Direct Comparison of Brookhaven Reflectivity Measurements with Free-Electron Theory
The reflectivity at normal incidence of copper and aluminum samples was recently measured over a large frequency range at Brookhaven by one of us (JT). Then using the Kramers-Kroning integrals, and assuming the free-electron model of conductivity, the dependence of conductivity on frequency was obtained. The results seemed to suggest, for example, that the dc conductivities of the copper and evaporated aluminum samples are a factor of 3 lower than expected. We propose in this report, instead, directly fitting the free-electron model to the low frequency end of the reflectivity data. This fitting does not depend on the higher frequency results and on Kramers-Kronig integrations, but it does assume that the data at the low frequency end is sufficiently accurate. Note that for our LCLS wakefield studies, it is only over these (relatively) low frequencies that we need to know the electrical properties of the metals. The equations that relate reflectivity R with the free electron parameters dc conductivity {sigma} and relaxation time {tau} are: (1) {tilde {sigma}} = {sigma}/1-ikc{tau}; (2) {tilde n} = {radical} {tilde {epsilon}} = {radical}(1+4{pi}i{tilde k}c/{omega}); and (3) R = |{tilde n}-1/{tilde n} + 1|{sup 2}. The parameters are ac conductivity {tilde {sigma}}, index of refraction {tilde n}, dielectric constant {tilde {epsilon}}, and wave number k = {omega}/c, with {omega} frequency and c the speed of light. In Fig. 1 we show the ideal behavior of R for a reasonably good conducting metal, where {sigma} = 0.12 x 10{sup 17}/s and {tau} = 0.55 x 10{sup -14} s (solid line); these parameters are, respectively, 2% ({sigma}) and 20% ({tau}) of the nominal values for copper. The parameters were chosen so that the important features of R(k) could be seen easily in one plot. We see 3 distinct regions: (1) for low frequencies, k {approx}< 1/c{tau}, R continually decreases, with positive curvature, and with a low frequency asymptote of (1 - {radical}2kc/{pi}{sigma}); (2) for intermediate frequencies the reflectivity is nearly constant, R {approx} (1 - {radical}1/{pi}{sigma}{tau}); (3) for k {approx}> k{sub p} = {radical}4{pi}{sigma}/c{sup 2}{tau}, the plasma frequency of the metal, R quickly drops to zero. The dashed lines in Fig. 1 give the analytic guideposts for the 3 regions. Note that it is only in the first and part of the second region that we can expect the free electron model to have validity in real metals; at higher frequencies the effects of absorption bands and other physics will distort the R(k) curve. In principle, knowing R accurately in the entire 1st region suffices for obtaining the free-electron parameters {sigma} and {tau}; in practice, however, knowing it also in the 2nd region gives us more confidence in the model and especially in the value of {tau}.
- Research Organization:
- SLAC National Accelerator Lab., Menlo Park, CA (United States)
- Sponsoring Organization:
- USDOE
- DOE Contract Number:
- AC02-76SF00515
- OSTI ID:
- 1000387
- Report Number(s):
- SLAC-TN-10-091; TRN: US201101%%176
- Country of Publication:
- United States
- Language:
- English
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