Superconducting RF Systems I

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1 Superconducting RF Systems I RF basics & principles of RF Superconductivity Erk JENSEN, CERN

2 Hendrik A. Lorentz Lorentz force A charged particle moving with velocity Ԧv = vacuum experiences the Lorentz force dp dt Ԧp m γ through an electromagnetic field in = q E + v B. The total energy of this particle is W = is W kin = mc 2 γ 1. mc pc 2 = γ mc 2, the kinetic energy The role of acceleration is to increase W. Change of W (by differentiation): WdW = c 2 Ԧp d Ԧp = qc 2 Ԧp E + Ԧv B dt = qc 2 Ԧp Edt dw = qv Edt Note: Only the electric field can change the particle energy! 2

James Clerk Maxwell 1831 1879 Maxwell s equations in vacuum Source-free: B 1 c 2 t E + t E = 0 B = 0 B = 0 E = 0 curl (rot, ) of 3 rd equation and of t 1st equation:

3 James Clerk Maxwell Maxwell s equations in vacuum Source-free: B 1 c 2 t E + t E = 0 B = 0 B = 0 E = 0 curl (rot, ) of 3 rd equation and of t 1st equation: E c 2 E = 0. t2 Using the vector identity E = E 2 E and the 4 th Maxwell equation, this yields: 2 E 1 2 c 2 E = 0, t2 i.e. the 4-dimensional Laplace equation. 3

4 Homogeneous plane wave E u y cos ωt k Ԧr B u x cos ωt k Ԧr k Ԧr = ω c z cos φ + x sin φ x Wave vector k: the direction of k is the direction of propagation, the length of k is the phase shift per unit length. k behaves like a vector. k = ω c c k = ω c E y φ z k z = ω c 1 ω c ω 2 4

5 Superposition of 2 homogeneous plane waves E y z x + = This way one gets a hollow rectangular waveguide. Metallic walls may be inserted where E y 0 without perturbing the fields. Note the standing wave in x-direction! 5

Rectangular waveguide Fundamental (TE 10 or H 10 ) mode in a standard rectangular waveguide. Example 1: S-band : 2.6 GHz... 3.95 GHz, electric field power flow Waveguide type WR284 (2.

6 Rectangular waveguide Fundamental (TE 10 or H 10 ) mode in a standard rectangular waveguide. Example 1: S-band : 2.6 GHz GHz, electric field power flow Waveguide type WR284 (2.84 wide), dimensions: mm x mm. y cut-off: f c = GHz. Example 2: L-band : 1.13 GHz GHz, Waveguide type WR650 (6.5 wide), dimensions: mm x mm. cut-off: f c = GHz. x z magnetic field power flow Both these pictures correspond to operation at 1.5 f c. power flow: 1 2 Re E H d ԦA 6

Note also that at f = f c, v φ,z =! With f, v φ,z c! 1: a = 52 mm, fτf c = 1.

7 Waveguide dispersion phase velocity v φ,z The phase velocity v φ,z is the speed at which the crest (or zero-crossing) travels in z-direction. Note on the 3 animations on the right that, at constant f, v φ,z λ g. Note also that at f = f c, v φ,z =! With f, v φ,z c! 1: a = 52 mm, fτf c = 1.04 f = 3 GHz k c y x k = ω c z 2: a = mm, fτf c = k z = 2π = ω λ g c 1 ω c ω ω 2 = ω v φ,z 3: a = mm, fτf c = ω c cutoff: f c = c 2a, v φ,z = 7

8 Radial waves Also radial waves may be interpreted as superposition of plane waves. The superposition of an outward and an inward radial wave can result in the field of a round hollow waveguide. 8

9 Round waveguide f/f c = 1.44 TE 11 fundamental TM 01 axial field TE 01 low loss f c 87.9 f c f c GHz a / mm GHz a / mm GHz a / mm 9

10 Waveguide perturbed by notches perturbations ( notches ) Signal flow chart Reflections from notches lead to a superimposed standing wave pattern. Trapped mode 10

