(a) Show that for an n-type semiconductor, the Fermi level lies midway between the donor states and the conduction band edge at low temperature (assuming Eᵥ = 0). 20 marks

(b) The wavelength of a prominent X-ray line from a copper target is 0·1512 m

Question

(a) Show that for an n-type semiconductor, the Fermi level lies midway between the donor states and the conduction band edge at low temperature (assuming Eᵥ = 0). 20 marks

(b) The wavelength of a prominent X-ray line from a copper target is 0·1512 m. The radiation, when diffracted with (111) plane of a crystal with fcc structure, corresponded to a Bragg angle of 20·2°. If the density of the crystal is 2698 kg/m³ and atomic weight is 26·98 kg/k mol, calculate the Avogadro number. 15 marks

(c) Which of the following decays are allowed and which are forbidden? If the decay is allowed, state which interaction is responsible. If it is forbidden, state which conservation law its occurrence would violate.
(a) n → p + e⁻ + ν̄ₑ
(b) Λ° → π⁺ + π⁻
(c) π⁻ → e⁻ + γ
(d) π° → e⁻ + e⁺ + νₑ + ν̄ₑ
(e) π⁺ → e⁻ + e⁺ + μ⁺ + νᵤ 15 marks

UPSC Answer Check · Accepted Answer

Derive the Fermi level position for n-type semiconductor in part (a) using charge neutrality and appropriate approximations at low temperature. For part (b), apply Bragg's law to find interplanar spacing, then use fcc geometry and density formula to calculate Avogadro number. For part (c), analyze each decay using conservation laws (charge, baryon number, lepton number, strangeness, energy-momentum) to determine allowed/forbidden status and identify the responsible interaction. Allocate approximately 40% time to (a), 30% to (b), and 30% to (c) based on mark distribution.
- Part (a): Derivation showing E_F = (E_C + E_D)/2 using charge neutrality condition n_0 = n_D+ at low T, with proper assumptions about donor ionization and negligible intrinsic carriers
- Part (b): Application of Bragg's law (2d sin θ = nλ), calculation of d_111 for fcc, relation between lattice parameter and atomic density, and final calculation of N_A ≈ 6.02 × 10^23 mol^-1
- Part (c)(i): n → p + e⁻ + ν̄ₑ is allowed weak decay (β-decay) conserving all quantum numbers
- Part (c)(ii): Λ° → π⁺ + π⁻ is forbidden as it violates baryon number conservation (B=1 → B=0)
- Part (c)(iii): π⁻ → e⁻ + γ is forbidden as it violates lepton number conservation (L=0 → L=1) and angular momentum
- Part (c)(iv): π° → e⁻ + e⁺ + νₑ + ν̄ₑ is forbidden as it violates energy conservation (m_π° < 2m_e + E_ν)
- Part (c)(v): π⁺ → e⁻ + e⁺ + μ⁺ + νᵤ is forbidden as it violates charge conservation (+1 → +1-1+1+0 = +1 actually check: +1 → -1+1+1+0 = +1, but violates lepton family number and energy)

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	25%	12.5	Correctly identifies Fermi level behavior in extrinsic semiconductors, properly applies Bragg's law and crystal structure concepts, and accurately applies all conservation laws (charge, baryon number, lepton numbers, strangeness, energy-momentum) for particle decays with correct interaction identification	Basic understanding of semiconductor doping and Fermi level shift; applies Bragg's law with minor errors; identifies some conservation laws correctly but misses subtle violations like lepton family number or energy constraints	Confuses Fermi level position in n-type vs p-type semiconductors; misapplies Bragg's law or confuses crystal structures; fails to identify basic conservation law violations or misidentifies interaction types
Derivation rigour	20%	10	Rigorous step-by-step derivation for (a) starting from charge neutrality, explicit low-temperature approximations, clear algebraic manipulation to E_F = (E_C + E_D)/2; systematic derivation of N_A in (b) with all intermediate steps shown	Derivation present but skips key steps or assumptions; arrives at correct result but with gaps in logical flow; missing explicit justification for approximations used	Missing derivation entirely or presents circular reasoning; jumps to final formula without justification; major logical gaps that invalidate the proof
Diagram / FBD	10%	5	Clear energy band diagram for part (a) showing E_C, E_V, E_D, E_F with proper relative positions at low T; crystal structure sketch for (b) showing (111) planes in fcc; Feynman-style diagrams or decay schematics for (c) illustrating particle interactions	Basic band diagram present but lacking labels or detail; minimal or no crystal structure visualization; textual descriptions substitute for diagrams in particle physics section	No diagrams despite clear need for visual representation; incorrect diagram showing wrong energy level ordering or wrong crystal structure
Numerical accuracy	25%	12.5	Correct unit conversion (0.1512 nm not m - candidate must spot the error), precise calculation of d_111 = a/√3, accurate lattice parameter from density, final N_A within 1% of accepted value 6.022 × 10^23 mol^-1 with proper significant figures	Correct method but arithmetic errors; uses given wavelength as 0.1512 m without correction; final answer order of magnitude correct but significant deviation from true N_A	Major calculation errors; wrong formula for interplanar spacing; completely unrealistic numerical result without comment; missing numerical part entirely
Physical interpretation	20%	10	Explains physical significance of Fermi level position (donor electrons partially fill conduction band at low T); interprets Bragg diffraction as probe of crystal structure; explains why forbidden decays cannot occur and connects allowed weak decays to W/Z boson exchange; notes the 0.1512 m vs nm discrepancy as typographical error	Minimal physical interpretation beyond mathematical results; some connection between calculation and physical reality; basic statements about conservation laws without deeper explanation	Purely mathematical treatment with no physical insight; fails to recognize unrealistic values; no discussion of underlying physics mechanisms

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