(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) Use Bragg’s law: nλ = 2d sin θ. The same set of planes means the interplanar spacing d is the same for both reflections.

For the first beam: 1 × λ₁ = 2d sin 30° = 2d × (1/2) = d. So λ₁ = d.

For the second beam: 3 × 97 nm = 2d sin 60° = 2d × (√3/2) = d√3. Therefore d = (3 × 97)/√3 nm = 97√3 nm.

Since λ₁ = d, λ₁ = 97√3 nm ≈ 168.0 nm.

Condition: Both reflections occur from the same set of crystallographic planes, so d is unchanged.

(b) The Einstein internal energy is U = 3N ħω/(e^(ħω/k_BT) − 1).

Let x = ħω/(k_BT). Then U = 3N ħω/(e^x − 1).

The specific heat at constant volume is C = (∂U/∂T)_V.

Differentiate U with respect to T: dU/dT = 3N ħω d/dT[(e^x − 1)⁻¹] = 3N ħω × [−(e^x − 1)⁻² e^x dx/dT].

Now x = ħω/(k_BT), so dx/dT = −ħω/(k_BT²) = −x/T.

Therefore dU/dT = 3N ħω × (e^x − 1)⁻² e^x × (x/T) = 3N ħω × (ħω/(k_BT²)) × e^x/(e^x − 1)² = 3N k_B × (ħω/(k_BT))² × e^x/(e^x − 1)².

For one mole, N = N_A and N_A k_B = R. Hence C = 3R (ħω/(k_BT))² e^(ħω/k_BT)/(e^(ħω/k_BT) − 1)².

At very low temperature, T ≪ ħω/k_B. Then x = ħω/(k_BT) ≫ 1, so e^x ≫ 1 and e^x − 1 ≈ e^x.

Thus C ≈ 3R x² e^x/(e^x)² = 3R x² e^(−x) = 3R (ħω/(k_BT))² e^(−ħω/k_BT).

Therefore, apart from the algebraic factor (ħω/k_BT)², the Einstein specific heat varies exponentially as C ∝ e^(−ħω/k_BT) at very low temperature.

(c) The ρ⁰ meson has quark content uū − dd̄ type combination; it has zero strangeness and no quantum number forbidding its decay into two pions. The decay ρ⁰ → π⁺ + π⁻ is allowed by the strong interaction. Strong interactions are fast, with characteristic times of order 10⁻²³ s. Hence the mean lifetime of ρ⁰ is about 10⁻²³ s.

The K⁰ meson has nonzero strangeness. For example, K⁰ = d s̄ (or d anti-s) and has strangeness S = +1. The final pions π⁺ and π⁻ have zero strangeness. Therefore the decay K⁰ → π⁺ + π⁻ changes strangeness by one unit. Strong and electromagnetic interactions conserve strangeness, so they cannot cause this decay. It must proceed through the weak interaction, which violates strangeness. Weak decays are much slower than strong decays, giving lifetimes of order 10⁻¹⁰ s or longer.

Thus the large lifetime difference arises because ρ⁰ decays strongly, while K⁰ must decay weakly because strangeness is not conserved in the final π⁺π⁻ state.

(d)

• (d)(a) u d̄: Quark charges are u = +2/3, d̄ = +1/3. Total charge Q = +1. Baryon number B = 0. Strangeness S = 0. This is the π⁺ meson in its ground pseudoscalar state, with J^P = 0⁻, isospin I = 1, I₃ = +1, mass ≈ 139.6 MeV/c², mean lifetime ≈ 2.60 × 10⁻⁸ s. It decays weakly. The vector excitation with the same quark content is ρ⁺, J^P = 1⁻, mass ≈ 775 MeV/c², decaying strongly with lifetime ≈ 10⁻²³ s.

• (d)(b) ū d: ū has charge −2/3, d has charge −1/3. Total charge Q = −1. B = 0, S = 0. This is the π⁻ meson, J^P = 0⁻, I = 1, I₃ = −1, mass ≈ 139.6 MeV/c², mean lifetime ≈ 2.60 × 10⁻⁸ s. It decays weakly. Its vector counterpart is ρ⁻.

• (d)(c) d d s: Charges: d = −1/3, d = −1/3, s = −1/3, so Q = −1. Baryon number B = 1. Strangeness S = −1. This is the Σ⁻ baryon, J^P = (1/2)⁺, isospin I = 1, I₃ = −1, mass ≈ 1197.4 MeV/c², mean lifetime ≈ 1.48 × 10⁻¹⁰ s. It decays weakly.

• (d)(d) u s s: Charges: u = +2/3, s = −1/3, s = −1/3, so Q = 0. Baryon number B = 1. Strangeness S = −2. This is the Ξ⁰ baryon, J^P = (1/2)⁺, isospin I = 1/2, I₃ = +1/2, mass ≈ 1314.9 MeV/c², mean lifetime ≈ 2.90 × 10⁻¹⁰ s. It decays weakly.

(e) A chain reaction is a self-sustaining sequence of nuclear fissions in which neutrons released in one fission cause further fissions in nearby fissile nuclei. For example, when a slow neutron is absorbed by ²³⁵U, the nucleus splits and emits 2–3 neutrons plus about 200 MeV energy. If at least one of these neutrons causes another fission on average, the process continues as a chain reaction.

The multiplication factor k is defined as k = (number of neutrons causing fission in the next generation)/(number causing fission in the present generation).

• If k < 1, the reaction is subcritical and dies out.
• If k = 1, the reaction is critical and is self-sustaining at constant power.
• If k > 1, the reaction is supercritical and grows rapidly.

The critical size of the core is the minimum size of the fissile core for which k = 1. In a small core, many neutrons leak out through the surface before causing further fission. If the core is smaller than the critical size, neutron leakage and parasitic absorption exceed neutron production, so the chain reaction cannot be sustained. If the core is larger than the critical size, the reaction becomes supercritical.

The critical size depends on the fissile material, its purity and enrichment, density, shape, and the presence of a reflector. A sphere has the smallest surface-to-volume ratio, so it gives the smallest critical size for a given material. A neutron reflector surrounding the core scatters some escaping neutrons back, reducing the critical size. The corresponding mass is called the critical mass, M_c = ρV_c.

In one-speed diffusion theory for a bare spherical core, the critical condition is approximately (π/R_c)² = (νΣ_f − Σ_a)/D, so R_c = π√[D/(νΣ_f − Σ_a)], where R_c is the critical radius, D is the diffusion coefficient, ν is the average number of neutrons per fission, Σ_f is the macroscopic fission cross-section, and Σ_a is the macroscopic absorption cross-section. This expression is valid for a homogeneous bare spherical core under one-speed diffusion theory.