Accelerator Physics Homework #3 P470 (Problems: -5). Particle motion in the presence of magnetic field errors is (Sect. II.2) y + K(s)y = B Bρ, where y stands for either x or z. Here B = B z for x motion, and B = B x for z motion. (a) Define a new coordinate η = y/ β with φ = (/ν) s 0 ds/β as the independent variable, show that the equation of motion becomes η + ν 2 η = ν 2 3/2 B β Bρ, where the overdot is the derivative with respect to the independent variable φ. Show that Eq. (2.60) is a solution of the above equation for the dipole field error, where B/Bρ is only a function of the independent variable s. (b) Show that the Green function G(φ, φ ) in is ( d 2 dφ 2 + ν2 ) G(φ, φ ) = δ(φ φ ) G(φ, φ ) = cos ν(π φ φ ). 2ν sin πν Use the Green function to verify the solution given by Eq. (2.60). Use Eq. (2.63) for the Fourier expansion of the dipole field error β 3/2 B/Bρ show that the closed orbit arising from the dipole error is y co (s) = β(s) where f k are integer stopband integrals. k= ν 2 f k ν 2 k 2 ejkφ, (c) Using a single stopband approximation. and limiting the closed-orbit deviation to less than 20% of the rms beam size, show that the integer stopband width Γ [ν] is given by Γ [ν] 5ν f [ν] / ɛ rms, where [ν] is the integer nearest the betatron tune. 2. Consider an accelerator made of 24 FODO cells with focal length f for focusing and f defocusing quadrupoles. (a) If one of the focusing quadrupoles is misaligned by 0 cm in the +ˆx direction, what is the maximum closed orbit excursion in the accelerator for phase advance per cell at 85.
(b) Now, if one of the focusing quadrupoles is misaligned by 0 cm in the +ẑ direction, what is the maximum closed orbit excursion in the accelerator for phase advance per cell at 85. What is the difference between this case and the previous problem? 3. Consider an accelerator made of 24 FODO cells with focal length f for focusing and f defocusing quadrupoles. If the focal length of one of the focusing quadrupoles is increased by 5%, what are its effects on the accelerator? 4. A localized rf dipole provides a dipole kick to beam motion with kick angle θ = θ 0 sin ωt. The equation of motion becomes d 2 y ds 2 + K(s)y = θ 0 sin(ωt + ξ) n δ(s s 0 nc), where θ 0 = B kick l kick /Bρ, ξ is an arbitrary phase, and C is the circumference of the accelerator. Find the beam motion in the accelerator in terms of the betatron amplitude function. 5. The dispersion function in a combined-function dipole satisfies the equation D + K x D = /ρ. (a) Show that the solution for constant K x = K > 0 is D = a cos Ks + b sin Ks + /ρk. Let D 0 and D 0 be the dispersion function and its derivative at s = 0. Show that the solution can be expressed as D(s) D 0 D (s) = M D 0, where the transfer matrix is cos Ks K sin Ks ( cos Ks) ρk M = K sin Ks cos Ks ρ sin Ks. K 0 0 (b) Show that the transfer matrix for constant K x = K < 0 is M = cosh K sinh K s sinh K s ( + cosh K s) K ρ K K s cosh K s sinh K s ρ K. 0 0 (c) Find the transfer matrix of a sector magnet. 2
(d) For a rectangular magnet, show that the horizontal transfer matrix is (see Exercise 2.2.3) ρ sin θ ρ( cos θ) M rectangular dipole = 0 2 tan(θ/2), 0 0 where ρ and θ are the bending radius and the bending angle. (e) In thin-lens (small-angle) approximation, show that the transfer matrices M for quadrupoles and dipoles become 0 0 l lθ/2 M quad = /f 0, M dipole = 0 θ, 0 0 0 0 where f is the focal length, and l and θ are the length and bending angle of the dipole. 6. Multiple scattering from gas molecules inside the vacuum chamber can cause beam emittance dilution, particularly at high-β locations. This effect can also be important in the strip-injection process. This exercise estimates the emittance dilution rate based on the multiple scattering formula (see the particle properties data) for the rms scattering angle ( ) 2 θ 2 = 2θ0 2 2 3.6[MeV]zp x, βcp X 0 where p, βc and z p are momentum, velocity, and charge number of the beam particles, X 0 is the radiation length, and x is the target thickness. The radiation length is X 0 = 76.4A Z(Z + ) ln(287/ Z) [g/cm2 ] where Z and A are the atomic charge and the mass number of the medium. (a) Using the ideal gas law, P V = nrt, where P is the pressure, V is the volume, n is the number of moles, T is the temperature, and R = 8.34 [J ( K mol) ], show that the equivalent target thickness in [g/cm 2 /s] at room temperature is x =.64 0 6 βp g [ntorr]a g [g/cm 2 /s], where βc is the velocity of the beam, P g is the equivalent partial pressure of a gas at room temperature T = 293 K, and A g is the gram molecular weight of a gas. Show that the emittance growth rate is = ( dɛ τ ɛ ɛ dt = 2.345 γ β [m] βɛ N [π mm mrad] z p pc [GeV] ) 2 P g [ntorr]a g X 0g [g/cm 2 ] [h ], where β is the average transverse betatron amplitude function in the accelerator, X 0g is the radiation length of the gas, γ is the Lorentz relativistic factor, z p is the charge of the projectile, and p is the momentum of the beam. Because the emittance growth is proportional to the betatron function, better vacuum at high-β location is useful in minimizing the multiple scattering effects. 3
