Kinetic Energy Of Atoms Research Articles

Decompositions of electron density and the shell problem in Hohenberg-Kohn functional theory

The earlier proposed theory of constructing orthonormal orbitals of a gas of noninteractive particles with given electron density (density decomposition) is shown to yield the solution of the shell problem well-known in the density functional or statistical methods. Kinetic energies of atoms are dwelt upon. The means and ways of constructing density decompositions for periodic systems are discussed.

A new formulation of scaled atoms-in-molecules theory

In scaled atoms-in-molecules theory, variational scaling parameters are introduced into the A- and B-atom eigenfunctions from which molecular basis functions are constructed. All intraatomic contributions to matrix elements are determined rigorously in terms of scaling parameters and experimental atomic state energies. Approximate atomic state eigenfunctions are used to compute interatomic contributions. Ground and excited state energy roots of the resulting molecular eigenvalue problem can be independently minimized with respect to the scaling parameters. The original formulation of theory (SAIM.V) employed atomic potential energy operators as part of the intraatomic correction due to scaling. A new formulation (SAIM.T) using atomic kinetic energy operators in this role is tested in calculations on H2, HeH+, He+2, and HeH. The simpler SAIM.T formulation is found to give results of the same quality as the original SAIM.V version.

Molecular dynamics simulation of low energy displacement cascades in Cu

Molecular dynamics computer simulations have been performed to study properties of low-energy (< 500 eV) displacement cascades in Cu. Various aspects of the time development of cascades are considered including instantaneous number of Frenkel pairs, partitioning of kinetic and potential energies, distribution of atom kinetic energies, cascade expansion rate, and Frenkel pair distributions. The anisotropy of the threshold energy for Frenkel-pair production is interpreted in terms of “branching”. Replacement sequences and the damage function are discussed based on analysis of events corresponding to 18 recoil directions. The damage function exhibits a plateau at v ~ 0.5 Frenkel pairs extending from 25–150 eV; at higher recoil energies the onset of multiple defect production is much slower than predicted by the modified Kinchin-Pease model.

A comment on the derivation of the atomic kinetic energy functionals with full Weizsäcker correction

In a recent paper [J. Chem. Phys. 76, 1467 (1982)], Gázquez and Robles derive a kinetic energy functional of the type T[ρ]=TW[ρ]+[1−(2/N)][1−(C0/N1/3) +(C1/N2/3)−⋅⋅⋅]TTF[ρ] (TW and TTF are the Weizsäcker and the Thomas–Fermi functionals) by considering the sine waves of free particles in a cube. We get different C0 and C1 values from those of Gázquez and Robles by considering periodic complex exponential waves instead of the sine waves.

Atomic kinetic energy functional in density functional theory

With the purpose of finding a better kinetic energy functional in density functional theory, the first-order reduced density matrix for atoms is considered and its approximate form which is composed of the exponentially decreasing functions is proposed as follows: ρ(r‖r′=Σnρn(r‖r′), (A) where n is the principal quantum number ρn′s are the first-order reduced density matrices and assumed as ρn(r‖r′)=An exp [−(Bn+iCnfn)r −(Bn−iCnfn)r′], (B) An, Bn, and Cn being the slowly varying functions of r and r′, and fn the function depending on the principal quantum number. From Eqs. (A) and (B), a reasonable kinetic energy functional is derived as T[ρ]=Σn1/8ℱ∇ρn∇ρn/ρndr +Σn3(3π2)2/3/10 f2nℱρn5/3 dr. (C) It is shown that the functional is adequate for describing the long-range behavior as well as the nuclear-cusp behavior of ρ(r), and also for the prediction of the electronic kinetic energies for atoms.

Analysis of substrate heat fluxes encountered during aluminum magnetron sputter deposition

The nature of the heat fluxes and charged particle currents at the substrate plane were analyzed for two different types of planar magnetron cathode. The first of these, a standard planar magnetron, was shown to deliver a substantial heat flux, concentrated beneath the “racetrack” regions of the cathode. This heat flux was associated with an electron current flowing to the substrate. The practice of electrically isolating the substrate had the effect of substantially reducing the heat flux. The second cathode was equipped with an electron capture shield. This cathode type delivered a reduced heat flux which was composed primarily of ion-electron recombination energy, sputtered atom heat of condensation and an unidentified neutral particle energy flux, believed to be sputtered atom kinetic energy.

Self-consistent potential for a two-level atom in a standing-wave field

It is shown that the effects of a standing electromagnetic wave of cosinusoidal form on a neutral two-level atom can be represented by an effective static potential, which is obtained by a local unitary transformation to dress the atom by the field. The form of the effective potential is obtained self-consistently by minimizing fluctuations of the atomic kinetic energy induced by the locality of the dressing transformation. Moreover this effective potential is shown to depend on the atom-field detuning and on the field intensity, and to exhibit discontinuities in some ranges of these parameters. The band structure which characterizes the eigenvalue spectrum of the dressed Hamiltonian is discussed in the spirit of the tight-binding approximation, and the fluctuations of the kinetic energy near resonance are shown to induce splitting of the bands; a physical interpretation of the atomic dynamics associated with these splittings is presented. Off resonance the fluctuations are shown to induce exchange of real photons and variations of the atomic kinetic energy. The results are compared with those of previous theories, leading to an improved understanding of the dynamics of a two-level atom in a standing-wave field.

