Structure of Crystal Lattice
Cubic crystal lattice, formed by the orderly arrangement of atoms, ions, or molecules in space lattice or three-dimensional system. In learning chemistry or physics, Braggs developed the very simple but useful relation between the wavelengths of the electromagnetic spectrum of x-rays and the spacing between the crystals lattice planes, nλ = 2d sinθ, where n = 1,2,3, etc and d = distance between the lattice planes. Braggs also developed the character of the line spectrum formed by x-rays which use for crystallographic study in chemistry. For every crystal system, there is three or four number of lattice planes, whose spacing reveals the position or points of the constituents.
For example, cubic sodium chloride formed by ionic bonding, and (100), (110) or (111) planes give the required information about crystal lattice.
The space lattice is divided into a large number of small symmetrical units when similar points are connected by sets of parallel lines along the three dimensions coordinate axes. These basic units of the space lattice are known as unit cells.
Definition of Bravais Crystal system
Bravais in 1848 shows that all 14 classes of arrangement define all possible forms of the crystalline solid like borax, boric acid, diamond, metallic crystals, etc. The cubic, tetragonal, orthorhombic, and monoclinic systems have more than one kind of arrangement of the constituents atom, ion, or molecule. These 14 unit cells originated from seven crystal systems are known as the Bravais lattice. Among these 14 crystal systems, cubic crystals form three kinds of unit cells like primitive or simple, body-centered, and face-centered.
Simple or Primitive Cubic Crystal Lattice
In simple or primitive cubic crystal lattice where atoms are present at the corners only. Each atom is equally shared by eight atoms because each cube contains eight corners. Therefore, the contribution of each atom to the unit cell = 1/8. Hence the number of atoms per unit cell = 8 × (1/8) = 1 only.
Body-Centered Cubic Crystal Lattice
The alkali metals like lithium, sodium, potassium, rubidium, cesium in the periodic table are examples of the body-centered cubic lattice. This type of cubic crystal system posses one atom at the center and eight atoms at the corner of the cube. In cesium chloride, the body elements and corner elements are different. In polar cesium chloride, all the chlorine ions at the corner and cesium cation at the center of the cubic crystal or vice versa. The body-centered atom belongs exclusively to the unit cell. Therefore, the total number of atoms per unit cell = 1 (for the body) + 1 (for corner) = 2.
Face-Centered Cubic Crystal Lattice
The number of periodic tables chemical elements like copper, silver, gold, nickel, platinum and solidified inert gases (helium, neon, argon krypton, xenon) of our environment possess face-centered cubic crystal structures. In a face-centered cubic lattice where the atoms are present at the six faces and the eight corners. But face atoms are equally shared by two unit cells. Therefore, the total number of atoms per unit cell = 3 (for face) + 1 (for corner) = 4. Hence the element density of face-centered unit cells greater them the primitive or body-centered unit cell.
Law of Rational and Miller Indices
In order to describe the crystallographic structure of a crystalline solid, we need to discuss the orientation of the plane passing through the lattice point of the cubic crystal. But the orientation of the lattice plane describes by the intercepts of the plane on the three basis vectors of the lattices. The intercepts of any plane along the three crystallographic structures either equal to the ratio of the unit cell or some integral multiple of them. This is known as the law of rotational indices.
In learning chemistry, let one plane cut two three dimensional axes by a and b and the third axis in infinity. Hence the Weiss indices = 1, 1, ∞. For a better representation of the indexing plane, we avoided infinity because infinity is inconvenient for determination. Therefore, the reciprocal of miller indices taken as Weiss indices. For example, miller indices of (11∞) and (12∞) represented in Weiss indices as (110) and (210) respectively.
Lattice Planes in Cubic Crystal
A cubic system is the simplest type of crystal system. Hence the intercept on the three-axis is equal and all the angles equal to 90°. In the simple cubic lattice, the points are at the corner of the cube. Therefore, the planes that can pass through the lattice point have miller indices (100), (110), (111). The (100) planes cut only on the x-axis but are parallel to the y and z-axis but (110) planes cuts x and y-axis but parallel to the z-axis and (111) planes cuts x, y, z-axis.
In sodium chloride, the sodium atom losses one electron by ionization to form sodium ion, and the chlorine atom has electron affinity gain this electron particle to form chloride ion. They binding each other by the electrostatic force of attraction to form the ionic body-centered cubic sodium chloride crystal with the liberation of the lattice energy. But ice crystals formed by hydrogen bonding of water molecules and different types of force like Van der Waals, electrostatic, hydrogen bonding, etc are responsible for the formation of cubic ice crystals.