The physics behind European XFEL

I tried to find detailed data at the official site and learned that the linear accelerator is 1.7 Km long, the energy of the electrons is 17.5 GeV (35000$\times$ the electron rest energy), and the wavelength of the radiation is 0.07 to 4.5 nm.

[If I am not wrong, the final speed of the electron is 0. 999 999 999 6 $c$]

What I'd like to know is:

• what is the power radiated at the end of linear acceleration (there should be some, according to Larmor formula

• (Why does the wavelength vary by 65 times? is it only due to regulation of employed energy? If it is so and) if we consider that the 0.07 wavelength corresponds to max energy (17.5 GeV), the energy of the photons is about 4.3x10^18 h (0.035 1/29x electron rest energy) that is one-millionth of its KE: how do we get the radiated power value from Larmor? Recently I wrote to a similar plant in England and they replied that radiation is mostly dependent on the angle of curvature (angle theta in the picture), but Larmor does not contemplate any angle. Can you shed any light on this experiment?

Edit

Apparently the question is not clear enough, I'll try to explain it: an accelerated charge radiates according to Larmor formula. In this XFEL experiment charges are at first accelerated linearly and then undulated. The sites accounts only for the radiation after the undulator.

Can you explain how the value of the energy of the laser flashes is related to the formula, and compare it with the value of radiation emitted before the undulator, during the linear acceleration?

• ...why the downvote? what's wrong with this question? – user157860 Sep 6 '17 at 12:28
• could that be a selectable wavelength range in different operating modes - not a spread of energies in a single pulse? – Martin Beckett Sep 8 '17 at 16:40
• @MartinBeckett, that is just what I thought, and the given value corresponds to max power, right? – user157860 Sep 8 '17 at 16:44
• 4.3x10^18 what? I imagine you've been informed that reporting numbers without units renders your work meaningless, right? Also, the reported range is 0.05 to 4.7 nanometres in your source, which means that you got your numbers from some other source you don't cite, but more importantly the wavelength range would be 4.5/0.07$\approx$65, not 650. These are elementary errors that you're expected to fix yourself before complaining that others found your post deficient. – Emilio Pisanty Sep 8 '17 at 17:01
• @EmilioPisanty, that was rather snarky,: I did not complain, I just asked what is wrong because the the downvoter couldn't be bothered to comment. I wouldn't call a typo an error. – user157860 Sep 8 '17 at 17:14

FEL physics, beyond the basics, gets very complicated very fast; among other things, naive calculations based on the Larmor formula simply don't cut it. In any case, even a shallow reading of the Wikipedia page provides initial answers to your questions, which suggests your question had a far-from-sufficient prior research.

To do some of your homework for you, the FEL radiation wavelength is given by $$\lambda_r = \frac{\lambda_u}{2 \gamma^2}(1+K^2) ,$$ where $\lambda_u$ is the undulator wavelength, $\gamma$ is the electrons' Lorentz contraction factor, and $K={\frac {\gamma \lambda _{u}}{2\pi \rho }}={\frac {eB_{0}\lambda _{u}}{2\pi m_{e}c}}$ is known as the wiggler strength parameter. The factor of $1/\gamma^2$ is the important one: it tells you that the electrons radiate at the wavelength of the undulator as they perceive it ─ but that this is Lorentz contracted by a huge amount, from the one-meter range down into the subnanometer scales in your question. This is the main tuning knob in the machine: by changing the electrons' energy, the Lorentz contraction changes and so does the output radiation wavelength.

On a separate track, to fill in another apparent gap in your text: the photon energy is entirely independent of the radiated power. The former tells you the size of the energy chunks in the beam, while the later multiplies that by how many photons the beam has to give you the total energy it carries. You seem to be talking about both as if they were the same thing, which makes it impossible to know what you mean by "the value of the energy of the laser flashes".

So, on that line, to clarify:

• The photon energy is strictly proportional to the beam's frequency via the Planck relation $E_\mathrm{photon} = \hbar \omega = hc/\lambda$ (so for the upper end of the E-XFEL range, this is on the order of $20\:\mathrm{keV}$, give or take). Thus, this is fixed by the undulator geometry and the electron beam Lorentz factor, as explained above.

• The beam power, on the other hand, tells you how much energy is in each pulse (i.e. how many photons per pulse, times the photon energy), divided either by the pulse duration (~100 fs, giving you the mean pulse power) or by the pulse repetition rate (~27 kHz, giving you the average power).

If Larmor physics applied to FEL radiation, then it would apply to the mean pulse power, so it would be completely unrelated to the photon energy.

However, the physics of the Larmor formula do not apply, and it is essentially meaningless in this context. Instead, the beam power is governed by the quantum mechanics of the self-amplified spontaneous emission (SASE) feedback loop, coupled with interference effects between emissions in different parts of the undulator as well as absorption, the electron bunch length and density, the efficiency of microbunching, and so on. (In particular, the energy of each particular electron is pretty irrelevant for this, as the emission is a collective effect.) There simply isn't a clean formula to write it down, and you really need a detailed description of each FEL accelerator and undulator configuration to be able to speak about this.

Moreover, the beam power by itself (which is what Larmor would give you if it applied) really isn't that useful as a figure of merit of the light source; instead, you care about the pulse energy (how many photons per pulse) and the repetition rate, and your ability to focus it down to your needs ─ and this is why the Facts & Figures page quotes the brilliance rather than the beam power.

If you really need raw numbers, they're available at table 5.2.2 of the design report, which puts the (design; i.e. not yet achieved) average power between 65 and 500 W, giving somewhere on the order of $10^{12}$ to $10^{14}$ photons per pulse. However, as I said, unless you're looking to implement an experiment with that beam and you know exactly what you're looking for, the raw numbers on beam power are largely useless.