Karl L. Bane AC Impedance, [email protected] October 12-13, 2004 Resistive Wall Wakes in the Undulator Beam Pipe; Implications Karl Bane

Karl L. Bane

AC Impedance, Implications [email protected]

October 12-13, 2004

Resistive Wall Wakes in the Undulator Resistive Wall Wakes in the Undulator Beam Pipe; ImplicationsBeam Pipe; Implications

Resistive Wall Wakes in the Undulator Resistive Wall Wakes in the Undulator Beam Pipe; ImplicationsBeam Pipe; Implications

Karl Bane

FAC MeetingFAC MeetingOctober 12, 2004

• Work done with G. Stupakov

(see SLAC-PUB-10707/LCLS-TN-04-11Revised, Oct. 2004)

Karl L. Bane


October 12-13, 2004

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.

Karl L. Bane


October 12-13, 2004

99% from resistive wall wakefields

(tail) (head)

Rule of thumbRule 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 = 510-4)

Karl L. Bane


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 WakeResistive Wall Wake

Karl L. Bane


October 12-13, 2004

General solution

wake for dc conductivity

(for Cu with a= 2.5mm, s0= 8.1m)

long range wake:

Karl L. Bane


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%)

•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))

Karl L. Bane


October 12-13, 2004

Parameters

•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

Karl L. Bane


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= 1/0.1m h/2= 2eV, red light; we are interested in

1/20th this frequency

Karl L. Bane


October 12-13, 2004

Impedance

•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 for Cu ac, dc:

Karl L. Bane


October 12-13, 2004

•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= 1/0.02m; mode wave number kr= 1/5m, damping time cr= 32m

Karl L. Bane


October 12-13, 2004

ac wake with high approximation

s0= 8 m

Karl L. Bane


October 12-13, 2004

Point charge wake

Karl L. Bane


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

Karl L. Bane


October 12-13, 2004

•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

Karl L. Bane


October 12-13, 2004

impedance for a=2.5mm Cu tube including anomalous skin effect

= 3.4

wake:

Karl L. Bane


October 12-13, 2004

Flat ChamberFlat 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

Karl L. Bane


October 12-13, 2004

induced energy deviation for round chamber

induced energy deviation for flat chamber

Karl L. Bane


October 12-13, 2004

Figure of meritFigure 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.

Karl L. Bane


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

Karl L. Bane


October 12-13, 2004

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)

Documents

Karl L. Bane AC Impedance, [email protected] October 12-13, 2004 Resistive Wall Wakes in the Undulator Beam Pipe; Implications Karl Bane