Mean Square Displacement Values Research Articles

Encapsulation of catechin derivatives in single-walled carbon nanotubes

This study explores the encapsulation dynamics and interaction mechanisms of catechin derivatives within single-walled carbon nanotubes (SWNTs) using molecular dynamics simulations. We assessed the strength of π-π stacking interactions between catechin derivatives and SWNTs by analyzing the distance evolution between aromatic rings, mean square displacement (MSD), and diffusion coefficient (D) values. The results revealed that interaction strength varied depending on the specific derivative and concentration. At low and medium concentrations, weaker interactions were observed for catechin, epicatechin, and epigallocatechin, while stronger interactions were found for epicatechin gallate and epigallocatechin gallate. Furthermore, MSD values indicated normal diffusion trends for catechin, epicatechin, and epigallocatechin, suggesting free movement within SWNTs. In contrast, sub-diffusion trends were observed for epicatechin gallate and epigallocatechin gallate, indicating limited mobility. This study offers insights into the interactions of catechin derivatives with SWNTs, enriching our understanding of supermolecular structure development and enabling potential applications in drug delivery, and biosensors.

Molecular Dynamics Simulation to Evaluate the Stability of Tetra-n-butyl Ammonium/Phosphonium Bromide Semiclathrate Hydrates in the Presence and Absence of Methane, Carbon Dioxide, Methanol, and Ethanol Molecules

Stability of tetra-n-butyl ammonium bromide (TBAB) and tetra-n-butyl phosphonium bromide (TBPB) semiclathrate hydrates with a hydration number of 38 is studied by the molecular dynamics (MD) simulation in the absence and presence of methane, carbon dioxide, methanol, and ethanol molecules. All of the simulation runs are performed under NVT (constant number of atoms, volume, and temperature) and NPT (constant number of atoms, pressure, and temperature) conditions using optimized potentials for liquid simulations-all-atom (OPLS-AA) force field with nonpolarizable water models (including SPC, TIP3P, TIP4P, and TIP4P/Ice), where the operating conditions include a temperature range of 250–350 K and pressures of 0.1 and 50 MPa. The potential energy and structural analysis, mean square displacement (MSD), diffusion coefficient, radial distribution function (RDF), lattice parameter, and the average number of hydrogen bonds between water–water molecules are evaluated by employing MD strategy. According to the potential energy results, the order for the water models in both TBAB and TBPB hydrate stabilities is as follows: TIP4P/Ice > TIP4P ≈ SPC > TIP3P. This order agrees well with the previous studies of gas hydrates and ices. In the absence and presence of methane and carbon dioxide guest molecules, the height of the peaks in the RDF of oxygen–oxygen atoms decreases with lowering pressure and increasing temperature; greater MSD, diffusion coefficient, and lattice parameter values are also achieved. Thus, in the absence and/or presence of guest gas molecules, the stability of TBAB and TBPB hydrate cages decreases with increasing temperature and reducing pressure. A good match is noticed between the results of MD simulation and previous experimental and theoretical studies, confirming the accuracy and validation of the simulation method. Reduction of the hydrogen bond balance between water molecules and formation of new hydrogen bonds between the water and the hydroxyl groups of methanol and/or ethanol molecules in the cages indicate that methanol and ethanol molecules have an inhibitory effect on TBAB and TBPB hydrates. Due to different hydrophilic versus hydrophobic behaviors of alcohols, the extent of disruption of hydrogen bonds between water–water molecules of TBAB and TBPB hydrates in the presence of methanol is larger than that in the presence of ethanol, which is consistent with the MD simulation results of clathrate hydrate. This study can help to further understand the semiclathrate hydrate stability or dissociation conditions at a molecular scale for better design and operation of corresponding processes.

