-Write the lattice parameters of the following unit cells, and define the name of the bravais lattice for each? b C a b a a a

Use the e-S definition of infinite limits to prove the statement. lim -00 %3D X - X→5 f(x) 1 is defined for all x # Let N 0 such that f(x) ---Select--- V < ---Select--- v Thus lim = -0, |x - 5| х — 5 |x – 5| X→5- X

For each of the following sets of primitive lattice vectors, identify the Bravais lattice type and give the dimensions of the conventional unit cell in terms of a, b, and c: (a) (a/2)& + (a/2)§, ay, (a/V2)2; (b) (a/2)âx + (a/2)ŷ, aỹ, aĉ; (c) ax + 2bŷ, bŷ, c2; (d) ļax + }bŷ, bŷ, cê.

Given position-space lattice vectors a, b and c, show that the reciprocal lattice vectors (see figure) a* = 2pi b x c / a . (b x c), b* = 2pi c x a/ a . (b x c), c* = 2pi a x b/ a . (b x c) are such that a. a* = 2pi, a.b* = 0

How "density of states" is defined and used in physics? Draw (schematically) the number of occupied energy levels for Fermi Dirac statistics, Maxwell Boltzmann statistics, and Bose Einstein statistics.

The dispersion relation for a one dimensional monatomic crystal with lattice spacing a, which interacts via nearest neighbor harmonic potential is given by: w=A Asin (), where A is a constant of appropriate unit. The group velocity at the boundary of the first Brillouin zone is: A. 0 B. 1 C. D. 2 Aa² 2 www. Aq² QUESTION 24 For a diatomic linear chain, the phonon dispersion relation (k) has two branches corresponding to + and sign respectively: 1 1 (M₁₂ + M₁₂) ± P | (M₁ + M₁₂ 1 There are two atoms in the unit cell with masses and M2 and the force constant of nearest neighbor interaction is F and the effective mass, sound will be: A) NA B) = 2 C) a M₁ + M₁ 2(M₁+M₁) w(k) = f M₁ + M₁ 2 4 M₂)²³ - M₁M₂ M₁M₂ M₁+M₂ sin² (29/1/² is kept constant. The velocity of

x 23 a2 y The image above shows the positions of the points lattice of a particular lattice. a. Express the primitive lattice vectors a1, a2, and a3 in the unit vectors x, y, and z. b. Determine the primitive vectors in the lattice space (reciprocal lattice) b1,b2, and b3 of the lattice

Question 3 Consider the simple and close-packed hexagonal direct lattices. (a) Find out the general expression for the structure factor for each lattice type. In this context, what is the significance of a complex structure factor? (b) Without resorting to the structure factor, determine in which structure(s) X-ray Bragg reflections arise from the lattice planes (0001), (0002), and (1010). Justify your answer by the use of the above determined structure factor or otherwise and explain the meaning of the four indices describing the above cited lattice planes.

Consider the plane makes intercepts of 2 A, 3A, 4A on the coordinate axis of the crystal with a:b:c=4:3:2. The crystal is orthorhombic crystal. Calculate the miller indices of the given plane of the crystal. (A) (4,1,1) (B) (4,3,2) (C) (4,2,1) (D) (4,4,4)

The interplanar spacing of (121) planes is 4.5Å for a cubic crystal. Find the atomic radius for a face-centred cubic close packed structure.