10 Lattice Vibrations and Thermal Properties 1

Mahavir Singh

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Learning Outcomes

 

The objective of the module is to

  • Get an understanding of interatomic interactions that exist in crystals.
  • Relate the idea of potential energies due to repulsive and attractive forces that exist between like and unlike constituent particles.
  • Understand the stability of the crystalline structure and its dependence on the cohesive energy of the crystals.
  • Dependence of crystal properties on cohesive energies of the crystal.

    1.Introduction:

 

There exists forces of interaction between the constituents (atoms, ions or molecules) when they are at an appreciable proximity to each other. This causes the matter to exist in condensed state which is one of the three states of occurrence of matter in nature. The atoms in solids and liquids are at a distance of a few Angstroms and the density is about 10 22 – 1023 atoms per cm3.

 

The crystalline solids, the atoms form a three dimensional pattern. When the atomic periodicity extends throughout the material then it is called a single crystal. In case the periodicity is ruptured at the grain boundaries, such materials are called polycrystalline solids and when the size of the crystallites become comparable to the pattern unit size, the materials possess amorphous structure.

 

So far with the existence of solids the following points cannot be ignored

  • There must exist attractive forces between atoms or molecules to hold them together.
  • There must exist repulsive force as large external forces are required to compress a solid.

   But the next query arises is to what causes the interaction to exist amongst the crystal constituents? The answer in a preliminary attempt is’ Cohesive Energy’ or the Binding Energy, which can be held responsible for the above said phenomenon. The cohesive energy of a metal is the amount of work required to break its individual atoms to ground state and the cohesive energy of molecular solids is the energy required to dissociate the solid into its constituent molecules well separated from one another.

 

2. Interatomic forces:-

 

The atoms in a crystal are held together by cohesive forces which are electrical in nature and the electrostatic force between electrons and nucleus is responsible for this binding force. When two atoms placed in close proximity to one another, the valence electrons interact and consequently a change in energies occur in such a way that the energy of the valence electrons in the composite system is lower than that in the isolated atoms thus energetically favoring bonding in crystals.

 

The approaching positive (nucleus) and negative (electrons) charges exert force of attraction or repulsion such that the total potential energy is

 

 

 

 

 

 

 

 

 

 

At equilibrium, the attractive and repulsive forces are equal but the corresponding energies are not as nm ,as is evident from eqn (2)

 

Imposing the condition to get the minimum in the energy curve the only condition satisfying is m > n, implies that, for the crystal bonding, the essential requirement is that the repulsive force component be of smaller order than that of attractive force component. Depending upon the charge distribution, bonding occurs when atoms lose, gain or share electrons, classifying the basic solids into two categories i.e,

 

1.  primary bonds which are generally inter atomic in nature and

2. secondary bonds which are mostly intermolecular bonds.

 

Ionic bonds, covalent bonds and metallic bonds are studied under the former category while molecular bonds and hydrogen bonds follow the later one. However the distinction between these is not sharp and many crystals belong to more than one class.

 

 

Our concern in this section will be inter- atomic bonds and a detailed study of nature of ionic and covalent bonds, however the other bonds are discussed briefly as under.

 

3 Secondary Bonds: an over view

 

3.1 Metallic Bonds:-

 

In metals there are more than one free electron per atom available which are assumed to compose electron cloud. The metal crystal may be considered as an array of positive charges embedded in a sea of electron gas. The electrostatic interaction among positive ions and negative electrons holds the metal together. Metallic bonding usually is favored in low electron valence cases.

 

Fig (2)   Metallic Bonding. Figure shows cloud of electrons and ions

 

Metallic compounds have crystalline structure and exhibit high electrical and thermal conductivity due to availability of free electrons. Metals are opaque and have lustrous appearance due to absorption and emission of electromagnetic radiations by the electrons.

 

Van der Waal’s Bonds:-

 

These are weal intermolecular bonds resulting due to dipole attractions. These dipoles may be permanent ( eg H2O or HCl) or momentarily produced due to non uniformity in the charge distribution. There is no sharing or transfer of electrons and hence these bonds are independent of valence electrons. The crystals so formed are known as molecular crystals. Inert gas atoms also have this type of bonding.

