Chapter 50: Nuclear Physics and Radioactivity

50.1 Inside the Atomic Nucleus

The nucleus is the dense, positively charged core of the atom, occupying a sphere of

radius approximately r = r_0 A^{1/3}, where r_0

1.2 × 10 ¹⁵ m and A is the mass

number. For a typical nucleus (A

50), the radius is about 4.5 × 10 ¹⁵ m = 4.5 fm

(femtometres). This is roughly 100,000 times smaller than the atom itself. Nuclear matter is extraordinarily dense. If the nucleus of a hydrogen atom were scaled up to the size of a marble (1 cm), the atom would be 1 km across. The density of

nuclear matter is approximately 2.3 × 10¹⁷ kg/m³ — about 230 million tonnes per cubic

centimetre. A teaspoon of nuclear material would weigh more than a mountain. Despite this density, nuclei do not simply crush together under gravity or fly apart under electrostatic repulsion. They are held together by the strong nuclear force, one of the four fundamental forces of nature. Understanding this force, and why some nuclei are stable while others are not, is the central problem of nuclear physics. Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

50.2 Protons, Neutrons, and Nuclear Forces

The nucleus contains protons (charge +e, mass 1.6726 × 10 ²⁷ kg

1.007 u) and

neutrons (no charge, mass 1.6749 × 10 ²⁷ kg

1.009 u), collectively called nucleons. The atomic mass unit u is defined as 1/12 of the mass of a carbon-12 atom: 1 u =

1.6605 × 10 ²⁷ kg.

Protons repel each other through the electromagnetic force (like charges repel). In a large nucleus, where many protons are packed together, this repulsion is enormous. Yet nuclei hold together. The explanation is the strong nuclear force: an attractive force between all nucleons (proton-proton, neutron-proton, and neutron-neutron) that is much stronger than the electromagnetic force at short distances (< 2–3 fm) but falls off extremely rapidly and becomes negligible beyond about 3 fm. The strong force acts equally between all nucleon pairs (charge-blind) and dominates at nuclear distances, easily overcoming electromagnetic repulsion. But because it has such a short range, each nucleon can only "feel" its nearest neighbours. For large nuclei (many protons), the long-range electromagnetic repulsion of protons at opposite ends of the nucleus accumulates without a compensating increase in strong-force attraction, eventually making large nuclei unstable. Force Acts on Range Relative strength at 1 fm Role in nucleus Strong nuclear All nucleons ~1–3 fm 1 (reference) Holds nucleus together Electromagnetic Charged particles

Infinite (1/r²)

~1/137 Repels protons from each other Weak nuclear Quarks, leptons < 0.1 fm ~10 ⁶ Drives beta decay Gravity All mass-energy

Infinite (1/r²)

~10 ³⁷

Negligible at nuclear scale

Table 50.1

Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

50.3 Isotopes

The atomic number Z determines the element (number of protons). The mass number

A = Z + N, where N is the number of neutrons. Atoms of the same element (same Z)

but different mass numbers (different N) are called isotopes. Most elements have several naturally occurring isotopes. Carbon has three: carbon-12

(¹²C, 6 protons, 6 neutrons, 98.9% of natural carbon), carbon-13 (¹³C, 6 protons, 7

neutrons, 1.1%), and carbon-14 (¹⁴C, 6 protons, 8 neutrons, trace amounts from cosmic-

ray production). Hydrogen has three isotopes: ¹H (protium, 99.98%), ²H (deuterium,

0.015%), and ³H (tritium, radioactive, trace amounts).

Isotopes are written as X^A_Z or, more commonly, as the element name followed by

the mass number: carbon-14, uranium-238, iodine-131. Since Z is implied by the

element symbol, the mass number alone is usually sufficient: ¹⁴C, ²³⁸U, ¹³¹I.

