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Lecture 2 Aspects of Transverse Beam Dynamics

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Lecture 2 Aspects of Transverse Beam Dynamics Lecture 2 Aspects of Transverse Beam Dynamics Chandra Bhat 1 Accelerator and Beamline Magnets Dipole Magnet: Dipole magnet is a device used to bend the path of charged particles during beam transport. The radius of curvature of a charged particle in a constant magnetic field perpendicular to its path is, 1 1 eB 0.2998 B[T ] 0.04 I [amp].n Iron Yoke = [m-1] = 0 = ;B = total R ρ p p[GeV / c] 0 h[cm] 1 n=number of turns h=pole gap Quadrupole Magnet: A device used to focus charged particle beam during beam transport. Particle trajectory Let us see what is the relationship between focal length, f, and the in a magnetic field A quadrupole strength. Fig. A shows bending of a charged particle in a magnetic field perpendicular to the plane of the paper and “B” l α ρ shows optical analogue of focusing. Then the deflection angle, l r eB eB α = − = − = φ l = − φ l l 2 α = − ρ f p βE ρ But the total bending field Bφ is given by, B dB Optics B = φ r = gr φ dr Then, egrl r 1 eg eg α = − = − or = kl; k = = 3 βE f f βE p f= focal length Quad strength 2 0.2998 g[Tesla / m] 2µ nI k[m−2 ] = ; g = 0 2 4 Field free region βE[GeV / c] R The quadrupole magnets provide material free aperture and focusing. A conventional quadrupole magnet used in synchrotrons has four iron poles with hyperbolic contours. By = −gx Bx = −gy Interesting features: The horizontal force component depends only on the horizontal position of the particle trajectory. Similarly Quadrupole Field for the vertical axis. Configuration y Fx = −evgx and Fy = evgy 4 x A single “quad” can provide focusing only in one plane because of the magnetic field configuration. Hence, one needs at least two consecutive quadrupole magnets to Defocusing get overall focusing of all charged particles. st Focusing 1 1 1 L f1= focal length of 1 quad = + − f nd 5 f f f f f 2= focal length of 2 quad Optics 1 2 1 2 L = separation between two quads Linear Machine: Contains only dipoles and quadrupoles. In these machines the horizontal and vertical motions are “completely decoupled” f1 f2 3 Sextupole Magnet: These are used for chromatic corrections Iron Yoke during beam transport, storing or beam acceleration. The sextupole magnets generate non-linear magnetic field and introduce horizontal and vertical motion of the beam 1 B = g ' ( x 2 − y 2 ), B = g ' xy y 2 x 6µ nI -1 g ' = 0 [m ] R 3 The momentum-independent sextupole strength is given by, eg' 0.2998 g'[Tesla / m2 ] 6 m = , m[m−3 ] = p p[GeV / c] 1 2 3 B x = gy + sxy + o ( 3 x y − y ) + ... Multipole Field Expansion: General 6 (Vertical bending field) 1 multipole field expansion is given by, B = B + gx + s ( x 2 − y 2 ) + ... y 0 y 2 7 (Horizontal bending field) In an un-coupled case, 1 1 B = 0 and B = B + gx + sx2 + sx3 + ... g, s, o are quadrupole, sextupole and octupole x y 0 y 2 6 strength parameters, respectively Thus, eBy eB0 y eg es 2 = + x + x + ... 8 p p p 2 p Dipole Quadrupole Sextupole contributions 4 Beam Transport and FODO Lattice A beam transport comprises of magnetic elements that form a lattice which guides the charged particle beam from one point to another. Such a beam generally traverses through a vacuum beam pipe. Lattice may be for FODO Period – Beam transfer line – Circular beam storage ring or accelerator – Linear accelerator 1/2QF QD 1/2QF FODO Lattice: Nomenclature A lattice comprising symmetric quadrupole triplets that k(s) Symmetric FODO can focus beam in both X and Y planes or a lattice comprising of Focusing-Drift-Defocusing-Drift with quad strengths. 