Читать книгу Introduction to Solid State Physics for Materials Engineers - Emil Zolotoyabko - Страница 14

Crystal symmetry imposes tight restrictions on its physical properties. Term “properties” relates to those that can be probed by regular (macroscopic) optical, mechanical, electrical, and other measurements, averaging over the actual atomic-scale periodicity of physical characteristics. Note that complete spatial symmetry of the crystal is revealed in diffraction measurements using quantum beams (X-rays, neutrons, electrons) with wavelengths comparable with translational periodicity. Note that crystal characteristics, even averaged over many translation periods, show anisotropy which is dictated by the crystal point group. Within this averaged approach, the symmetry constraints are formulated by means of the so-called Neumann's principle: the point group of the crystal is a sub-group of the group describing any of its physical properties. In simple words, the symmetry of physical property of the crystal cannot be lower than the symmetry of the crystal: it may be only equivalent or higher.

In practical terms, it means that if physical property is measured along certain direction within the crystal and then the atomic network is transformed according any symmetry element of its point group and measurement repeats, we expect to obtain the measurable effect of the same magnitude and sign as before. Any deviation will contradict particular crystalline symmetry and, thus, the Neumann's principle. Using mathematical language, physical properties are, generally, described by tensors of different rank, for which the transformation rules under local symmetry operations are well-known. Tensor rank defines the number of independent tensor indices, i, k, l, m,…, each of them being run between 1 and 3, if the 3D space is considered. In most cases, physical property is the response to external field applied to the crystal. Note that external fields are also described by tensors, which are called field tensors to distinguish them from crystal (material) tensors.

Figure 1.12 Illustration of the Biot–Savart law (Eq. (1.7)).

Tensors of zero rank are scalars. It means that they do not change at all under coordinate transformations related to symmetry operations. As an example of scalar characteristics, we can mention the mass density of a crystal. Tensor of rank one is a vector. It has one index i = 1,2,3, which enumerates vector projections on three mutually perpendicular coordinate axes within Cartesian (Descartes) coordinate system. It is easy to point out field vectors, for example, an applied electric field, ℰ_i, or electric displacement field, D_i. As crystal vector, existing with no external fields, one can recall the vector of spontaneous polarization, , in ferroelectric crystals (see Chapter 12). Spontaneous polarization, as well as polarization, P_i, induced by external electric field, is defined as the sum of elementary dipole moments per unit volume. Note that polarization P is polar vector having three projections, P_i, as e.g. radius-vector r (with projections, x_i). There exist also axial vectors (or pseudo-vectors), i.e. vector products (cross products) of polar vectors, which are used to describe magnetic fields and magnetic moments. In fact, magnetic field, ΔH, produced by the element Δl of a conducting wire carrying electric current, I_c, is described by the Biot–Savart law:

(1.7)

where r is the radius-vector connecting the element Δl and the observation point (see Figure 1.12). In turn, magnetic dipole moment, μ_d, is defined as an integral over the volume containing the current density distribution J:

(1.8)

Axial vectors are considered when analyzing magnetic symmetry and magnetic symmetry groups (Chapter 11).

Tensor of rank 2 has two independent indices i, k = 1, 2, 3. As a rule, it linearly connects two vectors, e.g. the vectors of the electric displacement field, D_i, and external electric field, ℰ_k, i.e. , as tensor of dielectric permittivity, ε_ik, does (see Chapter 8). Another example is the density of electric current, J_i, and electric field, ℰ_k, connected by the electrical conductivity tensor ρ_ik, i.e. (see Chapter 4). In further analyses, we will omit the summation symbols and use the reduced record (according to the Einstein convention) for tensor relationships, e.g.

(1.9)

(1.10)

There are two important field tensors of second rank, which are in common use. These are the stress and strain tensors. Stress tensor, σ_ik, connects vector of external force, F_i, applied to a certain crystal area, ΔS, and unit vector, , normal to this area:

(1.11)

Based on the mechanical equilibrium of the stressed solid, it is possible to prove that stress tensor (Eq. (1.11)) is symmetric one, i.e. σ_ik = σ_ki. Regarding strain tensor, it connects the deformation vector, u_i, in the vicinity of a given point and the radius-vector of this point, x_i. Deformation vector determines the difference in the distances between closely located points near x_i in the deformed and non-deformed states of the crystal. To provide local information on the deformed state, strain tensor, e_ik, is defined in the differential form:

(1.12)

Evidently, the strain tensor, defined by Eq. (1.12), is symmetric one, i.e. e_ik = e_ki.

