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The essence of Einstein's idea is like this: if a system is in some bound state with energy $-E_b$ with $E_b> 0$ (the threshold of the continuum band is taken as zero), and we drive the system somehow with frequency $\omega $, then the transition into the continuum is possible only if $\omega > E_b$.

But this is apparently wrong in the general case. For example, take the hydrogen atom in the ground state. If we apply an oscillating electric field $E \cos \omega t $, then if the strength of the electric field $E $ is strong enough, the electron can get ionized however small $\omega $ is. In the extreme limit, $\omega = 0 $, and surely the electron can tunnel into the continuum!

Anyway, the conjecture is that, possibly if the intensity of the light is strong enough, we can still get photoelectric effect with $\omega < \omega_c$?

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    $\begingroup$ ...I don't see what the problem is? The point of the photoelectric effect isn't explaining all ionization and transitions, it's showing that photons are real. $\endgroup$ – Jerry Schirmer Mar 20 '15 at 14:30
  • $\begingroup$ No. My point is that, in principle, there might be no $\omega_c$ at all. The experiments lead to the existence of $\omega_c $ possibly because the light field was too weak. This fortunate situation helped Einstein explain the photoelectric effect using the light quanta concept. $\endgroup$ – Jiang-min Zhang Mar 20 '15 at 14:39
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    $\begingroup$ if you do the experiment, the effect isn't sensitive to the intensity of the light (at least until you reach some saturation point, I"m sure). $\endgroup$ – Jerry Schirmer Mar 20 '15 at 15:20
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    $\begingroup$ @JerrySchirmer, photoelectric effect has no point - it is a natural phenomenon. It surely does not show that some concept, such as photon, is "real". There are other explanations of the effect that do not use a photon concept. $\endgroup$ – Ján Lalinský Mar 21 '15 at 15:25
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Anyway, the conjecture is that, possibly if the intensity of the light is strong enough, we can still get photoelectric effect with $\omega < \omega_c$?

The frequency threshold for the photoelectric effect is observed with light of common intensities; even very low intensity light will work.

There are ways to ionize atoms without such high-frequency light, for example with a strong enough electric field - this is how sparks and thunderstorm flashes occur.

There is also thermo-emission (or thermoionic emission), where electrons get ejected from a piece of metal with high temperature. This occurs in gas-discharge lamps, such as fluorescent mercury lamps.

These other ways of ionizing atoms and producing electricity use phenomena that are not called photoelectric effect, though - the quasistatic electric field from the power grid is usually not thought of as light, as it has very low frequency.

Since ionization due to strong electric field works with low-frequency sources, it is natural to assume that with strong enough laser, one could ionize the hydrogen or other atoms with light of frequency lower than the threshold frequency observed in common photoelectric effect.

I do not recall if such experiment has been done, though - if you find some reference, it could be interesting to read.

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By absorbing more than one photon an electron can still be detached. However, the probability will be small. On the other hand, using a focussed intense laser beam, the situation improves. In fact, isolated atoms can be ionised in this way. The process is called multi-photon ionisation and has been verfied experimentally.

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