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Showing papers on "Mean free path published in 1972"


Journal ArticleDOI
TL;DR: In this article, a formal deduction of the Boltzmann equation from the Liouville equation is presented for the case of rigid spheres, and the main result is that, under the assumptions of a sufficiently smooth N-particle distribution function for a smooth limit to exist and of an initial datum which satisfies (at least for N → ∞) the chaos assumption.
Abstract: A formal deduction of the Boltzmann equation from the Liouville equation is presented for the case of rigid spheres. The main result is that, in the Boltzmann limit (number of molecules tending to infinity, diameter tending to zero, finite mean free path), the Boltzmann equation follows under the assumptions of a sufficiently smooth N-particle distribution function for a smooth limit to exist and of an initial datum which satisfies (at least for N → ∞) the chaos assumption. Possible extensions to molecules interacting with central forces and to dense gases are briefly discussed.

94 citations


Journal ArticleDOI
TL;DR: In this paper, the thermoelectric power, resistivity and TCR of vacuum evaporated copper films were measured and the energy dependence of the electron mean free path and the area of the constant energy surface were obtained as 0.7 and -2.1 respectively.

49 citations


Journal ArticleDOI
TL;DR: In this article, the scattering from impurities in copper is analyzed in terms of s, p and d partial waves, and the anisotropic transition probability between points on the Fermi surface is calculated based on scattering phase shifts characterizing the impurity.
Abstract: The scattering from impurities in copper is analysed in terms of s, p and d partial waves. The (anisotropic) transition probability between points on the Fermi surface is calculated in terms of scattering phase shifts characterizing the impurity. Maps of the calculated scattering are presented for each phase shift and the corresponding Dingle temperatures are calculated for a variety of orbits seen in the de Haas-van Alphen effect. A procedure developed to calculate the impurity resistivity and anisotropy of the corresponding relaxation time demonstrated the need to use a vector mean free path in such a calculation. The calculated scattering is compared with experimental values in the Cu(Ni), Cu(Fe) and Cu(Al) systems and values derived for the scattering phase shifts.

49 citations


Journal ArticleDOI
TL;DR: The electrical properties of liquid mercury are reviewed in this article, where it is concluded that at normal densities, liquid mercury is a metal which can be described by the Ziman weak-scattering theory and that the resistivity of amalgams is well described by Evans and his collaborators.
Abstract: The electrical properties of liquid mercury are reviewed. It is concluded (a) That mercury at normal densities is a metal which can be described by the Ziman weak-scattering theory and that the resistivity of amalgams is well described by the work of Evans and his collaborators (1969, 1970). (b) There is nonetheless, even at normal densities, considerable evidence for a ‘pseudogap’ (g = N(EF)/N(E F)free<1), which fills up on increasing the temperature or on alloying. This has little or no effect on the conductivity because of the Edwards cancellation theorem. g does not affect the Hall coefficient when the mean free path L is greater than a, the interatomic distance. (c) At densities below about 10 g/cm3, but before the material becomes a semiconductor, the mean free path is of order a and the conductivity is proportional to g 2 and the Hall coefficient is given by the Friedman formula C/necg. (d) Special consideration is given to the results on the thermopower of amalgams at 1000°C due to Crisp,...

43 citations


Journal ArticleDOI
TL;DR: In this paper, a detailed account of phonon generation by superconducting tunnel junctions (Sn-I-Sn) and by Sn and Pb 0.5 Tl0.5 films pumped by a heat pulse was presented.
Abstract: We present a detailed account of phonon generation by superconducting tunnel junctions (Sn-I-Sn) and by Sn and Pb 0.5 Tl 0.5 films pumped by a heat pulse. The spectrum and propagation characteristics of the generated phonons is studied through the resonance absorption by Sb-donor levels in uniaxially compressed Ge. At all values of generator power (up to a few watts) the emitted phonons show a large density at a value of energy equal to the superconducting energy gap (2Δ). In contrast, the spectrum emitted by a constantan heater is shown to be in quantitative agreement with the blackbody-radiation model. The experiments show that the mean free path of longitudinal and transverse phonons in the superconductor changes discontinuously when hω=2Δ. Theoretical calculations show that significant reabsorption of phonons of energy 2Δ results in a nonlinear buildup of the intensity of the 2Δ phonons. The propagation characteristics of these phonons in Ge: Sb show a frequency and polarization dependence in excellent agreement with the Griffin-Carruthers theory of resonance-fluorescence phonon scattering by donor levels. The magnetic field tunability of the generated 2Δ phonons in Sn is utilized to study the ground state of V 3+ in Al 2 O 3 . Transverse phonons of energy 1.02 meV propagating along the c axis are resonantly absorbed as expected according to theory.

