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USPAS Accelerator Physics 2013 Duke University Magnets And USPAS Accelerator Physics 2013 Duke University Magnets and Magnet Technology Todd Satogata (Jefferson Lab) / [email protected] Waldo MacKay (BNL, retired) / [email protected] Timofey Zolkin (FNAL/Chicago) / [email protected] http://www.toddsatogata.net/2013-USPAS T. Satogata / January 2013 USPAS Accelerator Physics 1 Outline § Section 4.1: Back to Maxwell § Parameterizing fields in accelerator magnets § Symmetries, comments about magnet construction § Sections 4.2-3: Relating currents and fields § Equipotentials and contours, dipoles and quadrupoles § Thin magnet kicks and that ubiquitous rigidity § Complications: hysteresis, end fields § Section 4.4: More details about dipoles § Sector and rectangular bends; edge focusing § Extras: Superconducting magnets § RHIC, LHC, the future § Section 4.5: Ideal Solenoid (homework!) T. Satogata / January 2013 USPAS Accelerator Physics 2 Other References § Magnet design and a construction is a specialized field all its own § Electric, Magnetic, Electromagnetic modeling • 2D, 3D, static vs dynamic § Materials science • Conductors, superconductors, ferrites, superferrites § Measurements and mapping • e.g. g-2 experiment: 1 PPM field uniformity, 14m SC dipole § Entire USPAS courses have been given on just superconducting magnet design § http://www.bnl.gov/magnets/staff/gupta/scmag-course/ (Ramesh Gupta and Animesh Jain, BNL) T. Satogata / January 2013 USPAS Accelerator Physics 3 EM/Maxwell Review I § Recall our relativistic Lorentz force d(γmv) = q E + v B dt × § For large γ common in accelerators, magnetic fields are much more effective for changing particle momenta § Can mostly separate E (RF, septa) and B (DC magnets) • Some places you can’t, e.g. plasma wakefields, betatrons § Easiest/simplest: magnets with constant B field § Constant-strength optics • Most varying B field accelerator magnets change field so slowly that E fields are negligible • Consistent with assumptions for “standard canonical coordinates”, p 49 Conte and MacKay T. Satogata / January 2013 USPAS Accelerator Physics 4 EM/Maxwell Review II § Maxwell’s Equations for B, H and magetization M are ∂E B =0 H = j + = j H B/µ M ∇ · ∇× 0 ∂t ≡ − § A magnetic vector potential A exists B = A since A =0 ∇× ∇ · ∇× § Transverse 2D (Bz=Hz=0), paraxial approx (px,y<<p0) § Away from magnet coils ( j =0 , M =0 ) § Simple homogeneous differential equations for fields ∂B ∂B ∂B ∂B x + y =0 x y =0 ∂x ∂y ∂y − ∂x T. Satogata / January 2013 USPAS Accelerator Physics 5 Parameterizing Solutions ∂B ∂B ∂B ∂B x + y =0 x y =0 ∂x ∂y ∂y − ∂x § What are solutions to these equations? 