Vector model of the atom

In this volume, principal consideration is given to the lighter elements, so that the Russell-Saunders (549) vector model of the atom is used. In this model a multielectron atom is assumed to have the quantum numbers n, L = lif Ml, 8 = siy (or n, L, J = L + S, Mj). This implies stronger and Si-Sj coupling than U-Si coupling. It follows from Pauli s principle that for a closed shell =... 

For a detailed discussion of spectroscopic nomenclature and the vector model of the atom see Pauling and Goudsmit The Structure of Line Spectra. The triplet levels of helium were long called doublets, complete resolution being difficult. Their triplet character was first suggested by J. C. Slater, Proc. Nat. Acad. Set. 11, 732 (1925), and was soon verified experimentally by W. V. Houston, Phys. Rev. 29, 749 (1927). The names parhelium and orthohelium were ascribed to the singlet and triplet levels, respectively, before their nature was understood. 

Section 4.4 describes how the vector model of the atom can predict overall symmetries of many-electron atoms. A similar approach can be used to determine overall symmetries of molecular electronic states. 

The Hypothesis of Electron Spin, 124. Electronic States of Complex Atoms, 128. The Pauli Exclusion Principle, 129. The Calculation of Energy Levels, 132. Angular Momenta, 133. Multiplet Structure, 135. Calculation of the Energy Matrix, 143. Fine Structure, 151. The Vector Model of the Atom, 155. Selection Rules for Complex Atoms, 159. The Radial Portion of the Atomic Orbitals, 162. The Hartree Method, 163. The Periodic System of the Elements, 167. 

A brief review is given here of the spectroscopic vector model of an atom or ion. In crystal-field theory, the wave function of the isolated ion is taken as the unperturbed state, and the perturbing effect of the electric and magnetic fields is computed. Thus crystal field theory uses the language, nomenclature, and methods employed in the theory of atomic spectra. A complete discussion of these methods can be found in books by Condon and Shortley (5) and by Griffith (/). 

Figure 9.1 Vector model of the electron spin. Using atomic units, the spin is represented by a vector s of length y/3/2 which precesses around the z-axis. By looking at the respective projections of the precessing spin vector, the model provides two important properties the projection onto the z-axis leads to a sharp value, ms = 1/2 in the case shown (ms = —1/2 for a precession around the negative z-axis), but no sharp values exist for the projections in the xy-plane, i.e., for the projections onto the x- or y-axis one finds with equal probability...

Interpretation of interatomic vectors. Use of known atomic positions for an initial trial structure (a preliminary postulated model of the atomic structure) can be made, by application of Equations 6.21,4 and 6.21.5 (Chapter 6), to give calculated phase angles. Methods for obtaining such a trial structure include Patterson and heavy-atom methods. Such methods are particularly useful for determining the crystal structures of compounds that contain heavy atoms (e.g., metal complexes) or that have considerable symmetry (e.g., large aromatic molecules in which the molecular formula includes a series of fused hexagons). The Patterson map also contains information on the orientation of molecules, and this may also aid in the derivation of a trial structure. 

Many-Electron Spatial Wavefunctions 157 TOOLS OF THE TRADE Photoeiectron Spectroscopy 163 Approximate Solution to the Schrodinger Equation 164 BIOSKETCH Syivia Ceyer 179 Spin Wavefunctions and Symmetrization 179 Vector Model of the Many-Electron Atom 186 Periodicity of the Elements 190 Atomic Structure The Key to Chemistry 191... 

In the vector model o the atom the total angular momentum J is inclined at a constant angle to the direction of the field B, determined by the magnetic quantum number Mj, and precesses about B at the Larmor frequency... 

Even better agreement is observed between calorimetric and elastic Debye temperatures. The Debye temperature is based on a continuum model for long wavelengths, and hence the discrete nature of the atoms is neglected. The wave velocity is constant and the Debye temperature can be expressed through the average speed of sound in longitudinal and transverse directions (parallel and normal to the wave vector). Calorimetric and elastic Debye temperatures are compared in Table 8.3 for some selected elements and compounds. 

Our model of positive atomic cores arranged in a periodic array with valence electrons is shown schematically in Fig. 14.1. The objective is to solve the Schrodinger equation to obtain the electronic wave function ( ) and the electronic energy band structure En( k ) where n labels the energy band and k the crystal wave vector which labels the electronic state. To explore the bonding properties discussed above, a calculation of the electronic charge density... 

S is the scattering vector, Mj is the atomic displacement parameter in this simplified notation assumed to be isotropic, 6 is the scattering angle, and 1 the wavelength of the incident radiation. The atomic displacement depends on the temperature, and hence so does the Debye-Waller factor. If an atom is modeled by a classical oscillator, then the atomic displacement would change linearly with temperature ... 

The Stokes parameters for the polarization of an electron beam can be represented in a Cartesian basis which also provides a convenient pictorial view for the polarization state of an electron beam. Since the polarization of an ensemble of electrons requires the determination of spin projections along preselected directions, the classical vector model of a precessing spin will first be discussed. Here the spin is represented by a vector s of length 3/2 (in atomic units) which processes around a preselected direction, yielding as expectation values the projections (in atomic units, see Fig. 9.1)... 

Let s - 2, Ns be a set of isotopomers of a parent molecule 5=1, and let a = 1, Na, enumerate the atoms in the molecule. (Eventually, only substituted atoms will be relevant.) In the present notation, the sites of the atoms are referred to the PAS of the parent 5=1, and are hence defined by the position vectors (in the rigid mass point model). Let the mass change upon substitution of atom a be Ama(s) for isotopomer 5, we then have ... 

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