By Harold Hilton
Ufie of English readers these result of the mathematicdl conception of crystallography which aren't proved within the modem textbooks on that topic in theE nglish language. No declare to originality is made; the theorems proved are quite often to be present in the works of Schoenflies, Liebisch,
(Typographical error above are because of OCR software program and do not happen within the book.)
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Additional resources for Mathematical crystallography and the theory of groups of movements
In reality, the electrons Hamiltonian is: Hqb(Rl " ' " RN) = ( i=l ~ Ti + Vext(R) + Vint(R1 ' " ' RN)) qb(R1 " " " RN) = E ~ ( R I ' " " RN) (17) where dp(R 1 . . RN) and V i n t ( R i . . RN) are many-body functions, depending on the 1023 positions of the electrons, and Vext(R ) is the external potential set-up by the cores (see Fig. 12). The problem is to reduce (17) to (11), with appropriate approximations, though preserving information about the interactions within the cloud. We will examine in this subsection the methods employed to this task.
We have described in some detail these two properties. Measurements of the magnetic susceptibility and of the electronic specific heat give very clear information, from N(~tF) , on the presence of an f narrow band. They will be discussed in more detail in Chap. D. Spectroscopic methods, especially photoelectron spectroscopy, permit to obtain the same information directly. They will be discussed in Chap. E. The One-Electron Hamiltonian and the Local Density of State Approximation a. I n t r o d u c t i o n All the discussion of the preceding subsection has supposed the Hamiltonian (11) to be a good description of the energy of the electron population in the solid.
G. oxides) the problem becomes important: ~1 becomes very high, due to the charge transfer and stabilization of open structures, with consequent very little f-f-overlapping. This leads to a second remark, which concerns hybridization, of which we have also shortly spoken. In choosing the ~pi's in (23), we may "mix" to the 5 f wave functions the 6 d, 7 s, wave functions by taking "hybridized" orbitals wave functions as a starting point. Hybridization will be discussed in more detail later on. The point we want to make here is that, due to hybridization, the bandwidth W of the bands having f-character may be greatly increased.
Mathematical crystallography and the theory of groups of movements by Harold Hilton