11 Short-circuited waveguide TM 010 (no axial dependence) TM 011 TM

12 Single WG mode between two shorts short circuit a e jk zl short circuit 1 Signal flow chart 1 Eigenvalue equation for field amplitude a: e jk zl l ae jk zl a = e jk z2l a Non-vanishing solutions exist for 2k z l = 2πm: With k z = ω c 1 ω c ω 2, this becomes f0 2 = f c 2 + c m 2l 2. 12

13 Simple pillbox (only 1/2 shown) TM 010 -mode electric field (purely axial) magnetic field (purely azimuthal) 13

14 Pillbox with beam pipe TM 010 -mode One needs a hole for the beam passage circular waveguide below cutoff (only 1/4 shown) electric field magnetic field 14

15 A more practical pillbox cavity Rounding of sharp edges (to reduce field enhancement!) TM 010 -mode (only 1/4 shown) electric field magnetic field 15

16 A (real) elliptical cavity TM 010 -mode (only 1/4 shown) electric field magnetic field 16

17 Acceleration voltage and Τ R Q I define V acc = න E z e j ω βc z dz. The exponential factor accounts for the variation of the field while particles with velocity βc are traversing the cavity gap. With this definition, V acc is generally complex this becomes important with more than one gap (cell). For the time being we are only interested in V acc. The square of the acceleration voltage V acc 2 is proportional to the stored energy W; the proportionality constant defines the quantity called R-upon-Q : 2 R Q = V acc 2ω 0 W. electric field Attention different definitions are used in literature! 17

The transit time factor of an ideal pillbox cavity (no axial field dependence) of height (gap length) h is:

18 Transit time factor The transit time factor is the ratio of the acceleration voltage to the (non-physical) voltage a particle with infinite velocity would see: TT = V acc E z dz = E j ω βc ze z dz E z dz. The transit time factor of an ideal pillbox cavity (no axial field dependence) of height (gap length) h is: TT = sin χ 01h 2a χ01h 2a TT Field rotates by 360 during particle passage. (remember: ω 0 = 2πc λ = χ 01c a ) h λ 18

19 Stored energy The energy stored in the electric field is W E = ම cavity The energy stored in the magnetic field is W M = ම cavity ε 2 E 2 dv. μ 2 H 2 dv W M H W E E Since E and H are 90 out of phase, the stored energy continuously swaps from electric energy to magnetic energy. On average, electric and magnetic energy must be equal. In steady state, the Poynting vector describes this energy flux. In steady state, the total energy stored (constant) is W = ම cavity ε 2 E 2 + μ 2 H 2 dv. 19

20 Stored energy and Poynting vector John Henry Poynting electric field energy Poynting vector magnetic field energy 20

21 Wall losses & Q 0 The losses P loss are proportional to the stored energy W. The tangential H on the surface is linked to a surface current ԦJ A = n H (flowing in the skin depth δ = 2 ωμσ ). This surface current ԦJ A sees a surface resistance R s, resulting in a local power density R s H t 2 flowing into the wall. R s is related to skin depth δ as δσr s = 1. Cu at 300 K has σ SΤm, leading to R s 8 mω at 1 GHz, scaling with ω. Nb at 2 K has a typical R s 10 nω at 1 GHz, scaling with ω 2. The total wall losses result from P loss = R s H 2 t da. wall The cavity Q 0 (caused by wall losses) is defined as Q 0 = ω 0W P loss. Typical Q 0 values: No! Anomalous skin effect! Cu at 300 K (normal-conducting): O , should improves at cryogenic only by Ta by factor a factor 10! RRR. Nb at 2 K (superconducting): O

22 Shunt impedance Also the power loss P loss is also proportional to the square of the acceleration voltage V acc 2 ; the proportionality constant defines the shunt impedance R = V acc 2 2 P loss. Attention, also here different definitions are used! Traditionally, the shunt impedance is the quantity to optimize in order to minimize the power required for a given gap voltage. Now the previously introduced term R-upon-Q makes sense: R Q = R Τ Q 22

