We need to be able to describe planes and directions.

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Lester Jenkins
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1 We need to be able to describe planes and directions. Miller Indices & XRD 1 2 Determining crystal structure and identifying materials (B) Plastic deformation Plastic deformation and mechanical properties Plastic (permanent) deformation in metals occurs by the slip of atoms past each other in the crystal in certain directions

2 Physical Properties In certain materials, the atomic structure in certain planes causes the transport of electrons and/or heat to be particularly rapid in that plane, and relatively slow away from the plane. No Example: Graphite conduction this way Dr Greg s Crystallography 5 6 conduction in these planes only Heat conduction 400X faster in this direction than this We need to be able to describe these planes Miller Indices STM Image of Pt (111) [ Eigler et al.] 7 8 2

Basic building blocks Lets look at cubic

3 Basic building blocks Lets look at cubic systems first Simple Cube Body centred cube Face centred cube (and even the tetragonal systems really) 9 10 Describe? Possibilities Lattice planes Use maths!! hx + ky + lz =1 Too complicated for me!

$Get rid of fractions 4.$ $brackets NO COMMAS 13 14 1. Intercepts 2. reciprocals 3. lose fractions 4.$

4 Possibilities Example, front face of cube Use miller indices easier 1. Find the x,y,z intercepts Intercepts: X Y Z Z 2. Calculate the reciprocals 3. Get rid of fractions 4. Place inside round brackets, no commas Reciprocals Miller indices X Y Round brackets NO COMMAS Intercepts 2. reciprocals 3. lose fractions 4. round brackets, no commas Top face? 1. Intercepts 2. reciprocals 3. lose fractions 4. round brackets, no commas RHS face? Z Z Y Y X X

$1. Intercepts 2. reciprocals 3. lose fractions 4.$ Summing Up Z Face Front Top RHS Miller Indices Two 0 s and a 1 Thus two 0 s and a 1

5 1. Intercepts 2. reciprocals 3. lose fractions 4. round brackets, no commas LHS face? Summing Up Z Face Front Top RHS Miller Indices Two 0 s and a 1 Thus two 0 s and a 1 represent all the faces of a cube Family of. LHS Back Y Bottom X The Diagonal planes? In English, the family of 1,1,1 planes Crystallographic Positions

$Move the origin, then vector algebra a) Again, no fractions Indices of direction$ Subtract co-ordinates of tail of arrow from co-ordinates of head) 1 Denotes directions

Subtract co-ordinates of tail of arrow from co-ordinates of head) 1 Denotes directions

6 Crystallographic Directions Vectors!!!!!! Two ways 1. Straight vector algebra 2. Move the origin, then vector algebra a) Again, no fractions Indices of direction Subtract co-ordinates of tail of arrow from co-ordinates of head) 1 Denotes directions z (1,1,1) (0,1,1) y Indices of direction OR move origin to tail of arrow x z x

7 Indices of direction What are the indices of direction of these? All Surprising? z Indices of direction all the vertical ones All Surprising? z (0,1,0) y y x (1,1,0) x Indices of direction all the vertical ones z All Surprising? Two 0 s and a 1 (again!) x y And, for opposite direction Family of directions

Indices of direction Move the origin -1-1 z 0 0 100 100 x 29

of crystals may be different in different directions Metal

6 76.1 Cu 66.7 130.3 191.1 Hexagonal close packed Fe 125.

8 Indices of direction Move the origin -1-1 z x Anisotropy Lets look at the hexagonal system Properties of crystals may be different in different directions Metal Modulus of Elasticity (GPa) [100] [110] [111] Al Cu Hexagonal close packed Fe (GPa) Measured values: Al: 69, Cu: 117, Fe:

Planes in HCP Miller Indices & Miller Bravais Indices Miller indices

(i.e. can be derived from the first two) (h+k) = i h i k 33 34 Miller

Miller Same as cubic system Try the front face.

9 Planes in HCP Miller Indices & Miller Bravais Indices Miller indices (hkl) l M B notation (hkil) the third index (i) is a redundant a 3 one (i.e. can be derived from the first two) (h+k) = i h i k Miller Bravais indices (hkil) for the hexagonal system. Miller Bravais indices (hkil) for the hexagonal system. Same as cubic system Try the front face. a 1 a 2 c a 1 a 2 a 3 c - ( ih k 1 i= - (h+k) a 1 a 2 a 3 c

10 Miller Bravais indices (hkil) for the hexagonal system. Try this one: Try the next face. a 1 a 2 a 3 c Example: What is the designation, using four Miller indices, of the lattice plane shaded pink in the figure? The plane intercepts the a 1, a 3, and c axes at,, and, respectively. Inspection shows that the plane also crosses the a 2 axis at -½ The reciprocals are 1/1= 1/-½ = 1/1 = 1/ = These are already in integer form. Knowing i = (h + k). We can check to see if our indices are correct The designation of this plane is (h k i l) = Example: What is the designation, using four Miller indices, of the lattice plane shaded khaki in the figure? The plane intercepts the a 1, a 3, and c axes at r =, u =, and t =, respectively. The reciprocals are 1/r = 1/u = 1/t = These are already in integer form. We can write down the Miller indices as (h k i l) = (1 k -1 1), where the k index is undetermined so far. Use i = (h + k). That is, k = 0. The designation of this plane is (h k i l) = Directions - A little more complicated We can use just Bravais style indices [uvw] M-B ones, [uvtw] often used We can show that Where the primes are the 3 axes integers (Miller)

11 3 or 4 indices??? Lets index this direction using 4 indices You will find both in the literature Some say 4 is better than 3, some say 3 is better than 4!!!! We shall work with an orthoganol system to start with & ignore the a3 axis [u v w ] [u v t w] Orthogonal primitive cell To get the M-B indices Where does come from? [u v w ] [u v t w]

w = Now we have to use the equations u =(

are found to have symmetry Rotation about

12 Let s do another Ignore a 3 axis Find vectors like we with cubic system u = v = w = Now we have to use the equations u =( v = ( t = -( w = Smallest set of integers that give a path on all 4 axes Symmetry Symmetry Operations Unit cells are found to have symmetry Rotation about an axis Reflection at a plane Translational Inversion through a point Glide (=reflection + translation) Screw (=rotation + translation)

Rotational symmetry Symmetry Elements 90 Symmetry 1 2 3

(4-fold) A6 (6-fold) Symmetry axis Angle of rotation 49

13 Rotational symmetry Symmetry Elements 90 Symmetry fold axis A1 (1-fold) A2 (2-fold) A3 (3-fold) A4 (4-fold) A6 (6-fold) Symmetry axis Angle of rotation Mirror Symmetry Mirror Symmetry Symmetry Planes

14 Mirror Symmetry Symmetry Planes Centre of Symmetry 1. The centre of symmetry i is a point in space such that if a line is drawn from any part (atom) of the molecule to that point and extended an equal distance beyond it, an analogous part (atom) will be encountered Cullity. Elements of

The structure of liquids and glasses. The lattice and unit cell in 1D. The structure of crystalline materials. Describing condensed phase structures

The structure of liquids and glasses. The lattice and unit cell in 1D. The structure of crystalline materials. Describing condensed phase structures Describing condensed phase structures Describing the structure of an isolated small molecule is easy to do Just specify the bond distances and angles How do we describe the structure of a condensed phase?