Paper No. : Crystallography & crystal growth

Module : Classification of Crystalline Solids

Prof. Vinay Gupta, Department of Physics and Astrophysics , University of Delhi, Delhi

Development Team

Principal Investigator

Paper Coordinator

Content Writer

Content Reviewer

Prof. P. N. Kotru ,Department of Physics, University of Jammu,

Jammu-180006

Prof. P. N. Kotru ,Department of Physics, University of

Jammu, Jammu-180006

Prof Mahavir Singh Department of Physics, Himachal Pradesh

University, Shimla

Material science

Description of Module

Subject Name Physics

Paper Name Crystallography & crystal growth

Module Name/Title Classification of Crystalline solids

Module Id M22

TABLE OF CONTENTS 22. Classification of Crystalline Solids. 22.1 Types of Bonding. 22.2 Bonding in Solids 22.2.1 Ionic Bond 22.2.2 Covalent Bond 22.2.3 Metallic Bond 22.2.4 Molecular Bond 22.2.5 Hydrogen Bond 22.3 Latice Energy of Ionic Crystals 22.3.1 Assumptions 22.3.2 Interacting forces 22.4 Calculation of Repulsive Exponent 22.5 Theoretical and Experimental Lattice Energies

LEARNING OBJECTIVES

1. In this module we will learn about the binding forces that hold a crystal together.

2. Solids are classified according to the type of bonding in them. 3. Ionic, covalent, metallic, molecular and hydrogen bonded crystals are described

4. Characteristics features of various crystals types are given 5. Various types of bonds viz, ionic bond, covalent bond, metallic bond, molecular

bond and hydrogen bond are discussed, providing understanding of how

crystalline solids are held together. 6. Interacting forces contributing to the lattice energy of crystals are explained.

7. Expression leading to determination of lattice energy of ionic crystals is derived. 8. Calculation of repulsive exponent from compressibility data is explained

22 Classification of Crystalline Solids.

The definition of a crystal in its simple form is that “a crystal is a solid with naturally occurring plane faces”. It has already been explained that the external morphology of a crystal is a reflection of the regularity of the internal arrangement of the atoms or molecules within it which one can easily confirm by the use of x-rays. So, the definition of a crystal can be extended by including any solid material which has the atoms, molecules or ions of which it is made in some regular order. We shall consider here the various types of bonds which exist in crystals and hold them together. The classification into bonds of different kinds is done for simplicity of description, but in practice the bonds in a real crystal may be a mixture of two or more of these types. What are the binding forces that hold a crystal together? The answer to this question lies in describing the types of binding forces and how in terms of these forces the solids can be classified is as follows:

22.1 Types of bin ding

The forces of interaction between the atoms and molecules in solids are almost entirely

electrostatic in nature and with almost negligible contributions from magnetic interactions .The

attractive force increases as the distance between the atoms decreases till a limit is reached and

beyond which short range repulsive force between the like charges of nuclei comes into play.

Depending on the binding energy, solids are classified into five groups. By and large the important

differences among the several types of crystal bonds may be attributed to qualitative differences in the

distribution of electrons around the atoms and molecules or on the distribution of outer electrons in

space. Based on the type of bind ing, the so lids are classified as g iven below:

a) Ionic Crystal (examples: NaCl, LiF, KF, MgO etc.)

b ) Covalent Crystal (examples: Diamond, Silicon carb ide, Germanium etc.)

c) Metallic crystal (examples: Copper, Silver, Sodium, Iron.)

d ) Molecu lar crystal (examples: Argon, Methane)

e) Hydrogen bonded crystal (examples: H2O (Ice), HF)

Now, let us explain what essentially we mean by binding energy. Bind ing energy is the

energy necessary to dissociate the solid into separate atoms, molecules or ions, as may be applicable.

It is taken at room temperatu re, except for the molecu lar crystals where it is taken at the melt ing point.

The b inding energy is expressed either in Kilocalories (Kcal) per mole or in electron volts

(eV) per molecule. It may be noted that 1 eV / molecule = 23.05 K cal/mole). The ionic, covalent and

metallic bonds are regarded as strong or primary bonds, and the hydrogen and molecular bonds as the

weak o r secondary bonds.

The classificat ion o f crystal types may be given here in a tabu lar form as follows:

TABLE: 22.1

CLASSIFICATION OF CRYSTA L TYP ES

S.no Crystal

type

Exa mples

B.E.

(Kcal/mole)

Characteristics

1.

Ion ic

Sodium

chloride(NaCl)

180

•• Strong in frared absorpt ion

•• Low elect rical conductiv ity at low

temperatures

•• Good ion ic conductiv ity at h igher

150

Lithium fluo ride(LiF)

240

temperatures

•• Good cleavage

•• Formed by a combination of

highly electropositive and highly

electronegative elements

2.

Covalent

Diamond 170 •• Poor cleavage

•• Poor thermal conductiv ity

•• High hardness

•• Low electrical conductivity at low

temperatures

•• These are formed by combination

of the higher elements in the

middle co lumns of the periodic

table

Silicon carb ide(SiC) 283

Germanium(Ge)

-

3.

Metallic

Sodium(Na)

26

•• Opaqueness

•• Duct ility

•• Malleab ility

•• High electrical and thermal

conductivity

•• Formed as a result of combinat ion

of atoms of e lectroposit ive

elements.

Iron (Fe)

94

4.

Molecu lar

Methane(CH4)

2.4

•• Compressible

•• Generally evaporate as stable

molecu les.

•• Low boiling and

melting po ints.

•• Formed by inactive atoms such as

the rare gases, and saturated

molecules such as hydrogen and

methane.

Argon (A)

1.8

5.

Hydrogen-

bonded

Ice(H2 O) 12

•• Tendency to form groups of many

molecu les(Po lymerization)

•• Higher cohesive energy of

molecules as compared to similar

molecules without hydrogen

bonds.

Hydrofluoric acid (HF)

07

22.2 Bonding in S olids

solids.

