USPAS Accelerator Physics 2017 University of California, Davis Chapter 9: RF Cavities and RF Linear Accelerators Todd Satogata (Jefferson Lab) / satogata@jlab.org Randika Gamage (ODU) / bgama002@odu.edu http://www.toddsatogata.net/2017-uspas Happy Birthday to Hideki Yukawa, Paul Langevin, Rutger Hauer, and Gertrude Elion! Happy Rhubarb Pie Day, National Handwriting Day, and Measure Your Feet Day! 1
Vertical Dispersion Correction The end of section 6.10 in CM notes how sextupoles at locations of horizontal dispersion compensate both horizontal and vertical chromaticity k(s) B0 (B ) x 0 =[k(s)ds] x =(kl) x = x [k 0 (s)ds] x(1 ) f I 1 x,n = x(s)k 0 (s) ds CM 6.126 4 Q H Normal sextupoles: b 2 (s) B00 (B ) Horizontal dispersion: x! x + x (s) First order in x, δ: x 0 [b 2 (s)ds](x 2 y 2 )(1 ) x 0 b 2 ds[x 2 +2 x (s) x + 2 ](1 ) 2[b 2 ds x (s) ]x 2
k(s) x,n = B0 (B ) Vertical Dispersion Correction x 0 =[k(s)ds] x =(kl) x = x f Vertically for quads, sign flips and x goes to y y,n = +1 I y(s)k 0 (s) ds 4 Q V [k 0 (s)ds] x(1 ) x 0 b 2 ds[x 2 +2 x (s) x + 2 ](1 ) 2[b 2 ds x (s) ]x Sext I 1 x(s)k 0 (s) ds x (sext) = +1 I x(s)b 2 (s) x (s) ds 4 Q H 4 Q H Sextupole effect is coupled in vertical plane, with horz dispersion: Quad CM 6.127b y 0 [b 2 (s)ds](2xy)(1 )=[b 2 (s)ds][2(x + x (s) )y](1 ) I 1 y (sext) = y(s)[2b 2 (s) x (s)] ds 4 Q V 3
RHIC FODO Cell V sext Horizontal H sext Vertical half quadrupole dipole dipole half quadrupole quadrupole Horizontal dispersion 4
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RF Concepts and Design Much of RF is really a review of graduate-level E&M See, e.g., J.D. Jackson, Classical Electrodynamics The beginning of this lecture is hopefully review But it s still important so we ll go through it Includes some comments about electromagnetic polarization We ll get to interesting applications later in the lecture (or tomorrow) Particularly important are cylindrical waveguides and cylindrical RF cavities Will find transverse boundary conditions are typically roots of Bessel functions TM (transverse magnetic) and TE (transverse electric) modes RF concepts (shunt impedance, quality factor, resistive losses) 6
9.1: Maxwell s Equations Electric charge density Electric current density 7
Constitutive Relations and Ohm s Law D = E = r 0 E 0 =8.85 10 12 C N m 2 ~D : Electric flux density ~E : Electric field density " : Permittivity 0µ 0 = 1 c 2 B = µ H = µ r µ 0 H µ 0 =4 10 7 N s2 C 2 1.257 cm G A ~H : Magnetic field density ~B : Magnetic flux density µ : Permeability J = E : conductivity 8
Boundary Conditions Boundary conditions on fields at the surface between two media depend on the surface charge and current densities: E k B? and are continuous D? changes by the surface charge density s (scalar) H k changes by the surface current density J s ˆn (H 1 H 2 )=J s C&M Chapter 9: no dielectric or magnetic materials µ = µ 0 = 0 9
Wave Equations and Symmetry Taking the curl of each curl equation and using the identity ( E) = ( E) 2 E = 2 E then gives us two identical wave equations r 2 E E = µ t + µ 2 E t 2 r 2 H H = µ t + µ 2 H t 2 These linear wave equations reflect the deep symmetry between electric and magnetic fields. Harmonic solutions: (r, t) = ˆ (r)e i t for = E,H ) r 2 ˆ = 2 ˆ where 2 =( µ 2 + iµ ) 10
Conductivity and Skin Depth For high conductivity,, charges move freely enough to keep electric field lines perpendicular to surface (RF oscillations are adiabatic vs movement of charges) Copper: 6 10 7 1 m 1 Conductivity condition holds for very high frequencies (10 15 Hz) For this condition r µ 2! (1 + i) Inside the conductor, the fields drop off exponentially ˆ (z) = (0) e z/ e.g. for Copper skin depth r 2 µ 11