42 citations



Journal ArticleDOI
TL;DR: In this article, the electronic transport phenomena of Se-Te melts are analyzed with two models of density of states, and the optical absorption results give the width of optical gap and its temperature dependence.
Abstract: The electronic transport phenomena of SeTe melts are analyzed with two models of density of states. At low temperatures ( T ∗ ) a semiconducting behaviour is observed and we calculate activation energies involved in conducting and thermopower when only hole transport is taken into account. The optical absorption results give the width of optical gap and its temperature dependence. At high temperatures ( T > T ∗ ) a metallic conduction, with a short mean free path occurs so that a pseudogap and onset of localization near the Fermi level are likely when the Mott's criterion is satisfied. The observation of switching phenomena in liquid Se-rich alloys strengthens the analogy with amorphous semiconductors.

30 citations


Journal ArticleDOI
01 Jun 1972-Carbon
TL;DR: In this article, the thermal diffusivity of three pyrolytic graphites was measured using the modulating electron beam method over the temperature range from 1300°C to 1900°C.

27 citations


Journal ArticleDOI
TL;DR: In this paper, the spatial distribution of secondary electrons generated by a primary electron beam incident at a point of the sample surface is calculated for samples of varying thicknesses, and the current density generated under these conditions is also calculated for several different angles of incidence.
Abstract: The spatial distribution of secondary electrons generated by a primary electron beam incident at a point of the sample surface is calculated for samples of varying thicknesses. The current density generated under these conditions is also calculated for several different angles of incidence. According to the calculations, which are based on a commonly used idealized model, most secondary electrons leave the sample within a mean free path of their point of excitation, which is 10–20 A for metals. Because thin metal coatings are used to prevent many insulating samples from charging, these results relate to the ultimate resolution that one can expect in secondary emission scanning electron microscopy.

26 citations


Journal ArticleDOI
TL;DR: In this article, a phenomenological theory for the frequency dependent conductivity sigma ( omega ) and dielectric constant epsilon( omega ) of a quasi one dimensional metallic conductor of this type is developed in terms of a model of linear metallic strands interrupted by insulating lattice defects.
Abstract: One dimensional metallic behaviour of a mixed valence square planar (MVSP) complex compound has been convincingly demonstrated by the work of Kuse and Zeller. A simple phenomenological theory for the frequency dependent conductivity sigma ( omega ) and dielectric constant epsilon ( omega ) of a quasi one dimensional metallic conductor of this type is developed in terms of a model of linear metallic strands interrupted by insulating lattice defects. Since the insulating nature of the defects gives rise to zero DC conductivity in the absence of interstrand carrier transitions, epsilon ( omega ) and sigma ( omega ) are found to deviate significantly from their respective Drude type counterparts for omega < nu F/l0, where nu F denotes the carrier Fermi velocity and l0 the mean distance between defects in a given strand. If l0 is small by comparison with the 'intrinsic' mean free path for carrier motion along the defect free strand, a large and real static dielectric constant of order (l0/a)2 is obtained, where a denotes the spacing of the metallic atoms comprising the strands. For the particular MVSP compound studied by Kuse and Zeller it is estimated that l0 approximately 100AA, which together with the small effective carrier mass deduced for the same compound, implies the existence of large spacings of the 'boxed' carrier energy levels.

24 citations


Journal ArticleDOI
TL;DR: In this article, the authors measured the interaction lengths of singly charged relativistic secondary particles emitted from interactions of very heavy cosmic ray nuclei in emulsions by systematic following of their tracks through the stack.
Abstract: Interaction lengths of singly charged relativistic secondary particles emitted from interactions of very heavy cosmic ray nuclei in emulsions were measured by systematic following of their tracks through the stack. The primary energies were determined by the "knock on" electron method. The results show that particles from interactions with primary energies between 1 and 5 GeV/nucleon emitted in the angular interval (θev is the expected RMS angle for evaporation protons) have significantly shorter mean free paths. The variation of the observed mean free paths with distance from the primary interactions shows a transition from short to normal interaction lengths at about 3 cm which indicates the presence of some anomalous component interacting with a very short mean free path of a few cm and relative intensity of order of 10%. This result cannot be accounted for by any known particle or interaction. Considerations of emission angles and types of stars produced by the secondary particles lead to a tentative ...

