Lattice dynamics of FCC transition metals: A pseudopotential approach

Simple pseudopotential model for the binding energy of transition metals is proposed. The contribution of thes-like electrons is calculated in the second-order perturbation theory for the local model pseudopotential while that of thed-like electrons is taken into account by introduction of repulsive short-range interatomic potential. Model parameters were determined for ten fcc transitions metals (Cu, Ni, Fe, Co, Ag, Pd, Rh, Au, Pt, and Ir). This model was used for the calculation of the phonon dispersion and the density of states, as well as for the elastic constants and their pressure derivatives. Good agreement with experimental data was achieved for the overall shape of phonon spectra and even for the position of the Kohn anomalies in Pd and Pt. Existence of such anomalies is also stated for predicted phonon spectra of rhodium and iridium.     

Structure factors and phonon dispersion in liquid Li<Subscript>0.61</Subscript>Na<Subscript>0.39</Subscript> alloy

The phonon spectra for liquid Li and Na have been computed through the phenomenological model of Bhatia and Singh for disordered systems like liquids and glasses and the obtained results have been compared with the available data obtained by inelastic neutron scattering (INS) and inelastic X-ray scattering (IXS) experiments. The effective pair potentials and their space derivatives are important ingredients in the computation of the dispersion curves. The pair potentials are obtained using the pseudo-potential theory. The empty core model proposed by Ashcroft is widely used for pseudo-potential calculations for alkali metals. But, it is thought to be unsuitable for Li because of its simple 1s electronic structure. However, it can be used with an additional term known as Born-Mayer (BM) core term. The influence of the BM core term on the phonon dispersion is discussed. The same pseudo-potential formalism has been employed to obtain the dispersion relation in liquid Li_0.61Na_0.39 alloy. Apart from the phonon spectra, the Ashcroft-Langreth structure factors in the alloy are derived in the Percus-Yevick approximation.     

Phonon spectra of Zn and Cd using a model pseudopotential approach

Recently Animalu has developed a model potential approach to study the electronic properties of transition metals. We have applied the transition metal model potential (TMMP) of Animalu in the local approximation to derive the phonon dispersion curves of Zn and Cd alongΓA,ΓM andΓKM symmetry directions. Our results differ widely from the experimental data. Further we have added the non-local contribution to the dynamical matrix following the scheme of Eschrig and Wonn and found the resuits comparable to the measured ones.     

Lattice dynamical study of alkali metals: An unified approach based on CGW model

The phonon dispersion relations for lithium, sodium, potassium, rubidium and cesium along the principal symmetry directions as well as their lattice specific heats have been deduced using Clark, Gazis and Wallis angular force model. This model which conforms to the translational symmetry of the lattice, reproduces the observed crossover in lithium along [ζ00] direction at ζ = 0·49, without producing any crossovers in other alkali metals. Besides, the theoretical dispersion curves of all alkali metals are in excellent agreement with the corresponding experimental or homologous dispersion relations and theirϑ _D values compare well with the experimental values over a wide temperature range. It is shown that the strength of electron-ion interactions plays a significant role in the success of any unified lattice dynamical study of alkali metals while the three-body interactions of thecgw model do not. The importance of umklapp processes, failure of the earlier models to produce a crossover and the experimentalϑ _D-T curve in lithium as well as the apparent variation in the nature and range of atomic interactions of alkali metals are discussed.     

Lattice Dynamics of Some Fcc F-Shell Metals

A model pseudopotential depending on an effective core radius but otherwise parameter free is used to study the binding energy, equation of state, ion-ion interaction, phonon dispersion curves (q-space and r-space analysis), phonon density of states, Debye temperature, mode Grüneisen parameters, dynamical elastic constants, Debye-Waller factor, mean-square displacement, Debye-Waller temperature parameter and propagation velocities of elastic waves of some fcc f-shell metals La, Yb, Ce, and Th. The contribution of the s-like electrons is calculated in the second-order perturbation theory for the model potential while d- and f-like electron is taken into account by introduction of repulsive short-range Born-Mayer term. Very recently proposed screening function due to Sarkar et al. has been used to obtain the screened form factor. The parameter of the potential is evaluated by zero pressure condition. Which is independent of any fitting procedure. An excellent agreement between theoretical investigations and experimental findings prove the ability of the potential for d- and f-shell metals exclusively.     