Here, we shall d iscuss different kinds of bonds which form the basis of classification of

22.2.1 Ionic Bon d.

The existence of ions in aqueous solutions is a reality which is commonly an acceptable

exp lanation of electrolysis and of the conductivity of a salt in solution or in a molten state .When a

simple salt, say for example, sodium chloride is dissolved in water; the heat of solution is relatively

small. Th is heat of solution is much smaller than would be expected if the salt consisted of molecules

which had to be broken up into ions as it went into solution. It indicates that the salt may not really

contain molecules at all but is already ionized even while it is a solid. Several simple salts occur as

ionic crystals, that is, there are no definite molecules but there is a collection of positive ions and a

collection of negative ions, each positive ion being surrounded by negative ions and each negative ion

being surrounded by positive ions. The whole structure is electrically neutral, there being an attraction

between ions with opposite charges and there being repulsion between ions with charges of the same

sign. The equilibrium of the crystal is maintained as a result of balance between these two effects.

However, it is an established fact of electrostatics that a system of stationary charges cannot be in

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equilibrium under their own electrostatic coulomb forces of attraction and repulsion. It is known as

Earnshaw’s theorem . If the electrostatic forces were the only forces in action then all the charges

would be attracted to those of opposite sign and the whole solid would collapse to occupy zero

volume, if it be assumed that the ions are just point charges and do not possess size of their own .How

is then an equilibrium is maintained in the solid?

Let us take sodium ion as an example. It has a nucleus of finite size (~ 10

─12 cm in rad ius)

carrying charge of +11 units and it is surrounded by ten electrons, thus giving a net positive charge of

+1 unit for the ion. Application of Bohr‘s atomic theory, these electrons are in orbits of radius ~ 10─8

cm., e.g., the radius of the 1s electron’s orbit in a hydrogen atom is ~ 0.53X10─8

cm (Bohr rad ius).

Quantum mechanical or wave mechanical treatment would say that each electron in the atom is described by a wave function which is found by finding the solution of Schrodinger equation, and that

a simple way of visualizing is to consider that the electrons exist as clouds of negative charge around

the nucleus. Now, applying this logic we may suppose that if two atoms are brought closer enough

than these charge-clouds will get in the way of each other and will eventually result into a repulsion

between the two atoms. This phenomenon persists in the case of ions and there will be repu lsion

between the charge-clouds of the outer electrons of two ions at small distances apart; this is quite

independent of the electrostatic force between the ions due to their actual net charges. So, if the result

of the net charges on various ions of an ionic structure is to create force of attraction between the ions,

this force of repulsion between the charge-clouds is required to counter balance the attraction and

maintain the ions apart. The quantitative assessment of the two contributions to the forces between the

ions in an ionic crystal is dealt with further in the text.

The forces which bind the atoms together in ionic crystals arise out of electrostatic

attraction between positive and negative ions. In ionic crystals electrons are transferred from atoms of

one type to atoms of second type, so that the crystalline solid is made up of positive and negative ions.

The ions arrange themselves so that the coulomb attraction between ions of opposite sign is stronger

than the coulomb repulsion between ions of the same sign. The ionic bond is thus essentially the bond

resulting from the electrostatic interaction of oppositely charged ions.

Let us understand ionic bond more closely. An ion, being a charged atom, has a stable

electronic configuration like that of an inert gas. The neutral atom attains this configuration either by

accepting or donating an electron, as for example, Li+, Cl

─ and so on. The charge distribution of such

ion is spherically symmetrical, i.e., the interaction of the ion with other ions is independent of

direction. The ionic bond thus is due to coulomb attraction between two spherically symmetrical

charges of opposite sign .NaCl is a typical example of ionic crystal, in which an ionic bond is formed

by mutual attraction of two heteropolar ions Na+

and Cl─

( shown in figure 22.1) having the following

electronic configuration: Na (1s2, 2s

2, 2p

6, 3s)

Cl (1s2, 2s

2, 2p

6, 3s

2, 3p

5)

Na+

(1s2, 2s

2, 2p

6), like Neon configuration

Cl─

(1s2, 2s

2, 2p

6, 3s

2, 3p

6), like Argon configurat ion.

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Figure 22.1: Format ion of ionic bond in NaCl

Ionic crystals are characterized by strength, hardness etc., as mentioned in the table.

Generally, the ionic bond is non-directional, but there are cases where the ions deviate from the above

picture of spherical symmetry and the bond is rather directional in natu re.

It is well known that the salts produced by combining highly electropositive metals and

highly electronegative elements such as the halogens, oxygen and sulphur are the ideal ionic crystals.

Other, more complex salts, such as metal carbonates and nitrates and ammonium halides, also may be

classified as ionic crystals.

Most ideal ionic salts are diamagnetic, the exceptions being salts of transition metals,

which generally are paramagnetic and sometimes are ferromagnetic. The halides and oxides of the

simpler metals generally have an electrolytic conductivity which increases as the temperature rises. It

has been shown that the high-temperature conductivity σ satisfies the relation :

σ =A e─α/T

, where A and α are practically constant.

22.2.2 Covalent bon ds

The name covalent bond, also known as electron pair or homopolar bond, was first

introduced by Langmuir in 1919. This bond is formed between two atoms by sharing a pair of

electrons or pairing two unpaired electrons of two atoms thus bonded. An atom is relatively more

stable when its outer shell has full quota of electrons. In ionic bond, the participating atoms attain

stable configuration by transference of an electron from one to the other, whereas in covalent bond the

stability is attained by sharing of the electrons. Let us take the example of chlo rine molecule (Cl2).

The electronic configuration of neutral Cl is (1s2, 2s

2, 2p

6, 3s

2, 3p

5). Here, Cl atom having seven

electrons (3s2, 3p

5) in its outer shell will require one more electron to satisfy the stability requ irement.

This can be achieved when the two chlorine atoms share one pair of electrons between themselves, as

shown in a schematic diagram of figure 22.2. In this process they acquire the closed shell

configuration (3s2, 3p

6) of Argon.

