On the stability of crystal lattices IX. Covariant theory of lattice deformations and the stability of some hexagonal lattices

1942 ◽  
Vol 38 (1) ◽  
pp. 82-99 ◽  
Author(s):  
M. Born
Keyword(s):  
Elastic Constants ◽  
Hexagonal Lattice ◽  
Covariant Theory ◽  
Tensor Calculus ◽  
Crystal Lattices ◽  
New Form ◽  
The Stability ◽  
Central Forces

The theory of lattice deformations is presented in a new form, using the tensor calculus. The case of central forces is worked out in detail, and the results are applied to some simple hexagonal lattices. It is shown that the Bravais hexagonal lattice is unstable but the close-packed hexagonal lattice stable. The elastic constants of this lattice are calculated.

On the stability of crystal lattices. I

1940 ◽  
Vol 36 (2) ◽  
pp. 160-172 ◽  
Author(s):  
Max Born
Keyword(s):  
Positive Definiteness ◽  
Deformation Energy ◽  
The Body ◽  
Simple Explanation ◽  
Crystal Lattices ◽  
Simple Lattice ◽  
Macroscopic Deformation ◽  
The Face ◽  
The Stability ◽  
Central Forces

The stability of lattices is discussed from the standpoint of the method of small vibrations. It is shown that it is not necessary to determine the whole vibrational spectrum, but only its long wave part. The stability conditions are nothing but the positive definiteness of the macroscopic deformation energy, and can be expressed in the form of inequalities for the elastic constants. A new method is explained for calculating these as lattice sums, and this method is applied to the three monatomic lattice types assuming central forces. In this way one obtains a simple explanation of the fact that the face-centred lattice is stable, whereas the simple lattice is always unstable and the body-centred also except for small exponents of the attractive forces. It is indicated that this method might be used for an improvement of the, at present, rather unsatisfactory theory of strength.

The Stability of Crystal Lattices

1941 ◽  
Vol 37 (1) ◽  
pp. 34-54 ◽  
Author(s):  
R. Fürth
Keyword(s):  
Melting Temperature ◽  
Elastic Constants ◽  
Crystal Lattices ◽  
Short Survey ◽  
The Stability ◽  
Good Agreement

A short survey of Born's theory of the thermodynamics and melting of crystals is given. It is shown that Lindemann's and Grüneisen's law for the normal melting temperature can be deduced from this theory, and that the dependence of the melting temperature on pressure, and of the compressibility and the elastic constants on pressure and temperature, as predicted by the theory, are in good agreement with experiment. Several connexions between breaking and melting, suggested by the fundamental ideas on melting and stability of crystals, are discussed and verified. Finally a relation between the heat of melting and the heat of sublimation is deduced and compared with experiment.

The thermodynamics of crystal lattices

1944 ◽  
Vol 40 (2) ◽  
pp. 151-166 ◽  
Author(s):  
Margaret M. Gow
Keyword(s):  
Equation Of State ◽  
Elastic Constants ◽  
Preceding Paper ◽  
Numerical Results ◽  
Crystal Lattices ◽  
Cubic Lattice ◽  
Central Forces

SummaryUsing the results of the preceding paper of this series (by M. Bradburn), the equation of state and the elastic constants for a monatomic face-centred cubic lattice are calculated. Central forces with a potential of the form −ar−m+br−n are assumed to act between the particles. Numerical results are obtained for five sets of values (m, n) and represented in tables and diagrams. The general features of these are discussed and compared with previous computations.I take this opportunity of thanking Prof. Born who suggested this problem to me, for his interest in my work and his advice on many occasions. I am also indebted to Dr R. Fürth for his numerous helpful suggestions.

