Lattice mechanics

Consider a rigid regular (or "crystalline"; not amorphous) lattice composed of N particles. (We will refer to these particles as "atoms". In a real solid these atoms may be ions.) N is some large number, say around 1023 (on the order of Avogadro's number) for a typical piece of solid. If the lattice is rigid, the atoms must be exerting forces on one another, so as to keep each atom near its equilibrium position. In real solids, these forces include Van der Waals forces, covalent bonds, and so forth, all of which are ultimately due to the electric force; magnetic and gravitational forces are generally negligible. The forces between each pair of atoms may be characterized by some potential energy function V, depending on the separation of the atoms. The potential energy of the entire lattice is the sum of all the pairwise potential energies:[3]



where is the position of the th atom, and is the potential energy between two atoms.
It is extremely difficult to solve this many-body problem in full generality, in either classical or quantum mechanics. In order to simplify the task, we introduce two important approximations. First, we perform the sum over neighboring atoms only. Although the electric forces in real solids extend to infinity, this approximation is nevertheless valid because the fields produced by distant atoms are screened. Secondly, we treat the potentials as harmonic potentials: this is permissible as long as the atoms remain close to their equilibrium positions. (Formally, this is done by Taylor expanding about its equilibrium value, which gives proportional to .)
The resulting lattice may be visualized as a system of balls connected by springs. The following figure shows a cubic lattice, which is a good model for many types of crystalline solid. Other lattices include a linear chain, which is a very simple lattice which we will shortly use for modelling phonons. Other common lattices may be found in the article on crystal structure.



The potential energy of the lattice may now be written as



Here, is the natural frequency of the harmonic potentials, which we assume to be the same since the lattice is regular. is the position coordinate of the th atom, which we now measure from its equilibrium position. The sum over nearest neighbors is denoted as "(nn)".

Lattice waves



Due to the connections between atoms, the displacement of one or more atoms from their equilibrium positions will give rise to a set of vibration waves propagating through the lattice. One such wave is shown in the figure to the right. The amplitude of the wave is given by the displacements of the atoms from their equilibrium positions. The wavelength is marked.
There is a minimum possible wavelength, given by the equilibrium separation a between atoms. As we shall see in the following sections, any wavelength shorter than this can be mapped onto a wavelength longer than a, due to effects similar to that in aliasing.
Not every possible lattice vibration has a well-defined wavelength and frequency. However, the normal modes (which, as we mentioned in the introduction, are the elementary building-blocks of lattice vibrations) do possess well-defined wavelengths and frequencies. We will now examine it in detail.
Acoustic and optical phonons
In solids with more than one atom in the smallest unit cell, there are two types of phonons: "acoustic" phonons and "optical" phonons. "Acoustic phonons", which are the phonons described above, have frequencies that become small at the long wavelengths, and correspond to sound waves in the lattice. Longitudinal and transverse acoustic phonons are often abbreviated as LA and TA phonons, respectively.
"Optical phonons," which also arise in crystals with more than one atom in the smallest unit cell, always have some minimum frequency of vibration, even when their wavelength is large. They are called "optical" because in ionic crystals (like sodium chloride) they are excited very easily by light (in fact, infrared radiation). This is because they correspond to a mode of vibration where positive and negative ions at adjacent lattice sites swing against each other, creating a time-varying electrical dipole moment. Optical phonons that interact in this way with light are called infrared active. Optical phonons which are Raman active can also interact indirectly with light, through Raman scattering. Optical phonons are often abbreviated as LO and TO phonons, for the longitudinal and transverse varieties respectively.

Kevin M Contreras H
Electrónica del Estado Sólido
http://en.wikipedia.org/wiki/Phonon

Phonon

Considering the regular lattice of atoms in a uniform solid material, you would expect there to be energy associated with the vibrations of these atoms. But they are tied together with bonds, so they can't vibrate independently. The vibrations take the form of collective modes which propagate through the material. Such propagating lattice vibrations can be considered to be sound waves, and their propagation speed is the speed of sound in the material.
The vibrational energies of molecules, e.g., a diatomic molecule, are quantized and treated as quantum harmonic oscillators. Quantum harmonic oscillators have equally spaced energy levels with separation E = h. So the oscillators can accept or lose energy only in discrete units of energy h.
The evidence on the behavior of vibrational energy in periodic solids is that the collective vibrational modes can accept energy only in discrete amounts, and these quanta of energy have been labeled "phonons". Like the photons of electromagnetic energy, they obey Bose-Einstein statistics.
Considering a solid to be a periodic array of mass points, there are constraints on both the minimum and maximum wavelength associated with a vibrational mode.





