Solid State Physics Lecture 6 of 20

May 18, 2012 by Sandro Scandolo

S. Scandolo, ICTP

This lesson continues with the study of the X-ray diffraction problem in the case of three dimension and it was introduced the concept of the reciprocal lattice. It can be mathematically proved that in three dimensions it can be obtained three independent vectors (called G1, G2, G3 in the present lesson) that satisfied the condition of maximum intensity of the emitted radiation and all the others vectors that satisfied this condition can be obtained as a linear combination of G1, G2, G3. These three vectors can be derived from the Bravais lattice primitive vectors having units of inverse length. As these vectors have the same properties that the Bravais lattice primitive vectors they generate a lattice in the space of inverse length, this lattice is called the reciprocal lattice.

Then the experimental conditions and problems to obtain the reciprocal lattice were discussed, for example in order to have a maximum intensity ones have to obtain that the different between the it the incident and the emitted wave vectors is equal to one of the G vectors. For this purpose the detector is moved in the surface of a sphere, but in theory just this procedure it is not enough one has to do something more. For example it is possible to change the wavelength of the incident beam till we "hit" one of the reciprocal lattice vectors but this mean must of the time to change the source of the emission. The second possibility is to keep the wave length fixed and to rotate the sample in such a way to may coincide the reciprocal lattice vectors with the surface of the sphere. This is the case for a sample made of a single crystal in practice what we have is a sample made of grains oriented in every direction thus there is no need for rotating the sample.

Thus X-ray diffraction experiment allows us to obtain the reciprocal lattice. But using the mathematical statement that the reciprocal lattice of the reciprocal lattice is the original Bravais lattice ones can determine the Bravais lattice of the sample. In other words the real lattice is the reciprocal lattice of the one you see in the experiment.

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