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Coordination Number

Matter can exist in three possible states. Solid, liquid and gaseous state. Out of these states solid state is the most common state as most of the substance, we use in our daily life are solids. Solids differ from other two physical states in nature of constituent particles and the types of binding forces operating between them. Therefore the study of structure of solids is important to explain their properties compare to liquid and gaseous state. Solid can be classified as crystalline and amorphous solids.

A solid in which constituent particles possess a regular order arrangement is known as crystalline solids. While in amorphous solids constituent particles do not possess a regular arrangement.

For example,
glass, rubber and starch are example of amorphous solids. In crystalline solids such as sodium chloride, sucrose, there is a certain regular pattern which generated from the repetition of some basic units like atoms, ions or molecules, called a unit cell.

Hence a unit cell can be defined as a three dimensional group of lattice points that generated the whole lattice by stacking. Unit cell represents by a certain pattern of points which are called as lattice points and each lattice point can be an ion or atom or molecule. The three dimensional ordered arrangement of points is called a crystal lattice. In crystal lattice each lattice point is surrounded by other points. The number of nearest neighbors in a packing is known as coordination number.

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Coordination Number Definition

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A unit cell of solid represents by points connected by lines where points represents the constituent particles and lines helps to visualized the symmetry of solid. The arrangement of particles can be explained in one, two and three dimensional. When the arrangement is extends to two or three dimensional, each lattice point is surrounded by a certain number of other spheres. The number of spheres which are touching a particular sphere is known as coordination number

For example, in one dimensional there is only one way to of arranging spheres in which each sphere is in contact with two other spheres, therefore in this close packing of sphere, coordination number is two.

Close packing of Spheres

Similarly in two dimensional close packing, there are two possible arrangements of sphere with 4 and 6 coordination numbers. In square close packing, each sphere is surrounded by four other spheres and show four coordination number while in hexagonal close packing, coordination number is six.

Square packing and Hexagonal Packing

As the coordination number increases, the packing efficiency of crystal structure increases which increase the stability of crystal. For example, in hexagonal close packing 60.4% of available space is occupied by spheres whereas in square close packing only 52.4 % of space is occupied.

Determining Coordination Number

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In solid structures, for the determination of coordination number, we have to use simple models of crystals as bonds between spheres are not clearly defined in solid-state structures of crystals. In crystal models of solids, the number of nearest neighbor for a constituent particle is defined as coordination number. In other words the number of spheres which are touching a particular sphere is known as its coordination number.

The model of solid structure shows a fine picture of arrangement of constituent particles in three dimensional which provide a clear idea of coordination number for any constituent particle.The ratio of cation and anion radius is called as radius ratio which gives preferred coordination number and packing of crystal lattice.

Radius Ratio and Coordination Number Charts
Coordination Number 4

In two dimensional close packing of spheres, the rows of spheres are stacked over each other. If the spheres of second rows are sited in such a way that placed exactly above the spheres of first row, than such type of arrangement is known as square close packing. There is a vertical and horizontal alignment of particles in two rows. Suppose first row is marked as 'A' type of row than second row will be exactly same and must be 'A' type. 

Therefore such type of arrangement is also known as AAAA type of arrangement. In square close packing, each sphere is surrounded by four other spheres, hence coordination is four.

Square Close Packing

Coordination Number 6

In case of two dimensional close packing, apart from square close packing, there is one more possible way to arrange spheres which shows six coordination number and called as hexagonal close packing. In such type of close packing, the spheres of second row are seated in the depression between the spheres of first row. The spheres of third row are vertically aligned with the spheres in the spheres of first row and similar pattern if followed throughout the crystal. 

Suppose first row is marked as 'A' row and second row as 'B' row than the sphere of third row are seated in the depression of second row, hence third is exactly similar to first row and, can be marked as 'A' type. Therefore such type of arrangement is called as ABAB type. If we observe any one of the sphere of second row, its surrounded by six other spheres; two of upper layer, two of lower layer and two of same layer. Hence the coordination number will be six. The line joints centre of these six spheres makes a hexagon, therefore such type of packing known as hexagonal close packing.

Hexagonal Close Packing

Coordination Number 8

When each sphere is in contact with other eight spheres, it shows coordination number 8. For example; in body centered cubic unit cell, each lattice point is surrounded by eight other lattice points, hence show coordination number 8. Such type of unit cells form body centered cubic structure. Compare to HCC and FCC, it is less close packed structure and show only 68% packing efficiency.
Body Centered Cubic Unit Cell

FCC Coordination Number

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Face centred cubic (FCC) packing is a three dimensional packing of spheres obtained from two dimensional hexagonal close packing. This is one of the most efficient close packing with twelve coordination number and 74% packing efficiency. 

In two dimensional hexagonal packing the second layer placed on half of the voids of first layer. Now on second layer; there are two types of voids, one is tetrahedral voids and another is octahedral voids. Tetrahedral voids are in contact with four other spheres and Octahedral voids are surrounded by six spheres.
Face Centered Cubic
In three dimensional packing of spheres, the third layer is placed in octahedral voids and created a layer which is different from first (A) and second (B) layer, therefore marked as 'C' layer. This pattern is known as face centered cubic (FCC) or cubic close packing (CCP). Metals like iron, gold, silver, platinum, copper, nickel show such type of close packing. In FCC, a sphere is in contact with six other spheres in its own layer and also touch three sphere of above layer and three spheres of below layer. Hence the coordination number is 12.

Face Centred Cubic Structure

Coordination Number BCC

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The coordination number of BCC that is body centred cubic packing is eight. It is a three dimensional arrangement found in some metals like sodium, potassium, rubidium, and cesium. The packing efficiency of this packing is 68 % only. In this packing the first layer of sphere is expanded in all directions and second layer is placed in depression of first one. Here spheres are in contact with each other along the body diagonal. 

Body Centered Cubic Unit CellThree Dimensional Body Centered Cubic

Coordination Number of Simple Cubic

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When two dimensional square close packing extends to three dimensional packing, it results simple cubic lattice with coordination number six. In square close packing, first and second row of spheres are built up exactly in same way. When third layer of spheres will build in such a way that all layers are horizontally as well as vertically aligned with each other, than such type of arrangement results a simple cubic lattice. 

Since all layers arranged in the same manner, this arrangement is also known as AAAA type of arrangement. It forms with the primitive cubic unit cell. The packing efficiency of simple cubic is only 52%, therefore such crystal structure is adopted by few metals only.
Simple Cubic ArrangementSimple Cubic Arrangement
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