成功案例设置

A: The Physical Properties of
Anisotropic Materials

B: The Dynamics of Atoms in Crystals

C: Equations of State

2

GEOL0026: Session A; The Physical Properties of Anisotropic Materials

Books
There is a truly great, in all senses of the word, book on this subject; The Physical
Properties of Crystals Their Representation by Tensors and Matrices, by J F Nye, pub.
Oxford Science. No scientist working in this area should be without a copy.

1/. Introduction to Isotropic and Anisotropic Materials
Even quite advanced solid-state physics courses tend to assume that materials are
isotropic – whereas, in real life, they are not. What does this word mean in this context?

Isotropic materials
Suppose we apply an electric field, E, of so many Volts per metre, to our sample; if it is
made of an isotropic material (e.g. copper), we will make a current density, j, of so
many Amps per metre2 flow in a direction parallel to E. Provided that E is not too large,
the magnitude of j will be proportional to the magnitude of E and so we may write

j = σE

where σ is the electrical conductivity of the material.

Diagrammatically, we can picture this as shown
on the right

Anisotropic materials
Now consider the case where we apply our electric field to a crystalline substance such
as graphite. Graphite is anisotropic; the atoms are arranged in planar layers and there is
no reason for us to suppose that the electrical conductivity parallel to the layers will be
the same as that perpendicular to the layers; indeed, there is every reason for us to
suppose that it will not be the same. Dutta (Phys. Rev. B, 90, 187-192, 1953) measured
the conductivity of graphite and found that the conductivity in a direction perpendicular
to the hexagonal axis is about 104 times greater than that along the hexagonal axis.

The consequence of this is that if we apply our field, E, to a graphite crystal in a general
direction, j will tend to be directed towards the “direction of easy flow of current”, i.e.
towards the planar layers. Diagrammatically, we usually now have j no longer parallel
to E. Clearly the constant of proportionality, σ, between j and E will have to vary also.

Note: (a) this figure is only schematic not quantitative; (b) I have greatly diminished the
amount of anisotropy actually present in this material.
3

We find that for some directions (those perpendicular to and parallel to the layers) j and
E are parallel, but with a very different constant of proportionality (σ) between them. In
all other directions, j and E are no longer parallel, and the value of σ varies with
orientation, from a minimum value (in this case when E is perpendicular to the layers)
to a maximum value (in this case when E is parallel to the layers).

2/. Scalars, Vectors and Tensors
How can we describe behaviour such as that shown above by graphite quantitatively?
We shall continue to use our example of electrical conductivity (σ) to illustrate how this
may be done.

For an isotropic substance, σ is a scalar quantity; it has magnitude but no directional
dependence.

E and j are both vectors (you may have noticed that I have been writing them in bold
type) – and so they have both magnitude and direction; note that when referring to just
the magnitude of the vector I shall use normal italic type, i.e. E, j.

Now suppose that we choose three orthogonal reference axes in our crystal (it doesn’t
matter how we orient these axes at this stage – we just suppose that they exist);
following the notation in Nye’s book, we will refer to these axes as Ox1, Ox2, Ox3
(note that I shall sometimes abbreviate this to x1, x2, x3 when there is no danger of
confusing the axes with the coordinates of a point in space).

We can then fully describe both E and j,
in the usual way, in terms of their
components resolved onto these
three axes. Thus

E = [E1, E2, E3] and j = [j1, j2, j3]

with E = (E1
2 + E2
2 + E3
2)1/2 etc.

As our axes are, as yet, in no way constrained to be oriented with respect to the
crystallographic axes of the material we are studying (we shall see later that it is usually
highly convenient if they are!), there is every reason to suppose that the components of
our electric field, E1 etc., will each produce a current density in a general orientation.
Thus, in general, we must write

j1 = σ11E1 + σ12E2 + σ13E3
j2 = σ21E1 + σ22E2 + σ23E3
j3 = σ31E1 + σ32E2 + σ33E3

i.e., each component of E contributes to each component of j. The nine coefficients, σij,
now needed to describe our electrical conductivity may be written in the form of a
matrix

留学ICU™️ 留学生辅助指导品牌

在线客服 7*24 全天为您提供咨询服务

咨询电话(全球): +86 17530857517

客服QQ：2405269519

微信咨询：zz-x2580

服务项目

服务流程

客户保障

关于我们

微信订阅号