Usual Relaxation Research Articles

Measurement of the internal stress in metals by the decremental unloading technique

The validity of the decremental unloading (stress dip) technique for measuring the internal stress was carefully examined for Cu, Ti, and Fe polycrystals at 77 – 650 K. For Ti and Fe the critical stress, σ c, for zero relaxation depended on the relaxation time after dipping and generally showed a higher temperature dependence than the modulus. Besides, in the b.c.c. and h.c.p. metals in which the thermally-activated barries such as the Peierls hills or short range obstacles due to interstitial impurity atoms are high enough, the relaxation curves after dipping to within a certain stress level exhibited a transition from negative-to-positive relaxation. At temperatures below about 300 K, the critical stress, σ c∞, at which the time for the transition from negative-to-positive relaxation was very large, was in accord with the internal stress, σ μ , obtained from the usual stress relaxation and back-extrapolation techniques, and exhibited the same temperature dependence as the shear modulus. At higher temperatures, σ c∞ was smaller than σ μ . It is concluded that in this higher temperature regime the dislocation structure changes during unloading and relaxation so that the dip technique, which requires a long relaxation time, is not as reliable for measuring the internal stress. In this temperature region, the stress which first yielded negative relaxation at the very beginning of the relaxation curve was in accord with the internal stress obtained by the back-extrapolation technique.

The infrared absorption spectrum of n‐type GaAs

AbstractThe optical conductivity of free electrons in polar semiconducting compounds has recently been calculated by use of a generalized Boltzmann equation derived from the equation of motion of the quantum density matrix. This reduces to the quasi‐classical Boltzmann transport equation in the low frequency limit. In this paper, the optical conductivity is calculated for GaAs as a function of carrier concentration in terms of a frequency dependent relaxation time which reduces to the usual relaxation time in the limit of low frequencies and an elastic scattering mechanism. The relation of the frequency dependent relaxation time to the electron cross section is discussed. The optical constants and the frequency dependent relaxation time are given for GaAs at 298 K over the spectral region from 45 to 3.5 × 103 cm−1 for carrier concentrations from 3.4 × 1015 to 8.7 × 1018 cm−3. The real and imaginary parts of the complex refractive index and the reflectivity are calculated from the complex dielectric constant in terms of the frequency dependent relaxation times. The frequency at the minima of the reflectivity curve and at the plasma edge, where the real part of the electric constant vanishes, are correlated with carrier concentration. Comparison is made with recent experimental results.

Einstein-Kanzaki model of static and dynamic lattice relaxation: Application to vacancies in metallic hydrogen

A method is proposed for calculating the formation energy of localized defects in crystalline solids with pair forces of arbitrary range. The theory is most useful in the cases of small mass or high temperature for which, in addition to the usual static relaxation, changes in the lattice vibrations make a significant contribution. Defect migration is not described however. A self-consistent Einstein approach is used, each particle in the crystal oscillating with its own frequency about an average position. The total free energy is minimized with respect to all of these frequencies and positions. This minimization is made tractable by the assumption that large changes in frequency and position occur only for a finite number of particles near the defect; the changes for all the other particles are treated linearly. The result is very similar to Kanzaki's $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}\ensuremath{-}$ space "lattice statics" formalism. However, instead of being 3 \ifmmode\times\else\texttimes\fi{} 3 the lattice Green's function becomes a 4 \ifmmode\times\else\texttimes\fi{} 4 matrix, thereby encompassing changes in Einstein frequencies as well as particle positions. The method is applied to calculate the free energy of vacancy formation in metallic hydrogen.

The optical conductivity of free carriers in GaAs

The optical conductivity of free electrons in polar semiconducting compounds has recently been calculated by use of a generalized Boltzmann equation derived from the equation of motion of the quantum density matrix. This reduces to the quasi-classical Boltzmann transport equation in the low frequency limit: the optical conductivity thus obtained spans a spectral range from around 30cm −1 to 1.2 × 10 4 cm −1 in GaAs. In this paper, the optical conductivity is calculated for GaAs as a function of carrier concentration in terms of a frequency dependent relaxation time which reduces to the usual relaxation time in the limit of low frequencies and an elastic scattering mechanism. The low frequency limit of the relaxation time is used to estimate the mobility as a function of carrier concentration. The frequency dependent relaxation time is given for GaAs at 298 K over the spectral region from 45 cm −1 to 2.3 × 10 3 cm −1 for carrier concentrations from 3.4 × 10 15 cm −3 to 8.7 × 10 18 cm −3.

