Problem 4 - Atomic Physics

Question

Ratio of hyperfine splittings The spin and magnetic moment of the proton are \(\left(1 / 2,2.79 \mu_{\mathrm{N}}\right)\), of the deuteron \(\left(1,0.857 \mu_{\mathrm{N}}\right)\) and of \({ }^{3} \mathrm{He}\left(1 / 2,-2.13 \mu_{\mathrm{N}}\right)\). Calculate the ratio of the ground-state hyperfine splittings of (a) atomic hydrogen and deuterium and (b) atomic hydrogen and the hydrogen-like ion \({ }^{3} \mathrm{He}^{+}\).

Step-by-Step Solution

Answer

(a) Ratio for hydrogen and deuterium is approximately 1.63. (b) Ratio for hydrogen and 3He+ is approximately 1.31.

Step 1: Understand the Hyperfine Splitting Formula

The hyperfine splitting energy, \( E_{hf} \), is given by the formula \( E_{hf} \propto \mu \cdot I \), where \( \mu \) is the magnetic moment and \( I \) is the nuclear spin. For calculative purposes, constants which remain same across comparisons can be ignored.

Step 2: Set Up Comparisons for Ratio A (Hydrogen and Deuterium)

For hydrogen, \( \mu_p = 2.79\mu_N \) and \( I = 1/2 \). For deuterium, \( \mu_d = 0.857\mu_N \) and \( I = 1 \). Thus, the hyperfine splitting energy ratio \( R_{H/D} \) is \[ R_{H/D} = \frac{(2.79 \times 1/2)}{(0.857 \times 1)} \].

Step 3: Calculate Ratio A

Substituting the values, we have \[ R_{H/D} = \frac{2.79 \times 1/2}{0.857 \times 1} = \frac{1.395}{0.857} \approx 1.63 \].

Step 4: Set Up Comparisons for Ratio B (Hydrogen and 3He+)

For hydrogen, \( \mu_p = 2.79\mu_N \) and \( I = 1/2 \). For \( ^3He^+ \), \( \mu_{3He} = -2.13\mu_N \) and \( I = 1/2 \). Thus, the hyperfine splitting energy ratio \( R_{H/3He+} \) is \[ R_{H/3He+} = \left|\frac{(2.79 \times 1/2)}{(-2.13 \times 1/2)}\right| \].

Step 5: Calculate Ratio B

Substituting the values, we have \[ R_{H/3He+} = \left|\frac{2.79 \times 1/2}{-2.13 \times 1/2}\right| = \left|\frac{1.395}{-1.065}\right| \approx 1.31 \]. The negative sign is ignored since we are considering magnitude.

Key Concepts

Magnetic Moment

The magnetic moment is a fascinating property that arises from the motion of electric charges. In nuclei, this is due to the movement of protons and neutrons. Think of it as a tiny magnet created by the atomic nucleus.
The strength and orientation of this tiny magnet depend on nuclear properties, such as how the nucleons are arranged and their respective spins.

• The magnetic moment (\(\mu\)\u00f7f) is measured in terms of the nuclear magneton (\(\mu_N\)\u00f7f).
• Each type of atom or isotope has its unique magnetic moment, which influences hyperfine interactions.

For instance, the proton has a magnetic moment of 2.79 \(\mu_N\)\u00f7f, indicating a strong interaction in the presence of a magnetic field, and this affects the energy levels when investigating hyperfine splitting.

Nuclear Spin

The concept of nuclear spin involves the quantum mechanical spin of protons and neutrons within an atomic nucleus. Consider it the intrinsic angular momentum of a nucleus.
This property is crucial in determining the magnetic moment mentioned earlier and is denoted by \( I \)\u00f7f.

• Spin can take half-integer values, such as \(1/2\)\u00f7f, \(1\)\u00f7f, etc.
• It plays a pivotal role in nuclear magnetic resonance (NMR) and the hyperfine structure of atoms.

In the case of hydrogen, the simple atomic structure gives it a nuclear spin of \(1/2\)\u00f7f, whereas more complex atoms like deuterium exhibit a spin value of \(1\)\u00f7f, reflecting its different nuclear interactions.

Hydrogen

Hydrogen is the simplest and most abundant element in the universe. It has one proton and one electron, thus providing a straightforward model for studying atomic structure.
In hyperfine splitting, the interactions between the magnetic moments and spins of the hydrogen atom's components redefine its energy levels.

• The hyperfine structure arises due to the interaction between the electron magnetic moment and the nuclear magnetic moment.
• For hydrogen, the significant magnetic moment (2.79 \(\mu_N\)\u00f7f) of the proton significantly impacts hyperfine energy splitting.

Hydrogen’s simplicity offers crucial insights into quantum mechanics and hyperfine interactions when exploring hyperfine splitting ratios.

Deuterium

Deuterium is an isotope of hydrogen with a core that contains one proton and one neutron. This slight variation gives it unique properties compared to regular hydrogen.
The presence of the neutron adds complexity to the nuclear interactions, which is observable in hyperfine structures.

• The deuterium nucleus possesses a nuclear spin \( I = 1 \)\u00f7f, contrasting with hydrogen's \(1/2\)\u00f7f.
• Deuterium’s lesser magnetic moment (0.857 \(\mu_N\)\u00f7f) impacts its hyperfine splitting differently.

Understanding deuterium is crucial in fields like nuclear physics and spectroscopy due to its influence on understanding atomic interactions.

Helium-3

Helium-3 is a rare isotope of helium that plays a vital role in nuclear research. It consists of two protons and one neutron, making it distinct from the more common helium-4.
\(^3\)He notably possesses nuclear properties which differ significantly, impacting hyperfine interactions.

• Its nuclear spin is \(1/2\)\u00f7f, similar to hydrogen, yet it carries a considerable negative magnetic moment of -2.13 \(\mu_N\)\u00f7f.
• This results in unique hyperfine splitting behavior, critical for specialized fields like low-temperature physics.

Helium-3’s characteristics make it vital for applications like neutron detection and nuclear magnetic resonance studies.