To describe crystal structures once it is known we need only direct lattice. But how to get the information about the crystal structure , i.e. the details of atomic arrangement? The ready answer is of course X-ray diffraction. Now X-ray diffraction is not a direct microscopic technique. We do not see the atoms directly by this technique. We only see Bragg peaks in different positions and directions after the X-ray is scattered from a single crystal (Debye rings for powder samples) and detected on a film or by single or multidetectors. So we can determine the angular positions and of course their intensities. We can collect these information and plot them somehow. What is the relationship between these information and the actual crystal structure? We do not see directly the atoms by X-ray diffraction. We have to live with the information we have obtained from X-ray diffraction experiments. Now it turns out that we can relate the Bragg spots (or Debye rings) with the atomic planes of the direct lattice that you obviously like very much. The atomic planes can be defined with the help of miller indices (hkl). So each Bragg peak we have detected and measured in the diffraction experiment has a particular set values (hkl). But we still do not know the relationship between the direct crystal and the Bragg position and intensity. The relationship is not obvious because the diffracted spots are arranged in lattice, albeit on a distorted one. It is not easy to look at and understand their arrangement. Now physicists (and crystallographers) being intelligent guys, thought of some tricks to transform the angular information of the Bragg spots (arranged on a distorted lattice) in terms of a regular arrangement of a lattice. This transformed arrangements bears a simple reciprocal relationship to the direct lattice. This is called the reciprocal lattice. We can arrange the Bragg spots on this lattice and even can ascribe intensity values to all of these spots. Now surprise surprise! Each set of direct atomic planes has become now a point in the reciprocal lattice and carry the same miller indices (hkl) as the corresponding atomic planes and we now even know the intensity. This reciprocal lattice has lot of symmetry that are related to the symmetry of the direct lattice. As long as we do not know the unknown crystal structure and analyze the diffraction data for solving the crystal structure it is convenient to stay in the space for which we have direct experimental information. Once we analyze the data and determine the crystal structure we can then transform the information to the direct space. You must know that these two spaces are related by Fourier transforms. The information of the reciprocal lattice gives only the symmetry and translational properties but not the actual atomic positions. For that we must use the intensity values corresponding to the each reciprocal lattice point (hkl). Here I have illustrated how the reciprocal space is used to analyze the crystal structure only. Now if you real the text book of Kittel then you will see that the same reciprocal space is the most convenient space for the calculation of solid state properties and electronic structure. You can do these calculations in the direct space as well but the calculations would be more difficult.

I hope that I have convinced you about the usefulness of the reciprocal space. If not then come back again to me and ask questions.

To describe crystal structures once it is known we need only direct lattice. But how to get the information about the crystal structure , i.e. the details of atomic arrangement? The ready answer is of course X-ray diffraction. Now X-ray diffraction is not a direct microscopic technique. We do not see the atoms directly by this technique. We only see Bragg peaks in different positions and directions after the X-ray is scattered from a single crystal (Debye rings for powder samples) and detected on a film or by single or multidetectors. So we can determine the angular positions and of course their intensities. We can collect these information and plot them somehow. What is the relationship between these information and the actual crystal structure? We do not see directly the atoms by X-ray diffraction. We have to live with the information we have obtained from X-ray diffraction experiments. Now it turns out that we can relate the Bragg spots (or Debye rings) with the atomic planes of the direct lattice that you obviously like very much. The atomic planes can be defined with the help of miller indices (hkl). So each Bragg peak we have detected and measured in the diffraction experiment has a particular set values (hkl). But we still do not know the relationship between the direct crystal and the Bragg position and intensity. The relationship is not obvious because the diffracted spots are arranged in lattice, albeit on a distorted one. It is not easy to look at and understand their arrangement. Now physicists (and crystallographers) being intelligent guys, thought of some tricks to transform the angular information of the Bragg spots (arranged on a distorted lattice) in terms of a regular arrangement of a lattice. This transformed arrangements bears a simple reciprocal relationship to the direct lattice. This is called the reciprocal lattice. We can arrange the Bragg spots on this lattice and even can ascribe intensity values to all of these spots. Now surprise surprise! Each set of direct atomic planes has become now a point in the reciprocal lattice and carry the same miller indices (hkl) as the corresponding atomic planes and we now even know the intensity. This reciprocal lattice has lot of symmetry that are related to the symmetry of the direct lattice. As long as we do not know the unknown crystal structure and analyze the diffraction data for solving the crystal structure it is convenient to stay in the space for which we have direct experimental information. Once we analyze the data and determine the crystal structure we can then transform the information to the direct space. You must know that these two spaces are related by Fourier transforms. The information of the reciprocal lattice gives only the symmetry and translational properties but not the actual atomic positions. For that we must use the intensity values corresponding to the each reciprocal lattice point (hkl). Here I have illustrated how the reciprocal space is used to analyze the crystal structure only. Now if you real the text book of Kittel then you will see that the same reciprocal space is the most convenient space for the calculation of solid state properties and electronic structure. You can do these calculations in the direct space as well but the calculations would be more difficult.

