Kevin M Contreras H
Electrónica del Estado Sólido
http://www.ece.rochester.edu/courses/ECE423/ECE223_423_MSC426%20Workshop06/term%20papers%2006/Mathew_06.pdf

Reciprocal Lattices

span a Bravais lattice, then

span the reciprocal lattice, which is also a bravais lattice.The reciprocal of the reciprocal lattice is the set of all vectors for any recprocal lattice vector it is the original lattice.

As we discussed above, a simple cubic lattice spanned by

Chapter 4: Broken Translational Invariance in the Solid State has the simple cubic reciprocal lattice spanned by:

An FCC lattice spanned by:

has a BCC reciprocal lattice spanned by:

Conversely, a BCC lattice has an FCC reciprocal lattice.

The Wigner-Seitz primitive unit cell of the reciprocal lattice is the first Brillouin
zone. In the problem set (Ashcroft and Mermin, problem 5.1), you will show that the
Brillouin zone has volumen if the volume of the unit cell of the original lattice is . The first Brillouin zone is enclosed in the planes which are the perpendicularbisectors of the reciprocal lattice vectors. These planes are called Bragg planes forreasons which will become clear below.

Kevin M Contreras H
Electrónica del Estado Sólido
http://www.physics.ucla.edu/~nayak/solid_state.pdf

More Realistic Phonon Spectra: Optical Phonons, van Hove Singularities

Although Debye's theory is reasonable, it clearly oversimpli_es certain aspects of the physics. For instance, consider a crystal with a two-site basis. Half of the phonon modes will be optical modes. A crude approximation for the optical modes is an
Einstein spectrum:

In such a case, the energy will be:

With ω max chosen so that

Another feature missed by Debye's approximation is the existence of singularities
in the phonon density of states. Consider the spectrum of the linear chain:

The minimum of this spectrum is at k = 0. Here, the density of states is well described by Debye theory which, for a 1D chain predicts g(ω) ~ const:. The maximum is at k = π/a. Near the maximum, Debye theory breaks down; the density of states is singular:

In 3D, the singularity will be milder, but still present. Consider a cubic lattice. The spectrum can be expanded about a maximum as:

Then (6 maxima; 1/2 of each ellipsoid is in the B.Z.)

In 2D and 3D, there can also be saddle points, where but the eigenvalues of the second derivative matrix have di_erent signs. At a saddle point, the phonon spectrum again has a square root singularity. van Hove proved that every 3D phonon spectrum has at least one maximum and two saddle points (one with one negative eigenvalue, one with two negative eigenvalues). To see why this might be true, draw the spectrum in the full k-space, repeating the Brillouin zone. Imagine drawing lines connecting the minima of the spectrum to the nearest neighboring minima (i.e. from each copy of the B.Z. to its neighbors). Imagine doing the same with the maxima. These lines intersect; at these intersections, we expect saddle points.

Kevin M Contreras H
Electrónica del Estado Sólido
http://www.physics.ucla.edu/~nayak/solid_state.pdf

Statistical Mechnics of a Linear Chain

Let us return to the case of a linear chain of masses m separated by springs of force
constant B, at equilibrium distance a. The excitations of this system are phonons
which can have momenta k corresponding to energies Phonons are bosons whose number is not conserved, so they obey the Planck distribution.
Hence, the energy of a linear chain at _nite temperature is given by:



Changing variables from k to



In the limit kBT we can take the upper limit of integration to 1 and
drop the x-dependent term in the square root in the denominator of the integrand:



Hence, Cv ~ T at low temperatures In the limit kBT we can approximate :



In the intermediate temperature regime, a more careful analysis is necessary. In
particular, note that the density of states this
is an example of a van Hove singularity. If we had alternating masses on springs,
then the expression for the energy would have two integrals, one over the acoustic
modes and one over the optical modes.

