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Collective Phonon Mixing and Eigenvector Transport Under Isotope Substitution

Jeff Armstrong, Hamish Cavaye, Pankaj Sharma, Matthew E. Potter

Abstract

Isotopic substitution modifies nuclear masses without altering the electronic potential energy surface to first order and is therefore often interpreted as a simple rescaling of vibrational frequencies. In solids with dense phonon manifolds, however, mass substitution acts as a parametric Hermitian deformation of the mass-weighted dynamical matrix, generating a continuous family of eigenproblems whose eigenvectors can undergo substantial rotation within coupled subspaces. Here we investigate protiated and deuterated ZIF-8 using inelastic neutron scattering and density functional theory lattice-dynamics calculations. While many vibrational modes exhibit near-ideal mass scaling and preserve their character across isotopic endpoints, modes embedded in spectrally congested regions display pronounced redistribution of vibrational character that cannot be inferred from frequency shifts alone. Because inelastic neutron scattering intensity is directly weighted by hydrogen displacement amplitude, spectral sparsity and congestion provide experimental indicators of predictable frequency renormalisation or susceptibility to qualitative eigenvector reorganisation under deuteration. To establish physically meaningful mode correspondence, we develop an adiabatic eigenvector-continuation framework with overlap-based tracking and explicit stability diagnostics. These results show that vibrational identity in complex framework materials is best understood as a continuous trajectory in eigenvector space and provide a general framework for analysing isotope-induced spectral flow in dense phonon systems.

Collective Phonon Mixing and Eigenvector Transport Under Isotope Substitution

Abstract

Isotopic substitution modifies nuclear masses without altering the electronic potential energy surface to first order and is therefore often interpreted as a simple rescaling of vibrational frequencies. In solids with dense phonon manifolds, however, mass substitution acts as a parametric Hermitian deformation of the mass-weighted dynamical matrix, generating a continuous family of eigenproblems whose eigenvectors can undergo substantial rotation within coupled subspaces. Here we investigate protiated and deuterated ZIF-8 using inelastic neutron scattering and density functional theory lattice-dynamics calculations. While many vibrational modes exhibit near-ideal mass scaling and preserve their character across isotopic endpoints, modes embedded in spectrally congested regions display pronounced redistribution of vibrational character that cannot be inferred from frequency shifts alone. Because inelastic neutron scattering intensity is directly weighted by hydrogen displacement amplitude, spectral sparsity and congestion provide experimental indicators of predictable frequency renormalisation or susceptibility to qualitative eigenvector reorganisation under deuteration. To establish physically meaningful mode correspondence, we develop an adiabatic eigenvector-continuation framework with overlap-based tracking and explicit stability diagnostics. These results show that vibrational identity in complex framework materials is best understood as a continuous trajectory in eigenvector space and provide a general framework for analysing isotope-induced spectral flow in dense phonon systems.
Paper Structure (24 sections, 52 equations, 9 figures)

This paper contains 24 sections, 52 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. Inelastic Neutron Spectroscopy Measurements
  4. Density functional theory calculations
  5. Results
  6. Experimental validation of Lattice Dynamics and direct observations
  7. Direct comparison of H and D eigenvectors
  8. Perturbative interpretation of frequency shifts, mixing, and spectral intensity
  9. Adiabatic eigenvector continuation under mass deformation
  10. Continuity Diagnostic Results
  11. Why a two-level ($2\times 2$) crossing picture is insufficient in dense manifolds
  12. Collective coupling, nonlocal mixing, and multi-band collisions
  13. Conclusions and predictive outlook
  14. Acknowledgments
  15. Derivation of main equations
  16. ...and 9 more sections

Figures (9)

  • Figure 1: Experimental INS spectra (top) and neutron-weighted calculated spectra (bottom). Protiated ZIF-8 is shown in black and deuterated ZIF-8 in blue.
  • Figure 2: Mass-weighted overlap map for modes between 300 and 1200 cm$^{-1}$. $O_{ij}$ is represented by the color axis of the plot. Only pairs with $O_{ij} > 0.45$ are shown. Fundamental mode neutron projected DFT spectra for H and D are shown on the x and y axis respectively to aid visual band assignment.
  • Figure 3: Mass-weighted displacement patterns for representative modes. Panels a), b) & c) show protiated ZIF-8 modes at 416, 639, and 734 cm$^{-1}$, respectively. Panels d), e) & f) show the corresponding deuterated modes at 374, 546, and 553 cm$^{-1}$.
  • Figure 4: Normalized probability distributions $P(\Delta O)$ for interpolation grids $N_s = 100$, $1000$ and $10000$. Histograms are shown on a logarithmic scale to resolve the low-gap tail. The red bin marks the $\sim$0.1st percentile.
  • Figure 5: Grid-resolution dependence of spurious mode crossings and overlap robustness. The number of detected crossings (red, left axis) decreases with increasing grid density $N_s$, while the $0.1^{\mathrm{st}}$ percentile of $P(\Delta O)$ (blue, right axis) increases toward unity, indicating improved continuity in mode tracking. Solid lines show empirical fits to guide the eye.
  • ...and 4 more figures