Rutherford Scattering and Bohr Model of the Atom

Rutherford Scattering Experiment

The Rutherford scattering experiment, conducted by Ernest Rutherford and his colleagues Hans Geiger and Ernest Marsden in 1909, was pivotal in understanding the structure of the atom. Before this experiment, the prevailing model was J.J. Thomson's "plum pudding" model, which suggested that the atom was a uniform sphere of positive charge with electrons embedded in it.

In the experiment, a beam of alpha particles (which are helium nuclei, consisting of two protons and two neutrons, hence positively charged) was directed at a thin gold foil. A detector screen, coated with zinc sulfide, was used to observe the scattering of these alpha particles. When an alpha particle struck the screen, it produced a tiny flash of light (scintillation).

The observations were quite surprising and contradicted the plum pudding model:

• Most alpha particles passed straight through the gold foil with very little deflection. This indicated that the atom is mostly empty space.
• A small fraction of alpha particles were deflected by small angles.
• A very small number of alpha particles (about 1 in 8000) were deflected by large angles, even bouncing back in the direction from which they came.

Rutherford famously remarked, "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."

Rutherford's Nuclear Model of the Atom

Based on the experimental results, Rutherford proposed a new model of the atom in 1911, often called the nuclear model or the planetary model:

• The atom consists of a central, dense, positively charged nucleus. All the positive charge and almost all the mass of the atom are concentrated in this nucleus.
• The nucleus is extremely small compared to the overall size of the atom.
• Electrons, which are negatively charged, revolve around the nucleus in well-defined orbits, much like planets revolve around the Sun.
• The electrostatic force of attraction between the positive nucleus and the negative electrons provides the centripetal force required for the orbital motion.
• The atom as a whole is electrically neutral because the total positive charge of the nucleus is equal to the total negative charge of the electrons.

The radius of the nucleus is of the order of 10^-15 meters, while the radius of the atom is of the order of 10^-10 meters. This means the nucleus is about 100,000 times smaller than the atom.

Limitations of Rutherford's Model

Despite its revolutionary nature, Rutherford's model had significant flaws according to classical physics:

• Instability of the atom: According to classical electromagnetic theory, an accelerating charged particle must radiate energy. The electrons revolving around the nucleus are constantly accelerating (changing direction). Therefore, they should continuously lose energy by radiating electromagnetic waves. This loss of energy would cause the electrons to spiral inwards and eventually fall into the nucleus, making the atom unstable. However, atoms are observed to be stable.
• Atomic spectra: Rutherford's model could not explain the discrete line spectra observed when atoms emit light. If electrons lose energy continuously, they should emit radiation over a continuous range of frequencies, producing a continuous spectrum, not distinct lines.

Bohr Model of the Atom

Niels Bohr, a Danish physicist, proposed a model of the atom in 1913 that addressed the shortcomings of Rutherford's model by incorporating ideas from quantum theory. The Bohr model is particularly successful for hydrogen-like atoms (atoms with only one electron).

Postulates of Bohr's Model

Bohr's model is based on the following postulates:

1. Stationary Orbits: Electrons revolve around the nucleus in certain specific, stable orbits called stationary orbits or stationary states. While in these orbits, electrons do not radiate energy, despite being accelerated. These orbits are associated with definite energies.
2. Quantization of Angular Momentum: The angular momentum of an electron in a stationary orbit is quantized. It can only take discrete values that are integer multiples of h/2π, where h is Planck's constant.

   Mathematically, the condition is: L = mvr = n(h/2π)

   Where:

   • m is the mass of the electron.
   • v is the velocity of the electron.
   • r is the radius of the stationary orbit.
   • n is a positive integer called the principal quantum number (n = 1, 2, 3, ...).
3. Quantum Jumps and Energy Radiation: An electron can transition from one stationary orbit to another, but it cannot exist in between these orbits. When an electron jumps from a higher energy orbit (E₂) to a lower energy orbit (E₁), it emits energy in the form of a photon. The energy of the photon (hν) is equal to the difference in energy between the two orbits.

   hν = E₂ - E₁

   Conversely, when an electron absorbs a photon of appropriate energy, it can jump from a lower energy orbit to a higher energy orbit.