By associating a phonon energy with the modes and summing over the modes, Debye was able to find an expression for the energy as a function of temperature and derive an expression for the specific heat of the solid. In this expression, vs is the speed of sound in the solid

Debye Specific Heat

By associating a phonon energy



with the vibrational modes of a solid, where vs is the speed of sound in the solid, Debye approached the subject of the specific heat of solids. Treating them with Einstein-Bose statistics, the total energy in the lattice vibrations is of the form



This can be expressed in terms of the phonon modes by expressing the integral in terms of the mode number n.



Here the factor 3/2 comes from three considerations. First, there are 3 modes associated with each mode number n: one longitudinal mode and two transverse modes. Then you get a factor of 42 from integrating over the angular coordinates, treating the mode number n as the radius vector. Finally you constrain the integral to the quadrant in which all the components of n are positive, giving a factor of 1/8: the product of those is 3/2.
The usual form of the integral is obtained by making the substitution



and the limit on the integral in terms of x is obtained from



where the constant TD is here introduced. It is called the Debye temperature and is a constant associated with the highest allowed mode of vibration.
When all the constants are put in, the integral takes the form



The Debye specific heat expression is the derivative of this expression with respect to T. The integral cannot be evaluated in closed form, but numerical evaluation of the integral shows reasonably good agreement with the observed specific heats of solids for the full range of temperatures, approaching the Dulong-Petit Law at high temperatures and the characteristic T3 behavior at very low temperatures.
The specific heat expression which arises from Debye theory can be obtained by taking the derivative of the energy expression above.



This expression may be evaluated numerically for a given temperature by computer routines.



The data for silver shown at left is from Meyers. It shows that the specific heat fits the Debye model at both low and high temperatures.
Since the Debye specific heat expression can be evaluated as a function of temperature and gives a theoretical curve which has a specific form as a funtion of T/TD, the specific heats of different substances should overlap if plotted as a function of this ratio. At left below, the specific heats of four substances are plotted as a function of temperature and they look very different. But if they are scaled to T/TD, they look very similar and are very close to the Debye theory.



Kevin M Contreras H
Electrónica del Estado Sólido
http://hyperphysics.phy-astr.gsu.edu/Hbase/Solids/phonon.html

Bravais lattice

In geometry and crystallography, a Bravais lattice, studied by Auguste Bravais (1850),[1] is an infinite set of points generated by a set of discrete translation operations described by:



where ni are any integers and are known as the primitive vectors which lie in different planes and span the lattice. For any choice of position vector R , the lattice looks exactly the same.
A crystal is made up of one or more atoms (the basis) which is repeated at each lattice point. The crystal then looks the same when viewed from any of the lattice points.
Two Bravais lattices are often considered to be equivalent if they have isomorphic symmetry groups. In this sense, there are 14 possible Bravais lattices in three-dimensional space. The 14 possible symmetry groups of Bravais lattices are 14 of the 230 space groups.

Bravais lattices in at most 2 dimensions

In each of 0-dimensional and 1-dimensional space there is just one type of Bravais lattice.
In two dimensions, there are five Bravais lattices. They are oblique, rectangular, centered rectangular, hexagonal, and square.[2] There are 4 lattice systems, as the centered rectangular and rectangular lattices are in the same lattice system.



The five fundamental two-dimensional Bravais lattices: 1 oblique, 2 rectangular, 3 centered rectangular, 4 hexagonal, and 5 square

Bravais lattices in 3 dimensions

The 14 Bravais lattices in 3 dimensions are arrived at by combining one of the seven lattice systems (or axial systems) with one of the lattice centerings. Each Bravais lattice refers to a distinct lattice type.
The lattice centerings are:
• Primitive centering (P): lattice points on the cell corners only
• Body centered (I): one additional lattice point at the center of the cell
• Face centered (F): one additional lattice point at center of each of the faces of the cell
• Centered on a single face (A, B or C centering): one additional lattice point at the center of one of the cell faces.
Not all combinations of the crystal systems and lattice centerings are needed to describe the possible lattices. There are in total 7 × 6 = 42 combinations, but it can be shown that several of these are in fact equivalent to each other. For example, the monoclinic I lattice can be described by a monoclinic C lattice by different choice of crystal axes. Similarly, all A- or B-centered lattices can be described either by a C- or P-centering. This reduces the number of combinations to 14 conventional Bravais lattices, shown in the table below.




