13.1 Atomic Masses, Composition and Size of the Nucleus

A drop of blood from an athlete, a fragment of ice drilled from two kilometres under the Antarctic surface, a grain of meteorite: laboratories put all of them through the same machine, a mass spectrometer. It turns atoms into ions and lets a magnetic field sort them by mass, so precisely that it can tell oxygen-16 from oxygen-18, or a natural hormone from a synthetic one whose atoms differ by a single neutron. The ratio of heavy to light oxygen in old ice tells climate scientists the temperature of the Earth hundreds of thousands of years ago; the same kind of measurement tells geologists the age of a rock.

In Chapter 12 — Atoms we met the nucleus as a tiny, heavy, positively charged centre of the atom. In this section we look at it more closely. First we need a unit small enough to weigh atoms in, and a way of measuring their masses. Then we ask what the nucleus is made of — protons and neutrons — and how to label and classify nuclei. Finally we ask how big a nucleus is, and find a simple law, $R=R_0A^{1/3}$, with a striking consequence: nuclear matter has the same density in every nucleus, about $10^{14}$ times the density of water.

Atomic Masses and the Atomic Mass Unit

An atom of carbon has a mass of about $2\times10^{-26}$ kg. Numbers like that are awkward, so atomic and nuclear masses are measured in a unit chosen to make them close to whole numbers.

On this scale the hydrogen atom has mass $1.007825$ u, the proton $m_p=1.007276$ u, the neutron $m_n=1.008665$ u and the electron only $m_e=0.000549$ u. Notice that $1\ \text{u}\approx$ the mass of one nucleon, and that $N_A$ atoms of mass $1$ u have a mass of $1$ g: $(6.022\times10^{23})(1.66\times10^{-24}\ \text{g})\approx1$ g. That is why the atomic mass in u and the molar mass in g/mol are the same number.

Measuring atomic masses — the mass spectrometer. Atomic masses are measured by turning atoms into positive ions, accelerating them, and sending them into a uniform magnetic field at right angles to their velocity. As you saw in Section 4.1, an ion of mass $m$ and charge $q$ moving with speed $v$ then moves on a circle of radius $$r=\frac{mv}{qB}.$$ Ions with the same speed and charge but different masses follow circles of different radii and land at different places on a detector. The positions give the masses; the number of ions arriving at each place gives the relative amounts. With ions accelerated through a potential difference $V$, $\tfrac12mv^2=qV$, and the radius becomes $r=\dfrac{1}{B}\sqrt{\dfrac{2mV}{q}}$ — heavier ions still swing wider. When J. J. Thomson and F. W. Aston sent neon ions through such an instrument (1912–1919) they found two landing places, at masses close to $20$ u and $22$ u: atoms of the same element can have different masses. Natural neon is a mixture.

The atomic mass printed in a periodic table is the average over the natural mixture, weighted by the fraction (the natural abundance) of each kind of atom: $$\bar m=\sum_i f_i\,m_i,\qquad \sum_i f_i=1.$$

The Composition of the Nucleus

Rutherford's experiments showed that the nucleus carries a positive charge $+Ze$, where $Z$ is the number of electrons in the neutral atom. The obvious guess — that the nucleus is just $Z$ protons — fails at once: helium has $Z=2$ but a mass of about $4$ u, and uranium has $Z=92$ but a mass of about $238$ u. Something neutral and roughly as heavy as a proton must make up the rest.

In 1932 James Chadwick found it. When beryllium was bombarded with alpha particles it gave off a very penetrating, uncharged radiation that could knock protons out of paraffin wax with energies far too large for gamma rays. Chadwick showed, using conservation of energy and momentum in these collisions, that the radiation consisted of neutral particles with almost the same mass as the proton: neutrons. A free neutron is not stable — on its own it changes into a proton, an electron and an antineutrino in about a quarter of an hour on average — but inside a stable nucleus it lasts indefinitely.

Isotopes have the same number of electrons and therefore the same chemistry; they can be separated only by physical methods that depend on mass, such as the mass spectrometer. Isobars are different elements and behave completely differently in chemical reactions, even though their masses are almost equal.

Why not electrons in the nucleus? Before the neutron was known, some physicists suggested that a nucleus of mass number $A$ held $A$ protons and $A-Z$ electrons. Two facts rule this out. Confining an electron to a region $10^{-14}$ m across would, by the uncertainty principle, give it a kinetic energy of many MeV, far more than any nucleus could hold; and the spins of nuclei such as nitrogen-14 come out wrong. Electrons that emerge from nuclei in some decays are created at the moment of emission, not stored inside.

The Size of the Nucleus

The alpha-scattering experiments of Chapter 12 — Atoms only gave an upper limit on the size of a nucleus: an alpha particle of a few MeV is turned back by the Coulomb repulsion of a gold nucleus while still a few times $10^{-14}$ m from its centre, so the nucleus must be smaller than that. To probe further you need projectiles that get closer. In the 1950s beams of fast electrons — which feel only the electric force and are not repelled by the nucleus — were scattered from many different nuclei, and the pattern of scattering revealed how the charge is spread out. Nuclear sizes are measured in femtometres (also called fermis): $1\ \text{fm}=10^{-15}$ m.

Because $A^{1/3}$ grows so slowly, nuclear radii span only a small range: from $1.2$ fm for a single proton to about $7.4$ fm for uranium-238, even though the number of nucleons increases more than two-hundredfold. For two nuclei, $\dfrac{R_1}{R_2}=\left(\dfrac{A_1}{A_2}\right)^{1/3}$.

Exercises