Differential relaxation times and diffusivities of hot carriers in isotropic semiconductors

The differential relaxation time τ (k,F) defined in a previous paper for isotropic semiconductors in the hot-carrier range is shown to depend on the orientation with respect to the field force F. τ (k,F) is expressed using the collision operator and the distribution function. In the Ohmic range τ (k,F=0) is found to be equal to the usual relaxation time τ (k) related to transition probabilities per unit time. Longitudinal τN(k,E) and transverse τn(k,E) differential relaxation times are numerically computed for p-type germanium and are found to be actually different although of the same order of magnitude, namely 10−12–10−13 sec. It is proved that the distribution function for hot carriers may be a displaced Maxwellian only if τ (k) is k independent, which does not hold for most of the situations of physical interest. It is shown that the differential relaxation times involved in longitudinal D∥(E) and transverse D⊥(E) diffusion coefficients are τ∥(k,E) and τ⊥(k,E). D∥(E) and D⊥(E) are numerically computed for p-type germanium and are found to be in excellent agreement with experimental results. Longitudinal and transverse noise temperatures are identical to the electronic temperature for displaced Maxwellian distribution functions.

Formulation and use of a model for the phonon Umklapp collision operator

A new derivation is given of the Callaway model for the linearized phonon N-process collision operator. This is extended to cover the case of U processes, and the model is then applied in a simple case to calculate the thermal conductivity of a dielectric. The result obtained is about one half of that given by the usual relaxation time approximation, and this agrees with a simple qualitative analysis of the situation. A discussion is given of the difference between the present approach and that of Hamilton (1973).

Numerical solution of the viscous flow in the entrance region of parallel plates

The numerical solution to the flow in the entrance region of a semiinfinite parallel channel was obtained for Reynolds numbers between zero and 2000. The governing equations, which are the steady-state, incompressible Navier-Stokes equations, were first quasilinearized. The resultant linear equations were then approximated by finite-difference formulas, which were judiciously selected in order for the coefficient matrix of the finite-difference equations to be diagonally dominant. This system of the difference equations was solved by a Gaussian elimination process instead of the usual relaxation techniques. A convergence to solution of the original nonlinear equations was reached in three to four repeated Gaussian elimination processes. The numerical results are in good agreement with other known numerical solutions. Generally, four point formulas were employed in the finite-difference approximation, although some three point formulas were used near the boundary. The proposed numerical procedure was stable for all the Reynolds numbers considered.

Theory of E.P.R. of 3dimpurities in metals

Abstract The application to the E.P.R. behaviour of dilute alloys of the generalized s-d interaction model appropriate to a 3d impurity with orbital degrees of freedom is considered. It gives rise in some cases to a large spin-lattice relaxation rate in addition to the usual cross relaxation to the conduction-electron spin. Our calculation shows why bottlenecking is effective for some 3d impurities and ineffective for others, leading to qualitative differences in the observed E.P.R. behaviour. It is emphasized that, as a result of the bottlenecking phenomenon, E.P.R. is a uniquely powerful method for determining the electronic states of 3d impurities in metals.

Paramagnetic Relaxation in Some Nickel Tutton Salts and Cupric Nitrate Salt

The magnetic field and concentration dependences of the paramagnetic spin relaxation in some nickel tutton salts and a cupric nitrate salt are studied by non-resonant type experiment. A modified Hartshorn bridge operating up to 100 kHz was used inside a high field superconducting magnet of the maximum field of 60 kOe. At temperature region from 4.2 K to 1.5 K, high frequency susceptibilities show that there are several relaxation processes at some special values of magnetic field, of which relaxation times are faster than of usual spinlattice relaxation. The observed relaxation phenomena have been successfully explained by various types of multiple spin flip cross relaxation. The cross relaxation phenomena among unlike spins are also observed in the diluted mixed crystals of (Ni-Mn)-, (Ni-Fe)-, and (Ni-Co)- ammonium tutton salts. Qualitative explanations are given of the anomalous fild dependence of the spin-lattice relaxation time of Co(NH 4 ) 2 · (SO 4 ) 2 ·6H 2 O from the view point of the cross relaxa...