The thermodynamics of crystal lattices

1943 ◽  
Vol 39 (2) ◽  
pp. 100-103 ◽  
Author(s):  
Max Born
Keyword(s):  
Melting Point ◽  
Crystal Lattice ◽  
Temperature Dependence ◽  
Elastic Constants ◽  
Original Problem ◽  
Lattice Stability ◽  
Chemical Physics ◽  
Crystal Lattices ◽  
Fundamental Difficulty ◽  
The Stability

In 1939 I published a paper in the American Journal of Chemical Physics (Born(1)) in which I tried to develop the thermodynamics of a crystal lattice in the domain of classical (Boltzmann) statistics. Definite formulae and numerical tables for the temperature dependence of the elastic constants up to the melting point were obtained; nevertheless the work was unsatisfactory not only because the approximations used were rough and their accuracy not known, but because a fundamental difficulty with respect to lattice stability turned up. This led to a series of investigations by my collaborators and myself, published in these Proceedings under the title ‘On the stability of crystal lattices’ (quoted here as S I to SIX), by which the difficulty mentioned has been removed. It is now possible to return to the original problem, to which the present series of papers is devoted. In this introduction I wish to recapitulate the whole situation and to explain the plan of the following papers which will be published by my collaborators.

The stability of hexagonal lattices with a simple law of force

1952 ◽  
Vol 48 (2) ◽  
pp. 316-328 ◽  
Author(s):  
F. R. N. Nabarro ◽  
J. H. O. Varley
Keyword(s):  
Binding Energy ◽  
Lower Energy ◽  
Lattice Spacing ◽  
Hexagonal Lattice ◽  
Equilibrium Structure ◽  
Hexagonal Structure ◽  
Free Electrons ◽  
Nearest Neighbours ◽  
The Stability ◽  
Central Forces

AbstractA hexagonal lattice bound by central forces between nearest neighbours is always close-packed. If there is in addition an energy which, like the Fermi energy of free electrons, depends only on the volume, a close-packed equilibrium structure is still possible, but there may be another hexagonal structure of lower energy. A numerical example is given in which this occurs, the lattice spacing, binding energy and elastic constants being comparable with those of Zn, and the law of force showing no obvious peculiarities.

Pressure dependent ultrasonic properties of hcp hafnium metal

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ramanshu P. Singh ◽  
Shakti Yadav ◽  
Giridhar Mishra ◽  
Devraj Singh
Keyword(s):  
Elastic Constants ◽  
Pressure Range ◽  
Room Temperature ◽  
Ultrasonic Attenuation ◽  
Coupling Constants ◽  
Ultrasonic Properties ◽  
Acoustic Coupling ◽  
Third Order Elastic Constants ◽  
The Stability ◽  
The Given

Abstract The elastic and ultrasonic properties have been evaluated at room temperature between the pressure 0.6 and 10.4 GPa for hexagonal closed packed (hcp) hafnium (Hf) metal. The Lennard-Jones potential model has been used to compute the second and third order elastic constants for Hf. The elastic constants have been utilized to calculate the mechanical constants such as Young’s modulus, bulk modulus, shear modulus, Poisson’s ratio, and Zener anisotropy factor for finding the stability and durability of hcp hafnium metal within the chosen pressure range. The second order elastic constants were also used to compute the ultrasonic velocities along unique axis at different angles for the given pressure range. Further thermophysical properties such as specific heat per unit volume and energy density have been estimated at different pressures. Additionally, ultrasonic Grüneisen parameters and acoustic coupling constants have been found out at room temperature. Finally, the ultrasonic attenuation due to phonon–phonon interaction and thermoelastic mechanisms has been investigated for the chosen hafnium metal. The obtained results have been discussed in correlation with available findings for similar types of hcp metals.

On the stability of crystal lattices. II

1940 ◽  
Vol 36 (2) ◽  
pp. 173-182 ◽  
Author(s):  
Rama Dhar Misra
Keyword(s):  
Potential Energy ◽  
The Body ◽  
Complete Agreement ◽  
Stability Conditions ◽  
Crystal Lattices ◽  
Cubic Lattices ◽  
Simple Lattice ◽  
The Face ◽  
The Law ◽  
The Stability

On the assumption that the potential energy of the three cubic lattices of the Bravais type consists of two terms, an attractive one proportional to r−m and a repulsive one proportional to r−n, n > m, stability conditions are expressed in the form that two functions of the number n should be monotonically increasing. These functions have been calculated numerically for n = 4 to 15, and are represented as curves with the abscissa n. The result is that the face-centred lattice is completely stable, that the body-centred lattice is unstable for large exponents in the law of force, and that the simple lattice is always unstable,—in complete agreement with the results of Part I.