23 Geometric factor With ω 0 W Q 0 = R s H 2 t da, wall and assuming an average surface resistance R s, one can introduce the geometric factor G as ω 0 W G = Q 0 R s = H 2 t da. wall G has dimension Ohm, depends only on the cavity geometry (as the name suggests) and typically is O 100 Ω. Note that R s R = G G is only used for SC cavities. Τ R Q (dimension Ω 2, purely geometric) 23

24 Cavity resonator equivalent circuit Simplification: single mode I G V acc I B Generator Z P loss Beam β: coupling factor R: shunt impedance Τ R β LΤ C = R : R-upon-Q Q C L R Cavity 24

25 Power coupling - Loaded Q Note that the generator inner impedance also loads the cavity for very large Q 0 more than the cavity wall losses. To calculate the loaded Q (Q L ), losses have to be added: 1 = P loss + P ext + Q L ω 0 W = 1 Q Q ext + 1. The coupling factor β is the ratio With β, the loaded Q can be written Τ P ext P loss. Q L = Q β. For NC cavities, often β = 1 is chosen (power amplifier matched to empty cavity); for SC cavities, β = O

26 Z ω R Τ Q Resonance Q 0 = 1000 Q 0 = 100 Q 0 = 10 Q 0 = 1 ω While a high Q 0 results in small wall losses, so less power is needed for the same voltage. On the other hand the bandwidth becomes very narrow. Note: a 1 GHz cavity with a Q 0 of has a natural bandwidth of 0.1 Hz! to make this manageable, Q ext is chosen much smaller! ω 0 26

27 Summary: relations V acc, W and P loss Attention different definitions are used in literature! R Q = V acc 2ω 0 W k loss = ω R Q = V acc 4W 2 V acc Accelerating voltage R = V acc 2 2P loss = R Q Q 0 W Energy stored Q 0 = ω 0W P loss P loss wall losses 27

28 Beam loading The beam current loads the cavity, in the equivalent circuit this appears as an impedance in parallel to the shunt impedance. If the generator is matched to the unloaded cavity (β = 1), beam loading will (normally) cause the accelerating voltage to decrease. The power absorbed by the beam is 1 2 R V acci B. For high power transfer efficiency RF beam, beam loading must be high! For SC cavities (very large β), the generator is typically matched to the beam impedance! Variation in the beam current leads to transient beam loading, which requires special care! Often the impedance the beam presents is strongly reactive this leads to a detuning of the cavity. 28

29 Multipactor The words multipactor, to multipact and multipacting are artificially composed of multiple impact. Multipactor describes a resonant RF phenomenon in vacuum: Consider a free electron in a simple cavity it gets accelerated by the electric field towards the wall when it impacts the wall, secondary electrons will be emitted, described by the secondary emission yield (SEY) in certain impact energy ranges, more than one electron is emitted for one electron impacting! So the number of electrons can increase When the time for an electron from emission to impact takes exactly ½ of the RF period, resonance occurs with the SEY>1, this leads to an avalanche increase of electrons, effectively taking all RF power at this field level, depleting the stored energy and limiting the field! For this simple 2-point MP, this resonance condition is reached at 1 e V = fd 2 or 4π m V 112 V = f d 2. There exist other resonant bands. MHz m courtesy: Sarah Aull/CERN 29

30 Multipactor (contd.) Unfortunately, good metallic conductors (Cu, Ag, Nb) all have SEY>1! 1-point MP occurs when the electron impact where they were emitted Electron trajectories can be complex since both E and B influence them; computer simulations allow to determine the MP bands (barriers) To reduce or suppress MP, a combination of the following may be considered: Use materials with low SEY Optimize the shape of your cavity ( elliptical cavity) Conditioning (surface altered by exposure to RF fields) Coating (Ti, TiN, NEG, amorphous C ) Clearing electrode (for a superimposed DC electric field) Rough surfaces 30