Surface Resistance and Power Losses There is still non-zero power loss for finite resistivity Surface resistance: resistance to current flow per unit area r µ surface resistance R s 2 = 1 surface power loss P loss = R Z s H k 2 ds 2 This isn t just applicable to RF cavities, but to transmission lines, power lines, waveguides, etc anywhere that electromagnetic fields are interacting with a resistive media Deriving the power loss is part of your homework! We ll talk more about transmission lines and waveguides after one brief clarification S 12
Anomalous Skin Effect Conductivity really depends on the frequency and mean time between electron interactions (or inverse temperature) ( ) =!! 0 Classical limit is, adiabatic field wrt electron interactions For non-classical limit, J = E no longer applies since electrons see changing fields between single interactions 0 (1 + i )! Anomalous skin effect Mean free path Asymptotic value Normal skin effect Skin depth Reuter and Sondheimer, Proc. Roy. Soc. A195 (1948), from Calatroni SRF 11 Electrodynamics Tutorial 13
E( x, t) = E 0 e i k x i t Plane Wave Properties k = 2 (wave number) rf B( x, t) = B 0 e i k x i t v ph = k (phase velocity) Generally F (z,t) = Z 1 1 A( )e i( t k( )z) d v g = d dk (group velocity) The source-free divergence Maxwell equations imply k E 0 = k B 0 =0 so the fields are both transverse to the direction (This will be the ẑ direction in a lot of what s to come) Faraday s law also implies that both fields are spatially transverse to each other k B 0 = k E 0 = nˆn E 0 c ˆn k k index of refraction 14
Standing Waves E( x, t) = E 0 e i k x i t B( x, t) = B 0 e i k x i t Note that Maxwell s equations are linear, so any linear combination of magnetic/electric fields is also a solution. Thus a standing wave solution is also acceptable, where there are two plane waves moving in opposite directions: E(x, t) =E 0 e i k x i t +e i k x i t B(x, t) =B 0 e i k x i t +e i k x i t 15
Polarization E( x, t) = E 0 e i k x i t B( x, t) = B 0 e i k x i t As long as the E and B fields are transverse, they can still have different transverse components. So our description of these fields is also incomplete until we specify the transverse components at all locations in space This is equivalent to an uncertainty in phase of rotation of E and B around the wave vector k. The identification of this transverse field coordinate basis defines the polarization of the field. 16
Linear Polarization E( x, t) = E 0 e i k x i t B( x, t) = B 0 e i k x i t So, for example, if E and B transverse directions are constant and do not change through the plane wave, the wave is said to be linearly polarized. E/B field oscillations are IN phase Fields each stay in one plane 17
Circular Polarization E(x, t) =E 0 e i k x i t +e i k x i t+ /2 B(x, t) =B 0 e i k x i t +e i k x i t+ /2 If E and B transverse directions vary with time, they can appear as two plane waves traveling out of phase. This phase difference is 90 degrees for circular polarization. Two equal-amplitude linearly polarized plane waves 90 degrees out of phase 18
9.2: Cylindrical Waveguides Consider a cylindrical waveguide, radius a, in z direction E = E(r, )e i( t k gz) k g can be imaginary for attenuation down the guide We ll find constraints on this cutoff wave number Maxwell in cylindrical coordinates (math happens) gives k (and the same for H z ) (free space wave number) c k 2 c k 2 k 2 g 19
Transverse Electromagnetic (TEM) Modes Various subsets of solutions are interesting For example, the E z,h z =0nontrivial solutions require k 2 c = k 2 k 2 g =0 The wave number of the guide matches that of free space Wave propagation is similar to that of free space This is a TEM (transverse electromagnetic) mode Requires multiple separate conductors for separate potentials or free space Sometimes similar to polarization pictures we had before Coaxial cable 20