Furthermore, inter-atomic distances within a crystal are also changed upon heating (see Chapter 3). In that sense, a crystal heated up to some temperature, T₁, is in different “deformation” state as compared with its initial state at temperature, T₀. Thus produced relative change in lattice parameters is mathematically equivalent to strain (Eq. (1.12)). Tensor of second rank, which relates e_ik to the temperature increase, ΔT = T₁ − T₀ (tensor of rank zero, i.e. scalar), is called as tensor of linear expansion coefficients, α_ik:

(1.13)

Note that both crystal states, at T = T₀ and T = T_1, are thermodynamically equilibrium states at respective temperatures, and, therefore, no elastic energy is stored in such “deformed crystal,” whenever the temperature change is homogeneous across the crystal. The only energy difference between these two states is in free energy, which is temperature dependent.

Tensor of second rank may also connect a scalar and two vectors, as tensor of dielectric permittivity, ℰ_ik, does for energy density, W_e, of electromagnetic field within a crystal:

(1.14)

By using tensor representation for the electric displacement field (see Eq. (1.9)), we find that the energy density is quadratic with respect to the applied electric field, ℰ_i.

Tensor of third rank has three indices i, k, l = 1, 2, 3. It connects tensor of second rank and vector, e.g. stress, σ_ik, and induced electric polarization, P_i:

(1.15)

as for direct piezoelectric effect, or strain, e_ik, and applied electric field, ℰ_i:

(1.16)

for converse piezoelectric effect, both discussed in detail in Chapter 12. Another example is tensor, r_lik, of the linear electro-optic effect (the Pockels effect, also mentioned in Chapter 12). This tensor of third rank connects the change, Δn_ik, of refractive index, n, (which can be described in terms of the second rank tensor) under applied electric field, with the electric field vector, ℰ_l:

(1.17)

For the fourth rank tensor, there are several optional ways for its construction. It may connect two tensors of rank 2, e.g. stress, σ_ik, and strain, e_lm, as the stiffness tensor, C_iklm (tensor of elastic modules used in Chapter 3), does:

(1.18)

Similar tensor object, π_iklm, is used to describe the photo-elastic effect in crystals, which provides the change of refractive index under applied stress:

(1.19)

Another possibility is to connect tensor of second rank (e.g. strain tensor, e_ik) and two vectors (e.g. quadratic form of electric field, ℰ_l ℰ_m) as for electrostriction effect, g_iklm:

(1.20)

or changes in refractive index, as a function of quadratic form of electric filed, as for quadratic electro-optic effect, r_iklm (see Chapter 12):

(1.21)

Eqs. (1.20, 1.21) describe the second order (quadratic) effects in the induced strain and change of refractive index, respectively, as a result of electric field application to a crystal. Tensor of rank 4 may also interconnect scalar quantity with two tensors of the second rank, as the stiffness tensor does when one calculates the density of elastic energy, W_el, stored within a crystal:

(1.22)

Therefore, using tensor representation of applied stress via induced strain (Eq. (1.18)), we find the density of elastic energy to be quadratic with respect to the induced strain. Tensors of rank higher than 4 describe high-order effects in the interaction between external fields and materials. These effects are regularly weak and, hence, are not discussed here.

Tensors of different ranks are appropriately transformed under local symmetry operations. All these operations can be exemplified as certain rotations of coordinate system, in which tensors are defined. Transformed tensor forms are compared with the initial ones, and, on this basis, symmetry restrictions on physical properties are imposed, to be in accordance with Neumann's principle. Based on this comparison, the zero tensor components can be determined, as well as symmetry-mediated relationships between non-zero tensor components. More information on symmetry aspects in crystals can be found in the dedicated crystallography books.

Additional interesting and important physical phenomenon, also related to symmetry operations, is twinning in crystals. For example, it stands behind the crystallography of ferroelectric domains (see Chapter 12) and is one of the channels of plastic deformation in crystals being competitive with dislocation glide. We stress that in terms of crystallography, twinning always is the result of symmetry operations, but those not belonging to the point group of a specific crystal. More information about twinning in crystals is given in Appendix 1.B.