Resistive Wall Wakes in the Undulator Beam Pipe; Implications

1. Resistive Wall Wakes in the Undulator Beam Pipe; Implications Karl Bane FAC Meeting October 12, 2004 • Work done with G. Stupakov • (see SLAC-PUB-10707/LCLS-TN-04-11 • Revised, Oct. 2004)

2. Introduction • In the LCLS, the relative energy variation (within the bunch) induced within the undulator must be kept to a few times the Pierce parameter; if it becomes larger, part of the beam will not reach saturation • the largest contributor to energy change in the undulator is the (longitudinal) resistive wall wakefield • calculations until now have included only the so-called dcconductivity of the metal (beam pipe) wall • we here present calculations including the ac conductivity wake, and show that the effect is larger; we also investigate the anomalous skin effect, and show that it is negligible, and can be ignored • we show that the wake effect can be ameliorated by going to aluminum, and to a flat chamber.

3. 99% from resistive wall wakefields (tail) (head) Rule of thumb • (from LCLS FEL simulations, presented in talk by S. Reiche, Aug. 2003, Zeuthen workshop) • from 1nC results (since E=14 GeV, • L= 130 m) estimate total acceptable • induced energy variation ~0.2% (Pierce parameter = 510-4)

4. impedance (see A. Chao): • with • inverse Fourier transform to find wake • general solution is composed of a resonator term and a diffusion term Resistive Wall Wake • Dc conductivity

5. Generalsolution (for Cu with a= 2.5mm, s0= 8.1m) long range wake: wake for dc conductivity

6. resistive wall wake is a limiting effect in the LCLS undulator, with the induced ΔE~  the Pierce parameter (=0.05%) • can add effects of ac conductivity (see K. Bane and M. Sands, SLAC-PUB-95-7074) to resistive wall model Free electron model of conductivity(see e.g. Ashcroft and . Mermin, Solid State Physics) • Drude free-electron model of conductivity (1900): conduction electrons are treated as an ideal gas, whose velocity distribution in equilibrium is given by the Maxwell-Boltzmann distribution. Sommerfeld (1920’s) replaced the distribution by the Fermi-Dirac distribution. • Drude-Sommerfeld free-electron model correctly describes many electrical and thermal properties of metals through the infrared • Ac conductivity

7. density of conduction electrons n (~1022/cm3) • collision time (or mean free time, or relaxation time)  (~10-14 s) • dc conductivity = ne2/m • ac conductivity • => for short bunches need both dc, ac conductivities • Fermi velocity vF (~0.01c) • mean free path l= vF • note that /, l/ nearly independent of temperature Parameters

8. Im() for Cu Im() for Ag (Ashcroft/Mermin, p. 297) • note: so • k= 1/0.1m h/2= 2eV, red light; we are interested in • 1/20th this frequency How good is the free electron model for real metals? Im() from reflectivity measurements

9. new parameter = c/s0. • for Cu with beam pipe radius a= 2.5 mm, s0= 8 m, c= 8 m, = 1.0; for Al, s0= 9.3 m, c= 2.4 m, = 0.26. • for ac conductivity replace  with in parameter ; then again take inverse Fourier transform of Z for wake Impedance impedance for Cu ac, dc:

10. wake again Wz(z) is composed of a resonator and a diffusion component • for  1, can approximate • with the plasma frequency • for LCLS with Cu, plasma wave number kp= 1/0.02m; mode wave number kr= 1/5m, damping time cr= 32m

11. s0= 8 m ac wake with high  approximation

12. Point charge wake

13. Induced energy change (top) for LCLS bunch shape (bottom). Note that peak current between horns is 3 kA. charge—1 nC, energy—14 GeV, tube radius—2.5 mm, tube length—130 m Induced energy change for uniform bunch with peak current 3 kA and total length 60 m

14. 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= 0.04m and for k= 1/20m, = 0.04m ASE parameter =1.5l2/2; normalized parameter = /kc; for Cu at room temperature = 3.4 we can solve R-S’s formulas for surface impedance, to obtain the impedance and wake including ASE • Anomalous skin effect(Reuter and Sondheimer)

15. = 3.4 impedance for a=2.5mm Cu tube including anomalous skin effect wake:

16. keeping the vertical aperture fixed, flattening the beam pipe cross-section reduces the impedance (on average, the wall is further from the beam) • W(0) reduces by 2/16; for large s, W(s) is the same • Henke and Napoli have found dc wake for flat chamber; their impedance: • ( as defined before) • inverse Fourier transform to find the ac wake of flat chamber Flat Chamber

17. induced energy deviation for round chamber induced energy deviation for flat chamber

18. Figure of merit • I suppose we want to keep the maximum number of beam particles in resonance (keep within a window E/E that is a few times ) • particles in “horns” may not contribute, because of large energy spread/emittance • figure of merit: minimum total energy variation over 30 m stretch of beam, E.

19. Table I: Figure of merit, E, for different assumptions of beam pipe shape and material. Nominally, vertical aperture is 2a= 5 mm. Note that Cu-dc results are not physically realizable. Table II: Figure of merit, E, for beam pipes made of various metals

20. Discussion and conclusions • for the present design of LCLS undulator beam pipe—round, radius a=2.5 mm, Cu—, a large energy variation will be induced in the beam due to the resistive wall wakefield, => a good fraction of beam will not reach saturation • the energy variation can be reduced by going to a flat (keeping the vertical aperture fixed), Al chamber: over 30 m of beam, a minimum total energy variation of 0.6% becomes 0.2%. • the anomalous skin effect is not important and can be ignored • the wake effect can be reduced by reducing “horns” in bunch distribution or increasing beam pipe aperture • impurities, oxides, etc. in surface layer may increase wake effect • FEL simulations need to be performed to verify our conclusions and accurately quantify the results (see talk by H.-D. Nuhn)