I hope that I have convinced you about the usefulness of the reciprocal space. If not then come back again to me and ask questions.

Dear Sir,

Thank you very much for your useful answer. Now I can understand the role of reciprocal lattice.I will be very happy If i get the answer from you with an example (i mean any one of metals or semiconductors). I am waiting for your reply

I try to give short answer.

Using of recirpocal lattice is very convinient becouse condition of difraction reflex consist in coincidence of radius-vector of reflex and vector of reciprocal lattice. So, from experimental radius-vectors of reflexes one can easy to determine vectors of reciprocal lattice. then one can to built reciprocal lattice and to determine direct lattice.

Let me recall some properties of the direct and reciprocal spaces. The reciprocal relationship between the two spaces ensure that the reciprocal space of the reciprocal space is the direct space. For experimentalists who regularly use diffraction methods the reciprocal space is not just an imaginary space it is the space (albeit often distorted) where they measure and record their data. There are techniques viz. Buerger precession camera technique where you can get rid of the distortion and record the undistorted reciprocal space directly. It is done by precessing the film appropriately. This technique is now seldom used because the necessary transformation can be done easily by a computer. I still recall my wonderful feeling when I looked at the precession of several running Buerger precession cameras for the first time in a big experimental hall in Max-Planck-institute for Solid State Research in Stuttgart. Once you develop such a film then you realize that you have photographed the reciprocal space directly. So that was in fact more reality for me than the direct space of the material about which we knew very little at that stage.

For theoretical solid state physicists although they use the same reciprocal space (with an extra factor 2*\pi) as the experimental crystallographer, the reciprocal space is the Fourier or momentum space where the calculations can be done more conveniently. They seldom use the direct space.

Dear Tapan Chatterji,

Thank you very much for sharing your nice experience. It makes me to learn something new.

In short the crystal structure which you wish to know is in the direct space. But the X-ray (or neutron or electron) diffraction data are in the reciprocal space, distorted or undistorted. You have get the information of the direct space from the data you have in the reciprocal space. It would be extremely easy to get the information of the direct space if you knew not just the positions and intensities at all reciprocal points but also their phases. But you cannot record the phases in your conventional diffraction experiments and therefore you have to live with whatever you have. Now the aim of the crystallographic methods is to get the correct information from the direct space by some means despite the missing information of the phases. This is called the phase problem of crystallography. There are crystallographic methods which manages to overcome this difficulty.

Note that some modern microscopic techniques that allow you to glimpse the atomic world (direct space) directly. But these techniques are difficult and still are in the development stages.

