Single-frequency Model Research Articles

Thermal Shift Calculations (I): The Theory for Thermal Shift of Crystal Field Spectra of Rare Earth or Transition Metal Ions

The thermal shift of crystal field spectra of rare earth or transition metal ions includes four contributions of different characteristics. In this paper we develop a theory of calculating all those contributions in detail. The contributions to thermal shift consist of that of thermal expansion and those of electron-phonon interaction. The thermal shift caused by thermal expansion is interpreted and calculated by making use of theory of pressure-induced spectral shift. The contribution of phonons of optical branches is given by using single frequency model. The contributions of phonons of acoustic branches include two terms, which are derived by taking into account contributions of all the in electron-phonon interaction. The theoretical formulas for parameters in the most important term are given.

Theory of spectral thermal shift and R-line thermal-shift calculations of MgO:Cr3+

The thermal shift of the crystal-field spectra of 3d or 4f ions includes contributions of different characters. In this paper we develop an approach to calculating all those contributions in detail. The thermal shift caused by thermal expansion is interpreted and calculated by use of the theory of pressure-induced spectral shift. Contributions of phonons of optical branches are given using a single frequency model. Taking into account contributions of all the \ensuremath{\Gamma}\ensuremath{\gamma} in the electron-phonon interaction, two terms coming from phonons of acoustic branches are derived in detail, and theoretical formulas for the parameters in the most important term are given. By adequately taking into account all the contributions, we calculated the R-line thermal shift of MgO:${\mathrm{Cr}}^{3+}$ by fitting the experimental data. The results show good agreement with thermal-shift experimental data and a series of other experimental data of MgO:${\mathrm{Cr}}^{3+}$. Values of the most important parameters are calculated theoretically with wave functions obtained from diagonalization of complete ${\mathit{d}}^{3}$ energy matrices. They are in good agreement with those from fitting the experimental data. We found that it is the mixing of wave functions caused by the Coulomb interaction and/or the spin-orbital interaction that makes D(${\mathrm{\ensuremath{\Gamma}}}_{6}$) and D(${\mathrm{\ensuremath{\Gamma}}}_{8}$) nonzero. Our investigation shows that among all the contributions to thermal shifts, the ${\mathit{t}}_{2}^{32}$${\mathit{T}}_{1}$ term (resulting in redshift) is the most important one; the Raman term (resulting in blueshift) eliminates part of the ${\mathit{t}}_{2}^{32}$${\mathit{T}}_{1}$ term; their algebraic sum gives the contribution of electron-phonon interaction of the acoustic branches; contributions of optical branches (resulting in redshift) increases rapidly with temperature, and exceeds that of the acoustic branches when T\ensuremath{\ge}175 K; contribution by thermal expansion (resulting in blueshift) holds about 30% of total thermal shift in absolute value.

Theoretical study of free-electron lasers with reversed guide field

Free-electron laser (FEL) amplifier performance with a reversed axial guide field is studied theoretically and numerically. The theoretical analysis focuses on an analytic nonlinear calculation of the particle motion at negative resonance which predicts the amplitude of oscillation as a function of guiding center location. Numerical studies based on a new simulation code, which follows the three-dimensional particle motion and the self-consistent amplitude and phase of the rf field, reproduce the high efficiency results recently reported by Conde and Bekefi [1] for a reversed field FEL. These studies indicate that the saturated power is rather sensitive to the model used for the space-charge forces. The reported results for Group I orbits can be reproduced only by assuming an anomalously large emittance, while the Group II results are not consistent with the single-frequency model.

On the theory of spin-lattice relaxation due to the hopping motion of light interstitials; the role of excited states

