States of Matter SM VIII (post) Crystallography Experimental Basis Crystal Systems Closed Packing Ionic Structures Ref 12: 8 22-1 Experimental Basis is X-ray diffraction; see HT Fig. 21.1, Pet. Fig. 12.43 & 12.44. Recall: diffraction & interference occur if slits (openings) in a barrier and wavelength are of similar size. This requirement is satisfied when considering distances between planes in crystals & (similar to distances between atoms) wavelength of X-rays: 22-2 Prob in-text: 12: 9, 10 λ of X-rays atomic distances end: 12: 59, 61-63, 65a,b, 70 ~ 10 10 m ~ 100 pm Adv Rdg 16: 1,2, 4,5, 8,9 0.1 nm 0.1 nm HT Fig. 22.1 Block Diagram of X-Ray Crystallography 22-3 22-4 Pet. Fig. 12.43 Experimental Basis of Crystallography
Pet. Fig. 12.44 Diffraction by crystal planes 22-5 General Comments 22-6 crystals have 3D repeating pattern of molecular arrangements unit cell is smallest repeating unit whole crystal can be built by stacking unit cells in all directions, without gaps/voids Illustration: Crystal Systems 22-7 HT Fig. 22.2 Crystal Systems 22-8 7 systems exist see HT Fig. 22.2 all unit cells are parallelepipeds, (six faces, opposite faces parallel) but differ in length/angle relationships in addition, different lattice types may exist for each system, e.g., simple, body centered, face centered altogether: 14 lattice types
Crystal systems 22-9 Pet. Fig. 12.38 Cubic Crystal Systems 22-10 In CHEM 101/3: deal mostly w/ cubic system; (once w/ hexagonal system) See Pet. Fig. 12.38 Distinguish simple cubic body centered cubic, bcc face centered cubic, fcc 22-11 Counting Particles in a Unit Cell (see Pet. Fig. 12.42) basic assumption: atoms/molecules are hard spheres in touching contact counting particles in unit cell: body center: 1 (not shared) face center: 1 2 (shared by 2 cells) Pet. Fig. 12.42 Counting in Unit Cells 22-12 edge center: 1 4 ( shared by 4 cells) corner : 1 (shared by 8 8 cells) Total count of particles per unit cells : Simple cubic: 8 x 1/8 = 1 bcc : (8 x 1/8) + 1 = 2 fcc: (8 x 1/8) + (6 x 1/2) = 4
Coordination Number 22-13 Close Packing 22-14 = # of nearest neighbors, in touching contact = max. occupation of space by spherical objects, atoms in particular. Practice: Analyze Pet. Fig. 12.38 & (see Pet. Fig. 12.39) 12.42 1 st layer ( a ) of spheres: 6 spheres surround a central sphere simple cubic: 6 (bad mistake in class) in hexagonal fashion bcc: 8 fcc 12 Pet. Fig. 12.39 Close Packing 22-15 close packing 22-16 2 nd layer (b) spheres center into dips (dimples, indentations) of first layer notice: only 1/2 of dips are filled leaving 2 types of dips in the 2 nd layer: type c, can see through layer a (octahedral in Pet.) type h, can see tops of layer a (tetrahedral in Pet.)
close packing 3 rd layer (c) 22-17 blank, deliberately 22-18 if spheres centers on type c dips: layers repeat abcabcabc = cubic closest packing (ccp), has face centered cubic (fcc) unit cell see Pet. Fig. 12.40 coordination # = 12, # per unit cell = 4 if spheres center on type h dips: layers repeat ababab = hexagonal closest packing (hcp), has body centered hexagonal (bch) unit cell (afterthought: this statement is not quite true; the central yellow atom is somewhat off center; as a matter of fact, bch does not exist as a lattice type) see Pet. Fig. 12.41 coordination # = 12, # per unit cell = 2 Pet. Fig. 12.40 Cubic Closest Packing (ccp) 22-19 Pet. Fig. 12.41 Hexagonal Closest Packing (hcp) 22-20
Packing Efficiency 22-21 HT Fig. 22.3 PE in Simple Cubic Unit Cell 22-22 unit cell is only partially occupied by atoms ( hard spheres ) the rest is empty space (voids) PE = space occupied vol. of unit cell x 100% can be determined by simple geometric calculations; see HT Fig. 22.3, 22.4, 22.5 Summary: system PE (%) simple cubic 52.4 bcc 68.0 closest packing ccp (= fcc) 74.0 hcp (= bch) HT Fig. 22.4 PE in bcc Unit Cell 22-23 HT Fig. 22.5 PE in fcc Unit Cell 22-24
Applied Problems 22-25 Practice 22-26 vol. of unit cell vol. of particles PE crystal type # of particles per unit cell mass of unit cell = (# of particles) x (mass of 1 particle) (m = MM/N A ) density = mass of unit cell volume of unit cell Practice: Sample Final, #11 Pet. 12: 9, 10, 62-66 Practice... 22-27 Ionic Crystal Structures 22-28 type of structure depends, to large extent, on ratio of ionic radii (usually cation/anion) 2 important cases: 1.) if ratio between 0.4-0.7, get rock- salt (=NaCl) structure, e.g., NaCl, RbI, CaO, AgCl 2.) if ratio between 0.7 1.0, get CsCl structure, e.g., CsCl, CsI
NaCl Structure 22-29 Pet.Fig.12.48 NaCl Unit Cell 22-30 see Pet. Fig. 12.48 anions occupy ccp (fcc; abcabc...) positions; somewhat expanded (puffed up); no longer touching each other octahedral holes become large enough to accommodate cations for a tight fit (the center of the fcc cell is such an octahedral hole ) can also be seen as 2 interpenetrating fcc systems unit cell has 4 Cl & 4 Na + ions coordination # (# of cations touching anions & vv.): = 6 i.e., Na + has 6 Cl neighbors; Cl has 6 Na + neighbors CsCl Structure 22-31 Pet.Fig. 12.49 CsCl Unit Cell 22-32 see Pet. Fig. 12.49 anions occupy simple cubic ( primitive ) positions; somewhat expanded cations fit into body center positions can also be perceived as 2 inter-penetrating simple cubic networks unit cell has 1 Cl & 1 Cs + ion (mistake in pre notes) coordination # = 8