Accelerator Physics Homework #7 P470 (Problems: -4) This exercise derives the linear transfer matrix for a skew quadrupole, where the magnetic field is B z = B 0 a z, B x = B 0 a x, B s = 0; with B 0 a = ( Bx x B ) z, z x=z=0 where B 0 is the main dipole field strength, and a is the skew quadrupole coefficient in multipole expansion of the magnetic field Apparently, the skew quadrupole field satisfies Maxwell s equation B z / z + B x / x = 0 The vector potential is A s = B 0 a xz, A x = 0, A z = 0 (a) Show that the equation of motion in a skew quadrupole is x + qz = 0, z + qx = 0, where q = B z Bρ z = a ρ (b) Show that the transfer matrix of a skew quadrupole is where C + S + / q C S / q qs M = C + qs + C C S / q C + S + / q qs + C qs C + C + = S + = cos θ + cosh θ cos θ cosh θ, C =, sin θ + sinh θ sin θ sinh θ, S =, θ = ql, and L is the length of the skew quadrupole (c) The coordinate rotation from (x, z) to ( x, z) by an angle φ is x x cos φ 0 sin φ 0 x z = R(φ) x z, R(φ) = 0 cos φ 0 sin φ sin φ 0 cos φ 0 z z 0 sin φ 0 cos φ Show that the transfer matrix of a skew quadrupole is M skew quad = R( 45 )M quad R(45 ), where M quad is the transfer matrix of a quadrupole This means that a skew quadrupole is equivalent to a quadrupole rotated by 45
(d) In the thin-lens limit, ie L 0 and ql /f, where f is the focal length, show that the 4 4 coupling transfer matrix reduces to M = + ( ) 0 U f U, U =, U = U 0 ( ) 0 0 0 Linear transfer Matrix of a Solenoid: The particle equation of motion in an ideal solenoidal field is x + gz + g z = 0, z gx g x = 0, where the solenoidal field strength is g = eb (s) p (a) Show that the coupled equation of motion becomes y jgy jg y = 0, where y = x + jz, and j is the complex imaginary number (b) Transforming coordinates into rotating frame with ȳ = ye jθ(s), where θ = s 0 gds, show that the system is decoupled, and the equation of motion becomes ȳ + g ȳ = 0 Thus both horizontal and vertical planes are focused by the solenoid (c) Show that the transfer matrix in the rotating frame is where θ = gs cos θ sin θ 0 0 g g sin θ cos θ 0 0 M = 0 0 cos θ sin θ, g 0 0 g sin θ cos θ (d) Transforming the coordinate system back to the original frame, ie y = e jθ ȳ, show that the transfer matrix for the solenoid becomes cos θ sin θ cos θ sin θ cos θ g g sin θ M = g sin θ cos θ cos θ g sin θ sin θ cos θ sin θ cos θ g sin θ cos θ sin θ cos θ g g sin θ sin θ cos θ g sin θ cos θ cos θ Note here that the solenoid, in the rotating frame, acts as a quadrupole in both planes The focusing function is equal to g In small rotating angle approximation, the corresponding focal length is f = g L = Θ /L, where L is the length of the solenoid, and Θ = gl is the rotating angle of the solenoid
3 Show that the dispersion relation of the longitudinal collective instability for a coasting beam is = j ei 0n ω 0 (Z /n) Ψ 0 ω πβ E (Ω nω) δ dδ, where Ψ 0 (δ) is the unperturbed distribution, Z is the longitudinal impedance evaluated at the coherence frequency Ω, n is the mode number, ω 0 is the angular revolution frequency of the on-momentum reference particle, and ω is the angular revolution frequency for a particle with a fractional off-momentum δ = p/p 0 4 The solution of Eq (49) with initial condition y 0 = ẏ 0 = 0 is y = ˆF ω β ω ( sin ωt ω ω β sin ω β t (a) Plot y(t) as a function of t for ω =, ˆF = 00 with three particles at ωβ = 08, 09, and 099 (b) Let ω β = ω + ɛ, where ɛ is a small frequency deviation Show that y(t) ˆF [ cos(ɛt) sin ωt sin(ɛt) ] cos ωt ω ɛ ɛ Show that the first term in square brackets does not absorb energy