Condensed Matter 1) Obtain expressions for the heat capacity due to longitudinal vibrations of a chain of identical atoms; (a) in the Debye approximation; (b) using the exact density of

states (Eq. below). With the same constants K and M, which expression gives the greater heat capacity and why? Show that at low temperatures both expressions give the same heat capacity, proportional to T. g(w) Hint: L (M\1/2 ла K &F sec (ka) 2N (4K 1/² - 0²) M 2) Estimate the Fermi temperatures of: (a) liquid ³He (density 81 kg m-³), and (b) the neutrons in a neutron star (density 10¹7 kg m−³). <-1/2 h² (3π²N\2/3 2m N(M¹/2 2(K/M)¹/2 T K (4K/M-²)¹/2 3) The Bragg angle for a certain reflection from a powder specimen of copper is 47.75° at a temperature of 293 K and 46.60° at 1273 K. Calculate the coefficient of linear thermal expansion of copper. 4) The crystal basis of graphene and of diamond is composed of two carbon atoms in nonequivalent position. Thus, their dispersion curves are composed of acoustical branches and optical branches and their acoustical branches are assumed to obey to the Debye approximation: = vs|k| and their optical branches are assumed to obey to the Einstein model @= @E = Cst. Deduce the numerical values of their Debye and Einstein temperatures from their crystal structure and their common sound. velocity, Vs = 18,000 m/s with also VE (Einstein frequency) at about 4 *10¹3 Hz. From the Cv graph shown in below, evaluate the specific heat of diamond at room temperature, 290 K. (h, kB) 100 75 50 25 0 C, (%) x 3;2;1 NkB unit 0 1D -2D -3D T/0₂ 1 Heat capacity, C, as a function of T/05 for 1D, 2D, and 3D solids. The vertical scale is in Nkg unit to multiply by 1, 2, or 3 as a function of the degree of freedom for the atom vibrations. Note the initial evolution in T, T2, or 73 as a function of the dimensionality of the solid. 5) (1) Knowing that lithium crystallizes in a cubic system with lattice considering its atomic mass (7) and its volumetric mass (546 kg⋅m-³), find which one is it ([simple cubic, body- centered cubic (bcc), or face-centered cubic (fcc)]? (2) Knowing that the valence electrons of this metal (1 per atom) behave as free electrons, find the shape of the Fermi surface and its expression and then calculate its characteristic dimension KF. (3) Compare KF obtained in (2) to the distance dm, which in reciprocal space separates the origin from the first boundary of the first Brillouin zone nearest the origin. (Evaluate dm using simple geometric considerations without having to sketch the first Brillouin zone.) (4) Find the Fermi energy of lithium EF, the Fermi temperature TF, and the speed of F of the fastest free electrons. (5) Knowing that the resistivity p of lithium is of the order 10-5 cm at ambient temperature, find the time of flight, t, and the mean free path A of conduction electrons. (6) Find the drift velocity vd of conduction electrons subject to a electric field of 1 V/m and compare it with the Fermi velocity VF. (7) Starting from the relation ke =1/3 CeVFA (or with the help of the Wiedemann-Franz expression), find the thermal conductivity due to electrons Ke of lithium at ambient temperature T = 300 K.