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The classic demonstration of the photoelectric effect is discharging a negatively charged electroscope by illuminating zinc or aluminium connected to the electroscope with 254 nm light from a mercury lamp. Their threshold wavelength is ~290 nm. The demonstration with a germicidal lamp is like this: youtube video

Dressed as a bank robber and wearing glasses, I repeated the demo with copper and iron, but with these metals the electroscope did not discharge. Why are copper and iron, in practice, apparently unsuitable for this demonstration? Their threshold wavelength is ~270 nm, so 254 nm light should induce the photoelectric effect in these metals. Is the relation between the current and $\Delta\lambda = (\lambda-\lambda_{\text{Threshold}})$ very nonlinear?

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See the quantum efficiency (QE) dependence on the light photon energy for copper at https://www.osti.gov/servlets/purl/944127 (Fig.2, insert at the top right corner). QE may be indeed low when the photon energy is barely above the work function. One can also speculate that surface condition may be important for your negative results.

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Copper absolutely does experience the photoelectric effect. In particle accelerators that create electron beams, a very common source of electrons is a copper photocathode (PDF describing some technology). At one end of an accelerating cavity, a copper plate is illuminated by a UV laser to release electrons via the photoelectric effect. The electric fields in the accelerating cavity take those released electrons and send them on their way down the beam pipe at speeds approaching the speed of light.

From the linked article (fifth page, third paragraph on the right):

Metallic cathodes like copper and magnesium are robust in the vacuum environment of RF systems. And despite their low quantum efficiency, typically 0.001% at the UV wavelengths necessary for photoemission, metallic photocathodes mounted in RF photoinjectors are the predominant technology ... for light sources require=ing high peak brightness.

Here, quantum efficiency is the number of electrons released from the surface per photon hitting the surface. Copper has a rather low quantum efficiency in that only 0.001% of photons (one in a hundred thousand) results in a released electron. Therefore, much more intense light sources (like lasers) are needed to see a substantial effect. The material is used in accelerators because, after cleaning, it does not pollute the vacuum environment required by particle accelerator. Plus, copper is cheap compared to other cathode materials.

You can find a list of work functions of metals (including copper and iron) here.

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