 

 

Fig(3)    Vander Waals’s Bonds

 

The binding energy of inert gas crystals is of the order of 0.1 ev per molecule and hence have low melting and boiling points. Van der Waal’s forces are held responsible for properties like friction, cohesion and adhesion, surface tension, viscosit , condensation of gases and freezing of liquids .The compounds exist in crystalline as well as amorphous states.

 

These crystals have very high resistivity, low refractive index and low dielectric constant. These are generally transparent as there are no electrons available to absorb light energy.

 

Hydrogen Bonds:-

 

Hydrogen bond is formed under certain conditions when an atom of hydrogen gets attracted by strong forces of two atoms. This bond is of the order of 0.1ev.

 

The bond is largely ionic in nature as it is formed between two most electronegative atoms( F,O,N atoms). Hydrogen atom loses its electron to other atom and the atom adjacent to bare proton combines with it and hydrogen bond is formed. As shown in figure (4).There exists a strong permanent dipole that links with other dipoles with a force nearly of the order of that in ionic bonds. Thus hydrogen bond connects both the atoms.

 

Fig(4) Hydrogen bond formed between electronegative atoms

 

4.Primary Bonds: A detailed Analysis

 

4.1 Ionic bonds:-

 

Ionic (electrovalent) bonding takes place when one or more electrons get transferred from one atom to the other. The ions so formed are held together by electrostatic forces of attraction which is greater than the forces of repulsion. Alternately, ionic bonding results when an element of relatively low ionization energy combines with an element of high electron affinity. NaCl,CaF2,MgO etc are a few examples of ionic crystals.

 

The Na atom has a tendency to lose one electron (Na+) and Cl atom has a tendency to gain one electron(Cl-) in order to attain stable configuration. The electrostatic field extends in all directions and the process of ionic bonding of Na+ and Cl- ions extends to a number of ions. A deeper analysis is required for better understanding of the concept which can be explained in terms of cohesive energy of ionic bonds discussed as under.

 

4.2 Cohesive Energy:-

 

To get an estimate of cohesive energy of an ionic crystal, we continue to consider the simple case of NaCl.

 

For interaction between i and j ions of the crystal, let the interaction energy be represented as Uij .

 

The interaction of I ions with all other ions can be expressed as

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+ or – sign are used for like and unlike charges respectively.

 

Also earlier in this section, in quation (1), the expression for repulsive term has been replaced by an exponential expression in order to get a better representation of repulsive interaction, as for r= ρ , repulsive interaction reduces to 1/e of its value at r = 0.

 

Total lattice energy of the crystal having 2N ions or N molecules thus becomes

 

Utotal = NUi ———(4)

 

Considering repulsive interaction among immediate neighbors only , distance rij can be modified as

 

rij =pijR, where pij is a dimensionless quantity representing the distance between the ions in terms of nearest neighbor distance R.

 

If there are z number of nearest neighbors to ith ion then

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The cohesive energies for crystals formed by inert gases are calculated in a similar manner taking into consideration Vander Waal or London interaction energies that vary as rij-6 and are short range interactions.

 

The repulsive interactions vary as rij-12 . The expressions for net interactions are called Lennard-Jones potentials which provide information of Utotal at equilibrium distance Ro.

 

The stability of the crystalline structure so formed depends upon the cohesive energy of the crystals and since strong electrostatic forces exist between ions, these have large cohesive energies, resulting in high melting and boiling points. They are usually brittle also since the slipping of atoms over one another (that accounts to ductility in metals) is prevented by the ordering of positive and negative ions due to the nature of the bonds formed.

 

So ionic solids are reported to have the following characteristics

  • Possess large binding energies e.g, for NaCl it is ~ 7.8eV
  • Due to large cohesive energies, these solids have high melting and boiling points
  • Ionic solids generally possess crystalline structure.
  • These are generally poor conductors of electricity at normal temperatures, however electrical conductivity increases with rise in temperature.
  • These are usually transparent to visible light but characteristic peaks are obtained in far infra red region.
  • Soluble in polar solvents but insoluble in non polar solvents.

