Variational thermodynamic calculations for some liquid sd metals: II

1986 ◽  
Vol 64 (7) ◽  
pp. 852-856 ◽  
Author(s):  
D. H. Li ◽  
R. A. Moore ◽  
S. Wang
Keyword(s):  
Free Energy ◽  
Transition Metals ◽  
Valence Electron ◽  
Helmholtz Free Energy ◽  
Liquid Metals ◽  
Variational Calculation ◽  
Nonlocal Model ◽  
Pseudopotential Theory ◽  
Valence Electrons ◽  
Model Pseudopotential

A procedure employing a generalized, nonlocal, model-pseudopotential theory for the calculation of the total valence-electron energy in liquid metals was presented earlier and shown to be suitable for use in a variational calculation of the Helmholtz free energy, and hence also for other properties, of sd and the early 3d transition metals. In the first part of this paper we show that the same procedure also works well for the first four of the 4d transition metals. However, the accuracy of the calculations decreases with increasing number of d-like valence electrons. This is attributed to narrow valence d bands. Thus, in the second part of this paper we revise and generalize the earlier procedure to consider explicitly the localization of some of the d-like valence electrons on the ions. The validity of the revisions is shown by calculating a number of the properties of liquid metallic Cr, Mn, and Fe.

Variational thermodynamic calculations for some liquid sd metals

1986 ◽  
Vol 64 (1) ◽  
pp. 75-83 ◽  
Author(s):  
D. H. Li ◽  
R. A. Moore ◽  
S. Wang
Keyword(s):  
Ab Initio ◽  
Valence Electron ◽  
Helmholtz Free Energy ◽  
Liquid Metals ◽  
Model Potential ◽  
Variational Calculation ◽  
Hard Sphere Model ◽  
Sphere Model ◽  
Pair Potentials ◽  
The One

A tractable and reliable expression for the one valence-electron eigenenergies, required in calculating the total energy of a disordered sd-type metal, is formulated in the context of the model-potential theory. With the aid of this expression, the variational calculation of the Helmholtz free energy using the hard-sphere model as a reference system, as employed in ab initio calculations of the thermodynamic properties for the nearly-free-electron-like (NFE) liquid metals, can now be extended with reasonable accuracy to those liquid sd metals in which the d-like valence-electron states below the Fermi level are not very localized. Also, the ab initio-type pseudopotential calculation of the interionic pair potentials, as carried out for the NFE-like metals in the literature, is made practical for these sd metals in their disordered states.

Pseudopotential Theory for Atoms and Molecules

1977 ◽  
Vol 32 (8) ◽  
pp. 829-839 ◽  
Author(s):  
Levente Szasz
Keyword(s):  
Wave Function ◽  
Arbitrary Number ◽  
Valence Electron ◽  
Variation Method ◽  
Exact Equation ◽  
Pseudopotential Theory ◽  
Atoms And Molecules ◽  
Orthogonality Conditions ◽  
Valence Electrons ◽  
The Many

Abstract An exact pseudopotential theory is presented for atoms and molecules with arbitrary number of valence and core electrons and arbitrary number of nuclei. Using the variation method an equation is derived for the best many-valence-electron wave function which is orthogonalized to the core orbitals. Using this equation the exact equation is derived for the many-valence-electron pseudo-wavefunction which does not have to satisfy any orthogonality conditions. The Hamiltonian of the pseudopotential equation is of surprisingly simple structure and does not depend on the energy and/or on the wave function of the valence electrons. It is shown that the simple model Hamiltonian which is used in many pseudopotential calculations can be derived from the exact equation by two plausible approximations. The theory is elucidated on the example of atoms with two valence electrons.

scholarly journals Investigation of some basic thermodynamic properties of Na-K alloy

