Dynamic Hyperfine Interactions

• Dynamic HFIs are time-dependent couplings between nuclear and electronic magnetic moments observed in post-electron capture decay, characterized by transient electric field gradients.
• The integration of TDPAC spectroscopy with advanced ab initio DFT simulations precisely quantifies fluctuating EFG tensors and defect energetics during electronic recovery.
• The Bäverstam–Othaz and Lupascu approaches model stochastic electron-nuclear interactions, linking temperature-dependent damping to charge-state dynamics.

Dynamic hyperfine interactions (HFIs) refer to time-dependent couplings between nuclear and electronic magnetic moments in condensed matter and molecular systems, especially as experienced by a localized atomic nucleus subject to rapidly evolving electronic surroundings. These dynamic regimes are particularly prominent following nuclear events such as electron-capture (EC) decay, where the post-decay electronic environment undergoes rapid temporal evolution, manifesting as fluctuating electric field gradients (EFGs) or magnetic fields at the nuclear site. Dynamic HFIs are directly observable by techniques such as time-differential perturbed angular correlation (TDPAC), and their accurate interpretation now relies on integrating advanced ab initio density functional theory (DFT) calculations, explicit defect energetics, and a rigorous treatment of the stochastic processes governing electron-nuclear coupling.

1. Dynamic HFIs and Electron-Capture Decay Aftereffects (ECAE)

In EC decay, a parent atom (here, $^{111}$In) captures an orbital electron, transforming into a daughter atom (here, $^{111}$Cd) often in a multiply ionized state with numerous electron holes, particularly in the K, L, and M shells. These holes trigger a cascade of Auger processes, resulting in a highly non-equilibrium, fluctuating electronic environment near the newly formed nucleus. These electronic fluctuations give rise to non-static hyperfine interactions, most notably dynamic EFGs at the probe nucleus, which persist until electronic recovery (re-filling of electron holes) is complete.

In $\alpha$-Al$_2$O$_3$ doped with $^{111}$In($\to$)$^{111}$Cd, this process generates two distinct classes of TDPAC-detectable HFIs:

• HFI$_u$ ("undisturbed"): The probe experiences a final, well-defined, static EFG corresponding to complete electronic recovery (impurity level fully occupied, charge state $q = 1^-$).
• HFI$_d$ ("disturbed"): The probe traverses a fluctuating, non-equilibrium regime (partially filled impurity state, $q \approx 0.65^-$), resulting in a rapid, temperature-dependent damping in the TDPAC signal before eventual stabilization.

This temporal evolution is governed by the rate of electron-hole recombination and the nature of transient electronic configurations.

2. Experimental and Computational Methodologies

The detailed quantification of dynamic HFIs is achieved by a synergistic use of advanced ab initio DFT (here, WIEN2k with FP-APW+lo basis) and high-resolution TDPAC measurements:

• Ab initio/DFT Simulation: The EFG tensor at the Cd site is computed for a suite of charge states, modeling fractional electron filling in increments (e.g., steps of 0.05 $e^{-}$). This allows mapping $V_{33}$ and $\eta$ as a function of the electronic configuration, directly reflecting the fluctuating environments experienced by the probe nucleus.
• Defect Formation Energy Analysis: The relative stability of different charge states of Cd in $\alpha$-Al$_2$O$_3$ is assessed by formation energy calculations as a function of Fermi level (with all relevant chemical potentials duly included), revealing which electronic configurations are energetically accessible during the recovery process.
• TDPAC Spectroscopy: Time-resolved R(t) spectra encode the time evolution of the hyperfine interaction. Two interactions (HFI$_u$, HFI$_d$) reflect the coexistence of distinct charge-state trajectories. Dynamic signatures are correlated with specific electronic configurations by matching observed static EFGs to DFT-computed values.

