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5. The Value of a Physical Property in a Given Direction

One frequently finds data in the literature which give the value of some physical property in a given direction. In this chapter the concept of the magnitude of a physical property in a given direction, and also the relationship between this value and the respective tensor components will be illustrated on the examples of the direction dependence of the electrical conductivity and Young's modulus respectively.

The specific conductivity in the direction of the electric field is defined as the ratio of the component parallel with the electric field of the current density ($j_{\parallel}$) and the magnitude of the electric field (E), i.e. $j_{\parallel}/E$. Let the components of the electric field be E_i = En_i, where n_i denotes the i-th component of the unit vector ($\={n}$) pointing into the direction of the electric field. The component of the j current density parallel with $\=E$ is in tensor notation

$$j_{\parallel} = \frac{j_iE_i}{E}.$$ (5.1)

Consequently the specific electric conductivity in the direction given by the unit vector $\={n}$ will be

$$\sigma_{\={n}}=\frac{j_{\parallel}}{E}=\frac{(j_iE_i)}{E^2}=\sigma_{ij}\frac{E_i E_j}{E^2}$$ (5.2)

for which one has

$$\sigma_{\={n}} = \sigma_{ij} \cdot n_i \cdot n_j.$$ (5.3)

Thus formula (5.3) yields the relationship between the value of the electrical conductivity in the direction of $\={n}$ and the components of the electrical conductivity tensors. Similar relationships give the value of the physical properties which can be represented by a second-rank tensor (e.g. thermal conductivity, dielectric permittivity, thermal expansion) in a given direction of a crystalline medium.

Equation (5.3) may be applied in two ways. One possibility is to calculate the tensor components from the measured conductivity values and the corresponding direction cosines. For this purpose one should measure the electrical conductivity in different directions, which are not connected by symmetry, as many times as the number of the independent components. Another possibility of applying eq. (5.3) is quite opposite to the first one. With the aid of the already known tensor components the conductivity value can be computed for any direction.

Equation (5.3) becomes considerably simplified for crystals of the tetragonal, trigonal and hexagonal systems which have only two independent tensor-components ($\sigma_{11} = \sigma_{22}$ and $\sigma_{33}$)

$$\sigma_n = \sigma_{11}n^2_1 + \sigma_{22}n^2_2 + \sigma_{33}n^2_3 = \sigma_{11}(1-n^2_3) + \sigma_{33}n^2_3.$$ (5.4)

If the angle between the $\={n}$ vector and the x₃ principal axis of the crystal is denoted by $\theta$ the following equation is obtained

$$\sigma_{\={n}} = \sigma_{11}\sin^2 \theta + \sigma_{33}\cos^2 \theta.$$ (5.5)

The component $\sigma_{33}$ is frequently denoted as $\sigma_{\parallel}$ and the component of $\sigma_{11}$ as $\sigma_{\perp}$ with reference to the conductivity values parallel with the main axis of the crystal (i.e. with the three-, four- or six-fold axis) and vertical to this axis resp. With these notations Eq. (5.5) may be rewritten to obtain

$$\sigma_{\={n}}=\sigma_{\perp}\sin^2 \theta + \sigma_{\parallel}\cos^2 \theta.$$ (5.6)

As another example we will study the direction dependence of Young's modulus. To begin with it should be stated that Young's modulus in the pulling direction is defined as the ratio of the longitudinal stress ($\sigma_{ii}$) and the longitudinal strain ($\varepsilon_{ii})$. If the $x^{\prime}_3$ axis of the coordinate system is placed in the direction of the $\={n}$ unit vector Young's modulus in this direction will apparently be

$$E_{\={n}{\parallel}x^{\prime}_3} = \frac{\sigma^{\prime}_{33}}{\varepsilon^{\prime}_{33}}.$$ (5.7)

According to eq. (3.19) (if no external field exists)

$$\varepsilon^{\prime}_{33} = s^{\prime}_{3333}\sigma^{\prime}_{33}$$ (5.8)

and one obtains

$$E_{\={n}{\parallel}x^{\prime}_3} = \frac{1}{s^{\prime}_{3333}}.$$ (5.9)

Consequently in order to find the direction dependence of Young's modulus it is necessary to know the change of the tensor-component $s^{\prime}_{3333}$ in the various directions. This dependence, however, is given by the (4.5) transformation equation of the $s^{\prime}_{3333}$ tensor component.

$$s^{\prime}_{3333} = a_{3i} \cdot a_{3j} \cdot a_{3k} \cdot a_{3l} \cdot s_{ijkl}$$ (5.10)

where a_3i, a_3j, a_3k, a_3l denote the direction cosines of the $x^{\prime}_3$ axis parallel with the $\={n}$ unit vector with respect to the crystalphysical co-ordinate system; consequently

$$s^{\prime}_{3333} = n_i \cdot n_j \cdot n_k \cdot n_l \cdot s_{ijkl}.$$ (5.11)

From this equation and eq. (5.9) one obtains Young's modulus of the crystals belonging to the cubic system

$$E_{\={n}} = \frac{1}{s_{1111} -2(s_{1111} -s_{1122} - 2s_{2323})(n^2_1n^2_2+n^2_2n^2_3+n^2_3n^2_1)}.$$ (5.12)

This means that even in the case of cubic crystals Young's modulus is direction dependent.

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