I agree withTapan

Dear Sridhar, the reciprocal space (including reciprocal lattice concept) is related to Quantum Mechanics formalism (based on Schrodinger equtions). It's a way to easier understand some fenomena (much more than "just for read" some diffraction data). :) "The reciprocal lattice of a reciprocal lattice is the original lattice." Then, why ?!!! - you've asked a question somekind like this.... Try to read what's on the following links (at least)... http://en.wikipedia.org/wiki/Position_and_momentum_space ; http://en.wikipedia.org/wiki/Reciprocal_lattice

The idea of reciprocal lattice is the invention of crystallographers like Ewald. The motivation was to understand X-ray diffraction results. It was done before the Schrodinger's formulation of wave mechanics. When wave mechanics became known then there was interests to study (1) electron diffraction and also (2) quantum mechanics of electrons in periodic potentials. The former (1) was treated by Bethe and the later (2) was studied by Bloch, Pierels and others. These two problems are found to be identical and both needed reciprocal lattice concept because the main problem was after all just electron diffraction from the periodic potential and is similar to X-ray diffraction. So the concept of reciprocal lattice is common to all these problems which are in fact identical. You do not need to read wikipedia for this. In fact you should not encourage any scientist to read wikipedia and learn from there rather than from text books and original papers.

Tapan, it's there some paradox in your words ? :) "You do not need to read wikipedia for this. In fact you should encourage any scientist to read wikipedia and learn from there rather than from text books and original papers."

Sorry I droped not. So please read "should not" instead of "should". I am extremely for that.

:) It was not an encouragement, I also dislike questionable sources when talking about scientific research... It was just that in our friend I saw still a young student with some divergent questions (like on Phonon, General Relativity, Materials Physics, Astroparticle Physics domains) - that fact beeing sure a big plus for him - but there is not much time for a PhD student to read entire books and articles on collateral scientific interests... Wikipedia is not necessarily and always a bad online encyclopedia - it can be a good reminder, or abstract on some theories, giving also links to books and scientific papers.

Surely Wikipedia is a ready source of information. I use it often when I wish know about something a bit out of my research field. You get some useful references there. But if you encourage the students to learn from Wikipedia then the next generation of researchers will not read the excellent text books available. There exist another source of evil already, black box programs, which the students use without understanding. They even need a program to calculate the density from the structural information (there was such a question in Research Gate recently: where can I find a program that calculates the density from the cif file? There were immediate offers of several programs for calculating density. Can you imagine that someday that guy will be a responsible teacher!

The reciprocal lattice emerges naturally if you go through the mathematics of kinematical diffraction (basically Fourier transforming the electron density for a lattice). It can be worked out pretty easily (I do so in my book on XAFS). The diffraction condition is met when the photon momentum transfer vector k'-k lies on a point in the reciprocal lattice.

The reciprocal lattice to the direct space x-lattice obtained by Fourier transform into the k-space , is a basis for construction of the theory of condensed matter physics. It enables to formulate Bloch theory of electrons and of other quasiparticles by diagonalization the lattice Hamiltonians in k-space. It enables to find band energy spectra (Bloch scheme), quasiparticle dispersion laws, effective masses and simple scheme for interaction of quasipatricles. Since 50-ties (H. Froehlich) it is also a basis for introduction of field-theoretical methods into CM effective especially in problems with interactions.

However, when the translation invariance is lost (disordered systems, inhomogeneous materials) working in real space is more convenient and appropriate (for theoretical studies) . One will also be able to extract the dispersions and quasiparticles,.. but the quasi particle peak will have now a finite linewidth.

It is but questionable whether the quasiparticles could be defined at all. Wave fuctions of localized states exponentially fall down on the localization length.

A quasi particle can have a finite linewidth and still describe a well defined excitation ( extended state) ; a naive criterion would be if the width of the quasi particle peak is smaller that the its energy then the excitation is well defined...concerning the localization length,..in the Anderson model of disorder, if the disorder strength is not too strong a mobility edge separates localized from extended states (in 3D), ..in 1D all states are localized for an arbitrary small amount of disorder.

