ATOMIC RADIUS OF A CUBIC LATTICE
Half the distance between the centres of the two immediate neighbours in a unit cell, is called atomic radius (r) ; while the distance between the centres of two corner atoms of the cube, is called length of the cube edge(a).
(i) Simple cubic cell (SC)
The atomic radius (r)and length of the cube(a) are related as:
a = 2r
r = a/2
Fig Simple cubic cell
(ii) Body-centred cubic cell
From Fig14 , it is clear that :
Fig 14 Body-centred cubic cell
(iii) Face-centred cubic cell : From Fig 15 , it is clear that,
Fig 15 Face-centred cubic cell
CLOSE PACKING IN CRYSTALS
In order to understand the packing of the constituent particles in a crystal , it is assumed that these particles are hard spheres of identical size. The packing of these spheres takes place in such a way that they occupy the maximum available space and hence the crystal has maximum density. This type of packing is called close packing.
The packing of spheres of equal size takes place as follows :
Fig 16 Packing of spheres using available space efficiently
(a) Edge formation (b) Two modes of plane formation
(1) When the spheres are placed in horizontal row, touching each other, an edge of the crystal is formed (Fig a).
(2) When the rows are combined, touching each other, the crystal plane is obtained. The rows can be combined in two different ways :
(i) The particles when placed in the adjacent rows, show a horizontal as well as vertical alignment and form squares (Fig b). This type of packing is called square close packing.
(ii) The particles in every next row are placed in the depressions between the particles of the first row. The particles in the third row will be vertically aligned with those in the first row. This type of packing gives a hexagonal pattern and is called hexagonal close packing.
The second mode of packing (i.e. hexagonal close packing) is more efficient as more space is occupied by the spheres in this arrangement.
In square close packing, a central sphere is in contact with four other spheres whereas in hexagonal close packing , a central sphere is in contact with six other spheres ( Fig 16 b)
(3) Three dimensional packing : For two dimensional packing , a more efficient packing is given by hexagonal close packing. Let us consider a three dimensional packing keeping a hexagonal close packed pattern for layers. In the base layer shown in Fig (a) , the spheres are marked as A and the two types of voids between the spheres are marked as ‘a’ and ‘b’.
When a second layer is placed with spheres vertically aligned with those in the first layer, its void will come above the voids in the first layer. This is an inefficient way of filling the space.
When the second layer is placed in such a way that its spheres find place in the ‘a’ voids of the first layer, the ‘b’ voids will be left unoccupied since under this arrangement no sphere can be placed in them (Fig 17 b).
Now there are two types of voids in the second layer. These are marked as voids of ‘c’ and voids of ‘d’ . The voids ‘c’ are ordinary voids which lie above the spheres of first layer whereas voids ‘d’ lie on the voids of the first layer and hence are combinations of two voids one of the first layer and second of the second layer.
Solid circles represent layer A and dotted circles represent layer B
Layers of close packing of spheres (a) Hexagonal close packed base layer ( b) Two layers.
The ‘a’ and ‘b’ voids of the first layer are both triangular while only ‘c’ voids of the second layer are triangular. The ‘d’ voids of the second layer are combinations of two triangular voids (one each of first layer and second layer) with the vertex of one triangle upwards and the vertex of the other triangle downwards.
A simple triangular void in a crystal is surrounded by four spheres and is called tetrahedral void or hole (Fig 18 a)
Fig 18 Tetrahedral void
A double triangular void like ‘d’ is surrounded by six spheres and is called octahedral void (Fig 18 b).
Fig 18 b Octahedral void
Now there are two ways to build up the third layer.
(i) When the third layer is placed over the second layer in such a way that the spheres cover the tetrahedral or ‘c’ voids, a three dimensional closest packing is obtained where the spheres in every third or alternate layers are vertically aligned (i.e., third layer is directly above the first, the fourth above the second layer and so on). Calling the first layer as layer A and second layer B, the arrangement is called ABABAB…. pattern or hexagonal close packing (hcp) [Fig 19 (a) or Fig 20 (i) ]
Molybdenum, magnesium and beryllium crystallise in hcp (hexagonal close packing) structure.
