Unit 4: Atomic Models

Rutherford's Gold Foil Experiment

In 1911, Ernest Rutherford, along with Hans Geiger and Ernest Marsden, performed the seminal alpha-particle scattering experiment to probe the inner structure of the atom.

Experimental Setup

A narrow beam of high-energy alpha particles (He²⁺ ions) emitted from a radioactive source (such as radium or bismuth) was directed toward a extremely thin gold foil (thickness of approximately 10^-7 m). The scattered alpha particles were detected using a movable zinc sulfide (ZnS) screen paired with a microscope. Whenever an alpha particle struck the ZnS screen, a tiny flash of light (scintillation) was produced and observed.

Key Observations

• Most Alpha Particles Pass Unscattered: About 99% of the incident alpha particles passed straight through the gold foil without experiencing any significant deflection.
• Small Angle Deflection: A small fraction of alpha particles (about 1 in 8000) were deflected through small angles.
• Large Angle Deflection & Backscattering: An extremely small number of alpha particles were deflected by large angles (> 90°), and occasionally an alpha particle rebounded backward (scattering angle θ ≈ 180°).

Inferences and Conclusions

• Since most alpha particles pass undeflected, most of the space inside an atom is empty.
• To cause large-angle deflection of positively charged alpha particles, there must exist a concentrated, dense, positively charged region inside the atom, termed the nucleus.
• Almost the entire mass of the atom is concentrated within this tiny central nucleus.

Derivation of Rutherford's Differential Scattering Cross Section Formula

The differential scattering cross section quantifies the probability of an incident particle being scattered into a specific solid angle dΩ.

1. Physical Assumptions

• The target nucleus has mass far greater than the alpha particle and remains stationary during scattering.
• The alpha particle (charge q₁ = +2e, mass m) and nucleus (charge q₂ = +Ze) interact solely via electrostatic Coulomb repulsion.
• The collision is elastic, conserving both kinetic energy and angular momentum.

2. Geometric and Kinematic Definitions

• Impact Parameter (b): The perpendicular distance between the initial velocity vector of the alpha particle and the target nucleus.
• Scattering Angle (θ): The angle between the initial direction and final direction of the alpha particle trajectory.
• Distance of Closest Approach (r₀): The minimum distance between the alpha particle and the nucleus during a head-on collision.

3. Step-by-Step Derivation

Step A: Coulomb Repulsive Force

The repulsive force between the alpha particle and the nucleus at separation distance r is given by Coulomb's law:

F = (1 / 4πε₀) × (2Ze² / r²)

Step B: Conservation of Angular Momentum

Since Coulomb force is a central force, orbital angular momentum L is conserved throughout the motion:

L = m v₀ b = m r² (dφ / dt)

where v₀ is the initial velocity of the alpha particle at infinity, and φ is the instantaneous angular coordinate.

Rearranging yields:

dt = (m r² / m v₀ b) dφ = (r² / v₀ b) dφ

Step C: Integration of Change in Momentum

By symmetry, the net change in linear momentum Δp lies along the axis of symmetry (bisecting the angle π - θ). The component of force along this symmetry axis is F cos φ, where φ ranges from -(π - θ)/2 to +(π - θ)/2.

Δp = ∫ F cos φ dt

Substitute F and dt into the integral:

Δp = ∫ [ (1 / 4πε₀) × (2Ze² / r²) ] × cos φ × [ (r² / v₀ b) dφ ]

Simplifying the terms inside the integral:

Δp = [ 2Ze² / (4πε₀ v₀ b) ] ∫_-(π-θ)/2^+(π-θ)/2 cos φ dφ

Evaluating the definite integral:

∫_-(π-θ)/2^+(π-θ)/2 cos φ dφ = [ sin φ ]_-(π-θ)/2^+(π-θ)/2 = 2 sin[ (π - θ) / 2 ] = 2 cos(θ / 2)

Therefore, the momentum change from force integration is:

Δp = [ 2Ze² / (4πε₀ v₀ b) ] × 2 cos(θ / 2) = [ 4Ze² cos(θ / 2) ] / (4πε₀ v₀ b)

Step D: Vector Relationship of Momentum Change

In elastic collision, the initial momentum p_i and final momentum p_f have equal magnitude m v₀. The vector difference gives:

Δp = 2 m v₀ sin(θ / 2)

Step E: Expressing Impact Parameter b

Equating the two momentum expressions:

2 m v₀ sin(θ / 2) = [ 4Ze² cos(θ / 2) ] / (4πε₀ v₀ b)

Solving for b:

b = (Ze² / 2πε₀ m v₀²) × cot(θ / 2)

Since initial kinetic energy E = (1 / 2) m v₀², we substitute m v₀² = 2E:

b = [ Ze² / (4πε₀ E) ] × cot(θ / 2)

Step F: Differential Scattering Cross Section (dσ/dΩ)

Alpha particles approaching through an annular area dσ = 2π b |db| will be scattered into a solid angle dΩ = 2π sin θ dθ.

