jueves, 18 de marzo de 2010

Reciprocal space and the reciprocal lattice

By analogy with our definition of a one-dimensional Fourier transform, the three-dimensional Fourier transform can be defined as:

The only differences between this and the one-dimensional equivalent are that F and f are functions of three-dimensional vectors K and r respectively, and the product in the exponential is now a vector dot product. (Again, I am not worrying about minus signs and other factors in the exponential.) When we were thinking about how to perform a one-dimensional transform on a one-dimensional grating function, we used a diagram like this:

We multiplied our grating function by a stationary corkscrew function, and then integrated the resulting product over the entire extent of x. In two dimensions, the corkscrew function eiK.r is just a two-dimensional wave, as we described earlier.

Consider the distribution of atoms shown in the next diagram:

I've deliberately made the atom positions random, except for the fact that they are all roughly arranged on the blue lines.

Now when we take the Fourier transform of this two-dimensional distribution, we multiply the object function by all possible two-dimensional corkscrew functions, each with a different two-component K-vector. Here are some examples, for different K-vectors, coloured brown:

Remember that the green lines are where the corkscrew function has a particular phase, say zero. At zero phase, the real (cosine) part of the corkscrew function has its maximum value. At points on lines between the green lines, the corkscrew function has a maximum negative value (cos π = -1). Lets imagine superposing one of these two-dimensional waves upon our object function, as shown below:

In the next diagram, I draw red lines between the green lines: on these red lines the corkscrew function has a negative value. When we form the product between the atom distribution and the corkscrew function, atoms that lie on a green line give a positive resultant (coloured dark in the diagram below). Atoms that arrive on the red lines give a negative resultant (coloured red below). Atoms that lie between these positive and negative lines give a small resultant (coloured grey below).

For nearly all orientations and periodicities of the two-dimensional corkscrew function (nearly all values of the vector K in eiK.r) there are as many positive (dark) atoms as negative (red) atoms. However, like in the one-dimensional example we discussed before, when the green lines are in synchrony with the blue lines (the only clear periodicity in this array of atoms) then the integral of f(r) times eiK.r is very large. Two example K vectors that satisfy this condition are shown below:

In reciprocal space (or 'k-space'), the two brown vectors correspond to positions plotted from the origin where F(K) is very large, like this:

The cross is the origin of 2D reciprocal space. The black dots are where F(K) has a very large value, at the positions of brown K-vectors in the previous diagram. In fact, corkscrew functions pointing in the opposite direction to these K-vectors will also give large Fourier integrals. Indeed, any multiple length of the brown vector also gives a peak. In the next diagram, we draw the main features of reciprocal space for this partly-ordered structure.

Spots (which in a real electron diffraction pattern would appear as bright points of light on the phosphor screen) represent prevalent periodicities in the object. I've drawn a grey circle on this picture to suggest a 'diffuse ring': it would be wrong to suggest that from such an object, disordered in all but one direction, there would be no scattering at all at K-vectors that do not correspond to the periodic planes. There is in fact weak diffraction, around several rings in reciprocal space (only one of which is drawn) caused by prevalent average separations of the atoms.

For an entirely crystalline object (and most materials are crystalline) then it stands to reason that there is an array of sharp peaks in reciprocal space. Consider the following two-dimensional crystal:

Reciprocal space will now consist of only a series of points. The points will be at K-vectors which are perpendicular to the blue lines (planes in three-dimensions) and with lengths which are multiples of 1/(the distance between the planes). If you are new to reciprocal space, then this is the point where most people's brains reach overload. In the next diagram, I draw the reciprocal vectors corresponding to the blue lines in brown.

I have also drawn two pink vectors, labelled a and b: these two vectors can be used to generate the real-space distribution of atoms. The position of any of the atoms can be described by r = na + mb, where n and m are integers. Crystallography starts with these pink vectors, and then, rather mysteriously, defines the reciprocal (brown) vectors, or at least a subset of the brown vectors from which can be generated all reciprocal lattice points.

It is important to understand that the brown vectors do not point in the same direction as the pink vectors unless the pink vectors are perpendicular to one another. Can you believe the right hand lattice is the reciprocal of the left hand lattice in the diagram below?

