Calculate the relative number of "F nuclei in the higher and...

Calculate the relative number of "F nuclei in the higher and lower magnetic states at $25^{\circ} \mathrm{C}$ in magnetic fields of (a) $2.4 \mathrm{T},(\mathrm{b}) 4.69 \mathrm{T},$ and $(\mathrm{c})$ 7.05 $\mathrm{T}$

Key Concepts

Thermal Equilibrium

Thermal equilibrium is a condition where the system's state populations remain constant over time, meaning that the spins have exchanged energy with the surroundings sufficiently to reach a stable distribution described by the Boltzmann statistics. This concept is essential when calculating the relative number of spins in different magnetic states, as it ensures that the calculated populations accurately reflect the equilibrium conditions under which the experiment or measurement is performed.

Zeeman Energy Splitting

Zeeman energy splitting is the process by which the degeneracy of nuclear spin states is lifted when an external magnetic field is applied. This splitting results from the interaction of the nuclear magnetic moment with the magnetic field, giving rise to distinct energy levels for spins aligned with and against the field. The magnitude of the splitting depends on the strength of the magnetic field and the magnetic moment of the nucleus, and it is fundamental to understanding how different magnetic states are populated in a magnetic resonance experiment.

Boltzmann Distribution

The Boltzmann distribution describes how particles, such as nuclear spins, are statistically distributed among different energy levels at thermal equilibrium. It indicates that the ratio of populations in different energy states depends exponentially on the negative of the energy difference divided by the thermal energy (kT). In the context of magnetic states in an external field, this distribution explains why lower energy (aligned) states are more populated than higher energy (anti-aligned) states, particularly at finite temperatures.

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