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I am studying photoelectric effect and it is known that if the irradiated light on metal surface has a lower frequency than that of work function the electrons do not eacape, no matter how intense the light is. However, I imagined a thought experiment where work function is W=hυ and light consisting of photon with a frequency υ/2 is irradiated on the metal surface. Suppose the electron absorbs 2 (or more than 2) photons of energy hυ/2 each. It then has an energy equal to (or greater than) the work function and hence can escape the surface. Then we have a case where electrons are knocked out by light of lower frequency than the work function. In a similar way, we can have an electron knocked out of the metal surface by light of lower frequency but sufficient intensity so that there are many photons which the electron can absorb to gain enough energy to escape. Why can't we have this situation?

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  • $\begingroup$ The electron is stable at its initial lower energy level and only at the next energy level which is this case is a free electron. There is no halfway intermediate level for the electron to "hang out" or exist so it will never absorb 1/2 the energy. $\endgroup$ Apr 15, 2023 at 14:27
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    $\begingroup$ The probability of interaction between photons and electrons is governed by the fine structure constant (squared). Which means that per every 10000 possible interactions, only one is likely to happen. When you want to simultaneously absorb two photons, the interaction probability drops by another such factor. It is just too small a probability to observe in experiments unless you specifically search for it. Not to mention that the transition probability for this also depends upon a much smaller matrix element than the usual one. Note that $\nu/2$ is likely still UV, not visible light. $\endgroup$ Apr 15, 2023 at 14:38
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    $\begingroup$ Google for two photon absorption $\endgroup$ Apr 15, 2023 at 15:10
  • $\begingroup$ See e.g. accelconf.web.cern.ch/ipac2018/papers/tupml026.pdf for a research paper. Metals are harder systems than e.g. organic molecules or nano-particles because the effective interaction time with light is much smaller (femto-second or sub-femtoseconds, I believe), which needs very high light intensity, but it can be done and has been done. Search for "multi photon absorption photocathodes". $\endgroup$ Apr 15, 2023 at 16:23

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Multi-photon ionisation is possible, but the probability of it occurring is very low unless extremely strong fields are applied. But it was definitely a topic of interest in the 1960s: see for example this article deriving the expression for the rates, this article observing it in Cs$_3$Sb. For whatever reason interest seemed to wane after the 1960s, but recently with the increased accessibility of strong field sources there has been somewhat of a revival of interest in multi photon processes such as this. See for example these slides for a nice overview

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  • $\begingroup$ This answer is basically correct, but multiphoton ionization is much more of an active field (and much more of a routine process) than the text makes it sound. I don't know where the impression that interest waned after the 1960s comes from, though - the interest never dropped and has only increased after the creation of CPA for laser amplification. $\endgroup$ Aug 28, 2023 at 18:17
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Usually this is not possible. As described in the comments, a photon can be absorbed when an atom has two energy levels, and the photon contains just enough energy to promote an electron from one to the other.

Another way a photon can be absorbed is if it contains so much energy that the electron escapes entirely from the atom. In that case, any (high) photon energy will do. The extra energy goes into the kinetic energy of the photon plus recoil of the atom.

But there are special cases where two (or more) photons can be absorbed in two separate steps. This is sometimes taken advantage of in lasers to double or triple the frequency of light. If there are two sets of energy levels with just the right separation, a photon can promote an electron to one higher energy, and the second can promote it again. See this article on Upconversion.

Another way involves crystals with nonlinear response to electromagnetic waves. In a classical picture, electromagnetic waves exert forces on electrons, making them oscillate. Usually atoms are like springs. $F = kx$. Double the force to double the amplitude. But in some crystals there is a quadratic term. $F = k_1x + k_2x^2$. See Nonlinear Index.

This can cause higher frequencies to be emitted. Some lasers take advantage of this to emit higher frequency light. The non-linear component is usually weak. It takes a very intense light to excite it. These lasers often concentrate a lot of energy in a very short pulse. See this article on Frequency Doubling.

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  • $\begingroup$ Multiphoton processes are much more routine than the description in this answer. For an everyday example of the sources described as "some lasers" here, see green laser pointers, which are frequency-doubled IR lasers. $\endgroup$ Aug 28, 2023 at 18:19

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