Nuclides and isotope notation

A nuclide is a type of atomic nucleus with a particular number of protons and neutrons. It is written as `

`

The neutron number is

An isotope is one of two or more atoms of the same element with the same proton number but different neutron numbers. Isotopes therefore have the same chemical behaviour, because chemistry is governed mainly by electron arrangement, but their nuclear stability can be very different.

Carbon-12, carbon-13 and carbon-14, for example, all have $Z=6$. Their neutron numbers are 6, 7 and 8 respectively. Carbon-12 and carbon-13 are stable; carbon-14 is radioactive. Keep this distinction clear: isotopes are not automatically radioactive, and “isotope” is not a synonym for “unstable nucleus”.

Binding energy

Nuclear binding energy is the energy needed to split a nucleus completely into its separate protons and neutrons. When nucleons join to make a nucleus, energy is released. To separate that nucleus again, the same amount of energy has to be supplied.

Take deuterium as a simple model: one proton and one neutron make a bound nucleus. A bound deuterium nucleus has less mass than the proton and neutron on their own, because energy is released when the bound state forms.

Mass defect

The mass defect is the difference between the total mass of the separated nucleons and the mass of the nucleus formed from them. For a nucleus,

You may see nuclear masses written in kilograms, in unified atomic mass units $u$, or in $\mathrm{MeV\,c^{-2}}$. A unified atomic mass unit is a unit of mass equal to one twelfth of the mass of a neutral carbon-12 atom. For data-booklet style calculations, $1\,u\approx1.66\times10^{-27}\,\text{kg}\approx931.5\,\mathrm{MeV\,c^{-2}}$.

Atomic masses include electrons, so be careful when using them. In many nuclear calculations with neutral atomic masses, the electron masses cancel if the same number of electrons appears on both sides. If they don’t cancel, you must account for them carefully.

Why per nucleon matters

The total binding energy usually gets larger as nuclei get larger, for the simple reason that more nucleons are interacting. To compare one nucleus with another fairly, we use the average binding energy per nucleon: the binding energy divided by the number of nucleons,

A higher binding energy per nucleon tells us that each nucleon is, on average, held more tightly. So the useful idea is not “bigger nucleus means more stable”. It is “higher on the binding-energy-per-nucleon graph means more tightly bound”.

Reading the binding-energy curve

For light nuclei, the curve rises sharply. It reaches a peak around nucleon number 60, then falls slowly for heavier nuclei. That pattern explains two main ways energy can be released:

• light nuclei can release energy by fusion, moving up the curve towards higher binding energy per nucleon;
• very heavy nuclei can release energy by fission or alpha decay, moving towards products with greater binding energy per nucleon.

Helium-4 is unusually stable for a light nuclide. Alpha particles therefore appear often in the decay of heavy nuclei: a tightly bound group of two protons and two neutrons is energetically favourable.

Energy from missing mass

In nuclear reactions, the mass change can be big enough to show up through the energy transferred. Einstein’s mass–energy equivalence is

For a reaction, we usually use $\Delta E=c^2\Delta m$, where $\Delta E$ is the energy transferred (J).

When the products of a nuclear reaction have less total mass than the reactants, energy is released. When the products have more mass, energy has to be supplied. For calculations in MeV, the cleanest shortcut is usually $1\,u\approx931.5\,\text{MeV}\,c^{-2}$: a mass defect of $x\,u$ corresponds to about $931.5x\,\text{MeV}$.

Binding-energy method

Another way to calculate the energy released is to compare total binding energies. If the products have a greater total binding energy than the reactants, the difference is released, usually as kinetic energy of the products and/or photons. This is the same physics as the mass-defect method, just looked at from the energy side.

Why another force is needed

A strong nuclear force is a short-range attractive interaction between nucleons that can overcome the electrostatic repulsion between protons inside a nucleus. Without it, Coulomb repulsion would tear apart any nucleus containing more than one proton. Gravity is much too weak for this.

The strong nuclear force works over distances of the order of a femtometre. A femtometre is a unit of length equal to $10^{-15}\,\text{m}$. At typical nucleon separations, the strong force is very large and attractive. At larger separations it drops off quickly, which is why it doesn’t affect everyday objects.

Neutrons matter in larger nuclei. They add to the attractive strong interaction without adding proton–proton electrostatic repulsion. That is one reason heavy stable nuclei need more neutrons than protons.

So the stability of a nucleus comes from a balance, not from a quiet absence of forces. A star is held in a kind of balance too, but in that case inward gravity competes with outward thermal or radiation pressure. Nuclear stability is a microscopic balance between strong attraction and electric repulsion.

