Tensors and Anisotropic physical properties of rocks Part I : Tensor representation of the physical properties of single crystals

Transcription

1 Tensors and Anisotropic physical properties of rocks Part I : Tensor representation of the physical properties of single crystals David Mainprice Lab. Tectonophysique, Université Montpellier II France DFH-UFA Summer School Textures and Microstructures in the Earth Sciences Freiberg, June 29 - July 8, 2005

2 Plan of this Talk Introduction 2 nd Rank Tensors - Thermal Conductivity, Thermal Diffusivity and Thermal expansion 4 th Rank Tensors - Elasticity, Seismic Velocity Phase Transitions - SiO 2 Post-Perovskite

3 Introduction

4 Why are we interested in Single Crystals? To understand the anisotropic physical properties of polycrystalline rocks caused by crystal preferred orientation (CPO) it is important to know the about the simplest case, the single crystal. The single orientation (single crystal), has a perfectly defined ODF (orientation distribution function), PFs (pole figure) or IPFs (inverse pole figure). To understand the how crystal symmetry, sample symmetry, CPO and single crystal properties combined to produce anisotropic rock properties

5 Physical properties of crystals Thermal conductivity and diffusivity ( 2 th rank tensor) can be calculated from CPO Thermal expansion ( 2 th rank tensor) can be calculated from CPO Electrical conductivity, electrical polarization and dielectric properties) can be calculated from CPO, BUT may not be relevant if conductivity controlled by high conductivity phases in the grain boundaries (e.g. water or carbon) Piezoelectricity ( 3 rd rank tensor) can be calculated from CPO, if we can determine the CPO of the Left- and Right-handed crystals Elasticity ( 4 th rank tensor) seismic (elastic) properties, can be calculated from CPO

6 Anisotropic Properties Calcite optical properties : 2 nd Rank Tensor

7 Tensor rank of physical properties Physical Property (rank) Driving Force (rank) Response (rank) Density (0) Mass (0) Volume (0) Pyroelectricity (1) Temperature (0) Electric Field (1) Electric conductivity (2) Electric Field (1) Electric Current Density (1) Electric Permitivity (2) Electric Field (1) Dielectric Displacement (1) Dielectric Susceptibility (2) Electric Field (1) Polarization (1) Chemical Diffusivity (2) Potential Gradient ve (1) Chemical Flux (1) Thermal Conductivity (2) Temperature Gradient -ve (1) Heat Flux (1) Thermal Expansion (2) Temperature (0) Strain (2) Magnetic Susceptibility (2) Magnetic Field (1) Magnetisation Intensity (1) Magnetic Permeability (2) Magnetic Field (1) Magnetic Induction (1) Piezolectricity (3) Electric Field (1) Strain (2) Elastic Compliance (4) Stress (2) Strain (2) Elastic Stiffness (4) Strain (2) Stress (2)

8 Thermodynamically reversible changes For example, the elastic, thermal, electric and magnetic effects on strain (ε ij ), to first order can be written as a function of the independent variables (stress σ kl, electric field E k, magnetic field H l and temperature gradient ΔT) and their corresponding tensors as ε ij = S ijkl σ kl + d kij E k + q lij H l + α ij ΔT where S ijkl are the elastic compliance, d kij piezo-electric, q lij piezo-magnetic and α ij thermal expansion tensors.

9 Transformation of axes old axes x1 x2 x3 x 1 a11 a12 a13 new x 2 a21 a22 a23 x 3 a31 a32 a33 a ij is a 3 by 3 transformation matrix relating two orthogonal reference axes (x and x ). A major concept in defining the properties of tensors.

10 Cartesian Reference Frame I Measurement of anisotropic physical properties are reported in the literature as components of tensors, typically as tabulated values. The reference frame most commonly used is a right-handed Cartesian (also called orthonormal) system. For cubic, tetragonal and orthorhombic the obvious choice is to use the orthogonal lattice basis vectors a[100], b[010] and c[001] of the crystal axes. However, for most general case of triclinic crystal symmetry where a, b, and c are not orthogonal, there are many possible choices and no general convention. The choice of a specific reference frame is often guided by presence of cleavage or well developed crystal faces that allow for an easy determination of the crystal orientation.

