scholarly journals A new equation for period vectors of crystals under external stress and temperature in statistical physics: mechanical equilibrium condition and equation of state

2021 ◽  
Vol 136 (1) ◽  
Author(s):  
Gang Liu
Keyword(s):  
Equation Of State ◽  
Equilibrium Condition ◽  
Statistical Physics ◽  
Structural Phase ◽  
External Pressure ◽  
External Stress ◽  
Mechanical Equilibrium ◽  
Dependent Form ◽  
New Equation ◽  
External Forces

AbstractStarting with the rigorous derivation of the work done on the center cell by external forces, a new equation is derived for the period vectors (cell edge vectors) in crystals under external stress and temperature. Since the equation is based on the principles of statistical physics, it applies to both classical and quantum systems. The existing theory for crystals under external pressure is covered as a special case. The new equation turns out to be the mechanical equilibrium condition and the equation of state for crystals under external stress and temperature. It may be used to predict crystal structures and to study structural phase transitions and crystal expansions. For linear elastic crystals, it takes the microscopic and temperature-dependent form of the generalized Hooke’s law, and may therefore be used to calculate the corresponding elastic constants. It should be helpful in studying piezoelectric and piezomagnetic materials, as the period vectors change with external stress. It is also consistent and can be combined with the previously derived corresponding one for Newtonian dynamics.

scholarly journals Crystal Period Vectors under External Stress in Statistical Physics

2020 ◽  
Author(s):  
Gang Liu
Keyword(s):  
Statistical Physics ◽  
External Pressure ◽  
External Stress ◽  
Temperature Dependent ◽  
Classical Systems ◽  
Linear Elastic ◽  
Dependent Form ◽  
New Equation ◽  
Elastic Crystals ◽  
Special Case

For crystals under external stress and temperature, a general equation to determine their period vectors (cell edge vectors) was derived based on the principles of statistical physics. This equation applies to both classical systems and quantum systems. It is consistent and can be combined with the previously derived one in the Newtonian dynamics. The existing theory for crystals under external pressure is covered as a special case. The new equation is also the mechanical equilibrium condition and the equation of state for crystals under external stress and temperature. It should be helpful in studying piezoelectric and piezomagnetic materials, since the period vectors change with external stress. For linear elastic crystals, it is the microscopic and temperature-dependent form of the generalized Hooke's law, therefore, it can be used to calculate the corresponding elastic constants, for given temperatures.

scholarly journals Crystal Period Vectors under External Stress in Statistical Physics

2019 ◽  
Author(s):  
Gang Liu
Keyword(s):  
General Equation ◽  
Statistical Physics ◽  
Continuous Media ◽  
External Pressure ◽  
External Stress ◽  
Cell Edge ◽  
Newtonian Dynamics ◽  
Work Done ◽  
Crystal Cells

A basic and general equation to determine their period vectors (cell edge vectors) is necessary in physics, especially when crystals are under external stress. It has been derived in Newtonian dynamics in these years. Since statistical physics should also generate such, here we derive it. By extending the normal way for crystals under external pressure, regarding crystal cells as being filled with continuous media, writing the work done by the external stress on the crystal explicitly, and deriving the forces on the surfaces of the cells by the external stress, we arrived at the equation for the period vectors, which is in principle the same as the above mentioned counterpart achieved in Newtonian dynamics. It should be applicable when crystals are under different pressures in different directions, like in piezoelectric and piezomagnetic phenomena.

scholarly journals Crystal Period Vectors under External Stress in Statistical Physics

2019 ◽  
Author(s):  
Gang Liu
Keyword(s):  
General Equation ◽  
Statistical Physics ◽  
Continuous Media ◽  
External Pressure ◽  
External Stress ◽  
Cell Edge ◽  
Newtonian Dynamics ◽  
Work Done ◽  
Crystal Cells ◽  
Special Case

A basic and general equation to determine period vectors (cell edge vectors) is necessary in physics, especially when crystals are under external stress. It has been derived in Newtonian dynamics. Since statistical physics should also generate such equation, we will provide a derivation. By extending the normal derivation for crystals under external pressure, regarding crystal cells as being filled with continuous media, formulating the work done by the external stress on the crystal explicitly, and deriving the forces on the surfaces of the cells by the external stress, we arrived at the equation for the period vectors, which is in principle the same as the above mentioned counterpart achieved in Newtonian dynamics. Everything also restores when the external stress reduces to the special case of external pressure. It should be applicable when crystals are under different pressures in different directions, like in piezoelectric and piezomagnetic phenomena.

scholarly journals Crystal Period Vectors under External Stress in Statistical Physics

2020 ◽  
Author(s):  
Gang Liu
Keyword(s):  
Structure Prediction ◽  
Statistical Physics ◽  
Quantum Physics ◽  
External Stress ◽  
Temperature Dependent ◽  
Classical Physics ◽  
Linear Elastic ◽  
Dependent Form ◽  
New Form ◽  
Elastic Crystals

For crystal periodic structure prediction, a new and concise approach based on the principles of statistical physics was employed to derive a new form of the equation to determine their period vectors (cell edge vectors), under general external stress. Then the new form is applicable to both classical physics and quantum physics. It also turned out to be the equation of state and the specific explicit equilibrium condition for crystals under external stress and temperature. It should be helpful in piezoelectric and piezomagnetic studies, as the period vectors were changed by the external stress. For linear elastic crystals, it is actually also the microscopic but temperature-dependent form of the generalized Hooke's law, then can be used to calculate the corresponding elastic constants of the law, for given temperatures.

scholarly journals Crystal Period Vectors under External Stress in Statistical Physics