Isotopes of an element have identical chemical properties (because chemical behaviour depends on electron structure, not the nucleus) but different physical properties, particularly density and radioactive behaviour. This is why isotope separation requires physical methods (centrifuges, diffusion) rather than chemical reactions. Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

50.4 Nuclear Stability

Not all combinations of protons and neutrons form stable nuclei. The stable nuclides cluster along a narrow band in the Z-N plane called the valley of stability. For light nuclei (Z ≤20), the stable ratio is approximately N/Z 1 (equal numbers of protons and neutrons). For heavier nuclei, the stable ratio increases to about N/Z 1.5 at Z = 80, because extra neutrons are needed to dilute the growing proton-proton repulsion with additional strong-force binding. Binding energy is the energy required to completely separate a nucleus into its constituent protons and neutrons. It is the measure of nuclear stability: a larger binding energy means a more tightly bound, more stable nucleus. The binding energy per nucleon peaks at around 8.7 MeV for iron-56 (⁵⁶Fe), which is the most stable nucleus per nucleon. Nuclei lighter than iron can release energy by fusion; nuclei heavier than iron can release energy by fission. Both processes move toward the iron peak. The mass defect is the difference between the actual nuclear mass and the total mass

of the separate nucleons. By E = mc², this mass defect equals the binding energy: a

bound nucleus is lighter than the sum of its parts because energy was released when it formed.

E_binding = Δm × c² = (Z m_p + N m_n −m_nucleus) × c²

Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

50.5 Radioactive Decay

A radioactive nucleus is one that spontaneously transforms into a different nuclear configuration, emitting particles or radiation in the process. This is radioactive decay. It is a quantum mechanical process: each unstable nucleus has a fixed probability of decaying in any given time interval, independent of its past history, the temperature, the chemical environment, or any external influence. Radioactive decay is governed by the law of radioactive decay:

N(t) = N₀ e^(−λt)

or

A(t) = A₀ e^(−λt)

₀

The decay constant λ is a property of each specific nuclide. A large λ means rapid

decay; a small λ means slow decay. The half-life T½ is the time for half the nuclei in a

sample to decay:

T½ = ln 2 / λ = 0.693 / λ

₀ ₀

There are three main types of radioactive decay: alpha (α), beta (β), and gamma (γ).

Each involves a different physical process and produces different radiation with different properties. Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

50.6 Alpha Decay

Alpha decay occurs when a heavy nucleus emits an alpha particle (α particle), which is a helium-4 nucleus: 2 protons and 2 neutrons, charge +2e, mass 4 u. Alpha decay reduces the atomic number by 2 and the mass number by 4:

²⁴ₐX

→

²´₈₂Y

+

⁴₂He

Alpha particles are emitted with kinetic energies typically in the range 4–9 MeV and travel at about 5–7% of the speed of light. Despite their high energy, alpha particles are relatively heavy and doubly charged, so they lose energy rapidly through ionisation of surrounding matter. Alpha particles from nuclear decay are stopped by a few centimetres of air or a thin sheet of paper. Alpha decay occurs by quantum tunnelling: classically, the alpha particle is trapped inside the nucleus by the potential energy barrier created by the strong force. Quantum mechanically, the alpha particle's wave function extends through the barrier, giving a non-zero probability of escape. The probability of escape determines the

decay constant λ and hence the half-life. This explains why alpha emitters have half-

lives ranging from microseconds to billions of years depending on the nuclear energy and barrier width.

²³₈U (uranium-238) undergoes alpha decay: ²³⁸₉²U

→²³⁴₉⁰Th + ⁴₂He. The daughter nucleus is thorium-234. Check: Z: 92 = 90 + 2 ✓; A:

238 = 234 + 4 ✓. Half-life of ²³⁸U: 4.47 × 10⁹ years — comparable to the age of Earth,

so roughly half the original ²³⁸U from Earth's formation is still present.

Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

50.7 Beta Decay

Beta decay involves the transformation of a nucleon via the weak nuclear force. There are two types: beta-minus (β ) and beta-plus (β ). Beta-minus decay (β ) A neutron in the nucleus transforms into a proton, emitting an electron (the beta

particle) and an antineutrino (ν̅_e):

n → p + e +

ν̅_e

(β decay) Beta-minus decay occurs in neutron-rich nuclei (too many neutrons relative to protons). It moves the nucleus toward the valley of stability by converting a neutron to a proton. The electron emitted has kinetic energy up to the maximum allowed by energy conservation, but shares the energy with the antineutrino, so the electron energy spectrum is continuous (not discrete, as it would be without the neutrino). Beta-plus decay (β ) A proton in the nucleus transforms into a neutron, emitting a positron (e , the

antiparticle of the electron) and a neutrino (ν_e):

p → n + e +

ν_e

(β decay) Electron capture is an alternative to β decay: the nucleus absorbs an inner-shell electron, converting a proton to a neutron and emitting a neutrino. Both β decay and electron capture are used in PET (positron emission tomography) scanners.

decay: ¹⁴₆C →¹⁴₇N +

e

+ ν̅_e. Carbon-14 →nitrogen-14. Z: 6 + 1 = 7 = nitrogen ✓; A: 14 unchanged ✓.