from the external force but the second term can The first term corresponds to a reactive coupling and the second term is related to a resistive coupling (c) If a beam has a distribution function given by ρ(ξ) with ρ(ξ)dξ =, discuss the centroid of beam motion, ie y(t) = y(t)ρ(ξ)dξ For example, we choose ξ = ɛ and ρ(ɛ) = /, if ɛ ; and 0, otherwise 5 Show that the equations of motion in the presence of skew quadrupoles and solenoids are x + K x (s)x + gz (q g )z = 0, z + K z (s)z + gx (q + g )x = 0 where g = B (s)/bρ and q = ( B z / z)/bρ = a /ρ (a) Show that the perturbation potential due to skew quadrupoles and solenoids is V lc = a ( pz xz + g(s) ρ p x p ) x p z (b) Expand the perturbation potential in Fourier series and find the coupling coefficient G,,l for the l-th harmonic (c) If the accelerator lattice has P superperiods, show that G,,l = 0 unless l = 0 (Mod P ) 3 )
6 The Hamiltonian H = ν x J x + ν z J z + G,,l J x J z cos(φ x φ z + χ) for a single linear coupling resonance can be transformed to the normalized phase-space coordinates by { X = Jx cos(φ x + χ x ), P x = J x sin(φ x + χ x ), Z = J z cos(φ z + χ z ), P z = J z sin(φ z + χ z ), where χ x χ z = χ is a constant linear coupling phase (Mod π) that depends on the location in the ring (a) Show that the Hamiltonian in the new phase-space coordinates is H = ν x(x + P x ) + ν z(z + P z ) + G,,l(XZ + P x P z ) (b) Show that the eigen-frequency of the Hamiltonian is ν ± = (ν x + ν z ) ± λ, λ = (ν x ν z ) + G,,l (c) Solve X and Z in terms of the normal modes, and show that X = A + cos(ν + ϕ + ξ + ) G,,l λ + δ A cos(ν ϕ + ξ ), Z = G,,l λ + δ A + cos(ν + ϕ + ξ + ) + A cos(ν ϕ + ξ ), where A ±, ξ ± are obtained from the initial conditions Particularly, we note that the horizontal and vertical betatron oscillations carry both normal-mode frequencies 7 Consider the sum resonance driven by skew quadrupoles, where the coupling constant G,,l is given by Eq (357) The Hamiltonian in the action-angle variables is given by H = ν x J x + ν z J z + G,,l J x, J z cos(φ x + φ z lθ + χ + ), where χ + is the phase of the coupling constant, θ is the independent variable serving as the time coordinate, and l is an integer near ν x + ν z, ie ν x + ν z l (a) Show that the difference of the actions, J x J z, is a constant of motion (b) Let g = G,,l Show that J x = (g δ )J x + constant J z = (g δ )J x + constant where overdots are derivative with respect to the independent variable θ, and the resonance proximity parameter is δ = ν x + ν x l This means that if the tunes of a particle is within the sum resonance stopband width, ie δ < g, the actions of the particle will grow exponentially at a growth rate of g δ 4
8 Show that the 3ν x = l resonance strength is given by Eq (38) in the first-order perturbation approximation (a) Show that, for systematic resonances, G 3,0,l = 0 if l 0 (Mod P ), where P is the superperiodicity of the machine (b) Show that the resonance strength of the third-order resonance at 3ν x = l due to two sextupoles at s and s is proportional to [β x (s )] 3/ [S(s ) s] + [β x (s )] 3/ [S(s ) s] e j[3ψ (3ν x l) θ], where ψ = s s ds/β x is the betatron phase advance, θ = (s s )/R 0, and R 0 is the average radius of the accelerator Show that, at the 3ν x = l resonance, the geometric aberrations of these two sextupoles cancel each other if ψ = π and [β x (s )] 3/ [S(s ) s] = [β x (s )] 3/ [S(s ) s] (c) Based on the above result, show that the geometric aberration of two chromatic sextupoles located in the arc of FODO cells separated by 80 in phase advance cancel each other 5