2 Crystalline Solids-1

Amarjeet Singh

Learning Outcomes

After studying this module, you shall be able to

• Differentiate between amorphous solids and crystalline solids from the point of view of atomic arrangements and orders at different length scales.
• Understand the mathematical description of crystals in terms of unit cells. You will be able to differentiate between primitive cell and unit cells.
• Perform different symmetry operations on crystalline unit cells in 2 and three dimension.

    1.   INTRODUCTION:

Solid state is an aggregation form of matter characterized by strong interaction forces between constituent particles (atoms, ions, or molecules). As a result, a solid state material has an independent geometric form in contrast to liquids, which take the form of the container and an invariant volume in contrast to gases and vapors) in given temperature and pressure conditions. As temperature increases, a solid state material can evolve into another aggregation form (liquid or gas). Solid state physics studies the structural, mechanical, thermodynamic, electrical, magnetic, and optical properties of (poly-) crystalline and non-crystalline solids (for example, amorphous materials, such as glass). The properties of crystalline solids are determined by the symmetry of the crystalline lattice, because both electronic and phononic systems, which determine, respectively, the electric, magnetic and thermal response of solids, are very sensitive to the regular atomic order of materials and to any (local or non- local) perturbation of it.

Figure 1.1: Some of the examples of crystals

The crystalline structure can be revealed by the macroscopic form of natural or artificially- grown crystals as seen in the pictures above, or can be inferred from cleaving a material. Non-crystalline materials have no long-range order, but at least their optical properties are similar to that of crystalline materials because the wavelength of the incident photons (of the order of 1 µm) is much larger than the lattice constant of crystals and so, photons sense an effective homogeneous medium. Other properties of non-crystalline materials are derived from concepts related to crystalline solids and, therefore, the crystal structure is extremely important in understanding the properties of solid state materials. The macroscopic, perfect crystal is formed by adding identical building blocks consisting of atoms or groups of atoms. These building block is called unit cell which is the smallest component of the crystal that, when stacked together with pure translational repetition, reproduces the whole crystal.

2.   BASCIC CONCEPTS:

Unit Cells, Direct Lattice and Symmetry Operations:

A group of points arranged in a periodic pattern constitute of lattice (so called Bravais lattice) which is purely a mathematical concept. When we associate an atom or a group of atoms to a lattice point, the atoms will also be arranged a regular periodic pattern. Thus a crystal is lattice plus basis. The concept is nicely illustrated in figure 1.2. It is like regular and periodic arrangement of flower plants in a garden. The basis can be single atom (entire flower plant) or group of atoms (a group of different petals of a flower).

Figure 1.2: Geometrical illustration of Crystalline structure in 2-dimension (2D)

Although usually the basis consists of only few atoms, it can also contain complex organic or inorganic molecules (for example, proteins) of hundreds and even thousands of atoms. In two dimensions, all Bravais lattice points

R_mn = m a₁ + n a₂                                                                              (1)

can be obtained as superpositions of integral multiples of two non-collinear vectors a1 and a2 (m and n are arbitrary integers). A basis consisting of s atoms is then defined by the set of vectors rj = m ja1 + nja2 ,  j = 1,2,…,s, that describe the position of the centers of the basis atoms with respect to one point of the Bravais lattice. In general mj = 0, nj = 1. Every point of a Bravais lattice is equivalent to every other point, i.e. the arrangement of atoms in the crystal is the same when viewed from different lattice points. The Bravais lattice defined by (1) is invariant under the operation of discrete translation Tpq = pa1 + qa2 along integer multiples p and q of vectors a1 and a2, respectively, because

T_pq ( R_mn ) = T_pq + R_mn = R_{p + m , q + n}                              (2)

is again a Bravais lattice point. The translation operation has the following propert ies:

Additive: T_pq T_uv = T_{p+ u , q + v}

Associative: T_pq ( T_uv T_mn ) = ( T_pq T_uv )T_mn

Commutative: T_pq T_uv = T_uv T_pq

Inverse Law: T_pq ( – 1) = T _{– p , – q} and T_pq T_{– p , – q} = I

where I the identity transformation, it follows that the translations form an abelian (commutative) group. Because condition (2) is satisfied for all Bravais lattice points, a1 and a2 are called primitive translation vectors, and the unit cell determined by them is called primitive unit cell. The modulus of these vectors, a1 = | a1 | and a2 = | a2 |, are the lattice constants along the respective axes, and the area of the unit cell in two dimensions is S = | a1 × a2 |. It is important to notice that the set of vectors a1 and a2 is not unique (see the figures below), but all primitive unit cells have the same area.