31 Many gaps 31

32 What do you gain with many gaps? The Τ R Q of a single gap cavity is limited to some 100 Ω. Now consider to distribute the available power to n identical cavities: each will receive PΤn, thus produce an accelerating voltage of Τ 2RP n. (Attention: phase important!) The total accelerating voltage thus increased, equivalent to a total equivalent shunt impedance of nr. Τ P n Τ P n Τ P n Τ P n V acc = n 2R P n = 2 nr P n 32

33 Standing wave multi-cell cavity Instead of distributing the power from the amplifier, one might as well couple the cavities, such that the power automatically distributes, or have a cavity with many gaps (e.g. drift tube linac). Coupled cavity accelerating structure (side coupled) The phase relation between gaps is important! 33

The elliptical cavity The elliptical shape was found as optimum compromise between maximum

multi-cell elliptical cavity is typically operated in π-mode, i.e. cell length is exactly βλτ2.

3 GHz, 9-cell cavity SNS (805 MHz) SPL/ESS (704 MHz) LHC (400 MHz) *): http://accelconf.web.

34 The elliptical cavity The elliptical shape was found as optimum compromise between maximum gradient (E acc /E surface ) suppression of multipactor mode purity machinability A multi-cell elliptical cavity is typically operated in π-mode, i.e. cell length is exactly βλτ2. It has become de facto standard, used for ions and leptons! E.g.: ILC/X-FEL: 1.3 GHz, 9-cell cavity SNS (805 MHz) SPL/ESS (704 MHz) LHC (400 MHz) *): At 1.3 GHz: R 0 = mm, 2L = mm. D. Proch, 1993 *) 35

35 Elliptical cavity the de facto standard for SRF FERMI 3.9 GHz TESLA/ILC 1.3 GHz LEP GHz S-DALINAC 3 GHz SNS β = 0.61, 0.81, GHz CEBAF 1.5 GHz HERA 0.5 GHz cells HEPL 1.3 GHz CESR 0.5 GHz KEK-B 0.5 GHz TRISTAN 0.5 GHz 36

Paul Drude 1863 1906 Drude model: In a conductor, the electrons move through an ion lattice, bounce off the ions and get slowed in the process.

36 Paul Drude Drude model: In a conductor, the electrons move through an ion lattice, bounce off the ions and get slowed in the process. In the presence of E fields only, this is governed by the equation d p = ee p/τ. dt How does electrical conduction work? In steady state (DC), one gets v = e τ E, which is m Ohm s law: j n = nev = e2 τn m E = σe with σ DC = e2 τn m. Typical scattering time τ s. The electrons (blue) slowly drift to the right under the influence of a DC electric field 37

37 Ƹ Superconducting RF Systems I CAS Zurich Drude model extended for RF The equation d p = ee p can of course be solved for an excitation by dt τ a time-varying field E = R{ E e jωt }. Solution: Assume p(t) to be of the same time-dependence as RHS: p t = R pe jωτ e jωτ and solve: jω + 1 τ p Ƹ = e E or p Ƹ = τe E 1+jωτ. This naturally results in a complex, f-dependent σ: 1 σ = σ DC 1 + jωτ = σ DC 1 + ωτ 2 1 jωτ With τ s, ωτ < 1 for frequencies up to many THz! 38

38 Good (normal-)conductor boundary skin depth A good (normal-)conductor: inside the metal, the surface field H gives rise to a damped wave, which propagates into the metal with a propagation constant of k = jωμσ. The skin-depth is the inverse of the damping constant, the real part of k : Skin depth: δ = 1 α = 1 R{ jωμσ} = 2 ωμσ The wave impedance in the metal is Z = R n = losses. ωμ 2σ jωμ σ, its real part is the surface resistance and can be used to determine the δr n σ 1 I use the term R n for the surface resistance for normal conductors, R s for superconductors. 39

