Stress theorem

Stress Theorem in the Determination of Static Equilibrium by the Density Functional... 

The general stress theorem may be used to estimate temperature rises, if the coil energy is dumped uniformly throughout the low-temperature structure. Rewriting equation (1),... 

Thus, if the average a is --150MN/m (20,000psi), E/V ISOMJ/rn, i.e., a maximum temperature of --120K. In principal, any coil can probably be made self-protecting by this technique, except the energy store, which has cheated the stress theorem. 

The interpretation of the terms in the stress theorem are discussed in more detail in Ref. 13 and in papers to be published in Ref. 11. The basic idea is that the kinetic term is the quantum analogue of classical gas theory. It represents momentum p transferred by a flux p /m. The diagonal aa terms are always positive, i.e., expansive terms. The virial term is associated with potential forces in the a direction between particles displaced in the p direction. The diagonal terms are an energy density and must be negative (attractive) for system which is in equilbrium ( = 0) in order to cancel the expansive kinetic term, The off-diagonal terms may, of course, have either sign. 

The stress theorem is very useful in large macroscopic systems. In particular, since the structure of a crystal is completely specified by the size and shape of the unit ceil and positions of the atoms in the unit cell, the force and stress theorems give all the generalized forces conjugate to these variables. If the stress theorem is expressed in terms of relative coordinates and proper care is taken of the Coulomb interactions, then the stress theorem, Eqs. (8) and (8a), give the total stress in terms of the intrinsic bulk Hamiltonian and wave functions in the bulk of the crystal. Thus, the stress and force are sufficient to construct the complete equation of the state of any crystal. In general, the more complex and the lower the symmetry of the crystal, the more useful are the forces and stresses in the calculations. This is further described by Nielson in this proceedings and later in the present paper. 

It is most natural to consider next the elastic properties of the crystal because this is the dependence of the total energy upon the size and shape of the unit cell. As was discussed above, it is a great advantage to use the direct calculation of stress from the "stress theorem." This has been applied to accurate calculations of the elastic properties of the crystals Si, Ge, and GaAs, using methods which have general applicability to all solids. 

We have also investigated the stability of C in diamond, BC8 and another structure (MSG) discussed in Ref. 30. in Fig. 7 is shown the enthalpy H = E + PV for C in the region of the transition. Since P ia given by the stress theorem, H may be calculated directly. The main point is that for... 

The stress theorem determines the stress from the electronic ground state of any quantum system with arbitrary strains and atomic displacements. We derive this theorem in reciprocal space, within the local-density-functional approximation. The evaluation of stress, force and total energy permits, among other things, the determination of complete stress-strain relations including all microscopic internal strains. We describe results of ab-initio calculations for Si, Ge, and GaAs, giving the equilibrium lattice constant, all linear elastic constants Cy and the internal strain parameter t,. 

A different type of structural parameter was considered recently by the present authors, namely X representing a homogeneous macroscopic strain defined as the linear scaling of all particle positions as x- 1+e)x. The e is a constant 3x3 strain tensor, and e=0 corresponds to some reference configuration. The conjugate force is in this case defined as the macroscopic stress a, and an explicit general expression denoted the "stress theorem" is derived by Nielsen and Martin (1983). The result is a generalization of the quantum virial theorem (Born et al., 1926),... 

The combined force and stress theorems are necessary and sufficient to describe the general equation-of-state of a quantum system, i.e. the relations of force and stress to displacements and strain, and thus constitute a powerful tool in the study of structural and dynamic properties of matter. For example, phonon properties can be studied in great detail by imposing regular nuclear displacements and calculating the restoring forces. The reader is referred to the papers by Martin and by Kune (this volume). 

One application of the stress theorem is the study of elastic properties of solids, which becomes straightforward when a suitable finite macroscopic strain is applied to the solid. When the wavefunctions of the distorted solid are known, the stress tensor is evaluated with the stress theorem. In the harmonic approximation elastic constants are defined as the ratio of stress to strain, and it is furthermore possible to go to large strains to obtain all nonlinear elastic properties. In general it is necessary to be concerned with internal strains that may appear microscopically owing to the lower symmetry of the strained solid. In section 6 we show in detail how this problem is solved by combining the stress and force theorems. 