The dipolar contribution to the spin-lattice relaxation rate Gamma 1 of spin-carrying light interstitials is calculated assuming nearest-neighbour hopping among tetrahedral sites of a BCC lattice. As a new element, which was missing in previous treatments, the authors took into account explicitly the existence of localized excited states of the interstitials. The model considers two states per site, i.e. the ground state and one excited state, and takes into account transitions between the ground states, the excited states, and between an excited level and a ground state of two neighbouring interstitial sites as well as intrasite transitions between ground and excited states of the same site. If the intrasite transitions from the ground to the excited state are not fast compared with the tunnelling rate among the ground states of neighbouring tetrahedral sites, which is likely to be the case for hydrogen in BCC metals like Nb and Ta, the extended model predicts deviations from the results obtained for the usually considered model which describes the interstitial motion by a single effective hopping frequency among tetrahedral sites. The product alpha = Gamma 1D, in which D is the diffusivity of the interstitial, may be considerably larger for the extended model than for the single-hopping-frequency model. Moreover the various hopping frequencies do not drop out from alpha as does the effective hopping frequency in the latter case. This may lead to a strong increase of alpha with temperature in contrast to the temperature independence of alpha for the single-frequency model. The theoretical results are compared qualitatively with NMR experiments performed on the alpha -phases of NbHx and TaHx.

Multiphonon Nonradiative Transition of RE3+ Ions in Ten Host Crystals11This work was partly supported by the National Natural Science Foundation of China

By using the formula of RE3+ multiphonon nonradiative transition (MPNRT) probability Wp0 derived on the basis of single-frequency model, the dependence of the Wp0 for the various RE ions in 10 hosts at the energy gap ΔE is analyzed and calculated systematically. The results agree well with a great deal of experimental data taken from the references and show clearly the characteristic of MPNRT under the condition of weak electron-phonon interaction: Wp0 decreases rapidly with the increase of order p which is the leading factor to determine Wp0; the values of K. HUANG-Rhys factor S progressively increase in the order of halide-oxide-fluoride, and the values of S for the various hosts in the same group are close.

Multiphonon Nonradiative Transition of Rare-Earth Ions

Based on the general theory, the formula of multiphonon nonradiative transition probability is derived by adopting simplified approximations involving single-frequency model. Satisfactory agreement is obtained through the comparison of theory and experiment.The principal differences between the present paper and relevant literatures as well as the validity of simplified approximation are discussed.

The multiple-frequency Chandler wobble.

The problem of the excitation and damping of the Chandler wobble is reviewed and the conventional models analyzed. Using an axially near-symmetric and slightly triaxial Earth Model, a solution of the Liouville equation reveals that the Chandler wobble has multiple frequency and is rather a quasipermanent phenomenon in the Earth. The multiple Chandler frequency consists of a fundamental frequency attributable to the Earth's triaxiality, as well as series of small feedback frequencies arising from the Earth's instantaneous inertia, relative angular momentum, and inertia variation. After the introduction of the anelasticity of the mantle into the model, the Chandler wobble is found to be a combination of the Kelvin-Voigt and Maxwell oscillations with the former mechanism dominating the motion. The Chandler feedback comes mainly from the anelastic upper mantle, but it does not rule out fractional contributions from the oceans and atmosphere, the mantle-core coupling, or even large earthquakes; the Chandler wobble may behave like a multiple-coupling differential oscillation with the mantle as the major coupler. The feedback mechanism or frequency modulation gives the Chandler wobble a "beat" appearance. The quasi-permanency or the relaxation time of a multiple-frequency Chandler wobble is of the order of 104 to 106 years or higher, up to the contribution from the Maxwell feedback. The multiple-frequency model explains the Chandler amplitude and frequency variations as well as the persistence of the wobble beyond the damping relaxation time predicted by the single-frequency model.

The photoionisation cross section of deep-level impurities in semiconductors

A simple analytic expression for the photoionisation cross section of deep-level impurities in semiconductors is obtained in a form that is explicitly related to the electron-scattering strength of the impurity. It also includes the dependence on size and charge, and the character of the localised-state Bloch functions. All of these quantities affect the spectral dependence immediately above threshold. The model introduces a simple billiard-ball wavefunction and the shape functions near threshold are derived, for neutral and charged (plus or minus) centres which are either donor-like or acceptor-like. The effect of phonon coupling is discussed in the context of a single-frequency model. At low temperatures the spectral shape near the shoulder of the curve is determined by the charge, and the effect of phonon coupling is to raise the apparent threshold.