153

Figure 22.2: pair of chlorine atoms forming a covalent bond

It is worth noting here that only two electrons having same orbital quantum numbers e.g.,

total quantum number (n), azimuthal quantum number ( l) and magnetic quantum number (m), but

opposite or antiparallel spin (s) can pair up. Here, chlorine atom has only one unpaired electron and it

can effectively pair up with the unpaired electron of another chlorine atom to form a single covalent

bond. The number of such bonds in an atom depends on the number of unpaired elect rons. The charge

distribution in this case is not spherically symmetrical. The covalent band has directional propert ies

and may be visualized as the overlapping of electron cloud of two atoms, as is shown in figure

22.2.According to the suggestion of Pauling and Slater the strength of the bond is greatest when there

is maximum overlapping. The covalent crystals have characteristic properties as is given in the tab le

22.1. It should be noted that atoms which are close to the inert gas configuration tend to be ionic but

those far away from inert gases in the periodic table have a tendency to be covalent.

Ideal valence or covalent crystals are monoatomic non-conducting substances which have

high cohesive energies and high value of hardness. Diamond is the prototype of this class just in the

same way as the alkali halides are the prototype of ionic crystals. Diamond structure has the

characteristic feature of having four number of nearest neighbours of each atom which is equal to the

ordinary valence of carbon.

The covalent bond is the normal electron-pair bond of chemistry, encountered

particularly in organic chemistry. It is characterized by a high density of electrons between the ions

and also by marked directional properties. The carbon bond is a familiar example o f the d irect ional

properties of the covalent bond. Carbon atoms quite often prefer to join onto each other or to other

atoms by four bonds making tetrahedral angles with each other. It means that each carbon atom will

be at the centre of the tetrahedron formed by the nearest neighbour atoms. Diamond and methane are

typical examples of the tetrahedral covalent bond. The Diamond structure is loosely packed in a

geometrical sense. In this case the tetrahedral bond permits only four nearest neighbours, whereas a

close-packed structure would require twelve nearest neighbour atoms. The covalent bond is believed

to be usually formed from two electrons, one from each atom part icipating in the bond. The spins of

the two electrons in the bond are antiparallel. The carbon atom (2s2, 2p

2) has a tendency, in a way, to

fill up the 2p6

electron shell by sharing electrons with four neighbours.

There appears to be a continuous range of crystals between the ionic and the covalent

limits. It is both interesting and important to estimate the extent to which a given bond is ionic or

covalent. It appears to be difficult to do so with certainty. One may take NaF as an ionic crystal and

think possibility of InSb as largely covalent, but it is difficult to talk about the nature of the bonding

of Zns or Pbs. However, Pauling’s semi-empirical formula suggests that atoms with nearly filled

shells (Na ,Cl ) tend to be ionic , whereas atoms not close in the periodic table to the inert gases tend

to be covalent, as for example, C , Ge , Si , Te.

Let us try to understand formation of covalent bond in terms of wave mechan ics.

This bond is formed when two atoms each contribute one electron to form an electron pair which is

then shared by these two atoms and in some way this electron pair manages to form a bond which

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holds the two atoms together. According to Bohr theory of the atom, or the old quantum theory,

electrons within an atom are arranged into shells and the atoms tend to acquire complete shells of

electrons, if possible. We have already described the case of Na and Cl as shown in figure 22.1. By

losing its outer electron the Na atom acquires a complete outer shell of electrons by donating this

electron to the Cl atom which then acquires a complete outer shell of electrons. This way both Na and

Cl are satisfied and acquire complete outer shells, thereby becoming ions. Covalent bonds result in

case of atoms which would have to lose a rather large number of electrons if they were to acquire a

complete shell of electrons. Carbon atom is one such atom which finds it more convenient to share its

four outer electrons with four electrons from other atoms than to form a carbon ion. So, it forms four

covalent bonds with other atoms. The Bohr theory is silent about how this sharing of electrons takes

place. It is, therefore, necessary to know if more modern version of quantum theory or wave

mechanics can provide satisfactory exp lanation of this.

According to wave mechanics, an electron in an atom has associated with it a wave

function Ψ which depends on the position of the electron and on time. This wave function Ψ is

determined by solving Schrodinger’s equation or wave equation , for the atom in which the electron

happens to find itself .In fact, any particle be it electron , proton, neutron , meson or any other, in a

particular situation will have associated with it a wave function. This function does describe the

observable properties of the particle which in this case applies to electron, such as its position, energy,

momentum and so on. Let us talk about the position of the electron. Let electron’s position

coordinates be (x, y, z) then Ψ is a function of x, y and z. So we write it as Ψ (x, y, z) which may be

real or complex. │ Ψ (x, y, z) │2dxdydz is the probability that the electron in question would be found

in a small box of sides of length dx, dy and dz. It is possible to calculate the probability as a

percentage that a particle will be in a part icular place at a particular time. However, it is not always

possible to get exact informat ion about the behaviour of the members of such a small system as an

atom. Using the wave function one is able to say that an electron spends a certain fraction of its time

in one region and some other fraction of its time somewhere else. If one does large number of

determinations of the position of the electron, one can at best say that the fractional number of times

that one could find the electron in a particular position will become closer and closer to │ Ψ (x,y,z)│2

as the total number o f t imes that one performs the experiment is increased. One interprets as

imagin ing the electron as being like a cloud of electric charge , the density of the cloud at the point

(x,y,z) being determined by │ Ψ (x,y,z)│2.A lso, the shape of the electron cloud is different for

different electrons within the atom. For example, fo r an s-electron the charge cloud is spherical in

shape, whereas for a p-electron its shape is totally different as shown in figures 22.3 (a,b).

a b

Figure 22.3(a, b): shapes of electron clouds for s- and p- electrons

The p-type charge clouds are also sometimes named as p-type atomic orbitals, have a

tendency to form highly directed bonds between two atoms. It could be that two electrons may be in

each of these atomic orb itals provided these two electrons are with opposite spins, but the Pauli

Exclusion Princip le does not permit higher population of the orbitals. How it is then that we have six

p-electrons in an atom, say like Ne? It becomes possible on account of the fact that there are three

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distinct p-type orbitals in an atom, one of them d irected along the x-axis as shown in figure 22.3 (b).

