Injection mismatch type of injection mismatch will lead to an - PDF document

Injection mismatch: As a rule, proton/ion accelerators need their full aperture at injection, thus avoiding mismatch allows a beam of larger normalized emittance ε * and containing more Protons. In proton/ion ring accelerators any Injection mismatch type of injection mismatch will lead to an emittance blow-up. Off axis injection can be detected easily by (????) turn-by-turn BPMs in the ring (before Landau damping occurs). by Kay Wittenburg –DESY- The orbit mismatch can be corrected by a proper setup of the steering magnets, kickers and septas. However, any mismatch of the optical parameters α, β (and therefore γ) will also lead to an emittance blow-up (and beam losses) and is not detectable by BPMs. Fig. 1a shows the phase ellipse at a certain location in a circular accelerator. The ellipse is defined by the optics of the accelerator with the emittance ε and the optical parameters β = beta function, γ = (1 + α)/β and the slope of the beta function α = - β '/2. Fig. 1b-d shows the process of filamentation after some turns. Fig M1a: A phase space ellipse of a circular accelerator, defined by α, β, γ, ε Fig. M1 b-d: Filamentation of an unmatched beam (from Ref. 2) Assuming a beam is injected into the circular machine, defined by β 0 and α 0 (and therefore γ 0 ) with a given emittance ε 0 . For each turn i in the machine the three optical parameters will be transformed by During 1 turn the whole ellipse rotates with Q, but the projection on the x-axis oscillates with 2 Q. One turn ⎛ β ⎞ ⎛ 2 − 2 ⎞ ⎛ β ⎞ 2 C SC S ⎜ ⎟ ⎜ + 1 ⎟ ⎜ ⎟ i i gives two periods. ( ) α = ⎜ − + − ⎟ ⋅ α = ⎜ ⎟ ' ' ' ' ⎜ ⎟ 0 CC SC S C SS Starting with i + 1 i ⎜ ⎟ i ⎜ ⎟ ⎜ ⎟ γ γ ' 2 − 2 ' ' ' 2 ⎝ ⎠ ⎝ C S C S ⎠ ⎝ ⎠ + 1 i i where C and S are the elements of the Twiss matrix ( μ = 2 π q, q = tune, see B. Holzer’s talk): ⎛ ⎞ ⎛ μ + α ⋅ μ β ⋅ μ ⎞ cos sin sin C S ⎜ ⎟ = ⎜ 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ − γ ⋅ μ μ − α ⋅ μ ' ' sin cos sin ⎝ C S ⎠ ⎝ ⎠ (1) 0 0 and γ = (1+α 2 )/β Without any mismatch, the three parameters will be constant while a mismatch will result in an oscillation of the parameters. Exercise M1: Show the constant β without mismatch and the oscillation of β for the mismatch. What is the oscillation frequency? Explain by formula (resolving β i+1 ) and by picture Exercise M2: Discuss how to measure a 10% betatron mismatch at injection between a transport line β = β − α + γ 2 2 2 C SC S and a storage ring , for example in the HERAp accelerator. How large is the emittance blow-up? + 1 i i i i α 2 β − α β α + β 2 + α 2 β ⋅ 2 μ = β ⋅ 2 μ ( 2 ( 1 ) / ) sin sin 0 i 0 0 i 0 i i 0 Some important HERAp parameters Circumference circ = 6.3 km with (1) and some transformations Tune q = 0.31 or f = 13.8 kHz β = β ⋅ 2 μ + α 2 β − α β α + β 2 γ ⋅ 2 μ + α β − β α ⋅ ⋅ μ ⋅ μ cos ( 2 ) sin (( ) 2 sin cos Momentum E p = 40 GeV/c at injection + i 1 i 0 i 0 0 i 0 i 0 i 0 i Normalized emittance ε n = 20 π mm mrad, ε 0 = 5 ∙ 10 -7 β 0 = 238 m, α 0 = -2.2, => γ 0 = 0.0245 at the injection point ( βγ−α 2 =1 ). => Parameters of the ring with sin μ ⋅ cos μ = ½ sin |2 μ |, cos 2 μ = ½ (1+cos2 μ ), sin 2 μ = ½ (1-cos2 μ ) β = 214 m, α 0 = α , => γ = 0.0272 at the injection point. (10% mismatch)=> parameters of the injected beam one gets twice the betatron tune. Without mismatch β i+1 = β i The emittance blow-up due to the betatron mismatch ( α 0 = α ) can be calculated with the following FFT formula derived from Ref. 2, 3 (gaussian beams): 8000 0.5 * f rev = 23.65 kHz α − α β − β ⎛ ⎞ 7000 2*q = 29.2 kHz ε = ε ⋅ + ⋅ Δ Δ = ⎜ 0 0 ⎟ f rev -(2q-f rev ) = 18 kH z ( 1 0 . 5 det( ) ) J with J ⎜ ⎟ 0 filamented − ( γ − γ ) − ( α − α ) 6000 ⎝ ⎠ 0 0 5000 Amplitude |det Δ J| = ( α 0 − α ) 2 + (γ 0 − γ)⋅(β 0 − β) = 0.066 4000 3000 In this example a 10% β -mismatch leads to an emittance blow up 2000 Δε = (ε filamented – ε 0 )/ ε 0 ∙ 100 % = 3.3%. 1000 Fig. M2: β -oscillation amplitudes 0 0.185 1.48 2.775 4.07 5.365 6.66 7.955 9.25 10.545 11.84 13.135 14.43 15.725 17.02 18.315 19.61 20.905 22.2 23.495 24.79 26.085 27.38 28.675 29.97 31.265 32.56 33.855 35.15 36.445 37.74 39.035 40.33 41.625 42.92 44.215 45.51 46.805 Frequency 1

