Packing of atoms in solids

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1 MME131: Lecture 6 Packing of atoms in solids A. K. M. B. Rashid Professor, Department of MME BUET, Dhaka Today s topics Atomic arrangements in solids Points, directions and planes in unit cell References: 1. Callister. Materials Science and Engineering: An Introduction.. Askeland. The Science and Engineering of Materials. Lec 06, Page 1/19

2 Atomic arrangement in solids No order No regular relationship between atoms Atoms randomly fill up the space to which the gas is confined Example: monatomic gases Ar no order in argon gas Short-range order special arrangement of atoms extends only to the atom s nearest neighbours Each water molecule has a short-range order due to covalent bonding between hydrogen and oxygen atoms at an angle of 104.5, but each water molecule has no special arrangement but instead randomly fill up the space available to them. Similar situation exists in ceramic glasses and non-crystalline polymers H O short range order in steam short range order in glass Lec 06, Page /19

3 Long-range order Atoms rearrange themselves in an identical, regular, repetitive three dimensional grid-like pattern, called a lattice. Lattice is a collection of points, arranged in a periodic pattern so that the surroundings of each lattice point in the lattice are identical. Each atom has both short-range order, since the surroundings of each lattice points are identical, and long-range order, since the lattice extends periodically throughout the entire material. A lattice differs from material to material in both shape and size, depending on the size of atoms and the type of bonding between the atoms. regular long range order in metal Concept of ordering Crystalline both short- and long-range ordered, repetitive three-dimensional, geometric arrangement common in metallic, ceramic and some polymeric materials Non-crystalline, or Amorphous random, short-range ordered, non-repetitive common in glassy and some metallic materials due to complex chemistry and rapid cooling ordered and disordered form of silica Lec 06, Page 3/19

4 silicon single crystal Liquid crystal display amorphous but can undergo localized crystallization in response to an external electric field polycrystalline stainless steel showing grains and grain boundaries Classification of materials based on the type of atomic order Monatomic Gases No order Example: Argon gas Liquid Crystals Long-range order and short-range order in small volumes Examples: LCD Polymers Crystalline Materials Short and long-range order Amorphous Materials No long-range order, only short-range order Examples: Amorphous Si, Glasses, Plastics Single Crystals Examples: Si, GaAs Poly Crystals Examples: Metals & alloys, and most Ceramics Lec 06, Page 4/19

5 Lattice and the unit cell Lattice - A collection of points that divide space into smaller equally sized segments. Unit cell - A sub-division of the lattice that still retains the overall characteristics of the entire lattice, and stacked together endlessly to form the lattice. Atomic radius - The apparent radius of an atom, typically calculated from the dimensions of the unit cell, using close-packed directions (depends upon coordination number). Packing factor - The fraction of space in a unit cell occupied by atoms unit cell Unit cell parameters dimensions (a, b, c) angles (a, b, g) Lec 06, Page 5/19

6 The fourteen types of Bravais lattices grouped in seven crystal systems. Lec 06, Page 6/19

7 Unit cells commonly found in metallic materials The cubic unit cells Simple (SC) Body Centered (BCC) Face Centered (FCC) Example: Po Example: a-fe, Cr, Mo, V Example: g-fe, Al, Cu, Ni The hexagonal unit cell Example: Mg, a-ti, Zn, Zr c c/ a Hexagonal Close-Packed (HCP) True HCP unit cell Lec 06, Page 7/19

8 Illustration showing sharing of face and corner atoms Illustration of co-ordinations in (a) SC and (b) BCC unit cells. Six atoms touch each atom in SC, while the eight atoms touch each atom in the BCC unit cell. Example Atomic packing factor (APF) for Body-centred cubic (BCC) closed-packing direction D r C A B a CD = AD + AC CD = AD + (AB + BC ) (4r) = a 0 + (a 0 + a 0 ) = 3a 0 a 0 = (4/3) r Lec 06, Page 8/19

9 Atomic Packing Factor (APF) = atom volume cell volume = ( # atom / cell ) x ( volume / atom ) cell volume APF for BCC Unit Cell = () x (4/3 p r 3 ) () x (4/3 p r3 ) = a 3 (4r/3) 3 = 0.68 The relationships between the atomic radius and the lattice parameter in cubic systems Lec 06, Page 9/19

10 Unit cell properties: A summary Lattice Parameter Co-ordination Number Number of Atoms per Unit Cell Atomic Packing Factor SC a 0 = r BCC a 0 = (4/3) r FCC a 0 = () r HCP a 0 = r c = a Lec 06, Page 10/19

11 Points, directions, and planes in the unit cell Miller indices - A shorthand notation to describe certain crystallographic directions and planes in a material. Distance is measured in terms of lattice parameters along x, y and z coordinates using right-hand coordinate system. Planes Crystallographic planes are denoted by first ( ) brackets. A negative number is represented by a bar over the number. Directions - Crystallographic directions are denoted by square [ ] brackets. A negative number is represented by a bar over the number. Linear density - The number of lattice points per unit length along a direction. Packing fraction - The fraction of a direction (linear-packing fraction) or a plane (planar-packing factor) that is actually covered by atoms or ions. Lec 06, Page 11/19

