Bohr's Model & Physics of the Atom

What is inside an atom? It took a century of clever experiments, ending with alpha particles bouncing back off a sheet of gold, to find out that an atom is almost entirely empty space around a tiny, heavy nucleus.

From atomism to electrons#

The idea of atoms is ancient. With Robert Boyle (1627–1691), who studied how air compresses and expands, it moved from reasoning to experiment: compressible air must be tiny particles with empty space between them. An atom is the smallest unit of an element that carries all its properties. Discharge tubes and Thomson’s e/m measurement showed atoms contain negative electrons, so, since matter is neutral, there must be positive charge too. The question was how it is arranged.

Three models#

Thomson (1898)

A solid sphere of positive charge with electrons embedded in it, like cherries in a cake. It explained why heating a metal ejects only negative particles, and how ions and ionic compounds form.

Lenard (1903)

Cathode rays pass through thin foils almost undeviated, so atoms can’t be solid: they must be mostly empty. Lenard proposed atoms made of electrons and similar tiny positive particles. But then why would heating never eject the positive ones?

Rutherford (1906–1911)

Rutherford fired alpha particles at a thin gold foil. Most went straight through or were barely deflected, as expected for a heavy alpha brushing past tiny particles. But about one in 8000 was turned back by more than 90°.

• An alpha particle is about 7350 times heavier than an electron, so neither electrons nor electron-sized positive particles could throw it back. It must have hit something with a mass comparable to the whole atom.
• Since most alphas pass undisturbed, that heavy object must be tiny.

The scattering pattern showed this nucleus is positive and holds nearly all the atom’s mass, with a size of about 10 fm (1 fm = 10⁻¹⁵ m): some 10⁻⁵ of the atom’s diameter and 10⁻¹⁵ of its volume.

Rutherford’s nuclear atom: a tiny, massive, positive nucleus at the centre, with electrons moving round it at a distance; the space between is empty and sets the atom’s size; the nuclear charge equals the electrons’ total charge. It explains neutrality, the empty space, and why only electrons are easily ejected (the heavy nucleus barely notices heating). The electrons must move: at rest they would fall into the nucleus, but in circular orbits the Coulomb force just bends their path. That motion, as the next lessons show, brought its own trouble.

A hot solid glows with every colour. Hot hydrogen glows with just a few sharp ones, the same everywhere in the universe. Those lines were a fingerprint no classical theory could explain.

Line spectra#

A lamp filament, sent through a prism, gives a continuous spectrum. Hydrogen heated in a sealed tube gives a line spectrum: only certain sharp wavelengths, such as 656.3 nm and 486.1 nm, with nothing in between. Hydrogen also emits lines in the ultraviolet and infrared.

The lines fall into series. Within each series the gaps shrink toward shorter wavelengths, approaching a series limit: the Lyman series (ultraviolet), the Balmer series (visible), the Paschen series (infrared), and so on. All the wavelengths fit one formula:

Why Rutherford’s atom can’t be right#

Hydrogen is one electron and one proton. In Rutherford’s picture the electron circles the proton, the Coulomb force supplying the centripetal force. Mechanically fine, but Maxwell’s equations say an accelerating charge must radiate continuously. So the electron should lose energy, spiral inward, and crash into the proton, emitting a smoothly changing range of wavelengths on the way.

Reality: hydrogen at room temperature is perfectly stable and emits nothing. When excited by heat or a discharge it emits only those sharp wavelengths, always the same, from a lab in Delhi to the Sun. Classical physics had no explanation: something new was needed.

In 1913 Bohr broke with classical physics in a few bold assumptions, and out came the hydrogen spectrum, number for number.

Bohr’s postulates#

1. The electron moves round the nucleus in circular orbits.
2. Only orbits of certain radii are allowed. In these stationary orbits the electron does not radiate, despite Maxwell.
3. Each stationary orbit has a definite energy. An electron jumping from energy E₂ to a lower E₁ emits one photon with E₂ − E₁ = hν = hc/λ; it can also absorb energy and jump up.
4. In a stationary orbit the angular momentum is a whole multiple of h/2π: Bohr’s quantization rule.

Orbits and energies#

For hydrogen and hydrogen-like ions (He⁺, Li²⁺, … one electron, nucleus of charge Ze): the Coulomb force gives the centripetal force,

Combining with the quantization rule:

Radii

For hydrogen the smallest orbit is the Bohr radius, a₀ = ε₀h²/πme² ≈ 0.053 nm. In general

Energy levels

E₁ = −13.6 eV, E₂ = −3.4 eV, E₃ = −1.5 eV… The lowest level is the ground state; the others are excited states. Negative energy means the electron is bound; zero is the electron and nucleus far apart.

The spectrum explained#

At room temperature nearly every atom is in its ground state, with nowhere lower to go: no radiation. Supply energy and some electrons jump up; dropping back from level m to level n they emit

Every jump ending on n = 1 gives a line between 121.6 nm (2 → 1) and 91.2 nm (∞ → 1): the Lyman series. Ending on n = 2: 656.3 nm down to 365.0 nm, the Balmer series. Ending on n = 3: 1875 nm to 822 nm, the Paschen series.

Ionization, binding and excitation#

• Give a ground-state hydrogen atom more than 13.6 eV and its energy becomes positive: the electron is free. The minimum is the ionization energy (13.6 eV); the voltage that gives an electron that energy is the ionization potential (13.6 V).
• The binding energy is the energy released in assembling the atom from a far-apart electron and proton: also 13.6 eV for hydrogen.
• The excitation energy of a state is the energy to lift the atom to it from the ground state: 10.2 eV for hydrogen’s first excited state (excitation potential 10.2 V).