There is also a possibility to have p-type orbitals directed along the other remaining axes i.e., the y

and z axes. In this way, there are in all three p-type orbitals which together have the capacity to

accommodate a total of six electrons .Now, if it be considered that two atoms are placed near together,

there exists a possibility for the electrons’ charge clouds to rearrange themselves so that at least one of

these clouds embraces both the atoms and as a result, usually taking one electron from each of the two

atoms, develops a bond that holds the two atoms together in a molecule. We may extend this

argument by considering , in a similar way , that when a large number of atoms are placed together to

form a solid , these charge clouds may arrange themselves in order that any two neighbouring atoms

are held together by a charge cloud which goes around them both. In this process the whole solid is

held together; the bonds being covalent bonds and the substance is named as covalent solid.

Now let us take up what binds the ionic crystals to be held together .What are those

forces which hold the constituents of an ionic crystal together ? In case of ionic crystals, it is said that

starting from neutral atoms some of these atoms loose one or more of their electrons making them

positively charged while the other atoms got these electrons and in the process become negatively

charged. So, the crystal is then held together mainly by electrostatic forces. On the other hand, in the

covalent solid neighbouring atoms share some of their electrons with each other and they all remain

electrically neutral. In practice, real crystals are seldom purely ionic with no sharing of electrons and

with complete transfer of electrons from atoms of one kind to the atoms of the other kind. They are

also not purely covalent with complete sharing of electrons between atoms and no electrical charge at

all on any of the ions. If the crystal is perfect ionic, it has no directional bonds at all in it, and the

perfect covalent crystal has no electrical charge on any of its atoms. However, in practice there is no

crystal which is perfectly of either type. Even a crystal like NaCl which is generally known as an ionic

crystal can be shown to have its ions partially distorted and tend to form d irected bonds between two

adjacent ions. In fact, there are several structures which are covalent but do have some s mall posit ive

and negative charges developing on the atoms composing the structure.

Diamond is a very popular covalent bonded crystal. We do meet some examples of a

mixtu re of ionic and covalent bonding. Some of the atoms are bound together by covalent bonds and

form a large ion, and these ions are held together to build up an ionic crystal. We may take the

example of CaCO3. The C and O atoms are held together largely by covalent bonds to make up the

carbonate ion CO32─

and then these carbonate ions and the Ca2+

ions are held together by electrostatic

forces to build up an ionic crystal.

22.2.3 Metallic Bond

Metals are characterized by such distinctive properties such as high electrical and thermal

conductivities, malleability, ductility, opaqueness and so on. The bond which binds the atoms in

metals is typical of this group and is known as metallic bond. What holds a metal together? Since

metals are characterized by high conductivity, both electrical and thermal, it is just usual as being due

to free or almost free electrons within the metal, i.e., a metal consists of positive ions embedded in a

sea of free , or almost free electrons. The metal is thought of as being held together by the electrostatic

interactions between the ions and these electrons. The electrons available to participate in the

conductivity are called conduction electrons. Also part of the bonding is due to what are called

“exchange interactions” among the conduction electrons in a metal. The term of “exchange

interactions” appears in quantum mechanics which treats electrons as being identical particles.

Quantum mechanical t reatment gives an extra contribution to the energy of the electrons in the metal

and this energy is called as “exchange energy”. In fact, it is exchange interactions which lead to the

exp lanation of ferromagnetis m of iron or steel; the exchange interaction between the electrons of

neighbouring atoms is the cause behind their preference to get lined up parallel to each other. To

exp lain the forces which hold a metal together is rather difficult and involves many quantum–

mechanical concepts. Detailed discussion of these concepts is beyond the scope of this section which

may be found in reference like “The Wave Mechanics of Electrons in Metals” by S.Raimes (North -

Holland; Amsterdam).

In some metals such as the alkali metals the interaction of the ion cores with the

conduction electrons is believed to be largely responsible for the binding energy. One may think of an

alkali metal crystal as an array of positive ions embedded in a more-or-less uniform sea of negative

charge. In some metals such as the transition metals, it is suggested that there may also be bind ing

156

effects from covalent type bonds among the inner electron shells. Transition group elements have

incomplete d-electron shells and are characterized by h igh binding energy. The binding energy of an alkali metal crystal is much less as compared to that of an

alkali halide crystal; as such the bond formed by a quasi-free conduction electron is rather weak. It is

partly exp lained to be due to large interatomic distances in alkali metals on account of the fact that the

kinetic energy of the conduction electrons favours large interatomic d istances which leads to weak

binding. In the transition metals such as iron and tungsten the inner electronic shells contribute

substantially to the binding; the binding energy of tungsten, as for example, being 210 Kcal/mole.

As said above, it is rather difficult to give unquestionable explanation of bonding in

metals based on classical concepts. The answer to the problem lies in considering quantum

mechanical concepts. Metallic bond may be exp lained by ‘Free Electron Theory’. Here, the metal

atoms let their valence electrons get loose and move throughout the volume of the crystal in a manner

the gas molecules move in gas within a container. So, the term electron gas or electron cloud is used

for the free electrons in a metal .It is the attraction between the positively charged metal ions and the

negatively charged electron gas that binds the atoms together to form the metal crystal. This picture

which is more of ionic nature does explain satisfactorily several of the properties of metals, but it fails

to account for the observed specific heat of metals; the theoretically predicted value being 50% h igher

than the observed value.

A relatively more recent picture is based on quantum mechanics. Metals are electron

deficient substances i.e., the number of valence electrons is less than the number of orbitals. Modern

picture of metallic bond is to consider it to be more of a covalent bond than of an ionic type and is

referred to as electron deficient covalent bond. It may be made clear by taking an example o f sodium

(1s2, 2s

2, 2p

6, 3s). It has one unpaired electron (3s) in the outer shell. Th is unpaired electron pairs up

with that of another Na atom. As the third unpaired electron of another Na atom approaches this pair it

gets repelled because the Pauli Exclusion Principle does not permit an s–state to accommodate more

than two electrons. The energy of 3s state being very close to 3p, this electron goes to 3p state. The

central sodium atom with only one unpaired electron may thus form covalent bond with a number of

surrounding sodium atoms. In pract ice, therefore, the sodium atom makes a fraction of electron pair

bond with each surrounding atom. So, as such, we may treat this as an electron deficient bond.

