Low-temperature transport properties of the alkali metals. I. The electron-phonon interaction

1961 ◽  
Vol 263 (1315) ◽  
pp. 531-544 ◽  
Keyword(s):  
Fermi Surface ◽  
Alkali Metals ◽  
Plane Waves ◽  
Wave Functions ◽  
Electron Wave ◽  
Thermo Electric ◽  
Fermi Surfaces ◽  
Electron Phonon Interaction ◽  
Symmetry Properties ◽  
Umklapp Scattering

The form of the electron-phonon matrix element is calculated for metals with non-spherical Fermi surfaces by using electron wave functions, which are linear combinations of two plane waves, as in the model of nearly free electrons. Numerical calculations are made with the use of the ‘twelve cone’ approximation to the Brillouin zone. It is shown that, if the Fermi surface bulges towards the zone faces, there is a significant increase in the probability of Umklapp scattering of electrons, the increase depending on the amount of distortion of the Fermi surface, and on the symmetry properties of the electron wave functions. The increase in Umklapp scattering has important consequences for calculations of the resistivities of metals, and particularly for calculations of the ‘phonon drag’ contribution to the thermo-electric power.

The electronic energy bands of the alkali metals and metallic beryllium

1961 ◽  
Vol 261 (1307) ◽  
pp. 551-564 ◽  
Keyword(s):  
Alkali Metals ◽  
Plane Waves ◽  
Potential Method ◽  
Electronic Energy ◽  
Wave Functions ◽  
Electron Wave ◽  
Energy Bands ◽  
Fermi Surfaces ◽  
Electronic Energy Bands ◽  
Pseudo Potential

The applicability of various electronic energy band interpolation schemes to the alkali metals and metallic beryllium is discussed. An ‘ l -dependent’ pseudo-potential method using only a very small number of plane waves in the expansion of the valence electron wave functions is then applied to these metals. The density of states curves and the Fermi surfaces are calculated. It is found that for lithium the Fermi surface may touch the Brillouin zone boundaries. The calculated and observed low temperature electronic specific heat coef­ficients and valence band widths are compared.

The de Haas–van Alphen effect in alkali metals

1964 ◽  
Vol 281 (1384) ◽  
pp. 62-91 ◽  
Keyword(s):  
Fermi Surface ◽  
Field Dependence ◽  
Alkali Metals ◽  
Spherical Shape ◽  
Orientation Dependence ◽  
Sodium Potassium ◽  
Fermi Surfaces ◽  
Fixed Orientation ◽  
Absolute Amplitude ◽  
Structure Calculations

A new method for studying the de Haas–van Alphen effect in steady magnetic fields has been developed in which the field is modulated at frequency ω and a signal at frequency 2 ω is generated in a pick-up coil round the specimen because of the non-linear field dependence of magnetization. The rectified 2 ω signal is proportional to d 2 M /dH 2 and so shows de Haas–van Alphen oscillations either when H is varied for fixed orientation or when the orientation is varied in fixed H if the Fermi surface is anisotropic. Because the phase of oscillation is very high (of order 10 4 π ) even very slight anisotropy will produce a few oscillations when the orientation is varied and the method is therefore particularly sensitive for studying very nearly spherical Fermi surfaces. From the oscillations with H , values of the frequency F were found for sodium, potassium, rubidium and caesium which were close to those predicted for a free-electron sphere containing 1 electron per atom, though some small systematic deviations of order ½ % were observed which may be significant. From detailed study of the oscillations produced by rotation of single crystals in fixed H it was found possible to describe the orientation dependence of F (proportional to the area of cross-section of the Fermi surface) for potassium and rubidium consistently by a series of cubic harmonics and hence to deduce the small departures of the Fermi surfaces from spherical shape. The deviations from a sphere were found to be of the order of 1 part in 10 3 for potassium and a little less than 1 part in 10 2 for rubidium; these deviations are compared with those predicted by band structure calculations. Preliminary results for sodium suggest that it is appreciably less anisotropic than potassium. Some results are also reported on the temperature and field dependence and the absolute amplitude of the de Haas-van Alphen effect, and it is also shown how the effect can be used to measure very small variations of field with position.

The dependence of the hyperfine interaction on the cellular potential in Na and K. II

1969 ◽  
Vol 47 (13) ◽  
pp. 1331-1336 ◽  
Author(s):  
R. A. Moore ◽  
S. H. Vosko
Keyword(s):  
Fermi Surface ◽  
Electron Density ◽  
Wave Functions ◽  
Electron Wave ◽  
Surface Electron ◽  
Conduction Electrons ◽  
Hartree Fock ◽  
A Value ◽  
Dependent Potential ◽  
Conduction Electron Density

The dependence of the Fermi surface electron wave functions in Na and K on (i) an L-dependent effective local cellular potential constructed to simulate Hartree-Fock theory and (ii) the inclusion of the Hartree field due to the conduction electrons in the cellular potential is investigated. All calculations are performed using the Wigner–Seitz spherical cellular approximation and the Schrödinger equation is solved by the Kohn variational method. It is found that to ensure a value of the Fermi surface electron density at the nucleus accurate to ~5%, it is necessary to use the L-dependent potential along with the Hartree field due to a realistic conduction electron density.

scholarly journals Calculation of One-Electron Wave Functions and Energy Levels of N-Butane Molecule on the Basis of Slater Atomic Orbitals

2021 ◽  
Vol 75 (3) ◽  
pp. 229-233
Author(s):  
Faig Pashaev ◽  
Arzuman Gasanov ◽  
Musaver Musaev ◽  
Ibrahim Abbasov
Keyword(s):  
Group Theory ◽  
Energy Levels ◽  
Wave Functions ◽  
Electron Wave ◽  
Hamilton Operator ◽  
Atomic Orbitals ◽  
Symmetry Properties ◽  
Symmetry Transformations ◽  
Space Symmetry ◽  
Polyatomic Systems

Abstract It is known that the application of the group theory greatly simplifies the problems of polyatomic systems possessing to any space symmetry. The symmetry properties of such systems are their most important characteristics. In such systems, the Hamilton operator is invariant under unitary symmetry transformations and rearrangements of identical particles in the coordinate system. This allows to obtain information about the character of one-electron wave functions — molecular orbitals — the considered system, i.e. to symmetrise the original wave functions without solving the Schrödinger equation.

Low-temperature transport properties of the alkali metals II. The transport coefficients

1961 ◽  
Vol 264 (1316) ◽  
pp. 60-87 ◽  
Keyword(s):  
Fermi Surface ◽  
Alkali Metals ◽  
Transport Coefficients ◽  
Direct Calculation ◽  
Zone Boundary ◽  
Phonon Drag ◽  
Drag Effect ◽  
Thermo Electric ◽  
Sodium And Potassium ◽  
The Ideal

The theory of part I (Collins 1961) is applied to the direct calculation of the ‘ideal’ electrical and thermal resistivities and ‘phonon drag’ thermo-electric power, of the alkali metals. All three coefficients depend, in magnitude and as functions of temperature, on the shape of the Fermi surface and on the lattice spectrum. If it is assumed that the latter is identical in form for all metals in the group, the observed transport coefficients are consistent with a Fermi surface which is quite distorted in lithium, becomes nearly spherical in sodium and potassium, and is again distorted in rubidium and caesium. The argument is not sufficiently accurate to discriminate between s -like and p -like symmetry in each case, nor to decide whether the Fermi surface actually touches the zone boundary; the phonon drag effect is also very sensitive to the purity of the specimen.

scholarly journals Fermi Radii of Lithium by Positron Annihilation

1971 ◽  
Vol 49 (24) ◽  
pp. 3227-3233 ◽  
Author(s):  
J. J. Paciga ◽  
D. Llewelyn Williams
Keyword(s):  
Fermi Surface ◽  
Positron Annihilation ◽  
Wave Functions ◽  
Scattering Data ◽  
Electron Wave ◽  
A Value ◽  
Band Structure Calculations ◽  
Detector Geometry ◽  
Point Detector ◽  
Structure Calculations

