Hydrogen atoms are placed in an external magnetic field. The protons can make transitions between states in which the nuclear spin component is parallel and antiparallel to the field by absorbing or emitting a photon. What magnetic-field magnitude is required for this transition to be induced by photons with frequency $22.7$ MHz?

Verified step by step guidance

1. Understand the problem: The energy difference between the parallel and antiparallel spin states of the proton in a magnetic field is related to the frequency of the photon that induces the transition. This relationship is governed by the equation ΔE = hν, where ΔE is the energy difference, h is Planck's constant, and ν is the frequency of the photon.
2. Relate the energy difference to the magnetic field: The energy difference between the spin states in a magnetic field is given by ΔE = 2μ_BB, where μ_B is the magnetic moment of the proton and B is the magnetic field magnitude. Combine this with the photon energy equation to get 2μ_BB = hν.
3. Rearrange the equation to solve for the magnetic field magnitude B: B = (hν) / (2μ_B). This equation shows that the magnetic field magnitude is directly proportional to the photon frequency and inversely proportional to the proton's magnetic moment.
4. Substitute the known values into the equation: Use Planck's constant h = 6.626 × 10⁻³⁴ J·s, the given photon frequency ν = 22.7 MHz (convert to Hz: 22.7 × 10⁶ Hz), and the magnetic moment of the proton μ_B = 1.41 × 10⁻²⁶ J/T.
5. Perform the calculation: Plug the values into the equation B = (hν) / (2μ_B) to determine the magnetic field magnitude. Ensure all units are consistent (e.g., Hz for frequency, J·s for Planck's constant, and J/T for the magnetic moment).

Key Concepts

Nuclear Spin

Nuclear spin refers to the intrinsic angular momentum of atomic nuclei, which can exist in different orientations relative to an external magnetic field. For hydrogen, the proton can have its spin aligned parallel (lower energy state) or antiparallel (higher energy state) to the magnetic field. Transitions between these states involve the absorption or emission of energy, typically in the form of photons.

Photon Energy and Frequency

The energy of a photon is directly related to its frequency through the equation E = hν, where E is energy, h is Planck's constant, and ν is frequency. In the context of nuclear magnetic resonance, the energy difference between the parallel and antiparallel spin states corresponds to the energy of the photon required to induce a transition. Thus, knowing the frequency of the photon allows us to calculate the energy needed for the transition.

Magnetic Field Strength

The strength of an external magnetic field influences the energy levels of nuclear spin states. The energy difference between the two spin states is proportional to the magnetic field strength, described by the equation ΔE = γB, where ΔE is the energy difference, γ is the gyromagnetic ratio, and B is the magnetic field strength. To induce a transition at a specific photon frequency, the magnetic field must be adjusted to match the energy difference corresponding to that frequency.