SOLID STATE PHYSICS 2 ASSIGNMENT 5

EXAM QUESTIONS ASSIGNMENT 5 SOLVED SOLID STATE PHYSICS 2

QUESTION NO 1 :. Give relationship between magnetic susceptibility, magnetization and magnetic field.?

1. Relationship between Magnetic Susceptibility, Magnetization, and Magnetic Field:The magnetic susceptibility (χ) is a dimensionless proportionality constant that indicates the degree of magnetization (M) of a material in response to an applied magnetic field (𝐻).
   • The relationship is given by:𝑀=𝜒𝐻
   • where:
     • M is the magnetization, the magnetic moment per unit volume.
     • χ is the magnetic susceptibility.
     • H is the applied magnetic field.

QUESTION NO2 Explain Larmor angular frequency and give its value for a field induction B.

• Larmor Angular Frequency:
• Larmor angular frequency (𝜔𝐿​) describes the precession of magnetic moments in the presence of an external magnetic field. It is defined as the rate at which the magnetic moment of a charged particle precesses around the direction of the applied magnetic field. For a field induction 𝐵B, the Larmor frequency is given by:𝜔𝐿=𝑒𝐵2𝑚
• where:
• 𝑒 is the charge of the particle (e.g., the electron charge).
• 𝐵 is the magnetic flux density (magnetic field strength).
• 𝑚 is the mass of the particle (e.g., the electron mass).

QUESTION NO 3 :

Explain the origin of diamagnetism in a free atom

• Origin of Diamagnetism in a Free Atom:
• Diamagnetism arises in atoms due to the change in the orbital motion of electrons when an external magnetic field is applied. According to Lenz’s Law, an induced magnetic moment is created in a direction opposite to that of the applied field. This effect occurs because the applied magnetic field changes the velocity of the orbiting electrons, thus inducing a magnetic moment that opposes the change.

QUESTION NO 4 :Drive an expression of diamagnetism on the basis of classical theory.

• Expression of Diamagnetism Based on Classical Theory:
• In classical theory, diamagnetism can be explained by considering the effect of an applied magnetic field on the orbital motion of electrons in an atom.
  • Consider an electron of charge 𝑒e and mass 𝑚m moving in a circular orbit of radius r. When a magnetic field 𝐵 is applied perpendicular to the plane of the orbit, it exerts a Lorentz force on the electron.This force alters the velocity of the electron and thus induces a current in the loop.The induced current 𝐼I produces a magnetic moment 𝜇 opposing the applied magnetic field.
  For a simple circular orbit, the induced magnetic moment 𝜇 due to the applied field 𝐵 can be expressed as:
• 𝜇=−𝑒²𝑟²𝐵/4𝑚
• ​For a material with 𝑛n atoms per unit volume, the total magnetization M (magnetic moment per unit volume) is
• :𝑀=−𝑛𝑒²𝑟²𝐵/6𝑚
• ​The magnetic susceptibility 𝜒 is given by the ratio of magnetization 𝑀M to the applied magnetic field H:
• 𝜒=𝑀𝐻=−𝑛𝑒²𝑟²/6𝑚
• Here, the factor of 1/𝜇₀​ (where 𝜇0​ is the permeability of free space) is included to convert between 𝐵 and H

QUESTION NO 5 : Distinguish between diamagnetism, Paramagnetism and ferromagnetism.

• Diamagnetism:
  • Origin: Arises from the induced currents due to changes in the orbital motion of electrons in response to an external magnetic field.
  • Behavior: Creates an induced magnetic moment that opposes the applied field.
  • Susceptibility: Small and negative, indicating a weak repulsion by the magnetic field.
  • Temperature Dependence: Generally independent of temperature.
  • Example Materials: Bismuth, copper, gold.
• Paramagnetism:
  • Origin: Due to the presence of unpaired electrons that have intrinsic magnetic moments. These moments tend to align with the external magnetic field.
  • Behavior: Magnetic moments align with the applied field, enhancing it.
  • Susceptibility: Small and positive, indicating a weak attraction to the magnetic field.
  • Temperature Dependence: Decreases with increasing temperature, following Curie’s law (𝜒∝1𝑇χ∝T1​).
  • Example Materials: Aluminum, platinum, some salts of transition metals.
• Ferromagnetism:
  • Origin: Arises from strong interactions between atomic magnetic moments, causing them to align parallel to each other in regions called domains.
  • Behavior: Exhibits spontaneous magnetization even in the absence of an external field. The magnetic moments within domains are aligned, and domains can align with an external field.
  • Susceptibility: Large and positive, showing a strong attraction to the magnetic field.
  • Temperature Dependence: Strongly temperature-dependent; above the Curie temperature, ferromagnetic materials become paramagnetic.
  • Example Materials: Iron, nickel, cobalt.