Einfluß von Quanteneffekten der Methylgruppenrotation auf die Kernrelaxation in Festkörpern

Abstract By treating the hindered methyl group rotation in solids quantum-mechanically the proton spin lattice relaxation at low temperatures is calculated in first and second order. For low hindering barriers (smaller than 1 kcal/mole) significant differences from the usual relaxation theory are obtained. The inverse correlation time 1/τ0 is essentially given by the rate of non magnetic transitions between the zeroth and first torsional state; and as a consequence the apparent activation energy at low temperatures is given by the difference between these levels. For a given To the relaxation is by a factor ω0/Λ0 (Λ0 is the tunneling frequency) less effective than in the classical case. Good agreement between earlier measurements and this theory is obtained by considering at low efficiency factors also the magnetic interaction between the methylprotons and the surrounding protons. In this way the experimentally observed two relaxation maxima and the ω0-dependence can be explained too.

Space charge waves in solids

The conventional boltzmann-vlasov equations are solved for the case of a hot plasma with collisions such as may correspond to the conduction electrons in a semiconductor. A collision term is used which contains two relaxation times corresponding approximately to the usual momentum and energy relaxation times. This collision term is so constructed that it does not give rise to a spurious generation of particles by collisions. A general dispersion relation is obtained for space charge waves propagating in such a plasma. In the limit of a cold plasma this dispersion relaxation reduces to results familiar from cold plasma theory.

Relaxation Spectra and Association Constants of 1:1 Charge-Transfer Complexes

The possibility of determining the association constant of a 1:1 charge-transfer complex AB in solution from the concentration dependence of the relaxation time is discussed. If small quantities of termolecular complexes A2B and AB2 are also present there will be two additional relaxation times corresponding to the decomposition rate constants of these complexes. It is shown that if A and B are aromatic molecules whose two sides act completely independently so as to form an infinite series of sandwich complexes (model 3), then the usual relaxation techniques would detect only one relaxation time whose concentration dependence would be indistinguishable from that of a system containing only one complex, AB. If such a system were mistakenly assumed to contain only one complex AB, relaxation methods would lead to the correct value for the dissociation rate constant of AB but to an association rate constant equal to half the true value. The association constant found would thus be equal to half its true value, just as if it had been determined spectrophotometrically. The complete relaxation spectrum of model 3 and the corresponding normal reaction coordinates were calculated; the reciprocal relaxation times form three infinite series, the spacing in each series being twice the dissociation rate constant of AB. If the rate constants of reactions involving isolated molecules of A or B differ slightly from those of the corresponding reactions involving complexes of A and B, then perturbation theory shows that in principle three relaxation times, instead of one, should be observable; in practice, difficulties would arise since these relaxation times are all of similar magnitude.

Analysis of thermal relaxation processes

A method has been developed for analysing single relaxation processes. By a transformation of the usual relaxation equations functions of sound absorption and velocity are found which are linearly related to a simple function of the sound frequency/pressure ratio. The relaxation time of the process may be deduced from the slope and the intercept. It is shown how the method may be extended to analyse double relaxation processes.

Irreversibility in Small Stellar Dynamical Systems.

view Abstract Citations (157) References (6) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Irreversibility in Small Stellar Dynamical Systems. Miller, R. H. Abstract Irreversibility in stellar dynamical systems was studied by means of numerical experiments using n-body calculations carried out in a computer for n up to 32. In the experiments, two similar systems evolve simultaneously and the separation of their representative points in phase space was observed to grow exponentially with the time variable because of the effects of encounters. The time constant of this exponential growth defines a relaxation time different from the usual relaxation time of stellar dynamics. Certain properties of the calculation and comparisons of the observed values of this relaxation time with values expected on the basis of binary collisions led to the conclusion that n-body systems with inverse-square law forces behave as tightly coupled systems. The time constant is surprisingly short- about 4r/n, where r is the mean time between binary collisions of one particle. Hard spheres or potentials do not show the collective behavior of inverse-square law systems. Errors in calculations of n-body systems grow exponentially with the time through this mechauism, and may therefore invalidate the results of n-body calculations. Publication: The Astrophysical Journal Pub Date: July 1964 DOI: 10.1086/147911 Bibcode: 1964ApJ...140..250M full text sources ADS |

