Atomic Nuclei and Radioactive Decay

Lessons

Senior 6 · Physics

Atomic Nuclei and Radioactive Decay

5 lessons

Lesson 1 of 5

Nuclei, Nuclides, Mass and Energy

25 min

The densest matter on Earth

In Senior 5 you met Rutherford's discovery: nearly all the mass of an atom sits in a nucleus about $10^{-14}$ m across, while the atom itself is about $10^{-10}$ m. Squeeze that much mass into so small a space and you get a density near $2 \times 10^{17}\,\mathrm{kg\,m^{-3}}$ : a teaspoon of nuclear matter would have a mass of about a billion tonnes. Two more surprises follow in this unit. Every nucleus, from hydrogen to uranium, has nearly the same density. And every nucleus has LESS mass than the protons and neutrons it is made of. The "missing" mass is the energy that holds the nucleus together, and it is the source of the energy of the Sun and of nuclear power. This lesson sets up the language and units you need.

What a nucleus is made of

A nucleus contains protons (charge $+ e$ ) and neutrons (no charge), together called nucleons. Three numbers describe it:

the atomic number (proton number) $Z$ , which fixes the element;
the neutron number $N$ ;
the mass number (nucleon number) $A = Z + N$ .

A nucleus is written ${}^{A}_{Z}\mathrm{X}$ , where X is the chemical symbol, or simply as "X-A": uranium-235 is ${}^{235}_{92}\mathrm{U}$ .

Figure 1.1: Nuclide notation.

Term	Same	Different	Example
isotopes	$Z$	$N$ and $A$	${}^{12}_{6}\mathrm{C}$ , ${}^{13}_{6}\mathrm{C}$ , ${}^{14}_{6}\mathrm{C}$
isotones	$N$	$Z$ and $A$	${}^{14}_{6}\mathrm{C}$ , ${}^{15}_{7}\mathrm{N}$ , ${}^{16}_{8}\mathrm{O}$ (all $N = 8$ )
isobars	$A$	$Z$ and $N$	${}^{14}_{6}\mathrm{C}$ , ${}^{14}_{7}\mathrm{N}$ , ${}^{14}_{8}\mathrm{O}$

Figure 1.2: A corner of the chart of nuclides. Physicists plot every known nucleus on such a chart, about 3300 of them; fewer than 300 are stable.

Definition

A nuclide is a particular kind of nucleus, fixed by its values of $Z$ and $N$ , written ${}^{A}_{Z}\mathrm{X}$ . Isotopes are nuclides of the same element: the same $Z$ but different $N$ . They have the same chemistry but different nuclear properties. Example: ${}^{12}_{6}\mathrm{C}$ is stable, while ${}^{14}_{6}\mathrm{C}$ is radioactive.

How big is a nucleus?

Scattering experiments with fast electrons and alpha particles show that the nuclear radius grows with mass number as

r = r_0\sqrt[3]{A}, \qquad r_0 \approx 1.2 \times 10^{-15}\,\mathrm{m} = 1.2\,\mathrm{fm}

(1 femtometre, fm, is $10^{-15}$ m.) Since the volume $\frac{4}{3}\pi r^3 = \frac{4}{3}\pi r_0^3A$ is proportional to $A$ , each nucleon takes up the same volume, like marbles packed in a bag. The density is therefore the same for all nuclei:

\rho = \frac{\text{mass}}{\text{volume}} \approx \frac{A \times (1.66 \times 10^{-27}\,\mathrm{kg})}{\frac{4}{3}\pi r_0^3A} = \frac{3 \times 1.66 \times 10^{-27}}{4\pi(1.2 \times 10^{-15})^3} = 2.3 \times 10^{17}\,\mathrm{kg\,m^{-3}}

The $A$ cancels. For comparison, water is $1.0 \times 10^{3}\,\mathrm{kg\,m^{-3}}$ .

Worked example 1: two nuclei compared

Find the radii of the nuclei ${}^{56}_{26}\mathrm{Fe}$ and ${}^{238}_{92}\mathrm{U}$ , taking $r_0 = 1.2\,\mathrm{fm}$ . How many times larger is the volume of the uranium nucleus?

Solution

$r_{\text{Fe}} = 1.2 \times \sqrt[3]{56} = 1.2 \times 3.83 = 4.6\,\mathrm{fm}$ ; $r_{\text{U}} = 1.2 \times \sqrt[3]{238} = 1.2 \times 6.20 = 7.4\,\mathrm{fm}$ .

Volume is proportional to $A$ , so $\frac{V_{\text{U}}}{V_{\text{Fe}}} = \frac{238}{56} = 4.3$ . The uranium nucleus has 4.3 times the volume but only $\frac{7.4}{4.6} = 1.6$ times the radius, because $r \propto \sqrt[3]{A}$ .

