A quantitative study of the reflexion of X-rays from crystals of aluminium

1. It has already been shown that the observed variations with temperature of the intensity of reflexion of X-rays from crystals of rock-salt and sylvine agree closely with those predicted by the theory of Debye as modified by Waller, from the lowest temperature at which experiments have been made, that of liquid air, up to about 500° abs. Moreover, the absolute intensities of reflexion agree closely with those calculated theoretically, if the atomic scattering factors, F, are calculated from the Schrödinger charge-distributions for the atoms, obtained by the method due to Hartree. To obtain agreement, it is necessary to assume the existence of zero-point energy of an amount half a quantum for each degree of freedom, and the experiments may perhaps do considered as furnishing direct confirmation of such energy, since the differences between the intensities calculated with and without it are considerable. The work to do described in this paper was undertaken with a view to extending the investigations to a crystal of a metallic element. Aluminium at ones suggested itself as suitable for this purpose, since it can do obtained in largo single crystals, and since its coefficient of adsorption for the X-rays employed in the experiments, Mo K a , is small.

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An investigation into the existence of zero-point energy in the Rock-Salt Lattice by an X-Ray Diffraction Method

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1928.0055 ◽

1928 ◽

Vol 118 (779) ◽

pp. 334-350 ◽

Cited By ~ 41

Keyword(s):

Rock Salt ◽

Diffraction Method ◽

Zero Point ◽

Temperature Factor ◽

X Rays ◽

Zero Point Energy ◽

Chlorine Atoms ◽

Point Energy ◽

X Ray ◽

State Of Rest

In the present paper we shall attempt to collate the results of four separate lines of research which, taken together, appear to provide some interesting checks between theory and experiment. The investigations to be considered are (1) the discussion by Waller* and by Wentzel,† on the basis of the quantum (wave) mechanics, of the scattering of radiation by an atom ; (2) the calculation by Hartree of the Schrödinger distribution of charge in the atoms of chlorine and sodium ; (3) the measurements of James and Miss Firth‡ of the scattering power of the sodium and chlorine atoms in the rock-salt crystal for X-rays at a series of temperatures extending as low as the temperature of liquid air ; and (4) the theoretical discussion of the temperature factor of X-ray reflexion by Debye§ and by Waller.∥ Application of the laws of scattering to the distribution of charge calculated for the sodium and chlorine atoms, enables us to calculate the coherent atomic scattering for X-radiation, as a function of the angle of scattering and of the wave-length, for these atoms in a state of rest, assuming that the frequency of the X-radiation is higher than, and not too near the frequency of the K - absorption edge for the atom.¶ From the observed scattering power at the temperature of liquid air, and from the measured value of the temperature factor, we can, by applying the theory of the temperature effect, calculate the scattering power at the absolute zero, or rather for the atom reduced to a state of rest. The extrapolation to a state of rest will differ according to whether we assume the existence or absence of zero point energy in the crystal lattice. Hence we may hope, in the first place to test the agreement between the observed scattering power and that calculated from the atomic model, and in the second place to see whether the experimental results indicate the presence of zero-point energy or no.

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A quantitative study of the reflexion of X-rays by sylvine

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1928.0188 ◽

1928 ◽

Vol 121 (787) ◽

pp. 155-171 ◽

Cited By ~ 56

Keyword(s):

Quantum Mechanics ◽

Theoretical Result ◽

Zero Point ◽

Coherent Scattering ◽

X Rays ◽

Point Energy ◽

X Ray ◽

Quantitative Results ◽

Theory And Experiment ◽

Quantitative Treatment

In recent papers on the intensity of reflexion of X-rays from rocksalt crystals it has been shown that, from the temperature of liquid air up to about 500° abs., the dependence of the intensity of reflexion upon temperature is in accord quantitatively with a formula of the type originally deduced by Debye, if the modification suggested later by Waller is introduced, although the decrease of intensity for higher temperatures is much greater than that indicated by the law. In dealing with quantitative results of experiments on reflexion from crystals, it is convenient to consider the quantity usually denoted by F, which is a measure of the scattering power, in a given direction, of an atom for X-rays. In the course of experiments with rocksalt it has been possible to determine F both for Na and Cl, and the values so obtained, when corrected for temperature, agree very closely with the F factors calculated from Hartree’s Schrodinger density-distribution for the ions Cl - and Na + . The calculation is based upon the theoretical result, due to Wentzel and Waller that, to obtain the coherent scattering from an electron in an atom, the electron must be represented by its corresponding Schrodinger charge-density, each element of which must be supposed to scatter classically. In order to get agreement between the calculated and observed F curves, it is necessary to assume the existence of zero-point energy of amount half a quantum per degree of freedom, which is of course required by the new quantum mechanics. The agreement between theory and experiment in the case of rocksalt is extremely interesting, but, in order to place the quantitative treatment of X-ray reflexion on an entirely satisfactory basis, it appears to be of some importance to see whether a similar agreement can be obtained with other crystals. This is not quite so easy as might be supposed.

