Bethe notation

Another common notation for the irreducible representations is the so-called Bethe notation, in which the representations are denoted by E, symbols (i = 1,2,...), where the subscript i denotes the dimension. It is not simple to establish an equivalence between these two types of notation, since it depends on the symmetry group. For the moment, we will just mention both notations so that readers will be famihar with any character table. 

The Zeeman effect in the cubic 17 representation (f8 in the Bethe notation) is quite exceptional in that the symmetrized square of U contains the 7 symmetry of the magnetic field twice. 

Pi irreducible representation of the double group (Bethe notations)... 

Relative to the 5f - ground term. The values of J are indicated. Relative to the 5f J-9/2 ( l) ground level. Bethe notation is used for these Kramer s doublets. 

Fig. 3.5/ and 6d manifolds in an octahedral field and their correlation with free ion one-electron levels. Note that the crystal field lowers the energy of the 6d levels (relative to 5J) prior to splitting. Bethe notation is used for the Kramers s doublets and Mulliken notation for the spin-free Hamiltonian states, as usual. 

Rhenium (+4) has a 5d electronic configuration. When the ion is placed in an octahedral field the following terms in order of increasing energy result within the t g configuration A2g, Eg, T,g and T2g. These terms are further split by spin-orbit interaction into states denoted in the Bethe notation as T (i = 6,7,8)... 

We will now summarize the conclusion of the Kirkwood-Bethe theory. Fig 15 shows the computed peak pressure and computed reduced tunc constant for TNT plotted VS the inverse reduced distance. The dotted lines are a power function fit thru the computed peak pressures. The x s are drawn in by the writer to compare computed and measured reduced time constants (taken from Fig 7.9, p 240 of Ref 1). Comparison of other computed and measured shock parameters on the basis of the power functions shown below (in Cole s notation and in English units) is made in Table 11 (from p 242 of Ref 1)... 

The notation for the various kinds of d wave functions is that of Bethe,7 viz,... 

The IRs are labeled using both Bethe and Mulliken notation. 

Other groups may be handled in a similar manner to O in Example 8.1-1. For improper rotations, the two rules formulated previously hold also for double groups (Box 8.1). If the group contains the inversion operator, even or odd parity is indicated by a superscript of + or —in Bethe s notation and by a subscript g or u in Mulliken-Herzberg notation. 

The crystal-field multiplets are classified according to the irreducible representations of the respective double-group where both, the Bethe and the Mulliken (in parentheses) notations are written. DsH means the spin-Hamiltonian D-value accounting for all excitations AmIi - the lowest energy levels difference using the model-Hamiltonian in the first iteration Affl - the... 

Though the Mulliken-Griffith nomenclature is used here, the sub-scripts 3 and 5 on various quantities are reminescences of Bethe s original notation y3 and y5 for the two sub-shells. 

Oi contains the identity operation k2 contains three 180° rotations about x, y, z axes, respectively k3 contains six 90° rotations (+ and —) about the x, yt z axes k4 contains six 180° rotations about the six (110) axes k6 contains eight 120° rotations (4- and —) about the four (111) axes. The d wave functions are even and therefore operations involving inversion provide a redundant set. The degeneracy within a representation is given by ci. The Bethe (66) and Mulliken (457a) notations are compared.]... 

To generate an irreducible G subspace, for particular cases, f needs to be chosen with care. In the case of the kubic harmonics, first defined by Bethe in 1929 suitable functions are the mononomials x y"zP, which we identify in Elert s notation as (mnp). The kubic harmonics up to level 4 and their maps onto the irreducible representations of the cubic groups are listed in Table 3.9. 

The different notations for the IRs of Oh and Td are given in Appendix B. For the acceptors, the Bethe-Koster notation is used and for donors, the Mulliken s one. 

Actually, different B values were found from interconfigurational versus intra-configurational transitions. To account for this, the notations B y B33, emd Bgi have been introduced (19), the subscripts being related to Bethe s notation of d orbitals in cubic symmetry, ysfe) and ysffs). The transitions Bg(t%g) of a [Pg.177]

The a — / — 7 notation refers to the seminal big-bang paper under the names Alpher, Bethe, and Gamow (1948), published on 1 April. 

Group-theoretical labels for the eigenstates may be written in several forms, the most common usage being A or B for singlets, E for doublets, and T for triplets (Hamermesh, 1962) in which subscripts, primes, etc. are used in addition to the letters to distinguish- different irreducible representations. Another notation due to Bethe (1929) is Fj, where j runs over the various representations of the group (Koster et al., 1963). 

Using Bethe s T-notation for the underlying double group Cf, that is isomorphous to D. ... 

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