The volume of the unit cell can be calculated by evaluating where , and are the lattice vectors. The volumes of the Bravais lattices are given below:



Bravais lattices in 4 dimensions

In four dimensions, there are 52 Bravais lattices. Of these, 21 are primitive and 31 are centered.

Kevin M Contreras H
Electrónica del Estado Sólido
http://en.wikipedia.org/wiki/Bravais_lattice

Red recíproca

Paralelamente a la definición de red cristalina (denominada también red directa) puede definirse para cada cristal otro tipo de red que recibe el nombre de red recíproca.



La red recíproca no es mas que la red que se construye sobre el espacio vectorial dual del espacio vectorial asociado a la red directa. En la notación empleada en esta monografía, en la definición de dicha red recíproca se incluye un factor 2p. Según esto, si los vectores característicos de la red directa son , los de la red recíproca, , satisfacen la relación:



Y esto equivale a definir:



La red recíproca tiene importantes propiedades, entre las que cabe citar la de que sus puntos están íntimamente relacionados con la difracción de rayos X producida por la red cristalina.
Si los vectores de traslación de la red cristalina son ortogonales también lo serán los de la red recíproca.
La celda unidad de la red recíproca es el paralelepípedo formado por los vectores y su volumen viene dado por (2p)3/Vc siendo Vc el volumen de la celda unidad de la red cristalina.

Zonas de Brillouin

Las zonas de Brillouin son regiones de la red recíproca que tienen las siguientes propiedades:
- Cada zona de Brillouin tiene un volumen igual al volumen de la celda unidad de la red recíproca.

- Cada zona de Brillouin puede reducirse a cierta zona, llamada primera zona de Brillouin, por traslación de sus posiciones mediante vectores de la red recíproca hacia la primera zona.

- La primera zona de Brillouin, que coincide con la celda unidad de Wigner – Seitz de la red recíproca, llena todo el espacio bajo la acción de traslaciones definidas por la ecuación:



Donde h1, h2 y h3 son números enteros.

Las zonas de Brillouin se construyen mediante planos bisectrices perpendiculares a todos los vectores de la red recíproca. La primera zona es el volumen más pequeño alrededor de un origen arbitrario limitado por dichos planos. La segunda zona es el volumen entre la primera zona y el nuevo conjunto de planos y así sucesivamente.
La primera zona de Brillouin de una red cristalina cúbica centrada en caras es un octaedro truncado de volumen 4(2p/a3).
Si los vectores fundamentales de la red cristalina cúbica centrada en las caras son:



Los vectores fundamentales de la red recíproca, dados por (3) son:



Kevin M Contreras H
Electrónica del Estado Sólido
http://www.matematicasypoesia.com.es/monografias/solidos.htm

Crystal momentum

In solid-state physics crystal momentum or quasimomentum is a momentum-like vector associated with electrons in a crystal lattice. It is defined by the associated wave vectors k of this lattice, according to



(where is the reduced Planck's constant). Like regular momentum, crystal momentum frequently exhibits the property of being conserved, and is thus extraordinarily useful to physicists and materials scientists as an analytical tool.
Lattice Symmetry Origins
A common method of modeling crystal structure and behavior is to view electrons as quantum mechanical particles traveling through a fixed infinite periodic potential V(x) such that



where a is an arbitrary lattice vector. Such a model is sensible because (a) crystal ions that actually form the lattice structure are typically on the order of tens of thousands of times more massive than than electrons, making it safe to replace them with a fixed potential structure, and (b) the macroscopic dimensions of a crystal are typically far greater than a single lattice spacing, making edge effects negligible. A consequence of this potential energy function is that it is possible to shift the initial position of an electron by any lattice vector a without changing any aspect of the problem, thereby defining a discrete symmetry. (Speaking more technically, an infinite periodic potential implies that the lattice translation operator T(a) commutes with the Hamiltonian, assuming a simple kinetic-plus-potential form.)
These conditions imply Bloch's theorem, which states in terms of equations that



or in terms of words that an electron in a lattice, which can be modeled as a single particle wave function ψ(x), finds its stationary state solutions in the form of a plane wave multiplied by a periodic function u(x). The theorem arises as a direct consequence of the aforementioned fact that the lattice symmetry translation operator commutes with the system's Hamiltonian.
One of the notable aspects of Bloch's theorem is that it shows directly that steady state solutions may be identified with a wave vector k, meaning that this quantum number remains a constant of motion. Crystal momentum is then conventionally defined by multiplying this wave vector by Planck's constant:



While this is in fact identical to the definition one might give for regular momentum (for example, by treating the effects of the translation operator by the effects of a particle in free space), there are important theoretical differences. For example, while regular momentum is completely conserved, crystal momentum is only conserved to within a lattice vector, i.e., an electron can be described not only by the wave vector k, but also with any other wave vector k' such that

k' = k + K,

where K is an arbitrary reciprocal lattice vector. This is a consequence of the fact that the lattice symmetry is discrete as opposed to continuous, and thus its associated conservation law cannot be derived using Noether's theorem.

Kevin M Contreras H
Electrónica del Estado Sólido

http://en.wikipedia.org/wiki/Crystal_momentum

BRAGG'S LAW

In physics, Bragg's law states that when X-rays hit an atom, they make the electronic cloud move as does any electromagnetic wave. The movement of these charges re-radiates waves with the same frequency (blurred slightly due to a variety of effects); this phenomenon is known as the Rayleigh scattering (or elastic scattering). The scattered waves can themselves be scattered but this secondary scattering is assumed to be negligible. A similar process occurs upon scattering neutron waves from the nuclei or by a coherent spin interaction with an unpaired electron. These re-emitted wave fields interfere with each other either constructively or destructively (overlapping waves either add together to produce stronger peaks or subtract from each other to some degree), producing a diffraction pattern on a detector or film. The resulting wave interference pattern is the basis of diffraction analysis. Both neutron and X-ray wavelengths are comparable with inter-atomic distances (~150 pm) and thus are an excellent probe for this length scale.



X-rays interact with the atoms in a crystal.
The interference is constructive when the phase shift is a multiple of 2π; this condition can be expressed by Bragg's law,



where n is an integer determined by the order given, λ is the wavelength of the X-rays (and moving electrons, protons and neutrons), d is the spacing between the planes in the atomic lattice, and θ is the angle between the incident ray and the scattering planes.



According to the 2θ deviation, the phase shift causes constructive (left figure) or destructive (right figure) interferences.
Note that moving particles, including electrons, protons and neutrons, have an associated De Broglie wavelength.
Bragg's Law is the result of experiments into the diffraction of X-rays or neutrons off crystal surfaces at certain angles, derived by physicist Sir William Lawrence Bragg in 1912 and first presented on 11 November 1912 to the Cambridge Philosophical Society. Although simple, Bragg's law confirmed the existence of real particles at the atomic scale, as well as providing a powerful new tool for studying crystals in the form of X-ray and neutron diffraction. William Lawrence Bragg and his father, Sir William Henry Bragg, were awarded the Nobel Prize in physics in 1915 for their work in determining crystal structures beginning with NaCl, ZnS, and diamond.
Bragg scattering of visible light by colloids
A colloidal crystal is a highly ordered array of particles which can be formed over a very long range (from a few millimeters to one centimeter) in length, and which appear analogous to their atomic or molecular counterparts. The periodic arrays of spherical particles make similar arrays of interstitial voids, which act as a natural diffraction grating for visible light waves, especially when the interstitial spacing is of the same order of magnitude as the incident lightwave.
Thus, it has been known for many years that, due to repulsive Coulombic interactions, electrically charged macromolecules in an aqueous environment can exhibit long-range crystal-like correlations with interparticle separation distances often being considerably greater than the individual particle diameter. In all of these cases in nature, the same brilliant iridescence (or play of colors) can be attributed to the diffraction and constructive interference of visible lightwaves which satisfy Bragg’s law, in a matter analogous to the scattering of X-rays in crystalline solids.

Kevin M Contreras H
Electrónica del Estado Sólido

http://en.wikipedia.org/wiki/Bragg%27s_law