39 ρ [Ωm] Superconducting RF Systems I CAS Zurich Temperature dependence of resistivity of metals ρ T = 1 σ T = ρ 0 + A T Θ D 5 Θ D T x 5 ඵ 0 e x dx 1 1 e x Cu resistivity vs. T, assuming ρ 20 = nωm, RRR = 300 and Θ D = K (Debye T). [Bloch-Grüneisen formula] T [K] 40

40 R n [Ω] Superconducting RF Systems I CAS Zurich Anomalous skin effect The relation between skin depth δ and surface resistance R n, δr n σ 1, is valid only while the mean free path l of the electrons is smaller than the skin depth, l δ. If the skin depth gets smaller (e.g. at low T, high ω), R n will be dominated by l and be limited to R n 3π l σ μω 4 π 2 1Τ3 Prediction of surface resistance R n (T) of Cu at 400 MHz with RRR = 300 (blue) and correction for anomalous skin effect (ρl = Ωm 2 ) (orange). Not even a factor 10! T [K] 41

Superconductivity Heike Kammerling Onnes (1853 1926) Kammerling

temperatures and noted in 1911, when measuring the resistivity of

41 Superconductivity Heike Kammerling Onnes ( ) Kammerling Onnes investigated the behaviour of metal conductivity at low temperatures and noted in 1911, when measuring the resistivity of Mercury at 4.2 K: Kwik nagenoeg nul (meaning mercury almost zero ). 42

42 Critical temperatures of superconductors 43

43 Phase diagram of a SC 44

44 Perfect conductor With a magnetic field B 0 at the transition vacuum/superconductor, the following equations hold: 3 rd Maxwell equation: 1 st London equation: E = B t j s t = n se 2 m E, where j s = n s e Ԧv. This results in the equation j t s + n se 2 B = 0, or (slightly m transformed) in 2 B μ n se 2 B = 0 t m This could be solved by either a time-independent field, 0 t or by a field satisfying 2 B μ n se 2 B = 0. m λ L 2 45

A non-vanishing, time-independent field is not observed and

45 Meissner effect (flux expulsion) Observation: During cool-down, when T passes T c, the magnetic field gets completely expulsed. A non-vanishing, time-independent field is not observed and thus can be excluded as non-physical. Walther Meißner

46 Supported by Meissner s observation, the time-independent (non-trivial) solution of the equation j s + n se 2 B = 0 can be excluded as non-physical. t m London equations (London penetration depth: λ L = 1. (zero resistance): London equations j s t = n se 2 m E = 1 μ λ L 2 E 2. (Meissner effect): j s + n se 2 B = 0 m or 2 1 λ2 B = 0 L m μn s e 2) London brothers Heinz ( ), Fritz ( ) vacuum superconductor 47

47 The coherence length ξ Pippard found out that λ L depends on the purity of the material and therefore on the electron mean free path l. In 1953, he proposed the coherence length ξ as a new parameter to better describe the characteristic dimension of the electrons wave function in a superconductor. Brian Pippard In a pure metal, one finds ξ = ξ 0 ħv F k T c, with impurities one can approximate ξ = Clean limit: l ξ, dirty limit: l ξ. 1 ξ l 1 and λ l = λl ξ 0 ξ. 48

48 Characteristic lengths: London penetration depth λ L = μn s e2: the distance up to which magnetic field penetrates into a superconductor if placed in a magnetic field Electron mean free path l. gets smaller with impurities. m Coherence length ξ = : the distance over which the electron ξ 0 l wave function extends the size of Cooper pairs (see below). Orders of magnitude: λ L and ξ: tens of nm. (E.g. Nb: ξ 0 = 39 nm, Pb: ξ 0 = 83 nm) l: nm (dirty) to μm (clean) Sometimes used: κ λ L Τξ 49

49 BCS Theory Bardeen Cooper Schrieffer (BCS) In 1958, Bardeen, Cooper and Schrieffer proposed a theory of superconductivity in which there exists an attractive interaction between electrons, forming Cooper pairs. 50

Cooper pairs Positively charged wake due to moving electron attracting nearby atoms (electron-phonon interaction) this wake can attract another nearby electron and a Cooper pair is formed.

