Nuclear Physics: theory | Physics Circle

CONCEPT

Nucleus: Composition, Size & Nuclear Force

In one line
A nucleus is a tiny core of protons and neutrons. Its radius grows as A^1/3, so its density is the same for all nuclei.

COMPOSITION

26.1.1 What a nucleus contains

Protons and neutrons are called nucleons

Z = atomic number = number of protons.
N = number of neutrons.
A = mass number = number of nucleons.

A = Z + N

A nuclide is written AZX.
A nucleus has no electrons.

TYPES OF NUCLEI

26.1.2 Isotopes, isobars and isotones

Type	Same	Example
isotopes	Z	11H, 21H, 31H
isobars	A	31H, 32He
isotones	N = A − Z	31H, 42He

Isotopes have the same chemical properties.

NUCLEAR SIZE

26.1.3 Radius from mass number

R = R₀A^1/3

R₀ = 1.2 × 10⁻¹⁵ m = 1.2 fm.
Volume ∝ R³ ∝ A.

R₁R₂ = (A₁A₂)^1/3

R grows with A; density does not change

NUCLEAR DENSITY

26.1.4 Same for all nuclei

ρ = massvolume = mA(4π ÷ 3) R₀³A = 3m4πR₀³

m = mass of a nucleon = 1.67 × 10⁻²⁷ kg.
A cancels: ρ does not depend on A.
ρ ≈ 2.3 × 10¹⁷ kg m⁻³.

BREAK-UP AT REST

26.1.5 Velocity and radius ratios

m₁v₁ = m₂v₂

R₁R₂ = (m₁m₂)^1/3 = (v₂v₁)^1/3

NUCLEAR FORCE

26.1.6 What holds nucleons together

Strongest force in nature.
Short range: acts only up to about 10⁻¹⁵ m.
Attractive, and it makes the nucleus stable.
Charge independent: same for p–p, n–n and p–n.
Non-central; does not obey the inverse square law.

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The neutron

Discovered by Chadwick. Charge zero, mass 1.675 × 10⁻²⁷ kg.
Present in every nucleus except ordinary hydrogen.
Free neutron is unstable: half-life about 12 minutes.
Decay: n → p + e⁻ + ν̄.

Thermal neutrons

A moderator (heavy water, graphite, paraffin wax) slows fast neutrons.
Thermal neutron: energy about 0.025 eV, speed about 2.2 km s⁻¹.

Mirror nuclei

Same A, with Z and N interchanged.
Example: 31H and 32He.

Exchange of mesons

Nuclear force is an exchange force (Yukawa).
p ↔ n: exchange of a charged π-meson (π⁺, π⁻).
n–n and p–p: exchange of a neutral π-meson (π⁰).

Practise PYQs on this topicNucleus: Composition, Size & Nuclear Force · 47 questions with solutions

CONCEPT

Mass Defect & Binding Energy

In one line
A nucleus weighs less than its separate nucleons. The missing mass is the energy that binds it.

ATOMIC MASS UNIT

26.2.1 Mass and energy

1 u = 112 × mass of 126C atom = 1.66 × 10⁻²⁷ kg

E = mc² · 1 u = 931 MeV

Particle	Mass (u)
electron	0.0005486
proton	1.007276
neutron	1.00865

MASS DEFECT

26.2.2 Nucleus is lighter than its parts

Δm = Zm_p + (A − Z)m_n − M

M = mass of the nucleus.
Using atomic masses, use m_H in place of m_p.

BINDING ENERGY

26.2.3 Energy needed to pull a nucleus apart

B.E. = Δm c²

B.E. (in MeV) = Δm (in u) × 931

B.E. per nucleon = Δm × 931A MeV

The same energy is released when the nucleus forms.
Larger B.E. per nucleon ⇒ more stable nucleus.