This is the decay used in radiocarbon dating. Half-life: 5730 years. Fluorine-18, used in PET imaging, undergoes β decay: ¹⁸₉F →¹⁸₈O + e

+ ν_e. Half-life: 110 minutes.

Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

50.8 Gamma Decay

Gamma decay occurs when a nucleus in an excited nuclear energy state (a metastable state produced by a previous alpha or beta decay, or by neutron capture) drops to a

lower energy state by emitting a gamma photon (γ). The gamma ray is a high-energy

photon with energy typically in the range 0.1–10 MeV.

ᴮX* →ᴮX + γ

Gamma rays are the most penetrating form of ionising radiation: they have no charge and no mass, so they interact rarely with matter. They are attenuated (reduced in intensity) exponentially as they pass through material, but a significant fraction can pass through many centimetres of lead or metres of concrete. Gamma shielding requires dense, thick materials. Gamma-emitting isotopes are used extensively in medicine (technetium-99m for diagnostic imaging, cobalt-60 for cancer radiotherapy), in industry (radiographic inspection of welds and pipelines), and in food irradiation (sterilisation). Gamma cameras detect gamma photons emitted from inside the body to produce images of organ function. Radiation type Particle/wave Charge Typical energy Penetration Stopped by α (alpha) He-4 nucleus +2e 4–9 MeV Very low Paper, 3–5 cm air β (beta-minus) Electron e 0 to ~3 MeV Moderate Few mm aluminium β (beta-plus) Positron +e 0 to ~3 MeV Moderate Few mm aluminium

γ (gamma)

Photon 0.1–10 MeV Very high Several cm lea metres concrete Neutron Neutron Varies High (indirect) Hydrogen-rich mater (water, polyethylene)

Table 50.2

Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

50.9 Half-Life

The half-life T½ of a radioactive nuclide is the time required for exactly half of any

given number of radioactive nuclei to decay. It is a fixed property of each nuclide,

ranging from 10 ²² seconds (some nuclear isomers) to 10²² years (bismuth-209, the

rarest stable nuclide). Importantly, the half-life does not depend on how many nuclei are present: whether you have one gram or one tonne of a substance, the fraction that decays in one half-life is always exactly 1/2.

After n half-lives: N(t) = N₀ × (1/2)^n, where n = t/T½. The fraction remaining is always

(1/2)^n regardless of N₀. The activity (decay rate) also halves every T½.

Iodine-131 has T½ = 8.02 days. A hospital receives

a 5.0 GBq sample. Activity after 24 days (3 half-lives): A = 5.0 × (1/2)³ = 5.0/8 = 0.625

GBq. Activity after 40.1 days (5 half-lives): A = 5.0 × (1/2)⁵ = 5.0/32 = 0.156 GBq.

How long until a sample of ¹³¹I decays

to 10% of its initial activity? A/A₀ = 0.10 = e^( λt). Take natural log: ln(0.10) =

λt. λ =

ln2/T½ = 0.693/8.02 = 0.0864 day ¹. t =

ln(0.10)/0.0864 = 2.303/0.0864 = 26.6 days.

Nuclide Half-life Decay type Application Uranium-238

4.47 × 10⁹ yr

α Geological dating, nuclear fuel Carbon-14 5730 yr β Archaeological/biological radiocarbon dating Radium-226 1600 yr α Historical medical use; radon source Iodine-131 8.02 days

β , γ

Thyroid cancer treatment and imaging Technetium-99m 6.01 h

γ (isomer)

Most widely used nuclear medicine diagnostic agent Fluorine-18 110 min β PET scanning (FDG-PET) Polonium-214

164 μs

α Decay product of uranium series

Table 50.3

Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

50.10 Nuclear Fission

Nuclear fission is the splitting of a heavy nucleus into two lighter nuclei (fission fragments) plus several neutrons and a large amount of energy. Fission can occur spontaneously (spontaneous fission) but is much more commonly induced by neutron absorption. For uranium-235, neutron-induced fission is the key reaction in nuclear reactors and

uranium bombs. When ²³₅U absorbs a slow (thermal) neutron, it becomes the highly

unstable ²³⁶U*, which splits in one of many possible ways. A typical fission:

¹₀n + ²³₅₉²U →¹⁴₆₆Ba + ₉°³₆Kr + 3 ¹₀n + energy

The energy released per fission event of ²³₅U is about 200 MeV

3.2 × 10 ¹¹ J.