Exploring the Utility of Diffusing Wave Spectroscopy (DWS) as a Novel Tool for Early Detection of Stability Issues in Cosmetic Emulsions

In the current cosmetic and personal care industry, it is of great importance to have a technique that detects instabilities quickly and effectively, as consumers are demanding more innovative and sustainable ingredients. Diffusing wave spectroscopy (DWS) is a potential solution as it is a modern optical technique that can measure the spatial movement of particles or droplets in an emulsion—i.e., the mean square displacement (MSD), as a function of time. In the current investigation, systematic visual and diffusion behavior emulsion stability studies over a 3-h period on jojoba and avocado oil emulsions containing varying equal percentages of cocamidopropyl betaine (CAPB) and sodium lauryl ether sulfate (SLES) were conducted. The turbid emulsions studied had differing stabilities with unknown instability mechanisms to further explore if diffusing wave spectroscopy can offer a fast and early identification of problem cosmetic formulations. It was observed that, for emulsions displaying instability from 4 to 123 h, the greater the change in the MSD values over a 3-h period, the greater the instability of the emulsion. For all systems, the MSD values lowered and shifted to the right from hour 0 to hour 3. We conjecture that the emulsion droplets began to aggregate, potentially growing and giving rise to larger particles. The increasing particle size was the cause for the slowing down of the dynamics and thus diffusion, giving rise to the lowering of the MSD values. Our findings indicate that by testing an emulsion over a 3-h period, it is possible to determine whether it will be a problem formulation using DWS. Studies into this technology should be continued on a wider range of emulsions with known instability mechanisms to further our understanding of using DWS as a vital emulsion instability detector.

Interaction between Graphene-Based Materials and Small Ag, Cu, and CuO Clusters: A Molecular Dynamics Study.

The excessive use of antibiotics has contributed to the rise in antibiotic-resistant bacteria, and thus, new antibacterial compounds must be developed. Composite materials based on graphene and its derivatives doped with metallic and metallic oxide nanoparticles, particularly Ag, Cu, and Cu oxides, hold great promise. These materials are often modified with polyethylene glycol (PEG) to improve their pharmacokinetic behavior and their solubility in biological media. In this work, we performed molecular dynamics (MD) simulations to study the interaction between small Ag, Cu, and CuO clusters and several graphene-based materials. These materials include pristine graphene (PG) and pristine graphene nanoplatelets (PGN) as well as PEGylated graphene oxide (GO_PEG) and PEGylated graphene oxide nanoplatelets (GO-PEG_N). We calculated the adsorption energies, mean equilibrium distances between the nanoparticles and graphene surfaces, and mean square displacement (MSD) of the nanoclusters. The results show that PEGylation favors the adsorption of the clusters on the graphene surfaces, causing an increase in adsorption energies and a decrease in both distances and MSD values. The strengthening of the interaction could be crucial to obtain effective antibacterial compounds.

Mean Apparent Propagator MRI Is Better Than Conventional Diffusion Tensor Imaging for the Evaluation of Parkinson's Disease: A Prospective Pilot Study.

