Cumulant expansion approach to the decay dynamics of interacting Mössbauer nuclei after strong impulsive excitation

Abstract

Recent progress in accelerator-based x-ray sources brings higher excitation of ensembles of Mössbauer nuclei closer to experimental feasibility. Yet, a theoretical modeling of the decay dynamics of the interacting nuclear ensemble after the impulsive excitation is still an open challenge. Here, we derive a set of nonlinear equations which is capable of efficiently modeling large nuclear ensembles for arbitrary degrees of excitation. As key signature for higher excitation, we identify a non-linear time-evolution of the nuclear dipole phase, which can be tuned via the scattering geometry, and interferometrically be measured. Furthermore, we identify interesting finite-size effects in the nuclear dynamics of small ensembles. Our results provide important guidance for future experiments aiming at the non-linear excitation of nuclei. We further envision the exploration of finite size-effects in Mössbauer spectroscopy with highest spatial resolution, i.e., small sample volumes.

Figures (7)

• Figure 1: Coupling parameter $K=K^R+i K^I$ as function of the incident angle $\theta_\mathrm{in}$. For the calculation, Eq. (\ref{['eq:K_infiniteLinearChain']}) is used with dipole orientation $\theta_d=\pi/2$.
• Figure 2: Nuclear dipole phase evolution as function of incidence angle and degree of excitation. For a fixed incident angle $\theta_\mathrm{in}=5\,$mrad the different colors of the solid lines indicate different degrees of initial excitation $\mathcal{A}$. The transition to a non-linear phase evolution is clearly visible. For comparison, the shaded lines show the results for the central nucleus calculated on a finite chain with $N=3000$ nuclei. In addition, in the low-excitation regime $\mathcal{A}=10^{-5}\pi$, the different line styles of the blue lines show the results for different incident light angles $\theta_\mathrm{in}$. The dipole moment is set perpendicular to the chain of nuclei ($\theta_d=\pi/2$).
• Figure 3: Time-dependent intensity of the x-ray scattered by two nuclear chains. The reference chain has a detuning of the nuclear resonance of $\Delta=-3\Gamma$., such that the interference between the chains leads to pronounced minima in the intensity. The non-linear phase evolution can then be observed via shifts of these minima in time. The different lines illustrate the dependence on the different degrees of excitation $\mathcal{A}$. For the sample chain the dipole moment is set perpendicular to the chain of nuclei ($\theta_d=\pi/2$) and the x-ray incidence angles for the sample [reference] chain is $5\,$mrad [0.22 rad]. The inset shows the signal intensity as a function of time over a larger time interval. The gray area marks the time region shown in the main panel.
• Figure 4: Comparison of the coupling parameter $K_l$ being evaluated on the finite chain according to Eq. (\ref{['eq:K_finite']}) (solid) with that of the infinite chain according to Eq. (\ref{['eq:K_infiniteLinearChain']}) (dashed). In (a) the coupling parameter $K$ evaluated at the central nucleus $l=N/2$ is shown as a function of the number of nuclei $N$. In (b) for a fixed number of nuclei $N=500$ the coupling parameter is displayed as function of the nucleus' position in the chain $l$. The calculations are performed for dipole moment perpendicular to the chain ($\theta_d=\pi/2$) and a small incident light angle of $\theta_\mathrm{in}=50$ mrad.
• Figure 5: Comparison of the phase evolution computed with a finite chain evaluated at the central nucleus (dashed) and the translational invariant model (dotted). In (a) the chain has a length of $N=362$ nuclei, which corresponds to a point of maximal deviation of $K$ in Fig. \ref{['fig:convergenceOfK']}, in (b) a length of $N=257$ nuclei corresponding to a point of minimal deviation in $K$ is chosen. The different colors indicate different initial excitations $\mathcal{A}$ according to the respective legend. The calculations are performed for dipole moment perpendicular to the chain ($\theta_d=\pi/2$) and a small incident light angle of $\theta_\mathrm{in}=50$ mrad.