Translational invariance Lattice

Mixed valence is usually investigated in intermetallic compounds where the lanthanide atoms are arranged in high concentration in a translationally invariant lattice. Mixed valence however occurs also in dilute lanthanide impurities in an otherwise normal matrix. Lm measurements were applied on dilute Ce, Pr, Nd, Sm,... 

The first summation incorporates the (non-random) translational invariance, while the second includes random deviations from the lattice on a site-by-site basis. Note that the second summation explicitly indicates that all randomness or disorder is diagonal, not off-diagonal. The corresponding exact GF G = (lu — Hyl satisfies the matrix equation... 

It may come as a surprise to some that two commensurate surfaces withstand finite shear forces even if they are separated by a fluid.31 But one has to keep in mind that breaking translational invariance automatically induces a potential of mean force T. From the symmetry breaking, commensurate walls can be pinned even by an ideal gas embedded between them.32 The reason is that T scales linearly with the area of contact. In the thermodynamic limit, the energy barrier for the slider to move by one lattice constant becomes infinitely high so that the motion cannot be thermally activated, and hence, static friction becomes finite. No such argument applies when the surfaces do not share a common period. 

Now we want to apply the above described formalism, Section 5.1, to fractal lattices. Here one has to be careful when performing ensemble averages. Due to the lack of translational invariance, the ensemble averaged concentration (n(f, t)) on a fractal is no longer independent of the position vector r, as it is in normal dimensions. In order to avoid a further complication of the formalism, however, we neglect in the following analytical approach this... 

Charged point defects on regular lattice positions can also contribute to additional losses the translation invariance, which forbids the interaction of electromagnetic waves with acoustic phonons, is perturbed due to charged defects at random positions. Such single-phonon processes are much more effective than the two- or three phonon processes discussed before, because the energy of the acoustic branches goes to zero at the T point of the Brillouin zone. Until now, only a classical approach to account for these losses exists, which has been... 

If the potential energy of a lattice 0 is expanded in a number of components of small displacements of atoms from their equilibrium position Uj up to a cubic term, then the temperature dependence of the principal values of the LTEC tensor ,(T )are, in view of the translational invariancy of the lattice, described by the following expression ... 

Till now, we have only considered a mathematical set of points. However, a material, in reality, is not merely an array of points, but the group of points is a lattice. A real crystalline material is constituted of atoms periodically arranged in the structure, where the condition of periodicity implies a translational invariance with respect to a translation operation, and where a lattice translation operation, T, is defined as a vector connecting two lattice points, given by Equation 1.1 as... 

Figure 4.6. The CPA consists in extending the translational invariance of the mean lattice around a single site, occupied either by an A or by a B molecule. The condition of self-consistency of the mean lattice is obtained from the average of the propagations around this single site, which must be identical to the propagation in the mean lattice.

For the absorption spectrum, since the translational invariance of the initial lattice is lost, the wave vector is defined modulo a vector of the reciprocal superlattice thus, we must bring the absorption wave vector k (k = 0 and k = (2n/b)h) to a vector of the first zone of the superlattice ... 

This approximation consists in replacing the real disordered lattice by a crystal, translationally invariant, with molecules of average polarizability < > ... 

Matter is composed of spherical-like atoms. No two atomic cores—the nuclei plus inner shell electrons—can occupy the same volume of space, and it is impossible for spheres to fill all space completely. Consequently, spherical atoms coalesce into a solid with void spaces called interstices. A mathematical construct known as a space lattice may be envisioned, which is comprised of equidistant lattice points representing the geometric centers of structural motifs. The lattice points are equidistant since a lattice possesses translational invariance. A motif may be a single atom, a collection of atoms, an entire molecule, some fraction of a molecule, or an assembly of molecules. The motif is also referred to as the basis or, sometimes, the asymmetric unit, since it has no symmetry of its own. For example, in rock salt a sodium and chloride ion pair constitutes the asymmetric unit. This ion pair is repeated systematically, using point symmetry and translational symmetry operations, to form the space lattice of the crystal. 

It is of fundamental importance that these two expressions for the virial do not coincide under periodic boundary conditions. The reason is that the expression rn = r, — r, for the distance is no longer valid. In general, it has to be corrected by a suitable lattice shift to obtain the actual minimum image expression. The correct expression to use is the double sum, since it is manifestly translationally invariant, like the pressure. A thorough discussion is given in Appendix B of Ref. 39. 

Let us now consider a ferromagnetic system made of spins a (r) attached to the lattice sites. We shall admit that the spin-spin interaction is given by a Hamiltonian JV a) which will be assumed to be translationally invariant. At equilibrium, the weight of a configuration of the system is given by the Boltzmann s factor... 

The average environment that the reacting species encounter in gas phase and condensed fluid environments is isotropic and translationally invariant. This is not true for rigid environments with well-defined lattice sites, e.g. the average environment that a reacting species sees near a lattice site is very different from that near an interstitial site. 

The translation invariance of k) is obvious because the sum is extended to all cells in the crystal. In fact, if a translation by lattice vector I is applied... 

M (r k) is verified to be periodic throughout the direct lattice (the equivalence of the sum over lattice vectors m = g -I-1 and the sum over g originates from translation invariance and the periodic boundary conditions). 

Every matrix element is properly identified by the three indices p, v, and g, which specify the two AOs and the direct lattice vector g labeling the cell where the v-th AO is centered because, in principle, the origin of Xv(r — fy) can be anywhere in the crystal. The possibility of always referring to the 0-cell is because of translation invariance of the integrals in the local basis, for example ... 

The total energy is important and useful to us for answering this question. As discussed, the total energy of an infinite crystal, like in our model, is infinite. Therefore, the total energy per cell is definitely a preferable choice, for it is a finite well-defined property, because of translation invariance. For the sake of clarity, we remind the reader that, although the total energy per cell is defined within the direct lattice context (Eq. [39]), its calculation depends on knowing the density matrix, which in our scheme is obtained from Eq. [38]. 

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