Considering metal an array of positive ions and a cloud of free electrons forming

some kind of glueing effect of holding the metal together, Drude somewhere in the year 1900 t reated

these electrons as a classical gas. It was later modified by Sommerfeld who treated these electrons as

quantum mechanical part icles.

22.2.4 Molecu lar Bond

Molecular crystals are the solids formed by inactive atoms such as the rare gases and

saturated molecules like hydrogen and methane. They are characterized by low boiling and melt ing

points and they generally evaporate in the fo rm of stable molecules.

We have discussed ionic and covalent crystals. It was also explained that how some

crystals may be bound together by a mixture of these two types of bond. A molecular crystal is one

more example of a crystal in which there are several different types of bond present. In ionic crystals

there is nothing that exists as individual molecule whereas a molecular crystal is a collection of

individual molecules; the atoms that compose the molecule are held together by ordinary chemical

bonds, ionic bonds or covalent bonds. Usually, molecular crystals are formed by organic chemical

compounds. Obviously, the question arises as to what holds these molecules together to form

crystalline solids. Mostly it is Van der Walls’ forces and hydrogen bonds, but some small amount of

ionic or covalent bonding may be existing. It means that if various parts of the molecule develop

small positive or negative charges, electrostatic forces between the neighbouring molecules in a

molecular solid will exist which may contribute to holding the molecules together.

Inert gas atoms and saturated molecules are bound together in a solid phase by weak

electrostatic forces known as Van der Walls’ forces. Van der Walls’ forces are like ion ic forces,

electrical in origin. However, their origin is different. Ordinary ionic bonds arise because of the forces

between single charges, the van der walls’ forces arise as a result of the interactions of electric

dipoles. If the dipoles formed are weak and temporary, the resultant bond is called a Van der Walls’

bond .This type of weak force which bonds atoms was first discovered in gas atoms by Van der Walls.

The Van der Walls’ bond is responsible for binding atoms in crystals of inert gases such as Argon,

Krypton, Neon and Xenon which crystallize at very low temperatures. An electric d ipole , which is a

157

combination of two equal and opposite charges located at some fixed distance , say d , apart does not

carry any net electric charge , but the electrostatic force of interaction between it and another electric

dipole or even just an ordinary single charge does not disappear. One can judge this situation from

figure 22.4, and the actual force, in this case of repulsion, between the dipoles can be calculated.

Suppose now we reverse the direction of one of the dipoles in figure 22.4 then what happens? In that

case the net force between the two dipoles shall be of attractive type instead of repulsive type. May be

a molecule has a permanent dipole moment, but there is still a possibility of having van der Walls’

forces between molecules without a permanent dipole moment. These forces arise in the following

way: Even in an atom or molecule which has on the average an electric d ipole moment of zero, there

will be a fluctuating dipole moment associated with the instantaneous position of the electrons in the

atom .The instantaneous electric field associated with the moment will induce a dipole moment in

neighbouring atoms. The average interaction of the original moment and the induced moment result

into an attractive force between the atoms . Forces of this origin are also called dispersion forces.

Many organic solids are held together by Van der Walls’ fo rces.

Figure 22.4: An electric d ipo le

An atom or molecule which has no permanent dipole moment will still have

fluctuating dipole moment. What is the cause of these fluctuating dipole moments? Actually, the

electrons within the atom or molecule are in constant motion so that at any given instant the centre of

their charge distribution will not necessarily coincide with the positive charge of the nucleus ; there is,

therefore, a dipole moment which will be changing its direction and magnitude continuously. Even if

the average value of the dipole moment may be zero on account of the fluctuations, it can be shown

that the average value of force between two neighbouring molecules with these fluctuating dipole

moments does not come out, in general, to be zero. Further, it can also be shown that it is a force of

attraction proportional to 1/r6

as is required of Van der Walls’ fo rces.

Molecular crystals are characterized by weak b inding, with low melt ing and

boiling points. The crystal structures are often those with dense packing. The inert gas crystals

crystallize with cubic close packing.

22.2.5 Hydrogen Bond .

As neutral hydrogen has only one electron, it should form a covalent bond with only one

other atom. It is known, however, that under certain conditions an atom of hydrogen is attracted by

rather stronger forces to two atoms, thus forming what is called a hydrogen-bond between them, with

bond energy of about 5 kcal/mole.

Hydrogen bond may be visualized as a d irectional ionic bond. Though

hydrogen atom has one electron, it may under favourable conditions be attracted to other atoms and

the bond thus formed is known as hydrogen bond. Hydrogen atom with only one unpaired electron

cannot form more than one pure covalent bond. Obviously, the question arises as to how hydrogen

atom can form bonds between two other atoms. A simplified mechanism of this is that the hydrogen

atom shares its only valence electron with another atom and the bare proton left is attracted to the

158

outer unshared electrons of the other atom. The formation of hydrogen bond thus appears to be due

largely to ionic fo rces.

Now, let us take the example of water molecules. A molecu le of water (H2O)

may be visualized as being formed between an O and two H atoms by covalent bond and the

association of water molecules by hydrogen bonds as illustrated in figures 22.5 and 22.6.

Figure 22.5 and 22.6: For mat ion of hydrogen bonds

The hydrogen bond plays a very important role both in solids and liquids because of the

small energy involved in its formation and rupture. The stability of protein chains by hydrogen bonds

is a typical example.

The hydrogen bond is an important interaction between H2O molecules and is responsible,

together with the electrostatic attraction of electric dipole moments, for the striking physical

properties of water and ice. The hydrogen bond restrains protein molecules to their normal

geometrical arrangements. It is also responsible for the polymerization of hydrogen fluoride and

formic acid. It is important in certain ferroelectric crystals, such as potassium d ihydrogen phosphate.

