### β-NQR Spectroscopy

The measurement is performed using β-NQR (β-detector Nuclear Quadrupole Resonance) spectroscopy. In this technique, a spin-polarized beam of the radioactive nuclide of interest is embedded in a crystalline sample inside a region of near-zero DC magnetic field. As the nuclide undergoes beta-decay, the polarization of its nuclear spin induces an asymmetry in the angular distribution of the emitted beta-particles which can be measured using two detectors on either side of the sample, along the axis of polarization. However, if an AC magnetic field is applied across the sample and tuned to the precise resonant frequency of the isotope-crystal system, these nuclei will undergo energy transitions that destroy their polarization and thus the asymmetry of their beta-spectrum. Scanning the frequency of the AC field while continuously comparing the number of observed beta-particles from the two detectors then allows one to determine the nuclear resonant frequency to great precision.

The full Hamiltonian describing the interaction between the atomic nucleus and the fields inside a crystalline sample is given by

where *I* is the scalar nuclear spin and *Î* is the nuclear spin operator. However, by zeroing the DC magnetic field *H*_{0} and choosing a cubic crystal (in this case SrTiO_{3}) for which the crystalline asymmetry parameter η vanishes, this can be reduced to

This allows the quadrupole term to dominate the Hamiltonian—a great advantage of β-NQR over β-NMR spectroscopy, which uses dynamic modulations of a large constant magnetic field so that the quadrupole must be treated as a perturbation to the much stronger magnetic dipole term. The magnetic sublevels m_{I} are now easily calculable as a function of the nuclear spin and quadrupole moment, and therefore so is the base resonant frequency.

The resulting formulae are:

for ^{8}Li (spin-2) and for ^{9,11}Li (spin-3/2).

In the ideal case, the ratio of any two quadrupole moments should then simply become the ratio of their resonant frequencies multiplied by some fraction which depends on which isotopes are being compared. In the real-world case, the ambient DC magnetic field can only be reduced to the order of mG, so the Zeeman (DC-field) term in the Hamiltonian must be treated as a perturbation. This can be accounted for by making two measurements with opposite polarizations, so that any perturbation on one measurement should have an equal and opposite effect on the other and the average of the two should be bias-free.

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