2020 ◽  
Author(s):  
Gang Liu
Keyword(s):  
Structure Prediction ◽  
Statistical Physics ◽  
Quantum Physics ◽  
External Stress ◽  
Temperature Dependent ◽  
Classical Physics ◽  
Linear Elastic ◽  
Dependent Form ◽  
New Form ◽  
Elastic Crystals

For crystal periodic structure prediction, a new and concise approach based on the principles of statistical physics was employed to derive a new form of the equation to determine their period vectors (cell edge vectors), under general external stress. Then the new form is applicable to both classical physics and quantum physics. It also turned out to be the equation of state and the mechanical equilibrium condition for crystals under external stress and temperature. It should be helpful in piezoelectric and piezomagnetic studies, as the period vectors were changed by the external stress. For linear elastic crystals, it is actually also the microscopic but temperature-dependent form of the generalized Hooke's law, then can be used to calculate the corresponding elastic constants of the law, for given temperatures.

scholarly journals Crystal Period Vectors under External Stress in Statistical Physics

2020 ◽  
Author(s):  
Gang Liu
Keyword(s):  
Equation Of State ◽  
General Equation ◽  
Structure Prediction ◽  
Statistical Physics ◽  
Quantum Physics ◽  
External Stress ◽  
Classical Physics ◽  
Cell Edge ◽  
Newtonian Dynamics ◽  
New Form

In crystal periodic structure prediction, a basic and general equation is needed to determine their period vectors (cell edge vectors), especially under arbitrary external stress. It was derived in Newtonian dynamics years ago, which can be combined with quantum physics by further modeling. Here a new and concise approach based on the principles of statistical physics was employed to derive it into a new form, then applicable to both classical physics and quantum physics by its own. The new form also turned out to be the specific explicit equilibrium condition and the equation of state for crystals under external stress and temperature. This work was also compared with the elasticity theory.

scholarly journals Crystal Period Vectors under External Stress in Statistical Physics

2020 ◽  
Author(s):  
Gang Liu
Keyword(s):  
Structure Prediction ◽  
Statistical Physics ◽  
Quantum Physics ◽  
External Stress ◽  
Temperature Dependent ◽  
Classical Physics ◽  
Linear Elastic ◽  
Dependent Form ◽  
New Form ◽  
Elastic Crystals

For crystal periodic structure prediction, a new and concise approach based on the principles of statistical physics was employed to derive a new form of the equation to determine their period vectors (cell edge vectors), under general external stress. Then the new form is applicable to both classical physics and quantum physics. It also turned out to be the equation of state and the specific explicit equilibrium condition for crystals under external stress and temperature. It should be helpful in piezoelectric and piezomagnetic studies, as the period vectors were changed by the external stress. For linear elastic crystals, it is actually also the microscopic but temperature-dependent form of the generalized Hooke's law, then can be used to calculate the corresponding elastic constants of the law, for given temperatures.

scholarly journals Crystal Period Vectors under External Stress in Statistical Physics

2020 ◽  
Author(s):  
Gang Liu
Keyword(s):  
Structure Prediction ◽  
Statistical Physics ◽  
Quantum Physics ◽  
External Stress ◽  
Temperature Dependent ◽  
Classical Physics ◽  
Linear Elastic ◽  
Dependent Form ◽  
New Form ◽  
Elastic Crystals

For crystal periodic structure prediction, a new and concise approach based on the principles of statistical physics was employed to derive a new form of the equation to determine their period vectors (cell edge vectors), under general external stress. Then the new form is applicable to both classical physics and quantum physics. It also turned out to be the equation of state and the specific explicit equilibrium condition for crystals under external stress and temperature. It should be helpful in piezoelectric and piezomagnetic studies, as the period vectors were changed by the external stress. For linear elastic crystals, it is actually also the microscopic but temperature-dependent form of the generalized Hooke's law, then can be used to calculate the corresponding elastic constants of the law, for given temperatures.

Algorithms for Density, Potential Temperature, Conservative Temperature, and the Freezing Temperature of Seawater

2006 ◽  
Vol 23 (12) ◽  
pp. 1709-1728 ◽  
Author(s):  
David R. Jackett ◽  
Trevor J. McDougall ◽  
Rainer Feistel ◽  
Daniel G. Wright ◽  
Stephen M. Griffies
Keyword(s):  
Thermal Expansion ◽  
Equation Of State ◽  
Thermodynamic Potential ◽  
Potential Temperature ◽  
Inverse Function ◽  
Freezing Temperature ◽  
Ocean Models ◽  
The Difference ◽  
New Equation ◽  
Density Equation

Abstract Algorithms are presented for density, potential temperature, conservative temperature, and the freezing temperature of seawater. The algorithms for potential temperature and density (in terms of potential temperature) are updates to routines recently published by McDougall et al., while the algorithms involving conservative temperature and the freezing temperatures of seawater are new. The McDougall et al. algorithms were based on the thermodynamic potential of Feistel and Hagen; the algorithms in this study are all based on the “new extended Gibbs thermodynamic potential of seawater” of Feistel. The algorithm for the computation of density in terms of salinity, pressure, and conservative temperature produces errors in density and in the corresponding thermal expansion coefficient of the same order as errors for the density equation using potential temperature, both being twice as accurate as the International Equation of State when compared with Feistel’s new equation of state. An inverse function relating potential temperature to conservative temperature is also provided. The difference between practical salinity and absolute salinity is discussed, and it is shown that the present practice of essentially ignoring the difference between these two different salinities is unlikely to cause significant errors in ocean models.

A New Equation of State for Fluids

1928 ◽  
Vol 63 (5) ◽  
pp. 229 ◽  
Author(s):  
James A. Beattie ◽  
Oscar C. Bridgeman
Keyword(s):  
Equation Of State ◽  
New Equation
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