Homework # 2Chapter 2 KittelProb # 1, 2Phys 175ADr. Ray KwokSJSUConsider a plane (hkl) in a crystal latticea1a2a3ABCProb # 1 – Interplanar SeparationJohn Anzaldo(a) Prove that that the reciprocal lattice vector G = hb1+ kb2+ lb3is perpendicular to this plane We know that plane (hkl) gives axis intersections at the reciprocals to the values of the plane index, giving us intersections at (1/h,0,0), (0,1/k,0) and (0,0,1/l) labeled points A, B and C. If the vectors made by the connection of these points are perpendicular to G, their dot product will be zero.Prove that that the reciprocal lattice vector G = hb1+ kb2+ lb3is perpendicular to this planeAB= AC= CB= for an a1, a2, a3basisG•AB=Similarly for AC and BC, hence demonstration of perpendicularity.(b) Prove that the distance between two adjacent parallel planes of the lattice is  If you scale the intercepts by integer values you will find that there is always a plane repeated from a perfectly translated origin (i.e. T=u1a1+u2a2+u3a3 , where uiis an integer) The closest distance between a point and a plane iswhere is the unit normal to the plane and a vector from some point to the plane. For our purposes we will use point (0,0,0) and a point on the plane (1/h,0,0) to find vector .Prove that the distance between two adjacent parallel planes of the lattice is  Vector = , and = This gives which proves the relationship in question.(c) Show for a simple cubic lattice that  For a simple cubic we know for i≠j, and Usingwe find thatwhich gives , which yieldsProb # 2 – Hexagonal Space LatticeDaniel WolpertGiven: Primitive translation vectors of the hexagonal space latticea1= (3½a/2)ẋ + (a/2)ẏa2= -(3½a/2)ẋ + (a/2)ẏa3= cẑA) Find the Volume. Triple product: |a1· a2x a3| =cẑ000(a/2)-(3½a/2)0(a/2)(3½a/2)= 3½/2 a2cB) Find the primitive translations of the reciprocal latticeB1= 2π (a2x a3)/ V B2= 2π (a3x a1)/ V B3= 2π (a1x a2)/ V 2πcẑ000(a/2)-(3½a/2)ẑẏẋ= 2π/a3½ẋ + 2π/a ẏ0(a/2)(3½a/2)cẑ00ẑẏẋ2π 2π0(a/2)(3½a/2)cẑ00ẑẏẋ= -2π/a3½ẋ + 2π/a ẏ = 2π/c ẑSketch the first Brillouin zone of the hexagonal space