First-order crystal field functions

Strictly speaking, this is the correct form of the expression giving the first-order crystal-field effects. In practice, the matrix elements of Hcf between functions have... 

We first note that an isolated atom with an odd number of electrons will necessarily have a magnetic moment. In this book we discuss mainly moments on impurity centres (donors) in semiconductors, which carry one electron, and also the d-shells of transitional-metal ions in compounds, which often carry several In the latter case coupling by Hund s rule will line up all the spins parallel to one another, unless prevented from doing so by crystal-field splitting. Hund s-rule coupling arises because, if a pair of electrons in different orbital states have an antisymmetrical orbital wave function, this wave function vanishes where r12=0 and so the positive contribution to the energy from the term e2/r12 is less than for the symmetrical state. The antisymmetrical orbital state implies a symmetrical spin state, and thus parallel spins and a spin triplet. The two-electron orbital functions of electrons in states with one-electron wave functions a(x) and b(x) are, to first order,... 

Ehrenfest s concept of the discontinuities at the transition point was that the discontinuities were finite, similar to the discontinuities in the entropy and volume for first-order transitions. Only one second-order transition, that of superconductors in zero magnetic field, has been found which is of this type. The others, such as the transition between liquid helium-I and liquid helium-II, the Curie point, the order-disorder transition in some alloys, and transition in certain crystals due to rotational phenomena all have discontinuities that are large and may be infinite. Such discontinuities are particularly evident in the behavior of the heat capacity at constant pressure in the region of the transition temperature. The curve of the heat capacity as a function of the temperature has the general form of the Greek letter lambda and, hence, the points are called lambda points. Except for liquid helium, the effect of pressure on the transition temperature is very small. The behavior of systems at these second-order transitions is not completely known, and further thermodynamic treatment must be based on molecular and statistical concepts. These concepts are beyond the scope of this book, and no further discussion of second-order transitions is given. 

It must be kept in mind, that S represents just a first-order approximation of the distribution function, and this under the additional premise of complete cylindrical symmetry only. It might be an acceptable measure when comparing cases for which a mean-field model applies. However, comparing the order parameters of liquid crystals with those of other partially ordered phases, such as stretched polymers or tribological samples can be misleading due to possibly different types of distribution functions. 

Fig. 2 Dependence of the T2g term fine structure for the Cr ion in KMgp3. The curves are the splitting as the functions of the Ham reduction factor y calculated from the first and second order Ham theory. The open circles correspond to the energy of spinors in a static crystal field (y = 1.0), and the filled circles are observed experimental energies. The best fit is obtained for y = 0.31. AH four curves are merged into two (if y = 0 extremely strong Jahn-Teller interaction) with the separation of2 k + p)...

The zero-order product functions, IMq Lq) for the ground state and IM, Lq) for the excited d-d or f—f electronic state of the metal ion in the complex, require an augmentation with additional functions of the set for correction to first-order. The latter are either other metal ion functions, e.g. IM Lq), which is the course adopted in crystal field theory, or other ligand functions, e.g. Mq Lj) and IMg L ), as is assumed in the ligand polarization treatment. The two procedures are mutually exclusive to first-order, on account of the one-electron character of the transition moment operators, although the results of the two treatments are additive in a general independent-systems representation. 

Reis et al. report theoretical studies of the urea250 and benzene251 crystals. Their calculations start from MP2 ab initio data for the frequency-dependent molecular response functions and include crystal internal field effects via a rigorous local-field theory. The permanent dipolar fields of the interacting molecules are also taken into account using an SCF procedure. The experimental linear susceptibility of urea is accurately reproduced while differences between theory and experiment remain for /2). Hydrogen bonding effects, which prove to be small, have been estimated from a calculation of the response functions of a linear dimer of urea. Various optoelectronic response functions have been calculated. For benzene the experimental first order susceptibility is accurately reproduced and results for third order effects are predicted. Overall results and their comparison with studies of liquid benzene show that for compact nonpolar molecules environmental effects on the susceptibilities are small. 

The most precise measurements of the fine-structure parameters D and E have in fact been carried out using zero-field resonance. Figure 7.6 shows the three zero-field transitions in the Ti state of naphthalene molecules in a biphenyl crystal at T = 83 K. In these experiments, the absorption of the microwaves was detected as a function of their frequency [5]. The lines are inhomogeneously broadened and nevertheless only about 1 MHz wide. Owing to the small hnewidth of the zero-field resonances, the fine-structure constants can be determined with a high precision. This small inhomogeneous broadening is due to the hyperfine interaction with the nuclear spins of the protons (see e.g. [M2] and [M5]). For triplet states in zero field, the hyperfine structure vanishes to first order in perturbation theory, since the expectation value of the electronic spins vanishes in all three zero-field components (cf Sect. 7.2). The hyperfine structure of the zero-field resonances is therefore a second-order effect [5]. 

In a weak field, atomic considerations dictate which orbitals are filled, as was the case in the rare earths. Atomic states described by term symbols of total L and S describe the states for d orbital occupation these are spht by the d—d electrostatic repulsions. Spin orbital interaction is small in the transition metals and usually neglected in the first-order description of the levels. A weak crystal field shifts the levels and effects a spfitting which occurs because the crystal field removes the degeneracy of an L level the L level splits into its components. Atomic free ion wave functions having the S5unmetry of the crystal field are used to calculate the splittings. 

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