Splitting Bethe

Bethe Splitting of terms in crystals. Annalen der Physik 3, 133 (1929). Cotton Chemical applications of group theory (John Wiley). 

Thus in crystals where unit cells contain a molecules, to any single nondegenerate excited state of a free molecule in the crystal corresponds not one, but a bands of excited states and correspondingly several absorbtion lines. Such a splitting was first discussed by Davydov ((9)—(11)) and is usually called the Davydov splitting,8 to distinguish it from the Bethe splitting (14). 

The degeneracy of a molecular term is usually related to a sufficiently high symmetry of molecules. In crystals, however, the symmetry of fields in the place where the molecule a is located depends on the intermolecular interactions and can be lower than the symmetry of isolated molecules. In this case the degeneracy can be removed and a splitting can appear, which is just the Bethe splitting. 

To demonstrate the mechanism of Bethe splitting in molecular crystals we assume that a unit cell contains only one molecule but the term / under consideration is ro-fold degenerate. In the case being considered, quite analogous to that of the equation system (2.19), we obtain a system of ro equations for coefficients u r The solution of the secular equation shows that in crystals with one molecule in the unit cell ro excitonic bands appear which correspond to an ro-fold degenerate molecular term. 

If an impurity molecule is characterized by a degenerate term, the Bethe splitting removes partially or totally the degeneracy and instead of one term a multiplicity of terms appears. In crystals, due to their translation symmetry single degenerate terms expand to several excitonic bands. 

In crystals having unit cells with a molecules with an ro-fold degenerate term oto excitonic bands appear. In this case the splitting results from the simultaneous influence of many factors leading both to Davydov and Bethe splittings. The analysis of the splitting in this case becomes more cumbersome, although it remains quite elementary. 

In this case the Davydov splitting is absent, and when molecular terms are degenerate, a splitting occurs which is analogous to the Bethe splitting (see Section 2.1). 

If the reducing difference procedure is applied on the out-of-plane IR bands (Figure 2.22), a total disappearance of the subcomponent absorption maxima at 878 cm- and 838 cm of Bj symmetry class (Figure 2.15) is observed. According to the Bethe theory, subsequently described as Bethe splitting (BS), the IR bands are split in the solid-state into submaxima, whose number is equal to the number of the molecules in the unit cell [128]. In the crystal of the 4 -cyanophenyl-4-n-pentylbenzoate [128], the molecule is flat and the two molecules in the unit cell are mutually disposed in a coplanar manner, thus leading to a colinear orientation of the... 

Crystal field theory has its origins in Hans Bethe s famous 1929 paper Splitting of terms in crystals. In that paper Bethe demonstrated what happens to the various states of an ion when it is placed in a crystalline environment of definite symmetry. Later, John Van Vleck showed that the results of that investigation would apply equally well to a transition-metal compound if it could be approximated as a metal ion surrounded by ligands which only interact electrostatically with the... 

The second part of Bethe s paper describes a method by which the magnitudes of the splittings of the free-ion states may be calculated, assuming... 

The spin-orbit interaction, which couples L and S to give a total angular momentum J, splits the multiplets into their components labeled 2,v +1X /, where J is the total angular momentum quantum number. The spin-orbit splitting is given by (Bethe (1964))... 

Due to the so-called /-mixing within the crystal field, multiplets with different / values are coupled. However, similar to the free-ion case, the levels are still designated by the principal 25+1L j component of the crystal-field wavefunction. For the further labeling of levels split by the crystal field, either the irreducible representation /j (Bethe, 1929) to which the particular wavefunction belongs or the crystal quantum number /i defined by Hellwege (1949) are most commonly used. 

Bethe, H. (1929) Splitting of terms in crystals. (Termsaufspaltung in Kristallen.) Ann. Phys., 3, 133-206. [Transl. Consultants Bureau, New York.]... 

The crystal field theory. The basics of the CFT were introduced in the classical work by Bethe [150] devoted to the description of splitting atomic terms in crystal environments of various symmetry. The splitting pattern itself is established by considering the reduction in the symmetry of atomic wave functions while the spatial symmetry of the system goes down from the spherical (in the case of a free atom) to that of a point group of the crystal environment. It is widely described in inorganic chemistry textbooks (seee.g. [152]). 

The proof runs as follows Relativistic splitting can only occur between degenerate sets of orbitals and is therefore most important in the high symmetries. In On, the tjg sub-shell is split into e"g(callcd y7 by Bethe) at + d and u g(Bethe y8) at — Cnd- Now, the ligands only contribute to first-order splitting when the re sub-... 

In order to determine into how many Stark terms a given energy level splits when put into a ligand field without making a detailed calculation of the values, the group-theoretical methods of Bethe (66) are convenient. In this method it is noted that the spherical harmonics transform according to the Ith irreducible repre-... 

In the case of two beams Bethe obtains the following result. The intensity of the secondary beam does not merely depend on the direction of the primary beam, i.e. on the inaccuracy of the excitation, but under certain circumstances the other feeble beams may add to it appreciably. The i/r-functions of the beams are multiplied by the higher lattice potentials, so that the latter have an effect on the intensity, and, conversely, may theoretically be determined by measurements of intensity. For very fast electrons, i.e. when the refractive index is very small or the spherical shell of propagation very thin, the number of the weak reflections falling within the shell is not large, and the effect of these additional terms on the intensity of the reflected beam vanishes as compared with the intensity directly split off from the primary beam. Theory shows that the existence... 

An array of charges in a crystal produces an electric field at any one ion, the so-called crystalline electric field. The presence of this field causes a Stark splitting of the free ion energy levels which results in a substantial modification of magnetic, electrical and thermal properties of the material. The theory of the crystal field and its interpretation in terms of group theory are originally due to Bethe (16). 

---