The Zeeman effect is the modification of an atomic spectrum by the application of a strong magnetic field. It arises from the interaction between applied magnetic fields and the magnetic moments due to orbital and spin angular momenta (recall the evidence provided for electron spin by the SternGerlach experiment, Section 9.8). To gain some appreciation for the so-called normal Zeeman effect, which is observed in transitions involving singlet states, consider a pelectron, with \(l=1\) and \(m_{l}=0, \pm 1\). In the absence of a magnetic field, these three states are degenerate. When a field of magnitude \(B\) is present, the degeneracy is removed and it is observed that the state with \(m_{l}\) \(=+1\) moves up in energy by \(\mu_{\mathrm{B}} B\), the state with \(m_{l}=0\) is unchanged, and the state with \(m_{l}=-1\) moves down in energy by \(\mu_{\mathrm{B}} B\), where \(\mu_{\mathrm{B}}=e \hbar / 2 m_{e}=9.274 \times\) \(10^{-24} \mathrm{~J} \mathrm{~T}^{-1}\) is the Bohr magneton (see Section 15.1). Therefore, a transition between a \({ }^{1} \mathrm{~S}_{6}\) term and a \({ }^{1} \mathrm{P}_{1}\) term consists of three spectral lines in the presence of a magnetic field where, in the absence of the magnetic field, there is only one. (a) Calculate the splitting in reciprocal centimetres between the three spectral lines of a transition between a \({ }^{1} \mathrm{~S}_{0}\) term and a \({ }^{1} \mathrm{P}_{1}\) term in the presence of a magnetic field of \(2 \mathrm{~T}\) (where I \(\left.\mathrm{T}=1 \mathrm{~kg} \mathrm{~s}^{-2} \mathrm{~A}^{-1}\right)\), (b) Compare the value you calculated in (a) with typical optical transition wavenumbers, such as those for the Balmer series of the \(\mathrm{H}\) atom. Is the line splitting caused by the normal Zeeman effect relatively small or relatively large?

Step by step solution

01

Understand the normal Zeeman effect

The normal Zeeman effect is the splitting of degenerate spectral lines of an atom when an external magnetic field is applied. This occurs due to the different energy levels the electron states attain because of their magnetic moments interacting with the magnetic field. For a p-electron with orbital quantum number l = 1, states with magnetic quantum numbers ml = 0, +1, -1 are degenerate (have the same energy) without an external magnetic field. When a magnetic field is present, the energy shifts for ml = +1 and ml = -1 states, while ml = 0 state remains unchanged.

02

Calculate the energy shift

The energy shift due to the Zeeman effect can be calculated using the formula \(\Delta E = m_l \mu_B B\), where \(\Delta E\) is the change in energy, \(\mu_B\) is the Bohr magneton, \(\mu_B = 9.274 \times 10^{-24} \text{ J T}^{-1}\), \(\text{m}_l\) is the magnetic quantum number, and \(\text{B}\) is the magnetic field strength. Using \(\text{B} = 2 \text{ T}\) and the given value of the Bohr magneton, calculate the energy shift for each state with \(\text{m}_l = +1\) and \(\text{m}_l = -1\). The state \(\text{m}_l = 0\) remains unchanged, so its energy shift is zero.

03

Calculate the frequency shift

The energy shift can be converted to a frequency shift using the relationship \(\Delta E = h \Delta \u\), where \(\text{h}\) is Planck's constant, \(\text{h} = 6.626 \times 10^{-34} \text{ J s}\). Solving for \(\Delta \u\) gives \(\Delta \u = \frac{\Delta E}{h}\).

04

Calculate the wavenumber shift

The frequency shift can be related to the wavenumber shift using the relationship \(\Delta \u = c \Delta \bar{\u}\), where \(\text{c}\) is the speed of light in vacuum, \(\text{c} = 2.998 \times 10^{8} \text{ m s}^{-1}\), and \(\Delta \bar{\u}\) is the wavenumber shift in \(\text{cm}^{-1}\). The wavenumber shift can be calculated by dividing the frequency shift by the speed of light and converting meters to centimeters, \(\Delta \bar{\u} = \frac{\Delta \u}{c} \times 10^{2}\).

