Elementary Crystallography : The essenals 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 paerns the posion of atoms in a crystal • The posion of atoms in crystal coordinates, their local point symmetry, their crystal symmetry, their diffracon 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 operaons • 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 posions 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 Fraconal Coordinates

Axes x,y,z are right-handed Symmetry operaons

• Identy, unit or null • Inversion or center of symmetry • Rotaon • Reflecon 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 direcons Lace – planes and direcons

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. Direcons and Symmetry • [uvw] is a direcon e.g. [111] or [001] • is a family of direcons related by symmetry e.g. <111> with symmetry equivalent direcons [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 are direcons associated with a translaon in this direcon 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 direcons [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 = <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 Direcons – direct space Plane normals – reciprocal space Cubic Symmetry Lace Triclinic 3D Lace Triclinic lace - direcons

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 convenonal 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. • Databases accessed by EBSD systems and some will propose Pnma, the American Mineralogy database (hp://rruff.geo.arizona.edu/AMS/amcsd.php) will propose Pbnm Space groups – different unit cell orientaons II • For Pbnm for Forsterite a=4.7534 Å b=10.1902 Å c=5.9783 Å Fugino, K., S. Sasaki, Y Takeuchi, and R. Sadanaga (1981) X-ray determinaon of electron distribuons in forsterite, fayalite. and tephroite, Acta Cryst., B37, 513-518. • For Pnma for Forsterite a=10.1902 Å b=5.9783 Å c=4.7534 Å • If you measure pole figures by diffracon (Bragg’s law) or orientaons from Kikuchi lines in EBSD data depend on diffracon data related to the d-space between planes and hence the lengths of a,b,c of Forsterite. • The pole figures measured with Pbnm (100),(010) & (001) corresponds to (001),(100),(010) with Pnma. • So beware, always check which cell orientaon you are using Pbnm or Pnma. The Symmetry command in MTEX

• ‘crystal symmetry system’ e.g. Laüe class • [a,b,c] – unit cell lengths (Å, nm, rao of axes) • [alpha,beta,gamma]*degree - cell angles • 'X||a*','Z||c' - Two orthogonal reference direcons from X,Y or Z parallel to two orthogonal crystallographic direcons from a,a*,b,b* and c,c* MTEX examples of Crystal Symmetry (CS) cs = symmetry('triclinic',[5.29,9.18,9.42],[90.4,98.9,90.1]*degree,... 'X||a*','Z||c','mineral','Talc'); cs = symmetry('mmm', [4.7646 10.2296 5.9942], 'mineral', 'olivine');! cs = symmetry('quartz.cif'); cs = {... symmetry('mmm', [4.756 10.207 5.98], 'mineral', 'Forsterite'),... symmetry('mmm', [18.2406 8.8302 5.1852], 'mineral', 'Enstatite'),... symmetry('2/m', [9.746 8.99 5.251], [90,105.63,90]*degree, 'X||a*', 'Y||b*', 'Z||c', 'mineral', 'Diopside'),... symmetry('m-3m', 'mineral', 'Chromite')}; Frequency of different point groups

metals Crystallographic calculaons MTEX crystallographic Vectors

Crystallographic.direction.–.[uvw].. Family.of.symmetrically.equivalent.directions.. a.=ua.+.vb.+wc.=.a.+.0.+.0. . MTEX:.a.=.Miller(1,0,0,CS,'uvw','phase','Forsterite')

Crystallographic.pole.to.plane.–.(hkl).. Family.of.symmetrically.equivalent.planes.{hkl} a*.=ha*.+.kb*.+lc*.=.a*.+.0.+.0. . .MTEX: a*.=.a_star.=.⟘(100).=.Miller(1,0,0,CS,'hkl','phase','Forsterite'). Crystallographic angles in MTEX Editor Window

%% Angles between crystallographic vectors in MTEX M1=Miller(1,1,1,CS,'Forsterite','hkl') M2=Miller(1,2,-1,CS,'Forsterite','hkl') % Smallest angle between M1 and M2 using all symmetrically equivalent % directions of M2 Minimum_angle_M1_to_M2 = angle(M1,M2)/degree % Restrict angle to be between vectors (hkl) or [uvw] of M1 and M2 % using conversion to specimen vectors by command 'vector3d' Angle_M1_to_M2 = angle(vector3d(M1),vector3d(M2))/degree

