The Mean Amplitudes of Thermal Vibrations in Polyatomic Molecules. I. CF2=CF2 and CH2=CF2

1952 ◽  
Vol 20 (4) ◽  
pp. 726-733 ◽  
Author(s):  
Yonezo Morino ◽  
Kôzo Kuchitsu ◽  
Takehiko Shimanouchi
Keyword(s):  
Polyatomic Molecules ◽  
The Mean ◽  
Thermal Vibrations

The structure of ice

1958 ◽  
Vol 247 (1251) ◽  
pp. 424-434 ◽  
Keyword(s):  
Hydrogen Bond ◽  
Hydrogen Atoms ◽  
Cell Parameters ◽  
Bond Lengths ◽  
Hexagonal Ice ◽  
A Value ◽  
Air Temperatures ◽  
The Mean ◽  
Thermal Vibrations ◽  
Low Pressures

A consistent set of unit cell parameters at various temperatures is not yet available for ordinary ice, but the mean of the most precise measurements leads to a density of 0·9164 g/cm 3 at 0°C (atmospheric pressure) with a cubical expansion coefficient of 11 x 10 -5 , increasing to 0·9414 and 21 x 10 -5 at liquid air temperatures. Corresponding figures for heavy ice are 1·0172 g/cm 3 and 12 x 10 -5 at 0°C, 1·0449 and 18 x 10 -5 at -180°C. The hydrogen-bond lengths are not significantly different for ordinary and heavy ice, but in both cases the mirror-symmetric bond (along the principal axis) is about 0·01 Å shorter than the centro-symmetric bond at 0°C. At low temperatures the bond lengths tend to equalize at a value some 1% lower than at 0°C. The hexagonal (tridymite-type) and cubic (cristobalite-type) forms of ice have approxi­mately the same density and hydrogen-bond lengths at —130°C, and both appear to have a statistical randomness of the water-molecule orientation, consistent with there being one hydrogen only (nearly or exactly) along each bond. The thermal vibrations of the hydrogen atoms in hexagonal ice are anisotropic, those of the oxygen atoms nearly spherical. The ranges of stability of hexagonal, cubic and amorphous ice are not exactly known, but cubic ice is only formed at low rates of deposition, low pressures and at temperatures of about -80 to -140°C.

scholarly journals Melting Criterion for Cubic Metals Based on a Modified Angular Force Model

1974 ◽  
Vol 27 (1) ◽  
pp. 129 ◽  
Author(s):  
Jyoti Prakash ◽  
MP Hemkar
Keyword(s):  
Melting Point ◽  
Periodic Table ◽  
Lattice Dynamics ◽  
Force Model ◽  
Mean Square ◽  
Interatomic Spacing ◽  
Cubic Metals ◽  
The Mean ◽  
Thermal Vibrations

A modified angular force model for the lattice dynamics of metals is used to test Lindemann's melting criterion by computing the ratio Ym of the mean square amplitude of thermal vibrations to the square of the interatomic spacing at the melting point for a number of cubic metals belonging to different groups in the periodic table.

Refinement of the structure of chrysene

1960 ◽  
Vol 257 (1291) ◽  
pp. 491-514 ◽  
Keyword(s):  
Carbon Atom ◽  
Rigid Body ◽  
Least Squares ◽  
Theoretical Consideration ◽  
Three Dimensional ◽  
Hydrogen Atoms ◽  
The Mean ◽  
The Individual ◽  
Difference Fourier ◽  
Thermal Vibrations

The co-ordinates of the carbon and hydrogen atoms and the anisotropic thermal vibrations of the carbon atoms of chrysene have been determined by three-dimensional least-squares refinement and by three-dimensional Fourier, and difference Fourier, syntheses. The value of R = Ʃ || F o | - | F c ||/Ʃ| F o for over 1000 planes observed with Cu Kα radiation is 0·076. The standard deviations of the carbon atom co-ordinates are less than 0·0033 Å. One C–C bond is significantly longer than the mean and longer than would be expected by any theoretical consideration. The value 1·465 Å can be explained by overcrowding of the hydrogen atoms attached to the carbon atoms concerned. The ‘K’ bond of the phenanthrene nucleus has a length of 1·365 Å and this is in the same benzene ring and diametrically opposite to the very long bond mentioned above. It is shown that the thermal vibrations obtained for the individual carbon atoms can be well represented by considering the molecule to vibrate as a rigid body.