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An X-ray study of the heat motions of the atoms in a rock-salt crystal

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1927.0165 ◽

1927 ◽

Vol 117 (776) ◽

pp. 62-87 ◽

Cited By ~ 54

Keyword(s):

Fourier Analysis ◽

Rock Salt ◽

Low Temperatures ◽

X Rays ◽

Salt Crystal ◽

X Ray ◽

Salt Crystals ◽

Different Temperatures ◽

Atomic Scattering ◽

Actual Amplitude

1. The present paper may be divided into two parts. In the first, some experiments on the intensity of reflexion of X-rays by rock-salt crystals at low temperatures are described. The results of these experiments, when combined with data obtained previously at high temperatures, are compared with the theoretical formulæ of Debye and Waller for the temperature factor of X-ray reflexion. In the second part of the paper we have attempted to get some idea of the actual amplitude of the heat-motions of the atoms in the rock-salt lattice, by analysing the F curves, or curves showing the variation of the atomic scattering power with angle of scattering, using the method of Fourier analysis introduced by Duane and Havighurst, so as to obtain the distribution of electrons in the crystal unit at different temperatures. In connection with this work a new set of absolute determinations of intensity of reflexion has been made, and, from these, the F factors at different temperatures have been calculated, using the results of the experiments described in the first part of the paper.

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On the Use of Quantum Thermal Bath in Unimolecular Fragmentation Simulations

10.26434/chemrxiv.8870594.v2 ◽

2019 ◽

Author(s):

Riccardo Spezia ◽

Hichem Dammak

Keyword(s):

Degrees Of Freedom ◽

Molecular Simulations ◽

Zero Point ◽

Thermal Bath ◽

Unimolecular Dissociation ◽

Point Energy ◽

Rotational Degrees Of Freedom ◽

The Difference ◽

Nuclear Effects

<div> <div> <div> <p>In the present work we have investigated the possibility of using the Quantum Thermal Bath (QTB) method in molecular simulations of unimolecular dissociation processes. Notably, QTB is aimed in introducing quantum nuclear effects with a com- putational time which is basically the same as in newtonian simulations. At this end we have considered the model fragmentation of CH4 for which an analytical function is present in the literature. Moreover, based on the same model a microcanonical algorithm which monitor zero-point energy of products, and eventually modifies tra- jectories, was recently proposed. We have thus compared classical and quantum rate constant with these different models. QTB seems to correctly reproduce some quantum features, in particular the difference between classical and quantum activation energies, making it a promising method to study unimolecular fragmentation of much complex systems with molecular simulations. The role of QTB thermostat on rotational degrees of freedom is also analyzed and discussed. </p> </div> </div> </div>

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Modulation of optical absorption in m-Fe1−xRuxS2 and exploring stability in new m-RuS2

Scientific Reports ◽

10.1038/s41598-021-86181-7 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

H. Joshi ◽

M. Ram ◽

N. Limbu ◽

D. P. Rai ◽

B. Thapa ◽

...

Keyword(s):

Optical Absorption ◽

Phonon Dispersion ◽

Applied Pressure ◽

Orthorhombic Phase ◽

Zero Point ◽

Computational Method ◽

Crystal Field Theory ◽

Point Energy ◽

Practical Applications ◽

New Phase

AbstractA first-principle computational method has been used to investigate the effects of Ru dopants on the electronic and optical absorption properties of marcasite FeS2. In addition, we have also revealed a new marcasite phase in RuS2, unlike most studied pyrite structures. The new phase has fulfilled all the necessary criteria of structural stability and its practical existence. The transition pressure of 8 GPa drives the structural change from pyrite to orthorhombic phase in RuS2. From the thermodynamical calculation, we have reported the stability of new-phase under various ranges of applied pressure and temperature. Further, from the results of phonon dispersion calculated at Zero Point Energy, pyrite structure exhibits ground state stability and the marcasite phase has all modes of frequencies positive. The newly proposed phase is a semiconductor with a band gap comparable to its pyrite counterpart but vary in optical absorption by around 106 cm−1. The various Ru doped structures have also shown similar optical absorption spectra in the same order of magnitude. We have used crystal field theory to explain high optical absorption which is due to the involvement of different electronic states in formation of electronic and optical band gaps. Lӧwdin charge analysis is used over the customarily Mulliken charges to predict 89% of covalence in the compound. Our results indicate the importance of new phase to enhance the efficiency of photovoltaic materials for practical applications.

