Canonical band theory

To summarize we have reproduced the intricate structural properties of the Fe-Co, Fe-Ni and the Fe-Cu alloys by means of LMTO-ASA-CPA theory. We conclude that the phase diagram of especially the Fe-Ni alloys is heavily influenced by short range order effects. The general trend of a bcc-fcc phase transition at lower Fe concentrations is in accordance with simple band Ailing effects from canonical band theory. Due to this the structural stability of the Fe-Co alloys may be understood from VGA and canonical band calculations, since the common band model is appropriate below the Fermi energy for this system. However, for the Fe-Ni and the Fe-Cu system this simple picture breaks down. 

In 1973 Andersen and Woolley [1.25] extended the LCMTO method to molecular calculations. At the end of their paper they introduced that choice of MTO tail, i.e. proportional to J = 9i/j/9E, which in a natural fashion ensured orthogonality to the core states and at the same time led to an accurate and elegant formulation of linear methods. The resulting, technique was immediately developed in a paper by Andersen [1.26] which, in a condensed form, contains most of what one need know about the simple concepts of linear band theory. Thus, we find here the KKR equation within the atomic-sphere approximation at this stage is called ASM the LCMTO secular matrix, latter called the LMTO matrix the energy-independent structure constants and the canonical bands and the Laurent expansion of the logarithmic-derivative function and the corresponding potential parameters. 

In the following sections we shal] discuss the structure- and potential-dependent parts of the energy-band problem separately and introduce the concepts of canonical band theory. 

Fig.2.10. Self-consistent, fully hybridised energy-band structure for nonmagnetic chromium obtained as for tungsten and platinum, Figs.1.4,5. Comparison with the unhybridised bands in the previous figure gives a feeling for the accuracy one may obtain within the very simple unhybridised canonical band theory. It also gives an idea of the effect and importance of hybridisation, defined essentially as the difference between the two figures...

Expression (7.43) is the pressure relation we shall use in the complete calculations of pressure-volume curves. It will prove useful, however, to use an expression which is less accurate but more directly related to simple potential parameters such as band centres and masses in order to understand the more complete calculations. In the following we shall derive such an expression from canonical band theory. 

We have now discussed the equation of state in chromium within the framework of canonical band theory. The usefulness of this picture is not, however, restricted to a description of the more elaborate calculations because once we have understood the physics of the problem, we may change the external circumstances and use the first-order relations (7.46-48) to estimate the corresponding changes in the equation of state. An example of such use of the theory is given in [7.8] where the effect of antiferromagnetism in the 3d monoxides is judged from the ferromagnetic results. 

J. Madsen, O.K. Andersen, U.K. Poulsen, 0. Jepsen "Canonical Band Theory of the Volume and Structure Dependence of the Iron Magnetic Moment", in Magnetism and Magnetic Materials 1975 (AIP, New York 1976) p.328... 

Zhukov et al (1989) calculated the band structure, lattice constants, bulk modulus, cohesive energy and the hydrostatic breakdown tension for the hypothetical CrC monocarbide and eompared them with the values for TIC, VC, TiN and VN. In the series TiC -> VC -> CrC (,ohl decreases, but B and P increase as in the series ZrC - NbC WC, see Section 2.1. A qualitative explanation of such a behaviour based on the canonical band theory of Andersen (1975) was given in the review by Zhukov et al (1989). It was shown that Cr carbide with NaCl-type structure probably contains a considerable number of carbon vacancies - see Chapter 4. Later... 

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...

Andersen, O.K., Jepsen, O. and Glotzel, D. (1985). Canonical description of the band structures of metals. In Highlights of Condensed-Matter Theory, Soc. Ital. Fis. Corso 89, 59-176. 

In Chap.2 I deal with the simplest aspects of the LMTO method based upon the KKR-ASA equations. The intention is to familiarise the reader with the concepts and language used in linear theory. This is where I introduce structure constants, potential functions, canonical bands, and potential parameters, and where it is shown that the energy-band problem may be separated into a potential-dependent part and a crystal-structure-dependent part. 

As will be discussed in chapter 6, of fundamental importance in the theory of unimolecular reactions is the concept of a microcanonical ensemble, for which every zero-order state within an energy interval AE is populated with an equal probability. Thus, it is relevant to know the time required for an initially prepared zero-order state j) to relax to a microcanonical ensemble. Because of low resolution and/or a large number of states coupled to i), an experimental absorption spectrum may have a Lorentzian-like band envelope. However, as discussed in the preceding sections, this does not necessarily mean that all zero-order states are coupled to r) within the time scale given by the line width. Thus, it is somewhat unfortunate that the observation of a Lorentzian band envelope is called the statistical limit. In general, one expects a hierarchy of couplings between the zero-order states and it may be exceedingly difficult to identify from an absorption spectrum the time required for IVR to form a micro-canonical ensemble. 

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