Calculated effects of charge fluctuations on the phonon dispersion of YBa2Cu3O6 and YBa2Cu3O7. II. Results

Using the model presented in the preceding paper we investigate the effects of charge fluctuations (CF) on the phonon dispersion of YBa₂Cu₃O₆ (O₆) and YBa₂Cu₃O₇ (O₇). Starting from an ab-initio rigid-ion model as a reference system, CF are allowed for at the copper- and oxygen ions. The CF are treated as adiabatic electronic degrees of freedom. Within the rigid-ion model (RIM) the structural parameters are calculated by minimization of the energy. The results agree reasonably well with the experiment, indicating the suitability of the ionic model as a starting point and the importance of ionic forces for the properties of the high-temperature superconductors (HTSC) in general. Next, the phonon dispersion is calculated in the RIM as well as including CF additionally and the renormalization of the individual modes is discussed. By restricting the CF optionally to the planes, effects arising specifically from CF in the planes on the one hand and from CF in the chain as well as at the axial bridging oxygens (O4) on the other hand can be separated. We find the oxygen axial modes at the γ- and Z point (A_1g/A_2μ in O₆, A_g/B_1μ in O₇) particularly interesting. Most of these modes show considerable renormalizations. Moreover, the γ/Z-axial modes are characterized by the possibility of having CF of the same sign in the whole CuO planes what distinguishes them from the modes at other symmetry points. In particular, the Z-point axial modes are singular in having CF of alternating sign in consecutive structural units in c direction. Such a “c-direction-charge-transfer” has been shown previously to be an effective screening mechanism in La₂CuO₄. Indeed, we find a drastic renormalization of the plane-oxygen A_g mode at the Z point (A_g(O23;Z)) in O₇ (oxygen ions in neighboring planes vibrating in-phase), at least in the adiabatic approximation used here. In the insulating phase this mode exhibits, on the other hand, very large changes of the potential at the ion sites, whereas its renormalization is moderate only. The reason for this behaviour is that in the insulating phase in case of a two-dimensional electronic structure the charge transfer (screening) is restricted locally in the structural unit and long-range charge transfer is not possible as in the metal. However, a strong suppression of screening for this mode can also be expected for the metallic phase in O₇ in case non-adiabatic electron-phonon coupling would be important. The A_g (O23;Z)-mode thus seems to be by far the most interesting mode in O₇. These features are directly related to the layered structure of the HTSC compounds considered here. The O4-axial-breathing modes show significant renormalizations too, and are characterized by plane-chain charge transfer. Moreover, besides the O23- and O4-modes, the yttrium modes appear to be important too. In addition to the phonon-dispersion curves, we present values for the CF amplitudes and screened site-potential changes at the copper-and oxygen ions. Finally, we give transverse effective charges and dielectric constants for the insulating phase (O₆) as calculated within our formalism.     

Lattice vibronics of bcc phase metals (Ti,Zr and Hf) at higher temperatures

Ashcroft’s analytic bare ion pseudopotential form factor with a modified Hartree dielectric function has been employed to represent the temperature dependent interionic potential. This potential includes both direct ion-ion interaction and indirect ion-electron-ion interaction with and without the effects of ‘d’ bands, in some scantily studied complexbcc metals vizbcc Ti, Zr and Hf. The ab initio radial and tangential force constants extending out to 15th nearest neighbours are computed for the metals. The said potential is used for predicting the binding energy, elastic constants and phonon dispersion of the above mentioned metals and the results are satisfactorily compared with the corresponding measured data.     

