Question

Section 106 notes that as causes of resistance, ionic vibrations give way lo lattice imperfections at around \(10 \mathrm{~K}\). A typical spring constant between atoms in a solid is of order of megmitude \(10^{3} \mathrm{~N} / \mathrm{m}\), and typical spacing is nominully \(10^{-10} \mathrm{~m}\). Estimate how much the vibrating atoms" locations might deviate, as a fraction of their nominal separation, at \(10 \mathrm{~K}\)

Short answer

The vibrating atoms' locations deviate by about 1.17% of their nominal separation at 10 K.

Step-by-step solution

Understand the problem and given information

We have been given that the spring constant between atoms in a solid is of the order of \(10^{3} ~N/m\) and the typical atomic spacing is \(10^{-10} ~m\). We need to find out how much the atoms deviate from their normal position at a temperature of \(10 K\).

Relate the physical quantities

According to the Equipartition Theorem, each degree of freedom contributes \(1/2 k_BT\) to the average energy, where \(k_B\) is the Boltzmann constant and \(T\) is the temperature. For an atom displacing along one direction, the average potential energy will be \(1/2 k \langle x^2 \rangle\), where \(k\) is the spring constant and \(\langle x^2 \rangle\) is the mean square displacement. Since these two potential energies must be equal, we have \(1/2 k \langle x^2 \rangle = 1/2 k_B T\). Therefore, \(\langle x^2 \rangle = (k_B T) / k\). Thus, the deviation from the equilibrium position of the atoms, or the root mean square displacement x, is: \( x = \sqrt{\langle x^2 \rangle} \), which gives \( x = \sqrt {(k_B T) / k}\).

Substitute the given values and calculate the deviation

Substitute the values of \(k_B = 1.38 \times 10^{-23} ~J/K\), \(T = 10K\), and \( k = 10^3 ~N/m \) into the equation. This gives \( x = \sqrt {(1.38 \times 10^{-23} ~J/K \times 10~K) / (10^3~N/m)} = 1.17 \times 10^{-12} m\).

Calculate the deviation as a fraction of nominal separation

The deviation as a fraction of the nominal separation \(10^{-10} m\) is \( x/ 10^{-10} m = 1.17 \times 10^{-12} / 10^{-10} = 0.0117, \) or about 1.17%.

Key concepts

Equipartition Theorem and Atomic Displacement

The equipartition theorem is a fundamental principle in statistical mechanics that allows us to determine how energy is distributed among various degrees of freedom in a system at thermal equilibrium. When it comes to the behavior of atoms within solids, this theorem helps us understand how much energy is involved in the motion of the atoms, or 'atomic displacement'.

With temperature as a measure of the system's thermal energy, the theorem posits that at equilibrium, each degree of freedom associated with the atomic motion will, on average, have energy equal to \(\frac{1}{2} k_B T\), where \(k_B\) is the Boltzmann constant and \(T\) is the temperature in kelvins. This understanding is pivotal when calculating how much atoms deviate from their equilibrium positions due to their thermal energy. In solids, these atomic displacements lead to effects like thermal expansion and can influence material properties such as conductivity and elasticity.

Root Mean Square Displacement

The 'root mean square displacement' (RMSD) represents the standard deviation of the atoms' positional deviation from their average position in a solid. In the context of our problem, it quantifies the magnitude of deviation that occurs in a vibrating lattice as the atoms are subject to temperature-induced movement.

To calculate the RMSD, one squares the atomic displacements, averages these square values (hence the 'mean square'), and then takes the square root of that average (which is the 'root' part). Mathematically, RMSD is given by \(\sqrt{\langle x^2 \rangle}\), where \(\langle x^2 \rangle\) is the mean square displacement. Understanding RMSD is important because it's indicative of the degree to which atoms vibrate around their fixed points in the lattice and thus can be linked to factors such as material strength and thermal conductivity.

Boltzmann Constant

The 'Boltzmann constant' (\(k_B\)) is a fundamental constant in physics that connects the macroscopic world (where we talk about temperature and volume) with the microscopic world (where we talk about energy in terms of particles). \(k_B\) essentially translates temperature into energy and is a key component in various physical laws and formulas, including the aforementioned equipartition theorem.

Its value is approximately \(1.38 \times 10^{-23} J/K\), and it allows us to scale the thermal energy per degree of temperature for a single particle. When we probe into questions about atomic movement in a lattice, as in our exercise, \(k_B\) becomes the bridge between what we know (the temperature of the material) and what we want to find out (the atomic displacement or motion within that material).

Lattice Imperfections

Lastly, let's talk about 'lattice imperfections'. A crystalline solid is thought to have atoms arranged in a virtually perfect periodic structure called a lattice. But in reality, no crystalline solid is perfect; there are always lattice imperfections, which include point defects like vacancies, line defects such as dislocations, and planar defects including grain boundaries.

These imperfections play a significant role in the material's properties, affecting mechanical strength, electrical properties, and thermal transport. At low temperatures, the significance of atomic vibration in resistance decreases, while the influence of these imperfections becomes more pronounced. In our exercise, around 10K, we see a shift from resistance primarily due to ionic vibrations to resistance influenced by these imperfections, underscoring their importance in different temperature regimes.