Random does not mean lawless

Random does not mean lawless

Radioactive decay is a nuclear process where an unstable nucleus changes into a more stable arrangement by emitting particles and/or electromagnetic radiation. It is random: we cannot predict which particular nucleus will decay next. It is spontaneous: ordinary external conditions such as temperature, pressure or chemical state do not change the decay rate.

This feels awkward at first, until you think about sample size. Even a tiny source contains an enormous number of nuclei, so statistical averages are very reliable. One nucleus is unpredictable. A large sample is not.

A parent nuclide is the original unstable nuclide before decay. A daughter nuclide is the nuclide produced by the decay. If the daughter is also unstable, further decays may occur; this sequence is a decay chain, a series of radioactive decays leading eventually towards greater stability.

Alpha, beta and gamma changes

An alpha particle is a $^{4}\text{He}$ nucleus made of two protons and two neutrons. During alpha decay, the parent nucleus loses two protons and two neutrons, so its proton number and neutron number both decrease by 2.

A beta-minus particle is an electron emitted from a nucleus during beta-minus decay. Here, a neutron in the nucleus changes into a proton. The proton number increases by 1, the neutron number decreases by 1, and the nucleon number stays unchanged.

A beta-plus particle is a positron emitted from a nucleus during beta-plus decay. A positron is the antiparticle of the electron; it has the same mass as an electron but a positive charge. In beta-plus decay, a proton changes into a neutron, so the proton number decreases by 1 and the neutron number increases by 1. The nucleon number is unchanged.

A gamma photon is a high-frequency photon emitted by a nucleus when it moves from an excited nuclear energy state to a lower energy state. Gamma emission changes the nucleus’s energy state, but not its proton number or nucleon number.

The guide does not require the weak nuclear force as an explanatory mechanism. For beta decay, then, use the observed nuclear changes and conservation rules rather than the deeper interaction.

Balancing nuclear equations

A nuclear equation has to conserve nucleon number and charge number. The quick check is simple: add the top numbers on each side, then do the same with the bottom numbers.

For alpha decay, use this pattern:

For beta-minus decay:

For beta-plus decay:

For gamma emission:

The star shows that the nucleus is in an excited state. Some textbooks write ${}^{4}_{2}\text{He}$ rather than ${}^{4}_{2}\alpha$; both refer to the same alpha particle.

What the equations do not show

Decay equations often leave out the energy released. In alpha decay, most of that energy becomes kinetic energy of the alpha particle, while the daughter nucleus gets a smaller recoil energy. In gamma emission, the energy leaves as a photon. Beta decay is a bit less neat, since a neutrino or antineutrino takes part in the energy sharing.

Why beta decay needs another particle

A neutrino is a neutral elementary particle with extremely small mass that interacts only very weakly with matter. An antineutrino is the corresponding antiparticle of a neutrino. In this course, we write a neutrino as $u$ and an antineutrino as $\bar{ u}$.

In beta-minus decay, the electron is emitted together with an antineutrino. In beta-plus decay, the positron is emitted together with a neutrino. Adding these particles keeps energy, momentum and charge-related conservation rules valid in beta decay.

For practical purposes, neutrinos are extremely difficult to detect. They can travel through huge thicknesses of matter with almost no interaction, so physicists inferred their existence from conservation arguments before it was confirmed experimentally.

Ionizing and penetrating power

Ionizing radiation is radiation that can remove electrons from atoms or molecules, producing ions. Alpha, beta and gamma radiations are all ionizing, but they don’t behave the same way in matter.

Alpha particles ionize strongly because they are massive and carry charge $+2$. They lose energy fast, so their penetration is low: paper or a few centimetres of air usually stops them. Outside the body, this is manageable. Inside the body, an alpha-emitting source is dangerous because it deposits its energy in a small volume of tissue.

Beta particles are moderately ionizing and moderately penetrating. They pass through paper, but a few millimetres of aluminium stops them. Gamma photons ionize weakly but penetrate strongly, so substantial absorption needs thick lead or concrete.

Comparison of alpha, beta and gamma radiation for ionization, penetration and uses.

Choosing radiation for real uses

The useful isotope depends on both half-life and radiation type. In medical imaging, gamma emitters work well because photons can escape the body and be detected; the half-life has to be long enough for the procedure but short enough to reduce dose. For treating tissue, stronger localized absorption may be useful. To check the thickness of aluminium sheet, beta radiation is suitable because its absorption changes noticeably with thickness. Alpha would be stopped too easily, while gamma might pass through with too little change. For underground pipe leaks or river-flow tracing, the source should be detectable and should have a short enough half-life to avoid long-term contamination. Radioactive dating also depends on choosing a nuclide with a half-life comparable to the age range being investigated.

Activity and count rate are not the same

Activity means the number of nuclei in a radioactive sample that decay each unit time. For activity, $A$ is measured in becquerel (Bq), with $1\,\text{Bq}=1\,\text{decay s}^{-1}$. The same letter also appears as the nucleon number in nuclide notation, so use the context.

