Studies are reviewed that employ the effect of an anomalously strong shift of the fundamental $K$-series x-ray lines associated with a change in the number of 4$f$ electrons in a given atom for investigating the phenomenon of variable valency in compounds of the rare-earth elements in general, and the so-called isomorphous phase transitions in these compounds in particular. The mechanism of the shift effect and its theory are described. The available data on the shifts of the $K$-lines of the different compounds of all the rare-earth elements (except promethium) are discussed. A summary table gives the valencies (including the anomalous ones), the occurrence of which has been proved by this method. Application of the $K$-shift method for studying isomorphous (or isostructural) transitions allows one to establish unambiguously that the mechanism of these transitions involves a change in the number of 4$f$ electrons, and to measure the magnitude of these changes. The review concludes with a brief description of the principles and design of the required experimental apparatus-crystal diffraction spectrometers.

@article{Sumbaev:1978,
	author = {O. I. Sumbaev},
	title = {Shift of K x-ray lines associated with valency change and with isomorphous phase transitions in rare earths},
	publisher = {Physics-Uspekhi},
	year = {1978},
	journal = {Phys. Usp.},
	volume = {21},
	number = {2},
	pages = {141-154},
	url = {https://ufn.ru/en/articles/1978/2/c/},
	doi = {10.1070/PU1978v021n02ABEH005519}
}