Burning a single carbon atom in chemical combustion releases about 4 eV. Nuclear fission of uranium releases about 50 million times more energy per atom. The key to a sustained fission reaction is the chain reaction: each fission event releases

2–3 neutrons, which can induce further fissions in other ²³₅U nuclei. If on average at

least one neutron from each fission causes another fission, the chain reaction is self- sustaining (critical). In a nuclear reactor, control rods (made of neutron-absorbing materials such as boron or cadmium) regulate the neutron population to maintain criticality at a controlled, steady rate. In a nuclear bomb, the chain reaction is allowed to become supercritical, amplifying exponentially in microseconds.

and only 0.71% ²³₅U (which does). Reactor-grade uranium is typically enriched to 3–5%

²³₅U. Weapons-grade uranium is enriched to > 90% ²³₅U. This difference in enrichment

level makes the civilian and weapons programmes physically distinct. Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

50.11 Nuclear Fusion

Nuclear fusion is the combination of two light nuclei to form a heavier nucleus. Because the binding energy per nucleon increases as nuclei become heavier (up to iron), fusion releases energy for nuclei lighter than iron. Fusion releases more energy per unit mass than fission: the deuterium-tritium fusion reaction releases 17.6 MeV for nuclei with total mass number 5, compared to 200 MeV for fission of a mass-235 nucleus — about 3.5 times more energy per nucleon.

²₁H + ³₁H →⁴₂He + ¹₀n + 17.6 MeV

Fusion powers the Sun and all main-sequence stars. In the Sun's core (temperature

1.5 × 10⁷ K, density

150,000 kg/m³), the dominant process is the proton-proton

chain: four protons fuse (in several steps) to produce a helium-4 nucleus, releasing 26.7 MeV plus neutrinos. The Sun converts 600 million tonnes of hydrogen to helium every second, with a mass-to-energy conversion of about 4 million tonnes per second

via E = mc².

Achieving controlled fusion on Earth requires confining a plasma at temperatures of 10⁸ K or higher for long enough for the reaction rate to produce net energy. Current approaches include magnetic confinement (tokamak reactors, such as ITER under construction in France) and inertial confinement (high-power laser compression). As of 2025, sustained net energy gain remains a major engineering challenge, though scientific breakeven has been achieved.

The first thermonuclear weapon (hydrogen bomb) was tested by the United States in 1952 (Ivy Mike test). It used a fission bomb to trigger fusion of lithium deuteride, releasing 10.4 megatons of TNT equivalent — about 450 times more powerful than the Hiroshima bomb. The destructive power of fusion weapons comes primarily from the energy released in the fusion reaction, supplemented by fast-

neutron fission of a ²³⁸U tamper in most designs.

Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

50.12 Mass-Energy Equivalence: E = mc²

Einstein's special theory of relativity (1905) showed that mass and energy are

equivalent: a body with rest mass m at rest contains energy E = mc², where c = 3.00 ×

10⁸ m/s is the speed of light in vacuum.

E = mc²

In nuclear physics, E = mc² explains where the energy in nuclear reactions comes from.

In fission and fusion, the total mass of the products is slightly less than the total mass of the reactants. This mass difference (mass defect) is converted to kinetic energy of

the products, gamma radiation, and neutrinos. The conversion factor is c² = (3 × 10⁸)²

= 9 × 10¹⁶ J/kg: even a small mass defect corresponds to an enormous energy release.

Masses: ²H = 2.01410 u, ³H = 3.01605 u, ⁴He = 4.00260 u, n = 1.00866 u. Total

reactants: 2.01410 + 3.01605 = 5.03015 u. Total products: 4.00260 + 1.00866 =

5.01126 u. Mass defect: 5.03015

5.01126 = 0.01889 u. Energy: 0.01889 u × 931.5

MeV/u = 17.6 MeV. Confirmed.