Figure 1.3: Representation of a primitive unit cell

The primitive unit cell covers the whole lattice once, without overlap and without leaving voids, if translated by all lattice vectors. An equivalent definition of the primitive unit cell is a cell with one lattice point per cell (each lattice point in the figures above belong to four adjacent primitive unit cells, so that each primitive unit cell contains 4 × (1/4) = 1 lattice point). Non-primitive unit cells are larger than the primitive unit cells, but are sometimes useful since they can exhibit more clearly the symmetry of the Bravais lattice. Besides discrete translations, the Bravais lattice is invariant also to the point group operations, which are applied around a point of the lattice that remains unchanged. These operations are:

• Translation by unit vectors or with Bravais lattice vectors leaves the lattice invariant. A system possesses translation invariance if the property or the structure remains unaltered if a point is translated by a vector.
• Rotations by an angle 2π/n about a specific axis, denoted by Cn, and its multiples, Cjn = ( Cn)j . Geometric considerations require that n = 1, 2, 3, 4 and 6, and that repeating the ro tation n times one obtains Cnn = E, where E is the identity operation, which acts as r → r. Moreover, C1 = 2π = E does not represent a symmetry element. Examples of two-dimensional figures with different rotation symmetries:

Figure 1.4: Geometrical structures with different rotational symmetry

• Inversion I, which is defined by the operation r → – r if applied around the origin.
• Reflection σj, which can be applied around the horizontal plane, the vertical plane, or the diagonal plane.
• Imprope r rotation Sn , which consists of the rotation Cn followed by reflection in the plane normal to the rotation axis. Note that S2 = I .

   When we combine the point group symmetry with the translational symmetry, we obtain the space-group symmetry. The figure bellow represents several symmetry operations.

Figure 1.5: Symmetry operations (a) translations, (b) rotation, (c) inversion, and reflection with respect to a (d) vertical, and (e) horizontal plane.

In crystallography, symmetry is used to characterize crystals, identify repeating parts of molecules, and simplify both data collection and nearly all calculations. Also, the symmetry of physical properties of a crystal such as thermal conductivity and optical activity must include the symmetry of the crystal. Thus, a thorough knowledge of symmetry is essential to a crystallographer. An object is described as symmetric with respect to a transformation if the object appears to be in a state that is identical to its initial state, after the transformation. In crystallography, most types of symmetry can be described in terms of an apparent movement of the object such as some type of rotation or translation. The apparent movement is called the symmetry operation. The locations where the symmetry operations occur such as a rotation axis, a mirror plane, an inversion center, or a translation vector are described as symmetryelements.

    Value Addition:

Do You Know?

It is the arrangement of atoms and molecules that strongly influences the physical properties of the materials. The examples are graphite and diamond. Both are a collection of carbon atoms but only atomic arrangements are different. Diamond shines when illuminated with light whereas graphite does not reflect. Graphite can conduct electricity whereas the diamond is insulator. Diamond is so hard and graphite can be peeled easily. We have seen school children preparing the tip of the pencil. Only the arrangement of atoms can be so crucial for its scientific applications. Such materials are also called allotropes.

Amorphous and crystalline materials are distinguished by short range order and long range order. When we talk about crystalline materials with long range order, it is difficult to have the long range order up to infinite length. The order of long range materials is broken at some length scale. It may be from few microns to some hundreds of centimeters. A piece of crystal where the order is maintained within the physical boundaries is called a single crystal. A crystalline material with several crystallites within geometrical boundary of materials is called polycrystalline material. Most of the crystalline material we encounter in our daily life is poly crystalline. It is usually hard and costly affair to grow single crystals which are prepared for some specific purposes.

1. Suggested Reading Lattice is a mathe matical concept

When we say lattice it can be constructed purely on mathematical ground. All theorems of geometry and its principle can be applied to construct various kinds of lattice. To make it a physical reality we add basis to the lattice. A basic can be a single atom or group of atoms. That makes real crystals found in nature.

For More Details ( on this topic and othe r topics discussed in Text Module) See

1. Neil W. Ashcroft and N. David Mermin, Solid State Physics, Thomson Brooks/Cole, Eastern Press Bangalore (India) 2005
2. Charles Kittel, Introduction to Solid State Physics, John Wiley & Sons, Singapore 1999
3. W. K. Chen, Electrophyisics, NCTU

    Glossary:

Amorphous:

Amorphous means a material where the atomic order is short range, that is, up to 2-3 atomic radii.

Crystalline :

A material where atoms or a group of atoms are arranged in a particular order at large length scale, that is, up to several hundreds of atomic radii.

Unit Cell:

It is a building block of a crystalline material. Just as a wall can be constructed by stacking bricks, one can construct a crystal of any large dimension by stacking unit cells. We have a choice to construct the unit cell in a system as large as we wish.

Primitive Cell:

It is the smallest unit cell possible to construct. No smaller unit cell then primitive cell would exist.

Symmetry Ope ration:

Some mathematical operations such as translation, rotation, reflection, inversion leaves the arrangement invariant, such operations are called symmetry operations.