Elementary Crystallography : The essen3als for using EBSD & MTEX Institute of Geosciences Shizuoka University 21st-23th November 2014 David Mainprice (Géosciences Montpellier, Montpellier, France) Number of students falling a sleep Typical EBSD paerns of silicate minerals zone Axis = direction planes Paern symmetry analysis : Silicon Zone axes with 4mm, 3m and 2mm point group symmetries Identification as m3m J.R. Michael (2000) Higher Order Laue Zone (HOLZ) Rings Ewald’s sphere in reciprocal space Radius = 1/λ J.R. Michael (2000) HOLZ rings - Variaon with accelerang voltage C 5kV X 10kV 20kV 25kV C X 30kV N.B. “zoom” effect band spacings are larger and smaller solid angle of crystal space at low kV J.R. Michael (2000) Basic Crystallography - Needed for EBSD and MTEX • Understanding of planes, plane normals, direcons and zone axis, Miller or Bravais-Miller indices for different crystal symmetries for describing diffracon pa7erns the posi5on of atoms in a crystal • The posi3on of atoms in crystal coordinates, their local point symmetry, their crystal symmetry, their diffrac3on Laüe symmetry. Where to find informaon to describe a mineral structure not present in your EBSD system and how to enter it. • Once in your EBSD base calculate d-spacing and the diffracted intensity of each plane Plan • Direct space – the real world • Fraconal coordinates • Symmetry opera5ons • Lace planes and direcons • Point groups • Relaonship between direct and reciprocal laces • Crystallographic calculaons (structure factor, angles, d-space, reciprocal lace, metric tensor and orthogonal basis) The direct space • The posions of atoms in crystals is the most fundamental informaon we need. • Atoms in crystals physically exist in real or direct space where the atoms posi3ons are described relave to an origin, typically at a corner, in 3-D volume called the “unit cell”. • The 3 axes of the unit cell x, y and z from a right- handed set with angles alpha (α), beta (β) and gamma(γ) between y & z, x & z and x & y respecvely. The lace parameters, a, b, c describe the dimensions of the unit cell along x, y and z. Right hand rule Unit Cell Frac3onal Coordinates Axes x,y,z are right-handed Symmetry operaons • Iden3ty, unit or null • Inversion or center of symmetry • Rotaon • Reflec3on or mirror • Improper rotaon or rotaon-inversion Inversion symmetry is implemented in MTEX in specimen coordinates (i.e. pole figures) as an option called ‘antipodal’ Monoclinic • A mirror normal to the b-axis results from the combinaon of 2-fold b-axis and inversion. {2[010]} * {-1} = {m[010]} Orthorhombic • A perpendicular 2-fold c-axis results from the combinaon of 2-fold a-axis 2-fold b-axis . {2[100]} * {2[010]} = {2[001]} Four types of symmetry operaon Translation – need for Space Groups 3D and Stereogram for Cubic symmetry A direcon [UVW] : Ua + Vb + Wc Axes x,y,z (a,b,c) are not always orthogonal as α,β,γ may not be equal 90° Typical lace direc3ons Lace – planes and direc3ons William Hallowes Miller (1801–1880). English crystallographer who published a “Treatise on Crystallography” in 1839, using crystallographic reference axes that were parallel to the crystal edges and using reciprocal indices. Direc3ons and Symmetry • [uvw] is a direc3on e.g. [111] or [001] • <uvw> is a family of direc3ons related by symmetry e.g. <111> with symmetry equivalent direc3ons [111, [1-11], [-1-11], [1-11], [-1-1-1], [1-1-1], [-1-1-1] ,[1-1-1] depending of the crystal symmetry, for cubic there are 8. • ½ [uvw] or ½ <uvw> are direc3ons associated with a translaon in this direc3on of ½, ¼,etc. of the lace spacing (e.g. Burgers vector of dislocaon) Planes and symmetry • (hkl) is plane or general form e.g. (111). • {hkl} is family of planes or forms related by symmetry e.g. {hkl} ; (110),(1-10),(-110), (-110), members of the family are specified by the crystal symmetry, for cubic there are 4. • {0kl} is a special family of planes or forms e.g. {0kl} in orthorhombic ; (011),(012),(013) … MTEX and symmetry equivalent direc3ons [uvw] and planes (hkl) A plane normal m = Miller(1,0,1,cs,’hkl’) m = symmetrise(m) generate all Miller indices symmetrically equivalent to m A direcon n = Miller(0,1,0,cs,’uvw’) n = symmetrise(n) generate all Miller indices symmetrically equivalent to n cs = crystal symmetry = ‘mmm’ e.g. ‘mmm’ = orthorhombic Laue class symmetry MTEX – Three reference frames associated with crystals Direct lattice a,b,c Reciprocal Lattice a*,b*,c* Cartesian frame x,y,z For directions For pole to planes For Euler angles ϕ1Φ ϕ2 u = ua + vb + wc h = ha* + kb* + lc* and Tensors r = ix + jy +kz MTEX – symmetrise : (hkl) %% Forsterite (101) plane % define crystal symmetry as orthorhombic CS = crystalSymmetry('mmm', [4.756 10.207 5.98], 'mineral', 'Forsterite', 