Performances of the Lamb Model to Describe the Vibrations of Gold Quantum-Sized Clusters

Abstract

Lamb modes describe the vibrations of an object as a whole from the stellar scale to the nanometer one. Lamb description has been built from the linear elasticity theory and considers a homogeneous elastic sphere. Our work tries to determine the minimum scale where this description remains valid by studying the vibration of quantum sized gold clusters (Au6, Au9 and Au25) stabilized by organic molecules. First, our work shows that experimental frequencies of small functionalized gold clusters obtained using Low Frequency Raman Spectroscopy (LFRS) can be interpreted with density-functional theory (DFT) calculations. Moreover, the Lamb model broadly succeeds in predicting these Raman acoustic modes only if a correction considering the mass of the surrounding ligands is added. Ligands affect vibration modes of the core by their mass but also by their covalent bond with the core. The unexpected consequence of this electronic stabilization by the ligands is the sustainability of the Lamb description for clusters as small as 6 atoms. Finally, the limit of the Lamb model can be reached out at low temperature, where the vibration modes spectrum presents a sub-structuration that the Lamb description, developed for a homogeneous sphere, is unable to predict.

Ligand-dependent nano-mechanical properties of CdSe nanoplatelets: calibrating nanobalances for ligand affinity monitoring

“…First, the size of the nano-object must be small to observe a measurable effect of the ligands. 13,15,18 Second, the study of small nanoparticles requires to study the assemblies of nanoparticles presenting a small size and shape dispersion. Indeed, the width of the measured low frequency Raman peak is rapidly enlarged with these morphological dispersion parameters 19 and the effect of the ligands might be obscured.…”

Raman Spectroscopic Fingerprints of Atomically Precise Ligand Protected Noble Metal Clusters: Au38(PET)24 and Au38−xAgx(PET)24

“…However, these vibrations in general correspond to the bending modes of the staples (the band at around 67 cm −1 , see Figure S12, Supporting Information) and stretching of some of AuAu bonds (the band at around 70 cm −1 , see Figure S12, Supporting Information) in the core. Therefore, we cannot attribute all of these low frequency bands to the core alone in contrary to the previous work on Au 25 (SR) 18 clusters, [ 28 ] wherein ligands were considered only as a “modifier” of these bands. Our results indicate that the low frequency vibrations are altered by metal atoms and the ligands.…”

“…[26] Recently, Kato et al, and Martinet et al, observed the low frequency vibrations of Au 8 and Au 25 (SR) 18 clusters using Raman spectroscopy. [27,28] However, vibrational characteristics of the structurally diverse metal cores in ligand-protected metal clusters remain poorly understood.…”

A Review of State of the Art in Phosphine Ligated Gold Clusters and Application in Catalysis

“…The mass load model approximation is added to Lamb's system of equations to identify breathing‐like and quadrupolar‐like modes for small Au clusters. [ 333 ] Their results were compared to DFT and previous experimental work. [ 313 ] Only recently, Kato et al reported the terahertz Raman spectrum of [Au 8 X 2 (dppp) 4 ] 2+ in the gold cluster fingerprint region from 50 to 150 cm −1 .…”