Journal ArticleDOI
TL;DR: In this article, a new approximation based on treating, self-consistently, clusters of sites is introduced to describe the details of the energy spectrum of a disordered alloy when the mean free path is short.

Journal ArticleDOI
TL;DR: In this article, a discussion of the effect of lattice vibrations on the effective electron-ion-core elastic scattering vertex is given using the model of Duke and Laramore.
Abstract: A discussion of the effect of lattice vibrations on the effective electron-ion-core elastic scattering vertex is given using the model of Duke and Laramore. The renormalization introduced by the lattice vibrations can substantially increase the number of partial waves necessary to describe this scattering. The vertex approximation using the $s$-wave part of the phonon renormalization factor and phase shifts from a realistic potential to describe the electron-rigid-ion scattering is compared with the vertex approximation using the full phonon renormalization factor and the constant phase shift $s$-wave model to describe the electron-rigid-ion scattering. Model calculations of low-energy-electron-diffraction (LEED) intensity profiles are presented for a system having the geometrical parameters of Al(100), and effective Debye temperatures are obtained for the Bragg peaks in the intensity profiles. The dependence of these effective Debye temperatures on the inelastic-collision mean free path and on the characteristic falloff of the vibrational amplitude of the ion cores with distance from the surface is investigated. Even for a constant mean free path the ${{\ensuremath{\Theta}}_{D}}^{\mathrm{eff}}$ exhibit a pronounced energy dependence. By comparing the calculated ${{\ensuremath{\Theta}}_{D}}^{\mathrm{eff}}$ with the experimental measurements of Quinto et al., a crude estimate of the inelastic-collision mean free path in the surface region of aluminum is obtained.

Journal ArticleDOI
TL;DR: In this article, the spectral distribution of laser light scattered by dilute binary gas mixtures is analyzed using a 1/r4 repulsive potential model, which yields density correlation functions which are valid for arbitrary ratios of wavelength to mean free path.
Abstract: Time‐dependent correlation functions in a binary mixture of gases are studied on the basis of kinetic theory. The problem is formulated as an initial‐value solution to the linearized Boltzmann equation for the mixture. The analysis is made tractable by the use of an appropriate kinetic model which is developed for Maxwell molecules (1/r4 repulsive potential). The resulting description yields density correlation functions which are valid for arbitrary ratios of wavelength to mean free path. In particular, the results can be used to discuss deviations from hydrodynamic behavior. With the present theory a detailed analysis of the spectral distribution of laser light scattered by dilute binary gas mixtures is made. Explicit results for xenon‐helium mixtures are shown to be in very good agreement with the recent measurements of Clark and Benedek.

Journal ArticleDOI
TL;DR: In this paper, the phonon limited mean free path has been measured in cadmium using the limiting point method and a model has been proposed, which accounts for the experimental results over much of the (1120) plane.
Abstract: The phonon limited mean free path has been measured in cadmium using the limiting point method. The measurements are on the lens part of the Fermi surface and extend from (0001) to (1010) in the (1120) plane. The mean free path varies approximately as T5 except near (1010) where the temperature dependence is T6 or even T7. A model has been proposed, which accounts for the experimental results over much of the (1120) plane. This model takes account of elastic anisotropy and distinguishes between electrons rendered ineffective by a single phonon scattering event and those rendered ineffective by multiple scattering events. The model should be applicable to any metal over a wide temperature range.

Journal ArticleDOI
TL;DR: In this article, electron-optical phonon scattering in the thin SiO2 layer before one Si-emitted electron escapes into the vacuum was explained by electron−optical PHONON scattering, and an approximate value of the phonon energy at the center of the Brillouin zone was given.
Abstract: In vacuo photoemission from SiO2‐covered Si shows that the quantum yield and the energy threshold depend on the SiO2 layer in the 10‐ to 120‐A thickness range These effects have been explained by electron‐optical phonon scattering in the thin SiO2 layer before one Si‐emitted electron escapes into the vacuum An approximate value of the phonon energy at the center of the Brillouin zone (ΔE = 83 meV) and a mean free path for phonon scattering (l = 35 A) are given

Journal ArticleDOI
TL;DR: In this article, the electrical resistivity of thin annealed polycrystalline films of Ag has been measured at room temperature (RT) and liquid nitrogen temperature (77 °K).


