SJSU PHYS 175A - Crystal Structure (29 pages)

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Crystal Structure



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Crystal Structure

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29
School:
San Jose State University
Course:
Phys 175a - Solid State Physics
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Chapter 1 Crystal Structure Phys 175A Dr Ray Kwok SJSU Ideal Crystal Infinite periodic structure No edge No impurity Translational invariance for all observables Crystal Lattice Basis basis lattice crystal Translational invariance a With a set of primitive vectors a such that r r u1a1 u2a2 u3a3 where the u s are integers and a s are Primitive Translational Vectors If the origin is chosen at a lattice point all other lattice points is given by this Translational Vectors Translational Operator T u1a1 u2a2 u3a3 Primitive Translational Vectors are not unique Many choices eg 2D lattice Crystal axes Crystal axes are not necessarily the same as translational vectors Sometimes it s more convenience to use axes with more symmetrical properties such as rotation reflection called Point Group Symmetry crystal translational Crystallography Characterization and enumeration of crystal structure Require knowledge of Group Theory symmetry operators eg The rotational group in the Rubic Cube s solution Space Group Point Group Unit Cell Somewhat arbitrary Defined by T only or multiples Can have more than 1 lattice point per cell Primitive Lattice Cells defined by the parallelepiped of a one Lattice point per primitive cell volume a1 a2 x a3 or its cyclic permutation Wigner Seitz Cell bisectors planes of the nearest neighbors unique does not depend on choice of a Common Crystals simple cubic all a s are equal are 90 deg Tetragonal 2 a s are equal are 90 deg Orthorhombic all different a are 90 deg Trigonal all a s are equal are equal but not 90 deg Hexagonal 2 a s are equal are 90 deg is 120 deg not 60 convention Cubic conventional cells sc bcc fcc Not primitive cells of atoms per cell is 1 2 4 bcc fcc Primitive Cell fcc bcc Conventional Primitive Vectors r a 1 12 a x y z r 1 a 2 2 a x y z ar 1 a x y z 3 2 r a 1 12 a x y r 1 a 2 2 a y z ar 1 a z x 3 2 Basis Crystal Structure lattice point basis Basis rj xja1 yja2 zja3 for the jth atom in the cell where x y z 1 Examples are NaCl Fig 17 18 Space lattice is fcc basis Cl at 000 and Na In primitive cell Cl at 000 0 0 0 fcc Na at 0 0 0 0 0 0 Point Group Rotation Reflection Most lattice has 2 3 4 or 6 fold rotational symmetry Basis can be a lot more complicated twist two corners flip two edges Crystal Plane Index System for Crystal Planes See Fig 15 Take intercepts on a s eg 3 2 2 Inverse of them 1 3 multiply with LCD 6 to get 233 It describes the set of planes e g 100 set 0 0 set 012 etc 100 200 300 00 etc Miller Indices In cubic primitive cell this is the direction of the normal 233 of the 223 planes For a hkl plane hkl is the direction vector perpendicular to plane normal Use top bar for negative Family of planes with crystal symmetry is denoted by hkl e g 100 means 100 010 001 00 0 0 00 in sc structures Example simple cubic 233 plane Example fcc 233 plane fcc 233 plane intercepts are 3 2 2 With unit vectors defined in Fig 13 for example r a 1 12 a x y r 1 a 2 2 a y z ar 1 a z x 3 2 the normal 233 is calculated to be r r r r 3 3 N 2a1 3a2 3a3 a x y a y z a x z 2 2 r a N 5 x 5 y 3z 2 Closed Packed Structures hcp is basically hexagonal with off set layers For identical spheres fcc hcp are the most dense packing 0 74 of volume occupied These are the closest packed structures should try to convince yourself that this is true HCP FCC Fig 21 offset hexagonal every other layer ABABAB hcp Offset hexagonal twice ABCABCABC fcc A C A B Coordination number of nearest neighbors touching spheres As in Table 2 coordination 6 for sc 8 for bcc 12 for fcc Other structures Diamond structure Space lattice is fcc 2 pt basis 000 Fig 25 No primitive cell of single atom is possible Zinc Blend etc Example simple cubic Sketch 201 planes Calculate the shortest distance between planes Intercepts 1 infinity 2 Crystal planes z d a sin a a 5 a d a 5 a a 5 a d x a Direction of normal 201 d a 2 a 2 Example fcc planes Sketch the 111 planes in the standard fcc primitive vectors representation What s the normal of the planes in Cartesian coordinates What s the shortest distance between planes Solution from symmetry Normal 1 1 1 in Cartesian The adjacent planes are shown in the diagram left The planes are re drawn to show the symmetry of the cube The body diagonal is separated by the 2 planes so the shortest distance is d 3a 3 a 3 Solution from calculation a1 a 2 1 1 0 a2 a 2 0 1 1 a3 a 2 1 0 1 in Cartesian Coordinates Line 12 a 2 1 0 1 Line 13 a 2 0 1 1 The normal is given by the cross product of these 2 vectors x y z 1 0 1 x y z 0 1 1 Un normalized normal vector 1 1 1 in Cartesian Solution from calculation cont a1 a 2 1 1 0 a2 a 2 0 1 1 a3 a 2 1 0 1 Equation of planes Ax By Cz k constant Point 1 a 2 a 2 0 A B 2k a Point 2 0 a 2 a 2 B C 2k a so A C Point 3 a 2 0 a 2 A C 2k a so A B C k a Therefore equation of the first plane blue plane in previous diagram x y z a and normal A B C 1 1 1 un normalized The next plane has 3 corners at a a 0 0 a a and a 0 a and the corresponding coefficients A B C k 2a The equation of the red plane is x y z 2a or x y z na for all the planes The points at which the normal go thru these planes from origin are a 3 1 1 1 and 2a 3 1 1 1 The distance between these 2 points is just d a 3 3 a 3 Homework Ch 1 Problem 1 2 3 Proof of Table 2


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