The present paper is organized as follows Section 2 deals with the pseudopotential technique for solving the Schrddinger equation, section 3 derives the stress theorem expressed in reciprocal space within the local-density approximation. Section 4 comments on a number of technical but nevertheless important points in ab-initio calculations. Section 5 deals with calculations on the semiconductors Si, Ge, and GaAs, whose elastic properties is the topic of section 6. 

The stress theorem relies upon the variational principle applied together with a strain-scaling of the quantum system, as discussed in detail by the present authors elsewhere (Nielsen and Martin, to be published). The strain scales particle positions as x- (1 + e)x, and by definition the macroscopic stress per volume n (a and B denote cartesian coordinates) is derived from the total energy by... 

The equilibrium structure is diamond for Si and Ge, and zincblende for GaAs, as was verified in recent theoretical work (Yin and Cohen, 1982 Froyen and Cohen, 1982). With the given structure as the only input we have calculated the lattice constant, a, using the stress theorem. A first calculation of... 

The combined calculation of stress, forces and total energy have thus been shown to constitute a powerful and complete method for the study of structural, elastic and dynamic properties of solids. Using the stress theorem permits rapid determination of lattice parameters, for cubic as well as non-cubic crystals. This leads to large savings in computational effort, compared to direct calculations of the total energy with subsequent numerical differentiation. 

Stresses are usually related to strains through an effective modulus. If the components of stress are nondimensionalized by a suitable scalar modulus c, then they are also of order c. Using (A.94), (A.lOl), and the binomial theorem in (A.39), the relation between the normalized spatial stress s = s/c and the normalized referential stress S = S/c becomes... 

Whitney and Pagano [6-32] extended Yang, Norris, and Stavsky s work [6-33] to the treatment of coupling between bending and extension. Whitney uses a higher order stress theory to obtain improved predictions of a, and and displacements at low width-to-thickness ratios [6-34], Meissner used his variational theorem to derive a consistent set of equations for inclusion of transverse shearing deformation effects in symmetrically laminated plates [6-35]. Finally, Ambartsumyan extended his treatment of transverse shearing deformation effects from plates to shells [6-36]. 

The end effects have been neglected here, including in the expression for change in reservoir entropy, Eq. (178). This result says in essence that the probability of a positive increase in entropy is exponentially greater than the probability of a decrease in entropy during heat flow. In essence this is the thermodynamic gradient version of the fluctuation theorem that was first derived by Bochkov and Kuzovlev [60] and subsequently by Evans et al. [56, 57]. It should be stressed that these versions relied on an adiabatic trajectory, macrovariables, and mechanical work. The present derivation explicitly accounts for interactions with the reservoir during the thermodynamic (here) or mechanical (later) work,... 

In principle this integral could be applied directly to the Maxwell model to predict the decay of stress at any point in time. We can simplify this further with an additional assumption that is experimentally verified, i.e. that the function in the integral is continuous. The first value for the mean theorem for integrals states that if a function f(x) is continuous between the limits a and b there exists a value f(q) such that... 

It is important to stress the fact that in the proof of the MPC theorem, the laws of classical dynamics are never violated. One could summarize the signihcance of the MPC theorem by saying that, for a well-defined class of dynamical systems, the new formulation lays bare the arrow of time that is hidden in the illusorily deterministic formulation of these unstable systems. 

Up to now, potential energies were at the center of our arguments. Little attention was paid to the electronic kinetic energy. This situation arose from our application of the Helhuann-Feynman theorem with intent to stress the role of the potential... 

See Contact stress Hexagonal symmetry 132 Hohenberg-Kohn theorem 113 Hydrogen molecular ion 173 history 173 repulsive force 185 resonance interaction 177 van der Waals force 175 Hydrogen on silicon 336 Image force 56—59, 72, 93 concept 56 effect on tunneling 74 field emission, in 56 jellium model, in 93 observability by STM 72... 

The evaluation of elements such as the M n,fin s is a very difficult task, which is performed with different levels of accuracy. It is sufficient here to mention again the so called sudden approximation (to some extent similar to the Koopmans theorem assumption we have discussed for binding energies). The basic idea of this approximation is that the photoemission of one-electron is so sudden with respect to relaxation times of the passive electron probability distribution as to be considered instantaneous. It is worth noting that this approximation stresses the one-electron character of the photoemission event (as in Koopmans theorem assumption). 

Eunctionals p(D) and p(D) are defined for all measurable D. We should stress that such extensions are not unique. Extension of density (24) using the Hahn-Banach theorem for picking up a random integer was used in a very recent work by Adamaszek (2006). 

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