = s → x'(s) = x'(0) x'(s) 0 1 x'(0) By combining these elements, we can represent an arbitrarily complex ring or line as the product of matrices. M = M N ...M 2 M 1 13 Example: FODO Cell At the heart of every beam line or ring is the FODO cell, consisting of a focusing and a defocusing element, separated by drifts: L -L f -f The transfer matrix is then 2 L L L2 − − + 1 L 1 01 L 1 0 1 2L 1 1 f f f ⇒ M = + − = 0 1 1 0 1 1 L L f f − + 2 1 f f We can build a ring out of N of these, and the overall transfer matrix will be N 14 M = M FODO Betatron Motion Skipping a lot of math, we find that we can describe particle motion in terms of initial conditions and a “beta function” β(s), which is only a function of location in the nominal path. x 1/ 2 Lateral deviation x(s) = A[β (s)] sin(ψ (s) +δ ) in one plane s s ds The “betatron function” β(s) is Phase advance ψ (s) = ∫ effectively the local wavenumber 0 β (s) and also defines the beam envelope. Closely spaced strong quads -> small β -> small aperture, lots of wiggles Sparsely spaced weak quads -> large β -> large aperture, few wiggles Minor but important note: we need constraints to define β(s) For a ring, we require periodicity (of β, NOT motion): β(s+C)=β(s) For beam line: matched to ring or source 15 Betatron Tune As particles go around a ring, they will undergo a number of betatron oscillations ν given by 1 s+C ds ν = Ideal orbit π ∫ β 2 s (s) Particle This is referred to as the “tune” (horizontal and vertical) trajectory If the tune is an integer, or lower order rational number, the effect of any imperfection or perturbation will be reinforced on subsequent orbits If we also consider coupling between the planes, in general, we want to avoid k ν ± k ν = integer ⇒ (resonant instability) x x y y Avoid lines in “small” integers the “tune plane” fract.part Y tuneof 16 fract. part of X tune Twiss Parameters: α, β, γ As a particle returns to the same point s on subsequent x' A β A γ γ (s)x 2 (s) + 2α(s)x(s)x′(s) + β (s)x′(s) 2 = A2 x Twiss Parameters β = (betatron function) 1 dβ α = − 2 ds 1+ α 2 γ = β As we examine different locations on the ring, the parameters will change, but the area A remains constant 17 Emittance If each particle is described by an ellipse with a particular x' amplitude, then an ensemble of particles will always remain γε within a bounding ellipse of a particular area: βε x γ (s)x 2 + 2αxx′ + βx′2 = ε Area = ε Since these distributions often have long tails, we typically define the “emittance” as an area which contains some specific fraction of the particles. σ 2 Typically, ε = contains 39% of Gaussian particles rms β ε 95 = 6 ε rms contains 95% of Gaussian particles 18 Dispersion and Chromaticity Up until now, we have assumed that momentum is constant Real beams will have a distribution of momenta Two most important parameters describing the behavior of off-momentum particles are: Dispersion: describes the position dependence on momentum Most important in the bend plane ∆x D ≡ x (∆p / p) Chromaticity: describes the tune dependence on momentum ∆ν ∆ν /ν ξ ≡ OR ξ ≡ x (∆p / p) x (∆p / p) 19 Longitudinal Motion We generally accelerate particles using structures that generate time- varying electric fields (RF cavities), either in a linear arrangement or located within a circulating ring cavity 0 cavity 1 cavity N E = E(t) = E0 sinϖt +δ 0 E = E(t) = E0 sinϖt +δ1 E = E(t) = E0 sinϖt +δ N In both cases, we phase the RF cavities so that a nominal arriving particle will see the same accelerating voltage and therefore get same boost in energy V (t) Nominal Energy ∆E s = eV sinφ ∆n 0 s φ = tωRF φs 20 Examples of Accelerating RF Structues Jlab C100 cavity: string of eight 7-cell cavities, 1497MHz, design gradient 19.2MW/m average.
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