(a) An X-ray beam of wavelength (λ₁) undergoes a first order Bragg reflection at a Bragg angle of 30°. X-ray of wavelength 97 nm…

(a) Bragg’s law: nλ = 2d sin θ, with the same d for both beams.

First beam: λ₁ = 2d sin 30° = d. Second beam: 3 × 97 nm = 2d sin 60° = d√3, so d = 97√3 nm.

Hence λ₁ = d = 97√3 nm ≈ 168 nm.

(b) Let x = ħω/(k_BT). Then U = 3Nħω/(e^x − 1) and C = (∂U/∂T)_V.

Since dx/dT = −x/T = −ħω/(k_BT²), dU/dT = 3Nħω × e^x/(e^x − 1)² × ħω/(k_BT²) = 3Nk_B (ħω/(k_BT))² e^x/(e^x − 1)².

For one mole, Nk_B = R, so C = 3R (ħω/k_BT)² e^(ħω/k_BT)/(e^(ħω/k_BT) − 1)².

At very low temperature, x ≫ 1, so e^x − 1 ≈ e^x and C ≈ 3R x² e^(−x) = 3R (ħω/k_BT)² e^(−ħω/k_BT). Thus the exponential factor dominates and C ∝ e^(−ħω/k_BT); C falls to zero as T → 0, in agreement with the third law.

(c) ρ⁰ ≈ (uū − dd̄)/√2 has strangeness 0. The decay ρ⁰ → π⁺π⁻ conserves charge, baryon number, strangeness and isospin, so it proceeds by the strong interaction, whose characteristic time is about 10⁻²³ s.

K⁰ (d s̄) has strangeness S = +1, while π⁺π⁻ has S = 0. The decay K⁰ → π⁺π⁻ therefore changes strangeness by one unit (ΔS = 1). Strong and electromagnetic interactions conserve strangeness, so the decay must go through the weak interaction, which is far slower. Hence the lifetime is about 10⁻¹⁰ s (the short-lived K⁰ₛ component).

(d)

• u d̄: Q = 2/3 + 1/3 = +1, B = 0, S = 0. This is the π⁺ meson, J^P = 0⁻, I = 1, I₃ = +1, mass ≈ 139.6 MeV/c², lifetime ≈ 2.6 × 10⁻⁸ s (weak decay).
• ū d: Q = −1, B = 0, S = 0. This is the π⁻ meson, the antiparticle of π⁺, with the same J^P, mass and lifetime, and I₃ = −1.
• d d s: Q = −1/3 − 1/3 − 1/3 = −1, B = 1, S = −1. This is the Σ⁻ baryon, J^P = 1/2⁺, I = 1, I₃ = −1, mass ≈ 1197 MeV/c², lifetime ≈ 1.5 × 10⁻¹⁰ s (weak decay).
• u s s: Q = 2/3 − 1/3 − 1/3 = 0, B = 1, S = −2. This is the Ξ⁰ baryon, J^P = 1/2⁺, I = 1/2, I₃ = +1/2, mass ≈ 1315 MeV/c², lifetime ≈ 2.9 × 10⁻¹⁰ s (weak decay).

(e) A chain reaction is a self-sustaining series of fissions in which neutrons released by one fission induce further fissions. For example, a slow neutron absorbed by ²³⁵U gives fission fragments, 2–3 neutrons and about 200 MeV.

The multiplication factor k is the ratio of fission-causing neutrons in one generation to those in the previous generation. If k < 1 the reaction dies out (subcritical), if k = 1 it is steady (critical), and if k > 1 it grows (supercritical).

The critical size is the minimum size of the fissile core for which k = 1. In a smaller core, neutrons leak through the surface and are lost to parasitic absorption faster than they are produced, so the chain cannot be sustained. The critical mass is M_c = ρV_c. For a bare homogeneous sphere in one-speed diffusion theory, R_c = π√[D/(νΣ_f − Σ_a)]. The critical size is lowered by a spherical shape, higher enrichment or density, and a neutron reflector.