Nuclear quadrupole moments (Q), a measure of charge asymmetry in nuclei, are experimentally accessible via the quadrupole interaction frequency ν_Q=eQV_zz/h. To extract Q from this information one needs to knowV_zz, the electric-field gradient at the nuclear site. This must come from theory, to date only precise enough for free atoms or small molecules. Experimental ν_Q data in such environments are, however, only available for nuclear ground states.

Studies of short-lived nuclear excited states had up to now only been possible in solids, where theory is yet not sufficiently accurate. Using the pure isotope beams available at ISOLDE [1] and transforming them into molecules, the solid-state technique of g-g perturbed angular correlation (PAC) (Fig. 1 shows the table-top set-up), has now been used for the first time in project IS640 to measure ν_Q for nuclear excited states in isolated molecules [2].

Fig.2: Schematic explanation of PAC measurement on a free molecule. The Cd nucleus (dark blue) with quadrupole moment Q interacts with the electric-field gradient of the electrostatic potential at the nucleus Vxy = ∂2Φr/∂x∂y  (EFG) created by the Cd electron distribution (grey) modified by bonding to the bromine atoms (brown). This is observed by the modulation of the time-distribution of the two g rays emitted in decay of ^111mCd. The energy splitting of the nuclear intermediate state is directly seen as transition frequencies (ω) in the Fourier tranformation.

The basic idea of the new approach is the fact that in a linear molecule the principal axis of the EFG is by symmetry along the molecular axis and the rotational angular momentum J is perpendicular to this for all states. The component of the EFG tensor along J is thus equal to -1/2 times that along the molecular axis. The nuclear spin I being much smaller than J for practically all thermally populated rotational states leads to a total F=J+I essentially aligned with J.

The resulting perturbation function is then expected to be very similar for all J states, highly resembling the well-known one for a randomly oriented axially symmetric EFG. The measured time patterns for the cases of HgCl₂ and HgI₂ (Fig 2) demonstrate these facts clearly: As expected, the period for the free molecule is approximately half the one for the molecular solid, and the addition of gas to the molecule sample destroys the periodic behavior due to collisions changing the axis of molecular rotation during the measuring time.

Fig. 3: Observed PAC patterns for Hg halides at room temperature (top), in the gas phase (middle), and with additional other gas (bottom) (blue: data, red: theory).

Since modern quantum chemistry techniques allow to calculate the EFG for small molecules with high accuracy, the nuclear quadrupole moment Q has been obtained from the measured ν_Q for two excited states, ¹¹¹Cd(5/2⁺) and ¹⁹⁹Hg(5/2^-). These then allow to normalize a number of quadrupole moments of other Cd and Hg isotopes, using the known frequency ratios from solid state experiments.

The achievement of the primary goal of the experiment is demonstrated in Fig.3. The values of Q for ¹¹¹Cd(5/2⁺) determined from the data for three molecules (CdCl₂, CdBr₂, CdI₂) essentially coincide, and considerably higher accuracy than available hitherto has been achieved. Accurate nuclear quadrupole moment values are essential benchmarks for nuclear structure models. There are obviously further elements where the developed techniques will be used to obtain precise values for Q, the closed-shell nuclei Pb and Sn being the most important ones from the nuclear physics standpoint.

Fig. 4: Resulting Q for ¹¹¹Cd(5/2⁺) from recent EFG calculations using various methods; a: Ref. [4],  b: Ref. [5],  c: Ref. [6],  d: Ref. [7],  e: Present work

In addition, the fact that now precise values for the EFG can be quoted for the numerous solids previously investigated by PAC and other solid-state methods is of great relevance. One might hope that the presently not very accurate solid-state theory calculations can be improved by comparison with reliable experimental data.

The fact that in some cases precise ν_Qvalues are now available both for the free molecule and the corresponding molecular solid is an additional relevant outcome of the experiments. The changes due to the molecule-molecule interaction allow a sensitive test of the corresponding electronic structure changes. This interaction, typical for soft matter, is at present not well reproduced yet by theory.

The newly developed experimental techniques will in future experiments also permit a quantitative way to investigate the molecular rotation axis change by gas collisions. Only for a few light molecules have such results to date become available from conventional NMR experiments. They form a critical test for the theoretical treatment of very-long-range interaction between atoms and molecules. First data for the collisions of HgI₂ with noble gas atoms have already been obtained and are in the process of quantitative analysis.

References

[1] R. Catherall et al., The ISOLDE facility. J. Phys. G 44, 094002 (2017)

[2] H. Haas et al., Free molecule studies by perturbed g-g angular correlation: A new path to accurate nuclear quadrupole moments. Phys. Rev. Letters 125, 253001 (2021)

[3] H. Frauenfelder and R. M. Steffen, Angular correlations. In: Alpha-, beta- and gamma-ray spectroscopy, K. Siegbahn, Ed., pp 997-1198, NORTH HOLLAND, Amsterdam (1965)

[4] H. Haas and J. G. Correia, The quadrupole moments of Cd and Zn isotopes – an update. Hyperfine Interactions 198, 133 (2010)

[5] L. Errico et al., Precision of Electric-Field Gradient Predictions by Density Functional Theory and Implications for the Nuclear Quadrupole Moment and Its Error Bar of the ¹¹¹Cd 245 keV 5/2⁺ Level. J. Phys. Chem. C 120, 23111 (2016)

[6] H. Haas, M. B. Barbosa and J. G. Correia, The quadrupole moments of Cd and Zn isotopes – an apology. Hyperfine Interactions 237, 115 (2016)

[7] H. Haas et al., Quadrupole moments of Cd and Zn nuclei: When solid-state, molecular, atomic, and nuclear theory meet. Euro. Phys. Lett. 117, 62001 (2017)