Thermal Vibration Amplitude Research Articles

Thermal Vibrations of Atoms in Cubic Crystals II: The Amplitude of Atomic Vibrations

The Lindemann relationship between the Debye characteristic temperature and the melting point may also be obtained from a consideration of the temperature variation of thermal diffuse x-ray scattering. The amplitude of thermal vibrations of the atoms in cubic crystals can be expressed in terms of the distance between neighbouring atomic positions and the melting point. This leads to a correlation of the relative increases of disorder on melting of body-centred cubic and face-centred cubic crystals with the thermal behaviour of the elastic constants, electrical resistance, atomic diffusion, specific heat, thermal expansion and thermal conductivity. A `law of corresponding states' may be said to exist where, at corresponding temperatures, the amplitudes of atomic vibration are the same fraction of the distance between neighbouring atomic positions in the solid. The Debye temperature factor is found to be inversely proportional to the melting point for cubic structures.

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

A normal coordinate treatment based on the potential function of Urey-Bradley type has been applied to CF2=CF2 and CF2=CH2 molecules by taking into account the repulsions between nonbonded atoms in addition to the valence force field. The mean-square amplitudes of thermal vibration of the atomic distances in these molecules have been calculated by the use of the normal coordinates and compared with those obtained by I. Karle and J. Karle from the electron diffraction investigation.

The Effect of Temperature on the Intensity of Reflection of X-Rays from Zinc Crystals

Measurements have been made of the intensity of x-rays reflected from powdered zinc crystals at room temperature and at the temperature of liquid air. The atomic structure factors for the zinc atom at these temperatures are found to depend on the orientation of the particular plane from which the reflection of the x-rays takes place. Planes whose normals make small angles $\ensuremath{\psi}$ with the $c$ axis of the crystal give abnormally low values of the structure factor, while planes for which $\ensuremath{\psi}$ is large give high values. From the measurements at the two temperatures the temperature factor for the various planes has been determined and with the aid of this factor the structure factor for the atom at rest has been calculated. The structure factor data for the atom at rest do not lie on a smooth curve as one might expect, but are also found to depend on the value of $\ensuremath{\psi}$. This indicates that the zinc atoms in the crystal do not possess spherical symmetry but are drawn out in the direction of the $c$ axis. The average structure factor curve obtained experimentally has, however, been found to be in satisfactory agreement with a similar curve obtained from the theoretical charge distribution in the atom. The amplitude of thermal vibration of the atoms along various directions in the crystal has been determined. A comparison is made of the root mean square displacement as determined from the x-ray measurements with the average value obtained from specific heat data and a good agreement is found.

Effect of Temperature on the Intensity of X-Rays Scattered by Powdered Sodium Fluoride

The ratio of the integrated intensities of a number of regularly reflected x-ray lines from powdered sodium fluoride crystals at room and liquid air temperatures has been studied by means of a photographic method. The results of the experiments show that the ratio varies with the angle as predicted by the Debye-Waller formula. The characteristic temperature of sodium fluoride was found to be equal to 442\ifmmode^\circ\else\textdegree\fi{}K. This value agrees as well as one could expect with values calculated by the Lindemann melting-point formula and the known characteristic temperatures of sodium chloride or potassium chloride crystals. Using the known structure factor of sodium fluoride atoms at room temperature and assuming zero-point or no zero-point energy the structure factors of the atoms at rest have been calculated. A comparison with the theoretical structure factors of sodium fluoride and neon (sodium fluoride atoms are neon-like) and with the known experimental structure factor of gaseous neon atoms indicates that the assumption of zero-point energy is correct. The square root of the mean square amplitude of thermal vibration of the sodium and fluorine ions at room temperature was found to be equal to 0.182A.