Table of Contents
From Rutherford to Bohr–Sommerfeld
Rutherford’s nuclear model (tiny, positive nucleus; electrons around it) explained scattering experiments but not how electrons are arranged or why atoms emit line spectra.
The Bohr and later Bohr–Sommerfeld models are early attempts to answer:
- Why do atoms have stable electron orbits?
- Why are only certain energies (and thus certain spectral lines) observed?
- How can we connect atomic structure with observed spectra, especially of hydrogen?
They introduce quantized electron orbits and form an important step between classical physics and the modern quantum mechanical model.
The Bohr Model: Quantized Circular Orbits
Basic Assumptions
For the hydrogen atom (one electron, one proton), Bohr proposed:
- Electrons move in circular orbits around the nucleus, similar to planets around the sun.
- Only certain orbits are allowed: In these orbits the electron does not emit radiation, even though it is accelerating. These are the stationary states.
- Angular momentum is quantized:
$$
m_e v r = n \hbar \quad\text{with}\quad n = 1,2,3,\dots
$$
where
$m_e$ = electron mass,
$v$ = orbital speed,
$r$ = orbit radius,
$n$ = principal quantum number,
$\hbar = \dfrac{h}{2\pi}$, $h$ = Planck’s constant.
- Light is emitted or absorbed only when the electron jumps between orbits. The energy of the photon is
$$
\Delta E = E_{\text{final}} - E_{\text{initial}} = h\nu
$$
$E$ = energy of the electron in an orbit, $\nu$ = frequency of radiation.
These assumptions are not derived from classical physics; they are postulates introduced to fit experimental observations (especially spectral lines).
Allowed Radii and Energies in the Bohr Model
Balancing Coulomb attraction and centripetal force for a circular orbit, and using the angular momentum condition, Bohr derived for hydrogen-like atoms (one electron, nuclear charge $+Ze$):
- Radius of the $n$-th orbit:
$$
r_n = a_0 \,\frac{n^2}{Z}
$$
where $a_0$ is the Bohr radius,
$$
a_0 \approx 0.529\,\text{Å} = 0.529\times 10^{-10}\,\text{m}
$$
- Energy of the electron in the $n$-th orbit (for hydrogen, $Z=1$):
$$
E_n = - \frac{13.6\,\text{eV}}{n^2}
$$
The negative sign indicates the electron is bound (lower energy than a free electron at infinite distance, defined as $E=0$).
Key points:
- Each orbit has a fixed radius and fixed energy.
- The ground state is $n=1$; excited states are $n=2,3,\dots$
- The electron can only have these discrete energies, not values in between.
Explaining Line Spectra (Hydrogen as Example)
Hydrogen’s emission spectrum consists of lines (Balmer series, Lyman series, etc.). Each line corresponds to a transition between two allowed energy levels.
For a transition from level $n_i$ to $n_f$ ($n_i > n_f$), Bohr’s model gives:
$$
\Delta E = E_{n_f} - E_{n_i} = h\nu
$$
Using $E_n = -\dfrac{13.6\,\text{eV}}{n^2}$, the frequency and wavelength of emitted light are:
- Frequency:
$$
\nu = \frac{13.6\,\text{eV}}{h}\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)
$$
- Wavelength (using $c = \lambda \nu$):
$$
\frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)
$$
where $R_H$ is the Rydberg constant for hydrogen.
This formula reproduces the empirical Rydberg formula for hydrogen’s spectral lines and gives a strong justification for energy quantization.
Shortcomings of the Bohr Model
While very successful for hydrogen, the Bohr model has limitations:
- Works quantitatively only for one-electron systems (H, He$^+$, Li$^{2+}$, …).
- Cannot correctly describe spectra of multi-electron atoms.
- Treats electrons as small particles on well-defined circular orbits, which conflicts with later quantum principles.
- Cannot explain fine structures in spectra (small splittings of lines).
- Angular momentum quantization is too simple (only one quantum number, $n$).
To address some of these issues—especially the detailed structure of spectral lines—Sommerfeld extended Bohr’s model.