05

Compute the splitting in reciprocal centimeters

Using the derived formula from the previous steps, compute the wavenumber shift for \(\text{m}_l = +1\) and \(\text{m}_l = -1\) states. This will give the splitting in reciprocal centimeters between the three spectral lines.

06

Compare the splitting with typical transition wavenumbers

To determine whether the Zeeman effect causes a relatively small or large split, compare the calculated wavenumber splitting with typical optical transition wavenumbers such as those for the Balmer series of hydrogen. In general, the Balmer series transitions are on the order of thousands of \(\text{cm}^{-1}\), so if the Zeeman splitting is much smaller than this, it is considered relatively small.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Atomic Spectrum

The atomic spectrum refers to the distinct lines of color or wavelengths that appear when an atom's electrons transition between energy levels. In effect, this colorful barcode is unique to each element, much like a fingerprint. Every emission or absorption line corresponds to a specific energy change within the atom, and these spectral lines are observable both in the laboratory and in celestial objects.

When subjected to a magnetic field, such as in the case of the Zeeman effect, the spectral lines can split or shift, revealing further underlying structure that isn't apparent in the absence of the magnetic field. This phenomenon underlines the importance of the atomic spectrum in understanding atomic structure and quantum mechanics.

Magnetic Moments

Magnetic moments are fundamental to the understanding of the Zeeman effect. They're essentially a vector quantity representing the strength and direction of an atom's magnetism. These moments arise from the motion of charged particles such as electrons within atoms, contributing to both orbital and spin angular momenta. The interaction with external magnetic fields alters their usual energy states.

During the Zeeman effect, the magnetic field interacts with the magnetic moments of the electrons, resulting in the energy level shifts that lead to the spectral line splitting. The precise way in which these lines split depends on the orientation of the magnetic moments relative to the field, illustrating the connection between electron configuration and the atom's response to a magnetic environment.

Orbital Quantum Number

The orbital quantum number, denoted by the symbol 'l', is integral to specifying the shape of an electron's orbit in an atom and its angular momentum. It can take on integer values ranging from 0 to (n-1), where 'n' is the principal quantum number representing the main energy level of an electron.

In the context of the Zeeman effect, the value of the orbital quantum number plays a key role in determining the magnetic moment associated with an electron's orbital angular momentum. This, in turn, influences how much each energy state will shift when an external magnetic field is applied. For instance, a p-electron with an orbital quantum number of 1 can have three possible magnetic quantum numbers, leading to the distinctive triplet split observed in the normal Zeeman effect.

Bohr Magneton

The Bohr magneton, symbolized as \( \mu_B \), is a physical constant that enables the quantification of the electron's magnetic moment. It is defined as \( \mu_B = \frac{e\hbar}{2m_e} \), where 'e' is the elementary charge, '\(\hbar\)' is the reduced Planck constant, and 'm_e' is the electron mass. With a value of approximately \(9.274 \times 10^{-24} \text{ J T}^{-1}\), the Bohr magneton serves as a unit for expressing magnetic moments in atomic-scale systems.

In practical terms, it's this very constant that sets the scale for the energy shifts seen in the Zeeman effect. For each electron state with a specific magnetic quantum number, the change in energy due to an external magnetic field is a whole-number multiple of \(\mu_B\). This provides a straightforward way to predict how energy levels and spectral lines will split in a given magnetic field.

Spectral Lines Splitting

Spectral lines splitting is the observable manifestation of the Zeeman effect in the atomic spectrum. When an external magnetic field is not present, certain energy states in the atom are degenerate, meaning they have the same energy. However, once a magnetic field is introduced, these states can split into different energy levels based on their magnetic quantum numbers.

For a transition involving a singleton state, like the \(p\)-electron with \(l=1\) in the exercise, this split results in one spectral line branching out into three. This splitting allows physicists to study the intricate magnetic interactions within atoms and even to determine properties of the magnetic field itself. When examining Zeeman splitting in an atomic spectrum, it not only provides insights into the electronic structure of atoms but also into the nature of light and the quantum mechanical rules governing atomic behavior.