Rutherford-Bohr Atomic Model & Old Quantum Theory

1. J.J. Thomson's Plum-Pudding Model (1904)

Following his discovery of the electron (1897), J.J. Thomson proposed that an atom consists of a uniform sphere of positive electrostatic charge of atomic radius $R \sim 10^{-10}\,\text{m}$ (1 Å), within which tiny negative electrons are embedded like plums in a pudding.

Classical Predictions: Because positive charge is diffuse across the entire atomic volume, the maximum electric field inside the atom is weak ($E_{max} = \frac{Q}{4\pi\varepsilon_0 R^2} \sim 10^{11}\,\text{V/m}$). An energetic $\alpha$-particle ($q = +2e$, mass $m_\alpha \approx 7300 m_e$, $E_k \sim 5\text{ to }8\,\text{MeV}$) passing through a thin gold foil would experience only tiny deflections ($\theta < 1^\circ$). Multiple scattering could produce at most average deflections of a few degrees ($< 3^\circ$).

2. The Geiger-Marsden Experiments (1909–1911)

Under Ernest Rutherford's direction, Hans Geiger and Ernest Marsden directed a collimated beam of $5.5\,\text{MeV}$ $\alpha$-particles from a bismuth-214 radioactive source through an ultra-thin gold foil (thickness $\sim 400\,\text{nm}$, roughly $1000$ atoms thick) and recorded scintillations on a zinc sulfide ($\text{ZnS}$) phosphorescent screen.

The Shocking Discovery: While the vast majority of $\alpha$-particles passed straight through undeflected, roughly 1 in 8,000 $\alpha$-particles was deflected by angles greater than $90^\circ$, and some bounced directly backward ($\theta \approx 180^\circ$)! Rutherford famously recounted: "It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you."

Thomson's diffuse atom could never produce the colossal electrostatic field required to reverse a massive energetic $\alpha$-particle.

1. Coulomb Hyperbolic Trajectory Kinematics

Rutherford (1911) proposed that all the positive charge $+Z e$ and virtually all the mass of the atom are concentrated in a tiny central core termed the nucleus, surrounded by a cloud of orbiting electrons.

Consider an $\alpha$-particle (charge $q_1 = +2e$, mass $m_\alpha$) fired with initial speed $v_0$ and impact parameter $b$ (the perpendicular distance from the nucleus to the initial asymptotic velocity line) toward a stationary gold nucleus of charge $q_2 = +Z e$. The repulsive Coulomb force is:

Because the force is central, angular momentum $L = m_\alpha v_0 b = m_\alpha r^2 \dot{\phi}$ is strictly conserved. Integrating Newton's Second Law along the axis of symmetry yields the fundamental relation between impact parameter $b$ and scattering deflection angle $\theta$:

where $K = \frac{1}{2} m_\alpha v_0^2$ is the kinetic energy of the incoming particle.

2. Derivation of the Rutherford Differential Cross-Section

Particles incident with impact parameters between $b$ and $b + db$ through an annular ring of area $d\sigma = 2\pi b |db|$ are scattered into solid angle $d\Omega = 2\pi \sin\theta d\theta$:

Differentiating $b(\theta)$:

Substituting into the cross-section and using $\sin\theta = 2 \sin(\theta/2) \cos(\theta/2)$:

This is the celebrated Rutherford Scattering Formula. Key testable predictions verified experimentally by Geiger and Marsden:

1. Scattering intensity scales inversely as the fourth power of the sine of half the angle: $N(\theta) \propto \csc^4(\theta/2)$.
2. Scattering scales inversely with the square of incident kinetic energy: $N \propto K^{-2}$.
3. Scattering scales with the square of target nuclear charge: $N \propto Z^2$.
4. Scattering scales linearly with foil thickness $t$.