If you understand this fully, then you have understood one of the (several, but rather abstract) essential concepts in crystallography. I think the most general way of thinking about reciprocal space is as the Fourier transform of real space, which is the way I have described it on this page. This model can easily be extended to understand diffraction from thin and distorted crystals, as often encountered in electron microscopy. But this is not the way crystallographers generally think (or teach) the subject.

Most textbooks state a relationship between the two lattices above in terms of vector algebra. For the sake of completeness, let's derive this in simple steps, because it is important. Consider a bit of the real space distribution of atomic positions:

We know that the length of a*, which the reciprocal vector corresponding to a, must have a length of 1/d, where d is the spacing between the blue lines, marked as green in the diagram above. It must also be pointing in direction perpendicular to the blue planes. So we have to do two things: work out a way of constructing a unit vector (a vector of length 1) pointing in the direction of a* (call this a'), and work out a way of deriving the distance d.

The first step is easiest if we can imagine a third real-space vector, called c, which sticking into the paper (computer screen). By definition, the cross product of b with c is a vector pointing in the direction of a*: remember, the cross product, which we'll write as bxc, is a vector at right angles to both b and c. To get a unit vector, we must divide whatever we end up with by its length. So that means that

To work out d, we can take the dot product of a with a': that is, we pick out the component of a lying in the a' direction. We multiply a' by 1/d, to get

so that finally

There are similar expressions for b* and c*. The triple product a.bxc is the volume of the parallelepiped of sides a, b and c, which is the volume of the unit cell in the real space. This relationship is tricky to think of three-dimensional space. In general, c does not point perpendicularly into the paper (computer screen) but can be at any angle. In that case, the blue Bragg planes do not cut the paper (computer screen) at right angles, and so neither a* nor the green line d are in the plane of the paper. The above relationship still holds true. To get of picture of this in mind's eye, think our three-dimensional corkscrew made of sheets of cardboard:

You have to orientate this, altering also the separation of the sheets, until the sheets line up with a series of planes of atoms. You can try doing this if you have access to a ball-and-spoke model of a crystal. The brown pencil, of length 1/d, is then pointing from the origin of reciprocal space to some point in reciprocal space.

There are many planes which satisfy this condition, each accounting for one point in reciprocal space. Like the atom positions in real space, all the points in reciprocal space can be generated from just three of these k-vector (a*, b* and c*).

Clearly, if a* is the brown pencil in the diagram above, then b and c are bound to lie in the plane of the sheet of cardboard, whatever their direction or length. That is why the cross product, bxc can be used to generate a*. Reciprocal vectors are perpendicular to planes of atoms.

Kevin M Contreras H
Electrónica del Estado Sólido




The electron microscope does not have good enough resolution for accurate direct determination of unknown crystal structures. Diffraction patterns provide the most accurate data about crystal structures.

A sample of the crystal being studied is bombarded with photons, electrons or neutrons with an associated wavelength comparable to the interatomic spacing. A single atom (theoretically) scatters the incident waves equally in all directions, but in a crystal cancellation due to destructive interference gives zero in most directions. In certain directions constructive interference gives maxima of intensity, producing a pattern characteristic of the crystal structure. The problem for crystallographers to solve is how the positions of the peaks observed can be converted into useful information about the crystal structure.

2. The Bragg Law

This law was derived by the English physicists Sir W.H. Bragg and his son Sir W.L. Bragg in 1913 to explain why the cleavage faces of crystals appear to reflect X-ray beams at certain angles of incidence.

Derivation of the Bragg law; notice that reflection is specular

The Bragg law is based upon 2 assumptions:

1) Diffraction peaks are caused by constructive interference of waves reflected from parallel planes of atoms

2) The incident waves are reflected specularly, that is the angle of incidence is equal to the angle of reflection

For constructive interference, the path difference between waves reflected from the 2 planes must be an integer number of wavelengths. As can be seen in figure 11 above, the path difference is 2dsinq and so the Bragg law is
nl = 2dsinq (2)

There are many possible planes of atoms and so there are many peaks in the diffraction pattern. Only a few are shown on the diagram below, which shows a plane of atoms perpendicular to the c axis in an arbitrary crystal.