While electrons are fermions, Cooper pairs are bosons.

50 Cooper pairs Positively charged wake due to moving electron attracting nearby atoms (electron-phonon interaction) this wake can attract another nearby electron and a Cooper pair is formed. Cooper pairs are formed by electrons with opposite momentum and spin. Cooper pairs belong all to the same quantum state and have the same energy. While electrons are fermions, Cooper pairs are bosons. When carrying a current, each Cooper pair acquires a momentum which is the same for all pairs, The total momentum of the pair remains constant. It can be changed only if the pair is broken, but this requires a minimum energy ΔE. While NC electrons are scattered by the ion lattice (cf. Drude model) leading to resistive losses, Cooper pairs are not scattered by ion lattice. 51

51 Energy gap Normal Conductor Superconductor Levels empty Fermi Level Levels occupied Padamse, Knobloch, Hays: RF Superconductivity for Accelerators, Wiley 2008 The electron-phonon interaction changes the density of states. Near the Fermi surface an energy gap forms and energy states below get denser the new ground state (with Cooper-pairs) has a lower energy than the NC ground state. For Nb, Δ 0 Τ k B T c = 1.9, for Pb, Δ 0 Τ k B T c =

BCS theory GL theory Superconducting RF Systems I CAS Zurich The microscopic BCS theory The superconducting state consists of electron pairs and elementary excitations, quasiparticles, behaving

52 BCS theory GL theory Superconducting RF Systems I CAS Zurich The microscopic BCS theory The superconducting state consists of electron pairs and elementary excitations, quasiparticles, behaving almost as free electrons The energy gap Δ separates the energy levels of elementary excitations from the ground state level. At 0 K, only the ground state is occupied. Density of elementary excitations. There are no states within the energy gap Δ. Temperature dependence of Δ: Δ T Δ 0 K cos π 2 T T c 2 Original BCS theory has been derived using mean field approximation valid for 0 T < T c. Ginzburg-Landau theory is valid for T T c. Gor kov showed that lim T Tc (BCS) = GL. 53

Classification of superconductors Type-I

incomplete Two critical fields H c1 and H c2 κ

53 Classification of superconductors Type-I superconductor: Meissner effect complete Sharp transition NC/SC at field H c κ = λ L Τξ < 1Τ 2 Type-II superconductor: Meissner effect incomplete Two critical fields H c1 and H c2 κ > Τ 1 2 Examples: Pb, Sn, Hg, Cr, Al Examples: Nb, alloys 54

Flux tubes are repulsive creating, therefore the vortex lattice STM image

54 Flux quantization In Type-II superconductors flux tubes are created each carrying one flux quantum (the minimal flux allowed by quantum mechanics) Flux tubes are repulsive creating, therefore the vortex lattice STM image of Vortex lattice, 1989 H. F. Hess et al. Phys. Rev. Lett. 62, 214,

Field limits for type-ii superconductors H < H c1 : perfect Meissner state H c1 < H < H c2 : penetration and oscillation of vortices give rise to strong dissipation not useable for RF.

55 Field limits for type-ii superconductors H < H c1 : perfect Meissner state H c1 < H < H c2 : penetration and oscillation of vortices give rise to strong dissipation not useable for RF. But The Meissner state can remain metastable for H c1 < H < H sh, if vortices can be prevented from entering (Bean-Livingston barrier). H c2 0 H sh 0 H c1 0 Type-II superconductor 56

56 The superheating field The superheating field H sh is set by the competition between magnetic pressure (imposed by the external magnetic field), the energy cost to destroy superconductivity, and the attractive force due to the zero-current boundary condition at the interface. H c1 is the field where it is energetically favourable for the flux to be in the superconductor. H sh is the field where the Bean-Livingston barrier for flux entry disappears Defects can serve as entry points for flux preventing superheating Suggested further reading: B. Liarte et al. - Supercond. Sci. Technol. 30 (2017)