1/2QF QD 1/2QF Beamline and Circular Machine: Nomenclature FODO lattice can be repeated as many times as needed k(s) Dipole to form a beamline or a circular machine. QF QD 5 Weak Focusing Accelerators So far we have assumed that the trajectory of the path of any particle in a beam is always perpendicular to the dipole magnetic field in a circular accelerator. Suppose a particle has a vertical N angle ≠ 0. Then the particle follows a spiral path and gets lost. Therefore we need an additional focusing force to keep the particle stable. We know that for an ideal orbit S mv2 mv2 = qvB ⇒ Restoring Force = − qvB = 0 9 R y R y mv2 For any other particle the restoring force is given by, F = − qvB x R y 10 To keep the beam particle focused one needs the gradient component in the magnetic field. dBy R dBy x By = B0 y + x + ... = B0 y (1+ + ...) dx B0 y dx R v x R dBy ≈ B 1− n with n = − ⇐ Field Index r m 0 y +B R B0 y dx 11 R Then one can show that if 0<n<1 (“Steenbeck’s criterion”) particle can be focused in both x and y planes and beam X= can be made stable. (r-R) Reference orbit Particle being A circular accelerator with this stability criterion is focused called a “Weak Focusing Accelerator” 6 Weak Focusing Accelerators (cont.) An optical analogue of weak focusing accelerator is shown here Closed orbit for p >p0 Central Design orbit (Closed orbit for p =p0) ρ ∆p x0 = D(s) p0 Closed orbit for p <p0 Strong Focusing Accelerators The circular accelerators with magnetic n>>1 field gradients y y S S n << −1 and n >> 1 12 n<<-1 n>>1 used alternatively also provide very high stability to the beam. These are called n<<-1 “Strong Focusing Accelerators” N N Christofilos (1950) n>>1 n<<-1 Alternating gradient focusing Courant, Livingston and Snyder (1952) Combined function magnets Combination of focusing and de-focusing B quads shown in Fig. B with f1=-f2 will give 1 1 1 L L = + − = 2 ∴Always focusing f f1 f2 f1 f2 f1 Optical analogue of such an accelerator with the combination of focusing and de-focusing quads isshown in Fig. B . Closed Orbit for the strong focusing lattice Earlier strong focusing accelerators are Accelerators with combined built with combined function magnets. function magnets Recent strong focusing accelerators used ≈± separated function magnets; CERN PS n 288 • Dipoles for bending Fermilab Booster n=165, -207 •Quads for focusing Fermilab Recycler n=620, -598 • Sextupoles for chromatic corrections 8 Coordinate System Reference orbit Y We use an orthogonal right handed coordinate system (Center of the beam) s σ Or Ideal orbit x (y,x,s) that follows the reference orbit particle. Or Design orbit X ρ y ∴This is a moving coordinate system with x, y – deviation of particle trajectory from reference O* orbit at the point of interest “O* ” Trajectory of a particle s – tangential vector at “O* ” σ - individual particle trajectory. This will be our coordinate system in the rest of our analysis Hill’s Equation: From this picture, 2 Horizontal Plane 2 2 2 2 x dx (dσ ) = []dφ0 (ρ 0 + x ) + (dx ) ≈ (ds ) 1 + ; 4 dφ0 ρ 0 = ds x'= ρ 0 dx dσ ds Small angle x approximation or dσ = ds 1 + x 13 ρ 0 σ Part. Traj. The equation of motion is going to be with respect to the x’ s Reference reference orbit. orbit Note that in the deflection plane ρ 0 d 2 x dφ dφ dφ dx' = dφ − dφ ⇒ x'' = = 0 − dφ 0 2 14 0 ds ds ds Substituting ds dσ ds x dφ0 = , dφ = = 1+ 15 O’ O ρ0 ρ ρ ρ0 9 We get the equation of motion as, 1 1 x x' ' = − 1 + 16 ρ 0 ρ ρ 0 For the monochromatic beam with momentum p0, the curvature of the charged particle in an electromagnetic field is given by, 1 qBy q s 2 2 = = B0 y + gx + (x − y ) + ... ρ p0 p0 2 1 ≈ + kx 17 ρ 0 Then, 1 1 x 1 k 2 very small so we x' ' = − + kx 1 + = − k + 2 x − x ρ 0 ρ 0 ρ 0 ρ 0 ρ 0 can neglect Thus, the equation of motion in horizontal plane becomes 1 x' '+ Kx = 0; with K = k + 2 ρ 0 18a Similarly in vertical plane q dB Notice that this equation a) has not got dipole x''−kx = 0; with k = 0 x p dy strength, b) change in sign 0 18b x The term 2 in the above equation describes the “weak focusing” of a bending ρ0 magnet. In a large accelerator this term can be neglected. 10 Momentum Dispersion: In reality, the particles in a beam are not monochromatic. Now let us look at a particle that has a small momentum offset, i.e., ∆p p → p0 + ∆p = p0 (1+ δ ); δ = 19 p0 Then the curvature is rewritten as, 1 eBy −1 e 2 = (1 + δ ) = []B0 y + gx + ... []1 − δ + δ − ... ρ p0 p0 1 δ ≈ + kx − 20 ρ 0 ρ 0 Thus, the equations of motion become δ x' '+ Kx = ρ This has bending term 0 One can combine δ u' '+ K (s)u = ρ(s) and focusing term δ these two into one ρ k(s). K(s) is called spring y ' '+ ky = equation as, 0 ρ y 0 constant 21 These are equations of motion for “strong focusing” charged particle beam accelerators (and for the beam transport lines). This equation is called “Hill’s equation”. Notice that the magnitude of the focusing strength is a free parameter.
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