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Transformations

Robert E. Newnham

in Title Pages

Many physical properties depend on direction and the resulting anisotropy is best described with the use of tensors. Tensors are classified according to how they transform from one coordinate ... More

Many physical properties depend on direction and the resulting anisotropy is best described with the use of tensors. Tensors are classified according to how they transform from one coordinate system to another. Therefore, we begin by describing transformations. There are several reasons why we want to do this: (1) transformations help us define tensors, (2) these tensors can be used to describe physical properties, (3) the effects of symmetry on physical properties can be determined by howthe tensor transforms under a symmetry operation, (4) the magnitude of a property in any arbitrary direction can be evaluated by transforming the tensor, (5) using these numbers, we can draw a geometric representation of the property, and (6) the transformation procedure provides a way of averaging the properties over direction. This is useful when relating the properties of polycrystalline materials to those of the single crystal. Mathematically, there is nothing fancy about these transformations. We are simply converting one set of orthogonal axes (Z1, Z2, Z3) into another [math]. The two sets of axes are related to one another by nine direction cosines: a11, a12, a13, a21, a22, a23, a31, a32, and a33. Collectively all nine can be written as aij where i, j = 1, 2, 3. The axes and direction cosines are illustrated in Fig. 2.1. It is important not to confuse the subscripts of the direction cosines. As defined in the drawing, a12 is the cosine of the angle between [math] and Z2, whereas a21 is the cosine of the angle between [math] and Z2. The first subscript always refers to the “new” or transformed axis. The second subscript is the “old” or original axis. The original or starting axes is usually a right-handed set, but it need not be. The transformed “new” axes may be either right- or left-handed, depending on the nature of the transformation. This will become clearer when we look at some transformations representing various symmetry operations. In any case, both the old and new axes are orthogonal. Less

Tensors and physical properties

Robert E. Newnham

in Properties of Materials: Anisotropy, Symmetry, Structure

In this chapter we introduce the tensor description of physical properties along with Neumann’s Principle relating symmetry to physical properties. As pointed out in the introduction, many ... More

In this chapter we introduce the tensor description of physical properties along with Neumann’s Principle relating symmetry to physical properties. As pointed out in the introduction, many different types of anisotropic properties are described in this book, but all have one thing in common: a physical property is a relationship between two measured quantities. Four examples are illustrated in Fig. 5.1. Elasticity is one of the standard equilibrium properties treated in crystal physics courses. The elastic compliance coefficients relate mechanical strain, the dependent variable, to mechanical stress, the independent variable. For small stresses and strains, the relationship is linear, but higher order elastic constants are needed to describe the departures from Hooke’s Law. Thermal conductivity is typical of the many transport properties in which a gradient leads to flow. Here the dependent variable is heat flow and the independent variable is a temperature gradient. Again the relationship is linear for small temperature gradients. Hysteretic materials such as ferromagnetic iron exhibit more complex physical properties involving domain wall motion. In this case magnetization is the dependent variable responsive to an applied magnetic field. The resulting magnetic susceptibility depends on the past history of the material. If the sample is initially unmagnetized, the magnetization will often involve only reversible domain wall motion for small magnetic fields. In this case the susceptibility is anhysteretic, but for large fields the wall motion is only partly reversible leading to hysteresis. The fourth class of properties leads to permanent changes involving irreversible processes. Under very high electric fields, dielectric materials undergo an electric breakdown process with catastrophic current flow. Under small fields Ohm’s Law governs the relationship between current density and electric field with a well-defined resistivity, but high fields lead to chemical, thermal, and mechanical changes that permanently alter the sample. Irreversible processes are sometimes anisotropic but they will not be discussed in this book. Measured quantities such as stress and strain can be represented by tensors, and so can physical properties like elastic compliance that relate these measurements. This is why tensors are so useful in describing anisotropy. Less

Thermodynamic relationships

Robert E. Newnham

in Properties of Materials: Anisotropy, Symmetry, Structure

In the next few chapters we shall discuss tensors of rank zero to four which relate the intensive variables in the outer triangle of the Heckmann Diagram to the extensive variables in the inner ... More