Background and PurposeMean apparent propagator (MAP) MRI is a novel diffusion imaging method to map tissue microstructure. The purpose of this study was to evaluate the diagnostic value of the MAP MRI in Parkinson’s disease (PD) in comparison with conventional diffusion tensor imaging (DTI).Methods23 PD patients and 22 age- and gender-matched healthy controls were included. MAP MRI and DTI were performed on a 3T MR scanner with a 20-channel head coil. The MAP metrics including mean square displacement (MSD), return to the origin probability (RTOP), return to the axis probability (RTAP), and return to the plane probability (RTPP), and DTI metrics including fractional anisotropy (FA), and mean diffusivity (MD), were measured in subcortical gray matter and compared between the two groups. The receiver operating characteristic (ROC) curve was used to analyze the diagnostic performance of all the metrics. The association between the diffusion metrics and disease severity was assessed by Pearson correlation analysis.ResultsFor MAP MRI, the mean values of MSD in the bilateral caudate, pallidum, putamen, thalamus and substantia nigra (SN) were higher in PD patients than in healthy controls (pFDR ≤ 0.001); the mean values of the zero displacement probabilities (RTOP, RTAP, and RTPP) in the bilateral caudate, pallidum, putamen and thalamus were lower in PD patients (pFDR < 0.001). For DTI, only FA in the bilateral SN was significantly higher in PD patients than those in the controls (pFDR < 0.001). ROC analysis showed that the areas under the curves of MAP MRI metrics (MSD, RTOP, RTAP, and RTPP) in the bilateral caudate, pallidum, putamen and thalamus (range, 0.85–0.94) were greater than those of FA and MD of DTI (range, 0.55–0.69) in discriminating between PD patients and healthy controls. RTAP in the ipsilateral pallidum (r = −0.56, pFDR = 0.027), RTOP in the bilateral and contralateral putamen (r = −0.58, pFDR = 0.019; r = −0.57, pFDR = 0.024) were negatively correlated with UPDRS III motor scores.ConclusionMAP MRI outperformed the conventional DTI in the diagnosis of PD and evaluation of the disease severity.

Gelation mechanism and network structure of mixed kappa carrageenan/lambda carrageenan gels studied by macroscopic and microscopic observation methods

Dynamic rheology and particle tracking experiments were carried out to investigate the gelation mechanism and network structure of mixed κ-carrageenan (KC) and λ-carrageenan (LC) gels on the macroscopic and microscopic levels. Mixtures of KC and LC (total carrageenan concentration = 1.5%, K+ concentration = 15 mM) were prepared at different mixing ratios. The storage modulus (G′) was highest for pure KC and decreased monotonously with decreasing KC content. The mean square displacement (MSD) of probe particles was used to characterize the network on cooling and after one day of storage. On cooling, a single step decrease in the ensemble-averaged MSD was observed, in agreement with the rheological measurements. Time-averaged MSD values of individual particles showed a large distribution after one day in storage, especially for gels with low content of KC. This heterogeneity reflects microstructural heterogeneity, suggesting phase separation into KC-rich and LC-rich domains. Individual particle MSD values showed a bimodal distribution. The MSD could be described as a power-law, MSD~τα, with τ the lag time. The exponent α also showed a bimodal distribution. These findings suggest that two groups of probe particles with distinctly different mobilities are present in the mixed gels with particles of higher and lower mobilities present in the LC-rich and KC-rich phases. Based on the rheological results, as well as the particle tracking experiments, we propose that mixtures of KC and LC separate into a continuous phase and a non-continuous filler phase, with the continuous phase being made up of the more concentrated carrageenan. When the concentrations of the two carrageenans are roughly equal, we suggest that a bicontinuous phase-separated network is formed.

Approach to the inverse problem of superdiffusion on finite systems based on time-dependent long-range navigation.

This work addresses the superdiffusive motion of a random walker on a discrete finite-size substrate. It is shown that, with the inclusion of suitably tuned time-dependent probability of large distance jumps over the substrate, the mean square displacement (MSD) of the walker has a power-law dependence on time with a previously chosen exponent γ>1. The developed framework provides an exact solution to the inverse problem, i.e., an adequate jump probability function leading to a preestablished solution is evaluated. Using the Markov Chain (MC) formalism, an exact map for the time dependence of the probability function is derived, which depends on the topology of the substrate and on the chosen value of γ. While the formalism imposes no restriction on the substrate, being applicable from ordered Euclidean lattices to complex networks, results for the cycle graph and two-dimensional torus are highlighted. It is also shown that, based on the previously derived probability function, MSD values resulting from direct numerical simulations agree quite well with those solely obtained within the MC framework.