Differentiating b with respect to θ:

|db / dθ| = [ Ze² / (4πε₀ E) ] × (1 / 2) csc²(θ / 2)

The differential cross section per unit solid angle is:

dσ / dΩ = (2π b |db|) / (2π sin θ dθ) = (b / sin θ) × |db / dθ|

Using standard trigonometric identity sin θ = 2 sin(θ / 2) cos(θ / 2):

dσ / dΩ = [ (Ze² / (4πε₀ E)) cot(θ / 2) / (2 sin(θ / 2) cos(θ / 2)) ] × [ (Ze² / (4πε₀ E)) × (1 / 2) csc²(θ / 2) ]

Combining terms simplifies to Rutherford's famous differential scattering formula:

dσ / dΩ = [ Ze² / (16πε₀ E) ]² × [ 1 / sin⁴(θ / 2) ]

Rutherford's Model of Atom and Its Limitations

Postulates of Rutherford's Model

• Nuclear Structure: Atom consists of a dense, positively charged nucleus containing nearly all atomic mass.
• Planetary Analogy: Negatively charged electrons revolve around the central nucleus in circular orbits, akin to planets revolving around the Sun.
• Electrostatic Attraction: The required centripetal force for orbital motion is provided by electrostatic attraction between nucleus and electrons.
• Charge Neutrality: Total negative charge of electrons equals total positive charge of the nucleus.

Limitations of Rutherford's Model

Sommerfeld's Atom Model and Fine Structure of Hydrogen Atom

To resolve the limitations of Bohr's circular orbit model (such as fine-structure splitting of spectral lines), Arnold Sommerfeld extended Bohr's theory in 1915 by introducing elliptical orbits and relativistic velocity corrections.

Key Postulates of Sommerfeld Model

• Elliptical Orbits: Electrons move in elliptical orbits with the nucleus located at one of the foci. Circular orbits are treated as a special degenerate case of ellipses.
• Two Degrees of Freedom: The position of an electron in an elliptical orbit is determined by two coordinates: radial distance (r) and azimuthal angle (φ).
• Dual Quantization Rule: Sommerfeld applied Wilson-Sommerfeld quantization rule ∮ p dq = n h to both coordinates:
  • Azimuthal Quantization: ∮ p_φ dφ = k h (where k is the azimuthal quantum number, k = 1, 2, 3... n)
  • Radial Quantization: ∮ p_r dr = n_r h (where n_r is the radial quantum number, n_r = 0, 1, 2...)
• Total Quantum Number: n = n_r + k. The ratio of minor axis (b) to major axis (a) of the ellipse is given by:

b / a = k / n

Explanation of Fine Structure of Hydrogen Atom

When hydrogen spectral lines were analyzed with high-resolution spectrographs, single lines were found to consist of multiple closely spaced components (fine structure).

• Relativistic Mass Variation: Near perihelion (closest approach to nucleus), the electron's speed increases significantly, becoming a non-negligible fraction of the speed of light c.
• According to Einstein's relativity, mass varies with speed: m = m₀ / √(1 - v²/c²).
• This mass variation causes precessional motion of the elliptical orbit (precession of perihelion), resulting in a rosette-shaped trajectory.
• This precession slightly modifies the energy of each state depending on the eccentricity (governed by azimuthal quantum number k for a given n).
• Thus, states with same principal quantum number n but different k values split into slightly different energy sublevels, successfully explaining fine-structure spectral lines.

Limitations of Sommerfeld's Atom Model

• Inability to Predict Intensities: Sommerfeld model could predict the position/wavelength of fine-structure lines, but could not predict their relative intensities.
• Complex Spectra Failure: Failed to explain energy levels and spectra of multi-electron systems.
• Incomplete Anomalous Zeeman and Stark Effects: Could not fully account for anomalous Zeeman effect and Stark effect without introducing electron spin.
• Arbitrary Ad-hoc Quantization: Quantum conditions were imposed arbitrarily onto classical mechanics without a unified quantum mechanical framework.

Vector Atom Model and Quantum Numbers

To overcome limitations of classical and early quantum models, the Vector Atom Model was introduced. It incorporates two major advanced concepts:

1. Spatial Quantization: The orientation of orbital angular momentum vectors in space cannot be arbitrary; it is restricted to discrete direction angles relative to an external magnetic field.
2. Electron Spin: Proposed by Uhlenbeck and Goudsmit (1925), the electron possesses an intrinsic spinning motion about its own axis in addition to orbital motion.

Seven Quantum Numbers associated with Vector Atom Model

Gyromagnetic Ratio and Bohr Magneton

Gyromagnetic Ratio (γ)

The gyromagnetic ratio is defined as the ratio of the magnetic dipole moment (μ) of a particle to its orbital angular momentum (L).

Derivation for Orbital Motion of Electron:

Consider an electron of mass m and charge -e revolving in a circular orbit of radius r with speed v and time period T = 2π r / v.

Electric current I produced by moving electron:

I = e / T = e v / (2π r)

Magnetic dipole moment μ_l associated with current loop of area A = π r²:

μ_l = I × A = [ e v / (2π r) ] × (π r²) = (e v r) / 2

Orbital angular momentum L of electron:

L = m v r → v r = L / m

Substituting v r into magnetic moment equation:

μ_l = (e / 2m) × L

The Gyromagnetic Ratio γ is defined as:

γ = μ_l / L = e / (2m)

Important Note: Vectorially, because electron charge is negative, magnetic dipole moment is oriented opposite to angular momentum: μ_l = - (e / 2m) L.

Bohr Magneton (μ_B)

The Bohr Magneton is the fundamental natural unit of magnetic dipole moment associated with atomic orbital or spin angular momentum.

According to Bohr's postulate, angular momentum L is quantized as:

L = n ℏ = n h / (2π)

Substitute L into magnetic moment equation for ground state (n = 1):

μ_l = (e / 2m) × (h / 2π) = (e h) / (4π m)

This fundamental atomic constant is defined as the Bohr Magneton μ_B:

μ_B = (e ℏ) / (2m) = (e h) / (4π m)

Standard Values and Units

• e = 1.602 × 10^-19 C
• h = 6.626 × 10^-34 J·s
• m = 9.109 × 10^-31 kg
• μ_B Value: 9.274 × 10^-24 J/T (or A·m²)