Units for nuclear masses and energies

Nuclear masses are tiny in kilograms, so physicists use the unified atomic mass unit u, defined as one twelfth of the mass of a neutral carbon-12 atom:

1\,\mathrm{u} = 1.660\,54 \times 10^{-27}\,\mathrm{kg}

Nuclear energies are tiny in joules, so they are given in electronvolts: $1\,\mathrm{eV} = 1.602 \times 10^{-19}\,\mathrm{J}$ , and $1\,\mathrm{MeV} = 10^{6}\,\mathrm{eV} = 1.602 \times 10^{-13}\,\mathrm{J}$ .

Einstein's special relativity showed that mass and energy are equivalent. A mass $m$ is equivalent to an energy

E = m c^{2}

where $c = 3.00 \times 10^{8}\,\mathrm{m\,s^{-1}}$ is the speed of light. Any change in energy $\Delta E$ comes with a change in mass $\Delta m = \frac{\Delta E}{c^2}$ . In chemistry this change is far too small to notice; in nuclear physics it is about a million times larger and easily measured.

Particle	Mass / kg	Mass / u	Rest energy / MeV
proton	$1.672\,62 \times 10^{-27}$	1.007 276	938.27
neutron	$1.674\,93 \times 10^{-27}$	1.008 665	939.57
electron	$9.109\,38 \times 10^{-31}$	0.000 549	0.511

Values from CODATA 2018 (NIST), rounded.

Definition

The unified atomic mass unit (u) is $\frac{1}{12}$ of the mass of a neutral atom of carbon-12: $1\,\mathrm{u} = 1.660\,54 \times 10^{-27}\,\mathrm{kg}$ , equivalent to 931.5 MeV. The electronvolt (eV) is the energy gained by one electron accelerated through a potential difference of 1 V: $1\,\mathrm{eV} = 1.602 \times 10^{-19}\,\mathrm{J}$ . Example: a proton's rest energy is 938.3 MeV.

Worked example 2: the energy equivalent of 1 u

Show that 1 u is equivalent to about 931.5 MeV.

Solution

E = mc^2 = (1.660\,54 \times 10^{-27}\,\mathrm{kg})(2.997\,92 \times 10^{8}\,\mathrm{m\,s^{-1}})^2 = 1.492\,42 \times 10^{-10}\,\mathrm{J}

In MeV: $\frac{1.492\,42 \times 10^{-10}}{1.602\,18 \times 10^{-13}} = 931.5\,\mathrm{MeV}$ . So a mass change of 1 u releases or absorbs 931.5 MeV. Use this conversion whenever masses are given in u: it saves converting to kilograms and back.

Worked example 3: how fast does the Sun lose mass?

The Sun radiates energy at $3.828 \times 10^{26}\,\mathrm{W}$ (the IAU nominal solar luminosity). Find the rate at which its mass decreases.

Solution

\frac{\Delta m}{\Delta t} = \frac{P}{c^2} = \frac{3.828 \times 10^{26}}{(2.998 \times 10^{8})^2} = 4.26 \times 10^{9}\,\mathrm{kg\,s^{-1}}

More than four million tonnes every second. Even so, over its 4.6-billion-year life the Sun has lost well under 0.1% of its mass this way. You will see in Lesson 5 that this energy comes from nuclear fusion.

Warning

Common mistakes. (1) Confusing the mass number $A$ (a whole number of nucleons) with the atomic mass in u (for example 235.044 u for U-235). (2) Forgetting that tables usually give the mass of the neutral ATOM, including $Z$ electrons, not the bare nucleus. (3) Using $c = 3 \times 10^{8}$ and then quoting six significant figures: match your constants to the precision you need. (4) Writing $r \propto A$ : the radius goes as the CUBE ROOT of $A$ .

Tip

Keep one conversion in your head: 1 u = 931.5 MeV. Nuclear energy problems then become: find the mass change in u, multiply by 931.5, done.

The Real Meaning of E=mc² (PBS Space Time). Watch for the idea that mass is a measure of the energy a system contains, which is why a bound nucleus weighs less than its separate nucleons.