Structure, microparameters and properties of crosslinked DGEBA/MTHPA: A molecular dynamics simulation

Investigating the relationship between microstructure and macroscopic properties of epoxy resin (EP) materials for high-voltage insulation at the molecular level can provide theoretical guidance for the synthetic design of EP. Here, using diglycidyl ether (DGEBA) as the resin matrix and methyl tetrahydrophthalic anhydride (MTHPA) as the curing agent, a set of crosslinked EP molecular models at different curing stages were constructed based on the proposed crosslinking method. We studied the influences of crosslinking density on micro-parameters and macro-properties employing molecular dynamics (MD) simulations. The results indicate that crosslinking of DGEBA/MTHPA is a contraction and exothermic process. The structural parameters and macroscopic properties are closely related to the degree of crosslinking. With the increase of crosslinking density, the mean square displacement (MSD) of the system decreases, and the segment motion in the models is weakened gradually, while, the fractional free volume (FFV) first decreases and then increases. In addition, the thermal and mechanical properties of DGEBA/MTHPA have a significant dependence on the crosslinking density. Increasing crosslinking density can improve the glass transition temperature (Tg), reduce the coefficient of thermal expansion (CTE), and enhances the static mechanical properties of DGEBA/MTHPA system. Furthermore, the relationship between microparameters and properties has been fully investigated. Free volume is an important factor that causes thermal expansion of DGEBA/MTHPA. Moreover, there is a negative correlation between MSD and mechanical moduli. By elevating temperature, the decline in mechanical moduli may be due to the exacerbated thermal motion of the molecules and the increasing MSD values.

Application of Optical Tweezers to Understand the Effect of Renal Ciliary Length Alterations on Ciliary Biomechanics

A renal primary cilium act as a cellular mechanosensor. Fluid flow causes bending of renal cilia resulting in the increased intracellular concentration calcium ion, a second messenger which initiates downstream cell signaling pathways. Better understanding of ciliary biomechanics can help to interpret how ciliary mechanotransduction may be impaired in ciliopathies, such as polycystic kidney disease. Fluid flow in kidney tubules can generate three separate forces: shear and stretch acting on renal tubule wall or epithelial cell layers, and hydrodynamic drag which acts on protruding structures i.e., primary cilia. As compared to direct fluid flow experiments, optical trapping method has the advantage that it can isolate and evaluate only the effects of hydrodynamic drag on ciliary biomechanics, avoiding any background effects of shear and stretch forces acting on cellular layers. Optical tweezers technique is non‐invasive and non‐contact measurement and allow accurate measurement due to the ability to control a trapped primary cilium at a time. Thus, using optical tweezers biomechanical properties of an individual cilium can be correlated with bio‐physiological features of the cilium, such as length and spatial distribution.As reported previously by other research groups, kidney transplantation and ischemia‐reperfusion injury can affect renal ciliary lengths. Previous cell culture studies have shown presence or absence of fluid flow affects ciliary lengths of mouse kidney cortical duct epithelial cells. Stabilization of hypoxia inducible factor (HIF) 1α by cobalt chloride (CoCl2) can affect renal ciliary lengths [1] and can also alter bending modulus of the cilia [2]. We aim to understand the significance of altered ciliary length on the kidney cellular function and diseases. In this study, using optical trapping method we aim to understand if ciliary lengths are related to ciliary biomechanics and flow sensing. We used Mardin‐Darby Canine Kidney (MDCK) epithelial cells which were differentiated by serum starvation to generate primary cilia. We determined the mean square displacement (MSD) values of optically trapped cilia and correlated with ciliary lengths. The MSD value measures the displacement of a particle (i.e., tip of a cilium) from the initial position and long‐time section of MSD relies on the spring constant. MSD values can represent ability of cilia to deflect in response to fluid flow. Currently, we are trying to manipulate ciliary lengths using HIF stabilizer CoCl2 in the presence or absence of HIF inhibitors: chetomin and echinomycin. We anticipate that chetomin or echinomycin may restrict alterations of ciliary length by CoCl2 and may maintain the MSD value of cilia to a base level i.e., without any HIF stabilization. These findings can establish a direct connection between the ciliary length and ciliary flow‐sensing. Thus, this study aims to test whether renal epithelial ciliary lengths are altered in order to adjust the ciliary biomechanical properties or flow‐sensing function as a cellular response to adapt adverse physiological conditions, such as lack of fluid flow and localized hypoxia.This abstract is from the Experimental Biology 2018 Meeting. There is no full text article associated with this abstract published in The FASEB Journal.