Journal ArticleDOI
TL;DR: In this article, a simple transit-time theory is used in which nonmagnetic inclusions in polycrystalline YIG limit the mean path length of spin waves, and agreement between theory and experiment is found for inclusions of sizes 25 μm and 12 μm respectively.
Abstract: A simple transit-time theory is used in which nonmagnetic inclusions in polycrystalline YIG limit the mean path length of spin waves. Agreement between theory and experiment is found for inclusions of sizes 25 μm and 12 μm respectively. The results suggest that dipole-dominated spin waves can propagate across grain boundaries. Previous attempts to fit experimental results to the grain-size transit-time theory failed, because the pores rather than the grain boundaries limit the mean free path in low-anisotropy materials.

Journal ArticleDOI
TL;DR: In this article, a kinetic theory for the mass flux to a liquid droplet surrounded by its pure vapor is presented, where the coefficients of the first three terms of this expansion are derived by solving the Boltzmann equation using a modified Knudsen number iteration procedure.
Abstract: A kinetic theory is presented for the mass flux to a liquid droplet surrounded by its pure vapor. When the mass flux Γ is expanded in terms of a parameter α which is the ratio of the droplet size to the mean free path (inverse Knudsen number), one obtains a series of the form Γ=Γ(0)+Γ(1)α +Γ(2)α2 lnα +···. The coefficients of the first three terms of this expansion are derived by solving the Boltzmann equation using a modified Knudsen number iteration procedure. It is shown that the coefficients are determined by integrals associated with sequences of successive collisions among a number of vapor molecules and the droplet. These collision integrals bear a close similarity to the collision integrals derived earlier from the generalized Boltzmann equation for the density dependence of the transport properties of gases.

Journal ArticleDOI
TL;DR: In this article, resistivity measurements have been made on Nb cavities, as well as on Pb and Cu, at 296, 77, and 4.2 °K by means of a contactless induced current method.
Abstract: Resistivity measurements have been made on Nb cavities, as well as on Pb and Cu, at 296, 77, and 4.2 °K by means of a contactless induced‐current method. For superconductors, a constant magnetic field drives the material normal below the transition temperature. These measurements provide a simple means for initial material evaluation as well as a direct means of monitoring the effects of material parameters (purity, heat treatment, gas incorporation, etc.) on the electron mean free path. Approximate determinations of Hc, Hc1, and Hc2 can also be derived from these measurements. Normal‐state thermal conductivity and the Ginzburg‐Landau parameter κ are calculated from the resistivity measurements.

Journal ArticleDOI
TL;DR: In this article, a microwave cryostat is described, which enables us to measure the surface resistance of superconducting films in metastable states, and the mean free path, the penetration depth and the coherence length were determined in the metastable and the annealed state.
Abstract: A microwave cryostat is described, which enables us to measure the surface resistance of superconducting films in metastable states. In metastable disordered films often elevated transition temperatures are found. Surface resistance measurements can give information about electronic parameters of such films. In connection withdc-conductivity and critical magnetic field measurements, the mean free path, the penetration depth and the coherence length were determined in the metastable and the annealed state. These values have been compared with those expected from a free electron model.


Journal ArticleDOI
TL;DR: In this article, Andreev's theory of sound attenuation in the intermediate state was extended to allow for arbitrary values of $\mathrm{ql}$, the product of sound propagation vector and electron mean free path.
Abstract: We have measured the ultrasonic attenuation in high-purity Sn cylinders in the intermediate state. The laminar intermediate-state structure was set up with a rotating external field. Andreev's theory of sound attenuation in the intermediate state was extended to allow for arbitrary values of $\mathrm{ql}$, the product of sound propagation vector and electron mean free path. It was then possible to determine the size of the normal regions as a function of applied magnetic field from the experimental data. The intermediate-state structure thus determined agreed reasonably well with the Landau model. The analysis suggests, however, that in the intermediate state the attenuation in the high-frequency limit is not increasing linearly with applied field as simple considerations would suggest. The reason for this is not properly understood at the present time.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the case in which the mean free path of the conduction electrons is comparable to or greater than the film thickness, with particular attention to the transition between the dc limit and the anomalous skin effect limit.
Abstract: Numerical calculations are presented for the impedance per square of metallic films. We consider in detail the case in which the mean free path of the conduction electrons is comparable to or greater than the film thickness (thin films at low temperatures), with particular attention to the transition between the dc limit and the anomalous skin‐effect limit.