(ii) When the third layer is placed over the second layer in such a way that spheres cover the octahedral or ‘d’ voids, a layer different from layers A and B is produced. Let us call it as layer C. Continuing further a packing is obtained where the spheres in every fourth layer will be vertically aligned. This pattern of stacking spheres is called ABCABC …. pattern or cubic close packing (ccp). It is similar to face centred cubic (fcc) packing [Fig 19 b or Fig 20 (ii)]
Iron, nickel, copper, silver , gold and aluminium crystallise in ccp (cubic close packing) structures.
Fig 19 Actual view of (a) hexagonal close packing (b) Cubic close packing
Fig20 A simplified view of (i) Hexagonal close packing (ABAB…. Pattern) (ii) Cubic close packing (ABCABC ….. pattern)
Both the above patterns of stacking spheres, though different in form, are equally efficient. They occupy the maximum possible space which is about 74% of the available volume (empty space being only 26%). Hence they are called close packings.
Further in both hcp and ccp methods of stacking, a sphere is in contact with 6 other spheres in its own layer. It directly touches 3 spheres in the layer above and three spheres in the layer below. Thus a sphere has 12 close neighbours. It is said to have a co-ordination number of 12. The number of closest neighbours of any constituent particle is called co-ordination number. The common coordination numbers in different types of crystals are, 4, 6, 8 and 12.
To understand as to why they are called hexagonal close packing and cubic close packing, the arrangement of these layers may be represented as shown in Fig 21.
Hexagonal close packing (hcp)
Cubic close packing (ccp) Face-centred cubic (fcc)
Fig 21 (a) ABABAB…… type is hexagonal close packing (hcp)
(b) ABCABC….. type is cubing close packing.
In addition to the above two types of arrangements , a third type of arrangement found in metals is body-centred cubic (bcc) in which space occupied is about 68%.
Lithium, sodium, potassium, rubidium and caesium crystallize in the bcc structure.
The co-ordination number of each atom in the bcc structure is 8. Further it is not as closely packed as the first two, the empty space is about 32%.
Interstitial sites in Close packing of spheres
Even in the closest packing of spheres there is some empty space left between the spheres. This is known as interstitial site. For example, when the spheres are closely packed in a single plane, the centres of three adjacent spheres lie at the vertices of an equilateral triangle. In between these three spheres, there is some empty space. This empty space ( hole or void) is called triangular or trigonal site. In addition to this , there are two more common interstitial sites, viz., tetrahedral and octahedral sites, in closely packed lattices. These are discussed below.
If one sphere is placed upon three other spheres which are touching one another, tetrahedral structure results. This is shown in Fig 22.
Fig 22 Tetrahedral void
Since four spheres touch each other at one point only, they leave a small space in between which is called a tetrahedral site. The size of the site is much smaller than that of the sphere. However, the bigger the spheres , the larger is the size of the tetrahedral site formed by them. In hexagonal close packing (hcp) as well as in cubic close packing (ccp) arrangement, each sphere is in contact with three spheres in the layer above and three spheres in the layer below. Thus there would be two tetrahedral sites associated with each sphere. One of these is immediately above and the other immediately below the sphere. It may be noted that the tetrahedral sites does not mean that the shape of the site is tetrahedral, but it indicates that this site is surrounded by four spheres and the centres of these four spheres lie at the apices of a regular tetrahedron. It is this tetrahedral arrangement of the spheres which gives tetrahedral name to this site. Radius of the tetrahedral void relative to the radius of the sphere is 0.225, i.e., for tetrahedral void:
Another type of interstitial site formed by close packing of spheres in hcp as well as in ccp system is called an octahedral site. This interstitial site is formed at the centre of six spheres, the centres of which lie at the apices of a regular octahedron, as illustrated in Fig 23 .
The Fig 23. shows two layers of close packed spheres. The full circles represent the spheres in one plane while the dotted circles represent those in second plane. Two triangles have been drawn. One of these joins the centres of three spheres in one plane, while the second (dotted lines) joins three spheres in second plane. The octahedral sites marked by x, are formed where two triangles of different layers are superimposed one above the other. In this position the apices of these triangles point in opposite direction, as shown. Thus each octahedral site is created by two equilateral triangles with apices in opposite direction. The interstitial void formed by the combination of two triangular voids of the first and second layer is called octahedral void because this is enclosed between six spheres, centres of which occupy corners of a regular octahedron. The octahedral void is shown in the Fig 23. In a close packing of spheres, the number of octahedral voids is equal to the number of spheres. Thus there is one octahedral void associated with each sphere. Radius of the octahedral void relative to the radius of the sphere is 0.414 i.e., for an octahedral void :
Octahedral void is larger than a tetrahedral void.
