The Application Of Brillouin Zone In High Pressure Physics

The research on the structure and phase transitions of matters under pressure is an active branch of physics and has been achieved much in the last decade. Thanks for the advancement of high pressure experimental techniques, some sophisticated experimental methods, such as powder diffraction with the third generation of synchrotron radiation setup and the state-of-the-art single crystal diffraction technique, it allows scientists to have further insight into the world under high pressure.Alkali metals and alkali earth metals are considered as typical metals in ambient pressure, which have rich high pressure phase transitions. Unlike closed-packed structures with higher symmetries and higher coordination numbers, many high pressure phase structures exhibit lower symmetries and lower coordination numbers. Moreover, these changes are not isolated. There may be some interrelationships amongst these structural phase transitions.In this thesis, the stabilities of potassium's high pressure symmetry-breaking phases, such as oP8 structure and oC16 structure, are analyzed and also the stabilities of calcium's simple cubic (sc) structure and tI4 structure. We use the Brillouin-zone and Fermi-surface (BF-FS) interaction concept, Hume-Rothery theory, and the first-principle method based on density functional theory (DFT) as the keys. We use"REFLEX"package for simulating powder diffraction patterns and the functions of 3D graphing in MATLAB programming to apply Hume-Rothery theory. Additionally, the first-principle calculations are completed by"CASTEP"software package.Hume-Rothery theory was initially proposed as an empirical principle and mainly be applied in two-component alloys containing noble metals. Hume-Rothery theory builds a relation between the structures of alloys and the average number of valence electrons belonging to one atom (e/a). Further study shows that this relation can be explained with BF-FS interactions. On the one hand, e/a value is calculated by virtue of the mixture ratio of two components and then the Fermi sphere can be plotted with is radius kF calculated according to the free-electron model. On the other hand, in the powder diffraction pattern, we choose some strong peaks around 2kF. With all bisectors of vectors in reciprocal space indicated by selected peaks, a Brillouin zone can be constructed. According to band theory, the continuous energy levels of electrons will open a pseudogap when they come across the Brilloun zone, some electrons will aggregate to lower energy bands, then the total electronic energy will be lowered and this favors the stability of the structure. Additionally, the ratio of the volume of Fermi sphere and that of Brillouin zone (VFS/VBZ) can also be considered as a qusi-quantitative criterion, which describes the filling of electronic states in Brillouin zone. There are still many important issues in the research on the high pressure structures and phase transitions of alkali metals and alkali earth metals. How to understand the stabilities of some high pressure symmetry-breaking phases is a challenging problem. As a way of analyzing the stabilities of crystal structures, Hume-Rothery theory could be extended to study the stabilities of high pressure phases of alkali metals and alkali earth metals. Unlike two-component alloys, in this case, the factor that controls e/a value is not the mixture ratio of elements but the high pressure applied. The effect of high pressure is likely to compress the distances among atoms and the core electron orbital may be overlapped, and the core electrons will be involved in the physical processes of valence electrons and the valence of the element should be revised.We do the first-principle calculation with the"CASTEP"package to optimize the experimental structure with the geometry optimizing function to obtain a reasonable structure at a specified pressure. By calculating and analyzing the density of states (DOS) of electrons, we can substantiate the existence of BF-FS interactions within the framework of quantum mechanics.Recently the existence of oP8 structure as a high pressure phase of potassium was substantiated both experimentally and theoretically. This discovery contradicts a previous theoretical work, which suggested the only existence of tI4 structure of potassium but not oP8 structure. The oP8 structure is less symmetric than tI4 structure and its BZ-FS interactions may be much stronger. So Hume-Rothery theory can be applied to analyze their stabilities. According to the relation between structure and e/a, we choose AuGa andβ-Sn, which also have oP8 structure and tI4 structure respectively, as a reference and then obtain the values of e/a for oP8 structure and tI4 structure in potassium that are 2 and 4. In addition, by some calculations and selecting strong diffraction peaks appropriately, the 3D visualization of BZ-FS interactions can be plotted by virtue of MATLAB programming and VFS/VBZ could be obtained at the same time. After data analysis, we find oP8 structure in potassium is more stable than tI4 structure at the chosen pressure 58 GPa. This result is consistent with experiment. Besides the DOS of oP8 structure is also calculated and has a peak just below the Fermi energy. We do not see this in tI4 structure, which shows that electrons move in the direction of lower energy and the electronic energy is also lowered. Next we study the stability of oC16 structure. The transition from tI4 structure to oC16 structure is a typical high pressure symmetry-breaking phase transition, the application of Hume-Rothery to analyze their stabilities will make sense. With the relation between crystal structure and e/a, BF-FS interactions and VFS/VBZ for oC16 structure and tI4 structure can be obtained. However, our results cannot explain the stability of oC16 structure in potassium and are inconsistent with experiment, although, the DOS of oC16 structure indicates that BF-FS interactions exist indeed. We believe that the problem is due to the process of evaluating e/a value. When evaluating the e/a value of oC16 structure in potassium, we choose its isostructure BiSn and the e/a value is 4.5. Comparing with e/a=1 at ambient pressure for potassium, the change is hard to understand from the aspect of high pressure. So we cut into the problem by taking account of the Coulombic interactions between the core electrons and the nuclei. After making an amendment to the way of evaluating e/a value, 2.1218 and 1.7794 is obtained for oC16 structure and tI4 structure in potassium. With the amendment, they seem more reasonable than 4.5 and 4 without amendment. By plotting BZ-FS interactions and calculating VFS/VBZ, our results are in favor of the stability of oC16 structure in potassium at the chosen pressure 98 GPa and consistent with experiment. This amendment is not only applicable to oC16 structure in potassium but also to oP8 structure in both potassium and sodium. All our results appropriately explained the stabilities of these structures. As a conclusion, there are limitations for the extending process of Hume-Rothery theory to potassium. However, our amended method could still correctly analyze the stabilities of high pressure phases. So the physical model based on Hume-Rothery theory is applicable to alkali metal potassium. The question of the structure of calcium in the range 32-119 GPa is unresolved yet. Not completely consistent with the experimentally observed sc structure, one recent theoretical work suggested a symmetry-breaking phase tI4 structure in the range 34-78 GPa. By adopting amended Hume-Rothery theory, we chooseα-Po andβ-Sn, which also have sc structure and tI4 structure respectively, as a reference and then obtain the e/a values of sc structure and tI4 structure in potassium that are equal to 4.1224 and 2.7870. After some calculations, BZ-FS interactions and VFS/VBZ can also be obtained. Unfortunately, our results fail to explain stabilities of either structures and contradict to both the previous experimental and theoretical work. In conclusion, the physical model based on Hume-Rothery theory is not perfect. It is not able to appropriately describe the complex behaviors of electrons in calcium under high pressure. It shows that Hume-Rothery theory cannot be applied to multi-valence-electron metals with simple amendment.