VIEWS: 8 PAGES: 29 POSTED ON: 8/9/2012
1 <role> <requirements, Chem 111, Geol 101 or 111> <syllabus> <hard & best> <Lab meets in Rm 220> Mineralogy (Sp' 93) Part 1: Crystallography I. INTRODUCTION Mineralogy = the study of minerals Mineral = A naturally occurring homogeneous solid with a definite (but generally not fixed) chemical composition and a highly ordered atomic arrangement, generally formed by inorganic processes. atomic arrangement -> crystallography (Part I) chemical composition + structure -> crystal chemistry (Part II) analytical methods (throughout course) X-rays -> to see structure, d(Si-O) = 1.6A Electron microscopes -> Light spectroscopy (elements in the stars) * microscopy 2 II A. CRYSTALLOGRAPHY = study of crystals Why study crystallography? 1. Means to an end (e.g., mineral identification) ? -> The physical properties of a mineral are directly related to its crystal structure. Minerals are identified based upon their physical properties. Positive identification based upon analytical methods. 2. An end in itself (e.g., crystal structures <show ortho 8 -> 80 atoms in opx> .. How minerals are classified crystal = homogeneous (scale {micro-, crypto-) solid possessing long-range (?) three-dimensional internal order (a single piece of a mineral) crystalline amorphous eu-, sub-, an-hedral B. Symmetry operations Motivation: All minerals belong to 1 of 6 crystal systems. 1 of 32 point groups, and 1 of 230 space groups, based upon their symmetry (i.e., the number and type of the following symmetry operations) 1. Point symmetry = symmetry that keeps at least one point fixed (rotation & reflection) A. Rotation axes - keeps line fixed (1, 2, 3, 4, 6 )(add symbols) symmetry by rotation - with respect to a line 360/x degrees Repeat 2-fold = 360/2 = 180, 6 = 360/6 = 60 <show overhead note symbols <add projections> <use smiley and frowney faces (for more examples> no 5-, 7-, 8- etc do not tessellate space B. Mirror plane (m) - keeps plane fixed symmetry by reflection - with respect to a plane <show overhead> <use football field example with parallel and cross mirrors> C. Center of symmetry (i) - keeps point fixed symmetry w/respect to a point (an object's geometric center A point is i if all points equidistant from it , but in opposite directions, are equivalent. <show overhead - passout handout add projections> D. Rotoinversions - keeps point fixed symmetry with respect to a point and an axis, 3 combines rotations and inversions ( i (=-1), m(=-2,) -3, -4, -6 , (add symbols) <show overheads and note symbols-handouts add projections> Combination of symmetry operations (1) show & tell, we'll do! (2) precise mathematical methods <Gibbs book> Minerals are arranged into crystal systems based upon number and type of symmetry elements found by (1) observation (i.e.,nAr, ,nm, i, and nAri) lab (2) in structural determination lec Matrix methods: introduce matrices <go over handout> <draw x, y, z system in stick, show model, stereoplot> z z y y x x label x = [100], y = [010], z = [001] define rows of matrix as "x" "y" "z" and look at where x, y, & z go under the symmetry operation rotation -> [001]2 <draw circle and show axes move, add a random point> -1 0 0 0.577 -0.577 [001]2 = 0 -1 0 0.577 = -0.577 0 0 1 0.577 0.577 mirror -> [010]m <draw circle and show axes move, add a random point> 4 1 0 0 0.707 0.707 [001]m = 0 -1 0 0.707 = -0.707 0 0 1 0 0 center of symmetry -> i identity matrix (I) = an n x n matrix, with cij = 1, where i = j and cij = 0, where i /= j.