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This configuration means that the integration over the whole domain of the Brillouin zone can be reduced to a th part of the whole Brillouin zone. As a crystal structure periodic table shows, there are many elements with a FCC crystal structure, like diamond , silicon and platinum and their Brillouin zones and dispersion relations have this fold symmetry.


Two other familiar crystal structures are the body-centered cubic lattice BCC and hexagonal closed packed structures HCP with cubic and hexagonal lattices, respectively. The BCC structure has the fold pyritohedral symmetry of the point group T h. The HCP structure has the fold prismatic dihedral symmetry of the point group D 3h. A complete list of symmetry properties of a point group can be found in point group character tables.

Statistical Physics of Nanoparticles in the Gas Phase

In general it is easier to calculate a DOS when the symmetry of the system is higher and the number of topological dimensions of the dispersion relation is lower. The DOS of dispersion relations with rotational symmetry can often be calculated analytically. This result is fortunate, since many materials of practical interest, such as steel and silicon, have high symmetry.

In anisotropic condensed matter systems such as a single crystal of a compound, the density of states could be different in one crystallographic direction than in another. These causes the anisotropic density of states to be more difficult to visualize, and might require methods such as calculating the DOS for particular points or directions only, or calculating the projected density of states PDOS to a particular crystal orientation. The density of states is dependent upon the dimensional limits of the object itself. The referenced volume is the volume of k -space; the space enclosed by the constant energy surface of the system derived through a dispersion relation that relates E to k.

An example of a 3-dimensional k -space is given in Fig. It can be seen that the dimensionality of the system confines the momentum of particles inside the system. The volume, area or length in 3, 2 or 1-dimensional spherical k -spaces are expressed by. The 1, 2 and 3-dimensional density of wave vector states for a line, disk, or sphere are explicitly written as. The wavelength is related to k through the relationship. V k is the volume in k-space whose wavevectors are smaller than the smallest possible wavevectors decided by the characteristic spacing of the system.

The general form of DOS of a system is given as.

The scheme sketched so far only applies to monotonically rising and spherically symmetric dispersion relations. If the dispersion relation is not spherically symmetric or continuously rising and can't be inverted easily then in most cases the DOS has to be calculated numerically. More detailed derivations are available. The dispersion relation for electrons in a solid is given by the electronic band structure. The kinetic energy of a particle depends on the magnitude and direction of the wave vector k , the properties of the particle and the environment in which the particle is moving. For example, the kinetic energy of an electron in a Fermi gas is given by.

The dispersion relation is a spherically symmetric parabola and it is continuously rising so the DOS can be calculated easily. For longitudinal phonons in a string of atoms the dispersion relation of the kinetic energy in a 1-dimensional k -space, as shown in Figure 2, is given by. This expression is a kind of dispersion relation because it interrelates two wave properties and it is isotropic because only the length and not the direction of the wave vector appears in the expression.


The magnitude of the wave vector is related to the energy as:. Accordingly, the volume of n-dimensional k -space containing wave vectors smaller than k is:. Differentiating this volume with respect to the energy gives an expression for the DOS of the isotropic dispersion relation. The density of states plays an important role in the kinetic theory of solids. The product of the density of states and the probability distribution function is the number of occupied states per unit volume at a given energy for a system in thermal equilibrium.

This value is widely used to investigate various physical properties of matter. The following are examples, using two common distribution functions, of how applying a distribution function to the density of states can give rise to physical properties. Fermions are particles which obey the Pauli exclusion principle e. The distribution function can be written as. Bose—Einstein statistics : The Bose—Einstein probability distribution function is used to find the probability that a boson occupies a specific quantum state in a system at thermal equilibrium.

Bosons are particles which do not obey the Pauli exclusion principle e. The density of states appears in many areas of physics, and helps to explain a number of quantum mechanical phenomena. Calculating the density of states for small structures shows that the distribution of electrons changes as dimensionality is reduced. For quantum wires , the DOS for certain energies actually becomes higher than the DOS for bulk semiconductors, and for quantum dots the electrons become quantized to certain energies.

The photon density of states can be manipulated by using periodic structures with length scales on the order of the wavelength of light. Some structures can completely inhibit the propagation of light of certain colors energies , creating a photonic band gap: the DOS is zero for those photon energies.

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Other structures can inhibit the propagation of light only in certain directions to create mirrors, waveguides, and cavities. Such periodic structures are known as photonic crystals. Interesting systems are in general complex, for instance compounds, biomolecules, polymers, etc. Because of the complexity of these systems the analytical calculation of the density of states is in most of the cases impossible. Computer simulations offer a set of algorithms to evaluate the density of states with a high accuracy.

One of these algorithms is called the Wang and Landau algorithm.

Statistical Physics of Nanoparticles in the Gas Phase door Klavs Hansen (Boek) -

Within the Wang and Landau scheme any previous knowledge of the density of states is required. One proceeds as follows: the cost function for example the energy of the system is discretized. As soon as each bin in the histogram is visited a certain number of times , the modification factor is reduced by some criterion, for instance,. The Wang and Landau algorithm has some advantages over other common algorithms such as multicanonical simulations and parallel tempering.

For example, the density of states is obtained as the main product of the simulation. Additionally, Wang and Landau simulations are completely independent of the temperature. This feature allows to compute the density of states of systems with very rough energy landscape such as proteins. Log in to your account.

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