0 0 § Constant field: B = Bxx xˆ + Byy yˆ • Dipole fields, usually either only Bx or By • 360 degree (2 π) rotational “symmetry” § First order field: B =(Bxxx + Bxyy)ˆx +(Byxx + Byyy)ˆy • Maxwell gives Bn=Bxx=-Byy and Bs=Bxy=Byx B = B (xxˆ yyˆ)+B (xyˆ + yxˆ) n − s • Quadrupole fields, either normal Bn or skew Bs • 180 degree (π) rotational symmetry • 90 degree rotation interchanges normal/skew § Higher order… T. Satogata / January 2013 USPAS Accelerator Physics 6 Visualizing Fields I Dipole and “skew” dipole Quad and skew quad Sextupole and skew sextupole T. Satogata / January 2013 USPAS Accelerator Physics 7 Visualizing Dipole and Quadrupole Fields II LEP LHC quadrupole dipole field field § LHC dipole: By gives horizontal bending § LEP quadrupole: By on x axis, Bx on y axis § Horizontal focusing=vertical defocusing or vice-versa § No coupling between horizontal/vertical motion • Note the nice “harmonic” field symmetries T. Satogata / January 2013 USPAS Accelerator Physics 8 General Multipole Field Expansions § Rotational symmetries, cylindrical coordinates § Power series in radius r with angular harmonics in θ x = r cos θ y = r sin θ ∞ r n B = B (b cos nθ a sin nθ) y 0 a n − n n=0 ∞ r n B = B (a cos nθ + b sin nθ) x 0 a n n n=0 § Need “reference radius” a (to get units right) § (bn,an) are called (normal,skew) multipole coefficients § We can also write this succinctly using de Moivre as n ∞ x + iy B iB = B (a ib ) x − y 0 n − n a n=0 T. Satogata / January 2013 USPAS Accelerator Physics 9 But Do These Equations Solve Maxwell? § Yes J Convert Maxwell’s eqns to cylindrical coords ∂B ∂B ∂B ∂B x + y =0 x y =0 ∂x ∂y ∂y − ∂x ∂(ρB ) ∂B ∂(ρB ) ∂B ρ + θ =0 θ ρ =0 ∂ρ ∂θ ∂ρ − ∂θ § Aligning r along the x-axis it’s easy enough to see ∂ ∂ ∂ 1 ∂ ∂x ⇒ ∂r ∂y ⇒ r ∂θ § In general it’s (much, much) more tedious but it works ∂r 1 ∂θ 1 ∂r 1 ∂θ 1 = , = − , = , = ∂x cos θ ∂x r sin θ ∂y sin θ ∂y r cos θ ∂B ∂B ∂r ∂B ∂θ x = x + x ... ∂x ∂r ∂x ∂θ ∂x T. Satogata / January 2013 USPAS Accelerator Physics 10 Multipoles -4 (b,a)n “unit” is 10 (natural scale) (b,a)n (US) = (b,a)n+1 (European)! S N N S S Elettra magnets N T. Satogata / January 2013 USPAS Accelerator Physics 11 Multipole Symmetries § Dipole has 2π rotation symmetry (or π upon current reversal) § Quad has π rotation symmetry (or π/2 upon current reversal) § k-pole has 2π/k rotation symmetry upon current reversal § We try to enforce symmetries in design/construction § Limits permissible magnet errors § Higher order fields that obey main field symmetry are called allowed multipoles RCMS half-dipole laminations “H style dipoles” (W. Meng, BNL) (with focusing like betatron) T. Satogata / January 2013 USPAS Accelerator Physics 12 Multipole Symmetries II § So a dipole (n=0, 2 poles) has allowed multipoles: § Sextupole (n=2, 6 poles), Decapole (n=4, 10 poles)... § A quadrupole (n=1, 4 poles) has allowed multipoles: § Dodecapole (n=5, 12 poles), Twenty-pole (n=9, 20 poles)… § General allowed multipoles: (2k+1)(n+1)-1 § Or, more conceptually, (3,5,7,…) times number of poles § Other multipoles are forbidden by symmetries § Smaller than allowed multipoles, but no magnets are perfect • Large measured forbidden multipoles mean fabrication or fundamental design problems! § Better magnet pole face quality with punched laminations § Dynamics are usually dominated by lower-order multipoles T. Satogata / January 2013 USPAS Accelerator Physics 13 4.2: Equipotentials and Contours Field-defining Iron (high permeability) Current carrying conductors (coils) y position [mm] Field lines perpendicular to iron surface Z. Guo et al, NIM:A 691, x position [mm] pp. 97-108, 1 Nov 2012. T. Satogata / January 2013 USPAS Accelerator Physics 14 4.2: Equipotentials and Contours § Let’s get around to designing some magnets § Conductors on outside, field on inside § Use high-permeability iron to shape fields: iron-dominated • Pole faces are very nearly equipotentials, ⊥ B,H field • We work with a magnetostatic scalar potential Ψ • B, H field lines are ⊥ to equipotential lines of Ψ H = Ψ ∇ ∞ a r n+1 Ψ = [F cos((n + 1)θ)+G sin((n + 1)θ)] n +1 a n n n=0 where G B b /µ ,F B a /µ n ≡ 0 n 0 n ≡ 0 n 0 This comes from integrating our B field expansion. Let’s look at normal multipoles Gn and pole faces… T. Satogata / January 2013 USPAS Accelerator Physics 15 Equipotentials and Contours II § For general Gn normal multipoles (i.e. for bn) Ψ(equipotential for b ) rn+1 sin[(n + 1)θ] = constant n ∝ § Dipole (n=0): Ψ(dipole) r sin θ = y ∝ § Normal dipole pole faces are y=constant § Quadrupole (n=1): Ψ(quadrupole) r2 sin(2θ)=2xy ∝ § Normal quadrupole pole faces are xy=constant (hyperbolic) § So what conductors and currents are needed to generate these fields? T. Satogata / January 2013 USPAS Accelerator Physics 16 Dipole Field/Current § Use Ampere’s law to calculate field in gap § N “turns” of conductor around each pole § Each turn of conductor carries current I (C-style dipole) § Field integral is through N-S poles and (highly permeable) iron (including return path) NI µ NI 2NI = H dl =2aH H = ,B= 0 · ⇒ a a BL § NI is in “Amp-turns”, µ ~1.257 cm-G/A ∆x = 0 (Bρ) § So a=2cm, B=600G requires NI~955 Amp-turns T. Satogata / January 2013 USPAS Accelerator Physics 17 Wait, What’s That Δx’ Equation? BL Field, length: Properties of magnet ∆x = (Bρ) Rigidity: property of beam (really p/q!) § This is the angular transverse kick from a thin hard- edge dipole, like a dipole corrector § Really a change in px but paraxial approximation applies § The B in (Bρ) is not necessarily the main dipole B § The ρ in (Bρ) is not necessarily the ring circumference/2π § And neither is related to this particular dipole kick! ∆p F = x = q(βc)B ∆t = L/(βc) p0 x ∆t y ∆px ∆px = qLBy ∆px q ByL B ,L ∆x = LB = y ≈ p p y (Bρ) T. Satogata / January 2013 USPAS Accelerator Physics 18 Quadrupole Field/Current § Use Ampere’s law again § Easiest to do with magnetic potential Ψ, encloses 2NI a B b Ψ(a, θ)= 0 1 sin(2θ) 2 µ0 aB b 2NI = H dl = Ψ(a, π/4) Ψ(a, π/4) = 0 1 · − − µ0 2NI Ψ = NI sin(2θ)= xy a2 2NI 2µ NI H = Ψ = (yxˆ + xyˆ) B = 0 (yxˆ + xyˆ) ∇ a2 a2 § Quadrupole strengths are expressed as transverse gradients ∂By ∂Bx 2µ0NI BL B y=0 = = 2 ∆x = x ≡ ∂x | ∂y a (Bρ) (NB: Be careful, ‘ has different meaning in B’, B’’, B’’’…) T. Satogata / January 2013 USPAS Accelerator Physics 19 Quadrupole Transport Matrix § Paraxial equations of motion for constant quadrupole field d2x d2y + kx =0 ky =0 s βct ds2 ds2 − ≡ B 2µ NI q k = 0 ≡ (Bρ) a2 p § Integrating over a magnet of length L gives (exactly) cos(L√k) 1 sin(L√k) Focusing x √k x0 x0 = = MF Quadrupole x √k sin(L√k) cos(L√k) x0 x0 − cosh(L√k) 1 sinh(L√k) Defocusing y √k y0 y0 Quadrupole = = MD y √k sinh(L√k) cosh(L√k) y0 y0 T.
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