E = mc² also explains pair production and annihilation. A photon with energy ≥2m_e

c²

= 1.022 MeV can spontaneously convert to an electron-positron pair (pair production). When a positron meets an electron, they annihilate: their rest mass converts to two 511 keV gamma photons (positron annihilation). This is the physical basis of PET scanning. Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

50.13 Nuclear Technology and Applications

Nuclear power generation Nuclear power plants use controlled fission chain reactions to generate heat, which produces steam to drive turbines connected to electrical generators. Most operating

reactors use enriched uranium fuel (²³₅U) in a moderator (usually water, which slows

neutrons to thermal energies where fission is more likely) controlled by neutron- absorbing control rods. As of 2025, nuclear power generates about 10% of the world's electricity from around 440 operating reactors. France generates about 70% of its electricity from nuclear power. Nuclear medicine: diagnostics Single-photon emission computed tomography (SPECT) and PET scanning use radioactive tracers administered to patients. SPECT uses gamma emitters such as technetium-99m; PET uses positron emitters such as fluorine-18 (FDG, a glucose analogue). The spatial distribution of the tracer reveals metabolic activity — cancer cells metabolise glucose at elevated rates, making FDG-PET effective for tumour detection and staging. Iodine-131 is used both diagnostically and therapeutically for thyroid disorders, because the thyroid gland selectively concentrates iodine. Nuclear medicine: therapy Radiation therapy uses high-energy radiation (X-rays, gamma rays, proton beams, or electron beams) to destroy cancer cells by damaging their DNA. External beam radiotherapy uses a linear accelerator to direct a shaped beam at the tumour from multiple angles. Brachytherapy implants radioactive seeds directly into or near the tumour. Proton beam therapy deposits most of its energy at a precise depth (the Bragg peak), minimising damage to surrounding tissue. Radiocarbon dating Carbon-14 is produced continuously in the atmosphere by cosmic-ray neutrons

reacting with nitrogen-14: ¹⁴N + n →¹⁴C + p. Living organisms continuously exchange

carbon with the atmosphere, maintaining a constant ¹⁴C/¹²C ratio. At death, exchange

ceases and ¹⁴C decays with T½ = 5730 years. Measuring the residual ¹⁴C ratio allows

the age of organic material to be determined up to about 50,000 years with ±100 year precision. For older materials, longer-lived isotopes (potassium-40, uranium-238, rubidium-87) are used. Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

50.14 Radiation Safety and Biological Effects

Ionising radiation — radiation energetic enough to remove electrons from atoms — can damage biological tissue by breaking chemical bonds, generating free radicals, and directly damaging DNA. The biological effects depend on the type of radiation, the energy deposited, the organs irradiated, and the time over which the dose is received. The absorbed dose (D) measures energy deposited per unit mass of tissue: 1 gray (Gy) = 1 J/kg. But equal doses of different radiation types cause different biological damage: alpha particles, being heavy and highly ionising, cause about 20 times more biological damage per unit absorbed dose than gamma rays or X-rays. The equivalent dose H =

D × W_R (where W_R is the radiation weighting factor) is measured in sieverts (Sv): 1

Sv = 1 J/kg × (dimensionless weighting factor). For gamma and X-rays, W_R = 1 Gy =

1 Sv; for alpha particles, W_R = 20.

Dose (Sv) Typical effect < 0.1 No acute clinical effect; long-term cancer risk increase is statistically very small 0.1–0.5 Temporary blood cell count changes in sensitive individuals; no acute symptoms 1–2 Mild acute radiation syndrome (nausea, fatigue); ~10% mortality without treatment 4–5 Severe acute radiation syndrome; ~50% mortality without treatment (LD50) > 10 Fatal within days to weeks even with treatment Annual background (global avg)

0.0024 Sv = 2.4 mSv from natural sources (radon, cosmic rays,

food, soil)

Table 50.4

Radiation protection follows the ALARA principle: doses should be As Low As Reasonably Achievable. The three practical means of reducing dose are: (1) distance —

radiation intensity falls as 1/r² for a point source; (2) time — reduce exposure time; (3)

shielding — use appropriate materials. Alpha emitters are dangerous mainly if inhaled or ingested (internal exposure); externally they are stopped by skin. Beta emitters require light shielding. Gamma emitters require heavy shielding but are primarily an external hazard. Radon gas (from uranium decay in soil and rock) is the largest source Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity of natural radiation exposure.