3. Theoretical Models for the Dynamic Regime: BO and L Approaches

The stochastic nature of dynamic HFIs is described by two principal theoretical frameworks:

• Bäverstam-Othaz (BO) Approach: Treats the dynamic regime as a stochastic "on-off" process, characterized by:
  • A recovery constant $2g$ ($t_g = 1/(2g)$), the mean lifetime of the residual electron holes,
  • A relaxation constant $A_r$, quantifying the width of the initial EFG distribution (spread in fluctuating configurations).
  • The perturbation factor for the TDPAC signal is approximated as:

$$G_{22}(t) = \sum_j f_j G_{22}^{(j)}(t) e^{-(A_{r,j} + 2g_j)t}$$

• Lupascu (L) Approach: Implements a stochastic simulation (numerically integrating over possible EFGs) as unidirectional relaxation from a broad initial distribution of EFGs (width $S_i$) to a single static EFG (recovery rate $I_r \equiv T_r$). The resulting damping is governed by both $S_i$ and $I_r$.

The equivalence between the two is analytically established when the mean initial EFG is approximately equal to the final EFG ($\langle EFG_i \rangle \approx EFG_f$) and for small-magnitude sine contributions, yielding the identification $A_r \leftrightarrow S_i$ and $2g \leftrightarrow I_r$:

$G_{22}(t) \approx e^{-(A_r + 2g)t}$

4. Electronic Configurations, Charge States, and Energetics

Ab initio calculations demonstrate that the EFG at the Cd site is acutely sensitive to the electronic occupancy of the impurity level. Adding fractional electrons (simulating progressive refilling post-EC decay) leads to significant, often discontinuous shifts in $V_{33}$. Specifically:

• At $q = 1^-$ (fully filled state), the DFT-computed $V_{33}$ matches the experimental HFI$_u$ EFG.
• At $q \approx 0.65^-$, the computed $V_{33}$ matches HFI$_d$, indicating stabilization at a fractional filling determined by intrinsic carrier concentration and electron mobility.

Defect formation energies show that, for most Fermi levels in undoped or moderately p-doped $\alpha$-Al$_2$O$_3$, the ground state is $q = 1^-$. Only when the system is electron-deficient can intermediate charge states be transiently stabilized within the TDPAC time window. States with $q > 1^-$ (electron excess) or significant unfilled levels are energetically disfavored.

5. Temperature Dependence and Fluctuation Dynamics

The detailed temperature evolution captured by dynamic HFI parameters is as follows:

• High temperature: Rapid electron-hole recombination (large $2g$), narrow initial EFG distribution ($A_r$ small), resulting in minimally damped TDPAC anisotropy.
• Intermediate temperature: Increased hole lifetimes and broader EFG distributions, corresponding to strong dynamic relaxation. The number of fluctuating configurations (inferred from fit parameters) increases, producing pronounced damping in R(t).
• Low temperature: Electronic recovery is significantly slower, a larger subset of Cd probes remain in partially filled (HFI$_d$) states for longer periods, and dynamic damping dominates until eventual relaxation is observed.

The subtle linear reduction of $V_{33}$ for HFI$_u$ with increasing temperature is attributed to lattice expansion.

6. Implications and Unified Understanding

By systematically unifying experiment (TDPAC), DFT (electronic structure and defect energetics), and analytic stochastic modeling (BO and L approaches), this work demonstrates that dynamic hyperfine interactions in the ECAE regime arise from and sensitively probe the kinetics and energetics of transient electronic configurations during post-decay electronic recovery. The key conclusions and implications:

• Both models yield equivalent physical parameters (hole lifetime, EFG distribution spread) under standard experimental conditions, allowing a robust extraction of dynamic rates and distribution statistics from TDPAC data.
• The energetic analysis tightly connects observed static and dynamic hyperfine interactions to specific charge states of the probe impurity, validating the scenario in which HFI$_d$ is realized when a significant fraction of probes stabilize in incompletely filled configurations due to electron scarcity.
• The methodological framework supports detailed, temperature-resolved monitoring of defect and charge-state dynamics, offering a high-precision probe of local conductivity, electronic transport, and defect recovery processes in wide-gap oxides and semiconductors.

This comprehensive, integral approach to dynamic HFIs is broadly applicable across materials where nuclear decay aftereffects induce transient electronic environments, providing a quantitative pathway to relate nuclear spectroscopic observations with atomic-scale electronic structure and defect physics (Darriba et al., 3 Sep 2025).