Substrate-induced spin-orbit splitting of quantum-well and interface states in Au,Ag, and Cu layers of different thicknesses on W(110) and Mo(110) surfaces

The substrate-induced spin-orbit splitting of interface and quantum-well states formed in Au, Ag, and Cu layers on W(110) and Mo(110) surfaces has been revealed using angle- and spin-resolved photoelectron spectroscopy. It has been shown that the magnitude of the splitting depends noticeably on the atomic number of the substrate material and is markedly larger for layers of these metals on W(110), i.e., on the surface of a metal with a larger atomic number (Z _W = 74), than on the surface of Mo(110), i.e., an element with a smaller atomic number (Z _Mo = 42), while depending only weakly on the atomic number of the adsorbed metal. Measurements of the dispersion of the formed quantum-well states have shown that the substrate-induced spin-orbit splitting increases with increasing parallel component of the photoelectron momentum (which correlates with the Rashba model) for all thicknesses of deposited films (up to 10 ML). The magnitude of induced spin-orbit splitting of the interface states evolving in monolayer Au, Ag, and Cu coatings on W(110) and Mo(110) decreases with increasing parallel component of the excited photoelectron momentum.     

Vibrational properties of vacancy in bcc transition metals using embedded atom method potentials

The embedded atom method (EAM) potentials, with the universal form of the embedding function along with the Morse form of pair potential, have been employed to determine the potential parameters for three bcc transition metals: Fe, Mo, and W, by fitting to Cauchy pressure (C ₁₂???C ₄₄)/2, shear constants $G_\textrm{v} =({C_{11} -C_{12} +3C_{44}})/5$ and C ₄₄, cohesive energy and the vacancy formation energy. The obtained potential parameters are used to calculate the phonon dispersion spectra of these metals. Large discrepancies are found between the calculated results of phonon dispersion using the EAM and the experimental phonon dispersion results. Therefore, to overcome this inadequacy of the EAM model, we employ the modified embedded atom method (MEAM) in which a modified term along with the pair potential and embedding function is added in the total energy. The phonon dispersions calculated using potential parameters obtained from the MEAM show good agreement with experimental results compared to those obtained from the EAM. Using the calculated phonons, we evaluate the local density of states of the neighbours of vacancy using the Green’s function method. The local frequency spectrum of first neighbours of vacancy in Mo shows an increase at higher frequencies and a shift towards the lower frequencies whereas in Fe and W, the frequency spectrum shows a small decrease towards higher frequency and small shift towards lower frequency. For the second neighbours of vacancy in all the three metals, the local frequency spectrum is not much different from that of the host atom. The local density of states of the neighbours of the vacancy has been used to calculate the mean square displacements and the formation entropy of vacancy. The calculated mean square displacements of the first neighbours of vacancy are found to be higher than that of the host atom, whereas it is lower for the second neighbours. The calculated results of the formation entropy of the vacancy compared well with other available results.     

Calculated effects of charge fluctuations on the phonon dispersion of YBa2Cu3O6 and YBa2Cu3O7. I. The model

In this paper we present a microscopic model that allows us to study the effects of charge fluctuations on the phonon dispersion of the high-temperature superconductor YBa₂Cu₃O₇ and its insulating counterpart, YBa₂Cu₃O₆. An ab-initio rigid-ion model with pair potentials calculated by the Gordon-Kim method from the free-ion charge densities is used as a reference system. Starting from this reference system, charge fluctuations at the copper- and oxygen ions are introduced into the model. The charge fluctuations are treated as adiabatic degrees of freedom in a non-phenomenological way. The parameters entering the model are estimated consistently with the reference system from first principles rather than refering to the experimentally determined phonon dispersion. In addition to the metallic behavior (appropriate to YBa₂Cu₃O₇) obtained in this way, insulating behavior (appropriate to YBa₂Cu₃O₆) is simulated by requiring the polarizability function to fulfill a corresponding long. wavelength sum rule. Screened site-potential changes are defined that (besides the charge fluctuations) constitute a qualitative measure of the electron-phonon-interaction potential. Furthermore we investigate the long-wavelength limiting behavior of the most important quantities occurring in our formalism. We derive formulae that allow us to calculate the contribution of the charge fluctuations to the macroscopic dielectric constant and the transverse effective charges in the insulating phase.     