Multiphonon, non-radiative transition rate for electrons in semiconductors and insulators

A quantum-mechanical calculation of the multiphonon, non-radiative transition rate for electrons in solids is carried out without assuming the Condon approximation. A relatively simple result is obtained by adopting a single-frequency model. An expression valid at low and intermediate temperatures (0<T<or approximately=300K) is derived. The enhancement of rate over that derived in the Condon approximation is roughly (p/2)2, where p is the depth of the level in units of the phonon energy. It arises from a combination of enhanced mixing of initial and final electronic states and the introduction of three- and four-phonon processes within a single vibrational mode. The result is shown to agree very well with experimentally determined temperature dependence of the electron capture cross section for the B centre in GaAs. An expression valid at very high temperatures is shown to have the same form as the semi-classical rate derived by Henry and Lang (Phys. Rev. B., vol.15, p.989 (1977)).

Measurements of acoustoelectric gain and lattice attenuation in n-GaAs

Measurements of the non-ohmic behaviour associated with acoustoelectric amplification were employed to derive the frequency dependence of the small signal electronic gain parameter alpha 0 and the lattice attenuation alpha l in n-GaAs. Samples of different carrier concentrations at different temperatures were used to vary the frequency of the amplified flux. The use of a single frequency model in the derivation of alpha 0 and alpha 1 is justified in the text.

A check on oto-chek.

Introduction During the 1956 meeting of the American Academy of Ophthalmology and Otolaryngology, in the session on new instruments, Dr. Howard House presented the Oto-Chek, which he described as compact, battery-powered, transistorized, screening audiometer... [of which] there are two basic models.... One is a single-frequency model designed to test hearing at 4000 cycles, with fixed output levels of 20, 35 and 50 The other is a double-frequency model designed to test hearing at 2000 cycles and at 4000 cycles with fixed output levels of 15 and 35 decibels.... 1 In January, 1957, Dr. House and Dr. Aram Glorig collaborated in reiterating the merits of Oto-Chek audiometry at the meeting of the Western Section of the American Laryngological, Rhinological and Otological Society, in San Francisco. The paper they presented there was published the following July in The Laryngoscope 2 shortly after the appearance of Dr. House's first paper, in the

Application of the Method of Generating Function to Radiative and Non-Radiative Transitions of a Trapped Electron in a Crystal

Using the method of generating function, we have discussed the shape of the absorption band and the probability for non-radiative transition of a trapped electron in insulating or semiconducting crystal, especially their temperature dependence. We have thereby chosen a model for the vibrational motion of the lattice as general as possible, the normal modes for any two electronic states being different as regards not only their equilibrium positions but also the principal axes and frequencies. For non-radia­ tive transition, we have derived, in comparatively general cases, the high and low temperature behaviors of the probability which correspond to the process through activated states and the tunneling of the lattice co-ordinates, respectively. The result is applied to calculations of the probability for thermal ionization of trapped electrons (or holes) and the capture cross-section of impurities for free electrons (or holes) in non-polar semiconductors. Further, the high temperature expansion of a density matrix is used to discuss the transitions on a most general model in which lattice vibration is no longer of harmonic type. Two problems related to the degeneracy of electronic states are discussed briefly. ting or semiconducting crystal has been studied by a number of authors. The simplest but most essential explanation of the broadening effect is found in the text-book of Mott and Gurney/l where they take a one-dimensional model for the lattice configuration space to discuss the breadth of the F-absorption band. More general discussion has been given by Muto2l on a quantum mechanical basis. Inui and Uemura, aJ with one dimensional model, explained the shift of the F-absorption peak by the thermal expansion of the lattice, taking advantage of Ivey's empirical law. Quantum mechanical calculation of adiabatic potentials was first carried out by Williams4l for KCl : Tl crystal, with one dimensional model. Huang and Rhys5l discussed the shape of the F-absorption band and the thermal ionization pro­ bability of the F-electron, taking into account all the longitudinal waves of optical modes of vibration which interact strongly with the electron. Their mathematical technique, though very ingenious in itself, is confined to a single frequency model, and can hardly be genera­ lized to a many-frequency model which involves, for instance, the acoustical modes. Under these circumstances it is very desirable to have a mathematical tool which enables one to discuss radiative and non-radiative transitions on a general model of lattice vibration. One of the present authors6l developed the method of generating function several years ago,