The model of the monatomic chain (Section 8.3) can be extended by including a local interaction for each atom of the form $$ E_{\text {local }}(u)=\frac{1}{2} \alpha u^{2} $$ Show that the effect of an additional local potential energy is to shift the dispersion curve for the case with nearest-neighbour interactions to $$ m \omega^{2}=\alpha+4 J \sin ^{2} k a / 2 $$

Local potential energy is a concept that describes the energy stored within an atom due to its displacement. In the context of a monatomic chain model, this is represented by the equation: \[ E_{\text{local}}(u) = \frac{1}{2} \alpha u^2 \]where \(E_{\text{local}}\) represents the local potential energy, \(\alpha\) is a constant that measures the stiffness of the potential, and \(u\) is the displacement of the atom from its equilibrium position. This equation is known as a harmonic potential energy equation. The term \(\frac{1}{2} \alpha u^2\) indicates that the energy increases quadratically as the displacement increases. This means that any deviation of the atom from its equilibrium position will result in an increase in the local potential energy.

The nearest-neighbor interactions describe the energy between an atom and its immediate neighbors in a monatomic chain. The potential energy due to nearest-neighbor interactions is given by: \[ E_{\text{neighbor}} = \frac{1}{2} J (u_{i+1} - u_i)^2 \]here, \(E_{\text{neighbor}}\) represents the interaction energy, \(J\) is the coupling constant, and \(u_i\) and \(u_{i+1}\) are the displacements of neighboring atoms. The total potential energy in the monatomic chain is the sum of the local potential energy and the nearest-neighbor interaction energy: \[ E_{\text{total}} = \frac{1}{2} \alpha u_i^2 + \frac{1}{2} J (u_{i+1} - u_i)^2 \]This equation indicates that the total energy of an atom in the chain is influenced both by its own displacement and the displacement of its nearest neighbors.

The dispersion relation in the context of a monatomic chain model relates the frequency of vibration (\(\omega\)) of atoms to their wave vector (\(k\)). By including local potential energy, we get the dispersion relation as: \[ m \omega^2 = \alpha + 4J \sin^2\left(\frac{ka}{2}\right) \]where \(m\) is the mass of the atom, \(\alpha\) is the local potential energy constant, \(J\) is the nearest-neighbor interaction constant, and \(a\) is the lattice constant. This equation shows how the local potential energy shifts the dispersion curve. The term \(4J \sin^2(ka/2)\) comes from the nearest-neighbor interactions, and the term \(\alpha\) shifts this curve upwards. The shape of the dispersion relation indicates the way the vibration frequency varies with the wave vector and highlights the influence of both local and nearest-neighbor interactions.

A harmonic potential is a type of potential energy that follows Hooke's Law, where the force exerted by a spring is directly proportional to its displacement but acts in the opposite direction. In the monatomic chain model, a harmonic potential is used to describe the local potential energy of an atom displaced from its equilibrium position: \[ E_{\text{local}}(u) = \frac{1}{2} \alpha u^2 \]This quadratic dependence on displacement signifies that small displacements result in energies proportional to the square of the displacement. This makes it easier to calculate the forces and results in simple harmonic motion where the atoms oscillate about their equilibrium positions. The harmonic potential ensures the stability of the monatomic chain model as it imposes a restoring force that resists displacement of atoms, thereby maintaining the structural integrity of the chain.