BINDING ENERGY CURVE

26.2.4 B.E. per nucleon against A

Peak near A = 56; light nuclei fuse, heavy nuclei split

Rises sharply for light nuclei.
Maximum 8.8 MeV per nucleon at ⁵⁶Fe.
Falls slowly to 7.5 MeV at ²³⁸U.
Nearly constant for A between 30 and 170.
⁴He, ¹²C, ¹⁶O lie above their neighbours.

STABILITY

26.2.5 Neutrons against protons

Heavy stable nuclei have N > Z

Light stable nuclei: N ≈ Z.
Heavy stable nuclei: N > Z; extra neutrons balance the electric repulsion.
No stable nuclide has Z > 83.

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Pair production

A γ-photon near a nucleus creates e⁺ and e⁻

hν = e⁺ + e⁻ + kinetic energy

Rest energy of each particle: 0.51 MeV.
Minimum photon energy: 2 × 0.51 = 1.02 MeV.
Reverse process, pair annihilation: e⁺ + e⁻ → 2γ.

Packing fraction

f = ΔmA = M − AA

f is zero near A = 16 and A = 240

Can be positive, negative or zero.
Smaller f ⇒ more stable nucleus.

Even and odd numbers

Even Z, even N: most stable (about 60% of stable nuclides).
Odd Z, odd N: least stable; only five such stable nuclides.

Practise PYQs on this topicMass Defect & Binding Energy · 34 questions with solutions

CONCEPT

Nuclear Reactions & Q-value

In one line
In a nuclear reaction A and Z are conserved. The change in mass appears as the energy Q.

NUCLEAR REACTION

26.3.1 Writing a reaction

X + a → Y + b + Q

Short form: X(a, b)Y.
X = target, a = projectile, Y = product, b = emitted particle.

CONSERVATION LAWS

26.3.2 What stays the same

Total mass number A.
Total charge number Z.
Momentum and total energy.

Particle	Symbol
proton	11p
neutron	10n
deuteron	21H
alpha	42He
electron	0−1e
positron	0+1e

Q-VALUE

26.3.3 Energy of the reaction

Q = (mass of reactants − mass of products) c²

Q = Δm (in u) × 931 MeV

Q = B.E. of products − B.E. of reactants

Q > 0: exothermic, energy released.
Q < 0: endothermic, energy absorbed.
A free proton or neutron has zero binding energy.

DECAY AT REST

26.3.4 Sharing of energy

Parent at rest: momenta are equal and opposite

m₁v₁ = m₂v₂

K = p²2m ⇒ K₁K₂ = m₂m₁

Lighter particle takes more kinetic energy.
Equal p ⇒ equal de Broglie wavelength.

α-decay: K_α = Q × A − 4A · v_recoil = 4vA − 4

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Recoil on emitting a photon

p = hνc · K = h²ν²2Mc²

The nucleus loses a little more than hν of internal energy.

Reactions with a moving projectile

K_products = K_p + Q

A reaction with Q < 0 needs K_p > |Q|.

Common reactions

(α, p): ₂He⁴ + ₇N¹⁴ → ₈O¹⁷ + ₁H¹ (Rutherford, 1919).
(p, n): ₁H¹ + ₅B¹¹ → ₆C¹¹ + ₀n¹.
(n, p): ₀n¹ + ₇N¹⁴ → ₆C¹⁴ + ₁H¹.

Practise PYQs on this topicNuclear Reactions & Q-value · 28 questions with solutions

CONCEPT

Nuclear Fission & Fusion

In one line
Energy is released when nuclei move up the binding energy curve: heavy nuclei split, light nuclei join.

TWO WAYS TO RELEASE ENERGY

26.4.1 Fission and fusion

	Fission	Fusion
what happens	heavy nucleus splits	light nuclei join
example	U²³⁵ + n	²H + ²H
condition	slow neutrons	very high temperature
energy from	mass defect	mass defect

Q = (B.E. per nucleon of products − of parent) × nucleons

Mass of products is less than the mass of reactants.

