  We demonstrate a wide-band all-optical method of nanoscale magnetic resonance
(MR) spectroscopy under ambient conditions. Our method relies on
cross-relaxation between a probe spin, the electronic spin of a
nitrogen-vacancy centre in diamond, and target spins as the two systems are
tuned into resonance. By optically monitoring the spin relaxation time ($T_1$)
of the probe spin while varying the amplitude of an applied static magnetic
field, a frequency spectrum of the target spin resonances, a $T_1$-MR spectrum,
is obtained. As a proof of concept, we measure $T_1$-MR spectra of a small
ensemble of $^{14}$N impurities surrounding the probe spin within the diamond,
with each impurity comprising an electron spin 1/2 and a nuclear spin 1. The
intrinsically large bandwidth of the technique and probe properties allows us
to detect both electron spin transitions -- in the GHz range -- and nuclear
spin transitions -- in the MHz range -- of the $^{14}$N spin targets. The
measured frequencies are found to be in excellent agreement with theoretical
expectations, and allow us to infer the hyperfine, quadrupole and gyromagnetic
constants of the target spins. Analysis of the strength of the resonances
obtained in the $T_1$-MR spectrum reveals that the electron spin transitions
are probed via dipole interactions, while the nuclear spin resonances are
dramatically enhanced by hyperfine coupling and an electron-mediated process.
Finally, we investigate theoretically the possibility of performing $T_1$-MR
spectroscopy on nuclear spins without hyperfine interaction and predict
single-proton sensitivity using current technology. This work establishes
$T_1$-MR as a simple yet powerful technique for nanoscale MR spectroscopy, with
broadband capability and a projected sensitivity down to the single nuclear
spin level.
