Problem 22 - A Modern Approach to Quantum Mechanics

Question

Show that the Curie constant for an ensemble of \(N\) spin-l particles of mass \(m\) and charge \(q=-e\) immersed in a uniform magnetic field \(\mathbf{B}=B \mathbf{k}\) is given by $$ C=\frac{2 N \mu^{2}}{3 k_{\mathrm{B}}} $$ where \(\mu=g e \hbar / 2 m c\). Compare this value for \(C\) with that for an ensemble of spin- \(\frac{1}{2}\) particles, as determined in Example \(5.6 .\)

Step-by-Step Solution

Answer

The Curie constant for both spin-l and spin-\(\frac{1}{2}\) particles results in \(C = \frac{2 N \mu^2}{3 k_B}\).

Step 1: Understand the Problem

We need to find the Curie constant, \(C\), for a system of spin-l particles and compare it to the Curie constant for spin-\(\frac{1}{2}\) particles. The expression for \(\mu\) is given, and \(B\) is the magnitude of the magnetic field.

Step 2: Identify Relevant Formulae

The Curie constant \(C\) in a magnetic system is given by:\[C = \frac{N \mu^2}{k_B} \sum_m g_m^2 m^2,\]where \(g_m\) is the degeneracy and \(m\) ranges over the allowed magnetic quantum numbers. For spin-\(l\), \(m = -l, -l+1, \ldots, l-1, l\).

Step 3: Calculate \(\mu^2\) for Spin-\(l\) Particles

The magnetic moment \(\mu\) is given as:\[\mu = \frac{g e \hbar}{2 m c}.\]Squaring this expression gives:\[\mu^2 = \left(\frac{g e \hbar}{2 m c}\right)^2.\]

Step 4: Sum Over Magnetic Quantum Numbers

For spin-\(l\), there are \((2l+1)\) possible quantum number states. We set \(g_m = 1\) for simplicity:\[\sum_{m=-l}^{l} m^2 = l(l+1)(2l+1)/3.\]

Step 5: Derive Curie Constant for Spin-\(l\) Particles

Plug the value \(\sum m^2\) into the equation for \(C\):\[C = \frac{N \mu^2}{k_B} \cdot \frac{l(l+1)(2l+1)}{3(2l+1)} = \frac{N \mu^2 l(l+1)}{3 k_B}.\]When \(l = \frac{1}{2}\), the Curie constant simplifies to:\(C = \frac{2 N \mu^2}{3 k_B}.\)

Step 6: Compare With Spin-\(\frac{1}{2}\) Particles

The Curie constant for spin-\(l = \frac{1}{2}\) is already given as \(\frac{2 N \mu^2}{3 k_B}\), matching the derived \(C\) for an ensemble of spin-\(\frac{1}{2}\) particles.

Key Concepts

Magnetic Moment

The magnetic moment is a crucial concept when studying magnetism and quantum mechanics. It explains how particles interact with magnetic fields. Essentially, the magnetic moment is a vector quantity that indicates a dipole's alignment tendency relative to a magnetic field. Electric charges moving through space create a magnetic field. In quantum mechanics, even particles like electrons possess intrinsic magnetic moments due to their charge and spin. The magnitude of this magnetic moment, for a particle with mass \(m\) and charge \(-e\), is given by the expression \(\mu = \frac{g e \hbar}{2 m c}\). Here:- \(g\) is the gyromagnetic ratio- \(e\) is the charge of an electron- \(\hbar\) represents the reduced Planck's constant- \(c\) is the speed of lightUnderstanding these constants helps us comprehend the influence of the magnetic moment in different systems, such as a gas of spin particles in a magnetic field.

Spin Particles

Spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei, creating a magnetic field. Think of spin like a little internal compass needle that the particle carries. Despite the term, spin does not mean particles are physically spinning; it’s a quantum mechanical property signifying a type of inherent angular momentum.Particles such as electrons, protons, and neutrons have spin \(\frac{1}{2}\), meaning they can exist in two possible spin states: "spin up" and "spin down." These states contribute to the particle's overall magnetic properties and affect how they align in a magnetic field.For particles with spin \(l\), the possible values of magnetic quantum number \(m\) vary from \(-l\) to \(l\) in integer steps, leading to \((2l+1)\) possible states. Each of these states contributes to the system's overall magnetic behavior, captured by the summation \(\sum_{m=-l}^{l} m^2 = l(l+1)(2l+1)/3\). In this way, spin particles play a critical role in defining the magnetic features of matter.

Quantum Mechanics

Quantum mechanics revolutionized our understanding of microscopic systems by introducing concepts like wave-particle duality and uncertainty. It provides the framework to describe the intricate behaviors of particles at the smallest scales.One of the fundamental principles is the quantization of certain properties, such as energy, angular momentum, and magnetic moment. Quantum mechanics introduces discrete values for these properties, contrary to classical mechanics that allows for continuous values.In the context of magnetic systems, quantum mechanics helps us calculate parameters like the Curie constant. The Curie constant \(C\) defines how magnetization and magnetic susceptibility of a paramagnetic material depend on temperature. Using quantum mechanics, we can derive formulas like \(C = \frac{2 N \mu^2}{3 k_B}\) for spin-\(\frac{1}{2}\) particles, which underline the pivotal role of quantum insights in explaining macroscopic magnetic phenomena.Understanding these quantum mechanical principles is essential for exploring the magnetic properties of materials and their practical applications in technology and materials science.