Pauli magnetism

Pauli magnetism is induced in a free electron gas by the application of an external magnetic field H. This magnetic field, acting as a perturbation, creates different energy states for electrons with spin aligned to the field, the concentration of which is n+, and electrons with spin opposite to the field, the concentration of which is n. Also the density of state function will be different for the two populations (see Fig. 13) (N+(E) and N (E)). 

Our model is thus of a metal, with a small number of carriers in the conduction band and 3d- or 4f-moments antiferromagnetically coupled to each other by direct exchange. The carriers may be either inserted by doping or by overlap from the lower Hubbard band (cf. Chapter 4, Section 3). The most striking prediction of the model, however, is that the degenerate electron gas should have a much enhanced Pauli magnetism. Suppose that EF is the Fermi... 

To derive the Pauli magnetic susceptibility we start with the total energy Utat in the following way ... 

In normal metals the Pauli susceptibility is constant with temperature, but in normal metals we are dealing with bandwidths in the order of eV and the density of states is practically constant near the Fermi level. In heavy fermions the density of states changes very rapidly with energy which produces drastic effects in the temperature dependence of the susceptibility. The formulae which we adopt for the derivation of the Pauli susceptibility are eqs. (17-19) where the int ral in eq. (18) is the well known expression of the Pauli magnetization. When a peak in the density of states occurs at the Fermi level, this term shows a Curie-Weiss dependence with temperature and the susceptibility decreases more rapidly as the peak is enhanced and the bandwidth is reduced. We can also explain the susceptibility of heavy fermions in different, simpler terms. With Eji = and the carrier concentration N being... 

Figure5.4 Density of states versus energy for the two spin components. In the nonmagnetic states (a), the occupancy by spin-up and spin-down electrons is the same. In Pauli magnetism at absolute zero when a magnetic field is applied, the spin-down and spin-up electrons are moved away from each other, which leads to unparity in that the number of electrons with one type of...

Quite separate and distinct from this kind of science was the large body of research, both experimental and theoretical, which can be denoted by the term technical magnetism. Indeed, I think it is fair to say that no other major branch of materials science evinces so deep a split between its fundamental and technical branches. Perhaps it would be more accurate to say that the quantum- and statistical-mechanical aspects have become so ethereal that they are of no real concern even to sophisticated materials scientists, while most fundamental physicists (Neel is an exception) have little interest in the many technical issues their response is like Pauli s. 

Although it is required to refine the above condition I in actuality, this rather simple but impressive prediction seems to have much stimulated the experiments on the electrical-conductivity measurement and the related solid-state properties in spite of technological difficulties in purification of the CNT sample and in direct measurement of its electrical conductivity (see Chap. 10). For instance, for MWCNT, a direct conductivity measurement has proved the existence of metallic sample [7]. The electron spin resonance (ESR) (see Chap. 8) [8] and the C nuclear magnetic resonance (NMR) [9] measurements have also proved that MWCNT can show metallic property based on the Pauli susceptibility and Korringa-like relation, respectively. On the other hand, existence of semiconductive MWCNT sample has also been shown by the ESR measurement [ 10], For SWCNT, a combination of direct electrical conductivity and the ESR measurements has confirmed the metallic property of the sample employed therein [11]. More recently, bandgap values of several SWCNT... 

Pauli spin susceptibility for the aligned CNTs has been measured and it is reported that the aligned CNTs are also metallic or semimetallic [30]. The temperature dependence of gn and gx s plotted in Fig. 5(a). Both values increase with decreasing temperature down to 40 K. A similar increase is observed for graphite. The g-value dependence on the angle 0 at 300 K is shown in Fig. 5(b) (inset). The g-value varies between gn = 2.0137 and gx= 2.0103 while the direction of magnetic fields changes from parallel to perpendicular to the tubes. These observed data fit well as... 

Hund s rule, like the Pauli exclusion principle, is based on experiment It is possible to determine the number of unpaired electrons in an atom. With solids, this is done by studying their behavior in a magnetic field. If there are unpaired electrons present the solid will be attracted into the field. Such a substance is said to be paramagnetic. If the atoms in the solid contain only paired electrons, it is slightly repelled by the field. Substances of this type are called diamagnetic. With gaseous atoms, the atomic spectrum can also be used to establish the presence and number of unpaired electrons. 

Pauli justified the identification of four quantum numbers with each electron with the following apparently clever argument. He supposed that if a strong magnetic field is applied, the electrons are decoupled and so do not interact, and can be said to be in individual stationary states. Of course, the periodic table arrangement must also apply in the absence of a magnetic field. 

The spins of two electrons are said to be paired if one is T and the other 1 (Fig. 1.43). Paired spins are denoted Tl, and electrons with paired spins have spin magnetic quantum numbers of opposite sign. Because an atomic orbital is designated by three quantum numbers (n, /, and mt) and the two spin states are specified by a fourth quantum number, ms, another way of expressing the Pauli exclusion principle for atoms is... 

Associated with the spin of an electron is a magnetic moment, which can be expressed by a quantum number of + or —5. According to the Pauli principle, any two electrons occupying the same orbital must have opposite spins, so the total magnetic moment is zero for any species in which all the electrons are paired. In... 

The magnetic forces between electrons are negligibly small compared to the electrostatic forces, and they are of no importance in determining the distribution of the electrons in a molecule and therefore in the formation of chemical bonds. The only forces that are important in determining the distribution of electrons in atoms and molecules, and therefore in determining their properties, are the electrostatic forces between electrons and nuclei. Nevertheless electron spin plays a very important role in chemical bonding through the Pauli principle, which we discuss next. It provides the fundamental reason why electrons in molecules appear to be found in pairs as Lewis realized but could not explain. 

In 1925, Wolfgang Pauli gave chemists what they wanted from the physicists a physical principle underlying electron-pair valency. Pauli built on the fact that in addition to the continuous, line, and band spectra, there is a fine structure of doublets, triplets, and multiple lines, some of which are split in a magnetic field (Zeeman effect). 

Pauli proposed the use of a fourth quantum number, which could have two values, thereby explaining why it is that electrons with identical energies behave differently in a strong magnetic field. If it is assumed that no two electrons in an atom may occupy the same atomic state, meaning that no two electrons can have the same four quantum numbers, then there might be two, but no more than two, 5 electrons for each principal quantum number. Six different... 

In late fall 1925, the Dutch physicists G. Uhlenbeck and Samuel Goudsmit gave a physical interpretation to Pauli s postulate of a fourth quantum number. The electron, they proposed, may spin in one of two directions. In a given atom, a pair of electrons having three identical quantum-number values must have their spin axes oriented in opposite directions, and if paired oppositely in a single orbital, they neutralize each other magnetically. 22... 

The Pauli Exclusion Principle. No two electrons in an atom may have the same four quantum numbers n, t, m, ms where m and ms are respectively the magnetic and spin quantum numbers. 

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