Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Accelerators - Lectures I and II by Ted Wilson (John Adams Institute for Accelerator Science) The history of accelerators Linear accelerator Dispersion in a waveguide Slowing down the wave Quality factor and Shunt Impedance Cyclotron and Magnetic Rigidity The Synchrotron Magnet types and their multipole field shapes Transverse coordinates Quadrupoles and AG focusing Equation of motion in transverse co-ordinates The lattice and Beam sections Emittance , Beam Size, Q and Beta Phase stability, and Closed orbit Dispersion, and Synchrotron motion Chromaticity Luminosity Final focus optics Beam Beam Force and other LC limits Instability and the impedance of the wall How to organize a Design Study Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 1 Reference Material This talk http://acceleratorinstitute.web.cern.ch/acceleratorinstitute/Ambleside%20Wilson.pdf A text book “An Introduction to Particle Accelerators” – E. Wilson - OUP http://ukcatalogue.oup.com/product/9780198508298.do A general introduction – “Engines of Discovery” A. Sessler and E. Wilson - WSP http://www.worldscibooks.com/physics/6272.html Lattice design Rossbach ,J. and Schmüser, P. (1992). Basic course on accelerator optics. Proceedings of the 1986 CERN Accelerator School, Jyvaskyla, Finland, CERN 87-1 http://doc.cern.ch/yellowrep/2005/2005-012/p55.pdf Errors and Corrections http://preprints.cern.ch/cgi-bin/setlink?base=cernrep&categ=Yellow_Report&id=95-06_v1 Magnet and power Supply http://preprints.cern.ch/cgi-bin/setlink?base=cernrep&categ=Yellow_Report&id=92-05 Radio Frequency System and Cavities http://preprints.cern.ch/cernrep/2005/2005-003/2005-003.html Instabilities http://doc.cern.ch/yellowrep/2005/2005-012/p139.pdf Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 2 The history of accelerators Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 3 Linear accelerator Particle gains energy at each gap Lengths of drift tubes follow increasing velocity Spacing becomes regular as v approaches c Wideroe’s first linac: Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 4 Inside the Fermilab linac Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 5 The International Linear Collider (ILC) TESLA technology: these superconducting accelerator structures are built of niobium, and are the crucial components of the International Linear Collider. Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 6 Transverse magnetic (E) modes Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 7 Dispersion in a waveguide c /ω = λ k2 = c / ω c = λc 2 ω ⎛ ⎞ ⎝ c⎠ v ph = λ f = ω / k Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 8 − k = 2π / λ g 2 ω ⎛ c⎞ ⎝ c⎠ dω vg = dk Two travelling waves in a guide. Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 9 Slowing down the wave Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 10 Cavity resonators • n x E= 0 – because the E field should be normal to the perfectly conducting walls. Assume we can separate out a time dependent solutions aM = e leaving − ωM 2Q t {A1 cos Ω M t + A2 sin Ω M t} a spatial solution: Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 11 Quality factor Energy stored and dissipated per cycle Us Us Q = 2π =ω Ud W 1 ˆ2 Us = ∫ εE dv 2 where W is the power dissipated 2 V0 1 Isurf W = Rs = ∫ dA 2 2 σδ Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 12 Cyclotron ev × B = mv 2 ρ ρ mv p Bρ = = e e LIV1_4(Cyclotron_Side).PCT, LIV1_4(Cyclotron_Top).PCT,force,gi, NEWCLASSCYC.AD5 Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 13 Magnetic Rigidity from resolution of momenta that: the magnitude of the force may be written: Equating the right hand sides of the two expressions above, we find we can define a quantity known as magnetic rigidity: A common convention in charged particle dynamics is to quote pc in units of electron–volts. Whereupon: Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 14 Components of a synchrotron Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 15 Magnet types By x Dipoles bend the beam B y x Quadrupoles focus it 0 0 Sextupoles correct chromaticity Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 16 Multipole field expansion Scalar potential obeys Laplace 1 ∂ 2φ 1 ∂ ⎛ ∂φ ⎞ ∂ 2φ ∂ 2φ + 2 = 0 or 2 2 + ⎜r ⎟ = 0 2 r ∂θ r ∂r ⎝ ∂r ⎠ ∂x ∂y whose solution is ∞ φ = ∑φn r n sin nθ n=1 Example of an octupole whose potential oscillates like sin 4θ around the circle Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 17 Taylor series expansion Field in polar coordinates: To get vertical field Taylor series of multipoles Bz = φ0 + φ2 .2 x + φ3 .3 x 2 + φ4 .4 x 3 + ...... 