Ernest Rutherford

Ernest Rutherford's nuclear architecture rests upon four foundational paradigms:

• The Nuclear Atom: The positive charge and virtually all mass of an atom ($>99.95\%$) is concentrated in an ultra-dense, positively charged central core (the nucleus), whose diameter ($\sim 10^{-14}\text{ m}$) is approximately 10,000 times smaller than the total atomic radius ($\sim 10^{-10}\text{ m}$).
• Exponential Law of Radioactive Decay: Radioactivity is an intrinsic, stochastic intra-atomic disintegration, where the decay rate is strictly proportional to the number of remaining parent nuclei, characterized by an invariant half-life $t_{1/2}$.
• Particulate Nature of Nuclear Radiations: Classification of ionizing emissions into heavy, positively charged alpha ($\alpha$) particles (helium-4 nuclei: $\text{He}^{2+}$) and light, negatively charged beta ($\beta$) particles (relativistic electrons: $e^-$).
• Artificial Nuclear Transmutation: Heavy charged particle impact can penetrate the nuclear Coulomb barrier, disintegrating stable target nuclei and transmuting one elemental species into another.

+--------------------------------------------------------------+
|                RUTHERFORD SCATTERING ARCHITECTURE            |
|                                                              |
|        Alpha Source (Ra-226) -> Collimated Beam (E_k ~ 5 MeV)|
|                               |                              |
|                               v                              |
|        [ THIN GOLD FOIL (Au-197, Thickness ~ 400 nm) ]       |
|                               |                              |
|       +-----------------------+-----------------------+      |
|       |                                               |      |
|  99.99% of Alphas:                               0.01% of:   |
|  Traverse Foil Undeviated                        Backscatter |
|  (\theta \approx 0 degrees)                      (\theta>90) |
|       |                                               |      |
|  Empty Atomic Volume                             Dense Mass  |
|  Electron Cloud                                  Nucleus     |
|       |                                               |      |
|       +-----------------------+-----------------------+      |
|                               |                              |
|                               v                              |
|             [ RUTHERFORD DIFFERENTIAL CROSS SECTION ]        |
|             d\sigma/d\Omega \propto 1 / \sin^4(\theta/2)     |
+--------------------------------------------------------------+

2.1 The Rutherford Scattering Formula (1911)

Consider an alpha particle of mass $m_\alpha$, charge $q_1 = 2e$, and initial velocity $v_0$ colliding with a stationary target nucleus of charge $q_2 = Ze$.

The trajectory is a classical Keplerian hyperbola governed by repulsive Coulomb potential: $$V(r) = \frac{1}{4\pi \epsilon_0} \frac{2 Z e^2}{r}$$

By conservation of angular momentum and energy, the scattering angle $\theta$ is related to the impact parameter $b$: $$b = \frac{Z e^2}{2\pi \epsilon_0 m_\alpha v_0^2} \cot\left(\frac{\theta}{2}\right) = \frac{d}{2} \cot\left(\frac{\theta}{2}\right)$$ where $d$ is the distance of closest approach in a direct head-on collision ($\theta = 180^\circ$): $$d = \frac{2 Z e^2}{4\pi \epsilon_0 E_k}, \quad E_k = \frac{1}{2} m_\alpha v_0^2$$

The differential cross section per unit solid angle $\Omega$: $$\frac{d\sigma}{d\Omega} = \left( \frac{1}{4\pi \epsilon_0} \frac{Z e^2}{4 E_k} \right)^2 \frac{1}{\sin^4(\theta/2)}$$

2.2 Law of Radioactive Decay and Half-Life (Rutherford & Soddy, 1902)

The rate of disintegration is directly proportional to the number of parent nuclei $N(t)$: $$\frac{dN}{dt} = -\lambda N(t) \implies N(t) = N_0 e^{-\lambda t}$$ where $\lambda$ is the decay constant ($\text{s}^{-1}$).

The radioactive half-life $t_{1/2}$ (time for $N = N_0 / 2$): $$t_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.69315}{\lambda}$$

Mean lifetime $\tau$: $$\tau = \frac{1}{\lambda} \implies t_{1/2} = \tau \ln 2$$

Activity $A(t)$ (measured in Becquerels / Curies): $$A(t) = -\frac{dN}{dt} = \lambda N_0 e^{-\lambda t} = A_0 e^{-\lambda t}$$

2.3 The First Artificial Transmutation Reaction (1919)

Collision of 5.3 MeV alpha particles ($^4_2\text{He}$) with nitrogen gas ($^{14}_7\text{N}$): $$^{14}_7\text{N} + ^4_2\text{He} \longrightarrow \left[ ^{18}_9\text{F}^* \right] \longrightarrow ^{17}_8\text{O} + ^1_1\text{H}$$ - Reactants: Nitrogen-14 ($Z=7$) + Helium-4 ($Z=2$) - Product: Oxygen-17 ($Z=8$) + Proton ($Z=1$) Demonstrated exact conservation of baryon number ($A = 14 + 4 = 17 + 1 = 18$) and nuclear charge ($Z = 7 + 2 = 8 + 1 = 9$).

