Nuclear Physics

01

Evidence for the nucleus

Rutherford scattering connects a pattern of observations to an atomic model. Most alpha particles pass almost straight through a thin foil; rare large deflections reveal a small concentration of positive charge and mass.

A narrow beam of positively charged alpha particles encounters a thin metal foil, and a detecting screen records their outgoing directions. Distinguish the observed deflection pattern from the explanation inferred from it.

Large deflections are not explained by alpha particles striking individual light electrons. Nor does this experiment by itself identify the neutron or establish all electronic energy levels. Its central result is the existence of a nucleus much smaller than the whole atom.

Read nuclide notation

A nucleus is represented by AZX. The nucleon number A, also called mass number, counts protons plus neutrons. The proton number Z, also called atomic number, counts protons and identifies the element.

Here N denotes the neutron number of one nucleus. When N later denotes a sample population, the stated context changes its meaning.

Isotopes have the same Z but different neutron numbers. For example, chlorine-35 and chlorine-37 both have 17 protons, but contain 18 and 20 neutrons respectively.

For a sample population, N = (m/M)N_A, with sample mass m and supplied molar mass M in matching units. N_A = 6.02 × 10²³ mol^-1; a few times 10^-4 mol therefore contain order 10²⁰ entities. Name those entities: one atom has one nucleus, but usually many nucleons. The particle-counting explanation gives the mole and relative-mass foundations.

02

Alpha, beta and gamma

Radiation type determines what is emitted and how it interacts. Penetration and ionisation comparisons also depend on its energy and the material through which it travels.

A beta electron is not a pre-existing orbital electron simply leaving the atom. Gamma emission reduces nuclear energy without changing its nucleon or proton count. Zero photon rest mass does not mean zero photon energy or momentum.

Charged alpha and beta particles also experience electric forces in opposite directions in a given electric field. Gamma has no electric charge force. Charged-particle speeds depend on their kinetic energies and are below c; gamma travels at c in vacuum.

A penetration statement must name the radiation energy and absorber context. Gamma is not completely stopped by one universal thickness, and short alpha range does not imply weak ionisation. For living tissue, an internal alpha-emitting material can deposit energy densely close to its location.

03

Random decay and measured counts

An individual decay is unpredictable, while a large collection has a statistical trend. A detector records counts, which must be interpreted using the observation time, its response and the background.

Spontaneous means nuclear decay does not need an external triggering event. Random means we cannot predict the exact time at which a particular unstable nucleus will decay. Ordinary changes of temperature, pressure or chemical state do not provide a way to set that time in this model.

Activity is not automatically count rate

Activity is the rate of nuclear decays. Its unit is the becquerel:

Some emissions miss the detector or are not registered, while background adds other counts. An activity and a count rate therefore need not be numerically equal. Bq is a decay-rate unit, not a radiation-dose unit. Here activity A is distinct from a nuclide's nucleon number A.

Identify the background contribution

Background includes cosmic radiation and natural radionuclides in the environment, materials and bodies. Artificial sources can also contribute. A comparable background measurement excludes the investigated source while retaining the relevant detector and surroundings.

Fluctuations between equal intervals are evidence of the random nature of decay, not proof that the detector is broken. A smooth expected decay curve does not mean each nucleus decays on schedule.

Evaluate the counting method

Compare rates with source-detector positions, absorber, detector window and timing method fixed. A longer observation can reduce fractional random scatter when the expected rate is sufficiently stable, but it can blur a rapid decay change. Repetition does not correct an omitted background, incorrect time interval or changed geometry.

Changing detector efficiency and dead time at high rates can limit the result. Background can fluctuate or drift too. Retain the supplied records and explain a specific limitation rather than deleting points simply to obtain a smoother pattern.

04

Half-life and corrected curves

Half-life is the time for the expected number of undecayed parent nuclei, or the activity, to fall to half its initial value. For a detector reading, use the source contribution and check that the conditions make it proportional to the decaying population.

For a single unchanged radionuclide, the expected number of undecayed parent nuclei and the activity halve over the same interval. A proportional net source count rate does too when detection conditions remain fixed. Half-life is not the lifetime of every individual nucleus.

After one, two and three half-lives, the remaining fractions are 1/2, 1/4 and 1/8. A few half-lives leave a fraction, not zero. Match time units before comparing: 0.2 min is 12 s, while 2 min is 120 s.

Separate total rate from the source trend

In this supplied model, net source rate starts at 80.0 counts/s and halves every 120 s. Background remains 4.0 counts/s. These are instantaneous expected rates at the named times, not long-window measured averages.