(the one of matrix algebra) 1 0 0 0 1 0 I= 0 0 1 -1 0 0 0 -1 0 - I = i = 0 0 -1 <draw circle and show axes move, add a random point> rotoinversion [001] bar 4 a combination of a rotation & inversion multiply i by [001] 4 <draw circle and show axes move, add a random point> 0 -1 0 -1 0 0 0 1 0 0.707 0 [001]4 = 1 0 0 0 -1 0 = -1 0 0 = 0 = -0.707 0 0 1 0 0 -1 0 0 -1 0.707 -0.707 differences between here and book, cw vs ccw rotations In book there are 4 commas. Why? [001]-4 = {1, 2, -4, -4-1} many operations are composed of others (e.g.) final example [001]6 = 0, 60, 120, 180, 240, 300, deg rotation {1, 6, 3, 2, 3-1, 6-1} 2. Translational symmetry = symmetry created by displacing space. point symmetry keeps one point invariant. <do smiley faces and show symmetry> glide plane = combination of a mirror with translation <turkey tracks in the snow separated by t vectors> Statement: Combine (1) point group operations and translations to form 230 space groups & every crystal structure. Only have the point groups until translation is added. 5 C. Symmetry: axes & faces Nomenclature for crystallography 1. Crystallographic axes & crystal systems <cute way to introduce building blocks, Bravis lattices-14> <cube & halite & the diamond and tetrahedron> Six crystal systems are based upon the basis vector sets defining them. Minimum number of vectors (or axes) to define the system. 1-D -> number line 2-D -> x-y axes set, choices 3-D -> x-y-z axes set, sketch & net, choices, Life in a Cartesian world. or Six crystal systems: axes sets defined by crystal's structure, the crystal defines its own basis vector set. 1. cubic <show model> <make sketch stick & net> a=b=c = = = 90 (define , , ) 2. tetragonal <show model> <make sketch stick & net> a = b /= c = = = 90 3. hexagonal <show model> <make sketch stick & net> a = b /= c = = 90, = 120 6 4. orhtorhombic <show model> <make sketch stick & net> c<a<b = = = 90 5. monoclinic <show model> <make sketch stick & net> c<a<b = = 90, >90 6. triclinic <show model> <make sketch stick & net> c<a<b & > 90 < 90 Above six crystal systems are the skeletons for ALL crystalline materials. 7 2. Crystal faces: naming of faces Motivation: We use x-ray diffraction for (1) identifying minerals (powder diffraction) d-spacing = distance from origin to an (hkl) planes, in A (A = 1 * 10-10m) (2) determining crystal structures (single crystal diff) measure I and d(hkl) -> mathematically determine structure (x-ray pattern = Fourier transform of crystal structure) unit cell = smallest building block of a crystal (crystal's molecule <show OPX & halite> a. Weiss parameters = intercepts of crystal faces with crystal axes b. Miller indices (hkl) = inverse of Weiss parameters w/o fractions (inverse because of interactions of x-rays) GO TO 3-D version of above - overhead or board 8 <show a cube first w/ 010, 020, 111> <change x,y,z to a,b,c> <Give me an (hkl) -> cut a h times, b k times and c l times <clarify (234) as 1/2 1/3 1/4 or 1 2/3 2/4> <draw tetragonal sketch & net label (110) & (-110) <note (110):(-110) = 90> c (1-10) (1-10) 90 b b (110) (110) a (110) a <draw orthorhombic sketch & net, label (110) & (-110) <note (110):(-110) /= 90> c (-110) (-110) b b (110) (110) a (110) a <show in net what a [001]2 does to (100), (110) and (hkl)> 9 -1 0 0 h -h [001] 2 = 0 -1 0 k = -k 0 0 1 l l 3. Zones zone = a direction in a crystal represented by [uvw] zone axis, originally based on intersection of faces in crystal morphology The a,b,c crystallographic axes are [100], [010], [001] <Draw an a,b,c set with (hkl) and [uvw] labelled> 10 D. Crystal projections <already been doing them> Quantify projections by use of a stereonet: <draw> 270 = -90 180 90 90 Plot points based on & coordinates: <do example with overhead> & 10, 10 10, 80 140, 45 270, 90 Concept: 3-D -> 2-D, thus lines plot as points and planes plot as lines (curves). Uses: plot crystal faces & axes determine angles between faces & axes study symmetry <structure geology, sedimentology, directional data in general> 11 <Plot a crystal: axes & miller indices ask> <measure angles between faces & axes> <Plot symmetry elements> <measure angles between symmetry elements> E. Point groups <p 37> point group = creates a finite number of points in space, they are based upon the combinations of rotations & inversions. Set theory and geometrical rules are applied to generate 32 and only 32 point groups. Point groups named by Hermann-Maugoin symbols. By crystal systems <refer to handout for all elements, positions, naming p. 29,30 & 37 triclinic: 1, -1 -1 monoclinic: 2, m, 2/m 2/m symmetry element along b = [010] <differs from book drawing Why?> order of point group = # of symmetry elements 2/m = {1, i, [010]2, [010]m} = 4 orthorhombic: mm2, 222, 2/m 2/m 2/m 12 mm2 symbols refer to [100], [010], [001] mm2 = {1, [100]m, [010]m, [001]2 tetragonal: 4, -4, 4/m, 4mm, -42m, 422, 4/m 2/m 2/m -42m symbols refer to [001], [100], [110] hexagonal: 3, -3, 3m, 32, -32/m 6, -6, 6/m, 6mm, 622, -6m2, 6/m 2/m 2/m b -6m2 a symbols refer to [001], [100], [120] = normal to [100] cubic: 23, 2/m-3, -43m, 432, 4/m -3 2/m 13 4/m -3 2/m make 6 points around each three found, top & bottom symbols refer to [001] = [010]=[100], [111], [110] F. Crystal intergrowths - read it twice (97-103) Twinning: Twin laws: 14 G. Optical symmetry motivation 1. light A. What is light? particle (photons) vs wave (below) Ray path - wave normal - wave front Wave interference and phases - rocks in the pond B. Definitions and formulas n= c vo where, n = refractive index of material c = speed of light in vacuum = 3.0 x 108 m/s (1.0003) vo = speed of light in material o C. Polarized light -> description of the vibration direction of a light wave unpolarized, linear, elliptical, circular linear = light forced to vibrate in one plane. 15 2. indicatrix theory Indicatrix = a 3-D surface that represents the speed of light in material; its shape is proportional to the refractive index isotropic vs anisotropic Optical class Indicatrix symmetry Crystal system isotropic ∞/m ∞/m ∞/m cubic (sphere) uniaxial hexagonal positive ∞/m 2/m 2/m tetragonal negative ellipsoid prolate vs oblate biaxial orthorhombic positive 2/m 2/m 2/m monoclinic negative triaxail ellipsoid triclinic A. Isotropic = cubic minerals and amorphous junk Different sizes of spheres Relate n - WF - VD - WN with circle 16 RP and VD are conjugate radii 17 <start> B. Anisotropic materials = velocity of light varies within the crystal as a function of orientation. Anisotropic behavior (show calcite) Ellipse with N and n axis: <draw an ellipse w/N-n> N = slow VD n = fast VD privileged directions = directions in an anisotropic material along which light is constrained to vibrate = vibration directions (VD) 1. Uniaxial materials = minerals with 2 principal refractive indices, e & w, and one optic axis. Uniaxial indicatrix = prolate or oblate ellipsoid whose surface is the refractive index <add optic axis> Two types of unaxial indicatrix: Prolate (positive, ) and Oblate (negative, ) 2. Biaxial materials = minerals with three principal refractive indices, , , and two optic axes (willy nilly two circular sections). Biaxial indicatrix = triaxial ellipsoid < ' < < ' and X = , Y = , Z = (plus other symbols) Build indicatrix <use models, and sketch> 18 Z OA2 OA1 X Y 19 20 Chapter 3 -> Space groups & crystal structures H. Lattices (Ch 3) Order 1. 