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.

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.

scholarly journals Thermal scattering of X-rays by crystals II. The thermal scattering of the face-centred cubic and the close-packed hexagonal lattices

1947 ◽  
Vol 188 (1013) ◽  
pp. 189-208 ◽  
Keyword(s):  
Elastic Constants ◽  
Hexagonal Lattice ◽  
Preceding Paper ◽  
Scattering Matrix ◽  
Lattice Points ◽  
Dynamical Matrix ◽  
X Rays ◽  
Cubic Lattice ◽  
The Face ◽  
Thermal Scattering

The results of the preceding paper (Part I) are here applied to two special lattices, the face-centred cubic and the close-packed hexagonal lattices. In both cases the assumption is made that only next-neighbour atoms act on one another. In the case of the cubic lattice the number of atomic constants turns out to be equal to that of the elastic constants, so that the dynamical matrix can be expressed in terms of the latter. Numerical calculations are performed taking for the elastic constants those of potassium chloride; the results are compared with those obtained by Iona who used the correct ionic forces. Then the scattering matrix is calculated and a diagram of equi-diffusion lines is drawn which covers a part of the reciprocal space containing nine lattice points. In the case of the hexagonal lattice the number of atomic constants is greater than that of the elastic constants. The dynamical matrix is given in terms of the former; but equi-diffusion lines are constructed only for the vicinity of the selective reflexions (Jahn case) where the scattering can be expressed in terms of the elastic constants.

scholarly journals On the determination of molecular fields. —II. From the equation of state of a gas

1924 ◽  
Vol 106 (738) ◽  
pp. 463-477 ◽  
Keyword(s):  
Equation Of State ◽  
Molecular Model ◽  
Virial Coefficients ◽  
Preceding Paper ◽  
Molecular Field ◽  
Inverse Power ◽  
Theoretical Expression ◽  
Inverse Power Law ◽  
Special Cases ◽  
Molecular Fields

The investigation of a preceding paper has shown that the temperature variation of viscosity, as determined experimentally, can be satisfactorily explained in many gases on the assumption that the repulsive and attractive parts of the molecular field are each according to an inverse power of the distance. In some cases, in argon, for example, it was further shown that the experimental facts can be explained by more than one molecular model, from which we inferred that viscosity results alone are insufficient to determine precisely the nature of molecular fields. The object of the present paper is to ascertain whether a molecular model of the same type will also explain available experimental data concerning the equation of state of a gas, and if so, whether the results so obtained, when taken in conjunction with those obtained from viscosity, will definitely fix the molecular field. Such an investigation is made possible by the elaborate analysis by Kamerlingh Onnes of the observational material. He has expressed the results in the form of an empirical equation of state of the type pv = A + B/ v + C/ v 2 + D/ v 4 + E/ v 6 + F/ v 8 , where the coefficients A ... F, called by him virial coefficients , are determined as functions of the temperature to fit the observations. Now it is possible by various methods to obtain a theoretical expression for B as a function of the temperature and a strict comparison can then be made between theory and experiment. Unfortunately the solution for B, although applicable to any molecular model of spherical symmetry, is purely formal and contains an integral which can be evaluated only in special cases. This has been done up to now for only two simple models, viz., a van der Waals molecule, and a molecule repelling according to an inverse power law (without attraction), but it is shown in this paper that it can also be evaluated in the case of the model, which was successful in explaining viscosity results. As the two other models just mentioned are particular cases of this, the appropriate formulæ for B are easily deduced from the general one given here.

Equation of State and Elastic Constants of Compressed fcc Cu

2010 ◽  
Vol 27 (3) ◽  
pp. 036403 ◽  
Author(s):  
Bai Li-Gang ◽  
Liu Jing
Keyword(s):  
Equation Of State ◽  
Elastic Constants

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.

