Determine octupole deformation contribution to Coulomb energy and α-sensitivity in 229Th

Determine the sign and quantify the magnitude of the change in the axially symmetric nuclear octupole deformation (parameter β3) between the 229Th ground state and the 229mTh isomer, and derive its resulting contribution to the Coulomb energy difference ΔEC = EC(isomer) − EC(ground) and to the enhancement factor K for the fine-structure-constant sensitivity of the 229Th nuclear clock transition, within the geometric nuclear-shape framework r(θ) = Rs[1 + Σn βn Yn0(θ)].

Background

The enhancement factor K for sensitivity of the 229Th nuclear clock transition to variation of the fine-structure constant α is K = ΔEC/E, where ΔEC is the Coulomb energy difference between the isomer and ground state. Using measured values of the mean-square charge radius difference Δ⟨r^2⟩ and the quadrupole moment ratio Q0m/Q0, the authors find strong cancellation between the ⟨r^2⟩ and Q0 terms, yielding K = 5900(2300) and exposing extreme sensitivity to nuclear-shape details.

The theoretical treatment up to this point omits contributions from nuclear octupole deformation. The authors explicitly state that the contribution of the currently unknown nuclear octupole deformation to the Coulomb energy EC is not accounted for, and show that even a 1–5% differential change in β3 between the ground and isomeric states can shift K by thousands. Therefore, establishing the sign and magnitude of the octupole deformation change and its impact on ΔEC and K is essential for a precise α-sensitivity determination.