Longitudinal composite oscillators for measuring internal friction, piezoelectric modulus, and strain modulation effects are usually limited to a frequency range of 30 to 200 kHz. If the same crystals are vibrated in flexure, a longitudinal strain can be introduced with the resonance frequency below 3 kHz while at the same time keeping the inherent high Q of the composite system. This paper develops the theory for the strain amplitude and damping for the flexural composite oscillator made up of two quartz crystals plus specimen and, if appropriate, spacers. This high Q technique of vibrating in flexure has applications for strain modulation and damping experiments.

A Green function procedure for calculating the inner displacement about an impurity in a crystal lattice has been developed. The application to a substitutional impurity at the metal ion position in the fluorite lattice shows that both for the rigid ion model and for the shell model, the inner displacement is identical in form to that of the perfect lattice. However, the changes in the equilibrium positions of the neighbours of the impurity need to be taken into account when evaluating atomic displacements of the strained crystal.

In a crystal containing paramagnetic impurities the parameters that specify mean squares and products of the intrinsic strain components at the impurity sites can in principle be obtained from EPR linewidths provided that the spin-strain coupling tensor is known. It is shown that if the strain-induced defects are distributed with the symmetry of the crystal lattice the number of such parameters is restricted by the relation ( epsilon lambda ri epsilon mu sj)= delta lambda mu delta ij( epsilon lambda r epsilon lambda s) where epsilon lambda ri is the ith component of the rth repetition of the irreducible representation lambda of the crystal's point group. Some strain-broadening measurements in MgO are considered. Ruby linewidths have been measured and an expression containing four such strain parameters, a mean-square crystallite misalignment, and an expression for the linewidth arising from Cr3+ interaction with Al nuclei fitted to them to obtain meaningful strain parameters.

The inner displacements in a stressed crystal of the fluorite structure are calculated from the parameters of both rigid-ion and shell lattice dynamic models by considering the response of the lattice to long-wavelength phonons. The general results are applied to the three crystals CaF2, SrF2 and BaF2. The magnitudes of the inner displacements are found to be model dependent and are significant compared with the uniform strain displacements.

Difficulties that arise in applying the superposition model of crystal fields to non S-state ions are discussed with reference to recent measurements on Cr3+. The effects of including all d3 levels in a perturbation calculation of the spin-strain coupling tensor in ruby are considered. The results differ from a calculation using the 4F levels in the relative weight of the two terms and in the introduction of a new term when the crystal field contains components of E symmetry.

Expressions for the elements Gpq of the spin-strain coupling tensor have been derived for ruby (Al2O3:Cr3+) in terms of the superposition-model parameters describing the axially symmetric field of each ligand. Four intrinsic parameters, A2, A4 and their radial derivatives, provide a reasonable fit to the ten Gpq but the zero-field splitting D cannot be fitted and must be largely due to other mechanisms. The parameters obtained are used to estimate the elements of the G tensor in MgO:Cr3+ and these are compared with measured values.

The inner-strain tensor describing the departure from uniform strain within the unit cell of a stressed crystal has been calculated for the Al2O3 corundum structure from the parameters of the lattice dynamic model of Kappus (1975) using a technique based on the long-wave approach. The departure from uniform strain is significant particularly for the oxygen atoms. The major components of the tensor for the oxygen atoms approximate to those obtained using a pseudo-HCP model of the anion structure.

Strain measurements are made on the − radical in sodium bromate using the strain modulation technique. The strain dependence of the EPR lines is proportional to the departure of the g value from the free-spin value. This can be understood qualitatively in terms of the change of the energy of the excited state which is mixed into the ground state by spin-orbit coupling.

The strain-dependent line-shifts of EPR transitions in ruby have been measured using the UMER (ultrasonically-modulated EPR) technique. Results were obtained for four transitions for a range of magnetic field orientations perpendicular to an applied uniaxial stress and for angles of 0 degrees , 45 degrees and 90 degrees between stress and three-fold axis. These have been fitted by a perturbation Hamiltonian GijklSiSj epsilon kl yielding values for the spin-strain tensor G. Some combinations of the tensor elements of a small delta g term were also obtained. The G tensor obtained for Cr3+ in a trigonal distortion is similar to that obtained by rotating values for Cr3+ in Oh symmetry (MgO) into the same coordinates.