Relative positions of tetrahedral voids and octahedral voids in a face-centred cubic arrangement are shown in the figure 24. In the figure , hollow circles represent the tetrahedral or octahedral sites, while face-centred cubic arrangement is formed by spheres shown by dark circles.
Fig 24 Relative positions of (a) tetrahedral void (b) octahedral void
In a ccp structure , if there are N spheres(atoms or ions), there are 2N tetrahedral voids and N octahedral voids. In a face-centred cubic (fcc) unit cell there are four atoms / ions , therefore there are 4 octahedral voids and 8 tetrahedral voids. A tetrahedral void can accommodate an ion of radius r+= 0.225 r and an octahedral void can accommodate an ion of radius r+= 0.414 r .
In a ccp structure , there is 1 octahedral void in the centre of the body and 12 octahedral voids on the 12 edges of the cube. Each octahedral void on the edge is common to four other unit cells (Fig 24 b above). Each octahedral void on the edge is common to four other unit cells . Thus , in cubic close packed structure :
In ccp structure, there are 8 tetrahedral voids. In close packed structure, there are eight spheres in the corner of the unit cell and each sphere is in contact with three others giving rise to eight tetrahedral voids.
CALCULATION OF THE SPACES OCCUPIED -PACKING FRACTIONS
The ratio of the volume occupied by the atoms to the volume of the unit cell , is called atomic packing factor.
(i) Simple cubic cell (SCC)
Since there is only one atom per unit cell,
so volume occupied by atom = (4/3) r3
and volume of the cube = a3
= (2r)3 (since a = 2 r)
= 8 r3.
(ii) Body-centred cubic cell ( BCC)
Since each unit has 2 atoms, so
(ii) Face-centred cubic (FCC)
Since there are four atoms per unit cell, so
CALCULATION OF DENSITY OF A CUBIC CRYSTAL
Let Z be the number of formula units(atoms in case of an element or molecules in case of ionic solid) ; M be the (atomic mass / molecular mass of crystalline substance ) and ‘a’ be the edge length of the crystal. Then,
Volume of unit cell = a3
Mass of unit cell = [Number of formula units in unit cell ]
x [ Mass of each formula unit]
Note : In case of cubic crystal of an element, M stands for atomic mass ; while for ionic substance, M is its molecular mass and N Avogadro number.
Sizes of interstitial sites in ionic crystals - Limiting radius ratio
In general the size of a trigonal site is smaller than the size of a tetrahedral site which in turn is smaller than the size of an octahedral site. The size of the site varies with size of the spheres surrounding it. In case of ionic crystals, the radius of a cation which may occupy a given site created by anionic spheres can be calculated by simple geometry. This can been done in terms of the ratio of the radius of cation to that of anion (r+/ r) , also known as radius ratio.
In order to acquire minimum energy, the ions are close packed in such a way that each ion is surrounded by as many ions as possible. The number of ions surrounding an oppositely charged ion is known as co-ordination number. The higher the co-ordination number, the greater would be the stability of the ionic compound, in general. Only that particular structure will be adopted by ionic compound which gives the lowest energy or maximum stability. The energy of an ionic crystal is determined by the resultant of energy of attraction between cations and anions and energy of repulsion between the anions surrounding the cation. Energy of attraction increase with the co-ordination number of the cation. The energy of repulsion would increase when anions of the same size and charge are squeezed together because the electron charge clouds of the squeezed ions would repel one another.
The inter nuclear distance r is taken as the sum of the radii of cations and anions. If the cations and anions do not touch each other (i.e., the interstitial site so formed by anions is too big for cations, then the internuclear distance would increase and energy of attraction would decrease.
Thus, the optimum arrangement of cations and anions would be the one in which required number of anions touch a cation without being squeezed. Such an arrangement would be of minimum energy and maximum stability. The limiting radius ratios for various types of ionic arrangements in crystals are determined by keeping the above principle in view.
a. Limiting radius ratio for trigonal site
A trigonal site is formed when three anions represented by spheres lie at the vertices of an equilateral triangle and a cation, represented by a small sphere, occupies the trigonal site( Fig 25).
