Crystal structure

In mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry. Patterns are located upon the points of a lattice, which is an array of points repeating periodically in three dimensions. The points can be thought of as forming identical tiny boxes, called unit cells, that fill the space of the lattice. The lengths of the edges of a unit cell and the angles between them are called the lattice parameters. The symmetry properties of the crystal are embodied in its space group. A crystal's structure and symmetry play a role in determining many of its physical properties, such as cleavage, electronic band structure, and optical transparency.

Insulin crystals.

Unit cell

The crystal structure of a material or the arrangement of atoms within a given type of crystal structure can be described in terms of its unit cell. The unit cell is a tiny box containing one or more atoms, a spatial arrangement of atoms. The unit cells stacked in three-dimensional space describe the bulk arrangement of atoms of the crystal. The crystal structure has a three dimensional shape. The unit cell is given by its lattice parameters, the length of the cell edges and the angles between them, while the positions of the atoms inside the unit cell are described by the set of atomic positions (xi , yi , zi) measured from a lattice point.

Miller indices

Planes with different Miller indices in cubic crystals

Vectors and atomic planes in a crystal lattice can be described using a three-value Miller index notation (ℓmn). The ℓ, m and n directional indices are separated by 90°, and are thus orthogonal. In fact, the ℓ component is mutually perpendicular to the m and n indices.

By definition, (ℓmn) denotes a plane that intercepts the three points a1/ℓ, a2/ℓ, and a3/ℓ, or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell (in the basis of the lattice vectors). If one or more of the indices is zero, it simply means that the planes do not intersect that axis (i.e. the intercept is "at infinity").

Considering only (ℓmn) planes intersecting one or more lattice points (the lattice planes), the perpendicular distance d between adjacent lattice planes is related to the (shortest) reciprocal lattice vector orthogonal to the planes by the formula:

Planes and directions

The crystallographic directions are fictitious lines linking nodes (atoms, ions or molecules) of a crystal. Similarly, the crystallographic planes are fictitious planes linking nodes. Some directions and planes have a higher density of nodes. These high density planes have an influence on the behavior of the crystal as follows:

• Optical properties: Refractive index is directly related to density (or periodic density fluctuations).

• Adsorption and reactivity: Physical adsorption and chemical reactions occur at or near surface atoms or molecules. These phenomena are thus sensitive to the density of nodes.

• Surface tension: The condensation of a material means that the atoms, ions or molecules are more stable if they are surrounded by other similar species. The surface tension of an interface thus varies according to the density on the surface.

Dense crystallographic planes

• Microstructural defects: Pores and crystallites tend to have straight grain boundaries following higher density planes.

• Cleavage: This typically occurs preferentially parallel to higher density planes.

• Plastic deformation: Dislocation glide occurs preferentially parallel to higher density planes. The perturbation carried by the dislocation (Burgers vector) is along a dense direction. The shift of one node in a more dense direction requires a lesser distortion of the crystal lattice.

Cubic structures

For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length (usually denoted a); similarly for the reciprocal lattice. So, in this common case, the Miller indices (ℓmn) and [ℓmn] both simply denote normals/directions in Cartesian coordinates. For cubic crystals with lattice constant a, the spacing d between adjacent (ℓmn) lattice planes is (from above):

Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes:

• Coordinates in angle brackets such as <100> denote a family of directions which are equivalent due to symmetry operations, such as [100], [010], [001] or the negative of any of those directions.

• Coordinates in curly brackets or braces such as {100} denote a family of plane normals which are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.

For face-centered cubic (fcc) and body-centered cubic (bcc) lattices, the primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions.

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

Lattices

Lattices in two dimensions: detailed discussion

There are five 2D lattice types as given by the crystallographic restriction theorem. Below, the wallpaper group of the lattice is given in parentheses; note that a pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. If the symmetry group of a pattern contains an n-fold rotation then the lattice has n-fold symmetry for even n and 2n-fold for odd n.

• a rhombic lattice, also called centered rectangular lattice or isosceles triangular lattice (cmm), with evenly spaced rows of evenly spaced points, with the rows alternatingly shifted one half spacing (symmetrically staggered rows):

• a hexagonal lattice or equilateral triangular lattice (p6m)

• a square lattice (p4m):

• a rectangular lattice, also called primitive rectangular lattice (pmm):

• more generally, a parallelogrammic lattice, also called oblique lattice (p2)(with asymmetrically staggered rows):

For the classification of a given lattice, start with one point and take a nearest second point. For the third point, not on the same line, consider its distances to both points. Among the points for which the smaller of these two distances is least, choose a point for which the larger of the two is least. (Not logically equivalent but in the case of lattices giving the same result is just "Choose a point for which the larger of the two is least".)