BIBECHANA ◽  
2021 ◽  
Vol 18 (2) ◽  
pp. 1-8
Author(s):  
Rajesh C Malan ◽  
Aditya M Vora
Keyword(s):  
Free Energy ◽  
Thermodynamic Properties ◽  
Local Field ◽  
Alkali Metals ◽  
Helmholtz Free Energy ◽  
Correlation Effect ◽  
The Other ◽  
Pseudopotential Theory ◽  
Local Field Correction ◽  
Energy Components

Investigation of thermodynamic of liquid binary alloys using pseudopotential theory is reported. The potential suggested by Fiolhais et al. with its individual parameters is used for the entire calculation. A transferability of the potential from the solid to liquid medium is achieved for the presently reported binary alloy. The internal energy components, Helmholtz free energy, entropy, and total energy at various proportions of the participating alkali metals are included in the study. The comparison with the other data has been shown in the present article. Exchange and correlation effect is also tested with the help of various local field correction functions. BIBECHANA 18 (2) (2021) 1-8

A canonical ensemble derivation of the McMillan-Mayer solution theory

1983 ◽  
Vol 48 (10) ◽  
pp. 2888-2892 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Energy Function ◽  
Special Function ◽  
Helmholtz Free Energy ◽  
Canonical Ensemble ◽  
Free Energy Function ◽  
Solution Theory ◽  
Pure Solvent ◽  
The Common

A special free energy function is defined for a solution in the osmotic equilibrium with pure solvent. The partition function of the solution is derived at the McMillan-Mayer level and it is related to this special function in the same manner as the common partition function of the system to its Helmholtz free energy.

scholarly journals Free Energy of Metals from Quasi-Harmonic Models of Thermal Disorder

Metals ◽  
2021 ◽  
Vol 11 (2) ◽  
pp. 195
Author(s):  
Pavel A. Korzhavyi ◽  
Jing Zhang
Keyword(s):  
Free Energy ◽  
First Principles ◽  
Density Functional ◽  
Helmholtz Free Energy ◽  
Complex Materials ◽  
Predictive Capability ◽  
Temperature Dependent ◽  
Disordered Alloys ◽  
Thermal Disorder ◽  
Structure Calculations

A simple modeling method to extend first-principles electronic structure calculations to finite temperatures is presented. The method is applicable to crystalline solids exhibiting complex thermal disorder and employs quasi-harmonic models to represent the vibrational and magnetic free energy contributions. The main outcome is the Helmholtz free energy, calculated as a function of volume and temperature, from which the other related thermophysical properties (such as temperature-dependent lattice and elastic constants) can be derived. Our test calculations for Fe, Ni, Ti, and W metals in the paramagnetic state at temperatures of up to 1600 K show that the predictive capability of the quasi-harmonic modeling approach is mainly limited by the electron density functional approximation used and, in the second place, by the neglect of higher-order anharmonic effects. The developed methodology is equally applicable to disordered alloys and ordered compounds and can therefore be useful in modeling realistically complex materials.

Thermodynamics Energy for Both Supervised and Unsupervised Learning Neural Nets at a Constant Temperature

1999 ◽  
Vol 09 (03) ◽  
pp. 175-186 ◽  
Author(s):  
HAROLD SZU
Keyword(s):  
Free Energy ◽  
Constant Temperature ◽  
Helmholtz Free Energy ◽  
Principal Component ◽  
Neural Nets ◽  
Mixing Condition ◽  
Smart Cameras ◽  
Minimization Principle ◽  
Output Entropy ◽  
Artificial Neural Network Ann

Unified Lyaponov function is given for the first time to prove the learning methodologies convergence of artificial neural network (ANN), both supervised and unsupervised, from the viewpoint of the minimization of the Helmholtz free energy at the constant temperature. Early in 1982, Hopfield has proven the supervised learning by the energy minimization principle. Recently in 1996, Bell & Sejnowski has algorithmically demonstrated. Independent Component Analyses (ICA) generalizing the Principal Component Analyses (PCA) that the continuing reduction of early vision redundancy happens towards the "sparse edge maps" by maximization of the ANN output entropy. We explore the combination of both as Lyaponov function of which the proven convergence gives both learning methodologies. The unification is possible because of the thermodynamics Helmholtz free energy at a constant temperature. The blind de-mixing condition for more than two objects using two sensor measurement. We design two smart cameras with short term working memory to do better image de-mixing of more than two objects. We consider channel communication application that we can efficiently mix four images using matrices [AO] and [Al] to send through two channels.

