Nuclides and isotope notation

A nuclide is a type of atomic nucleus with a particular number of protons and neutrons. It is written as ({}^{A}_{Z}X), where (A) is the nucleon number or mass number (unitless), (Z) is the proton number or atomic number (unitless), and (X) is the chemical symbol. The neutron number is (N=A-Z), where (N) is the number of neutrons in the nucleus (unitless).

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, since chemistry depends mainly on electron arrangement, but their nuclear stability can be very different.

Take carbon-12, carbon-13 and carbon-14. They all have (Z=6). Their neutron numbers are 6, 7 and 8 respectively. Carbon-12 and carbon-13 are stable; carbon-14 is radioactive. The key point: 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 come together to form a nucleus, energy is released. To pull that nucleus apart again, you have to supply the same amount of energy.

A simple example is deuterium: one proton and one neutron joined as a bound nucleus. The bound deuterium nucleus has less mass than the separate proton and neutron, since energy has been released as 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 they form. For a nucleus, (\Delta m=Zm_p+Nm_n-m_{ ext{nucleus}}), where (\Delta m) is the mass defect (kg, u, or MeV c⁻²), (m_p) is the mass of a proton (kg, u, or MeV c⁻²), (m_n) is the mass of a neutron (kg, u, or MeV c⁻²), and (m_{ ext{nucleus}}) is the mass of the nucleus (kg, u, or MeV c⁻²).

You may see nuclear masses given in kilograms, in unified atomic mass units (u), or in ( ext{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. In data-booklet style work, (1,u\approx1.66 imes10^{-27}, ext{kg}\approx931.5, ext{MeV c}^{-2}).

Take care when atomic masses are used, because an atom includes electrons. In many nuclear calculations with neutral atomic masses, 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

Total binding energy usually increases as nuclei become larger, since more nucleons are interacting. For a fair comparison between nuclei, we use the average binding energy per nucleon: the binding energy divided by the number of nucleons, \( ext{BE per nucleon}=BE/A\), where \(BE\) is the total binding energy of the nucleus (J or MeV).

A larger binding energy per nucleon tells us that each nucleon is, on average, held more tightly. So the useful interpretation 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 steeply. It peaks around nucleon number 60, then decreases slowly for heavier nuclei. This gives two main ways for nuclei to release energy:

• 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. That helps explain why alpha particles occur so 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 (E=mc^2), where (E) is the energy equivalent of a mass (J), (m) is mass (kg), and (c) is the speed of light in a vacuum (m s⁻¹). For a reaction we usually write (\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. If the products have more mass, energy must be supplied. In MeV calculations, (1,u\approx931.5, ext{MeV c}^{-2}) is usually the simplest route: a mass defect of (x,u) corresponds to about (931.5x, ext{MeV}).

Binding-energy method

Another way to find the energy released is to compare total binding energies. If the total binding energy of the products is greater than that of the reactants, the difference is released, usually as kinetic energy of the products and/or photons. It’s 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 with more than one proton. Gravity is much too weak for that role.

The strong nuclear force works over distances of about a femtometre. A femtometre is a unit of length equal to (10^{-15}, ext{m}). At typical nucleon separations, the strong force is extremely large and attractive; at larger separations, it drops away quickly, so it does not affect everyday objects.

Neutrons matter a lot 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.

A stable nucleus is not sitting there because no forces act on it. It is a balance. A star is held in a kind of balance too, but in that case inward gravity competes with outward thermal or radiation pressure. In a nucleus, the balance is microscopic: strong attraction against electric repulsion.

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 can’t 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.

That can feel awkward at first, but sample size matters. Even a tiny source contains an enormous number of nuclei, so statistical averages are very reliable. One nucleus cannot be predicted; the behaviour of a large sample can.

A parent nuclide is the original unstable nuclide before decay. A daughter nuclide is the nuclide produced by the decay. If that daughter is also unstable, more decays may follow in a decay chain, a series of radioactive decays leading eventually towards greater stability.

Alpha, beta and gamma changes

An alpha particle is a helium-4 nucleus made of two protons and two neutrons. During alpha decay, the parent nucleus loses two protons and two neutrons, so the proton number and neutron number both decrease by 2.

A beta-minus particle is an electron emitted from a nucleus in 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 in 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 released by a nucleus when it drops from an excited nuclear energy state to a lower energy state. Gamma emission changes the nucleus’s energy state, but it does not change its proton number or nucleon number.

The guide does not require the weak nuclear force as an explanatory mechanism, so for beta decay we focus on 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 for the bottom numbers.

For alpha decay, use this pattern:

[{}^{A}{Z}X\rightarrow{}^{A-4}{Z-2}Y+{}^{4}_{2}\alpha]

For beta-minus decay:

[{}^{A}{Z}X\rightarrow{}^{A}{Z+1}Y+{}^{0}_{-1}\beta^-+\bar{ u}]

For beta-plus decay:

[{}^{A}{Z}X\rightarrow{}^{A}{Z-1}Y+{}^{0}_{+1}\beta^++ u]

For gamma emission:

[{}^{A}{Z}X^*\rightarrow{}^{A}{Z}X+\gamma]

The star shows that the nucleus is in an excited state. Some books write ({}^{4}{2} ext{He}) instead of ({}^{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 takes a smaller recoil energy. In gamma emission, the energy is carried away as a photon. Beta decay is less straightforward, because a neutrino or antineutrino is involved.

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.

Neutrinos are very hard to detect. They can travel through huge thicknesses of matter with almost no interaction, so physicists inferred their existence from conservation arguments before confirming it 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 matter affects them in very different ways.