Some Miller planes which could cause diffraction

3. Experimental Methods

Peaks of intensity are hence obtained for an appropriate combination of l, d and q.

Since d is fixed, it is necessary to scan in either wavelength or angle.

In the Laue Method, a single crystal is stationary in a beam of x-ray or neutron radiation of continuous wavelength. Diffraction only occurs at the appropriate discrete values of l for which planes exist of spacing d and incidence angle q satisfying the Bragg law.

In the rotating crystal method, a single crystal is rotated about a fixed axis in a beam of monoenergetic x-rays or neutrons. The variation in angle brings different planes into the appropriate position for reflection.

In the powder method monochromatic radiation strikes a fine powder of the specimen. In this form the crystallites will be present in virtually every possible orientation. Diffracted rays are reflected from individual crystallites that happen to be in the appropriate orientation. The reflected beam is detected at an angle 2q to the original beam.

Example of an electron diffraction pattern obtained by the Laue method; this was taken using silicon with the electron beam parallel to [1 1 1]

Kevin M Contreras H
Electrónica del Estado Sólido


miércoles, 17 de marzo de 2010

Bravais Lattices

In two dimensions, periodic unit cells can have one of five basic shapes: general parallelogram, general rectangle, square, 60-120 degree rhombus, and generic rhombus. The last is often described as a "centered" lattice, a rectangle with an extra point in the middle, to bring out the rectangular nature of the pattern.

It's not too hard to see the rectangular pattern in a rhombic lattice, but it can be very hard to see the patterns in three dimensional lattices. For this reason, three-dimensional lattices must often be described as unit cells with additional points.

There are 14 basic unit cells in three dimensions, called the Bravais Lattices.


• P - Primitive: simple unit cell
• F - Face-centered: additional point in the center of each face
• I - Body-centered: additional point in the center of the cell
• C - Centered: additional point in the center of each end
• R - Rhombohedral: Hexagonal class only

Isometric Cells

The F cell is very important because it is the pattern for cubic closest packing. There is no C cell because such a cell would not have cubic symmetry.

Tetragonal Cells

A C cell would simply be a P cell with a smaller cross-section. An F cell would reduce to a network of I cells.

Hexagonal Cells

The R cell is unique to hexagonal crystals. The two interior points divide the long diagonal of the cell in thirds. This is the only Bravais lattice with more than one interior point. A rhombohedron can be thought of as a cube distorted along one of its diagonals.

Orthorhombic Cells

The orthorhombic class is the only one with all four types of Bravais lattice

Monoclinic and Triclinic Cells

Monoclinic F or I cells could also be represented as C cells. Any other triclinic cell can also be represented as a P cell.

Kevin M Contreras H
Electrónica del Estado Sólido


Van Hove singularity

A Van Hove singularity is a kink ("discontinuity") in the density of states (DOS) of a solid. The wavevectors at which Van Hove singularities occur are often referred to as critical points of the Brillouin zone. (The critical point found in phase diagrams is a completely separate phenomenon.) The most common application of the
Van Hove singularity concept comes in the analysis of optical absorption spectra.

The occurrence of such singularities was first analyzed by the Belgian physicist Léon Van Hove in 1953 for the case of phonon densities of states.[1]


Consider a one-dimensional lattice of N particles, with each particle separated by distance a, for a total length of L = Na. A standing wave in this lattice will have a wave number k of the form

where λ is wavelength, and n is an integer. (Positive integers will denote forward waves, negative integers will denote reverse waves.) The smallest wavelength possible is 2a which corresponds to the largest possible wave number kmax = π / a and which also corresponds to the maximum possible n: nmax = L / 2a. We may define the density of states g(k)dk as the number of standing waves with wave vector k to k+dk:[2]

Extending the analysis to wavevectors in three dimensions the density of states in a box will be

where d3k is a volume element in k-space, and which, for electrons, will need to be multiplied by a factor of 2 to account for the two possible spin orientations. By the chain rule, the DOS in energy space can be expressed as

is the gradient in k-space.