The hydrogen bond plays a major ro le in the bonding of atoms in protein molecules, as it holds these

molecules to their normal geometrical arrangements. The hydrogen bond also plays an important ro le

in atomic bondings in the molecules responsible for life’s processes, as for example, it controls the

pairing between the two strands of a DNA molecu le.

22.3 Latt ice Energy of Ionic Crystals

Crystals grow in a number of ways. Several materials can be grown as crystals in the

laboratory by slowly cooling the molten material below the melt ing point. In these cases, the

crystalline solid fo rms when intermolecu lar forces of attraction overcome thermal agitation and a

condensed phase results. Heat is given off in the process, and this heat is directly related to what is

known as cohesive energy of the solid. The cohesive energy is a negative quantity as usually defined ,

and it arises largely from the fo rces which hold the atoms of the solid together.

One of the important and fundamental problems in the theory of crystallography of

solids is the calculation of the binding energy of a crystal. This cannot be done unless we have

knowledge of the forces between the composing particles. The simplest group of crystals to deal with

in this respect are the ionic crystals, for which calculations of the cohesive energy were made in 1910

by Born and Madelung.

Since the atoms composing a crystalline solid are arranged in a regular periodic lattice,

the energy responsible for b inding them together is known as “Lattice Energy” or, “Crystal Energy”.

Lattice energy plays a pre-eminent role in crystallography, solid state physics and materials science in

general. Because of their simple nature and of the applicability of relatively simple mathemat ical

treatment, the lattice energy of ionic crystals has been the subject matter of much theoretical

calculation by many workers e .g., Madelung, Born, Ewald, Evjen etc.

22.3.1 Assumptions

In the calculat ion of lattice energy, we shall have to make certain assumptions which are

given as follows:

i) Ionic crystals are made up of positive and negative ions, the charge distribution of which is

159

spherically symmetrical. In other words, the force between two such ions depends only on their

distance apart and is independent of direct ion.

This is a good example of the simplification of a problem resulting from considering certain

groups of elementary particles as units, the calculation being carried out for these units rather than for

the elementary particles themselves. For example, in sodium chloride it is assumed that these units are

the Na+

ion, with an electron configuration 1s2, 2s

2,2p

6and the Cl

─ ion , with an electron ic

configuration 1s2

, 2s2

,2p6

,3s2, 3p

6 .In the theory one works with these ions as charged particles,

forgetting to a large extent about their internal constitution. The influence of the latter is, however,

introduced in the form of refinements later.

ii) All these ions are located at proper lattice sites, as it should be in a perfect crystal. It may, however,

be noted that perfect crystals do not exist, and even , if a crystal is ‘perfectly grown’ and chemi-

pure there are always a ( relatively s mall ) number of lattice defects present.

22.3.2 In teract ing Forces

The interacting forces which might contribute to the lattice energy of crystals are:

i) Attractive force due to coulomb interaction between positive and negative ions.

ii) Attraction due to polarizat ion of individual ions in the field of other ions.

iii) Short range repulsive force due to overlapping of the electron density of the ions. This force

comes into play when the ions come closer than the closest distance for coulomb attract ion.

The main interaction, according to the theory, is the ordinary electrostatic or

coulomb force between the ions (as mentioned in the first one given above), which accounts for

the large cohesive energies of the crystals. The electrostatic forces, which tend to contract the

dimensions of the crystal, are balanced by repulsive forces which, from the classical viewpoint,

have uncertain origin and which vary much more rapid ly with intrinsic distance than do the

coulomb forces between charges.

iv) Van der Walls’ interactions between the ions.

In actual calculation, however, contributions from forces given under (i) and (ii) are

taken into account whereas those under (iii) and (iv), being very small in comparison to the former,

are neglected. Let us see how far the calculated result, based on the above mentioned assumptions

and contributions, agree with the experimental results in the ionic crystal NaCl.

Sodium chloride crystallizes in the structure as already described earlier in the text. The

space lattice is face centred cubic and Na+

and Cl─

ions arranged alternately at the lattice points.

We construct the sodium ch loride crystal structure by arranging alternately Na+

and Cl─

ions at the lattice points of a simple cubic crystal. In the crystal each ion is surrounded by six nearest

neighbours of the opposite charge and twelve next nearest neighbours of the same charge as the

reference ion. We suppose that the Na+

ion carries a single positive charge so that the electronic

configuration is identical with neon, and the Cl─

ion carries a single negative charge (argon

configuration ).

Taking any Na+

ion inside the crystal as the reference point, it will have:

(i) Six (6) ions of opposite sign (Cl─) as the nearest neighbour at a distance say, r.

(ii) Twelve (12) ions of the same sign (Na+) as the second nearest neighbour at a distance √2.r,

(iii) Eight (8) ions of opposite sign again ( Cl─

) at a d istance √3.r,

(iv) Twenty four (24) ions of opposite sign (Cl─

) ions at a d istance √5.r, and so on.

This ion will attract or repel one another, as the case may be, by Coulomb interact ion.

The electrostatic energy of the Na+

ions with each of the surrounding ions can be written by following

the method developed for ion pair.

Sodium and chlorine ions have obtained the configurations of the nearest inert gas; sodium ion having

the configuration of neon and the chlorine ion that of argon. The bonding is as a result of a coulomb-

type attraction between the positive and negative ions. Taking a general case, the force of attraction

FA between the ions is given by Coulomb ’s law:

FA = ─ (+ Z1e) (─ Z2e)

r2

where Z1 and Z2 are the ionic charges, e is the electronic charge and r is the distance between the ions.

