#### Simple lattices, Physics tutorial

Introduction:

The most highly symmetrical lattices that take place naturally are the cubic structure. These are, thus, of some practical interest and as well give useful simple examples that assist in visualizing the more general cases. Around 90% of metallic crystal structure crystallizes into three (3) densely packed crystal structures vis-a-vis Body-Centered Cubic cell (BCC), Face-Centered Cubic cell (FCC) and Hexagonal Close-Packed (HCP).

Definition of Simple lattices:

The simple lattices are crystalline solids which comprise of a small group of atoms (that is, unit cells) that includes unique characteristics.

Simple lattices:

The simple lattices encompass the given elementary properties:

1) Effective number of atoms or unit cell, 'Z' that states the number of atom per primitive cell

2) Atomic radius, 'R' generally defines in terms of the lattice constant (that is, length of a side of unit cell), 'a'.

3) Nearest neighbor distance that defines the closest distance between the atomic centers.

4) Coordinate number that defines the number of nearest neighbor of the atom.

5) Atomic Packing Fraction (APF) stated as the fraction of volume in a crystal structure which is occupied through atoms.

The simple cubic lattice:

The simple cubic unit cell is a cube (that is, all sides of the similar length and all face perpendicular to one other) having an atom at each and every corner of the unit cell.

The unit cell fully explains the structure of the solid that can be regarded as an almost endless repetition of the unit cell.

Simple or primitive cubic lattice (sc or cubic-P) consists of one lattice point at the each and every corner of the unit cell. It consists of unit cell vectors a = b = c and interaxial angels α = β = γ = 90°.

The volume of the unit cell is readily computed from its shape and dimensions. This computation is specifically simple for a unit cell that is cubic. In this illustration, the atoms are in contact with each other all along the edges of the unit cell. Therefore the side of the unit cell consists of a length of 2r, where 'r' is the radius of an atom.

The simple cubic lattice consists of basis vectors a1 = ai, a2 = aj, a3 = ak and the unit cell is a simple cube. The simplest crystal based on this lattice consists of single atoms at the lattice points. Each and every atom consists of six identical nearest neighbors.

Some of the metals adopt the simple cubic structure because of inefficient use of space. The density of a crystalline solid is associated to its 'percent packing efficiency'. The packing efficiency of a simple cubic structure is only around 52%. (That is, 48% is empty space!)

% Packing Efficiency = (Volume of atoms in a unit cell/Volume of unit cell) x 100 %

Body-Centered cubic Lattice:

Body centered cubic or BCC signifies to a crystal structure in which the atoms are positioned at the corners of a cubic cell having one atom at the cell center position.

The body-centered cubic unit cell is a cube (that is, all sides of the similar length and all face perpendicular to one other) with an atom at each and every corner of the unit cell and an atom in the center of the unit cell.

The unit cell fully illustrates the structure of the solid which can be considered as an almost endless repetition of the unit cell.

The volume of the unit cell is readily computed from its shape and dimensions. This computation is particularly simple for a unit cell that is cubic. In case of the body-centered cubic unit cell, the atoms lying all along the main diagonal of the cube are in contact with one other. Therefore the diagonal of the unit cell consists of a length of 4 r, where 'r' is the radius of an atom.

Atoms, obviously, don't have well-defined bounds, and the radius of an atom is rather ambiguous. In the context of crystal structures, the diameter (2r) of an atom can be stated as the center-to-center distance between two atoms packed as tightly altogether as possible. This gives an effective radius for the atom and is at times termed as the atomic radius.

A more challenging job is to find out the number of atoms which lie in the unit cell. As illustrated above, an atom is centered on each and every corner and in the middle of the unit cell. The atom at the center of the unit cell lies entirely in the unit cell. The atoms positioned on the corners, though, exist partially within the unit cell and specifically outside the unit cell. In finding out the number of atoms within the unit cell, one should count only that part of an atom that in reality lies in the unit cell.

The density of solid is the mass of all the atoms in the unit cell divided by the volume of the unit cell.

The packing efficiency of a 'BCC' lattice is considerably higher than that of the simple cubic:   69.02%

The higher coordination number and packing efficiency signify that this lattice employs space more efficiently than the simple cubic.

The bcc lattice encompass alkali metals like Na, Li, K, Rb, Cs, magnetic metals like Cr and Fe, and refractory metals like Nb, W, Mo, Ta.

Face-Centered Cubic Lattice:

The face-centered cubic unit cell is the cube (that is, all sides of the similar length and all face perpendicular to one other) having an atom at each and every corner of the unit cell and an atom positioned in the middle of each and every face of the unit cell.

Face-centered cubic lattice (that is, fcc or cubic-F), similar to all lattices, consists of lattice points at the eight corners of the unit cell plus additional points at the centers of each face of the unit cell. It consists of unit cell vectors a = b = c and interaxial angles α = β = γ = 90°.