The set of points in k-space which correspond to a particular energy E form a surface in k-space, and the gradient of E will be a vector perpendicular to this surface at every point.[3] The density of states as a function of this energy E is:

where the integral is over the surface
of constant E. We can choose a new coordinate system such that is perpendicular to the surface and therefore parallel to the gradient of E. If the coordinate system is just a rotation of the original coordinate system, then the volume element in k-prime space will be

We can then write dE as:

and, substituting into the expression for g(E) we have:

where the
term is an area element on the constant-E surface. The clear implication of the equation for g(E) is that at the k-points where the dispersion relation has an extremum, the integrand in the DOS expression diverges. The Van Hove singularities are the features that occur in the DOS function at these k-points.

A detailed analysis[4] shows that there are four types of Van Hove singularities in three-dimensional space, depending on whether the band structure goes through a local maximum, a local minimum or a saddle point. In three dimensions, the DOS itself is not divergent although its derivative is. The function g(E) tends to have square-root singularities (see the Figure) since for a spherical free electron Fermi surface

so that .

In two dimensions the DOS is logarithmically divergent at a saddle point and in one dimension the DOS itself is infinite where
is zero.

A sketch of the DOS g(E) versus energy E for a simulated three-dimensional solid.

The Van Hove singularities occur where dg(E)/dE diverges.

Experimental observation

The optical absorption spectrum of a solid is most straightforwardly calculated from the electronic band structure using Fermi's Golden Rule where the relevant matrix element to be evaluated is the dipole operator
where is the vector potential and is the momentum operator. The density of states which appears in the Fermi's Golden Rule expression is then the joint density of states, which is the number of electronic states in the conduction and valence bands that are separated by a given photon energy. The optical absorption is then essentially the product of the dipole operator matrix element (also known as the oscillator strength) and the JDOS.

The divergences in the two- and one-dimensional DOS might be expected to be a mathematical formality, but in fact they are readily observable. Highly anisotropic solids like graphite (quasi-2D) and Bechgaard salts (quasi-1D) show anomalies in spectroscopic measurements that are attributable to the Van Hove singularities. Van Hove singularities play a significant role in understanding optical intensities in single-walled nanotubes (SWNTs) which are also quasi-1D systems.

Kevin M Contreras H
Electrónica del Estado Sólido


Debye versus Einstein

Debye vs. Einstein. Predicted heat capacity as a function of temperature.

So how closely do the Debye and Einstein models correspond to experiment?

Surprisingly close, but Debye is correct at low temperatures whereas Einstein is not.

How different are the models? To answer that question one would naturally plot the two on the same set of axes... except one can't. Both the Einstein model and the Debye model provide a functional form for the heat capacity. They are models, and no model is without a scale. A scale relates the model to its real-world counterpart.

One can see that the scale of the Einstein model, which is given by

is ε / k. And the scale of the Debye model is TD, the Debye temperature. Both are usually found by fitting the models to the experimental data. (The Debye temperature can theoretically be calculated from the speed of sound and crystal dimensions.)

Because the two methods approach the problem from different directions and different geometries, Einstein and Debye scales are not the same, that is to say

which means that plotting them on the same set of axes makes no sense. They are two models of the same thing, but of different scales. If one defines Einstein temperature as

then one can say

and, to relate the two, we must seek the ratio

The Einstein solid is composed of single-frequency quantum harmonic oscillators,

That frequency, if it indeed existed, would be related to the speed of sound in the solid. If one imagines the propagation of sound as a sequence of atoms hitting one another, then it becomes obvious that the frequency of oscillation must correspond to the minimum wavelength sustainable by the atomic lattice, λmin.

which makes the Einstein temperature

and the sought ratio is therefore

Now both models can be plotted on the same graph. Note that this ratio is the cube root of the ratio of the volume of one octant of a 3-dimensional sphere to the volume of the cube that contains it, which is just the correction factor used by Debye when approximating the energy integral above

Kevin M Contreras H
Electrónica del Estado Sólido