Key Results of Bohr's Model for Hydrogen Atom

Bohr applied his postulates to the hydrogen atom (with one proton and one electron) and derived several important results:

1. Radius of Stationary Orbits

The radius of the n-th stationary orbit for a hydrogen atom is given by:

r_n = n²a₀ / Z

Where:

• n is the principal quantum number.
• Z is the atomic number (for hydrogen, Z = 1).
• a₀ is the Bohr radius, the radius of the first orbit (n=1) for hydrogen.

The value of the Bohr radius (a₀) is approximately 0.529 Å (Angstroms) or 5.29 × 10^-11 meters.

So, for hydrogen: r_n = n²a₀. This shows that the radii of the allowed orbits increase as the square of the principal quantum number.

2. Energy of Stationary Orbits

The total energy of an electron in the n-th orbit of a hydrogen atom is given by:

E_n = - (Z² * 13.6 eV) / n²

Where:

• Z is the atomic number (Z=1 for hydrogen).
• n is the principal quantum number.
• 13.6 eV is the ground state energy of the hydrogen atom (energy of the first orbit, n=1).

So, for hydrogen: E_n = -13.6 eV / n².

The negative sign indicates that the electron is bound to the nucleus. The energy is minimum (most negative) for n=1 (the ground state), meaning it's the most stable state. As n increases, the energy becomes less negative, approaching zero as n approaches infinity. When E = 0, the electron is free from the nucleus.

3. Velocity of Electrons

The velocity of the electron in the n-th orbit of a hydrogen atom is given by:

v_n = (2.18 × 10⁶ m/s) * Z / n

For hydrogen: v_n = (2.18 × 10⁶ m/s) / n.

This shows that the velocity of the electron decreases as n increases. The electron moves fastest in the ground state (n=1).

4. Frequency and Wavelength of Spectral Lines (Rydberg Formula)

When an electron transitions from an initial state n_i to a final state n_f (where n_i > n_f), a photon is emitted with energy hν = E_ni - E_nf.

hν = (-13.6 eV / n_f²) - (-13.6 eV / n_i²)

hν = 13.6 eV (1/n_f² - 1/n_i²)

The frequency ν is given by ν = (13.6 eV / h) (1/n_f² - 1/n_i²).

Since c = νλ, the wave number (1/λ) is given by:

1/λ = ν/c = (13.6 eV / (hc)) (1/n_f² - 1/n_i²)

The constant (13.6 eV / (hc)) is known as the Rydberg constant (R_H) for hydrogen.

R_H ≈ 1.097 × 10⁷ m^-1

So, the Rydberg formula is: 1/λ = R_H (1/n_f² - 1/n_i²)

Spectral Series of Hydrogen

The transitions between different energy levels lead to different spectral series:

Limitations of Bohr's Model

Despite its successes, Bohr's model also had limitations:

• It could only accurately predict the spectrum of hydrogen and hydrogen-like ions (e.g., He⁺, Li²⁺). It failed to explain the spectra of multi-electron atoms.
• It could not explain the fine structure of spectral lines (the splitting of spectral lines into several closely spaced lines).
• It could not explain the relative intensities of spectral lines.
• It violated the Heisenberg Uncertainty Principle (which was formulated later), as it assumed precise values for both the position and momentum of the electron simultaneously.
• It did not account for the wave nature of electrons (which was proposed by de Broglie).

These limitations paved the way for the development of the more sophisticated quantum mechanical model of the atom.

Rutherford Scattering Formula

Rutherford also derived a formula that describes the number of alpha particles scattered per unit time per unit solid angle (i.e., the differential scattering cross-section). This formula is based on the assumption that the scattering is due to the electrostatic Coulomb repulsion between the positively charged alpha particles and the positively charged nucleus.

The number of alpha particles scattered into a solid angle dΩ at an angle θ is proportional to:

N(θ) dΩ ∝ (Z₁ Z₂)² / (r² sin⁴(θ/2)) dΩ

Where:

• Z₁ is the atomic number of the projectile (alpha particle, Z₁=2).
• Z₂ is the atomic number of the target nucleus (gold, Z₂=79).
• θ is the scattering angle.
• r is the distance of the detector from the scattering center.

This formula shows that the scattering intensity is proportional to the square of the product of the atomic numbers of the projectile and target, and it increases rapidly as the scattering angle θ approaches zero (forward scattering). The 1/sin⁴(θ/2) dependence explains why very few particles are scattered at large angles.