9780030839931, Chapter 7, Problem 6P

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Chapter 7, Problem 6P Problem (a) Show that there is family of lattice planes perpendicular to any n-fold rotation axis of a Bravais lattice, 3. (The result is also true when but requires a somewhat more elaborate argument (Problem 7).) (b) Deduce from (a) that an n-fold axis cannot exist in any three dimensional Bravais lattice unless can exist in some two-dimensional Bravais lattice (c) Prove that no two-dimensional Bravais lattice can have an -fold axis with 7. (Hint: First show that the axis can be chosen to pass through lattice point. Then argue by reduction ad using the set of points into which the nearest neighbor of the fixed point taken by the rotations to find pair of points closer together than the assumed nearest neighbor distance (Note that the case requires slightly different treatment from the others).) Step-by-step solution Step (a) A symmetry operation is that which transforms the crystals to itself That is crystal remains invariant under symmetry These operations are rotation, reflection and inversion The translation operation applies to lattices only while all the remaining operations and their combinations apply to all objects and are collectively known as point symmetry operations The inversion operation is applicable only to three dimensional crystals Step of 6 A lattice is said to possess the rotation symmetry its rotation by an angle about an axis (or a point in lattice) transforms the lattice to Also since the lattice always remains invariant by rotation of the angle must be an integral multiple of Θ That is, 2π Or, 2π Step The factor takes integral values and is known as multiplicity of rotation axis The possible values of that are compatible with the requirement of translation symmetry are 1,2,3,4 and A two-dimensional square lattice has 4-fold rotation The 5-fold rotation is not compatible with translation symmetry operation and that only 1,2,3,4 and fold rotations are As the new points will be produced by successive rotations in 1and 2-fold rotation, therefore there is family of lattice planes perpendicular to any fold rotation axis of Bravais lattice, Step (b) In three fold axis operation denoted by 3, rotation through 120° brings the point a to the point b as shown the below figure-1 Further rotation through 120° brings the point b to the position The next application of rotation operation through 120° brings the point back to In performing the rotation again, we arrive at points a,b and So the no new points will be produced by successive We say multiplicity of operation From the above observation we conclude that, an -fold axis cannot exist in any three-dimensional Bravais lattice unless can exist in some two-dimensional Bravais lattice Figure 1 c b a 120° 3-fold Comments (3) Anonymous Not helpful all Anonymous gabbage Anonymous garbage Step of 6 (c) Consider row of lattice points A,B,C and D as shown in below figure-2 Figure 2 B Θ A B C D Step of 6 Let T be the lattice translation vector and let the lattice have n-fold rotational symmetry with rotation axes passing through the lattice point's perpendicular to the plane of paper. Rotations by an angle 2π about points B and C in the clockwise and anticlockwise directions respectively give points B' and which must be identical to B and C. Thus the points B' and C' must also be lattice points and should follow lattice translation Hence must be some integral multiple of BC So, B'C" (BC) Or, mT 2 Here, m an Since the allowed values of m are 3,2,1,0 and -1 These correspond to the allowed values of as 0° or 360°, 60°, 120° and 180° Thus from equation 2π the permissible values of n are 1,6,4,3 and 2. Thus we concludes that 5-fold rotation is not permissible because it is not compatible with lattice translation other rotations, such as rotation, are also not From the above observation we conclude that, no two-dimensional Bravais lattice can have an n-fold axis with 5 or