11 Cartesian Reference Frame II Here are 3 common choices, for the tensor Cartesian reference frame (e 1,e 2,e 3 ) ; a) e 3 = c[001], e 2 = b* (010) and hence for a right-handed system e 1 = e 2 x e 3 b) e 3 = c[001], e 1 = a* (100) and e 2 = e 3 x e 1 c) e 3 = c* (001), e 1 = a[100] and e 2 = e 3 x e 1 Reference Frames a), b) and c) for triclinic plagioclase Labradorite An66 ( a=0.817 nm b=1.287 nm c=1.420 nm α=93.46 β= γ=90.51 )

12 Cartesian Reference Frame III You should also be aware that the tensor Cartesian reference frame used for a tensor may not be the same as the reference frame with respect to the crystal axes used for the Euler angles to define your orientation data. For example the commonly used BearTex Texture package uses (a) and the EBSD software from HKLtechnologies uses (b). In general a transformation matrix (tij) will be needed to bring the tensor reference frame into the correct orientation for a specific Euler angle frame. For example g (i,j)=g(k,i)*t(k,j) where g(k,i)=g(φ1,φ,φ2) is Euler angle matrix, g (i,j) is orientation matrix of the tensor and t(k,j) the transformation matrix. Examples of reference frames for elastic constants, but check for yourself! a,b/b*,c* (monoclinic alkali feldspar) (frame c) a*,b/b*,c (monoclinic diopside,augite,hornblende,muscovite) (frame b) a,(c* x a),c* (triclinic plagioclase) (frame c)

13 Transformation of components of a vector A vector P with components (p1,p2,p3) with respect to reference axes x1,x2,x3 is transformed to P = (p 1,p 2,p 3) with respect to reference axes axes x 1,x 2,x 3 using the cosines of the angles between the reference frames. p 1 = p1cos(x1^x 1) + p2cos(x2^x 1)+ p3cos(x3^x 1) etc. or p 1 = p1*a11 + p2*a12+ p3*a13 p 2 = p1*a21 + p2*a22+ p3*a23 p 3 = p1*a31 + p2*a32+ p3*a33 or in dummy suffix notation p i = pj*aij

14 Transformation laws for tensors Rank of Name Tensor New in terms of old Old in terms of new Scalar 0 X =X X =X Vector 1 P i=aijpj Pi=AjiP j 2 nd Rank 2 T ij=aikajltkl Tij=AkiAljT kl 3 rd Rank 3 T ijk=ailajmakntlmn Tijk=AliAmjAnkT lmn 4 th Rank 4 T ijkl= AimAjnAkoAlpTmnop Tijkl= AimAnjAokAplT mnop Difference between a transformation matrix (A ij ) and a 2 nd rank tensor (T ij ) Aij is an 3 by 3 matrix relating two (right-handed) reference frames. Tij is a physical quantity (2 nd rank tensor) that for a given set of reference axes is represented by 9 numbers (a 3 by 3 table).

15 Zero and First Rank Tensors Zero Rank Tensor ( 1 component) Scalars example density (kg/m3) First Rank Tensor (3 components) Vectors example electric field E = [E1 E2 E3] = E i

16 2 nd Rank Tensors

17 2 nd Rank Tensors - important for geophysics Typically relates 2 vectors for example thermal conductivity : applied vector (negative) temperature gradient and resulting vector heat flow density, exception thermal expansion relates temperature (0) and strain (2). " % T ij = $ T 11 T 12 T 13' $ T 21 T 22 T 23' # $ T 31 T 32 T 33& ' 9 components The generic 2 nd rank tensor T is the relation between an applied vector p and resultant vector q. We can write relation between p and q as a tensor equation p = T q or p i = T ij q j (i=1,2,3 ; j=1,2,3) In general the vectors p and q are not parallel.

18 The representation quadric for 2 nd Rank Tensors Geometrical representation of symmetrical second-rank tensors as a second-degree surface ( called a quadric). The quadric may be an ellipsoid or a hyperboloid. Most common second-rank tensors are symmetric (Tij = Tji) and when the 3 principal coefficients are all positive then the property is represented by an ellipsoid with axes 1/ T1, 1/ T2 and 1/ T3, which is the case for electric polarization, electrical and thermal conductivity and optical properties.

19 The representation hyperboloid for 2 nd Rank Tensors If one of the principal coefficients is negative then surface is a hyperboloid of one sheet (e.g. thermal expansion of plagioclase feldspar). If two of the principal coefficients are negative then surface is a hyperboloid of two sheets or caps, this the case for the thermal expansion of calcite with contraction the basal plane. If all three of the principal coefficients are negative then surface is an imaginary ellipsoid, this is the case for many susceptibilities of paramagnetic and diamagnetic minerals, such as quartz, calcite and aragonite.