The effect of surface temperature on H2/D2(v = 0, j = 0)–Ni(100) scattering processes

We carry out both four-dimensional (4D×2D) and six-dimensional (6D) quantum dynamics on a parametrically time- and temperature-dependent effective Hamiltonian for H₂/D₂(v = 0,j = 0)–Ni(100) collision process. Such an effective potential was derived within a theoretical framework of mean-field approximation by considering weakly correlated interaction between molecular degrees of freedom, phonon modes and electron– hole pair (elhp) coupling through a Hartree-product-type wave function, where the initial state distribution of the surface modes and elhp coupling were introduced through Bose– Einstein and Fermi– Dirac probability factor, respectively. The temperature-dependent dissociation and state-to-state transition probabilities for H₂/D₂(v = 0,j = 0)–Ni(100) system are depicted as a function of initial kinetic energ of the incoming diatom. Though such effect appears negligibly small for H₂(v = 0,j = 0)–Ni(100) system, it is prominent in the case of D₂(v = 0,j = 0)–Ni(100) collision. It appears that the change of dissociation and transition probabilities of D₂ with the increase of surface temperature is exclusively dictated by the phonon modes directed along Z-axis, but the effect of elhp coupling particularly for transition probabilities is insignificant.     

Characteristic features of anharmonic effects in the lattice dynamics of fcc metals

The dispersion in the entire Brillouin zone and the temperature dependence (right up to the melting temperature) of the anharmonic frequency shift and phonon damping in a number of fcc metals is investigated on the basis of microscopic calculations. It is found that the anharmonic effects depend sharply on the wave vector in the directions Г-X, X-W, and Г-L and, in contrast to bcc metals, the magnitude of the effects is not due to the softness of the initial phonon spectrum. It is shown that the relative frequency shifts and the phonon damping near melting do not exceed 10–20%. The relative role of various anharmonic processes is examined, and the relation between the results obtained and existing experimental data is discussed. Pis’ma Zh. éksp. Teor. Fiz. 69, No. 9, 649–652 (10 May 1999)     

Structure of Some 4f Rare Earth Liquid Metals — A Charged Hard Sphere Approach

A well-established pseodopotential is used to study the structure of some 4f rare earth liquid metals (Ce, Pr, Eu, Gd, Tb, and Yb). The structure factor S(q), pair distribution function g(r), interatomic distance r₁, and coordination number n₁ are calculated using Charged Hard Sphere (CHS) reference system. To introduce the exchange and correlation effects, the local field correction due to Sarkar et al. (S) is applied. The present investigation is successful in generating the structural information of Ce, Pr, Eu, Gd, Tb, and Yb 4f rare earth liquid metals.     

有限温度下二维Heisenberg铁磁系统的声子衰减

在二维正方Heisenberg铁磁系统的基础上建立了磁振子-声子相互作用模型. 利用松原格林函数理论研究了系统的声子衰减，计算了布里渊区的主要对称点线上的声子衰减曲线. 发现在第一布里渊区，在Δ线上，横向声频支声子无衰减，在Z线上，纵向声频支声子无衰减；横向声频支声子衰减比纵向声频支声子衰减至少大一个数量级，并讨论了各项参数的变化对横向声频支声子衰减与纵向声频支声子衰减的影响. 根据声子衰减与声子寿命的关系，声子衰减与声子态密度的关系，可以讨论横向声频支声子与纵向声频支声子的寿命与态密度. 关键词： 磁振子-声子相互作用 横向声频支声子衰减 纵向声频支声子衰减 声子寿命     

Structure of Some 4f Rare Earth Liquid Metals - A Charged Hard Sphere Approach

A well-established pseodopotential is used to study the structure of some 4f rare earth liquid metals (Ce,Pr, Eu, Gd, Tb, and Yb). The structure factor S(q), pair distribution function g(r), interatomic distance r1, and coordination number n1 are calculated using Charged Hard Sphere (CHS) reference system. To introduce the exchange and correlation effects, the local field correction due to Sarkar et al. (S) is applied. The present investigation is successful in generating the structural information of Ce, Pr, Eu, Gd, Tb, and Yb 4f rare earth liquid metals.     

Static and dynamic properties of KCN<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>Cl<Subscript>1−<Emphasis Type="Italic">x</Emphasis></Subscript>

An extended three-body force shell model (ETSM) has been applied to investigate the static and dynamic properties of KCN _x Cl_1−x for the compositionx = 0.56 and 1.0 at 300 K. The phonon dispersion curves computed by us are compared with the single crystal neutron diffraction data. The unusual features of these curves are the upward curvature seen in some of the acoustic branches. This is a result ofK-dependent softening of the phonon due to translation-rotation coupling. The transverse acoustic branch is more soft near the zone centre.     