57 RF case The time-dependent magnetic field in the penetration depth λ L will generate an electric field E = B t. At T > 0 K, there will exist a fraction of unpaired electrons: n n T e Δ k B T Since Cooper pairs have inertia, they cannot shield these NC electrons from E, hence R s > 0. 58

electron density Superconducting RF Systems I CAS Zurich Two-fluid model Proposed by Gorter and Casimir already in 1943: Charge carriers are divided in two subsystems,

58 electron density Superconducting RF Systems I CAS Zurich Two-fluid model Proposed by Gorter and Casimir already in 1943: Charge carriers are divided in two subsystems, superconducting carriers of density n s and normal electrons of density n n. Assume n s n = 1 T T c 4, n n n = T T c 4, ns + n n = n. The total current results from J = J s + J n n s n n T c T 59

59 RF surface impedance of SC From 1 st London equation: t Ԧ J s = E 1 μ λ2, J s = j L ωμλ2 E, or J s = j n se 2 E. L mω For the unpaired electrons (Ohm s law): For the total current: J = J n + J s = σ n jσ s E = n n e 2 τ m j n se 2 m ω J n = n n e 2 τ m E E. Surface impedance: Z s = R s + jx s = jωμ σ n jσ s. With the 2-fluid model this results in: R s = 1 2 μω2 σ n λ L 3 X s = ω μ λ L 60

60 RF surface resistance of a SC We found R s = 1 2 μω2 σ n λ L 3 what does this mean? Frequency dependence: R s ω 2 : use low frequency cavities to reduce power dissipation! Temperature dependence: from two-fluid model: σ n n n e Δ k B T R s exp Δ k B T There are better approximations around, most famous the formulae developed by Halbritter (1970), which approximate R s as a function of ω, T, ξ 0, λ L, T c, Δ, and l, see e.g. here: 61

61 Halbritter approximation example Be carefureful here. The website suggests 40 nm. The input required is πξ 0 Τ2, while ξ 0 38 nm for Nb. 62

62 We had introduced above: λ l = λ L 1 + ξ 0 l σ 1 l R s dependence on material purity It follows that R s has a local minimum at Τ l ξ 0 = Τ 1 2. This is remarkable: at some point, increasing l (cleaner material) makes things worse! Nb on Cu,1.5 GHz, 4.2 K clean C. Benvenuti et al., Physica C 316 (1999) 153. dirty R s 1 + ξ 0 l Τ 3 2 l ξ 0 η η This example: Nb films sputtered on Cu By changing the sputtering species, the mean free path was varied. RRR of niobium on copper cavities can be tuned for lowest R S. 3Τ2 η 63

63 BCS resistance R s,bcs A 2 ω2 μ 2 λ L 1 + ξ l T c T RRR ρ n 300 K e f GHz Δ k B Tc 2 Tc T e 1.92T c T 64

64 Residual resistance In real, technical superconductors, the observed surface resistance deviates from the BCS prediction and can be written as R s = R BCS + R res. Possible contributions to R res : Trapped magnetic flux and thermal currents Lossy oxides, metallic hydrides Normal-conducting precipitates Grain boundaries Interface losses Magnetic impurities Subgap states T = 2 K B. Aune et al., PR-STAB 3 (2000) Nb, 1.3 GHz For Nb, R res 1 10 nω often dominates R s at low f (< 1 GHz) and low T (< 2 K). 65

CERN Accelerator School. RF Cavities. Erk Jensen CERN BE-RF

CERN Accelerator School. RF Cavities. Erk Jensen CERN BE-RF CERN Accelerator School RF Cavities Erk Jensen CERN BE-RF CERN Accelerator School, Varna 010 - "Introduction to Accelerator Physics" What is a cavity? 3-Sept-010 CAS Varna/Bulgaria 010- RF Cavities Lorentz