(b) During the H strip-injection process, the H passes through a thin foil of thickness t foil [µg/cm 2 ]. Show that the emittance growth per passage is ɛ = 7.8 β,foil t foil [µg/cm 2 ] β 2 (pc[mev]) 2 X 0 [g/cm 2 ] [π mm mrad], where β,foil is the betatron amplitude function at the stripper location, p is the momentum of the injected beam, βc is the velocity of the beam, and X 0 is the radiation length. Estimate the emittance growth rate per passage through carbon foil with H beams at an injection energy of 7 MeV if β,foil = 2 m and t foil = 4 [µg/cm 2 ]. 7. In the presence of gradient error, the betatron amplitude functions and the betatron tunes are modified. We define the betatron amplitude deviation functions A and B as A = α β 0 α 0 β, B = β β 0, β0 β β0 β where β 0 and β are respectively the unperturbed and the perturbed betatron amplitude functions associated with the gradient functions K 0 and K, and α 0 and α are related to the derivatives of the betatron amplitude functions. Thus β 0 and β satisfy the Floquet equation: β 0 = 2α 0, α 0 = K 0β 0 γ 0, dψ 0 /ds = /β 0, β = 2α, α = K β γ, dψ /ds = /β, where ψ 0 and ψ are the unperturbed and the perturbed betatron phase functions. (a) Show that ( db ds = A + ), β 0 β ( da ds = +B + ) + β 0 β K, β 0 β where K = K K 0. (b) In a region with no gradient error, show that A 2 + B 2 = constant, i.e. the phasespace trajectory of A vs B is a circle. (c) In thin-lens approximation, show that the change of A at a quadrupole with gradient error is A = β 0 β K ds β 0 g, where g = + K ds is the integrated gradient strength of the error quadrupole, and β 0 is the averaged value of betatron function in the quadrupole. See also H. Zgngier, LAL report 77-35, 977; B.W. Montague, CERN 87-03, 75-90 (987). 4
(d) In thin-lens approximation, show that the change of A in a sextupole is A β 0 g eff, where g eff = (B 2 s/bρ)x co is the effective quadrupole strength, (B 2 s/bρ) is the integrated sextupole strength, and x co is the closed-orbit deviation from the center of the sextupole. (e) If we define the average betatron phase function as where φ = 2 ν show that the function B satisfies ( s s 0 β 0 + β ) ds, ν = ( s0 +C + ) ds, 4π s 0 β 0 β d 2 B d φ 2 + 4 ν2 B = 4 ν 2 (β 0β ) 3/2 β 0 + β K. Find the equation for the betatron amplitude function in the limit of small gradient error. 8. When the betatron tune ν of a particle sits at p/2, show that the action of the particle will grow exponentially as exp{2π G p n} where n is the revolution number, where G p = 2π β(s) K(s)e ipφ(s) ds is the stop-band width, defined as the Fourier amplitude of quadrupole-field error at the pth harmonic, K(s) is the random quadrupole error, β(s) is either the horizontal or vertical betatron amplitude function, φ(s) = (/ν) s 0 (ds/β(s), and s is the longitudinal coordinate. 9. Show that the evolution of the H-function is H = H 0 + 2(α 0 D 0 + β 0 D 0 )[M 23M M 3 M 2 ] +2(γ 0 D 0 + α 0 D 0)[M 3 M 22 M 23 M 2 ] + β 0 [M 3 M 2 M 23 M ] 2 +γ 0 [M 3 M 22 M 23 M 2 ] 2 2α 0 [M 3 M 2 M 23 M ][M 3 M 22 M 23 M 2 ] where H 0 = γ 0 D 2 0 + 2α 0D 0 D 0 + β 0D 2 0 ; M ij is a matrix element of the transfer matrix; and α 0, β 0, and γ 0 are Courant-Snyder parameters at the initial location. (a) Using the M ij, Find H in a sector dipole. (b) Find H in a rectangular dipole (use the result of Exercise 2.4.). 5
0. A set of four rectangular dipoles with zero net bending angle B[ρ, θ] O[l ] B[ ρ, θ] O[l 2 ] B[ ρ, θ] O[l ] B[ρ, θ] has many applications. It can be used as a beam translation (chicane) unit to facilitate injection, extraction, internal target operation, etc. It can also be used as one unit of the wiggler magnet for modifying electron beam characteristics or for producing synchrotron radiation. (a) Show that the rectangular magnet beam translation unit is achromatic to all orders, and show that the R 56 element of the transport matrix, in small angle approximation, is R 56 = 2θ 2 (l + 2 3 ρθ). (b) A simplified compact geometry with l = l 2 = 0 (shown in the figure above) is often used as a unit of the wiggler magnet in electron storage rings. Assuming that D 0 = D 0 = 0, show that the dispersion function created by the wiggler magnet is 2 D(s) ( cos φ), 0 < s < L w = ( cos ρ φ) ( cos θ) cos φ w [sin θ + 2 tan θ( cos θ)] sin φ, L w < s < 2L w where φ = s/ρ w and φ = (s L w )/ρ w, and sin φ, 0 < s < L D sin θ 2 tan θ( cos θ), s = L (s) = w+ sin φ + ( cos θ) sin φ [sin θ + 2 tan θ( cos θ)] cos φ, L w < s < 2L w. Show that cos θ D(s = 2L w ) = 2ρ w, D (s = 2L w ) = 0. cos θ Since D = 0 at the symmetry point, the wiggler is an achromat. In small bendingangle approximation, show that the dispersion function becomes { s D(s) = 2 /2ρ w, 0 < s < L w (2L 2 w (2L w s) 2 )/2ρ w, L w < s < 2L w, { D s/ρw, 0 < s < L (s) = w (2L w s)/ρ w, L w < s < 2L w. 2 Be sure to take the edge focusing into account. 6