(a) (i) What is diamagnetism ? Why do diamagnetic materials have negative magnetic susceptibility ? (ii) Discuss the limitations…

(a) Diamagnetism (20 marks)

(i) Diamagnetism is the weak magnetisation that a material acquires in a direction opposite to the applied field, so that it is slightly repelled. It is present in all matter and dominates when all electrons are paired. When a field B is applied, the orbits of the electrons precess about B at the Larmor frequency ω_L = eB/2mₑ; by Lenz's law the induced circulation produces a moment opposing B. For Z electrons per atom, with ⟨r²⟩ the mean square radius of the orbits, each atom acquires a moment -Ze²B⟨r²⟩/6mₑ, so for N atoms per unit volume the Langevin-Larmor susceptibility is χ = -μ₀NZe²⟨r²⟩/6mₑ. Every factor is positive, so χ is negative and very small, about -10⁻⁵ to -10⁻⁶, independent of temperature.

(ii) Classically there would be no diamagnetism at all (Bohr-van Leeuwen theorem), so it is a quantum effect, but the simple quantum theory has limits. It treats each atom or ion independently and applies to closed-shell systems such as noble gases and ionic solids; where atoms carry a permanent moment the paramagnetic term overwhelms it. The value of ⟨r²⟩ needs accurate wave functions, so for many-electron atoms only estimates are possible. Second-order perturbation theory adds a positive, temperature-independent Van Vleck term from mixing with excited states, which can be comparable to the diamagnetic term. It also ignores the conduction electrons of metals, which give Landau diamagnetism, χ_L = -⅓χ_Pauli, and it cannot explain the large anisotropic diamagnetism of bismuth and graphite, which depends on band structure and the Fermi surface. Finally, it assumes weak fields; in strong fields at low temperature oscillatory effects (de Haas-van Alphen) appear.

(iii) The diagram plots χ against T on a signed logarithmic scale, so that phenomena differing by up to ten orders of magnitude can be compared. The diamagnet is a flat line at about -10⁻⁵. The paramagnet follows Curie's law χ = C/T, with values of 10⁻⁵ to 10⁻³, rising as T falls. The ferromagnet follows the Curie-Weiss law χ = C/(T - T_C) above the Curie temperature T_C and, below it, has very large values of about 10³ to 10⁵. The antiferromagnet shows a cusp (maximum) at the Néel temperature T_N, with Curie-Weiss behaviour above it. A superconductor below T_c is a perfect diamagnet with χ = -1, by far the strongest response.

(b) Comparator (10 marks)

In the circuit the junction of R₁ and R₂ goes to the inverting input and the signal V_in to the non-inverting input, so the op-amp is an open-loop comparator with no feedback and no hysteresis. The reference is the divider voltage:

V_ref = 15 V × R₂/(R₁ + R₂) = 15 × 1.0/(8.2 + 1.0) ≈ 1.63 V.

The output saturates at +12 V when V_in > 1.63 V and at -12 V when V_in < 1.63 V. The input is V_in = 5 sin ωt (peak 5 V). The output switches where 5 sin θ = 1.63, that is sin θ = 0.326, so θ ≈ 19° and 180° - 19° = 161°. The output is therefore a square wave that is at +12 V for 19° < ωt < 161° (142° per cycle, a duty cycle of about 39%) and at -12 V for the remaining 218° (about 61%), switching exactly at the instants the sine wave crosses the dashed V_ref level.

(c) Criticality of a nuclear reactor (20 marks)

A self-sustained reactor is critical when the effective multiplication factor k_eff, the ratio of neutrons in one generation to those in the previous one, equals 1: each fission leads on average to exactly one further fission, so the chain reaction is steady. Reactivity ρ = (k_eff - 1)/k_eff is zero at criticality, positive when supercritical and negative when subcritical.

In an infinite medium the chain is governed by four processes, in the four-factor formula k∞ = ηεpf: reproduction factor η, the number of fission neutrons produced per thermal neutron absorbed in the fuel; fast fission factor ε, the enhancement of fission by fast neutrons; resonance escape probability p, the fraction of neutrons that slow down to thermal energy without being captured in resonances (chiefly of ²³⁸U); and thermal utilisation f, the fraction of thermal neutrons absorbed in the fuel rather than in moderator, coolant and structure. In a finite reactor neutrons also leak out, and k_eff = k∞ P_FNL P_TNL, the six-factor formula, where P_FNL and P_TNL are the non-leakage probabilities during slowing down (fast) and diffusion (thermal). A small fraction of neutrons, about 0.0065 for ²³⁵U, is delayed and makes control possible.

Criticality requires the material buckling B_m² = (k∞ - 1)/M², fixed by composition (M² is the migration area), to equal the geometrical buckling B_g², fixed by the shape and size. For a bare sphere B_g² = (π/R)², so R_c = π/B_m. For a finite cylinder of radius R and height H, B_g² = (2.405/R)² + (π/H)², where 2.405 is the first zero of the Bessel function J₀. For a rectangular box, B_g² = (π/a)² + (π/b)² + (π/c)²; an infinite slab has (π/a)². The dimensions used are extrapolated dimensions, slightly larger than the physical ones, to allow for the flux extending beyond the boundary. For a given B_m² the critical size, and hence the critical mass, is smallest for the sphere, which has the least leakage per unit volume.