3 Crystalline Solids-2

Amarjeet Singh

    Learning Outcomes

 

After studying this module, you shall be able to

  • Perform different symmetry operations on crystalline unit cells in 2 and three dimension.
  • Learn the classification of crystal type in two and three dimension with appropriate mathematical description using Bravais lattice.
  • Calculate packing fraction of crystalline unit cells with understanding how certain geometrical arrangements leads to closed packing fraction.

    1. BRAVAIS LATTICE

 

1.1 2D:

 

Crystal lattices are classified according to their symmetry properties at point group operations. The five Bravais lattice types in two dimensions are shown in the figure below. These are:

  • Square lattice , for which | a1 | = | a2 |, and γ = 90°, where γ is the angle between a1 and a2,
  • Rectangular lattice , for which | a1 | ≠ | a2 |, and γ = 90°,
  • Centered rectangular lattice , which is a rectangular lattice with an additional lattice point in the center of the rectangle,
  • Hexagonal lattice , for which | a1 | = | a2 | , and γ = 60° (or 120° for a different choice of the origin),
  • Oblique rectangular lattice (called also oblique lattice), for which | a1 | ≠ | a2 |, and γ ≠ 90°, 60° (or 120°).

 

Figure 2.1: Bravais lattices in 2-dimension

 

With the exception of the centered rectangular lattice, all unit cells in the figure above are primitive unit cells. The primitive cell for the centered rectangular lattice is a rhombus (see figure below) and therefore this Bravais lattice is also called rhombic lattice, case in which its primitive unit cell has | a1 | = | a2 |, and γ ≠ 90°, 60° (or 120°). Each lattice type has a different set of symmetry operations. For all Bravais lattice types in two dimensions, the rotation axes and/or reflection planes occur at lattice points.

 

Figure 2.2: Rhombic Lattice and Wigner Seitz cell

 

There are also other locations in the unit cell with comparable or lower degrees of symmetry with respect to rotation and reflection. These locations are indicated in the figure above. In order to incorporate the information about the point group symmetry in the primitive cell, the Wigner-Seitz cell is usually employed. This particular primitive unit cell is constructed by first drawing lines to connect a given lattice point to all nearby lattice points, and then drawing new lines (or planes, in three-dimensional lattices) at the middle point and normal to the first lines. The Wigner Seitz cell is the smallest area (volume) enclosed by the latter lines (planes). An example of the construction of a Wigner-Seitz cell for a two- dimensional oblique lattice is illustrated in the figure above. For a two-dimensional square lattice the Wigner-Seitz cell is also a square. The Wigner-Seitz cell is always centered on a lattice point and incorporates the volume of space which is closest to that lattice point rather than to any other point. In other words, the faces of the Wigner- Seitz cell are determined by the intersection between equal-radius spheres centered at the nearest-neighbor points of the Bravais lattice.

 

1.2 BRAVAIS LATTICE IN 3D:

 

In a similar manner, in three dimensions , all Bravais lattice points

 

Rmnp = ma1 + na2 + pa3                                                                         (3)

 

can be obtained as superpositions of integral multiples of three non-coplanar primitive translation vectors a1 , a2 and a3 (m, n, and p are arbitrary integers), and the point group operations are defined identically. The volume of the primitive unit cell, which in this case is a parallelepiped, is O = | (a1 × a2). a3 |. There are 14 three-dimensional Bravais lattices, which belong to 7 crystal· systems, as can be seen from the figure below, where the primitive translation vectors are denoted by a, b, c (with respective lengths a, b, and c), and a , ß , γ are the angles between b and c, c and a, and a and b, respectively. These crystal systems, which are different point groups endowed with a spherical symmetric basis, are:

  • Cubic, for which a = b = c, a = ß = γ = 90°. It consists of three non-equivalent space- group lattices: simple cubic, body-centered cubic and face-centered cubic. This is the crystal system with the highest symmetry and is characterized by the presence of four C3 axes (the diagonals of the cube)
  • Te tragonal, for which a = b ≠ c, a = ß = γ = 90°. It encompasses the simple and body- centered Bravais lattices and contains one C4 symmetry axis.
  • Orthorhombic, for which a ≠ b ≠ c, a = ß = γ = 90°. It incorporates the simple, body- centered, face centered, and side-centered lattices and has more than one C 2 symmetry axis or more than one reflection plane (actually, three such axes/planes, perpendicular to each other).
  • Monoclinic, for which a ≠ b ≠ c, a = γ = 90°≠ ß. It includes the simple and side- centered lattices, and has one C2 symmetry axis and/or one reflection plane perpendicular to this axis.
  • Rhombohedral (Trigonal), for which a = b = c, a = ß = γ ≠ 90°. It contains a single C3 axis.
  • Hexagonal, for which a = b ≠ c, a = ß = 90°, γ = 120°. It is characterized by the existence of a single C6 symmetry axis. The conventional hexagonal unit cell (see the figure at right) is composed of three primitive cells.
  • Triclinic, for which a ≠ b ≠ c, a ≠ ß ≠ γ ≠ 90°. This is the crystal system with the lowest symmetry. It is not symmetric with respect to any rotation axis or reflection plane

Figure 2.3: Bravias Lattices in 3-dimension

 

The relations between these lattices can be summarized in the figure at the right. The different crystal systems have different numbers of unit cell types because other possible unit cell types cannot represent new Bravais lattices.