On the atomic kinetic energy functionals with full Weizsacker correction

The functional proposed by Acharya, Bartolotti, Sears, and Parr for representing the kinetic energy of the ground state of an N electron atom or ion in terms of its electron density, namely T[ρ] = Tw[ρ] +γ(N, Z) TTF[ρ] [TW is the Weizsacker term, TTF is the Thomas–Fermi term and γ(N, Z) = 1−C0/N1/3] is derived by: (1) considering explicitly the variation of the charge density in an atom, and (2) approximating the pair correlation function for parallel spin electrons by that of a uniform free-electron gas, but including the corrections to the momentum at the Fermi level and the appropriate boundary conditions that result from taking into account that the number of electrons in an atom is finite. The first consideration leads in a natural way to the full Weizsacker correction, while the second consideration yields γ(N, Z) = (1−2/N)(1−C0/N1/3−C1/N1/3) times the Thomas–Fermi term. Thus, the functional obtained is exact for one-electron atoms and two-electron Hartree–Fock atoms, yields an adequate functional derivative, is in agreement with the leading correction to Thomas–Fermi, and provides an excellent representation of the kinetic energy of atoms 2⩽Z⩽54 when Hartree–Fock densities are used.

Cohesive Energies of Simple Metals as Determined from Atomic Kinetic Energies

Cohesive Energies of Simple Metals as Determined from Atomic Kinetic Energies

An atomic kinetic energy functional with full Weizsacker correction

A functional is proposed for representing the electronic kinetic energy of the ground state of an N-electron atom or ion in terms of its electron density, [Formula: see text] Here T(w) is the Weizsacker quantity ((1/8))integral(nablarho.nablarho/rho)dtau and T(0) is the Thomas-Fermi quantity C(F) integral rho(5 / 3)dtau. From Hartree-Fock data on 55 neutral atoms, C = 1.412 +/- 0.033; for 1200 atoms and ions, C = 1.332 +/- 0.053. The proposed functional gives the derivative deltaT/deltarho its most important correct properties. The term T(w) is shown to give the kinetic energy of the K shell, whereas the term (C/N((1/3)))T(0) gives an incorrect statistical estimate of that energy. An alternative correction -(C/N((1/3)))T gives even better results.

Comparative study of the gradient expansion of the atomic kinetic energy functional–isoelectronic series

In previous work, it was shown that, for the ground state of neutral atoms, the truncated gradient expansion of the kinetic energy functional T[ρ] can give very good kinetic energy values when Hartree–Fock densities are used. In this work, we use the Hodges expansion formula: T[ρ]=T0[ρ]+T2[ρ]+T4[ρ], with T0[ρ]=3/10 (3π2)2/3∫ρ5/3dτ, T2[ρ] = (1/72) ∫[(∇ρ)2/ρ]dτ, etc., to calculate the kinetic energies of heliumlike, neonlike, and argonlike atoms and ions. The errors in the values obtained depend significantly on the number of electrons, N, of the system when N is small. As N increases, the error gradually decreases.

Comparative study of the gradient expansion of the atomic kinetic energy functional-neutral atoms

In a previous work, it has been shown that for the noble gas atoms, the ground state kinetic energy functional T[ρ], where ρ is the electronic density, can be computed to surprising accuracy from the truncated gradient expansion: T[ρ]=T0[ρ]+T2[ρ]+T4[ρ], with T0[ρ] =3/10(3π2) ℱρ5/3dτ, T2[ρ]=1/72ℱ (∇ρ)2/ρdτ, and T4[ρ] given by the formula of Hodges. In this work, we report systematic calculations for improved near-Hartree–Fock atomic densities up to Z=54. We find that the truncated gradient expansion of T[ρ] can also lead to very good kinetic energies for open-shell and unpaired-electron atoms. The value given by T[ρHF] generally gets better, as compared with the Hartree–Fock result, as Z increases. For the Xe atom, T0+T2+T4 is within 0.08% of the exact Hartree–Fock kinetic energy with T2/T0=0.048, T4/T2=0.16. It is interesting to note that some minima appear in the deviation curves for T0, T0+T2, and T0+T2+T4, as compared with Hartree–Fock results. For the near-degenerate ground and lower excited states of some atoms, the truncated kinetic energy expansion can reproduce the relative kinetic energies qualitatively. On the other hand, because the cusp condition at the atomic nucleus and the exponentially vanishing density are valid for every Hartree–Fock density, it may not be possible to reproduce the Hartree–Fock or exact results with the gradient expansion in a systematic way.

