Karl Bane AC Wake--Measurement, April 7-8, 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 Revised, Oct SLAC-LCLS-TN-05-6, Feb. 2005

Karl Bane AC Wake--Measurement, April 7-8, Outline Review ac resistive wall wake calculations given last time Analyze reflectivity measurements performed by J. Tu at Brookhaven Discuss implications

Karl Bane AC Wake--Measurement, April 7-8, 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 dc conductivity 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.

Karl Bane AC Wake--Measurement, April 7-8, 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  /s,  = 2.52  s; Al --  = 3.35  /s,  = 0.75  s if it is valid, the wake in a round beam pipe can be approximated: with the plasma frequency

Karl Bane AC Wake--Measurement, April 7-8, Point charge wake

Karl Bane AC Wake--Measurement, April 7-8, induced energy deviation for round chamber induced energy deviation for flat chamber charge—1 nC, energy—14 GeV, tube radius—2.5 mm, tube length—130 m

Karl Bane AC Wake--Measurement, April 7-8, 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

Karl Bane AC Wake--Measurement, April 7-8, FTIR Reflectivity Measurements We can measure Kramers-Kronig relation: causality (J. Tu talk)

Karl Bane AC Wake--Measurement, April 7-8, Measured conductivity compared with calculation --  1 (0) is dc conductivity --Kramer’s Kronig analysis of measurements gives dc conductivity factor 1/2.5 of expected value Kramers Kronig analysis of measurements Calculation using Ashcroft-Mermin , 

Karl Bane AC Wake--Measurement, April 7-8, Frequency k vs. reflectivity R for a metallic conductor, assuming the free electron model (solid line). For this example  =1.2  /s and  =5.4  s. Analytic guideposts are also given (dashes). --for LCLS bunch interested in : [10,100]  m, or k: [0.06,0.6]  m -1 Behavior of R for free-electron model

Karl Bane AC Wake--Measurement, April 7-8, 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 Brookhaven reflectivity data for Cu and evaporated Al range of interest

Karl Bane AC Wake--Measurement, April 7-8, Al survey of measurements, Shiles, et al, also slope in R(  ), but only ½ as steep --clear absorption resonance, only hinted at in Brookhaven results range of interest 1ev  k= 5  m -1

Karl Bane AC Wake--Measurement, April 7-8, 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 region of interest

Karl Bane AC Wake--Measurement, April 7-8, 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 region of interest

Karl Bane AC Wake--Measurement, April 7-8, For Al, how does the fit taken affect  E for the LCLS? Consider ( ,  ): nom.– (3.35  /s, 0.75  s) fit– (2.12  /s, 0.58  s) model2– (1.68  /s, 0.48  s)

Karl Bane AC Wake--Measurement, April 7-8, 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)

Karl Bane AC Wake--Measurement, April 7-8, 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.