(a) By considering the three-dimensional harmonic oscillator potential, explain the energy levels of nucleons inside the nucleus…

(a) Harmonic oscillator shell model (20 marks)

Take the nucleons to move independently in the central potential V(r) = ½mω²r². Separating the Schrödinger equation in Cartesian coordinates gives three one-dimensional oscillators, so E = (nₓ + n_y + n_z + 3/2)ħω = (N + 3/2)ħω, where N = 2(n - 1) + l, n = 1, 2, … is the radial quantum number and l the orbital angular momentum. The zero-point energy is the ground state N = 0: each of the three degrees of freedom contributes ½ħω, so E₀ = (3/2)ħω, with ħω ≈ 41 A^(-1/3) MeV. Levels of given N are degenerate, with g_N = (N + 1)(N + 2) including spin, and shells close at cumulative occupancies 2, 8, 20, 40, 70 and 112.

The model reproduces only the first three observed magic numbers, 2, 8 and 20. It fails for 28, 50, 82 and 126 because a purely central potential gives the wrong gaps between levels; the harmonic well has no mechanism that separates the states of a shell by total angular momentum. Mayer and Jensen (1949) added a strong spin-orbit term -a l·s, which lowers the j = l + 1/2 level (the opposite ordering to atoms). It pulls the high-j intruder levels 1f₇/₂, 1g₉/₂, 1h₁₁/₂ and 1i₁₃/₂ down from the shell above into the shell below, opening the large gaps at 28, 50, 82 and 126.

(b) Packing fraction and binding energy (10 marks)

The packing fraction is f = (M - A)/A × 10⁴, where M is the atomic mass in u and A the mass number, that is the mass excess per nucleon. The binding energy is the energy needed to disassemble a nucleus into its free protons and neutrons, E_b = [Zm_H + (A - Z)mₙ - M]c², with 1 u = 931.5 MeV; f_b = E_b/A is the binding energy per nucleon.

The f-A curve starts at large positive values for light nuclei (about +70 for ²H), passes through zero at ¹²C, reaches a broad minimum of about -11.6 near ⁵⁶Fe (A ≈ 50-60), and rises again to small positive values for the heaviest nuclei (about +2 for ²³⁸U). The f_b-A curve rises steeply from 1.11 MeV for ²H, shows a peak at ⁴He (7.07 MeV), reaches a broad maximum of about 8.8 MeV near ⁵⁶Fe, and falls slowly to about 7.6 MeV for ²³⁸U.

The two graphs are complementary because f measures the mass excess and f_b the mass defect, which differ essentially in sign: the minimum of f coincides with the maximum of f_b. Nuclei near A ≈ 56 are the most tightly bound, so fusion of light nuclei and fission of heavy nuclei both move towards this region and release energy.

(c) X-ray diffraction and the Laue method (20 marks)

(i) A single fixed crystal in a monochromatic beam reflects only when some plane happens to satisfy Bragg's law 2d sinθ = nλ. Three methods raise the probability that planes of the right orientation are exposed. The Laue method keeps the crystal fixed but uses a continuous ("white") spectrum, so that some wavelength fits every plane. The rotating-crystal method uses monochromatic X-rays and rotates the crystal, so that different planes in turn satisfy the law. The Debye-Scherrer powder method uses monochromatic X-rays on a powder of randomly oriented crystallites, so that some always satisfy it.

(ii) In the Laue method a narrow beam of continuous X-rays, of wavelengths from λ_min = 12.4/V(kV) Å upwards, from a high-voltage tube passes through a pin-hole collimator onto a stationary single crystal, and the diffracted beams are recorded on a flat film placed beyond the crystal (transmission) or between the source and the crystal (back reflection). Each set of planes selects from the spectrum the wavelength that satisfies Bragg's law for its spacing and angle, equivalently the three Laue conditions a·(s - s₀) = hλ, b·(s - s₀) = kλ, c·(s - s₀) = lλ.

Origin of the spots: in the Ewald construction, a reflection occurs whenever a reciprocal lattice point lies on the Ewald sphere; because the radiation is continuous, every reciprocal lattice point between the limiting spheres of radii 1/λ_min and 1/λ_max is on some sphere, giving a spot. Spots from planes of one zone lie on an ellipse in transmission and a hyperbola in back reflection. A spot may combine higher orders (λ, λ/2, λ/3).

Utility: the pattern shows the symmetry of the crystal, with only the 11 Laue groups distinguishable because of Friedel's law, and it fixes the orientation of the crystal, which is used to cut and mount crystals. Streaked spots (asterism) reveal strain and imperfection. Because the wavelength of each spot is unknown, the method does not give lattice parameters or the full structure.