3.1 The Geiger-Marsden Alpha Scattering Apparatus (1909)

        +-------------------------------------------------------+
        | Airtight Cylindrical Brass Chamber (Evacuated)       |
        |                                                       |
        | [ Radium Source R ] ---> Collimator Slit S            |
        |                               |                       |
        |                               v                       |
        |                      [ Gold Foil F ]                  |
        |                             / \                       |
        |       (Forward Beam)       /   \ (Deflected Alphas)   |
        |             |             /     \                     |
        |             v            /       v                    |
        |       [ Screen 0 deg ]  /      [ Zinc Sulfide (ZnS) ] |
        |                        /       [ Screen M (150 deg) ] |
        |                       /                  |            |
        |                      /                   v            |
        |      (Deflection Angle \theta)      [ Microscope ]    |
        |                                     [ Observer Eye ]  |
        +-------------------------------------------------------+

3.2 Nuclear vs. Plum-Pudding Scattering Trajectories

THOMSON 'PLUM-PUDDING' MODEL:
   Alpha Particle (Straight-line penetration, \theta < 1 deg)
   ------------------------------------------------------------>
     ( + ) diffuse positive charge cloud with tiny electrons

RUTHERFORD NUCLEAR MODEL:
     Alpha 1 (Remote b):    -----------------------------------> Undeviated
     Alpha 2 (Medium b):    -------------\
                                          \-----> Deflection (\theta ~ 30 deg)
     Alpha 3 (Near direct): -------------\
                                         /-----> BACKSCATTERED! (\theta > 90 deg)
                                       * [ Nucleus: Mass=99.95%, Radius=10^-14 m ]

• Experimental Cross Section Validation (Geiger & Marsden, 1913): Counted over 100,000 alpha scintillations across angles from $5^\circ$ to $150^\circ$, verifying the $1/\sin^4(\theta/2)$ dependence with an error margin under 1.5%.
• Determination of Nuclear Charge $Z$: Confirmed that the scattering cross section scales as $Z^2$, proving gold nuclear charge $Z \approx 79$.
• Proof that Alpha Particles are Helium (1908): Rutherford and Thomas Royds collected alpha particles in an evacuated thin-walled glass tube; after electrical discharge, spectroscopic analysis revealed the characteristic spectral emission lines of pure Helium.
• Radius of the Atomic Nucleus: Calculated the upper bound of the gold nucleus radius: $$r_{\text{nucleus}} < d_{\min} \approx 3.2 \times 10^{-14}\text{ m} = 32\text{ fm}$$

5.1 Classical Electrodynamic Instability of Nuclear Orbitals

Rutherford's model placed electrons in mechanical orbits around the nucleus. Under classical Maxwellian electrodynamics, orbiting electrons must continuously accelerate and radiate energy, causing the atom to collapse within $10^{-11}\text{ seconds}$. Rutherford acknowledged this failure mode, leaving its resolution to his postdoctoral pupil Niels Bohr (1913).

5.2 Breakdown of Coulomb Law at High Energy (Nuclear Size Limit)

The classical Rutherford scattering formula assumes a point-like Coulomb potential. When alpha particle energies exceed $\sim 25\text{ MeV}$, the distance of closest approach penetrates the nuclear surface ($r < R_0 A^{1/3}$), triggering the attractive strong nuclear force and causing the observed cross section to deviate violently from $1/\sin^4(\theta/2)$.

5.3 Assumption of Pure Electron-Proton Nuclear Composition

Before Chadwick's discovery of the neutron in 1932, Rutherford hypothesized that the nucleus was composed of $A$ protons and $A-Z$ 'nuclear electrons' binding them together. This hypothesis violated Heisenberg's uncertainty principle (confining an electron within $10^{-14}\text{ m}$ requires impossible relativistic kinetic energies) and spin statistics.

• The Cause and Nature of Radioactivity (with F. Soddy), Philosophical Magazine, Vol. 4, pp. 370–396 & 569–585, 1902.
• Radioactive Transformations, Charles Scribner's Sons, New York, 1906.
• The Scattering of $\alpha$ and $\beta$ Particles by Matter and the Structure of the Atom, Philosophical Magazine, Series 6, Vol. 21, pp. 669–688, May 1911 (the nuclear atom paper).
• The Laws of Deflexion of $\alpha$ Particles through Large Angles (by H. Geiger and E. Marsden), Philosophical Magazine, Vol. 25, pp. 604–623, 1913.
• Collision of $\alpha$ Particles with Light Atoms: IV. An Anomalous Effect in Nitrogen, Philosophical Magazine, Vol. 37, pp. 581–587, 1919 (first artificial transmutation).
• Radioactive Substances and their Radiations, Cambridge University Press, 1913.