Resistive wall wake with ac conductivity and the anomalous skin effect

Transcription

1 Resistive wall wake with ac conductivity and the anomalous skin effect Karl Bane and Gennady Stupakov August 6, 2004

2 Introduction resistive wall wake is a limiting effect in the LCLS undulator, with the induced E~ ρ, the Pierce parameter can add effects of ac conductivity (see K. Bane and M. Sands, SLAC- PUB ) and anomalous skin effect (Reuter and Sondheimer,, Proc Royal Soc., 1948) to resistive wall model Ac conductivity Drude free-electron model of conductivity (1900): conduction electrons are treated as an ideal gas, whose velocity distribution was given in equilibrium at temperature T by the Maxwell-Boltzmann distribution Sommerfeld (1920 s) replaced the distribution by the Fermi-Dirac distribution this free-electron model correctly describes many electrical and thermal properties of metals

3 Parameters density of conduction electrons n (~10 22 /cm 3 ) collision time (or mean free time, or relaxation time) τ (~10-14 s) dc conductivity σ= ne 2 τ/m ac conductivity Fermi velocity v F (~0.01c) mean free path l= v F τ note that σ/τ, l/τ nearly independent of temperature

4 Relaxation times τ in [10-14 s] (Ashcroft and Mermin, p. 10)

5 How good is the free electron model for real metals? Im(ε) for Cu Im(ε) for Ag Im(ε) from reflectivity measurements (Ashcroft/Mermin, p. 297) note: so k= 1/0.1µm }ω=2ev, red light; (also a/γ= 1/0.1µm)

6 Calculation of wake--dc conductivity impedance (see A. Chao): with inverse Fourier transform to find wake general solution is composed of a resonator term and a diffusion term

7 General solution (for Cu with a= 2.5mm, s0= 8.1µm) W/(Z0 c/πa2) wake for dc conductivity s/s0 long range wake:

8 Ac conductivity new parameter Γ= cτ/s 0. for Cu with beam pipe radius a= 2.5 mm, s 0 = 8 µm, cτ= 8 µm, Γ= 1.0; for Al, s 0 = 9.3 µm, cτ= 2.4 µm, Γ= for ac conductivity replace σ with in impedance; then again take inverse Fourier transform for wake Z(4πa 2 /Z 0 s 0 ) 10 5 Re(Z) dc Re(Z) ac note: Re(Z)~0 for k 1/2µm 0 5 Im(Z) ac Im(Z) dc impedance ks 0

9 wake W z (z) is composed of a resonator and a diffusion component for Γ 1, can approximate with the plasma frequency

10 for the resonator component of the wake: the frequency κ r, damping factor α r, and amplitude f r ; the large Γ analytical formula is given by dashes (from K. Bane and M. Sands).

11 W/(Z 0 c/πa2) high Γ approx ac wake dc wake W/(Z 0 c/πa2) s/s high Γ approx s 0 = 8 µm dc wake ac wake s/s 0 ac wake with high Γ approximation

12 W/(Z 0 c/πa2) Cu-dc Al-ac Cu-ac s/µm Point charge wake

13 E/% Cu-ac E/% 0 Al-ac 0.2 Cu-dc Al-ac 0.5 Cu-dc Cu-ac s/µm Induced energy change for rectangular bunch with full length of 65 µm λz s/µm charge 1 nc, energy 14 GeV, tube radius 2.5 mm, tube length 130 m (head) s/µm (tail) Induced energy change (top) for LCLS bunch shape (bottom).

14 Anomalous skin effect (Reuter and Sondheimer) when l> the skin depth, the anomalous skin effect occurs, the fields don t drop exponentially with distance into metal in principle this can happen at low temperatures or high frequencies; nevertheless, It is evident that no appreciable departure from the classical behaviour is to be expected at ordinary temperatures, so that the anomalous skin effect is essentially a low-temperature phenomenon Reuter and Sondheimer. for Cu at room temperature,l= 0.04µm and for k= 1/20µm, δ= 0.04µm

15 sketch of solution method consider semi-infinite metal, with surface in xy plane, and positive z into metal the distribution function of electrons is written in form: f= f 0 +f 1 (v,z), with f 0 the Fermi-Dirac distribution combining Maxwell s equation and Boltzmann s equation, obtain without first term get classical solution also classical if ; at high frequencies is classical if path travelled by electron during one period of the field is small compared to the penetration depth [c 1 is a constant]

16 Anomalous skin effect results given in terms of the surface impedance Z= R+i X

17 α=1.5l 2 /δ 2 ; normalized parameter Λ= α/kcτ; for Cu at room temperature Λ= 3.4 R/R cl 3 (X/X cl ) 2 Λ= Λ= log 10 (kcτ) sensitivity of anomalous skin effect to frequency; given are R/R cl (blue) and X/X cl (red). note: for Λ= 3.4, peak of R/R cl = 1.2 to find impedance, set ; for wake again take inverse Fourier transform

18 Z(4πa 2 /Z 0 s 0 ) 5 Re(Z) anom Re(Z) ac 0 Λ= Im(Z) anom Im(Z) ac ks 0 impedance for a=2.5mm Cu tube including anomalous skin effect W/(Z 0 c/πa2) ac wake dc wake anom. wake s/s 0 wake