The unit cell fully explains the structure of the solid that can be regarded as an almost endless repetition of the unit cell.

The volume of the unit cell is readily computed from its shape and dimensions. This computation is particularly simple for a unit cell that is cubic. In case of the face-centered cubic unit cell, the atoms lying all along the diagonal of each face are in contact with one other. Therefore the diagonal of each face consists of a length of 4r, where 'r' is the radius of an atom.

Atoms, obviously, don't have well-defined bounds, and the radius of an atom is rather ambiguous. In the context of crystal structures, the diameter (2r) of an atom can be stated as the center-to-center distance among the two atoms packed as tightly together as possible. This gives an effective radius for the atom and is at time termed as the atomic radius.

A more challenging job is to find out the number of atoms which lie in the unit cell. As illustrated above, an atom is centered on each and every corner and in the middle of each face of the unit cell. None of such atoms lies fully in the unit cell. Each and every atom exists partly within the unit cell and partly outside the unit cell. In finding out the number of atoms within the unit cell, one should count only that part of an atom which in reality lies in the unit cell.

The significant characteristic of a crystal structure is the nearest distance among the atomic centers nearest-neighbor distance) and for the face-centered cubic this distance is a/√2.

The density of a solid is the mass of all the atoms in the unit cell divided by the volume of the unit cell.

FCC systems contain an APF (that is, Atomic packing factor of 0.74, the maximum packing for a system in which all the spheres contain equal diameter).

Hexagonal Close-Packed (HCP):

Assume that you are given a large number of tennis balls and asked to pack them altogether in the most efficient manner. What is the most efficient packing strategy? One could toss all the balls altogether in a box and shack the box to induce the balls to settle. The resultant packing of the balls is termed as a random closest-packed structure. Not shockingly it is not the most efficient approach to pack the tennis balls.

However there are a variety of factors which affect how atoms pack altogether in crystals, atoms usually search for the most efficient packing structure in order to maximize the intermolecular attractions. Metals give the simplest packing case, because these atoms can usually be regarded as the uniform spheres.

The two most efficient packing arrangements are the hexagonal closest-packed structure (hcp) and the cubic closest-packed structure (ccp).

In case of a hexagonal closest packed structure, the third layer consists of the similar arrangement of spheres as the first layer and covers all the tetrahedral holes. As the structure repeats itself after each and every two layers, the stacking for hcp might be illustrated as 'a-b-a-b-a-b'. The atoms in a hexagonal closest packed structure efficiently engage 74% of space whereas 26% is the empty space.

In a crystal, the atoms are arranged in a regular repeating model. The smallest repeating unit is termed as the unit cell. The whole structure can be reconstructed from the knowledge of the unit cell. The unit cell is characterized through three lengths and three angles. The quantities a and b are the lengths of the sides of the base of the cell and γ is the angle among these two sides. The quantity c is the height of the unit cell. The angles α and β illustrate the angles between the base and the vertical sides of the unit cell.

In the hexagonal closest-packed structure, a = b = 2r and c = 4(2/3)1/2 r, here r is the atomic radius of the atom. The sides of the unit cell are perpendicular to the base, therefore α = β = 90o. The base consists of a diamond (hexagonal) shape corresponding by γ = 120o.

Coordination number and APF for HCP are precisely similar as those for FCC: 12 and 0.74 correspondingly.

The hcp structure is extremely common for the elemental metals, comprising: Beryllium, Cadmium, Magnesium, Titanium, Zinc and Zirconium

Closed-packed Structures:

The arrangement in a cubic closest packing as well efficiently fills up 74% of space. Similar to the hexagonal closest packing, the second layer of spheres is positioned onto of half of the depressions of the first layer. The third layer is fully different than those first two layers and is stacked in the depressions of the second layer, therefore covering all of the octahedral holes. The spheres in the third layer are not in line with such in layer A, and the structure doesn't repeat until a fourth layer is added. The fourth layer is similar as the first layer; therefore the arrangement of layers is 'a-b-c-a-b-c'.

In crystal, the atoms are arranged in a regular repeating model. The smallest repeating unit is termed as the unit cell. The whole structure can be reconstructed from knowledge of the unit cell. The unit cell is characterized through three lengths and three angles. The quantities a and b are the lengths of the sides of the base of the cell and γ is the angle among these two sides. The quantity c is the height of the unit cell. The angles α and β illustrate the angles between the base and the vertical sides of the unit cell.

In cubic closest-packed structure, a = b = c = 2 (2)1/2 r, here r is the atomic radius of the atom. The sides of the unit cell are all mutually perpendicular, therefore α = β = γ = 90o. The unit cell for the cubic closest-packed structure is the face-centered cubic unit cell (fcc).

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