Command Window

M1 = Miller (show methods, plot) size: 1 x 1 mineral: Forsterite (mmm) h 1 k 1 l 3

M2 = Miller (show methods, plot) size: 1 x 1 mineral: Forsterite (mmm) h 1 k 2 l -1

Minimum_angle_M1_to_M2 = 53.6606

Angle_M1_to_M2 = 96.3997 Dot, scalar or inner product of vectors a and b Dot, scalar or inner product of vectors • Angle between a and b = cos-1(a.b/|a||b|) = acosd(dot(a,b)/norm(a)/norm(b)) • If two direcons a,b are perpendicular then dot(a,b)=0 • Dot product is equivalent to cosine of the angle between two vectors. No need to actually calculated the angles between, n slip plane normal, b the slip direcon and the r extension direcon. For Schmid factor SF = cos(angle n to r) cos(angle b to r), SF = dot(n,r)*dot(b,r) (N.B. r extension direcon should be in crystal coordinates) Critical resolved shear stress for slip - τc

r Schmid’s law (1924) - resolved shear stress (τ) σ for slip in direction (b) on the slip plane with normal (n) τ = (F/A) cos λ cos φ = σ cos λ cos φ = σ S

where cos θ cos φ is called the Schmid factor (0≥S≤0.5), n The minimum resolved shear stress to activate b slip (yielding) is called the Critical Resolved Shear Stress or CRSS or τc

Hence at yielding σ = τc/S and τc = σ S

Schmid factor, S = (cos λ)*(cos φ) = (b.r) * (n.r) where σ r=stress axis, n=slip plane normal and b= slip direction

N.B. The Schmid factor is defined in crystal coordinates or inverse pole figure space

Weiss Zone Law : dot product h.v= 0 The Weiss zone law states that if the direction [uvw] lies in the plane (HKL), then: Hu + Kv + Lw = 0 From the Weiss zone law the following rule can be derived: The direction, [uvw], of the intersection of 2 planes (H1K1L1) and (H2K2L2) is given by: u = K1L2 − K2L1 v = L1H2 − L2H1 w = H1K2 − H2K1 As it is derived from the Weiss zone law, this relation applies to all crystal systems, including those that do not have orthogonal a, b and c axes.

Hexagonal a-axes direcons in the basal plane a3[-1-120]

+2

-1 +2 -1 -1 a2[-12-10]

+2 -1 unit direction 1/3[2-1-10] a1[2-1-10] -1 a2[-12-10] a1[2-1-10] a3[-1-120] MTEX – reference frame with hexagonal axes

X X

Y Z -Z Y Z -Z

X ll a* Z ll c X ll a Z ll c Resumé HKL and hkil UVW and uvtw

• HKL and hkil i=-(h+k) and h+k+i=0 • H=h; K=k; L=l • h=H; k=K; i=-(h+k); l=L • e.g. (1 0 0) = (1 0 -1 0) = (10.0) • UVW and uvtw, u+v+t=0 • U=(u-t); V=(v-t); W=w • u=(2U-V)/3; v=(2V-U)/3; t=-(U+V)/3; w=W • e.g. [100] = [2-1-10] = [2-1.0] or [2-1*0] • Weiss Zone Law hu + kv + it + lw = 0 Vector cross product Vector cross product

• Calculang the vector that is normal to v1 and v2. v3 = v1 x v2 , v3 = cross(v1,v2), this very useful. • Area of plane containing v1 and v2 Vector Scalar Triple Product Vector Scalar Triple Product

• Test if your reference frame (v1,v2,v3) is right- handed, if dot(v1,cross(v2,v3))>0 it is right- handed • Calculate the unit cell volume = dot(a,cross(b,c)) • Calculate the unit reciprocal cell volume = dot(a*,cross(b*,c*)) Structure factor - amplitude and phase of diffracted wave • The structure factor is a mathemacal funcon describing the amplitude and phase of a wave diffracted from crystal lace planes characterised by Miller indices (hkl) • The structure factor may be expressed as

where the sum is over all atoms in the unit cell, xj,yj,zj are the posional coordinates of the jth atom, fj is the scaering factor of the jth atom (is measure of the scaering power of an isolated atom and is specific for each atom Mg, Ca, Fe etc, their charge state Fe2+, Fe3+, and

radiaon, x-rays, neutrons or electrons) αhkl is the phase of the diffracted beam.