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21 What is thermal diffusivity k ij? Fourier s equation - relates the rate of change of temperature with time at given position in an object, Note k is 2 nd order tensor, hence k can be anisotropic. dt/dt = k 2 T k= Thermal diffusivity = K/ρ C p units [m 2 /s] K=Thermal conductivity ρ=density C p = specific heat at constant pressure T= temperature t = time

22 Thermal conductivity of olivine single crystals

23 Thermal diffusivity of Olivine 2 nd order Tensor Diffusivité thermique (mm2.s-1) 3 2,5 2 1,5 1 0,5 [001] [100] [010] Température (K) Gibert et al, GRL, 2003

24 Radius-normal property and melting wax experiment Consider a red hot arrow touching the second order prism plane of a single crystal of quartz and creating a point heat source. A (negative) thermal gradient will result in heat flow away from the hot heat source (arrow tip) to the colder regions of the crystal. As the thermal resistivity (reciprocal of thermal conductivity) varies with direction the heat flow will not be equal in all radial directions, i.e. heat flow is anisotropic.

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26 The stimulus vector and response vectors for 2 nd rank tensors

27 The effect of symmetry on symmetric 2 nd rank tensors Cubic (Sphere) Tetragonal, Hexagonal, Trigonal (Uniaxial Ellipsoid) Orthorhombic Monoclinic Triclinic < Triaxial Ellipsoids >

28 Effect of symmetry on Physical Properties : Neumann s Principle F.E. Neumann s principle (1885) states that symmetry elements of any physical property of a crystal must include ALL the symmetry elements of the point group of the crystal. This implies that that a given physical property may possess a higher symmetry than that possessed by the crystal and it cannot be of a lower symmetry than that of the crystal. Some physical properties are inherently centrosymmetric (all symmetric second order tensors and elasticity) which will add a center of symmetry in many minerals (e.g. quartz) and result in a higher symmetry than the possessed by the crystal.

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30 The magnitude of a 2 nd rank property in a given direction Let a given direction (x) be specified by the direction cosines (x 1,x 2,x 3 ). Referred to general axes T(x) = T ij x i x j From the magnitude T(x) in a given direction one can calculate the radius vector r(x) of the representation quadric. When T(x) is negative, which occurs for example for thermal expansion in calcite, radius vector 1/ T(x), is an imaginary value. R(x) = 1/ T(x) Although the quadric representation is a hyperboloid when one or more principal axes are negative, the representation is quite straight forward as a contoured equal area projection (pole figure).

31 Example of values of a tensor

32 4 th Rank Tensors

33 4 th Rank Tensors - important for geophysics In any crystalline material there is balance between Coulomb attractive forces between oppositely charge ions and Born repulsive forces due to the overlap of electron shells. At any given thermodynamic state the crystal will tend toward an equilibrium structure. For a change in hydrostatic or non-hydrostatic stress, the crystal structure will adjust at the atomic level to the new thermodynamic state. The fundamental nature of atomic forces in the determination of elastic properties has been illustrated by the emergence of first principles atomic modeling to predict single crystal elastic tensors of geophysical importance at lower mantle PT conditions.

34 Hooke s Law σ = c ε and ε = s σ where c = stiffness coefficients (dimensions of stress) s = compliance coefficients (dimensions of 1/stress) σ = stress tensor (second order tensor) ε = deformation tensor (second order tensor) or " ij = c ijkl " ij = s ijkl # kl # kl i,j,k,l can have the values 1,2 or 3 so 3 x 3 x 3 x 3 = 3 4 = 81 coefficients. But due to the symmetry of the deformation and stress tensors the 81 coefficients are not independent. In Voigt notation we can write the Cijkl tensor as 6 by 6 symmetric tensor Cij with 21 independent values for a triclinic crystal.

35 The Voigt Notation Voigt Tensors p,q = 1 to 6 i,j,k,l = 1 to 3 Matrix (p,q) Tensor (ij or kl) or or or 21 #" 11 " 12 " 13 " ij = % %" 12 " 22 " 23 $ %" 13 " 23 " 33 #" 11 " 12 " 13 " ij = % %" 21 " 22 " 23 $ %" 31 " 32 " 33 & ( ( ' ( & ( ( ' ( ))> i # 1 " = % % 6 $ % # " " 6 " 5 " " 2 " 4 " 5 " 4 " 3 " 6 " 5 %" ))> " i = %" 6 " 4 2 " 2 2 %" 5 " 4 % $ " & ( ( ' ( & ( ( ( '(

