Class 12 Physics · Chapter 12 NotesAtoms

By the end of the nineteenth century, J. J. Thomson's experiments on electric discharge through gases showed that all atoms contain identical negatively charged electrons, while the atom as a whole is electrically neutral. In 1898 Thomson proposed that the positive charge is spread uniformly throughout the atom with electrons embedded in it, like seeds in a watermelon. This was called the plum pudding model. Later scattering experiments showed that the positive charge is not spread out at all, but concentrated in a very small central region, so this model was abandoned.

• Electrons are identical for all atoms and carry negative charge
• An atom is electrically neutral, so positive charge must also exist
• Thomson's modelpositive charge distributed uniformly, electrons embedded in it

In the Geiger-Marsden experiment, a beam of 5.5 MeV alpha-particles from a radioactive source was directed at a thin gold foil of thickness 2.1 × 10⁻⁷ m inside a vacuum chamber. Scattered particles were detected using a zinc sulphide screen and microscope, which produced scintillations. Most alpha-particles passed straight through, about 0.14% scattered by more than 1°, and roughly 1 in 8000 deflected by more than 90°. Rutherford argued that a large backward deflection requires a strong repulsive force from a tiny, massive, positively charged centre. This led to the nuclear model: all positive charge and most of the mass in a nucleus of size about 10⁻¹⁵ m to 10⁻¹⁴ m, with electrons revolving around it, while the atom itself is about 10⁻¹⁰ m across.

• Coulomb force between alpha-particle and nucleusF = (1/4πε₀)(2e)(Ze)/r²
• Nucleus size ≈ 10⁻¹⁵ m to 10⁻¹⁴ m; atom size ≈ 10⁻¹⁰ m
• Most of the atom is empty space

The path followed by an alpha-particle depends on its impact parameter b, the perpendicular distance between the initial velocity direction and the centre of the nucleus. A particle with a small impact parameter passes close to the nucleus and is deflected strongly; in a head-on collision (b minimum) it rebounds almost straight back, with scattering angle near 180°. A particle with a large impact parameter passes far from the nucleus and is hardly deflected. Since only a very small fraction of particles rebound, head-on collisions are rare, which means the nucleus occupies a very small volume. This makes Rutherford scattering a way to set an upper limit on nuclear size.

• Small b → large scattering angle; large b → small scattering angle
• Head-on collisionθ ≈ π (particle returns along its path)
• Distance of closest approachd = 2Ze²/(4πε₀K)

Rutherford's model has two serious problems. First, an electron moving in a circle is accelerating, so according to classical electromagnetic theory it should continuously radiate energy, spiral inward and make the atom unstable. Second, the frequency of revolution would change continuously, giving a continuous spectrum instead of the observed line spectrum. In 1913 Niels Bohr proposed three postulates. First, electrons revolve only in certain stable orbits, called stationary states, without radiating energy. Second, the angular momentum of the electron is quantised: L = nh/2π, where n = 1, 2, 3, ... Third, a photon is emitted when an electron jumps from a higher energy state Eᵢ to a lower state E_f, with hν = Eᵢ − E_f.

• Stationary statesno radiation while the electron stays in an allowed orbit
• QuantisationL = nh/2π, n is the principal quantum number
• Photon emissionhν = Eᵢ − E_f

Combining the quantisation condition with the Coulomb force providing the centripetal force gives the allowed orbit radii rₙ = (n²/m)(h/2π)²(4πε₀/e²). The total energy of the electron is negative, showing that it is bound to the nucleus, and is given by Eₙ = −me⁴/(8n²ε₀²h²) = −13.6/n² eV. The lowest state, n = 1, is the ground state with energy −13.6 eV, so 13.6 eV is needed to ionise hydrogen. Higher n values give excited states: n = 2 has −3.40 eV, n = 3 has −1.51 eV, and so on. The energy levels get closer together as n increases, and E = 0 corresponds to a free electron at rest.

• rₙ ∝ n², so the n = 1 orbit is the smallest (Bohr radius ≈ 5.3 × 10⁻¹¹ m)
• Eₙ = −13.6/n² eV
• Excitation from n = 1 to n = 2 needs 10.2 eV; to n = 3 needs 12.09 eV

When an electron falls from a higher state nᵢ to a lower state n_f, a photon of frequency ν is emitted such that hν = Eₙᵢ − Eₙ_f. Because the energies take only discrete values, the emitted light contains only certain wavelengths, producing an emission line spectrum of bright lines on a dark background. Conversely, when white light passes through a gas, the atoms absorb photons of exactly the energies needed for upward transitions, giving dark absorption lines at the same wavelengths as the emission lines. Each element has its own characteristic spectrum, which acts like a fingerprint for identifying the gas. Bohr's model successfully explained the observed hydrogen spectrum.

• Emission spectrumbright lines on a dark background
• Absorption spectrumdark lines on a continuous bright background
• Line wavelengths correspond to energy differences between stationary states

Bohr's second postulate seemed arbitrary: why should angular momentum be an integral multiple of h/2π? In 1923 Louis de Broglie explained it using the wave nature of matter. An electron in a circular orbit behaves like a wave, and a stable orbit forms only when a whole number of de Broglie wavelengths fits into the circumference: 2πrₙ = nλ, with λ = h/mv. Substituting λ gives 2πrₙ = nh/mv, which rearranges to mvrₙ = nh/2π. This is exactly Bohr's quantisation condition. The allowed orbits are therefore those supporting standing waves, and only these resonant orbits can persist.

• de Broglie wavelengthλ = h/mv
• Standing wave condition2πrₙ = nλ
• This yields mvrₙ = nh/2π, Bohr's quantisation condition

Bohr's model works well for hydrogenic atoms, that is, atoms with a single electron such as hydrogen, singly ionised helium or doubly ionised lithium. It cannot be extended even to helium with two electrons, because it ignores the electrical repulsion between electrons, which is comparable in size to the electron-nucleus attraction. The model also cannot explain the relative intensities of spectral lines, since some transitions are more favoured than others. A more complete description of atomic structure requires quantum mechanics, in which Bohr's orbits are replaced by regions where the electron is likely to be found.

• Valid only for single-electron (hydrogenic) atoms
• Cannot explain relative intensities of spectral lines
• Replaced by quantum mechanics, which gives a more complete picture