50.15 Common Mistakes in Nuclear Physics

•

Confusing mass number and atomic number. Mass number A = protons +

neutrons; atomic number Z = protons only. In decay equations, both A and Z

must balance separately. A balanced decay equation satisfies ΣA_reactants =

ΣA_products AND ΣZ_reactants = ΣZ_products simultaneously.

• Forgetting that gamma decay does not change Z or A. Gamma emission changes only the energy state of the nucleus, not its proton or neutron count. The daughter nuclide is the same element with the same mass number as the parent. •

Confusing activity and number of nuclei. Activity A = λN — they are

proportional but not equal. A nucleus can have a large N but low activity (small

λ, long half-life) or small N but high activity (large λ, short half-life). Units: N is

dimensionless; A is in becquerel (Bq = decays/second).

•

Applying n = t/T½ when t is not a whole multiple of T½. For non-integer n, use

the exponential formula N = N₀ e^(-λt). Using (1/2)^n only works cleanly when

n is an integer. •

Using the wrong mass-energy conversion factor. The conversion is: 1 u = 931.5

MeV/c². A mass defect of Δm in unified mass units gives energy E = Δm × 931.5

MeV. If Δm is in kg, use E = Δm × c² with c in m/s. Mixing units gives errors of

many orders of magnitude. • Thinking radioactive decay is triggered or accelerated by conditions. It is not. Heating, pressurising, dissolving, or chemically reacting a radioactive substance does not change its half-life. Decay is a quantum probability inherent to the nucleus, independent of external conditions (with the tiny exception of electron capture in exotic environments).

In decay equation problems, check both Z and A balance separately. A common mistake is balancing A correctly but forgetting that the daughter nucleus must also have the correct Z (and therefore be the right element in the periodic table). Write out the full nuclide notation — symbol, mass number superscript, atomic number subscript — for every particle to make the balance check explicit. Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

Chapter Summary

Atomic nuclei contain protons and neutrons bound by the strong nuclear force. Nuclei outside the valley of stability undergo radioactive decay — alpha, beta, or gamma — releasing energy and transforming to more stable configurations. Nuclear reactions release millions of times more energy per atom than chemical reactions. •

Nucleus: protons (Z) + neutrons (N); mass number A = Z + N; radius ∼1.2

A^{1/3} fm; density ∼2.3 × 10¹⁷ kg/m³.

• Nuclear stability: determined by N/Z ratio and binding energy; most stable nuclide per nucleon is iron-56. •

Radioactive decay law: N = N₀ e^(-λt); half-life T½ = ln2/λ = 0.693/λ; activity A

= λN.

•

Alpha decay: emits ⁴He; Z

2, A

4. Beta-minus: emits e ; Z + 1, A unchanged.

Beta-plus: emits e ; Z 1, A unchanged. Gamma: emits photon; Z and A unchanged. • Fission: heavy nucleus splits into lighter fragments + neutrons + ~200 MeV. Chain reaction sustained in reactors and bombs. • Fusion: light nuclei combine to form heavier nucleus + ~17.6 MeV (D-T). Powers stars; pursued for terrestrial energy generation. •

E = mc²: mass defect converts to energy. 1 u = 931.5 MeV; small mass defects

yield enormous energies. • Applications: power generation, medical imaging and therapy, radiocarbon dating, food irradiation. Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

Key Terms

nucleon

A proton or neutron; the constituents of atomic nuclei.

atomic number (Z)

Number of protons in the nucleus; defines the element.

mass number (A)

Total number of nucleons (protons + neutrons) in a nucleus; A = Z

+ N.

isotopes

Atoms of the same element (same Z) with different numbers of neutrons

(different A); same chemical properties, different nuclear properties.

strong nuclear force

The short-range (< 3 fm) attractive force between all nucleons

that overcomes electromagnetic repulsion and holds the nucleus together.

binding energy

The energy required to completely separate a nucleus into its

constituent nucleons; equals the mass defect times c².

mass defect

The difference between the sum of separate nucleon masses and the

actual nuclear mass; this deficit represents the binding energy.

radioactive decay

The spontaneous transformation of an unstable nucleus to a more

stable state by emission of particles or radiation.

decay constant (λ)

The probability of decay per unit time for a single nucleus; λ =

ln2/T½; units s ¹.

half-life (T½)