scholarly journals Structural simplicity and complexity of compressed calcium: electronic origin

2014 ◽  
Vol 70 (3) ◽  
pp. 423-428 ◽  
Author(s):  
Valentina F. Degtyareva
Keyword(s):  
Valence Electron ◽  
Interaction Model ◽  
Outer Core ◽  
Electron Number ◽  
Fermi Sphere ◽  
Simple Cubic ◽  
Valence Electrons ◽  
Cubic Structure ◽  
Structural Simplicity ◽  
Tetragonal Cell

A simple cubic structure with one atom in the unit cell found in compressed calcium is counterintuitive to the traditional view of a tendency towards densely packed structures with an increase in pressure. To understand this unusual transformation it is necessary to assume electron transfer from the outer core band to the valence band, and an increase of valence electron number for calcium from 2 to ∼ 3.5. This assumption is supported by the Fermi sphere–Brillouin zone interaction model that increases under compression. The recently found structure of Ca-VII with a tetragonal cell containing 32 atoms (tI32) is similar to that in the intermetallic compound In5Bi3with 3.75 valence electrons per atom. Structural relations are analyzed in terms of electronic structure resemblance. Correlations of structure and physical properties of Ca are discussed.

Thermodynamics of Bidisperse Ferrofluids in Zero External Magnetic Field: Theory and Simulations

2015 ◽  
Vol 233-234 ◽  
pp. 331-334
Author(s):  
Anna Yu. Solovyova ◽  
Ekaterina A. Elfimova
Keyword(s):  
Magnetic Field ◽  
Field Theory ◽  
Free Energy ◽  
External Magnetic Field ◽  
Volume Concentration ◽  
Helmholtz Free Energy ◽  
Hard Spheres ◽  
Particle Volume ◽  
Simulation Data ◽  
Theoretical Predictions

The thermodynamic properties of a ferrofluid modeled by a bidisperse system of dipolar hard spheres in the absence of external magnetic field are investigated using theory and simulations. The theory is based on the virial expansion of the Helmholtz free energy in terms of particle volume concentration. Comparison between the theoretical predictions and simulation data shows a great agreement of the results.

scholarly journals Hamiltonian Thermodynamics

2020 ◽  
Vol 16 (4) ◽  
pp. 557-580
Author(s):  
S.A. Rashkovskiy ◽  
Keyword(s):  
Free Energy ◽  
Hamiltonian Systems ◽  
Thermal Energy ◽  
Helmholtz Free Energy ◽  
Random Processes ◽  
Carnot Cycle ◽  
Thermodynamic Process ◽  
Thermodynamic Cycles ◽  
Thermodynamic State ◽  
Laws Of Thermodynamics

It is believed that thermodynamic laws are associated with random processes occurring in the system and, therefore, deterministic mechanical systems cannot be described within the framework of the thermodynamic approach. In this paper, we show that thermodynamics (or, more precisely, a thermodynamically-like description) can be constructed even for deterministic Hamiltonian systems, for example, systems with only one degree of freedom. We show that for such systems it is possible to introduce analogs of thermal energy, temperature, entropy, Helmholtz free energy, etc., which are related to each other by the usual thermodynamic relations. For the Hamiltonian systems considered, the first and second laws of thermodynamics are rigorously derived, which have the same form as in ordinary (molecular) thermodynamics. It is shown that for Hamiltonian systems it is possible to introduce the concepts of a thermodynamic state, a thermodynamic process, and thermodynamic cycles, in particular, the Carnot cycle, which are described by the same relations as their usual thermodynamic analogs.

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