Figure 2.4: definition of lattice vectors in BCC and FCC unit cell

 

For example, both the body-centered and the face-centered monoclinic lattices can be reduced to the side-centered lattice by appropriately choosing the primitive translation vectors. Examples of two sets of primitive translation vectors for a body centered cubic (bcc) lattice are represented in the figure below at left and center, while the figure at right displays a set of primitive translation vectors for a face-centered cubic (fcc) lattice.

 

The primitive translation vectors for the left figure above can be expressed as

 

a1 = ( a / 2 )( x + y – z ) , a2 = ( a / 2 )( – x + y + z ) , a3 = ( a / 2 )( x – y + z ) , (4)

 

while those for the right figure are

 

a1 = ( a / 2 )( x + y ) , a2 = ( a / 2 )( y + z ) , a3 = ( a / 2 )( z + x ) (5)

 

A simple lattice has lattice points only at the corners, a body-centered lattice has one additional point at the center of the cell, a face-centered lattice has six additional points, one on each side, and a side-centered lattice has two additional points, on two opposite sides. The simple lattices are also primitive lattices and have one lattice point per cell, since the eight sites at the corners are shared by eight adjacent unit cells, so that 8×(1/8) = 1.  The non-simple lattices are non-primitive. The volume of the primitive unit cell in these lattices is obtained by dividing the volume of the conventional unit cell by the number of lattice points. In particular, the body-centered (BCC) lattices have two points per unit cell: the eight at the corners which contribute with 8×(1/8) = 1, and the one in the center, which belongs entirely to the unit cell. The face-centered (FCC) lattices have 4 lattice points per cell: those in the corners contribute with 8×(1/8) = 1, and those on the faces contribute with 6×(1/2) = 3, since they are shared by two adjacent cells. Finally, the hexagonal closed pack (HCP) lattices have six lattice points per cell: the points at the corner contribute with 12×(1/6) = 2, and those on the faces with 8×(1/2) = 4.

 

Figure 2.5: Unit cells of BCC, FCC and HCP structures with a basis of single atom of radius R

 

PACKING FRACTION:

 

If each lattice point is expanded into a sphere with a radius equal to half of the distance between nearest neighbors, such that adjacent spheres touch each other, then a packing fraction can be defined as the fraction between the volume of the spheres contained in the conventional unit cell and the volume of the unit cell. Note that in the volume between the spheres one can always insert smaller spheres, which can stand for other atom types. The table below (table 1) gives the number of atoms per unit cell and packing fraction of the given structure. Larger the packing fraction, the higher is the packing density. There are two closed packed structures FCC and HCP which we shall describe is little more detail below.

 

TABLE 1:

S. No. Lattice Type Number of atoms per unit cell Packing fraction:
1 SC 1 0.52
2 BCC 2 0.68
3 FCC 4 0.74
4 HCP 6 0.74

    CLOSE PACKED STRUCTURES:

 

FCC and hexagonal crystal structures are most highly packed with packing efficiency of 74% (APF= 0.74). Such structures can be described in terms of close-packed atomic planes. In FCC, {111} planes are close-packed and the basal plane (0001) is the close-packed one in hexagonal close-packed (HCP) system. Therefore, both of these structures can be generated by stacking of these planes. A portion of such a stack is shown in the picture below.

  • There are two types of voids between the atoms – vertex up (b), and vertex down (c).
  • The atoms in the next layer sit on the b sites (See animation below). In FCC, atoms in the third layer sit over the c sites and this repeats giving rise to ABC ABC ABC type of stacking.
  • In HCP system, centers of atoms of the third layer lie directly over centers of atoms of the first layer (a positions) giving rise to AB AB AB type of stacking.

 

Figure 2.6: Stacking of FCC and HCP structures

 

Value Addition:

 

Do You Know?

 

The 14 Bravais lattices incorporate all possible crystalline structures; they result by taking into consideration the space-group symmetry, i.e. the symmetry at translations and the point group symmetry of the lattice (the symmetry with respect to rotation, reflection or inversion). When the basis consists of only one atom, the Bravais lattice is identical with the crystalline structure. But when the basis is complex and consists of several atoms, say s, the crystalline structure can be seen as formed by the interpenetration of s Bravais lattices. The Bravais lattices have always an inversion center in one of the lattice points, whereas such an inversion center can lack in crystals with complex bases. By counting the point groups of the possible different crystals (which have bases with different symmetries), one ends with 32 crystalline classes that can be accommodated by the 7 crystal systems. Also, there are 230 space groups that result from the combination of the 32 crystalline structures with the translational symmetry.

  1. Suggested Reading

    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:

 

Bravais Lattice:

 

A lattice which obeys certain rules for particular set of symmetry operations.

 

Wigne r-Seitz cell:

 

This particular primitive unit cell is constructed by first drawing lines to connect a given lattice point to all nearby lattice points, and then drawing new lines (or planes, in three-dimensional lattices) at the middle point and normal to the first lines.

 

Coordination number:

 

The number of nearest neighbours

 

Packing fraction:

 

How much mass is packed in a given volume is all about packing fraction. Depending on different geometrical arrangement of atoms, the packing fraction can be varied.