1-D: rows of things (smiley faces, atoms, etc) separated by translation (t 2. 2-D: planes of things, can contain rotations (1,2,3,4,6), m, and t lattice = a set of points where each point is surrounded by identical points, points are atoms & atom complexes (get lost) 5 2-D lattices or plane lattices (define unit cells) oblique: t1 /= t2, /= 90 t2 t1 rectangular: t1 /= t2, = 90 t2 t1 21 centered rectangular: t1 = t2, /= 60, 90, or 120 t2 t1 (primitive-P vs centered ) hexagonal: t1 = t2, = 60 or 120 t2 t1 square: t1 = t2, = 90 t2 t1 3. 3-D 14 3-D lattices -> Bravis lattices (p 117) & show (define unit cells) primitive (P) = atoms on corners, n=8 body centered (I) = atoms on corners plus one in center, n=9 face centered (F) = atoms on corners plus one on each face, n=14 side centered (A,B, or C) = atoms on corners plus on two parallel faces, n10 triclinic: P monoclinic: P, I orthorhombic: P, I, F, C (=A & B) tetragonal: P, I hexagonal: P, R -> ==/=90, a=b=c cubic (isometric): P, I, F Space symmetry: 22 screw axes = rotation axis plus a translation = [uvw]rn where, r= rotation n = translation component along [uvw] Screw axes symbols -> point group symbols + barbs Translation determined by-> n/r -> (examples) [001]4 as 1/4 translation along c -> <3-D axis & 2-D stereo> 1 [010]2 = 1/2 translation along b -> <3-D axis & 2-D lattice> 1 Barbs point in direction of rotation (they "dig-in") For cw rotations translation = r-n/r: 43 -> 4-3/4 =1/4 # of barbs related to translation <show 6, 61 - 65, 66, stereo 61, 64> glide planes = reflection plus a translation axial glides = a/2 or b/2 or c/2 b a n-glides = a/2 & b/2,& c/2 2 or 3 of them d-glides = a/4 & b/4 & c/4 2 or 3 of them Glide symbols m = <parallel and perpendicular> axial = <parallel mirror with arrow perpendicular a&b ---, c .... n = <parallel mirror with 45 bard perpendicular _._._._. d = <parallel mirror with 45 bard perpendicular _._>._._. 23 I. Space groups Goal: Use space group operations and atomic positions to represent crystal structures: Space groups = combination of 32 pt groups & 14 Bravis lattices to yield 230 space groups. Composed of point group operation (R) and a translation (t). Space group symbols: letter + "point group symbol" (rotations, m, i, or screws, glides). P 1 (#1) P 2/m P 21/m C 2/c P 2/m 2/m 2/m (Pmmm) P 2/b 2/c 2/n (Pbcn) I 41/a -3 2/d (Ia3d) (#230) space group operation (S) = R & t = where, R = 3 x 3 point group operation t = 3 x 1 translation and its location in space l11 l12 l13 t1 l21 l22 l23 t2 S= l31 l32 l33 t3 t = (I3 - R)e + d where, I3 = 3x3 identity R = point group operation e = position of symmetry operation d = displacement (translation) of symmetry operation example:[001]41 at 1/4a + 1/4b 1. Write R 24 0 -1 0 [001] 4= 1 0 0 0 0 1 2. Write e - location 1 1 e1 4 4 e = = 1 = 1 e2 4 4 e3 e3 0 3. Write d - displacement 0 d1 d= d2 = 0 d3 1 4 4. Use t = (I3 - R)e + d 1 1 0 1 0 0 0 -1 0 4 2 t= 1 0 1 0 - 1 0 0 + 0 = 0 4 0 0 1 0 0 1 0 1 1 4 4 thus, 1 -y + 1 0 -1 0 2 x 2 S = 1 0 0 0 y = x z 0 0 -1 1 z +1 4 4 <go back to lattice and move [0,0,0] and [1/8 1/4 1/2]> Special vs general positions: special position