How do we measure the crystal momenta and the corresponding energies?

Determination of crystal momentum and the corresponding energy seems to be essential for drawing the band structure. What is the principle that is used to measure the crystal momentum and the corresponding energy? Do these experimental values of the crystal momentum lie anywhere between $$-\infty$$ and $$\infty$$, and then folded back into the first Brillouin zone (using the fact that it can be defined modulo a reciprocal lattice vector)?

Please support your answer with mathematical equations, and keep the discussion as simple as possible by adhering to one dimension only, if possible.

Part of the point of crystal momentum is that is hard to measure all the different forces acting on the electron in the crystal. It would be nice to simplify the effects of the crystal into something like a billiard ball model where F=ma or its equivalents and treat the electron as a particle in the crystal. So thinking of the electron wave-packet having a crystal momentum $$\hbar k$$ and the electrons having an 'effective mass' helps do that. Essentially we are wrapping all the dirty complicated stuff into the momentum and saying we have an effective particle like a plane waves with a crystal momentum, and are just treating it as a particle. Then we can think of things as an electron gas, or particles as having properties like mobility, and measure the response to applied electric or magnetic field in ways that can be consistent with our macroscopic measurements. Of course the system is quantum mechanical and electrons follow Femi-Dirac statistics and that has to be considered.

When we consider the crystal that the electron as a wave can travel through, the periodic nature of the potential implies that the solution needs to be periodic, (Bloch's theorem). If we send the electron waves in different directions we have different solutions. However in practice, the momentum space we can explore in experiments, and the number of allowed states that are filled in a band are somewhat limited. This is convenient since it often allows us to assume all the action is happening near the minimum of a conduction band or near the maximum of a valence band. But from a theoretical point of view we can solve the crystal for a wide range of energies and for a wide range of momentum, even if it is hard for an electron to fill a state in experiment.

The point about calculation of the band-structure is somewhat important. It is a known problem that depending on the technique that while the dispersion E(k) can be calculated, where along the energy axis the curves should be placed can be a problem. So experimental data is used to help lock down the band structure so it corresponds to reality. Similarly, some things like the ordering of the valence bands and their properties can be hard to figure out, so experimental measurements including those as a function of temperature are useful for seeing how good the band structure model is. The free electron band structure model, being different than a tight binding model, being different than something that includes spin orbit coupling, or more sophisticated density functional models etc.

For example: Photoluminescence can give the energy of band edge one way, but a transmission measurement may give the absorption edge corresponding to the band gap a different way. Hall mobility as a function of temperature, device mobility, or conductivity measurements (thermal or electric) all give information that could inform the nature of the band structure. Again usually around local minimums. Cyclotron resonance for example can map out the fermi-levels and map out the fermi-surfaces as a function of frequency helping to understand the band structure of metals. Similarly different techniques to find where defect states are located are mostly energy dependent measurements rather than measuring momentum in the crystal.

One exception is angle resolved photo-emission spectroscopy. There using a tunable light source changing the energy, and angle you can have a probe of the surface up to a certain depth and fill different states. Then measuring energy of the emitted electron and the direction to get its momentum (in the vacuum where is free of the influences of the surrounding crystal) where you can map out both energy and momentum as a function of angle. If this is plotted with E on the vertical axis and k (momentum) on the horizontal axis this corresponds to the band structure.

Currently there is a lot of interest in topological materials and APERS is pretty useful experimentally seeing how the calculated band structure matches real material properties.

You can also have other band structure like photonic materials, plasmonic or metamaterials and probe them as function of energy, by changing the frequency of the light or electromagentic wave, and by changing the angle change the momentum. Doing this you can also try to map out the dispersion curve of E(k) corresponding to the 'band structure'.