36 The Voigt Notation In addition we need to introduce factors of 2 and 4 into the equations relating compliance in tensor and matrix notations Sijkl = Spq for p=1,2,3 and q = 1,2,3 2Sijkl = Spq for either p= or q = 4,5,6 4Sijkl = Spq for either p= and q = 4,5,6 However Cijkl = Cpq for all i,j,k,l and all p and q (for i,j,k,l = 1,2,3 and p,q = 1 to 6) Transformation law p = δij.i +(1-δij)(9-i-j) q = δkl.k +(1-δkl)(9-k-l) with the Kronecker delta, δij = 0 when i and δij = 1 when i=j

37 Cij Effect of Crystal Symmetry

38 Typical Cij data set From J.D.Bass (1995) Elasticity of Minerals, Glasses and Melts Mineral Physics and Crystallography A Handbook of Physical Constants AGU Reference Shelf 2 pp

39 Young s Modulus No single surface can represent the elastic behaviour of a single crystal. The surface of the Young s modulus is often used. Young s modulus for a given direction is defined as the ratio of the longitudinal stress to the longitudinal elastic strain.

40 The origin of seismic anisotropy Horizontal layering - sediments, metamorphic layering.. Upper and lower crust, transition zone, D Vertically aligned cracks in the crust filled with gas, liquid or solid CPO in lower crust, upper mantle & inner core

41 Christoffel Tensor T ik The calculation of the three seismic phase velocities from the Christoffel tensor T ik = C ijkl n j n l where n is the plane wave propagation direction and C ijkl the elastic constants of the crystal. The three eigenvalues E i of the symmetric Christoffel tensor (T ik ) are related to three seismic phase velocities V i (qp,qs1,qs2) by V i = (E i /ρ) 1/2, where ρ is the density and qs1 > qs2. The three eigenvectors of Christoffel tensor are the particle motion vectors of qp,qs1,qs2. Why qp and not Vp? For a triclinic elastic body the plane wave propagation direction (n) is not parallel to the particle motion vector for P, hence it is called quasi-p or qp. Similarly the n is not perpendicular to particle motion vector for S1 and S2, hence qs1,qs2.

42 Wavefronts

43 Group and phase velocity

44 Group velocity and slowness surfaces N.B. These surfaces are not generally an ellipse!

45 Polarisation in anisotropic medium 1. Polarisation = displacement of material 2. Vector qvp is closest to propagation direction 3. Vectors qvp,qvs1 and qvs2 are perpendicular 4. Vp>Vs1>Vs2 in anisotropic media

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49 Olivine

50 Olivine [100] [010]

51 Olivine Vp (km/s) dvs(%) Vs1 polarisation [100] [100] [010]

52 Calcite [c] [a] [m] [m] [a]

53 Calcite Vp (km/s) dvs(%) Vs1 polarisation [m] [a]

54 Alpha Quartz [c] [a] [m] [m] [a]

55 Alpha Quartz Vp (km/s) dvs(%) Vs1 polarisation [m] [a]

56 Muscovite Vp (km/s) dvs(%) Vs1 polarisation a* b

57 Mainprice et al 1993 J.Struc.Geol. Vol 15, pp

58 References Tensors and Crystal Physics Bhagavantam, S., Crystal Symmetry and Physical Properties, London, New York, Academic Press (1966) Cady, W. G. Piezoelectricity: An Introduction to the Theory and Applications of Electromechanical Phenomena in Crystals, New rev. ed., 2 vols. New York: Dover, Curie, P., Oeuvres, pp. 118, Paris, Société Français de Physique (1908) Hartmann,E. 1984, An Introduction to Crystal Physics, International Union of Crystallography Teaching Pamphlet 18 D.R.Lovett Tensor properties of Crystals, Institute of Physics Publishing, Bristol and Philadelphia, 2nd Edition 1999 D.McKie and C.McKie Crystalline Solids Nelson, 1974 (Chapter 11) Nye, J. F., Physical Properties of Crystals, Oxford, Clarendon Press (1957) A.Putnis, Introduction to Mineral Sciences, Cambridge University Press (Chapter 2) Wenk, H.-R. and Bulakh,A., Minerals Their Constitution and Origin, Cambridge University Press. (Chapter 8) Wooster, W. A., Tensors and Group Theory for the Physical Properties of Crystals, Oxford, Clarendon Press (1973).

