Crystlline Structures The sics Crystl structure of mteril is wy in which toms, ions, molecules re sptilly rrnged in 3-D spce. Crystl structure = lttice (unit cell geometry) + bsis (tom, ion, or molecule positions plced on lttice points within the unit cell). A lttice is used in context when describing crystlline structures, mens 3-D rry of points in spce. Every lttice point must hve identicl surroundings. Unit cell: smllest repetitive volume which contins the complete lttice pttern of crystl. A unit cell is chosen to represent the highest level of geometric symmetry of the crystl structure. It s the bsic structurl unit or building block of crystl structure. 7 crystl systems in 3-D Ech crystl structure is built by stcking unit cells nd plcing objects (motifs, bsis) on the lttice points: 14 crystl lttices in 3-D, b, nd c re the lttice constnts, b, g re the interxil ngles 1
Metllic Crystl Structures (the simplest) Recll, tht ) coulombic ttrction between deloclized vlence electrons nd positively chrged cores is isotropic (non-directionl), b) typiclly, only one element is present, so ll tomic rdii re the sme, c) nerest neighbor distnces tend to be smll, nd d) electron cloud shields cores from ech other. For these resons, metllic bonding leds to close pcked, dense crystl structures tht mximize spce filling nd coordintion number (number of nerest neighbors). Most elementl metls crystllize in the FCC (fce-centered cubic), CC (body-centered cubic, or HCP (hexgonl close pcked) structures: Room temperture crystl structure Crystl structure just before it melts 2
Recll: Simple Cubic (SC) Structure Rre due to low pcking density (only -Po hs this structure) Close-pcked directions re cube edges. A hrd sphere unit cell (ech sphere represents n ion core with metllic bonding in between): Reducedsphere unit cells (8 totl: 2 x 2 x 2) Coordintion # = 6 (# of nerest neighbors) Group V elements (As, Sb nd i) ll crystllize in structures tht cn be thought of s distorted versions of 1 2 3 the SC rrngement trigonl (rhombohedrl) 1 Three nerest-neighbors re close nd three SC: 2 bodydigonl 3 digonl re further wy. These structures re more body- 1 ccurtely described s 3 + 3 rrngements. 3 3 (red line) 2 (green line)
Simple Cubic (SC) Atomic Pcking Fctor (APF) APF = Volume of toms in unit cell* Volume of entire unit cell APF for simple cubic structure = 0.52 Adpted from Fig. 3.42, Cllister & Rethwisch 3e. *ssume hrd spheres R=0.5 close-pcked directions=cube edges contins 8 x 1/8 = 1 tom/unit cell (see next slide for clcultion) toms unit cell APF = 1 volume 4 3 p (0.5) 3 tom 3 volume unit cell 4
Number of Lttice points (e.g. toms/ions) per Unit Cell +. N e = number of lttice points on cell edges (shred by? cells) Exmple: e = edge e 5
Recll: ody Centered Cubic (CC) Structure Atoms touch ech other long cube (body) digonls. --Note: All toms re identicl; the center tom is shded differently only for ese of viewing. Exmples: Cr, W, -Fe, T, Mo Coordintion # = 8 Adpted from Fig. 3.2, Cllister & Rethwisch 3e. Hrd sphere unit cell Reduced-sphere unit cell 2 toms/unit cell = 1 center + 8 corners x 1/8 Extended unit cells In ddition to the 8 N.N., there re 6 next N.N. (N.N.N.) only 15% (2/ 3) further wy. So this nerly 14 coordinte structure could lso be described s hving 8 + 6 coordintion. 8 unit cells: 6
Recll: CC APF APF for body-centered cubic structure = 0.68 Q 3 Q N 2 P N Adpted from Fig. 3.2(), Cllister & Rethwisch 3e. toms O unit cell APF = R 2 4 3 p ( 3/4 )3 3 P Close-pcked directions (cube digonls): length = 4R= 3 volume unit cell volume tom 7
ody Centered Tetrgonl (CT) Recll CC: or extend CC lttice: The fct tht the 6 next N.N. re so close suggests tht tetrgonl (=b c) distortions could led to the formtion of 10 nd 12 coordinte structures: -If we compress the CC structure long the c-xis, the toms long the verticl xis become N.N., i.e. CN=10 (CT) when c/= 2/ 3: -Likewise if we shrink nd b with respect to c, then the in-plne or equtoril toms become N.N., i.e., CN=12 when c/= 2. -The equtoril toms re now on the fce centers (since height of cell is = digonl of squre bse) the repet unit is now identicl to the Fce Centered cubic (FCC) or cubic close pcked (CCP): b Only Protctinium (P) crystllizes in the CT (CN=10) structure, but mny crystl structures hve CT lttice. c 8