Half of the initial raw total is 42 counts/s, but the total at one half-life is 44 counts/s: 40 from the source plus 4 background. Read half-life from the corrected quantity. The falling trend becomes less steep as fewer parents remain; it is not a constant rate of loss.

The exponential model relates this shape to decay constant, activity and a logarithmic data fit.

05

Activity and exponential decay

The decay constant describes the chance of decay per nucleus per unit time. A large population follows a predictable expected trend even though each individual decay remains random.

Let N be the number of undecayed parent nuclei and λ the decay constant. For a sufficiently short interval Δt, the probability that one parent nucleus decays is approximately λΔt. The unit of λ is s^-1 when time is in seconds.

Activity is the expected decay rate, in Bq: one decay per second. A constant λ means a constant fractional decay rate, not a constant number of decays per second. As N decreases, A decreases. The cumulative number that have decayed is N₀ - N, which increases.

These equations describe one parent radionuclide with constant decay constant. They do not make its individual nuclei decay at predetermined times, and a mixture need not have one exponential total activity.

Connect decay constant and half-life

At t = t_1/2, half the original parent population remains:

A half-life of order 100 s suggests λ of order 10^-2 s^-1. For the supplied 120 s half-life, λ = 0.00577623 s^-1. Convert minutes to seconds before using that value with a time.

Use ratios and logarithms

With constant detection conditions and correctly subtracted background, the source count rate is proportional to activity:

The logarithm's argument is a dimensionless ratio. A plot of ln(r/r₀) against t has gradient -λ. For the 120 s model, the time to reach 10.0% is:

Zero and negative corrected rates have no real logarithm. A positive value very close to background can also have large relative uncertainty; being mathematically loggable does not make it a reliable fitting point.

06

Choose radiation for a task

A useful source must interact in the required way and remain suitable for the time involved. Explain both its penetration or ionisation and its half-life, using the stated task.

Measure sheet thickness

Place a radiation source and detector on opposite sides of a sheet. For a suitable beta source and sheet material, some particles are absorbed and some are transmitted. Greater thickness then produces a lower corrected detector rate when the geometry and other conditions stay fixed.

A source absorbed almost completely by even the thinnest sheet gives little useful contrast. A source that passes through all the relevant thicknesses almost unchanged also gives poor sensitivity. Choose radiation that is partially transmitted over the required thickness range.

The source half-life should be long enough for useful output over the operating and calibration period, or the expected decay must be allowed for. A falling rate caused by source decay must not be interpreted automatically as a thicker sheet.

Detect an internal tracer

For external detection of a tracer within the body, suitable emitted photons must penetrate tissue and reach the detector. The distribution of the tracer provides information about the stated organ or process; its chemical suitability also matters.

The half-life must allow preparation and measurement while avoiding unnecessarily prolonged radioactive activity. Physical half-life describes nuclear decay; it does not by itself give the time for biological clearance from the body.

Deliver radiation to a treatment target

In external treatment, sufficiently penetrating radiation can reach a target within tissue. Ionisation deposits energy and can damage cells, which is the intended effect in the target but also a concern for surrounding tissue. Direction and exposure distribution matter alongside radiation type.

The source must maintain suitable output over its operating period, with decay allowed for. A long half-life reduces the rate of output change but also means the source remains radioactive for longer. Half-life alone does not determine a treatment's suitability.

Sterilise items through packaging

Penetrating gamma radiation can reach microorganisms inside suitable final packaging. Ionisation damages them without requiring the package to be opened. The source must provide useful output over the processing interval, and its changing activity must be allowed for as it ages.

Explain the hazard in context

Ionisation can damage living cells and genetic material. Relevant factors include activity, emitted energy and radiation type, exposure time, distance, shielding, and whether radioactive material enters the body. Short alpha range can limit an external exposure while an internal alpha emitter deposits energy densely near its location.

Irradiation means exposure to radiation. Contamination means radioactive material is present on or inside an object. Ordinary irradiation does not necessarily make the object radioactive; do not confuse the radiation reaching an object with the source material moving onto it.

07

Balance nuclear equations

A nuclear equation accounts for nucleon number and electric charge. Balance both before identifying an unknown product; energy and momentum must also be conserved in the complete process.

In AZX, A counts nucleons and Z counts protons. For an emitted particle, the lower number records its charge in units of +e. An electron therefore has lower number -1 even though it contains no negative proton.

Alpha emission

An alpha particle carries four nucleons and charge +2e. The daughter therefore has A reduced by 4 and Z reduced by 2. Here 238 = 234 + 4 and 92 = 90 + 2.

