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

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):  
Quantum Mechanics ◽  
Periodic Structure ◽  
General Equation ◽  
Structure Prediction ◽  
Statistical Physics ◽  
Quantum Physics ◽  
External Stress ◽  
Classical Physics ◽  
Cell Edge ◽  
Newtonian Dynamics

In crystal periodic structure prediction, a general equation is needed to determine the period vectors (cell edge vectors), especially when crystals are under arbitrary external stress. It has been derived in Newtonian dynamics years ago, which can be combined with quantum mechanics by further modeling. Here we derived such an equation in statistical physics, applicable to both classical physics and quantum physics by itself.

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 ◽  
Harmonic Oscillators ◽  
General Belief ◽  
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. Contrary to a general belief, the new form also concluded that harmonic oscillators can cause crystal thermal expansion.

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 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.

Surfaces, Interfaces, and Changing Shapes in Multilayered Films

MRS Bulletin ◽  
1999 ◽  
Vol 24 (2) ◽  
pp. 39-43 ◽  
Author(s):  
Daniel Josell ◽  
Frans Spaepen
Keyword(s):  
Free Energy ◽  
Excess Free Energy ◽  
External Pressure ◽  
Electronic Density ◽  
Spherical Drop ◽  
Crystal Anisotropy ◽  
Fundamental Quantity ◽  
The Difference ◽  
Image Position ◽  
Work Done

It is generally recognized that the capillary forces associated with internal and external interfaces affect both the shapes of liquid-vapor surfaces and wetting of a solid by a liquid. It is less commonly understood that the same phenomenology often applies equally well to solid-solid or solid-vapor interfaces.The fundamental quantity governing capillary phenomena is the excess free energy associated with a unit area of interface. The microscopic origin of this excess free energy is often intuitively simple to understand: the atoms at a free surface have “missing bonds”; a grain boundary contains “holes” and hence does not have the optimal electronic density; an incoherent interface contains dislocations that cost strain energy; and the ordering of a liquid near a solid-liquid interface causes a lowering of the entropy and hence an increase in the free energy. In what follows we shall show how this fundamental quantity determines the shape of increasingly complex bodies: spheres, wires, thin films, and multilayers composed of liquids or solids. Crystal anisotropy is not considered here; all interfaces and surfaces are assumed isotropic.Consideration of the equilibrium of a spherical drop of radius R with surface free energy γ shows that pressure inside the droplet is higher than outside. The difference is given by the well-known Laplace equation:This result can be obtained by equating work done against internal and external pressure during an infinitesimal change of radius with the work of creating a new surface.

Isochronous anharmonic oscillations

1998 ◽  
Vol 76 (8) ◽  
pp. 645-657 ◽  
Author(s):  
Pirooz Mohazzabi
Keyword(s):  
General Equation ◽  
Vertical Plane ◽  
One Dimensional ◽  
One To One ◽  
Anharmonic Oscillations ◽  
Special Case

The problem of a particle oscillating without friction on a curve in a vertical plane (referred to as a vertical curve) is addressed. It is shown that there are infinitely many asymmetric concave vertical curves on which oscillations of a particle remain isochronous. The general equation of these curves is derived, and a one-to-one correspondence between these curves and one-dimensional potentials is established. The results are compared with the existing literature, and an interesting nontrivial special case is discussed. Some issues regarding interpretation of the results in the context of action and angle variables are also addressed. PACS No. 03.20

Asymptotic solutions of the Orr—Sommerfeld equation

1975 ◽  
Vol 280 (1295) ◽  
pp. 271-316
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansions ◽  
Asymptotic Properties ◽  
Matched Asymptotic Expansions ◽  
Asymptotic Solutions ◽  
Asymptotic Approximations ◽  
General Kind ◽  
Work Done ◽  
Sommerfeld Equation ◽  
Special Case

A rigorous justification is given of work done by Eagles (1969), in which he applied the method of matched asymptotic expansions to the Orr-Sommerfeld equation to obtain formal uniform asymptotic approximations to a certain pair of solutions. (Somewhat more polished formal expansions of the same general kind were subsequently obtained by Reid (1972).) First, a study is made of the asymptotic properties of solutions of a certain differential equation which admits the Orr—Sommerfeld equation as a special case. Previous work on this differential equation by Lin & Rabenstein ( i960, 1969) is extended to develop a theory suited to our main purpose: to prove the validity of Eagles’s approximations. It is then shown how this theory can be used to prove the existence of actual solutions of the Orr—Sommerfeld equation approximated by these formal expansions. In addition, it is verified that these solutions have the properties assumed by Eagles (1969).

The Current-Voltage Relationships Due to Transit-Time Effects in an Intrinsic Semiconductor

1975 ◽  
Vol 12 (1) ◽  
pp. 37-43
Author(s):  
Y. W. Lam
Keyword(s):  
Transit Time ◽  
General Equation ◽  
Practical Importance ◽  
Small Signal ◽  
Time Effects ◽  
Saturation Velocity ◽  
Intrinsic Semiconductor ◽  
Current Voltage ◽  
Terminal Voltage ◽  
Special Case

The analysis is based on the small-signal theory whereby all variables are linearized to a d.c. component and an a.c. component at the fundamental frequency. A general equation is then obtained for the a.c. terminal voltage in terms of the a.c. current and transit time. A special case of practical importance is also considered in which the electric field throughout the semiconductor is sufficiently high so that carriers move at their saturation velocity.

Sign in / Sign up

Export Citation Format

Share Document