ANISOTROPY Preamble

1.9 ANISOTROPY Preamble

For many physical properties of a solid, the anisotropic properties are very important. This anisotropy arises out of the “changes” of any property along ‘three different spatial directions’, which are discussed in terms of ‘tensors’. This anisotropy is described with respect to the magnetic properties in the section - 5.5, without invoking the concept of tensors, but it is discussed only in terms of an ‘anisotropic constant’ for nano-crystalline particles of magnetite, which shows that this concept is quite important. For other physical properties to be discussed, this concept is again not used within the avail- able scope of this book. But, nevertheless, this particular aspect should be explored very briefly so that the readers may use the concept or appreciate the problem of anisotropy, if encountered in any topic dealing with the ‘anisotropy’ of the nano-materials or materials containing nano-particles with interest- ing properties and applications [38].

1.9.1 Anisotropy in a Single Crystal

All the crystals have three directions in the three spatial coordinates in the Cartesian sense in the Eucleadian space. If the physical properties of solids do not vary along these directions, then the crystal is called isotropic. But in many cases, various physical properties do vary in different Cartesian coordi- nates, including the single crystals. In this case, the crystals are called anisotropic, which is the subject matter of tensoral analysis [39, 40].

As an example of anisotropy in a single crystal, let us consider the ‘electrical conductivity’ wherein the electric field (E) gives rise to a current (I), which is a vector that will have the same direc- tion as the electric field vector. From the linear relation between the cause and effect (i.e., the causality), the current components relative to the arbitrarily chosen Cartesian coordinate system are given as :

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where, the quantities σ ik are the components of the ‘conductivity tensor’, which is a symmetric tensor, i.e., σ ik = σ ki . We can make use of this symmetry property and we can multiply the above three equa-

tions by E x ,E y ,E z respectively, and upon adding, we simply get the relation as :

xx E x + σ yy E y 2 + σ 2 zz E z +2 σ xy E x E y +2 σ yz E y E z +2 σ zx E z E x It is seen that the right hand side represents a quadratic surface. If we choose the coordinates along the ‘principal’ axes of the surface, then the mixed terms vanish, and we get the new coordinate system as :

I x E x +I y E y +I z E z = σ

where, σ 1 , σ 2 and σ 3 are the ‘principal’ conductivities. Therefore, the electrical conductivities of any crystal (even possessing lower symmetry) may be characterized by the above three conductivities, or rather by three specific resistivities. It has to be clearly noted that both the current and the field have same direction only when the applied field falls along any one of the three ‘principal’ axes.

In cubic crystal, the three quantities are equal, the conductivity does not vary with the direction. In hexagonal, trigonal and tetragonal crystals, the conductivity depends only on the angle ( φ) between the direction in which σ is measured and the hexagonal, trigonal or tetragonal axis, since in these crys- tals, two out of the three quantities are equal. In such a situation, we can write the conductivity equation as :

σ(φ) = σ ⊥ /sin 2 φ+σ || /cos 2 φ

where, the signs with σ refer to the directions perpendicular and parallel to the axis. Now, starting with the above relations, let us describe different properties of solids in terms of the effect of two terms at a time, i.e., scalars, vectors and tensors.

1. Vector - Vector Effect. The above effect can be called “vector-vector” effect, since an elec- tric current (i.e., a vector quantity) is produced by an applied electric field, which is also a vector quantity. The relations deduced above can also be applied to the other physical proper- ties, like thermal conductivity. Here, a thermal current vector is created by a ‘thermal gradi- ent’, which is also a vector quantity. Another example is the ‘diffusion’ under the influence of

a ‘concentration gradient’, and here both the variables are vector quantities.

2. Scalar-Tensor Effect. Similar relation as above can be deduced for this effect as well. As an example, the case of a “deformation” (i.e., a tensor quantity) of a solid can be cited, which results from a change in temperature, which is a scalar quantity. This may be characterized by

the three ‘principal’ expansion coefficients : α 1 , α 2 and α 3 . Here again, in cubic crystals, α 1 = α 2 = α 3 , and such crystals are obviously isotropic in this context. The angular dependence of α for hexagonal, trigonal and tetragonal crystals is also given by the last equation, as shown above, for the electrical conductivity.

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3. Vector-Tensor Effect.

A typical example is piezoelectricity, i.e., the creation of an electrical polarization, which is equal to the atomic displacement or deformation (i.e., a tensor quantity) with the application of an electric field vector or vice-verse. This is an important property with so many important applications in human life. It is better to visualize a piezoelectric material (e.g., a gas lighter) as a vector (cause) and a tensor (effect).

4. Tensor - Tensor Effect. In the mechanics of solids, the most important property studied so far is the ‘elastic deformation or strain’, which is a tensor quantity, but this effect can usually take place under the influence of some form of ‘mechanical stress’ that has to be applied, which is again a tensor quantity. This tensor-tensor effect may require many more compo- nents (stress or strain components) than appeared in a rather simpler case of a vector-vector effects, as described above.

Well, this particular short discussion sums up the applications of “tensor” for some of the impor- tant physical properties of solids along with vector and scalar quantities to give a more or less compre- hensive outlook on this type of mathematical technique, although there are numerous other examples on the applications of “tensor” in materials science and solid state physics with a very interesting math- ematical treatment. All these aspects could be important for nano-materials.