The time for half of any given number of radioactive nuclei to decay;

T½ = ln2/λ = 0.693/λ.

activity

(A)

The rate of radioactive decays; A = λN; unit: becquerel

(Bq) = 1

decay/second.

alpha decay

Emission of a helium-4 nucleus (⁴₂He); parent nucleus: Z

2, A 4.

beta-minus decay (β )

Conversion of a neutron to a proton with emission of an

electron and antineutrino; Z + 1, A unchanged.

beta-plus decay (β )

Conversion of a proton to a neutron with emission of a positron

and neutrino; Z 1, A unchanged.

gamma decay

Emission of a gamma photon from a nucleus transitioning from an

excited state to a lower energy state; Z and A unchanged.

nuclear fission

Splitting of a heavy nucleus into lighter fragments with release of

neutrons and large amounts of energy (~200 MeV per fission of ²³₅U).

chain reaction

A self-sustaining sequence of fission events where each fission releases

neutrons that induce further fissions.

nuclear fusion

The combination of light nuclei into a heavier nucleus with release of

Physics

Foundations of Physical Science

Chapter 50 — Nuclear Physics and Radioactivity energy; powers stars; requires extremely high temperatures.

mass-energy equivalence

E = mc²; mass can be converted to energy and vice versa; 1

u = 931.5 MeV/c².

absorbed dose

Energy deposited by radiation per unit mass of tissue; unit: gray (Gy) =

1 J/kg.

equivalent dose

Absorbed dose weighted by radiation type; H = D × W_R; unit:

sievert (Sv).

Physics

Foundations of Physical Science

Chapter 50 — Nuclear Physics and Radioactivity

Concept Review Questions

Answer in complete sentences. No calculations required unless stated.

1. Explain why a nucleus requires neutrons as well as protons in order to be stable.

Why does the N/Z ratio for stable nuclei increase above 1 for heavy elements?

2. Describe the strong nuclear force. How does it differ from the electromagnetic

force in terms of range and the types of particles it acts on? Why is the nuclear force responsible for both nuclear stability and nuclear decay?

3. Explain the concept of half-life. Why is the half-life of a radioactive substance

constant regardless of the initial quantity? How does the exponential decay law express this property mathematically?

4. Distinguish between alpha, beta, and gamma radiation in terms of their

composition, charge, penetrating power, and the change they produce in the parent nucleus.

5. Explain the conditions needed for a self-sustaining fission chain reaction. What is

the role of the moderator in a nuclear reactor? What is the role of the control rods?

6. Explain why fusion requires temperatures of millions of degrees. What must be

overcome for two nuclei to fuse? Why is fusion more difficult to achieve on Earth than fission?

7. Explain how E = mc² accounts for the energy released in nuclear reactions. How

does the binding energy per nucleon curve explain why both fission (of heavy nuclei) and fusion (of light nuclei) release energy?

8. Describe two specific applications of radioactive isotopes in medicine. For each,

explain which isotope is used, what type of radiation it emits, and what property makes it suitable for that application. Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

Radioactive Decay Problems

9. Write the complete balanced decay equations for the following: (a) Alpha decay

of radium-226 (²²⁶Ra). (b) Beta-minus decay of carbon-14 (¹⁴C). (c) Beta-plus

decay of nitrogen-13 (¹³N). (d) Alpha decay followed by beta-minus decay of

thorium-234 (²³⁴Th).

10. Identify the unknown nuclide X in each equation: (a) ²³₈U →X + ⁴He. (b) X →

²³₄Th + ⁴He. (c) ³₂P →³²S + X + ν̅_e.

11. Thorium-230 decays by alpha emission to radium-226, which decays by alpha

emission to radon-222. Write both equations and confirm that the products are correct by checking Z and A.

12. Uranium-238 undergoes a decay series, ultimately producing lead-206. The

series involves 8 alpha decays and 6 beta-minus decays. Show that these

numbers are consistent with the change from ²³⁸U (Z=92) to ²⁰⁶Pb (Z=82): (a)

account for the change in A, (b) account for the change in Z. Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity Half-Life Calculations

13. Phosphorus-32 (used in cancer treatment) has T½ = 14.3 days. (a) Find the

decay constant λ in day ¹. (b) Starting with 80 MBq, find the activity after 42.9

days. (c) Find the time for the activity to fall to 5 MBq.

