The stability of crystal lattices. III

Author(s):  
M. Born ◽  
R. Fürth

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.

Author(s):  
Rama Dhar Misra

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.


Author(s):  
S. C. Power

It is shown that the theorem stated in Born's paper, and proved for the case of a linear lattice of N equal particles under certain restrictions concerning the forces between the particles, that macroscopic stability (stability for long waves) implies microscopic stability, may be extended to three dimensions for the particular case of a face-centred cubic lattice, where the effects of all neighbours, other than the first twelve neighbours, are neglected.I take this opportunity of expressing my sincere thanks to Prof. Born for much valuable advice.


Author(s):  
Max Born

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.


Author(s):  
H. W. Peng ◽  
S. C. Power

The main purpose of the paper is an investigation of the stability of a certain class of Bravais lattices, namely, those with a rhombohedral cell of arbitrary angle. The potential energy is assumed to consist of two terms, each proportional to a reciprocal power of the distance. In the continuous series of lattices obtained by changing the rhombohedral angle, there are included the three cubic Bravais lattices, the simple (s), the face-centred (f) and the body-centred (b) lattices. It is shown that (f) and (b) correspond to a minimum of the potential energy, and (s) to a maximum. A method for calculating the potential energy for the intermediate rhombohedral lattices is developed, and, with the help of a certain characteristic function, it is shown by numerical calculation that the (f) lattice corresponds to the absolute minimum of potential energy, and that no extrema, other than (f), (s) and (b), exist. In the last section, the case of a compound (non-Bravais lattice) is considered, and it is shown that the equilibrium and stability conditions for the law of force assumed can be divided into one set for change of volume, and an independent set for change of shape.We take this opportunity of expressing our sincere thanks to Prof. Born for his interest in our work, and for much valuable advice.


Author(s):  
Max Born ◽  
Rama Dhar Misra

In order to explain the phenomena of melting, tensile strength, etc., we have to investigate the stability of crystals for finite deformations, for which deviations from Hooke's law occur. Although these deviations are in most cases of an irreversible character, it is necessary, for a systematic study, to develop mathematical methods for treating the mechanical (reversible) case of a highly strained crystal lattice, where terms of higher order than the second in the deformation energy must be taken into account.


2016 ◽  
Vol 72 (2) ◽  
pp. 243-249 ◽  
Author(s):  
Hamzeh Mujahed ◽  
Benedek Nagy

The Wiener index of a connected graph, known as the `sum of distances', is the first topological index used in chemistry to sum the distances between all unordered pairs of vertices of a graph. The Wiener index, sometimes called the Wiener number, is one of the indices associated with a molecular graph that correlates physical and chemical properties of the molecule, and has been studied for various kinds of graphs. In this paper, the graphs of lines of unit cells of the face-centred cubic lattice are investigated. This lattice is one of the simplest, the most symmetric and the most usual, cubic crystal lattices. Its graphs contain face centres of the unit cells and other vertices, called cube vertices. Closed formulae are obtained to calculate the sum of shortest distances between pairs of cube vertices, between cube vertices and face centres and between pairs of face centres. Based on these formulae, their sum, the Wiener index of a face-centred cubic lattice with unit cells connected in a row graph, is computed.


Author(s):  
Alokananda Kar ◽  
Shouvik Sadhukhan ◽  
Surajit Chattopadhyay

In this paper, we study two different cases of inhomogeneous EOS of the form [Formula: see text]. We derive the energy density of dark fluid and dark matter component for both the cases. Further, we calculate the evolution of energy density, gravitational constant and cosmological constant. We also explore the finite time singularity and thermodynamic stability conditions for the two cases of EOS. Finally, we discuss the thermodynamics of inhomogeneous EOS with the derivation of internal energy, Temperature and entropy and also show that all the stability conditions are satisfied for the two cases of EOS.


