Question

The well-known sodium doublet is two yellow spectral lines of very close wavelength. \(589.0 \mathrm{nm}\) and \(589.6 \mathrm{nm} .\) lt is caused by splitting of the \(3 p\) energy level. due to the spin-orbit interaction. In its ground state, sodium's single valence electron is in the \(3 s\) level. It may be excited to the next higher level, the \(3 p\), then emit a photon as it drops back to the \(3 s\). However. the \(3 \rho\) is actually two levels. in which \(L\) and \(S\) are aligned and antialigned. IIn the notation of Section 8.7 these are. respectively. the \(3 p_{3 / 2}\) and the \(3 p_{1 n}\) ) Because the transitions stan from slightly different initial energies yet have identical final energies(the \(3 s\) having no orbital angular momentum to lead to spin- orbit interaction), there are two differenl wavelengths possible for the emitted photon. Calculate the difference in energy between the two photons. From this, obtain a rough value of the average strength of the internal magnetic field experienced by sodium's valence electron.

Short answer

To calculate the energy difference between the two photons emitted when a sodium's valence electron transitions from the \(3p\) (split into \(3p_{3/2}\) and \(3p_{1/2}\)) to the \(3s\) level, use the photon energy formula with the given wavelengths of the spectral lines. Then, use this energy difference to calculate the average strength of the internal magnetic field experienced by the electron, applying the relationship between energy difference and magnetic field strength.

Step-by-step solution

Calculate the Energy of the Photons

Firstly, apply the formula for energy of photons \(E = h\nu\), where \(h\) is the Planck's constant (\(6.63 \times 10^{-34}\) Joule second) and \(\nu\) is the frequency of the photon. However, as the exercise provided the wavelength instead of the frequency, convert frequency to wavelength using the formula \(\nu = c / \lambda\), where \(c\) is the speed of light (\(3.00 \times 10^{8} m/s\)) and \(\lambda\) is the wavelength. Consequently, the formula for energy becomes \(E = hc/ \lambda\). Plug the given wavelengths into this formula to calculate the energy for each photon.

Calculate the Energy Difference

Secondly, calculate the energy difference between the two photon by subtracting the energy of the photon with shorter wavelength from that with longer wavelength.

Calculate the Magnetic Field

Finally, use the computed energy difference to calculate the average strength of the magnetic field. The energy difference is equal to \(\Delta E = g\mu_B B\), where \(\mu_B\) is the Bohr magneton (\(9.27 \times 10^{-24} J/T\)), \(B\) is the internal magnetic field, and \(g\) is the Landé g-factor (approximately 1 for many electron transitions). Thus, \(B = \Delta E/ (g\mu_B)\). Substitute the calculated energy difference and given values into this formula to obtain the average strength of the internal magnetic field.

Key concepts

Spectral Lines

Spectral lines are like fingerprints for elements; they are unique patterns of light emitted or absorbed by an element. Each element produces a distinct set of lines across the spectrum as a result of electrons transitioning between energy levels within an atom. In the case of the sodium doublet, we observe two close, but distinct yellow spectral lines.

These lines are created when an electron falls from a higher energy level to a lower one, releasing energy in the form of light. The specific wavelengths of these lines (589.0 nm and 589.6 nm for the sodium doublet) depend on the energy difference between the two levels. As the energy difference falls within a very narrow range, the resulting spectral lines are very close to each other in the visible spectrum. Analyzing such spectral lines provides valuable information about the atomic structure and the processes occurring within an atom.

Spin-Orbit Interaction

Spin-orbit interaction is a quantum mechanical phenomenon that describes the coupling between an electron's spin and its orbital motion around the nucleus. This interaction causes energy levels to split into sub-levels. For sodium, the '3p' energy level splits into two close sub-levels: '\(3 p_{3/2}\) and '\(3 p_{1/2}\).'

This splitting occurs because the electron behaves like a tiny magnet due to its spin. The orbital motion of the electron generates a magnetic field, and the interaction between the magnetic dipole of the electron spin and this magnetic field leads to the energy level splitting, resulting in the two closely spaced spectral lines seen as the sodium doublet. The degree of splitting is influenced by the strength of the internal magnetic field experienced by the valence electron, which we can infer from the sodium doublet's spectral lines.

Planck's Constant

Planck's constant is a fundamental constant denoting the smallest action that can occur in nature. Its value is approximately \(6.63 \times 10^{-34}\text{ Js}\). In the context of spectral lines, Planck's constant relates to the energy of photons emitted or absorbed during electronic transitions.

When electrons jump between two energy levels, they absorb or emit energy equivalent to the product of Planck's constant and the frequency of the associated photon: \(E = hu\). Thus, when studying spectral lines, Planck's constant provides a bridge between the observed wavelengths of light and their corresponding energies. The smaller the action, such as an electron transitioning between two very close energy levels, the finer the spectral details that can be observed, which is crucial for understanding the sodium doublet.

Bohr Magneton

The Bohr magneton, denoted by \(\mu_B\), is a physical constant that describes the magnetic moment of an electron resulting from its angular momentum. It is defined as \(\mu_B = \frac{e\hbar}{2m_e}\), where \(e\) is the elementary charge, \(\hbar\) is the reduced Planck's constant, and \(m_e\) is the electron mass. The value of the Bohr magneton is approximately \(9.27 \times 10^{-24} J/T\).

The significance of the Bohr magneton becomes evident when investigating the spin-orbit interaction, as it helps calculate the difference in energy caused by this interaction. By measuring the energy difference between the spectral lines of the sodium doublet and using the Bohr magneton, we can estimate the strength of the internal magnetic field that the valence electron is subjected to. This allows us to explore the intrinsic magnetic properties of electrons and their interactions within atoms, providing a deeper understanding of atomic physics and quantum mechanics.