    4.3 Covalent Bonds:-

 

Covalent bonding results due to the mutual sharing of one or more valence electrons (rather than by electron transfer as in ionic bonds) of similar (or dissimilar) atoms in order to attain a stable configuration. On single electron sharing the bond so formed is single covalent bond, while on sharing of two , three or more, the bonds are double , triple and so on. H2 and Cl2 have single covalent bonds and carbon atom has four covalent bonds.

 

To understand the dynamics of covalent bonding let us consider H2  molecule.

 

For two hydrogen atoms to combine, each has a single 1s electron to be accommodated in the lowest energy level of the molecule, must have anti parallel spins and there is no space for accommodating a third electron as Pauli’s exclusion principle will come in picture and the same will be sent to next level as shown in figure 5. There is strictly no sharing of three electrons.

 

 

The covalent bonds are directional i.e the electron pairs are concentrated along the line joining the two atoms.

 

In case of diamond atoms, the there are only two unpaired electrons in the ground state( 1s22s22p2 ). The excited state has four electrons available for sharing (1s22s12 ) called sp3 hybrid state.

 

 

Purely covalent crystals are relatively few in number , some examples other than diamond are, silicon, germanium and silicon carbide (SiC) . Covalent crystals usually have higher cohesive energies than ionic crystals making them hard and possess high melting points.

 

Some characteristics of covalent solids are listed as

  • Have comparable or even higher binding energies than ionic solids.
  • High melting and boiling points.
  • Generally all covalent solids are insulators because of non availability of loose electron, some are even perfect insulators. However conductivity of some crystals like Ge can be enhanced by doping.
  • Soluble in non polar solvents but insoluble in polar solvents
  • These are opaque to shorter wavelengths but transparent to longer ones.

As a matter of fact a continuous range of crystals is found to exist between the ionic and covalent limits, and it becomes important to estimate the extent over which a given bond may be considered ionic or covalent. A semiempirical theory developed by J.C Phillips on fractional ionic or covalent character provides a successful solution to the dilemma.

 

you can view video on Lattice vibration and Thermal properties 1

 

Summary:

 

After the completion of this module we are able to understand  the following

  • Various types of inter atomic forces exist in solids
  • The charge distribution in the outermost shells of the atoms plays an important role in deciding the nature of the bonding and hence the type of crystal.
  • The potential energies due to attraction and repulsion in a crystal decide its stability as compared to its constituent atoms or molecules.
  • The cohesive energy plays a vital role in deciding various physical and chemical properties in solids.

    Value Addition:

 

Do You Know?

 

The physical definition of solids has several ingredients. If restricting the study to solids having regular arrangement exhibiting translational symmetry, i.e perfectly crystalline solids, the atoms in such solids occupy equilibrium positions in an assumed lattice such that with the rise in temperature the crystal behavior starts showing deviations from normal behavior.

 

Metal crystals shear very easily as they are held together not by covalent bonds but by a cloud of free electrons. The energy of these electrons are affected due to change in volume , so these exibit good cleavage strength under the change of volume , giving them malleability and ductility.

 

Ionic crystals are hard because of strong attractive(coulomb) forces between anions and cation and on the application of a shearing forcethe ions tend to slip past over one another with quite ease as ionic bonding is non directional. At a stage when the strong repulsive forces originating due to violation of the Pauli’s principle , the crystal fractures.

 

For More Details ( on this topic and other topics discussed in Text Module) See

 

James D Patterson Bernard C Bailey, Solid State Physics, Introduction To Theory

Adrianus J Dekker,Solid State Physics

Charles Kittel, Introduction to Solid State Physics.

 

Solid State Physics

 

Lattice Vibrations and Thermal Properties

 

Glossary:

 

Bond energy :

 

The minimum requirement of energy for the formation of a particular chemical bond.

 

Non directional bonds:

 

Bonds that have no preferred bond direction due to the uniform spread of electrons throughout the crystal these donot exert directional influence, as in metallic bonds. but in covalent bonds, the electron pairing is concentrated along the line joining the two atoms hence are directional in nature.

 

Cohesive energy:

 

It is the amount of work required to be done to completely disperse a crystal into its constituent particles.

 

Ion core:

 

An atom without its valence electrons forms an ion core.

 

Zero point vibrations:

 

At absolute zero lattice vibrations do not freeze as suggested by classical models, instead there exist non zero amplitude vibrations called zero point vibrations.