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1: Given that the primitive basis vectors of a lattice are a = (a/2)(x+y). = (a/2)(y + 2), b c = (a/2)(z + x). where x, y, and "Z are unit vectors in the X-, y- and 2- directions of a Cartesian coordinate system. A-Determine the Bravais lattice. B-Calculate the volume of the primitive unit cell. C- Show that the reciprocal lattice to the reciprocal lattice is the real lattice. (12 pt) Q2: (8 pt) Show that the volume of the primitive reciprocal cell is Q=&V, where Vis the volume of the corresponding cell in real space. Recall the vector identity: (CX A) X (AX B)-(C. AX B)A. (8 pt) Azv k₂a 03: Use the Debye model to calculate the heat capacity of a monatomic lamice in one dimension at temperatures small compared with the Debye temperature &- where w is the sound velocity, a is the lattice spacing and ka is Boltzmann's constant. [hint: in Debye model. the density of states in on dimension is p(a)=- -]. (14pt.) Q4: Show that the Madelung constant for a one-dimensional array of ions of alternating sign with equal distance between successive ions is equal to 2 In 2. (5 pt) Q5: (5 pt) Consider a longitudinal wave which propagates in a monatomic linear lamice of atoms of mass M, spacing a, and nearest-neighbor interaction C. , = cos(wt-ska) a- Show that the total energy of the wave is E ==MΣ(da, / d)² + CΣ(2,-*..)³ Where s runs over all atoms b- By substitution of u, in this expression, show that the time-average total energy per atom is (hint: you may use this dispersion realation: w = (4C/M) | sin_ka|} Mw²²+Cl-cos Kap² - Mw (16 pt.)