'color', 'dark green') % plane(101) m = Miller(1,0,1,CS,'Forsterite','hkl') % plot plot(m,'labeled') % save file saveFigure('/MatLab_Programs/Forsterite_plane_101.pdf') %% Symmetry equivalent planes to (101) m_equivalent = symmetrise(m) % plot plot(m_equivalent,'labeled') % save file saveFigure('/MatLab_Programs/Forsterite_planes_101_equivalents.pdf') ! u p p er lower u p p er lower ( 101) ( 101) ( 10 1)¯ ( 1¯ 01) ( 1¯ 01¯ ) Single hkl in upper hemisphere With symmetrise command Four hkls 2 in upper and 2 lower hemisphere MTEX – symmetrise : [uvtw] %% Ice Ih direction a1 [2-1-10] % crystal symmetry CS = crystalSymmetry('6/mmm', [4.5181 4.5181 7.3560],... [90.0 90.0 120.0]*degree,'X||a*','Z||c','mineral','Ice Ih') % direction a1 = [2-1-10] m = Miller(2,-1,-1,0,CS,'Ice Ih','uvtw') % plot plot(m,'labeled') % save file saveFigure('/MatLab_Programs/Ice_direction_a1.pdf') %% Ice Ih symmetry equivalent directions <a> = <2-1-10> m_equivalent = symmetrise(m) % plot plot(m_equivalent,'labeled') % save file saveFigure('/MatLab_Programs/Ice_direction_a1_equivalents.pdf') upper ! lower upper lower [21¯10]¯ [21¯10]¯ [1120]¯ [21¯10]¯ [1120]¯ [21¯10]¯ [12¯ 10]¯ [1210]¯ [12¯ 10]¯ [1210]¯ [2110]¯ [1¯120]¯ [2110]¯ [1¯120]¯ SAME uvtw in upper & lower With symmetrise command hemispheres. 6 uvtws in upper and lower hemispheres Special case in basal (0001) plane Special case in basal (0001) plane When are directions [uvw] parallel to poles to planes ⊥(hkl) with the same indices ? MTEX – When are directions [uvw] parallel to poles to planes ⊥(hkl) with the same indices ? %% Conversion between hkl and uvw % N.B. converted uvw and hkl not always integer values h = Miller(1,1,1,CS,'Forsterite','hkl') % change hkl (h) to uvw uvw = h.uvw u = Miller(1,1,1,CS,'Forsterite','uvw') % change uvw (u) to hkl hkl = u.hkl For (hkl) = (111) the [uvw] =[ 0.0442 0.0096 0.0280] so the indices are NOT the same in this case, but for (hkl) = (100) the [uvw] =[100] they are the same. Example Miller planes 2D Spacing between planes x ll a < y ll b Examples of planes and their hkls Direc3ons – direct space Plane normals – reciprocal space Cubic Symmetry Lace Triclinic 3D Lace Triclinic lace - direc3ons First and higher order planes are parallel, but have different spacing Symmetry operaons From Putnis (2002) 2D points groups: operaons about a point From Putnis (2002) p = primitive : origin at 0,0 Examples of 2D spaces groups: a collecon of points with symmetry operaons c = centred : origin at 1/2, 1/2 How much symmetry do you need ? May be possible with additional symmetry Analysis in some cases Possible with additional symmetry analysis Classical EBSD More than 7 Crystal Symmetry systems You need more than this ! Crystal Symmetries in MTEX : looks great ! EBSD data Laüe class MTEX recognises Laüe classes Laue class -3m1 #1 0 0 0 – null operation 0° around c-axis Euler_list_3m = rotaon (show methods, plot) #2 120 0 0 – 120° around c-axis size: 6 x 1 #3 240 0 0 – 240° around c-axis Bunge Euler angles in degree phi1 Phi phi2 #4 15 180 75 – 180° around a1-axis 0 0 0 #5 15 180 195 – 180° around a2-axis 120 0 0 #6 15 180 315 – 180° around a3-axis 240 0 0 15 180 75 15 180 195 … And if want your pole figures to have the center 15 180 315 of symmetry of a Laue class you need to used the ‘antipodal’ option. We will discuss this when we plot pole figures MTEX4 recognises Point Groups Triclinic Tetragonal 1 4 -1 Laüe class -4 4/m Laüe class Monoclinic 422 211 2-fold (2) parallel to a 4mm m11 mirror (m) normal to a -42m 2/m11 Laüe class(2 & m =2/m) a-axis -4m2 121 2-fold (2) parallel to b 4/mmm Laüe class 1m1 mirror (m) normal to b 12/m1 Laüe class(2 & m =2/m) b-axis Hexagonal 112 2-fold (2) parallel to c 6 11m mirror (m) normal to c -6 112/m Laüe class(2 & m =2/m) c-axis 6/m Laüe class 622 Orthorhombic 6mm 222 -62m 2mm -6m2 m2m 6/mmm Laüe class mm2 mmm Laüe class Cubic 23 Trigonal m-3 Laüe class 3 432 -3 Laüe class -43m 321 m-3m Laüe class 3m1 -3m1 Laüe class 312 -31m Laüe classes (11) <- Point groups (32) <- Space groups (230) * * * * * * α-quartz : Space Groups #152 P3121 #154 P3221 * Point Group 32 Laüe Class -3m * Ice Ih : Space Groups #194 P63/mmc Point Group 6/mmm Laüe Class 6/mmm * * * * High Laüe * Low Laüe http://en.wikipedia.org/wiki/Space_group Symmetry elements of the 11 Laüe classes rotaonal elements Monoclinic sengs In MTEX as 12/m1 [a b c] In MTEX as 112/m [a b c] Space groups – different unit cell orientaons I • Case of Olivine orthorhombic space group #62 with 6 different orientaons of the unit cell • The conven3onal a,b,c can be assigned in 6 ways; Pnam(abc), Pmnb(cab), Pcmn (bca), Pnma (a-cb), Pbnm(ba-c), Pmcn (-cba) • Although Pnma is the standard, but rarely used in mineralogy, where Pbnm is classically used.