Three-dimensional dynamical symmetries

Oglesby DD, Archuleta RJ, Nielsen SB (1998) Earthquakes on dipping faults the effects of broken symmetry. Science 280 1055-1059 Oglesby DD, Archuleta RJ, Nielsen SB (2000) The three-dimensional dynamics of dipping faults. Bull Seismol Soc Am 90 616-628... 

The most general Hamiltonian with dynamic symmetry (II) again has a form similar to Eq. (2.43), with both linear and quadratic terms. This is a peculiar feature of one-dimensional problems. In order to simplify the discussion of three-dimensional problems, we prefer to consider a Hamiltonian with only quadratic terms... 

Dynamical symmetries for three-dimensional problems can be studied by the usual method of considering all the possible subalgebras of U(4). In the present case, since one wants states to have good angular momentum quantum numbers, one must always include the rotation algebra, 0(3), as a subalgebra. One can show then that there are only two possibilities, corresponding to the chains... 

The eigenvalue problem for the Hamiltonian H [Eq. (2.92)] can be solved in closed form whenever H does not contain all the elements but only a subset of them, the invariant or Casimir operators. For three-dimensional problems there are two such situations corresponding to the two chains discussed in the preceding sections. We begin with chain (I). Restricting oneself only to terms up to quadratic in the elements of the algebra, one can write the most general Hamiltonian with dynamic symmetry (I) as... 

Diffraction by an ideal mosaic crystal is best described by a kinematical theory of diffraction, whereas diffraction by an ideal crystal is dynamical and can be described by a much more complex theory of dynamical diffraction. The latter is used in electron diffraction, where kinematical theory does not apply. X-ray diffraction by an ideal mosaic crystal is kinematical, and therefore, this relatively simple theory is used in this book. The word "mosaie" describes a crystal that consists of many small, ideally ordered blocks, which are slightly misaligned with respect to one another. "Ideal mosaic" means that all blocks have the same size and degree of misalignment with respect to other mosaic blocks. Most of this chapter is dedicated to conventional crystallographic symmetry, where three-dimensional periodicity is implicitly assumed. 

There are many two- and three-dimensional versions of the periodic chart of the elements (Van Spronsen 1969 Mazurs 1974). There are short charts, long charts, and charts based on the symmetry considerations of group dynamics (Barut 1972 Rumer and Fet 1972). Given that the chart of the elements is to be a template for the molecular periodic system, it follows that the choice of the former will greatly influence the appearance of the latter. The third assumption made by a designer of physical periodic systems, then, has to be that one certain two-dimensional chart is the best template for his or her molecular system. 

In the quest for a universal feature in the short-to-intermediate time orientational dynamics of thermotropic liquid crystals across the I-N transition, Chakrabarti et al. [115] investigated a model discotic system as well as a lattice system. As a representative discotic system, a system of oblate ellipsoids of revolution was chosen. These ellipsoids interact with each other via a modified form of the GB pair potential, GBDII, which was suggested for disc-like molecules by Bates and Luckhurst [116]. The parameterization, which was employed for the model discotic system, was k = 0.345, Kf = 0.2, /jl= 1, and v = 2. For the lattice system, the well-known Lebwohl-Lasher (LL) model was chosen [117]. In this model, the particles are assumed to have uniaxial symmetry and represented by three-dimensional spins, located at the sites of a simple cubic lattice, interacting through a pair potential of the form... 

The situation at surfaces is more complicated, and richer in information. The altered chemical environment at the surface modifies the dynamics to give rise to new vibrational modes which have amplitudes that decay rapidly into the bulk and so are localized at the surface [33]. Hence, the displacements of the atoms at the surface are due both to surface phonons and to bulk phonons projected onto the surface. Since the crystalline symmetry at the surface is reduced from three dimensions to the two dimensions in the plane parallel to the surface, the wavevector characterizing the states becomes the two-dimensional vector Q = qy). (We follow the conventional notation using uppercase letters for surface projections of three-dimensional vectors and take the positive sense for the z-direction as outward normal to the surface.) Thus, for a given Q there is a whole band of bulk vibrational frequencies which appear at the surface, corresponding to all the bulk phonons with different values of (which effectively form a continuum) along with the isolated frequencies from the surface localized modes. 

The first step is to understand how a dynamical symmetry approach can lead to practical results regarding uncoupled anharmonic oscillators. This will be accomplished in two distinct subsections, addressing the one-and three-dimensional problems, respectively. (Two-dimensional questions are presently under study and are not considered in this paper. We, do, however, provide some iiiformation for those two-dimensional situations of direct interest to our immediate goals whenever the opportunity will arise.)... 

To reiterate, we prefer to describe the one-dimensional model first because of its mathematical simplicity in comparison to the three-dimensional model. From a strictly historical point of view, the situation is slightly more involved. The vibron model was officially introduced in 1981 by lachello [26]. In his work one can find the fundamental idea of the dynamical symmetry, based on U(4), for realizing an algebraic version of the three-dimensional Hamiltonian operator of a single diatomic molecule. After this work, many other realizations followed (see specific... 

For a given irreducible representation of U(4) we are left with three quantum numbers to label the physical states. This is the natural outcome of the quantum mechanical treatment of a three-dimensional system, in which, besides j and one has to deal with a radial quantum number [tip in chain (a) and a> in chain (b)]. The advantage of the dynamical symmetry approach is found in the fourth quantum number N. Such an extra quantum number has the important role of allowing access to entire families of distinct physical situations. In the specific three-dimensional case, we will see how the number N spans situations characterized by different anharmonicities or, equivalently, by a different number of bound states. 

Eq. (2.112) represents an anharmonic expansion in the quantum number V. We insist, of course, on the essential difference between the one- and three-dimensional case, as in the latter case, we also obtain the rotational contribution to the energy spectrum. As a matter of fact, the final outcome of the dynamical symmetry 0(4) is the (well-approximated) rovibrational spectrum of a three-dimensional Morse oscillator. Its physical parameters can easily be related to the algebraic quantities A, N, Eg, and B by means of the following relations ... 

To begin with, we recall that in certain cases, the algebraic model has been already put in a one-to-one correspondence with a specific potential function for the usual space coordinates. We have already studied dynamic symmetries providing exact solutions for the one-, two-, and three-dimensional truncated harmonic oscillators, the Morse and Poschl-Teller potential functions. When we consider more complicated algebraic expansions in terms of Casimir operators, or when we deal with coupled... 

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