Count rate means the number of radiation detections an instrument records per unit time. We can write the count rate as $R$, where $R$ is the detector count rate ($\text{s}^{-1}$ or counts $\text{s}^{-1}$). Count rate is usually lower than activity: the detector only covers part of the full solid angle, some radiation is absorbed before it reaches the detector, and the detector is not perfectly efficient.

A half-life is the time taken for the number of undecayed parent nuclei in a sample to drop to half its initial value. It is also the time taken for the activity of a sample to halve. We write half-life as $T_{1/2}$, where $T_{1/2}$ is the half-life (s).

Measuring a half-life

A typical school experiment uses a Geiger–Müller tube and counter for beta or gamma radiation, or a spark counter for alpha radiation. Start by measuring the background count rate with no source present, ideally over several minutes. Then put the source at a fixed distance from the detector and record the count rate at regular time intervals. Subtract the background to get the corrected count rate before using the graph to find the half-life.

Radioactive counting is statistical. If $C$ counts are recorded

the random uncertainty is roughly

. Longer counting intervals therefore reduce the percentage uncertainty, but don't make the interval so long that the source changes significantly during one reading.

Halving step by step

For a whole-number count of half-lives, just halve repeatedly. If

then after $n$ complete half-lives the fraction left is $(1/2)^n$.

In count-rate questions with constant geometry, use

The same halving pattern applies to the number of undecayed parent nuclei, the activity, and the corrected count rate.

Exact halving of parent amount and corrected count rate at integer half-lives.

For example, after 1, 2, 3 and 4 half-lives, the remaining parent fraction is $1/2$, $1/4$, $1/8$ and $1/16$. If the daughter is stable, its amount increases, but in the mathematical model the parent never becomes exactly zero.

Correcting the count

Background radiation is ionizing radiation detected even when no intended radioactive source is present. It can come from cosmic rays, rocks and building materials, radon gas, food, medical sources, and past artificial releases.

The corrected count rate is

Background matters most near the end of a decay experiment, when the source count rate has become small. So half-life estimates from a corrected-count-rate graph are usually most reliable if they are taken from the earlier part of the curve, where the source signal is much larger than background. Use several independent halving intervals, then average them.

Scattering evidence

Alpha-particle scattering shows that electrostatic repulsion isn’t the whole story at very small nuclear separations. At fairly large separations, the scattering agrees with the Coulomb prediction. But when alpha particles get close enough to some nuclei, the measured scattering stops fitting a purely electrostatic model. That mismatch points to an additional short-range nuclear interaction.

Electron scattering adds another line of evidence. Accelerated electrons behave like waves, so they can diffract from nuclei. A diffraction minimum occurs approximately when $\theta\approx\lambda/b$, where $\theta$ is the scattering angle of the first minimum (rad), $\lambda$ is the de Broglie wavelength of the electron (m), and $b$ is the nuclear diameter (m). From this, nuclear size can be investigated. At sufficiently high energies, scattering reveals internal nucleon structure, supporting the idea that nuclear behaviour is controlled by short-range interactions inside the nucleus.

The binding-energy curve gives evidence too. If every nucleon attracted every other nucleon equally over long ranges, binding energy per nucleon would keep increasing strongly with nucleon number. It doesn’t. The approximate saturation shows that each nucleon mainly interacts strongly with nearby nucleons.

The zone of stability

The zone of stability is the region on a neutron-number versus proton-number plot where stable nuclides are found. In small nuclei, stable nuclides tend to have $N\approx Z$. Larger nuclei need $N>Z$, since extra neutrons increase strong-force attraction without adding proton–proton electrostatic repulsion.

Nuclides above the zone of stability are neutron-rich. They usually undergo beta-minus decay, where a neutron changes into a proton, so the nuclide moves towards the zone. Nuclides below the zone are proton-rich. These tend to undergo beta-plus decay or electron capture, converting a proton into a neutron. Very heavy nuclides often undergo alpha decay; losing two protons and two neutrons shifts them towards a more stable region and makes the nucleus smaller.

That’s why the neutron-to-proton ratio matters. Nuclear stability depends not just on total size, but on the balance between strong attraction and electric repulsion.

Saturation of binding

Above about nucleon number 60, the binding energy per nucleon stays roughly constant, though it decreases slowly for heavier nuclei. This near-flat part of the curve is strong evidence that the strong nuclear force acts over a short range.

A nucleon in a heavy nucleus does not feel a strong attraction from every other nucleon there. Its near neighbours matter most. When extra nucleons are added at the surface of an already large nucleus, the average binding per nucleon shifts only a little.