• The intensity of a diffracted beam is directly related to the amplitude of the structure factor 2 2 1/2 IFhklI = ((Ahkl) + (Bhkl) ) , where as the phase is given by tan (phase angle) = Bhkl /Ahkl

Most recent reference for electrons is Electron diffracon. C. Colliex, J. M. Cowley, S. L. Dudarev, M. Fink, J. Gjønnes, R. Hilderbrandt, A. Howie, D. F. Lynch, L. M. Peng, G. Ren, A. W. Ross, V. H. Smith Jr, J. C. H. Spence, J. W. Steeds, J. Wang, M. J. Whelan and B. B. Zvyagin. Internaonal Tables for Crystallography (2006). Vol. C, ch. 4.3, pp. 259-429 doi: 10.1107/97809553602060000593

Scaering Factor Amplitude – schemac x-rays & electrons

- Stronger diffraction at smaller 2θ, where scattered waves are in phase (constructive interference) - Stronger diffraction for heavier atoms, because an increasing fraction of the scattering is elastic (no energy loss) and hence stronger diffraction - Scattered intensity is given by (scattering amplitude)2 Scaering factor amplitude : anions and caons Atomic scaering amplitudes for Cu X-rays & Electrons a strong variation in scattering as a function of 2θ because their wavelengths are similar to inter-atomic distances.

They also have similar behaviour because they interact with the outer (shell) electron cloud of the atoms.

The electrons scattering decreases more rapidly than x-rays with increasing 2θ, because electrons also interact with the atom nuclei.

The neutrons scattering is almost constant with 2θ, because neutrons interact mainly with the nucleus (except for magnetic materials) and the size of the nucleus is very small compared to the wavelength of neutrons. Example alpha-Quartz

• REFLEXIONS • • ------• ref no. h k l d [A] d* [1/A] F(Real) F(Imaginary) |F| Phase [deg] I/Imax • ------• [ 1] -1 1 -1 3.3434 0.29909 -2.0643e+05 -3.5754e+05 4.1285e+05 240.00 100.000% • [ 2] 1 -1 1 3.3434 0.29909 -2.0643e+05 3.5754e+05 4.1285e+05 120.00 100.000% • [ 3] 0 1 1 3.3434 0.29909 4.1285e+05 5.7203e-02 4.1285e+05 0.00 100.000% • [ 4] 0 -1 -1 3.3434 0.29909 4.1285e+05 -5.7203e-02 4.1285e+05 0.00 100.000% • [ 5] 1 0 -1 3.3434 0.29909 -2.0643e+05 3.5754e+05 4.1285e+05 120.00 100.000% • [ 6] -1 0 1 3.3434 0.29909 -2.0643e+05 -3.5754e+05 4.1285e+05 240.00 100.000% • [ 7] 1 -1 -1 3.3434 0.29909 -1.3703e+05 -2.3735e+05 2.7407e+05 240.00 44.067% • [ 8] -1 1 1 3.3434 0.29909 -1.3703e+05 2.3735e+05 2.7407e+05 120.00 44.067% • [ 9] 0 -1 1 3.3434 0.29909 2.7407e+05 1.9396e-02 2.7407e+05 0.00 44.067% • [ 10] 0 1 -1 3.3434 0.29909 2.7407e+05 -1.9396e-02 2.7407e+05 0.00 44.067% • [ 11] -1 0 -1 3.3434 0.29909 -1.3703e+05 2.3735e+05 2.7407e+05 120.00 44.067% • [ 12] 1 0 1 3.3434 0.29909 -1.3703e+05 -2.3735e+05 2.7407e+05 240.00 44.067% • [ 13] 0 2 -3 1.3749 0.72732 2.2390e+05 -6.1543e-03 2.2390e+05 0.00 29.401% • [ 14] 0 -2 3 1.3749 0.72732 2.2390e+05 6.1543e-03 2.2390e+05 0.00 29.401% • [ 15] 2 -2 -3 1.3749 0.72732 2.2390e+05 2.1561e-02 2.2390e+05 0.00 29.401% • [ 16] -2 2 3 1.3749 0.72732 2.2390e+05 -2.1561e-02 2.2390e+05 0.00 29.401% • [ 17] 2 0 3 1.3749 0.72732 2.2390e+05 2.1592e-02 2.2390e+05 0.00 29.401% • [ 18] -2 0 -3 1.3749 0.72732 2.2390e+05 -2.1592e-02 2.2390e+05 0.00 29.401%