Beta-minus emission

This displays the nucleon and charge account: 14 = 14 + 0 and 6 = 7 - 1. Within the nucleus, a neutron changes into a proton. The total nucleon number stays the same, while the daughter's proton number rises by one.

The emitted electron is produced by the nuclear process, rather than removed from an atomic shell. The displayed A/Z account is not a claim that only two products carry energy and momentum: an additional uncharged particle also participates.

The beta-decay evidence explains why the electron and daughter alone cannot account for the observations.

Gamma emission

The star indicates a higher-energy nuclear state. A gamma photon carries away energy and momentum; the nucleus has the same A and Z afterwards. The energy change does not require a change of element or isotope.

A reaction can have an incoming particle

Both sides contain 18 nucleons and charge +9e. A and Z balance across all incoming and outgoing particles, not just the two nuclei that look most prominent.

Conserved nucleon number does not mean conserved rest mass. A change of total rest mass can accompany a change in kinetic or radiation energy while total mass-energy remains conserved.

08

What beta decay reveals

Beta decay gives evidence for an additional particle through both the spread of electron energies and the momentum that is missing from an electron-and-daughter account.

Test the two-product prediction

Consider a parent nucleus initially at rest and a fixed decay branch to a specified daughter state. If only an electron and that daughter were produced, momentum conservation would require equal and opposite momenta. Together with the fixed energy available, this would fix the electron's energy.

Instead, beta electrons from that branch have a continuous range of energies up to an endpoint. The energy varies because it is shared with another emitted particle as well as the recoiling daughter. The extra particle is uncharged and difficult to detect; in beta-minus decay it is an antineutrino.

The energy distribution and momentum imbalance support an additional carrier. They are not a reason to abandon energy or momentum conservation. Balancing nucleon number and charge alone would not have revealed the missing energy and momentum.

09

Mass defect and binding energy

A bound nucleus has less rest mass than its separated nucleons. The corresponding energy difference is the energy required to separate the nucleus completely.

Use mass units consistently

The unified atomic mass unit u is defined as one twelfth of the mass of a neutral carbon-12 atom. For these supplied calculations use 1 u = 1.66 × 10^-27 kg. Nucleon number A is dimensionless; it is not an exact mass in u.

The rounded nuclear masses used below contain no orbital electrons. Proton and neutron masses are close to, but not exactly, 1 u.

Convert an energy difference from mass

Mass and energy are equivalent through E = mc², where c is the speed of electromagnetic waves in vacuum; use the supplied value 3.00 × 10⁸ m/s. A change of rest mass Δm corresponds to an energy change ΔE = Δmc². For a complete nuclear process, a decrease in total rest mass can supply kinetic energy and radiation while total mass-energy is conserved.

One electronvolt is the energy transferred to a charge of magnitude e through a potential difference of 1 V. With supplied e = 1.60 × 10^-19 C:

A mass difference of order 10^-3 u gives an energy of order 1 MeV, about 10^-13 J per nucleus. This is a useful scale check before a precise subtraction. An eV is a unit of energy, not electric potential.

Binding energy is a positive separation energy

For a nucleus with Z protons and A - Z neutrons, the positive mass defect is:

The separated nucleons have greater rest energy. Forming the bound nucleus releases this energy difference; separating it completely requires the same amount to be supplied. A nucleus does not keep radiating its binding energy merely because it remains bound.

10

Energy from fission and fusion

A reaction releases energy when its products have less total rest mass than its reactants. The binding-energy-per-nucleon curve helps explain why both the fusion of light nuclei and the fission of heavy nuclei can do this.

Read the binding-energy curve

Binding energy per nucleon rises rapidly across the light nuclei, reaches a broad maximum near nucleon numbers 50-60 in the iron/nickel region, then decreases gradually for heavier nuclei. It measures an average binding energy, not the total binding energy of a nucleus.

Products with greater total binding energy have lower total rest energy for the same complete nucleon account. Multiply each nucleus's binding energy per nucleon by its own A before comparing totals. Local isotope differences exist, and this curve alone does not determine a reaction rate or every kind of nuclear stability.

Explain fission and fusion in mark-scheme order

A common 3-mark question asks why both fission and fusion release energy. Write the links in this order:

1. Fusion: light nuclei are on the rising part of the curve. When they join, the product has a higher binding energy per nucleon.
2. Fission: a heavy nucleus is past the peak. When it splits into two medium nuclei, the products are nearer the peak and have a higher binding energy per nucleon.
3. In both cases the total binding energy of the products is greater than that of the reactants. So the products have less total rest mass. This mass difference Δm is released as energy, E = Δmc², mostly as kinetic energy of the products.

