Optical activity and enantiomorphism

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.

Thermal expansion

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

Transformation operators for symmetry elements

Since the physical property tensors depend on symmetry, mathematical methods for determining the influence of symmetry are needed. More specifically, transformation matrices are required for the symmetry elements that generate the crystal class. As explained earlier, some classes require only one symmetry element while others may require two or three. None require more than three. This chapter also includes a brief discussion of the seven Curie groups used for textured solids and liquids. As explained in Chapter 2, transformations from one coordinate system to another can be specified by a set of nine direction cosines, aij where i, j = 1, 2, 3. The first index, i, refers to the “new” or transformed axis, the second index j to the “old” or reference axis. There are four types of symmetry elements that require discussion: rotation axes, mirror planes, inversion centers, and inversion axes in which rotation is accompanied by inversion. All rotations are assumed to be in the counterclockwise direction. Other symmetry elements such as rotoreflection axes are not needed. The fourteen symmetry elements in Table 4.1 generate the 32 crystal classes. No proof is offered here but this statement can be verified geometrically using the stereographic projections in Chapter 3. There are two final points concerning these transformation matrices. First, keep in mind that they must obey the orthogonality conditions described in Chapter 2. This provides a useful way of avoiding mistakes. The second point concerns handedness. Symmetry elements involving reflection or inversion reverses the handedness of the coordinate system. This of course includes inversion axes such as 1 ̅̅, 2 ̅ = m, 3 ̅, 4 ̅, and 6 ̅. The handedness change can be verified by showing that the determinant of the transformation matrix is −1. For ordinary rotation axes such as 2, 3, 4, and 6, there is no change in handedness, and the determinant is +1. In Table 4.2 we list the minimum symmetry requirements for each of the 32 crystal classes. These are the transformation operations needed to develop the physical property matrices for single crystals. These techniques will be described in the next chapter.

Electro-optic phenomena

Optical beams can be controlled by manipulating the refractive indices and absorption coefficients with applied electric fields. In communication systems electro-optic effects are used in phase and amplitude modulation, in beam deflectors, and in tunable filters. Three such effects are illustrated in Fig. 28.1. Lead lanthanum zirconate titanate (PLZT) is a transparent electroceramic that can be prepared in several different ferroelectric forms with large electro-optic coefficients. When prepared in a normal ferroelectric form it can be used in two different ways. A light-tunable shutter is constructed by coating a multidomain ceramic of PLZT with a photoconducting layer and transparent electrodes. A bias voltage on the electrodes is transferred to the ceramic when the photoconductor is illuminated. The electric field alters the domain structure and the degree of light scattering, controlling the intensity of light. Fully poled ferroelectric ceramics exhibit the linear electro-optic effect Using planar electrodes the PLZT is poled perpendicular to the optical beam. Polarizer and analyzer are positioned in the ±45◦ positions, and light intensity is controlled by altering the birefringence with an electric field. The third experiment utilizes a pseudo-cubic PLZT composition with a large quadratic electro-optic effect. No poling is required in this case. With polarizer and analyzer again in the ±45◦ positions, the transmitted light intensity is proportional to E2 rather than E. Linear and quadratic electro-optic coefficients are defined in terms of the fieldinduced changes in the optical indicatrix: . . . Bij(E) − Bij(0) = Δ

Galvanomagnetic and thermomagnetic phenomena

The Lorentz force that a magnetic field exerts on a moving charge carrier is perpendicular to the direction of motion and to the magnetic field. Since both electric and thermal currents are carried by mobile electrons and ions, a wide range of galvanomagnetic and thermomagnetic effects result. The effects that occur in an isotropic polycrystalline metal are illustrated in Fig. 20.1. As to be expected, many more cross-coupled effects occur in less symmetric solids. The galvanomagnetic experiments involve electric field, electric current, and magnetic field as variables. The Hall Effect, transverse magnetoresistance, and longitudinal magnetoresistance all describe the effects of magnetic fields on electrical resistance. Analogous experiments on thermal conductivity are referred to as thermomagnetic effects. In this case the variables are heat flow, temperature gradient, and magnetic field. The Righi–Leduc Effect is the thermal Hall Effect in which magnetic fields deflect heat flow rather than electric current. The transverse thermal magnetoresistance (the Maggi–Righi–Leduc Effect) and the longitudinal thermal magnetoresistance are analogous to the two galvanomagnetic magnetoresistance effects. Additional interaction phenomena related to the thermoelectric and piezoresistance effects will be discussed in the next two chapters. In tensor form Ohm’s Law is . . .Ei = ρijJj , . . . where Ei is electrical field, Jj electric current density, and ρij the electrical resistivity in Ωm. In describing the effect of magnetic field on electrical resistance, we expand the resistivity in a power series in magnetic flux density B. B is used rather than the magnetic field H because the Lorentz force acting on the charge carriers depends on B not H.

Magnetic phenomena

In this chapter we deal with a number of magnetic properties and their directional dependence: pyromagnetism, magnetic susceptibility, magnetoelectricity, and piezomagnetism. In the course of dealing with these properties, two new ideas are introduced: magnetic symmetry and axial tensors. Moving electric charge generates magnetic fields and magnetization. Macroscopically, an electric current i flowing in a coil of n turns per meter produces a magnetic field H = ni amperes/meter [A/m]. On the atomic scale, magnetization arises from unpaired electron spins and unbalanced electronic orbital motion. The weber [Wb] is the basic unit of magnetic charge m. The force between two magnetic charges m1 and m2 is where r is the separation distance and μ0 (=4π×10−7 H/m) is the permeability of vacuum. In a magnetic field H, magnetic charge experiences a force F = mH [N]. North and south poles (magnetic charges) separated by a distance r create magnetic dipole moments mr [Wb m]. Magnetic dipole moments provide a convenient way of picturing the atomistic origins arising from moving electric charge. Magnetization (I) is the magnetic dipole moment per unit volume and is expressed in units of Wb m/m3 = Wb/m2. The magnetic flux density (B = I + μ0H) is also in Wb/m2 and is analogous to the electric displacement D. All materials respond to magnetic fields, producing a magnetization I = χH, and a magnetic flux density B = μH where χ is the magnetic susceptibility and μ is the magnetic permeability. Both χ and μ are in henries/m (H/m). The permeability μ = χ + μ0 and is analogous to electric permittivity. χ and μ are sometimes expressed as dimensionless quantities (x ̅ and μ ̅ and ) like the dielectric constant, where = x ̅/μ0 and = μ ̅/μ0. Other magnetic properties will be defined later in the chapter. A schematic view of the submicroscopic origins of magnetic phenomena is presented in Fig. 14.1. Most materials are diamagnetic with only a weak magnetic response induced by an applied magnetic field.

Elasticity

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 . . . .

Pyroelectricity

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).

Tensors and physical properties

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.

Chemical anisotropy

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.