The technique of using ultrasonic modulation to measure strain effects in solids has been extended to the study of the quadrupole splitting of 27Al in Al2O3 by NMR. The advantages and limitations of the strain modulation technique for NMR are discussed.

The difficulties of straining a thin sample such as mica have been overcome by applying to the sample a 40 kHz standing strain wave. ESR measurements have been made in mica using magnetic field modulations and ultrasonic strain modulation (UMER). By separating the contribution of different defects the UMER results the underlying complexity of the spectra.

The EPR longitudinal and shear strain parameters for tetragonal Gd3+ in CaF2 have been measured using the ultrasonic strain modulation technique (UMER). A comparison has been made with the results for the cubic Gd3+ site using the superposition model as a guide, but as the local elastic constants for the tetragonal defect are unknown, the conclusions are tentative.

The piezoelectric composite oscillator vibrating either in the longitudinal or torsional mode enables the mechanical damping, strain amplitude, elastic modulus, and piezoelectric modulus of a specimen to be determined rapidly and accurately. Previously the technique has been developed for specimens of a length equal to a mutliple of half-wavelengths requiring a specimen size of ∼80×3×3 mm for 40 kHz. In this paper we develop the theory of the longitudinal technique for sandwich specimens of ∼1 to 10×3×3 mm which are cemented between two fused quartz rods to form a one-half-wavelength unit. We demonstrate the sandwich specimen technique by determining the compliance (S1111) of a 〈100〉 KCl single crystal without using its density and find that this value is in agreement with the published result. The equations for the ideal transformer ratio and strain amplitude for the sandwich specimen are confirmed experimentally by measuring the voltage from an α quartz specimen cemented between rods of α quartz, 〈110〉 CaF2, and 〈100〉 CaF2, respectively.

The EPR strain parameters of cubic Gd3+ in CaF2 have been measured using the ultrasonic strain modulation technique (UMER) in which 40 kHz strain modulation replaces the magnetic field modulation in conventional EPR. The results are consistent with, but more extensive than, those obtained by static stress measurements of the shift in the resonant field. An analysis of the results shows that the nearest-neighbor superposition model is unable to account for the low value of the fourth-order shear spin-lattice relaxation coefficient Gs.), but for the dilatation and tetragonal spin-lattice relaxation coefficients, the model works satisfactorily.

By replacing the magnetic modulation in an ESR experiment with strain modulation it is possible to determine strain effects in ESR simply and directly. The apparatus producing the strain modulation consists of two piezoelectric quartz crystals and a specimen vibrating in a longitudinal mode at 40 kHz. The stress applied to the specimen is longitudinal ( sigma /sub /// only) while the strain amplitude can range from a thermal limit of 10-13 to a fracture limit of 10-3. Because ultrasonically modulated electron spin resonance (UMER) enables a spectrum change due to strain to be measured directly rather than as a small difference between two large magnetic fields, it has been found to be more sensitive than other methods. The equipment for conducting UMER experiments is described and the theoretical basis for the method is discussed.

Our technique of replacing the magnetic field modulation in an ESR experiment with an ultrasonic strain modulation in single crystal specimens of one half wave length (∼ 80 × 3 × 3 mm) has been extended to small single crystal samples 3–6 mm long. Using this technique we have obtained values for the second- and fourth-order parameters which describe the change of the crystal field with strain for a sample of tetragonal gadolinium in CaF2.

The one phonon coherent inelastic neutron scattering cross section is analysed in terms of the transformation properties of the phonon polarization vectors. An expression involving characters of point group representations is derived which is zero when the scattering conditions are such that a particular branch of the phonon dispersion curve has zero scattering cross section. This expression is related to Elliott and Thorpe's expression for sums of the scattering over branches with polarization vectors transforming as the same irreducible representation. The selection rules for LiNbO3 are derived as an example.

Some multiphonon selection rules for infrared absorption and Raman scattering have been evaluated for ferroelectric and paraelectric sodium nitrite at symmetry points in the Brillouin zone. As the results for the paraelectric phase depend on the mechanism for the re-orientation of the nitrite ion, the normal modes are discussed in some detail for each case.