In the next few chapters we shall discuss tensors of rank zero to four which relate the intensive variables in the outer triangle of the Heckmann Diagram to the extensive variables in the inner triangle. Effects such as pyroelectricity, permittivity, pyroelectricity, and elasticity are the standard topics in crystal physics that allow us to discuss tensors of rank one through four. First, however, it is useful to introduce the thermodynamic relationships between physical properties and consider the importance of measurement conditions. Before discussing all the cross-coupled relationships, we first define the coupling within the three individual systems. In a thermal system, the basic relationship is between change in entropy δS [J/m3] and change in temperature δT [K]: . . . δS = CδT, . . . where C is the specific heat per unit volume [J/m3 K] and T is the absolute temperature. S, T, and C are all scalar quantities. In a dielectric system the electric displacement Di [C/m2] changes under the influence of the electric field Ei [V/m]. Both are vectors and therefore the electric permittivity, εij, requires two-directional subscripts. Occasionally the dielectric stiffness, βij, is required as well. . . . Di = εijEj Ei = βijDj. . . . Some authors use polarization P rather than electric displacement D. The three variables are interrelated through the constitutive relation . . . Di = Pi + ε0Ei = εijEj. . . . The third linear system in the Heckmann Diagram is mechanical, relating strain xij to stress Xkl [N/m2] through the fourth rank elastic compliance coefficients sijkl [m2/N]. . . . xij = sijklXkl. . . . Alternatively, Hooke’s Law can be expressed in terms of the elastic stiffness coefficients cijkl [N/m2]. . . Xij = cijklxkl. . . . When cross coupling occurs between thermal, electrical, and mechanical variables, the Gibbs free energy G(T, X, E) is used to derive relationships between the property coefficients. Temperature T, stress X, and electric field E are the independent variables in most experiments. Less

Specific heat and entropy

Robert E. Newnham

in Properties of Materials: Anisotropy, Symmetry, Structure

Before beginning the discussion of directional properties, we pause to consider specific heat, an important scalar property of solids which helps illustrate the important thermodynamic ... More

Before beginning the discussion of directional properties, we pause to consider specific heat, an important scalar property of solids which helps illustrate the important thermodynamic relationships between measured properties. Heat capacity, compressibility, and volume expansivity are interrelated through the laws of thermodynamics. Based on these ideas, similar relationships are established for other electrical, thermal, mechanical, and magnetic properties. Several atomistic concepts are introduced to help understand the structure–property relationships involved in specific heat measurements. The heat capacity or specific heat is the amount of heat required to raise the temperature of a solid by 1K. It is usually measured in units of J/kg K. Theorists prefer to work in J/mole K, and older scientists sometimes use calories rather than joules. One calorie is 4.186 J. For solids and liquids, the specific heat is normally measured at a constant pressure: where ΔQ is the heat added to increase the temperature by ΔT. Measurements on gases are usually carried out at constant volume: Electrical methods are generally employed in measuring specific heat. A heating coil is wrapped around the sample and the resulting change in temperature is measured with a thermocouple. If a current I flows through a wire of resistance R, the heat generated by the wire in a time Δt is given by . . . ΔQ = I2R Less

Pyroelectricity

Robert E. Newnham

in Properties of Materials: Anisotropy, Symmetry, Structure

As the name implies, pyroelectricity is a first rank tensor property relating a change polarization P to a change in temperature δT. The defining relation can also be written in terms of the ... More

As the name implies, pyroelectricity is a first rank tensor property relating a change polarization P to a change in temperature δT. The defining relation can also be written in terms of the electric displacement D since no field is applied: . . . Pi = Di = piδT [C/m2]. . . Pyroelectricity is a first rank polar tensor because of the way it transforms. Being polar vectors, Pi and Di transform as . . . D'i = aijDj . . . whereas the temperature change transforms as a zero rank tensor, or a scalar: . . . δT' = δT. . . . Transforming the defining relation for pyroelectricity we get . . . D'i = aijDj = aijpjδT = aijpjδT' = p'iδT'. . . . Both the independent variable δT and the dependent variable Di have now been transformed to the new coordinate system. The property relating D'i to δT' is the transformed pyroelectric coefficient p'i = aijpj. Thus the pyroelectric coefficient is a polar first rank tensor property. In Sections 6.1 and 7.3 it was shown that the electrocaloric effect and the pyroelectric effect are governed by the same set of coefficients pi. The change in entropy per unit volume caused by an electric field is . . . δS = piEi [J/m3]. The pyroelectric (=electrocaloric coefficient) coefficient is usually expressed in units of μC/m2 K and can be either positive or negative in sign depending on whether the spontaneous (built-in) polarization is increasing or decreasing with temperature. Pyroelectricity disappears in all centrosymmetric materials. The proof follows. For a first rank tensor there are, in general, three nonzero coefficients p1, p2, and p3 representing the values of the pyroelectric coefficient along property axes Z1, Z2, and Z3, respectively. The principal axes are perpendicular to each other and are chosen in accordance with the IEEE convention (Section 4.3). Less

