Basic Concepts in Crystallography - Emil Zolotoyabko

8/2/2019 Basic Concepts in Crystallography - Emil Zolotoyabko

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A Crystal

Crystallography is based on a specific symmetry which is called translational

symmetry. The exact meaning of translational symmetry will be clearer when reading

this book. On an intuitive level we can define a crystal as a 3D object densely built ofrepeating structural units located at fixed distances from their neighbors, the

distances being unchanged across the crystal volume. This picture is a result of

scientific developments over hundreds of years.

In fact, attempts to understand the internal structure of materials had been

undertaken already in ancient times. Most renowned is the speculative atomic

theory of ancient Greeks (primarily by Leucippus and Democritus). At the end of the

middle ages and, especially, during the Renaissance period, people began to

recognize the importance of measurements for establishing scientific concepts.

Perhaps Dutch jewelers were the first to understand that diamond stone is built

differently from a piece of amber. However, the first quantitative results are dated by1669, when Danish professor of anatomy Nicolas Stenon, the court physician of

Duke of Toscana, published a paper, in which he established the first law of

crystallography the law of constant angles. While crystals of the same material can

differ by their external shape and size, the angles between the corresponding crystal

faces are always identical. In his study, Stenon had measured the angles between

faces forming the habits (typical shapes) of large natural crystals of quartz (SiO2)

and hematite (Fe2O3).

The next important step was accomplished only in 1784, by French mineralogist

Rene Hauy, who established the second law of crystallography the law of rational

parameters. According to Hauy, if certain edges of a crystal are taken as the axes ofcoordinate system, than the ratios of segments cut by two dissimilar crystal faces on

any axis are always found to be rational fractions. In other words, the segments

mentioned can be expressed as integer numbers of some elementary lengths. In fact,

the exact meaning of this law was understood later in the context of so-called Miller

indexes (see Chapter 3), which were introduced only in 1839. Anyway, it was a great

discovery, since on that basis one canfirmly conclude that the crystal consists of some

repeating blocks, the integer number of which within crystalline specimens provides

Basic Concepts of Crystallography, Emil Zolotoyabko. 2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

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the law of rational parameters. The presence of elementary blocks assembled without

voids assure 3D periodicity on a scale dictated by the block size. Hauy assumed that

elementary blocks, which form crystals, are small polyhedrons. Therefore, long

before modern powerful methods of structural analysis, by measuring the crystal

habits only, thespatial periodicity of crystalline structures was deduced. Nowadays,by

using high-resolution transmission electron microscopy or scanning tunneling

microscopy, the periodic atomic networks within crystals or on crystal surface can

be directly visualized (see Figures 1.1 and 1.2, respectively).

In fact, it is amazing that such giant numbers of atoms or molecules ($1023) are

disposed at equal distances from each other, and it is worth discussing very brieflythe

reasons behind that. Strictly speaking we have no comprehensive quantitative

description of such cooperative phenomenon. We know only that the interatomic

potential consists of two terms of opposite signs reflecting short-distance repulsion

between ion cages and long-distance attraction between ions via electron glue.

Competition between these two contributions provides theminimum of thepotential

at some fixed distance, rd (see Figure 1.3), which could be considered as the

interatomic distance in a crystal. However, crystal formation is a cooperative action

of billions and billions of atoms and we are still far from being able to calculate these

collective effects with high enough accuracy.

Nevertheless, thermodynamics predicts that, at least at zero absolute temperature,

T 0, the structurally ordered state is most favorable. According to the third law of

thermodynamics, the entropy, S, of the system tends to zero, when T! 0. In turn,

Figure 1.1 High-resolution transmission electron microscopy image of atomic columns in the

InAs/GaSb superlattice.

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Figure 1.2 Scanning tunneling microscopy image (29.7 29.7 nm) of the periodically arrangedatomic rows on the cleavage surface of the InAs/GaSb superlattice.

Repulsion

Potential

rd

Distance, r

Attraction

Figure 1.3 Typical behavior of interatomic potential as a function of the interatomic distance, r.

Sharp ascent left-side from the equilibrium point, rd, is due to repulsive forces between ions, while

attractive forces result in the tile slowly approaching zero potential at large distances, r) rd.

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entropy, S, is related to the number of possible system configurations, N, by the

Boltzmann equation:

S k ln N 1:1

where k is the Boltzmann constant. It follows from Equation 1.1 that in order to have

S 0 atT 0, the value ofNshould be one, that is, the single (unique) configuration

of the system is realized. It is evident that this corresponds to the ordered state, since

any deviation from order towards disordered state could be realized in many

equivalent ways. In fact, if we have a number of spatially ordered atoms in our

system, tiny shift of each of them means certain disorder and thus the number of

disordered configurations grows rapidly with the number of atoms involved.

If temperature T> 0, thermodynamics allows different configurations of the

system to exist, including disordered ones. Practically, we know that at room

temperature some solid materials exist in the amorphous (i.e. structurally disor-

dered) form, which is possible, at least, from the entropy point of view. However, the

amorphous form of a material is the exception rather than the rule, since it is

assumed that even atfinite temperature there are crystalline forms with lower free

energy. If so, the amorphous state is always metastable, that is, it seemingly exists

because of the insignificant diffusion rates or due to some stabilizing factors, such as

for example, a high concentration of impurity atoms or other lattice defects,

inhomogeneous strain fields, and so on.

Nonetheless, most solid materials are crystals, and we begin by describing their

symmetry properties.

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Basic Concepts in Crystallography - Emil Zolotoyabko