An atomic kinetic energy functional with full Weizsacker correction

An atomic kinetic energy functional with full Weizsacker correction

Substrate heating in cylindrical magnetron sputtering sources

The magnetic plasma confinement in magnetron sputtering sources makes the plasma ionization process very efficient and allows high sputtering rates to be achieved with reduced substrate heating, particularly in the cylindrical-post magnetron geometry where a virtual anode separates the substrates from the plasma. Heating flux measurements at typical substrate locations surrounding cylindrical-post magnetron sputtering sources were made for the argon sputtering of seventeen metals, representing a range of atomic masses, crystallographic structures and sputtering yields. The heating flux data are combined with deposition rate measurements and are reported as energy per coating atom deposited. They are consistently interpreted in terms of four basic heating processes: (1) heat of condensation; (2) sputtered atom kinetic energy; (3) plasma radiation; (4) ion neutralization and reflection at the cathode. Ion reflection becomes increasingly important for the high mass materials. The energy per atom deposited is independent of the deposition rate and the argon pressure and varies from 15 to 25 eV atom −1 for light metals to over 50 eV atom −1 for heavy metals with moderate sputtering yields. The sputtering efficiency is shown to be a useful concept in accounting for ion reflection. Comparative data are reported for sputtering with nitrogen and oxygen and for cylindrical-hollow magnetron sources. The energy per atom is relatively high in nitrogen and oxgyen sputtering because of a high sputtering efficiency and a reduced metal sputtering yield. The closer proximity of the plasma in hollow cathodes introduces a plasma bombardment contribution to the heating flux.

Bound on lowest-order gradient correction in kinetic energies of atoms

The lowest-order gradient correction in the kinetic energy functional is shown to be less than 1 9 of the total kinetic energy. The numerical calculations of Wang et al. are referred to as an example.

Gradient expansion of the atomic kinetic energy functional

It is demonstrated that the ground-state atomic kinetic energy functional T[ϱ], where ϱ is the electron density, can be computed to surprising accuracy from the truncated gradient expansion: T[ϱ] = + T 2[ϱ] + T 4[ϱ], with T o[ϱ] = 3 10 (3π 2) 2 3 ∫ ϱ 5 3 d τ, T 2 [ϱ] = 1 72 ∫ (▿ϱ) 2ϱ −1 d τ, and T 4 [ϱ] given by the formula of Hodges. Calculations of T 0, T 2 and T 4 are reported for He with ϱ both the Hartree—Fock and a very accurate density, and for Ne, Ar and Kr with ϱ the Hartree—Fock density. For Kr, T 0 + T 2 + T 4 is within 0.3% of the exact Hartree—Fock T, with T 2/ T 0 = 0.05, T 4/ T 2 = 0.17.

Computer simulation of atomic-displacement cascades in solids in the binary-collision approximation

A comprehensive computer program has been developed for the simulation of atomic-displacement cascades in a variety of crystalline solids, using the binary-collision approximation to contruct the projectile trajectories. The atomic scattering is governed by the Moli\'ere potential.Impact-parameter-dependent inelastic losses are included using Firsov's theory. Thermal vibrations of the target atoms and crystal surfaces may be included. Permanent displacement of lattice atoms may be based on either an energy-threshold criterion or a Frenkel-pair-separation criterion. An extensive series of calculations has been made for cascades in the simple metals Cu, Fe, and Au, to test the effects on the results of many of the model parameters. When a displacement-threshold energy is used, the number of Frenkel pairs is found to be a linear function of that part of the primary recoil energy which remains as the kinetic energy of atoms. This result is independent of target temperature, of the presence or absence of inelastic energy losses, and of various details of the model. In contrast, when a separation criterion is used, the number of defects increases less rapidly than linearly. This effect is caused by increased recombination in the highly disturbed tracks of the energetic recoils. Agreement between theoretical and experimental estimates of the radiation damage produced by neutron irradiation of Cu is substantially improved in the latter model.

Inferring the Distribution of Atomic Kinetic Energies from Neutron-Scattering Experiments

It is shown that the scattering of neutrons at large momentum transfers can provide information on the detailed shape of the distribution of atomic kinetic energies. An expansion of the kinetic-energy distribution function $n(E)$ leads to a corresponding expansion of the large -$k$ limit of the scattering function $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, \ensuremath{\omega})$. Deviations of the latter from a Gaussian shape reflect deviations of $n(E)$ from a Maxwellian. Corrections for finite $k$ are also derived. An application to the scattering data of Harling for liquid $^{4}\mathrm{He}$ leads to estimates of the condensate fraction and average kinetic energy for liquid temperatures in the range 1.27 to 4.2 \ifmmode^\circ\else\textdegree\fi{}K.

Atomic kinetic energies in liquid 4He

Atomic kinetic energies in liquid 4He are computed using Sposito's empirical potential. Results are in closer agreement with a neutron scattering experiment than are similar results based on the Lennard-Jones potential.