PH-409 (2015) Tutorial Sheet No. 1 * problems shall be discussed in tutorial class 1. Consider the following infinite 2- dimensional pattern. ..... ....qp db qp db qp db qp.... ....db qp db qp db qp db.... ....qp db qp db qp db qp.... ..... Indicate: (a) The primitive unit cell (b) Bravais lattice with smallest basis (c) A rectangular cell with smallest basis (d) The primitive vectors (e) The conventional vectors. 2. A crystal has a one atom basis and a set of primitive translation vectors (in Å ). ⃗ ⃗ = 3𝑘̂ and 𝑑 ̂ ) , 𝑏 ⃗ = 3𝑗̂ =, 𝑐 ⃗ = 1.5(𝑗̂ + 𝑘̂ + 𝑙 ̂ are unit vectors (a) What is the Bravais lattice? (b) What are the where 𝑗̂, 𝑘̂ and 𝑙 volumes of the primitive and the conventional unit cells? 3*. Show that the relation between atomic radius ' r ' (half the distance of closest approach) and lattice constant ' a ' is given as follows assuming one atom hard sphere per lattice point. a a 3 a 3 a 2 sc : r ; fcc: r ;bcc : r ; diamond: r 4 8 2 2 4*. Show that the c/a ratio for an ideal hcp structure is 8 3 / = 1.633. If c/a ratio is significantly larger than this value, the crystal structure may be thought of as composed of planes of closely packed atoms, the planes being loosely stacked. 5*. Show that the ratio of the volume of the spheres to that of the crystal ( called packing fraction ) is 0.74 for fcc and hcp, 0.68 for bcc, 0.52 for sc and 0.34 for diamond structure. 6*. Copper crystallizes in fcc structure. Calculate the edge of the conventional cubic unit cell and atomic radius of Cu. ( Density of Cu= 8.96 gm/cc and atomic weight = 63.55 ) 7. Find the distance between the alkali and halogen ions for KBr and KCl given their densities to be 2750 and 1984 kg/m 3 .( At. wt. of K =39.1, Br =79.9 and Cl =35.5 )
8*. Calculate the density of packing in atoms per square meter for the following planes: (a) (110) in Fe ( bcc with a= 2.87 Å ) (b) (111) in Cu ( fcc with a= 3.61 Å ) (c) (110) in Cu. 9. Find the miller indices for planes which intersect the primitive vector directions at a distance of (i) 6a,2b,3c (ii) a,2b, (iii) 2a,-b,2c ; where a,b,c are the lattice constants in the three directions. 10*. Show that in a cubic crystal (i) angle between [ u 1 v 1 w 1 ] and [ u 2 v 2 w 2 ] directions is given by 𝑣 1 𝑣 2 + 𝑤 1 𝑤 2 + 𝑥 1 𝑥 2 cos 𝜄 = 2 + 𝑤 1 2 + 𝑥 1 2 + 𝑤 2 2 + 𝑥 2 2 )(𝑣 2 2 ) √(𝑣 1 (ii) [ hkl ] is normal to (hkl) (iii) the distance 'd' between adjacent planes of (hkl) is given by 𝑏 𝑒 = √ℎ 2 + 𝑙 2 + 𝑚 2 11*. Consider the planes with indices (100) and (011). The lattice is fcc and the indices refer to the conventional cubic cell. What are the indices of these planes when referred to the following primitive axes? ⃗ = 𝑏 ⃗ = 𝑏 ⃗ = 𝑏 ̂); 𝑐 ⃗ ̂ + 𝑗̂); 𝑑 2 (𝑗̂ + 𝑘̂) 𝑏 2 (𝑘̂ + 𝑙 2 (𝑙 _ 12*. Consider a cubic crystal. (a) Show that [1 1 0] direction lies in (111) plane. (b) _ Find the directions that are perpendicular to [1 1 0] direction and also lie in (111) plane. (c) Find the angle that [100] direction makes with [111] direction. (d) Find _ _ the plane that contains these two directions. (e) Does [1 1 1 ] also lie in this plane? (f) What are the directions at which this plane intersects the (111) plane. 13*. What are the indices of the plane that contains the three lattice points, given by the following position vectors? ⃗ ⃗); 𝑠 2 ⃗ ⃗ + 𝑑 ⃗⃗ = (2𝑏 ⃗); 𝑠 3 ⃗ 𝑠 ⃗ ⃗⃗ = (𝑏 ⃗ − 𝑐 ⃗ ⃗ ⃗ + 𝑑 ⃗ ⃗ ⃗⃗ = (3𝑐 ⃗) 1 14. Show that the angle between any two lines (bonds) joining a site of the 1 diamond structure is cos −1 (− 3 ) = 109 o 28'.
15. Consider a bcc structure with one atom per lattice point. Assume the atoms to be hard spheres of 1 Å radius. (i) Find the maximum radius that a sphere can 1 1 have if it is to occupy a position in the lattice such that its center is at ( 2 , 2 , 0) . The original structure should not be distorted. (ii) Find the maximum radius if the 1 1 center of the sphere is to occupy ( 2 , 4 , 0) position instead of the earlier one. 16*. Find the critical ratio of radii of ions A and B for which the B ions would start touching other B ions, in case AB crystallizes in (i) NaCl,(ii) CsCl structure. What role does this ratio plays in determining the structure of alkali halides. 17. The primitive unit cell of a compound is described below: The atom B is at the center, while atoms A occupy the corners of a cube. Atoms X are at the face centers of the cube. (i) What is the formula of the compound? (ii) What is the Bravais lattice type and basis? (iii) Describe the nearest neighbors of atoms A, B and X. (iv) Find the critical ratio of radii of A, B and X atoms when atoms A and B touch X atoms and simultaneously X atoms touch X atoms. 18. What is the Bravais lattice formed by all points with Cartesian coordinates ( n 1 ,n 2 ,n 3 ) if (a) all n i are either even or odd integers, (b) the sum of the integer n i is required to be even? 19. Show geometrically that the face centered tetragonal structure is equivalent to a body centered tetragonal lattice in which the sides of the unit basis unit cell 1 is √2 times what it is for the face centered arrangement. Why are the fcc and bcc structure distinct then?
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