Additional: Limitations of Bohr Model

Failure for Multi-electron Atoms

Bohr's model was remarkably successful in explaining the spectrum of the hydrogen atom and hydrogen-like ions (like He$^+$, Li$^{++}$), which have only a single electron. However, it faced significant difficulties when applied to atoms with more than one electron (multi-electron atoms).

Limitations in Explaining Multi-electron Spectra

When Bohr attempted to apply his model to the helium atom (Z=2, with two electrons), even the simplest multi-electron atom, the calculations for the energy levels and spectral lines did not agree with the experimental observations. The complexity of the interactions between multiple electrons posed a challenge that the simple Bohr model could not handle.

In multi-electron atoms, the motion of each electron is influenced not only by the nucleus but also by the electrostatic repulsion from all other electrons. These electron-electron interactions are complex. Furthermore, the inner electrons "screen" the nuclear charge from the outer electrons, so the effective nuclear charge experienced by an outer electron depends on its position and the distribution of other electrons.

Bohr's model did not have a mechanism to account for these electron-electron interactions or the screening effect in a quantitative way. It treated the electron as moving in a fixed, central potential created only by the nucleus, which is a valid approximation only for a single-electron system where there are no other electrons to interact with or screen the nucleus.

While modifications were attempted (like using an effective nuclear charge for outer electrons), these were largely empirical adjustments and lacked the rigorous theoretical foundation that the model provided for hydrogen. The spectra of multi-electron atoms are much more complex than hydrogen's, with a larger number of lines, and Bohr's model was unable to predict these complex spectra accurately.

The development of quantum mechanics was necessary to provide a proper theoretical framework for understanding the structure and spectra of multi-electron atoms, where the concept of electron orbits is replaced by electron probability distributions (orbitals), and electron-electron interactions are handled through more sophisticated approximations.

Inability to Explain Fine Structure

Even for the hydrogen atom, upon closer examination of the spectral lines with high-resolution spectrometers, it was discovered that some lines which appeared as single lines under low resolution were actually composed of two or more closely spaced lines. This splitting of spectral lines is called the fine structure of the atomic spectrum.

Illustration of fine structure, where a single line splits into multiple closely spaced lines.

Origin of Fine Structure (Beyond Bohr's Model)

Bohr's model predicted that the energy levels of the hydrogen atom depend only on the principal quantum number $n$. This would lead to spectral lines with precisely defined wavelengths corresponding to energy differences $E_{n_i} - E_{n_f}$. The observed fine structure indicated that the energy levels of hydrogen are slightly split, meaning there are subtle energy differences within a given principal quantum number $n$ that Bohr's model did not account for.

The fine structure splitting is due to relativistic effects and the intrinsic angular momentum of the electron, called electron spin.

• Relativistic Correction: At the high speeds that electrons achieve in inner orbits, relativistic effects become significant and cause small corrections to the energy levels. These corrections depend on the orbital angular momentum quantum number, not just the principal quantum number.
• Spin-Orbit Interaction: The electron has an intrinsic angular momentum (spin) and an associated magnetic dipole moment. As the electron orbits the nucleus, it experiences a magnetic field due to its motion in the nucleus's electric field (from the electron's perspective, the nucleus is orbiting it). The interaction between the electron's spin magnetic moment and this orbital magnetic field leads to a small energy splitting depending on the relative orientation of the spin and orbital angular momenta (parallel or anti-parallel). This is called spin-orbit coupling.

These effects cause the energy levels for $n>1$ to split into multiple sub-levels that are very close in energy. Transitions between these sub-levels result in the fine structure of the spectral lines.

Bohr's model, being non-relativistic and not incorporating electron spin, was fundamentally unable to explain the fine structure of the hydrogen spectrum. Explaining fine structure required relativistic quantum mechanics and the inclusion of electron spin.

Zeeman and Stark Effect

Further experiments revealed that atomic spectral lines exhibit splitting when the atoms are subjected to external electric or magnetic fields. These effects are known as the Zeeman effect and the Stark effect, respectively.

Zeeman Effect

The Zeeman effect is the splitting of spectral lines when the light-emitting (or absorbing) atoms are placed in an external magnetic field. A single spectral line is observed to split into several closely spaced lines.

Splitting of a spectral line in the presence of an external magnetic field (Zeeman effect).

Explanation: The energy levels of an atom are affected by an external magnetic field because the orbiting electrons (and their spins) possess magnetic dipole moments. When these magnetic dipoles are placed in a magnetic field, they interact with the field, and their energy depends on their orientation relative to the field ($U = -\vec{M} \cdot \vec{B}$). This interaction energy shifts the original energy levels, causing them to split into sub-levels. Transitions between these split energy sub-levels result in the splitting of the spectral lines. The number and spacing of the split lines depend on the strength of the magnetic field and the magnetic properties of the atomic states involved (orbital and spin magnetic moments).

The simple Bohr model predicted discrete energy levels but did not include the concept of electron magnetic moments or how they would interact with an external magnetic field to split the energy levels. Therefore, it could not explain the Zeeman effect. Explaining the Zeeman effect requires the quantisation of angular momentum and associated magnetic moments, which is handled correctly by quantum mechanics.

Stark Effect

The Stark effect is the splitting of spectral lines when the light-emitting (or absorbing) atoms are placed in an external electric field. Similar to the Zeeman effect, a single spectral line splits into multiple components.

Explanation: Atoms have electric dipole moments (though for a neutral atom like hydrogen, the average dipole moment is zero in the absence of an external field). An external electric field induces or interacts with the atom's electric dipole moment (or the quadrupole moment, etc.), perturbing the electron's motion and causing small shifts and splitting of the energy levels. Transitions between these split energy sub-levels lead to the splitting of the spectral lines. The magnitude of the splitting depends on the strength of the electric field and the electric properties of the atomic states.

Bohr's model also failed to explain the Stark effect. The interaction of the electron's orbit with an external electric field leads to changes in energy levels that are not accounted for in the basic Bohr theory. Quantum mechanics, using perturbation theory, successfully explains both the Zeeman and Stark effects.

Overall Limitations

In summary, while Bohr's model was a crucial step in the development of atomic theory and successfully explained the hydrogen spectrum by introducing quantised energy levels and orbits, its limitations became apparent when:

• Applied to multi-electron atoms.
• Examining the fine details of the hydrogen spectrum (fine structure).
• Considering the behaviour of atoms in external electric and magnetic fields (Zeeman and Stark effects).
• It also did not fully explain why the electron does not radiate in stationary orbits, other than stating it as a postulate.
• It did not incorporate the wave nature of the electron in its fundamental postulates (although de Broglie later provided an interpretation for the quantisation of angular momentum).

These limitations highlighted the need for a more comprehensive and fundamental theory of atomic structure, which was provided by quantum mechanics, based on the wave nature of matter, probability distributions, and sophisticated mathematical tools.