I would like to complete, that the quasiparticle concept based on the band index k, the wave number which spans over a Brillouin zone , can not be used for disordered solids. One exception is if one works within a model of CPA (coherent potential approximation) which is a kind of a mean field. Otherwise the wave number is not a good quantum number. and the quasiparticles in the sense like in periodic lattices does not apply. The problem of another quasiparticles can be even more delicate. E.g, internal motion of Wannier-Mott exciton is easily dissociated by the disorder. .

What you measure in XRD are the data in reciprocal space but the atoms of the sample that scatter X-rays are in the direct space. The two spaces are related by Fourier transformation. If you Fourier transform the direct space you get the reciprocal space and if you inverse transform the reciprocal space you get back the direct space. Now the  in X-ray diffraction data that you measure are information in reciprocal space. But you wish to know the structure of your sample. So you must inverse Fourier transform the data in reciprocal space. To do that you need amplituses and also the phases. But you have already lost the phase information in the experiment; You only know the amplitudes from the measured intensities. That is why structure analysis is not just a straight forwad direct mathematical process. The phases must be guessed or deterlmined or found by trial and error with the crystal chemical knowledge. These methods are described in the text books and literature and also incorporated in black box programs which do things for you and you remain a routine user of the programs. I advise you not to be such a user but try to understand the physics and mathematics involed in structural analysis.

How did I miss this nugget? I got a lot of catching-up to do.

I continue to get better, learning from you folks :-) Thanks!

https://www.flickr.com/photos/85210325@N04/14348488010/in/album-72157645018820696/

https://www.flickr.com/photos/85210325@N04/10515372183/in/album-72157645018820696/

Every crystal has both direct crystal lattice and reciprocal lattice. A plane in the direct lattice is represented by a point in reciprocal lattice. A microscope image is of a direct lattice while X-ray diffraction pattern (SAED) is a picture of reciprocal lattice.

Thanks Tapan Chatterji   for giving me a way how i understand reciprocal lattice 

THANKS AGAIN SIR

"We do not see the atoms directly by this technique." Exactly!

However, it is possible to image the strain fields around defects in the atomic arrangement in near perfect single crystals using XRD in reciprocal space. The spatial resolution would depend on the minimum voxel size that may be examined. Each Laue spot is afterall a "miniature" topograph of the specific crystallographic plane corresponding to the (hkl). The topograph, Id(x, y) is the diffraction result of the 2D "incident beam" Io (x, y) interacting with the voxel "electron density" matrix.

The measurements in reciprocal space make "small big and big small" i.e., the values are inverted :-) Thereby, making it even easier to measure miniscule distortions in the REAL/DIRECT space through the reciprocal space created by the Fourier interaction in XRD of real space and the X-rays wave. That is quite an unique NDE tool without the stringent environment  (vacuum etc.) needed in electron microscopy! The spatial resolution  in XRD imaging is presently limited by the pixel size available (1-5µm presently) and the optics utilized. See below for a method to enhance this current spatial resolution limit dramatically beyond 25X (Review of Scientific Instruments, 50(1), 26 (1979); "X-ray magnifier"; Boettinger, W.J.; Burdette, H.E.; Kuriyama, M.).

https://www.researchgate.net/post/OAT-Oblique_Angle_Topography_Is_this_the_first_in_the_world_Do_you_know_of_anyone_else_in_literature_that_has_used_this_method_for_magnification

https://www.ncbi.nlm.nih.gov/pubmed/18699332

Thanks Tapan Chatterji for the wonderful explanation. Now I am clear about it. I enjoyed reading how the image of atoms in direct lattice are constructed using the information from XRD and by using transformation.Someday I'd like to do a project on structural analysis of any sample. Is it possible for me to do that?

Nice conceptual explanation. Dear Prof. Tapan, can you recommend some best books to read about this from your experience?