Krebs's model and Lindemann's melting law

1968 ◽  
Vol 46 (15) ◽  
pp. 1677-1679 ◽  
Author(s):  
A. K. Singh ◽  
P. K. Sharma
Keyword(s):  
Melting Temperature ◽  
Periodic Table ◽  
Lattice Dynamics ◽  
Mean Square Displacement ◽  
Mean Square ◽  
Interatomic Spacing ◽  
Cubic Metals ◽  
The Mean ◽  
Thermal Vibrations

Lindemann's melting criterion for cubic metals is examined using Krebs's model for the lattice dynamics of metals by computing the ratio, xm, of the mean square displacement of thermal vibrations and the square of interatomic spacing at the melting temperature for a number of cubic metals belonging to different groups of the periodic table. This ratio has nearly the same value for elements of one particular group, but exhibits wide variation from one group to another. It is concluded that Lindemann's melting law is inadequate.

Predicting some Effects of Lattice Vibrations on Proton Scattering Patterns

1970 ◽  
Vol 14 ◽  
pp. 1-10
Author(s):  
C. S. Barrett
Keyword(s):  
Temperature Factor ◽  
Lattice Vibrations ◽  
Proton Scattering ◽  
Mean Square ◽  
X Ray Diffraction ◽  
X Ray ◽  
Structure Factors ◽  
Integrated Intensity ◽  
The Mean ◽  
Thermal Vibrations

AbstractA method of predicting the approximate relative intensities of lines in proton blocking patterns recently proposed, which is based on summing the squares of structure factors for the various orders of reflection of a plane, is found to predict certain effects of lattice vibrations on the lines in some recently reported patterns. The mean square amplitude of vibration enters the calculations through a Debye-Waller temperature factor like that used in X-ray diffraction. When patterns are compared for groups of crystals that are nearly identical except for this temperature factor, the qualitative predictions by this method agree with the observations. If it is also arbitrarily assumed that the integrated intensity dip at a spot where lines intersect is approximated by summing the calculated Integrated intensity dips for all of the lines crossing at the spot, one has a simple and convenient method of predicting relative spot intensities. Such calculations have been successful in establishing the order of decreasing intensity for most of the spots along a given line, with several different kinds of crystals. This method also serves to predict qualitatively how prominent the spots appear relative to the lines, in general, in patterns of crystals that differ appreciably only in the amplitude of the thermal vibrations.

The Structure and Ionic Conduction in Fluorite-Type Solid Solutions

1992 ◽  
Vol 293 ◽  
Author(s):  
Yoshiaki Ito
Keyword(s):  
Ionic Conduction ◽  
Diffraction Method ◽  
X Ray Diffraction ◽  
X Ray ◽  
Temperature Range ◽  
Fluorite Type ◽  
The Mean ◽  
Fluorine Ions ◽  
Thermal Vibrations ◽  
Regular Site

AbstractThe crystal structure of β-Pb0.9Bi0.1F2.1 has been investigated in the temperature range 294 to 710K by single crystal X-ray diffraction method. Phase transition is observed at about 700K and the mean square displacements (MSD) of metal and fluorine ions show a rapid increase near the transition temperature. The fluorine ions do not show anharmonic thermal vibrations in the regular site in the temperature range.

scholarly journals The theory of metals. —I

1932 ◽  
Vol 138 (836) ◽  
pp. 594-606 ◽  
Keyword(s):  
Free Path ◽  
Mean Free Path ◽  
Present Theory ◽  
Point Of View ◽  
Mathematical Treatment ◽  
Constant Field ◽  
Free Electrons ◽  
The Mean ◽  
Thermal Vibrations ◽  
Good Agreement

1. The quantum theory of electrical conduction in a solid has two main problems to face, the number of “free electrons” and the “mean free path.” Of these the first is the simpler and has, to a certain extent, been solved. The evaluation of the mean free path, on the other hand, has given rise to some controversy and cannot be regarded as satisfactory. In his original paper on conduction Bloch gave a theory of the interaction of the electrons and the thermal vibrations in a metal which leaves much to be desired from the point of view of rigour, but which leads to results in good agreement with experiment. Peierls criticised this treatment and gave a new one, which, if correct, would considerably alter the theory.§ Peierls omitted most of the calculations, which are difficult, and based his treatment on physical arguments, which are by no means easy to follow, and which require justification. Recently L. Brillouin|| has given an extended mathematical treatment of the points in dispute, and obtains results which differ considerably from those of both Bloch and Peierls. None of these calculations is really satisfactory, the main objection being that the physical assumptions have not been made sufficiently precise. A method is given here which treats consistently the interaction of the electrons and the lattice, and which enables the assumptions to be clearly seen. It also has the advantage that it can be extended quite naturally to deal with the problems of the dispersion and absorption of light in metals, which will be treated in subsequent papers. In this paper the general theory will be developed, and applied to the discussion of the debatable points in the theories of Bloch and Peierls. Although the general opinion seems to be that Peierls’ criticisms are correct, the opposite view is arrived at here, and so, if the present theory is correct, the anomalous processes ” introduced by Peierls have little impor­tance for the electrical conductivity in a constant field.

Sign in / Sign up

Export Citation Format

Share Document