Unprecedented mechanical response of the lattice thermal conductivity of auxetic carbon crystals

Lattice thermal conductivity (κ) of bulk materials usually increases under compression and decreases under tension, while there are still some unusual systems, exhibiting reduced κ when compressed. However, to date it has never been reported for a bulk material, whose κ is substantially enhanced under tensile strain. In this paper, we have studied thermal transport of three auxetic carbon crystals: cis-C, trans-C and hin-C for short, and their strain responses by performing first-principles calculations. It is intriguing to find that their κ are much lower than those of their allotropes, and further decrease abnormally under compression. More strikingly, κ of trans-C (cis-C) anomalously increases with tensile strain up to 7% (6%) with maximum κ of almost 7 (5) times larger than the unstrained value. The abnormal strain dependent κ are attributed to the dominant role of the enhancement of phonon lifetime under stretching, which can be further explained from the unique atomic structure of the main chain of polydiacetylene in trans-C and cis-C. The weakening of phonon anharmonicity is reflected by the enhancement of root mean-square displacement values. The reported giant augmentation of κ may inspire intensive research on auxetic carbon crystals as potential materials for emerging nanoelectronic devices.

Breakdown of universal Lindemann criterion in the melting of Lennard-Jones polydisperse solids

It is commonly believed that melting occurs when mean square displacement (MSD) of a particle of crystalline solid exceeds a threshold value. This is known as the Lindemann criterion, first introduced in the year of 1910 by Lindemann. However, Chakravarty et al., demonstrated that this common wisdom is inadequate because the MSD at melting can be temperature dependent when pressure is also allowed to vary along the coexistence line of the phase diagram [Chakravarty C, Debenedetti P G and Stillinger F H 2007 J. Chem. Phys. 126 204508]. We show here by extensive molecular dynamics simulation of both two and three dimensional polydisperse Lennard-Jones solids that particles on the small and large limits of size distribution exhibit substantially different Lindemann ratio at melting. Despite all the dispersion in MSD, melting is found to be first order in both the dimensions at 5–10% dispersity in size. Sharpness of the transition is incommensurate with the different rate of growth of MSD. The increased MSD values of smaller particles play a role in the segregation of them prior to melting.

Glycerol revisited molecular dynamic simulations of structural, dynamical, and thermodynamic properties

We performed molecular dynamics simulations to investigate the properties of glycerol for a wide range of temperatures at standard pressure. We calculated structural (radial distribution functions and pair potential of mean force), dynamical (mean square displacement and transport properties), and thermodynamic (density, thermal expansion, and Hildebrand solubility parameter) properties of glycerol. The results of structural properties showed that the correlation between glycerol atoms weakens as temperature increases. The values of mean square displacement showed that changing temperature has a strong influence on mobility of glycerol atoms. The values of diffusion coefficient and viscosity are remarkably close to the experimental values over the whole range of temperatures studied. The simulation results provide a reasonable estimation of density with percent error of 0.40 %. The simulated values of Hildebrand solubility parameter of glycerol decrease with raising temperature because the cohesive forces weaken. To the best of our knowledge, this work for the first time calculates the potential of mean force, viscosity, and Hildebrand solubility parameter of glycerol by MD simulation.