Spin-Lattice Relaxation

The measurement of electron spin-lattice relaxation times for paramagnetic crystals at low temperatures is complicated by the fact that the spin specific heat can be much larger than the lattice specific heat. At low temperatures direct processes dominate in the spin-lattice relaxation mechanism and evidence exists that indicates that only a narrow portion of the phonon spectrum takes part in the relaxation processes. This situation is not encompassed by usual treatments of the spin-lattice problem and a microscopic treatment is presented which allows for this selective excitation of the phonon spectrum. It is pointed out that phonon relaxation times can be the dominant quantity measured in the usual saturation spin-lattice relaxation measurements. The analysis indicates how pulse measurements may be used to evaluate the actual spin-lattice relaxation time independent of the phonon relaxation time. A discussion of some conditions under which the concept of temperature may be applied to quantum-mechanical systems interacting with electromagnetic fields, such as in solid-state amplifiers or absorbers, is given.

Mechanical Resonance Dispersion in Crystalline Polymers at Audio-Frequencies

Measurements of the complex shear compliance (J*=J′—iJ″) of crystalline polymers (polytetrafluorethylene, linear polyethylene, and polyvinyl stearate) at closely spaced frequencies in the audio-frequency range have resulted in the discovery of multiple frequency dispersions of the resonance type. Values of the loss compliance (J″) are found to have sharply defined maxima in the range from 100 to 5000 cps; values of the storage compliance (J′) rise to a maximum and then drop to a minimum as a narrow (50 to 100 cps wide) dispersion region is traversed in the direction of increasing frequency. The results indicate the presence of a dispersion mechanism different from those leading to the usual relaxation dispersions observed in connection with the mechanical properties of polymers. The magnitudes of the mechanical loss maxima and the frequencies at which the effect is observed depend on the static stress history and state of the sample material as well as temperature. Similar results obtained for pure polycrystalline metals (lead, indium, and aluminum) lead to speculation that stress-induced motions of dislocations or other imperfections in the crystalline structure may be important in producing this type of dispersion. At present, however, there is no satisfactory explanation for these phenomena.

A Note on the Numerical Solution of Fourth Order Differential Equations

SummaryAn old numerical method of solving fourth order differential equations is put in relaxation form. The higher order correction terms are included and the technique is illustrated by an example. The method has the advantage of being more rapidly convergent than the usual relaxation procedure for fourth order equations. Some comments are made on the numerical solution of the viscous flow equation.

The Mechanism of Orientation in i-Butyl and i-Amyl Bromide Glasses

The relaxation times of two typical, unassociated polar molecules have been calculated from experimental dispersion data. Both Stokes' law and absolute reaction rate theories have been applied to show that the inner friction constant in the usual Debye relaxation formula must be considered in terms of a mechanism involving higher activation energies than those obtaining in ordinary viscous flow. Hence, the macroscopic viscosity will generally be inadequate as an inner friction constant for relaxation. Slight alterations in the structure of these molecules are found apparently to change their relative orientations and hence energies of interaction. Rotational motion in viscous flow of lesser degree but similar kind to that of relaxation is indicated by the parallelism between the respective energies in i-butyl and i-amyl bromides. Possible generalizations have been extended to relaxation and flow phenomena in liquid crystals, where effectively the co-existence of phases derived from a single component over a temperature range makes the structure considerations simpler than in the usual case of solutions.

Preliminary report on the effects of vagus stimulation on the dog's stomach and the influence of asphyxia on these effects

Summary1. In the dog stimulations of the vagus repeated at short intervals are followed by contractions of the stomach of approximately uniform amplitude.2. Prolonged stimulation of the vagus is followed by a single contraction of the stomach and its usual subsequent relaxation in spite of continued stimulation. Failure of cardiac inhibition occurs at about the same time as relaxation of the stomach begins.3. Reduction of the body temperature to 25° C. to 23° C. is followed by decreased response to vagus stimulation but no indications of either failure of gastric response to repeated vagus stimulation or tetanus. At 21° C. vagus stimulation is without effect on the stomach but stimulation of the stomach wall is followed by a contraction.4. Partial asphyxia of the stomach by shutting off the blood supply to the part is followed by rapid failure of gastric response to repeated vagus stimulations similar to that observed in the turtle. Removal of the asphyxia is followed by recovery of the gastric response t...