Recll: Fce Centered Cubic (FCC) Structure Atoms touch ech other long fce digonls. --Note: All toms re identicl; the fce toms re shded differently only for ese of viewing. Exs.: Al, Cu, Au, Pb, Ni, Pt, Ag Coordintion # = 12 12-fold coordintion of ech lttice point (sme tom) is identicl. Adpted from Fig. 3.1, Cllister & Rethwisch 3e. Hrd sphere unit cell Reduced-sphere unit cell 4 toms/unit cell = 6 fce x 1/2 + 8 corners x 1/8 2 unit cells (1 x 2) Extended unit cells 9
APF for fce-centered cubic structure = 0.74 which is mximum chievble APF for sme dimeter spheres, known s Kepler conjecture. Adpted from Fig. 3.1(), Cllister & Rethwisch 3e. 2 Recll: FCC APF FCC lso known s cubic close pcked (CCP) structure. Close-pcked directions: length = 4R = 2 toms unit cell APF = 4 4 3 p ( 2/4 )3 3 volume tom volume unit cell 10
Recll: Hexgonl Closed Pcked (HCP) Not ll metls hve unit cells with cubic symmetry, mny metls hve HCP crystl structures. z sl plne A A y G H E F x A A A exs: Zn, Mg, -Ti (room temp) Alterntively, the unit cell of HCP my be specified s prllelepiped defined by toms lbeled A through H with J tom lying in unit cell interior (not t body center), it s t (1/3 x, 2/3 y, ½ z). CN = 12 Atoms/unit cell = 6 APF = 0.74 c/ = 1.633 11
Recll: Idel c/ rtio for HCP is 1.633 A sketch of one-third of n HCP unit cell is shown: Consider the tetrhedron lbeled s JKLM, which is reconstructed s: The tom t point M is midwy between the top nd bottom fces of the unit cell--tht is MH = c/2. And, since toms t points J, K, nd M, ll touch one nother, JM=JK=2R=, where R is the tomic rdius. Furthermore, from tringle JHM, (JM) 2 =(JH) 2 +(MH) 2 or 2 2 2 c 2 JH Now, we cn determine the JH length by considertion of tringle JKL, n equilterl tringle, cos 30 o / 2 JH 3 2 Substituting this vlue for JH in the bove expression yields or JH 3 2 2 solving for c/ c 8 1. 633 3 4 3 2 c 12
Stcking of Metllic Crystl Structures How cn we stck metl toms to minimize empty spce? Which plne is more close-pcked? 2-dimensions vs. Now stck these 2-D lyers to mke 3-D structures 13
Stcking of HCP nd FCC Most crystl structures in this clss re described by compring them to 1 of the 2 C.N.=12 HCP or FCC It is esiest to view HCP nd FCC structures s stcked, close pcked lyers: Single closest pcked lyer: HCP Unit Cell: 3D Projection 3-D structure is formed by stcking these lyers upon one nother. Atoms in the second lyer fit into the vlleys formed by 3 toms of 1 st lyer: A A... Stcking Sequence A sites Shded spheres fit on top of the unshded lyer to mximize pcking. There re two possible positions for the 3 rd lyer: #1: if the toms tke the positions lbeled 1, then they re directly bove the toms in the 1 st lyer thus the 3 rd lyer reproduces the 1 st. If this pttern continues we hve the A A stcking sequence of the HCP structure. 2D Projection equivlent Top lyer 3 rd lyer 1 st lyer 2 nd lyer c sites Middle lyer A sites Adpted from Fig. 3.3(), Cllister & Rethwisch 3e. ottom lyer 14
FCC Unit Cell: FCC Stcking Sequence #2: if, on the other hnd, the toms in the 3 rd lyer occupy the positions lbeled 2, then this lyer is distinct from the 1 st nd the 2 nd. The 4 th lyer must then repet either 1 st or the 2 nd lyer. If it repets the 1 st, then we hve the AC AC stcking sequence of the FCC (CCP) structure: If it repets the 2 nd, then AC AC sequence such s in L, Nd, Pr nd Pm (recll clss7/slide 2: hc-pcked) A C equivlent A is the 1 st lyer is the 2 nd lyer C is the 3 rd lyer 3 rd lyer 1 st lyer 2 nd lyer Closed pcked plnes {111} of toms Corner hs been removed to show stcking of close pcked toms AC AC... Stcking Sequence Tringle represents (111) plne 15
FCC Stcking Sequence (continued) 2D Projections: C A A sites C C sites C sites 16
Compring HCP nd FCC FCC: The similrity of pcking in these 2 structures is noteworthy, both hve 12 CN, APF=0.74 nd identicl densities. The only significnt difference between the structures is in the stcking sequence (AC A), i.e. where the third close pcked lyer in FCC is positioned. Exmple, coblt undergoes mrtensitic trnsformtion from FCC to HCP t ~695 K. The lttice constnt of FCC Co is 3.5446 Å, with nerest-neighbor distnce of 2.5064 Å. While for HCP Co, the lttice constnts nd c re 2.5071 nd Equivlent 4.0695 Å, respectively, its nerestneighbor distnce is 2.4971 Å, nd the to bove but rotted rtio c/ equls 1.623. 17 Note tht A in FCC would be equivlent to A in HCP, without the third close pcked lyer. HCP: