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.     

The Effect of Temperature on Lattice Mechanical Properties of Noble and Transition Metals

A simple pseudopotential model is used for the calculation of the temperature dependence of lattice mechanical properties which also depend on the phonon density of states such as lattice heat capacity C _V , Debye temperature _D, harmonic contribution to free energy, thermal pressure, isothermal bulk modulus corrected to the fourth order, volume thermal expansion coefficient , Debye-Waller factor, mean-square displacement, Debye-Waller temperature parameter, and X-ray characteristic temperature _M of Cu, Ag, Au, Ni, Pd, Pt, Rh, and Ir. The contribution 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 show the ability of our model potential to reproduce wide class of properties in noble and transition metals.     

Atom scattering as a probe of the surface electron-phonon interaction at conducting surfaces

An atomic projectile colliding with a surface at kinetic energies in the thermal or hyperthermal range interacts with and is reflected by the electronic density well in front of the first layer of target atoms, and it is generally accepted that the repulsive interaction potential is proportional to the density of electrons extending outside the surface. This review develops a complete treatment of the elastic and inelastic scattering of atoms from a conducting surface in which the interaction with the electron density and its vibrations is treated using electron-phonon coupling theory. Starting from the basic principles of formal scattering theory, the elastic and inelastic scattering intensities are developed in a manner that identifies the small overlap region in the surface electron density where the projectile atom is repelled. The effective vibrational displacements of the electron gas, which lead to energy transfer through excitation of phonons, are directly related to the vibrational displacements of the atomic cores in the target crystal via electron-phonon coupling. The effective Debye-Waller factor for atom-surface scattering is developed and related to the mean square displacements of the atomic cores. The complex dependence of the Debye-Waller factor on momentum and energy of the projectile, including the effects of the attractive adsorption well in the interaction potential, are clearly defined. Applying the standard approximations of electron-phonon coupling theory for metals to the distorted wave Born approximation leads to expressions which relate the elastic and inelastic scattering intensities, as well as the Debye-Waller factor, to the well known electron-phonon coupling constant λ. This treatment reproduces the previously obtained result that the intensities for single phonon inelastic peaks in the scattered spectra are proportional to the mode specific mass correction components λ_Q,ν defined by the relationship λ = 〈λ_Q,ν〉. The intensities of elastic diffraction peaks are shown to be a weighted sum over the λ_Q,ν, and the Debye-Waller factor can also be expressed in terms of a similar weighted summation. In the simplest case the Debye-Waller exponent is shown to be proportional to λ and for simple metals, metal overlayers, and other kinds of conducting surfaces values of λ are extracted from available experimental data. This dependence of the elastic and inelastic scattering, and that of the Debye-Waller factor, on the electron-phonon coupling constant λ shows that measurements of elastic and inelastic spectra of atomic scattering are capable of revealing detailed information about the electron-phonon coupling mechanism in the surface electron density.     

Lattice mechanical properties of some fcc f-shell metals

A pseudopotential depending on an effective core radius is proposed to study the binding energy, equation of state, ion-ion interaction, phonon dispersion curves (q-space and r-space analysis), mode Grüneisen parameters and dynamical elastic constants 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 potential while d and f-like electron is taken into account by introducing repulsive short-range Born-Mayer term. The parameter of the potential is evaluated by zero pressure condition. An excellent agreement between theoretical investigations and experimental findings is achieved which confirms the present formalism     

Multiphonon contribution to the structure factor of metals

The structure factor of Na is calculated including two-phonon terms and the Debye-Waller factor. The result is compared with the one-phonon approximation usually employed to evaluate the electronic transport coefficients. This multiphonon contribution can amount to 13 per cent at the melting point and 10 per cent at room temperature in the transport sensitive region of wave vector 1.5 k_F < q < 2 k_F, where k_F is the Fermi wave vector. We conclude that calculations of electronic transport coefficients of metals intended to attain a precision better than 10 per cent above the Debye temperature must take into account the contributions of the Debye-Waller factor and the two phonon terms.     

Debye-Waller factor in atom-surface scattering: Elastic and inelastic channels

Explicit expressions for the Debye-Waller factor for the elastic and one-phonon channels are presented to lowest order in the phonon displacement, using a hard wall model to represent the atom-surface interaction. The periodicity of the crystal is accounted for; thus we explicitly generalize to all elastic channels the reflectance result found by Garcia et al. within the plane-surface model, and we include the contribution of the umklapp processes to the inelastic channels. We show how for high incident energy of the atom all Debye-Waller factors reduce to the standard result.     

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.     

Network topology and subgap resonances observed by Fourier transform scanning tunnelling microscopy of cuprate high-temperature superconductors

Fourier transform scanning tunnelling microscopy (STM) on Bi₂Sr₂CaCu₂O_{8+ δ} (BSCCO) subgap resonances has deciphered an octet of ‘quasiparticle’ states that are consistent with the Fermi surface and energy gap observed by angle-resolved photoemission spectroscopy (ARPES), but the origin of the high-intensity k-space octets and the sharply defined r-space chequerboard is unexplained. The filamentary ferroelastic nanodomain model that predicted the r-space chequerboard also explains the k-space octets and the origin of the apparent anisotropic surface d-wave gap by using strong electron–phonon interactions outside the CuO₂ planes. The topological model identifies the factors that stabilize high-intensity k-space octets in the presence of a very high level of irregular r-space chequerboard noise.     

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.     

Phonon dispersion in thorium: A pseudopotential approach

A one parameter model pseudopotential is used in investigating the phonon dispersion in thorium. A very good agreement has been obtained between theory and experiment, especially, in transverse branches. This is attributed to the removal of the discontinuity in the pseudopotential in r-space.     

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.     

On the neutron diffraction in crystals in the field of a sound wave

Diffraction of neutrons in crystals under influence of a sound wave is considered. The probability of scattering of neutrons at the elastic interaction with the crystal is calculated. On the contrary, scattering of neutrons by an acoustical phonon has inelastic character. The possibility to control the Debye-Waller factor is shown.     

A semiclassical approach to inelastic scattering from solid surfaces and to the Debye-Waller factor

A semiclassical formulation of inelastic atom-surface scattering is presented. This formulation is a mixture of classical S-matrix theory and a classical path model. A Debye-Waller factor enters this theory very naturally as the probability of elastic reflection in the presence of inelastic channels. Because of its importance the Debye-Waller factor is discussed in some detail. Finally, assuming a simplified model of the gas-surface system, the whole scattering problem is solved analytically.     

Renormalization group method based on the ionization energy theory

Proofs are developed to explicitly show that the ionization energy theory is a renormalized theory, which mathematically exactly satisfies the renormalization group formalisms developed by Gell–Mann–Low, Shankar and Zinn-Justin. However, the cutoff parameter for the ionization energy theory relies on the energy-level spacing, instead of lattice point spacing in k-space. Subsequently, we apply the earlier proofs to prove that the mathematical structure of the ionization-energy dressed electron–electron screened Coulomb potential is exactly the same as the ionization-energy dressed electron–phonon interaction potential. The latter proof is proven by means of the second-order time-independent perturbation theory with the heavier effective mass condition, as required by the electron–electron screened Coulomb potential. The outcome of this proof is that we can derive the heat capacity and the Debye frequency as a function of ionization energy, which can be applied in strongly correlated matter and nanostructures.     

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.     

Messung der thermisch diffusen Elektronenstreuung in polykristallinen Aluminiumschichten

The diffuse elastic scattering of 51 kev electrons of a polycrystalline aluminium foil was measured by means of a retarding field apparatus in the temperature range 150 –800 °K. The temperature diffuse scattering intensityJ _TDS was separated from the temperature independent partJ _DS of the background intensity. The measured intensityJ _TDS shows good agreement with the theoretical angular distribution of the temperature diffuse scattering due to one- and two phonon processes. The value of the mean square displacement\(u^{\bar 2} (T)\), which is required for the theoretical calculation as well as for the experimental determination ofJ _TDSwas taken from the measured Debye-Waller factor exp(?2M) for the kinematic reflections of the same foil. The temperature independent partJ _DS of the background intensity is mainly due to elastic scattering of electrons in the amorphous oxide layer of the aluminium foil.     

High-Pressure Vibrational Studies of HCP Metals by Raman Spectroscopy

The pressure response of Raman phonons, determined for several hcp metals, includes positive pressure shifts as well as anomalies like mode softening in connection with phase transitions. It is shown that the phonon frequencies and their pressure dependences are related to macroscopic elastic parameters. More general, these results show that the measurement of Raman-active phonons in metals provides a direct probe of bonding, and agreement with theoretical models gives additional confidence in ab initio techniques.     

Quantum 1/f noise associated with ionized impurity scattering and electron-phonon scattering in condensed matter

Scattering of charged particles is accompanied by the emission of soft photons. Handel's theory of 1/f noise, based on the infrared quasi-divergent coupling of the system to the electromagnetic field, indicates that the current associated with a beam of scattered particles will exhibit 1/f noise. His derivation is valid in a vacuum. Here we extend his results and obtain the fluctuation spectrum for the fluctuations in cross-section and for the scattering rates w _kk′ in k-space, using the Born approximation. Next we consider mobility fluctuations due to these scattering rates, employing the relaxation time solutions of the Boltzmann transport equation, valid in non-degenerate semiconductors. Explicit results are obtained for the mobility-fluctuation noise caused by ionized impurity scattering, acoustic phonon scattering, optical phonon scattering, polar optical phonon scattering, and intervalley scattering. We derive Hooge's law, and the Hooge parameters for the above-mentioned processes are obtained in detail. This is then applied to n-type silicon and n-type gallium arsenide; the overall Hooge parameter, which is a weighted average of the partial α-parameters, is computed as a function of temperature and compared with experiment. For silicon, good agreement is obtained with available data. As a byproduct we also find the mobilities as function of temperature for these materials. Excellent agreement with the well-known experimental data is observed.

We still note that this is the first theoretical derivation of Hooge's law and that the magnitude of the noise is obtained in detail without adjustable parameters. We believe that quantum 1/f noise gives the limiting value of 1/f noise that can be observed.     

Theory of inelastic atom-surface scattering: Examples of energy distributions

The theory of the energy distribution of atoms scattered inelastically by solid surfaces which was developed previously is applied to various examples. The dependence of the results on a number of parameters is studied in detail. The importance of many phonon contributions as compared to the validity of first order distorted wave Born approximation is considered in particular. It turns out that low energy He atoms scattered by heavy transition metals provide a good example for which one phonon emission (or absorption) dominates. All other noble gases show appreciable many phonon contributions increasing, of course, with increasing mass of the noble gas and temperature of the solid. For heavy noble gases such as Kr and Xe the energy distribution approaches a gaussian, the width of which is due to the thermal and zero-point motion of the lattice. This width is quite large and thus probably masks most of the fine structure of the energy distribution occuring in classical trajectory calculations. We have also tried to apply the theory to light diatomic molecules. Although the results are less certain, partly because of the neglect of the internal motion of the molecules and partly because of uncertainties in the interaction parameters, one probably can expect appreciable many phonon effects already for H₂ and, of course, more so for N₂ and O₂. Recent experimental results on the Debye-Waller factor of Ne/Cu can be reproduced with reasonable potential parameters.     

Intermolecular repulsive-dispersive potentials explain properties of impurity spectra in soft solids

Behavior of optical impurity spectra in van der Waals glasses is rationalized with the aid of a two-particle Lennard-Jones model of intermolecular interactions. Simple mathematical manipulations with the 6-12 potential yield inhomogeneous distribution functions (IDFs) of zero phonon lines (ZPLs) at different compressions, and the expressions for wavelength dependent pressure shift coefficients of ZPLs (or holes), local phonon frequencies, and linear and quadratic coupling constants. Experimentally, the ZPL to phonon wing intensity ratios (Huang-Rhys or Debye-Waller factors) are measured for bacteriochlorophyll a in glass-forming triethylamine 5 K. Enhancement of coupling strength with increasing transition wavelength is observed, in qualitative agreement with the model.