Section 02_X-ray_Diffraction

Section 02_X-ray_Diffraction - Physics 927 E.Y.Tsymbal...

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Physics 927 E.Y.Tsymbal Section 2: X-ray Diffraction and Reciprocal Lattice Bragg law . Most methods for determining the atomic structure of crystals are based of the idea of scattering of radiation. X-rays is one of the types of the radiation which can be used. The wavelength of the radiation should have a wavelength comparable to a typical interatomic distance which is in solids of a few angstroms (10 -8 cm). The x-ray wavelength O can be estimated as follows 12.4 (Å) () QO O ± hc Eh EkeV . (2.1) Therefore, x-rays of energy 2-10 keV are suitable for studying the crystal structure. X-rays interact with electronic shells of atoms in a solid. Electrons absorb and re-radiate x-rays which can then be detected. Nuclei are too heavy to respond. The reflectivity of x-rays is of the order of 10 -3 - 10 -5 , so that the penetration in the solid is deep. Therefore, x-rays serve as a bulk probe. In 1913 Bragg found that crystalline solids have remarkably characteristic patterns of reflected x-ray radiation. In crystalline materials, for certain wavelengths and incident directions, intense peaks of scattered radiation were observed. Bragg accounted for this by regarding a crystal as made out of parallel planes of atoms, spaced by distance d apart. The conditions for a sharp peak in the intensity of the scattered radiation were that: (1) the x-rays should be specularly reflected by the atoms in one plane; (2) the reflected rays from the successive planes interfere constructively. Fig.1 dsin T T d Fig.1 shows x-rays which are specularly reflected from adjacent planes. The path difference between the two x-rays is equal to 2 d sin T . For the x-rays to interfere constructively this difference must be an integer number of wavelengths. This leads to the Bragg condition: 2 d sin T = m O (2.2) The integer m is known as the order of the corresponding reflection (or order of interference). There are a number of various setups for studying crystal structure using x-ray diffraction. In most cases, the wavelength of radiation is fixed, and the angle is varied to observe diffraction peaks corresponding to reflections from different crystallographic planes. Using the Bragg law one can then determine the distance between the planes. The Bragg law is greatly oversimplified (but it works!). It (i) says nothing about intensity and width of x-ray diffraction peaks; (ii) neglects differences in scattering from different atoms; (iii) neglects distribution of charge around atoms. 1
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Physics 927 E.Y.Tsymbal Diffraction condition and reciprocal lattice . Later von Layer introduced a different approach for x- ray diffraction. He regarded a crystal as composed of identical atoms placed at the lattice sites T and assumed that each atom can reradiate the incident radiation in all directions. Sharp peaks are observed only in the directions and at wavelengths for which the x-rays scattered from all lattice points interfere constructively.
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This note was uploaded on 03/11/2012 for the course PHYSICS 927 taught by Professor Staff during the Fall '11 term at UNL.

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Section 02_X-ray_Diffraction - Physics 927 E.Y.Tsymbal...

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