Chap 1 Structure of Crystals1.1. Show that the face-centered tetragonal lattice is equivalent to thebody-centered tetragonal lattice.1.2. Show that the spacing d(hkl ) between adjacent lattice planes with Millerindices (hkl ) is equal to 222l k h /++a for cubic Bravais lattices and to 222222////1c l b k a h ++for orthorhombic Bravais lattices.1.3. Calculate the densities of lattice points in the (100), (110), (111), and (hkl )lattice planes of a simple cubic lattice.1.4. Write a computer program that will determine the distance d(n) form a givenatom to the n th nearest neighbor(NN) in a Bravais lattice. Also computer N(n ) ,the number of n th NNs .Carry out the calculation for the SC, BBC ,and FCC lattices.1.5. Calculate the packing fractions for the following crystal structures: FCC,HCP, and diamond .1.6. Show that the B atoms in an A-B 8 bounding unit come into contact with eachother when r B =1.366r A [i.e., when r A= (3-1)r B ] .Here r A and r B are the radii of the hard-sphere A and B atoms, respectively .Find the analogous conditions on the radii for the A-B 6 and A-B 4 bonding units.1.7. Prove for hard-sphere atoms in the HCP crystal structure that c/a=3/8=1.633.1.8. Assuming that the atoms in the CC crystal structure are hard spheres of radiusR in contact with each other, calculate the maximum radii r of the smaller hard-sphere atoms that could occupy the octahedral interstitial sites in the FCC crystal structure.Chap 2 Bonding in Solids1.1. Compute the cohesive energies for monatomic crystals of atoms bondstogether by the Lennard-Jones potential U(r) given in Eq.(2.3). Express thecohesive energy, c H ∆(0K) = n (atoms)(CN/2))(0r U , in terms of the parameter ε and the equilibrium interatomic distance r 0 in terms of the parameter σ. Here n (atom) is the concentration of atoms and CN is the coordination number. Carry out the calculations for the SC, BCC ,and FCC crystal structures. Eq.(2.3): ])()[(4)(612612rr r C r B r U σσε-=-+= 1.2. Given the following lattice constants for crystals with the NaCl crystalstructure, a(NaCl)=0.563nm, a(KCl)=0.629nm, a(NaF)=0.462nm, and a(KF)=0.535nm, show that these data are not sufficient to obtain a self-consistent set of ionic radii for the Na +, K +, Cl -, and F- ions .Why is it not possible to determine a completely self-consistent set of radii from the data given?1.3. Use the cohesive energy c H ∆ (see below)o f cubic β-SiC with the zincblendecrystal structure to determine the bond energy E (Si-C).1.4. Calculate the potential energy U of an anion-cation(Na +-Cl -) pair resultantingfrom their mutual Coulomb attraction and then compare the result with the cohesive energy c H ∆ of NaCl listed in the above Table. The lattice constant of NaCl is a = 0.563 nm .(Hint: Take into account the fact that each Na + ion in NaCl interacts with six NN Cl - ions, and vice versa.)1.5. In the structural change from BCC a-Fe to FCC r-Fe at T=912℃ thelattice constant change from a(BCC)=0.290nm to a(FCC)=0.364nm. Assuming that the Fe atoms act as hard spheres, which is more nearly constant in a-Fe and r-Fe ---the radius r met or the atomic volume V met ?3、Diffraction and Reciprocal Lattice3.1 Prove that ∑Rexp(iq·R)=0, where {R} is a set of Bravais lattice vectors andq （≠0）lies within the primitive unit cell of the reciprocal lattice. Also prove the orthogonality identity appearing in Eq.