On the cubic and hexagonal close-packed lattices

1955 ◽  
Vol 227 (1171) ◽  
pp. 447-465 ◽  
Keyword(s):  
Solid Helium ◽  
Zero Point ◽  
Lattice Energy ◽  
Close Packing ◽  
Inert Gases ◽  
Zero Point Energy ◽  
Point Energy ◽  
Hexagonal Close Packed ◽  
The Difference ◽  
Central Forces

The present paper was stimulated by the discovery by Dugdale & Simon (1953) of a polymorphic transition in solid helium. A discussion is given of the relative stability of the cubic and hexagonal close-packed lattices assuming central forces of the Mie—Lennard-Jones type. Taking static lattice energy alone into account the usual laws of force favour the hexagonal close-packed lattice, the difference in energy being about 0·01%. However, lattice dynamics indicates that the equivalent Debye Θ at the absolute zero is smaller for the cubic lattice, the difference being about 1%. Hence, ignoring zero-point energy, we should expect a transition to occur from hexagonal to cubic at an elevated temperature. The estimated temperature and energy of the transition are of the same order of magnitude as those observed experimentally in solid helium. An estimate is made of the effect of zero-point energy; the results can be applied with confidence to the heavier inert gases, but can only be considered as giving a qualitative indication for helium, since anharmonic effects are of great importance in this case. For the other inert gas solids it is concluded that the experimentally observed cubic close-packing at all temperatures must be due to non-central forces.

HYDROGEN PEROXIDE AND ITS ANALOGUES: III. ABSORPTION SPECTRUM OF HYDROGEN AND DEUTERIUM PEROXIDES IN THE NEAR ULTRAVIOLET

1951 ◽  
Vol 29 (6) ◽  
pp. 490-493 ◽  
Author(s):  
M. K. Phibbs ◽  
Paul A. Giguère
Keyword(s):  
Hydrogen Peroxide ◽  
Absorption Spectrum ◽  
Ultraviolet Light ◽  
Heavy Water ◽  
Zero Point ◽  
Zero Point Energy ◽  
Point Energy ◽  
Near Ultraviolet ◽  
The Difference ◽  
High Concentrations

The absorption of ultraviolet light between 3000 and 4000 Å by solutions of hydrogen peroxide in water and of deuterium peroxide in heavy water has been measured at various concentrations. Both peroxides show slight but real deviations from Beer's law at high concentrations. Substitution of hydrogen by deuterium shifts the absorption continuum by about 390 cm.−1 towards shorter wave lengths. This shift is of the same order as that calculated from the difference in zero-point energy of the two isotopic molecules.

Thermodynamic properties and melting of solid helium

1953 ◽  
Vol 218 (1134) ◽  
pp. 291-310 ◽  
Keyword(s):  
Experimental Data ◽  
Solid Helium ◽  
Characteristic Temperature ◽  
Melting Curve ◽  
Zero Point ◽  
Point Energy ◽  
Melting Temperatures ◽  
Wide Range ◽  
Melting Properties ◽  
The Difference

The melting properties and thermodynamic functions of solid helium have been determined at temperatures from 4 to 26° K and at pressures up to 3000 atm. The upper temperature corresponds to about five times the critical temperature of helium; it was therefore possible to measure properties of the solid state in a range which has not yet been attained for any other substance. The melting curve shows no signs of an approach to a solid-fluid critical point; in fact, the difference between the phases becomes more pronounced at higher melting temperatures. The internal energy at 0° K was calculated from the experimental data and was found to be in good agreement with the theoretical values based on the Slater-Kirkwood potential, using 9/8 Rθ as an estimate of the zero-point energy ( θ being the Debye characteristic temperature). A first-order transition in the solid was revealed; its equilibrium line cuts the melting curve at 14.9° K and moves to higher temperatures at higher densities. The heat of transition is very small, about 0.08 cal/mole. The transition is assumed to correspond to a change of crystal structure from hexagonal to cubic close-packed. At the highest pressure solid helium is compressed to less than half its volume under equilibrium conditions at absolute zero, and the Debye θ is increased five times. It was hence possible to test the Lindemann melting formula for a single substance over a very wide range. The formula was found to fit the experimental data satisfactorily, although the value of the constant in it differed somewhat from the classical value.

scholarly journals The sorption of hydrogen and deuterium by copper and palladium I-The behaviour of copper and copper oxides

