The reference-probe model for a robust and optimal radical-pair-based magnetic compass sensor

Radical-pair reactions have been suggested to be sensitive to the direction of weak magnetic fields, thereby providing a mechanism for the magnetic compass in animals. Discovering the general principles that make radical pairs particularly sensitive to the direction of weak magnetic fields will be essential for designing bioinspired compass sensors and for advancing our understanding of the spin physics behind directional effects. The reference-probe model is a conceptual model introduced as a guide to identify radical-pair parameters for optimal directional effects. Radical pairs with probe character have been extensively shown to enhance directional sensitivity to weak magnetic fields, but investigations on the role of the reference radical are lacking. Here, we evaluate whether a radical has reference character and then study its relevance for optimal directional effects. We investigate a simple radical-pair model with one axially anisotropic hyperfine interaction using both analytical and numerical calculations. Analytical calculations result in a general expression of the radical-pair reaction yield, which in turn provides useful insights into directional effects. We further investigate the relevance of the reference character to robustness against variations of earth-strength magnetic fields and find that the reference character captures robust features as well. Extending this study to radical pairs with more hyperfine interactions, we discuss the interplay between reference character and optimal and robust directional effects in such more complex radical pairs.