A collinear-point detector geometry has been used to study positron annihilation in single-crystal lithium. The results are interpreted in terms of the higher momentum components of the electron wave functions and the lithium Fermi surface. A consistent interpretation favors a value for the Fourier component V110 of the lattice potential of close to 0.10 Ry and a Fermi surface whose radius in the [110] direction is 2.9% greater than in the [100] direction. This latter result is consistent with Compton-scattering data and both results are in close agreement with the recent band-structure calculations of Rudge.

Multiple-plane wave calculation of the electron–phonon interaction in Al

1976 ◽  
Vol 54 (15) ◽  
pp. 1585-1599 ◽  
Author(s):  
H. K. Leung ◽  
J. P. Carbotte ◽  
D. W. Taylor ◽  
C. R. Leavens
Keyword(s):  
Plane Wave ◽  
Fermi Surface ◽  
Plane Waves ◽  
Spectral Weight ◽  
Model Fit ◽  
Final States ◽  
Electron Phonon Interaction ◽  
Self Energy ◽  
Multiple Plane ◽  
Migdal's Theorem

We have computed from first principles the electron–phonon spectral weight αk2(ω)Fk(ω) for many points k on the aluminum Fermi surface (FS). According to Migdal's theorem this spectral weight completely determines the self-energy of the electrons. The calculations involve an integral over final states on the Fermi surface which we calculate from Ashcroft's 4-plane wave pseudopotential model fit to the de Haas–van Alphen data. For the electron–phonon matrix element 15 plane waves are included. The phonons are taken from a Born–von Karman model fit to the measured dispersion curves. From the spectral weights we compute the Fermi surface variation of the electron–phonon effective mass and the quasiparticle lifetimes at various temperatures.

Magnetotunneling spectroscopy imaging of electron wave functions in self-assembled InAs quantum dots

2001 ◽  
Vol 171 (12) ◽  
pp. 1365
Author(s):  
E.E. Vdovin ◽  
Yu.N. Khanin ◽  
Yu.V. Dubrovskii ◽  
A. Veretennikov ◽  
A. Levin ◽  
...  
Keyword(s):  
Quantum Dots ◽  
Wave Functions ◽  
Electron Wave ◽  
Inas Quantum Dots ◽  
Self Assembled

Quantum Boxes, Stringed Instruments

2017 ◽  
Author(s):  
Frank S. Levin
Keyword(s):  
Energy Level ◽  
Schrödinger Equation ◽  
Quantum System ◽  
Schrodinger Equation ◽  
Probability Distributions ◽  
Wave Functions ◽  
Energy Level Diagram ◽  
One Dimensional ◽  
Stringed Instruments ◽  
Symmetry Properties

Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.

A free-electron theory of conjugated molecules

1953 ◽  
Vol 49 (4) ◽  
pp. 650-658 ◽  
Author(s):  
J. Stanley Griffith
Keyword(s):  
Free Electron ◽  
Wave Functions ◽  
Electron Charge ◽  
Algebraic Equations ◽  
Electron Wave ◽  
Electron Theory ◽  
Conjugated Molecules ◽  
Electron Charge Density ◽  
Linear Polyenes ◽  
Alternant Hydrocarbons

ABSTRACTThe values of a free-electron eigenfunotion at the carbon nuclei of a conjugated hydrocarbon are found to satisfy a system of algebraic equations. These equations are similar in form to those obtained in the method known as the linear combination of atomic orbitale but only coincide with them for linear polyenes and benzene. The symmetry, degeneracy and energy of the eigenvectors of these free-electron equations correspond exactly to those of the free-electron wave functions found by the usual methods. From this correspondence, a theorem is deduced about the free-electron charge density in alternant hydrocarbons.

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