Journal ArticleDOI
TL;DR: The metal base triode sandwich was used to measure the current transfer ratio in a Schottky barrier structure and the transfer ratio was discussed considering two distinct current types, that through base metal and that through pinholes as mentioned in this paper.
Abstract: The metal‐base triode sandwich was used to measure the current transfer ratio in a Schottky barrier structure. The transfer ratio is discussed considering two distinct current types—that through base metal and that through pinholes. An electron‐phonon mean free path of 125 A and an electron‐electron mean free path of 230 A (at 0.7 eV) were measured. An investigation of the structure of very thin Al and Au films deposited on Al2O3 was made by transmission electron microscopy. Al films were found to be more continuous than Au films of comparable thickness.

Journal ArticleDOI
TL;DR: In this article, it is shown that when a constant field in the 10V cm−1 range is applied to annealed anodic oxide films, the ion current builds up in accelerating fashion.
Abstract: When a constant field in the 10 V cm−1 range is applied to annealed anodic oxide films the ion current builds up in accelerating fashion. It is shown that a model in which the ions move through channels in the oxide and in which the moving ions clear the initially blocked channels reproduces the general form of the ion current vs. time behavior. Dignam's theory of this process is discussed. A recent claim that considerations of mean free path render inapplicable to these films models in which the concentration of ions changes with field due to homogeneous generation of Frenkel defects is shown to be unjustified.A further observed phenomenon in which the latent ionic conductivity of the films decays at zero field is also discussed in terms of the competing models.

Journal ArticleDOI
TL;DR: In this paper, a general method for calculating the electron mean free path of small-band-gap multivalent metals via a nearly free-electron model (NFEM) in which only one pair of Brillouin-zone planes is considered at a time.
Abstract: We have constructed a general method for calculating the electron mean free path of small-band-gap multivalent metals via a nearly-free-electron model (NFEM) in which only one pair of Brillouin-zone planes is considered at a time. The expansion coefficients of the appropriate Bloch wave functions are concentrated about the zone planes with discontinuities in value or slope at the planes themselves. Our procedure is to postulate trial solutions for the Boltzmann equation explicitly containing these concentrated functions as well as functions smoothly distributed over the Fermi surface. The ratio of band-gap energy to Fermi energy forms a natural expansion parameter. Form factors for Fermi-surface scattering are expressed as power series in the square of the scattering vector; only a few terms suffice to give quite satisfactory representations for most form factors. Analytic solutions are developed for a two-term expansion. As pointed out by Ziman, the $s$-type and $p$-type states on opposite sides of the zone gap have markedly different mean free paths. The mean free path is not only dependent on $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$, but also direction dependent; the component in the zone plane is less affected by the discontinuities than the component perpendicular to the zone plane. It follows that the relaxation time is both $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ dependent and a tensor.

Journal ArticleDOI
TL;DR: In this article, the effects of collisions on the scattering function of a fully ionized gas, and on related functions describing conductivity and energy loss of a test particle are discussed using a simplified Fokker-Planck collision model which preserves certain important features of the long-range small angle soft collisions.
Abstract: The effects of collisions on the scattering function of a fully ionized gas, and on the related functions describing conductivity and energy loss of a test particle are discussed using a simplified Fokker‐Planck collision model which preserves certain important features of the long‐range small angle soft collisions. The dispersion relations are discussed in detail, and explicit calculations are carried out for the scattering function and the energy‐loss function in the long‐wavelength approximation when the wavelength is longer than the mean free path and the Debye length. The damped plasma oscillation poles dominate the energy‐loss function while for the scattering function the ambipolar diffusion dominates the low‐frequency region.

Journal ArticleDOI
TL;DR: In this article, the escape length of photoelectrons in PbTe films was determined by measuring the quantum yield as a function of film thickness, and the results were shown to be useful in estimating the mean free path for electron•electron scattering in pbTe.
Abstract: The escape length of the photoelectrons in PbTe films was determined by measuring the quantum yield as a function of film thickness. The data indicate that the escape length of the electrons excited by photons of energies 7.7–11 eV is approximately 20–40 A. The results are shown to be useful in estimating the mean free path for electron‐electron scattering in PbTe.