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Fission of uranium

Liquid drop picture: the nucleus stretches and splits

23592U + 10n → 14156Ba + 9236Kr + 310n + Q

Energy per fission ≈ 200 MeV, about 0.8 MeV per nucleon.
Average 2.5 neutrons per fission, each about 2 MeV (fast).
U²³⁵ fissions with slow neutrons; most of the energy is KE of the fragments.
Explained by Bohr and Wheeler (liquid drop model).

Chain reaction

Each fission can start three more

k = rate of production of neutronsrate of loss of neutrons

k	Chain reaction	Size
k = 1	steady (reactor)	critical
k > 1	accelerates (atom bomb)	super-critical
k < 1	dies out	sub-critical

Nuclear reactor

Parts of a reactor

Part	Job	Example
fuel	undergoes fission	U²³⁵, Pu²³⁹
moderator	slows neutrons	graphite, heavy water
control rods	absorb neutrons	cadmium
coolant	carries away heat	water, CO₂
shield	stops radiation	thick concrete

Fusion and stars

Nuclei must overcome electric repulsion

Needs about 10⁷ K to 10⁸ K and high pressure: thermonuclear reaction.
Proton-proton chain in the Sun:

411H → 42He + 2e⁺ + 2ν + 2γ + 26.7 MeV

Atom bomb	Hydrogen bomb
fission	fusion
critical size matters	no limit of size
normal temperature and pressure	high temperature and pressure
less energy	more energy

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CONCEPT

Radioactive Decays: Alpha, Beta, Gamma

In one line
An unstable nucleus emits α, β or γ on its own. Each emission changes A and Z in a fixed way.

RADIOACTIVITY

26.5.1 A property of the nucleus

Spontaneous emission of radiation by heavy nuclei (Becquerel, 1896).
A nuclear event: the electrons play no part.
Not affected by pressure, temperature, fields or chemical state.
Elements with Z > 82 are naturally radioactive.

THE THREE RADIATIONS

26.5.2 α, β and γ

In a magnetic field α and β bend opposite ways; γ goes straight

Feature	α	β	γ
nature	He nucleus	fast electron	photon
charge	+2e	−e	0
speed	about 10⁷ m s⁻¹	up to 0.99c	c
penetration	1	100	10,000
ionisation	10,000	100	1
in E or B field	deflected	deflected	not deflected
energy spectrum	line	continuous	line

Penetrating power: γ > β > α.
Ionising power: α > β > γ.

DISPLACEMENT LAWS

26.5.3 Change in A and Z

Emission	A	Z	N
α	A − 4	Z − 2	N − 2
β⁻	A	Z + 1	N − 1
β⁺	A	Z − 1	N + 1
γ	A	Z	N

α-decay: AZX → A−4Z−2Y + 42He

β⁻-decay: AZX → AZ+1Y + e⁻ + ν̄

β⁺-decay: AZX → AZ−1Y + e⁺ + ν

USEFUL RESULTS

26.5.4 Isotopes and isobars

One α + two β⁻ ⇒ isotope of the parent (A falls by 4).
One β⁻ or one β⁺ ⇒ isobar of the parent.
γ emission ⇒ same nucleus at a lower energy.

COUNTING α AND β

26.5.5 Parent to final nucleus

n_α = A − A′4

n_β = 2n_α − (Z − Z′)

Find n_α first; only α changes A.
Example: ²³⁸U → ²⁰⁶Pb gives 8α and 6β.

β-DECAY

26.5.6 Where the electron comes from

β⁻: n → p + e⁻ + ν̄

β⁺: p → n + e⁺ + ν

The electron is created in the nucleus; it was not there before.
A free neutron decays because m_n > m_p.
A proton turns into a neutron only inside a nucleus.
ν and ν̄ have no charge and almost no mass.

β energies are continuous up to a maximum; the neutrino shares the energy

γ-DECAY

26.5.7 Nucleus falls to a lower level

Follows α or β decay, from an excited nucleus.
Nuclear levels are about 1 MeV apart; atomic levels about 1 eV.
So γ-rays have far smaller wavelength than visible light.