Relaonship direct to reciprocal lace

Crystal and Sample Symmetry

1. Crystal symmetry : often defined by the Laue class for CPO measurements based on Electron, X-ray or Neutron diffraction. The Laue class may not reflect the full characteristics of the mineral, e.g. right- and left- hand quartz are in the same Laue class.

2. Sample symmetry : for geological samples use Triclinic symmetry. In materials science and experimentally deformed samples Monoclinic (simple shear deformation), Orthorhombic (rolling or plane strain deformation) and Axial (axial deformation) symmetry may be justified. Sample co-ordinates Sample co-ordinates: axial symmetry about Z-axis 7 Curie liming groups of specimen symmetry

• ∞ (4,6) ∞ II Z • ∞ m (4mm,6mm) ∞ II Z, m⊥X • ∞ 2 (422,622) ∞ II Z, 2 II X • ∞ / m ∞ II Z, m⊥Z • ∞ / mm ∞ II Z, m⊥Z, m⊥X • ∞ ∞ ∞ II Z, ∞ II X • ∞ ∞ m ∞ II Z, ∞ II X, m⊥X Piezoelectric groups Crystal symmetry with equivalent piezoelectric properes Appendix

• Reciprocal Lace Concept • Metric Tensor & Reciprocal Metric tensor • Crystallographic calculaons The reciprocal lace concept : 1 A concept is a cognive unit of meaning - an abstract idea or a mental symbol The noon of reciprocal vectors was introduced into vector analysis by J. W. Gibbs (1881). The concept of reciprocal space or reciprocal lace was adapted by P. P. Ewald to interpret the diffracon paerns of an orthorhombic crystal (1913). The concept is very useful for metric calculaons and diffracon geometry. Let a, b, c be the elementary translaon vectors (i.e. associated with alpha, beta, gamma) of real space or direct lace. The reciprocal lace is defined by translaon vectors a*,b* and c*, which sasfy the following condions related to the direct space: a*.b = a*.c = 0 (a* is normal to b and c) b*.a = b*.c = 0 (b* is normal to a and c) c*.a = c*.b = 0 (c* is normal to a and b) and a*.a = 1 (a=1/a* these relaons fix the direcon and magnitude of a* etc) b*.b = 1 (b=1/b*) c*.c = 1 (c=1/c*)

From these relaons it follows that reciprocal vectors are a* = (b x c) / a. (b x c) b*=(c x a) / b. (c x a) c*=(a x b) / c.(a x b) As volume of the unit cell , V = a. (b x c) = b. (c x a) = c.(a x b) a* = (b x c) / V b*=(c x a) / V c*=(a x b) / V

With magnitudes Ia*I = bc sin alpha / V Ib*I = ca sin beta / V Ic*I = ab sin gamma / V where Ia*I is equal reciprocal of the spacing of (100) planes, similarly for Ib*I and Ic*I for (010) and (001) .

The roles of direct and reciprocal space many be interchanged: that is the “reciprocal” of the reciprocal lace is the direct lace ! Therefore a = (b* x c*) / V b=(c* x a*) / V c=(a* x b*) / V It may be easily verified that that reciprocal of direct triclinic lace also has triclinic symmetry, but not is true for all lace symmetry classes.