Thermal expansion

Robert E. Newnham

in Properties of Materials: Anisotropy, Symmetry, Structure

When a material is heated uniformly it undergoes a strain described by the Relationship … xij = αijΔT, … where αij are the thermal expansion coefficients and

Piezoelectricity

Robert E. Newnham

in Properties of Materials: Anisotropy, Symmetry, Structure

The prefix “piezo” (pronounced pie-ease-o) comes from the Greek word for pressure or mechanical force. Piezoelectricity refers to the linear coupling between mechanical stress and electric ... More

The prefix “piezo” (pronounced pie-ease-o) comes from the Greek word for pressure or mechanical force. Piezoelectricity refers to the linear coupling between mechanical stress and electric polarization (the direct piezoelectric effect) or between mechanical strain and applied electric field (the converse piezoelectric effect). The equivalence between the direct and converse effects was established earlier using thermodynamic arguments (Section 6.2). The principal piezoelectric coefficient, d, relates polarization, P, to stress, X, in the direct effect (P = dX) and strain, x, to electric field E (x = dE). Thus the units of d are [C/N] or [m/V] which are equivalent to one another. Typical sizes for useful piezoelectric materials range from about 1 pC/N for quartz crystals to about 1000 pC/N for PZT (lead zirconate titanate) ceramics. To understand how the piezoelectric effect varies with direction and how it is affected by symmetry, it is necessary to determine how piezoelectric coefficients transform between coordinate systems. Since polarization is a vector and stress a second rank tensor, the physical property relating these two variables must involve three directions: … Pj = djklXkl … . In the new coordinate system … P'i = aijPj = aijdjklXkl … . Transforming the stress to the new coordinate system gives … P'i= aijdjklamkanlX'mn = d'imnX'mn…. Thus piezoelectricity transforms as a polar third rank tensor… . d'imn = aijamkanldjkl … . In general there are 33 = 27 tensor components, but because the stress tensor is symmetric (Xij = Xji), only 18 of the components are independent. Therefore the piezoelectric effect can be described by a 6 × 3 matrix. Less

Elasticity

Robert E. Newnham

in Properties of Materials: Anisotropy, Symmetry, Structure

All solids change shape under mechanical force. Under small stresses, the strain x is related to stress X by Hooke’s Law (x) = (s)(X), or the converse relationship (X) = (c)(x). The elastic ... More

All solids change shape under mechanical force. Under small stresses, the strain x is related to stress X by Hooke’s Law (x) = (s)(X), or the converse relationship (X) = (c)(x). The elastic compliance coefficients (s) are generally reported in units of m2/N, and the stiffness coefficients (c) in N/m2. For a fairly stiff material like a metal or a ceramic, c is about 1011 N/m2 = 1012 dynes/cm2 = 100 GPa = 0.145 × 108 PSI. Hooke’s Law is a linear relation between stress and strain, and does not describe the elastic behavior at high stress levels that requires higher order elastic constants (Chapter 14). Irreversible phenomena such as plasticity and fracture occur at still higher stress levels. Two directions are needed to specify stress (the direction of the force and the normal to the face on which the force acts), and two directions are needed to specify strain (the direction of the displacement and the orientation of the measurement axis). Thus there are four directions involved in measuring elastic stiffness, which is therefore a fourth rank tensor: … Xij = cijklxkl …. Less

Diffusion and ionic conductivity

Robert E. Newnham

in Properties of Materials: Anisotropy, Symmetry, Structure

The phenomenon of atomic and ionic migration in crystals is called solidstate diffusion, and its study has shed light on many problems of technological and scientific importance. Diffusion is ... More

The phenomenon of atomic and ionic migration in crystals is called solidstate diffusion, and its study has shed light on many problems of technological and scientific importance. Diffusion is intimately connected to the strength of metals at high temperature, to metallurgical processes used to control alloy properties, and to many of the effects of radiation on nuclear reactor materials. Diffusion studies are important in understanding the ionic conductivity of the materials used in fuel cells, the fabrication of semiconductor integrated circuits, the corrosion of metals, and the sintering of ceramics. When two miscible materials are in contact across an interface, the quantity of diffusing material which passes through the interface is proportional to the concentration gradient. The atomic flux J is given by where J is measured per unit time and per unit area, c is the concentration of the diffusing material per unit volume, and Z is the gradient direction. The proportionality factor D, the diffusion coefficient, is measured in units of m2/s. This equation is sometimes referred to as Fick’s First Law. It describes atomic transport in a form that is analogous to electrical resistivity (Ohm’s Law) or thermal conductivity. There are several objections to Fick’s Law, as discussed in Section 19.5. Strictly speaking, it is valid only for self-diffusion coefficients measured in small concentration gradients. Since J and Z are both vectors, the diffusion coefficient D is a second rank tensor. As with other symmetric second rank tensors, between one and six measurements are required to specify Dij, depending on symmetry. The relationship between structure and anisotropy is more apparent in PbI2. Lead iodide is isostructural with CdI2 in trigonal point group.m. The self-diffusion of Pb is much easier parallel to the layers where the Pb atoms are in close proximity to one another. Diffusion is more difficult along Z3 = [001] because Pb atoms have a very long jump distance in this direction. The mineral olivine, (Mg, Fe)2SiO4, is an important constituent of the deeper parts of the earth’s crust. Less

