Problem 3 - University Physics with Modern Physics

Question

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 \(\mathrm{MHz} ?\)

Step-by-Step Solution

Answer

Required magnetic field is approximately 0.53 Tesla.

Step 1: Understand the Problem Context

We need to determine the magnetic field strength required to induce transitions in hydrogen nuclei when exposed to photons of a specific frequency (22.7 MHz). The transitions occur between states where the nuclear spin is parallel and antiparallel to the field.

Step 2: Identify Relevant Physics

The energy difference between the two spin states of the proton in a magnetic field can be expressed using the equation: \(\Delta E = h u\), where \( u \) is the frequency of the associated photon. Also, the energy difference is related to the magnetic field \(B\) by \(\Delta E = 2\mu_B B\), where \(\mu_B\) is the magnetic moment of the proton.

Step 3: Relate Energy Difference and Magnetic Field

Since \(\Delta E = h u = 2 \mu_B B\), we can express the magnetic field \(B\) in terms of \(u\): \[ B = \frac{h u}{2 \mu_B} \]We know \(h\) (Planck's constant) is approximately \(6.626 \times 10^{-34} \text{ J s}\), and \(\mu_B\) is approximately \(1.4105 \times 10^{-26} \text{ J T}^{-1}\).

Step 4: Calculate the Magnetic Field Strength

Insert the given photon frequency (\(22.7 \text{ MHz} = 22.7 \times 10^6 \text{ Hz}\)) into the equation:\[ B = \frac{6.626 \times 10^{-34} \times 22.7 \times 10^6}{2 \times 1.4105 \times 10^{-26}} \]Perform the calculation to find:\[ B \approx 0.53 \text{ T} \]

Step 5: Conclusion

The magnetic field required to induce the proton transition with a 22.7 MHz frequency photon is approximately 0.53 Tesla.

Key Concepts

Hydrogen Atom Transitions

In the study of atomic physics, understanding how hydrogen atoms transition between different energy states is fundamental. These transitions occur when photons - light particles - interact with the atoms.

When a hydrogen atom is in a magnetic field, its nucleus, particularly the protons, can exist in different spin states. The two main states include:

• Parallel spin state: where the nuclear spin aligns with the direction of the magnetic field.
• Antiparallel spin state: where the nuclear spin opposes the direction of the field.

The process of flipping between these states involves transitions that require or release energy. This energy corresponds to the absorption or emission of photons, often of specific frequencies.

Such transitions are not just theoretical; they have practical applications. For instance, magnetic resonance imaging (MRI) exploits these principles to visualize internal structures of the human body.

Nuclear Spin States

Nuclear spin states refer to how the nucleus's intrinsic angular momentum is oriented, specifically in the presence of an external magnetic field. Protons, being particles with a spin, will naturally orient themselves in relation to the magnetic lines of force.

In the context of a magnetic field, a hydrogen nucleus can exhibit:

• Spin parallel to the magnetic field, which is a lower energy state.
• Spin antiparallel to the magnetic field, which is a higher energy state.

These spin states are quantized, meaning they only take on specific values. Transitions between these states necessitate the absorption of energy (via photons) when moving to a higher energy state or emission of energy when falling to a lower energy state. Understanding these spin states is crucial, as it forms the basis for technologies like Nuclear Magnetic Resonance (NMR) used in chemistry and medicine.

Photon Frequency

Photon frequency is a crucial concept when discussing transitions in atomic systems like hydrogen. A photon has both energy and a corresponding frequency, and these properties are directly linked through Planck's law, expressed as \[E = h \cdot u\]where \(E\) is energy, \(h\) is Planck's constant, and \(u\) (or \(u\)) is the frequency.

The frequency of a photon determines whether its energy is suitable for interacting with particles like protons, inducing transitions. In the case of our hydrogen atom example, a photon with a frequency of 22.7 MHz is necessary to cause a transition between spin states.

Higher frequency photons carry more energy, suitable for inducing changes in atomic structures, whereas lower frequency photons have less energy. This relationship underscores many fields, such as spectroscopy, where the frequency of absorbed or emitted light gives clues about atomic structures and interactions.

Energy Difference

The energy difference between two nuclear spin states is at the core of understanding atomic transitions in a magnetic field. In a magnetic field, the energy difference \(\Delta E\) between the parallel and antiparallel spin states is related to the magnetic field strength and the magnetic moment of the proton.

This relationship can be captured through this equation:\[\Delta E = 2 \mu_B B\]Here, \(\Delta E\) represents the energy needed to flip the spin state, \(B\) is the magnetic field, and \(\mu_B\) is the magnetic moment.

The necessity for a photon with a specific frequency arises from this energy difference. The photon's energy must match \(\Delta E\) to facilitate a transition.

This linkage forms the basis for determining the magnetic field required for specific transitions, exemplified by our exercise which calculates a magnetic field strength allowing for transitions with photons of 22.7 MHz frequency. Understanding these energy differences is vital in manipulating and measuring atomic states through technologies like spectroscopy and magnetic imaging.