The slight downward slope is still worth noticing. Very heavy nuclei have a little less binding energy per nucleon than nuclei near iron and nickel, so energy can be released when heavy nuclei split or are rearranged. That connects to fission in Topic E.4; here, the main point is the curve’s near-constancy as evidence for short-range nuclear attraction.

Discrete spectra mean discrete levels

A discrete spectrum contains only certain separated energies, not a continuous range. In a particular alpha decay, the emitted alpha particles often have one or a few well-defined kinetic energies. Gamma photons from a specific nuclear de-excitation also come out with specific energies.

This gives evidence for discrete energy levels in nuclei. A nucleus can move from one allowed energy state to another; the energy difference may appear as a gamma photon, or it may change the kinetic energy of an emitted alpha particle.

Atomic photons versus nuclear photons

Physically, photons are distinguished by their energy, not by the label we attach to them. An X-ray photon and a gamma photon can have the same energy. The name usually points to the origin: X-rays normally come from electron processes in atoms or from decelerating high-energy electrons, while gamma rays come from nuclear energy transitions. Because nuclear energy gaps are commonly much larger than atomic electron energy gaps, nuclear photons are often in the keV to MeV range.

The beta spectrum problem

In alpha decay, only two main products come out. If the parent nucleus starts at rest, conservation of energy and momentum fixes the kinetic energies for a given transition. So alpha energies are discrete.

In beta decay, the emitted beta particle does something different: it appears with a continuous range of energies, from near zero up to a maximum. If the decay left only the daughter nucleus and the beta particle, energy and momentum conservation would seem to fail.

The missing piece is the neutrino or antineutrino. The available energy is shared between the beta particle, the daughter nucleus and the neutrino or antineutrino. With three particles involved, the energy can be split in many different ways, which produces the continuous beta spectrum.

This is a neat example of conservation laws steering physics. Physicists did not give up conservation of energy and momentum; they proposed a neutral particle that would be hard to detect. Later experiments detected neutrino interactions, supporting the conservation-based hypothesis.

From proportional decay to an exponential law

For a radioactive sample, decay happens at a rate proportional to the number of undecayed nuclei still present. In decay-law equations, $N$ is reused for the number of undecayed nuclei remaining (unitless), while $N_0$ is the initial number of undecayed nuclei (unitless).

The radioactive decay law is

Use this equation when the time interval is not a whole number of half-lives; it works for arbitrary times.

Taking natural logarithms gives $\ln(N/N_0) = -\lambda t$. In experiments, that’s handy: for corrected count rate, a graph of $\ln R$ against time is a straight line, because count rate is proportional to the number of undecayed nuclei when the detector geometry is fixed.

The small-interval condition

The decay constant is fixed for a particular nuclide. It gives the probability per unit time that one nucleus will decay, but that phrase has to be used carefully. Over a short time interval, $P\approx\lambda\Delta t$, where $P$ is the probability that a nucleus decays during the interval (unitless) and $\Delta t$ is the time interval (s).

This approximation only works when $\lambda\Delta t$ is sufficiently small. Then the chance of decay during the interval is small, and the number of undecayed nuclei changes very little during that interval.

So don’t just say, loosely, “$\lambda$ is the probability of decay” unless the time unit and the small-interval condition are clear. A safer version is: $\lambda$ is the probability of decay per unit time in the limiting case of a very small time interval.

Activity law

Activity is the rate at which nuclei decay. Written in calculus form,

Using the decay constant,

$A=\lambda N=\lambda N_0e^{-\lambda t}$

So $A=A_0e^{-\lambda t}$ as well, where $A_0$ is the initial activity (Bq). Corrected count rate follows the same exponential curve, as long as the detector arrangement stays fixed.

Applications with arbitrary times

For medical isotopes and dating work, times often are not tidy multiples of a half-life. That’s when the exponential form is used. In radiocarbon dating, for example, the remaining fraction of carbon-14 is linked to the time since the organism stopped exchanging carbon with the environment. In medical imaging, the activity left after transport, preparation and scan time decides whether the isotope is still useful and what dose it gives.

Deriving the link

At one half-life, half of the original nuclei are still present. Put $N=N_0/2$ and $t=T_{1/2}$ into the decay law:

So,

Determining half-life in practice

For a short or medium half-life, first measure the background count. Then record observed count rate against time and subtract the background. From there, either read off repeated halving intervals from the corrected-count-rate curve or plot $\ln(R_{\text{corrected}})$ against time. The straight-line method is usually more reliable: its gradient is $-\lambda$, so the half-life comes from $T_{1/2}=\ln 2/\lambda$.

For a very long half-life, the activity changes far too slowly for you to watch it halve. Instead, find the number of nuclei in a known sample, measure its activity, and then use $A=\lambda N$ and $T_{1/2}=\ln 2/\lambda$.