Elastic anomalies of Hg2X2 compounds

1992 ◽  
Vol 70 (9) ◽  
pp. 745-751
Author(s):  
K. S. Viswanathan ◽  
J. C. Jeeja Ramani
Keyword(s):  
Elastic Constants ◽  
Fourth Order ◽  
Order Parameter ◽  
Zero Point ◽  
Stability Conditions ◽  
Point Energy ◽  
The Third ◽  
Limit Formula ◽  
The Stability ◽  
Elastic Anomalies

The anomalies of the second-, third-, and fourth-order elastic constants are considered for the phase transition of Hg2X2 type of compounds. Expressions are obtained for the equilibrium values of the order parameters in the ferroelastic phase from the stability conditions. The fluctuation in the order parameter is evaluated from the Landau–Khalatnikov equation. An expression is derived for the shift in the zero-point energy in the low-temperature ferroelastic phase and the specific heat anomaly. It is shown that these are proportional to (T − T)2 and (T − Tc), respectively. All the anomalies of the second-order elastic (SOE) constants are obtained from a single general formula, and relations among them are established. The temperature variation of the SOE constants in the limit [Formula: see text] is discussed. Similarly, expressions are derived for the anomalies of the third- and fourth-order elastic constants. In the limit [Formula: see text] it is shown that these constants diverge as [Formula: see text] and [Formula: see text], respectively.

The stability of crystal lattices. III

1940 ◽  
Vol 36 (4) ◽  
pp. 454-465 ◽  
Author(s):  
M. Born ◽  
R. Fürth
Keyword(s):  
Tensile Strength ◽  
Energy Density ◽  
Numerical Calculations ◽  
Stability Conditions ◽  
Equilibrium Conditions ◽  
Crystal Lattices ◽  
Cubic Lattice ◽  
Special Assumption ◽  
The Face ◽  
The Stability

The energy density of a cubic lattice, homogeneously deformed by a force acting in the direction of one axis, is calculated, and the equilibrium conditions and the stability conditions for any arbitrary small additional deformations are derived. A special assumption is made as to the law of force between the atoms, and the numerical calculations are performed for the face-centred lattice. In this way the strain as a function of the deformation is calculated and, from the stability conditions, the tensile strength is determined. The results are not in agreement with the experimental facts, and the possible reasons for this disagreement are discussed.

Elastic constants and anisotropic pair correlations in solid hydrogen and deuterium

1979 ◽  
Vol 57 (2) ◽  
pp. 136-146 ◽  
Author(s):  
S. Luryi ◽  
J. Van Kranendonk
Keyword(s):  
Elastic Constants ◽  
Phonon Dispersion ◽  
Debye Model ◽  
Elasticity Tensor ◽  
Pair Correlations ◽  
Nearest Neighbours ◽  
Einstein Model ◽  
The One ◽  
Central Forces ◽  
Phonon Dispersion Relations

The anisotropic displacement–displacement correlation function for the two types of pairs of nearest neighbours in solid hep hydrogen and deuterium is studied. Two mechanisms contributing to the deviation of the pair distribution function from axial symmetry around the pair axis are identified. The one is due to the anisotropy of the phonon dispersion relations and is treated in a generalized Debye model parameterized in terms of the elastic constants. The elasticity tensor is decomposed into rotationally irreducible parts, and certain new relations between the elastic constants of hep crystals with central forces are derived. The other mechanism arises from the immediate, anisotropic environment of a pair and is treated using a generalized Einstein model. The relevance of these results for the interpretation of the microwave spectrum of pairs of orthohydrogen molecules in parahydrogen is also discussed.

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