Iron and nickel sit at the peak. Neither fusing nor splitting them raises the binding energy per nucleon, so neither process releases energy from them.

Fusion combines light nuclei

In the supplied deuterium-tritium reaction, a hydrogen-2 nucleus and a hydrogen-3 nucleus form an alpha particle and a neutron:

Both sides have five nucleons and charge +2e. Use all products when comparing rest masses. The supplied nuclear masses, in u, are 2.01355321, 3.01550072, 4.00150618 and 1.00866492 in the displayed order.

Fission splits a heavy nucleus

One possible neutron-induced fission channel is:

The totals are A = 236 and charge +92e on both sides. Other fragment combinations are possible; this is one supplied channel, not a unique outcome of every fission event.

Free neutrons contribute no nuclear binding energy. This is an estimate from the supplied rounded averages, not a precision energy value for the channel. The products are more tightly bound overall, so the decrease in total rest energy is available as kinetic energy and radiation.

For both fusion and fission, compare the complete initial and final accounts. Subtracting two per-nucleon values without weighting by nucleon number does not give the reaction energy.

Revision

Nuclear Physics at a glance

Identify what is being counted, what is actually detected and which energy account is complete. Those decisions come before substituting numbers.

Structure and evidence

Most alpha particles passing a thin foil are little deflected; rare large deflections support a small concentration of positive charge and mass. A nucleus has Z protons and A - Z neutrons. A neutral atom has Z electrons. Isotopes have the same Z and different neutron counts; ionisation changes electrons without changing the nucleus.

Radiation and detection

Alpha is a helium-4 nucleus, beta-minus an electron from a nuclear process, and gamma an electromagnetic photon. Alpha and beta have opposite electric-force directions; gamma is uncharged. Penetration and ionisation comparisons need an energy and absorber context. Dense internal ionisation can be hazardous even for short-range radiation.

Activity in Bq counts decays per second. Detector rate counts recorded events per second. They are related only through the stated emission and detection conditions. For unequal durations in a stable-rate series, pool counts over the total time; do not give every interval equal weight.

Decay is spontaneous and random for an individual nucleus. A large population has a predictable statistical trend, with fluctuations in finite counts. Repeats do not remove an incorrect background or changing geometry.

Half-life and time dependence

After one, two and three half-lives, the expected parent population, activity or proportional corrected rate is 1/2, 1/4 and 1/8 of its initial value. Halve a background-subtracted rate. The raw total tends towards background rather than zero.

Applications need two kinds of reasoning

Explain the penetration or ionisation that makes the task possible, then relate half-life to its preparation, measurement or operating interval. A thickness gauge needs partial transmission over the useful range. An externally detected internal tracer needs radiation that escapes tissue. Treatment and sterilisation use damage from ionisation, with suitable penetration and controlled exposure.

Longer half-life is not universally preferable. Physical half-life is distinct from biological clearance. Irradiation is exposure to radiation; contamination is the presence of radioactive material.

Conservation and binding

Balance total nucleon number and charge across all particles. Alpha emission changes the daughter by ΔA = -4 and ΔZ = -2. Beta-minus leaves A unchanged and raises daughter Z by one. Gamma leaves A and Z unchanged while reducing nuclear energy.

For a fixed beta-decay branch, a two-product model would fix the electron energy. Its continuous energy spectrum and a missing momentum vector support an additional emitted particle. Energy and momentum conservation apply to the complete account.

Keep atomic and nuclear mass conventions consistent and retain digits until after subtraction. Binding energy is the positive energy required for complete separation. A fusion or fission energy account must include every reactant and product, including free neutrons.

The binding-energy-per-nucleon curve rises steeply across light nuclei, with helium-4 (about 7.1 MeV) above its neighbours. It peaks at about 8.8 MeV near A = 56 (iron and nickel) and falls slowly to about 7.6 MeV for uranium. Fusion of light nuclei and fission of heavy nuclei both move towards the peak. The products then have greater total binding energy and less rest mass, and the mass difference is released as energy. Compare total binding energies for a reaction by multiplying each per-nucleon value by its own nucleon count.

A denotes a nucleon count in nuclide notation and activity in a decay-rate equation; identify the context and unit. N can denote neutron number A - Z or the remaining parent population. Dimensionless counts can also be labelled n or m under a stated convention; in this chapter n denotes amount in mol and m denotes mass. Read the definition and unit before substituting.

Before reporting an answer, check the named entity, time unit, background correction, mass convention and whether the requested energy is total or per nucleon.