Collective dynamics of Zr-based bulk metallic glasses

The collective dynamics for the longitudinal as well as the transverse modes of the phonon eigenfrequencies are carried out for three Zr-based bulk metallic glasses (BMGs) in the present paper by employing pseudopotential theory. The theoretical phonon models proposed by Hubbard-Beeby (HB), Takeno-Goda (TG) and Bhatia-Singh (BS) are used to generate the phonon dispersion curves (PDC). The elastic and thermodynamic properties have also been investigated for the said BMGs by using the elastic limit of the phonon dispersion curves. The screening impact was also observed by using five different forms of local field functions due to Hartree (H), Taylor (T), Ichimaru and Utsumi (IU), Faridet al. (F) and Sarkaret al. (S) for the aforesaid properties. The derived results are seen to be in qualitative agreement with the available experimental and theoretically calculated data, as they confirm the applicability of Shaw's constant core model potential and self-consistent phonon theory for such studies.     

Lattice mechanical properties of noble and transition metals

A model pseudopotential depending on an effective core radius but otherwise parameter free is used to study the interatomic interactions, phonon dispersion curves (inq and r-space analysis), phonon density of states, mode Grüneisen parameters, dynamical elastic constants (C ₁₁,C ₁₂ andC ₄₄), bulk modulus (B), shear modulus (C′), deviation from Cauchy relation (C ₁₂–C ₄₄), Poisson’s ratio (σ), Young’s modulus (Y), behavior of phonon frequencies in the elastic limit independent of the direction (Y ₁), limiting value in the [110] direction (Y ₂), degree of elastic anisotropy (A), maximum frequencyω _max, mean frequency 〈ω〉, 〈ω ²〉^1/2=(〈ω〉/〈ω ⁻¹〉)^1/2, fundamental frequency 〈ω ²〉, and propagation velocities of the elastic constants in Cu, Ag, Au, Ni, Pd, and Pt. The contribution of s-like electrons is calculated in the second-order perturbation theory for the model potential while that of d-like electrons is taken into account by introducing repulsive short-range Born-Mayer like term. Very recently proposed screening function due to Sarkar et al. has been used to obtain the screened form factor. The theoretical results are compared with experimental findings wherever possible. A good agreement between theoretical investigations and experimental findings has proved the ability of our model potential for predicting a large number of physical properties of transition metals.     

Lattice mechanical properties of alkaline earth metals in bcc and fcc phases

A model pseudopotential depending on an effective core radius treated as a parameter is used for alkaline earth metals in bcc and fcc phases to study the Binding energy, Interatomic interactions, phonon dispersion curves, Phonon density of states, Debye-Waller factor, mean square displacement, Debye-Waller temperature parameters, dynamical elastic constants (C₁₁, C₁₂ and C₄₄), bulk modulus (B), shear modulus (C′), deviation from Cauchy relation (C₁₂−C₄₄), Poisson's ratio (σ), Young's modulus (Y), behavior of phonon frequencies in the elastic limit independent of the direction (Y₁), limiting value in the [1 1 0] direction (Y₂), degree of elastic anisotropy (A) and propagation velocities of the elastic waves. The contribution of s-like electrons is incorporated through the second-order perturbation theory due to model potential. The theoretical results are compared with the existing experimental data. A good agreement between theoretical investigations and experimental findings has confirmed the ability of our potential to yield large numbers of lattice mechanical properties of certain alkaline earth metals.     

Theoretical study of structural energy, phonon spectra, and elastic constants of Rh and Ir

The transition metal pair potential (TMPP) is used to study band structure energy of Rh and Ir. Both metals are found to be most stable in fcc structure down to atomic volume 0.5V ₀. The pressure at 0.5V ₀ is found to be 5.235 Mbar and 9.216 Mbar in Rh and Ir, respectively. The TMPP is also used to study other properties of these metals like cohesive energy, phonon frequencies at observed volume. The bulk moduli and elastic constants of these metals at observed volume are calculated by including the volume contribution.