✦

Practice

State the numbers of protons, neutrons and nucleons in ${}^{60}_{27}\mathrm{Co}$ and ${}^{131}_{53}\mathrm{I}$ .
From the list ${}^{40}_{18}\mathrm{Ar}$ , ${}^{40}_{19}\mathrm{K}$ , ${}^{40}_{20}\mathrm{Ca}$ , ${}^{39}_{19}\mathrm{K}$ , ${}^{41}_{20}\mathrm{Ca}$ , pick out (a) a pair of isotopes, (b) three isobars, (c) a pair of isotones.
Find the radius of a ${}^{4}_{2}\mathrm{He}$ nucleus and of a ${}^{197}_{79}\mathrm{Au}$ nucleus ( $r_0 = 1.2\,\mathrm{fm}$ ).
Show that the density of a nucleus does not depend on its mass number.
Convert a mass of 0.0304 u into MeV and into joules.
Find the energy equivalent of 1.0 g of matter, in joules and in kilowatt-hours.
The mass number of chlorine is often quoted as 35.5. Explain why this is not a mass number of any nucleus.
Keza says: "The nucleus of a uranium atom is much denser than the nucleus of a helium atom, because uranium is a much heavier element." Evaluate her statement.

Show answersHide answers

Co-60: 27 protons, 33 neutrons, 60 nucleons. I-131: 53 protons, 78 neutrons, 131 nucleons.
(a) ${}^{39}_{19}\mathrm{K}$ and ${}^{40}_{19}\mathrm{K}$ (or ${}^{40}_{20}\mathrm{Ca}$ and ${}^{41}_{20}\mathrm{Ca}$ ); (b) ${}^{40}_{18}\mathrm{Ar}$ , ${}^{40}_{19}\mathrm{K}$ , ${}^{40}_{20}\mathrm{Ca}$ ; (c) ${}^{40}_{18}\mathrm{Ar}$ and ${}^{41}_{20}\mathrm{Ca}$ have $N = 22$ and $21$ , so no; ${}^{39}_{19}\mathrm{K}$ ( $N = 20$ ) and ${}^{40}_{20}\mathrm{Ca}$ ( $N = 20$ ) are isotones (so are ${}^{40}_{19}\mathrm{K}$ and ${}^{41}_{20}\mathrm{Ca}$ , both $N = 21$ ).
He: $1.2 \times \sqrt[3]{4} = 1.9\,\mathrm{fm}$ ; Au: $1.2 \times \sqrt[3]{197} = 7.0\,\mathrm{fm}$ .
Mass $\approx Au$ ; volume $= \frac{4}{3}\pi r_0^3A$ ; $\rho = \frac{3u}{4\pi r_0^3}$ , with no $A$ .
$0.0304 \times 931.5 = 28.3\,\mathrm{MeV}$ ; $28.3 \times 1.602 \times 10^{-13} = 4.54 \times 10^{-12}\,\mathrm{J}$ .
$E = 1.0 \times 10^{-3} \times (3.00 \times 10^{8})^2 = 9.0 \times 10^{13}\,\mathrm{J}$ ; $\frac{9.0 \times 10^{13}}{3.6 \times 10^{6}} = 2.5 \times 10^{7}\,\mathrm{kWh}$ .
35.5 is the average relative atomic MASS of natural chlorine, a mixture of about three parts Cl-35 to one part Cl-37. Each nucleus has a whole-number mass number (35 or 37).
Wrong. Both volume and mass are proportional to $A$ , so all nuclei have about the same density, $2.3 \times 10^{17}\,\mathrm{kg\,m^{-3}}$ . The uranium nucleus is heavier because it is BIGGER, not denser.

Key points

${}^{A}_{Z}\mathrm{X}$ : $Z$ protons, $N = A - Z$ neutrons. Isotopes share $Z$ , isotones share $N$ , isobars share $A$ .
$r = r_0\sqrt[3]{A}$ with $r_0 \approx 1.2\,\mathrm{fm}$ ; all nuclei have density $\approx 2.3 \times 10^{17}\,\mathrm{kg\,m^{-3}}$ .
$E = m c^{2}$ ; $1\,\mathrm{u} = 1.660\,54 \times 10^{-27}\,\mathrm{kg} = 931.5\,\mathrm{MeV}$ ; $1\,\mathrm{eV} = 1.602 \times 10^{-19}\,\mathrm{J}$ .

Sources and credits

10.2: Properties of Nuclei, from University Physics Volume 3 by Samuel J. Ling, Jeff Sanny and William Moebs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Constants: CODATA 2018 recommended values, National Institute of Standards and Technology (NIST). Solar luminosity: IAU 2015 Resolution B3 (nominal value), International Astronomical Union.
Video: The Real Meaning of E=mc², PBS Space Time, embedded from YouTube.

The two diagrams, the worked examples and the practice exercises are written by Ganzaa.

1The densest matter on Earth

2What a nucleus is made of

3How big is a nucleus?

4Units for nuclear masses and energies

The densest matter on Earth

What a nucleus is made of

How big is a nucleus?

Units for nuclear masses and energies