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Absorption of γ-rays

I = I₀e^−μx

x = thickness, μ = absorption coefficient.

Electron capture

An inner electron joins a proton in the nucleus: p + e⁻ → n + ν.
Happens when β⁺ emission is not energetically possible.

Radioactive series

Series	Parent	End
4n (thorium)	²³²Th	²⁰⁸Pb
4n + 1 (neptunium)	²³⁷Np	²⁰⁹Bi
4n + 2 (uranium)	²³⁸U	²⁰⁶Pb
4n + 3 (actinium)	²³⁵U	²⁰⁷Pb

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CONCEPT

Law of Radioactive Decay: Half-life & Mean Life

In one line
The rate of decay is proportional to the number of nuclei left, so N falls exponentially and halves every T.

DECAY LAW

26.6.1 Rutherford and Soddy

dNdt = −λN

N = N₀e^−λt · M = M₀e^−λt

N₀ = nuclei at t = 0; N = nuclei left at time t.
λ = decay constant: probability of decay per unit time.
Decayed nuclei = N₀ − N.
λ depends only on the nuclide, not on N or on conditions.

HALF-LIFE

26.6.2 Time to reduce to half

N halves in every interval T

T = ln 2λ = 0.693λ

N = N₀(12)ⁿ · n = tT

Works for mass and activity too.
n need not be a whole number: n = 32 gives 12√2.

Time	Fraction left	Fraction decayed
T	12 (50%)	12 (50%)
2T	14 (25%)	34 (75%)
3T	18 (12.5%)	78 (87.5%)
4T	116 (6.25%)	1516 (93.75%)

TIME BETWEEN TWO STAGES

26.6.3 Use the fraction left

t₂ − t₁ = T × log₂(N₁N₂)

Always turn "% decayed" into "% left" first.
Example: 40% decayed to 85% decayed means 60% left to 15% left, a factor of 4, so 2T.
If e^−λt = x, then after half the time the fraction is √x.

MEAN LIFE

26.6.4 Average life of a nucleus

τ = sum of the lives of all nucleitotal number of nuclei = 1λ

τ = T0.693 = 1.44 T · T = 0.693 τ

After one mean life 37% is left

At t = τ: N = N₀e = 0.37 N₀ (37% left, 63% decayed).
At t = 2τ: N = N₀e² = 0.135 N₀.
Mean life is greater than half-life.

STRAIGHT-LINE GRAPH

26.6.5 Finding λ

ln(NN₀) = −λt

Plot of ln of (N ÷ N₀) against t: slope −λ

τ = size of 1slope.

Practise PYQs on this topicLaw of Radioactive Decay · 126 questions with solutions

CONCEPT

Activity, Comparing Samples & Dating

In one line
Activity is the decay rate A = λN. It falls just like N, and it lets us compare samples and find ages.

ACTIVITY

26.7.1 Decays per second

A = |dNdt| = λN

A = A₀e^−λt = A₀(12)^t/T

Units: 1 Bq = 1 decay per second.
1 Rd = 10⁶ decays per second.
1 Ci = 3.7 × 10¹⁰ decays per second.
Between two times: A₁A₂ = e^{λ(t₂ − t₁)}.

from a graph: ln R = ln R₀ − λt

USEFUL RESULTS

26.7.2 Asked often

Activity falls to 1x in time t ⇒ it falls to 1xⁿ in time nt.
A₀A = e at t = t′ ⇒ λ = 1t′, T = t′ ln 2.
Rate of change of activity ∝ λ² ∝ T⁻².

N = mM N_A · A = 0.693T × mM N_A

m = mass of the sample, M = atomic weight, N_A = Avogadro number.
Use T in seconds for an answer in Bq.

