The nucleus, discovered by Rutherford, is a dense, positively charged core at the centre of an atom. Nuclear physics deals with the composition, size, binding energy, radioactive decay and fission/fusion of nuclei.
Nuclear Composition
A nucleus consists of protons (charge +e, mass 1.007276 u) and neutrons (no charge, mass 1.008665 u) collectively called nucleons.
- Atomic number Z = number of protons
- Mass number A = number of protons + neutrons = Z + N
- Notation: ZA X (e.g., 612 C)
- Isotopes: same Z, different A (same element, different mass)
- Isobars: same A, different Z
- Isotones: same N, different Z
Nuclear Size
Empirical relation: R = R0 A1/3, where R0 = 1.2 x 10-15 m = 1.2 fm.
Nuclear density is approximately constant: rhonucleus = 2.3 x 1017 kg/m3 (same for all nuclei!).
Mass Defect and Binding Energy
Mass defect: Delta m = Z mp + (A-Z) mn - mnucleus
Binding energy: B.E. = Delta m x c2 (in joules) = Delta m x 931.5 MeV/u
- Binding energy per nucleon (B.E./A) is a measure of nuclear stability. The graph of B.E./A vs A shows:
- Peak around iron (A = 56), with B.E./A = 8.75 MeV — most stable.
- Lighter and heavier nuclei have lower B.E./A — they can gain stability by fusion and fission respectively.
Radioactive Decay
Radioactivity (Becquerel, 1896): unstable nuclei spontaneously emit radiation.
- Types of decay:
- Alpha (alpha): emits 24 He. Z decreases by 2, A by 4.
- Beta-minus (beta^-): emits electron + antineutrino. Z increases by 1, A unchanged.
- Beta-plus (beta^+): emits positron + neutrino. Z decreases by 1, A unchanged.
- Gamma (gamma): emits high-energy photon. No change in Z or A (nucleus goes to lower energy state).
Radioactive Decay Law:
N(t) = N0 e-lambda t
Activity A = lambda N = A0 e-lambda t
Half-life: T1/2 = ln2 / lambda = 0.693 / lambda
Mean life: tau = 1/lambda = T1/2 / 0.693
Number of half-lives in time t: n = t / T1/2. Remaining nuclei = N0 / 2n.
Becquerel (Bq): 1 disintegration per second. 1 Curie (Ci) = 3.7 x 1010 Bq.
Nuclear Reactions
Q-value = (mass of reactants - mass of products) x c2
If Q > 0, energy is released (exothermic — fission and fusion).
If Q < 0, energy must be supplied (endothermic).
Nuclear Fission: Heavy nucleus splits into lighter nuclei releasing large energy.
e.g., 92235 U + 01 n → 56141 Ba + 3692 Kr + 3 01 n + Q (200 MeV)
Chain reaction controlled in nuclear reactors.
Nuclear Fusion: Light nuclei combine to form heavier nucleus, releasing energy.
e.g., 12 H + 13 H → 24 He + 01 n + 17.6 MeV
Requires extremely high temperature (millions of Kelvin) — basis of stars and hydrogen bombs.
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Find mass defect of 24 He. (mHe = 4.002602 u, mp = 1.007276 u, mn = 1.008665 u)
Delta m = 2(1.007276) + 2(1.008665) - 4.002602 = 2.014552 + 2.017330 - 4.002602 = 0.029280 u.
Find binding energy of 24 He using Delta m = 0.029280 u.
B.E. = 0.029280 x 931.5 = 27.27 MeV. B.E. per nucleon = 27.27/4 = 6.82 MeV.
A radioactive sample has a half-life of 20 years. How much remains after 60 years?
Number of half-lives n = 60/20 = 3. Remaining = N0 / 23 = N0/8 = 12.5% of original.
The decay constant of a nucleus is 0.002 s-1. Find its half-life.
T1/2 = 0.693 / lambda = 0.693 / 0.002 = 346.5 s.
The activity of a sample at t=0 is 1000 Bq. After 4 hours the activity is 250 Bq. Find the half-life.
Activity falls to 1/4 in 4 hours => 2 half-lives have passed. T1/2 = 4/2 = 2 hours.
Write the alpha decay equation for 92238 U.
92238 U → 90234 Th + 24 He. Atomic number decreases by 2, mass number by 4.
Calculate Q-value for the reaction: 36 Li + 01 n → 13 H + 24 He. (masses: Li-6 = 6.01513 u, n = 1.00867 u, H-3 = 3.01605 u, He-4 = 4.00260 u)
Mass of reactants: 6.01513 + 1.00867 = 7.02380 u. Mass of products: 3.01605 + 4.00260 = 7.01865 u.
Delta m = 7.02380 - 7.01865 = 0.00515 u. Q = 0.00515 x 931.5 = 4.8 MeV.
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Common mistakes
- Forgetting the unit conversion: 1 u = 931.5 MeV/c2 (not 931.5 MeV directly — need to multiply by c2 which gives MeV when u is used).
- In alpha decay, reducing both Z and A incorrectly.
- Confusing half-life T1/2 = 0.693/lambda with mean life tau = 1/lambda.
Summary
Nuclei consist of protons and neutrons; nuclear radius R = R0 A1/3. Binding energy per nucleon peaks at iron. Radioactive decay follows N = N0 e-lambda t; T1/2 = 0.693/lambda. Fission releases ~200 MeV; fusion of light nuclei powers stars.