Transverse Magnetic (TM) Modes Maxwell s equations are linear so superposition applies We can break all fields down to TE and TM modes TM: Transverse magnetic (H z =0, E z 0) TE: Transverse electric (E z =0, H z 0) TM provides particle energy gain or loss in z direction! Separate variables E z (r, )=R(r) ( ) Boundary conditions E z (r = a) =E (r = a) =0 Maxwell s equations in cylindrical coordinates give d 2 R dr 2 + 1 dr r dr + kc 2 n 2 r 2 R =0 d 2 d 2 + n2 =0 Bessel equation SHO equation => n integer, real 21
TM Mode Solutions The radial equation has solutions of Bessel functions of first J n (k c r) and second N n (k c r) kind Toss N n since they diverge at r=0 Boundary conditions at r=a require J n (k c a)=0 This gives a constraint on k c k c = X nj /a where X nj is the j th nonzero root of J n Xnj ) E z (r,,t)=(c 1 cos n + C 2 sin n )J n a r e i( t k gz) This mode of this field is commonly known as the TM nj mode The first index corresponds to theta periodicity, while the second corresponds to number of radial Bessel nodes TM 01 is the usual fundamental accelerating mode 22
TM Mode Visualizations Xnj E z (r,,t)=(c 1 cos n + C 2 sin n )J n a r e i( t k gz) Java app visualizations are also linked to the class website For example, http://www.falstad.com/emwave2 (TM visualization) Scroll down to Circular Modes 1 or 2 in first pulldown 23
Cavity Modes Dave McGinnis, FNAL 24
E z J 0 u 02 r R TM020 Mode B r J 1 u 02 R Beam R L Peggs/Satogata 25
(a) E z interacting with head TM110 Mode (b) ~B(r, ) excited by head ~F on tail Beam Tail, Q= Ne/2 Head, Q= Ne/2 ~B R L Peggs/Satogata 26
Cutoff Frequency It s easy to see that there is a minimum frequency for our waveguide that obeys the boundary conditions Modes at lower frequencies will suffer resistance Dispersion relation in terms of radial boundary condition zero 2 2 k 2 kg 2 = kg 2 = k 2 Xnj c = c a A propagating wave must have real group velocity kg 2 > 0 This gives the expected lower bound on the frequency c X nj a ) cuto frequency c = cx nj a Wavelength of lowest cutoff frequency for j=1 is c = 2 k c 2.61a 27
Dispersion or Brillouin Diagram Propagating frequencies > c ph > 45, v ph >c Circular waveguides above cutoff have phase velocity >c Cannot easily be used for particle acceleration over long distances 28
Iris-Loaded Waveguides Solution: Add impedance by varying cylinder radius This changes the dispersion condition by loading the waveguide 29
Cylindrical RF Cavities What happens if we close two ends of a waveguide? With the correct length corresponding to the longitudinal wavelength, we can produce longitudinal standing waves Can produce long-standing waves over a full linac Modes have another index for longitudinal periodicity TM010 mode Simplified tuna fish can RF cavity model 30
Boundary Conditions Additional boundary conditions at end-cap conductors at z=0 and z=l 31
TE nmj Modes E z =0everywhere Longitudinal periodicity X 0 nj is j th root of J 0 n f nmj = c 2 s X 0 nj a 2 + m l 2 32
TM nmj Modes H z =0everywhere E z (r,,z) J n (k c r)(c 1 cos n + C 2 sin n ) cos m z l Longitudinal periodicity J n (k c a)=0 X nj is the j th root of J n f nmj = c 2 s Xnj a 2 + m l 2 TM 010 is (again) the preferred acceleration mode 33
Equivalent Circuit A TM 010 cavity looks much like a lumped LRC circuit Ends are capacitive Stored magnetic energy is inductive Currents move over resistive walls E, H fields 90 degrees out of phase Stored energy C&M 9.82 34
TM 010 Average Power Loss ends sides Compare to stored energy Both vary like square of field, square of Bessel root Dimensional area vs dimensional volume in stored energy U = 0 2 E2 0 a 2 l[j 1 (X 01 )] 2 35
Quality Factor Ratio of average stored energy to average power lost (or energy dissipated) during one RF cycle How many cycles does it take to dissipate its energy? High Q: nondissipative resonator Copper RF: Q~10 3 to 10 6 SRF: Q up to 10 11 Q = U hp loss i U(t) =U 0 e 0t/Q Pillbox cavity Q = al (a + l) 36
Niobium Cavity SRF Q Slope Problem 37