Band theory dispersion

Figure6.6. (a) ARUPS spectra(F 150K,n 20eV)ofTTF-TCNQalongthe chain direction. The dashed line outlines the dispersion of the main spectral feature, (b) ARUPS spectra along the perpendicular a-direction. (c) Selected spectra from (a), after background subtraction. The asterisks mark the main dispersive peak, and the arrows mark emission not accounted for by band theory. Energies are referred to E-p, determined to 1 meV accuracy on an evaporated, polycrystalline silver film. Reprinted with permission from F. Zwick, D. Jdrome, G. Margaritondo, M. Onellion, J. Voit and M. Grioni, Physical Review Letters, 81, 2974 (1998). Copyright (1998) by the American Physical Society.

The reader shall be familiar with the concepts of band-theory (which are illustrated in excellent textbooks ) such as the band dispersion E(k) and the Bloch-state solutions of the equation of motion... 

If one sets the zero of energy at the top of the valence band, then, according to band theory, the dispersion relations (i.e., the - k relations) for the conduction band Ec k) and the valence band Ev k) are... 

Fig. 3.25 presents some dispersion curves by Brooks et al for NpN, PuN and AmN. The most pronounced changes in this series are observed for the Aj band of 5/ origin. This band becomes more and more narrow when going along this series of compounds. Finally the bands of M5/ and of higher M5/, 6d and N2p states become separated. However, the hybridisation of 5/ and N2p states remains considerable. For example, the state Tjs of NpN consists of 47% df states and 53% N2p states, while Fj of AmN consists of 46% 5/ states and 50% N2p states. The calculated and experimental values of the lattice constant versus the atomic number of the actinides are presented in Fig. 3.26. As can be seen, the experimental dependence exhibits a minimum for UN and is very different from the dependence for rare earth nitrides. The latter is monotonic and exhibits an anomaly for CeN, where Ce has an anomalous valency. While the dependence observed for rare earth nitrides can easily be explained by lanthanide compression, in the case of actinide nitrides the interpretation of such a dependence is far from trivial. The explanation proposed by Brooks et al (1984) is based on a simplified equation of state using canonical band theory. The equation takes into account only /-/ and f-p...

Another important concept in band theory is that of effective mass. In a semiconductor, most of the charges reside at the edge of the conduction or valence band. Band edges can be approximated to parabolic bands, by analogy with the free electron dispersion law, Eq. (3)... 

Recalling that a separation is achieved by moving the solute bands apart in the column and, at the same time, constraining their dispersion so that they are eluted discretely, it follows that the resolution of a pair of solutes is not successfully accomplished by merely selective retention. In addition, the column must be carefully designed to minimize solute band dispersion. Selective retention will be determined by the interactive nature of the two phases, but band dispersion is determined by the physical properties of the column and the manner in which it is constructed. It is, therefore, necessary to identify those properties that influence peak width and how they are related to other properties of the chromatographic system. This aspect of chromatography theory will be discussed in detail in Part 2 of this book. At this time, the theoretical development will be limited to obtaining a measure of the peak width, so that eventually the width can then be related both theoretically and experimentally to the pertinent column parameters. 

Rate Theory to describe the mechanism of band dispersion and which will be the major topic of Part 2 of this book. 

Unfortunately, any equation that does provide a good fit to a series of experimentally determined data sets, and meets the requirement that all constants were positive and real, would still not uniquely identify the correct expression for peak dispersion. After a satisfactory fit of the experimental data to a particular equation is obtained, the constants, (A), (B), (C) etc. must then be replaced by the explicit expressions derived from the respective theory. These expressions will contain constants that define certain physical properties of the solute, solvent and stationary phase. Consequently, if the pertinent physical properties of solute, solvent and stationary phase are varied in a systematic manner to change the magnitude of the constants (A), (B), (C) etc., the changes as predicted by the equation under examination must then be compared with those obtained experimentally. The equation that satisfies both requirements can then be considered to be the true equation that describes band dispersion in a packed column. 

At this point, it is important to stress the difference between separation and resolution. Although a pair of solutes may be separated they will only be resolved if the peaks are kept sufficiently narrow so that, having been moved apart (that is, separated), they are eluted discretely. Practically, this means that firstly there must be sufficient stationary phase in the column to move the peaks apart, and secondly, the column must be constructed so that the individual bands do not spread (disperse) to a greater extent than the phase system has separated them. It follows that the factors that determine peak dispersion must be identified and this requires an introduction to the Rate Theory. The Rate Theory will not be considered in detail as this subject has been treated extensively elsewhere (1), but the basic processes of band dispersion will be examined in order to understand... 

My interest at that time revolved around evaluating optical rotary dispersion data [12]. The paired values of optical rotation vs. wavelength were used to fit a function called the Drude equation (later modified to the Moffitt equation for William Moffitt [Harvard University] who developed the theory) [13]. The coefficients of the evaluated equation were shown to be related to a significant ultraviolet absorption band of a protein and to the amount of alpha-helix conformation existing in the solution of it. 

Chapter 3 is devoted to dipole dispersion laws for collective excitations on various planar lattices. For several orientationally inequivalent molecules in the unit cell of a two-dimensional lattice, a corresponding number of colective excitation bands arise and hence Davydov-split spectral lines are observed. Constructing the theory for these phenomena, we exemplify it by simple chain-like orientational structures on planar lattices and by the system CO2/NaCl(100). The latter is characterized by Davydov-split asymmetric stretching vibrations and two bending modes. An analytical theoretical analysis of vibrational frequencies and integrated absorptions for six spectral lines observed in the spectrum of this system provides an excellent agreement between calculated and measured data. 

The theory of band structures belongs to the world of solid state physicists, who like to think in terms of collective properties, band dispersions, Brillouin zones and reciprocal space [9,10]. This is not the favorite language of a chemist, who prefers to think in terms of molecular orbitals and bonds. Hoffmann gives an excellent and highly instructive comparison of the physical and chemical pictures of bonding [6], In this appendix we try to use as much as possible the chemical language of molecular orbitals. Before talking about metals we recall a few concepts from molecular orbital theory. 

In previous chapters, liquid chromatography column theory has been developed to explain solute retention, band dispersion, column properties and optimum column design for columns that are to be used for purely analytical purposes. The theories considered so far, have assumed that solute concentrations approach (for all practical purposes) infinite dilution, and, as a consequence, all isotherms are linear. In the specific design of the optimum preparative column for a particular preparative separation, initially, the same assumptions will be made. 

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