2020 ◽  
Vol 2 (1) ◽  
pp. 32-55
Author(s):  
Natalia Christofoletti Barrenha
Keyword(s):  
The Face ◽  

This text seeks to explore the Argentine films Castro (Alejo Moguillansky, 2009) and El asaltante (Pablo Fendrik, 2007) from within the displacement of their characters through the city. This transit configures the organising element of the plots, determining the direction and rhythm of events. The escape motto will structure the film analyses, which are also twinned by the sensory apprehension that comes from the spaces they travel through. The notion of escape, as explored by Esteban Dipaola in Argentine cinema of the 1990s, continues to throb in mid-to-late 2000s production, and in these films represents the means by which the protagonists deploy critical attitudes—sometimes radical and explosive, sometimes silent—in the face of fixed notions, suggesting some scepticism about the “stability” and “order” that they (dis)encounter in normality. RESUMEN Este texto busca explorar los largometrajes argentinos Castro (Alejo Moguillansky, 2009) y El asaltante (Pablo Fendrik, 2007) a partir del desplazamiento de sus personajes por la ciudad. El transitar se configura como elemento organizador de las tramas, determinando la dirección y el ritmo de los acontecimientos. El tema de la fuga irá estructurando los análisis de las películas, las cuales también están relacionadas por la aprehensión sensorial que hacen de los espacios que recorren. La noción de fuga, tal y como fue explorada por Esteban Dipaola en el cine argentino de los años 90, continúa vigente en la producción de mediados/fines de la primera década del siglo XXI, y en estas películas es el recurso por medio del cual los protagonistas despliegan actitudes críticas – a veces radicales y explosivas, y a veces silenciosas – frente a nociones convencionales, lo cual hace pensar que existe un cierto escepticismo con relación a la “estabilidad” y al “orden” que ellos (des)encuentran en la normalidad. RESUMO Este texto busca explorar os longas-metragens argentinos Castro (Alejo Moguillansky, 2009) e El asaltante (Pablo Fendrik, 2007) a partir do deslocamento de seus personagens pela cidade. O transitar configura-se como elemento organizador das tramas, determinando a direção e o ritmo dos acontecimentos. O mote da fuga estruturará as análises dos filmes, os quais também se irmanam pela apreensão sensorial que fazem dos espaços que percorrem. A noção de fuga, conforme explorada por Esteban Dipaola no cinema argentino da década de 1990, continua a pulsar na produção de meados/fins dos anos 2000, e é, nestes filmes, o recurso através do qual os protagonistas desdobram atitudes críticas – às vezes radicais e explosivas, às vezes silenciosas – diante de noções fixas, sugerindo certo ceticismo em relação à “estabilidade” e à “ordem” que eles (des)encontram na normalidade.


2020 ◽  
Vol 21 (8) ◽  
pp. 741-747
Author(s):  
Liguang Zhang ◽  
Yanan Shen ◽  
Wenjing Lu ◽  
Lengqiu Guo ◽  
Min Xiang ◽  
...  

Background: Although the stability of proteins is of significance to maintain protein function for therapeutical applications, this remains a challenge. Herein, a general method of preserving protein stability and function was developed using gelatin films. Method: Enzymes immobilized onto films composed of gelatin and Ethylene Glycol (EG) were developed to study their ability to stabilize proteins. As a model functional protein, β-glucosidase was selected. The tensile properties, microstructure, and crystallization behavior of the gelatin films were assessed. Result: Our results indicated that film configurations can preserve the activity of β-glucosidase under rigorous conditions (75% relative humidity and 37°C for 47 days). In both control films and films containing 1.8 % β-glucosidase, tensile strength increased with increased EG content, whilst the elongation at break increased initially, then decreased over time. The presence of β-glucosidase had a negligible influence on tensile strength and elongation at break. Scanning electron-microscopy (SEM) revealed that with increasing EG content or decreasing enzyme concentrations, a denser microstructure was observed. Conclusion: In conclusion, the dry film is a promising candidate to maintain protein stabilization and handling. The configuration is convenient and cheap, and thus applicable to protein storage and transportation processes in the future.


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