Let r+ and r be radii of cation and anion respectively. As is evident from the figure, in equilateral triangle EBC,
BC = CE = BE = 2 r
If A is the centre of the triangle ( and hence the cation) , then,
BD = r and AB = r+ + r
In right angled BDA ,
Thus, the limiting radius ratio (r+/ r) for trigonal site is 0.155. This means that a trigonal site is only about 0.155 of the size of the surrounding spheres, i.e., anions.
b. Limiting radius ratio for tetrahedral site
A tetrahedral site is formed by placing one sphere over three spheres which are touching one another. In an ionic crystal, such a site can be created by placing the anions at the alternate corners of a cube. Let the cation occupy the tetrahedral site. Suppose a is the length of each side of the cube and r and r+ are the radii of anion and cation, respectively. For simplicity, the cation and anions are shown by separated spheres ( Fig 26).
A tetrahedral site. The spheres actually touch one another, though not shown so in the figure for the sake of simplicity
Actually the cation and anion are touching one another so that,
AC = 2 r and AD = 2 AO = 2( r + r +)
In the figure, the face diagonal,
Since the spheres are actually touching each other, hence
AC = r + r = 2 r
2 r = 2 a
As is evident from figure, the body diagonal
The cation is present at the centre of the body diagonal AD so that half the length of this diagonal is equal to the sum of the radii of the anion and cation. Thus,
Dividing equation (iii) by (i) we get:
Thus the limiting r+/ r ratio for tetrahedral site is 0.225.
If the radius ratio is less than 0.225 , then either the anion is too big or the cation is too small in size. In either case, the anions will have to be squeezed so that the anions and the cation just touch one another. The squeezing of anions would increase the repulsion between them and as a result, the tetrahedral arrangement would become less stable. The squeezing of anions can be avoided by decreasing the number of anions surrounding the cation, i.e., by decreasing the co-ordination number of the cation. This would change the structure of the crystal from tetrahedral to planar trigonal. Thus, the crystal would have a trigonal structure if the radius ratio is less than 0.225.
c. Limiting radius ratio for octahedral site
An octahedral site is shown in Fig 27 .
An octahedral site. Only a cross-section
through octahedral site is shown.
A cross-section through an octahedral site shows that the four anions are at the corners of a square and cation is at the centre. One anion at the top and one anion at the bottom of the plane formed by the four anions also touch the cation. Suppose each side of the square is ‘a’ and r and r+ are the radii of the anion and cation respectively. In right angled ABC,
But, AC = r + 2 r+ + r = 2 r + 2 r+
2 r + 2 r+ = 2 a
As is evident from the figure, AB = a = 2 r
r = a / 2 …….. (ii)
Dividing Eq. (i) by (ii) , we get
Thus , the limiting r+/ r ratio for octahedral site is 0.414 .
If the radius ratio is less than 0.414, then either the anion is too big or the cation is too small in size. In either case, the anions shall have to be squeezed so that the cation and the anions just touch one another. This would increase repulsion between the anions and destabilise the octahedral arrangement of anions. Thus, ionic crystals with radius ratio less than 0.414 would tend to have a tetrahedral structure.
d. Limiting radius ratio for cubic site
Consider a cubic site, as shown in Fig 28.
A cubic site. The spheres are not shown
touching ( for simplicity)
The spheres are not shown touching one another for the sake of simplicity. Suppose the length of each side of the cube is ‘a’ and r and r+ are the radii of anion and cation respectively. In the figure, AB = 2 r = a
or r = a / 2 ……..(i)
Length of the face diagonal
Length of the body diagonal
But, body diagonal
AD = r + 2 r+ + r = 2 r + 2 r+
Dividing Eq. (ii) by (i) , we get
Thus, the limiting r+/ r ratio for cubic site is 0.732.
If the radius ratio is less than 0.732, then the arrangement would become unstable and the co-ordination number of the cation would tend to decrease to stabilise the system. In other words, the system would prefer to change from cubic to octahedral arrangement of anions. Thus, ionic crystals with radius ratio less than 0.732 would tend to adopt an octahedral geometry.
ATOMIC RADIUS OF A CUBIC LATTICE