Resistive Wall Wakes in the Undulator Beam Pipe – Theory, Measurement

1. Resistive Wall Wakes in the Undulator Beam Pipe – Theory, Measurement Karl Bane FAC Meeting April 7, 2005 • Work done with G. Stupakov, Jiufeng Tu • see: • SLAC-PUB-10707/LCLS-TN-04-11 Revised, Oct. 2004 • SLAC-LCLS-TN-05-6, Feb. 2005

2. Outline • Review ac resistive wall wake calculations given last time • Analyze reflectivity measurements performed by J. Tu at Brookhaven • Discuss implications

3. 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 • earlier calculations included only the so-called dcconductivity of the metal (beam pipe) wall; we showed last time that: • --if the ac conductivity is included the wake effect is larger • --that the anomalous skin effect is negligible, and can be ignored • --that the wake effect can be ameliorated by going to aluminum, and to a flat chamber.

4. the free-electron model of conductivity is expected to be valid at low frequencies; it has two parameters: dc conductivity , and relaxation time  • at 295K: Cu--= 5.261017/s, = 2.5210-14 s; • Al --= 3.351017/s, = 0.7510-14 s • if it is valid, the wake in a round beam pipe can be approximated: • with the plasma frequency

5. Point charge wake

6. charge—1 nC, energy—14 GeV, tube radius—2.5 mm, tube length—130 m induced energy deviation for round chamber induced energy deviation for flat chamber

7. Table I: Figure of merit, E (minimum total energy variation over 30 m stretch of beam), 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. --figure of merit only gives rough idea of effect in LCLS; better to use analytic model of Z. Huang and G. Stupakov or simulations

8. FTIR Reflectivity Measurements (J. Tu talk) Kramers-Kronig relation: causality We can measure

9. Measured conductivity compared with calculation Kramers Kronig analysis of measurements Calculation using Ashcroft-Mermin ,  • -1(0) is dc conductivity • -Kramer’s Kronig analysis of measurements gives dc conductivity factor 1/2.5 of expected value

10. Behavior of R for free-electron model Frequency k vs. reflectivity R for a metallic conductor, assuming the free electron model (solid line). For this example =1.21016 /s and =5.410-15 s. Analytic guideposts are also given (dashes). --for LCLS bunch interested in : [10,100] m, or k: [0.06,0.6]m-1

11. Brookhaven reflectivity data for Cu and evaporated Al range of interest Measurement results from Brookhaven: reflectivity R vs. frequency k for copper and evaporated aluminum. --Cu absorption band at 10 m-1 also seen in plot in Ashcroft-Mermin

12. Al survey of measurements, Shiles, et al, 1980 range of interest 1evk= 5 m-1 --also slope in R(), but only ½ as steep --clear absorption resonance, only hinted at in Brookhaven results

13. region of interest Al fit Aluminum reflectivity: comparison of measurements (blue) with calculations using nominal ,  (green); and fitted values: 0.63 nominal , 0.78 nominal  (red). The position of k= 1/c for the fit is also shown. --in literature no data below 0.2 m-1

14. region of interest Cu fit Copper reflectivity: comparison of measurements (blue) with calculations using nominal ,  (green); and fitted values: 0.66 nominal , 0.67 nominal  (red). The position of k= 1/c for the fit is also shown. --in literature no data below 0.3 m-1

15. For Al, how does the fit taken affect E for the LCLS? Consider (,): nom.– (3.351017/s, 0.7510-14 s) fit– (2.121017/s, 0.5810-14 s) model2– (1.681017/s, 0.4810-14 s)

16. Conclusion and Discussion --The resistive wall wakefield in the beam pipe of the LCLS undulator will induce a significant energy variation within the bunch; it seems that this will inhibit much of the beam from lasing. --This effect can be ameliorated by going from a pipe made of Cu to one of Al (by ~2), and from one with a round to a flat cross-section (by ~30%). --To really quantify this, one should go to analytical calculation of Z. Huang and G. Stupakov, or simulations (S. Reichle and W. Fawley)

17. Brookhaven Measurements --fits (/nom, /nom) are approx. (0.65, 0.80) for Al, and (0.65, 0.65) for Cu; fit is reasonably good, but does not cover desired frequency (k) range for Cu (only 40%) --At higher frequencies the measured R varies linearly with frequency, unlike the constant dependence expected by the free-electron model. This point should be understood before giving too much credence to our results --The deviation in R for Cu above 0.3 m-1 is small and may not have much effect. However, it is difficult to know precisely the implication for the wakefield of a copper pipe: reflectivity at normal incidence alone, without a model, is not enough information to make such a calculation. --Measurements of more Al samples are in the works.