PH-409 (2015) Tutorial Sheet No. 2 * Problems shall be discussed in tutorial class ˆ ˆ ˆ 20*. A two dimensional lattice has basis vectors 2 ; 2 . Find the a i b i j basis vectors of the reciprocal lattice. 21. (a) Show that the reciprocal lattice of the reciprocal lattice is the original direct lattice. (b) Find the reciprocal lattice of a one dimensional lattice with spacing 'a'. Also find the first Brillouin Zone. Show that the reciprocal lattice to orthorhombic 22. a ; 90 o c -face centered lattice, having the following a b c primitive vectors, is another orthorhombic c -face centered lattice. a b ˆ ˆ ˆ ˆ ; ; a ai b i j c ck 2 2 23. Draw the first four Brillouin zones of a two dimensional square lattice and show that they are of equal area. 24*. Consider a plane in a crystal lattice. hk (a) Prove that the reciprocal lattice vector G is normal to this hA kB C plane. (b) Show that the distance between two adjacent (ℎ𝑙𝑚) planes of the lattice is given by the following. 2 d hk G (c) Using (b) show for a simple cubic lattice the following relationship. a d 2 2 2 h k 25. A simple orthorhombic lattice is characterized by following primitive vectors. ˆ ˆ ˆ ; ; ; a b c a a ai b bj c ck Find the angle between [ℎ𝑙𝑚] direction and normal to the (ℎ𝑙𝑚) planes.
26*. (i) Consider (100) plane of a bcc lattice. What would be the miller index of this plane when referring to the primitive vectors of the bcc lattice given below? ⃗ = 𝑏 ⃗ = 𝑏 ⃗ = 𝑏 ̂); 𝑐 ⃗ ̂); 𝑑 ̂) 𝑏 2 (−𝑗̂ + 𝑘̂ + 𝑙 2 (𝑗̂ − 𝑘̂ + 𝑙 2 (𝑗̂ + 𝑘̂ − 𝑙 (ii) Find the ‘d’ value for (hkl) planes, when the miller indices refer to the primitive vectors of the bcc lattice given in part (a) of this problem. (iii) Using the formula obtained in part (b) find the d value for the plane obtained as a result of part (a) of this problem and explain the answer. 27. What could be the maximum wavelength of a monochromatic x-ray beam such that when incident on a simple cubic material with lattice constant ' a ', at least one Bragg reflection is observed? For this wavelength what should be the direction of the incident beam and corresponding to which planes reflection will be seen. 28. The edge of a unit cell in a cubic crystal is 2.62 Å . Find the Bragg angle corresponding to reflection from (100), (110) and (111) planes, given the monochromatic X-ray beam has a wavelength = 1.54 Å . 29*. A cubic crystal is mounted with a [100] direction parallel to the incident X-ray beam. What are the positions of the beam diffracted from (110) and (111) planes. What would be the wavelengths for such a diffraction to occur? 30*. Consider a two dimensional square lattice of lattice constant ‘ a ’. A monochromatic beam of x-ray having the same wavelength as the lattice constant is incident in [10] direction. Draw an Ewald sphere and clearly show the reflections that would be seen. What are the reciprocal vectors involved in these reflections? What are the glancing angles and what are the indices of the direction of outgoing x-ray beams? 31. Which of the following reflections would be missing in a bcc lattice? (100), (110), (111), (200), (210), (211), (220) Answer a similar question for an fcc lattice. 32*. Iron Pyrite has an fcc structure with a=5.42 Å . If Iron K radiation is used, at what angle do we expect a (300) line? On an actual photograph a line occurs at this angle if radiation is unfiltered. How may this be accounted for? ( for Fe K =1.937 Å , K =1.757 Å ) 33. A bcc metal ( atomic weight 55.85 ) shows first ring in the powder photograph at =22.3 o . Find the density of the metal and the values for which next two rings would be observed. (λ=1.54 Å )
34*. A single crystal of fcc Al, with nearest neighbor distance = 2.8 Å is mounted in such a way that x-ray beam is incident normally on the (100) plane. Assume that x-ray beam has all the wavelengths greater than 2.5 Å . Which of the reflections will be observed and at what angles? What will be the wavelengths for which these reflections will be observed? 