|"... The nature is created so that all the simple in it is true,|
and all the complicated is false."
Grigory Skovoroda.(Ukrainian philosopher, XVIII century)
As the title figure of this page shows, you will find below a lecture by an imaginary professor (in the middle) about the peculiar ways in which scientists extract atomic level information about the structure of crystals. In particular we will concentrate on the reciprocal lattice and its relationship with the real lattice. The reciprocal lattice can be observed if we shine X-rays or other short wave radiation onto the real lattice. Unlike the real lattice, the reciprocal lattice can be confusing and needs definite knowledge to be interpreted. However even at this point we can state that the things which are larger in real space are smaller in reciprocal space by definition.
:Before we go any further, let me introduce you to the student.
P: Reciprocal space is also called Fourier space, k- space, or momentum space in contrast to real space or direct space.
The concept of the reciprocal lattice was devised to tabulate two important properties of crystal planes: their slopes and their interplanar distances.
The reciprocal space lattice is a set of imaginary points constructed in such a way that the direction of a vector from one point to another coincides with the direction of a normal to the real space planes and the separation of those points (absolute value of the vector) is equal to the reciprocal of the real interplanar distance.
P: Nota bene!!!
It is convenient to let the reciprocal lattice vector be 2p times the reciprocal of the interplanar distance. This convention converts the units from periods per unit length to radians per unit length. Radians per centimeter (cm-1) are widely used units. It simplifies comparison of different periodic phenomena. For instance k , the wavevector, has an absolute value of 2p/l. If we choose the above convention, we are able to compare the two values directly. Henceforth this convention will be used. All it does is expand the size of the reciprocal lattice.
S:If the reciprocal lattice is imaginary, how is it described?
P:The points of the reciprocal lattice have the same meaning as the points defined in geometry, - there is nothing specific located there. However a spot of diffracted light could be observed at that location. We can describe the reciprocal lattice in the same way we describe the real ones, but we should keep in mind the difference: one indicates the location of real objects (atoms) and has dimensions of m, the other indicates the positions of abstract points (magnitude and direction of momentum ) and has dimensions of m-1.
P: If you wish to see how the reciprocal space looks like, you can use the visual aids available in our class.
P:Of course... The relationship between the period and frequency is similar to that of the reciprocal and the direct lattice. Therefore Fourier transformation is used in the studies of the real lattice to yield the reciprocal lattice in the same fashion as with the studies of any other periodic function, therefore the reciprocal space is also called Fourier space.
S:How can I construct a reciprocal lattice from a direct one?
P:It is easy - Pick some point as an origin, then:
a) from this origin, lay out the normal to every family of parallel planes in the direct lattice;
b) set the length of each normal equal to 2p times the reciprocal of the interplanar spacing for its particular set of planes;
c) place a point at the end of each normal.
P: You can better understand the procedure with the use of the same visual aids as before.
P: The direct lattice is the reciprocal lattice of its reciprocal lattice in the same way as 1/(1/a) = a, and so the rules are the same as written above.
S: It looks quite difficult to lay out normals to millions of parallel planes of the real crystal?
P: Yes, but fortunately the reciprocal lattice can be described by just one unit cell which can be multiplied by translation along all the coordinate axes in the same fashion as a direct lattice.
S: Ok, now I know how to build the reciprocal lattice from the direct one. How can I apply my knowledge in real life?
P: If you are familiar with the reciprocal lattice, you can understand and interpret the results of diffraction experiments, and obtain useful information about the internal structure of crystalline matter.
S: Oh, that is interesting! So how is it done?
P: It is accomplished using the so called Ewald construction which allows one to put the information about the wavelength and the direction of the incident radiation into the reciprocal lattice and determine the diffraction pattern in a relatively straightforward way.
S: Ok, we can study the periodicity of solids using the above approach... A solid is impossible without a surface... The surface has periodicity as well... How can we apply the gained knowledge to study the surface?
P: At the surface the translational symmetry of the bulk of a crystal breaks down. A surface has finite translations in two dimensions which lay in its plane and the absence of a translation along the normal can be envisaged as a translation to infinity in that direction. The reciprocal lattice and the Ewald construction of the surface is built following the same rules as for the bulk. However since the reciprocal of infinity is equal to zero, it has a multitude of points laying infinitely close to each other in the direction of the normal. The collection of these points which spread from the two-dimensional surface lattice to infinity are called Bragg rods. Because the rods are infinitely dense lattice points, the diffraction from a surface occurs continuously with changes in the direction and the magnitude of the incident wavevector as long as the wave is short enough to be diffracted.
P: Ok, I'm tired, but fortunately that is all I wanted to say today. Do you have any questions?
S: Yes Sir. It is still not completely clear why the reciprocal space is called momentum space.
P: Well that is easy,- because momentum is directly proportional to wavevector and is inversely proportional to the wavelength. Therefore each point of the reciprocal lattice represents the direction and the magnitude of momentum.
S: Ok, it seems not too difficult to follow all the material covered. It is definitely more easy than I thought. But I think, or rather feel that there should be some point which people find challenging. What did you find most difficult in understanding the reciprocal lattice?
P: This is the secret of course, but despite lecturing on reciprocal lattice for years, I still cannot imagine the origin at infinity. This will unfortunately not be only instance when the best reference point is so far from our laboratory...
We all are learning something every day to satisfy our natural curiosity. But we should not go too far....
|Conclusion: Never press this button!!!|