Where Bohr’s model falls short#

A huge success in 1913, and still how many people picture atoms. But it couldn’t be extended to atoms with more than one electron (not even helium), and precise measurements show the 656.3 nm line is really at least seven close lines. And its postulates are a patch: electrons are simply excused from obeying electromagnetism in certain orbits.

Quantum mechanics replaced Bohr’s orbits with waves of probability. It reproduces his energies exactly, but tells a very different story about where the electron is.

The wave function#

Light needs both pictures: the field wave E⃗(r⃗, t) and the photon. Where |E|² is large, photons are likely; yet a detector always catches a whole photon at one spot, never a fraction. An electron is described the same way, by a wave function ψ(r⃗, t), found by solving Schrödinger’s equation. ψ may be spread over a large region, but the electron is always detected whole at a single point; the probability of finding it in a small volume dV is |ψ|²dV. All its properties, energy included, are contained in ψ.

The hydrogen atom in quantum mechanics#

Its solutions ψ_{n l ml} are labelled by three numbers:

• n, the principal quantum number: 1, 2, 3, …
• l, the orbital angular momentum quantum number: 0, 1, …, n − 1.
• m_l, the magnetic quantum number: −l, …, +l (2l + 1 values).

No orbits, only probabilities

The ground state is

The chance of finding the electron between r and r + dr is P(r)dr, with P(r) = (4/a₀³)r²e^−2r/a₀. It peaks at r = a₀, where Bohr put the electron, but the electron can be found closer or farther. For n = 2, l = 0, P(r) has two humps, near a₀ and near 5.4a₀, while Bohr would put every n = 2 electron at exactly 4a₀.

Spin, shells and Pauli’s principle#

Every electron also has a built-in spin angular momentum, whose component along any direction is +h/4π or −h/4π, described by a fourth quantum number m_s = ±½. A set (n, l, m_l, m_s) is one quantum state. Each n has 2n² states, forming a major shell: K (n = 1), L (n = 2), M (n = 3)…

Values of l are written with letters s, p, d, f for l = 0, 1, 2, 3. States with the same n and l form a subshell: 1s, 2s, 2p, 3s… In atoms with many electrons the energy depends on both n and l, and electrons fill the states from the lowest energy upward.

Ordinary light sources are a crowd of atoms each doing its own thing. In a laser, one photon persuades atoms to emit identical copies of itself, giving light that is coherent, pure in colour and almost perfectly parallel.

Why ordinary light is messy#

In a normal source atoms emit independently: the photons have random phases (incoherent light), slightly different energies (a spread Δλ) and random directions. LASER, Light Amplification by Stimulated Emission of Radiation, gives light with all photons in phase, Δλ as small as about 10⁻⁶ nm (about a thousand times purer than the krypton-86 standard), and a beam so parallel it can go to the Moon and back.

Three ways an atom meets light#

Take an atom with levels E₁ (lower) and E₂ (higher).

• Stimulated absorption: a photon of energy E₂ − E₁ is absorbed, lifting the atom from E₁ to E₂.
• Spontaneous emission: left alone, an atom in E₂ falls back by itself, emitting a photon. A typical excited state lasts about 10 ns (its lifetime); some last about a millisecond, 10⁵ times longer: metastable states.
• Stimulated emission: a photon of energy E₂ − E₁ passing an atom already in E₂ can trigger it to drop, emitting a second photon with exactly the same energy, phase and direction.

Making light grow#

Suppose many more atoms are in a metastable state E₂ than in E₁. One photon stimulates an emission: now two identical photons. They stimulate two more atoms, and so on: an avalanche of identical photons. Two conditions are needed:

Population inversion and pumping

Normally the lower level is more crowded, and its atoms would just absorb the photons. The upper level must hold more atoms than the lower: a population inversion. Keeping it up means continually feeding atoms into E₂ and removing them from E₁: pumping.

Mirrors

Photons must stay around long enough to stimulate many emissions. Mirrors at the two ends of the tube or rod send them back and forth; one mirror lets through a little (about 1 %), which is the laser beam. Photons not travelling along the axis escape from the sides after a few reflections, so only light along the axis builds up: hence the beam’s directionality.

The helium–neon laser#

A glass tube holds about 90 % helium and 10 % neon at low pressure, with two electrodes connected to a high voltage and mirrors M₁ and M₂ (M₂ partly transmitting) at the ends. The laser transition is in neon, between E₂ = 20.66 eV and E₁ = 18.70 eV.

1. The discharge ionizes some atoms; the freed electrons, accelerated by the field, knock helium atoms into a metastable state at 20.61 eV.
2. Excited helium atoms collide with neon atoms and hand over their energy, lifting neon to E₂ (20.66 eV, very close): pumping.
3. Neon in E₁ lives only about 10 ns before dropping lower, so E₁ stays nearly empty: inversion between E₂ and E₁ is maintained.
4. Stimulated emission from E₂ to E₁ builds up between the mirrors.

Uses#

Since the first laser in 1960: microsurgery (a tightly focused spot), burning away tumours and cutting tissue with infrared lasers, re-attaching detached retinas, drilling metal and diamond, cutting 50 layers of cloth at once without fraying, telephone and internet signals through optical fibres, fusion research, surveying and aligning equipment, bouncing light off the Moon to measure its distance, holograms (3D images), and reading CDs and other optical discs.