Atom, nucleus, spectra and radioactivity

Fast positive particles are fired at thin foil. Almost all pass straight through, but a few turn sharply sideways or backwards. How can easy passage coexist with such strong deflection, and what does it reveal about the atom?

If positive charge were spread throughout the atom, the rare, very large deflections of fast alpha particles would be hard to explain. Thin-foil experiments gave a different picture: most particles pass almost straight through, while a small fraction approach a tiny region of strong repulsion. This led to the nuclear model: positive charge and nearly all atomic mass are concentrated in a very small nucleus. The conclusion links the model to the observed distribution of deflections.

An atom can be electrically neutral overall. Its positive nuclear charge is then balanced by negative electrons: a neutral atom has as many electrons as protons. Yet picturing electrons as tiny planets on precisely known orbits is insufficient. The modern model describes allowed energy states. This additional restriction is needed to explain spectra, which a simple 'nucleus plus electrons' picture alone cannot predict.

Ordinary nuclei contain positive protons and electrically neutral neutrons, together called nucleons. In a nucleus with several nucleons, nuclear interactions hold them together and can overcome proton repulsion at short distances. There is an important limiting case: the most common hydrogen nucleus contains one proton and no neutron. It would therefore be wrong to say that every nucleus must contain both kinds of particle.

How can we specify nuclear composition briefly? Atomic number Z counts protons and so identifies the chemical element. Mass number A counts all nucleons: protons plus neutrons. Both are dimensionless. Replacing one neutron with a proton leaves A unchanged but changes Z and the element. The name 'mass number' does not make A a mass in kilograms: it is an integer count of particles.

Subtract protons from the total nucleon count: A = Z + N, so N = A − Z is the number of neutrons in one nucleus. Equal Z with different N gives isotopes of the same element. Removing an electron, however, leaves the nucleus and isotope unchanged and makes the atom an ion. Comparing these two imagined changes helps distinguish nuclear composition from the electron cloud's electrical state.

Why does an atom not absorb any arbitrary amount of light energy? In such processes, light transfers energy in separate packets called photons. Higher light frequency means more energy per photon. This complements the wave picture without cancelling the observed interference fringes. A photon is not simply a cut-out piece of a sine wave: the concept describes how light and matter exchange energy.

Allowed atomic energies resemble marked levels: intermediate values are unavailable to a stationary state. In a transition to lower energy, the atom can emit a photon whose energy equals the level difference; moving upwards requires a suitable photon. This gives particular frequencies and separate spectral lines. Steps are a useful analogy for energy differences, not literal floors inside the atom between which an electron ball flies.

Changes are not confined to electrons. An unstable nucleus can spontaneously transform into another state or nucleus while emitting particles or photons: this is radioactivity. The decay time of one nucleus cannot be known in advance. 'Radioactive' does not imply a visible glow. Regularity emerges in a large group of identical nuclei, not in a precise timetable for each one.

One emitted particle is the alpha particle: a positively charged helium nucleus with two protons and two neutrons. Emitting it removes four nucleons from the parent nucleus: A decreases by 4 and Z by 2. Counting explains the changes in element and mass number without memorizing a separate diagram. This is not an ordinary neutral helium atom: an alpha particle has no electrons.

Beta-minus decay emits a negatively charged electron; beta-plus decay emits a positron with the electron's mass and a positive charge. These particles are produced in the nuclear transformation; a beta electron is not removed from the electron cloud as in ionization. A neutrino or antineutrino also participates and carries away some energy. Thus a diagram showing just one visible charged particle omits some decay products.

A nucleus can move to lower energy without changing its proton or neutron count. It then emits a gamma photon, an electrically neutral packet of electromagnetic radiation. Its energy state changes while A and Z stay the same. Compare alpha decay, which removes nucleons: here energy is released without that change of composition. A photon's neutrality does not prevent it interacting with matter and transferring energy.

How can we detect what our eyes cannot see? Radiation, or secondary particles it creates, can remove electrons from atoms, causing ionization. The charged atomic remnants and electrons can produce an electrical signal in a counter; a suitable chamber shows tracks. The instrument records effects of interaction, not necessarily the particle directly. The absence of a visible glow therefore proves nothing about whether radiation is present.

Take very many identical unstable nuclei. The half-life T is the time after which half the initial number remains on average. For one nucleus this is not a compulsory deadline: it may decay earlier or survive several half-lives. A large group makes the remaining fraction predictable, with small relative random fluctuations. Half-life belongs to the isotope, not to the mass of the sample chosen.

Start with N₀ parent nuclei. After T we expect N₀/2; after 2T, half the remainder gives N₀/4; after 3T, N₀/8. Each step reduces the current number rather than subtracting the same half of the initial number. The group therefore need not vanish after two half-lives. Imagining coins with roughly half randomly removed each round helps explain the statistics, although decay occurs in continuous time.

All intervals are covered by Nрад = N₀·2^(−t/T), the mean number of undecayed parent nuclei of one isotope when none are added to the sample. At t = 0, N₀ remain; at t = T, half remain; increasing t reduces the mean. Express t and T in the same units so the exponent is dimensionless. Nрад counts sample nuclei, whereas N = A − Z counts neutrons in one nucleus: these are different counting tasks.

A nucleus has Z = 6, A = 14. A large group of these unstable nuclei initially had N₀ = 8000. How many neutrons are in one nucleus, and how many nuclei are expected after two half-lives?

First consider one nucleus: Z = 6 means six protons and A = 14 means fourteen nucleons in total. Hence N = A − Z = 14 − 6 = 8 neutrons.

Now consider the size of the large group, not nuclear composition. Over two half-lives, the remainder is halved twice: 8000 → 4000 → 2000.

Check using the law: Nрад = 8000·2⁻² = 2000. This is the expected number of undecayed parent nuclei; the actual number fluctuates randomly around the mean.

The numbers 8 and 2000 refer to different things: neutrons inside one nucleus and nuclei throughout the sample.

Each such nucleus has 8 neutrons. After two half-lives, an average of 2000 parent nuclei remain, one quarter of the initial number.

Scattering indicates a small, massive nucleus; line spectra additionally require discrete energy states.

Z identifies the element, A counts nucleons, and N = A − Z gives neutrons; removing an electron changes ionization state, not the isotope.

One nucleus decays randomly, but a large group follows successive halving of the remainder.

Nuclear interactions are connected with stored energy and mass defect.