The five cases correspond to the triangle being equilateral, right isosceles, right, isosceles, and scalene. In a rhombic lattice, the shortest distance may either be a diagonal or a side of the rhombus, i.e., the line segment connecting the first two points may or may not be one of the equal sides of the isosceles triangle. This depends on the smaller angle of the rhombus being less than 60° or between 60° and 90°.

The general case is known as a period lattice. If the vectors p and q generate the lattice, instead of p and q we can also take p and p-q, etc. In general in 2D, we can take a p + b q and c p + d q for integers a,b, c and d such that ad-bc is 1 or -1. This ensures that p and q themselves are integer linear combinations of the other two vectors. Each pair p, q defines a parallelogram, all with the same area, the magnitude of the cross product. One parallelogram fully defines the whole object. Without further symmetry, this parallelogram is a fundamental parallelogram.

The fundamental domain of the period lattice.

The vectors p and q can be represented by complex numbers. Up to size and orientation, a pair can be represented by their quotient. Expressed geometrically: if two lattice points are 0 and 1, we consider the position of a third lattice point. Equivalence in the sense of generating the same lattice is represented by the modular group: represents choosing a different third point in the same grid, represents choosing a different side of the triangle as reference side 0-1, which in general implies changing the scaling of the lattice, and rotating it. Each "curved triangle" in the image contains for each 2D lattice shape one complex number, the grey area is a canonical representation, corresponding to the classification above, with 0 and 1 two lattice points that are closest to each other; duplication is avoided by including only half of the boundary. The rhombic lattices are represented by the points on its boundary, with the hexagonal lattice as vertex, and i for the square lattice. The rectangular lattices are at the imaginary axis, and the remaining area represents the parallelogrammetic lattices, with the mirror image of a parallelogram represented by the mirror image in the imaginary axis.

Lattices in three dimensions

The 14 lattice types in 3D are called Bravais lattices. They are characterized by their space group. 3D patterns with translational symmetry of a particular type cannot have more, but may have less symmetry than the lattice itself.

Lattices in complex space

A lattice in Cn is a discrete subgroup of Cn which spans the 2n-dimensional real vector space Cn. For example, the Gaussian integers form a lattice in C.

Every lattice in Rn is a free abelian group of rank n; every lattice in Cn is a free abelian group of rank 2n.

Kevin M Contreras H
Electrónica del Estado Sólido
http://en.wikipedia.org/wiki/Lattice_(group)

Classical mechanics

Classical mechanics is the physics of forces, acting upon bodies. It is often referred to as "Newtonian mechanics" after Newton and his laws of motion. Classical mechanics is subdivided into statics (which deals with objects in equilibrium) and dynamics (which deals with objects in motion). See also mechanics.

Classical mechanics produces very accurate results within the domain of everyday experience. It is superseded by relativistic mechanics for systems moving at large velocities near the speed of light, quantum mechanics for systems at small distance scales, and relativistic quantum field theory for systems with both properties.

Nevertheless, classical mechanics is still very useful, because (i) it is much simpler and easier to apply than these other theories, and (ii) it has a very large range of approximate validity. Classical mechanics can be used to describe the motion of human-sized objects (such as tops and baseballs), many astronomical objects (such as planets and galaxies), and even certain microscopic objects (such as organic molecules.)

Although classical mechanics is roughly compatible with other "classical" theories such as classical electrodynamics and thermodynamics, there are inconsistencies that were discovered in the late 19th century that can only be resolved by more modern physics. In particular, classical nonrelativistic electrodynamics predicts that the speed of light is a constant relative to an aether medium, a prediction that is difficult to reconcile with classical mechanics and which led to the development of special relativity. When combined with classical thermodynamics, classical mechanics leads to the Gibbs paradox in which entropy is not a well-defined quantity and to the ultraviolet catastrophe in which a blackbody is predicted to emit infinite amounts of energy. The effort at resolving these problems led to the development of quantum mechanics.

Description of the theory

We will now introduce the basic concepts of classical mechanics. For simplicity, we only deal with a point particle, which is an object with negligible size. The motion of a point particle is characterized by a small number of parameters: its position, mass, and the forces applied on it. We will discuss each of these parameters in turn.

In reality, the kind of objects which classical mechanics can describe always have a non-zero size. True point particles, such as the electron, are properly described by quantum mechanics. Objects with non-zero size have more complicated behavior than our hypothetical point particles, because their internal configuration can change - for example, a baseball can spin while it is moving. However, we will be able to use our results for point particles to study such objects by treating them as composite objects, made up of a large number of interacting point particles. We can then show that such composite objects behave like point particles, provided they are small compared to the distance scales of the problem, which indicates that our use of point particles is self-consistent.