3. Nuclear Dimension: Distance of Closest Approach ($d_{min}$)

In a head-on collision ($b = 0, \theta = 180^\circ$), the $\alpha$-particle decelerates until its entire initial kinetic energy is converted into electrostatic potential energy at the turning point $d_{min}$:

For a $7.7\,\text{MeV}$ $\alpha$-particle fired at Gold ($Z = 79$):

Since Coulomb's law held precisely down to $d_{min}$, the atomic nucleus must have radius $R_{nuc} < 30\,\text{fm} = 3 \times 10^{-14}\,\text{m}$—more than 10,000 times smaller than the overall atomic radius ($10^{-10}\,\text{m}$)! The atom is almost entirely empty space.

1. The Classical Instability of Rutherford's Planetary Atom

Although Rutherford's nuclear model triumphed in explaining scattering, it suffered from a fatal theoretical catastrophe under classical electrodynamics.

In Rutherford's model, electrons orbit the central nucleus like planets around the Sun, held by Coulomb attraction:

An orbiting electron undergoing centripetal acceleration is an accelerated electric charge. According to classical electrodynamics, any accelerated charge radiates electromagnetic power governed by the Larmor Radiation Formula:

2. Derivation of the Collapse Time ($\tau$)

The total mechanical energy of an electron in a circular orbit of radius $r$ is:

Differentiating with respect to time:

Solving for the orbital decay rate $dr/dt$:

Integrating from initial radius $r_0 \approx 0.53 \times 10^{-10}\,\text{m}$ down to $r = 0$:

The Classical Paradox: Classical electrodynamics predicts that every atom in the universe must collapse into its nucleus within a hundredth of a nanosecond, emitting a continuous burst of radiation! In reality, atoms are stable for billions of years and emit sharp, discrete line spectra.

1. Niels Bohr's Quantum Postulates (1913)

To overcome classical collapse, Niels Bohr introduced three bold quantum postulates for hydrogenic atoms (one electron orbiting a nucleus of charge $+Ze$):

1. Stationary States Postulate: Electrons move in discrete, non-radiating circular orbits called stationary states. While in these orbits, the electron accelerates but does NOT radiate electromagnetic energy.
2. Angular Momentum Quantization: The orbital angular momentum $L$ of the electron is quantized in integer multiples of $\hbar = h / (2\pi)$:
   $$L = m_e v r = n \hbar = n \frac{h}{2\pi} \quad (n = 1, 2, 3, \dots)$$
3. Frequency Postulate (Bohr Transition Rule): Radiation is emitted or absorbed only when an electron transitions discontinuously between two stationary states ($n_i \to n_f$). The photon frequency is:
   $$h\nu = E_i - E_f$$

2. Derivation of Orbit Radii and Speeds

Equating the Coulomb electrostatic force to the centripetal force:

Substituting $v$ into the quantization condition $m_e v r = n\hbar$ yields the quantized radius $r_n$:

where the Bohr radius of hydrogen ($Z = 1, n = 1$) is:

3. Quantized Energy Levels & Rydberg Formula

The total energy $E_n = K + U = \frac{1}{2} m_e v^2 - \frac{Z e^2}{4\pi\varepsilon_0 r_n} = - \frac{Z e^2}{8\pi\varepsilon_0 r_n}$:

For a transition from initial level $n_i$ to final level $n_f$ ($n_i > n_f$), the wavenumber $\bar{\nu} = 1/\lambda$ of the emitted photon is:

where the Rydberg constant for infinite mass is:

4. The Hydrogen Spectral Series

1. Finite Nuclear Mass & Reduced Mass Formulation

In Bohr's elementary derivation, the nucleus was assumed to have infinite mass ($M = \infty$) and remain strictly fixed at the center of the orbit. In reality, both the electron (mass $m_e$) and nucleus (mass $M$) orbit around their common center of mass.

To account for finite nuclear motion, the electron mass $m_e$ in all dynamical formulas is replaced by the two-body Reduced Mass $\mu$:

Because $\mu < m_e$, the actual Rydberg constant $R_M$ for an atom with nuclear mass $M$ is slightly smaller than the theoretical infinite-mass constant $R_\infty$:

For ordinary Hydrogen (${}^1\text{H}$, $M = m_p \approx 1836.15 m_e$):

2. Harold Urey's Discovery of Deuterium (1932)

Heavy hydrogen, or Deuterium (${}^2\text{H}$ or $\text{D}$), possesses a nucleus (deuteron) containing one proton and one neutron, giving it a nuclear mass approximately double that of protium: $M_D \approx 2 m_p \approx 3670.48 m_e$.

