Question

The Zeeman effect occurs in sodium just as in hydrogen - sodium's lone 3 s valence electron behaves much as hydrogen's 1.5. Suppose sodium atoms are immersed in a \(0.1 \mathrm{~T}\) magnetic field. (a) Into how many levels is the \(3 p_{1 / 2}\) level split? (b) Determine the energy spacing between these states. (c) Into how many lines is the \(3 p_{1 / 2}\) to \(3 s_{1 / 2}\) spectral line split by the field? (d) Describe quantitatively the spacing of these lines. (e) The sodium doublet \((589.0 \mathrm{nm}\) and \(589.6 \mathrm{nm}\) ) is two spectral lines. \(3 p_{3 n} \rightarrow 3 s_{1 / 2}\) and \(3 p_{1 / 2} \rightarrow 3 s_{1 / 2}\) which are split according to the two different possible spin-orbit ener gies in the \(3 p\) state (see Exercise 60 ). Detemine the splitting of the sodium doublet (the energy diff erence between the two photons). How does it compare with the line splitting of part (d), and why?

Short answer

The \(3 p_{1 / 2}\) level is split into two levels. The spectral line of \(3 p_{1 / 2}\) to \(3 s_{1 / 2}\) is split into three lines, symmetrically spaced. The splitting of the spectral line is much smaller than the splitting of the sodium doublet. The difference in the splitting is due to the difference in the cause of splitting, which is the interaction with a small magnetic field for the Zeeman splitting and the spin-orbit interaction for the sodium doublet.

Step-by-step solution

Splitting of energy levels

The splitting of energy levels depends on the Landé g-factor. For a \(3 p_{1 / 2}\) state, the Landé g-factor is 2. There are two possible orientations for the projection of the orbital angular momentum in presence of magnetic field, one parallel and one anti-parallel. So, the \(3 p_{1 / 2}\) level will be split into two levels.

Energy spacing calculation

The energy spacing between the states can be calculated with the formula \(\Delta E = g \mu_B B\), where g is the Landé g-factor, \(\mu_B\) is the Bohr magneton, and B is the magnetic field strength. Substituting the values, we can get the value of \(\Delta E\).

Splitting of spectral line

The spectral line of \(3 p_{1 / 2}\) to \(3 s_{1 / 2}\) will be split into three lines. This is because the \(3 s_{1 / 2}\) state has one value of magnetic quantum number and the \(3 p_{1 / 2}\) has two values.

Spacing of spectral lines

The spectral lines are symmetrically spaced. The middle line corresponds to the transition where there is no change in the magnetic quantum number (\(Δm = 0\)), and the other two lines correspond to the transitions where the magnetic quantum number changes by \(±1\). The energy difference between the lines is equal to \(ΔE\).

Splitting of sodium doublet

The splitting of the sodium doublet can be calculated by converting the wavelengths given to energy using the equation for the energy of a photon \(E = h c / λ\). The difference in energy is the splitting of the sodium doublet.

Comparison of line splitting

The splitting of the spectral line from step 3 seems much smaller than the splitting of the sodium doublet. This is because the line splitting in part (d) is the Zeeman splitting, which is caused by a small magnetic field, whereas the splitting of the sodium doublet is due to the spin-orbit interaction, which is a larger effect.

Key concepts

Spectral Line Splitting

When we observe the phenomenon of spectral line splitting, we are looking at the fascinating effect of how a magnetic field can influence the energy levels of atomic electrons. Think of spectral lines as unique fingerprints of elements that tell us about their electronic transitions. Normally, these lines are singular, but under the influence of a magnetic field, they can split into multiple components. This can tell us a lot about the structure of the atom and the interactions between its electrons and the external magnetic field.

For example, in the case of sodium within a 0.1 T magnetic field, the lone valence electron experiences a splitting of its energy levels, resulting in changes to the spectral lines that we can observe. Specifically, the transition from the 3p_1/2 to the 3s_1/2 level, instead of producing a single spectral line, creates a pattern of new lines—this is the observable Zeeman effect.

Landé g-factor

Dive into the Landé g-factor and you'll discover how this dimensionless number encapsulates the magnetic and angular momentum properties of an electron's orbit. It's a crucial player in the quantum world, providing insight into how those electron orbits are influenced by magnetic fields. The Landé g-factor varies depending on the electron's orbital and spin angular momentum states.

The g-factor is essential for calculating the Zeeman effect's impact on energy levels. For example, in sodium's 3p_1/2 state, the Landé g-factor of 2 is used to determine the energy shifts that occur due to the external magnetic field. This value helps us predict how much the spectral lines will split and the new energy levels the electrons will adopt.

Bohr Magneton

The Bohr magneton (\( \boldsymbol{\text{\textmu}_B} \)) is a key physical constant when it comes to atomic structure. It's named after Niels Bohr, and it quantifies the magnetic moment of an electron due to its orbital or spin motion. The Bohr magneton lets us calculate the energy differences that occur when an atom is exposed to a magnetic field.

In our textbook exercise, the Bohr magneton is instrumental in determining the energy spacing between new energy levels created by the Zeeman effect. The formula \( \boldsymbol{\text{\textDelta} E = g \text{\textmu}_B B} \) comes into play, linking the influence of a magnetic field with the resultant energy spacing.

Magnetic Quantum Number

In the quantum world, we use the magnetic quantum number to specify the orientation of an electron's angular momentum. This is like choosing a specific lane in a multi-lane highway; it distinguishes the different paths electrons can take around the nucleus. The number is related to the possible alignments of the angular momentum with a magnetic field, which can either be parallel or anti-parallel.

When dealing with the Zeeman effect, the magnetic quantum number informs us about the possible energy states. The sodium atom example shows that the magnetic quantum number, with its different values for the 3p_1/2 and 3s_1/2 levels, leads to the spectral line of a given transition being split into three distinct lines. Essentially, the magnetic quantum number helps predict the pattern of line splitting we can expect to see.

Spin-Orbit Interaction

The spin-orbit interaction is like a dance between an electron's spin and its motion around the nucleus, marrying the two to influence the energy levels within an atom. It accounts for fine structures in the spectral lines, giving rise to slight variations in energy that are not due to external factors like magnetic fields.

In our sodium doublet example, the spin-orbit interaction explains the energy difference between the two closely spaced spectral lines at 589.0 nm and 589.6 nm. This interaction is much stronger than the effect of a moderate external magnetic field, which is why the doublet splitting is significantly larger than what we see in the Zeeman effect's line splitting. It shows us how internal electron dynamics can have profound effects on an atom's energy landscape.