Exercise M2a: What is the beam size after filamentation? What kind of measurement will you propose to determine the β -mismatch? Which monitor do you propose to use for this measurement? β 1 =238 m, Normalized emittance ε n = 20 π mm mrad, ε 0 = 5 · 10 -7 , ε filamented = 5.162 · 10 -7 Δε = ( ε filamented – ε 0 )/ ε 0 · 100 % = 3.3%. Emittance blow up due to mismatch 16 A simple beam width measurement after filamentation at (for example) β 1 =238 m results in: 14 σ = ε ⋅ β = 10 . 91 mm 0 0 1 12 σ = ε ⋅ β = 11 . 08 mm emittance blow up [%] 0 filamented 1 10 filamented beta mismatch 8 alpha mismatch � The effect is hard to detect for a typical measurement (Take the resolution of the instrument into account) 6 4 A mismatch of the phase space will result in transverse shape oscillations, at least for some ten turns, before the filamentation 2 of the beam. => Observation of the width-oscillation at one location. 0 -20 -15 -10 -5 0 5 10 15 20 % of mismatch Fig. M3: Emittance blow up Δε = ( ε filamented - ε )/ ε · 100 % due to mismatch A measurement of width oscillations at injection is a very efficient method to detect an optical mismatch Oscillations of beam width due to mismatched injection that increases the emittance in the circular accelerator. Measurement of the turn-by-turn shape oscillation is possible with a fast (turn by turn) readout of: 12.5 12 Proposed Monitors? 1. Thin screen (OTR, Phosphor) 11.5 2. SEM grids, 11 3. IPM, 4. QP-Pickup Width x [mm] "+10% beta-mism." 10.5 5. Synchrotron Radiation (SR) -Monitor (electrons). "+10% alpha-mism." examples in Refs. 4-8 "-10% alpha-mism." 10 "no mism." 10% mism. after filamentation 9.5 Exersice M2b: What is the effect of the proposed monitor(s) on the beam? 9 •Screen/Grid: Emittance blow-up and losses 8.5 •IPM: Very small, a sufficient signal at each turn needs a pressure bump => emittance blow-up and losses •QP-Pickup: None (see Rodri's talk), but very difficult to suppress the dipole mode. 8 •SR-Monitior: None, but no light from protons! 1 2 3 4 5 6 7 8 9 10 11 Number of turns after Injection Figure M4: Oscillations of the beam width due to mismatched injection. Note also the small difference of the beam width after filamentation. Blow-up: 20 A screen/grid or IPM pressure bump will give an additional constant increase of the emittance, but it 18 can easily be separated from the oscillation observation. The protons receive a mean kick at each traverse through a screen resulting in an additional angle θ . 16 ⎡ ⎛ ⎞ ⎤ 0 . 014 1 14 d d θ = ⋅ ⋅ + ⎜ ⎟ ⎢ 1 log ⎥ Z in radians ⎜ ⎟ Beam width (2 σ ) [mm] ⋅ β 10 9 12 p l ⎣ ⎝ l ⎠ ⎦ rad rad HERA PETRA 10 HERA No mism. where p is the momentum in GeV/c and Z=1 the charge number of the proton, β = v/c the velocity, DESY III No mism. d the thickness of the foil and l rad the radiation length of the material of the foil. This formula 8 describes the gaussian approximation of the mean scattering angle of the protons after one traverse. Fig. M5: Emittance growth due to 6 The change of the emittance δε for every turn can be calculated by: a d = 10 μ m Titanium foil at 4 injection energy of HERA and PETRA ( α = –2 m, β = 40 m, ε n = δε = ⋅ π ⋅ θ 2 ⋅ β 2 2 14 π mm mrad, q = 0.14, p = 7.5 rms GeV/c) and DESY III (with a foil- 0 0 5 10 15 21 26 31 36 41 46 thickness of d = 1 μ m (!)) which adds quadratically to the 1 σ - emittance of the previous turn. Number of turns π 1 The emittance blow-up is shown in Fig. M5 for a 10 μ m thick titanium foil as the source of OTR 2 D. Möhl, P.J. Bryant,CAS: δε = ⋅ θ 2 ⋅ β δ ε = ⋅ θ ⋅ β radiation. In addition a betatron mismatch of 10% is assumed. The figure shows a small growth of the rms rms 2 rms x 2 beam width due to the foil, which does not affect the beam width oscillation. The growth rate is small compared to the oscillation amplitude. The faster growth rate in PETRA is a result of the smaller π momentum of the injected protons and therefore a larger scattering angle in the foil. This angle will M. Giovannozzi: CAS 2005: δε = ⋅ θ ⋅ β 2 become much larger in DESY III (p=310 MeV/c, β = 0.3), so that the beam width will become rms 4 unacceptably large within one turn and the loss rate will increase drastically (in Fig. M5 the line for DESY III extend the border of the figure within 3 turns even with a 1 μ m screen). 2