Position and its derivatives

The position of a point particle is defined with respect to an arbitrary fixed point in space, which is sometimes called the origin, O. It is defined as the vector r from O to the particle. In general, the point particle need not be stationary, so r is a function of t, the time elapsed since an arbitrary initial time. The velocity, or the rate of change of position with time, is defined as

The acceleration, or rate of change of velocity, is

The acceleration vector can be changed by changing its magnitude, changing its direction, or both. If the magnitude of v decreases, this is sometimes referred to as deceleration; but generally any change in the velocity, including deceleration, is simply referred to as acceleration.

Forces; Newton's Second Law

Newton's second law relates the mass and velocity of a particle to a vector quantity known as the force. Suppose m is the mass of a particle and F is the vector sum of all applied forces (i.e. the net applied force.) Then Newton's second law states that

The quantity mv is called the momentum. Typically, the mass m is constant in time, and Newton's law can be written in the simplified form

where a is the acceleration, as defined above. It is not always the case that m is independent of t. For example, the mass of a rocket decreases as its propellant is ejected. Under such circumstances, the above equation is incorrect and the full form of Newton's second law must be used.

Newton's second law is insufficient to describe the motion of a particle. In addition, we require a description of F, which is to be obtained by considering the particular physical entities with which our particle is interacting. For example, a typical resistive force may be modelled as a function of the velocity of the particle, say

with λ a positive constant. Once we have independent relations for each force acting on a particle, we can substitute it into Newton's second law to obtain an ordinary differential equation, which is called the equation of motion. Continuing our example, suppose that friction is the only force acting on the particle. Then the equation of motion is

This can be integrated to obtain

where v0 is the initial velocity. This means that the velocity of this particle decays exponentially to zero as time progresses. This expression can be further integrated to obtain the position r of the particle as a function of time.
Important forces include the gravitational force and the Lorentz force for electromagnetism. In addition, Newton's third law can sometimes be used to deduce the forces acting on a particle: if we know that particle A exerts a force F on another particle B, it follows that B must exert an equal and opposite reaction force, -F, on A.

Energy

If a force F is applied to a particle that achieves a displacement δr, the work done by the force is the scalar quantity

Suppose the mass of the particle is constant, and δWtotal is the total work done on the particle, which we obtain by summing the work done by each applied force. From Newton's second law, we can show that

where T is called the kinetic energy. For a point particle, it is defined as

For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the individual particles' kinetic energies.

A particular class of forces, known as conservative forces, can be expressed as the gradient of a scalar function, known as the potential energy and denoted V:

Suppose all the forces acting on a particle are conservative, and V is the total potential energy, obtained by summing the potential energies corresponding to each force. Then

This result is known as the conservation of energy, and states that the total energy, , is constant in time. It is often useful, because most commonly encountered forces are conservative.

Kevin M Contreras H
Electrónica del Estado Sólido
http://www.knowledgerush.com/kr/encyclopedia/Classical_mechanics/

Phonon

A phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. The study of phonons is an important part of solid state physics, because they contribute to many of the physical properties of materials, such as thermal and electrical conductivity. For example, the propagation of phonons is responsible for the conduction of heat in insulators, and the properties of long-wavelength phonons gives rise to sound in solids (hence the name phonon).

According to a well-known result in classical mechanics, any vibration of a lattice can be decomposed into a superposition of normal modes of vibration. When these modes are analysed using quantum mechanics, they are found to possess some particle-like properties (see wave-particle duality.) When treated as particles, phonons are bosons possessing zero spin.
The following article provides an overview of the physics of phonons.

Non-interacting phonons

Modelling a lattice

We begin our investigation of phonons by examining the mechanical systems from which they emerge. Consider a rigid regular (or "crystalline") lattice composed of N particles. We will refer to these particles as "atoms", though in a real solid they may actually be ions. N is some large number, say around 1023 (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:

where ri is the position of the ith atom, and V 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. Firstly, we only perform the sum over neighbouring atoms. Although the electric forces in real solids extend to infinity, this approximation is nevertheles valid because the fields produced by distant atoms are screened. Secondly, we treat the potentials V as harmonic potentials, which is permissible as long as the atoms remain close to their equilibrium positions. (Formally, this is done by Taylor expanding V about its equilibrium value.)

The resulting lattice may be visualized as a system of balls connected by springs. Two such lattices are shown in the figures below. The figure on the left shows a cubic lattice, which is a good model for many types of crystalline solid. The figure on the right shows a linear chain, 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, ω is the natural frequency of the harmonic potentials, which we assume to be the same since the lattice is regular. Ri is the position coordinate of the ith atom, which we now measure from its equilibrium position. The sum over nearest neighbours is denoted as "(nn)".