Therefore, the electrostatic energy of the Na+

ions with each of the surrounding ions may be g iven as:

Eattraction= Єe.( 6e2/√1r) (+ Z1) (─Z2) + (12e

2/√2.r)(+Z1)

2 +( 8e

2/√3.r) (+Z1)(─ Z2) +( 6e

2/2r)(+Z1)

2

+ ( 24e2/√5r)(+Z1)(─Z2) + ………………………

160

In the present example of NaCl, the two ions carry equal but opposite charges and hence

+ ( Z1 ) = ─ ( ─ Z2 ) , Or, Z1 = Z2 = │ Z │

Eattraction = Єe= ─ e

2 │Z│

2 ( 6/√1 ─ 12/√2 + 8/√3 ─6/√4 + 24/√5 ± …...) ……………(22.1)

r

The series within the bracket depends purely on the type and structure of the crystal. The

term within brackets is called the Madelung constant, and represents the geometrical arrangement of

ions. It can be seen that the convergence of this series is poor. It was first calculated by Madelung and

so is called Madelung constant A. e is the charge per ion and r is the shortest interionic distance. It

may be noted that because Coulomb fo rces decrease relatively slowly with distance, it is not sufficient

to consider only a few shells of ions around the central ion.

Evidently, the coefficient e2/r is a pure number determined only by the crystal

structure. Series of this type have been calculated by Madelung ,Ewald and Ev jen. For the NaCl the

result is:

Єe = ─ A e2/r , with A = 1.747568 …………………….( 22.2)

For other crystal structures composed of positive and negative ions of the same valency , the

Madelung constant are:

Cesium ch loride ( CsCl ): A = 1.762670

Zincblende( ZnS ) : A = 1.6381

Wurtzite (ZnS ) : A = 1.641

It may be noted that e in equation no.22.2 represents in general the electronic charge times the valence

of the ions under consideration. The minus sign in equation number 22.2 indicates that the average

influence of all other ions on the one under consideration is of attractive nature.

To prevent the lattice from collapsing, there must also be repulsive forces between the

ions. These repulsive forces become noticeable when the electron shells of neighbouring ions begin to

overlap, and they increase strongly in this region with decreasing values of r. These forces, as other

overlap forces, can best be described on the basis of wave mechanics, on account of the fact they are

of non-classical nature. Born in his early work assumed that the repulsive energy between two ions as

function of their separation could be expressed by a power law of the type B/rn. It may be expressed

as:

Єrep. = B/rn

…………………(22.3)

It suggests that the repulsive force has been assumed to be inversely proportional to some power of

the distance of separation between ions r. B represents a constant and n is a number known as Born

exponent. Both B and n are characteristic of the ions in the solid under consideration. Equation at

number (22.3) is an expression for the repulsive energy of one particular ion due to presence of all

other ions.

The repulsive term in the case of NaCl is written as:

Єrep. = B/rn

,

In view of the fact that repulsive forces depend so strongly on the distance between the particles, the

repulsive energy (Єrep.) is mainly determined by the nearest neighbours of the central ion.

Thus, the total energy (potential energy) of one ion due to the presence of all others is

then obtained by adding equations at (22.2) and (22.3),

Є = ─ A e2/r + B/ r

n ………....…………(22.4)

Assuming that the two types of forces just discussed above are the only ones involved in the subject

matter under discussion, we have to take into account and neglecting the surface effects, and proceed

161

2

further in the determination of total binding energy of a crystal containing N positive and N negative

ions:

E(r) = N (─ A e2/r + B/r

n)

= NЄ (r) …………………………… (22.5)

It may be noted that we have multip lied by N rather than 2N because otherwise the energy between

each pair of ions in the crystal would have been counted twice. The two contributions to the total

binding energy E(r) are represented schematically in figure 22.7. The figure 22.7 is a schemat ic

representation of the energy of attraction (I) and of repulsion (II) as a function of the lattice parameter.

The resultant (III) exh ibits a min imumfor a lattice constant r0 , corresponding to equilibrium.

Figure 22.7: Dependence of potential energy of ion ic crystal on the value of lattice constant

Considering the crystal at absolute zero the equilibrium conditions require that the potential

energy E should be minimum which will be the case for the equilibrium value r=r0 ( equilibrium

distance), where r0 represents the smallest interionic distance in the crystal at T=0. The equilibrium

distance r0 can be obtained by differentiating equation no. 22.5 with respect to r and equating it to zero

( the attractive force being balanced by the repulsive force at the distance).

i.e. (dE/dr) r=r0 = 0

Or, (dE/dr) r=r0 = Ae2/r0 ─ nB/r0

n+1 = 0 ………… (22.6)

We get from equation no. 22.6,

r0= (nB/Ae2

)1/(n─1)

…………………..(22.7)

= ( Ae

2/n ) r0

n ─ 1………....…………(22.8)

Substituting it into equation no. 22.5 y ields for the lattice energy EL,

EL = E ( r0 ) = N { ─ Ae

2/r0 + (A e

2/n)r0

n─1. 1/r0

n },

= ─ N Ae

2/r0( 1─ 1/n )

= N ЄL …………………………...(22.9)

Where , ЄL = Є ( r0 )

The interionic distance can be obtained from x-ray diffraction data; the charge per ion is also known,

and thus the lattice energy can be calculated if the repulsive exponent n is known. The question now

remains how to determine n. The discussion on that will follow now.

162

22.4 Calcu lat ion of Repulsive Exponent.

Repulsive exponent can be calculated from compressibility data. Born obtained the

unknown repulsive exponent ‘n’ from measurements of the compressibility of the crystal as fo llows:

The compressibility ‘K’ is defined as:

K = (─ 1/V )dV/dp…………………22.10

At very low temperatures we may neglect thermal effects and using first law of thermodynamics ,

express the equation as :

dE = ─ p dV

Or, ─ dE/dV = p

─ d/dV ( dE/dV ) = dp/dV

Or, ─ d

2E/dV

2= dp/dV ……………22.11

From equation no. 22.10, we have 1/K = ─ V dp/dV

Substituting for dp/dV from equation no. 22.11, we have

1/K = V. d2E/dV

2

At absolute zero, we represent for K as K0 and V as V0,

1/K0 = V0 (d

2E/dV

2) V=V0

Or, 1/K0 V0 = (d

2E/dV

2) V=V0 …………………22.12

Where V0 is the volume of the crystal corresponding to an interionic d istance r0, V corresponds to the

variable r.

