The Atom: Nucleus, Electrons, Isotopes

An atom is the smallest unit that still carries the character of a chemical element: a nucleus holding protons and neutrons, surrounded by electrons occupying the space around it. Almost all the mass sits inside the nucleus, and almost none of the volume does.

Two CODATA 2022 constants say that better than any drawing. The Bohr radius — the characteristic scale of the electron part of a hydrogen atom — is 5.291 772 105 44(82) × 10⁻¹¹ m; the rms charge radius of the proton, the nucleus of that same atom, is 8.4075(64) × 10⁻¹⁶ m. Divide the Bohr radius by the proton radius and you get about 63,000.

The nucleus: where the mass stays

A proton weighs 1.007 276 466 5789(83) u and a neutron 1.008 664 916 06(40) u, while an electron weighs only 5.485 799 090 441(97) × 10⁻⁴ u. The ratio of proton mass to electron mass is 1836.152 673 426(32).

Nor does a nucleus swell in step with the number of particles it holds: the IAEA charge-radius table gives ¹²C 2.4702(22) fm and ²³⁸U 5.8571(33) fm — nearly 20 times the nucleons, but only about 2.4 times the radius.

Electrons: shells and subshells, not orbits

NIST describes the state of an electron in an atom with the quantum numbers n, l, mₗ, mₛ — not with a path. Electrons sharing n belong to the same shell; those sharing both n and l belong to the same subshell, and electrons within a subshell are equivalent. The values l = 0, 1, 2, 3 are written as the letters s, p, d, f. The Pauli exclusion principle forbids two electrons from matching in all four quantum numbers, so a subshell holds at most 2(2l + 1) electrons: 2 for s, 6 for p, 10 for d.

That rule can be read straight off the data. The ground configurations NIST lists for the first ten elements fill in exactly the order 1s, 2s, then 2p: hydrogen 1s, helium 1s², lithium 1s²2s, up to neon 1s²2s²2p⁶.

The energy needed to strip the outermost electron shows why a "full shell" is a real thing. Neon, with 2p full, takes 21.564541(7) eV; lithium, with a single loose electron in 2s, takes 5.391 714 996(22) eV — nearly four times less. An electronvolt is a unit of energy that reduces to joules. These same levels are what produce light when an electron drops to a lower one, the mechanism behind the aurora.

Z and A: two numbers are enough to name a nucleus

The atomic number Z is the proton count, and it decides the element. The mass number A is the total nucleon count, protons plus neutrons; the neutron count is the difference A − Z. NIST's data tables mark each isotope by exactly those three things: atomic number, element symbol and mass number.

Change Z and you change the element. Change A while holding Z fixed and it is still the same element — and that is the definition of an isotope.

Isotopes: same element, different neutron count

Carbon always has Z = 6. ¹²C has 6 neutrons, ¹³C has 7, ¹⁴C has 8. According to NIST, the carbon usually met in the laboratory is 0.9893(8) ¹²C and 0.0107(8) ¹³C; ¹⁴C occurs only in traces. The first two are stable, while ¹⁴C has a half-life of 5700(30) years and decays 100% by β⁻, according to the IAEA's ENSDF data.

Uranium gives another example: 0.992742(10) is ²³⁸U and 0.007204(6) is ²³⁵U; both are α emitters, but their half-lives are 4.468(6) × 10⁹ years and 7.04(1) × 10⁸ years respectively — more than a sixfold gap, enough for the ratio between them to shift over time and become a clock. Measurements of isotope ratios like these are exactly what produce figures such as "tropical seawater warmed by about 8–10 °C" in the Permian extinction.

The number of nuclides with experimental data far exceeds the number of elements: the NUBASE2020 evaluation records ground-state properties for 3,340 nuclides, with data through 30 October 2020. Yet most light elements still have at least one sufficiently stable isotope — in CIAAW's table of radioactive elements, only two entries have Z below 83: technetium (Z = 43) and promethium (Z = 61).

Atomic mass is an interval, not a number

The atomic mass scale has been anchored on ¹²C since 1960: one dalton is 1/12 the mass of a ¹²C atom that is neutral, unbound, in its ground state and at rest. So ¹²C weighs exactly 12 u — by definition, not by measurement.

But a real carbon sample is a mixture of isotopes, and the proportions of that mixture depend on where the material came from. So NIST and IUPAC give carbon not a single standard atomic weight but an interval: [12.0096, 12.0116]. Hydrogen likewise: [1.00784, 1.00811]. The familiar 12.011 on the periodic table is one point inside that interval, not a constant of nature.

The mass that is not there

Adding up 6 protons, 6 neutrons and 6 electrons separately, using the CODATA 2022 values, gives 12.098940 u. A real ¹²C atom weighs exactly 12 u. About 0.098940 u — roughly 0.82% — never shows up in the total, and that shortfall corresponds to the binding energy holding the nucleus together.

That energy is not the same for every nucleus. The IAEA gives the binding energy per nucleon as 7073.9156 keV for ⁴He, 7680.1446 keV for ¹²C and 8790.3563 keV for ⁵⁶Fe: the closer to iron, the more tightly each nucleon is held. That difference is why fusion inside the Sun releases energy, and also why the chain stops at iron.