is on a symmetry element (no new ones) general positions is not on a symmetry element (new ones) 25 J. Crystal structures crystal structure = the atomic arrangement of atoms within a mineral, must know both atom location type. Process for crystal structure determination: 1. determine chemistry - electron microprobe 2. determine the shape of the unit cell (willy nilly Bravis lattice, and a,b,c and ,,) - x-ray diffraction, diffraction locations 3. determine the positions of the atoms (willy nilly space group) - x-ray diffraction - diffraction intensities To represent a structure we need a: 1. The space group #1 - #230 2. The unit cell parameters a,b,c & 3. Atomic positions, position parameters, or fractional cell coordinates x = x'/a, y = y'/b, z = z'/c (e.g.) andalusite Al2SiO5 -> #58 = Pnnm (P 21/n 21/n 2/m) a = 7.798, b = 7.903, c = 5.557, = 90 atom x y z Al1 0.0 0.0 0.24 Al2 0.37 0.14 0.5 Si 0.25 0.25 0.0 Oa 0.42 0.36 0.5 Ob 0.42 0.36 0.0 Oc 0.10 0.40 0.0 Od 0.23 0.14 0.24 Al 1, 0.24 b Od Si Al 2, 0.5 a Space groups operations -> 4 Si, Al1, Al2, Oa, Ob, Oc, 8 Od 26 Also look at bond lengths & angles bond length = center - center distance between to bonded atoms d(Si-Od) = 1.630 d(Al1-Od) = 2.086 d(Al2-Od) = 1.814 Thus 4, 6, 5 coordinated cations bond angle = angle made my three atoms <draw> (H-O-H) in water = 104.9 should 109.4 <data on andalusite> <data on clinoptiolite> Views of structures a. Ball & spoke (DRILL or ATOMS) b. polyhedral (make, ATOMS) c. Cork (make) d. 2-d projections (draw, ATOMS) e. stereo-graphic drawings (ATOMS) K. Polymorphism & twinning (read it twice) Structure types = isostructural = materials with different chemistry and the same structure halite (NaCl), KCl, MgO plus 100s polymorphism = different structures with the same chemistry diamond vs graphite low vs high quartz (P3221 to P6222) at 573C reconstructive = bond breaking structural reorganization (gra-dia) displacive = bond bending structural reorganization (low-high qtz) order-disorder = site occupancy of atoms (4-fold ring 1Al & 3Si) disorder higher symmetry, order lower symmetry polytypism = a subclass of polymorphism based upon different stacking arrangements (ZnS wurtzite(H .abcabc.) & sphalerite (C .abab.) structure defects twinning 27 L. X-ray diffraction 1. Theory and production x-rays = electro-magnetic radiation with = 1-1000A ( = 1/E) produced by inner electron transitions M L K Cu target = 1.54A, Mo = 0.707A 2. Mineralogical Applications a. Single crystal diffraction - structure determination measure d(hkl) & I(hkl) - 2000 of 'em <show overhead of 4-circle> <show photos, eudialyte, feldspar, andalusite, lead to reciprocal space> 28 i. Reciprocal space vs direct space inverse relationship between direct & reciprocal space (100) +b (010) 030 +a 200 +b* 210 +a* scale = x-rays used note: a* normal to (100) b* normal to (010) c* normal to (001) work with (100) backward for d-spacing a* = 1/a b* = 1/b c* = 1/c a* = 180 - b* = 180 - g* = 180- 29 ii. Calculating d-spacing a* sin * 0 -c* cos sin * h a* cos * b* c* cos * k = r* 1 l 0 0 c dhkl = 1 / r* b. Powder diffraction - mineral identification n = 2d sin Where, n = 1, 2, ...,n = wavelength (Cu K = 1.54A) d = d-spacing in crystal = angle of incidence of x-ray beam onto sample <draw diffraction setup below with lattice plans> <look at handouts> *index peaks, measure positions & intensities ->identify d hkl I/Ii 2th 4.26 100 35 20.8 3.34 101 100 26.7 2.46 110 12 36.5 2.28 102 12 39.5 etc 1.80 003 1 50.7 a=4.26, c=5.40