Now the relat ion between volume and interionic distance must of course be of the form:

V ∞ N r3,

Or, V = C N r3

.............................22.13

Where N is the total number of molecules

r is the nearest neighbour distance or interionic distance,

C is a constant and is characteristic of the type of latt ice.

For NaCl, as fo r example C = 2,

Now, dE/dV = (dE/d r).(d r/dV)……………22.14

From equation no.22.13 we have: r

3 = V/C.N,

Or, 3r

2dr = (1/CN) dV,

Or, dr/dV = 1 .................................22.15

3r2

CN

Substituting it in equation no. 22.14, we may write:

dE/dV = 1 . dE

3r2CN dr

& d

2 E /dV

2 = d/dV( 1/3r

2 CN. dE/dr)

= 1/3CN. d/dV ( 1/r

2. dE/dr)

163

d2E/dV

2 =

1 d (1/ r2.dE/dr)……………….(22.16)

9C2N

2r2

dr

0

0

0

0

0

r

0 0

0

0

= 1/3CN dr/dV{ d/dr(1/r2. dE/d r)}

Substituting for dr/dV from equation no. 22.15, we have:

d2E /dV

2 = (1/3CN). (I/3r

2CN){d/dr(1/r

2dE/dr)},

Or,

From equation no. 22.5 , we have:

dE/dr = N (+ A e

2/r

2 ─ nB/r

n+1)

Substituting for dE/dr in equation no. 22.16 we have:

d2

E/dV2

= 1 d { 1/r2( NAe

2/r

2 ─ NnB/r

n+1) }

9C2

N2

r2

dr

= 1 d ( NAe2/r

4 ─ NnB/r

n+3)

9 C2

N2

r2

dr

= 1 {─ 4NAe

2 /r

5+ Nn(n+3)B/r

n+4 }

9 C2

N2

r2

Now at r = r0 and using equation no. 22.12, we have:

1/K0 V0 = (d

2 E/d V

2) r0

= 1 {─ 4Ae2/r

5 +

9C2

N r 2

n (n+3) B/ r0 n+4

}

Using equation no. 22.13 for V=V0 and making substitution for B from equation no. 22.8,

1 = 1 ─ 4Ae2/r

5 + .n ( n+3)Ae

2 r0

n─1

K0CNr03

9C2Nr0

2 n (r0

n+4)

Or, 1 = Ae2

─ 4 + n+3

Or,

K0 r0 9C r 5 5

= Ae2

(n─1)

9Cr 5

1 + Ae2

= n. Ae2

K0r0 9Cr 5

9Cr 5

Or, n = 9Cr04

+ 1 ………………….22.17

K0Ae2

For NaCl, C = 2 .

So, n = 18 r 4

K0Ae2

+ 1 ………………….22.18

From equation no. 22.17 the parameter n can be calculated if K0 is known. For example, the

compressibility of sodium Chloride as measured by Slater is 9.4. There are some small variations in

the value of n as measured by different workers. However, in general, values of n computed from

equation no. 22.17 are as low as n = 6.0 for LiF, n = 10 for NaCl, n = 10.5 for CsCl and as high as

164

Material science

Crystallography & crystal growth

Classification of Crystalline Solids

n = 11.0 fo r RbI.

An improvement in the theory of cohesive energy is made by introducing Van der walls’ (dipole -

dipole) interactions as well as vibrational energy. An approximate value of zero-point lattice

vibrational energy per ion pair can be obtained by making a Debye approximation and using a

characteristic cut off frequency νm. The relative importance of the Coulomb, repulsive, Van der Walls’

and zero point terms for three materials of NaCl structure are given in table 22.1.

TABLE: 22.1

Various contributions to the binding (cohesive ) energy of some ionic

crystals*

Type of force Expression NaCl RbI AgBr

Coulomb -A e2/r -8.85 eV -6.79eV -8.65 eV

Repulsive B/2rn

1.02 0.67 1.37

Van der Walls’ C/r

6

-0.125

-0.175

-1.18

Zero point (9/4)hνm 0.075 0.0312 0.0375

Total

-7.90

- 6.26 -8.4

*Values based on N.F.Mott & R.W. Gurney , Electronic Processes in

Ionic Crystals ( Oxford Univ. Press, New York 1948 & also M.Born and

M.Goppert- Mayer, Handbuch der Physik, 24/2, 726 (1933); Z. Physik

75, 1 (1932)

22.5 Theoretical & Experimental Latt ice Energies

The lattice energy EL may be calculated from equation no. 22.9 on substitution by the

proper values for the charge of the ions, the interatomic distance and Born exponent ‘n’.

One can see from the expression that the lattice energy is smaller than coulomb energy by 1/n.

The repulsive potential exponent or the Born exponent n can be calculated from the

compressibility data of crystals. Its value has been found to be in the neighbourhood of 10 in

most of the ionic crystals. The lattice energy of NaCl calcu lated from expression no. 22.9 has

been found to be 8 eV whereas the experimental value is 7.9 eV. It is a reasonably good

agreement between theoretically calculated and the experimentally measured values of lattice

energy (cohesive energy) of ionic solids. The agreement could be more convincing when

contribution due to other forces of interaction, e.g., Van der Walls’ forces are taken into

account in addit ion to those given in equation no. 22.4.

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Material science

Crystallography & crystal growth

Classification of Crystalline Solids

SUMMARY

• Binding forces that hold a crystalline solid together are described.

• Classification of solid according to the type of bonding in them is discussed.

• Different types of bonding viz., Ionic, covalent, metallic, molecular and hydrogen

bonds in crystals are discussed, providing an understanding of how crystalline solids are held together.

• Interacting forces that contribute to the lattice energy of crystals are described.

• Expression leading to determination of lattice energy of ionic crystals is derived.

• Calculation of repulsive exponent from compressibility data is explained.

• The validity of expressions obtained on lattice energy is shown to get support by agreement of theoretical and experimental values of lattice energy of a crystal.