Phosphorus-31 Spin-½ Nucleus: NMR and Silicon Quantum Computing Applications — E8 Intelligence Research

FINDING: Phosphorus-31 (³¹P) is a spin-½ nucleus (I=1/2) with 15 protons and 16 neutrons, stable, and its spin coherence is central to NMR and silicon-based quantum computing (e.g., Kane qubit architecture). | MATH: Nuclear spin I = 1/2 → 2I+1 = 2 spin states (m_I = +1/2, −1/2). Nuclear g-factor for ³¹P ≈ 2.263 (relative to nuclear magneton). Larmor frequency ω = γB₀, where γ(³¹P) ≈ 17.235 MHz/T (at 11.74 T, ν ≈ 202.4 MHz). Zeeman splitting ΔE = ħγB₀. For phosphorus-doped silicon, hyperfine coupling A (³¹P electron-nuclear) ≈ 117 MHz (bulk Si) — this is the key coherence parameter. | CONNECTION: Spin-½ is the fundamental SU(2) doublet — the simplest non-trivial representation of the rotation group. The two-level system maps to the Bloch sphere (SU(2)/U(1) ≈ S²). The ratio of hyperfine coupling to Zeeman frequency (A/γB₀) is tunable — at resonance, this ratio = 1, a critical point. No direct golden-ratio or base-60 link in the raw data, but the spin-½ doublet is the root of all qubit sy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com