Karl L. Bane AC Impedance, Implications [email protected].edu October 12-13, 2004 Resistive Wall Wakes in the Resistive Wall Wakes in the Undulator Beam Pipe; Implications Undulator Beam Pipe; Implications Karl Bane FAC Meeting FAC Meeting October 12, 2004 • Work done with G. Stupakov (see SLAC-PUB-10707/LCLS-TN-04-11 Revised, Oct. 2004)
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Resistive Wall Wakes in the Undulator Beam Pipe; Implications
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). Introduction. - PowerPoint PPT Presentation
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IntroductionIntroduction• 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.
•Ac conductivityAc conductivity•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-(see K. Bane and M. Sands, SLAC-PUB-95-7074) 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
Free electron model of conductivity (see e.g. Ashcroft and (see e.g. Ashcroft and . Mermin, . Mermin, Solid State PhysicsSolid State Physics))
•Anomalous skin effectAnomalous skin effect (Reuter and Sondheimer)(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= 0.04m and for k= 1/20m, = 0.04m
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
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
Discussion and conclusionsDiscussion 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)