# X-ray Lasers and Forbidden Transitions

My notes from an introductory course about lasers say that

There does not exist a laser emitting in the X-ray because the spontaneous decay lifetime is too short to have stimulated emission. In fact, it goes with the inverse of the frequency of the transition, therefore being small for high frequency transitions.

I know that: $$τ_{sp} \propto \frac{1}{\omega_0^3 |μ_{12}|^2}$$ with $ω_0$ angular frequency associated with the transition and $μ_{12}$ expectation value of the transition operator. I also know that, for transition with a very low probability, such as magnetic dipole allowed (and electric dipole forbidden), this lifetime can significantly increase.

I also know that there are a lot of different selection rules (electric quadrupole, magnetic quadrupole, ...), each one less probable than the preceeding, for which the spontaneous decay lifetime could be higher.

Therefore, why don't x-ray lasers exist? Is it just that is still more convenient to develop synchrotrons or is there some other reason? What have been the scientific efforts in this direction?

• Somewhat related: physics.stackexchange.com/questions/296237/… – Emilio Pisanty Jul 5 '17 at 21:30
• @EmilioPisanty I was somehow thinking about an electronic transition, but the question and the related answers you linked to are very very interesting! – JackI Jul 6 '17 at 5:12
• There are always Auger transitions. Almost always this is by far the dominant decay mechanism of core holes. – Pieter Jul 6 '17 at 9:23
• @Pieter Auger transitions are non-radiative, though, so they're more of a radiation sink than a radiation source. – Emilio Pisanty Jul 6 '17 at 10:17
• soft X ray lasers are a reality since decades. – scrx2 Jul 12 '17 at 21:13

As mentioned in Semoi's answer, electronic transitions in the x-ray regime have the disadvantage that they will tend to be ionizing transitions, i.e. they will put one electron in the continuum, where it will tend to fly away and not come back.

However, in general, that is only true for neutral atoms, but once you remove one or a few electrons, the remaining electrons are much more tightly bound, which means that you can have transitions with a much higher energy difference that still remain within the bound-states manifold. Thus, if you're working in an ionized plasma, you have good chances of being able to implement a closed lasing cycle, which can be pumped externally or even via the plasma's own collisional excitations.

Of course, this will make for a challenging experiment. For one, free electrons can be highly dispersive in the soft x-ray regime, so phase-matching needs to be done carefully. More importantly, good optical elements, particularly in transmission and at normal-incidence reflection, are thin on the ground from the XUV upwards, so building a resonant cavity will be somewhere between hard and impossible (though as I explained in this related answer, the loss of the cavity is not completely fatal). Nevertheless, it can be done.

I first became aware that this sort of amplification is a possibility via the absolutely heroic experiment described in

Demonstration of circularly polarized plasma-based soft-x-ray laser. A. Depresseux et al. Phys. Rev. Lett. 115, 083901 (2015).

Multimillijoule, highly coherent x-ray laser at 21 nm operating in deep saturation through double-pass amplification. B. Rus et al. Phys. Rev. A 66, 063806 (2002).

which itself reviews many viable approaches to the problem.

Consider an electron, which is bound to an atom. If it absorbs an X-ray photon, the electron leaves the atom and becomes a free electron. Hence, the inverse process is "not possible" by stimulated emission -- the free electron does not belong to the atom. Furthermore, the X-rays would "immediately" be re-absorbed, because transitions of bounded electrons into the continuum are not constrained by transition rules.

• I see your point, and I think this could be a great limitation. On the other hand thinking for instance to the X-ray incoherent emission from an Al or Mg lamp, the transitions involved are electronic core ones. Instead of having photoemission due to the spontaneous decay of a certain hole in the core of an atom, why can't we have stimulated emission from the same process (but involving different levels, since the first transition is allowed)? – JackI Jul 6 '17 at 5:26
• A 2-level system can't form a laser, because the electrons would reabsorb the emitted light. That's why a laser is formed in 3-level systems. Here it is possible, to deplete the "ground state" $|0>$ and that the excited states do not reabsorb the light. For X-ray this is not fulfilled, because electrons from all atomic levels would reabsorb the light. This does not change, if the light is generated in the nucleus. Hence, one would need to use a plasma and deplete its "ground state". It won't be easy to control the energy levels in this hot and strongly interacting environment. – Semoi Jul 6 '17 at 10:08

The X-ray laser is more of a stimulated Brehmsstralung emission or synchrotron emission process. If you accelerate lasers with a frequency $\nu_a$ then the radiation emitted is approximately $\nu~\simeq~\gamma^2\nu_a$. The wiggler or free electron laser operates on this principle. An array of magnetic dipoles force a beam of electrons to wiggle or follow an undulating path. If the velocity of the electron is $v~-~0.999999c$, or $\gamma^2~=~5\times 10^5$. Now assume the dipole magnets are $1\:\mathrm{cm}$ apart. The frequency of oscillation for the electron beam will be about $\nu~=~3\times 10^{10}\:\mathrm{s}^{-1}$. The frequency of radiation emitted is then about $1.5\times 10^{15}\:\mathrm{Hz}$. This is in the low UV range. This requires the electrons be pushed to $\gamma~=~224$ or to $112 \:\mathrm{MeV}$.

• I'm sorry for my poor English (it's not my first language), but I am having difficulties in understanding your answer. Your point is, actually, that I can't compare an hypothetical X-ray laser and the Bremsstrahlung emission that we have in a synchrotron? – JackI Jul 6 '17 at 5:21
• @Crowell: As you said, the free electron laser (FEL) exist and thus an X-ray laser exists. However, I understood the initial question as "why does is it impossible to have a X-ray diode / solid state laser?" – Semoi Jul 6 '17 at 9:47
• To above, it was not clear in the question this was only for solid state lasers. As for Jack the point is that we can have a sort of stimulated emission of synchrotron or Brehmsstralung radiation. – Lawrence B. Crowell Jul 6 '17 at 15:53