The Rydberg constant for deuterium is:

Because $R_D > R_H$, every spectral line of deuterium is shifted slightly toward shorter wavelengths (higher frequencies) relative to hydrogen. For the red $H_\alpha$ Balmer line ($n = 3 \to n = 2$):

By evaporating liquid hydrogen down to its triple point to concentrate heavy isotopes, Harold Urey and collaborators photographed a faint satellite line shifted by precisely $1.79\,\text{\AA}$ from the $H_\alpha$ line, confirming the existence of deuterium and earning Urey the 1934 Nobel Prize in Chemistry.

1. Objective of the Franck-Hertz Experiment (1914)

While atomic emission spectroscopy revealed discrete light wavelengths, James Franck and Gustav Hertz provided the first direct non-optical proof that internal atomic energy states are quantized, demonstrating discrete energy absorption through inelastic electron collisions with mercury vapor atoms.

2. Experimental Apparatus & Operational Mechanics

The apparatus consists of a heated glass tube filled with mercury ($\text{Hg}$) vapor at low pressure ($P \approx 1\text{ to }10\,\text{Torr}$):

• Cathode (K): Heated filament emitting electrons with negligible initial thermal energy.
• Grid Anode (G): Fine mesh accelerated by variable forward voltage $V_a$ ($0\text{ to }30\,\text{V}$).
• Collector Plate (P): Positioned beyond the grid, biased with a small retarding reverse voltage $V_r \approx 1.0\text{ to }1.5\,\text{V}$ relative to the grid.

Electrons reaching the grid must possess kinetic energy $K > e V_r$ to overcome the retarding barrier and register as collector current $I_c$.

3. Elastic vs. Inelastic Collisions & Current Drops

• Low Accelerating Voltages ($V_a < 4.9\,\text{V}$): The kinetic energy of the electrons is insufficient to excite a mercury atom from its ground state ($6^1S_0$) to its first excited state ($6^3P_1$, which lies $4.9\,\text{eV}$ higher). The electrons undergo purely elastic collisions. Because a mercury atom is $\sim 360,000$ times heavier than an electron, the electron bounces off with essentially zero kinetic energy loss. As $V_a$ increases, collector current $I_c$ rises steadily.
• First Inelastic Threshold ($V_a = 4.9\,\text{V}$): As soon as $V_a$ reaches $4.9\,\text{V}$, electrons near the grid attain exactly $4.9\,\text{eV}$ of kinetic energy. They undergo inelastic collisions with mercury atoms, transferring their entire $4.9\,\text{eV}$ to excite mercury electrons. Left with near-zero kinetic energy ($K \approx 0$), these electrons cannot overcome the $1.5\,\text{V}$ retarding potential. Collector current $I_c$ plunges sharply!
• Periodic Successive Inelastic Drops ($V_a = 9.8\,\text{V}, 14.7\,\text{V}, 19.6\,\text{V}$): As $V_a$ increases further, electrons gain enough energy to undergo a second inelastic collision at $9.8\,\text{V}$ ($2 \times 4.9\,\text{V}$), a third at $14.7\,\text{V}$ ($3 \times 4.9\,\text{V}$), and so on.

The resulting $I_c$ versus $V_a$ curve exhibits a striking series of periodic peaks and valleys separated by exactly $\Delta V = 4.9\,\text{V}$.

Optical Confirmation: When mercury atoms de-excite back to the ground state, they emit ultraviolet photons of wavelength:

Franck and Hertz observed the simultaneous appearance of $253.7\,\text{nm}$ UV luminescence as soon as $V_a$ exceeded $4.9\,\text{V}$, clinching the 1925 Nobel Prize in Physics.