1.Bravais Lattices The Bravais lattices Bravais Lattice detail

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1 1.Brvis Lttices The Brvis lttices Brvis Lttice detil The Brvis lttice re the distinct lttice types which when repeted cn fill the whole spce. The lttice cn therefore be generted by three unit vectors, 1, 2 nd 3 nd set of integers k, l nd m so tht ech lttice point, identified by vector r, cn be obtined from: r r r r = k 1 + l 2 + m 3 (12.1.1) In two dimensions there re five distinct Brvis lttices, while in three dimensions there re fourteen. These fourteen lttices re further clssified s shown in the tble below where 1, 2 nd 3 re the mgnitudes of the unit vectors nd α, β nd γ re the ngles between the unit vectors. Nme Number of Brvis lttices Conditions Triclinic α β γ Monoclinic α = β = 90 γ Orthorhombic α = β = γ = 90 Tetrgonl 2 1 = 2 3 α = β = γ = 90 Cubic 3 1 = 2 = 3 α = β = γ = 90 Trigonl 1 1 = 2 = 3 α = β = γ < Hexgonl 1 1 = 2 3 α = β = 90, γ = 120 A12-1

2 12.2. Cubic lttices Cubic lttices re of interest since lrge number of mterils hve cubic lttice. There re only three cubic Brvis lttices. All other cubic crystl structures (for instnce the dimond lttice) cn be formed by dding n pproprite bse t ech lttice point to one of those three lttices. The three cubic Brvis lttices re the simple cubic lttice, the body centered cubic lttice nd the fce centered cubic lttice. A summry of some properties of cubic lttices is listed in the tble below: Lttice type Number of lttice points/toms per unit cell Nerest distnce between lttice points Mximum pcking density Exmple Simple cubic 1/1 π/6 = 52 % Phosphor Body centered cubic 2/2 3/2 π 3/8 = 68 % Tungsten Fce centered cubic 4/4 2/2 π 2/3 = 74 % Aluminum Dimond 4/8 2/2 Nerest distnce between toms: 3/4 π 3/16 = 34 % Silicon Cubic lttices hve the highest degree of symmetry of ny Brvis lttice. They belong to the (m3m) symmetry group, which contins the following symmetry groups nd opertions: Symmetry group Symbol Symmetry Opertions Identity 1 Three equivlent xis of two-fold Six equivlent xis of four-fold Six equivlent xis of two-fold Eight equivlent xis of three-fold 3[2 ] [100], [010], [001] 6[4 ] [100], [010, [001], [-100], [0-10], [00-1] 6[2] [110], [101], [011], [1-10], [10-1], [01-1] 8[3] [111], [11-1], [1-11], [-111], [-1-1-1], [-1-11], [- 11-1], [1-1-1] A12-2

3 11-1], [1-1-1] Inversion -1 Three equivlent mirror plnes 3[m ] [100], [010], [001] Six equivlent xis of four-fold with inversion 6[-4] [100], [010, [001], [-100], [0-10], [00-1] Six equivlent mirror plnes 6[m] [110], [101], [011], [1-10], [10-1], [01-1] Eight equivlent xis of three-fold with inversion 8[-3] [111], [11-1], [1-11], [-111], [-1-1-1], [-1-11], [- 11-1], [1-1-1] Note tht the (m3m) symmetry group is the highest possible symmetry group ssocited with cubic crystl. A limited symmetry of the bsis (the rrngement of toms ssocited with ech lttice point) cn yield lower overll symmetry group of the crystl Simple cubic lttice The simple cubic lttice consists of the lttice points identified by the corners of closely pcked cubes. ` Figure A The simple cubic lttice. The simple cubic lttice contins one lttice point per unit cell. The unit cell is the cube connecting the individul lttice points. The toms in the picture re shown s n exmple nd to indicte the loction of the lttice points. The mximum pcking density occurs when the toms hve rdius, which equls hlf of the side of the unit cell. The corresponding mximum pcking density is 52 %. A12-3

4 12.4. Body centered cubic lttice The body-centered lttice equls the simple cubic lttice with the ddition of lttice point in the center of ech cube. Figure The body-centered cubic lttice. The body centered cubic lttice contins two lttice point per unit cell. The mximum pcking density occurs when the toms hve rdius, which equls one qurter of the body digonl of the unit cell. The corresponding mximum pcking density is 68 % Fce centered cubic lttice The fce centered lttice equls the simple cubic lttice with the ddition of lttice point in the center of ech of the six fces of ech cube. A12-4

5 Figure The fce centered cubic lttice. The fce centered cubic lttice contins four lttice points per unit cell. The mximum pcking density occurs when the toms hve rdius, which equls one qurter of the digonl of one fce of the unit cell. The corresponding mximum pcking density is 74 %. This is the highest possible pcking density of ny crystl structure s clculted using the ssumption tht toms cn be treted s rigid spheres Dimond lttice The dimond lttice consists of fce centered cubic Brvis point lttice, which contins two identicl toms per lttice point. The distnce between the two toms equls one qurter of the body digonl of the cube. The dimond lttice represents the crystl structure of dimond, germnium nd silicon. A12-5

6 Figure A The dimond lttice of silicon nd germnium The dimond lttice contins lso four lttice points per unit cell but contins 8 toms per unit cell. The mximum pcking density occurs when the toms hve rdius, which equls one eighth of the body digonl of the unit cell. The corresponding mximum pcking density is 34 % Zincblende lttice The zincblende lttice consists of fce centered cubic Brvis point lttice, which contins two different toms per lttice point. The distnce between the two toms equls one qurter of the body digonl of the cube. The dimond lttice represents the crystl structure of zincblende (ZnS), gllium rsenide, indium phosphide, cubic silicon crbide nd cubic gllium nitride. Figure A The zinc-blende crystl structure of GAs nd InP Crystl models Mke your own model of cubic crystl nd C 60 (crbon 60 bucky bll) molecule. Print the pdf files nd follow the ssembly instructions. A12-6