59 Phase Transitions and Physical Properties

60 SiO 2 Phase Diagram

61 Alpha-Beta Quartz Transition

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63 Thermal Diffusivity of Quartz Single Crystal 7,0E-06 1,8E-06 6,0E-06 1,6E-06 diffusivité thermique [m2/s] 5,0E-06 4,0E-06 3,0E-06 2,0E-06 Direction c Diffusivité thermique [m2/s] 1,4E-06 1,2E-06 1,0E-06 8,0E-07 1,0E-06 Direction a 6,0E-07 0,0E Température [ C] 4,0E Température [ C] Benoit Gibert 2003

64 alpha Quartz to beta Quartz Transition alpha Quartz 32 Trigonal C 11 C 33 C 11 C 33 C ij (GPa) 50 C 44 C 66 C 66 C 13 C 12 0 C 13 C C 12 Temperature (K) beta Quartz 622 Hexagonal

65 Quartz alpha-beta transition in single crystal : Vp

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67 Alpha-beta quartz transition : Bulk and shear moduli alpha Quartz 32 Trigonal Bulk Modulus (GPa) Shear (Voigt) Shear (VRH) Shear (Reuss) Bulk Shear K voigt G voigt K reuss G reuss G vrh K vrh Temperature(K) beta Quartz 622 Hexagonal

68 High Pressure SiO 2 Polymorphs

69 SiO2 Polymorphs : Cij theory

70 Effect on single crystal elastic constants C ij using Landau theory 1200 Stishovite P4 2 /mnm C Tetragonal C 11 C 22 C ij (GPa) C 33 C 11 = C 22 C 66 C 12 C 44 = C 55 C 23 C 12 C 44 C C 13 = C 23 C Pressure (GPa) C 55 CaCl 2 -type Pnnm Orthorhombic Carpenter et al. 2000

71 Effect on isotropic elastic constants Stishovite P4 2 /mnm Tetragonal K, Bulk Modulus Modulus (GPa) Voigt VRH Reuss G, Shear Modulus 100 CaCl 2 -type Pnnm Orthorhombic Pressure (GPa)

72 But things are more Kendall (1998) wrote: LPO in lower-mantle minerals is an unlikely cause for the anisotropy. although we must always allow for the possibility that there is an as-yet unknown mineralogy that dominates D. complex.

73 The Post-Perovskite Story Motohiko Murakami, Kei Hirose et al (May 2004) announce new post-perovskite phase. Also found by Oganov & Ono (Nature, July 2004) Based on CaIrO 3 or UFeS 3 structure. Stable at P>120 GPa. D will not be made of perovskite at all!

74 History of Post- Perovskite (May to September 2004) 1. Murakami et al publish the first experimental evidence for Post-Perovskite Science 7 th May. 2. Tsuchiya et al. ab initio simulations of elastic constants at 0K. GRL 17 th July Iitakata et al ab initio simulations of structure, stability and elastic constants at 0K Nature 22 nd July. 4. Organov & Ono ab initio simulations of structure, stability and elastic constants at 0K & experimental data Nature 22 nd July. 5. Tsuchiya et al. ab initio phase transition perovskite-post perovskite and Clapeyron slope. EPSL August Stackhouse et al. ab initio simulations of elastic constants at 4000K EPSL January 2005.

75 Post Perovskite X-ray diffraction pattern

76 Crystal structure of post-perovskite layered structure large compressibility along b axis b c a Lattice system: Face-centered orthorhombic Space group: Cmcm Formula unit [Z]: 4 (4) Post-Perovskite Perovskite Lattice parameters[å] a: (4.286) [120 GPa] b: (4.575) c: (6.286) Volume [120 GPa] [Å 3 ]: (123.3)

77 Single Crystal Elastic Constants

78 Post-Perovskite Single Crystal Anisotropy Vp(km/s) AVs(%) Vs1 Polarisation

79 CAREWARE Presently available Unicef CAREWARE programs For Mac and Windows : (link : ftp://saphir.dstu.univ-montp2.fr/tphy/david/careware_unicef_programs/ Pole Figures Inverse Pole Figures Anisotropic Elastic and Seismic properties Second Order Symmetric Tensor Properties Field work (az/inc) pole figures All programs have a simple question and answer interface (no graphical interface with pull down menus) and are intended to produced publication or presentation ready graphics in Adobe Illustrator postscript.

80 Pole figure programs for data in Euler angle triplets PF2k - general purpose program for Euler angle triplets PFch5 - for Channel 5 ASCII text export files (*.ctf)

81 Seismic and elastic tensor properties for data in Euler angle triplets Anis2k - data in Euler angle triplets AnisCh5 - for Channel 5 ASCII text export file (*.ctf)