A very similar question was asked by S. Sampath a few months ago in RG. His question was:

What is the need of using reciprocal lattice? Why should we not use direct lattice?

Here were my answers:

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.

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 encourage any scientist to read wikipedia and learn from there rather than from text books and original papers.

Check out Kittel or Ashcroft's Solid State Physics man. It is hard to explain with a few words. Anyway, in short, it is the invert of real space and it measures how "fast" a point/plane repeats. Physicists use it to explore material properties in a periodic system, i.e., electrons/phonon dispersions, because of its convenience. It is like Fourier Transform which converts a quanity from time domain to frequency domain. The principles behind are the same.

A very similar question was asked by S. Sampath a few months ago in RG. His question was:

What is the need of using reciprocal lattice? Why should we not use direct lattice?

Here were my answers:

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.

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 encourage any scientist to read wikipedia and learn from there rather than from text books and original papers.

In order to describe a single crystal we take only direct lattice. But in order to describe the whole crystallographic structure(large number of crystal), in which there are lots of plane having same slope and dimensions so it is very difficult to study every plane separately we take reciprocal lattice. Which is easier method to study the properties of whole crystal.

This is a lattice in space of quasipulses and it reflects translatioal and other symmetries of real lattice.

those are the dimensions of the crystal in the k- space and used to simplify the study of the crystal structure and diffraction process , the we can convert them to real space

A point en reciprocal lattice represent a plane in real space

"A point in reciprocal lattice represent a plane in real space"...is not really correct since only a direction (hkl) in reciprocal space represents a plane in real space. The points in reciprocal space are described by Laue indices hkl which dont have to be coprime whereas lattice planes are described by Miller indices (hkl) where h,k and l must be coprime. The same happens with lattice directions [uvw] and lattive points :uvw:. First are defined to be coprime whereas the latter describe any lattice point by a vector and dont have to be coprime.