14. A sample of radon-222 (T½ = 3.82 days) has initial activity 2.00 × 10⁶ Bq. (a)

Find the initial number of nuclei. (b) Find the activity after 15.0 days. (c) How many nuclei remain after 10.0 days?

15. An archaeological sample of charcoal is found to have ¹⁴C activity 20% of the

activity of a modern sample. Find the age of the charcoal. (T½ of ¹⁴C = 5730

years.)

16. Two radioactive samples initially contain the same number of nuclei. Sample A

has T½ = 10 days; Sample B has T½ = 30 days. (a) Which has the higher initial

activity? (b) After 30 days, which has more nuclei remaining? (c) After 60 days,

find the ratio N_A/N_B.

Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

Nuclear Energy Analysis

17. Find the binding energy of helium-4 (⁴He). Masses: proton = 1.00728 u, neutron

= 1.00866 u, ⁴He = 4.00260 u. (a) Find the mass defect in u. (b) Find the binding

energy in MeV. (c) Find the binding energy per nucleon in MeV/nucleon.

18. The binding energy per nucleon of iron-56 is 8.79 MeV/nucleon. Calculate the

total binding energy of iron-56. How does this compare to carbon-12

(B.E./nucleon = 7.68 MeV)?

19. In the D-T fusion reaction, 17.6 MeV is released. (a) Calculate the energy

released per kilogram of reactant mixture (50% D by mass, 50% T by mass; masses 2 u and 3 u respectively). (b) Compare to burning coal (energy density ~30 MJ/kg).

20. Calculate the energy released when 1.0 kg of ²³₅U undergoes complete fission,

given that each fission event releases 200 MeV and the molar mass of ²³₅U is

235 g/mol. Express the answer in joules and in tonnes of TNT equivalent (1

tonne TNT = 4.2 × 10⁹ J).

Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

Applied Nuclear Technology Problems

21. A PET scanner uses fluorine-18 (¹⁸F, T½ = 110 min) as a tracer. A patient is

injected with 370 MBq at 09:00. (a) What is the activity at 14:30 the same day? (b) At what time does the activity fall below 10 MBq? (c) Why is a short half-life desirable for a diagnostic tracer?

22. A nuclear reactor produces 1000 MW of thermal power from ²³₅U fission, with

each fission releasing 200 MeV. (a) How many fissions per second are required?

(b) How many kilograms of ²³₅U are consumed per day? (c) How does this

compare to the coal (energy density 30 MJ/kg) required for a 1000 MW coal power plant per day?

23. An alpha particle from americium-241 (a common smoke detector source, T½ =

432 years) has kinetic energy 5.49 MeV. (a) How far does this alpha particle travel in air (assume stopping power ∼0.5 MeV/cm in air at STP)? (b) Why are americium-241 sources safe to handle with unprotected hands but dangerous if

ingested? (c) Find the activity of a 1.0 μg sample of ²⁴¹Am.

24. Radiocarbon dating of a sample gives an activity of 1.25 disintegrations per

minute per gram of carbon. Modern carbon gives 15.3 disintegrations per minute per gram. Find the age of the sample. Is this consistent with an early medieval origin (800–1200 CE)? Physics: Foundations of Physical Science Chapter 50 — Nuclear Physics and Radioactivity

Chapter 50 Checkpoint

Before concluding Part IX, confirm you can do each of the following. I can… See Section Describe nuclear structure: protons, neutrons, mass number, atomic number 50.1–50.2 Explain isotopes and why they have the same chemical but different nuclear properties 50.3 Explain nuclear stability, the valley of stability, and binding energy 50.4

Write and apply the radioactive decay law; relate T½ and λ

50.5–50.9 Write balanced equations for alpha, beta, and gamma decay; identify daughter nuclides 50.6–50.8 Perform half-life calculations using both the integer (1/2)ⁿ and exponential methods 50.9 Describe nuclear fission and explain the chain reaction mechanism in reactors and bombs 50.10 Describe nuclear fusion and explain why it requires extreme temperatures 50.11

Apply E = mc² and the mass defect to calculate

nuclear energy release 50.12 Describe four nuclear technology applications with specific isotopes and physics 50.13 Explain absorbed dose, equivalent dose, and the biological effects of ionising radiation 50.14 Identify and avoid common errors in nuclear decay equations and half-life calculations 50.15