# Is energy exchange quantized?

In the photoelectric effect there is a threshold frequency that must be exceeded, to observe any electron emission, I have two questions about this.

I) Lower than threshold: What happen with lesser frequency/energy photons? I mean there is no energy transferred? If some energy were transferred, (and it must be transfered quantized), how can be experimentally known it was quantized, if there is no electron emission?

II) Intensity dependence: There is any known dependency with intensity? (I mean a dependency not about the number of electrons (I am already aware of this) but about the energy of them) I thought energy of electron emited was independent with intensity, but then I found this link dependency with intensity (is a paper to buy that I haven't read) but relates energy with intensity.

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## 2 Answers

Energy exchange is quantized when moving a electron from one bound state to another bound state.

This isn't because the exchange is inherently quantized, but because the states the electron may occupy are quantized.

Thus the standard photo-electric effect in which a photon can not excite an atom unless it has a minimum energy.

However,...

There are multi-photon processes by which sub-threshold light can excite transitions.

Cross-section for them go by intensity-squared (or worse) and are very small for any reasonable light intensity. To study or employ them you get powerful, short pulsed laser systems. Where short pulsed means nano-second or faster pulses and powerful means "Do not look into beam with remaining eye". Even then you don't get a lot of rate.

These processes are utterly negligible for the kind of benchtop experiment we use to teach the photoelectric effect: you just can't get enough intensity. (See below for how negligible.)

The conceptual model here is that the first photon bumps the electron to a short-lived, unstable state without well defined quantum numbers, and the second comes along before that state decays and finishes the job.

We're currently exploring the application of such a process to calibrating light yields, opacities in a large volume of scintillating material.

From New J.Phys.12:113024, 2010:

For gases the one-photon absorption cross-section $\sigma_1$ is typically of the order of $10^{−17}\text{ cm}^2$, whereas the two-photon and the three-photon cross-sections are of the order of $\sigma_s = W/F_2 \approx 10^{−50}\text{ cm}^4\text{ s}$ and $\sigma_3 = W/F_3 \approx 10^{−83}\text{ cm}^6\text{ s}^2$, respectively.

Where $F$ is intensity in photons/second and W is excitation rate in reciprocal seconds.

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Hello @dmckee I think this statement "the first photon bumps the electron to a short-lived, unstable state without well defined quantum numbers, and the second comes along before that state decays and finishes the job" is very is bold but doubtful, if that is possible it seems to be a direct contradiction with the "photon concept", because you could gradually add energy without needing a single particle-like action. – HDE Apr 14 '11 at 18:13
@HDE: See for instance for an application of this technique. Though this paper actually uses three photons with an intermediate step at a well defined energy level on the way to ionization, and therefore goes by intensity cubed. – dmckee♦ Apr 14 '11 at 18:27
Further, this process could not be mimicked by a continuous process, as there is a energy threshold for the one process, a energy threshold for the two photon process, an so on; and the rate are determined in a quantum mechanical way. – dmckee♦ Apr 14 '11 at 18:33
Amazing, multiphoton ionization.. I didn't knew anything of this!( perhaps I have stop reading Louis de Broglie and get something newer haha..) I wonder if there is not a chance of subdividing again the threshold of every "photon process" into photons of successively small energy, if possible, where is the limit? – HDE Apr 14 '11 at 20:12

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@HDE: The limit is the cross-section. Even for immensely intense light sources the production rate is dropping like a stone. Each additional photon brings in another factor of the fine structure constant and another short lifetime for the unstable state (and the further they are from a stable state, the shorter the lifetime). Look at the behavior quoted above. If you need more than few photons, it just won't happen often enough to measure, much less use. – dmckee♦ Apr 14 '11 at 20:18

If the frequency of the incident radiation is lower than the threshold frequency then either a photon is fully absorbed or not absorbed at all. It is absorbed only if it has an energy which is just enough to excite the electron to the higher state but not enough (less than the work function) to make the electron to leave the surface of the material.

The only way we know that the energy of the electron is quantized in an atom is by analyzing its spectrum.

For a particular frequency above the threshold, the number of the electrons emitted will linearly increase with the intensity of the incident radiation.

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This is the first order effect, and is correct for essentially every purpose. However, there are contributions from higher order graphs which allow multi-photon effects. It takes a pulsed, high power laser to get the intensity needed to study them. – dmckee♦ Apr 14 '11 at 17:21
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