Question

\(^{14} \mathrm{N}\) is a spin 1 particle such that the energy levels are at 0 and \(\pm \gamma B \hbar,\) where \(\gamma\) is the magnetogyric ratio and \(B\) is the strength of the magnetic field. In a 4.8 T field, the energy splitting between any two spin states expressed as the resonance frequency is \(14.45 \mathrm{MHz}\). Determine the occupation numbers for the three spin states at \(298 \mathrm{K}\).

Short answer

The occupation numbers for the three spin states \(n_1\), \(n_2\), and \(n_3\) at \(298 \mathrm{K}\) can be calculated using Boltzmann distribution: - For state 1: \(n_1 = \frac{N e^{E_0 / 2k_BT}}{Z}\) - For state 2: \(n_2 = \frac{N}{Z}\) - For state 3: \(n_3 = \frac{N e^{-E_0 / 2k_BT}}{Z}\) where \(E_0 = \frac{1}{2}(9.57 \times 10^{-28} \mathrm{J})\), \(k_B\) is Boltzmann's constant, \(T = 298 \mathrm{K}\), and \(Z = e^{E_0 / 2k_BT} + 1 + e^{-E_0 / 2k_BT}\). The occupation numbers will be relative to each other since the total number of particles N is not provided. Plug in the values for \(E_0\), \(k_B\), and \(T\) to determine the ratio among the occupation numbers \(n_1\), \(n_2\), and \(n_3\).

Step-by-step solution

Determine the energy level spacings

From the information provided, the energy splitting between any two spin states is given by the resonance frequency: \(14.45 \mathrm{MHz} = 14.45 \times 10^6 \mathrm{Hz}\). Since the energy split between neighboring levels is related to the resonance frequency by \(\Delta E = h \cdot \Delta \! f\) (where \(h\) is Planck's constant), we have: \[ \Delta E = h \cdot (14.45 \times 10^6 \mathrm{Hz}) \] Calculate the energy level spacings: \[ \Delta E = (6.626 \times 10^{-34} \mathrm{J \cdot s}) \cdot (14.45 \times 10^6 \mathrm{Hz}) ≈ 9.57 \times 10^{-28} \mathrm{J} \]

Find the energy for each spin state

Now we need to find the energy of each spin state. The given energy levels are at \(0\) and \(\pm \gamma B \hbar\). Since the energy spacings are common, the spin states have energies \(-E_0\), \(0\), and \(+E_0\). Thus, the energy for each spin state is: - For state 1: \(E_1 = -E_0 = -\frac{1}{2} \Delta E = -\frac{1}{2}(9.57 \times 10^{-28} \mathrm{J})\) - For state 2: \(E_2 = 0\) - For state 3: \(E_3 = +E_0 = +\frac{1}{2} \Delta E = +\frac{1}{2}(9.57 \times 10^{-28} \mathrm{J})\)

Calculate the relative probabilities of each state

Using the Boltzmann distribution, the probability of finding a particle in a specific energy state is given by: \[ P_i = \frac{e^{-E_i / k_BT}}{Z} \] where \(E_i\) is the energy of the i-th state, \(k_B\) is Boltzmann's constant, \(T\) is temperature, and \(Z\) is the partition function, which normalizes the probabilities: \[ Z = \sum_{i} e^{-E_i/k_BT} \] Here, we have three states. Calculate the probabilities for each state: - For state 1: \(P_1 = \frac{e^{E_0 / 2k_BT}}{Z}\) - For state 2: \(P_2 = \frac{1}{Z}\) - For state 3: \(P_3 = \frac{e^{-E_0 / 2k_BT}}{Z}\) Find the partition function: \[ Z = e^{E_0 / 2k_BT} + 1 + e^{-E_0 / 2k_BT} \]

Normalize the probabilities to find the occupation numbers

To find the occupation numbers \(n_1\), \(n_2\), and \(n_3\), we multiply the probabilities by the total number of particles N: - For state 1: \(n_1 = \frac{N e^{E_0 / 2k_BT}}{Z}\) - For state 2: \(n_2 = \frac{N}{Z}\) - For state 3: \(n_3 = \frac{N e^{-E_0 / 2k_BT}}{Z}\) It should be noted that since we are not given an actual number of particles, the occupation numbers will be relative to each other, not absolute values. The ratio among the occupation numbers \(n_1\), \(n_2\), and \(n_3\) can be determined by plugging in the values for \(E_0\), \(k_B\), and \(T\).

Key concepts

Energy Level Spacing

In quantum physics, energy level spacing refers to the difference in energy between two successive energy levels for particles or atoms. Knowing the energy level spacing helps us understand how particles interact with external fields, such as magnetic fields. This understanding is essential for studying spin states, where different spin orientations have different energy levels.
In the original problem, energy level spacing was calculated using the resonance frequency of 14.45 MHz and Planck's constant (6.626 \times 10^{-34} \text{ J}\cdot\text{s}). The calculated spacing was approximately \(9.57 \times 10^{-28} \text{ J}\).
This spacing tells us how far apart the energy levels are for the spin states of ^{14} \text{N} in a magnetic field. By computing this, we can predict how particles might switch or occupy these states under different conditions.

Boltzmann Distribution

The Boltzmann Distribution is a fundamental concept in statistical mechanics. It provides a way to predict the distribution of particles among different energy states at thermal equilibrium. This distribution depends on the energy of the states and the temperature of the system.
The probability of finding a particle in a particular state with energy \(E_i\) is given by: \(P_i = \frac{e^{-E_i / k_BT}}{Z}\), where \(k_B\) is the Boltzmann constant, \(T\) is temperature, and \(Z\) is the partition function.
In the context of this problem, the Boltzmann distribution is used to determine the relative probabilities of the three spin states of ^{14} \text{N}. These probabilities are based on the energies calculated earlier and are critical for finding the occupation numbers, which show how many particles are in each state.

Magnetogyric Ratio

The magnetogyric ratio, often symbolized by \(\gamma\), is a property of particles that describe how their spins interact with a magnetic field. It is a key factor in nuclear magnetic resonance (NMR) because it influences the energy level spacing of spin states in a magnetic field.
This ratio defines how much the energy levels split when a particle with spin is placed in the magnetic field. For ^{14} \text{N}, it impacts the separation of the spin states at \(0\) and \(\pm \gamma B \hbar\), where \(B\) is the magnetic field's strength.
Understanding the magnetogyric ratio helps us precisely calculate the energy level spacing, which is vital in magnetic resonance imaging (MRI) and other applications requiring magnetic field interactions.

Partition Function

The partition function, denoted \(Z\), is a crucial part of statistical mechanics calculations. It acts as a normalization factor that ensures probabilities sum to one. This function is defined as the sum of exponentials of all possible energy states, each weighted by their Boltzmann factor:
\[Z = \sum_{i} e^{-E_i / k_BT}\]
For ^{14} \text{N} in the exercise, the partition function is calculated over the three energy states. It helps determine the probability and, eventually, the occupation numbers of particles in each state.
This function enables us to transform complex calculations into manageable probabilities by incorporating both temperature and energy, making it indispensable in understanding thermal systems and predicting state populations.