The Bohr–Sommerfeld Model: Elliptical Orbits and Additional Quantization
Extension to Elliptical Orbits
Sommerfeld generalized Bohr’s circular orbits to elliptical orbits, still with the nucleus at one focus. This allowed:
- Different possible shapes for orbits with the same principal quantum number $n$.
- A partial explanation of fine structure (small splittings) in hydrogen spectra.
In the Bohr–Sommerfeld model:
- The electron’s path can be a closed ellipse instead of just a circle.
- The shape of the ellipse (how elongated it is) is related to another quantum number.
Additional Quantum Numbers
Sommerfeld introduced additional quantization rules for motion in the radial and angular directions. These lead conceptually to multiple quantum numbers:
- Principal quantum number $n$: roughly related to the size and energy of the orbit (kept from Bohr).
- Azimuthal (or orbital) quantum number $l$: related to the shape (eccentricity) of the orbit.
In the full quantum theory (not yet here), $l$ can take values
$$
l = 0,1,2,\dots,(n-1)
$$
In the Bohr–Sommerfeld picture, each pair $(n,l)$ corresponds to a distinct elliptical orbit with a particular shape and angular momentum.
Later developments in quantum mechanics reinterpret these quantum numbers in terms of wavefunctions and orbitals, not classical orbits, but the labels $n$ and $l$ persist.
Relativistic Corrections and Fine Structure
Sommerfeld also:
- Included relativistic effects: electron mass increases slightly with speed (from special relativity).
- Noted that for more elliptical orbits, the electron comes closer to the nucleus and moves faster near perihelion, enhancing relativistic effects.
As a result:
- Orbits with different $l$ (different shapes) but the same $n$ have slightly different energies.
- This leads to a splitting of spectral lines (fine structure) that the simple Bohr model cannot explain.
The Bohr–Sommerfeld model thus improved the agreement between theory and observed spectra of hydrogen, though not completely.
Conceptual Features and Historical Role
Old Quantum Theory
Bohr’s and Sommerfeld’s models are part of the so-called old quantum theory:
- Classical mechanics is still used to describe the motion of electrons (orbits).
- Quantum features are added by imposing quantization conditions (allowed orbits, quantized angular momentum).
- There is no full wave description of electrons; particles still follow definite paths.
This approach is semi-classical: classical motion plus quantum “rules”.
Achievements
The Bohr–Sommerfeld model:
- Explained hydrogen’s line spectrum and gave a theoretical basis for the Rydberg formula.
- Introduced the idea of quantized energy levels in atoms.
- Suggested a hierarchy of quantum numbers ($n$, $l$), anticipating the quantum numbers in the modern model.
- Gave a partial explanation of fine structure in the hydrogen spectrum.
These achievements strongly supported the idea that energy and angular momentum in atoms are quantized.
Fundamental Limitations
Despite improvements over Rutherford’s purely classical picture, the Bohr–Sommerfeld model has serious problems:
- Orbits are not stable or well-defined in the full framework of quantum mechanics; the idea of exact classical paths is abandoned.
- Cannot handle spectra of more complex atoms in a consistent way.
- Cannot describe many phenomena (e.g., Zeeman effect in full generality, chemical bonding).
- Contains internal inconsistencies when pushed beyond hydrogen.
These limitations motivated the development of modern quantum mechanics, where electrons are described by wavefunctions and orbitals rather than classical orbits.
Connection to the Modern Quantum Mechanical Model
In the later quantum mechanical model:
- Electrons are not on fixed orbits but in orbitals (probability distributions).
- The quantum numbers $n$ and $l$ (and additional ones) appear again, but now with a clear wave-mechanical meaning.
- Energy levels and selection rules for transitions are derived from solving Schrödinger’s equation, not imposed as separate postulates.
The Bohr–Sommerfeld model is therefore best viewed as:
- A historical bridge between classical physics and quantum mechanics.
- The first model to quantitatively connect atomic structure with spectral lines.
- A model that introduces quantum numbers and energy quantization, concepts that remain fundamental even though the picture of electrons on orbits is replaced.