A band structure calculation can be used to calculate any property of a crystal, so any measurement of a property of a solid can be used to check a band structure calculation. After calculating the band structure and the electron density of states, it is possible to calculate the internal energy density. This can be done for a range of lattice parameters. There should be a minimum in the internal energy density of the observed lattice parameters. An x-ray diffraction experiment can be used to check that the calculation has been done properly.

A relatively simple experiment to check the band structure specific heat. Here, the energy needed to increase the temperature is measured. Abrupt changes in the specific heat are caused by structural phase transitions. To calculate the specific heat, it is necessary to calculate the electron component and the phonon component. To find the temperature of a phase transition, it is necessary to calculate the temperature dependence of the electron component and the phonon component of the Helmholtz free energy of both crystal phases. The phase transition will take place at the temperature where the free energies cross.

Optical absorption is often used to check band structure calculations. A photon will be absorbed if it has the correct energy to move an electron from an occupied state to an empty state. Both the initial and final electron state must have the same $\vec{k}$ value. If there are many occupied states separated from empty states by the same energy, there will be a peak in the optical absorption at that energy. If all of the peaks in the measured optical absorption match the positions that the band structure predicts they will occur, this is strong evidence that the band structure has been calculated properly. If a crystal is transparent to visible light (like quartz), there is no optical absorption and visible photons don't have enough energy to lift an electron from an occupied state to an empty state. This means there must be a band gap with an energy larger than a blue photon. The border between visible and ultraviolet light is about 3.25 eV, so transparent materials must be electrical insulators. Crystals that show a strong photoconductivity when visible or infrared light shines on them are semiconductors. The band gap must be smaller than the photon energy used in the photoconductivity experiment.

An experiment that more directly measures the band structure is photoemission spectroscopy (PES). Photons with an energy $hf$ eject electrons from some material, and the energy of the ejected electrons is used to obtain information about that material. The measurement is based on the photoelectric effect which was observed by Heinrich Hertz in 1887 and explained by Albert Einstein in 1905. Ultraviolet photons are typically used to investigate the valence electrons of molecules or solids, while x-ray photons are typically used to determine the chemical composition of a sample. There are different variants of photoemission.

When ultraviolet photons are used for photoemission, the measurement is called Ultraviolet Photoemission Spectroscopy (UPS). The light source is sometimes the spectral lines of a gas discharge lamp and sometimes synchrotron sources. With a synchrotron, it is possible to continuously sweep the photon energy. Ultraviolet photons can bring electrons in occupied states 0 eV to 100 eV below the Fermi energy into the vacuum above the sample. It can be used to measure the part of the electron density of states below the Fermi energy.

In a Angle Resolved Photoemission Spectroscopy (ARPES) experiment, the component of the electron $\vec{k}$ vector parallel to the surface is unchanged when the electron is ejected from the surface. By measuring the ejected electrons as a function of the angle, it is possible to determine the energy and the $\vec{k}$ vector parallel to the surface. If the electronic state that is being measured is two-dimensional, such as graphene or a topological material, ARPES measures the $E(\vec{k})$ dispersion relation near the Fermi energy of the material.

In Inverse Photoemission Spectroscopy (IPES), electrons with a well-defined energy are shot into the unoccupied states of the material above the Fermi energy. These electrons can emit photons when they fall to a lower energy level. By detecting these photons, it is possible to infer the electron density of states above the Fermi energy.

When x-ray photons are used in a photoemission experiment, it is called X-ray Photoemission Spectroscopy (XPS) In XPS, the core electrons are ejected. The ejected core electrons have characteristic energies for every atom. For instance, the 1s electrons of a carbon atom are ejected with an energy of 284.2 eV. Tables of the energies of the emitted electrons can be found in references [1] and [2]. XPS is primarily used to determine which atoms are present on a surface. It is also possible to infer which bonds an atom has formed with other atoms. When a core electron is missing, it is possible that the atom will eject an additional Auger electron. These Auger electrons also have characteristic energies that will be observed by the detector. For those who are going to perform XPS experiments, reference [3] is a good place to start learning about the technique.

1. ThermoFischer X-ray photoelectron spectroscopy of atomic elements
2. X-ray Photoelectron Spectroscopy (XPS) Reference Pages
3. Fred A. Stevie, and Carrie L. Donley, Introduction to x-ray photoelectron spectroscopy, J. Vac. Sci. Technol. A 38, 063204 (2020);