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Dynamical strengthening of covalent and non-covalent molecular interactions by nuclear quantum effects at finite temperature

Nature Communications ◽

10.1038/s41467-020-20212-1 ◽

2021 ◽

Vol 12 (1) ◽

Author(s):

Huziel E. Sauceda ◽

Valentin Vassilev-Galindo ◽

Stefan Chmiela ◽

Klaus-Robert Müller ◽

Alexandre Tkatchenko

Keyword(s):

Finite Temperature ◽

Electrostatic Interactions ◽

Zero Point ◽

Quantum Effects ◽

Van Der Waals Interactions ◽

Interatomic Distances ◽

Point Energy ◽

Molecular Bond ◽

Energy Surfaces ◽

Nuclear Quantum Effects

AbstractNuclear quantum effects (NQE) tend to generate delocalized molecular dynamics due to the inclusion of the zero point energy and its coupling with the anharmonicities in interatomic interactions. Here, we present evidence that NQE often enhance electronic interactions and, in turn, can result in dynamical molecular stabilization at finite temperature. The underlying physical mechanism promoted by NQE depends on the particular interaction under consideration. First, the effective reduction of interatomic distances between functional groups within a molecule can enhance the n → π* interaction by increasing the overlap between molecular orbitals or by strengthening electrostatic interactions between neighboring charge densities. Second, NQE can localize methyl rotors by temporarily changing molecular bond orders and leading to the emergence of localized transient rotor states. Third, for noncovalent van der Waals interactions the strengthening comes from the increase of the polarizability given the expanded average interatomic distances induced by NQE. The implications of these boosted interactions include counterintuitive hydroxyl–hydroxyl bonding, hindered methyl rotor dynamics, and molecular stiffening which generates smoother free-energy surfaces. Our findings yield new insights into the versatile role of nuclear quantum fluctuations in molecules and materials.

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Time-Resolved Measurements and Master Equation Modelling of the Unimolecular Decomposition of CH3OCH2

Zeitschrift für Physikalische Chemie ◽

10.1515/zpch-2020-0007 ◽

2020 ◽

Vol 234 (7-9) ◽

pp. 1233-1250 ◽

Cited By ~ 1

Author(s):

Arrke J. Eskola ◽

Mark A. Blitz ◽

Michael J. Pilling ◽

Paul W. Seakins ◽

Robin J. Shannon

Keyword(s):

Energy Transfer ◽

Master Equation ◽

Rate Coefficient ◽

Zero Point ◽

Buffer Gas ◽

Unimolecular Decomposition ◽

Time Resolved ◽

Point Energy ◽

Wide Range ◽

Transfer Parameters

AbstractThe rate coefficient for the unimolecular decomposition of CH3OCH2, k1, has been measured in time-resolved experiments by monitoring the HCHO product. CH3OCH2 was rapidly and cleanly generated by 248 nm excimer photolysis of oxalyl chloride, (ClCO)2, in an excess of CH3OCH3, and an excimer pumped dye laser tuned to 353.16 nm was used to probe HCHO via laser induced fluorescence. k1(T,p) was measured over the ranges: 573–673 K and 0.1–4.3 × 1018 molecule cm−3 with a helium bath gas. In addition, some experiments were carried out with nitrogen as the bath gas. Ab initio calculations on CH3OCH2 decomposition were carried out and a transition-state for decomposition to CH3 and H2CO was identified. This information was used in a master equation rate calculation, using the MESMER code, where the zero-point-energy corrected barrier to reaction, ΔE0,1, and the energy transfer parameters, ⟨ΔEdown⟩ × Tn, were the adjusted parameters to best fit the experimental data, with helium as the buffer gas. The data were combined with earlier measurements by Loucks and Laidler (Can J. Chem.1967, 45, 2767), with dimethyl ether as the third body, reinterpreted using current literature for the rate coefficient for recombination of CH3OCH2. This analysis returned ΔE0,1 = (112.3 ± 0.6) kJ mol−1, and leads to $k_{1}^{\infty}(T)=2.9\times{10^{12}}$ (T/300)2.5 exp(−106.8 kJ mol−1/RT). Using this model, limited experiments with nitrogen as the bath gas allowed N2 energy transfer parameters to be identified and then further MESMER simulations were carried out, where N2 was the buffer gas, to generate k1(T,p) over a wide range of conditions: 300–1000 K and N2 = 1012–1025 molecule cm−3. The resulting k1(T,p) has been parameterized using a Troe-expression, so that they can be readily be incorporated into combustion models. In addition, k1(T,p) has been parametrized using PLOG for the buffer gases, He, CH3OCH3 and N2.

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