How Good Could a Chemical Magnetometer Be? A Sensitivity Bound for Radical-Pair Magnetometry

Migratory birds are widely believed to sense the geomagnetic field through a radical-pair reaction whose product yield depends on field strength and direction. The mechanism is attractive as an engineering template because it operates at Earth's field strength, at ambient temperature, in a molecular medium — a combination no existing magnetometer offers. Kominis and Gkoudinakis have placed biological receptors on the energy-resolution scale, but no specification for an engineered device — a sensitivity in tesla per root hertz — has been derived, and no such device has been placed on the sensitivity-versus-volume plane against deployed instruments. This paper supplies both. Transduction slopes are computed from exact singlet-yield calculations rather than from a saturating parameterisation of the kind used to summarise high-field data. The difference at Earth's field is one to two orders of magnitude: exact slopes for a natural-abundance pair at 50 µT are several hundred per tesla and of opposite sign to the parameterised value, because Earth's field falls on the steep flank of the low-field effect rather than in the quadratic region. The practical consequence inverts the usual design intuition: natural-abundance chemistry is already within a factor of two of optimal, isotopic reduction of the hyperfine coupling buys little, and the binding constraint is instead to keep the low-field extremum — where sensitivity vanishes identically — away from the operating field. A tenfold increase in radical-pair lifetime, ordinarily regarded as beneficial, can move that null onto the operating field and cost more than an order of magnitude. Propagating the slope through radical-pair formation rate, shot noise, optical depth and photostability gives an idealised floor of 4.1 pT per root hertz in a 3.6 cubic millimetre sensing volume at 100 mW of pump power, with a factor-of-six spread across nuclear basis size, degrading to 41-414 pT per root hertz for realistic optical readout. On energy resolution the device reaches 2.4 x 10^8 hbar idealised, against 3.8 x 10^8 hbar for a DC nitrogen-vacancy ensemble and 9.2 x 10^8 hbar for a scalar atomic magnetometer: parity requires the readout to come within a factor of 1.3 to 2.0 of counting every recombination event individually, far beyond any demonstrated scheme. A structural obstacle is identified that no improvement in readout would remove. The applications for which native Earth-field operation is decisive — magnetic-anomaly navigation and geophysical survey — require absolute accuracy, which atomic and proton magnetometers obtain by referencing a Larmor frequency to a fundamental constant. A radical-pair sensor can be made absolute by reading a spin-resonance frequency instead of a yield, as demonstrated for organic thin-film devices, but resolving a Zeeman splitting of 1.4 MHz at 50 µT against a hyperfine linewidth of order 28 MHz is not possible; frequency-referenced operation therefore requires fields well above the hyperfine scale, and the two virtues are mutually exclusive. The niche that survives is not point magnetometry but large-area, flexible or biocompatible field imaging, where the sensing medium's being a liquid or a film is worth more than absolute accuracy. The archive contains the paper (PDF and DOCX), an exact singlet-yield solver, the sensitivity model, the figure generators, and a README.