1 ∂Bz ∂ 2 Bz 2 1 ∂ 3 Bz 3 = B0 + x+2 2 x + x + ... 3 1! ∂x 3! ∂x ∂x Dip. Quad Sext Octupole Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 18 Fig. cas 1.2c Multipole field shapes Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 19 Multipoles in the Hamiltonian We said contains x (and y) dependence We find out how by comparing the two expressions: We find a series of multipoles: For a quadrupole n=2 and: Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 20 Q diagram nQ = p , l QH + mQV = p , Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 21 Transverse coordinates ρ s Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 22 Gutter Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 23 Fields and force in a quadrupole No field on the axis Field strongest here (hence is linear) Force restores ∂B y Gradient Normalized: ∂x 1 ∂By . k =− ( B ρ ) ∂x POWER OF LENS Defocuses in vertical plane l ∂By 1 lk = − . = ( B ρ ) ∂x f Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 24 Fig. cas 10.8 Alternating gradients Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 25 Equation of motion in transverse coordinates Hill’s equation (linear-periodic coefficients) where at quadrupoles like restoring constant in harmonic motion Solution (e.g. Horizontal plane) y = β (s) ε sin[φ ( s) + φ0 ] Condition Property of machine Property of the particle (beam) ε Physical meaning (H or V planes) Envelope Maximum excursions Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 26 The lattice Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 27 Beam sections Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 28 after,pct Example of Beam Size Calculation Emittance at 10 GeV/c Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 29 Physical meaning of Q and β Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 30 Betatron phase space at various points in a lattice Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 31 Closed orbit of an ideal machine F F F In general particles executing betatron oscillations have a finite amplitude One particle will have zero amplitude and follows an orbit which closes on itself In an ideal machine this passes down the axis x′ x Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 32 Closed orbit Zero betatron amplitude Phase stability V = V 0 sin(2 π f a + φ s ) Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 33 PHS.AD5 Dispersion F Low momentum particle is bent more It should spiral inwards but: There is a displaced (inwards) closed orbit Closer to axis in the D’s Extra (outward) force balances extra bends F D(s) is the “dispersion function” F F F F Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 34 Fig. cas 1.7-7.1C Synchrotron motion F This is a biased rigid pendulum 2 2 π η V h f φ&& = − 0 2 (sinφ − sinφs ) E0 β γ F For small amplitudes 2 V h f π η 2 0 φ&& + φ =0 2 E0 β γ F Synchrotron frequency η hV0 cos φs fs = f . 2 2πE0β γ F Synchrotron “tune” η hV0 cos φs fs Qs = = . 2 f 2πE0β γ Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 35 Bucket and pendulum • θ θ F The “bucket” of synchrotron motion is just that of the rigid pendulum F Linear motion at small amplitude F Metastable fixed point at the top F Continuous rotation outside Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 36 Measurement of Chromaticity We can steer the beam to a different mean radius and a different momentum by changing the rf frequency and measure Q Since Hence Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 37 Correction of Chromaticity Parabolic field of a 6 pole is really a gradient which rises linearly with x If x is the product of momentum error and B" D Δp dispersion Δk = . ( B ρ) p The effect of all this extra focusing cancels chromaticity ⎡ 1 B"(s )β ( s )D( s)ds ⎤ dp ΔQ = ⎢ ∫ . ⎥ ( Bρ) ⎣ 4π ⎦ p Because gradient is opposite in v plane we must have two sets of opposite polarity at F and D quads where betas are different Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 38 Luminosity Imagine a blue particle colliding with a beam of cross section area - A σ Probability of collision is ⋅N A For N particles in both beams σ A Suppose they meet f times per second at the revolution frequency f rev = ⋅ N2 Event rate 2 f rev N ⋅σ A βc 2πR Make big e.g. 10 −25 Make