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Bragg's Diffraction.

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Published in: Physics
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Presentation On Bragg's Diffraction.

Akhilesh K / Lucknow

4 years of teaching experience

Qualification: M.Sc (NIT Rourkela - 2019)

Teaches: All Subjects, English, Mathematics, Science, Chemistry, Physics, Algebra, IIT JEE Mains, AIPMT, NEET

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  1. Bragg's diffraction,
  2. FCC Versus HCP FIRST LAYER AT 'A' SITE Leaves us with the possibility of putting second layer at 'B' site or 'C' site. A site B site C site SECOND LAYER AT 'B' SITE gives us the option of putting the third layer at 'C' site or at 'A' site 1 0
  3. If the third layer is put at 'A' site, then ABABAB... structure is formed. Hexagonal ABABAB.. If the third layer is put at 'C site, then ABABCABC... structure is formed. ABCABCABC.. FCC
  4. layer of spheres (a) (b) Hexagonal Close Packed Structure (c) Face Centered Cubic Stru ctu re
  5. Bragg's diffraction. So far we have seen the various facets of crystal structure. Since the atomic radius is of the order of Angstrom; a level much below our eyes resolution, how is it possible to comment so accurately about the atomic arrangements or crystal lattice. In recent years though we have developed many tools for direct visualization of the atoms e.g., electron microscope, atomic force microscope, etc, but these were not available some 50 years back. Still the crystal structure analysis has been going on for more than 50 years. Then how did people identified the structures so accurately, that even today those results are in agreement with the results obtained via latest techniques. For this the credit goes to two physicist, W.L.Bragg and W.H.Bragg. In 1913, they put forward a theory (commonly know as Bragg's theory) of x-ray diffraction by the crystals, which in turn may be used for their structure determination.
  6. According to Bragg, the periodic arrangement of atoms in the crystal acts like a set of discrete parallel planes separated by a distance 'd'. If a x-ray beam (having wavelength comparable to 'd') is made incident on this sample, then the x-rays will undergo diffraction from the various planes of the crystal and a corresponding diffraction pattern will be observed. There will be maxima in the pattern whenever, the following relation is satisfied = 2dsin9 where n is an integer, A is the wavelength of incident wave, d is the spacing between the planes in the atomic lattice, and 0 is the angle between the incident ray and the scattering planes. This Bragg's relation can be used only for X < 2d.
  7. The incident x-ray radiation would produce a Bragg peak if their reflections off the various planes, interfered constructively. The interference is constructive when the phase shift is a multiple of 21T or nÅ; this condition can be expressed by Bragg's law, Destructive interference 29 29 Constructive interference
  8. For a crystalline solid, the waves are scattered from lattice planes separated by the interplanar distance d. Whenever the scattered waves interfere constructively, they remain in phase since the path length of each wave is equal to an integer multiple of the wavelength. The path difference between two waves is given by 2dsine, where e is the scattering angle. Hence whenever this path difference is equal to nÅ, the constructive interference condition is satisfied. This leads to Bragg's law, = 2dsin6 d dsine A diffraction pattern is obtained by measuring the intensity of scattered waves as a function of scattering angle. Very strong intensities known as Bragg peaks are obtained in the diffraction pattern when scattered waves satisfy the Bragg condition.
  9. 100 80 60 40 20 111 10 (002) (102) A typical x-ray diffraction pattern (202) (112) 20 (204) (211) (114) 30 (220) 11 11 60 1 11 40 Scale 50 70 2-Theta - Few crystallographic planes in a 2D square lattice
  10. Reciprocal lattice: It is a lattice formed by the vectors which are reciprocal to the original (direct) lattice vectors. This lattice is also known as Fourier transform of the original lattice (or direct lattice). This space is also known as momentum space or less commonly k- space, due to the relationship between the momentum and position. Reciprocal Lattice Vectors 3 a3 X al 2 3 Need of reciprocal lattice: We have seen in Bragg's diffraction that each set of plane corresponds to one peak in the x-ray diffraction plot. Since each set of planes may be containing a very large (N 1023) atoms and it is very difficult to deal with such a large number of atoms in real space, hence a reciprocal space is created. Each point in reciprocal space corresponds to a set of planes.
  11. Reciprocal Lattice of FCC is a BCC lattice 2m volume = älü2 x ä3 a3 X al al X a 2m
  12. 1. 2. 3. 4. 5. Assignment [to be discussed on 26 march] Show that reciprocal Lattice of (i) BCC is a FCC and (ii) SC is SC Find the packing fraction of (i) simple cubic (ii) diamond lattice (iii) NaC1 The geometric arrangement of LiF is same as that of NaCl. If the unit cell of LiF is 0.402 nm, find the density of LiF. The crystal structure of calcium is face centered cubic. Its cell edge is 0.557 nm. Calculate the radius (in nm) of the calcium atoms, assuming they are spheres. [Ans: 0.197 nm] The crystal structure of calcium is face centered cubic. Its cell edge is 0.557 nm. Calculate the density of Ca in g /cm3 [Atomic mass, Ca, 40.08; Avogadro's number 6.022 x 1023] [Ans: 1.54g/cm3]

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