Triclinic crystal : Anorthite Calic-plagioclase

Smith ???? The reciprocal lace concept : 2

Some useful relaonships

1. A lace vector of the reciprocal lace r* r* = ha* + kb* + lc* is normal to the the planes with Miller indices (hkl) in the real or direct lace. If the h,k,l have small integer values the plane is said to be “low index” plane (e.g. 100).

2. The magnitude of r* (i.e. Ir*I = 1/dhkl) , is equal to the reciprocal of the spacing of the planes (dhkl) of the real lace with indices (hkl). The dhkl is called the “d-spacing” in angstroms or nanometres.

Proof: 1. The vectors (A-O),(B-O) and (C-O) define the intercepts of the plane (hkl) at A with a/h, B with b/k and C with c/l. The vectors along the triangular edges of plane (hkl) are therefore; (B-A) = b/k - a/h (C-A) = c/l - a/h (C-B) = c/l – b/k Hence r*.(B-A) = (ha*+kb*+lc*).(b/k - a/h) = h/ha.a*- k/kb.b* = 1 - 1 = 0 as a.a*=b.b*=1 and h/h=k/k= 1 and therefore r*.(B-A) = r*.(C-A) = r*.(C-B) =0 Vectors (B-A), (C-A) and (C-B) are in the plane (hkl) and perpendicular to r*

2. Remembering that a dot product AO.r* is equal to the projecon of AO on the direcon of r* mulplied by the magnitude of r* . If n=r*/Ir*I is a unit vector parallel to ON and r*, then OA.n is the projecon of AO in the direcon of n,

dhkl = OA.n = (a/h). (r*/Ir*I) = h a.a*/h Ir*I = 1/Ir*I as h/h =a.a*=1

The reciprocal lace concept : 3

Some useful relaonships (connued)

3. Weiss zone law r* = ha* + kb* + lc* and d = ua + vb + wc The plane (hkl) is in the zone [uvw] if r* is perpendicular to direcon d. (ha* + kb* + lc* ).(ua + vb + wc) = 0 hua*.a + kvb*.b + lwc*.c = 0 as a*.a=b*.b =c*.c =1 hu + kv+ lw = 0

Given two planes in the same zone (h1 k1 l1) and (h2 k2 l2) Zone direcon is u:v:w = (k1 l2 – k2 l1) : (l1 h2 –l2 h1) : (h1 k2 – h2 k1)

4. The angle “phi” between two planes (h1 k1 l1) and (h2 k2 l2) r1* = h1a* + k1b* + l1c* r2* = h2a* + k2b* + l2c* r1*.r2* = Ir1*I Ir2*I cos (phi) cos (phi) = r1*.r2* /Ir1*I Ir2*I = (h1a* + k1b* + l1c*).(h2a* + k2b* + l2c*)/(dhkl1 dhkl2) dhkl = 1/ I ha* + kb* + lc*I

2 2 2 2 2 dhkl = 1/ √(h a* +k b* +l c*+2klb*c*cos(alpha*)+2lhc*a*cos(beta*) +2hka*b*cos(gamma*))

Metric Tensor • The metric tensor is useful to make crystallographic calculaons on the direct lace on any symmetry using matrix methods • The metric tensor M = Metric Tensor 2

• The scalar product is two direcons is given by

This allows the calculaon of the angle between 2 direcons and the modulus of a direcon cos (angle) = [UVW1].[UVW2]/ IUVW1I IUVW2I Reciprocal Metric Tensor

• In a similar way we can construct a reciprocal metric tensor

• cos (angle) = [hkl1].[hkl2]/ Ihkl1I Ihkl2I Using reciprocal metric tensor Crystallographic to Cartesian axes • Many calculations are simple and more familiar in a Cartesian or orthogonal basis • A transformation matrix (CMT) Cartesian to Triclinic crystal.