12 Determining Miller indices of directions 1. Determine the coordinates of the two points (head and tail). Subtract coordinates of tail from the coordinates of head 3. Clear fractions and/or reduce results to the lowest integer 4. Enclose the number in square brackets. If a negative sign is resulted, show the negative sign with a bar over the number. Example Determining Miller indices of directions Determine the Miller indices of directions A, B, and C in the following figure. 1. Determine the coordinates of head and tail. Subtract coordinates of tail from that of head 3. Clear fractions and/or reduce to the lowest integer 4. Enclose the number in square brackets. Direction A 1. 1, 0, 0 and 0, 0, 0. 1, 0, 0-0, 0, 0 = 1, 0, 0 3. No fractions to clear or integers to reduce 4. [ ] Lec 06, Page 1/19

13 Direction B 1. 1, 1, 1 and 0, 0, 0. 1, 1, 1-0, 0, 0 = 1, 1, 1 3. No fractions to clear or integers to reduce 4. [ ] Direction C 1. 0, 0, 1 and 1/, 1, 0. 0, 0, 1-1/, 1, 0 = -1/, -1, 1 3. (-1/, -1, 1) = -1, -, 4. [ 1 ] family of directions Indicated by < > brackets Equivalency of crystallographic directions of a form in cubic systems. Some other points about direction: A direction and its negative are not identical: [100] is not equal to [100]. They represents the same but opposite directions. A direction and its multiple are identical: [100] is the same direction as [00]. We just forgot to reduce to lowest integers. Lec 06, Page 13/19

14 Example Determining repeat distance, linear density and packing fraction of [110] direction in FCC copper Distance of two corner atoms in a FCC structure is a 0. So the repeat distance (between two adjacent atoms) is = ½(a 0 ) = ½(3.6151x10 8 ) =.5563x10 8 cm The linear density is reciprocal of the repeat distance, i.e. linear density = 1 /.5563x10 8 = 3.91x10 7 lattice points/cm Packing fraction of a particular direction is the fraction actually covered by atoms. So here, packing fraction is the linear density times r, or Packing fraction = (3.91x10 7 lattice point/cm ) x (1.78x10 8 cm) = 1.0 i.e., atoms lie continuously along [110] direction (since this is the close-packed direction). Determining Miller indices of planes Identify the coordinate points at which the plane intercept the x, y, and z axes in terms of the number lattice parameters If the plane passes through the origin, the origin must be moved!!! Take reciprocals of these intercepts Clear fractions, but do not reduce them to lowest integers Enclose the resulting numbers in parentheses ( ). Negative numbers should be written with a bar over the number. Lec 06, Page 14/19

15 Example Determining Miller indices of planes Determine the Miller indices of planes A, B, and C in the following figure. Identify the coordinate points Take reciprocals of these intercepts Clear fractions Enclose the numbers in parentheses ( ). Plane A 1. x = 1, y = 1, z = 1. 1/x = 1, 1/y = 1, 1 /z = 1 3. No fractions to clear 4. (1 1 1) Plane B 1. The plane never intercepts the z axis, so x = 1, y =, z =. 1/x = 1, 1/y =1/, 1/z = 0 3. Clear fractions (by x): 1/x =, 1/y = 1, 1/z = 0 4. (10) Plane C 1. We must move the origin, since the plane passes through 0, 0, 0. Let s move the origin one lattice parameter right in the y-direction. Then, x =, y = -1, z =. 1/x = 0, 1/y = -1, 1/z = 0 3. No fractions to clear. 4. (0 1 0) Lec 06, Page 15/19

16 z z family of planes indicated by { } brackets [00] y y x x Some other points about planes: Planes and their negatives are identical: [00] is identical to [00]. Planes and their multiples are not identical: [100] is not same as [00]. Example Calculating the planar density and packing fraction Calculate the planar density and planar packing fraction for the (010) and (00) planes in simple cubic polonium, which has a lattice parameter of nm. Lec 06, Page 16/19

17 SOLUTION The total atoms on each face is one. The planar density is: Planar density (010) atom per face area of face 8.96 atoms/nm 1atom per face (0.334) atoms/cm The planar packing fraction is given by: area of atoms per face Packing fraction (010) area of face pr (r) 0.79 (1atom) ( pr ( a 0 ) ) However, no atoms are centered on the (00) planes. Therefore, the planar density and the planar packing fraction are both zero. The (010) and (00) planes are not equivalent! Example Drawing direction and plane Draw (a) [11] direction and (b) (10) plane in a cubic unit cell. Lec 06, Page 17/19

18 SOLUTION (a) [ 1 1 ] direction Because we know that we will need to move in the negative y-direction, let s locate the origin at 0, +1, 0. The tail of the direction will be located at this new origin. A second point on the direction can be determined by moving +1 in the x-direction, in the y-direction, and +1 in the z-direction. (a) ( 1 0 ) plane To draw the plane, first take reciprocals of the indices to obtain the intercepts, that is: x = 1/- = -1/ y = 1/1 = 1 z = 1/0 = Since the x-intercept is in a negative direction, and we wish to draw the plane within the unit cell, let s move the origin +1 in the x-direction to 1, 0, 0. Then we can locate the x-intercept at 1/ and the y-intercept at +1. The plane will be parallel to the z-axis. Lec 06, Page 18/19

19 Next Class MME131: Lecture 7 Packing sequences in crystals Lec 06, Page 19/19

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