Alpha particles ionize strongly because they are massive and have charge +2. They lose energy fast, so they penetrate poorly: paper, or a few centimetres of air, usually stops them. Outside the body this is usually 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 yet penetrate deeply, 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 must be long enough for the procedure but short enough to reduce dose. When 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 per unit time. For activity, (A) is measured in becquerel (Bq), where (1, ext{Bq}=1, ext{decay s}^{-1}). The same letter also stands for nucleon number in nuclide notation, so use the context.

Count rate means the number of radiation detections an instrument records per unit time. We may write a count rate as (R), where (R) is the detector count rate (s⁻¹ or counts s⁻¹). The count rate is usually lower than the activity: the detector only covers part of the full solid angle, some radiation gets 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 for several minutes. Next, 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, where (C) is the number of detected events (unitless), the random uncertainty is roughly (\pm\sqrt{C}). Longer counting intervals therefore reduce percentage uncertainty, but the interval must not be so long that the source changes significantly during one reading.

Halving step by step

For a whole-number of half-lives, just halve repeatedly. If (n=t/T_{1/2}), where (n) is the number of half-lives elapsed (unitless) and (t) is elapsed time (s), then after (n) whole half-lives the fraction left is ((1/2)^n).

In count-rate questions with constant geometry, use (R=R_0/2^n), where (R_0) is the initial corrected count rate (s⁻¹). The same repeated halving works for 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, the daughter amount increases, but in the mathematical model the parent never reaches exactly zero.

Correcting the count

Background radiation is ionizing radiation that is still detected 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

[R_{ ext{corrected}}=R_{ ext{observed}}-R_{ ext{background}}]

where (R_{ ext{corrected}}) is the count rate due to the source after subtraction (s⁻¹), (R_{ ext{observed}}) is the total measured count rate (s⁻¹), and (R_{ ext{background}}) is the background count rate (s⁻¹).

Background matters most near the end of a decay experiment, when the source count rate has become small. For that reason, half-life estimates from a corrected-count-rate graph are usually best taken from the earlier part of the curve, where the source signal is much larger than background. Take several independent halving intervals and average them.

Scattering evidence

Alpha-particle scattering shows that, at very small nuclear separations, electrostatic repulsion cannot be the whole story. At relatively large separations, the scattering agrees with the Coulomb prediction. But when alpha particles get close enough to some nuclei, the observed scattering moves away from the purely electrostatic model. That deviation points to an additional short-range nuclear interaction.

Electron scattering gives another line of evidence. Accelerated electrons behave like waves and can diffract from nuclei. A diffraction minimum occurs approximately when ( heta\approx\lambda/b), where ( heta) 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). So nuclear size can be investigated. At sufficiently high energies, scattering reveals internal nucleon structure, which supports the idea that nuclear behaviour is governed by short-range interactions inside the nucleus.

The binding-energy curve provides 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 does not. The approximate saturation tells us 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 usually 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. They usually undergo beta-plus decay or electron capture, which converts a proton into a neutron. Very heavy nuclides often undergo alpha decay, as losing two protons and two neutrons moves 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, decreasing only slowly for heavier nuclei. That near-flat part of the curve is good evidence that the strong nuclear force has a short range.

A nucleon in a heavy nucleus is not strongly attracted by every other nucleon in that nucleus. Its main interactions are with near neighbours. When extra nucleons are added at the surface of an already large nucleus, the average binding per nucleon changes only slightly.

The small downward slope still counts. Very heavy nuclei have a little less binding energy per nucleon than nuclei near iron and nickel, so splitting or rearranging heavy nuclei can release energy. This connects naturally to fission in Topic E.4, but the main point here 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, rather than a continuous range. In a given alpha decay, the alpha particles often leave with one or a few well-defined kinetic energies. Gamma photons from a particular nuclear de-excitation have specific energies too.

That pattern is evidence that nuclei have discrete energy levels. A nucleus can jump from one allowed energy state to another, and the energy difference either appears as a gamma photon or changes the kinetic energy of an emitted alpha particle.

Atomic photons versus nuclear photons

Physically, photons are identified 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. Since 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, there are two main products. When the parent nucleus starts at rest, conservation of energy and momentum fixes the kinetic energies for a given transition. So alpha energies come out discrete.

In beta decay, the emitted beta particle appears with a continuous range of energies, from near zero up to a maximum. If the daughter nucleus and beta particle were the only products after the decay, 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 divided in many different ways, giving a continuous beta spectrum.

This is a lovely example of conservation laws guiding physics. Physicists did not abandon conservation of energy and momentum; they proposed a neutral particle that was hard to detect. Later experiments detected neutrino interactions, giving experimental support to the conservation-based hypothesis.

From proportional decay to an exponential law

For a radioactive sample, the decay rate depends on how many undecayed nuclei are still present. In decay-law equations, (N) is reused for the number of undecayed nuclei remaining (unitless), while (N_0) means the initial number of undecayed nuclei (unitless).

The radioactive decay law is

[N=N_0e^{-\lambda t}]

where (\lambda) is the decay constant of the nuclide (s⁻¹). 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, this is useful: a graph of (\ln R) against time is a straight line for corrected count rate, since 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 statement needs a bit of care. For 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 only works when (\lambda\Delta t) is small enough. Then the chance of decay during the interval is small, and the number of undecayed nuclei barely changes in that time.

So don’t 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 tells us how quickly nuclei are decaying. Written with calculus,

[A=-\frac{dN}{dt}]

where (dN/dt) is the rate of change of the number of undecayed nuclei with time (s⁻¹). The minus sign is there because the number of undecayed nuclei goes down, while activity is given as a positive rate.

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 has the same exponential shape, as long as the detector arrangement stays fixed.

Applications with arbitrary times

Medical isotopes and dating work don’t always use times that are neat multiples of a half-life. That’s when the exponential form is the one to use. In radiocarbon dating, for example, the remaining fraction of carbon-14 links 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.