This part of the course starts with a microscopic picture of solids (Lecture 9). This is to get a theoretical strength for materials.
Then it looks at how atoms bond (Lecture 10),
then how atoms stack together (Lecture 11), and
then how brittle and plastic materials fail (Lecture 12).

In this lecture the following are introduced:
• Negative Potential Energy
• Potential Energy and Force
• The Lenard-Jones Potential Energy Function
• Equilibrium Separation
• Maximum Binding Energy
• The Electron Volt

Negative Potential Energy

Solids are three-dimensional arrays of vibrating atoms stacked ~0.1 nm apart. They gain their strength from cohesive electromagnetic forces. Without these forces we would simply sink through the floor. To understand these forces we need to look at negative electrical potential energy.

To explain the concept of negative potential energy in general, consider the gravitational case shown in the diagram to the right. It shows three red balls in three gravitational potential energy situations. The zero energy situation is a flat horizontal plane where the red ball will sit still and not move under gravity. It can, of course, move freely if it is pushed to one side and given some kinetic energy.

On top of the hill, the red ball has a positive potential energy because gravity can make it roll down the potential gradient at increasing speed.

In the well, the red ball is confined by the walls around it. A negative potential energy means that something is confined within some region of space. The depth of the well indicates the strength of the bonding. The red ball can bounce backwards and forwards if it has some kinetic energy, but can't get out of the well until it is either lifted out with potential energy, or given enough kinetic energy to make it bounce out.

For atoms in a solid, the negative electrical potential of the atoms is much larger than their kinetic energy, so the atoms are held in place, vibrating about, at the bottom of a potential well. The vibrations do not break the bonds holding them together unless the temperature approaches the melting point. At this point the kinetic energy becomes comparable with the potential energy.

Potential Energy and Force

Go here for a review of Work and Energy

The Work done by a system's force produces an increase in kinetic energy.
Work done against a system's force produces an increase in potential energy.
Re-writing the potential energy change shows that the size of the force equals the steepness of the potential gradient, and from the negative sign the force acts down the slope.

The Lenard-Jones Potential Energy Function

In solids, there are attractive forces pulling the atoms together and also repulsive forces that prevent the atoms from getting too close. If the repulsive force were not present then solids would collapse in on themselves. (Black holes are examples of what happens in stars when the repulsive force is overcome by gravity). To describe the forces between atoms, we need an electric potential energy function that gives a potential well with both attractive and repulsive terms.

It will have to look something like this:

Where the slope is positive it gives a force in the negative direction (attraction).
Where the slope is negative it gives a force in the positive direction (repulsion).

This shape is typical of Lenard-Jones Potential Energy Functions which are written mathematically as:

The negative potential term (positive slope) dominates on the right-hand side of the graph where
A gives the strength of the attractive potential, and
n gives the steepness of the attractive slope.
The positive potential term (negative slope) dominates on the left-hand of the graph where
B gives the strength of the repulsive potential, and
m gives the steepness of the repulsive slope.

The force equation derived from the Lenard-Jones potential is:	It looks somewhat similar to the potential graph.
These were drawn using the data. J	N
Notice similarities and differences. The repulsive terms are steep (m~9): the attractive terms are shallow (m~1). The bottom of the potential well is the zero of force. The bottom of the force well is a little further out and gives the point where the bond breaks.
The distance out to the bottom of the potential well is the equilibrium separation, i.e. the separation to which the atoms will naturally move.	The depth of the potential well gives the maximum binding energy, i.e. the maximum energy (not force) needed to break the bond.

The atoms will have negative potential energy but also some positive kinetic energy from heat in the form of vibrations. Their kinetic energy will make them vibrate and ride higher in the well, so the actual binding energy will be smaller than the maximum.

Finding the Equilibrium Separation

The force equation from the Lenard-Jones potential energy function is given by:

At equilibrium separation, the attractive and repulsive forces balance.

The empirical constants A and B have a strong influence on the equilibrium separation.
If the repulsion is bigger, i.e. B/A becomes larger, then the equilibrium separation r_o will increase and the system will be more loosely bound.
If the repulsion is smaller, i.e. B/A becomes smaller, then the equilibrium separation r_o will decrease and the system will be more closely bound.

Using the given data:

The equilibrium separation is: Which is around ~0.1nm.

Finding the Maximum Binding Energy

The maximum binding energy is the minimum potential energy. It is found by substituting the equilibrium separation, r_o, into the interatomic potential, i.e.

From the equation for equilibrium separation, , so substituting this

Two features should be noted from this.
The Maximum Binding Energy depends:
(1) directly on A, not B, i.e. the size of the constant for the attractive potential.
(2) inversely on r_o the equilibrium separation. As this is the bond length, the shorter the bond, the greater the binding energy.

Using the given data:

The Maximum Binding Energy is: This is 1.26 aJ. It is very small but in combination with millions of billions of other atoms becomes large on the macroscopic scale.

The Electron Volt

The binding energy above is so small that it is measured in attoJoules = 10^-18 Joules. A unit that is often used for these tiny energies is called the electron volt.

One electron volt (eV) is the energy acquired (or lost) by an electron in crossing through 1V.
Now, Work (Joule) = Charge (Coulomb) × Potential Difference (Volt)

Substituting the values, 1eV = 1.6×10^-19 (electron charge) × 1 (potential difference) = 1.6×10^-19 Joule

1eV = 1.6 x 10^-19 J = 160 zJ

The binding energy of 1.26 aJ above, when measured in eV, is given by:

A binding energy of 1.26 aJ is 7.9 eV.
Note: since electric potential is usually defined in relation to positive charges, an electron falls up electric potentials and has to be pushed downwards. However in dealing with semiconductor energy diagrams, the opposite is often used. The moral is: read the fine print for the diagram.

Example S1
The potential energy function for the force between two particular ions, carrying charges +e and —e respectively, may be written as,

(i) Find the equilibrium separation distance for these ions.
(ii) Find the potential energy at equilibrium separation.

Answer S1

At equilibrium separation, ro the cohesive force between the ions drops to zero. The ratio B/A is the dominating factor.

Substituting this value of ro back into the potential energy function gives: The constant A should dominate over the 8th root.

Example S2
Magnesium Oxide (Mg²⁺O^2-) and Sodium Chloride (Na⁺Cl^-) have the same form of interatomic potential.

The only difference is that, z=2 for Magnesium Oxide and z=1 for Sodium Chloride. Find the ratio of their equilibrium separations.

Answer S1

Find the general equilibrium separation from V Equilibrium separation is inversely proportional to the 4th root of the ionic charge z.

Find the ratio for the different z's. The Magnesium Oxide atoms are 84% closer together than Sodium Chloride.

Summarising:

Negative Potential Energy indicates a bound state.
Force is the negative of the potential gradient
The Lenard-Jones Potential Energy Function
Equilibrium Separation (or bond length):	•is the distance out to the bottom of the potential well, and •is found by setting the force equal to zero.
The Maximum Binding Energy:	•equals the minimum potential energy, •is found by substituting equilibrium separation into interatomic potential, •depends on the size of the attractive potential (A), and •is greater with shorter equilibrium separations.
One electron volt (eV) is the energy acquired (or lost) by an electron in crossing through 1V	1eV = 1.6 x 10-19 J = 160 zJ

*Acknowledgement: This part of the course was based on a course given by Dr.B.R.Lawn at UNSW.

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