• ucartesian = CMT . ucrystal - uvw -1 T • hcartesian = (CMT ) . hcrystal - hkl -1 • ucrystal-uvw = (CMT ) . ucartesian T • hcrystal-hkl = (CMT) . hcartesian u = direction, h = pole to plane Mt = transpose of M, M-1 = inverse of M Rhombhedral and hexagonal axes Summary of crystallographic calculaons

• There are many possible choices of a orthonormal or cartesian axes, X,Y and Z for a triclinic crystal, the most general case. • Here we will use the choice used W.L. Bond (1976) Crystal Technology, John Wiley & Son, (appendix pp.329-335) and M.B. Boisen, Jr. & G.V. Gibbs (1990) Mathemacal Crystallography, Reviews in Mineralogy , vol. 15, (revised edion), Mineraological Society of America (pp72-76) • Z is parallel to the c-axis, Y is perpendicular to the a and c axes, hence Y is parallel to [c x a] = b*, and X is in the plane containing the a and c direcons and is normal to c and b*, hence X is parallel to [b*x c] to make a right-handed cartesian basis • A transformaon matrix can be constructed with three column vectors that define the direcons X,Y and Z in the triclinic crystal basis defined by a,b,c,alpha,beta,gamma. Orthogonal-Crystal Transformations

⎛ asin β b u 0⎞ CMT = ⎜ 0 b v 0⎟ ⎜ ⎟ ⎝ a cosβ b cos α c⎠ where u = (cos γ - cos α cos β) / sin β , v = (sin2α - u2 )1/2 and a,b,c are the unit cell dimensions and α,β,γ are the inter-axial angles.

⎛ 1 / asinβ −u / a v sinβ 0 ⎞ TMC = ⎜ 0 1 / b v 0 ⎟ ⎜ ⎟ cot / c (ucot cos ) / cv 1 / c ⎝ − β β − α ⎠ Uses of the Orthogonal-Crystal Transforms

uc = CMT.ut ut = TMC.uc T hc = ht.TMC = (TMC) .ht T ht = hc.CMT = (CMT) .hc Crystal-Reciprocal Transforms (Metric Tensors)

Direct (or Real) Lattice Metric tensor, G 2 ⎛a.a a.b a.c⎞ ⎛ a abcosγ accosβ⎞ ⎜ ⎟ G = ⎜a.b b.b b.c⎟ = abcosγ b2 bccosα ⎜ ⎟ ⎜ ⎟ ⎜ 2 ⎟ ⎝a.c a.b c.c⎠ ⎝ accosβ bccosα c ⎠ Reciprocal Lattice Metric Tensor, G-1 2 ⎛a*.a* a*.b* a*.c*⎞ ⎛ (a*) a *b * cosγ * a * c * cosβ *⎞ ⎜ ⎟ G-1 = ⎜a*.b* b*.b* b*.c*⎟ = a *b * cosγ * (b*)2 b * c * cosα * ⎜ ⎟ ⎜ ⎟ ⎜ 2 ⎟ ⎝a*.c* a*.b* c*.c*⎠ ⎝ a * c * cosβ * b * c * cosα * (c*) ⎠ a,b,c direct lattice vectors a*,b*,c* reciprocal lattice vectors Unit cell Volumes Volume of unit cell V = G 1/2 1/2 Volume of reciprocal unit cell V * = G-1

Length of vectors u 2 = ut.G.u u = (ut.G.u)1/2 h 2 = ht.G-1.h u = (ht.G-1.h)1/2

Angles between vectors Angle between planes h and k cosθ = (ht.G-1.k)/( h.k ) Angle between directions u and v cosφ = (ut.G .v)/(u.v ) Angle between direction u and plane h cos = (ht.v)/( h.v ) δ Thank you

• Special thanks to Prof Katsuyoshi MICHIBAYASHI for invitaon to come here and Shizuoka University for funding my visit as invited Professor in November. • I thank you all for listening. • Most things I talked about (programs, pdf of publicaons, MTEX examples and link to the MTEX site) can accessed via my webpage hp://www.gm.univ-montp2.fr/PERSO/mainprice/ • Also look at NEW MTEX website hp://mtex-toolbox.github.io