Kevin M Contreras H
Electrónica del Estado Sólido
http://www.physics.ucla.edu/~nayak/solid_state.pdf

Continuum mechanics

Modeling an object as a continuum assumes that the substance of the object completely fills the space it occupies. Modeling objects in this way ignores the fact that matter is made of atoms, and so is not continuous; however, on length scales much greater than that of inter-atomic distances, such models are highly accurate. Fundamental physical laws such as the conservation of mass, the conservation of momentum, and the conservation of energy may be applied to such models to derive differential equations describing the behavior of such objects, and some information about the particular material studied is added through a constitutive relation.

Continuum mechanics deals with physical properties of solids and fluids which are independent of any particular coordinate system in which they are observed. These physical properties are then represented by tensors, which are mathematical objects that have the required property of being independent of coordinate system. These tensors can be expressed in coordinate systems for computational convenience.

The concept of a continuum

Materials, such as solids, liquids and gases, are composed of molecules separated by empty space. On a macroscopic scale, materials have cracks and discontinuities. However, certain physical phenomena can be modeled assuming the materials exist as a continuum, meaning the matter in the body is continuously distributed and fills the entire region of space it occupies. A continuum is a body that can be continually sub-divided into infinitesimal elements with properties being those of the bulk material.

The validity of continuum assumption may be verified by a theoretical analysis, in which either some clear periodicity is identified or statistical homogeneity and ergodicity of the microstructure exists. More specifically, the continuum hypothesis/assumption hinges on the concepts of a Representative Volume Element (RVE) and 'separation of scales' based on the Hill-Mandel condition. [Sometimes, in place of RVE, the term Representative Elementary Volume (REV) is used.] This condition provides a link between an experimentalist's and a theoretician's viewpoint on constitutive equations (linear and nonlinear elastic/inelastic or coupled fields) as well as a way of spatial and statistical averaging of the microstructure.When the separation of scales does not hold, or when one wants to establish a continuum of a finer resolution than that of the RVE size, one employs a Statistical Volume Element (SVE), which, in turn, leads to random continuum fields. The latter then provide a micromechanics basis for stochastic finite elements (SFE). The levels of SVE and RVE link continuum mechanics to statistical mechanics. The RVE may be assessed only in a limited way via experimental testing: when the constitutive response becomes spatially homogeneous.
In fluids, the Knudsen number is used to assess to what extent the approximation of continuity can be made.

Formulation of Model



Figure 1. Configuration of a continuum body

Continuum mechanics models begin by assigning a region in three dimensional Euclidean space to the material body modeled. The points within this region are called particles or material points. Different configurations or states of the body correspond to different regions in Euclidean space. The region corresponding to the body's configuration at time is labeled
.
A particular particle within the body in a particular configuration is characterized by a position vector


,
where are the coordinate vectors in some frame of reference chosen for the problem (See figure 1). This vector can be expressed as a function of the particle position in some reference configuration, for example the configuration at the initial time, so that


.
This function needs to have various properties so that the model makes physical sense. needs to be:
• continuous in time, so that the body changes in a way which is realistic,
• globally invertible at all times, so that the body cannot intersect itself,
• orientation-preserving, as transformations which produce mirror reflections are not possible in nature.
For the mathematical formulation of the model, is also assumed to be twice continuously differentiable, so that differential equations describing the motion may be formulated.

Kinematics: deformation and motion



Figure 2. Motion of a continuum body.

A change in the configuration of a continuum body results in a displacement. The displacement of a body has two components: a rigid-body displacement and a deformation. A rigid-body displacement consist of a simultaneous translation and rotation of the body without changing its shape or size. Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration to a current or deformed configuration (Figure 2).

The motion of a continuum body is a continuous time sequence of displacements. Thus, the material body will occupy different configurations at different times so that a particle occupies a series of points in space which describe a pathline.
There is continuity during deformation or motion of a continuum body in the sense that:
• The material points forming a closed curve at any instant will always form a closed curve at any subsequent time.
• The material points forming a closed surface at any instant will always form a closed surface at any subsequent time and the matter within the closed surface will always remain within.

It is convenient to identify a reference configuration or initial condition which all subsequent configurations are referenced from. The reference configuration need not be one that the body will ever occupy. Often, the configuration at is considered the reference configuration, . The components of the position vector of a particle, taken with respect to the reference configuration, are called the material or reference coordinates.

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