Dynamics of the Peripheral Membrane Protein P2 from Human Myelin Measured by Neutron Scattering—A Comparison between Wild-Type Protein and a Hinge Mutant

Myelin protein P2 is a fatty acid-binding structural component of the myelin sheath in the peripheral nervous system, and its function is related to its membrane binding capacity. Here, the link between P2 protein dynamics and structure and function was studied using elastic incoherent neutron scattering (EINS). The P38G mutation, at the hinge between the β barrel and the α-helical lid, increased the lipid stacking capacity of human P2 in vitro, and the mutated protein was also functional in cultured cells. The P38G mutation did not change the overall structure of the protein. For a deeper insight into P2 structure-function relationships, information on protein dynamics in the 10 ps to 1 ns time scale was obtained using EINS. Values of mean square displacements mainly from protein H atoms were extracted for wild-type P2 and the P38G mutant and compared. Our results show that at physiological temperatures, the P38G mutant is more dynamic than the wild-type P2 protein, especially on a slow 1-ns time scale. Molecular dynamics simulations confirmed the enhanced dynamics of the mutant variant, especially within the portal region in the presence of bound fatty acid. The increased softness of the hinge mutant of human myelin P2 protein is likely related to an enhanced flexibility of the portal region of this fatty acid-binding protein, as well as to its interactions with the lipid bilayer surface requiring conformational adaptations.

Relation between dynamics, activity and thermal stability within the cholinesterase family

Incoherent neutron scattering is one of the most powerful tools for studying dynamics in biological matter. Using the cold neutron backscattering spectrometer IN16 at the Institut Laue Langevin (ILL, Grenoble, France), temperature dependence of cholinesterases’ dynamics (human butyrylcholinesterase from plasma: hBChE; recombinant human acetylcholinesterase: hAChE and recombinant mouse acetylcholinesterase: mAChE) was examined using elastic incoherent neutron scattering (EINS). The dynamics was characterized by the averaged atomic mean square displacement (MSD), associated with the sample flexibility at a given temperature. We found MSD values of hAChE above the dynamical transition temperature (around 200K) larger than for mAChE and hBChE, implying that hAChE is more flexible than the other ChEs. Activation energies for thermodynamical transition were extracted through the frequency window model (FWM) (Becker et al. 2004) [1] and turned out to increase from hBChE to mAChE and finally to hAChE, inversely to the MSDs relations. Between 280 and 316K, catalytic studies of these enzymes were carried out using thiocholine esters: at the same temperature, the hAChE activity was systematically higher than the mAChE or hBChE ones. Our results thus suggest a strong correlation between dynamics and activity within the ChE family. We also studied and compared the ChEs thermal inactivation kinetics. Here, no direct correlation with the dynamics was observed, thus suggesting that relations between enzyme dynamics and catalytic stability are more complex. Finally, the possible relation between flexibility and protein ability to grow in crystals is discussed.

Raman scattering of rare earth hexaborides

Raman scattering spectra were measured for the rare-earth hexaborides RB6 (R = Ce, Gd, or Dy). All Raman-active phonons due to B6 vibrations were observed in the range 600 – 1400 cm-1. Anomalous peaks were detected below 200 cm-1, which correspond to vibrations of rare-earth ion excited by second-order Raman scattering process. The intensity and energy of the rare-earth mode decrease with decreasing temperature. This suggests that the rare-earth ion vibrates in a shallow and anharmonic potential due to the boron cage. Using the reported values of mean square displacement of rare-earth ion, we estimated the anharmonic contribution for the rare-earth vibrations.

