9780030839931, Chapter 5, Problem 3P

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Chapter Problem 3P Problem (a) Show that the density of lattice points area) lattice plane where the primitive cell volume and spacing between neighboring planes the family to which the given plane belongs (b) Prove that the lattice planes with the greatest densities points are the {111} planes in face cubic Bravais lattice and the planes body-centered cubic Bravais lattice. (Hint: This most easily done exploiting the relation between families lattice planes and reciprocal lattice vectors.) Step solution Step (a) The relationship between the normal to plane and the crystallographic axes and may be obtained by considering the primitive unit shown the figure below b As height this cell the area of the shaded base whose sides are and this gives volume of this cell Step Thus volume the unit 100 Step As there exactly one lattice point every primitive the density points (per unit area) lattice plane the primitive cell volume and the spacing between neigh aboring planes, the density lattice points (per unit area) the lattice plane will be Step (b) The relation signifies that the density of lattice points directly proportional the inter planer spacing between the planes This larger density lattice points will be the inter planer spacing between planes We know Here miller indices the plane are h. and and the reciprocal the reciprocal lattice set of vectors G Step The above equation d(hkl)= signifies that the value the planer spacing between planes s larger when the length of G that is the reciprocal lattice vector perpendicular d Substitute the value of equation G The above equation signifies that the density points will be maximum when in reciproca space having Face centered cubic lattice (fcc) primitive vectors are, a, And Step So reciprocal lattice b. a,(a,xa,) b Step of 10 The above equation signifies that the reciprocal lattice The vector from the of the cube to the center of the body is the smallest one The plane perpendicular to this G. will be From the above observation conclude that the lattice planes with greatest densities of points are the planes an (face centered cubic) lattice Step 8 Body-centered cubic lattice (bcc) Fo primitive vectors are And So reciprocal lattice will be b, b. Step The above equation signifies that the reciprocal lattice of The vector from the cube the center of the face smallest The plane that is perpendicular to this will be {110} From the above observation conclude that the lattice planes with the greatest densities of points the planes (body-centered cubic) Bravais lattice