Atom, vector model

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 ... 

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 (/). 

The second and third terms are the interaction terms that couple the atom, here modeled as a two-state system with Pauli matrices, to the electromagnetic field. We consider the momentum to be the operator p - - V and consider this operator as not only operating on the vector potential but on the wavefunction. Hence we find that... 

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)... 

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...

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 =... 

Of course, a vector model described above has strong limitations. It can be applied only in the case of large angular momentum quantum number nlues. To have a precise (luantum mechanical description of light interaction with atoms and molecules, one should use a quantum mechanical description. Usage of monochro-... 

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. 

Molecules or atoms are described as a system of interacting material points, whose motion is described d5mamically with a vector of instantaneous positions and velocities. The atomic interaction has a strong dependence on the spatial orientation and distances between separate atoms. This model is often referred to as the soft sphere model, where the softness is analogous to the electron clouds of atoms. 

The Rydberg atom experiments described above are well adapted to the study of the atomic observables via the very sensitive field ionization method. The observation of the field itself and its fluctuations would also be very interesting. (In the Bloch vector model, the field variables are associated to the pendulum velocity whereas the atomic ones are related to its position). It has recently been shown either by full quantum mechanical calculations or by the Bloch vector semi-classical approach that if the system is initially triggered by a small external field impinging on the cavity, the fluctuations on one phase of the field become at some time smaller than in the vacuum field. This is a case of radiation "squeezing" which would be very interesting to study on Rydberg atom maser systems. 

We now describe the system of atomic moments or spins 5j in more detail on a specific model, the n vector model. We assume that the magnetic atoms are located on a periodic lattice. Each magnetic atom (i) carries a spin S( this is a vector, with n components 5(1, Sa... 5( . In our considerations, we ignore all quantum effects the components Sia are just numbers. There is one constraint—i.e., the total len 5 of each spin is fixed. We choose the following normalization ... 

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