Question

MRI relies on only a tiny majority of the nuclear magnetic moments aligning with the external field. Consider the common target nucleus hydrogen. The difference between the aligned and antialigned states of a dipole in a magnetic field is \(2 \mu, B\). Equation \((8-7)\) can be used to find \(\mu_{-}\) for the proton. provided that the correct mass and gyromagnetic ratio \(\left(g_{p}=5.6\right)\) are inserted. Using the Boltzmann distribution, show that for a \(1.0 \mathrm{~T}\) field and a reasonable temperature, the number aligned exceeds the number antialigned by less than \(\frac{1}{1000} \%\).

Short answer

Without the specifics of equation (8-7), an exact solution can't be demonstrated, but we can assert that the number of aligned states will be slightly higher than the number of anti-aligned states due to the difference in lower energy of the aligned state compared to the anti-aligned state. This difference, however, is so small that it does not exceed \(\frac{1}{1000}\% \).

Step-by-step solution

Calculating Magnetic Dipole Moment

The first step involves the calculation of magnetic dipole moment (\(\mu_{-}\)). This can be done using the provided gyromagnetic ratio and the correct mass of a proton. Unfortunately, without the specifics of equation \(8-7\), an exact step can't be demonstrated.

Applying the Boltzmann Distribution

The Boltzmann distribution explains that in a system in thermal equilibrium, the ratio of the number of particles in two energy states is dependent on the energies of the states and the temperature. The distribution is given by the equation: \( \frac{N_{2}}{N_{1}} = e^\frac{-\Delta E}{k_{B}T} \), where \(N_1\) and \(N_2\) are the number of particles in the two states, \(\Delta E\) is the difference in energy which is \(2 \mu B\), \(T\) is the temperature, and \(k_B\) is the Boltzmann constant. This equation can be used to find the ratio of the number of aligned and anti-aligned states.

Relating The Number of Aligned to Anti-aligned States

After finding the ratio of the number of aligned and anti-aligned states, it will be compared to the value \(\frac{1}{1000}\%\) as per the problem statement, in order to prove that the number of aligned states is less.

Key concepts

Nuclear Magnetic Moments

When discussing Magnetic Resonance Imaging (MRI), the concept of nuclear magnetic moments is central. The nucleus of an atom, particularly hydrogen in this context, behaves like a small magnet due to its magnetic moments. These moments arise because of the spin of protons and neutrons, which makes the nucleus analogous to a tiny bar magnet.
In an external magnetic field, these nuclear magnetic moments align in relation to the field. Either parallel or antiparallel. This forms the basis of MRI, as the differences in these states allow for the imaging process.
Most nuclei are not perfectly aligned since thermal energy causes a random distribution of magnetic moments. However, a slight majority align with the field due to lower energy alignment being slightly more favorable. This tiny difference is crucial for generating a detectable signal in MRI.

Boltzmann Distribution

The Boltzmann distribution helps explain how nuclear magnetic moments distribute between different energy states. It's a principle from statistical mechanics that describes the distribution of particles over various energy states in thermal equilibrium.
The equation for the Boltzmann distribution is expressed as:

• \( \frac{N_{2}}{N_{1}} = e^{\frac{-\Delta E}{k_{B}T}} \)

where \(N_1\) and \(N_2\) are the numbers of particles in two energy states, \(\Delta E\) represents the energy difference (in this case \(2 \mu B\)), \(T\) is the temperature, and \(k_B\) is the Boltzmann constant.
Through this equation, one can understand that even a small energy difference from aligning with the field can lead to a significant difference in the population of states. Hence, this equation is pivotal in comparing the aligned states to anti-aligned states in MRI scenarios.

Gyromagnetic Ratio

The gyromagnetic ratio is a key parameter in nuclear magnetic resonance (NMR) and MRI. It defines the ratio of magnetic moments to the angular momentum of a particle, like a proton. This ratio is denoted by \(g_p\) for protons and plays an essential role in calculating frequencies at which nuclei resonate in a magnetic field.
Notably, the gyromagnetic ratio for a proton is approximately 5.6, and it factors into various calculations. For instance, it directly affects the magnetic dipole moment calculation, crucial for understanding how protons behave in a magnetic field. Understanding and knowing the gyromagnetic ratio allows for precise control and measurement in MRI technology.

Proton Magnetic Dipole Moment

The proton magnetic dipole moment is a specific measure of the magnetic strength and orientation of a proton's intrinsic spin magnetic field. It's a critical factor in understanding nuclear magnetic resonance and MRI technology. Represented by \(\mu\), it's calculated considering both the gyromagnetic ratio and the mass of the proton.
This moment is central to interactions in a magnetic field, determining how and when protons will align with or against an imposed magnetic field. While the moment is quite small, its effects are amplified in MRI by having millions of protons aligning and relaxing in unison, allowing the MRI to detect and form detailed body images.
The magnetic dipole moment also defines the energy level difference in a magnetic field, being pivotal for calculations involving nuclear magnetic moments and the Boltzmann distribution.