(3.12):G G WS G G i WS V dr e ''-=⎰,)(δ3.2 Find the Fourier coefficients Vn for the periodic functions)/2sin()(a x A x V π= and )/2cos()(a x B x V π=.3.3 Prove that the plane defined by the equation G·r = A lies a distance d=A/Gform the origin and that the normal to the plane is parallel to ∧G .3.4 Use the results of Problem 3.3 to generate formulas for the bounding planesof the first Brillouin zones for the FCC, BCC, and HCP crystal structures. 3.5 Determine the structure factor for the basis, Φ(q), defined in∑=Φj iqs j j eq f q )()(, for the cubic ZnS, CsCl, and NaCl crystal structures.3.6 Draw the x-ray ring patterns produced by diffractions from powders for theSC, FCC, BCC, and diamond crystal structures.3.7 Given an amorphous solid in which each atom has an electron densitydescribed by n(r)=A exp(-2r/a) and the pair distribution function is the unit step function g(r) =Ө(r-b), find the expected scattering intensity.3.8 Sketch the Wigner-Seitz cell for the HCP crystal structure.3.9 Find the distances from the center of the Wigner-Seitz cells for theBCC,FCC.3.10 The primitive translation vectors of the hexagonal lattice can be written as∧∧+=2231a j a i u , ∧∧+-=2232a j a i u , ∧=k c u 3(a) Show that the fundamental translation vectors of the reciprocal lattice aregiven bya j a i g ππ2321∧∧+= , a j a i g ππ2322∧∧+-= , ∧=k c g π23 (b) Describe and sketch the first Brillouin zone of the hexagonal lattice.(c) Prove that the perpendicular distance d(hkl) between adjacent parallel planes in the hexagonal lattice is222223)(41)(c l a k hk h hkl d +++=[Hint: Use )(/2)(hkl G hkl d π=.]3.11 Find the shortest G(hkl) for (a) the BCC crystal structure, and (b) the FCCcrystal structure.Chap 4 Order and Disorder in Solids4.1 Take a small box or cylindrical container and measure its volume. Pourmarbles or ball bearings into the box until it is full. Determine the volume occupied by the sphere. Compute the packing fraction. Repeat the experiment several times and average the results. Compute your result with the packing fraction for FCC and HCP, BCC, SC, and the value 0.64obtained for the random packing of hard spheres.4.2 Draw sketches of the two-atom Frenkel pair interstitial configurationsknown as “dumbbells ” in both FCC and BCC metals.4.3 Consider the equilibrium concentration of Frenkel defects in a solid. (a) Derive the value of N L (V) given in Eq.(4.7) ( ]2)()(exp[)]()([)(2/1Tk A G V G V N A N V N B I L I L L +-=]2)()(exp[)(2/1T k V G V G N N B I LI L +-≈) by first setting N L (A) = N L (V) in Eq.(4.6) and then minimizing the resulting Gibbs free energy with respect to NL(V).Eq.(4.6):)()()()()0(A G A N V G V N G G I I L L ++≈)(ln )()](ln[)]([ln {V N V N V N N V N N N N T k L L L L L L L L B -----)}(ln )()](ln[)]([ln A N A N A N N A N N N N I I I I I I I I ----+(b) Repeat the derivation using Lagrange multipliers to enforce the constraintsof Eq.(4.4)( )()(V N A N N L L L +=, )()(V N A N N I I I +=).4.4 Consider a monatomic solid consisting of N atoms. Determine thenumber of ways, W, that n of the atoms may be removed to form n vacancies. Compute the entropy, given by S=k B lnW. For the SC, BCC, and FCC crystal structures, compute the entropy for forming NN vacancies.4.5 Consider a one-dimensional monatomic solid with N atoms and N L (V)vacancies at temperature T>0K. Show that the fractional vacancy concentration n v (T)=N L (V)/N is given approximately by00//a a l l n v ∆-∆≈. Here l 0=Na 0 is the length of the solid at T=0 K, l ∆ is the change in length, a 0 is the lattice constant of T = 0 K, and a ∆ is the change in the lattice constant. (Hint: Write the change of length as vacancies thermal l l l l l ∆+∆=-=∆0)。