A common model of a solid assumes the atoms to be points executing simple harmonic motion about mean lattice positions. What would be the coefficient of linear expansion of such a lattice?

To determine the coefficient of linear expansion for a solid modeled with atoms as points executing simple harmonic motion around their mean lattice positions, we need to delve into some fundamental concepts of solid-state physics and thermodynamics.

Understanding the Basics

In a crystalline solid, atoms are arranged in a regular lattice structure. When the temperature of the solid increases, the kinetic energy of the atoms also increases, causing them to vibrate more vigorously about their equilibrium positions. This vibration can be modeled as simple harmonic motion, where the restoring force acting on the atoms is proportional to their displacement from the mean position.

Coefficient of Linear Expansion Defined

The coefficient of linear expansion, denoted as α, quantifies how much a material expands per unit length for a one-degree increase in temperature. Mathematically, it is expressed as:

• α = (1/L₀) * (ΔL/ΔT)

Here, L₀ is the original length, ΔL is the change in length, and ΔT is the change in temperature.

Deriving the Coefficient of Linear Expansion

To derive α for our model, we start by considering the potential energy of the atoms in the lattice. For simple harmonic motion, the potential energy U can be expressed as:

• U = (1/2) k x²

where k is the spring constant (related to the stiffness of the atomic bonds) and x is the displacement from the equilibrium position. As temperature increases, the average displacement of the atoms also increases due to enhanced thermal vibrations.

Relating Temperature to Displacement

The average energy of the atoms can be related to temperature through the equipartition theorem, which states that each degree of freedom contributes (1/2) k_B T to the total energy, where k_B is the Boltzmann constant. For one-dimensional motion, we have:

• E = (1/2) k_B T

As the temperature increases, the average displacement x can be shown to scale with the square root of temperature:

• x ∝ √T

Connecting Displacement to Linear Expansion

As the temperature changes, the change in length ΔL can be related to the change in average displacement. If we consider a one-dimensional lattice, the change in length due to thermal expansion can be approximated as:

• ΔL ≈ L₀ (Δx/L₀)

Substituting the relationship of displacement with temperature, we find:

• ΔL ∝ L₀ √T

Final Expression for Coefficient of Linear Expansion

Now, substituting this relationship into the expression for α gives:

• α = (1/L₀) * (ΔL/ΔT) ∝ (1/L₀) * (L₀ √T/ΔT)

As ΔT increases, the coefficient of linear expansion α can be approximated to be proportional to the square root of the temperature change, leading to:

• α ∝ 1/√T

Conclusion

In summary, for a solid modeled with atoms as points executing simple harmonic motion, the coefficient of linear expansion α is inversely proportional to the square root of the temperature. This means that as temperature increases, the coefficient of linear expansion decreases, reflecting the nature of atomic vibrations in the lattice. This relationship is crucial for understanding how materials behave under thermal stress and is fundamental in fields such as materials science and engineering.