# Reflectivity Measurements for Copper and Aluminumin the Far Infrared and the Resistive Wall Impedance in the LCLSUndulator Page: 3 of 4

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R

1 r0.8

0.6

0.4

0.2- ' 1-sqrt(1/Gti)

-1-sqrt(2kc/na)

k= 1/et k= kp

. . . . .. . . . . . . . . . . . . . . . .. . . . . . . . k [ m - 1]

5 10 15 20Figure 1: Reflectivity R vs. frequency k for an ideal con-

ductor, assuming the free-electron model (solid line). Ana-

lytic guideposts are also given (dashes).

Anomalous Skin Effect (ASE)

When the skin depth 6, becomes less than or compa-

rable to the mean free-path of electrons C the classical ac

model of conductivity no longer holds, and the (room tem-

perature) anomalous skin effect (ASE) applies [6,7]. [For

Cu, 6, ~ t when k (20 pm)-1.] Different expression

are known for the cases of specular and diffuse reflection of

electrons at the surface. Fitting these formulas to infrared

measurements for Cu, Ag, Au, Lenham and Treherne con-

cluded that the diffuse model is normally applicable, even

for well-prepared samples [4]. This is the model we use.

For diffuse reflection, the surface impedance is [7]

47rikr A ikcrA (4)

ZS= 1 + ikcr 3 'T, 0 t(1 + ikcr)3 10 n 1 + m d ( )

Ar 1 ThnA

t(t) = [(1 + t2) tan -- t] (6)

with ASE parameter A 37rCf2/c2T. Then = 47r/Z7c.

For given a, r, ASE's effect on R(k) will be small in

the lowest energy region (region 1); in region 2 (the flat

region), R(k) will be lower than for the classical, ac model.

MEASUREMENTS

Three samples were measured: an Al film, solid Al,

and solid Cu. The samples (thickness ~ few mm) were

mounted on an optically-black cone, and the room temper-

ature reflectivity was measured in a near-normal-incidence

arrangement from k ~ 0.02-10 pm-- on a Bruker IFS

66v/S and a Bruker IFS 113v Fourier transform infrared

(FTIR) spectrometer. By evaporating a thick gold film in

situ in ultra-high vacuum (< 1 x 10-8 Torr) over the sam-

ple, the precise ratio of sample reflectivity to the reflectiv-

ity of Au was measured. Knowing the reflectivity of Au,

the absolute reflectivity of the sample was thus determined.

The details of this technique have been described previ-

ously [8-10]. Using this in situ evaporation technique, the

errors associated with misalignments, window interference

and surface inhomogeneity can be eliminated. As a result,absolute reflectivity of the sample R(k) can be measured

to a precision of 0.1% or better.

Fig. 2 shows R(k) for Cu and evaporated Al over the

entire measured range. [The solid Al sample is consid-

ered a bad sample: it had poor reflectivity (R 0.95 at

k= 0.5 pm-1) and noticeable granularity. This sample's

data is considered no further.] Comparing the general fea-

tures of the data with the literature, we see for Cu the onset

to interband absorption at 10 pm-1 as in Ref. [11] (p. 297),

for Al a weakly suggested absorption spike at 7.5 pm-1

that is very pronounced in Ref. [5]. For Cu, the dip begin-

ning at 1.5 pm--1 is not seen in Ref. [11]. We assume the

differences are due to sample variability (finish, etc).

R0.9

07s

0.7Cu

2.5 5 7.5 10Figure 2: Measurement results: Reflectivity R vs. fre-

quency k for copper and evaporated aluminum.

Focusing on frequencies below 1 pm-1: we see in R a

possibly correct dependence at the very low end [(1- R) ~

Ik]. But this is followed, unexpectedly, by a linear de-

crease (not a constant). When comparing the Al curve with

reflectivity measurements performed in 1980 by Shiles, et

al [5], we can see that the earlier measurements also have a

slope, but that is a factor of 2 less steep. The non-zero slope

is not understood. The data is, nevertheless, smooth and

well-behaved at low k, to 0.5 pm for Al, to 0.3 pm-- for

Cu (representing, respectively, 85% and 50% of our range

of interest). We assume our models are valid in these re-

gions, and in these regions we will perform our fits.

Fitting to the Data

The aluminum comparison, over nearly twice the region

of interest, is given in Fig. 3. Blue repeats the measured

results. The result of the nominal free-electron model, with

a 3.35 x 1017/s and r 0.75 x 10-14 s (at 295 K)

[11], is given in green. The fit to the ac model (the red

curve) gives parameters, relative to their nominal values:

or =0.63, r= 0.78. The fit to the ASE model (t

0.016 pm; the dashed curve) gives: a 0.61, T-T 1.28.

We note that the fitted curves fit the data very well up to

0.5 pn 1, that a is less than ideal by a factor of 2/3 (which

is plausible), and that r is near nominal. Note that the ASE

model, which differs from the ac model only when k is not

too small, gives a fit that is unique only in r.

The copper comparison is shown in Fig. 4. Given are the

measured data (blue) and the nominal calculation, with a3

1

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Bane, K.L.F.; Stupakov, G.; /SLAC; Tu, J.J. & /City Coll., N.Y. Reflectivity Measurements for Copper and Aluminumin the Far Infrared and the Resistive Wall Impedance in the LCLSUndulator, article, June 27, 2006; [Menlo Park, California]. (digital.library.unt.edu/ark:/67531/metadc876471/m1/3/: accessed June 23, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.