Fermi gas. Consider an ideal Fermi gas whose atoms have mass m= 5 ×10^-24 grams, nuclear spin I=

step 1 As we know that since at absolute zero all energy level upto the Fermi level EF will be filled w

Fermi gas. Consider an ideal Fermi gas whose atoms have mass...

Fermi gas. Consider an ideal Fermi gas whose atoms have mass $m=$ $5 \times 10^{-24}$ grams, nuclear spin $I=\frac{1}{2}$, and nuclear magnetic moment $\mu=$ $1 \times 10^{-23} \mathrm{erg} /$ gauss. At $T=0 \mathrm{~K}$, what is the largest density for which the gas can be completely polarized by an external magnetic field of $10^5$ gauss? (Assume no electronic magnetic moment).

Key Concepts

Fermi Gas

A Fermi gas is a quantum-mechanical model of non-interacting fermions that obey the Pauli exclusion principle. At very low temperatures, the particles fill available quantum states up to a maximum energy value, known as the Fermi energy. This concept is important in understanding the behavior of electrons in metals, neutrons in neutron stars, and other systems of fermions.

Fermi Energy

The Fermi energy is the highest occupied energy level of a Fermi system at zero temperature. It is directly related to the particle density and plays a crucial role in determining the properties and behavior of a degenerate Fermi gas. In problems involving complete polarization, the Fermi energy must be compared with energy shifts induced by an external magnetic field.

Spin Polarization

Spin polarization refers to the degree to which the spins of particles in a system are aligned in a particular direction. In the context of a Fermi gas subject to an external magnetic field, complete spin polarization occurs when all the spins align with the field, often requiring the magnetic (Zeeman) energy to exceed the Fermi energy of the system.

Zeeman Effect

The Zeeman effect involves the splitting of energy levels of particles with magnetic moments when placed in an external magnetic field. This splitting results in an energy difference proportional to the magnetic moment and the strength of the field, which is a key factor in determining whether all spins in a Fermi gas can be aligned (polarized) by the field.

Pauli Exclusion Principle

The Pauli exclusion principle states that no two identical fermions can occupy the same quantum state simultaneously. This principle is fundamental to the behavior of a Fermi gas, as it causes the particles to fill available energy levels sequentially up to the Fermi energy at zero temperature, influencing phenomena such as complete spin polarization under an external magnetic field.

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