1. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 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. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 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 dc conductivity 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. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 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. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 impedance (see A. Chao): with inverse Fourier transform to find wake general solution is composed of a resonator term and a diffusion term Dc conductivityDc conductivity Resistive Wall Wake

5. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 General solution wake for dc conductivity (for Cu with a= 2.5mm, s 0 = 8.1  m) long range wake:

6. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 Ac conductivityAc conductivity resistive wall wake is a limiting effect in the LCLS undulator, with the induced ΔE~  the Pierce parameter (=0.05%) (see K. Bane and M. Sands, SLAC- PUB-95-7074)can add effects of ac conductivity (see K. Bane and M. Sands, SLAC- PUB-95-7074) to resistive wall model 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 (see e.g. Ashcroft and. Mermin, Solid State Physics) Free electron model of conductivity (see e.g. Ashcroft and. Mermin, Solid State Physics)

7. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 Parameters density of conduction electrons n (~10 22 /cm 3 ) collision time (or mean free time, or relaxation time)  (~10 -14 s) dc conductivity  = ne 2  /m ac conductivity => for short bunches need both dc, ac conductivities Fermi velocity v F (~0.01c) mean free path l= v F  note that  / , l/  nearly independent of temperature

8. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 Im(  ) for Cu Im(  ) for Ag (Ashcroft/Mermin, p. 297) How good is the free electron model for real metals? Im(  ) from reflectivity measurements note: so k= 1/0.1  m  h  /2  = 2eV, red light; we are interested in  1/20 th this frequency

9. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 Impedance new parameter  = c  /s 0. for Cu with beam pipe radius a= 2.5 mm, s 0 = 8  m, c  = 8  m,  = 1.0; for Al, s 0 = 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 for Cu ac, dc:

10. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 wake again W z (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 k p = 1/0.02  m; mode wave number k r = 1/5  m, damping time c  r = 32  m

11. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 ac wake with high  approximation s 0 = 8  m

12. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 Point charge wake

13. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 Induced energy change (top) for LCLS bunch shape (bottom). Note that peak current between horns is 3 kA. Induced energy change for uniform bunch with peak current 3 kA and total length 60  m charge—1 nC, energy—14 GeV, tube radius—2.5 mm, tube length—130 m

14. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 Anomalous skin effect (Reuter and Sondheimer)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= 0.04  m and for k= 1/20  m,  = 0.04  m ASE parameter  =1.5l 2 /  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

15. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 impedance for a=2.5mm Cu tube including anomalous skin effect  = 3.4 wake:

16. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 Flat Chamber 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

17. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 induced energy deviation for round chamber induced energy deviation for flat chamber

18. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 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. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 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. Karl L. Bane AC Impedance, Implicationskbane@slac.stanford.edu October 12-13, 2004 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)