COMPARING TWO SAMPLES

26.7.3 Ratio of activities

Larger λ: starts faster, falls sooner

A₁A₂ = λ₁N₁λ₂N₂ = N₁T₂N₂T₁

Half-life of a sample: T ∝ N ÷ A.

SAME START, DIFFERENT λ

26.7.4 Ratio of nuclei left

N₁N₂ = e^{−(λ₁ − λ₂)t}

Put the ratio as a power of e and compare the powers.
With half-lives: count half-lives of each sample separately.
Decayed numbers: use 1 − fraction left for each.
If T of A = τ of B, then λ_A = 0.693 λ_B: B decays faster.

PARALLEL DECAY

26.7.5 One nucleus, two processes

A can decay in either of two ways

λ = λ₁ + λ₂

1T = 1T₁ + 1T₂ · T = T₁T₂T₁ + T₂

SUCCESSIVE DECAY

26.7.6 A → B → C

A falls, B rises then falls, C keeps rising

net rate of B: dN_Bdt = λ₁N_A − λ₂N_B

N_B steady (equilibrium): λ₁N_A = λ₂N_B

So N_AN_B = λ₂λ₁ = T₁T₂.

DAUGHTER AND PARENT

26.7.7 Finding time from a ratio

N_YN_X = 2^t/T − 1

Y is stable and comes only from X.
Ratio X : Y = 1 : 7 ⇒ X is 18 of the start ⇒ t = 3T.

RADIOACTIVE DATING

26.7.8 Age from activity

t = T × ln(A₀ ÷ A)ln 2

Carbon-14 (T = 5730 years) for wood and bones.
A₀ = activity of a fresh sample of the same size.
Uranium and lead for rocks and the earth.

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Production at a steady rate

N = Pλ(1 − e^−λt)

P = nuclei produced per second; N starts from zero and tends to P ÷ λ.

Uses of radioactive isotopes

Medicine: cobalt-60 (cancer), iodine-131 (thyroid), sodium-24 (blood circulation).
Archaeology: carbon-14 dating; uranium-lead for rocks.
Agriculture: cobalt-60 preserves potato crops; phosphorus-32 in fertilisers.
Tracers: leaks in pipes, biochemical reactions.

Practise PYQs on this topicLaw of Radioactive Decay · 126 questions with solutions

EXAM-DAY CARD

Just see this before the exam

R = R₀A^1/3

size

B.E. = Δm × 931

binding energy

Q = Δm c²

reactions

n_α = A − A′4

decays

N = N₀(12)^t/T

half-life

A = λN

activity

FORMULA SHEET

Whole chapter on one page

Nucleus

A = Z + N

R = R₀A^1/3, R₀ = 1.2 fm

ρ ≈ 2.3 × 10¹⁷ kg m⁻³

R₁R₂ = (v₂v₁)^1/3

Binding energy

1 u = 931 MeV

Δm = Zm_p + Nm_n − M

B.E. = Δm × 931 MeV

peak 8.8 MeV at ⁵⁶Fe

Reactions

A and Z conserved

Q = (m_in − m_out) × 931

Q = B.E._out − B.E._in

K_α = Q(A − 4)A

Decays

α: A − 4, Z − 2

β⁻: Z + 1; β⁺: Z − 1

n_α = A − A′4

n_β = 2n_α − (Z − Z′)

Decay law

N = N₀e^−λt

N = N₀(12)^t/T

T = 0.693λ

τ = 1λ = 1.44 T

Activity

A = λN = A₀e^−λt

A₁A₂ = N₁T₂N₂T₁

N₁N₂ = e^{−(λ₁ − λ₂)t}

parallel: λ = λ₁ + λ₂

TOP MISTAKES

Avoid these = easy marks

MISTAKE ✗

Density depends on A

"% decayed" used as "% left"

β⁻ lowers Z

T = 1λ

CORRECT ✓

Mass and volume both ∝ A: ρ constant

Left = 100% − decayed

β⁻: Z + 1; β⁺: Z − 1; A same

T = 0.693λ; τ = 1λ