Problem 2 - University Physics with Modern Physics

Question

Hydrogen atoms are placed in an external \(1.65-\mathrm{T}\) magnetic field. (a) The protons can make transitions between states where the nuclear spin component is parallel and antiparallel to the field by absorbing or emitting a photon. Which state has lower energy: the state with the nuclear spin component parallel or antiparallel to the field? What are the frequency and wavelength of the photon? In which region of the electromagnetic spectrum does it lie? (b) The electrons can make transitions between states where the electron spin component is parallel and antiparallel to the field by absorbing or emitting a photon. Which state has lower energy: the state with the electron spin component parallel or antiparallel to the field? What are the frequency and wavelength of the photon? In which region of the electromagnetic spectrum does it lie?

Step-by-Step Solution

Answer

Proton spin parallel has lower energy; photon ~70 MHz in RF range. Electron spin antiparallel has lower energy; photon few GHz in microwave range.

Step 1: Understanding proton energy levels

In a magnetic field, the nuclear spin states of a proton can either align with or against the field. The state with the spin parallel (aligned) to the magnetic field has lower energy compared to the state that is antiparallel (opposing) the magnetic field.

Step 2: Calculating proton transition frequency

The energy difference between the two spin states of a proton in a magnetic field is given by the formula \( \Delta E = \gamma_p \hbar B \), where \( \gamma_p = 2.675 \times 10^8 \text{ rad/s/T} \) is the proton gyromagnetic ratio, \( \hbar \) is the reduced Planck's constant \( (1.055 \times 10^{-34} \text{ J⋅s}) \), and \( B = 1.65 \text{ T} \) is the magnetic field strength. This energy difference corresponds to the frequency \( u \) of the photon, calculated as \( u = \frac{\Delta E}{h} \), where \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ J⋅s}) \).

Step 3: Calculating proton transition wavelength

The wavelength \( \lambda \) of the photon can be found using the equation \( \lambda = \frac{c}{u} \), where \( c = 3 \times 10^8 \text{ m/s} \) is the speed of light.

Step 4: Identifying electromagnetic spectrum region for protons

Calculate \( u \) and substitute it in \( \lambda = \frac{c}{u} \) to find the wavelength. This wavelength will be compared with the typical ranges of electromagnetic spectra to determine the corresponding region (e.g., radio, microwave).

Step 5: Understanding electron energy levels

As with protons, electrons also have spin states in a magnetic field. However, due to their negative charge, the electron state with spin antiparallel to the magnetic field has lower energy.

Step 6: Calculating electron transition frequency

The energy difference for an electron transition can be calculated similarly to protons using \( \Delta E = \gamma_e \hbar B \), where \( \gamma_e = 1.760 \times 10^{11} \text{ rad/s/T} \). Find the frequency using \( u = \frac{\Delta E}{h} \).

Step 7: Calculating electron transition wavelength

Using the frequency calculated previously, find the wavelength for the electron transition with \( \lambda = \frac{c}{u} \).

Step 8: Identifying electromagnetic spectrum region for electrons

Determine the region of the electromagnetic spectrum corresponding to the calculated wavelength for electrons.

Key Concepts

Magnetic Field Effects

The presence of a magnetic field has a significant effect on atomic particles such as protons and electrons. When hydrogen atoms are placed in an external magnetic field, the spins of these particles can either align parallel or antiparallel to the magnetic field.

• The energy level of a particle is influenced by its spin orientation in the magnetic field.
• In the case of protons, those spins that are parallel to the magnetic field have lower energy.
• For electrons, the situation is opposite: antiparallel spins result in lower energy.

These transitions between spin states occur with the absorption or emission of photons, and these energy changes provide insights into the frequencies and wavelengths associated with specific transitions. Understanding these magnetic field effects allows us to explore deeper phenomena like magnetic resonance and spectroscopic techniques.

Nuclear Spin

Nuclear spin is a quantum mechanical property of protons, which are components of atomic nuclei. This intrinsic form of angular momentum plays a crucial role in determining the behavior of nuclei in magnetic fields. In a strong magnetic field:

• The nuclear spin of a proton can be parallel or antiparallel to the field.
• The state with the parallel spin has a lower energy level than the antiparallel one due to its alignment with the field.
• When a proton changes from one spin state to another, it either absorbs or emits a photon, which corresponds to a specific frequency and wavelength.

These transitions form the basis of nuclear magnetic resonance (NMR) techniques, widely used in medical imaging (MRI) and chemical analysis.

Electron Spin

Electron spin is another quantum property that affects how electrons behave in a magnetic field. Unlike protons, electrons have a negative charge, influencing their energy alignment:

• The spin of an electron can also be parallel or antiparallel to a magnetic field.
• In contrast to protons, the antiparallel spin state of an electron is lower in energy.
• Electrons transitioning between these spin states result in the absorption or emission of photons.

Understanding electron spin transitions is essential for explaining phenomena such as electron paramagnetic resonance (EPR) and various electronic applications, where the interplay of magnetic fields with electron spins enables high precision spectroscopy and analysis tools.

Electromagnetic Spectrum

The electromagnetic spectrum is the range of all types of electromagnetic radiation. Transition energies between spin states of protons and electrons in a magnetic field determine the frequency and wavelength of emitted or absorbed photons:

• The frequency of the photon's radiation can be calculated from the energy difference between spin states.
• This frequency dictates the position of the photon within the electromagnetic spectrum, ranging from radio waves to gamma rays.
• For proton transitions in a magnetic field, the resulting radiation typically falls within the radiofrequency range of the spectrum.
• Electron transitions often produce radiation in the microwave region due to their higher energy differences.

Recognizing these regions helps identify the applications and detection methods suitable for each type of spin transition, significantly impacting fields like telecommunications, medical imaging, and spectroscopy.