35. On an x-ray powder photograph of a cubic substance taken with Cu K radiation ( =1.542 Å ), lines are observed at Bragg angles of 12.3, 14.1, 20.2, 24.0, 25.1, 29.3, 32.2 and 33.1 degrees. Assign indices to these lines. Decide whether the lattice is primitive, face centered or body centered and calculate the cell size. 36*. Write the structure factor expression for NaCl. KCl has also NaCl structure, but why some lines that appear in NaCl are found to be absent in KCl? Show that in such cases structure factor S can be written as S (hkl)= S (fcc) . S (basis) 37. Find the structure factor for diamond. Find the zeros of S and show that the allowed reflections in this case satisfy h + k + l = 4 n , where all indices are even and n is any integer, or else all indices are odd. 38. What should be the indices of the plane (with reference to convectional cells), corresponding to first four rings on the powder photograph of a mono atomic crystal, if the crystal has a (a) bcc structure, (b) fcc structure (c) diamond structure. 39. Powder specimens of three monoatomic cubic crystals A, B, C are analyzed with a Debye Scherrer camera. It is known that one sample is face centered, one is body centered and one has a diamond structure. The approximate values of 2 of the first four diffraction rings in each case are given below. (a) Identify the crystal structures of A,B and C. (b) If the wavelength of the incident x-ray beam is 1.5 Å , what is the length of the side of the conventional cubic cell in each case. (c) If the diamond structure was replaced by the Zinc Blende structure with a cubic unit cell of the same side, at what angles the first four rings now occur. S. No. A B C 42.2 28.8 42.8 1 49.2 41.0 73.2 2 72.0 50.8 89.0 3 87.3 59.6 115.0 4
40. Cuprous Oxide has a cubic unit cell with oxygen atoms at (0,0,0) and at 𝑏 2 (±1, ±1, ±1) (all combinations permitted). The copper atoms are arranged 𝑏 𝑏 4 (1,1,1), 4 (1, −1, −1), in a tetrahedron around the oxygen atom at 𝑏 𝑏 4 (−1,1, −1) and at 4 (−1, −1,1) . (a) What is the Bravais lattice type, with as small a basis as possible? (b) How many atoms of Cu and O are there in the basis and what are their positions? (c) If f O and f Cu are the atomic form factors of O and Cu respectively, find the structure factor of Cuprous Oxide corresponding to first four non-zero intensity reflections. 41. Consider a simple tetragonal lattice (a=b≠c, = = =90 o ). Write the primitive vectors for the direct lattice and find the same for the reciprocal lattice. Find the distance between the nearest (hkl) planes in the direct lattice. Assume c=1.1 a. Find the h, k, l values corresponding to the first four planes in the order of increasing distinct values in x-ray diffraction, assuming one atom basis. Would the answer to last part change if at each of the lattice point, a basis of two identical atoms is put, with one atom at the origin and another at body centre? If the answer is different from the first one, what would now be the h, k, l values corresponding to first four reflections now? 42*. The primitive vectors of a hexagonal Bravais lattice are given as follows. 3 a a ˆ ˆ ˆ ˆ ; ; a ai b i j c ck 2 2 Find the (i) Primitive vectors of the reciprocal lattice, its type and lattice constant. (ii) Distance between the nearest planes in the direct lattice, where hk miller indices are defined with respect to the primitive vectors given above. (iii) Structure factor of an element which crystallizes in hcp structure in terms of atomic form factor. (iv) Which of the following reflections would be absent in hcp structure if 𝑑 8 𝑏 = √ 3 . Also arrange them in order of increasing θ . (100), (001), (101), (002), (102) (v) The first four reflections of an element crystallizing in hcp structure are observed at the glancing angles of 20.8 o ,22.4 o , 23.8 o and 31.4 o , when x-ray of wavelength 1.54 Å is used in a powder pattern. Evaluate the lattice constants ‘ a ’ and ‘ c ’ and index the lines. Compare c/a ratio with the ideal one.
43 . For the hydrogen atom in the ground state, the number density is 1 −2𝑠 𝑜(𝑠) = 3 𝑓 𝑏 0 𝜌𝑏 0 where a 0 is the Bohr radius. Show that the form factor is 16 𝑔 = 2 ) 2 (4 + 𝐻 2 𝑏 0 44. (a) If n(x) is a periodic function of x with periodicity 'a', it can be expanded as n p exp (i2 p x/a) n(x) = p Show that the Fourier coefficient n p is given by a n p = (1/a) n x ( )exp( i 2 px / a dx ) 0 (b) Show that the inversion of n( r )= n G exp( i G . r ) gives. n G = (1/V) n r i G.r dV ( )exp( ) cell where V is the volume of the unit cell.
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