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Chapter 7: Problem 1

Suppose that longitudinal vibrations in a linear chain of \(N\) identical atoms obey periodic boundary conditions, so that the motion of the first and last atom of the chain is identical. Find the number of normal modes of vibration for this system.

Short Answer

Expert verified
The number of normal modes of vibration is N.

Step by step solution

01

Understand Periodic Boundary Conditions

Periodic boundary conditions mean that the atom at position 1 and the atom at position N are identical in terms of their motion. This implies that the chain is effectively looped onto itself.
02

Identifying Motion in a Linear Chain of Atoms

In a linear chain where each atom can oscillate longitudinally, such a system can have distinct vibrational frequencies. These are known as normal modes.
03

Determine the Number of Normal Modes

For a linear chain with periodic boundary conditions and N atoms, the number of distinct normal modes will equal the number of atoms, which is N. This is because each atom can participate in a mode that satisfies the boundary condition.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Periodic Boundary Conditions
Imagine a chain of atoms arranged in a straight line. In real life, the atoms at the ends would experience different surroundings than those in the middle. Periodic boundary conditions simplify this by assuming that the first and last atoms are neighbors. So, this setup effectively loops the chain into a circle. Such conditions help in studying the vibrations as they provide a uniform environment for all atoms.
With periodic boundary conditions, if you move from atom 1 and keep moving in the same direction, after \(N\) steps, you will find yourself back at atom 1. This assumption is useful in various physical systems, like crystal lattices and polymers.
Longitudinal Vibrations
When talking about longitudinal vibrations, we are referring to oscillations that occur along the direction of the chain. Picture how a Slinky can be compressed and stretched; the coils move closer or further away from each other. Similarly, in a linear chain of atoms, each atom can move back and forth along the direction of the chain.
These vibrations create waves of motion that travel along the chain. The wave's length and the frequency can vary, leading to different patterns of oscillation. In physics, these specific patterns are known as the normal modes of vibration.
Identical Atoms
In this system, all atoms are identical, meaning they have the same mass and interact with neighboring atoms in the same way. This homogeneity makes it easier to analyze how the whole chain behaves. When atoms are identical, they will all move in a similar manner when subject to the same forces.
Identical atoms ensure that the vibrational properties are consistent throughout the chain. This uniformity is crucial for calculating normal modes since the simplifying assumption is that each atom will contribute equally to each mode.
Normal Modes of Vibration
The concept of normal modes in a linear chain is essentially about finding specific, stable patterns of motion. Each normal mode represents a distinct wave pattern where all atoms oscillate at the same frequency.
In a system with periodic boundary conditions and \(N\) identical atoms, there will be exactly \(N\) normal modes. This is because each atom contributes a degree of freedom to the system. The waves formed can be characterized by their wave numbers, ranging from \(0\) to \(N-1\).
Knowing the normal modes helps in understanding how energy is distributed in the system and how it can affect properties like heat and electrical conduction in materials.

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Most popular questions from this chapter

Discuss the vibrations of diatomic lattice and describe its optical and acoustic modes.

\text { Show that for a diatomic linear chain with two types of atoms of mass m and \(M\), the density of phonon modes per unit frequency interval diverges at the maximum frequency on either side of the gap: while it tends to acquire a constant value as \(\omega \rightarrow 0\). Assuming the nearest neighbour interaction described by a single force constant \(K\), for all interacting pair.

A more accurate description of lattice vibration is obtained if higher neighbour interactions are also included. Prove that the inclusion of nth neighbours modifies the dispersion relation of a one dimensional monoatomie system to $$ M \omega^{2}=2 \sum_{y=1}^{n} K_{2}[1-\cos (s k a)] $$
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