1935 ◽  
Vol 153 (878) ◽  
pp. 77-88 ◽  
Keyword(s):  
Zero Point ◽  
Heterogeneous Reactions ◽  
Energy Of Activation ◽  
Zero Point Energy ◽  
Point Energy ◽  
Activated State ◽  
Heterogeneous Processes ◽  
The Difference ◽  
State Of Affairs ◽  
Homogeneous Reactions

The investigation of homogeneous reactions of deuterium and of deuterides has shown that the heavy isotopic molecules react in general less readily than the light, owing to the greater energy of activation required in the deuterium reactions. This result is qualitatively in accordance with theoretical expectations, the reason for the difference being mainly due to the larger zero point energy of the hydrogen compounds. Quantitative analysis of reactions whose mechanism is well established has, however, shown that the ratio of the velocities is often considerably less than that expected if all the zero point energy is con­tributed to the energy pool in the activating collision. The smaIler ratio of velocity is due to the fact that the zero point energy of the activated state cannot be neglected, the difference for H and for D re­action often being considerable. For example, in the reaction Br + H 2 → HBr [DBr] + H [D] the difference in the energy of activation is 1·50 kg cal, and therefore the difference in zero point energies in the activated state is 0·3 kg cal; similarly for Cl + H 2 (D 2 ) → HCI (DCI) + H (D) the respective figures are 1·23 and 0·55 kg cal, while for the reaction H + H 2 (para) = H 2 (ortho) + H and D + D 2 (ortho) = D 2 (para) + D, the difference in the activated state is no less than 1·20 kg cal. The position with regard to heterogeneous reactions of hydrogen is much less satisfactory and therefore any further means of obtaining information in tin branch of kinetics is of value. Deuterium at first sight promised to be a useful tool in helping to discriminate between various hypotheses. Unfortunately, it is difficult to predict theoretically exactly what will happen in any well-defined instance even with a complete experimental knowledge of the mechanism of the reaction. The state of affairs is analogous to that obtaining with homogeneous reactions, it being necessary to determine experimentally the behaviour of the two isotopes in a variety of heterogeneous processes.

scholarly journals The replusive forces between isotopic molecules

1940 ◽  
Vol 174 (959) ◽  
pp. 504-509 ◽  
Keyword(s):  
Zero Point ◽  
Electron Shell ◽  
Intermolecular Forces ◽  
Zero Point Energy ◽  
Restoring Force ◽  
Point Energy ◽  
Nuclear Masses ◽  
Energy Of Interaction ◽  
The Difference ◽  
Made In

It is usually assumed that the forces of attraction and repulsion between two molecules depend only on their electronic structures and are independent of the nuclear masses. While this assumption is undoubtedly true to a first approximation it fails to take into account the zero-point energy associated with the nuclear vibrations, which will modify the charge distribution both of the nuclei and of the electronic shells. Since the zero-point energy depends upon the nuclear mass, this may lead to differences between the behaviour of a pair of isotopic molecules such as H 2 and D 2 . If the restoring force of the nuclear vibration is not directly proportional to the displacement of the nuclei from their mean position, then the mean internuclear distance will be different for the two isotopes. The magnitude of this anharmonic effect can be calculated from spectrum data, and it is found that for H 2 and D 2 the difference is less than 10 -11 cm., and hence negligible. However, the energy of interaction of two molecules will not be a linear function of the inter­ nuclear distance within the molecules, so that even for harmonic oscillations the observed interaction will depend upon the magnitude of the zero-point energy. Both the attractive and the repulsive intermolecular forces will be affected in this way, but it is difficult to treat the former owing to the absence of any exact treatment of exchange forces between molecules. The problem is more easily attacked in the case of the Coulomb forces (which have a net repulsive effect), and the present paper constitutes an attempt to estimate the isotope effect for this type of force. It will be necessary to neglect the mutual deformation of the charge distributions caused by the approach of the two molecules. This assumption is usually made in dealing with Coulomb forces, and it is unlikely to introduce serious error in calculating the isotopic difference. The total Coulomb interaction between two molecules can then be written as G = G n + G n c + G e (1) where G n is the interaction between the nuclei of the two molecules, G ne the interaction between the nuclei of one molecule and the electron shell of the other, and G e the interaction between the two electron shells. The principles involved are most easily seen by considering G n for two diatomic molecules.

On the Use of Quantum Thermal Bath in Unimolecular Fragmentation Simulations

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>

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

1928 ◽  
Vol 118 (779) ◽  
pp. 334-350 ◽  
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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