Lattice Dynamical Prediction of the Elastic Constants of Diamond

1996 ◽  
Vol 453 ◽  
Author(s):  
Robert R. Reeber
Keyword(s):  
Equation Of State ◽  
Elastic Constants ◽  
Thermophysical Properties ◽  
Room Temperature ◽  
Interatomic Potential ◽  
Dynamical Theory ◽  
Potential Models ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Metastable Material

AbstractThe thermophysical properties of diamond, a metastable material at room temperature, are difficult to measure at high temperatures. These properties are of interest for testing equation of state and interatomic potential models. Here we utilize a geometrical lattice transformation, one dimensional lattice dynamical theory, and the principle of corresponding states to calculate the elastic constants of diamond over an extended temperature range.

The Contact Problem for a Rigid Inclusion Pressed Between Two Dissimilar Elastic Half Planes

1981 ◽  
Vol 48 (1) ◽  
pp. 104-108
Author(s):  
G. M. L. Gladwell
Keyword(s):  
Contact Problem ◽  
Plane Strain ◽  
Integral Equations ◽  
Elastic Constants ◽  
Contact Stress ◽  
Rigid Inclusion ◽  
Numerical Results ◽  
Stress Distributions ◽  
Plane Strain Problem

Paper concerns the plane-strain problem of a rigid, thin, rounded inclusion pressed between two isotropic elastic half planes with different elastic constants. Required to find the extents of the contact regions between each plane and the inclusion, and the contact stress distributions. The governing integral equations are solved approximately by using Chebyshev expansions. Numerical results are presented.

Elastic Moduli and Elastic Constants of Alloy AuCuSi With FCC Structure Under Pressure

2019 ◽  
Vol 38 (2019) ◽  
pp. 264-272 ◽  
Author(s):  
Nguyen Quang Hoc ◽  
Bui Duc Tinh ◽  
Nguyen Duc Hien
Keyword(s):  
Elastic Constants ◽  
Elastic Moduli ◽  
Nearest Neighbor ◽  
Young Modulus ◽  
Numerical Results ◽  
Main Metal ◽  
Substitution Alloy ◽  
Rigidity Modulus ◽  
The Mean ◽  
Fcc Structure

AbstractThis paper studies on the dependence of the mean nearest neighbor distance, the Young modulus E, the bulk modulus K, the rigidity modulus G and the elastic constants C11, C12, C44 on temperature, pressure, the concentration of substitution atoms and the concentration of interstitial atoms for alloy AuCuSi (substitution alloy AuCu with interstitial atom Si) with FCC structure by the way of the statistical moment method (SMM). The numerical results for alloy AuCuSi are compared with the numerical results for main metal Au, substitution alloy AuCu, interstitial alloy AuSi, other calculated results and experiments.

Stability of finite-amplitude interfacial waves. Part 2. Numerical results

1985 ◽  
Vol 160 ◽  
pp. 317-336 ◽  
Author(s):  
D. I. Pullin ◽  
R. H. J. Grimshaw
Keyword(s):  
Lower Layer ◽  
Modulational Instability ◽  
Preceding Paper ◽  
Boussinesq Approximation ◽  
Finite Amplitude ◽  
Numerical Results ◽  
Linearized Stability ◽  
Local Wave ◽  
Truncated Fourier Series ◽  
Wave Induced

In the preceding paper (Grimshaw & Pullin 1985) we discussed the long-wavelength modulational instability of interfacial progressive waves in a two-layer fluid. In this paper we complement our analytical results by numerical results for the linearized stability of finite-amplitude waves. We restrict attention to the case when the lower layer is infinitely deep, and use the Boussinesq approximation. For this case the basic wave profile has been calculated by Pullin & Grimshaw (1983a, b). The linearized stability problem for perturbations to the basic wave is solved numerically by seeking solutions in the form of truncated Fourier series, and solving the resulting eigenvalue problem for the growth rate as a function of the perturbation wavenumber. For small or moderate basic wave amplitudes we show that the instabilities are determined by a set of low-order resonances. The lowest resonance, which contains the modulational instability, is found to be dominant for all cases considered. For higher wave amplitudes, the resonance instabilities are swamped by a local wave-induced Kelvin–Helmholtz instability.

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