![]() The position coordinates of the atoms in the unit cell are given as: (0 0 0) andįor identical atoms, f 1 = f 2 = f (say). The general form of the structure factor is given by The diamond structure is formed by the combination of two interpenetrating fcc sub- lattices: the basis being (0 0 0) and Find the structure factor of the basis and prove that if all indices are even, the structure factor of basis vanishes unless h + k + l = 4n, where n is an integer. Therefore,į(hkl) = 4f when h k l are odd or all even, and Now, we know that exp(πim) = (-1) m = (-1) if m is odd = (+1) if m is evenįor identical atoms, f 1 = f 2 = f 3 = f 4 = f (say). = f 1 + f 2 exp πi(h + k) + f 3 exp πi (h + l) + f 4 exp πi (k + l) Position coordinates of these atoms in the unit cell are: Substituting these values in equation (1), we get In an fcc structure, there four atoms in the unit cell. General form of the structure factor is given by Hence show that for the fcc lattice, no reflection can occur for the partly even and partly odd indices. Find the geometrical structure factor for an fcc structure in which all atoms are identical. Volume of the direct unit cell is given by Determine the primitive translation vectors of the reciprocal lattice. The primitive translation vectors of a hexagonal space lattice may be taken as where are unit vectors. Here, λ be maximum, sin θ must be maximum, i.e. We know that the Bragg’s equation is given by 2d sinθ =nλ Determine the longest wavelength that can be analyzed by a rock-salt-crystal with interplanar spacing 2.82Å in the first and the second orders of the X - ray diffraction. If the lattice parameter of the bcc iron is 2.87Å, determine the wavelength of the X - ray used. The first peak was observed for the (110) plane at 2θ = 44.70 0. X - rays of unknown wavelength are diffracted from an iron sample. Calculate the wavelength of X-ray if the lattice parameter of the crystal is 3.15Å.Ģd sinθ = nλ, n = 1, (110) plane, θ = 20.2 0, a = 3.15A 0 The first order Bragg’s angle corresponding to the (110) plane is 20.2 0. A bcc crystal is used to measure the wavelength of some X - ray. Given (hkl) = (220), structure is fcc, Bragg's angle θ = 38.2 0, λ = 1.54Å, a = ?įor a cubic crystal, we know that the interplanar spacing ' d' and the lattice parameter ' a' are related throughįurther, from Bragg’s equation, we have 2d hkl sinθ hkl = λ Determine the lattice parameter of a nickel (fcc) if Bragg's angle for its (220) reflection is 38.2 0 and the wavelength of the X - ray used is 1.54Å. Find the wavelength of the X - ray used.Ģd sinθ =nλ For 1 st order diffraction, n = 1 The first order peak from the (111) plane appears at an angle of 21.7 0. ![]() The lattice parameters of a copper (fcc) is 3.51Å. Similarly, for third order diffraction, n = 3, the Bragg's equation becomes 2d sinθ 3 = 3λ Given n = 1, θ = 15 0, λ = fixed, θ 2 = ?, θ 3 = ?įor first order, the Bragg’s equation is 2d sinθ 1 = λįor second order diffraction, n = 2, the Bragg's equation becomes d sinθ 2 = λ Determine the angle for second and third order when the same X - ray beam is used. When a crystal is subjected to a monochromatic X - ray beam, the first order diffraction is obtained at an angle of 15 0. ![]()
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