small LUMINOSITY ≈ 1030 to 1034 [cm-2 s-1 ] Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 39 Hour-glass effect β (s) = β * + s2 β* Luminosity has to be calculated in slices desirable to have σz ≤ βy ⇒ short bunch length Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 40 Final focus optics final doublet (FD) IP f 1 f (=L*) 2 f1 f2 f2 Need large demagnification of the (mainly vertical) beam size β y* of the order of the bunch length σz (hourglass effect) Need free space around the IP for physics detector Assume f2 = 2 m ⇒ f1 ¡ 600 m Can make shorter design but this roughly sets the length scale Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 41 Example of a final focus Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 42 Field around a moving cylinder of charge Take unit length of the bunch Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 43 Beam Beam Force Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 44 Beam-beam Q shift limit This is the (linear) defocusing effect of one beam on the other and since it is comparable to the non–linear effects, we take it as a practical limit for beam stability r p Nβ * dQ = ≤ 0.003 γA Make the β as small as possible The luminosity formula is similar, and to win luminosity for a fixed Q shift we only gain linearly with energy * γ f rev N 2 L= A We increase the value of N up to the limit imposed by instabilities It also helps to split into many bunches Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 45 Beam-Beam for linear colliders Luminosity where enhancement (pinch) D is “disruption” Beamstrahlung Make bunch length as large as possible but must be less than Courant’s β which is the length of the waist (hourglass effect) Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 46 Linear collider total power limit All the N particles accelerated f times per second must each be given energy U Prf = Pbeam η η With an efficiency, η, the power limits f and the luminosity – can be hundreds of MW fN L= A = NfeU 2 While N is limited by instabilities and A by focusing techniques Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 47 Impedance of the wall Wall current Iw due to circulating bunch Vacuum pipe not smooth, Iw sees an IMPEDANCE : Resistive = in phase. capacitive lags, inductive leads Impedance Z = Zr + iZi Induced voltage V ~ Iw Z = –IB Z V acts back on the beam Ö INSTABILITIES INTENSITY DEPENDENT Test: If Initial Small Perturbation is : INCREASED? INSTABILITY DECREASED? STABILITY Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 48 Stability diagram Keil Schnell stability criterion: Z Fm0 c 2 β 2γη ⎛ Δp ⎞ 2 ≤ ⎜ ⎟ n Io ⎝ p ⎠ FWHH Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 49 Accelerators - Lectures I and II by Ted Wilson (John Adams Institute for Accelerator Science) The history of accelerators Linear accelerator Dispersion in a waveguide Slowing down the wave Quality factor and Shunt Impedance Cyclotron and Magnetic Rigidity The Synchrotron Magnet types and their multipole field shapes Transverse coordinates Quadrupoles and AG focusing Equation of motion in transverse co-ordinates The lattice and Beam sections Emittance , Beam Size, Q and Beta Phase stability, and Closed orbit Dispersion, and Synchrotron motion Chromaticity Luminosity Final focus optics Beam Beam Force and other LC limits Instability and the impedance of the wall How to organize a Design Study Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 50 Reference Material This talk http://acceleratorinstitute.web.cern.ch/acceleratorinstitute/Ambleside%20Wilson.pdf A text book “An Introduction to Particle Accelerators” – E. Wilson - OUP http://ukcatalogue.oup.com/product/9780198508298.do A general introduction – “Engines of Discovery” A. Sessler and E. Wilson - WSP http://www.worldscibooks.com/physics/6272.html Lattice design Rossbach ,J. and Schmüser, P. (1992). Basic course on accelerator optics. Proceedings of the 1986 CERN Accelerator School, Jyvaskyla, Finland, CERN 87-1 http://doc.cern.ch/yellowrep/2005/2005-012/p55.pdf Errors and Corrections http://preprints.cern.ch/cgi-bin/setlink?base=cernrep&categ=Yellow_Report&id=95-06_v1 Magnet and power Supply http://preprints.cern.ch/cgi-bin/setlink?base=cernrep&categ=Yellow_Report&id=92-05 Radio Frequency System and Cavities http://preprints.cern.ch/cernrep/2005/2005-003/2005-003.html Instabilities http://doc.cern.ch/yellowrep/2005/2005-012/p139.pdf Lecture 1 and 2 - E. Wilson – 16-Oct08 - Slide 51