Kevin M Contreras H
Electrónica del Estado Sólido
http://www.knowledgerush.com/kr/encyclopedia/Phonon/

Brillouin Zones

In order to define the Brillouin zone we need to define first the reciprocal lattice. The set of all wave vectors K that yield plane waves with the periodicity of a given Bravais lattice is known as its reciprocal lattice. Analytically, K belongs to the reciprocal lattice of a Bravais lattice of points R, provided that the relation

holds for any r, and all R in Bravais lattice. Factoring out

we can characterize the reciprocal lattice as the set of wave vectors K satisfying

for all R in the Bravais lattice.

The reciprocal lattice is itself a Bravais lattice and its primitive vectors can be generated from the vectors of the direct lattice. Let a1, a2, a3 be a set of primitive vectors, then the reciprocal lattice can be generated by the three primitive vectors:

Using the relations between direct and reciprocal lattice it can be shown that the reciprocal lattice of sc is sc (at k space), the reciprocal of bcc is fcc, and reciprocal of fcc is bcc.

The first Brillouin zone is defined to be the Wigner-Seitz primitive cell of the reciprocal lattice, or it could be defined as the set of points in k space that can be reached from the origin without crossing any Bragg plane. The second Brillouin zone is the set of points that can be reached from the first zone by crossing only one Bragg plane. The (n + 1)th Brillouin zone is the set of points not in the (n - 1)th zone that can be reached from the nth zone by crossing n - 1 Bragg planes.

Alternatively, the nth Brillouin zone can be defined as the set of points that can be reached from the origin by crossing n - 1 Bragg planes, but no fewer.

The first three Brillouin zones for some lattices are illustrated below (they were constructed by the program BrillouinZone):

Kevin M Contreras H
Electrónica del Estado Sólido
http://phycomp.technion.ac.il/~nika/brillouin_zones.html

Statistical Mechanics/Debye Theory

Say we have a material with vibrating modes visually akin to that of jello. In other words, we are looking at the phononic modes. The energy equation for a specific mode of a phonon is identical to that of a photon. The only difference between the phonon and photon case is that the phonon case has a possible upper limit (because the highest frequency occurs at the shortest wavelength: when one molecule is at its amplitude and the neighbours are at their troughs), and that we have another possible mode of polarization (two transverse, one longitudinal), and that we are dealing with v (the speed of sound in the material) instead of c. Thus, when we transform the integral as in the photon derivation (another similar density of states problem) we get:

Where nD is this maximal case.

We define something now called the Debye Temperature (a physical property of a material):

Note: the kB is apparent because we've been working in energy-represented T, instead of the typical Kelvin-scaled T, kB effectively means the T's here-on are in Kelvin-units)

If we use this, and take the low-temperature limit of T << θ, we can approxiamated the upper limit as we continue the integration as ∞ (which physically means that there's a bunch of higher-energy levels states that the solid is not reaching) we can evaluate almost exactly as before:

U(T) = 3π4NkBT4 / 5θ3 and:

This is known as Debye's T3 law, and follows well with experiment.

Kevin M Contreras H
Electrónica del Estado Sólido
http://en.wikibooks.org/wiki/Statistical_Mechanics/Debye_Theory

Bragg scattering

As the incidence angle of the ERS SAR is oblique (23) to the local mean angle of the ocean surface, there is almost no direct specular reflection except at very high sea states.

It is therefore assumed that at first approximation Bragg resonance is the primary mechanism for backscattering radar pulses.

The Bragg equation defines the ocean wavelengths for Bragg scattering as a function of radar wavelength and incidence angle:

The short Bragg-scale waves are formed in response to wind stress. If the sea surface is rippled by a light breeze with no long waves present, the radar backscatter is due to the component of the wave spectrum which resonates with the radar wavelength.

The threshold windspeed value for the C-band waves is estimated to be at about 3.25 m/s at 10 meters above the surface. The Bragg resonant wave has its crest nominally at right angles to the range direction.

For surface waves with crests at an angle to the radar line-of-sight (see this figure ) the Bragg scattering criterion is

where:
is the wavelength of the surface waves propagating at angle to the radar line- of sight.

The SAR directly images the spatial distribution of the Bragg-scale waves. The spatial distribution may be effected by longer gravity waves, through tilt modulation, hydrodynamic modulation and velocity bunching.

Moreover, variable wind speed, changes in stratification in the atmospheric boundary layer, and variable currents associated with upper ocean circulation features such as fronts, eddies, internal waves and bottom topography effect the Bragg waves.

Kevin M Contreras H
Electrónica del Estado Sólido
http://earth.esa.int/applications/data_util/SARDOCS/spaceborne/Radar_Courses/Radar_Course_III/bragg_scattering.htm