Optical activity and enantiomorphism

Robert E. Newnham

in Properties of Materials: Anisotropy, Symmetry, Structure

When plane-polarized light enters a crystal it divides into right- and lefthanded circularly polarized waves. If the crystal possesses handedness, the two waves travel with different speeds, and ... More

When plane-polarized light enters a crystal it divides into right- and lefthanded circularly polarized waves. If the crystal possesses handedness, the two waves travel with different speeds, and are soon out of phase. On leaving the crystal, the circularly polarized waves recombine to form a plane polarized wave, but with the plane of polarization rotated through an angle αt. The crystal thickness t is in mm, and α is the optical activity coefficient expressed in degrees/mm. The polarization vector of the combined wave can be visualized as a helix, turning α ◦/mm path length in the optically-active medium. Because of the low symmetry of a helix, optical activity is not observed in many high symmetry crystals. Point groups possessing a center of symmetry are inactive. In relating α to crystal chemistry it is convenient to divide optically-active materials into two categories: Those which retain optical activity in liquid form, and those which do not. It has long been known that optically-active solutions crystallize to give optically-active solids. This follows from the fact that molecules lacking mirror or inversion symmetry can never crystallize in a pattern containing such symmetry elements. Thus one way of obtaining optically-active materials is to begin with optically-active molecules, as in Rochelle salt, tartaric acid and cane sugar. Few of these crystals are very stable, however, and the optical activity coefficients are usually small, typically 2◦/mm. The same is true of many inorganic solids, though they are seldom optically active in the liquid state. For NaClO3 and MgSO4·7H2O, α is about 3◦/mm. Quartz and selenium, however, have coefficients an order of magnitude larger, showing the importance of helical structures to optical activity. Both compounds crystallize as right- and left-handed forms in space groups P312 and P322, with helices spiraling around the trigonal screw axes. Quartz contains nearly regular SiO4 tetrahedra with Si–O distances of 1.61 Å. Levorotatory quartz belongs to space group P312 and contains right-handed helices; enantiomorphic dextrorotatory quartz crystallizes in P322. Trigonal selenium also contains helical chains. Less

Chemical anisotropy

Robert E. Newnham

in Properties of Materials: Anisotropy, Symmetry, Structure

Chemical anisotropy concerns the ways in which crystals grow or dissolve in different directions. It is an appropriate subject to end this book because it brings together the oldest and the newest ... More

Chemical anisotropy concerns the ways in which crystals grow or dissolve in different directions. It is an appropriate subject to end this book because it brings together the oldest and the newest parts of crystal physics. Long, long ago mineralogists described the shapes of natural crystals and noted correlations with cleavage, hardness, and other physical properties. Chemical etching was another favorite topic in classical crystal physics that has undergone a recent revival because of the interest in the micromachining of semiconductor devices. Chemical anisotropy involves the interaction of a crystal with a chemically active environment that promotes dissolution or growth. For this reason it is primarily a surface property, rather than a bulk property of the crystal. This is one of the reasons why chemical anisotropy is not normally included in crystal physics books. The other reason is that rates of growth and dissolution depend on the chemical nature of the environment much more than the bulk properties of crystals do. Nevertheless, this is an important subject in contemporary crystal physics. Surfaces become more and more important as the scale of engineered devices grows smaller. The crystal physics of surface properties is a natural extension of classical crystal physics. It is a topic still in its infancy. Under favorable conditions, crystal growth takes place in such a way that the external surface is bounded by a set of plane faces. The preferred shape of rocksalt family crystals is cube bounded by six symmetry-related {100} faces. For diamond, an octahedral shape with eight {111} faces often appears. Quartz, calcite, and rutile belong to lower symmetry crystal systems with more anisotropic morphologies. Quartz crystals are often elongated along the c-axis with a hexagonal cross-section bounded by six {100} faces while the ends are terminated by six {101} and six {011} faces. Calcite tends to form rhombohedra with six faces shaped like parallelograms. Rutile (TiO2) crystals are often elongated along the c-axis forming slender needles. Less