Rapid Determination of Nanodiamond Size Distribution and Impurity Concentration from Raman Spectra Using an Open Machine-Learning Toolbox

TL;DR

This work addresses the inverse problem of determining nanodiamond size distributions and impurity concentrations from Raman spectra by introducing an open, physics-grounded toolbox. A forward model combines size quantization, a microscopic dispersion via the Keating framework, and explicit impurity/disorder effects to synthesize spectra for arbitrary size distributions. It offers two inverse strategies: a neural-network regression trained on synthetic data and a Metropolis-style stochastic refinement, both validated on experimental spectra and showing concordant results with independent measurements. The open-source toolbox, including background-subtraction functionality, enables robust nanodiamond characterization in the 2–8 nm range and provides a practical platform for Raman spectrum inversion in diverse synthesis contexts.

Abstract

Ready-to-use numerical toolbox for nanodiamond Raman spectra calculation and fit is presented. The developed theoretical approach allows accounting for arbitrary nanoparticle size-distribution and the microscopic line broadening mechanisms for the optical phonons. The two tools for solving the inverse problem of the nanodiamond properties reconstruction using a known Raman spectrum are provided. The first one utilizes a dense neural network trained on a vast array of synthetic Raman spectra. The second approach is based on the stochastic Metropolis algorithm, which updates the ensemble parameters by small quantities, tending to the state with minimal error. Both methods are available thanks to the computationally instant elasticity theory-like model for optical phonon modes in diamond nanocrystals that accurately reproduces the results of the atomistic approaches. Using experimental Raman spectra for nanodiamonds prepared by various techniques, we tested our tools and observed a faithful agreement with the data as well as between the two methods. The open and documented software is accessible online (nanoraman.pythonanywhere.com) and as a Python module (github.com/KoniakhinSV/Nanoparticle_Raman).

Figures (6)

• Figure 1: Panel (a) shows Raman spectrum vector $\mathbf{X}$ derived using continuous bond polarization model and disorder theory for nanodiamond Raman spectra based on the size distribution and defect parameters encoded in $\mathbf{Y}$. (b) Nanoparticle size histogram is encoded in the first 55 bins (red dots). The disorder concentration is $C_{\rm imp} \approx 2.8$ (encoded as 0.2 times the largest element in green dot one-hot vector) and the intrinsic broadening $\Gamma_0 = 4.4$ cm$^{-1}$ (encoded as 1.28 to the power of the largest element in blue dot one-hot vector).
• Figure 2: Architecture of the neural network that was shown to efficiently encode the multidimensional function which maps from the Raman spectrum to the corresponding nanodiamond size distribution histogram and defect parameters.
• Figure 3: Schematics of Metropolis spectrum fitting procedure. Panel a) shows the current size distribution updated by a correction with the trial distribution as a result. Panel b) shows the spectra simulated for the current size distribution (and also $c_\text{imp}$ and $\Gamma$ parameters) and for the trial one. The trial correction is declined by the algorithm because it results in a worse spectrum fit.
• Figure 4: Result of the fits for nanodiamond samples reported in Ref. shenderova2011nitrogen. Panels a),b),c) are for the Raman spectra, and panels e),f),g) are for the reconstructed size distributions of detonation synthesis nanodiamond samples ND-TNT/RDXw, ND-TNT/RDXd, and ND-TNT/HNSw of typical size 4-5 nm, correspondingly. Histograms are given by the number of particles. Panels d) and h) show the Raman spectrum and the histogram of the sample ND-G/RDX40 obtained from a graphite/RDX mixture and having a size of 9 nm. For spectra (panels a-d), the black curves are the spectra, and the dotted green curves are the original spectra before background subtraction refinement. The blue dashed curves are the Raman spectra reconstructed from the neural network prediction, and the red dashed curves are for those obtained from the Metropolis approach. In the histograms (panels e-h), red bars give the predictions for the Metropolis (M) approach, and blue ones are for the neural network (NN) approach. Gray bars with dashed contours are for the unreliable parts of the neural network-based histograms (see text).
• Figure 5: Here, the developed software is benchmarked on samples of various origins. Panels a) and d) are for the Raman spectra and reconstructed size distribution, respectively, for the 4.3 nm detonation nanodiamonds studied in Ref. yoshikawa1995raman. Panels b) and e) are for the Raman spectra and reconstructed size distribution of the adamantane HPHT sample, reported in Ref. kudryavtsev2023raman. Finally, the Raman spectrum c) of the MSY18-O1 sample from Ref. stehlik2015size is analyzed with the respective size distribution shown in f). For the Metropolis fit, the defect concentration was manually set to zero, which resulted in the shift of the respective size histogram (orange bars in the histogram) to a smaller size, better coinciding with the HRTEM data. The rest of the notation follows that used in Fig. \ref{['fig_exp_shend']}.