Molecular Dynamics Simulation of Diffusion Behavior of Benzene/Water in PDMS-Calix[4]arene Hybrid Pervaporation Membranes

Molecular dynamics (MD) simulation was employed to investigate diffusion behavior of small penetrants in rubbery-polymer-based hybrid membranes, using pervaporative removal of benzene from its dilute solution by poly(dimethylsiloxane) (PDMS) membranes filled with calix[4]arene (CA) as the model system. In our previous experimental investigation, the normalized permeation rate of benzene (NPRb) and separation factor (benzene/water) through PDMS−CA hybrid membranes did not follow the usual monotonous or single peak/valley change, but accompanied minimum and maximum values instead. In the present study, nonbonding interaction energy between PDMS and CA, mean-square displacement (MSD), free volume characteristics, and diffusion coefficients of benzene and water in pure PDMS and hybrid membranes were analyzed by molecular dynamics simulation. The simulation results revealed that MSD and fractional free volume (FFV) values were closely dependent on interaction energy. Diffusion coefficients of benzene and water at “infinite dilution” and saturated condition displayed the same changing tendency, although the values at saturated condition were a bit larger. Moreover, it was observed that diffusion coefficients were not only related to FFV but also affected by the interaction between CA and the penetrants. Overall, the MD results agreed well with the experimental results.

Cumulant expansion analysis of thermal disorder in face centred cubic copper metal by molecular dynamics simulation

Molecular dynamics, using a previously validated semi-empirical potential, are used to explore thermal disorder in face centred cubic copper over the temperature range 100 K to 1100 K. The mean-square displacement (MSD) of atoms from their mean positions has been extracted from the simulation. Results compare favourably with experiment and are consistent with the predictions of a range of theoretical lattice dynamics models. The temperature dependence of the first four cumulants of each probability distribution function (PDF) for the four nearest neighbour shells is compared with the predictions of a simple one-dimensional anharmonic oscillator model. With the exception of the third cumulant for the second shell, the behaviour of the simulated cumulants is consistent with this model. Where direct comparison of the simulation and experimentally derived cumulants is possible the two are found to be in good agreement except at low temperatures where quantum effects (neglected in the classical molecular dynamics simulation) are most significant. The simulated PDF profiles are reproduced well across the temperature range by a reconstruction formula employing only the first four cumulants. The linear expansivitity calculated from the cumulants is in only modest agreement with experimental data. Analysis of the PDF of a shell in the high essentially uncorrelated isotropic vibration of the atoms, but demonstrates the difficulty of extracting MSD values from extended X-ray absorption fine structure (EXAFS) spectroscopy and correlating EXAFS and X-ray diffraction thermal parameters. r limit is consistent with

Extended x-ray-absorption fine-structure investigation of short-range order in a-Ge1-xSnx alloys.

The local environments of germanium and tin atoms in a-Ge 1-x Sn x alloys (0.0≤x≤0.2) have been investigated by measuring the extended x-ray-absorption fine structure (EXAFS) at the Ge K edge and at the Sn L III edge. The measured EXAFS at both edges could be described only by assuming the presence of a mixed first coordination shell composed of both Ge and Sn atoms. Coordination numbers, bond lengths, and relative mean-square displacement values characterizing the first coordination shell around Ge and Sn atoms were obtained

Debye-Waller factor for high-spin and low-spin iron/III/ dithiocarbamates

The absolute values of the Debye-Waller factor for high-spin Fe/pyrrdtc/3 and low-spin Fe/pyroldtc/3 iron/III/ complexes /pyrrdtc-pyrrolidyl-N-carbodithioato, pyroldtc-pyrrole-N-carbodithioato/ were determined at room temperature. The obtained values of mean square displacement are in very good agreement with those determined by X-ray diffraction methods.

Theoretical Limitations on the Temperature Dependence of Vibrational Properties of Harmonic Solids and Molecules

Upper and lower bounds have been found for the mean-square displacement and mean-square velocity of constituent atoms, and for the heat capacity associated with the atomic motions of a harmonic system. These limits are a direct consequence of the harmonic approximation and thermal equilibrium. These limits give the maximum and minimum allowable values of mean-square displacement, mean-square velocity, or heat capacity at any one temperature in terms of the value of the dynamic quantity at any other temperature. In many cases these limits are very restrictive, determining the value of a dynamic quantity at a second temperature to within a few percent.